Numerical Methods for Coupling the Reynolds-Averaged Navier-Stokes Equations with the Reynolds-Stress Turbulence Model

Patrik Rautaheimo and Time Siikonen

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Teknillinen korkeakoulu Helsinki University of Technology Energiatekniikan laitos Department of Energy Engineering Sovelletun termodynamiikan laboratorio Laboratory of Applied Thermodynamics

Helsinki University of Technology Laboratory of Applied Thermodynamics

Numerical Methods for Coupling the Reynolds-Averaged Navier-Stokes Equations with the Reynolds-Stress Tbrbulence Model

Patiik Rautaheimo1 and Timo Siikonen 2 Helsinki University of Technology, Espoo, Finland

Report No 81 1995

1995

Otaniemi

ISBN 951-22-2748-7 ISSN 1237-8372

^Research Scientist, Laboratory of Applied Thermodynamics 2 Associate Professor, Laboratory of Applied Thermodynamics » DISCLAIMER

Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. I

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Contents

Nomenclature 2

1 Introduction 4

2 Governing Equations 6 2.1 Flow Equations ...... 6 2.2 Reynolds-Stress Turbulence Model ...... 7 2.3 Dissipation Transport Equation ...... 10

3 Numerical Method 11 3.1 Spatial Discretization ...... 11 3.2 Diagonalization of the Flow Equations ...... 12 3.3 Rotation Operator ...... 17 3.4 Boundary Conditions ...... 19 3.5 Time Integration Method ...... 20

4 Test Calculations 22 4.1 Channel Flow ...... 22

5 Summary 27 » 2

Nomenclature

A Jacobian matrix dF/dU B Jacobian matrix dG/dU C Jacobian matrix dH/dU CFL Courant number D difference between e and e (= 2vk/y%)\ also Jacobian of the source term E total internal energy per unit volume F, G, H flux vectors in x-, y- and ^-directions H total entalphy (= E/p + p/p + u"u") I matrix of Reynolds stresses F flux in a given direction in space L right eigenvector matrix with the primitive variables M transformation matrix from the conservative to the primitive variables (= dU/dV) Ma Mach number P production of the kinetic energy of turbulence; also boundary condition matrix Pr Prandtl number Q source term R right eigenvector matrix with the conservative variables; also residual Re Reynolds number S strain-rate; also cell-face area T temperature; also rotation matrix; turbulent transport T rotation operator U vector of the conservative variables V cell volume; also vector of the primitive variables W vector of the characteristic variables Cp specific heat in a constant pressure Cy specific heat in a constant volume _ d diffusion of the Reynolds-stresses ; also cell thickness e internal energy per unit mass, i.e. specific internal energy i,j, k unit vectors in Cartesian coordinate system k heat conductivity; also kinetic energy of turbulence, (= u "u" /2); stabil­ ity factor n unit vector normal to a cell face p static pressure p* effective pressure p e derivative of pressure in constant density (= dp/de\p) pe derivative of pressure in constant internal energy (= dp/dp\e ) q heat flux t time -u, v, vj velocity components in x-, y- and ^-directions x, y, z Cartesian coordinates y+ non-dimensional normal distance from the surface A diagonal eigenvalue matrix $ the velocity pressure-gradient correlation characteristic variable 7 ratio of specific heats Cp/c* Sij Kronecker ’s delta e dissipation of kinetic energy of turbulence A eigenvalue of A p, dynamic viscosity v kinematic viscosity p density a Schmidt’s number t normal or shear stress w specific dissipation rate

Subscripts

T turbulent conditions i i-index; summation index ij ij-component of a matrix i, j, k grid coordinate directions v viscous x, y, z coordinate directions e dissipation value

Superscripts

l left-hand side of the cell face r right-hand side of the cell face local coordinates; also Roe averaging; corrected dissipation Favre time-averaging operator Time-averaging operator " Fluctuating component ' Fluctuating component 4

