Helsinki University of Technology Laboratory of Applied Thermodynamics

Transcription

1 Helsinki University of Technology Laboratory of pplied Thermodynamics umerical Methods for Coupling the Reynolds-veraged avier Stokes Equations with the Reynolds-Stress Turbulence Model Patrik Rautaheimo 1 and Timo Siikonen 2 Helsinki University of Technology, Espoo, Finland Report o taniemi ISB ISS Research Scientist, Laboratory of pplied Thermodynamics 2 ssociate Professor, Laboratory of pplied Thermodynamics

2 1 Contents omenclature 2 1 Introduction 4 2 Governing Equations 6 21 Flow Equations 6 22 Reynolds-Stress Turbulence Model 7 23 Dissipation Transport Equation 10 3 umerical Method Spatial Discretization Diagonalization of the Flow Equations Rotation perator Boundary Conditions Time Integration Method 20 4 Test Calculations Channel Flow 22 5 Summary 27

3 3 4 ; >? B C KL K 2 omenclature Jacobian matrix Jacobian matrix Jacobian matrix Courant number difference between and ; also Jacobian of the source term total internal energy per unit volume "$% flux vectors in -, - and ' -directions total entalphy *, matrix of Reynolds stresses flux in a given direction in space right eigenvector matrix with the primitive variables transformation matrix from the conservative to the primitive variables 4: Mach number production of the kinetic energy of turbulence; also boundary condition matrix ;=< Prandtl number source term right eigenvector matrix with the conservative variables; also residual?=@ Reynolds number strain-rate; also cell-face area temperature; also rotation matrix; turbulent transport rotation operator vector of the conservative variables 8 cell volume; also vector of the primitive variables D vector of the characteristic variables EGF specific heat in a constant pressure EIH specific heat in a constant volume diffusion of the Reynolds-stresses; also cell thickness K internal energy per unit mass, ie specific internal energy M unit vectors in Cartesian coordinate system heat conductivity; also kinetic energy of turbulence, ; stability factor K unit vector normal to a cell face + static pressure

4 B L L < M L 3 ' + effective + derivative of pressure in constant density + + derivative of pressure in constant internal energy +1 heat flux time velocity components in -, - and -directions ' Cartesian coordinates non-dimensional normal distance from the surface diagonal eigenvalue matrix the velocity pressure-gradient correlation characteristic variable ratio of specific heats E%H Kronecker s delta dissipation of kinetic energy of turbulence eigenvalue of dynamic viscosity kinematic viscosity density Schmidt s number normal or shear stress specific dissipation rate L M Subscripts I ' Superscripts turbulent conditions L -index; summation index M -component of a matrix grid coordinate directions viscous coordinate directions dissipation left-hand side of the cell face 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

5 4 1 Introduction In recent years, computational fluid dynamics CFD has begun to be an engineering design tool for practical flow problems Some complex aerodynamical configurations have been calculated, eg 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 DS gives an accurate 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 DS, LES requires a lot of computing resources and also the theory is not yet fully evaluated Thus, turbulence modeling for Reynolds-averaged avier Stokes equations still plays a very important role in CFD In a simple case, eg, in the case of a 2-D boundary layer flow at a constant pressure, there is no need to use a complicated turbulence model like the Reynoldsstress model RSM For example, the algebraic turbulence model of Baldwin and Lomax [2] has been successfully used for various airfoil sections [3] lgebraic models utilize 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, as and 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 the effect 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 Unfortunately, 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 complicated engineering flow problems However, RSM also has disadvantages because of complexity, terms that need modelling and many closure coefficients, and an increased number of differential equations In spite of the apparent difficulties, RSM has been used to calculate internal flows for a long time [4] Recently some calculations 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

6 compressible flow assumption, the Reynolds stresses act not only in the momentum equation, but also in the energy equation This makes the implementation more difficult 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 nisotropic coupling makes the implicit time integration and upwinding difficult to apply, but in some cases this approach may accelerate the convergence In this study, Shima s low-reynolds number RSM [7] is coupled with a compressible flow solver [8] based on Roe s method [9] 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 5

7 H H H > 6 2 Governing Equations 21 Flow Equations The Reynolds-averaged avier Stokes equations RS, and the equations for the Reynolds stresses and dissipation of turbulence can be written in the following form H H 1' H 21 where is vector of conservative variables,, and are the inviscid fluxes and, and are the viscous fluxes The source-term > has non-zero components for the Reynolds-stress equations Vectors and are % where is density, + is pressure,, and are velocity components in the Cartesian coordinate system,,,,, and are the Reynolds stresses, and the total internal energy defined L is the specific internal energy and is the summation index Here double prime is the Favre-averaged fluctuation component in the direction, tilde denotes Favre averaging, and the bar means ordinary time averaging Vectors and can be written similarly

