NASA Contractor Report 178242 

(NASA-CB-178242) 3EE IBbEi?!AL-VCBIEX 

EI;UAITIGYS final &sport (€la LCaiIiioo Univ-) 

17 P CSCL 01A 

THE THERMAL-VORTEX EQUATIONS 

John V. Shebalin 

OLD DOMINION UNIVERSITY 

Norfolk, Virginia 

Contract NAS1-17993 

February 1987 

National Aeronautics and 

Space Administration 

Langley Research Center 

Hampton,Virginia 23665 

NE7- 18536 

Unclas 

43765 
John V. Shebalin 

Department of Electrical and Computer Engineering 

Old Dominion University, Norfolk, VA 23508 

AB ST RACT 

The Boussinesq approximation is extended so as to explicitly account for 

the transfer of fluid energy through viscous action into thermal energy. 

Ideal and dissipative integral invariants are discussed, in addition to the 

general equations €or thermal-fluid motion. 

i 
-1019 

In this paper the equations of motion for a thermo-gravitationally 

stratified fluid are developed. These equations are pertinent to atmospheric 

and oceanic flows in which temperature variations affect fluid flow through 

boundary forces and in which fluid motion affects temperature variations 

through viscous dissipation (in addition to convection). The equations 

developed here are an extension of the Boussinesq approximation in that they 

explicitly account for the transfer of fluid energy through viscous action 

into thermal energy. These equations thus conserve physical energy, while the 

Boussinesq equations do not. 

This is not to say there are no conserved quantities in the Boussinesq 

approximation. In 2-D flows there are two: the first, sometimes called the 

energy, involves the velocity squared and the temperature (or density) 

perturbation squared. This is not strictly the physical energy, which is 

linear in the temperature perturbation. The second conserved quantity in 2-D 

flow, apparently unrecognized, is essentially the spatial correlation of the 

vorticity and temperature perturbation. 

Here the general equations for thermal-fluid motion will be discussed 

first. Then the two-dimensional version of these will be detailed. Next, the 

standard Boussinesq approximation will be discussed: in particular, the twodimensional form. Finally a summary and conclusion will be given. 

GENERAL BlpUATIaRS’ 

In this work, the fluid under consideration will be treated as 

incompressible, i.e. , density change with respect to pressure is negligible; 

however, density will change with temperature. 

written as 

The fluid density can then be 
P= PO+ P’ 

where p is a constant and p’ is the density variation. Similarly, the 

temperature is given by 

0 

T=T +T’ 0 

where T is a constant and T’ is the temperature variation. The relation 

between p’ and T’ is linear and is given by 

0 

p’ = -Porn’ 

where 8 is the thermal-expansion coefficient of the fluid (at T ). 

The pressure, in turn, is given by 

0 

P = Po + P’ 

Po=Pog. r 

where p’ is the pressure variation and 

++ 

+ + Here g is the acceleration due to gravity and I: is a position vector. 

In the present 

gravity : 

-P 

+v. RT= - av’ 

at 

(here v is the 

(3) 

(4) 

(5) 

case, the Navier-Stokes 

-1 4 + v *2=+ 9’ 

P 

equation must include the effects of 

(6) 

kinematic viscosity). If we now place (11, (31, (4) and (5) 

into (6) and neglect second and higher order quantities, we arrive at 

av’+ + 

- + v . w = - V(p’/p0) + v& - rn’; at 

This, along with the incompressibility condition 

+ 

v.v=o 

form the velocity equations of the Boussinesq approximation’. 

Now let us consider the temperature (and by eq. 3, the density) 

evolution. (It is here that we differ from the standard Boussinesq 

approximation and use a more general form). The general equation for 

temperature evolution in an incompressible fluid is 

2 
- m+v+. 

at 

where x = - 

K is the thermometric conductivity (cm2/sec), K is the thermal 

conductivity, and C 

fluid. The last term in (9) is 

is the specific heat (at constant pressure) of the P 

- where the summation convention is in effect (e.q., vi wi = vi w1 + v2 w2 + 

v3 w3) 

2.0. pv=o 

- ap + v . vp = -pv . v 

At this point, let us write the continuity equation: 

+ 

at 

Expanding the divergence term yields 

-b + 

at 

Using (1) and keeping only the lowest order terms (remember 

that V . v - 0) gives 

- + + ap’ + v . Vp’ = -pov . v 

at 

Now, because of (2), equation (9) may be written 

- 

aT’ + V %vi %k + v . b’ = xdT’ +- (- + -) 

at 2cp &k &i 

(11) 

(12) 

(14) 

The term within brackets in (15) and on the right hand side of (14) is small 

compared to the convective term (v . b’); also, the thermal expansion 

coefficient Bc10 . The product of Band the bracketed term in (15) is 

+ 

-2 

3 
therefore of a higher order than any terms being retained in this theoretical 

development. 

condition ( V ? = 0) t i.e., any possible contribution of v v to equation 

It is for this reason that we accept the incompressibility 

+ 

(7) is negligible. These assertions will be made quantitative in the next 

section. 

