Abstract:
A general, closed, anisotropic kinetic turbulence theory for gases and liquids is based on new solutions of the Maxwell moment equations of the Boltzmann equations. These solutions provide a closed initial equation set for the four time average fluid mechanic variables, the sixteen time average thermal motion correlation and the sixteen time average turbulent motion correlations listed in Table I.

Description:
RELATED APPLICATION 
   This application claims priority from now abandoned provisional application Ser. No. 60/152,391 filed Sep. 3, 1999. 

   BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   This invention relates generally to the field of fluid dynamics. More particularly, the invention is a method for accurately and efficiently analyzing anisotropic turbulent flows of both gases and liquids. 
   2. Background 
   The study and engineering application of hydraulics or fluid mechanics dates to the dawn of civilization. The first irrigation canals were constructed before 5,000 BC. Theoretical analysis of fluid motions began in the modem scientific sense with Newton&#39;s  Principia  (1687) and his laws of motion. Bernoulli (1738) established a relationship between the pressure and velocity of an ideal fluid which became his famous theorem. d&#39;Alembert (1744) deduced that the steady flow of an ideal fluid about a body produces no drag (d&#39;Alembert&#39;s Paradox). Euler (1755) founded modem fluid mechanic analyses by deriving the differential equations of motion (continuity and momentum conservation) for an ideal (inviscid) isotropic fluid. Navier (1822) and Stokes (1845) independently derived the viscous terms necessary to extend Euler&#39;s equations to isotropic, viscous fluids. The equation of state and the conservation of energy equation (First Law of Thermodynamics) written utilizing Fourier&#39;s (1822) law of thermal energy transfer completed the classical Navier Stokes equations governing isotropic fluid motions. 
   Reynolds (1883) experimentally showed that there are two modes of fluid motion, laminar and turbulent motion, which occurs at large values of the dimensionless Reynolds number. Reynolds (1894), by considering a time average motion and a time dependent turbulent motion, showed that turbulent motions introduce additional turbulent (Reynolds) stresses into the Navier Stokes equations, greatly increasing the number of flow properties required to describe turbulent flows. 
   Despite over a century of intense aircraft and aerospace developments, wind tunnel and laboratory experiments, theoretical analyses and numerical computations based on the Navier Stokes equations, the determination of a general theory of turbulent motion remains one of the last great unsolved problems of classical physics. 
   Maxwell (1858) founded modern kinetic theory when he introduced his new concept of the molecule velocity distribution function and deduced its equilibrium form. Maxwell (1867) derived the Maxwell transport equations and showed that the collision change integrals could be solved analytically without knowing the molecule velocity distribution function for Maxwell molecules. Boltzmann (1872) derived his integro-differential equation for Maxwell&#39;s molecular velocity distribution function and solved for his famous H-theorem, which proved that Maxwell&#39;s equilibrium form was correct. Chapman (1916), using Maxwell&#39;s transport equations, determined accurate general formulae for the gas transport coefficients. Enskog (1917) gave a general solution method for the Boltzmann equation, which showed that the Euler equations were the first and the Navier Stokes equations were the second approximate solutions to the Boltzmann equation and gave gas transport coefficients identical to Chapman&#39;s. Both the Chapman and Enskog analyses were for isotropic perfect gases, being based on perturbations of Maxwell&#39;s isotropic equilibrium solution. Their independent analyses cemented the belief in both the scientific and particularly the engineering professions that the Navier Stokes equations were generally and universally valid and that the correct way of analyzing both laminar and turbulent flows was through the Navier Stokes equations. This belief has been engraved in stone in both the scientific and engineering literature throughout the 20th century. 
   Burnett (1935) derived the third approximate solution to the Boltzmann equations using Enskog&#39;s solution method. Grad (1949) derived a thirteen moment method of solving Boltzmann&#39;s equation which closely followed Enskog&#39;s methods. Both of these analyses have been proven to lack generality and have not been significant advances of the Navier Stokes theory. Bird (1963) introduced the Direct Simulation Monte Carlo (DSMC) numerical method for calculating rarified gas flows. Yen (1966) showed that the directional thermal energies (temperatures) were vastly different in shock waves and that the longitudinal temperature overshot its downstream value for shock Mach numbers greater than 
               9   5       ⁢       (   1.34   )     .           
Elliot and Baganoff (1974) showed that the Navier Stokes normal stress relationship and the Fourier energy flux component ratio were valid only in sound waves and were invalid at the shock end points. Elliot (1975) showed that the Navier Stokes normal stress relationship was invalid everywhere in shock waves and incompatible with the directional thermal kinetic energy moments of the Boltzmann equation. DSMC numerical shock calculations confirmed all of these analytical predictions and numerically illustrated the anisotropic not isotropic fluid features of many simple gas flows.
 
   Kliegel (1990), following a suggestion of Maxwell (1867) that near equilibrium flows were anisotropic and should be represented by an anisotropic, not isotropic Maxwellian, performed the first general anisotropic solution analysis of the Boltzmann equation. Kliegel (1990) showed that the Euler-Navier Stokes-Burnett equation sequence was not the correct approximate solution sequence for the Boltzmann equation. He showed that the correct gas dynamic equation set was an anisotropic fluid seven equation set for the density, three fluid velocity components and three directional thermal kinetic energies (temperatures), not the classic isotropic fluid five equation Euler-Navier Stokes set for the density, three fluid velocity components and total temperature. He also gave shear and directional energy flux relationship having the correct Mach number dependence for both sound waves and weak shocks. Kliegel&#39;s anisotropic equations resolved d&#39;Alembert&#39;s paradox giving a profile pressure drag associated with pushing an ideal (shearless, energy fluxless) fluid about a body. They also correctly predicted the directional thermal energy separations and overshoots and the thermal energy flux component changes occurring in shock waves as predicted by Yen (1966) and numerical DSMC shock calculations. Bird (1994) summarizes recent DSMC calculational capabilities. Chen and Jaw (1998) presented a recent summary of classical isotropic fluid Navier Stokes based turbulent flow modelling. 
   The present disclosure correctly consolidates all previous fluid dynamic and kinetic theory analyses and extends it to define a new method of analyzing and computing anisotropic turbulent flows. 
   The prior art cited above may be found in the following references:
     1. 1687, Newton, I.  Philosophiae Naturalis Principia Mathematica , Oxford.   2. 1738, Bernoulli, D.,  Hydrodynamica sive de Viribus et Motibus Fluidorum Commentarii , Berlin.   3. 1744, d&#39;Alembert, J.,  Traite de l&#39;equilibre et du mouvement des fluides pour servir de suite au traité de dynamique , Paris.   4. 1755, Euler, L.,  Principles généreaux du mouvement des fluides , Histoire de l&#39;Academie de Berlin.   5. 1822, Navier, L. M. H.,  Mémoires de l&#39;Academie des Sciences de l&#39;Institut de France , V 6, p. 389, Paris.   6. 1822, Fourier, J.,  Théorie analytique de la chaleur , Paris.   7. 1845, Stokes, G. G.,  On the Theory of the Internal Friction of Fluids in Motion , Transactions of the Cambridge Philosophical Society, V8, p. 287, Cambridge.   8. 1858, Maxwell, J. C.,  Illustrations of the Dynamical Theory of Gases , Philosophical Magazine, V 19, p. 19 and V20, p. 21.   9. 1867, Maxwell, J. C.,  On the Dynamical Theory of Gases , Philosophical Transactions of the Royal Society, V 157, p. 49.   10. 1872, Boltzmann, L.,  Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen, Styungsberichte , Akad. Wiss, Vienna, Part II, V 66, p. 275.   11. 1883, Reynolds, O.,  An Experimental Investigation of the Circumstances which Determine whether the Motion of Water Shall Be Direct or Sinuous, and the Law of Resistance in Parallel Channels , Philosophical Transactions of the Royal Society, V174, p. 935.   12. 1895, Reynolds, O.,  On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion , Philosophical Transactions of the Royal Society, V186A, p. 123.   13. 1904, Prandth, L.,  Uber Flüssigkeitsbewegung bei sehr kleiner Reibung , Proceedings of the Third International Mathematical Conference, Heidelberg, Leipzig.   14. 1916, Chapman, S.,  The Kinetic Theory of Simple and Composite Monotonic Gases: Viscosity, Thermal Conduction, and Diffusion , Proceedings of the Royal Society London, A, V 93, p. 1.   15. 1917, Enskog, D.,  Kinetische Theory der Vorgänge in mässig verdunnten Gasen , Almquist &amp; Wilsells, Uppsala, Sweden.   16. 1935 Burnett, D.,  The Distribution of Molecular Velocities and the Mean Motion in a Non - Uniform Gas , Proceedings London Mathematical Society, V 40, p. 382.   17. 1949, Grad, H.,  On the Kinetic Theory of Rarefied Gases , Convention of Pure and Applied Mathematics, V 2, p. 331.   18. 1963, Bird, G. A.,  Approach to Translational Equilibrium in a Rigid Sphere Gas , Physics of Fluids, V 6, p. 1518.   19. 1966, Yen, S-M,  Temperature Overshoot in Shock Waves , Physics of Fluids, V9, p. 1417.   20. 1970, Chapman, S., Cowling, T. G.,  Mathematical Theory of Non - Uniform Gases , Third Edition, Cambridge University Press.   21. 1974, Elliot, J. P., Baganoff, D.,  Solution of the Boltzmann Equation at the Upstream and Downstream Singular Points in a Shock Wave , Journal of Fluid Mechanics, V 65, p. 603.   22. 1975, Elliot, J. P.,  On the Validity of the Navier - Stokes Relation in a Shock Wave , Canadian Journal of Physics, V 53, p. 583.   23. 1990, Kliegel, J. R.,  Maxwell Boltzmann Gas Dynamics , Proceedings of the 17 th  International Symposium on Rarefied Gas Dynamics, Aachen 1990, Edited by Alfred E. Beylich, VCH, New York.   24. 1994, Bird, G. A.,  Molecular Gas Dynamics and the Direct Simulation of Gas Flows , Oxford University Press, New York.   25. 1998, Chen, C-J. Jaw, S-Y,  Fundamentals of Turbulence Modeling , Taylor &amp; Francis, Washington.   

   SUMMARY OF THE INVENTION 
   This disclosure concerns a simple method of deriving the correct kinetic anisotropic fluid turbulent flow equations from the Boltzmann equation and the correct thermal and turbulent moment closures to yield a new, closed set of fluid dynamic equations describing anisotropic turbulent motions, solving the old quest for the solution to this classical problem. 
   Current turbulent flow analyses are based on the isotropic Navier Stokes equations, turbulent fluctuation moments of these equations and modeling of the unknown turbulent moments, identifying introduced unknown modeling coefficients by experiment. This results in a variety of semi-empirical turbulent flow analyses based on different Navier Stokes moment relationships satisfying mass, momentum and total energy conservation for an isotropic fluid, calibrated to approximately match a limited number of experiments. The limitations of current Navier Stokes based turbulent analyses are well known and appear to be irresolvable after a century of intense analytical and experimental effort. 
   The present invention provides a general, closed, anisotropic kinetic turbulence theory for gases and liquids based on new solutions of the Maxwell moment equations of the Boltzmann equations. These solutions provide a closed initial equation set for the four time average fluid mechanic variables, the sixteen time average thermal motion correlation and the sixteen time average turbulent motion correlations listed in Table I. Higher order closure sets involving higher order (fourth and fifth, etc.) thermal and turbulent moment sets are easily generated allowing solutions of turbulent flow problems to higher and higher accuracy if desired or needed for special problems. These higher order sets will not be further discussed since they add only unnecessary detail to the disclosure and reveal little that is new. 
   An algebraic differential equation solution method described in my co-pending application entitled “Method for Algebraically Solving Differential Equations, including Stiff Equations, to High Accuracy” may be used to reduce the thirty six governing differential equations to the solution of as few as six differential equations (continuity, momentum, total energy and turbulent energy) plus thirty algebraic relationships. The actual equation set solved depends on the problem and accuracy desired. 
   The presented analytic anisotropic turbulence theory is mathematically correct, involving no semi-empirical modeling. The even order higher order moment closures involved are general and correct to known accuracy. The analysis can be used to solve the infinite set of Maxwell moment equations to any level desired to verify the accuracy of the initial closure set. 
   The anisotropic kinetic turbulence theory presented allows the calculation of turbulent motions with the same accuracy and computer resources as current laminar flow calculations. 
   The new teachings of the present invention may be summarized as follows:
         1. A closed turbulence theory may be simply derived directly from the Boltzmann equation without turbulent correlation modelling or experimentally determined modelling coefficients.   2. Turbulent flows are totally inviscid flows, unchanged by molecular collisions.   3. Turbulence is universal, having the same governing equations and the same physical structure in both gases and liquids because turbulence is an inviscid flow phenomena described entirely by inviscid convective flow equations, independent of collisions and molecular interaction laws.   4. The fact that time average turbulent flow correlation between turbulent flow fluctuations and density or thermal velocity correlations are zero because there is no physical (collisional) connection between these fluctuations, since both the turbulent flow fluctuations and density are conserved in collisions.   5. A method of analyzing the Maxwell moment equations to obtain a closed set of equations which describe time average turbulent flows to any desired accuracy level. All higher moment equation above the closure level can be solved in even/odd sets of two, thus yielding solutions to all Maxwell moment equations with error of Burnett order or less.   6. Direct predictions of turbulent transition locations, turbulent energy growth rates, shears and other thermal and turbulent flow quantities of physical importance can now be derived or calculated from solution of the presented equations with the same accuracy and computer resources as current laminar flow analyses or calculations.   7. Universal even order thermal and turbulent moment closure relationships exist and are explicitly given, allowing the termination and solution of the Maxwell moment equations at any desired odd moment equation set, and solution of the Boltzmann equation to any desired approximation.       

   DETAILED DESCRIPTION OF THE INVENTION 
   In the following description, for purposes of explanation and not limitation, specific details are set forth in order to provide a thorough understanding of the present invention. However, it will be apparent to one skilled in the art that the present invention may be practiced in other embodiments that depart from these specific details. In other instances, detailed descriptions of well-known methods and devices are omitted so as to not obscure the description of the present invention with unnecessary detail. 
   The Boltzmann equation may be written in fluid dynamic collision rate form: [See Equation set 1]; 
   where 
             D   Dt     =     [     ρ   ⁢           ⁢     F   ⁡     (   v   )         ]           
is the Lagrangian convective motion derivative, ρ is the fluid density, υ is the molecular collision rate, F(v) is the molecular velocity (v) distribution function and F(v)* is the molecular velocity distribution function collided into F(v). There is no approximation in writing the Boltzmann collision integral in this format, since the required thermal velocity moments of F(v)* are correctly calculated from the full Boltzmann collision integral (Kliegel (1990)). The molecular collision rate is
 υ=ρ K    
for Maxwell molecules, where K is a molecular collision constant.
 
   The illustrative derivation will be done for Maxwell molecules for simplicity, since the collision change integrals are known exactly for Maxwell molecules. The same derivation can be performed for any molecular model, it is just more algebraically complex and adds nothing to the teachings. 
   Both the original Enskog (1917) solution as a perturbation about the equilibrium Maxwellian and the Kliegel (1990) solution as a perturbation about a non-equilibrium anisotropic Maxwellian are Navier Stokes order accurate 
           (       1     υ   2       ⁢           ⁢   Burnett   ⁢           ⁢   order   ⁢           ⁢   error   ⁢           ⁢   terms     )         
Burnett order error terms) and do not depend on density gradients. Thus, density gradient corrections to the molecular velocity distribution function and its moments belong to the Burnett order of corrections. This important fact, although present in the result of both analyses, is explicitly stated by neither author nor utilized to further extend or simplify the analyses.
 
   Multiplying the Boltzmann equation by the molecular velocity product v 1   l v 2   m v 3   n , where l, m, n are integers, and integrating over all possible velocities (indicated by the bar) yields the Maxwell molecular velocity moment equation: [See Equation Set 2]. Derivation of the anisotropic turbulent flow equations from this equation will be discussed later. 
   Consider the molecular velocity (v) to be composed of a mass mean fluid flow velocity (u) and a thermal motion velocity (c), then the Cartesian components of these velocities are related by
 
 v   1   =u   1   +c   1  
 
Multiplying the Boltzmann equation by the thermal velocity product c 1   l c 2   m c 3   n , where l, m and n are integers, and integrating over all possible thermal velocities (indicated by the bar) yields the Maxwell thermal velocity moment equation: [See Equation Set 3].
 
   For odd moment (l+m+n=2N+1, N integer) equations, the coefficients of the density gradient terms are even moments and products of lower even moments, which have equilibrium values. Thus these even moment coefficients must be zero to Burnett order for the density gradient terms to be of Burnett order. This requirement yields the general even order (fourth, sixth and higher) density gradient independent thermal moment closure relationships: [See Equation Set 4]. 
   The fourth order thermal moment closure relationships can also obtained from Kliegel (1990) by simple calculation but he did not state these results nor utilize them in any manner. Substituting these relationships into Equation Set 3, one obtains the closed odd order (third, fifth and higher) density gradient independent thermal moment equation. [See Equation Set 5]. 
   The classical Maxwell thermal velocity twenty moment equation set is: [See Equation Set 6]. 
   After using the fourth order moment density gradient independent closure relationships: [See Equation Set 7], the closed directional thermal energy flux and structure equations become: [See Equation Set 8] where the fourth order thermal moments have been replaced by products of second order moments through the fourth order density gradient independent closure relationships. 
   The above set of equations are the correct anisotropic gas dynamic flow equations with neglected terms of Burnett order in the convective terms of the closed thermal energy flux and structure equations. All other equations in the set are exact. These equations resolve the known deficiencies of the Euler-Navier Stokes-Burnett equation sets, d&#39;Alembert&#39;s paradox, the normal stress incompatibility, the incorrect Mach number dependency of the Navier Stokes normal stress and Fourier energy flux components, and the incorrect coordinate rotation variance of the Burnett equations. These equations are valid for both sound waves and weak shocks, and thus will accurately predict all flows except in those regions of high change rates in distances comparable to the mean free path length, such as occur in strong shock waves and at boundaries in very rarified flows. 
   The twenty moment equations may be further simplified by performing an Enskog-type ordering analysis on all terms to obtain the reduced equation set: [See Equation Set 9] 
   One can extend the above analysis to time average turbulent flows by considering the mass mean fluid flow velocity (u) to be composed of a time average velocity (ū), and a turbulent motion velocity (u′), like Reynolds (1894). The Cartesian components of these velocities are related by:
 
 u   1   =  u 1   +   u′   1  
 
where now
 
 v   1   =  u 1   +   u′   1   +c   1  
 
   The turbulent motion (u′) is inviscid, unaffected by collisions since the mean flow velocity (u) is conserved in collisions and only the thermal (c) velocities are changed in collisions. This important fact has not been previously recognized, explicitly stated or utilized in the scientific/engineering literature. Since both the density and turbulent motion are preserved in collisions, there is no collisional (physical) coupling between turbulent motion fluctuations, density fluctuations and thermal velocity correlation fluctuations. Thus, there is no time average correlation between these fluctuations. The averaging time required for the time average correlations between turbulent velocity fluctuations and molecular (number density and thermal motion correlation) fluctuations to be essentially zero is quite short, since the molecular fluctuations occur on a collisional time scale of nanoseconds and turbulent flow fluctuations occur on much larger fluid dynamic time scales (typically measured in milliseconds or greater). Thus, the time dependent time average anisotropic turbulent flow equations presented can be used to compute turbulent flows changing on fluid dynamic time scales for the vast majority of engineering flows. The time change adequacy of the time average anisotropic turbulent flow equations needs to be verified for extremely rapid flow changes such as those associated with shocks, flames and deteriorations. The only turbulent flow variables necessary to analyze anisotropic turbulent flows in the vast majority of physical cases are those given in Table I below, all other time average correlations being zero. 
   