8 H B B 7 The viscous flux in the L direction is J 24 1% where the viscous stress tensor is defined as 25 and J is the diffusion flux of the kinetic energy of turbulence Diffusion of the Reynolds stresses are included in the source vector > The diffusion coefficients of the dissipation as 26 where is the appropriate Schmidt s number, and is the turbulent viscosity of the fluid determined with the applied turbulence model Heat flux K is obtained by using Boussinesq s approximation and is written as K E F ;=< EGF ;=< 27 where EGF is specific heat The pressure is calculated from an equation of state +, which, for a perfect gas, is written + where is the ratio of specific heats E F E%H Reynolds-Stress Turbulence Model The Reynolds-stress transport can be written in the following Cartesian tensor form ; B J 29

9 8 where ;,, B, and J are the production, the velocity pressure-gradient correlation, the turbulent transport, the dissipation rate and the diffusion terms, respectively The exact form of the source-terms can be written as J is the mean strain-rate ; + + B s can be seen, the turbulent transport, the velocity pressure-gradient and the dissipation rate must be modelled The viscous diffusion takes a simpler form, if the flow is incompressible or weakly compressible [6] J 216 Shima s Model for the Velocity Pressure-Gradient and the Dissipation Rate Term In this study, the modelling is performed utilizing Shima s approach [7] However, the dissipation equation is adapted from Chien s 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 where ; 218 ;

10 ; 9 ; ; ; 219 ; 220 where ; is the production term of Eq210, and ; are 0 0 varies with the wall function as The multiplyer in Eq224 was, after test calculation, modified to be It should also be noted that in this study the tangential components of the velocity pressure-gradient wall term were omitted in order to get a correct velocity profile and shear-stress distribution close to the wall Diffusion Term In the diffusion term, the pressure diffusion is ignored or it is assumed to be part of the turbulent transport term In Shima s model, the diffusion term of Hanjalic and Launder s [11] is applied E Convergence problems were experienced using Eq225 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] B J 226 Diffusion in the normal direction is usually much more important than in the streamwise direction This can be taken into account using the model of Daly and Harlow s [13] + 0 bove turbulent eddy viscosity + 0 E 0 is defined as E 228

11 E where the kinetic energy of turbulence is The equation for contains empirical coefficients These are given by E E 0 23 Dissipation Transport Equation The dissipation transport equation was taken from Chien s $ model [10] because it was experienced to be stable and well behaved Chien s dissipation transport equation can be written as = E ; E where is the normal distance from the wall, and is defined by K 5 8 The production ; and is is a trace of the production tensor ; The relationship between where is kinematic viscosity With this assumption, has a value of zero at the wall and has an exact value with respect to However, does not have the desired behaviour near the wall This results in a non-zero diffusion flux at the wall The equation for contains empirical coefficients These are given by 234 where the turbulence Reynolds number is defined as?=@ 235 Chien proposed slightly different forms for E and E Since the computations performed 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 E and E

12 J J J C C 11 3 umerical Method The finite-volume CFD program for complex three-dimensional geometries [8] was used in the present calculations The program utilizes Cartesian velocity components in a cell-centred approach In the evaluation of the inviscid fluxes, Roe s method [9] is applied For a spatial discretization MUSCL-type TVD-scheme to approximate advective volume-face fluxes is applied The discretized equations are integrated in time by applying the 3 -factorization [14] The code utilizes a multigrid V-cycle for the acceleration of convergence Complicated geometries can be handled with multiblock grids 31 Spatial Discretization In the present solution, a finite-volume technique is applied The flow equations have an integral form J 8 K J K > J 8 31 for an arbitrary fixed region 8 a computational cell L yields 8 with a boundary Performing the integrations for J 8 > 32 where the sum is taken over the faces of the computational cell The flux for the face is defined 33 Here, and are the fluxes defined by Eqs22 to 24 in the, and ' directions, respectively In the evaluation of the inviscid fluxes, Roe s method [9] is applied rotation operator is used for the velocity components and also for the Reynolds stresses The flux is calculated as 34 where C is a rotation operator that transforms the dependent variables to a local coordinate system normal to the cell surface In this way, only the Cartesian form

