Abstract:
A method is provided to select appropriate material properties for turbulent friction drag reduction, given a specific body configuration and freestream velocity. The method is based on a mathematical description of the balance of energy at the interface between the viscoelastic surface and the moving fluid, and permits determination of the interaction of turbulent boundary layer fluctuations with a viscoelastic layer by solving two subtasks—i.e., a hydrodynamic problem and an elasticity problem, which are coupled by absorption and compliancy coefficients. Displacement, velocity, and energy transfer boundary conditions on a viscoelastic surface are determined, and a Reynolds stress type turbulence model is modified to account for redistribution of turbulent energy in the near-wall region of the boundary layer. The invention permits drag reduction by a coating with specified density, thickness, and complex shear modulus to be predicted for a given body geometry and freestream velocity. For practical applications, viscoelastic coatings may be combined with additional structure, including underlying wedges to minimize edge effects for coatings of finite length, and surface riblets, for stabilization of longitudinal vortices.

Description:
This application claims the benefit of No.  60 / 138 , 023 , filed Jun.  8 ,  1999 . 
    
    
     BACKGROUND OF INVENTION 
     Since M. O. Kramer reported successful experimental results in 1957, there have been repeated attempts to reduce frictional drag in turbulent fluid flow over a surface by applying a passive compliant coating. Experimental results in this area have been mixed. Most investigators have reported a drag increase, while only a few have claimed drag reduction for turbulent flow. A number of theoretical studies have characterized the stability of the laminar boundary layer over a deforming surface and other studies have characterized the reaction of a coating to a fluctuating load. However, no rigorous analytical technique has been previously reported that has been used to successfully design a drag-reducing coating for turbulent flow. 
     In the past, passive coatings were tested without specification and full characterization of critical physical parameters, such as the frequency dependent complex shear modulus, density, and thickness. In order to achieve and ensure drag reduction with a viscoelastic coating, a methodology is required for selecting appropriate material properties and for estimating anticipated drag reduction as a function of configuration and velocity. 
     Relevant background information for associated technical topics is available in the literature, and may be useful due to the technical complexity of this invention. A classical discussion of boundary layer theory, including formulation of Navier-Stokes and turbulent boundary layer equations, is provided in Boundary-Layer Theory, by Dr. Hermann Schlichting, published by McGraw Hill, New York, seventh edition, 1979. A discussion of structures and scales in turbulent flows can be found in Turbulence, 1975, McGraw Hill, written by J. O. Hinze, and in “Coherent Motions in the Turbulent Boundary Layer,” in Annual Review of Fluid Mechanics, 1991, volume 23, pp. 601-39, written by Steven K. Robinson. Background on Reynolds stress types of turbulence models is found in the chapter, “Turbulent Flows: Model Equations and Solution Methodology,” written by Tom Gatski, and included in the Handbook of Computational Fluid Mechanics, published by Academic Press in 1996. Equations in fluid and solid mechanics are often expressed in indicial, or tensor, notation, for compactness. Chapter 2 in the text A First Course in Continuum Mechanics, by Y. C. Fung, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1977, provides a brief introduction into tensor notation for mechanics equations. An introduction to finite difference methods, which are used to solve the system of momentum and continuity equations for a turbulent fluid, is provided in the text, Computational Fluid Dynamics for Engineers, written by Klaus Hoffman, and published in 1989 by the Engineering Education System i Austin  in Austin, Texas. Descriptions of measured and mathematically modeled physical properties of polymers are found in the text, Viscoelastic Properties of Polymers by J. D. Ferry, Wiley, New York, 1980, 3 rd  edition. The article, “Loss Factor Height and Width Limits for Polymer Relaxation,” by Bruce Hartmann, Gilbert Lee, and John Lee, in the Journal of the Acoustical Society of America Vol. 95, No. 1, January 1994, discusses mathematical characterization of shear moduli for real viscoelastic, polymeric materials, including those approximated by the Havriliak-Negami approach. 
     Recently in the international literature (K. S. Choi, X. Yang, B. R. Clayton, E. J. Glover, M. Atlar, B. N. Semonev, and V. M. Kulik, “Turbulent Drag Reduction Using Compliant Surfaces,” Proceedings of the Royal Society of London, A (1997) 453, pp. 2229-2240). Choi et al. reported experimental measurements of up to 7% turbulent friction drag reduction for an axisymmetric body coated with a viscoelastic material. These experiments were performed in the United Kingdom, using coatings designed and fabricated in Russia at the Institute of Thermophysics, Russian Academy of Sciences, Novosibirsk, by a team headed by B .N. Semenov. The basic design approach was outlined in “On Conditions of Modelling and Choice of Viscoelastic Coatings for Drag Reduction,” in Recent Developments in Turbulence Management, K. S. Choi, ed., 1991, pp. 241-262, Dordrecht, Kluwer Publishers. The Novosibirsk design approach is semi-empirical in nature, and does not take into account the full characterization of the complex shear modulus of the viscoelastic material, namely, the relaxation time of the material. The Novosibirsk design approach does take into account frequency-dependent material properties. Furthermore, the Novosibirsk concept is valid only for a membrane-type coating, such as a film which coats a foam-rubber saturated with water or glycerine, and where only normal fluctuations of the surface are considered. 
     The structure of coatings intended for drag reduction has been addressed in the international literature, starting with the 1938 patent No. 669-897, “An Apparatus for the Reduction of Friction Drag,” issued in Germany to Max O. Kramer. Kramer later received a patent in 1964, U.S. Pat. No. 3,161,385, and in 1971, U.S. Pat. No. 3,585,953 for coatings to extend laminar flow in a boundary layer. Soviet inventor&#39;s certificates, such as “A Damping Covering,” USSR patent 1413286, Publication 20.01.1974, Bulletin of the Inventions 14, by V. V. Babenko, L. F. Kozlov, and S. V. Pershin, “An Adjustable Damping Covering,” USSR patent 1597866, Publication 15.03.1978, Bulletin of the Inventions 110, by V. V. Babenko, L. F. Kozlov, and V. I. Korobov, and “A Damping Covering for Solid Bodies,” USSR patent 1802672, Publication 07.02.1981, Bulletin of the Inventions 15, by V. V. Babenko and N. F. Yurchenko, have also described the structure of drag-reducing coatings comprised of viscoelastic materials. These inventor&#39;s certificates identified the three-dimensional structure within a drag-reducing coating, but do not address the methodology for choosing appropriate parameters of the viscoelastic materials to be used in the manufacture of such coatings. Structural features include multiple layers of materials, longitudinal, rib-like inclusions of elastic, viscoelastic, or fluid materials, and heated elements. Viscoelastic coatings may be combined with other forms of structure, such as longitudinal riblets molded on or within the surface of the coating. As described in the international literature in publications such as “Secondary Flow Induced by Riblets,” written by D. B. Goldstein and T. C. Tuan, and published in the Journal of Fluid Mechanics, volume 363, May 25, 1998, pp. 115-152, two-dimensional, rigid riblets alone have been shown experimentally to reduce surface friction drag up to about 10%. 
     BRIEF SUMMARY OF THE INVENTION 
     The present invention enables the design of a passive viscoelastic coating for the reduction of turbulent friction drag. Coatings with material properties designed using the methodology described in this invention have reduced friction drag by greater than 10%. The methodology of the present invention permits, as a first object of the invention, the specification of the frequency dependent complex shear modulus, the density, and the thickness of an isotropic viscoelastic material which will reduce turbulent friction drag relative to specific flow conditions over a rigid surface. Quantitative levels of drag reduction can be estimated. Mathematical detail is provided for the cases of turbulent flow over a rigid flat plate as well as a viscoelastic flat plate, where the invention accounts for both normal and longitudinal oscillations of the surface. A second object of the invention is the specification of material properties for a coating composed of multiple layers of isotropic viscoelastic materials. A third object of the invention is the specification of material properties for a coating composed of an anisotropic material. A fourth object of the invention is the minimization of edge effects for coatings of finite length. A fifth object of the invention is the stabilization of longitudinal vortices through combination of viscoelastic coating design with additional structure, such as riblets. 
     The methodology used herein to describe the interaction of a turbulent boundary layer (TBL) with a viscoelastic (VE) layer involves two tasks, 1) a fluids task, involving the calculation of turbulent boundary layer parameters, given boundary conditions for a rigid, elastic, or viscoelastic surface (herein referred to as the TBL problem), and 2) a materials task, involving the calculation of the response of a viscoelastic or elastic surface to a periodic forcing function which approximates the loading of the turbulent boundary layer. The invention focuses upon cation of amplitudes of  layer. The invention focuses upon calculation of amplitudes of surface oscillations and velocities, and of the energy flux for a viscoelastic coating (hereinafter referred to as the VE problem). These two tasks are coupled by coefficients related to surface boundary conditions of energy absorption and surface oscillation amplitudes (hereinafter referred to as dynamic and kinematic boundary conditions, respectively). The TBL problem is first solved for a rigid surface, thus providing necessary input to describe the forcing function on the surface, and also providing baseline calculations of friction drag, for comparison. The VE problem is solved next, given a periodic forcing function that approximates the shear and pressure pulsations of a given boundary layer. Initial choices for material parameters are based on theoretical and empirical guidelines. Optimal material parameters are chosen, following a series of iterations, such that the following two criteria are met:
         1) the energy flux into the viscoelastic coating is maximum, and   2) the amplitudes of surface oscillation are less than the thickness of the viscous sublayer of the turbulent boundary layer over the coating.
 