     
       
             
           
             
             
             
           
             
             
             
             
             
             
           
             
             
             
           
         
             
               TABLE I 
             
           
           
             
                 
             
             
               Turbulent Flow Variables 
             
           
        
         
             
               Fluid Dynamic 
               Thermal Moments 
               Turbulent Moments 
             
             
                 
             
           
        
         
             
               
                 ρ 
               
               (1) 
               
                 c 1   2   
               
               (3) 
               
                 u′ 1   2   
               
               (3) 
             
             
                 
             
             
               
                 u 1   
               
               (3) 
               
                 c 1 c 2   
               
               (3) 
               
                 u′ 1 u′ 2   
               
               (3) 
             
             
                 
             
             
                 
                 
               
                 c 1 c 1   2   
               
               (3) 
               
                 u′ 1 u′ 1   2   
               
               (3) 
             
             
                 
             
             
                 
                 
               
                 c 1 c 2   2   
               
               (6) 
               
                 u′ 1 u′ 2   2   
               
               (6) 
             
             
                 
             
             
                 
                 
               
                 c 1 c 2 c 3   
               
               (1) 
               
                 u′ 1 u′ 2 u′ 3   
               
               (1) 
             
           
        
         
             
               Four Fluid 
               Sixteen Thermal 
               Sixteen Turbulent 
             
             
               Dynamic 
               Moment 
               Moment 
             
             
               Variables 
               Variables 
               Variables 
             
             
                 
             
           
        
       
     
   
   The time average turbulent flow variables are the flow density (  ρ ) where the bar indicates time averaging, the three mass mean velocity components (  u 1   ), the three anisotropic thermal kinetic energies (  c 1   2   , where the two bars indicate both time averaging and averaging over all thermal velocities), the three thermal shear components (  c 1 c 2   ) the nine thermal kinetic energy fluxes (  c 1 c 1   2    and  c 1 c 2   2   ) the thermal structure correlation (  c 1 c 2 c 3   ) the three anisotropic turbulent kinetic energies (  u′ 1   2   ) the three turbulent shear components (  u′ 1 u′ 2   ) the nine turbulent kinetic energy fluxes (  u′ 1 u′ 1   2    and  u′ 1 u′ 2   2   ) and the turbulent structure correlation (  u′ 1 u′ 2 u′ 3   ). 
   The instantaneous Maxwell molecular velocity (v) and the thermal velocity (c) moment equations (Equation Sets 2 and 3) are both valid at every instant of time and the instantaneous Maxwell thermal velocity moment equations can be derived from the instantaneous Maxwell molecular velocity moment equations. The time average Maxwell molecular velocity (v) and time average thermal velocity (c) moment equations are also both valid simultaneously. The time average thermal velocity equations yield equations governing the behavior of the anisotropic time average thermal directional kinetic energies, shears, directional kinetic energy fluxes and structure correlations: [See Equation Set 10]. 
   The time average molecular velocity equations yield equations governing mass and momentum conservation and the behavior of both the time average thermal and turbulent directional energies, shears, energy fluxes and structure correlations: [See Equations Set 11]. 
   After eliminating the collision change terms with the time average thermal Maxwell moment equations and other simplifications, one obtains the equations governing the behavior of the anisotropic time average turbulent directional kinetic energies, shears, directional energy fluxes and structure correlations: [See Equation Set 12]. 
   Both sets of third order directional kinetic energy fluxes and structure correlation equations are closed using the fourth order density gradient independent time average thermal and turbulent moment closure relationships: [See Equation Sets 13 and 14]. 
   One thus obtains the complete closed Maxwell twenty moment equation set governing anisotropic time average turbulent flows: [See Equation Set 15]. 
   By the same process, corresponding higher order time average turbulent moment equation sets may be derived using higher order closure relationships. The general density gradient independent time average thermal and turbulent even moment closure relationships are: [See Equation Sets 16 and 17]. 
   These closures allow the infinite set of time average Maxwell moment equations governing turbulent flows to be closed at any desired odd order level starting at the third order. If one closes at the third order (say), then one can solve all higher order moment sets (fourth and fifth, six and seventh, etc.) in order, where the errors are still of Burnett order in the convective terms of the highest order moment equations (fifth, seventh, etc.) considered in the set. The method thus allows solution of all the Maxwell moment equations, both those sets below and above the chosen closure level. 
   The above results may be simply extended in many ways. For example, the general even order density gradient independent moment closure relationships (Equation Set 4) are valid in equilibrium. Thus, [See Equation Set 18] and the even order collision integrals (  c 1   l+1 c 2   m c 3   n   *−  c 1   l+1 c 2   m c 3   n   , etc.) may be calculated for all molecular models. 
   It will be recognized that the above-described invention may be embodied in other specific forms without departing from the spirit or essential characteristics of the disclosure. Specifically, the anisotropic turbulent flow equations may be reformatted and/or approximated and computed in many different ways to solve specific problems. Obvious first steps would be to add gravitational forces or to sum the directional turbulent energy equations to obtain a total turbulent energy equation or to reduce the complexity of the equation set through a Prandtl (1904) boundary layer analysis. The analysis presented (valid for monatomic perfect gases) can be expanded to diatomic (and polyatomic) gases, gas mixtures, reacting gas mixtures, plasmas, etc. through known methods, Chapman and Cowling (1970). The analysis can be extended to isotropic fluids by setting the three directional thermal kinetic energies  c 1   2    or  c 1   2    equal, and solving only the summed total thermal energy equation, rather than the three directional thermal kinetic equations. Since this reduces the Equation Set 9 to the Navier Stokes equations, Kliegel (1990), the analysis can be extended to liquids by setting both the three directional thermal kinetic energies equal and the density constant. The closed anisotropic turbulent flow equation set can be utilized with existing Reynolds averaged Navier Stokes equation solvers to solve incompressible isotropic fluid problems. Thus, it is understood that the invention is not to be limited by the foregoing illustrative details, but rather is to be defined by the appended claims. 
   While all equation sets may be solved by straightforward computation, they may be more efficiently solved using the method described in my co-pending application entitled “Method for Algebraically Solving Differential Equations, Including Stiff Equations, to High Accuracy” Ser. No. 09/654,004 filed Sep. 1, 2000. 
   Equation Set 1 
   (1-1) 
   
     
       
         
           
             
               
                 
                   
                     D 
                     Dt 
                   
                   ⁡ 
                   
                     [ 
                     
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         F 
                         ⁡ 
                         
                           ( 
                           v 
                           ) 
                         
                       
                     
                     ] 
                   
                 
                 = 
                   
                 ⁢ 
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         t 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         ρ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           F 
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                       
                       ] 
                     
                   
                   + 
                   
                     
                       v 
                       1 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           ρ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             F 
                             ⁡ 
                             
                               ( 
                               v 
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       v 
                       2 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           ρ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             F 
                             ⁡ 
                             
                               ( 
                               v 
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                   + 
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     v 
                     3 
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         ρ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           F 
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   ρ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     υ 
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             F 
                             ⁡ 
                             
                               ( 
                               v 
                               ) 
                             
                           
                           * 
                         
                         - 
                         
                           F 
                           ⁡ 
                           
                             ( 
                             v 
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
   
   Equation Set 2 
   (2-1) 
                 ∂     ∂   t       ⁡     [     ρ   ⁢           ⁢         v   1   l     ⁢     v   2   m     ⁢     v   3   n       _       ]       +       ∂     ∂     x   1         ⁡     [     ρ   ⁢           ⁢         v   1     l   +   1       ⁢     v   2   m     ⁢     v   3   n       _       ]       +       ∂     ∂     x   2         ⁡     [     ρ   ⁢           ⁢         v   1   l     ⁢     v   2     m   +   1       ⁢     v   3   n       _       ]       +       ∂     ∂     x   3         ⁡     [     ρ   ⁢           ⁢         v   1   l     ⁢     v   2   m     ⁢     v   3     n   +   1         _       ]         =     ρ   ⁢           ⁢     υ   ⁡     [             v   1   l     ⁢     v   2   m     ⁢     v   3   n       _     *     -         v   1   l     ⁢     v   2   m     ⁢     v   3   n       _       ]               
where  v 1   l v 2   m v 3   n    is the instantaneous molecular velocity moment v 1   l v 2   m v 3   n  averaged over F(v) and  v 1   l v 2   m v 3   n   * is the instantaneous scattered molecular velocity moment v 1   l v 2   m v 3   n  averaged over F(v)*.
 
   Equation Set 3 
   (3-1) 
                 ∂     ∂   t       ⁡     [         c   1   l     ⁢     c   2   m     ⁢     c   3   n       _     ]       +       u   1     ⁢       ∂     ∂     x   1         ⁡     [         c   1   l     ⁢     c   2   m     ⁢     c   3   n       _     ]         +       u   2     ⁢       ∂     ∂     x   2         ⁡     [         c   1   l     ⁢     c   2   m     ⁢     c   3   n       _     ]         +       u   3     ⁢       ∂     ∂     x   3         ⁡     [         c   1   l     ⁢     c   2   m     ⁢     c   3   n       _     ]         -     l   ⁢           ⁢           c   1     l   -   l       ⁢     c   2   m     ⁢     c   3   n       _     ⁡     [         ∂     ∂     x   1         ⁡     [       c   1   2     _     ]       +       ∂     ∂     x   2         ⁡     [         c   1     ⁢     c   2       _     ]       +       ∂     ∂     x   3         ⁡     [         c   1     ⁢     c   3       _     ]         ]         -     m   ⁢           ⁢           c   1   l     ⁢     c   2     m   -   l       ⁢     c   3   n       _     ⁡     [         ∂     ∂     x   1         ⁡     [         c   1     ⁢     c   2       _     ]       +       ∂     ∂     x   2         ⁡     [       c   2   2     _     ]       +       ∂     ∂     x   3         ⁡     [         c   2     ⁢     c   3       _     ]         ]         -     n   ⁢           ⁢           c   1   l     ⁢     c   2   m     ⁢     c   3     n   -   l         _     ⁡     [         ∂     ∂     x   1         ⁡     [         c   1     ⁢     c   3       _     ]       +       ∂     ∂     x   2         ⁡     [         c   2     ⁢     c   3       _     ]       +       ∂     ∂     x   3         ⁡     [       c   3   2     _     ]         ]         +     l   ⁡     [             c   1   l     ⁢     c   2   m     ⁢     c   3   n       _     ⁢       ∂     u   1         ∂     x   1           +           c   1     l   -   l       ⁢     c   2     m   +   l       ⁢     c   3   n       _     ⁢       ∂     u   1         ∂     x   2           +           c   1     l   -   1       ⁢     c   2   m     ⁢     c   3     n   +   l         _     ⁢       ∂     u   1         ∂     x   3             ]       +     m   ⁡     [             c   1     l   +   l       ⁢     c   2     m   -   l       ⁢     c   3   n       _     ⁢       ∂     u   2         ∂     x   1           +           c   1   l     ⁢     c   2   m     ⁢     c   3   n       _     ⁢       ∂     u   2         ∂     x   2           +           c   1   l     ⁢     c   2     m   -   l       ⁢     c   3     n   +   l         _     ⁢       ∂     u   2         ∂     x   3             ]       +     n   ⁡     [             c   1     l   +   l       ⁢     c   2   m     ⁢     c   3     n   -   l         _     ⁢       ∂     u   3         ∂     x   1           +           c   1   l     ⁢     c   2     m   +   l       ⁢     c   3     n   -   l         _     ⁢       ∂     u   3         ∂     x   2           +           c   1   l     ⁢     c   2   m     ⁢     c   3   n       _     ⁢       ∂     u   3         ∂     x   3             ]       +       ∂     ∂     x   1         ⁡     [         c   1     l   +   l       ⁢     c   2   m     ⁢     c   3   n       _     ]       +       ∂     ∂     x   2         ⁡     [         c   1   l     ⁢     c   2     m   +   l       ⁢     c   3   n       _     ]       +       ∂     ∂     x   3         ⁡     [         c   1   l     ⁢     c   2   m     ⁢     c   3     n   +   l         _     ]       +       [           ⁢           c   1     l   +   l       ⁢     c   2   m     ⁢     c   3   n       _     -     l   ⁢       (     c   1   2     )     _     ⁢           ⁢         c   1     l   -   l       ⁢     c   2   m     ⁢     c   3   n       _       -       m   ⁡     [         c   1     ⁢     c   2       _     ]       ⁢         c   1   l     ⁢     c   2     m   -   l       ⁢     c   3   n       _       -       n   ⁡     [         c   1     ⁢     c   3       _     ]       ⁢         c   1   l     ⁢     c   2   m     ⁢     c   3     n   -   l         _         ]     ⁢     1   ρ     ⁢       ∂   ρ       ∂     x   1           +       [           ⁢           c   1   l     ⁢     c   2     m   +   l       ⁢     c   3   n       _     -     l   ⁢       (       c   1     ⁢     c   2       )     _     ⁢         c   1     l   -   l       ⁢     c   2   m     ⁢     c   3   n       _       -       m   ⁡     [       c   2   2     _     ]       ⁢         c   1   l     ⁢     c   2     m   -   l       ⁢     c   3   n       _       -       n   ⁡     [         c   2     ⁢     c   3       _     ]       ⁢         c   1   l     ⁢     c   2   m     ⁢     c   3     n   -   l         _         ]     ⁢     1   ρ     ⁢       ∂   ρ       ∂     x   2           +       [           ⁢           c   1   l     ⁢     c   2   m     ⁢     c   3     n   +   l         _     -       l   ⁡     [         c   1     ⁢     c   3       _     ]       ⁢         c   1     l   -   l       ⁢     c   2   m     ⁢     c   3   n       _       -       m   ⁡     [         c   2     ⁢     c   3       _     ]       ⁢         c   1   l     ⁢     c   2     m   -   l       ⁢     c   3   n       _       -       n   ⁡     [       c   3   2     _     ]       ⁢         c   1   l     ⁢     c   2   m     ⁢     c   3     n   -   l         _         ]     ⁢     1   ρ     ⁢       ∂   ρ       ∂     x   3             =     υ   ⁡     [             c   1   l     ⁢     c   2   m     ⁢     c   3   n       _     *     -         c   1   l     ⁢     c   2   m     ⁢     c   3   n       _       ]             
where  c 1   l c 2   m c 3   n    is the instantaneous thermal velocity moment c 1   l c 2   m c 3   n  averaged over F(v), and  c 1   l c 2   m c 3   n   * is the instantaneous scattered thermal velocity moment c 1   l c 2   m c 3   n  averaged over F(v)*. The latter may be calculated from the full Boltzmann collision change integrals, Kliegel (1990).
 
   Equation Set 4 
   (4-1); (4-2); (4-3) 
                       c   1     l   +   l       ⁢     c   2   m     ⁢     c   3   n       _     =       ⁢         l   ⁡     [       c   1   2     _     ]       ⁢         c   1     l   -   l       ⁢     c   2   m     ⁢     c   3   n       _       +       m   ⁡     [         c   1     ⁢     c   2       _     ]       ⁢         c   1   l     ⁢     c   2     m   -   l       ⁢     c   3   n       _       +       n   ⁡     [         c   1     ⁢     c   3       _     ]       ⁢         c   1   l     ⁢     c   2   m     ⁢     c   3     n   -   l         _                           c   1   l     ⁢     c   2     m   +   l       ⁢     c   3   n       _     =       ⁢         l   ⁡     [         c   1     ⁢     c   2       _     ]       ⁢         c   1     l   -   l       ⁢     c   2   m     ⁢     c   3   n       _       +       m   ⁡     [       c   2   2     _     ]       ⁢         c   1   l     ⁢     c   2     m   -   l       ⁢     c   3   n       _       +       n   ⁡     [         c   2     ⁢     c   3       _     ]       ⁢         c   1   l     ⁢     c   2   m     ⁢     c   3     n   -   l         _                           c   1   l     ⁢     c   2   m     ⁢     c   3     n   +   l         _     =       ⁢         l   ⁡     [         c   1     ⁢     c   3       _     ]       ⁢         c   1     l   -   l       ⁢     c   2   m     ⁢     c   3   n       _       +       m   ⁡     [         c   2     ⁢     c   3       _     ]       ⁢         c   1   l     ⁢     c   2     m   -   l       ⁢     c   3   n       _       +       n   ⁡     [       c   3   2     _     ]       ⁢         c   1   l     ⁢     c   2   m     ⁢     c   3     n   -   l         _                     
neglected terms being of Burnett order, where l+m+n=2 N+1, an odd number.
 
   Equation Set 5 
   (5-1) 
                 ∂     ∂   t       ⁡     [         c   1   l     ⁢     c   2   m     ⁢     c   3   n       _     ]       +       u   1     ⁢       ∂     ∂     x   1         ⁡     [         c   1   l     ⁢     c   2   m     ⁢     c   3   n       _     ]         +       u   2     ⁢       ∂     ∂     x   2         ⁡     [         c   1   l     ⁢     c   2   m     ⁢     c   3   n       _     ]         +       u   3     ⁢       ∂     ∂     x   3         ⁡     [         c   1   l     ⁢     c   2   m     ⁢     c   3   n       _     ]         +     l   ⁡     [             c   1   l     ⁢     c   2   m     ⁢     c   3   n       _     ⁢       ∂     u   1         ∂     x   1           +           c   1     l   -   l       ⁢     c   2     m   +   l       ⁢     c   3   n       _     ⁢       ∂     u   1         ∂     x   2           +           c   1     l   -   1       ⁢     c   2   m     ⁢     c   3     n   +   l         _     ⁢       ∂     u   1         ∂     x   3           +         c   1   2     _     ⁢       ∂     ∂     x   1         ⁡     [         c   1     l   -   l       ⁢     c   2   m     ⁢     c   3   n       _     ]         +           c   1     ⁢     c   2       _     ⁢       ∂     ∂     x   2         ⁡     [         c   1     l   -   l       ⁢     c   2   m     ⁢     c   3   n       _     ]         +           c   1     ⁢     c   3       _     ⁢       ∂     ∂     x   3         ⁡     [         c   1     l   -   l       ⁢     c   2   m     ⁢     c   3   n       _     ]           ]       +     m   ⁡     [             c   1     l   +   l       ⁢     c   2     m   -   l       ⁢     c   3   n       _     ⁢       ∂     u   2         ∂     x   1           +           c   1   l     ⁢     c   2   m     ⁢     c   3   n       _     ⁢       ∂     u   2         ∂     x   2           +           c   1   l     ⁢     c   2     m   -   l       ⁢     c   3     n   +   l         _     ⁢       ∂     u   2         ∂     x   3           +           c   1     ⁢     c   2       _     ⁢       ∂     ∂     x   1         ⁡     [         c   1   l     ⁢     c   2     m   -   l       ⁢     c   3   n       _     ]         +         c   2   2     _     ⁢       ∂     ∂     x   2         ⁡     [         c   1   l     ⁢     c   2     m   -   l       ⁢     c   3   n       _     ]         +           c   2     ⁢     c   3       _     ⁢       ∂     ∂     x   3         ⁡     [         c   1   l     ⁢     c   2     m   -   l       ⁢     c   3   n       _     ]           ]       +     n   ⁡     [             c   1     l   +   l       ⁢     c   2   m     ⁢     c   3     n   -   l         _     ⁢       ∂     u   3         ∂     x   1           +           c   1   l     ⁢     c   2     m   +   l       ⁢     c   3     n   -   l         _     ⁢       ∂     u   3         ∂     x   2           +           c   1   l     ⁢     c   2   m     ⁢     c   3   n       _     ⁢       ∂     u   3         ∂     x   3           +           c   1     ⁢     c   3       _     ⁢       ∂     ∂     x   1         ⁡     [         c   1   l     ⁢     c   2   m     ⁢     c   3     n   -   l         _     ]         +           c   2     ⁢     c   3       _     ⁢       ∂     ∂     x   2         ⁡     [         c   1   l     ⁢     c   2   m     ⁢     c   3     n   -   l         _     ]         +         c   3   2     _     ⁢       ∂     ∂     x   3         ⁡     [         c   1   l     ⁢     c   2   m     ⁢     c   3     n   -   l         _     ]           ]         =     υ   ⁡     [             c   1   l     ⁢     c   2   m     ⁢     c   3   n       _     *     -         c   1   l     ⁢     c   2   m     ⁢     c   3   n       _       ]             
where l+m+n=2N+1, an odd number and the neglected terms are of Burnett order in the convective terms (left hand side).
 