13 of the inviscid flux is needed This is calculated from < 35 where and are the solution vectors evaluated on the left and right sides of the cell surface, < is a the right eigenvector of the Jacobian matrix 1, the corresponding eigenvalue is, and is the corresponding characteristic variable obtained from?, where MUSCL-type discretization is used for the evaluation of and In the evaluation of and, primary flow variables 1, and conservative turbulent variables are utilized Jacobian matrix can be split in the following way?? 4 36 where? and? are the right and left eigenvector matrices of, and are the corresponding matrices with respect to the primitive variables, is the diagonal eigenvalue matrix, and 4 and 4 are the transformation matrices between the conservative and the primitive variables 32 Diagonalization of the Flow Equations coupling between the avier Stokes and the Reynolds-stress equations is introduced, L since the Reynolds stresses may be connected with the pressure [6] In the -momentum equation, the resulting effective pressure can be defined as + 37 In order to utilize Roe s method, the Jacobian of the flux vectors must be diagonalized 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 Eq37 is applied Since the anisotropic pressure field is difficult to handle, the turbulent pressure is usually approximated by the mean of three components 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 the turbulence ; in the vector The production term is exact in RSM and it can be included in vector This way, independent eigenvectors can be found In the following, both of the diagonalization approaches are described

14 @ > ; 13 Isotropic Diagonalization Vector can be divided into two parts where vector corresponds to the isotropic part of turbulence and anisotropic part contains the 310 The Jacobian of the vector can be diagonalized similarly to that of the model [8] The effect of on a solution is small, and, consequently, can be evaluated using central differences In this approach, there is no need to rotate Reynolds stresses into the local cell face coordinates nisotropic Diagonalization The second method of diagonalization utilizes the production term ; The production term is exact in RSM and it can be separated from the other source-terms > Production is included in vector This is not a conservative form of the vector 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 ote that bars and tildes are dropped out of the flow variables for simplicity fter some mathematical manipulation, the Jacobian 1 can be found to be in the

15 0 F E 14 primitive variable form as F F I H 1 In this case the eigenvalues, ie the characteristic speeds, are E where E is the speed of sound For an arbitrary equation of state, the speed of sound is E ++,+ otations + and + are % 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 D?

16 $ $ $ $ $ $, + ' $ $ $ $ $ $ ' ' ' 15 where % " % " " and matrix " * % $ * 0 $ $ $ " " " " " "- "- " is * " * " " " 0 0 ' 318

17 16 The right eigenvector matrix is " 0 0 * * 0 * * * * * 0 319

18 E E 17 where < Ë 5 E < Ë E E E * 320 In matrices and? there are terms that have to be limited to avoid unnatural behaviour between turbulent and laminar regions For example, the term or cannot get very large values This can be limited by using following inequality using this 2 can be limited as Proof for Eq321 can be obtained in the following way: In a 2-dimensional case fluctuating velocities and can be rotated into an other coordinate system and, which make an angle with the axes and The Reynolds stresses can be obtained in the new coordinate system as If we choose so that 0 0 so-called principal axes and after some calculation Eq321 is reduced to the following form 325 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

19 9 3 B 9 B Rotation perator 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 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 the normal component is The grid must be checked to be such that no singularities exist, ie, the normal vectors of the surface are not parallel to the vector rotation matrix is built by taking a vector product between the normal vector and the vector The resulting vector is normalized The third directional vector is obtained by a vector product of the two first ones The resulting rotation matrix B can be written as where and Velocity components are rotated from the global to local coordinate system as 329 where the hats denote local coordinates The Reynolds stresses are rotated from the global to the local coordinate system by the following formula [15] 3 B:3 B 330 where 331

20 3 B B ; 19 Transformation from the local to the global coordinate system is given by B 3 B 332 The rotation operator C is built by using this rotation matrix B for the velocities and the Reynolds stresses 34 Boundary Conditions The boundary values are given in ghost cells so that the actual boundary conditions are satisfied on the cell faces t the free-stream boundary, the values of the dependent variables are kept as constants In the flow field, and are limited to their free-stream values t the wall, the velocity components are set to zero In the present cases, the wall is also assumed to be adiabatic ll turbulent quantities are set to zero, also 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 velocity components into the local Cartesian coordinate system " 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 coordinate system This can be expressed in a mathematical form as B 5 B ; fter some manipulations the following equation is obtained: where vector is the normal vector of the symmetry plane The resulting velocity vector 1 is put into the ghost cell This formulation for the velocity components was also applied by Batina [16] In general, the symmetry conditions of the stress tensor are very complicated formulations In the local coordinate system where the L direction is normal to the symmetry plain, this condition can be written ; 335