Let us now examine the energy conservation implicit in (71, (8) and 

(14). To begin, we use the vector identity 

(16) 

to change (7) into 

(17) 

NOW, (17) is multiplied by v . , (14) by Cp 

$++ 2 + 

- = v x w - V(p’/p0 + v /2) + ”& - m’g at 

+ + + 

where the vorticity is w z V x v. 

and the resulting equations are added, resulting in 

2 

- a (I v2 + C T’) = -v . at2 P 2 P 

+ + C T’) - Cpxvl”] 

(Here a/&, 2 $ for convenience). 

bracketed terms on the right hand side of (18) in more detail. 

At this point we must treat the last two 

First, consider the term whose coefficient is v : 

(19) 

4 
+ + Here V . v = a v = 0 is the incompressibility condition. Using g = - V4 , kk 

the last term on the right hand side of (18) is 

+ + p’+ + 

- f3T’V. g =- v* g 

PO 

P’ + =- 2- *o 

Using the continuity condition, (20) becomes 

++ Pho -b P oap’ -BT’v.g=V. (2 

-b = - [$ (pOT’) + V . (pOT’ v)] (21) 

NOW, using (19) and (21), equation (18) becomes 

IC 

2 

+ d+V+ -- 

PO 2 -0. [v ( 

This energy conservation equation is accurate up to first order in terms such 

as p’ and T’. Multiplying both sides of (22) by p and integrating this 0 

expression over the volume of interest, VI we arrive at the following 

d 

dt 

- 

where the total energy in volume V is 

(23) 

(24) 

5 
where the viscosity is rl = po u. Note that, in (24) , 

1 2 

-p v: 

20 

p C T’: 

OP 

Kinetic energy density 

Thermal energy density 

6 poT’ = -p’gz: Gravitational potential energy density (26) 

Several points should be noted concerning the energy conservation 

formulas (23), (24) and (25). First, the reference point €or the total energy 

Second, if the volume V is large enough so that there is no energy flow 

through the bounding surface (or if the flow is periodic or balanced across 

the boundaries) then the surface integral in (23) is zero and dE/dt = 0. 

In the next section, the equations will be written in non-dimensional 

form. This will allow for the relative size of various terms to be determined 

and establish criteria for deciding on the similarity of flow in different 

physical situations. Application of these results to flows within the 

NASA/LaRC Vortex Research Facility will provide a quantitative example. 

At this point we will introduce the presence of a temperature gradient. 

The fluid, at rest, will have a uniform vertical temperature gradient 

y (OC/cm); the temperature variation will thus be composed of two parts: 

T’= yz+ T (27) 

Here T is the temperature anomaly field. Placing (27) into (14) gives 

6 

.’ 

6 
This, along with the equation for fluid motion (7) and incompressibility 

condition (8) are the equations of motion for our thermal-fluid system; these 

are the equations we wish to put in nondimensional form. 

In order to produce our nondimensional equations, we will use the 

following characteristic values: 

Table I Characteristic Values 

L: length 

V: velocity 

y: temperature gradient 

B: coefficient of thermal compressibility 

g: gravitational acceleration 

v: kinematic viscosity 

x: thermometric conductivty 

cp: 

N = Jky: Brunt-Vaisala frequency 

specific heat at constant pressure 

.. .. .. 

The quantities in Table I can be combined into nondimensional groups'; 

these groups have values for specific flows which allow them to be compared to 

other flows. The pertinent groups are given in Table 11. 

7 
Table I1 Dimensionless Groups 

VL 

V 

V 

X 

sr- =- NL N* 

Reynolds number: R;.- 

Prandtl number : P=- 

Strouhal number: VI 

V2 

Lg 

Froude number: F =- 

2 Grashof number: G = (RN*) 

DuLong V2 number : m=- 

Y.L cP 

We may now take equations (7) and (14) and transform them into 

In these equations, v will be measured in terms of + dimensionless equations. 

V, T’ in terms of YL, (pO/p 1 in terms of gL, and time t in terms of N-’. 