1 Introduction

In recent years, computational fluid dynamics (CFD) has begun to be an engineer ­ ing design tool for practical flow problems. Some complex aerodynamical config ­ urations have been calculated, e.g. a delta wing [1]. Unfortunately, many systems involve turbulent or transitional flow that cannot be fully understood, and thus, the case is not realistically simulated. Direct numerical simulation (DNS) gives an ac­ curate solution also in complex flow situations, but cannot be applied for practical calculations because of an enormous computing cost. Large eddy simulation (LES) has produced promising results, but like DNS, LES requires a lot of computing re­ sources and also the theory is not yet fully evaluated. Thus, turbulence modeling for Reynolds-averaged Navier-Stokes equations still plays a very important role in CFD. In a simple case, e.g., in the case of a 2-D boundary layer flow at a constant pres­ sure, there is no need to use a complicated turbulence model like the Reynolds-stress model (RSM). For example, the algebraic turbulence model of Baldwin and Lomax [2] has been successfully used for various airfoil sections [3]. Algebraic models uti­ lize universal near-wall functions that are not valid in more complex 3-D flows. The limitations of the algebraic models drive us to use more general and, unfortunately, also more complex turbulence models. Two-equation models, ask — e and k — u models are more general because they take the history of turbulence into account. The weakness of these Boussinesq-approximation-based models is that they do not include theeffect of different Reynolds stresses. Furthermore, the Reynolds-stress tensor is aligned with the mean strain tensor. If curvature, pressure change or rotation is introduced, the standard Boussinesq eddy-viscosity approximation fails if no ad hoc modifications are used. Unfortu ­ nately, these ad hoc modifications are not general. The main advantages of the RSM in comparison with the Boussinesq approximation can be found in the history effects of different stress components and in anisotropic assumption. RSM also provides more data on turbulence itself than the lower order models. From a theoretical point of view, RSM has the potential to solve many compli ­ cated engineering flow problems. However, RSM also has disadvantages because of complexity, terms that need modelling and many closure coefficients, and an in­ creased number of differential equations. In spite of theapparent difficulties, RSM has been used to calculate internal flows for a long time [4]. Recently some calcu­ lations have been made for external flow as flows over airfoils [5], RSM has mainly been used with incompressible flow. In this case, the Reynolds stresses act only in the momentum equations. However, in the present case with a 5 compressible flow assumption, the Reynolds stresses act not only in the momentum equation, but also in the energy equation. This makes the implementation more dif ­ ficult. Furthermore, with the Reynolds-stress equations the coupling with the flow equations is anisotropic. In general, an isotropic assumption is made [6] also with compressible flow. Anisotropic coupling makes the implicit time integration and up- winding difficult to apply, but in some cases this approach may accelerate the con ­ vergence. In this study, Shima’s low-Reynolds number RSM [7] is coupled with a com ­ pressible flow solver [8] based on Roe ’s method [9]. A new anisotropic coupling method that utilizes the exact equation for the production of turbulence is introduced. The method is applied and the results are presented for a flow in a plane channel. 6

2 Governing Equations

2.1 Flow Equations

The Reynolds-averaged Navier-Stokes equations (RANS), and the equations for the Reynolds stresses u" u'j and dissipation of turbulence e can be written in the follow ­ ing form au d{F-F„) d(G-G„) a(H-Hv) „ sT + _a^+ ay +~Tz------0 (Z1) where U is vector of conservative variables, F, G and H are the inviscid fluxes and Fv, Gv and Hv are the viscous fluxes. The source-term Q has non-zero components for the Reynolds-stress equations. Vectors U and F are

( p \ ( pu __ X pu pu 2 +p + pu'u pv puv + pu'v" pw puw + pu'w" u(E + p + pu'u") + pvu'v" + pwu'w" — ~Tt 'rz pu'u" puu u = 3 rF -- — ”7 ~i "n (2.2) pu'v" puu V pu'w" puu— II wII pv"v" puv---"» VII pv"w" puv w pw"w" puw" w" \ pe / V - pue J where p is density, p is pressure, u, v and w are velocity components in the Carte­ sian coordinate system, u"u", u"v", u"w", v"v", v"w" and w"w" are the Reynolds stresses, and E the total internal energy defined as

E = p~e + ^ + P

The viscous flux in the i direction is

i xy TXz V/Txx "i~ 'UT'xy *4~ WTXZ C[x “T~ dii 0 Fv = (2.4) 0 0 0 0 0 H£(de/dx) where the viscous stress tensor is defined as

dtij dui 2 duk + (2.5) Tij = M dxi dxj and da is the diffusion flux of the kinetic energy of turbulence. Diffusion of the Reynolds stresses are included in the source vector Q. The diffusion coefficients of the dissipation as . Mr He — /J, A----- (2.6 ) where ae is the appropriate Schmidt’s number, and /jl t is the turbulent viscosity of the fluid determined with theapplied turbulence model. Heat flux q is obtained by using Boussinesq ’s approximation and is written as

q = -(k + kT)VT = - + ht-5^) VT (2.7) where Cp is specific heat. The pressure is calculated from an equation of state p = p(A e), which, for a perfect gas, is written

u2 + v2 + w2 P — (t — 1 ){F - p -Ar-) = (7-i)/% (2.8) where 7 is the ratio of specific heats Cp/cy.

