W

AlAA 95-0860

A Reynolds Stress Equation Turbulence Model for Compressible Flows. Part I: Flat Plate Boundary Layers

M. Gnedin and D. Knight Rutgers University Piscataway, NJ I

L

33rd Aerospace Sciences Meeting and Exhibit

AIAA Paper No. 95-0860

A Reynolds Stress Equation Turbulence Model for Compressible Flows. Part I: Flat Plate Boundary Layers’>

v Marianna Gnedint and Doyle Knight$

Department of Mechanical and Aerospace Engineering Rutgers University - The State University of New Jersey

PO Box 909 . Piscataway NJ 08855-0909

Abstract

A Reynolds Stress Equation model has been de- veloped for compressible turbulent flows. The model is based upon a straightforward extension of a simplified Reynolds Stress Equation model for incompressible flows, with additional mod- elling for specific effects of compressibility. Re- sults are presented for incompressible and com- pressible flat plate turbulent boundary layers. The compressible computations include both adi- abatic and cold walls. The computed skin fric- tion, adiabatic wall temperature, heat transfer and velocity profiles are in good agreement with

W existing empirical correlations. h t h e r research in application of the model to complex compress- ible turbulent flows is in progress.

1 Introduction

The nature of turbulence is exceptionally com- plicated and a complete understanding is lacking despite more than a century of research. Current day computer power is still inadequate for direct numerical simulations of turbulent flows in prac- tical engineering design. Therefore, substantial research efforts are being invested in turbulence modelling. Models of turbulence range from sim- ple algebraic models to sophisticated second or- der closures [l], [2]. For a taxonomy ofturbulence models, see [3].

tGraduate student. tProfessor, Associate Fellow AIAA. Copyright @ 1994 by Marianna Gnedin and Doyle Knight. Published by the American Institute of Aeronautics and

W Astronautics, Inc. with permission.

The most popular two-equation models, as well as algebraic models, are intrinsically limited in ability to describe the turbulent flow behavior because they employ the Boussinesq hypothesis which assumes that the turbulent shear stress and heat flux are local in nature. For example, the accurate prediction for the surface heat trans- fer and skin friction has not been achieved using algebraic and two-equation models for complex 2-D and 3-D hypersonic shock - boundary layer interactions [4].

It is increasingly apparent that improvement in the accuracy of prediction of complex 3-D tur- bulent flows is more likely to be achieved using a full Reynolds Stress Equation (RSE) closure model than two-equation or algebraic eddy vis- cosity models [5] [6]. The RSE model includes transport equations for the components of the turbulent stress tensor and, therefore allows for a more realistic description of turbulence.

Virtually all RSE model development has been focused on incompressible flows [2] [7], with few extensions to compressible flows [8] [9]. Some as- pects of RSE modelling for incompressible flows may be extended to supersonic, adiabatic and near adiabatic turbulent boundary layers using Morkovin’s hypothesis [lo]. However, new con- cepts and modelling are needed for a significant portion of the compressible regime including non- adiabatic supersonic turbulent boundary layers, free shear flows and hypersonic boundary layers.

The objective of the present research is to ex- tend an existing RSE models, originally derived & for an incompressible flow, to compressible flow,

J

and to incorporate additional modelling of the ef- fects of compressibility on turbulence. This work is a part of a larger effort to develop a Reynolds Stress Equation model for complex compressible flows including 3-D shock wave turbulent bound- ary layer interactions. Results are presented for an incompressible boundary layer at Mach 0.1 and for a supersonic boundary layer at Mach 3.0 and 6.0.