0 

Remembering this, the equations are 

- &+N*V. s1+ %=-- + ’ ?‘d+ N* T’; FN* 

ar 1+ 1 Du 2 

k N* N*W +- v . PP’ = - 2”’ +- (a v + ai vk) 2N*R k i 

- 

(29) 

(30) 

In order to estimate the values of the coefficients in (29) and (30) for 

a practical situation, we have drawn the characteristic values from a typical 

experiment at the Vortex Research Facility at NASA/L~RC’. 

in Table 111, along with the associated dimensionless numbers. 

These are presented 

At this point we can estimate the various sizes of the terms in (29) and 

(30). The right hand sides of (29) and (30) are important in relation to the 

size of the v V terms on the left sides, which have coefficients of l/N*. 

+ 

a 
Table I11 Typical Values 

v = 50 cm/sec 

L=lOCm 

v = .145 cm2/sec 

)( = .202 cm2/sec 

y = .OlOC/cm 

g = 980 cm/sec2 

= 3.48 IO-~/OC 

c 

N = 0.185 sec" 

R = 3450 

= 1.012 x 107 cm2/(sec2'oC) P 

P = 0.72 

N* = 0.037 

Du = 0.0025 

F = 0.255 

G = 16300 

Thus, the relative sizes of the various terms to these "convective" or 

inertial terms is given by their coefficients with the 1/N* factor divided 

out. These relative coefficients are given in Table IV. 

Table IV Relative Coefficients 

+ 

v equation 

- 3.92 

= 2.9 xi0 

-- 

F 

- -4 

R 

(N*12 = 1.4 x 10 

-3 

T "equation 

1 -4 -= 4.0 x 10 

RP 

-= 7.2 

R 

9 
At this point, let us estimate the validity of assuming 

incompressibility. If we nondimensionalize (15) we have 

-8 1 2 = (1.4 IO-’) + (2.5 IO (ak vi + a ik 

Thus, in a dimensionless representation where all variables and their 

spatial derivatives are nominally of unit magnitude, the error in assuming 

incompressibility is about one part in ten million. 

¶!W3 DIHENSIONAL FUNI 

It is oCten useful (and computationally expedient) to approximate certain 

flows as being two dimensional. In particular, the evolution of vortex wakes 

trailing aircraft and submarines can be studied by examining the dynamics of 

the pertinent fluid in a cross-sectional plane transverse to the direction of 

motion. In this section we will reduce the general equations of motion (8), 

(17) and (28) to those appropriate to two-dimensional thermally-stratified 

flow. 

As is well known, the incompressibility condition (8) can be 

automatically satisfied in 2-D flow by expressing the velocity in terms of a 

stream function $ (y,z): 

-+ A 

v = VX(XJI) 

A 

=-XXVJI 

The vorticity, in turn, can also be expressed in terms of the stream function: 

10 
A + 

w= v xb= v x (V xx $1 

= v& . V$) - 329 

=-GPJ, 

The scalar vorticity is thus w = -9 J, where the operator 

9 z a2/ay2 + a2/aZ2 because there is no x dependence in the 2-D 

approximation. 

At this point let us take the curl of eq ( 17) : 

+ + 

= vx(; x i) + vV2w - f3W' x g at 

Similarly, the divergence of ( 17 ) gives 

++ + 

v2/2) = V . (v x w) - 8 Vl" . g 

(32) 

(33) 

(34) 

These equations, (33) and (341, are sti,, app Lca-,e to 3-D 

incompressible flow; in particular, they show that the pressure p0 is given, 

at each instant of time, by the instantaneous values of v and T', both of 

which obey non-linear coupled evolution equations. If we now put (271, (31) 

and (32) into (281, (33) and (341, we get a set of equations for studying the 

2-D flow of a thermally-stratified fluid: 

+ 

We also note that the quantity conserved by these equations, for a closed 

system, comes from (24); this is the energy of the system (leaving off time 

independent terms): 

11 
E= I 

V 

. rb 

(38) 

We will call this assembly of equations, (351, (36) and (371, the thermalvortex equations. 

Let us note here that the energy given by (38) is conserved in the 

presence of viscosity and thermal conductivity. As has already been pointed 

out, the energy as given by (38) is explicitly composed of a kinetic part, a 

thermal part, and a gravitational part. This conserved quantity (integral 

I invariant) E can thus be considered to be the actual physical energy of the 

fluid system we are modelling. 