2.2 Reynolds-Stress Turbulence Model

The Reynolds-stress transport can be written in the following Cartesian tensor form

P Dt = Pij + $ij — Tij — €ij — dij (2.9) 8 where Py, $y, Ty, ey and dy are the production, the velocity pressure-gradient cor ­ relation, the turbulent transport, the dissipation rate and the diffusion terms, respec­ tively. The exact form of the source-terms can be written as

_ a a duj _ "7/ 7 duz Pij = -pu, (2.10) iuk‘—dx~k

®ij = V I 4------(2.11) Vdxj dxi d 77 77 771 Tij PuiujUk\ (2.12) dxi

€ij = p (2.13) dxk dxk

dii ~ dxk [Sikp'u- + 8jkpu" - p (SikU- (2.14)

Sij is the mean strain-rate

dui duj\ 2 duk Sij = d^ + aFtj- 36iiaTh (2.15)

As can be seen, theturbulent transport, the velocity pressure-gradient and the dis ­ sipation rate must be modelled. The viscous diffusion takes a simpler form, if the flow is incompressible or weakly compressible [6]

a du-Uj [p (SikUj + f p (2.16) dxk dxk dxk

Shima’s Model for the Velocity Pressure-Gradient and the Dissi­ pation Rate Term

In this study, the modelling is performed utilizing Shima’s approach [7]. However, the dissipation equation is adapted from Chien’s k — e model [10], and also some modifications were needed in the velocity pressure-gradient term in order to produce reasonable results. In Shima’s model the velocity pressure-gradient and the dissipation rate term are connected as -*-SE +p’ (1^+fe) ~ ~l(lip€+++*«■» (2-17) where

(2.18) 9

(2.19)

a {Pij — 2 fiijP'j + ipkSij + /3 ^Aj — g<5y-Pj /i (2.20) where is the production term of Eq.(2.10), Aj and P are

_ « /, duk _ a nduk (2.21) A; = - ' dxj <9a:i rt — "h ii ^Uk p = (2.22)

varies with the wall function /to as

a = a [i - (i - i/a ) /j (2.23) .pVkyA fw = exp - 0.015- (2.24) P

The multiplyer in Eq.(2.24) was, after test calculation, modified to be 0.013. It should also be noted that in this study the tangential components of the velocity pressure- gradient wall term 4>tjtW were omitted in order to get a correct velocity profile and shear-stress distribution close to the wall.

Diffusion Term In thediffusion term, the pressure diffusion Sikp'u” + Sjkp'u'l is ignored or it is as­ sumed to be part of the turbulent transport term. In Shima’s model, the diffusion term of Hanjalic and Launder ’s [11] is applied

tt n n ,v I — // // vd_ ////,_ // // vd_ // u . — it it ~ n ii i ,r\ nc\ - mujuk = °s- (mui ^~u3uk + Pujui ^uiuk + Puk^i Q^uiuj I (2.25) dxi dxi

Convergence problems were experienced using Eq.(2.25) as a consequence of the destabilizing cross-derivatives. Because of this, two diffusion methods were tested. The first one is a scalar diffusion term [12] r\ II It d OUi Uj Tij + dij (p + Pr/cij,T) (2.26) dxk dxk

Diffusion in the normal direction is usually much more important than in the stream- wise direction. This can be taken into account using the model of Daly and Harlow ’s [13] ______

II II II . C I It , C I II ' — a It tt II pUi UjUk + OikP Uj + 5jkp Ui = -cs-pukut —Ui Uj (2.27)

Above turbulent eddy viscosity pr is defined as k2 P t = W— (2.28) where the kinetic energy of turbulence is k = ~u” u". The equation for u"u” contains empirical coefficients. These are given by

cs - 0.11 cs = 0.22 Ci = 1.5 C2 = 0.4 (2.29) c„ = 0.09(1 — e-00115y+) a^T = 0.82

2.3 Dissipation Transport Equation

The dissipation transport equation was taken from Chien’s k — e model [10] because it was experienced to be stable and well behaved. Chien’s dissipation transport equa­ tion can be written as

De d de + CiyP - C2^~ - 2[lAre y+/2 (n + fiT/ae) (2.30) dxj ■k k where yn is the normal distance from the wall, and y+ is defined by

1/2 . (Mr yjpTw p|V X V\ y = yn— = Vn------= yn (2.31) U- l1

The production P is a trace of the production tensor P^. The relationship between e and e is e = e + Z> (2.32)

D = 2 (2.33) % where v is kinematic viscosity. With this assumption, i has a value of zero at the wall and e has an exact value with respect to k. However, e does not have the desired 0(y2) behaviour near the wall. This results in a non-zero diffusion flux at the wall. Theequation for e contains empirical coefficients. These are given by

ci = 1.44 c2 = 1.92(1 - 0.226-^/36) (2.34) cr£ = 1.3 where the turbulence Reynolds number is defined as

(2.35)

Chien proposed slightly different forms for c% and c2. Since the computations per­ formed for the flat plate boundary layer [8] appeared to be insensitive to the mod ­ ifications, the formulas above were based on the most commonly used coefficients Ci = 1.44 and Ci = 1.92. 11

3 Numerical Method

The finite-volume CFD program for complex three-dimensional geometries [8] was used in the present calculations. The program utilizes Cartesian velocity compo ­ nents in a cell-centred approach. In the evaluation of the inviscid fluxes, Roe ’s meth­ od [9] is applied. For a spatial discretization MUSCL-type TVD-scheme to approx ­ imate advective volume-face fluxes is applied. The discretized equations are inte­ grated in time by applying the DD ADZ -factorization [14]. The code utilizes a multi- grid V-cycle for the acceleration of convergence. Complicated geometries can be handled with multiblock grids.