2 Nomenclature

L p T

Deflnition Mach number Turbulent Mach number Reynolds number, based on initial boundary layer thickness Reynolds number, based on momentum thickness Laminar Prandtl number Turbulent Prandtl number

Density X-velocity Y-velocity Pressure Temperature Total energy Total enthalpy Enthalpy Turbulence kinetic energy Dissipation rate of turbulence energy Dilatation dissipation Solenoidal dissipation Molecular viscosity Viscous stress tensor Heat-Flux vector Friction velocity, Adiabatic waU temperature Temperature at the wall Density at the wall Heat flux at the wall Molecular viscosity at the wall Shear stress at the wall

YU,lVW

3 Model Equations

3.1 Reynolds Averaged Navier-Stokes Equations

L/ The Reynolds-averaged equations for conserva- tion of mass, momentum and energy are,

where = 8/81, 8k = 8/8zk and the Einstein summation convention is employed. The overbar denotes ensemble average, Le.,

where f(") are the individual realizations of the variable f(+, y, L, t ) . The mass-averaged variable f is defined as the mass-weighted ensemble aver- age,

and the fluctuating variable f" in the mass- averaged expansion is

Alternately, the fluctuating variable f' in the un- weighted expansion is

f'= f - f (7)

2

In (1) to (3), p is the mean density, iri is the mass-averaged velocity, p is the mean pressure, and Z is the mass-averaged total energy per unit mass,

L

(8) - 1 - -

2 Z = c.T + -u;u~ + k

where k is the mass-averaged turbulence kinetic energy

1- p k = - p ~ i U; 2

The mean molecular viscous stress Tij is

(9)

where ji E p(f’). Similarly, the molecular heat flux is

where Pr is the molecular Prandtl number.

U

3.2 Reynolds Stress Equations with Closure Approximations

Eqs (1) - (3) are incomplete due to the presence of the Reynolds stress -- and turbulent heat flux -cppT”u:I. The equation for the Reynolds stress is.

-

The terms on the right hand side and their re- spective models are presented below.

Production The production of Reynolds stress is

This term requires no further modelling since the Reynolds stress and mean velocity are dependent

variables. J Diffusion A series of correlations can be grouped together in divergence form,

The model employed for this term is,

f -8, (pu’i’u;o)]

- + 8i (puyu;)}}

4 where fi E j i l p and Cd, is a constant. This model is a straightforward extension of the model of Launder, Reece and Rodi ill] to compressible flow.

Pressure-Rate of Strain The correlation of the instantaneous pressure and fluctuating rate- of-strain appear as,

cij = p (8,u:’ + BiU”)

The simple model of Rotta [121, developed and utilized for incompressible flows [13], is directly extended to compressible flows as,

where C,, and C,, are constants.

3

Dissipation The dissipation is,

Dij = rikaku; t Tjk8ku; (18)

kd Based on the Kolmogorov hypothesis of isotropy of the smallest scales [14], the dissipa- tion is modeled as

where c is the dissipation rate per unit mass of turbulence kinetic energy. According to Sarkar et a1 [15] and Zeman (161, the dissipation is sep- arated into solenoidal pe , and compressible j j ~ components,

pe = P ( e , +e,) (20)

where p, represents the dissipation attributable to the vorticity fluctuations, and p C is the dis- sipation associated with the dilatation fluctua- tions. According to Sarkar et al, cc = Cke,M: with Ck = 1. The following model was employed for this term.

2 - Dij = -;p, (1 4- c k h f : ) 6ij (21)

Here the turbulence Mach number is Mt = a/& Eqns (13), (15), (17), and (21) there- fore constitute a closure of the Reynolds Stress Equation (12) assuming development of a model equation for the solenoidal dissipation e,.

Rate of Dissipation The conventional equa- tion [17] is employed for e,,

a&# t %&pe.c& =

where C,, , C., and C,, are constants.

Turbulent Heat Flux The turbulent heat flux is modeled using a gradient diffusion hypothesis

where Ch is a constant. This approach has been

The boundary layer version of the equations is adopted by Speziale [a]. v

listed in [18].

3.3 Values of Constants

The Reynolds Stress Equation model incorpo- rates eight constants, whose values have been de- termined by comparison with experimental data for a series of simple turbulent flows including the decay of isotropic turbulence [19], homogeneous turbulent shear flow [20), the constant stress tur- bulent layer [21], and the flat plate turbulent boundary layer [22]. The values of the constants are listed below:

Table 1: Turbulence Model Constants

Quantity Value cdi 0.086 CP, 4.325 CP2 0.179 C. 1 1.01 c;, 1.80 c e , 0.10 Ck 1.0 ch 0.0857

4 Numerical Algorithm

A two-dimensional compressible boundary layer code was developed to enable evaluation of the model for unseparated incompressible and com- pressible boundary layers. The boundary layer code is based on the Box Scheme of Keller 1231. The method is a fully implicit, unconditionally stable, second order accurate algorithm employ- ing non-uniform grids.