In comparison, let us look at the system of equations which are obtained 

when we set v = x = 0: the inviscid equations. It is well known that in this 

case, (35) and (36) conserve a certain quantity which has been denoted the 

'energy12; this energy E' is 

This, however, is not the physical energy (although, in some computer 

models, it will be numerically close). In addition, with v = x = 0, it is 

easy to show that, for a closed system, there is also another conserved 

quantity : 

C = jv WT dydz (40) 

The presence of two conserved quantities fundamentally changes the 

evolutionary behavior of a dynamic system6. 

being modelled does not actually have two (or more) integral invariants, 

although our model system does, then the predictions we make with our model 

system must surely be suspect. 

If the physical system which is 

12 
These model systems arise in a number of applications, notably inviscid 

hydrodynamics and magnetohydrodynamics. These are interesting models; their 

numerical realizations indeed give rise to very interesting statistical 

systems whose fundamental behavior depends on the number of integral 

invariants’. However , viscosity (and thermal conductivity) are always present 

in real systems so that the accurate modelling of any real system requires the 

incorporation of the proper dissipative mechanisms (viscosity and 

conductivity). 

The presence of integral invariants allows us a numerical check of our 

simulations: we set u = x = 0 and see that the integral invariants are 

conserved: The real system has u # 0, x f 0; in this case there is only one 

“integral invariant“: the total energy. 

~SICne 

In this paper, the equations of motion suitable for studying the 

incompressible motion of a thermally stratified fluid were developed. It was 

shown that the two quantities which satisfy time evolution equations were the 

temperature anomaly and the vorticity. The equations were therefore termed 

the thermal-vortex equations. 

These expressions (for a closed system) contain one integral invariant, 

which is clearly the physical energy. In contrast, the quantity commonly 

called the energy of the non-dissipative equations (u = x = 0) is not, in 

fact, the physical energy. In addition, the non-dissipative equations contain 

a second integral invariant. The presence of two invariants, plus the fact 

that neither is the actual energy, divorces the non-dissipative model from 

physical reality. 

13 
1. 0. M. Phillips, The Dynamics of the Upper Ocean, Sec. 2.4, University 

Press, Cambridge, UK, 1969. 

2. Richard S. Hirsh, "A Numerical Simulation of Vortex Motion in a Stratified 

Environment and Comparison with Experiments", Johns Hopkins APL Technical 

Digest, 5, 3, pp. 203-210, 1984. 

3. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Chapter V, Pergamon, New 

York, 1959. 

4. R. C. Weast, ed., Handbook of Chemistry and Physics, 55th Ed., pp. F305- 

322, CRC Press, Cleveland, OH, 1974. 

5. Satran, et al, "Vortex Research Facility Improvements and Preliminary 

Density Stratification Effects on Vortex Wakes", AIAA Paper No. 85-0050, 

AIM 23th Aerospace Science Meeting, Jan. 1985, Reno, Nevada. 

6. R. H. Kraichnan and D. Montgomery, "Two Dimensional Turbulence", Rep. 

Prog. Phys., - 43, pp 547-619, 1980. 

7. Shebalin, et al., "Anisotropy in MHD Turbulence Due to a Mean Magnetic 

Field", J. Plasma, Phys. , 29, pp. 525-547, 1983. 

14 
Standard Bibliographic Page 

I II 

. Report No. 

NASA CR-178242 

The Thermal-Vortex Equations 

2. Government Acceasion No. 3. Recipient’s Catalog No. 

February 1987 

6. Performing Organization Code 

. Title and Subtitle 

. Author(s) 

John V. Shebalin 

5. Report Date 

8. Performing Organization Report No. 

19. Security Classif.(of this report) 20. Security Classif.(of this page) 21. No. of Pages 

Unclassified Unclassified 16 

. Performing Organization Name and Address 

Old Dominion University 

Department of Electrical and Computer Engineering 

Norfolk, VA 23508 

22. Price 

A0 2 

110. Work Unit No. 

11. Contract or Grant No. 

NAS1-17993 

13. Type of Report and Period Covered 

2. Sponsoring Agency Name and Addresa 

Contractor Report 

14. Sponsoring Agency Code National Aeronautics and Space Administration 

Washington, DC 20546 

505-60-21-03 I 

5. Supplementary Notes 

Langley Technical Monitor: Robert P. Weston 

Final Report - Task Assignment 27 

6. Abstract 

The Boussinesq approximation is extended so as to explicitly account for 

the transfer of fluid energy through viscous action into thermal energy. 

Ideal and dissipative integral invariants are discussed, in addition to the 

general equations for thermal-fluid motion. 

L7. Key Words (Suggested by Authors(s)) 

Vortex Wake 

Vortex Alleviation 

Vortex Flows 

18. Distribution Statement 

Unclassified-Unlimited 

b 

NASA Langley Form 63 (June 1985)