3.1 Spatial Discretization

In the present solution, a finite-volume technique is applied. The flow equations have an integral form

v s v for an arbitrary fixed region V with a boundary S. Performing the integrations for a computational cell i yields

(3.2) where the sum is taken over the faces of the computational cell. The flux for the face is defined F = nxF + nyG + nzH (3.3) Here F, G and H are the fluxes defined by Eqs.(2.2) to (2.4) in the x, y and z direc ­ tions, respectively. In the evaluation of the inviscid fluxes, Roe ’s method [9] is applied. A rotation operator is used for the velocity components and also for the Reynolds stresses. The flux is calculated as F = T~lF{TU). (3.4) where T is a rotation operator that transforms the dependent variables to a local co ­ ordinate system normal to the cell surface. In this way, only the Cartesian form of 12 the inviscid flux F is needed. This is calculated from

F{Ul, Ur) = \ \F(Ul) + F(Ur) 1 rW|AW|a(fc) (3.5) 2 2 k=i whereU l and UT are the solution vectors evaluated on the left and right sides of the cell surface, is a the right eigenvector of the Jacobian matrix A = dF/dU, the corresponding eigenvalue is \(k\ and is the corresponding characteristic vari ­ able obtained from R~l8U, where 8U = Ur — Ul. A MUSCL-type discretization is used for the evaluation of Ul and UT. In the evaluation of Ul and UT, primary flow variables (p, u, v, w, p), and conservative turbulent variables (pu"u”, pe) are utilized. Jacobian matrix A can be split in the following way

A = RAR-1 = MLAL^M'1. (3.6) where R and R~l are the right and left eigenvector matrices of A, L and IT1 are the corresponding matrices with respect to the primitive variables, A is the diago ­ nal eigenvalue matrix, and M and M-1 are the transformation matrices between the conservative and the primitive variables.

3.2 Diagonalization of the Flow Equations

A coupling between the Navier-Stokes and the Reynolds-stress equations is intro ­ duced, since the Reynolds stresses may be connected with the pressure [6]. In the i-momentum equation, the resulting effective pressure can be defined as

Pt=P+PUiu" (3.7)

In order to utilize Roe ’s method, the Jacobian of the flux vectors must be diagonal ­ ized. This requires that the Jacobian matrix of the flux vector has a complete set of eigenvectors. Unfortunately, linearly independent eigenvectors cannot be found if the anisotropic pressure field of Eq.(3.7) is applied. Since the anisotropic pressure field is difficult to handle, the turbulent pressure is usually approximated by the mean of three components

P*=P+ (3.8)

Using this, the flux vector can be divided into the isotropic and anisotropic parts, and the Jacobian of the isotropic part can be diagonalized. The second method of diagonalization utilizes the production of theturbulence Pij in the vector F. The production term is exact in RSM and it can be included in vector F. This way, independent eigenvectors can be found. In the following, both of the diagonalization approaches are described. 13

Isotropic Diagonalization Vector F can be divided into two parts

F = Fi + F2 (3.9) where vector Fi corresponds to the isotropic part of turbulence and F2 contains the anisotropic part.

( | puu — \pv"v" — | pw"w" ( pu \ pu'v" pu 2 +p+ | pk puw" puv puw pu u(E + p+ | pk) (2 u"u" — v"v" — w"w")+ "7/ 'r/ . — // // puu'u" pvu V + pwu w a n ,F2 = (3.10) -Fl = puu— V puu w 0 puv— II V II 0 puv''w" 0 puw'w" 0 0 pue / 0 0

The Jacobian of the vector Fi can be diagonalized similarly to that of the k — e model [8]. The effect of F2 on a solution is small, and, consequently, F2 can be evalu ­ ated using central differences. In this approach, there is no need to rotate Reynolds stresses into the local cell face coordinates.

Anisotropic Diagonalization The second method of diagonalization utilizes the production term Pij. The produc ­ tion term is exact in RSM and it can be separated from the other source-terms

Q = Q' + P (3.11)

Production is included in vector F. This is not a conservative form of the vector F but RSM is never in a strong conservation law form because of the source-terms. In many cases the Jacobian takes the simplest form if the primary variables are used. Here, the selected primary variables are

V=(p u v w e ttV' vFw" VV t/V' wV ef (3.12) Note that bars and tildes are dropped out of the flow variables for simplicity. After some mathematical manipulation, theJacobian (dF/dU) can be found to be in the 14 primitive variable form as

l p 0 0 0 0 0 0 0 0 0 °\ Se. + «"«" £« p ‘ p u 0 0 p 1 0 0 0 0 0 0 u"v" p 0 u 0 0 0 1 0 0 0 0 0 u" w" 0 0 u 0 0 0 1 0 0 0 0 2 0 p 0 0 u 0 0 0 0 0 0 0 0 2 uu" 0 0 u 0 0 0 0 0 0 0 u"v" u"u" 0 0 0 u 0 0 0 0 0 0 u "ui" 0__ u"u" 0 0 0 u 0 0 0 0 0 0 2 uv" 0 0 0 0 0 u 0 0 0 0 0 u"w" u"v" 0 0 0 0 0 u 0 0 0 0 0 2%"w" 0 0 0 0 0 0 u 0 V o 0 0 0 0 0 0 0 0 0 0 uj (3.13) In this case the eigenvalues , i.e. the characteristic speeds, are