   Equation Set 6 
   Twenty Moment Equations 
   Continuity (6-1) 
                 ∂   ρ       ∂   t       +       ∂     ∂     x   1         ⁡     [     ρ   ⁢           ⁢     u   1       ]       +       ∂     ∂     x   2         ⁡     [     ρ   ⁢           ⁢     u   2       ]       +       ∂     ∂     x   3         ⁡     [     ρ   ⁢           ⁢     u   3       ]         =   0         
Momentum (6-2)
 
                         ∂   ρ       ∂   t       ⁡     [     ρ   ⁢           ⁢     u   1       ]       +       ∂     ∂     x   1         ⁡     [     ρ   ⁡     [       u   1   2     +       c   1   2     _       ]       ]       +       ∂     ∂     x   2         ⁡     [     ρ   ⁡     [         u   1     ⁢     u   2       +         c   1     ⁢     c   2       _       ]       ]       +     ∂     ∂     x   3         +     [     ρ   ⁡     [         u   1     ⁢     u   3       +         c   1     ⁢     c   3       _       ]       ]       =   0                           
Directional Thermal Energy (6-3)
 
                       ∂     ∂   t       ⁡     [       c   1   2     _     ]       +       u   1     ⁢       ∂     ∂     x   1         ⁡     [       c   1   2     _     ]         +       u   2     ⁢       ∂     ∂     x   2         ⁡     [       c   1   2     _     ]         +       u   3     ⁢       ∂     ∂     x   3         ⁡     [       c   1   2     _     ]         +     2   ⁡     [           c   1   2     _     ⁢       ∂     u   1         ∂     x   1           +           c   1     ⁢     c   2       _     ⁢       ∂     u   1         ∂     x   2           +           c   1     ⁢     c   3       _     ⁢       ∂     u   1         ∂     x   3             ]       +       1   ρ     ⁡     [         ∂     ∂     x   1         ⁡     [     ρ   ⁢         c   1     ⁢     c   1   2       _       ]       +       ∂     ∂     x   2         ⁢           [     ρ   ⁢         c   2     ⁢     c   1   2       _       ]     +       ∂     ∂     x   3         ⁡     [     ρ   ⁢         c   3     ⁢     c   1   2       _       ]         ]         =     υ   ⁡     [         1   3     ⁢       c   2     _       -       c   1   2     _       ]                               
Thermal Shear (6-4)
 
   
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         t 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                         
                         _ 
                       
                       ] 
                     
                   
                   + 
                   
                     
                       u 
                       1 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       u 
                       2 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       u 
                       3 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           2 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           u 
                           1 
                         
                       
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         c 
                         2 
                         2 
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           u 
                           1 
                         
                       
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           2 
                         
                         ⁢ 
                         
                           c 
                           3 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           u 
                           1 
                         
                       
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         c 
                         1 
                         2 
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           u 
                           2 
                         
                       
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           2 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           u 
                           2 
                         
                       
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                     ⁢ 
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           3 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           u 
                           2 
                         
                       
                       
                         dx 
                         3 
                       
                     
                   
                   + 
                   
                     
                       1 
                       ρ 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             ∂ 
                             
                               ∂ 
                               
                                 x 
                                 1 
                               
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               ρ 
                               ⁢ 
                               
                                 
                                   
                                     c 
                                     2 
                                   
                                   ⁢ 
                                   
                                     c 
                                     1 
                                     2 
                                   
                                 
                                 _ 
                               
                             
                             ] 
                           
                         
                         + 
                         
                           
                             ∂ 
                             
                               ∂ 
                               
                                 x 
                                 2 
                               
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               ρ 
                               ⁢ 
                               
                                 
                                   
                                     c 
                                     1 
                                   
                                   ⁢ 
                                   
                                     c 
                                     2 
                                     2 
                                   
                                 
                                 _ 
                               
                             
                             ] 
                           
                         
                         + 
                         
                           
                             ∂ 
                             
                               ∂ 
                               
                                 x 
                                 3 
                               
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               ρ 
                               ⁢ 
                               
                                 
                                   
                                     c 
                                     1 
                                   
                                   ⁢ 
                                   
                                     c 
                                     2 
                                   
                                   ⁢ 
                                   
                                     c 
                                     3 
                                   
                                 
                                 _ 
                               
                             
                             ] 
                           
                         
                       
                       ] 
                     
                   
                 
                 = 
                 
                   
                     - 
                     υ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     _ 
                   
                 
               
             
             
               
                   
               
             
           
         
       
     
   
   Equation Set 6 
   Directional Thermal Energy Fluxes (6-5) 
   
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         t 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             1 
                             2 
                           
                         
                         _ 
                       
                       ] 
                     
                   
                   + 
                   
                     
                       u 
                       1 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               1 
                               2 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       u 
                       2 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               1 
                               2 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       u 
                       3 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               1 
                               2 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     3 
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               
                                 c 
                                 1 
                               
                               ⁢ 
                               
                                 c 
                                 1 
                                 2 
                               
                             
                             _ 
                           
                           ⁢ 
                           
                             
                               ∂ 
                               
                                 u 
                                 1 
                               
                             
                             
                               ∂ 
                               
                                 x 
                                 1 
                               
                             
                           
                           ⁢ 
                           
                             
                               
                                 c 
                                 2 
                               
                               ⁢ 
                               
                                 c 
                                 1 
                                 2 
                               
                             
                             _ 
                           
                           ⁢ 
                           
                             
                               ∂ 
                               
                                 u 
                                 1 
                               
                             
                             
                               ∂ 
                               
                                 x 
                                 2 
                               
                             
                           
                         
                         + 
                         
                           
                             
                               
                                 c 
                                 3 
                               
                               ⁢ 
                               
                                 c 
                                 1 
                                 2 
                               
                             
                             _ 
                           
                           ⁢ 
                           
                             
                               ∂ 
                               
                                 u 
                                 1 
                               
                             
                             
                               ∂ 
                               
                                 x 
                                 3 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   - 
                   
                     3 
                     ⁢ 
                     
                       
                         
                           c 
                           1 
                           2 
                         
                         _ 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               ∂ 
                               
                                 ∂ 
                                 
                                   x 
                                   1 
                                 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   c 
                                   1 
                                   2 
                                 
                                 _ 
                               
                               ] 
                             
                           
                           + 
                           
                             
                               ∂ 
                               
                                 ∂ 
                                 
                                   x 
                                   2 
                                 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     c 
                                     1 
                                   
                                   ⁢ 
                                   
                                     c 
                                     2 
                                   
                                 
                                 _ 
                               
                               ] 
                             
                           
                           + 
                           
                             
                               ∂ 
                               
                                 ∂ 
                                 
                                   x 
                                   3 
                                 
                               
                             
                             [ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 
                                   c 
                                   1 
                                 
                                 ⁢ 
                                 
                                   c 
                                   3 
                                 
                               
                               _ 
                             
                             ] 
                           
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             c 
                             1 
                             2 
                           
                           ⁢ 
                           
                             c 
                             1 
                             2 
                           
                         
                         _ 
                       
                       ] 
                     
                   
                   + 
                   
                     
                       
                         ∂ 
                         ` 
                       
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                     [ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           2 
                         
                         ⁢ 
                         
                           c 
                           1 
                           2 
                         
                       
                       _ 
                     
                     ] 
                   
                   + 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                     [ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           3 
                         
                         ⁢ 
                         
                           c 
                           1 
                           2 
                         
                       
                       _ 
                     
                     ] 
                   
                   + 
                   
                     [ 
                     
                       
                         
                           [ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 
                                   c 
                                   1 
                                   2 
                                 
                                 ⁢ 
                                 
                                   c 
                                   1 
                                   2 
                                 
                               
                               _ 
                             
                             - 
                             
                                 
                             
                             ⁢ 
                             
                               3 
                               ⁢ 
                               
                                 
                                   
                                     c 
                                     1 
                                     2 
                                   
                                   _ 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       c 
                                       1 
                                       2 
                                     
                                     _ 
                                   
                                   ] 
                                 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                           1 
                           ρ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ∂ 
                             ρ 
                           
                           
                             ∂ 
                             
                               x 
                               1 
                             
                           
                         
                       
                       + 
                       
                         
                           [ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 
                                   c 
                                   1 
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                                 ⁢ 
                                 
                                   c 
                                   1 
                                   2 
                                 
                               
                               _ 
                             
                             - 
                             
                               3 
                               ⁢ 
                               
                                 
                                   
                                     
                                       c 
                                       1 
                                     
                                     ⁢ 
                                     
                                       c 
                                       2 
                                     
                                   
                                   _ 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       c 
                                       1 
                                       2 
                                     
                                     _ 
                                   
                                   ] 
                                 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           1 
                           ρ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ∂ 
                             ρ 
                           
                           
                             ∂ 
                             
                               x 
                               2 
                             
                           
                         
                       
                       + 
                       
                         
                           [ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 
                                   c 
                                   1 
                                 
                                 ⁢ 
                                 
                                   c 
                                   3 
                                 
                                 ⁢ 
                                 
                                   c 
                                   1 
                                   2 
                                 
                               
                               _ 
                             
                             - 
                             
                               3 
                               ⁢ 
                               
                                 
                                   
                                     
                                       c 
                                       1 
                                     
                                     ⁢ 
                                     
                                       c 
                                       3 
                                     
                                   
                                   _ 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       c 
                                       1 
                                       2 
                                     
                                     _ 
                                   
                                   ] 
                                 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           1 
                           ρ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ∂ 
                             ρ 
                           
                           
                             ∂ 
                             
                               x 
                               3 
                             
                           
                         
                       
                     
                     ] 
                   
                 
                 = 
                 
                   
                     υ 
                     2 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                         
                         _ 
                       
                       - 
                       
                         3 
                         ⁢ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               1 
                               2 
                             
                           
                           _ 
                         
                       
                     
                     ] 
                   
                 
               
             
             
               
                   
               
             
           
         
       
     
   
   Equation Set 6 
   (6-6) 
   
     
       
         
           
             
               
                 ∂ 
                 
                   ∂ 
                   t 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       c 
                       1 
                     
                     ⁢ 
                     
                       c 
                       2 
                       2 
                     
                   
                   _ 
                 
                 ] 
               
             
             + 
             
               
                 u 
                 1 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                         2 
                       
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               
                 u 
                 2 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                         2 
                       
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               
                 u 
                 3 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                         2 
                       
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                     2 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     u 
                     1 
                   
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     2 
                     2 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     u 
                     1 
                   
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     3 
                   
                   ⁢ 
                   
                     c 
                     2 
                     2 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     u 
                     1 
                   
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
             + 
             
               2 
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         
                           c 
                           2 
                         
                         ⁢ 
                         
                           c 
                           1 
                           2 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           u 
                           2 
                         
                       
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           2 
                           2 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           u 
                           2 
                         
                       
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           2 
                         
                         ⁢ 
                         
                           c 
                           3 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           u 
                           2 
                         
                       
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                   
                 
                 ] 
               
             
             - 
             
               
                 
                   c 
                   2 
                   2 
                 
                 _ 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           c 
                           1 
                           2 
                         
                         _ 
                       
                       ] 
                     
                   
                   + 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                         
                         _ 
                       
                       ] 
                     
                   
                   + 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             3 
                           
                         
                         _ 
                       
                       ] 
                     
                   
                 
                 ] 
               
             
             - 
             
               2 
               ⁢ 
               
                 
                   
                     
                       c 
                       1 
                     
                     ⁢ 
                     
                       c 
                       2 
                     
                   
                   _ 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                     + 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             c 
                             2 
                             2 
                           
                           _ 
                         
                         ] 
                       
                     
                     + 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       c 
                       1 
                       2 
                     
                     ⁢ 
                     
                       c 
                       2 
                       2 
                     
                   
                   _ 
                 
                 ] 
               
             
             + 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       c 
                       1 
                     
                     ⁢ 
                     
                       c 
                       2 
                     
                     ⁢ 
                     
                       c 
                       2 
                       2 
                     
                   
                   _ 
                 
                 ] 
               
             
             + 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
               [ 
               
                   
               
               ⁢ 
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                   ⁢ 
                   
                     c 
                     2 
                     2 
                   
                 
                 _ 
               
               ] 
             
             + 
             
               
                 [ 
                 
                     
                 
                 ⁢ 
                 
                   
                     
                       
                         c 
                         1 
                         2 
                       
                       ⁢ 
                       
                         c 
                         2 
                         2 
                       
                     
                     _ 
                   
                   - 
                   
                     
                       
                         c 
                         1 
                         2 
                       
                       _ 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           c 
                           2 
                           2 
                         
                         _ 
                       
                       ] 
                     
                   
                   - 
                   
                     2 
                     ⁢ 
                     
                       
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                         
                         _ 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                 
                 ] 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 1 
                 ρ 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   ∂ 
                   ρ 
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 [ 
                 
                     
                 
                 ⁢ 
                 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         2 
                         2 
                       
                     
                     _ 
                   
                   - 
                   
                     3 
                     ⁢ 
                     
                       
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                         
                         _ 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             c 
                             2 
                             2 
                           
                           _ 
                         
                         ] 
                       
                     
                   
                 
                 ] 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 1 
                 ρ 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   ∂ 
                   ρ 
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 [ 
                 
                     
                 
                 ⁢ 
                 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                       ⁢ 
                       
                         c 
                         2 
                         2 
                       
                     
                     _ 
                   
                   - 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           3 
                         
                       
                       _ 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           c 
                           2 
                           2 
                         
                         _ 
                       
                       ] 
                     
                   
                   - 
                   
                     2 
                     ⁢ 
                     
                       
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                         
                         _ 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                 
                 ] 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 1 
                 ρ 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   ∂ 
                   ρ 
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
           
           = 
           
             
               υ 
               2 
             
             ⁡ 
             
               [ 
               
                 
                   
                     1 
                     3 
                   
                   ⁢ 
                   
                     
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     _ 
                   
                 
                 - 
                 
                   3 
                   ⁢ 
                   
                     
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         1 
                         2 
                       
                     
                     _ 
                   
                 
               
               ] 
             
           
         
       
     
   
   Equation Set 6 
   Thermal Structure (6-7) 
                 ∂     ∂   t       ⁡     [         c   1     ⁢     c   2     ⁢     c   3       _     ]       +       u   1     ⁢       ∂     ∂     x   1         ⁡     [         c   1     ⁢     c   2     ⁢     c   3       _     ]         +       u   2     ⁢       ∂     ∂     x   2         ⁡     [         c   1     ⁢     c   2     ⁢     c   3       _     ]         +       u   3     ⁢       ∂     ∂     x   3         ⁡     [         c   1     ⁢     c   2     ⁢     c   3       _     ]         +           c   1     ⁢     c   2     ⁢     c   3       _     ⁢       ∂     u   1         ∂     x   1           +           c   3     ⁢     c   2   2       _     ⁢       ∂     u   1         ∂     x   2           +           c   2     ⁢     c   3   2       _     ⁢       ∂     u   1         ∂     x   3           +           c   3     ⁢     c   1   2       _     ⁢       ∂     u   2         ∂     x   1           +           c   1     ⁢     c   2     ⁢     c   3       _     ⁢       ∂     u   2         ∂     x   2           +           c   1     ⁢     c   3   2       _     ⁢       ∂     u   2         ∂     x   3           +           c   2     ⁢     c   1   2       _     ⁢       ∂     u   3         ∂     x   1           +           c   1     ⁢     c   2   2       _     ⁢       ∂     u   3         ∂     x   2           +           c   1     ⁢     c   2     ⁢     c   3       _     ⁢       ∂     u   3         ∂     x   3           -           c   2     ⁢     c   3       _     ⁡     [         ∂     ∂     x   1         ⁡     [       c   1   2     _     ]       +       ∂     ∂     x   2         ⁡     [         c   1     ⁢     c   2       _     ]       +       ∂     ∂     x   3         ⁡     [         c   1     ⁢     c   3       _     ]         ]       -           c   1     ⁢     c   3       _     ⁡     [         ∂     ∂     x   1         ⁡     [         c   1     ⁢     c   2       _     ]       +       ∂     ∂     x   2         ⁡     [       c   2   2     _     ]       +       ∂     ∂     x   3         ⁡     [         c   2     ⁢     c   3       _     ]         ]       -           c   1     ⁢     c   2       _     ⁡     [         ∂     ∂     x   1         ⁡     [         c   1     ⁢     c   3       _     ]       +       ∂     ∂     x   2         ⁡     [         c   2     ⁢     c   3       _     ]       +       ∂     ∂     x   3         ⁡     [       c   3   2     _     ]         ]       +       ∂     ∂     x   1         ⁡     [         c   2     ⁢     c   3     ⁢     c   1   2       _     ]       +       ∂     ∂     x   2         ⁡     [         c   1     ⁢     c   3     ⁢     c   2   2       _     ]       +       ∂     ∂     x   3         ⁡     [         c   1     ⁢     c   2     ⁢     c   3   2       _     ]       +       [           c   2     ⁢     c   3     ⁢     c   1   2       _     -           c   2     ⁢     c   3       _     ⁡     [       c   1   2     _     ]       -     2   ⁢           c   1     ⁢     c   2       _     ⁡     [         c   2     ⁢     c   3       _     ]           ]     ⁢           ⁢     1   ρ     ⁢           ⁢       ∂   ρ       ∂     x   1           +       [           ⁢           c   1     ⁢     c   2     ⁢     c   3   2       _     -           c   1     ⁢     c   3       _     ⁡     [       c   2   2     _     ]       -     2   ⁢           c   1     ⁢     c   2       _     ⁡     [         c   2     ⁢     c   3       _     ]           ]     ⁢           ⁢     1   ρ     ⁢           ⁢       ∂   ρ       ∂     x   2           +       [           ⁢           c   1     ⁢     c   2     ⁢     c   3   2       _     -           c   1     ⁢     c   2       _     ⁡     [       c   3   2     _     ]       -     2   ⁢           c   1     ⁢     c   3       _     ⁡     [         c   2     ⁢     c   3       _     ]           ]     ⁢     1   ρ     ⁢       ∂   ρ       ∂     x   3             =       -     3   2       ⁢   υ   ⁢           ⁢         c   1     ⁢     c   2     ⁢     c   3       _             
where  c 2   =  c 1   2   +  c 1   1   +  c 3   2    and the collision change terms (right hand sides) have been calculated using the method of Kliegel (1990).
 
   Equation Set 7 
   (7-1; 7-2; 7-3; 7-4) 
   
     
       
         
           
               
           
           ⁢ 
           
             
               
                 
                   
                     
                       
                         c 
                         1 
                         2 
                       
                       ⁢ 
                       
                         c 
                         1 
                         2 
                       
                     
                     _ 
                   
                   = 
                     
                   ⁢ 
                   
                     3 
                     ⁢ 
                     
                       
                         
                           c 
                           1 
                           2 
                         
                         _ 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             c 
                             1 
                             2 
                           
                           _ 
                         
                         ] 
                       
                     
                   
                 
               
             
             
               
                 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         1 
                         2 
                       
                     
                     _ 
                   
                   = 
                     
                   ⁢ 
                   
                     3 
                     ⁢ 
                     
                       
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                         
                         _ 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             c 
                             1 
                             2 
                           
                           _ 
                         
                         ] 
                       
                     
                   
                 
               
             
             
               
                 
                   
                     
                       
                         c 
                         1 
                         2 
                       
                       ⁢ 
                       
                         c 
                         2 
                         2 
                       
                     
                     _ 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         
                           c 
                           1 
                           2 
                         
                         _ 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             c 
                             2 
                             2 
                           
                           _ 
                         
                         ] 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           _ 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 c 
                                 1 
                               
                               ⁢ 
                               
                                 c 
                                 2 
                               
                             
                             _ 
                           
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         3 
                         2 
                       
                     
                     _ 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                         
                         _ 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             c 
                             3 
                             2 
                           
                           _ 
                         
                         ] 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           _ 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 c 
                                 2 
                               
                               ⁢ 
                               
                                 c 
                                 3 
                               
                             
                             _ 
                           
                           ] 
                         
                       
                     
                   
                 
               
             
           
         
       
     
   
   Equation Set 8 
   Closed Directional Thermal Energy Fluxes (8-1; 8-2) 
   
     
       
         
           
             
               
                 ∂ 
                 
                   ∂ 
                   t 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       c 
                       1 
                     
                     ⁢ 
                     
                       c 
                       1 
                       2 
                     
                   
                   _ 
                 
                 ] 
               
             
             + 
             
               
                 u 
                 1 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         1 
                         2 
                       
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               
                 u 
                 2 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         1 
                         2 
                       
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               
                 u 
                 3 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         1 
                         2 
                       
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               3 
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           1 
                           2 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           u 
                           1 
                         
                       
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           2 
                         
                         ⁢ 
                         
                           c 
                           1 
                           2 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           u 
                           1 
                         
                       
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           3 
                         
                         ⁢ 
                         
                           c 
                           1 
                           2 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           u 
                           1 
                         
                       
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                   
                 
                 ] 
               
             
             + 
             
               3 
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         c 
                         1 
                         2 
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             c 
                             1 
                             2 
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           2 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             c 
                             1 
                             2 
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           3 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             c 
                             1 
                             2 
                           
                           _ 
                         
                         ] 
                       
                     
                   
                 
                 ] 
               
             
           
           = 
           
             
               υ 
               2 
             
             ⁡ 
             
               [ 
               
                 
                   
                     
                       c 
                       1 
                     
                     ⁢ 
                     
                       c 
                       2 
                     
                   
                   _ 
                 
                 - 
                 
                   3 
                   ⁢ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         1 
                         2 
                       
                     
                     _ 
                   
                 
               
               ] 
             
           
         
       
     
   
   
     
       
         
           
             
               
                 ∂ 
                 
                   ∂ 
                   t 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       c 
                       1 
                     
                     ⁢ 
                     
                       c 
                       2 
                       2 
                     
                   
                   _ 
                 
                 ] 
               
             
             + 
             
               
                 u 
                 1 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                         2 
                       
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               
                 u 
                 2 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                         2 
                       
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               
                 u 
                 3 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                         2 
                       
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                     2 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     u 
                     1 
                   
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     2 
                     2 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     u 
                     1 
                   
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     3 
                   
                   ⁢ 
                   
                     c 
                     2 
                     2 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     u 
                     1 
                   
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
             + 
             
               2 
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         
                           c 
                           2 
                         
                         ⁢ 
                         
                           c 
                           1 
                           2 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           u 
                           2 
                         
                       
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           2 
                           2 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           u 
                           2 
                         
                       
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           2 
                         
                         ⁢ 
                         
                           c 
                           3 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           u 
                           2 
                         
                       
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                   
                 
                 ] 
               
             
             + 
             
               
                 
                   c 
                   1 
                   2 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       c 
                       2 
                       2 
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       c 
                       2 
                       2 
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       c 
                       2 
                       2 
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               2 
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           2 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         c 
                         2 
                         2 
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           2 
                         