If the amplitude of oscillations exceeds the thickness of the viscous sublayer of the turbulent flow, then the oscillations effectively increase the roughness of the surface, thus leading to an increase in friction drag. Furthermore, as the phase speed of disturbances in the boundary layer exceeds the shear wave speed in the material, a resonant interaction with large amplitude waves occurs. These conditions are to be avoided. Moderate energy flux into the material, however, where energy is transformed into internal shear waves and eventually dissipated as heat, leads to a qualitative and quantitative change in the turbulent energy balance, with a consequent reduction in friction drag. Optimal physical properties of a viscoelastic material will vary with freestream velocity, position along a body, pressure gradient, and any other factors which influence the development of the boundary layer and the characteristics of local turbulent fluctuations.
       

     By solving the TBL equations, the turbulent friction drag over a viscoelastic, elastic, or rigid surface can be quantitatively evaluated. In the case of a viscoelastic surface, where energy is absorbed and surface oscillations are nonzero, both dynamic and kinematic boundary conditions are specified. These boundary conditions are derived directly from the solution of the VE equations for energy flux and surface oscillation amplitudes, and then transferred into a dissipation boundary condition and Reynolds stress boundary conditions for solution of the TBL equations. Vertical oscillations influence the effective roughness of the surface, and the root-mean square (rms) value of the vertical oscillation amplitude is classified as the dynamic roughness. If the oscillation amplitudes are lower than the viscous sublayer thickness, it is appropriate to estimate Reynolds stresses as zero. The equations for a turbulent boundary layer describe turbulent diffusion as a gradient approximation, which accommodates the dynamic boundary condition, and near-wall functions are introduced to describe for different surfaces the redistribution of turbulent energy in the near-wall region. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present invention will become more fully understood from the detailed description given below and the accompanying drawings, which are given by way of illustration only and thus are not limitative of the present invention, wherein: 
         FIG. 1a  is a schematic of the interaction of a passive viscoelastic surface with a turbulent boundary layer, where a low-amplitude traveling wave develops on the surface,  FIG. 1b  is an enlargement of the region within the circle shown in  FIG. 1a , and indicates displacement amplitudes, and 
         FIG. 2  is a flowchart of the basic methodology required to a) choose a viscoelastic material that reduces turbulent friction drag under given flow conditions, and b) to quantify the value of drag reduction relative to a rigid surface, and 
         FIG. 3  is a detail of a structure introduced at the intersection between a viscoelastic coating and a rigid surface in order to minimize local oscillation amplitude and edge effects. 
     
    
    
     DETAILED DESCRIPTION 
     The present invention identifies physical and geometric parameters of a viscoelastic coating that reduces turbulent friction drag under given flow conditions. Furthermore, the invention permits evaluation of the anticipated drag reduction effectiveness of a given material with known physical properties for a given body configuration and set of flow conditions. The methodology has been applied principally to the characterization of coatings for turbulent flow over flat plates and bodies of revolution, and can also be applied to more complex geometries having curvature and nonzero pressure gradients. 
     A fluid boundary layer is the very thin layer of fluid adjacent to a surface over which fluid is flowing. It is the region where frictional forces play a major role, and is where the flow adjusts from conditions at the surface to conditions in the freestream of the flow. The outer edge of the boundary layer is traditionally defined as that location where the ratio, β, of the mean velocity, U, to the freestream velocity, U 28  , is a constant which is approximately equal to 1: 
                   U     U   ∞                y   δ     =   1       =     β   ≈   1             (     Equation   ⁢           ⁢   1     )             
 
The value of the constant, β, chosen will depend upon configuration and numerical stability concerns. For the case of a flat plate, an appropriate value for this constant is 0.9975.
 
     A turbulent boundary layer is characterized by a spectrum of pressure and shear fluctuations, the frequency, phase speed, and amplitude characteristics of which are a function of such factors as freestream velocity, body configuration, surface conditions, and pressure gradient. With flow over a rigid surface, there is no motion of the surface. With an elastic or viscoelastic surface, the wall pressure and shear fluctuations act as a forcing function which can deform the surface, creating surface waves. With a viscoelastic surface, energy from the turbulent boundary layer may be absorbed and dissipated by the coating, thus necessitating proper specification of boundary conditions for both Reynolds stresses and energy absorption at the wall (i.e., kinematic and dynamic boundary conditions). 
       FIG. 1  is a schematic of a passive viscoelastic coating that interacts with a turbulent boundary layer of thickness, δ. For simplicity, the case of a flow with a freestream velocity of U ∞  over a coated flat plate is considered, where coordinates x, y, and z are in the longitudinal, normal, and transverse directions, respectively. Alternatively, x is represented as x 1 , y as x 2 , and z as x 3 . It is assumed that the fluid is viscous and incompressible, and that the coating material is viscoelastic (i.e., possesses a combination of elastic and viscous physical properties). If the surface is rigid, there will be no oscillation or energy absorption. If the surface is elastic or viscoelastic, there will be surface oscillation and, if the surface is viscoelastic, there will be energy exchange as well. The components of longitudinal and vertical surface displacement of the viscoelastic material are given by ξ 1  and ξ 2 , and the friction velocity is defined as: 
               u   *     =         τ   w     ρ               (     Equation   ⁢           ⁢   2     )               
where τ w  is the shear stress at the wall and ρ is the density of the fluid. The interaction of the turbulent flow with a viscoelastic coating leads to the formation of a quasi-periodic surface wave. The motion and energy absorption of the coating (kinematic and dynamic boundary conditions) in turn affect the energy balance in the turbulent boundary layer and the value of the friction drag, the latter of which is the surface integral of the wall shear stress.
 