= u,u + c,u — \juu', u — y/u"u", u + c, u,u + yju"u",u + y/u"u",u, u, u, u (3.14) where c is the speed of sound. For an arbitrary equation of state, the speed of sound is c2 = PeP/p 2 +Pp + 3u’V (3.15) Notations p e and p p are dp dp (3.16) Pe = Be' P p = a?1 It is seen that the Reynolds stresses have an effect on the characteristic speeds and on the definition of the speed of sound. Using the primitive variables, the characteristic variables are

SW = (a(i)) = L~l6V = BTXSU (3.17) where i; oft, o o o o o o + I (p/p 2)

SV i; 4:R 4 o o o o o o

= u?

ft ft

V Tk o o o o o ft I r ft ft ft

°o

— I

V &}! o o o o ° ii« 1

and I -> 11 o

matrix 4% 4% o o o o o o ft C: CN R s 9N o o o o CN o o I L I y

1 *4 ft lift ft

i is CUcN o o o o o s: I i; ft

ft}! ft QJ ft ft n % I ft: ft ft f lift f o o o o o o i; > ft sft ft: ft

ft ti; ft ft o o o o o o

Q.

ft Rift ft

J

ft lift 13 o o o o O o cl ft

=a= ===

o o o o o o cl

(3.18) 15 16

The right eigenvector matrix is

o o o o o o o o o o o o

o o o

O O rH

o o o

^ 5 o + + T—I p "9V OQ I

000

O O 1 o

03 (3.19) 17 where

752 = H + cu + 2 (u"v" + u'w" + cvu"v" + cwu'w")/(c2 — u"u") 7*55 = H — cu + 2 (u"v" + u'w" — cvu'v" — cwu"w")/(c2 — u"u") H = E/p + p/p + uFu" (3.20)

In matrices Zr1 and J? there are terms that have to be limited to avoid unnat ural behaviour between turbulent and laminar regions. For example, the term u" v" /\Ju'u or 5(u"v")/\Ju"u" cannot get very large values. This can be limited by using fol ­ lowing inequality V^u V a /^v V > \u"v"\~ (3.21) using this u"v" /y u"u" can be limited as

\u"v" I < \Jv'v" (3.22)

Proof for Eq.(3.21) can be obtained in the following way: In a 2-dimensional case fluctuating velocities u[ and u2 can be rotated into another coordinate system xl and x2, which make an angle 0 with the axes xx and x2 // //* , //* . , cos

The Reynolds stresses can be obtained in the new coordinate system as

uxux = cos

u'2u2 = u'l*u'{* sin 2 0 + u[*u2* sin 20 + u2u2 cos 2 0 (3.24)

If we choose 0 so that u[*u2* = 0 (so-called principal axes) and after some calcu­ lation Eq.(3.21) is reduced to the following form

sin 4 0 + cos 4 0 > — 2 sin 2 0 cos 2 0 (3.25) which proves the inequality. The eigenvectors and the characteristic variables have a fairly complex form. In a computational approach this form can be rearranged and simplified to some degree.

3.3 Rotation Operator

In the anisotropic case, the Reynolds stresses as well as velocity components, must be rotated into a local coordinate system. For this purpose a rotation operator has an 18

effect on the velocity components and also on the Reynolds stresses if the anisotropic diagonalization is applied. The normal of the cell face is known and two tangent directions must be determined. There are many ways to do this but all of them have singularity points. In the present calculations the singular direction of thenormal component is ±(1,1,1)T. The grid must be checked to be such that no singularities exist, i.e., the normal vectors of the surface are not parallel to the vector ±(1,1,1)T. A rotation matrix is built by taking a vector product between the normal vector and the vector (1,1, l)r. The resulting vector is normalized. The third directional vector is obtained by a vector product of the two first ones. The resulting rotation matrix T can be written as ^ Tl^ Tiy 71 z ^ T — 7121 n22 %23 (3.26) \ n32 7133 / where

7721 = ~{nz - ny)

if X 7122 — ~{nx ~ 7iz) if x 7123 — ~\ny ~ nx) Tisi = nyn23 — nzn22 7132 = 71,7121 - 7lz7123 7133 = nxn22 - 71Z7121 (3.27)

and a = \J(nz - ny)2 + (71* — nz)2 + (ny - nx)2 (3.28) Velocity components are rotated from the global to local coordinate system as f u\ ( v 1 = T j v (3.29) \wj \wj I where the hats denote local coordinates. The Reynolds stresses are rotated from the global to the local coordinate system by the following formula [15] i = TIT t (3.30) where f // n // It \ -u'iu'2 *77 ft *77 n 1 = ~u2^1 u2u2 —y^2us (3.31) it // n it \-%3%1 —Tig TI2 uzuz y Transformation from the local to the global coordinate system is given by