                         ⁢ 
                         
                           c 
                           3 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                 
                 ] 
               
             
           
           = 
           
             
               υ 
               6 
             
             ⁡ 
             
               [ 
               
                 
                   
                     
                       c 
                       1 
                     
                     ⁢ 
                     
                       c 
                       2 
                     
                   
                   _ 
                 
                 - 
                 
                   9 
                   ⁢ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                         2 
                       
                     
                     _ 
                   
                 
               
               ] 
             
           
         
       
     
   
   Equation Set 8 
   Closed Thermal Structure (8-3) 
                 ∂     ∂   t       ⁡     [         c   1     ⁢     c   2     ⁢     c   3       _     ]       +       u   1     ⁢       ∂     ∂     x   1         ⁡     [         c   1     ⁢     c   2     ⁢     c   3       _     ]         +       u   2     ⁢       ∂     ∂     x   2         ⁡     [         c   1     ⁢     c   2     ⁢     c   3       _     ]         +       u   3     ⁢       ∂     ∂     x   3         ⁡     [         c   1     ⁢     c   2     ⁢     c   3       _     ]         +           c   1     ⁢     c   2     ⁢     c   3       _     ⁢       ∂     u   1         ∂     x   1           +           c   3     ⁢     c   2   2       _     ⁢       ∂     u   1         ∂     x   2           +           c   2     ⁢     c   3   2       _     ⁢       ∂     u   1         ∂     x   3           +           c   3     ⁢     c   1   2       _     ⁢       ∂     u   2         ∂     x   1           +           c   1     ⁢     c   2     ⁢     c   3       _     ⁢       ∂     u   2         ∂     x   2           +           c   1     ⁢     c   3   2       _     ⁢       ∂     u   2         ∂     x   3           +           c   2     ⁢     c   1   2       _     ⁢       ∂     u   3         ∂     x   1           +           c   1     ⁢     c   2   2       _     ⁢       ∂     u   3         ∂     x   2           +           c   1     ⁢     c   2     ⁢     c   3       _     ⁢       ∂     u   3         ∂     x   3           +         c   1   2     _     ⁢       ∂     ∂     x   1         ⁡     [         c   2     ⁢     c   3       _     ]         +           c   1     ⁢     c   2       _     ⁢       ∂     ∂     x   2         ⁡     [         c   2     ⁢     c   3       _     ]         +           c   1     ⁢     c   3       _     ⁢       ∂     dx   3       ⁡     [         c   2     ⁢     c   3       _     ]         +           c   1     ⁢     c   2       _     ⁢       ∂     ∂     x   1         ⁡     [         c   1     ⁢     c   3       _     ]         +         c   2   2     _     ⁢       ∂     ∂     x   2         ⁡     [         c   1     ⁢     c   3       _     ]         +           c   2     ⁢     c   3       _     ⁢       ∂     ∂     x   3         ⁡     [         c   1     ⁢     c   3       _     ]         +           c   1     ⁢     c   3       _     ⁢       ∂     ∂     x   1         ⁡     [         c   1     ⁢     c   2       _     ]         +           c   2     ⁢     c   3       _     ⁢       ∂     ∂     x   2         ⁡     [         c   1     ⁢     c   2       _     ]         +         c   3   2     _     ⁢       ∂     ∂     x   3         ⁡     [         c   1     ⁢     c   2       _     ]           =       -     3   2       ⁢   υ   ⁢           ⁢         c   1     ⁢     c   2     ⁢     c   3       _             
the neglected terms are of Burnett order in the convective terms (left hand sides).
 
   Equation Set 9 
   Continuity (9-1) 
                   ⁢           ∂   ρ       ∂   t       +       ∂     ∂     x   1         ⁡     [     ρ   ⁢           ⁢     u   1       ]       +       ∂     ∂     x   2         ⁡     [     ρ   ⁢           ⁢     u   2       ]       +       ∂     ∂     x   3         ⁡     [     ρ   ⁢           ⁢     u   3       ]         =   0           
Momentum (9-2)
 
                 ∂     u   1         ∂   t       +       u   1     ⁢       ∂     u   1         ∂     x   1           +       u   2     ⁢       ∂     u   1         ∂     x   2           +       u   3     ⁢       ∂     u   1         ∂     x   3           +       1   ρ     ⁡     [         ∂     ∂     x   1         ⁡     [     ρ   ⁢           ⁢       c   1   2     _       ]       +       ∂     ∂     x   2         ⁡     [     ρ   ⁢           ⁢         c   1     ⁢     c   2       _       ]       +       ∂     ∂     x   3         ⁡     [     ρ   ⁢           ⁢         c   1     ⁢     c   3       _       ]         ]         =   0         
Directional Thermal Energy (9-3)
 
                 ∂     ∂   t       ⁡     [       c   1   2     _     ]       +       u   1     ⁢       ∂     ∂     x   1         ⁡     [       c   1   2     _     ]         +       u   2     ⁢       ∂     ∂     x   2         ⁡     [       c   1   2     _     ]         +       u   3     ⁢       ∂     ∂     x   3         ⁡     [       c   1   2     _     ]         +     2   ⁡     [           c   1   2     _     ⁢       ∂     u   1         ∂     x   1           +           c   1     ⁢     c   2       _     ⁢       ∂     u   1         ∂     x   2           +           c   1     ⁢     c   3       _     ⁢       ∂     u   1         ∂     x   3             ]       +       1   ρ     ⁡     [         ∂     ∂     x   1         ⁡     [     ρ   ⁢           ⁢         c   1     ⁢     c   1   2       _       ]       +       ∂     ∂     x   2         ⁡     [     ρ   ⁢           ⁢         c   2     ⁢     c   1   2       _       ]       +       ∂     ∂     x   3         ⁡     [     ρ   ⁢           ⁢         c   3     ⁢     c   1   2       _       ]         ]         =     υ   ⁡     [         1   3     ⁢       c   2     _       -       c   1   2     _       ]             
Thermal Shear (9-4)
 
                   ⁢         c   1     ⁢     c   2       =     -       1   υ     ⁡     [           c   2   2     _     ⁢       ∂     u   1         ∂     x   2           +         c   1   2     _     ⁢       ∂     u   2         ∂     x   1             ]                 
Directional Thermal Energy Fluxes (9-5)
 
   
     
       
         
           
               
           
           ⁢ 
           
             
               
                 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         1 
                         2 
                       
                     
                     _ 
                   
                   = 
                     
                   ⁢ 
                   
                     - 
                     
                       
                         
                           
                             c 
                             1 
                             2 
                           
                           _ 
                         
                         
                           2 
                           ⁢ 
                           υ 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             7 
                             ⁢ 
                             
                               
                                 ∂ 
                                 
                                   ∂ 
                                   
                                     x 
                                     1 
                                   
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     c 
                                     1 
                                     2 
                                   
                                   _ 
                                 
                                 ] 
                               
                             
                           
                           + 
                           
                             
                               ∂ 
                               
                                 ∂ 
                                 
                                   x 
                                   1 
                                 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   c 
                                   2 
                                   2 
                                 
                                 _ 
                               
                               ] 
                             
                           
                           + 
                           
                             
                               ∂ 
                               
                                 ∂ 
                                 
                                   x 
                                   1 
                                 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   c 
                                   3 
                                   2 
                                 
                                 _ 
                               
                               ] 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
             
             
               
                 
                   
                     
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         1 
                         2 
                       
                     
                     _ 
                   
                   = 
                     
                   ⁢ 
                   
                     - 
                     
                       
                         
                           
                             c 
                             2 
                             2 
                           
                           _ 
                         
                         
                           6 
                           ⁢ 
                           υ 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             5 
                             ⁢ 
                             
                               
                                 ∂ 
                                 
                                   ∂ 
                                   
                                     x 
                                     2 
                                   
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     c 
                                     1 
                                     2 
                                   
                                   _ 
                                 
                                 ] 
                               
                             
                           
                           + 
                           
                             3 
                             ⁢ 
                             
                               
                                 ∂ 
                                 
                                   ∂ 
                                   
                                     x 
                                     2 
                                   
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     c 
                                     2 
                                     2 
                                   
                                   _ 
                                 
                                 ] 
                               
                             
                           
                           + 
                           
                             
                               ∂ 
                               
                                 ∂ 
                                 
                                   x 
                                   2 
                                 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   c 
                                   3 
                                   2 
                                 
                                 _ 
                               
                               ] 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
             
           
         
       
     
   
   Equation Set 9 
   Thermal Structure (9-6)
 
  c 1 c 2 c 3   =0
 
neglected terms being of Burnett order in the thermal shear, directional energy fluxes and structure. The shear stress and thermal energy flux relationships are identical to the results Kliegel (1990) obtained by an Enskog-type solution of the Boltzmann equation.
 
   Equation Set 10 
   Time Average Directional Thermal Energy (10-1) 
                 ∂     ∂   t       ⁡     [       c   1   2       _   _       ]       +         u   1     _     ⁢       ∂     ∂     x   1         ⁡     [       c   1   2       _   _       ]         +         u   2     _     ⁢       ∂     ∂     x   2         ⁡     [       c   1   2       _   _       ]         +         u   3     _     ⁢       ∂     ∂     x   3         ⁡     [       c   1   2       _   _       ]         +     2   ⁡     [           c   1   2       _   _       ⁢       ∂       u   1     _         ∂     x   1           +           c   1     ⁢     c   2         _   _       ⁢       ∂       u   1     _         ∂     x   2           +           c   1     ⁢     c   3         _   _       ⁢       ∂       u   1     _         ∂     x   3             ]       +       1     ρ   _       ⁡     [         ∂     ∂     x   1         ⁡     [       ρ   _     ⁢           ⁢         c   1     ⁢     c   1   2         _   _         ]       +       ∂     ∂     x   2         ⁡     [       ρ   _     ⁢           ⁢         c   2     ⁢     c   1   2         _   _         ]       +       ∂     ∂     x   3         ⁡     [       ρ   _     ⁢           ⁢         c   3     ⁢     c   1   2         _   _         ]         ]         =       υ   _     ⁡     [         1   3     ⁢       c   2       _   _         -       c   1   2       _   _         ]             
Time Average Thermal Shear (10-2)
 
   
     
       
         
           
             
               
                 ∂ 
                 
                   ∂ 
                   t 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       c 
                       1 
                     
                     ⁢ 
                     
                       c 
                       2 
                     
                   
                   
                     _ 
                     _ 
                   
                 
                 ] 
               
             
             + 
             
               
                 
                   u 
                   1 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   u 
                   2 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   u 
                   3 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 
                   c 
                   2 
                   2 
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
             + 
             
               
                 
                   c 
                   1 
                   2 
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       2 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       2 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       2 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
             + 
             
               
                 1 
                 
                   ρ 
                   _ 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           ρ 
                           _ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               c 
                               1 
                               2 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       ] 
                     
                   
                   + 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           ρ 
                           _ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                               2 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       ] 
                     
                   
                   + 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           ρ 
                           _ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       ] 
                     
                   
                 
                 ] 
               
             
           
           = 
           
             
               - 
               
                 υ 
                 _ 
               
             
             ⁢ 
             
                 
             
             ⁢ 
             
               
                 
                   c 
                   1 
                 
                 ⁢ 
                 
                   c 
                   2 
                 
               
               
                 _ 
                 _ 
               
             
           
         
       
     
   
   Equation Set 10 
   Time Average Directional Thermal Energy Fluxes (10-3) 
   
     
       
         
           
             
               
                 ∂ 
                 
                   ∂ 
                   t 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       c 
                       1 
                     
                     ⁢ 
                     
                       c 
                       1 
                       2 
                     
                   
                   
                     _ 
                     _ 
                   
                 
                 ] 
               
             
             + 
             
               
                 
                   u 
                   1 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         1 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   u 
                   2 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         1 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   u 
                   3 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               3 
               ⁡ 
               
                 [ 
                 
                   
                     
                       [ 
                       
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             1 
                             2 
                           
                         
                         
                           _ 
                           _ 
                         
                       
                       ] 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           
                             u 
                             1 
                           
                           _ 
                         
                       
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           2 
                         
                         ⁢ 
                         
                           c 
                           1 
                           2 
                         
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           
                             u 
                             1 
                           
                           _ 
                         
                       
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                   
                   + 
                   
                     
                       c 
                       3 
                     
                     ⁢ 
                     
                       c 
                       1 
                       2 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           
                             u 
                             1 
                           
                           _ 
                         
                       
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                   
                 
                 ] 
               
             
             + 
             
               3 
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         c 
                         1 
                         2 
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             c 
                             1 
                             2 
                           
                           
                             _ 
                             _ 
                           
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           2 
                         
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             c 
                             1 
                             2 
                           
                           
                             _ 
                             _ 
                           
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           3 
                         
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             c 
                             1 
                             2 
                           
                           
                             _ 
                             _ 
                           
                         
                         ] 
                       
                     
                   
                 
                 ] 
               
             
           
           = 
           
             
               
                 υ 
                 _ 
               
               2 
             
             ⁡ 
             
               [ 
               
                 
                   
                     
                       c 
                       1 
                     
                     ⁢ 
                     
                       c 
                       2 
                     
                   
                   
                     _ 
                     _ 
                   
                 
                 - 
                 
                   3 
                   ⁢ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         1 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                 
               
               ] 
             
           
         
       
     
   
   Equation Set 10 
   Time Average Directional Thermal Energy Fluxes (10-4) 
   
     
       
         
           
             
               
                 ∂ 
                 
                   ∂ 
                   t 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       c 
                       1 
                     
                     ⁢ 
                     
                       c 
                       2 
                       2 
                     
                   
                   
                     _ 
                     _ 
                   
                 
                 ] 
               
             
             + 
             
               
                 
                   u 
                   1 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   u 
                   2 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   u 
                   3 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 [ 
                 
                   
                     
                       c 
                       1 
                     
                     ⁢ 
                     
                       c 
                       2 
                       2 
                     
                   
                   
                     _ 
                     _ 
                   
                 
                 ] 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     2 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 c 
                 3 
               
               ⁢ 
               
                 c 
                 2 
                 2 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
             + 
             
               2 
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         
                           c 
                           2 
                         
                         ⁢ 
                         
                           c 
                           1 
                           2 
                         
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           
                             u 
                             2 
                           
                           _ 
                         
                       
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           2 
                         
                         ⁢ 
                         
                           c 
                           1 
                           2 
                         
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           
                             u 
                             2 
                           
                           _ 
                         
                       
                       
                         ∂ 
                         
                           x 
                           
                             2 
                             ⁢ 
                             
                                 
                             
                           
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           2 
                         
                         ⁢ 
                         
                           c 
                           3 
                         
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           
                             u 
                             2 
                           
                           _ 
                         
                       
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                   
                 
                 ] 
               
             
             + 
             
               2 
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           2 
                         
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         c 
                         2 
                         2 
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                     ⁢ 
                     
                       
                         [ 
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                         
                         ] 
                       
                       
                         _ 
                         _ 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           2 
                         
                         ⁢ 
                         
                           c 
                           3 
                         
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                     ⁢ 
                     
                       
                         [ 
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                         
                         ] 
                       
                       
                         _ 
                         _ 
                       
                     
                   
                 
                 ] 
               
             
             + 
             
               
                 
                   c 
                   1 
                   2 
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       c 
                       2 
                       2 
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       c 
                       2 
                       2 
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       c 
                       2 
                       2 
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
           
           = 
           
             
               
                 υ 
                 _ 
               
               2 
             
             ⁡ 
             
               [ 
               
                 
                   
                     1 
                     3 
                   
                   ⁢ 
                   
                     
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                 
                 - 
                 
                   3 
                   ⁢ 
                   
                     
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         1 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                 
               
               ] 
             
           
         
       
     
   
   Equation Set 10 
   Time Average Thermal Structure (10-5) 
   
     
       
         
           
             
               
                 ∂ 
                 
                   ∂ 
                   t 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       c 
                       1 
                     
                     ⁢ 
                     
                       c 
                       2 
                     
                     ⁢ 
                     
                       c 
                       3 
                     
                   
                   
                     _ 
                     _ 
                   
                 
                 ] 
               
             
             + 
             
               
                 
                   u 
                   1 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   u 
                   2 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   u 
                   3 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     3 
                   
                   ⁢ 
                   
                     c 
                     2 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     3 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     3 
                   
                   ⁢ 
                   
                     c 
                     1 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       2 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       2 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     3 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       2 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     1 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       3 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       3 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       3 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
             + 
             
               
                 
                   c 
                   1 
                   2 
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   c 
                   2 
                   2 
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   c 
                   3 
                   2 
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
           
           = 
           
             
               - 
               
                 3 
                 2 
               
             
             ⁢ 
             
               υ 
               _ 
             
             ⁢ 
             
               
                 
                     
                 
                 ⁢ 
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
               
               
                 _ 
                 _ 
               
             
           
         
       
     
   
   Equation Set 11 
   Continuity (11-1) 
                       ∂     ρ   _         ∂   t       +       ∂     ∂     x   1         ⁡     [       ρ   _     ⁢           ⁢       u   1     _       ]       +       ∂     ∂     x   2         ⁡     [       ρ   _     ⁢           ⁢       u   2     _       ]       +       ∂     ∂     x   3         ⁡     [       ρ   _     ⁢           ⁢       u   3     _       ]         =   0                           
Momentum (11-2)
 
               ∂     ∂   t       ⁡     [       ρ   _     ⁢           ⁢       u   1     _       ]       +       ∂     ∂     x   1         ⁡     [       ρ   _     ⁡     [         u   _     1   2     +       u   1   ′2     _     +       c   1   2       _   _         ]       ]       +       ∂     ∂     x   2         ⁡     [       ρ   _     ⁡     [           u   1     ⁢     u   2       _     +         u   1   ′     ⁢     u   2   ′       _     +         c   1     ⁢     c   2         _   _         ]       ]       +       ∂     ∂     x   3         [       ρ   ⁡     [         u   1     ⁢     u   3       +         u   1   ′     ⁢     u   3   ′       _     +         c   1     ⁢     c   3         _   _         ]       =   0             
Turbulent Directional Energy (11-3)
 
   
     
       
         
           
             
               ∂ 
               
                 ∂ 
                 t 
               
             
             ⁡ 
             
               [ 
               
                 
                   ρ 
                   _ 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         u 
                         1 
                         2 
                       
                       _ 
                     
                     + 
                     
                       
                         u 
                         1 
                         ′2 
                       
                       _ 
                     
                     + 
                     
                       
                         c 
                         1 
                         2 
                       
                       
                         _ 
                         _ 
                       
                     
                   
                   ] 
                 
               
               ] 
             
           
           + 
           
             
               ∂ 
               
                 ∂ 
                 
                   x 
                   1 
                 
               
             
             ⁡ 
             
               [ 
               
                 
                   ρ 
                   _ 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         u 
                         1 
                         3 
                       
                       _ 
                     
                     + 
                     
                       3 
                       ⁢ 
                       
                         
                           u 
                           1 
                         
                         _ 
                       
                       ⁢ 
                       
                         
                           u 
                           1 
                           ′2 
                         
                         _ 
                       
                     
                     + 
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           1 
                           ′2 
                         
                       
                       _ 
                     
                     + 
                     
                       3 
                       ⁢ 
                       
                         
                           u 
                           1 
                         
                         _ 
                       
                       ⁢ 
                       
                         
                           c 
                           1 
                           2 
                         
                         
                           _ 
                           _ 
                         
                       
                     
                     + 
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           1 
                           2 
                         
                       
                       
                         _ 
                         _ 
                       
                     
                   
                   ] 
                 
               
               ] 
             
           
           + 
           
             
               ∂ 
               
                 ∂ 
                 
                   x 
                   2 
                 
               
             
             ⁡ 
             
               [ 
               
                 
                   ρ 
                   _ 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         
                           u 
                           2 
                         
                         ⁢ 
                         
                           u 
                           1 
                           2 
                         
                       
                       _ 
                     
                     + 
                     
                       
                         
                           u 
                           2 
                         
                         ⁢ 
                         
                           u 
                           1 
                           ′2 
                         
                       
                       _ 
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           
                             u 
                             1 
                           
                           ⁢ 
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             1 
                             ′ 
                           
                         
                         _ 
                       
                     
                     + 
                     
                       
                         
                           u 
                           2 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           1 
                           ′2 
                         
                       
                       _ 
                     
                     + 
                     
                       
                         
                           u 
                           2 
                         
                         _ 
                       
                       ⁢ 
                       
                         
                           c 
                           1 
                           2 
                         
                         
                           _ 
                           _ 
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           u 
                           1 
                         
                         _ 
                       
                       ⁢ 
                       
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                         
                         
                           _ 
                           _ 
                         
                       
                     
                     + 
                     
                       
                         
                           c 
                           2 
                         
                         ⁢ 
                         
                           c 
                           1 
                           2 
                         
                       
                       
                         _ 
                         _ 
                       
                     
                   
                   ] 
                 
               
               ] 
             
           
           + 
           
               
             
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       ρ 
                       _ 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               u 
                               3 
                             
                             ⁢ 
                             
                               u 
                               1 
                               2 
                             
                           
                           _ 
                         
                         + 
                         
                           
                             