     The methodology for this invention is schematically shown in FIG.  2 . The present invention includes solutions for: 1) turbulent boundary layer (TBL) parameters, including friction drag over rigid, viscoelastic, or elastic plates; and, 2) energy absorption and oscillation amplitudes of a viscoelastic (VE) plate excited by a periodic load which approximates that of a turbulent boundary layer. These two parts of the solution are coupled by boundary, conditions,  parts of the solution are coupled by boundary conditions, both for energy absorbed by the surface and for the amplitudes of surface motion, and are part of an overall methodology for selecting drag-reducing coatings and for quantifying drag reduction for given flow conditions. Below, the two parts of the solution, as well as the methodology that couples them, are described. 
     Characterization of a Turbulent Boundary Layer Over Rigid, Elastic, or Viscoelastic Surfaces (TBL Problem) 
     In  FIG. 2 , TBL parameters are first characterized for a rigid surface (step  1 ) and, once material properties are defined, for a viscoelastic material (step  3 ). The same basic approach is applied in step  3  as in step  1 , except the kinematic and dynamic boundary conditions are defined differently. For a rigid surface, there is no surface motion and no energy absorption. For an elastic surface, there is surface motion, but no energy absorption. For a viscoelastic surface, there is surface motion and energy absorption. 
     General System of Equations of Continuity, Motion and Energy: Turbulent flow parameters are obtained through the solution of a system of equations of continuity, motion, and energy, with accompanying boundary conditions. These equations are developed from principles of conservation of mass, conservation of momentum (Newton&#39;s second law), and energy balance (as based on the first law of thermodynamics). 
     In Cartesian coordinates, the general equation of continuity for a compressible fluid having a density of ρ, and velocity components U, V, and W in the streamwise, normal, and transverse directions, is given by Equation 3, below. 
                   ∂   ρ       ∂   t       +       ∂     (     ρ   ⁢           ⁢   U     )         ∂   x       +       ∂     (     ρ   ⁢           ⁢   V     )         ∂   y       +       ∂     (     ρ   ⁢           ⁢   W     )         ∂   z         =   0           (     Equation   ⁢           ⁢   3     )             
 
Alternatively, Equation 3 can be written in terms of indicial notation (as in Equation 4), where x, y, and z are represented by x 1 , x 2 , and x 3 , respectively, an where U, V, and W are represented by U 1 , U 2 , and U 3 , respectively. It is implied that the index, i, can have a value of 1, 2, or 3, and that a repeated index of i indicates summation. 
                   ∂   ρ       ∂   t       +       ∂     (     ρ   ⁢           ⁢     U   i       )         ∂     x   i           =   0           (     Equation   ⁢           ⁢   4     )             
 
     Equation (3) is further simplified for a fluid that is incompressible, i.e., where the density of the fluid is constant, the following applies: 
                 ∂     U   i         ∂     x   i         =   0           (     Equation   ⁢           ⁢   5     )             
 
     The generalized equations of motion, termed the Navier-Stokes equations, are expressed in Cartesian coordinates for the case of an incompressible fluid with constant viscosity as: 
                   ∂     U   i         ∂   t       +       U   j     ⁢       ∂     U   i         ∂     x   j             =         -     1   ρ       ⁢       ∂   ??       ∂     x   i           +     g   i     +     v   ⁢         ∂   2     ⁢     U   i           ∂     x   j       ⁢     ∂     x   j                       (     Equations   ⁢           ⁢   6   ⁢   a   ⁢     -     ⁢   6   ⁢   b   ⁢     -     ⁢   6   ⁢   c     )             
 
In Equations 6:
         P is the mean pressure,   g i  is the body force vector, due to external fields, such as gravity, which act on the element, and   ν is the kinematic viscosity (assumed to be constant).
 
Note that the above expression, written in indicial notation, actually represents three equations, for the three components of velocity in the x, y, and z directions.
       

     Turbulent velocity components may be described as the sum of the mean and fluctuating components, U i  and u′, respectively, where U 1 , U 2 , and U 3 , are equivalent to U, V, and W and where u′ 1 , u′ 2 , and u′ 3  are equivalent to u′, v′, and w′:
 
U i =U i +u i   (Equations 7a-7b-7c)
 
An overbar indicates time-averaging:
 
U i =Ū i   (Equations 8a-8b-8c)
 
Substituting Equations 7a-7c into Equations 6a-6c and time-averaging yields the following system of three complex nonlinear second-order partial differential equations of motion for turbulent flow: 
                 U   j     ⁢       ∂     U   i         ∂     x   j           =         -     1   ρ       ⁢       ∂   ??       ∂     x   i           +     g   i     +     v   ⁢         ∂   2     ⁢     U   i           ∂     x   j       ⁢     ∂     x   j             -       ∂         u   i   ′     ⁢     u   j   ′       _         ∂     x   j                   (     Equations   ⁢           ⁢   9   ⁢   a   ⁢     -     ⁢   9   ⁢   b   ⁢     -     ⁢   9   ⁢   c     )             
 
In Equations 9a-9c, the components  u′ 2   ,  v′ 2   , and  w′ 2    are termed normal Reynolds stresses, and the components in the form −  u′v′ , −  v′w′ , and −  u′w′  are termed Reynolds shear stresses. The turbulent kinetic energy, k, is defined as: 
             k   =       1   2     ⁢     (         u   ′2     _     +       v   ′2     _     +       w   ′2     _       )               (     Equation   ⁢           ⁢   10     )             
 
Closure of the generalized system of equations including the continuity equation (Equation 5) and the equations of motion (Equations 9a-9c) for a turbulent flow requires seven additional equations to characterize the six Reynolds stresses,  u i ′u j ′ , and the rate of transport of the turbulent kinetic energy, k. This invention solves for the isotropic dissipation rate, ε, which is related to energy transport through the fluid and at the fluid-surface interface. The energy transport equation is based upon the first law of thermodynamics, where heat, dQ, added to a volume during an element of time, dt, serves to increase internal energy, dE, and to perform work, dWK. 
                 ⅆ   Q           ⅆ   t     ︸     heat       =         ⅆ   E           ⅆ   t     ︸     energy       +       ⅆ   WK           ⅆ   t     ︸     work                 (     Equation   ⁢           ⁢   11     )             
 
     There exist multiple approaches within the literature for developing additional equations for Reynolds stress terms in turbulent flow, but this invention adopts a Reynolds-stress-transport-type methodology. In this methodology, equations for Reynolds stresses take the following general form: 
                 U   k     ⁢       ∂         u   i   ′     ⁢     u   j   ′       _         ∂     x   k           =       P   ij     -     Π   ij     -       ∂     J   ijk         ∂     x   k         -     2   ⁢     ɛ   ij                 (     Equations   ⁢           ⁢   12   ⁢   a   ⁢     -     ⁢   12   ⁢   f     )             
 
where P ij  is termed the production, Π ij  is termed the pressure-strain correlation tensor, J ijk  is termed the diffusive flux of the Reynolds stresses, and ε ij  is termed the dissipation tensor.
 