21 3 3 3 B 3 > 20 3 B 3 B B ; 3 ; B B ; B 3 B ; B ; 3 By applying Eqs330, 332 and 335, the symmetry formulation in the global coordinate system is 336 It is regonized that in this equation is the same as in Eq333 Therefore, the final form of the symmetry formula for the stress tensor is given by Time Integration Method The discretized equations are integrated in time by applying the 3 -factorization [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 ? 338 where and are first-order spatial difference operators in the L, M and directions,, and are the corresponding Jacobian matrices,, and? is the right-hand side of Eq32 The Jacobians are calculated as? 3? 339 where are diagonal matrices containing the positive and negative eigenvalues, and is a factor to ensure the stability of the viscous term [17] $ J 340 where J 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 L direction the tridiagonal

22 > > 21 equation set resulting from Eq338 is replaced by two bidiagonal sweeps and a matrix multiplication [8] The matrix inversion resulting from the source-term linearization is performed before the spatial sweeps In order to improve stability, only negative source-terms can be linearized lthough the form of the source-term indicates that equations may become stiff near the walls, the terms related to the walls are not linearized Beacause of the complexity of the source-terms, the matrix is approximated by using the following trick 341 In this way, the maximum change of caused by > is limited to value of is evaluated using the current values of as The 342 was set to after test calculations Calculation of the and is described in [8] 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 from a coarse to a finer grid level is recalculated using the current value of as 343 fter test calculations, and were assigned to be and, respectively Variable assures a possible change of sign in the tangential components of Reynolds stresses

23 22 4 Test Calculations 41 Channel Flow The model was checked by calculating a fully developed flow in a plane channel The results were compared with the DS data of Kim et al [20], and the Reynoldsstress budgets were compared with Mansour et al [21] data The DS data is at?=@ 5 where, and are the mean velocity, the channel halfwidth and molecular viscosity Because the flow solver uses compressible methods the Mach number was set to This introduced a change in density across the channel The mesh is rectangular The height of the first row of cells is or nly half of the channel is modelled The length of the computational mesh is The calculations were performed using cyclic boundary conditions fter having a converged result, the solution was taken from the downstream boundary and utilized as the upstream boundary condition of the next run Fully developed flow was obtained after computations, which corresponds to the length of 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 practically identical Several solution methods described earlier were compared The numbering of the test cases can be seen in Table 41 The calculated mean flow variables are presented in Table 42 s can be seen in Table 42, the difference between the isotropic and anisotropic flux-difference splittings, case and case, is small The displacement thickness and the momentum thickness are defined as J 41 Table 41 Description of the test cases Case Multigrid Flux splitting Diffusion 1 nisotropic Daly et al 2 nisotropic Scalar diffusion 3 Isotropic Scalar diffusion 4 Isotropic Daly et al

24 23 Table 42 Mean flow variables DS Case 1 Case 2 Case 3 E?=@ 5?=@?=@ =5 Fig 41 Convergence of the norm of the momentum and the J residuals 42 where is a half-width of the channel and velocity at the centre line The values of and are approximately smaller than in DS The calculations were performed with five grid levels at The multigrid corrections were omitted on the first two cells from the solid wall The secondorder upwind scheme was used First, iteration cycles were performed with the model at and the Reynolds stresses were uncoupled from the flow converged solution was obtained after to iteration cycles The convergence history of the norm of the momentum and the residuals are shown in Fig 41 In this case the difference in convergence rate between the isotropic and anisotropic methods is marginal The anisotropic method converged roughly in cycles faster than the isotropic one The computing time with RSM is increased by if the anisotropic coupling is applied The increased convergence rate was obtained by the multigrid acceleration, as can be seen in Fig41 Without the multigrid cycling it takes about iteration sweeps to get a converged solution The convergence rate is about the same with Chien s model and with RSM aturally, in the calculations with Chien s model there is no transient after cycles Chiens model runs two times faster

25 24 Fig 42 Mean velocity profiles in wall coordinates Fig 43 Comparision of the calculated Reynolds stresses and the DS-data in a plane channel per iteration sweep than RSM The velocity profiles are compared in Fig 42 in terms of, which is a universal dimensionless velocity defined as, where is a friction velocity The velocity profiles in a viscous sublayer agree well with DS and universal profiles The velocity profiles are not completely satisfactory in outer layers The Reynolds stresses can be seen in Fig43 where Turbulent intensities agree well with DS data except that the peak level is low and the near-wall values are not satisfactory The fluctuating component normal to the wall should be damped rapidly close to the wall, but the simulation entirely misses this effect This has also an effect on the source-term distributions The shear-stress distribution in Fig 43 agrees with the DS results The source-term distributions are presented in Fig 44 The source terms are