                               u 
                               3 
                             
                             _ 
                           
                           ⁢ 
                           
                             
                               u 
                               1 
                               ′2 
                             
                             _ 
                           
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                             
                               
                                 u 
                                 1 
                               
                               ⁢ 
                               
                                 u 
                                 1 
                                 ′ 
                               
                               ⁢ 
                               
                                 u 
                                 3 
                                 ′ 
                               
                             
                             _ 
                           
                         
                         + 
                         
                           
                             
                               u 
                               3 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               1 
                               ′2 
                             
                           
                           _ 
                         
                         + 
                         
                           
                             
                               u 
                               3 
                               ′ 
                             
                             _ 
                           
                           ⁢ 
                           
                             
                               c 
                               1 
                               2 
                             
                             
                               _ 
                               _ 
                             
                           
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                             
                               u 
                               1 
                             
                             _ 
                           
                           ⁢ 
                           
                             
                               
                                 c 
                                 1 
                               
                               ⁢ 
                               
                                 c 
                                 3 
                               
                             
                             
                               _ 
                               _ 
                             
                           
                         
                         + 
                         
                           
                             
                               c 
                               3 
                             
                             ⁢ 
                             
                               c 
                               1 
                               2 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       ] 
                     
                   
                   ] 
                 
               
               = 
               
                 
                   ρ 
                   _ 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     υ 
                     _ 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             c 
                             2 
                           
                           
                             _ 
                             _ 
                           
                         
                         3 
                       
                       - 
                       
                         
                           c 
                           1 
                           2 
                         
                         
                           _ 
                           _ 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
   
   Equation Set 11 
   Turbulent Shear (11-4) 
                 ∂     ∂   t       ⁡     [       ρ   _     ⁡     [           u   1     ⁢     u   2       _     +         u   1   ′     ⁢     u   2   ′       _     +         c   1     ⁢     c   2         _   _       +     ]       ]       +       ∂     ∂     x   1         ⁡     [       ρ   _     ⁡     [           u   2     ⁢     u   1   2       _     +         u   2     ⁢     u   1   ′2       _     +         u   2   ′     ⁢     u   1   ′2       _     +     2   ⁢         u   1     ⁢     u   1   ′     ⁢     u   2   ′       _       +         u   2     _     ⁢       c   1   2       _   _         +     2   ⁢       u   1     _     ⁢         c   1     ⁢     c   2         _   _           ]       ]       +       ∂     ∂     x   2         ⁡     [       ρ   _     ⁡     [           u   2     ⁢     u   2   2       _     +         u   1     ⁢     u   2   ′2       _     +     2   ⁢         u   2     ⁢     u   1   ′     ⁢     u   2   ′       _       +         u   1   ′     ⁢     u   2   ′2       _     +     2   ⁢       u   2     _     ⁢         c   1     ⁢     c   2         _   _         +         u   1     _     ⁢       c   2   2       _   _         +         c   1     ⁢     c   2   2         _   _         ]       ]       +       ∂     ∂     x   3         ⁡     [       ρ   _     ⁡     [           u   1     ⁢     u   2     ⁢     u   3       _     +         u   3     ⁢     u   1   ′     ⁢     u   2   ′       _     +         u   2     ⁢     u   1   ′     ⁢     u   3   ′       _     +         u   1   ′     ⁢     u   2   ′     ⁢     u   3   ′       _     +         u   1   ′     ⁢     u   2   ′     ⁢     u   3   ′       _     +         u   3     _     ⁢         c   1     ⁢     c   2         _   _         +         u   2     _     ⁢         c   1     ⁢     c   3         _   _         +         u   1     _     ⁢         c   2     ⁢     c   3         _   _         +         c   1     ⁢     c   2     ⁢     c   3         _   _         ]       ]         =       -     ρ   _       ⁢     υ   _     ⁢           ⁢         c   1     ⁢     c   2         _   _               
Turbulent Directional Energy Fluxes (11-5)
 
                       ∂     ∂   t       ⁡     [       ρ   _     ⁡     [           u   1     _     3     +     3   ⁢       u   1     _     ⁢       u   1   ′2     _       +         u   1   ′     ⁢     u   1   ′2       _     +     3   ⁢       u   1     _     ⁢       c   1   2       _   _         +         c   1     ⁢     c   1   2         _   _         ]       ]       +       ∂     ∂     x   1         ⁡     [       ρ   _     ⁡     [           u   1     _     4     +     6   ⁢         u   1     _     2     ⁢       u   1   ′2     _       +     4   ⁢       u   1     _     ⁢         u   1   ′     ⁢     u   1   ′2       _       +         u   1   ′2     ⁢     u   1   ′2       _       ]       ]       +       ∂     ∂     x   1         ⁡     [       ρ   _     ⁡     [       6   ⁢         u   1     _     2     ⁢       c   1   2     _       +     4   ⁢       u   1     _     ⁢         c   1     ⁢     c   1   2         _   _         +     6   ⁢       u   1   ′2     _     ⁢       c   1   2       _   _         +         c   1   2     ⁢     c   1   2         _   _         ]       ]       +       ∂     ∂     x   2         [           ⁢       ρ   _     [           ⁢             u   1     _     3     ⁢       u   2     _       +     3   ⁢         u   1     _     2     ⁢         u   1   ′     ⁢     u   2   ′       _       +     3   ⁢         u   1     ⁢     u   2       _     ⁢       u   1   ′2     _       +         u   2     _     ⁢         u   1   ′     ⁢     u   1   ′2       _       +     3   ⁢       u   1     _     ⁢         u   2   ′     ⁢     u   1   ′2       _       +         u   1   ′     ⁢     u   2   ′     ⁢     u   1   ′2       _       ]     ]     +       ∂     ∂     x   2         ⁡     [       ρ   _     ⁡     [       3   ⁢         u   1     _     2     ⁢         c   1     ⁢     c   2         _   _         +     3   ⁢       u   1     _     ⁢       u   2     _     ⁢       c   1   2       _   _         +         u   2     _     ⁢         c   1     ⁢     c   1   2         _   _         +     3   ⁢       u   1     _     ⁢         c   2     ⁢     c   1   2         _   _         +     3   ⁢       u   1   ′2     _     ⁢         c   1     ⁢     c   2         _   _         +     3   ⁢         u   1   ′     ⁢     u   2   ′       _     ⁢       c   1   2       _   _         +         c   1     ⁢     c   2     ⁢     c   1   2         _   _         ]       ]       +       ∂     ∂     x   3         [           ⁢       ρ   _     [           ⁢             u   1     _     3     ⁢       u   3     _       +     3   ⁢         u   1     _     2     ⁢         u   1   ′     ⁢     u   3   ′       _       +     3   ⁢         u   1     ⁢     u   3       _     ⁢       u   1   ′2     _       +         u   3     _     ⁢         u   1   ′     ⁢     u   1   ′2       _       +     3   ⁢       u   1     _     ⁢         u   3   ′     ⁢     u   1   ′2       _       +         u   1   ′     ⁢     u   3   ′     ⁢     u   1   ′2       _       ]     ]     +       ∂     ∂     x   3         ⁡     [       ρ   _     ⁡     [       3   ⁢         u   1     _     2     ⁢         c   1     ⁢     c   3         _   _         +     3   ⁢       u   1     _     ⁢       u   3     _     ⁢       c   1   2       _   _         +         u   3     _     ⁢         c   1     ⁢     c   1   2         _   _         +     3   ⁢       u   1     _     ⁢         c   3     ⁢     c   1   2         _   _         +     3   ⁢         u   1   ′     2     _     ⁢         c   1     ⁢     c   3         _   _         +     3   ⁢         u   1   ′     ⁢     u   3   ′       _     ⁢       c   1   ′2       _   _         +         c   1     ⁢     c   3     ⁢     c   1   2         _   _         ]       ]         =       ρ   _     ⁢           ⁢       υ   _     [       3   ⁢         u   1     _     [           c   2       _   _       3     -       c   1   2       _   _         ]       +       1   2     ⁡     [           c   1     ⁢     c   2         _   _       -     3   ⁢         c   1     ⁢     c   1   2         _   _           ]         ]                               
Turbulent Directional Energy Fluxes (11-6)
 
                                 ∂     ∂   t       ⁡     [       ρ   _     ⁡     [           u   1     ⁢     u   2   2       _     +         u   1     ⁢     u   2   ′2       _     +     2   ⁢         u   2     ⁢     u   1   ′     ⁢     u   2   ′       _       +         u   1   ′     ⁢     u   2   ′2       _     +         u   1     _     ⁢         c   2   2     _     _       +     2   ⁢       u   2     _     ⁢         c   1     ⁢     c   2         _   _         +           c   1     ⁢     c   2   2       _     _       ]       ]       +               ∂     ∂     x   1         ⁡     [       ρ   _     ⁡     [           u   1   2     ⁢     u   2   2       _     +         u   1   2     ⁢     u   1   ′2       _     +         u   2   2     ⁢     u   1   ′2       _     +     4   ⁢         u   1     ⁢     u   2     ⁢     u   1   ′     ⁢     u   2   ′       _       +     2   ⁢         u   1     ⁢     u   1   ′     ⁢     u   2   ′2       _       +     2   ⁢         u   2     ⁢     u   2   ′     ⁢     u   1   ′2       _       +         u   1   ′2     ⁢     u   2   ′2       _       ]       ]       +       ∂     ∂     x   1         ⁡     [       ρ   _     ⁡     [           u   1   2     _     ⁢       c   2   2       _   _         +         u   2   2     _     ⁢       c   1   2       _   _         +     4   ⁢         u   1     ⁢     u   2       _     ⁢         c   1     ⁢     c   2         _   _         +     2   ⁢       u   1     _     ⁢         c   1     ⁢     c   2   2         _   _         +         u   1   ′2     _     ⁢       c   2   2       _   _         +     2   ⁢       u   2     _     ⁢         c   2     ⁢     c   1   2         _   _         +         u   2   ′2     _     ⁢       c   1   2       _   _         +     4   ⁢         u   1   ′     ⁢     u   2   ′       _     ⁢         c   1     ⁢     c   2         _   _         +         c   1   2     ⁢     c   2   2         _   _         ]       ]       +       ∂     ∂     x   2         ⁡     [       ρ   _     ⁡     [           u   1     ⁢     u   2   3       _     +     3   ⁢         u   2   2     ⁢     u   1   ′     ⁢     u   2   ′       _       +     3   ⁢         u   1     ⁢     u   2     ⁢     u   2   ′2       _       +         u   1     ⁢     u   2   ′     ⁢     u   2   ′2       _     +     3   ⁢         u   2     ⁢     u   1   ′     ⁢     u   2   ′2       _       +         u   1   ′     ⁢     u   2   ′     ⁢     u   2   ′2       _       ]       ]       +       ∂     ∂     x   2         ⁡     [       ρ   _     ⁡     [       3   ⁢       u   2   2     _     ⁢         c   1     ⁢     c   2         _   _         +     3   ⁢         u   1     ⁢     u   2       _     ⁢       c   2   2       _   _         +         u   1     _     ⁢         c   2     ⁢     c   2   2         _   _         +     3   ⁢       u   2     _     ⁢         c   1     ⁢     c   2   2         _   _         +     3   ⁢       u   2   ′2     _     ⁢         c   1     ⁢     c   2         _   _         +     3   ⁢         u   1   ′     ⁢     u   2   ′       _     ⁢       c   2   2       _   _         +         c   1     ⁢     c   2     ⁢     c   2   2         _   _         ]       ]       +       ∂     ∂     x   3         [       ρ   _     [             u   1     ⁢     u   3     ⁢     u   2   2       _     +         u   1     ⁢     u   3     ⁢     u   2   ′2       _     +         u   2   2     ⁢     u   1   ′     ⁢     u   3   ′       _     +     2   ⁢         u   1     ⁢     u   2     ⁢     u   2   ′     ⁢     u   3   ′       _       +     2   ⁢         u   2     ⁢     u   3     ⁢     u   1   ′     ⁢     u   2   ′       _       +     2   ⁢         u   2     ⁢     u   1   ′     ⁢     u   2   ′     ⁢     u   3   ′       _       +         u   1     ⁢     u   3   ′     ⁢     u   2   ′2       _     +         u   3     ⁢     u   1   ′     ⁢     u   2   ′2       _     +         u   1   ′     ⁢     u   3   ′     ⁢     u   2   ′2       _     +       ∂     ∂     x   3         ⁡     [       ρ   _     ⁡     [           u   2   ′2     _     ⁢         c   1     ⁢     c   3         _   _         +           u   1     ⁢     u   3       _     ⁢       c   2   2       _   _         +     2   ⁢         u   1     ⁢     u   2       _     ⁢         c   2     ⁢     c   3         _   _         +     2   ⁢         u   2     ⁢     u   3       _     ⁢         c   1     ⁢     c   2         _   _         +         u   1     _     ⁢         c   3     ⁢     c   2   2         _   _         +     2   ⁢       u   2     _     ⁢         c   1     ⁢     c   2     ⁢     c   3         _   _         +         u   3     _     ⁢         c   1     ⁢     c   2   2         _   _           ]       ]       +       ∂     ∂     x   3         ⁡     [       ρ   _     ⁡     [           u   2   ′2     _     ⁢         c   1     ⁢     c   3         _   _         +     2   ⁢         u   1   ′     ⁢     u   2   ′       _     ⁢         c   2     ⁢     c   3         _   _         +           u   1   ′     ⁢     u   3   ′       _     ⁢       c   2   2       _   _         +     2   ⁢         u   2   ′     ⁢     u   3   ′       _     ⁢         c   1     ⁢     c   2         _   _         +         c   1     ⁢     c   3     ⁢     c   2   2         _   _         ]       ]         =         ρ   ⁢           ⁢   υ     _     ⁡     [           u   1     _     ⁡     [           c   2       _   _       3     -       c   1   2       _   _         ]       -     2   ⁢       u   2     _     ⁢         c   1     ⁢     c   2         _   _         +       1   6     ⁡     [           c   1     ⁢     c   2         _   _       -     9   ⁢         c   1     ⁢     c   2   2         _   _           ]         ]                             
Turbulent Structure (11-7)
 
   
     
       
         
           
             
               
                 ∂ 
                 
                   ∂ 
                   t 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     ρ 
                     _ 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             u 
                             1 
                           
                           ⁢ 
                           
                             u 
                             2 
                           
                           ⁢ 
                           
                             u 
                             3 
                           
                         
                         _ 
                       
                       + 
                       
                         
                           
                             u 
                             3 
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         
                           
                             u 
                             2 
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         
                           
                             u 
                             1 
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                           ⁢ 
                           
                             c 
                             3 
                           
                         
                         
                           _ 
                           _ 
                         
                       
                       + 
                       
                         
                           
                             u 
                             3 
                           
                           ⁢ 
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                         
                         _ 
                       
                       + 
                       
                         
                           
                             u 
                             2 
                           
                           ⁢ 
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             3 
                             ′ 
                           
                         
                         _ 
                       
                       + 
                       
                         
                           
                             u 
                             1 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             3 
                             ′ 
                           
                         
                         _ 
                       
                       + 
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             3 
                             ′ 
                           
                         
                         _ 
                       
                     
                     ] 
                   
                 
                 ] 
               
             
             + 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     ρ 
                     _ 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             u 
                             2 
                           
                           ⁢ 
                           
                             u 
                             3 
                           
                           ⁢ 
                           
                             u 
                             1 
                             2 
                           
                         
                         _ 
                       
                       + 
                       
                         
                           
                             u 
                             1 
                             2 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             3 
                             ′ 
                           
                         
                         _ 
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           
                             
                               u 
                               1 
                             
                             ⁢ 
                             
                               u 
                               2 
                             
                             ⁢ 
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           
                             
                               u 
                               1 
                             
                             ⁢ 
                             
                               u 
                               3 
                             
                             ⁢ 
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                           
                           _ 
                         
                       
                       + 
                       
                         
                           
                             u 
                             2 
                           
                           ⁢ 
                           
                             u 
                             3 
                           
                           ⁢ 
                           
                             u 
                             1 
                             ′2 
                           
                         
                         _ 
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           
                             
                               u 
                               1 
                             
                             ⁢ 
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                       
                       + 
                       
                         
                           
                             u 
                             2 
                           
                           ⁢ 
                           
                             u 
                             3 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             1 
                             ′2 
                           
                         
                         _ 
                       
                       + 
                       
                         
                           
                             u 
                             3 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             1 
                             ′2 
                           
                         
                         _ 
                       
                     
                     ] 
                   
                 
                 ] 
               
             
             + 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     ρ 
                     _ 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             3 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             1 
                             ′2 
                           
                         
                         _ 
                       
                       + 
                       
                         
                           
                             u 
                             1 
                             2 
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           
                             
                               u 
                               1 
                             
                             ⁢ 
                             
                               u 
                               3 
                             
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         
                           
                             
                               u 
                               2 
                             
                             ⁢ 
                             
                               u 
                               3 
                             
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             c 
                             1 
                             2 
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           
                             
                               u 
                               1 
                             
                             ⁢ 
                             
                               u 
                               2 
                             
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           
                             u 
                             1 
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         
                           
                             u 
                             2 
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               3 
                             
                             ⁢ 
                             
                               c 
                               1 
                               2 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         
                           
                             u 
                             3 
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               c 
                               1 
                               2 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                     
                     ] 
                   
                 
                 ] 
               
             
             + 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     ρ 
                     _ 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             u 
                             1 
                             ′2 
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         
                           
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             c 
                             1 
                             2 
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         
                           
                             c 
                             2 
                           
                           ⁢ 
                           
                             c 
                             3 
                           
                           ⁢ 
                           
                             c 
                             1 
                             2 
                           
                         
                         
                           _ 
                           _ 
                         
                       
                     
                     ] 
                   
                 
                 ] 
               
             
             + 
             
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       ρ 
                       _ 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               u 
                               1 
                             
                             ⁢ 
                             
                               u 
                               3 
                             
                             ⁢ 
                             
                               u 
                               2 
                               2 
                             
                           
                           _ 
                         
                         + 
                         
                           
                             
                               u 
                               2 
                               2 
                             
                             ⁢ 
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                             
                               
                                 u 
                                 1 
                               
                               ⁢ 
                               
                                 u 
                                 2 
                               
                               ⁢ 
                               
                                 u 
                                 2 
                                 ′ 
                               
                               ⁢ 
                               
                                 u 
                                 3 
                                 ′ 
                               
                             
                             _ 
                           
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                             
                               
                                 u 
                                 2 
                               
                               ⁢ 
                               
                                 u 
                                 3 
                               
                               ⁢ 
                               
                                 u 
                                 1 
                                 ′ 
                               
                               ⁢ 
                               
                                 u 
                                 2 
                                 ′ 
                               
                             
                             _ 
                           
                         
                         + 
                         
                           
                             
                               u 
                               1 
                             
                             ⁢ 
                             
                               u 
                               3 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′2 
                             
                           
                           _ 
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                             
                               
                                 u 
                                 2 
                               
                               ⁢ 
                               
                                 u 
                                 1 
                                 ′ 
                               
                               ⁢ 
                               
                                 u 
                                 2 
                                 ′ 
                               
                               ⁢ 
                               
                                 u 
                                 3 
                                 ′ 
                               
                             
                             _ 
                           
                         
                         + 
                         
                           
                             
                               u 
                               1 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′2 
                             
                           
                           _ 
                         
                         + 
                         
                           
                             
                               u 
                               3 
                             
                             ⁢ 
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′2 
                             
                           
                           _ 
                         
                       
                       ] 
                     
                   
                   ] 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       ρ 
                       _ 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′2 
                             
                           
                           _ 
                         
                         + 
                         
                           
                             
                               u 
                               2 
                               2 
                             
                             _ 
                           
                           ⁢ 
                           
                             
                               
                                 c 
                                 1 
                               
                               ⁢ 
                               
                                 c 
                                 3 
                               
                             
                             
                               _ 
                               _ 
                             
                           
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                             
                               
                                 u 
                                 2 
                               
                               ⁢ 
                               
                                 u 
                                 3 
                               
                             
                             _ 
                           
                           ⁢ 
                           
                             
                               
                                 c 
                                 1 
                               
                               ⁢ 
                               
                                 c 
                                 2 
                               
                             
                             
                               _ 
                               _ 
                             
                           
                         
                         + 
                         
                           
                             
                               
                                 u 
                                 1 
                               
                               ⁢ 
                               
                                 u 
                                 3 
                               
                             
                             _ 
                           
                           ⁢ 
                           
                             
                               c 
                               2 
                               2 
                             
                             
                               _ 
                               _ 
                             
                           
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                             
                               
                                 u 
                                 1 
                               
                               ⁢ 
                               
                                 u 
                                 2 
                               
                             
                             _ 
                           
                           ⁢ 
                           
                             
                               
                                 c 
                                 2 
                               
                               ⁢ 
                               
                                 c 
                                 3 
                               
                             
                             
                               _ 
                               _ 
                             
                           
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                             
                               u 
                               2 
                             
                             _ 
                           
                           ⁢ 
                           
                             
                               