An automatic grid adaption algorithm was de- veloped to accurately resolve the narrow turbu-

L 4

lent / non-turbulent interface at the edge of the boundary layer. This interface is a consequence of the nonlinearity of the RSE model and is com- mon to virtually all RSE and two-equation tur- bulence models [I]. Adequate resolution of this layer is essential for stable integration of the gov- erning boundary layer equations.

The method of wall functions, which consists in matching the solution to the Law of the Wall, is utilized for computing a compressible boundary layer (detailed description of the wall functions is provided in (241).

The matching occurs in a region neighboring the wall, referred to as the wall layer. This region is of the order of y+ z 100 where y+ is defined as

i /

where pu, pW and u, are respectively the den- sity, the molecular viscosity at the wall, and the friction velocity

u, = 6 (25)

where rw is the shear stress at the wall. Since the Law of the Wall represents the asymptotic be- havior of the flow near the boundary, the precise location of its application is irrelevant provided it lies within the general region of applicability.

'v

5 Application to Flat Plate Boundary Layer

A series of computations were performed for a turbulent flat plate adiabatic wallboundary layer at Mach numbers M , = 0.1, 3 and 6. The re- sults for M , = 0.1 where utilized to determine the value of the model constant Cd, by com- parison with the Law of the Wake. All other constants in the turbulence model were deter- mined by comparison with experimental data for a series of simple turbulent flows including decay of isotropic turbulence, homogeneous turbulent shear flow and the constant stress turbulent layer (see Section 3.3).

The computations at M, = 3 and M , = 6 were performed to evaluate the accuracy of th, RSE model for compressible boundary layerk J/ Both adiabatic wall and isothermal wall bound- ary layers were considered. For the isothermal boundary layers, the ratio of the wall tempera- ture to the adiabatic wall temperature is 0.4.

5.1 Adiabatic Wall Boundary Layer Computations

The M , = 0.1 computation is compared to the experiment of Weighardt and Tillmann [25] in Fig. 1. The computed and experimental results for skin friction coefficient c, and streamwise ve- locity profile are displayed. The velocity profile corresponds to Ree = 1 . 4 . IO'. Excellent agree- ment between the computation and experiment is observed.

. .I .I .- I

Y

" .. "

Figure 1: Flat plate Boundary layer at M , = 0.1 a) Skin friction coefficient e, us Ree. b) Velocity profile at Bee = 1.4.10'

The RSE model prediction for the skin friction coefficient c, at the three different Mach num- bers is plotted in Fig. 2 us Re# . The theoretical skin friction coefficient, using the Van Driest I1 theory [26], is also shown. The initial transient (e .q. , Reo 5 1.5. lo4 for M , = 3) is a conse- quence of the approximate nature of the initial boundary layer profile and can be ignored. Be- d

5 W

yond the initial transient, the agreement between the computed and theoretical skin friction coef- Scient is within the uncertainty(= 10%) of the

The RSE model prediction for the adia- batic wall temperature Tam-, normalized by the freestream static temperature T,, is presented in Fig. 3 at M , = 3 and 6. Downstream of the initial transient, the computed Taw is within 2% of the theoretical result

\ v a n Driest 11 theory. ".

1..

L!\b, ........... ..... ........................................... , , ,~

_. -.- . . ..._ " - t a m .

"I

- I- ,I u .m. - - .......

.............. ...... ......... ........ ,*p--. .. ....-a- ................ .... ...... ......... ........ . . . ................

Figure 2 Skin friction c, us Reo at a) M = 0.1 b) M = 3 and c) M = 6

The sensitivity of the computed C, and Taw to he vertical location y& of application of the wall

L

Figure 3: Adiabatic temperature Taw/T, us Reo at a) M = 3 and b) M = 6

functions is shown in Fig. 4 for the M, = 3 and in Fig. 5 for the M , = 6. Downstream of the initial transient, the computed value of e , and Taw are observed to be effectively insensitive to the value of y& for 20 5 y& 5 200 at M , = 3. The permissible range for y& decreases with increasing Mach number. As indicated in Fig. 5, the range at M , = 6 is approximately 30 5 y& 5 60.

