Helsinki University of Technology CFD-group/ The Laboratory of Applied Thermodynamics MEMO No CFD/TERMO-5-97 DATE: December 9,997 TITLE A comparison of complete vs. simplied viscous terms in boundary layer ow and lid-driven cavity ow. AUTHOR() Petri Majander ABTRACT Viscous terms are derived in curved co-ordinates using nite volume formulation. An approximation in common use called Thin hear Layer approximation is derived. Turbulent ow over a at plate with a k ; -model as well as a laminar lid-driven ow are computed and the results are compared between the full viscous terms and the TL approximation. MAIN REULT An implementation of full viscous terms in FINFLO solver. PAGE 4 KEY WORD Thin hear Layer, viscous ow, boundary layer ow, lid-driven cavity ow APPROVED BY Timo iikonen December 9,997
Introduction At high Reynolds numbers wall shear layers, wakes or free shear layers will be of limited size, and if the viscous region remains limited then the dominating inuence of the shear layers comes from the gradients transverse to the main ow direction. If we consider an arbitrary curvilinear system of co-ordinates with and along the surface and directed towards the normal of the surface, then the thin shear layer (TL) approximation of the Navier-tokes equations consists of neglecting all and -derivatives occurring in the turbulent and viscous shear stress terms. Also, at high Reynolds numbers the mesh is made dense in the direction normal to the shear layer, and therefore the neglected terms are computed with a lower accuracy []. In the following applications derivatives are computed in a simplied manner in all curved co-ordinade directions which is referred to as TL approximation. Finite Volume Formulation. Derivation of the viscous terms Transport equation in conservation form can be written as d dt V UdV + which in discretized form is written as X f aces d dt (V i iu i )+ The ux at the surface is written as k ~F (U) d~ V QdV () ^F (U) k V i Q i : () ^F (F ; F v )~n x +(G ; G v )~n y +(H ; H v )~n z () where the components of the unit normal vector are written as ~n x jj~i, x ~n y ~j y and ~n jj z jj~ z k. Each surface of the cell has a unit normal ~n (see Fig. ). Viscous uxes F v, G v and H v are " # T F v 0 xx xy xz u xx + v xy + w xz ; q x x " # T G v 0 yx yy yz u yx + v yy + w yz ; q y y " # T H v 0 zx zy zz u xz + v yz + w zz ; q z (4) z
n i,j+/ q i,j * qi+,j * i+/,j i,j / i+/,j Fig. : Two neighbouring gridcells in the -direction are shown to clarify the notations. where the last term is viscous part of generic scalar equation, including turbulent kinetic energy and dissipation of kinetic energy. The diusion coecient is approximated as + T, where T is the turbulent viscosity and is appropriate chmidt's number. Heat conduction ux is ~q ;(k + k T )rt ; c p Pr + c p T rt (5) Pr T where k, k T are laminar and turbulent thermal conductivities respectivily, Pr,Pr T are laminar and turbulent Prandl numbers respectivily and c p is specic heat at constant pressure. A stress tensor for newtonian viscous uid is written as ij u j + u i ; x i x j (r ~V ) ij ; u 0 iu 0 j (6) where for the Reynolds stresses Boussinesq's approximation is used u ;u 0 iu 0 j j T + u i ; x i x j (r ~V ) ij The stress tensor is symmetric ij ji. Viscous uxes are obtained from Eqs. () and (4) ^F v 0 ^F v ^F v ^F 4 v ^F 5 v ( xx n x + yx n y + zx n z )d ( xy n x + yy n y + zy n z )d ( xz n x + yz n y + zz n z )d [(u xx + v xy + w xz )n x +(u xy + v yy + w yz )n y (7)