                                 c 
                                 1 
                               
                               ⁢ 
                               
                                 c 
                                 2 
                               
                               ⁢ 
                               
                                 c 
                                 3 
                               
                             
                             
                               _ 
                               _ 
                             
                           
                         
                         + 
                         
                           
                             
                               u 
                               1 
                             
                             _ 
                           
                           ⁢ 
                           
                             
                               
                                 c 
                                 3 
                               
                               ⁢ 
                               
                                 c 
                                 2 
                                 2 
                               
                             
                             
                               _ 
                               _ 
                             
                           
                         
                         + 
                         
                           
                             
                               u 
                               3 
                             
                             _ 
                           
                           ⁢ 
                           
                             
                               
                                 c 
                                 1 
                               
                               ⁢ 
                               
                                 c 
                                 2 
                                 2 
                               
                             
                             
                               _ 
                               _ 
                             
                           
                         
                       
                       ] 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     ρ 
                     _ 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             u 
                             2 
                             ′2 
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             c 
                             2 
                             2 
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             3 
                           
                           ⁢ 
                           
                             c 
                             2 
                             2 
                           
                         
                         
                           _ 
                           _ 
                         
                       
                     
                     ] 
                   
                 
                 ] 
               
             
             + 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     ρ 
                     _ 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             u 
                             1 
                           
                           ⁢ 
                           
                             u 
                             2 
                           
                           ⁢ 
                           
                             u 
                             3 
                             2 
                           
                         
                         _ 
                       
                       + 
                       
                         
                           
                             u 
                             3 
                             2 
                           
                           ⁢ 
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                         
                         _ 
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           
                             
                               u 
                               2 
                             
                             ⁢ 
                             
                               u 
                               3 
                             
                             ⁢ 
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           
                             
                               u 
                               1 
                             
                             ⁢ 
                             
                               u 
                               3 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                       
                       + 
                       
                         
                           
                             u 
                             1 
                           
                           ⁢ 
                           
                             u 
                             2 
                           
                           ⁢ 
                           
                             u 
                             3 
                             ′2 
                           
                         
                         _ 
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           
                             
                               u 
                               3 
                             
                             ⁢ 
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                       
                       + 
                       
                         
                           
                             u 
                             2 
                           
                           ⁢ 
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             3 
                             ′2 
                           
                         
                         _ 
                       
                       + 
                       
                         
                           
                             u 
                             1 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             3 
                             ′2 
                           
                         
                         _ 
                       
                     
                     ] 
                   
                 
                 ] 
               
             
             + 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     ρ 
                     _ 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             3 
                             ′2 
                           
                         
                         _ 
                       
                       + 
                       
                         
                           
                             u 
                             3 
                             2 
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           
                             
                               u 
                               1 
                             
                             ⁢ 
                             
                               u 
                               3 
                             
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         
                           
                             
                               u 
                               1 
                             
                             ⁢ 
                             
                               u 
                               2 
                             
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             c 
                             3 
                             2 
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           
                             
                               u 
                               2 
                             
                             ⁢ 
                             
                               u 
                               3 
                             
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           
                             u 
                             3 
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         
                           
                             u 
                             2 
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               3 
                               2 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         
                           
                             u 
                             1 
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               c 
                               3 
                               2 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                     
                     ] 
                   
                 
                 ] 
               
             
             + 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     ρ 
                     _ 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             u 
                             3 
                             ′2 
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                           
                           _ 
                         
                         ⁢ 
                         
                           
                             c 
                             3 
                             2 
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       + 
                       
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                           ⁢ 
                           
                             c 
                             3 
                             2 
                           
                         
                         
                           _ 
                           _ 
                         
                       
                     
                     ] 
                   
                 
                 ] 
               
             
           
           = 
           
             
               ρυ 
               _ 
             
             ⁡ 
             
               [ 
               
                 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                   ⁢ 
                   
                     
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                 
                 + 
                 
                   
                     
                       u 
                       2 
                     
                     _ 
                   
                   ⁢ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                 
                 + 
                 
                   
                     
                       u 
                       3 
                     
                     _ 
                   
                   ⁢ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                 
                 + 
                 
                   
                     3 
                     2 
                   
                   ⁢ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                 
               
               ] 
             
           
         
       
     
   
   Equation Set 12 
   Directional Turbulent Energy (12-1) 
                 ∂     ∂   t       ⁡     [       u   1   ′2     _     ]       +         u   1     _     ⁢       ∂     ∂     x   1         ⁡     [       u   1   ′2     _     ]         +         u   2     _     ⁢       ∂     ∂     x   2         ⁡     [       u   1   ′2     _     ]         +         u   3     _     ⁢       ∂     ∂     x   3         ⁡     [       u   1   ′2     _     ]         +     2   ⁡     [           u   1   ′2     _     ⁢       ∂       u   1     _         ∂     x   1           +           u   1   ′     ⁢     u   2   ′       _     ⁢       ∂       u   1     _         ∂     x   2           +           u   1   ′     ⁢     u   3   ′       _     ⁢       ∂       u   1     _         ∂     x   3             ]       +       1     ρ   _       ⁡     [         ∂     ∂     x   1         ⁡     [         ρ   ⁢             _     ⁢         u   1   ′     ⁢     u   1   ′2       _       ]       +       ∂     ∂     x   2         ⁡     [         ρ   ⁢             _     ⁢         u   2   ′     ⁢     u   1   ′2       _       ]       +       ∂     ∂     x   3         ⁡     [       ρ   _     ⁢           ⁢         u   3   ′     ⁢     u   1   ′2       _       ]         ]         =   0         
Turbulent Shear (12-2)
 
   
     
       
         
           
             
               
                 ∂ 
                 
                   ∂ 
                   t 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       u 
                       1 
                       ′ 
                     
                     ⁢ 
                     
                       u 
                       2 
                       ′ 
                     
                   
                   _ 
                 
                 ] 
               
             
             + 
             
               
                 
                   u 
                   1 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         u 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         u 
                         2 
                         ′ 
                       
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   u 
                   2 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         u 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         u 
                         2 
                         ′ 
                       
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   u 
                   3 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         u 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         u 
                         2 
                         ′ 
                       
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     u 
                     1 
                     ′ 
                   
                   ⁢ 
                   
                     u 
                     2 
                     ′ 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 
                   u 
                   2 
                   ′2 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 
                   
                     u 
                     2 
                     ′ 
                   
                   ⁢ 
                   
                     u 
                     3 
                     ′ 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
             + 
             
               
                 
                   u 
                   1 
                   ′2 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       2 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 
                   
                     u 
                     1 
                     ′ 
                   
                   ⁢ 
                   
                     u 
                     2 
                     ′ 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       2 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 
                   
                     u 
                     1 
                     ′ 
                   
                   ⁢ 
                   
                     u 
                     3 
                     ′ 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       2 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
             + 
             
               
                 1 
                 
                   ρ 
                   _ 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           ρ 
                           _ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               1 
                               ′2 
                             
                           
                           _ 
                         
                       
                       ] 
                     
                   
                   + 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           ρ 
                           _ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′2 
                             
                           
                           _ 
                         
                       
                       ] 
                     
                   
                   + 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           ρ 
                           _ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                       
                       ] 
                     
                   
                 
                 ] 
               
             
           
           = 
           0 
         
       
     
   
   Equation Set 12 
   Directional Turbulent Energy Fluxes (12-3) 
   
     
       
         
             
           
               
             
                 
               
                   
                 
                     
                   
                     
                       
                         ∂ 
                         
                           ∂ 
                           t 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               1 
                               ′2 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                     + 
                     
                       
                         
                           u 
                           1 
                         
                         _ 
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             ∂ 
                             
                               x 
                               1 
                             
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 u 
                                 1 
                                 ′ 
                               
                               ⁢ 
                               
                                 u 
                                 1 
                                 ′2 
                               
                             
                             _ 
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         
                           u 
                           2 
                         
                         _ 
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             ∂ 
                             
                               x 
                               2 
                             
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 u 
                                 1 
                                 ′ 
                               
                               ⁢ 
                               
                                 u 
                                 1 
                                 ′2 
                               
                             
                             _ 
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         
                           u 
                           3 
                         
                         _ 
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             ∂ 
                             
                               x 
                               3 
                             
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 u 
                                 1 
                                 ′ 
                               
                               ⁢ 
                               
                                 u 
                                 1 
                                 ′2 
                               
                             
                             _ 
                           
                           ] 
                         
                       
                     
                     + 
                     
                       3 
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   u 
                                   1 
                                   ′ 
                                 
                                 ⁢ 
                                 
                                   u 
                                   1 
                                   ′2 
                                 
                               
                               _ 
                             
                             ⁢ 
                             
                               
                                 ∂ 
                                 
                                   
                                     u 
                                     1 
                                   
                                   _ 
                                 
                               
                               
                                 ∂ 
                                 
                                   x 
                                   1 
                                 
                               
                             
                           
                           + 
                           
                             
                               
                                 
                                   u 
                                   2 
                                   ′ 
                                 
                                 ⁢ 
                                 
                                   u 
                                   1 
                                   ′2 
                                 
                               
                               _ 
                             
                             ⁢ 
                             
                               
                                 ∂ 
                                 
                                   
                                     u 
                                     1 
                                   
                                   _ 
                                 
                               
                               
                                 ∂ 
                                 
                                   x 
                                   2 
                                 
                               
                             
                           
                           + 
                           
                             
                               
                                 
                                   u 
                                   3 
                                   ′ 
                                 
                                 ⁢ 
                                 
                                   u 
                                   1 
                                   ′2 
                                 
                               
                               _ 
                             
                             ⁢ 
                             
                               
                                 ∂ 
                                 
                                   
                                     u 
                                     1 
                                   
                                   _ 
                                 
                               
                               
                                 ∂ 
                                 
                                   x 
                                   3 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     - 
                     
                       3 
                       ⁢ 
                       
                         
                           
                             u 
                             1 
                             ′2 
                           
                           _ 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 ∂ 
                                 
                                   ∂ 
                                   
                                     x 
                                     1 
                                   
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     u 
                                     1 
                                     ′2 
                                   
                                   _ 
                                 
                                 ] 
                               
                             
                             + 
                             
                               
                                 ∂ 
                                 
                                   ∂ 
                                   
                                     x 
                                     2 
                                   
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     
                                       u 
                                       1 
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       u 
                                       2 
                                       ′ 
                                     
                                   
                                   _ 
                                 
                                 ] 
                               
                             
                             + 
                             
                               
                                 ∂ 
                                 
                                   ∂ 
                                   
                                     x 
                                     3 
                                   
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     
                                       u 
                                       1 
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       u 
                                       3 
                                       ′ 
                                     
                                   
                                   _ 
                                 
                                 ] 
                               
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       3 
                       ⁢ 
                       
                         
                           
                             c 
                             1 
                             2 
                           
                           
                             _ 
                             _ 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 δ 
                                 
                                   δx 
                                   1 
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     u 
                                     1 
                                     ′2 
                                   
                                   _ 
                                 
                                 ] 
                               
                             
                             + 
                             
                               
                                 ∂ 
                                 
                                   ∂ 
                                   
                                     x 
                                     2 
                                   
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     
                                       u 
                                       1 
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       u 
                                       2 
                                       ′ 
                                     
                                   
                                   _ 
                                 
                                 ] 
                               
                             
                             + 
                             
                               
                                 ∂ 
                                 
                                   ∂ 
                                   
                                     x 
                                     3 
                                   
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     
                                       u 
                                       1 
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       u 
                                       3 
                                       ′ 
                                     
                                   
                                   _ 
                                 
                                 ] 
                               
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       3 
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 u 
                                 1 
                                 ′2 
                               
                               _ 
                             
                             ⁢ 
                             
                               
                                 ∂ 
                                 
                                   ∂ 
                                   
                                     x 
                                     1 
                                   
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     c 
                                     1 
                                     2 
                                   
                                   
                                     _ 
                                     _ 
                                   
                                 
                                 ] 
                               
                             
                           
                           + 
                           
                             
                               
                                 
                                   u 
                                   1 
                                   ′ 
                                 
                                 ⁢ 
                                 
                                   u 
                                   2 
                                   ′ 
                                 
                               
                               _ 
                             
                             ⁢ 
                             
                               
                                 ∂ 
                                 
                                   ∂ 
                                   
                                     x 
                                     2 
                                   
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     c 
                                     1 
                                     2 
                                   
                                   
                                     _ 
                                     _ 
                                   
                                 
                                 ] 
                               
                             
                           
                           + 
                           
                             
                               
                                 
                                   u 
                                   1 
                                   ′ 
                                 
                                 ⁢ 
                                 
                                   u 
                                   3 
                                   ′ 
                                 
                               
                               _ 
                             
                             ⁢ 
                             
                               
                                 ∂ 
                                 
                                   ∂ 
                                   
                                     x 
                                     3 
                                   
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     c 
                                     1 
                                     2 
                                   
                                   
                                     _ 
                                     _ 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         
                           ∂ 
                           
                             ∂ 
                             
                               x 
                               1 
                             
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 u 
                                 1 
                                 ′2 
                               
                               ⁢ 
                               
                                 u 
                                 1 
                                 ′2 
                               
                             
                             _ 
                           
                           ] 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             ∂ 
                             
                               x 
                               2 
                             
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 u 
                                 1 
                                 ′ 
                               
                               ⁢ 
                               
                                 u 
                                 2 
                                 ′ 
                               
                               ⁢ 
                               
                                 u 
                                 1 
                                 ′2 
                               
                             
                             _ 
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               1 
                               ′2 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                     + 
                     
                         
                       
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       u 
                                       1 
                                       ′2 
                                     
                                     ⁢ 
                                     
                                       u 
                                       1 
                                       ′2 
                                     
                                   
                                   _ 
                                 
                                 - 
                                 
                                   3 
                                   ⁢ 
                                   
                                     
                                       
                                         u 
                                         1 
                                         ′2 
                                       
                                       _ 
                                     
                                     ⁡ 
                                     
                                       [ 
                                       
                                         
                                           u 
                                           1 
                                           ′2 
                                         
                                         _ 
                                       
                                       ] 
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               1 
                               
                                 ρ 
                                 _ 
                               
                             
                             ⁢ 
                             
                               
                                 ∂ 
                                 
                                   ρ 
                                   _ 
                                 
                               
                               
                                 ∂ 
                                 
                                   x 
                                   1 
                                 
                               
                             
                           
                           + 
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       u 
                                       1 
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       u 
                                       2 
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       u 
                                       1 
                                       ′2 
                                     
                                   
                                   _ 
                                 
                                 - 
                                 
                                   3 
                                   ⁢ 
                                   
                                     
                                       
                                         
                                           u 
                                           1 
                                           ′ 
                                         
                                         ⁢ 
                                         
                                           u 
                                           2 
                                           ′ 
                                         
                                       
                                       _ 
                                     
                                     [ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         u 
                                         1 
                                         ′2 
                                       
                                       _ 
                                     
                                     ] 
                                   
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               1 
                               
                                 ρ 
                                 _ 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ∂ 
                                 
                                   ρ 
                                   _ 
                                 
                               
                               
                                 ∂ 
                                 
                                   x 
                                   2 
                                 
                               
                             
                           
                           + 
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       u 
                                       1 
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       u 
                                       3 
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       u 
                                       1 
                                       ′2 
                                     
                                   
                                   _ 
                                 
                                 - 
                                 
                                   3 
                                   ⁢ 
                                   
                                     
                                       
                                         
                                           u 
                                           1 
                                           ′ 
                                         
                                         ⁢ 
                                         
                                           u 
                                           3 
                                           ′ 
                                         
                                       
                                       _ 
                                     
                                     ⁡ 
                                     
                                       [ 
                                       
                                         
                                           u 
                                           1 
                                           ′2 
                                         
                                         _ 
                                       
                                       ] 
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                               1 
                               
                                 ρ 
                                 _ 
                               
                             
                             ⁢ 
                             
                               
                                 ∂ 
                                 
                                   ρ 
                                   _ 
                                 
                               
                               
                                 ∂ 
                                 
                                   x 
                                   3 
                                 
                               
                             
                           
                         
                         = 
                         0 
                       
                     
                   
                 
               
             
           
         
       
     
   
   Equation Set 12 
   Directional Turbulent Energy Fluxes (12-4) 
   
     
       
         
           
             
               
                 ∂ 
                 
                   ∂ 
                   t 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       u 
                       1 
                       ′ 
                     
                     ⁢ 
                     
                       u 
                       2 
                       ′2 
                     
                   
                   _ 
                 
                 ] 
               
             
             + 
             
               
                 
                   u 
                   2 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         u 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         u 
                         2 
                         ′2 
                       
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   u 
                   1 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         u 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         u 
                         2 
                         ′2 
                       
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   u 
                   3 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         u 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         u 
                         2 
                         ′2 
                       
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               2 
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           2 
                           ′2 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           
                             u 
                             2 
                           
                           _ 
                         
                       
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           1 
                           ′2 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           
                             u 
                             2 
                           
                           _ 
                         
                       
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           2 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           3 
                           ′ 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           
                             u 
                             2 
                           
                           _ 
                         
                       
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                   
                 
                 ] 
               
             
             + 
             
               
                 
                   
                     u 
                     2 
                     ′ 
                   
                   ⁢ 
                   
                     u 
                     2 
                     ′2 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 
                   
                     u 
                     1 
                     ′ 
                   
                   ⁢ 
                   
                     u 
                     1 
                     ′2 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     u 
                     1 
                   
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 
                   
                     u 
                     3 
                     ′ 
                   
                   ⁢ 
                   
                     u 
                     1 
                     ′2 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     u 
                     1 
                   
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
             - 
             
               2 
               ⁢ 
               
                 
                   
                     
                       u 
                       1 
                       ′ 
                     
                     ⁢ 
                     
                       u 
                       2 
                       ′ 
                     
                   
                   _ 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             u 
                             2 
                             ′2 
                           
                           _ 
                         
                         ] 
                       
                     
                     + 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                     + 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   ] 
                 
               
             
             - 
             
               
                 
                   u 
                   2 
                   ′2 
                 
                 _ 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                         
                         _ 
                       
                       ] 
                     
                   
                   + 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           u 
                           1 
                           ′2 
                         
                         _ 
                       
                       ] 
                     
                   
                   + 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             3 
                             ′ 
                           
                         
                         _ 
                       
                       ] 
                     
                   
                 
                 ] 
               
             
             + 
             
               2 
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         c 
                         2 
                         2 
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           2 
                         
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           2 
                         
                         ⁢ 
                         
                           c 
                           3 
                         
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                 
                 ] 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       u 
                       2 
                       ′2 
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   c 
                   1 
                   2 
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       u 
                       2 
                       ′2 
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       u 
                       2 
                       ′2 
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               2 
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         u 
                         2 
                         ′2 
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           2 
                           ′ 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           2 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           3 
                           ′ 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                 
                 ] 
               
             
             + 
             
               
                 
                   
                     u 
                     1 
                     ′ 
                   
                   ⁢ 
                   
                     u 
                     2 
                     ′ 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       c 
                       2 
                       2 
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   u 
                   1 
                   ′2 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       c 
                       2 
                       2 
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     u 
                     1 
                     ′ 
                   
                   ⁢ 
                   
                     u 
                     3 
                     ′ 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       c 
                       1 
                       2 
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       u 
                       1 
                       ′ 
                     
                     ⁢ 
                     
                       u 
                       2 
                       ′ 
                     
                     ⁢ 
                     
                       u 
                       2 
                       ′2 
                     
                   
                   _ 
                 
                 ] 
               
             
             + 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       u 
                       1 
                       ′2 
                     
                     ⁢ 
                     
                       u 
                       2 
                       ′2 
                     
                   
                   _ 
                 
                 ] 
               
             
             + 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       u 
                       1 
                       ′ 
                     
                     ⁢ 
                     
                       u 
                       3 
                       ′ 
                     
                     ⁢ 
                     
                       u 
                       2 
                       ′2 
                     
                   
                   _ 
                 
                 ] 
               
             
             + 
             
               
                 [ 
                 
                   
                     
                       
                         u 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         u 
                         2 
                         ′ 
                       
                       ⁢ 
                       
                         u 
                         2 
                         ′2 
                       
                     
                     _ 
                   
                   - 
                   
                     3 
                     ⁢ 
                     
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                         
                         _ 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             u 
                             2 
                             ′2 
                           
                           _ 
                         
                         ] 
                       
                     
                   
                 
                 ] 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 1 
                 
                   ρ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ρ 
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 [ 
                 
                   
                     
                       
                         u 
                         1 
                         ′2 
                       
                       ⁢ 
                       
                         u 
                         2 
                         ′2 
                       
                     
                     _ 
                   
                   - 
                   
                     
                       
                         u 
                         1 
                         ′2 
                       
                       _ 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           u 
                           2 
                           ′2 
                         
                         _ 
                       
                       ] 
                     
                   
                   - 
                   
                     2 
                     ⁢ 
                     
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                         
                         _ 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                 
                 ] 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 1 
                 
                   ρ 
                   _ 
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ρ 
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 [ 
                 