     In the general case, equations for all six Reynolds stress terms, and for the energy dissipation rate must expressed. The equation for the isotropic dissipation rate, ε, is similar in structure to the equations for the transport of Reynolds stresses. Full mathematical expressions for the Reynolds stress and isotropic dissipation rate equations shall be expressed in the following section for the specific case of a two-dimensional turbulent boundary layer. 
     In summary, the equations which are solved to determine turbulent boundary layer parameters include:
         the continuity equation, Equation 5,   the three equations of motion, Equations 9a, 9b, 9c,   six equations for Reynolds shear and normal stresses, Equations 12a-12f, and   the equation for isotropic dissipation rate, ε (Equation 16 below).       

     The methodology for the solution of turbulent flow parameters involves a finite difference approximation of the system of equations of motion and continuity, with accompanying boundary conditions. 
     Turbulent Boundary Layer Equations: Complete mathematical formulations are provided for the specific case of a turbulent boundary layer with a steady, two-dimensional mean flow and a constant freestream velocity, U ∞ . Two-dimensional turbulent boundary layer equations, as termed in the literature, are derived from the general continuity equation (Equation 5) and equations of motion (Equations 9a-9c), given the assumptions that:
         The mean transverse velocity, W, is zero.   Gravitational forces can be neglected.   The pressure gradient is approximately zero in the y direction.   The mean velocity in the streamwise direction, U, is much greater than the mean velocity in the normal direction, V.   The rate of change of parameters in the x direction is much smaller than the rate of change of parameters in the y direction.
 
The above assumptions permit simplification of the set of equations required to solve for turbulent boundary layer parameters to include the revised continuity equation, Equation (13): 
                   ∂   U       ∂   x       +       ∂   V       ∂   y         =   0           (     Equation   ⁢           ⁢   13     )             
 
and the equation of motion for the U velocity component (Equation 14): 
                 U   ⁢       ∂   U       ∂   x         +     V   ⁢       ∂   U       ∂   y           =         -     1   ρ       ⁢       ∂   ??       ∂   x         +     v   ⁢         ∂   2     ⁢   U       ∂     y   2           -       ∂         u   ′     ⁢     v   ′       _         ∂   y                 (     Equation   ⁢           ⁢   14     )             
       

     Where transport equations for the six Reynolds stress components are required in the general case, the Reynolds shear stress components −  v′w′  and −  u′w′  are considered to be very small, so that only equations for  u′ 2   ,  v′ 2   ,  w′ 2   , and −  u′v′  are formulated in the format of Equation 12 (repeated below as Equations 15a-15d). 
                 U   k     ⁢       ∂         u   i   ′     ⁢     u   j   ′       _         ∂     x   k           =       P   ij     -     Π   ij     -       ∂     J   ijk         ∂     x   k         -     2   ⁢     ɛ   ij                 (     Equations   ⁢           ⁢   15   ⁢   a   ⁢     -     ⁢   15   ⁢   d     )             
 
where P ij  is the production term, Π ij  is the pressure-strain correlation tensor, J ijk  is the diffusive flux of the Reynolds stresses, and ε ij  is the dissipation tensor. A fifth equation for ε is: 
                 U   ⁢       ∂   ɛ       ∂   x         +     V   ⁢       ∂   ɛ       ∂   y           =         C     s   ⁢           ⁢   1       ⁢     f   1     ⁢     ɛ   k     ⁢     P   Σ       -       C     s   ⁢           ⁢   2       ⁢     f   2     ⁢       ɛ   k     ⁡     [     ɛ   -     v   ⁢         ∂   2     ⁢   k       ∂     y   2             ]         +       ∂     ∂   y       ⁢     (     2   ⁢     ɛ   t     ⁢       ∂   ɛ       ∂   y         )       +     v   ⁢         ∂   2     ⁢   ɛ       ∂     y   2                     (     Equation   ⁢           ⁢   16     )             
 
where the expression for viscous diffusion may alternatively be approximated as: 
                 [     [               ⁢   v   ⁢         ∂   2     ⁢   k       ∂     y   2           =     2   ⁢     v   ⁡     (       ∂     k     1   /   2           ∂   y       )                   (     Equation   ⁢           ⁢   17     )     ⁢         ]     ⁢         ]                   v   ⁢         ∂   2     ⁢   k       ∂     y   2           ≈     2   ⁢     v   ⁡     (       ∂     k     1   /   2           ∂   y       )                 (     Equation   ⁢           ⁢   17     )             
 
if required for numerical stability in solutions of viscoelastic, non-oscillating surfaces with limited grid points in the near-wall region.
 
     In Equations (15a-15d), the term P ij  may be expressed as: 
               P   ij     =         -         u   i   ′     ⁢     u   k   ′       _       ⁢       ∂     U   j         ∂     x   k           -           u   j   ′     ⁢     u   k   ′       _     ⁢       ∂     U   i         ∂     x   k                     (     Equations   ⁢           ⁢   18   ⁢   a   ⁢     -     ⁢   18   ⁢   d     )             
 
In Equation (16), the term P Σ  may be expressed as: 
               P   Σ     =       1   2     ⁢     P   ii               (     Equation   ⁢           ⁢   19     )             
 
     The pressure-strain correlation tensor, Π ij , which redistributes energy between different components of Reynolds stresses, may be expressed as: 
               Π   ij     =           C   1     ⁡     (     ɛ   k     )       ⁢     (           u   i   ′     ⁢     u   j   ′       _     -       δ   ij     ⁢     2   3     ⁢   k       )       +       C   2     ⁡     (       P   ij     -       δ   ij     ⁢     2   3     ⁢     P   Σ         )       +     π     ij   ,   1     ′     +     π     ij   ,   2     ′     +     π     ij   ,   3     ′               (     Equations   ⁢           ⁢   20   ⁢   a   ⁢     -     ⁢   20   ⁢   d     )             
 
where the π′ ij,1  terms represent near-wall redistribution of turbulent energy from the streamwise component to the normal and transverse components, the π′ ij,2  terms represent near-wall variation of the Reynolds stress tensor component production, and the π′ ij,3  terms represent near-wall redistribution of turbulent energy proportional to local vorticity: 
                     π     ij   ,   1     ′     =       -     C   1   ′       ⁢     ɛ   k     ⁢     (           v   ′2     _     ⁢     δ     i   ⁢   j         -       3   2     ⁢     (             v   ′     ⁢     u   i   ′       _     ⁢     δ   jl       +           v   ′     ⁢     u   j   ′       _     ⁢     δ   il         )         )     ⁢     f   (     l   y     )                                 (     Equations   ⁢           ⁢   21   ⁢   a   ⁢     -     ⁢   21   ⁢   d     )                 π     ij   ,   2     ′     =       -       C   2   ′     ⁡     (       P   ij     -       2   3     ⁢     δ     i   ⁢   j       ⁢     P   Σ         )         ⁢     f   ⁡     (     l   y     )                 (     Equations   ⁢           ⁢   22   ⁢   a   ⁢     -     ⁢   22   ⁢   d     )                   [     [               ⁢     π     ij   ,   3     ′       =       -       C   3   ′     ⁡     (       P   ij     -     D   ij       )         ⁢   f   ⁢           ⁢     (     l   y     )                 (     Equations   ⁢           ⁢   23   ⁢   a   ⁢     -     ⁢   23   ⁢   d     )     ⁢         ]     ⁢         ]                   π     ij   ,   3     ′     =         C   3   ′     ⁡     (       P   ij     -     D   ij       )       ⁢     f   ⁡     (     l   y     )                 (     Equations   ⁢           ⁢   23   ⁢   a   ⁢     -     ⁢   23   ⁢   d     )             
 