The streamwise velocity profile at M , = 0.1 is compared with the incompressible Law of the Wall and Wake in Fig 6. The theoretical velocity profile U is

/

d

u 1 YU, 211 x y - = -log(-)+B'+-~in*(--) (27) UT Y n 2 6

where u, E is the friction velocity, v is the kinematic viscosity, 6 is the local boundary layer thickness, and B' = 5.0 and II = 0.55 are constants [27].

In Fig. 6(a) the computed profile is compared

6

"I I

". I.

8..

.I

I

-

:I"------

Figure 4: Sensitivity of a) c,, and b) Taw to the initial value of y& at M- = 3.

I

Figure 5: Sensitivity of a) e,, and b) Taw to the initial value of y& at M, = 6.

Figure 6: Computed and theoretical velocity pro- files at M = 0.1. a)Law of the Wall, and b) defect layer.

with the Law of the Wall. Excehent agreement is observed. In Fig. 6(b), the computed velocity defect (Urn - V ) /u, is compared with the theo- retical profile

J

The agreement between the computed and the- oretical profile is excellent. As indicated previ- ously, this comparison was employed to deter- mine the value of C d , .

The transformed streamwise velocity profiles at M , = 3 and 6 are compared with the com- pressible Law of the Wall and Wake in Figs. 7 and 8. The transformed velocity is defined by

.J where A and B are defined by

7

is compared with the theoretical defect law

(30) A2 = P r , L - 1 2Tm

2 M m T ,

in Fig. 8 where u, is determined from application of the Law of the Wall and Wake at y = 6, with Prt = 0.9. The compressible Law of the

Wall and Wake is

The computed profiles at M , = 3 and 6 exhibit asymptotic agreement with the Law of the Wall (Fig. 7).

(35) with 6+ = b u , / v . Reasonable agreement is ob- served at both Mach numbers.

c

Figure 7: Computed and theoretical transformed velocity profiles at a) M = 3, and b) M = 6

The computed transformed defect velocity

)> (33) 2 A 2 ( . /Urn) - E J3Tm sin-' (

Figure 8: Computed and theoretical transformed defect velocity profiles at a) M = 3, and b) M = 6

5.2 Isothermal Wall Boundary Layer Computations

A series of computations were performed for an isothermal wall flat plate boundary layer. The wall temperature Tw = 0.4Taw. Mach numbers M , = 3 and 6 were considered.

The RSE model prediction for the skin fric- tion coefficient cf at two different Mach numbers and a range of values of Res is plotted in Figs. 9 and 10 vs Reo together with the theoretical skin friction coefficient based on Van Driest II theory. Beyond the initial transient, the agreement be- tween the computed and theoretical skin friction coefficient is within the uncertainty (a 10%) of the theory.

L'

Figure 9: Skin friction cf 'vs Res, Tw/Taw = 0.4, a) M = 3 and b) M = 6

The sensitivity of the computed cf to the ver- tical location y& of application of the wall func- tions is shown in Fig. 11 for the M , = 3 and 6. Downstream of the initial transient, the com- puted value of cf and the Reynolds analogy fac- tor is observed to be effectively insensitive to the value of y& for 50 5 y& 5 160 at M , = 3. As it was previously observed in the adiabatic wall case, the permissible range for y& decreases with increasing Mach number. As indicated in Fig. 11, the range at M , = 6 is approximately 60 5 y& 5 100.

Figure 10: Skin friction cf vs Res, Tw/Taw = 0.4, a) M = 3 and b) M = 6

Figure 11: Sensitivity of cf to the initial value of y& at a) M , = 3 and b)& = 6. .J

9

The Reynolds analogy factor 2 S t / c f , where Qw St z

pmUmcp(Tw - Taw)

I...*...

is presented in Fig. 12 as a function of Res at M , = 3 and 6. Downstream of the initial r.*

transient, the computed Reynolds analogy factor agrees with the theoretical result l / P r t within 3%.