26 25 Fig 44 Budgets of Reynolds stresses Symbols are from the DS calculation non-dimensionalized using, where is kinematic viscosity Close to the wall, the source terms do not agree with the DS data, but closer to the centre of the channel the agreement is good The velocity pressure-gradient term does not behave 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 Production of the shear stress ; is not correct close to the wall This is a consequence of the normal Reynolds-stress component perpendicular to the wall having too high values in the viscous sublayer in Fig 43 The dissipation term could have been modelled differently by using a Taylor series near the wall [22] That approach could have damped near the wall more efficiently lthough 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 convergence rate The convergence rate of the scalar diffusion case case 2 in Fig 41 is similar to that obtained using the Daly and Harlow method case 1 However, the mean flow variables are changed slightly in Table 42 Especially, the choice of the diffusion model changed the friction coefficient Some difference can also be seen

27 26 Fig 45 Comparision of the turbulent diffusion Symbols are from the DS calculation in the velocity profiles in Fig 42, and in the shear stresses in Fig 43 lthough there are differences in the velocity profiles between these methods, the turbulent diffusion B does not exhibit large differences in Fig 45 It can be seen from Fig 45 that the turbulent diffusion rates are not satisfactorily modelled The Daly and Harlow turbulent diffusion term of Eq227 is only slightly better than the scalar diffusion term of Eq226

28 27 5 Summary The Reynolds-averaged avier Stokes equations with a low-reynolds number RSM have been solved using an implicit method with a multigrid acceleration for convergence In the evaluation of fluxes the turbulence equations are coupled with the inviscid part of the flow equations, and Roe s method is applied new anisotropic coupling of the avier Stokes and the Reynolds-stress equations is introduced lso a new method of treating symmetry boundaries is presented The solution methods have been tested using Shima s low-reynolds number model 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 model Using a multigrid only a few hundred iteration cycles are required for the solution of a flow in a plane channel This paper has focused attention on the problem of coupling the Reynolds stresses with avier Stokes equation with a compressible flow assumption The applied closure model is a relatively old one Hence, a very good agreement with the DS data was not even expected The new coupling method introduced only small improvements in the convergence rate in the present incompressible case The differences between the coupling methods may become larger in a case of super or hypersonic flow

29 28 Bibliography [1] Siikonen, T, Kaurinkoski, P, and Laine, S, Transonic Flow over a Delta Wing Using a Turbulence Model, in Proceedings of the 19th ICS Congress, naheim, pp , Sept 1994 ICS Paper [2] Baldwin, B and Lomax, H, Thin Layer pproximation and lgebraic Model for Separated Turbulent Flows, Jan 1978 I Paper [3] Holst, T, Viscous Transonic irfoil Workshop Compendium of Results, Journal of ircraft, Vol 25, Dec 1988, pp [4] Leschziner, M, Modelling Engineering Flows with Reynolds Stress Turbulence Closure, Journal of Wind Engineering and Industrial erodynamics, Vol 35, 1990, pp [5] Davidson, L, Prediction of the Flow round an irfoil Using a Reynolds Stress Transport Model, Journal of Fluids Engineering, Vol 117, March 1995, pp [6] Vandromme, D, Turbulence Modeling for Turbulent Flows and Implementation in avier Stokes Solvers, in Introduction to the Modeling of Turbulence, von Karman Institute for Fluid Dynamics Lecture Series , 1991 [7] Shima,, Reynolds-Stress Model for ear-wall and Low-Reynolds- umber Regions, Journal of Fluids Engineering, Vol 110, 1988, pp [8] Siikonen, T, n pplication of Roe s Flux Difference Splitting for the Turbulence Model, Helsinki University of Technology, Laboratory of erodynamics, 1994 ISB [9] Roe, P, pproximate Riemann Solvers, Parameter Vectors, and Difference Schemes, Journal of Computational Physics, Vol 43, 1981, pp [10] Chien, K-Y, Predictions of Channel and Boundary-Layer Flows with a Low-Reynods-umber Turbulence Model, I Journal, Vol 20, Jan 1982, pp [11] Hanjalic, K and Launder, B E, Reynolds Stress Model of Turbulence and its pplication to Thin Shear Flows, Journal of Fluid Mechanics, Vol 52, o 4, 1972, pp

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