                     
                 
                 ⁢ 
                 
                   
                     
                       
                         u 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         u 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         u 
                         2 
                         ′2 
                       
                     
                     _ 
                   
                   - 
                   
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           3 
                           ′ 
                         
                       
                       _ 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           u 
                           2 
                           ′2 
                         
                         _ 
                       
                       ] 
                     
                   
                   - 
                   
                     2 
                     ⁢ 
                     
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                         
                         _ 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                 
                 ] 
               
               ⁢ 
               
                 1 
                 
                   ρ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ρ 
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
           
           = 
           0 
         
       
     
   
   Equation Set 12 
   Turbulent Structure (12-5) 
               ∂     ∂   t       ⁡     [         u   1   ′     ⁢     u   2   ′     ⁢     u   3   ′       _     ]       +         u   1     _     ⁢       ∂     ∂     x   1         ⁡     [         u   1   ′     ⁢     u   2   ′     ⁢     u   3   ′       _     ]         +         u   2     _     ⁢       ∂     ∂     x   2         ⁡     [         u   1   ′     ⁢     u   2   ′     ⁢     u   3   ′       _     ]         +         u   3     _     ⁢       ∂     ∂     x   3         ⁡     [         u   1   ′     ⁢     u   2   ′     ⁢     u   3   ′       _     ]         +           u   1   ′     ⁢     u   2   ′     ⁢     u   3   ′       _     ⁢       ∂       u   1     _         ∂     x   1           +           u   3   ′     ⁢     u   2   ′2       _     ⁢       ∂       u   2     _         ∂     x   2           +           u   2   ′     ⁢     u   3   ′2       _     ⁢       ∂       u   1     _         ∂     x   3           +           u   3   ′     ⁢     u   1   ′2       _     ⁢       ∂       u   2     _         ∂     x   1           +           u   1   ′     ⁢     u   2   ′     ⁢     u   3   ′       _     ⁢       ∂       u   2     _         ∂     x   2           +           u   1   ′     ⁢     u   3   ′2       _     ⁢       ∂       u   2     _         ∂     x   3           +           u   2   ′     ⁢     u   1   ′2       _     ⁢       ∂       u   3     _         ∂     x   1           +           u   1   ′     ⁢     u   2   ′2       _     ⁢       ∂       u   2     _         ∂     x   3           +           u   1   ′     ⁢     u   2   ′     ⁢     u   3   ′       _     ⁢       ∂       u   3     _         ∂     x   3           -           u   1   ′     ⁢     u   2   ′       _     ⁡     [         ∂     ∂     x   1         ⁡     [         u   1   ′     ⁢     u   3   ′       _     ]       +       ∂     ∂     x   2         ⁡     [         u   2   ′     ⁢     u   3   ′       _     ]       +       ∂     ∂     x   3         ⁡     [       u   3   ′2     _     ]         ]       -           u   1   ′     ⁢     u   3   ′       _     ⁡     [         ∂     ∂     x   1         ⁡     [         u   1   ′     ⁢     u   3   ′       _     ]       +       ∂     ∂     x   2         ⁡     [       u   2   ′2     _     ]       +       ∂     ∂     x   3         ⁡     [         u   2   ′     ⁢     u   3   ′       _     ]         ]       -           u   2   ′     ⁢     u   3   ′       _     ⁡     [         ∂     ∂     x   1         ⁡     [       u   1   ′2     _     ]       +       ∂     ∂     x   2         ⁡     [         u   1   ′     ⁢     u   2   ′       _     ]       +       ∂     ∂     x   3         ⁡     [         u   1   ′     ⁢     u   3   ′       _     ]         ]       +         u   1   ′2     _     ⁢       ∂     ∂     x   1         ⁡     [         c   2     ⁢     c   3         _   _       ]         +           u   1   ′     ⁢     u   2   ′       _     ⁢       ∂     ∂     x   2         ⁡     [         c   2     ⁢     c   3         _   _       ]         +           u   1   ′     ⁢     u   3   ′       _     ⁢       ∂     ∂     x   3         ⁡     [         c   2     ⁢     c   3         _   _       ]         +           u   1   ′     ⁢     u   2   ′       _     ⁢       ∂     ∂     x   1         ⁡     [         c   1     ⁢     c   3         _   _       ]         +         u   2   ′2     _     ⁢       ∂     ∂     x   2         ⁡     [         c   1     ⁢     c   3         _   _       ]         +           u   2   ′     ⁢     u   3   ′       _     ⁢       ∂     ∂     x   3         ⁡     [         c   1     ⁢     c   3         _   _       ]         +         
Equation Set 13 (13-1; 13-3; 13-4)
 
                       c   1   2     ⁢     c   1   2         _   _       =     3   ⁢         c   1   2       _   _       ⁡     [       c   1   2       _   _       ]                           c   1     ⁢     c   2     ⁢     c   1   2         _   _       =     3   ⁢           c   1     ⁢     c   2         _   _       ⁡     [       c   1   2       _   _       ]                           c   1   2     ⁢     c   2   2         _   _       =           c   1   2       _   _       ⁡     [       c   2   2       _   _       ]       +     2   ⁢           c   1     ⁢     c   2         _   _       ⁡     [         c   1     ⁢     c   2         _   _       ]                             c   1     ⁢     c   2     ⁢     c   3   2         _   _       =             c   1     ⁢     c   2         _   _       ⁡     [       c   3   2       _   _       ]       +     2   ⁢           c   1     ⁢     c   3         _   _       ⁡     [         c   2     ⁢     c   3         _   _       ]                       
neglected terms being of Burnett order.
 
Equation Set 14 (14-1; 14-2; 14-3; 14-4)
 
                       u   1   ′2     ⁢     u   1   ′2       _     =     3   ⁢         u   1   ′2     _     ⁡     [       u   1   ′2     _     ]                           u   1   ′     ⁢     u   2   ′     ⁢     u   1   ′2       _     =     3   ⁢           u   1   ′     ⁢     u   2   ′       _     ⁡     [       u   1   ′2     _     ]                           u   1   ′2     ⁢     u   2   ′2       _     =           u   1   ′2     _     ⁡     [       u   2   ′2     _     ]       +     2   ⁢           u   1   ′     ⁢     u   2   ′       _     ⁡     [         u   1   ′     ⁢     u   2   ′       _     ]                             u   1   ′     ⁢     u   2   ′     ⁢     u   3   ′2       _     =             u   1   ′     ⁢     u   2   ′       _     ⁡     [       u   3   ′2     _     ]       +     2   ⁢           u   1   ′     ⁢     u   3   ′       _     ⁡     [         u   2   ′     ⁢     u   3   ′       _     ]                       
neglected terms being of Burnett order.
 
   Equation Set 15 
   Time Average Continuity (15-1) 
                 ∂     ρ   _         ∂   t       +       ∂     ∂     x   1         ⁡     [       ρ   _     ⁢           ⁢       u   1     _       ]       +       ∂     ∂     x   2         ⁡     [       ρ   _     ⁢           ⁢       u   2     _       ]       +       ∂     ∂     x   3         ⁡     [       ρ   _     ⁢           ⁢       u   3     _       ]         =   0         
Time Average Momentum (15-2)
 
   
     
       
         
           
             
               
                 ∂ 
                 
                   ∂ 
                   t 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     ρ 
                     _ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 ] 
               
             
             + 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     ρ 
                     _ 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             u 
                             1 
                           
                           _ 
                         
                         2 
                       
                       + 
                       
                         
                           u 
                           1 
                           ′2 
                         
                         _ 
                       
                       + 
                       
                         
                           c 
                           1 
                           2 
                         
                         
                           _ 
                           _ 
                         
                       
                     
                     ] 
                   
                 
                 ] 
               
             
             + 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     ρ 
                     _ 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             u 
                             1 
                           
                           ⁢ 
                           
                             u 
                             2 
                           
                         
                         _ 
                       
                       + 
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                         
                         _ 
                       
                       + 
                       
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                         
                         
                           _ 
                           _ 
                         
                       
                     
                     ] 
                   
                 
                 ] 
               
             
             + 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     ρ 
                     _ 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             u 
                             1 
                           
                           ⁢ 
                           
                             u 
                             3 
                           
                         
                         _ 
                       
                       + 
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             3 
                             ′ 
                           
                         
                         _ 
                       
                       + 
                       
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             3 
                           
                         
                         
                           _ 
                           _ 
                         
                       
                     
                     ] 
                   
                 
                 ] 
               
             
           
           = 
           0 
         
       
     
   
   Equation Set 15 
   Time Average Directional Thermal Energy (15-3) 
                 ∂     ∂   t       ⁡     [       c   1   2       _   _       ]       +         u   1     _     ⁢       ∂     ∂     x   1         ⁡     [       c   1   2       _   _       ]         +         u   2     _     ⁢       ∂     ∂     x   2         ⁡     [       c   1   2       _   _       ]         +         u   3     _     ⁢       ∂     ∂     x   3         ⁡     [       c   1   2       _   _       ]         +     2   ⁡     [           c   1   2       _   _       ⁢       ∂       u   _     1         ∂     x   1           +           c   1     ⁢     c   2         _   _       ⁢       ∂     u   1         ∂     x   2           +           c   1     ⁢     c   3         _   _       ⁢       ∂     u   1         ∂     x   3             ]       +       1     ρ   _       ⁡     [         ∂     ∂     x   1         ⁡     [       ρ   _     ⁢         c   1     ⁢     c   1   2         _   _         ]       +       ∂     ∂     x   2         ⁡     [       ρ   _     ⁢         c   2     ⁢     c   1   2         _   _         ]       +       ∂     ∂     x   3         ⁡     [       ρ   _     ⁢         c   3     ⁢     c   1   2         _   _         ]         ]         =       υ   _     ⁡     [         1   3     ⁢       c   2       _   _         -       c   1   2       _   _         ]             
Time Average Thermal Shear (15-4)
 
   
     
       
         
           
             
               
                 ∂ 
                 
                   ∂ 
                   t 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       c 
                       1 
                     
                     ⁢ 
                     
                       c 
                       2 
                     
                   
                   
                     _ 
                     _ 
                   
                 
                 ] 
               
             
             + 
             
               
                 
                   u 
                   1 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   u 
                   2 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   u 
                   3 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 
                   c 
                   2 
                   2 
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
             + 
             
               
                 
                   c 
                   1 
                   2 
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       2 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       2 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       2 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
             + 
             
               
                 1 
                 
                   ρ 
                   _ 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           ρ 
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               c 
                               1 
                               2 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                       
                       ] 
                     
                   
                   + 
                   
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             ρ 
                             _ 
                           
                           ⁢ 
                           
                             
                               
                                 c 
                                 1 
                               
                               ⁢ 
                               
                                 c 
                                 2 
                                 2 
                               
                             
                             
                               _ 
                               _ 
                             
                           
                         
                         ] 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             ρ 
                             _ 
                           
                           ⁢ 
                           
                             
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   c 
                                   1 
                                 
                                 ⁢ 
                                 
                                   c 
                                   2 
                                 
                                 ⁢ 
                                 
                                   c 
                                   3 
                                 
                               
                             
                             
                               _ 
                               _ 
                             
                           
                         
                         ] 
                       
                     
                   
                 
                 ] 
               
             
           
           = 
           
             
               - 
               
                 v 
                 _ 
               
             
             ⁢ 
             
                 
             
             ⁢ 
             
               
                 
                   c 
                   1 
                 
                 ⁢ 
                 
                   c 
                   2 
                 
               
               
                 _ 
                 _ 
               
             
           
         
       
     
   
   Equation Set 15 
   Closed Time Average Directional Thermal Energy Fluxes (15-5) 
                 ∂     ∂   t       ⁡     [         c   1     ⁢     c   1   2         _   _       ]       +         u   1     _     ⁢       ∂     ∂     x   1         ⁡     [         c   1     ⁢     c   1   2         _   _       ]         +         u   2     _     ⁢       ∂     ∂     x   2         ⁡     [         c   1     ⁢     c   1   2         _   _       ]         +         u   3     _     ⁢       ∂     ∂     x   3         ⁡     [         c   1     ⁢     c   1   2         _   _       ]         +     3   ⁡     [         c   1     ⁢     c   1   2     ⁢       ∂     u   1         ∂     x   1           +       c   2     ⁢     c   1   2     ⁢       ∂     u   1         ∂     x   2           +       c   3     ⁢     c   1   2     ⁢       ∂     u   1         ∂     x   3             ]       +     3   ⁡     [           c   1   2       _   _       ⁢     ∂     ∂     x   1         ⁢       [     c   1   2     ]       _   _         +           c   1     ⁢     c   2         _   _       ⁢       ∂     ∂     x   2         ⁡     [       c   1   2       _   _       ]         +           c   1     ⁢     c   3         _   _       ⁢       ∂     ∂     x   3         ⁡     [       c   1   2       _   _       ]           ]         =         v   _     2     ⁡     [           c   1     ⁢     c   2         _   _       -     3   ⁢         c   1     ⁢     c   1   2         _   _           ]             
(15-6)
 
                 ∂     ∂   t       ⁡     [         c   1     ⁢     c   2   2         _   _       ]       +         u   1     _     ⁢       ∂     ∂     x   1         ⁡     [         c   1     ⁢     c   2   2         _   _       ]         +         u   2     _     ⁢       ∂     ∂     x   2         ⁡     [         c   1     ⁢     c   2   2         _   _       ]         +         u   3     _     ⁢       ∂     ∂     x   3         ⁡     [         c   1     ⁢     c   2   2         _   _       ]         +           c   1     ⁢     c   2   2         _   _       ⁢       ∂       u   1     _         ∂     x   1           +           c   2     ⁢     c   2   2         _   _       ⁢       ∂       u   1     _         ∂     x   2           +           c   3     ⁢     c   2   2         _   _       ⁢       ∂       u   1     _         ∂     x   3           +     2   ⁡     [             c   2     ⁢     c   1   2         _   _       ⁢       ∂       u   2     _         ∂     x   1           +           c   1     ⁢     c   2   2         _   _       ⁢       ∂       u   2     _         ∂     x   2           +           c   1     ⁢     c   2     ⁢     c   3         _   _       ⁢       ∂       u   2     _         ∂     x   3             ]       +     2   ⁡     [             c   1     ⁢     c   2         _   _       ⁢       ∂     ∂     x   1         ⁡     [         c   1     ⁢     c   2         _   _       ]         +         c   2   2       _   _       ⁢       ∂     ∂     x   2         ⁡     [         c   1     ⁢     c   2         _   _       ]         +           c   2     ⁢     c   3         _   _       ⁢       ∂     ∂     x   3         ⁡     [         c   1     ⁢     c   2         _   _       ]           ]       +         c   1   2       _   _       ⁢       ∂     ∂     x   1         ⁡     [       c   2   2       _   _       ]         +           c   1     ⁢     c   2         _   _       ⁢       ∂     ∂     x   2         ⁡     [       c   2   2       _   _       ]         +           c   1     ⁢     c   3         _   _       ⁢       ∂     ∂     x   3         ⁡     [       c   2   2       _   _       ]           =         v   _     2     ⁡     [         1   3     ⁢         c   2     ⁢     c   2         _   _         -     3   ⁢         c   2     ⁢     c   1   2         _   _           ]             
Closed Time Average Thermal Structure (15-7)
 
   
     
       
         
           
             
               
                 ∂ 
                 
                   ∂ 
                   t 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       c 
                       1 
                     
                     ⁢ 
                     
                       c 
                       2 
                     
                     ⁢ 
                     
                       c 
                       3 
                     
                   
                   
                     _ 
                     _ 
                   
                 
                 ] 
               
             
             + 
             
               
                 u 
                 1 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 u 
                 2 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 u 
                 3 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     3 
                   
                   ⁢ 
                   
                     c 
                     2 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     3 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     3 
                   
                   ⁢ 
                   
                     c 
                     1 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       2 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       2 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     3 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       2 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     1 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       3 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       3 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       3 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
             + 
             
               
                 
                   c 
                   1 
                   2 
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     2 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   c 
                   2 
                   2 
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     1 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     c 
                     3 
                   
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   c 
                   3 
                   2 
                 
                 
                   _ 
                   _ 
                 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ] 
                 
               
             
           
           = 
           
             
               - 
               
                 3 
                 2 
               
             
             ⁢ 
             
               v 
               _ 
             
             ⁢ 
             
               
                 
                   c 
                   1 
                 
                 ⁢ 
                 
                   c 
                   2 
                 
                 ⁢ 
                 
                   c 
                   3 
                 
               
               
                 _ 
                 _ 
               
             
           
         
       
     
   
   Equation Set 15 
   Time Average Directional Turbulent Energy (15-8) 
               ∂     ∂   t       ⁡     [       u   1   ′2     _     ]       +       u   1     ⁢       ∂     ∂     x   1         ⁡     [       u   1   ′2     _     ]         +       u   2     ⁢       ∂     ∂     x   2         ⁡     [       u   1   ′2     _     ]         +       u   3     ⁢       ∂     ∂     x   3         ⁡     [       u   1   ′2     _     ]         +     2   ⁡     [           u   1   ′2     _     ⁢       ∂       u   1     _         ∂     x   1           +           u   1   ′     ⁢     u   2   ′       _     ⁢       ∂       u   1     _         ∂     x   2           +           u   1   ′     ⁢     u   3   ′       _     ⁢       ∂       u   1     _         ∂     x   3             ]       +       1     ρ   _       [           ∂     ∂     x   1         ⁡     [       ρ   _     ⁢         u   1   ′     ⁢     u   1   ′2       _       ]       +       ∂     ∂     x   2         ⁡     [       ρ   _     ⁢         u   2   ′     ⁢     u   1   ′2       _       ]       +       ∂     ∂     x   3         ⁡     [       ρ   _     ⁢         u   3   ′     ⁢     u   1   ′2       _       ]         =   0             
Time Average Turbulent Shear (15-9)
 
   
     
       
         
           
             
               
                 ∂ 
                 
                   ∂ 
                   t 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       u 
                       1 
                       ′ 
                     
                     ⁢ 
                     
                       u 
                       2 
                       ′ 
                     
                   
                   _ 
                 
                 ] 
               
             
             + 
             
               
                 
                   u 
                   1 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         u 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         u 
                         2 
                         ′ 
                       
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   u 
                   2 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         u 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         u 
                         2 
                         ′ 
                       
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   u 
                   3 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         u 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         u 
                         2 
                         ′ 
                       
                     
                     _ 
                   
                   ] 
                 
               
             
             + 
             
               
                 
                   
                     u 
                     1 
                     ′ 
                   
                   ⁢ 
                   
                     u 
                     2 
                     ′ 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 
                   u 
                   2 
                   ′2 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 
                   
                     u 
                     2 
                     ′ 
                   
                   ⁢ 
                   
                     u 
                     3 
                     ′ 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       1 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
             + 
             
               
                 
                   u 
                   1 
                   ′2 
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       2 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
             
             + 
             
               
                 
                   
                     u 
                     1 
                     ′ 
                   
                   ⁢ 
                   
                     u 
                     2 
                     ′ 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       2 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
             
             + 
             
               
                 
                   
                     u 
                     1 
                     ′ 
                   
                   ⁢ 
                   
                     u 
                     3 
                     ′ 
                   
                 
                 _ 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     
                       u 
                       2 
                     
                     _ 
                   
                 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
             
             + 
             
               
                 1 
                 
                   ρ 
                   _ 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           ρ 
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               1 
                               ′2 
                             
                           
                           _ 
                         
                       
                       ] 
                     
                   
                   + 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           ρ 
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′2 
                             
                           
                           _ 
                         
                       
                       ] 
                     
                   
                   + 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           ρ 
                           _ 
                         
                         ⁢ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                       
                       ] 
                     
                   
                 
                 ] 
               
             
           
           = 
           0 
         
       
     
   
   Equation Set 15 
   Closed Time Average Turbulent Directional Energy Fluxes (15-10) 
   
     
       
         
           
             
               
                 
                   ∂ 
                   
                     ∂ 
                     t 
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         u 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         u 
                         1 
                         ′2 
                       
                     
                     _ 
                   
                   ] 
                 
               
               + 
               
                 
                   u 
                   1 
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       ∂ 
                       
                         x 
                         1 
                       
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           1 
                           ′2 
                         
                       
                       _ 
                     
                     ] 
                   
                 
               
               + 
               
                 
                   
                     u 
                     2 
                   
                   _ 
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       ∂ 
                       
                         x 
                         2 
                       
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           1 
                           ′2 
                         
                       
                       _ 
                     
                     ] 
                   
                 
               
               + 
               
                 
                   
                     u 
                     3 
                   
                   _ 
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       ∂ 
                       
                         x 
                         3 
                       
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           1 
                           ′2 
                         
                       
                       _ 
                     
                     ] 
                   
                 
               
               + 
               
                 
                   3 
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           1 
                           ′2 
                         
                       
                       _ 
                     
                     ] 
                   
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       
                         u 
                         1 
                       
                       _ 
                     
                   
                   
                     ∂ 
                     
                       x 
                       1 
                     
                   
                 
               
               + 
               
                 
                   