The sensitivity of the computed Reynolds anal- ogy factor to the vertical location y& of applica- tion of the wall functions is shown in Fig. 12 for the M , = 3 and 6. Downstream of the initial transient, the computed value of the Reynolds

b

1

'* ... I *,<:

_.I' ,..,_... ,." ..I ___.-- _.- . .

analogy factor is observed to be effectively insen- sitive to the value of y& .

,, - '* _ _ . -

Y I I -

Figure 13: Computed and theoretical velocity profiles at M = 3 and different y&.

- "'." - I.-"

.:- I*. I w.

1.. ...

Figure 12: Sensitivity of the Reynolds analogy factor to the initial value of y& at a) Mm = 3

w #

and b)M, = 6. ._..

The computed transformed streamwise veloc- ity profiles at M , = 3 and 6 and diferent vertical locations y& of application of the wall functions are compared with the compressible Law of the wall (27) in Fig 13 and 14. The files at M , = 3 and 6 exhibit asymptotic agree- rnent with the Law of the Wall.

. , ... #

Pro- Figure 14: Computed and theoretical velocity profiles at M = 6 and different y&.

c, 10

W

The computed transformed defect velocity (33) is compared with the theoretical defect law (34) in Figs. 15 and 16. Reasonable agreement is ob- served at both Mach numbers.

m. 50,

,'.I"

.. .... d T.L.0, .. ..

' . % I Y Y " " " " - ..

m - d ,'."

. . . . . . . .. .. - .. " " "

Figure 16: Computed and theoretical trans- formed defect velocity profiles at M = 6 and dif- ferent y&.

Figure 15: Computed and theoreticd trans- formed defect profiles at M = 3 and dif- ferent y&.

ter case. The computed values of the skin friction coefficient, velocity profiles, adiabatic wall tem- perature and the Reynolds analogy factor are in good agreement with empirical correlations.

6 Conclusion 7 Acknowledgments

A Reynolds Stress Equation model has been de- veloped for compressible flow. The model is based on a straightforward extension of a simpli- fied Reynolds Stress Equation model for incom- pressible flows, with additional modelling for spe- cific effects of compressibility. The values of the model constants have been determined by com- parison with experiment for a series of simple tur- bulent flows. Results have been obtained for an incompressible turbulent boundary layer at Mach 0.1 and for compressible .turbulent boundary lay- ers at Mach 3.0 and 6.0. Both adiabatic wall and isothermal wall boundary layers have been con- sidered, with the ratio of the wall temperature to the adiabatic wall temperature of 0.4 in the lat-

This research was sponsored by AFOSR Grant F49620-93-1-0005 monitored by Dr. Len Sakell. The computations have been performed at the Rutgers University Supercomputer Remote Ac- cess Center and at the DOD High Performance Computing Center USAE Waterways Experi- ment Station.

References

[I] Wilcox, D., Turbulence for CFD. La Canada, CA: DCW Industries, 1993.

11 U

(21 Launder, B., ”Phenomenological Modelling: Present . . .and Future ?”, Lecture Notes in Physics, Vol. 357, 1990, pp. 439-485.

[3] Reynolds, W.C., “Computation of Turbu- lent Flows”, Annual Review of Fluid Me- chanics, Vol. 8, 1976, pp. 183-208.

W

[4] Narayanswami, N., Horstman, C.C., Knight, D.D., “Computation of Crossing Shock/ Turbulent Boundary Layer Interaction at Mach 8.3”, AIAA J, Vol. 31, p. 1369-1376, 1993.

[5] Bushnell, D., “Hypervelocity Fluid Dynam- ics - Recent Challenges and Critical Is- sues,” Invited Lecture, Forty Third Annual Meeting of the Division of Fluid Dynamics, American Physical Society. Bulletin of the American Physical Society, Vol. 35, No. 10, 1990, p. 2314.

(61 Bushnell, D., “Turbulence Modelling in Aerodynamic Shear Flow - Status and Prob- lems”, AIAA Paper 91-021,1991.