+ (u xz + v yz + w zz )n z ; q x n x ; q y n y ; q z n z ]d ^F 6 v [u( xx n x + xy n y + xz n z )+v( xy n x + yy n y + yz n z ) + w( zx n x + zy n y + zz n z ) ; q x n x ; q y n y ; q z n z ]d x n x + y n y + z n z d: (8) After inserting the stress tensors dened by Eqs. (6) and (7) the uxes in the direction and on the surface i+ become " ^F ( + v i+ T ) u v w x ; y ; n z x i+ + u y + v n x y i+ w + x + u # n z z i+ i+ ^F v i+ ^F 4 v i+ " u ( + T ) y + v x v + z + w n y z i+ ( + T ) " u z + w x + w u v z ; x ; y n x i+ + v u w y ; x ; z # i+ n x i+ + v n z i+ # i+ z + w y n y i+ n y i+ ^F 5 v i+ ^F 6 v i+ u i+ ; qx i+ ^F v i+ + v i+ n x i+ ^F v i+ + w i+ + q y i+ n y i+ ^F 4 v i+ + q z i+ x n x i+ + y n y i+ + z n z i+ n z i+ i+ i+ (9) These equations are valid on every surface as corresbonding components of the surface areas l and normals n l are used. A detail concerning a computation of incompressible ow is mentioned here. The stress tensor of Eq. (6) can be divided into incompressible and compressible parts u j ij ( + T ) + u i ; x i x j (r ~V ) ij ij ; ( + T ) (r ~V ) ij : (0) Inserting this form to the uxes of Eq. (8) gives for ^F v ^F v xx ; ( + T ) (r ~V ) ^F 5 v u xx ; ( + T ) (r ~V ) + v xy n x + yy ; ( + T ) (r ~V ) and ^F 5 v n x + yx n y + zx n z d n x + xy n y + xz n z n y + yz n z
+ w zx n x + zy n y + zz ; ( + T ) (r ~V ) n z ; q x n x ; q y n y ; q z n z ] d () For an incompressible ow r ~ V 0 mathematically. In conservative equations the mass ux is in balance when R ~Vd 0. In a pseudo-compressibility method density is not necessarily constant. Even with a constant density, the mass ux balance does not indicate that r~v 0, since the uxes are numerically calculated from dierent formulae from r~v 0. Hence compressible viscous eects may be present to some extent. For truly incompressible i.e. constant density ows this eect is of the same order as the truncation order. In order to compute the cartesian derivatives a somewhat less known version of the generalized divergence theorem of Gauss is applied (see e.g. []), V rdv d~ () Integration is performed over the volume V ijk on the left side, V ijk rdv V ijk x ~ i + y ~ j + z ~ k and over the corresponding faces on the right side, d ~ x i+ j k i+ j k ; x i; j k i; j k ~ i + x i j+ x i j k+ y i+ y i j+ y i j k+ z i+ z i j+ ijk k i j+ ; k x i j; k i j; ~ i + k ij k+ ; x i j k; i j k; j k i+ j k ; y i; j k i; 4 () ~ i + j k ~ j + k i j+ ; k y i j; k i j; ~ j + k i j k+ ; y i j k; i j k; j k i+ j k ; z i; j k i; z i j k+ i j k+ k i j+ k ; z i j; k i j; ; z i j k; i j k; Hence the derivative of in x -direction can be solved V ijk x ijk ~ j + j k ~k + ~k + k ~k: (4) x i+ j k i+ j k ; x i; j k i; j k + x i j+ k i j+ k ; x i j; k i j; + k x i j k+ ij k+ ; x i j k; i j k; (5) If the variables are dened in the middle of volumes, then the values at surfaces must be interpolated as averages. These integrations can also be performed over a dierent volume e.g. V i+ j k which is equivalent to increasing all i -indeces by in Eqs. (4) and (5).