                     
                       u 
                       2 
                       ′ 
                     
                     ⁢ 
                     
                       u 
                       1 
                       ′2 
                     
                   
                   _ 
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       
                         u 
                         1 
                       
                       _ 
                     
                   
                   
                     ∂ 
                     
                       x 
                       2 
                     
                   
                 
               
               + 
               
                 
                   
                     
                       u 
                       3 
                       ′ 
                     
                     ⁢ 
                     
                       u 
                       1 
                       ′2 
                     
                   
                   _ 
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       
                         u 
                         1 
                       
                       _ 
                     
                   
                   
                     ∂ 
                     
                       x 
                       3 
                     
                   
                 
               
               + 
               
                 3 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         
                           u 
                           1 
                           ′2 
                         
                         _ 
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             ∂ 
                             
                               x 
                               1 
                             
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               u 
                               1 
                               ′2 
                             
                             _ 
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                         
                         _ 
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             ∂ 
                             
                               x 
                               2 
                             
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               u 
                               1 
                               ′2 
                             
                             _ 
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             3 
                             ′ 
                           
                         
                         _ 
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             ∂ 
                             
                               x 
                               3 
                             
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               u 
                               1 
                               ′2 
                             
                             _ 
                           
                           ] 
                         
                       
                     
                   
                   ] 
                 
               
               + 
               
                 3 
                 ⁢ 
                 
                   
                     
                       c 
                       1 
                       2 
                     
                     
                       _ 
                       _ 
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           ∂ 
                           
                             ∂ 
                             
                               x 
                               1 
                             
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               u 
                               1 
                               ′2 
                             
                             _ 
                           
                           ] 
                         
                       
                       + 
                       
                         
                           ∂ 
                           
                             ∂ 
                             
                               x 
                               2 
                             
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 u 
                                 1 
                                 ′ 
                               
                               ⁢ 
                               
                                 u 
                                 2 
                                 ′ 
                               
                             
                             _ 
                           
                           ] 
                         
                       
                       + 
                       
                         
                           ∂ 
                           
                             ∂ 
                             
                               x 
                               3 
                             
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 u 
                                 1 
                                 ′ 
                               
                               ⁢ 
                               
                                 u 
                                 3 
                                 ′ 
                               
                             
                             _ 
                           
                           ] 
                         
                       
                     
                     ] 
                   
                 
               
               + 
               
                 3 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         
                           u 
                           1 
                           ′2 
                         
                         _ 
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             ∂ 
                             
                               x 
                               1 
                             
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               c 
                               1 
                               2 
                             
                             
                               _ 
                               _ 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                         
                         _ 
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             ∂ 
                             
                               x 
                               2 
                             
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               c 
                               1 
                               2 
                             
                             
                               _ 
                               _ 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             3 
                             ′ 
                           
                         
                         _ 
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             ∂ 
                             
                               x 
                               3 
                             
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               c 
                               1 
                               2 
                             
                             
                               _ 
                               _ 
                             
                           
                           ] 
                         
                       
                     
                   
                   ] 
                 
               
             
             = 
             0 
           
           ⁢ 
           
             
 
           
         
       
     
   
   Equation Set 15 
   Closed Time Average Turbulent Directional Energy Fluxes (15-11) 
   
     
       
         
           
             
               ∂ 
               
                 ∂ 
                 t 
               
             
             ⁡ 
             
               [ 
               
                 
                   
                     u 
                     1 
                     ′ 
                   
                   ⁢ 
                   
                     u 
                     2 
                     ′2 
                   
                 
                 _ 
               
               ] 
             
           
           + 
           
             
               
                 u 
                 1 
               
               _ 
             
             ⁢ 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       u 
                       1 
                       ′ 
                     
                     ⁢ 
                     
                       u 
                       2 
                       ′2 
                     
                   
                   _ 
                 
                 ] 
               
             
           
           + 
           
             
               
                 u 
                 
                   2 
                   ⁢ 
                   
                       
                   
                 
               
               _ 
             
             ⁢ 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       u 
                       1 
                       ′ 
                     
                     ⁢ 
                     
                       u 
                       2 
                       ′2 
                     
                   
                   _ 
                 
                 ] 
               
             
           
           + 
           
             
               
                 u 
                 
                   3 
                   ⁢ 
                   
                       
                   
                 
               
               _ 
             
             ⁢ 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       u 
                       1 
                       ′ 
                     
                     ⁢ 
                     
                       u 
                       2 
                       ′2 
                     
                   
                   _ 
                 
                 ] 
               
             
           
           + 
           
             2 
             ⁡ 
             
               [ 
               
                 
                   
                     
                       
                         u 
                         2 
                         ′ 
                       
                       ⁢ 
                       
                         u 
                         2 
                         ′2 
                       
                     
                     _ 
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         
                           u 
                           2 
                         
                         _ 
                       
                     
                     
                       ∂ 
                       
                         x 
                         1 
                       
                     
                   
                 
                 + 
                 
                   
                     
                       
                         u 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         u 
                         2 
                         ′2 
                       
                     
                     _ 
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         
                           u 
                           2 
                         
                         _ 
                       
                     
                     
                       ∂ 
                       
                         x 
                         2 
                       
                     
                   
                 
                 + 
                 
                   
                     u 
                     1 
                     ′ 
                   
                   ⁢ 
                   
                     u 
                     2 
                     ′ 
                   
                   ⁢ 
                   
                     u 
                     3 
                     ′ 
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         
                           u 
                           2 
                         
                         _ 
                       
                     
                     
                       ∂ 
                       
                         x 
                         3 
                       
                     
                   
                 
               
               ] 
             
           
           + 
           
             
               
                 
                   u 
                   1 
                   ′ 
                 
                 ⁢ 
                 
                   u 
                   2 
                   ′2 
                 
               
               _ 
             
             ⁢ 
             
               
                 ∂ 
                 
                   
                     u 
                     1 
                   
                   _ 
                 
               
               
                 ∂ 
                 
                   x 
                   1 
                 
               
             
           
           + 
           
             
               
                 
                   u 
                   2 
                   ′ 
                 
                 ⁢ 
                 
                   u 
                   2 
                   ′2 
                 
               
               _ 
             
             ⁢ 
             
               
                 ∂ 
                 
                   
                     u 
                     1 
                   
                   _ 
                 
               
               
                 ∂ 
                 
                   x 
                   2 
                 
               
             
           
           + 
           
             
               
                 
                   u 
                   3 
                   ′ 
                 
                 ⁢ 
                 
                   u 
                   1 
                   ′2 
                 
               
               _ 
             
             ⁢ 
             
               
                 ∂ 
                 
                   
                     u 
                     1 
                   
                   _ 
                 
               
               
                 ∂ 
                 
                   x 
                   3 
                 
               
             
           
           + 
           
             2 
             ⁡ 
             
               [ 
               
                 
                   
                     
                       
                         u 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         u 
                         2 
                         ′ 
                       
                     
                     _ 
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                         
                         _ 
                       
                       ] 
                     
                   
                 
                 + 
                 
                   
                     
                       u 
                       2 
                       ′2 
                     
                     _ 
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                         
                         _ 
                       
                       ] 
                     
                   
                 
                 + 
                 
                   
                     
                       
                         u 
                         2 
                         ′ 
                       
                       ⁢ 
                       
                         u 
                         3 
                         ′ 
                       
                     
                     _ 
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                         
                         _ 
                       
                       ] 
                     
                   
                 
               
               ] 
             
           
           + 
           
             
               
                 u 
                 1 
                 ′2 
               
               _ 
             
             ⁢ 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     u 
                     2 
                     ′2 
                   
                   _ 
                 
                 ] 
               
             
           
           + 
           
             
               
                 
                   u 
                   1 
                   ′ 
                 
                 ⁢ 
                 
                   u 
                   2 
                   ′ 
                 
               
               _ 
             
             ⁢ 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     u 
                     2 
                     ′2 
                   
                   _ 
                 
                 ] 
               
             
           
           + 
           
             
               
                 
                   u 
                   1 
                   ′ 
                 
                 ⁢ 
                 
                   u 
                   3 
                   ′ 
                 
               
               _ 
             
             ⁢ 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     u 
                     2 
                     ′2 
                   
                   _ 
                 
                 ] 
               
             
           
           + 
           
             2 
             ⁡ 
             
               [ 
               
                 
                   
                     
                       
                         u 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         u 
                         2 
                         ′ 
                       
                     
                     _ 
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                         
                         
                           _ 
                           _ 
                         
                       
                       ] 
                     
                   
                 
                 + 
                 
                   
                     
                       u 
                       2 
                       ′2 
                     
                     _ 
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             c 
                             1 
                           
                           ⁢ 
                           
                             c 
                             2 
                           
                         
                         
                           _ 
                           _ 
                         
                       
                       ] 
                     
                   
                 
                 + 
                 
                   
                     
                       u 
                       2 
                       ′ 
                     
                     ⁢ 
                     
                       u 
                       3 
                       ′ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                         ] 
                       
                     
                   
                   _ 
                 
               
               ] 
             
           
           + 
           
             
               
                 u 
                 1 
                 ′2 
               
               _ 
             
             ⁢ 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     1 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     c 
                     2 
                     2 
                   
                   
                     _ 
                     _ 
                   
                 
                 ] 
               
             
           
           + 
           
             
               
                 
                   u 
                   1 
                   ′ 
                 
                 ⁢ 
                 
                   u 
                   2 
                   ′ 
                 
               
               _ 
             
             ⁢ 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     2 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     c 
                     2 
                     2 
                   
                   
                     _ 
                     _ 
                   
                 
                 ] 
               
             
           
           + 
           
             
               
                 
                   u 
                   1 
                   ′ 
                 
                 ⁢ 
                 
                   u 
                   3 
                   ′ 
                 
               
               _ 
             
             ⁢ 
             
               
                 ∂ 
                 
                   ∂ 
                   
                     x 
                     3 
                   
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     c 
                     2 
                     2 
                   
                   
                     _ 
                     _ 
                   
                 
                 ] 
               
             
           
           + 
           
             2 
             [ 
             
               
                 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                         
                         _ 
                       
                       ] 
                     
                   
                 
                 + 
                 
                   
                     
                       c 
                       2 
                       2 
                     
                     
                       _ 
                       _ 
                     
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                         
                         _ 
                       
                       ] 
                     
                   
                 
                 + 
                 
                   
                     
                       
                         c 
                         2 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                         
                         _ 
                       
                       ] 
                     
                   
                 
                 + 
                 
                   
                     
                       c 
                       1 
                       2 
                     
                     
                       _ 
                       _ 
                     
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           u 
                           2 
                           ′2 
                         
                         _ 
                       
                       ] 
                     
                   
                 
                 + 
                 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         2 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           u 
                           2 
                           ′2 
                         
                         _ 
                       
                       ] 
                     
                   
                 
                 + 
                 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         c 
                         3 
                       
                     
                     
                       _ 
                       _ 
                     
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           u 
                           2 
                           ′2 
                         
                         _ 
                       
                       ] 
                     
                   
                 
               
               = 
               0 
             
           
         
       
     
   
   Equation Set 15 
   Closed Time Average Turbulent Structure (15-12) 
   
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         t 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             3 
                             ′ 
                           
                         
                         _ 
                       
                       ] 
                     
                   
                   + 
                   
                     
                       
                         u 
                         1 
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         u 
                         2 
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         u 
                         3 
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           2 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           3 
                           ′ 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           
                             u 
                             1 
                           
                           _ 
                         
                       
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           3 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           2 
                           ′2 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           
                             u 
                             1 
                           
                           _ 
                         
                       
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           2 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           3 
                           ′2 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           
                             u 
                             1 
                           
                           _ 
                         
                       
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           3 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           1 
                           ′2 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           
                             u 
                             2 
                           
                           _ 
                         
                       
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           2 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           3 
                           ′ 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           
                             u 
                             2 
                           
                           _ 
                         
                       
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           3 
                           ′2 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           
                             u 
                             2 
                           
                           _ 
                         
                       
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           2 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           1 
                           ′2 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           
                             u 
                             3 
                           
                           _ 
                         
                       
                       
                         ∂ 
                         
                           x 
                           1 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           2 
                           ′2 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           
                             u 
                             3 
                           
                           _ 
                         
                       
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           2 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           3 
                           ′ 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           
                             u 
                             3 
                           
                           _ 
                         
                       
                       
                         ∂ 
                         
                           x 
                           3 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           3 
                           ′ 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           2 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           3 
                           ′ 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         u 
                         3 
                         ′2 
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       [ 
                       
                         
                           
                             u 
                             1 
                             ′ 
                           
                           ⁢ 
                           
                             u 
                             2 
                             ′ 
                           
                         
                         _ 
                       
                       ] 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         u 
                         2 
                         ′2 
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           2 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           3 
                           ′ 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         u 
                         1 
                         ′2 
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           2 
                           ′ 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           3 
                           ′ 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           3 
                         
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           2 
                         
                         ⁢ 
                         
                           c 
                           3 
                         
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         c 
                         3 
                         2 
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               2 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           2 
                         
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         c 
                         2 
                         2 
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           2 
                         
                         ⁢ 
                         
                           c 
                           3 
                         
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               1 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         c 
                         1 
                         2 
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           2 
                         
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         
                           c 
                           3 
                         
                       
                       
                         _ 
                         _ 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               u 
                               3 
                               ′ 
                             
                           
                           _ 
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           3 
                           ′ 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           2 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           3 
                           ′ 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         u 
                         3 
                         ′2 
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           2 
                           ′ 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         u 
                         2 
                         ′2 
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           2 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           3 
                           ′ 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               1 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         u 
                         1 
                         ′2 
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             1 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           2 
                           ′ 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                         ] 
                       
                     
                   
                   + 
                   
                     
                       
                         
                           u 
                           1 
                           ′ 
                         
                         ⁢ 
                         
                           u 
                           3 
                           ′ 
                         
                       
                       _ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             x 
                             3 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               c 
                               3 
                             
                           
                           
                             _ 
                             _ 
                           
                         
                         ] 
                       
                     
                   
                 
                 = 
                 0 
               
             
             
               
                   
               
             
           
         
       
     
   
   Equation Set 16 
   (16-1; 16-2; 16-3) 
                       c   1     l   +   l       ⁢     c   2   m     ⁢     c   3   n         _   _       =       ⁢         l   ⁡     [       c   1   2       _   _       ]       ⁢         c   1     l   -   l       ⁢     c   2   m     ⁢     c   3   n         _   _         +       m   ⁡     [         c   1     ⁢     c   2         _   _       ]       ⁢         c   1   l     ⁢     c   2     m   -   l       ⁢     c   3   n         _   _         +       n   ⁡     [         c   1     ⁢     c   3         _   _       ]       ⁢         c   1   l     ⁢     c   2   m     ⁢     c   3     n   -   l           _   _                             c   1   l     ⁢     c   2     m   +   l       ⁢     c   3   n         _   _       =       ⁢         l   ⁡     [         c   1     ⁢     c   2         _   _       ]       ⁢         c   1     l   -   l       ⁢     c   2   m     ⁢     c   3   n         _   _         +       m   ⁡     [       c   2   2       _   _       ]       ⁢         c   1   l     ⁢     c   2     m   -   l       ⁢     c   3   n         _   _         +       n   ⁡     [         c   2     ⁢     c   3         _   _       ]       ⁢         c   1   l     ⁢     c   2   m     ⁢     c   3     n   -   l           _   _                             c   1   l     ⁢     c   2   m     ⁢     c   3     n   +   l           _   _       =       ⁢         l   ⁡     [         c   1     ⁢     c   3         _   _       ]       ⁢         c   1     l   -   l       ⁢     c   2   m     ⁢     c   3   n         _   _         +       m   ⁡     [         c   2     ⁢     c   3         _   _       ]       ⁢         c   1   l     ⁢     c   2     m   -   l       ⁢     c   3   n         _   _         +       n   ⁡     [       c   3   2       _   _       ]       ⁢         c   1   l     ⁢     c   2   m     ⁢     c   3     n   -   l           _   _                       
neglected terms being of Burnett order.
 
   Equation Set 17 
   (17-1; 17-2; 17-3) 
                       u   1       ′   ⁢           ⁢   l     +   l       ⁢     u   2     ′   ⁢           ⁢   m       ⁢     u   3     ′   ⁢           ⁢   n         _     =       ⁢         l   ⁡     [       u   1     ′   ⁢           ⁢   2       _     ]       ⁢         u   1       ′   ⁢           ⁢   l     -   l       ⁢     u   2     ′   ⁢           ⁢   m       ⁢     u   3     ′   ⁢           ⁢   n         _       +       m   ⁡     [         u   1   ′     ⁢     u   2   ′       _     ]       ⁢         u   1     ′   ⁢           ⁢   l       ⁢     u   2       ′   ⁢           ⁢   m     -   l       ⁢     u   3     ′   ⁢           ⁢   n         _       +       n   ⁡     [         u   1   ′     ⁢     u   3   ′       _     ]       ⁢         u   1     ′   ⁢           ⁢   l       ⁢     u   2     ′   ⁢           ⁢   m       ⁢     u   3       ′   ⁢           ⁢   n     -   l         _                           u   1     ′   ⁢           ⁢   l       ⁢     u   2       ′   ⁢           ⁢   m     +   l       ⁢     u   3     ′   ⁢           ⁢   n         _     =       ⁢         l   ⁡     [         u   1   ′     ⁢     u   2   ′       _     ]       ⁢         u   1       ′   ⁢           ⁢   l     -   l       ⁢     u   2     ′   ⁢           ⁢   m       ⁢     u   3     ′   ⁢           ⁢   n         _       +       m   ⁡     [       u   2     ′   ⁢           ⁢   2       _     ]       ⁢         u   1     ′   ⁢           ⁢   l       ⁢     u   2       ′   ⁢           ⁢   m     -   l       ⁢     u   3     ′   ⁢           ⁢   n         _       +       n   ⁡     [         u   2   ′     ⁢     u   3   ′       _     ]       ⁢         u   1     ′   ⁢           ⁢   l       ⁢     u   2     ′   ⁢           ⁢   m       ⁢     u   3       ′   ⁢           ⁢   n     -   l         _                           u   1     ′   ⁢           ⁢   l       ⁢     u   2     ′   ⁢           ⁢   m       ⁢     u   3       ′   ⁢           ⁢   n     +   l         _     =       ⁢         l   ⁡     [         u   1   ′     ⁢     u   3   ′       _     ]       ⁢         u   1       ′   ⁢           ⁢   l     -   l       ⁢     u   2     ′   ⁢           ⁢   m       ⁢     u   3     ′   ⁢           ⁢   n         _       +       m   ⁡     [         u   2   ′     ⁢     u   3   ′       _     ]       ⁢         u   1     ′   ⁢           ⁢   l       ⁢     u   2       ′   ⁢           ⁢   m     -   l       ⁢     u   3     ′   ⁢           ⁢   n         _       +       n   ⁡     [       u   3     ′   ⁢           ⁢   2       _     ]       ⁢         u   1     ′   ⁢           ⁢   l       ⁢     u   2     ′   ⁢           ⁢   m       ⁢     u   3       ′   ⁢           ⁢   n     -   l         _                     
neglected terms being of Burnett order.
 
   Equation 18 
   (18-1; 18-2; 18-3) 
           t   ,     80   ⁢                     c   1     l   +   l       ⁢     c   2   m     ⁢     c   3   n       _     *     =       ⁢       l   ⁡     [         c   2     _     3     ]       ⁢         c   1     l   -   l       ⁢     c   2   m     ⁢     c   3   n       _         ,               ⁢       l   ⁢           ⁢   odd     ,     m   ⁢           ⁢   and   ⁢           ⁢   n   ⁢           ⁢   even                   =       ⁢   0               ⁢     l   ,     m   ⁢           ⁢   and   ⁢           ⁢   n   ⁢           ⁢   all   ⁢           ⁢   odd                             c   1   l     ⁢     c   2     m   +   l       ⁢     c   3   n       _     *     =       ⁢       m   ⁡     [         c   2     _     3     ]       ⁢         c   1   l     ⁢     c   2     m   -   l       ⁢     c   3   n       _         ,               ⁢       m   ⁢           ⁢   odd     ,     l   ⁢           ⁢   and   ⁢           ⁢   n   ⁢           ⁢   even                   =       ⁢   0               ⁢     l   ,     m   ⁢           ⁢   and   ⁢           ⁢   n   ⁢           ⁢   all   ⁢           ⁢   odd                             c   1   l     ⁢     c   2   m     ⁢     c   3     n   +   l         _     *     =       ⁢       n   ⁡     [         c   2     _     3     ]       ⁢         c   1   l     ⁢     c   2   m     ⁢     c   3     n   -   l         _         ,               ⁢       n   ⁢           ⁢   odd     ,     l   ⁢           ⁢   and   ⁢           ⁢   m   ⁢           ⁢   even                   =       ⁢   0               ⁢     l   ,     m   ⁢           ⁢   and   ⁢           ⁢   n   ⁢           ⁢   all   ⁢           ⁢   odd                       
neglected terms being of Burnett order, where l+m+n=2N+1, an odd number.