[7] Speziale, C., “Analytical Methods for the Development of Reynolds Stress Closure Models in Turbulence”, Annual Review of Fluid Mechanics, Vol. 23, 1991, pp. 107-157.

u

c

[8] Speziale, C., and Sarkar, S., “Second-Order Closure Models for Supersonic Turbulent Flows”, AIAA Paper 91-0217.

[9] Zeman, O., and Blaisdell, G., “New Physics and Models for Compressible Turbulent Flows”, Advances in Turbulence, Vol. 3, Springer-Verlag, Berlin, 1991, pp. 445-454.

[lo] Favre, A., ed., The Mechanics of Turbulence, Gordon and Breach, NY, 1964.

[ll] Launder, B., Reece, G., and Rodi, W., “Progress in the Development of a Reynolds Stress Turbulence Closure”, J. Fluid Me- chanics, Vol. 68, Part 3, 1975, pp. 537-566.

[12] Rotta, J., “Recent Attempts to Develop a Generally Applicable Calculation Method for Turbulent Shear Flows”, AGARD CP- 93, 1972.

[13] Mellor, G., Herring, H., “A Survey of Mean Turbulent Field Closure Models”, AIAA J, VO~. 11, p. 590-599, 1973.

[14] Tennekes, H., and Lumley, J., A First W

Course in Turbulence, The MIT Press, 1972.

1151 Sarkar, S., Erlebacher, G., Hussaini, M., and Kreiss, H., “The Analysis and Modelling of Dilatational Terms in Compressible Turbu- lence”, J. Fluid Mechanics, Vol. 227, 1991, pp. 473-493.

(161 Zeman, O., “Dilatational Dissipation: The Concept and Application in Modelling Com- pressible Mixing Layers”, Physics of Fluids

[17] Jones, W., and Launder, B., “The Predic- tion of Laminarization with a Two-Equation Model of Turbulence”, Int. J. Heat ‘and Mass Zhnsfer, Vol. 15, 1972, pp. 301-304.

A, 1990, pp. 178-188.

[18] Knight, D.D., “A Reynolds Stress Equation Model of Turbulence: Version No. 4”, In- ternal Report 14, Department of Mechanical and Aerospace Engineering, autgers Univer- sity, July 1993.

(191 Knight, D.D., “On the Decay of Isotropic Turbulence - Reynolds Stress Equation Model,” Internal Report 4, Department of Mechanical and Aerospace Engineering, Rutgers University, January 1993.

[20] Knight, D.D., “On Homogeneous Turbu- lent Shear Flow - Reynolds Stress Equa- tion Model,” Internal Report 5, Department of Mechanical and Aerospace Engineering, Rutgers University, January 1993.

(211 Knight, D.D., “Determination of Constants in Reynolds Stress Equation Model,” Inter- nal Report 6, Department of Mechanical and Aerospace Engineering, Rutgers University, February 1993.

(221 Gnedin, M., “Determination of the Con- stant cd, in the Reynolds Stress Equation Model,” Internal Report 15, Department of Mechanical and Aerospace Engineering, Rutgers University, August 1993.

/

W

L

12

[23] Keller, H., “Accurate Difference Methods for Nonlinear Two-Point Boundary Value Prob- lems”, SIAM Journal of Numerical Analysis, VOI. 11, pp. 305-320,1974.

.d [24] Knight, D.D., “Compressible Turbulent

WallLayer. Reynolds Stress Equation Model Version No. 4,” Internal Report 17, Depart- ment of Mechanical and Aerospace Engi- neering, Rutgers University, 1993.

1251 Wieghardt, K., T h a n n , W., “On the Tur- bulent Friction Layer for Rising Pressure”, Tech. Rep. TM 1314, NACA, 1951.

[26] Hopkins, E., and Inouye, M., “An Evalu- ation of Theories for Predicting Turbulent Skin Friction and Heat Transfer on Flat Plates at Supersonic and Hypersonic Mach Numbers”, AIAA J, Vol. 9, No. 6, 1971, pp. 993-1003.

[27] Coles, D., “The Young Person’s Guide to the Data”, AFOSR-IFP-Stanford Conference on Computation of Turbulent Boundary Lay- ers, 1968.

13