Here 
         f   ⁡     (     l   y     )       ⁢               
 
is a unique damping function for the near-wall region: 
               f   ⁡     (     l   y     )       =         R   t       R   k       ⁢     (     1   +       1   +     100     R   t             )               (     Equation   ⁢           ⁢   24     )             
 
where: 
               R   k     =         k     1   /   2       ⁢   y     v             (     Equation   ⁢           ⁢   25     )             
 
and 
               R   t     =       k   2       v   ⁢   ɛ               (     Equation   ⁢           ⁢   26     )             
 
Here, D ij  is a dissipation tensor: 
                     D   ij     =         -         u   i     ⁢     u   j       _       ⁢       ∂     U   i         ∂     x   j           -           u   j     ⁢     u   i       _     ⁢       ∂     U   i         ∂     x   i                           ∂     J   ijk         ∂     x   k                     (     Equations   ⁢           ⁢   27   ⁢   a   ⁢     -     ⁢   27   ⁢   d     )             
 
is the gradient of turbulent and viscous diffusive flux of the Reynolds stresses in the boundary layer, where only one component remains in the boundary-layer representation: 
                 -       ∂     J   ijk         ∂     x   2           ±           ∂     ∂   y       ⁢     (         (     A   ⁢   C     )     t     ⁢       k   ɛ     ⁡     [         v   ′2     _     ⁢       ∂         u   i   ′     ⁢     u   j   ′       _         ∂   y         ]         )       ︸       turbulent   ⁢           ⁢   diffusion         +       v   ⁢         ∂         u   i   ′     ⁢     u   j   ′       _         ∂     y   2         ︸         viscous   ⁢           ⁢   diffusion               (     Equations   ⁢           ⁢   28   ⁢   a   ⁢     -     ⁢   28   ⁢   d     )             
 
where A is 6 in the equation for  v′ 2   , 2 for  u′ 2    and  w′ 2   , and 4 for −  u′v′  (i.e., the effective gradients of turbulent diffusion are different for different components of Reynolds stress), and where the coefficient of turbulent diffusion, ε t , is: 
               ɛ   t     =       C   t     ⁢     k   ɛ     ⁢       v   ′2     _               (     Equation   ⁢           ⁢   29     )             
 
except for Equation (16), where:
 
C i =C ε   (Equation 30)
 
     The dissipation tensor, ε ij , is written as: 
               ɛ     i   ⁢   j       =       v   ⁢           ∂       u   i   ′     _         ∂     x   k         ⁢       ∂       u   j   ′     _         ∂     x   k           _       ≈         f   s     ⁢           u   i   ′     ⁢     u   j   ′       _       2   ⁢   k       ⁢   ɛ     +       (     1   -     f   s       )     ⁢     1   3     ⁢     δ     i   ⁢   j       ⁢   ɛ                 (     Equations   ⁢           ⁢   31   ⁢   a   ⁢     -     ⁢   31   ⁢   d     )             
 
where ƒ s  characterizes flow in the near-wall region: 
               f   s     =       1     1   +     0.06   ⁢     R   t           ⁢           ⁢   and             (     Equation   ⁢           ⁢   32     )                 R   t     =       k   2       v   ⁢   ɛ               (     Equation   ⁢           ⁢   33     )             
 
     Equation (16) includes two functions, ƒ 1  and ƒ 2 , which also introduce corrections for near-wall flows:
 
ƒ 1 =1+0.8e −R     t     (Equation 34)
 
ƒ 2 =1−0.2e −R     t     2   (Equation 35)
 
Values of constants for flow over a flat plate are as shown in Table 1:
 
                                                             C 1     C 2     C ε1     C a2     C t     C ε     C 1 ′   C 2 ′   C 3 ′                   1.34   0.8   1.45   1.9   0.12   0.15   0.36   0.45   0.036                    
For the case of a two-dimensional boundary layer, turbulent boundary layer parameters at different x and y locations are determined through solution of the continuity equation (Equation 13), the equation of motion in the x-direction (Equation 14), the transport equations for the Reynolds stresses  u′ 2   ,  v′ 2   ,  w′ 2   , and −  u′v′  (Equations 15a-15d), and the equation for the isotropic energy dissipation rate (Equation 16), given appropriate boundary conditions. The problem is solved numerically using a finite difference approximation. All equations are reduced to a standard type of parabolic equation in terms of a given function, and solution is obtained at designated grid points in an (x,y) coordinate system.
 
     Boundary Conditions: Boundary conditions are values of parameters at the limits of the boundary layer, i.e., at the surface and the freestream. The freestream velocity is defined as U ∞ . Boundary conditions at the surface are specified for Reynolds normal and shear stresses (kinematic boundary conditions), as well as for the isotropic dissipation rate (dynamic boundary condition). For an arbitrary geometry, the x and y coordinates of the surface must be specified. If the surface is a flat plate, the boundary will be along the line y=0. 
     Since oscillation amplitudes at the surface are small, linearized kinematic boundary conditions, where mean velocities at the surface are assumed to be zero, are appropriate. Boundary conditions for fluctuating velocity components at the surface of a flat plate are expressed as: 
                   u   ′            y   =   0       =           ∂     ξ   1         ∂   t       ⁢   cos   ⁢           ⁢   Θ     -       ξ   2     ⁢       u   *   2     v                 (     Equation   ⁢           ⁢   36     )                     v   ′            y   =   0       =       ∂     ξ   2         ∂   t               (     Equation   ⁢           ⁢   37     )                     w   ′            y   =   0       =         ∂     ξ   1         ∂   t       ⁢   sin   ⁢           ⁢   Θ             (     Equation   ⁢           ⁢   38     )             
 
where ξ 1  and ξ 2  are the longitudinal and vertical surface displacement components, respectively, u* is the friction velocity (as previously defined), and Θ is the angle of the longitudinal axis relative to the mean flow in the x 1 -x 3  plane. With linearized boundary conditions, mean velocities at the wall are assumed to be zero. Surface displacements are approximated by the first mode of a Fourier series: 
               ξ   i     =         ∑     j   =   1     ∞     ⁢           ⁢       a   ij     ⁢     ⅇ     i   ⁢           ⁢       α   j     ⁡     (     x   -   Ct     )               ≈       a   il     ⁢     ⅇ     i   ⁢           ⁢       α   e     ⁡     (     x   -   Ct     )                       (     Equation   ⁢           ⁢   39     )             
 
Here, α 3  is the wavenumber corresponding to the maximum turbulent energy in the boundary layer, and is given by: 
               α   e     =       ω   e     C             (     Equation   ⁢           ⁢   40     )             
 
where the energy-carrying frequency, ω e , is assumed to be: 
               ω   e     =       U   ∞     δ             (     Equation   ⁢           ⁢   41     )             
 
and the phase speed corresponding to energy-carrying disturbances in the boundary layer is assumed to be:
 
C≈0.8U ∞   (Equation 42)
 
Since there is a range of frequencies which carry energy, as reported within the scientific literature, it is advantageous to also perform calculations for the case where: 
               ω   e     =     2   ⁢       U   ∞     δ               (     Equation   ⁢           ⁢   43     )             
 