I = TtIT (3.32)

The rotation operator T is built by using this rotation matrix T for the velocities and the Reynolds stresses. 19

3.4 Boundary Conditions

The boundary values are given in ghost cells so that the actual boundary conditions are satisfied on the cell faces. At the free-stream boundary, the values of the depen ­ dent variables are kept as constants. In the flow field, u"u" and e are limited to their free-stream values. At the wall, the velocity components are set to zero. In the present cases, the wall is also assumed to be adiabatic. All turbulent quantities are set to zero, also i. Because of this, these variables within the ghost cells are set to be of the opposite sign to the values in the cell adjacent to the surface. The ghost cell values are applied for the calculation of the flux adjacent to the surface. For the calculation of the surface fluxes themselves a second-order extrapolation is applied for the evaluation of the wall pressure and one-sided formulas are used for the derivatives at the wall. Velocities at the symmetry walls can be calculated by first rotating the veloc ­ ity components (ui,u2,u3) into the local Cartesian coordinate system (xi,x2,x3) that has a coordinate direction normal to the boundary face. The sign of the velocity component in this direction is changed and the three velocity components are then rotated back into the original (x\, x2, x3) coordinate system. This can be expressed in a mathematical form as

u (-1 0 0\ ( u u ^ II V 0 1 0 T V = TtPT v (3.33) l 0 Wj mir 0 V \w W y After some manipulations the following equation is obtained:

( u ^ 2nxny 2nxnz\ ( u ^ V = 2?2-3;77 >y l-2< -2 nynz V (3.34) 1 . i-2^y \w) \Wy mir \-2nxnz 2TbyTtz where vector (nx, ny, nz)T is the normal vector of the symmetry plane. The result­ ing velocity vector (u, v, w)Pr is put into the ghost cell. Thisformulation for the velocity components was also applied by Batina [16]. In general, the symmetry conditions of the stress tensor are very complicated for ­ mulations. In the local coordinate system where the i direction is normal to the sym­ metry plain, this condition can be written

Z-l 0 0 0 0\ 0 10 1 0 = PIP (3.35) V 0 0 1 V o 0 1) By applying Eqs.(3.30), (3.32) and (3.35), the symmetry formulation in the global coordinate system is

Imir =T t ImirT = T t PIPT = T t PTIT t PT (3.36) 20

It is regonized that in this equation TT P T is the same as in Eq.(3.33). Therefore, the final form of the symmetry formula for thestress tensor is given by

(1 - 2 nl Q/lbxfby —2nxnz N ^1-2^ —2nxnz ^

d-mir — 2nxny 1-2 nl —2nynz I 2?tx^y i-2%; —2 nynz \ ‘2/Tlx'ft'z —2%2/72'z 1 _ 2n2z) \ 272-x ^z —27Zy72-2 1-2^WZ /y (3.37)

3.5 Time Integration Method

The discretized equations are integrated in time by applying the DDADI-factori ­ zation [14]. This is based on the approximate factorization and on the splitting of the Jacobians of the flux terms. In the implicit stage the factorization is done isotropicly. The implicit stage consists of a backward and forward sweep in every coordinate direction. The sweeps are based on a first-order upwind differencing. In addition, the linearization of the source-term is factored out of the spatial sweeps. The boundary conditions are treated explicitly, and a spatially varying time step is utilized. The implicit stage can be written after factorization as follows

\l + Si+i/2Af — df Si-ij2Ai )j x

[i + —{djSj+1/2Bf - df Sj-i/2Bj)] x

[i + yr(dkSk+l/2Cf - dfSk-l/2Ck)] x

[l-AtDi]AUi = ^Ri where dfj k and dfk are first-order spatial difference operators in the i, j and k di ­ rections, A, B and C are the corresponding Jacobian matrices, D = dQ/dU, and Ri is the right-hand side of Eq.(3.2). The Jacobians are calculated as

A± = EGA* l + /cl)#"1 (3.39) where A* are diagonal matrices containing the positive and negative eigenvalues, and & is a factor to ensure the stability of the viscous term [17]

k _ 2(/r + Hr) (3.40) pd where d is the height of the cell. The idea of the diagonally dominant factorization is to put as much weight on the diagonal as possible. In the i direction the tridiagonal equation set resulting from Eq.(3.38) is replaced by two bidiagonal sweeps and a matrix multiplication [8]. Thematrix inversion resulting from the source-term linearization is performed before the spatial sweeps. In order to improve stability, only negative source-terms 21 can be linearized. Although the form of the source-term indicates that equations may become stiff near the walls, the terms related to the walls are not linearized. Bea- cause of the complexity of the source-terms, the matrix D is approximated by using the following trick D = (3.41)