5. Thin hear Layer Approximation Computing derivatives accurately is computationally heavy and the required memory size is increased, which of course also depends on the structure of the code. At high Reynolds numbers only the terms in the direction of curved co-ordinates are preserved. This is obtained by dropping the cross derivatives from equation (5) []. The computational grid should be as orthogonal and dense as possible in the directions against the surfaces, where the dominating viscous eects are present. Also, in the following, the integration is performed over the volume V i+ j k, V i+ j k x i+ j k x i+ j k i+ j k ; x i j k i j k : (6) In the thin-layer approximation the surface components are replaced by the interpolated surface component between them. This is usually readily available and the volume V i+ j k can be approximated as an average (V i + V i+ ). The Eq.( 6) can be further modied as x j k x i+ ( i+ j k ; i j k ) V i+ j k i+ j k i+ j k V i+ j k n x i+ j k ( i+ j k ; i j k ) n x i+ j k ( i+ j k ; i j k ) j k : (7) As the approximate derivatives of Eq. (7) are inserted into the uxes given by Eqs.( 9) ^F v i+ ; 8 < : n z i+ d i+ + 4 n y i+ + 4 n x i+ 4 n x i+ 4 4 (w i+ ; w i ) (u i+ ; u i ) ; 5 nx i+ (u i+ ; u i )+ n x i+ n y i+ (v i+ ; v i ) (v i+ ; v i ) (w i+ ; w i )+ n z i+ (u i+ ; u i ) n x i+ + n y i+ + n z i+ + n x i+ n y i+ (v i+ ; v i ) 5 ny i+ 5 nz i+ (u i+ ; u i ) 9 i+
+ n x i+ n z i+ (w i+ ; w i ) 5 i+ Other momentum ux components are derived in a similar manner, ^F v i+ ^F 4 v i+ + n y i+ 4 n x i+ (u i+ ; u i ) 4 n + n + n y i+ x i+ z i+ + n z i+ n y i+ (w i+ ; w i ) + 4 n x i+ n z i+ 5 i+ (v i+ ; v i ) (u i+ ; u i )+ n y i+ n z i+ (v i+ ; v i ) 4 n z i+ + n x i+ + n z i+ (w i+ ; w i ) 5 i+ 6 (8) : (9) The TL approximation for the energy and the generic scalar equation becomes ^F 5 v i+ u i+ ^F + v v i+ i+ ^F + w v i+ i+ ^F 4 v i+ ; (k + k T ) T x n x i+ + T y n y i+ + T z n z i+ i+ ^F 6 v i+ u i+ ^F + v v i+ i+ c p ; + Pr T Prdi+ Pr T ^F + w v i+ i+ ^F 4 v i+ i+ (T i+ ; T i ) i+ x n x i+ + y n y i+ + z n z i+ ( i+ ; i ) i+ i+ i+ i+ The momentum uxes can also be written in a compact form ^F i+ v i+ i+ (u i+ ; u i )+ n x i+ (u i+ ; u i ) ^F (v v i+ i+ ; v i )+ n y i+ (u i+ ; u i ) ^F 4 v i+ i+ i+ (w i+ ; w i )+ n z i+ (u i+ ; u i ) (0) () where u i and u i+ are the scaled contravariant velocities in the direction of the surface ~ i+ u i n x i+ u i + n y i+ v i + n z i+ w i ()
7 Fig. : Grid of the at plate boundary ow. In the implicit phase the equations are linearized in a simplied manner. The eect of the viscous terms T is added on the Jacobian matrices A FU, B GU and C HU as A R A ( A + TI)R ; A () B R B ( B + TI)R ; B C R C ( C + TI)R ; C : Here R is the transformation matrix from conservative variables U to characteristic variables W, W R ; U. are the diagonal matrices containing the positive and the negative eigenvalues of the Jacobians. The diagonal weight factor T is computed as T ( + T ) (4) d where d is the lenght of the cell in the direction of the computational sweep. Test cases. Flat Plate The case is a ow over at plate with a high free-stream turbulent intensity.inlet velocity was 9.4 m/s and the pressure gradient was zero. Measurements have been made down to x.495 m that corresponds to Re x 940000. Upstream turbulence intensity is Tu 6.0%.Dissipation is set so that decay of free stream is in balance with production. The calculation is started 6 cm before the plate. The lenght of the at plate is.6 m. The height is0cmand the height of the rst row of cells was :5 0 ;5 [4] that is equal to y + 0:7 at most of the domain (at the leading edge y + :). Grid is clustered near the wall though the nearest three rows are kept constant. The ratio between neighbouring cells is y n+ y n :5. The grid size is 96 64 and the grid is seen in gure.