In the absence of resonance, it is appropriate to time-average components of the Reynolds stress at the wall: 
                     u   ′2     _            y   =   0       =       1   2     ⁢       ω   e   2     ⁡     (              ξ   1          2     +         2   ⁢     u   *   2           ω   e     ⁢   v       ⁢          ξ   1          ⁢          ξ   2          ⁢     sin   ⁡     (       φ   2     -     φ   1       )         +       1     ω   e   2       ⁢       (       u   *   2     v     )     2     ⁢            ξ   2          2         )                 (     Equation   ⁢           ⁢   44     )                       v   ′2     _            y   =   0       =       1   2     ⁢     ω   e   2     ⁢            ξ   2          2               (     Equation   ⁢           ⁢   45     )                       w   ′2     _            y   =   0       =       1   2     ⁢     ω   e   2     ⁢            ξ   1          2     ⁢     tan   2     ⁢   Θ             (     Equation   ⁢             ⁢             ⁢   46     )                     -         u   ′     ⁢     v   ′       _              y   =   0       =       1   2     ⁢     ω   e   2     ⁢          ξ   2          ⁢          ξ   1          ⁢     cos   ⁡     (       φ   2     -     φ   1       )                 (     Equation   ⁢           ⁢   47     )             
 
where |ξ i | is the rms amplitude of the displacement. For a passive isotropic viscoelastic coating excited by a forced load, the response takes the form of a traveling wave, so that the phase shift between normal and longitudinal displacements, φ 2 -φ 1 , will be approximately π/ 2 , and the  displacements, φ 2 -φ 1   , will be approximately π/ 2 , and the  Reynolds shear stresses at the surface will be approximately zero. For anisotropic materials, the phase shift can be different, so that negative Reynolds shear stresses can be generated at the wall. For a rigid wall, there will be no motion at the wall, so that Reynolds shear and normal stresses shall be equal to zero.
 
     The boundary condition for the isotropic dissipation rate is: 
                     ɛ          y   =   0       =       ∂     ∂   y       ⁡     [       (     v   +     ɛ   t       )     ⁢       ∂   k       ∂   y         ]                y   =   0             (     Equation   ⁢           ⁢   48     )             
 
where the first term reflects viscous dissipation and the second reflects absorption of energy by the viscoelastic material. For a rigid surface, there is no energy absorption at the wall, so that the second term equals zero. The absorption of turbulent energy by the coating is equivalent to −  p′v′ , and can be approximated by 
                 ɛ   t     ⁢       ∂   k       ∂   y                          y   =   0       ,       
 
which is the diffusive flux of energy across the boundary, characterized using a gradient mechanism for turbulent diffusion. This expression of the dynamic boundary condition is compatible with the Reynolds stress transport methodology of turbulence closure.
 
     Equations 13, 14, 15a-15d, and 16 are solved for mean velocity components, Reynolds normal and shear stresses, and energy dissipation, given the kinematic and dynamic boundary conditions (Equations (44) through (48)) based on the solution of the viscoelasticity problem (as described in the following section). The problem is solved numerically, using finite difference approximations of the parabolic equations. Friction drag for a body with a viscoelastic coating is calculated as the integral of wall shear stresses, τ w , over the surface of the body, where: 
                     τ   w     =     μ   ⁢       ∂   U       ∂   y                            y   =   0             (     Equation   ⁢           ⁢   49     )             
 
for a two-dimensional body, and where μ=ρν is the dynamic viscosity. Comparison of results with those calculated for a rigid body of identical geometry under identical flow conditions leads to an estimation of anticipated friction drag reduction.
 
     To reduce friction drag, it is necessary to minimize surface oscillation amplitudes, while maximizing the flux of turbulent energy from the flow into the coating, −  p′v′ . If the amplitudes of surface oscillation, ξ 2   + , do not exceed the thickness of the viscous sublayer, generally where: 
               y   +     =         yu   *     v     &lt;   ≈   7             (     Equation   ⁢           ⁢   50     )             
 
then the normal Reynolds stresses at the boundary (Equations (44)-(46)) may be approximated as zero. For a coating which absorbs energy, with low levels of oscillation, shear stresses in the near-wall region of the boundary layer decrease, as does the production of turbulence in the boundary layer. For a coating that oscillates at amplitudes greater than that of the viscous sublayer, the surface can act as a dynamic roughness element and thereby enhance the level of turbulence generated within the boundary layer.
 
Response of a Viscoelastic Material to a Turbulent Boundary Layer (VE Problem)
 
     The second part of the methodology determines the response of a viscoelastic material to a turbulent boundary layer (step  2  in FIG.  2 ). For a rigid surface, Reynolds stresses on the surface (Equations (44) to (47)) are zero, and the isotropic dissipation rate contains only the viscous term. However, for a viscoelastic material, the kinematic and dynamic boundary conditions are determined through solution of the two-dimensional conservation of momentum equation for a viscoelastic material: 
                 ρ   s     ⁢         ∂   2     ⁢     ξ   i         ∂     t   2           =       ∂     σ   ij         ∂     x   j                 (     Equations   ⁢           ⁢   51   ⁢   a   ⁢     -     ⁢   51   ⁢   b     )             
 
where ρ s  is the material density, ξ 2  and ξ 2  are the longitudinal  where ρ s    is the material density, ξ   1    and ξ   2    are the longitudinal  and normal displacements through the thickness of the coating, and σ ij  is the amplitude of the stress tensor. The stress tensor for a viscoelastic material is written for a Kelvin-Voigt type of material as:
 
σ ij =λ(ω)ε s δ ij +2μ(ω)ε ij   s   (Equations 52a-52d)
 
where ε ij   s  is the strain tensor: 
               ɛ   ij   s     =       1   2     ⁢     (         ∂     ξ   i         ∂     x   j         +       ∂     ξ   j         ∂     x   i           )               (     Equations   ⁢           ⁢   53   ⁢   a   ⁢     -     ⁢   53   ⁢   d     )             
 
and:
 
ε s =ε ii   s   (Equation 54)
 
λ(ω) is the frequency-dependent Lame constant, which is defined in terms of the bulk modulus, K(ω), which can be reasonably approximated as the static bulk modulus, K 0 , and the complex shear modulus, μ(ω): 
               λ   ⁡     (   ω   )       =       K   0     -       2   3     ⁢     μ   ⁡     (   ω   )                   (     Equation   ⁢           ⁢   55     )             
 
     Displacements, ξ i , are approximated as periodic, in the form of Equation (39), and can be expressed as a function of potentials of longitudinal and transverse (shear) waves: 
               ξ   _     =         ∇   ϕ     +       ∇     Ψ   →       ⁢     Ψ   →         =     {     0   ,   0   ,   Ψ     }               (     Equation   ⁢           ⁢   56     )             
 
where ∇ φ  is the gradient of φ and ∇×{right arrow over (Ψ)} is the curl of the vector {right arrow over (Ψ)}. Equation (54) can be rewritten as two decoupled equations for the two wave potentials: 
                 [       2   ⁢     μ   ⁡     (   ω   )         +     λ   ⁡     (   ω   )         ]     ⁢       ∇   2     ⁢   φ       =       ρ   s     ⁢         ∂   2     ⁢   φ       ∂     t   2                   (     Equation   ⁢           ⁢   57     )                   μ   ⁡     (   ω   )       ⁢       ∇   2     ⁢   Ψ       =       ρ   s     ⁢         ∂   2     ⁢   Ψ       ∂     t   2                   (     Equation   ⁢           ⁢   58     )             
 
     Equations (57) and (58) can be solved for the potentials, φ and Ψ, and hence for displacements, velocities, and stresses through the thickness of the coating, if boundary conditions are specified. The coating is fixed at its base, so that the longitudinal and normal displacements are zero, and the shear stress and pressure load on the surface is known. Pressure and shear pulsations on the coatings are approximated as periodic functions, with a form similar to that of the displacements in Equation (39), but with the following magnitudes, respectively:
 
τ ω =ρu* 2   (Equation 59)  
 
ρ rms =K p τ ω- K p ρu* 2   (Equation 60)  
 
I ω   =ρμ   φ   2   ( Equation  59   )
 
ρ rms   =K   p   I   (t)   =K   p   ρμ*   2   ( Equation  60   )
 
where K p  is the Kraichnan parameter, whose value is approximated as 2.5.
 