In this way, the maximum change of U caused by Q is limited to |A[7 max |. The value of |A[/max | is evaluated using the current values of pu'-u ” as

|A(p<«")max| = pu"u'-fCk (3.42)

Ck was set to 5 after test calculations. Calculation of the e and ht is described in [8]. A multigrid method is used to accelerate the convergence. The Jameson ’s method [18] with a simple V-cycle has been adopted. The implementation for the multigrid cycling is described in [19] and [8]. In order to enhance the stability of the multigrid cycling, the size of correction A(pu”u”) from a coarse to a finer grid level is recalculated using the current value of pu ” u'j as A(m<2 A(m4) (3.43) i [ JA(mm)l _ Ci\pu"u'j\ + C2\A(pu"u”)\ After test calculations, C\ and C2 were assigned to be 0.1 and 1.0, respectively. Vari­ able C2 assures a possible change of sign in the tangential components of Reynolds stresses. 22

4 Test Calculations

4.1 Channel Flow

The model was checked by calculating a fully developed flow in a plane channel. The results were compared with the DNS data of Kim et al. [20], and the Reynolds- stress budgets were compared with Mansour et al. [21] data. The DNS data is at Rem = pum8/p Rrf 2 800 where um, 8 and p are the mean velocity, the channel half­ width and molecular viscosity. Because the flow solver uses compressible methods the Mach number was set to 0.2. This introduced a 1% change in density across the channel. The meshis rectangular 48 x 32. The heightof the first row of cells is Ay = 0.0055 or Ay+ 0.9. Only half of the channel is modelled. The length of the computational mesh is 325. The calculations were performed using cyclic boundary conditions. After having a converged result, the solution was taken from the down ­ stream boundary and utilized as the upstream boundary condition of the next run. Fully developed flow was obtained after 4 computations, which corresponds to the length of 64 channel widths. The convergence of the results was checked by using a two-times denser grid. The results obtained with the two grid densities are practi­ cally identical. Several solution methods described earlier were compared. The numbering of the test cases can be seen in Table 4.1. The calculated mean flow variables are pre­ sented in Table 4.2. As can be seen in Table 4.2, the difference between the isotropic and anisotropic flux-difference splittings, case 2 and case 3, is small. The displace ­ ment thickness 5* and the momentum thickness 9 are defined as

Table 4.1: Description of the test cases.

Case CFL Multigrid Flux splitting Diffusion 1 25 5 Anisotropic Daly et al. 2 25 5 Anisotropic Scalar diffusion 3 25 5 Isotropic Scalar diffusion 4 200 1 Isotropic Daly et al. 23

Table 4.2: Mean flow variables. DNS Casel Case 2 Case 3 g*/g 0.141 0.130 0.130 0.133 4/6 0.087 0.077 0.079 0.080 H = 8*19 1.62 1.68 1.66 1.65 Cf = W(£p«m) 8.18 x IQ"* 8.35 x 10-3 8.65 x 10"3 8.65 x 10-3 EeT = puT8/p, 180 180 183 184 , k II II C 2800 2834 ? 2834 2832 i? Rec = puc8/p, 3250 3256 3261 3266 Urn/ UT 15.63 15.76 15.50 15.39

Case 1 CFl=25 MG=5 Case 1 CFL=25 MG=5 Case 2 CFl=25 MG=5 Case 2 Cfh=25 MG=5 Case 3 CF1=25 MG=5 Case S CFU=25 MG=5 Case 4 CFl=200 MG=1 Case 4 CFU=200 MG=1 Chien k-e

200,0 400.0 600.0 200.0 400.0 600.0 CYCLES CYCLES

Fig. 4.1: Convergence of the L% norm of the x momentum and the pu'u residuals.

where A is a half-width of the channel and Uc velocity at the centre line. The values of 8* and 9 are approximately 10% smaller than in DNS. The calculations were performed with five grid levels at CFL = 25. The multi­ grid corrections were omitted on the first two cells from the solid wall. The second- order upwind scheme was used. First, 40 iteration cycles were performed with the k—e model at CFL — 35 and the Reynolds stresses were uncoupled from the flow. A converged solution was obtained after 200 to 250 iteration cycles. The convergence history of the Lz norm of the x momentum and the pu'u residuals are shown in Fig. 4.1. In this case the difference in convergence rate between theisotropic and anisotropic methods is marginal. The anisotropic method converged roughly in 50 cycles faster than the isotropic one. The computing time with RSM is increased by 13% if the anisotropic coupling is applied. The increased convergence rate was obtained by the multigrid acceleration, as can be seen in Fig. 4.1. Without the multigrid cycling it takes about 2 000 iteration sweeps to get a converged solution. The convergence rate is about the same with Chien’s k—e model and with RSM. Naturally, in the calculations with Chien’s model there is no transient after 40 cycles. Chiens k — e model runs two times faster per 24

Ma=0.2 Re=2800 Daly and Harlow Scalar diffusion o Kim et al.

i i i i m i rn111 T i I I i 11

Fig. 4.2: Mean velocity profiles in wall coordinates.