8 u w Fig. : Grid of the lid driven cavity ow.. Lid-driven cavity ow The second test case is an isothermal cavity ow -problem. The grid size for the problem is 80 80 and the grid is seen in gure. The top wall is moving with velocity 0.00608 m/s and the resulting Reynolds number is 400. 4 Results 4. Flow over the Flat Plate The second-order upwind discretization with Roe's splitting was used for inviscid uxes. The case was scaled so that the free stream velocity U e was increased from 9:4ms to 94ms to ensure convergence. Computation with full uxes was done with four multigrid levels whereas the TL computation converged only with three levels maximum. These cases were iterated excessive 0000 iterations at the nest grid level, which wasduetopoorconvergence of some properties in the TL computation shown in Fig. (4). The computation of the full viscous uxes took about 5% more CPU -time/cycle and the memory need was.09 MB for the full terms and 9.08 MB for the TL approximation (9.47 MB with the same four levels). The results were nearly identical from both calculations. kin friction c f w U e, where w is a shear stress at the wall, is presented in Fig. 5. The velocity proles are compared in Fig. 6 in terms q of u +, which is a dimensionless velocity dened as u + uu, where u w is a friction velocity. Dimensionless distance
9 a) b) Fig. 4: Convergence histories of drag coecients, total mass and turbulent kinetic energy with full viscous terms (a) and with TL approximation (b). Fig. 5: kin friction coecient.
0 from the q wall is approximated as y + y n u. Turbulence level is dened as Tu ku e, where k is the turbulent kinetic energy. The turbulence level proles are presented in Fig. 7 In this case a pseudo-compressibility method was used with the second-order upwind discretization. 0 000 iterations were computed. The u/v-velocity distributions were plotted at y/x -positions 0:065L, 0:5L, 0:5L, 0:75L and 0:975L respectively, where L stands for the lenght and the height of the cube. The plots are represented in Fig. 8. If u-velocity results are compared with the reference solution [5] at y 0:5L, a suspection arised that numerical dissipation has spoilt the solution. The computatation with the full viscous terms was continued with 5000 iterations with the third-order upwind biased uxes and these results are closer to the reference solution. In this case the full viscous terms give results closer to the reference solution than the TL approximation. 5 Discussion In the boundary layer ow there was practically no dierence between the results computed with the TL approximation and the full viscous terms. In this case the dominating viscous forces might be those in the direction of the plate surface and hence in direction of the grid line. In the cavity ow there was a small dierence between the two cases and the full viscous terms produced slightly better results. This might result from the nature of the case, at small Reynolds numbers the TL approximation is not valid. If the last 'contravariant' terms are neglected in equations (), then correct terms are obtained for any orthogonal grid with incompressible ow. It was rst thought that this 'compressible' term might not vanish as well as rv -term in the full viscous terms, but both terms were of the same order of magnitude. Also, the pseudo-compressibility method was suspected to cause numerical dissipation in the solution.
Fig. 6: Dimensionless velocity proles at various downstream positions.
Fig. 7: Turbulence intensities at various downstream positions.
Fig. 8: caled u/v -velocity proles at dierent x/y -positions.
4 References [] Hirsch C., Numerical Computation of Internal and External Flows, Volume : Fundamentals of Numerical Discretization, John Wiley & ons Ltd. [] Arfken G., Mathematical Methods for Physicists, Academic Press Inc., 985, an Diego. [] iikonen T., Computation of Viscous Fluxes in Curved Co-ordinates., memo M-0-88, Laboratory of Aerodynamics, Helsinki University of Technology (in Finnish). [4] Rautaheimo P., Test Calculation of Various Two-equation Low Reynolds Number Turbulence Models,Memo CFD/TERMO-96, Laboratory of Applied Thermodynamics, Helsinki University of Technology. [5] Ghia, U. and Ghia, K.N. and hin, C.T.,High-Re olutions for Incompressible Flow Using the Navier-tokes Equations and a Multigrid Method, Journal of Computational Physics, 48, 98.