     If shear pulsations are included, a phase shift between shear and pressure pulsations must also be introduced. 
     If calculations are performed for a unit load, then surface displacements under actual load will be: 
                      ξ   i          ⁢     ❘     p     rms   =   actual           =             ρ   ⁢           ⁢     K   p     ⁢     u   *   2     ⁢          ξ   i            ⁢     ❘       p   rms     =   1         H     ⁢   H   ⁢       u   ~     *   2       =       C   ki     ⁢   H   ⁢       u   ~     *   2                 (     Equation   ⁢           ⁢   61     )             
 
Kinematic boundary conditions in Equations (44) to (47) for the turbulent boundary layer problem are rewritten in terms of output from the materials problem: 
                   u   ′2     _       U   ∞   2       =       1   2     ⁢       (     H   δ     )     2     ⁢       (       C   k1     +         u   ~     *     ⁢     Re   *     ⁢     C   k2         )     2     ⁢         u   ~     *   4     ⁡     (       Re   *     =         u   *     ⁢   δ     v       )                 (     Equation   ⁢           ⁢   62     )                     v   ′2     _       U   ∞   2       =       1   2     ⁢       (     H   δ     )     2     ⁢     C   k2   2     ⁢       u   ~     *   4               (     Equation   ⁢           ⁢   63     )                     w   ′2     _       U   ∞   2       =       1   2     ⁢       (     H   δ     )     2     ⁢     C   k1   2     ⁢       u   ~     *   4     ⁢     tan   2     ⁢   Θ             (     Equation   ⁢           ⁢   64     )                 -           u   ′     ⁢     v   ′       _       U   ∞   2         =   0           (     Equation   ⁢           ⁢   65     )             
 
     Dynamic boundary conditions are rewritten in the form: 
                     -               p   ′     ⁢     v   ′       _            y   =   0         ρU   ∞   3         ≈         ɛ   ~     t     ⁢       ∂     k   ~         ∂     y   ~                              y   =   0             (     Equation   ⁢           ⁢   66     )             
 
where:
 
C k3 =C k2 K p γ(ω)
 
and where γ(ω) is a dissipative function of the coating material.
 
     The flux of turbulent fluctuating energy through the surface can be solved for directly, as:
 
−  p′v′ | y=0 =−¼|p′|ω(−tξ 2 +tξ* 2 )  (Equation 67)
 
but the nondimensionalized flux can also be approximated as a diffusive flux term, using the gradient diffusion approach: 
                     -               p   ′     ⁢     v   ′       _            y   =   0         ρU   ∞   3         ≈         ɛ   ~     t     ⁢       ∂     k   ~         ∂     y   ~                              y   =   0             (     Equation   ⁢           ⁢   68     )             
 
where: 
               k   ~     =     k     U   ∞   2               (     Equation   ⁢           ⁢   69     )                 y   ~     =     y   δ             (     Equation   ⁢           ⁢   70     )                   ɛ   ~     t     =       ɛ   t         U   ∞     ⁢   δ               (     Equation   ⁢           ⁢   71     )             
 
     Equation 68 provides a basis for determining the value of the kinematic coefficient of turbulence diffusion, ε q , on an absorbing surface. Substituting Equation 68 into Equation  66  yields the following expression for ε q , defined as ε t | y=0 :   66  yields the following expression for {tilde over (ε)} q   , defined as {tilde over (ε)}   t | y=0 :
                 ɛ   ~     q     =           ɛ   t     ⁢     ❘     y   =   0             U   ∞     ⁢   δ       =       1   4     ⁢         C   k3     ⁡     (     H   δ     )       2     ⁢       u   ~     *     ⁢           u   ~     *   3     ⁢     Δ   max           k   max   +     -     k   q   +                     (     Equation   ⁢           ⁢   72     )             
 
Here, K +   max  is the maximum of turbulence kinetic energy, and K +   q  is the kinetic energy of the oscillating surface, both quantities nondimensionalized by U ∞   2 . y max   +  is defined as the normal distance from the surface to the maximum of turbulence energy, nondimensionalized as follows: 
               y   max   +     =         y   max     v     ⁢     u   *               (     Equation   ⁢           ⁢   73     )             
 
and: 
               Δ   max     =       y   +       Re   *               (     Equation   ⁢           ⁢   74     )             
 
where: 
               Re   *     =         u   *     ⁢   δ     v             (     Equation   ⁢           ⁢   75     )             
 
Thus we can determine the dissipation rate at the wall based on (Equation 48).
 
Methodology to Choose Properties of a Drag-Reducing Viscoelastic Material
 
     A methodology to choose properties of a viscoelastic coating that reduces turbulent friction drag necessarily requires both of the previously described solutions for turbulent boundary layer parameters and response of a viscoelastic material. 
     For the case of two-dimensional flow over a flat plate, the TBL problem is solved for a rigid plate, in order to determine the boundary layer thickness, δ, at a given freestream velocity, U ∞ , and location. The boundary layer thickness is determined from the finite difference solution of the seven equations of continuity, motion in the x-direction, transport equations for Reynolds normal and shear stresses, and the equation for dissipation rate, assuming no motion at the wall. The external limit of the boundary layer is defined as that location where the ratio of the mean velocity to the freestream velocity is a constant, β, between 0.95 and 1.0. 
     The frequency-dependent, complex shear modulus of a material, μ(ω), can be expressed in different mathematical forms, some of which approximate experimentally measured shear modulus data more accurately than others. A single relaxation time (SRT) material is one where the complex shear modulus is expressed using a single relaxation time, τ s , and a single value for the dynamic shear modulus, μ 2 . In  and a single value for the dynamic shear modulus μ 2   . In  Equation (76), an SRT material would be represented for the case of N=1. A multiple relaxation time (MRT) material is one where N&gt;1 in the representation for complex shear modulus in Equation (78). 
               μ   ⁡     (   ω   )       =       μ   0     +       ∑     j   =   1     N     ⁢           ⁢       μ   j     ⁡     [           (     ωτ   j     )     2       1   +       (     ωτ   j     )     2         +     ⅈ   ⁢       ωτ   j       1   +       (     ωτ   j     )     2             ]                   (     Equation   ⁢           ⁢   76     )             
 
The Havriliak-Negami (HN) representation for the complex shear modulus, is given by Equation 74. This equation is more complex, but often is more suitable for describing real materials: 
                 μ   -     μ   ∞           μ   0     -     μ   ∞         =     1       [     1   +       (     ⅈωτ   HN     )       α   HN         ]       β   HN                 (     Equation   ⁢           ⁢   77     )             
 
For a HN type of material, whose complex shear modulus is expressed in the form of Equation (77), μ ∞  is the limiting high-frequency modulus and α HN  and β HN  are constants. For the type of polymeric materials used for drag-reducing coatings, K(ω) is essentially constant, with a value of approximately 1×10 8  Pa.
 