Daly and Harlow Scalar diffusion o Kim et al.

100.0 200.0

Fig. 4.3: Comparision of the calculated Reynolds stresses and the DNS-data in a plane chan­ nel. iteration sweep than RSM. The velocity profiles are compared in Fig. 4.2 in terms of u+, which is a univer ­ sal dimensionless velocity defined as u+ = u/uT, where uT = yrw/p is a friction velocity. Thevelocity profiles in a viscous sublayer agree well with DNS and uni ­ versal profiles. The velocity profiles are not completely satisfactory in outer layers. The Reynolds stresses can be seen in Fig. 4.3 where uTms = yu"u"/v%. Turbu­ lent intensities agree well with DNS data except that the urms peak level is low and the vrms near-wall values are not satisfactory. The fluctuating component normal to the wall should be damped rapidly close to the wall (y+ < 10), but the simulation entirely misses this effect. This has also an effect on the source-term distributions. The shear-stress distribution in Fig. 4.3 agrees with the DNS results. The source-term distributions are presented in Fig. 4.4. The source terms are 25

0.0 50.0 100.0 150.0 0.0 50.0 100.0 150.0 y+ y+

Fig. 4.4: Budgets of Reynolds stresses. Symbols are from the DNS calculation. non-dimensionalized using where v is kinematic viscosity. Close to the wall, the source terms do not agree with the DNS data, but closer to the centre of the channel the agreement is good. The velocity pressure-gradient term <3?y does not be­ have correctly, and there are also problems in the dissipation term close to the wall. The wall correction in the velocity pressure-gradient was omitted because it gave a totally unsatisfactory distribution of the shear stress uv". Production of the shear stress P12 is not correct close to the wall. This is a consequence of the normal Reynolds-stress component perpendicular to thewall v"v" having too high values in the viscous sublayer in Fig. 4.3. The dissipation term could have been modelled differently by using a Taylor series near the wall [22]. That approach could have damped v"v" near the wall more efficiently. Although not shown, anisotropic and isotropic flux-difference splittings do not introduce changes in the distributions of the source-terms. The calculation method for diffusion, does not have a strong effect on the con ­ vergence rate. The convergence rate of the scalar diffusion case (case 2) in Fig. 4.1 is similar to that obtained using the Daly and Harlow method (case 1). However, the mean flow variables are changed slightly in Table 4.2. Especially, the choice of the diffusion model changed the friction coefficient. Some difference can also be seen 26

Poly ond Horiow Scalar diffusion Poly and Horiow v Kim et ol. Scalar diffusion - 9 ^ ? Kim et ol.

9 V

-0.05 „/—\ V ^VTTTTVTO 999999999W' V -0.15 0.0 50.0 100.0 150.0 y+

Daly ond Horiow Scalar diffusion Daly and Harlow 0.005 v Kim et ol. 0.005 Scalar diffusion ? Kim et al. tVVVVVVt VVV7VV 1 cs 0.000 "> 0.000

-0.005 -0.005

-0.010 -0.010 0.0 50.0 100.0 150.0 100.0 y+

Fig. 4.5: Comparision of the turbulent diffusion. Symbols are from the DNS calculation. in the velocity profiles in Fig. 4.2, and in the shear stresses in Fig. 4.3. Although there are differences in the velocity profiles between these methods, the turbulent diffusion does not exhibit large differences in Fig. 4.5. It can be seen from Fig. 4.5 that the turbulent diffusion rates are not satisfactorily modelled. The. Daly and Harlow turbulent diffusion term of Eq.(2.27) is only slightly better than the scalar diffusion term of Eq.(2.26). 27

5 Summary

The Reynolds-averaged Navier-Stokes equations with a low-Reynolds number RSM have been solved using an implicit method with a multigrid acceleration for conver ­ gence. In the evaluation of fluxes the turbulence equations are coupled with the in- viscid part of the flow equations, and Roe ’s method is applied. A new anisotropic coupling of the Navier-Stokes and the Reynolds-stress equations is introduced. Also a new method of treating symmetry boundaries is presented. The solution methods have been tested using Shima’s low-Reynolds number mod ­ el. The developed numerical scheme appears to be stable and efficient. Per iteration cycle the calculation with the Reynolds-stress model takes only about twice as long as the calculation with Chien’s low-Reynolds number k — e model. Using a multi- grid only a few hundred iteration cycles are required for thesolution of a flow in a plane channel. This paper has focused attention on the problem of coupling the Reynolds stresses with Navier-Stokes equation with a compressible flow assumption. The applied clo ­ sure model is a relatively old one. Hence, a very good agreement with the DNS data was not even expected. The new coupling method introduced only small improve ­ ments in the convergence rate in the present incompressible case. Thedifferences between thecoupling methods may become larger in a case of super or hypersonic flow. 28

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