     It is recommended to first determine an optimal SRT type of material, and then to choose an HN type of material, whose properties can be created with available polymer chemistry. 
     An SRT material can be adequately characterized by the material thickness, H, the density, ρ s , the static shear modulus, μ 0 , the dynamic shear modulus, μ s , and the relaxation time, τ. An appropriate density for the viscoelastic material, ρ s , is within 10% that of water. For an SRT material, the initial guess for the static shear modulus of the material, μ 0 , is:
 
μ 0 =ρ s   C   2   (Equation 78)
 
based on the criterion that the speed of shear waves in the material is approximately the same as the phase speed of the energy-carrying disturbances, C. This phase speed, C, is assumed to be 0.8 of the value of the freestream velocity, U ∞  (Equation 42). If the convective velocity exceeds the shear wave velocity, an instability occurs, and large waves appear on the surface of the material, leading to an increase of drag for the coating.
 
     An initial choice for thickness, H, for isotropic viscous materials, where 
           μ   s       μ   0       &gt;   1       
 
is: 
             H   =       3   ⁢   C       ω   e               (     Equation   ⁢           ⁢   79     )             
 
and for isotropic, low viscosity materials, where 
           μ   s       μ   0       &lt;   1       
 
is: 
             H   =       5   ⁢   C       ω   e               (     Equation   ⁢           ⁢   80     )             
 
     The optimal desired thickness for a coating may be greater than practical for a given application. While isotropic coatings thinner than recommended in Equations 79-80 can still be effective, anisotropic coatings that are stiffer in the normal dimension relative to the transverse and longitudinal dimensions can provide equivalent performance with significant reduction in thickness. 
     Given specified values of H, μ 0 , and ρ s , the VE problem, as expressed in Equations (57) and (58), is solved numerically for a matrix of values of τ s  and μ s  (i.e., for different values of the complex shear modulus) and for a range of wavenumbers. The wavenumber corresponding to the maximum turbulent energy in the boundary layer is: 
               α   e     =       ω   e     C             (     Equation   ⁢           ⁢   81     )             
 
where the frequency, ω e , for maximum energy-carrying disturbances is estimated by Equation (41). Calculations yield surface displacement amplitudes and the flux of turbulent fluctuating energy into the coating. The best combination of properties for a SRT material occurs where the surface displacement under actual load (Equation (61)) is less than the viscous sublayer thickness, and where the energy flux into the coating (Equation (57)) is at a maximum. Furthermore, it is desirable to maintain this criterion for a range of frequencies from approximately one decade below to one decade above the energy-carrying frequency, ω e .
 
     Once a set of optimal values of τ s  and μ s  are determined for a set of specified values of H, μ 0 , and ρ s , the calculations are iterated using slightly different values of thickness, H, and static modulus, μ 0 . From these calculations are chosen the optimal set of parameters for an SRT material (H, μ 0 , ρ s , τ s  and μ s ), given specified flow conditions and configuration. 
     The complex shear moduli of real polymeric materials, such as polyurethanes and silicones, which are candidates for viscoelastic coatings cannot be adequately described by the SRT representation. More complex MRT or HN representations of the shear modulus require multiple constants, and are less suitable for numerical parametric evaluation. Therefore, results for SRT materials are used to select candidate materials, such as described by the HN formulation (Equation (79)) which can be more readily fabricated in practice. As a guideline, it is desired to match the complex shear modulus curves of the target SRT material and the HN material (value and slope) over frequencies ranging from decade below to one decade above ω e , with the most important matching being in the immediate vicinity of ω e . 
     To design a multi-layer isotropic coating, properties of a complex shear modulus, density, and thickness are specified for individual layers, and non-slip boundary conditions between layers are imposed. The properties of the upper layer are specified according to the methodology for a single layer, and the lower layers will have progressively lower static shear moduli, as optimized for lower freestream velocities. Thus, well-designed multi-layer coatings can reduce drag over a range of freestream velocities. 
     In the design of an anisotropic coating, the complex shear modulus has different values in the normal direction relative to the longitudinal and transverse directions (hereinafter termed transversely isotropic). If the viscoelastic material follows a single-relaxation time model, then the static shear modulus, μ 0 , the dynamic shear modulus, μ s , and the relaxation time, τ s , will differ with direction, as expressed in Equations (82) and (83). The static shear modulus in the normal direction, μ 01 , will be greater than than in the longitudinal-transverse plane, μ 02 . The complex shear modulus in the normal direction is expressed as: 
                 μ   1     ⁡     (   ω   )       =       μ   01     +       μ   s1     ⁡     [           (     ωτ   s1     )     2       1   +       (     ωτ   s1     )     2         +     ⅈ   ⁢       ωτ   s1       1   +       (     ωτ   s1     )     2             ]                 (     Equation   ⁢           ⁢   82     )             
 
while the shear modulus in the streamwise and transverse directions is expressed as: 
                 μ   2     ⁡     (   ω   )       =       μ   02     +       μ   s2     ⁡     [           (     ωτ   s2     )     2       1   +       (     ωτ   s2     )     2         +     ⅈ   ⁢       ωτ   s2       1   +       (     ωτ   s2     )     2             ]                 (     Equation   ⁢           ⁢   83     )             
 
For a viscoelastic, transversely isotropic material, surface oscillation amplitudes can be reduced relative to an isotropic material, while the level of energy flux into the material is increased. Thus, well-designed anisotropic coatings will be significantly thinner than isotropic coatings associated with the same level of drag reduction.
 
Methodology to Choose Structure of a Drag-Reducing Viscoelastic Coating
 
     A further aspect of coating design is the choice of internal structure within the viscoelastic material. In practical applications of viscoelastic coatings, the coating will be finite in length, with leading, trailing, and side edges. The influence of the finite edges affects coating performance. Well posed edges can order and stabilize transverse and longitudinal vortical structures in the near-wall region of the flow and thereby delay the deformation of these vortical structures and enhance the stability of the flow. However, unstructured edges can accentuate the amplitude of oscillations of the viscoelastic material in this region. Local instabilities can degrade the performance of the coating, so that, even with a well-designed material, the influence of the edges can lead to a drag increase. Hence, the coating is structured in the vicinity of finite edges. The thickness of the coating is decreased to minimize such oscillations, using techniques such as a rigid wedge underneath the coating, or other localized structure near an edge (FIG.  3 ). For large bodies, a continuous coating may be impractical or difficult to fabricate. An alternative design is a piecewise continuous coating, composed of finite segments of coating, where both the longitudinal and transverse edges of the coating system are organized to stabilize flow structures and to minimize adverse effects at the edges of each segment. 
     In addition to well-posed edges, viscoelastic coatings may be combined with surface structure to enhance the stabilization of longitudinal vortices along the length of the coating, and hence to increase the level of drag reduction through multiple physical mechanisms. Structure can include the placement of riblets on top of the viscoelastic coating, or the creation of so-called “inverse” riblets. In the latter case, a viscoelastic coating may be molded over ribs or ridges of rigid material, so that longitudinal riblet structures form when fluid flows over the viscoelastic surface. 
     The dimensions (scales) of the segments and the dimensions of the structures within the coating are selected as multiples of the transverse and longitudinal scales in the near wall turbulent flow. These scales vary with body speed, position along the body and when non-Newtonian additives, such as dilute aqueous solutions of high-molecular weight polymers, are present.