LES at high Re numbers: a near-wall eddy-viscosity formulation

Transcription

1 a high Re numbers: a near-wall eddy-viscosiy formulaion Georgi Kalizin, Jeremy A. Templeon & Gorazd Medic Mechanical ngineering Deparmen, Sanford Universiy, Sanford, CA 943, USA Absrac A near-wall eddy-viscosiy formulaion for is presened. A RANS-like eddy-viscosiy is imposed in he near-wall region and is correced wih he resolved urbulen sress. The RANS eddy-viscosiy is obained from a resolved channel flow a = 39 and sored in a look-up able. This approach is used wih boh no-slip boundary condiions and a wall model on coarse grids. Resuls are presened for channel flow a = 39 wih no-slip boundary condiions, and up o = 1, using a wall model. 1 Inroducion Large-eddy simulaion () is a echnique whereby small urbulen scales are removed and heir effec on he reained scales is modeled. The main goal of his echnique is o allow urbulen simulaions o be performed efficienly a high Reynolds numbers in complex geomeries. However, is currenly limied o moderae Reynolds numbers due o he need o resolve he small, energy conaining scales near he wall []. I has been esimaed ha he he number of scales ha mus be resolved if curren sub-grid scale (SGS) models are used goes as Re τ [1]; almos as expensive as direc numerical simulaion (DNS). Due o he significance of his problem, many soluions have been proposed. One is he area of wall modeling, where an approximae wall sress is provided o he in order o compensae for he unresolved physics in he near wall region. Recen reviews can be found in [3] and [11]. The difficuly wih wall models is ha hey mus compensae no only for unresolved physics, bu also for numerical and SGS modeling errors ha occur on coarse grids []. This problem has led oher researchers o propose he use of he Reynolds-averaged Navier-Sokes (RANS) equaions near he wall (see [13] and he references herein). One such mehod is deached-eddy simulaion (DS) [14], which was designed for he compuaion of massively separaed flows. In DS, RANS urbulence models are modified by adjusing heir lengh scales o swich from he wall-normal disance near a wall o an filer widh away from he wall [1]. When used in plane channel flow, DS simulaions developed an arificial buffer layer where he model swiched coefficiens, resuling in an overprediced mass flow rae (underprediced skin fricion) [1]. Combining wall models wih near-wall RANS reamens could have compuaional advanages, however few such aemps have been made. An imporan example of one, however, was performed in [3]. A sandard wall model was used o provide wall sresses o he, while he viscosiy from he dynamic model was replaced in he firs cell wih a value exrapolaed from he channel inerior. This improved he predicion of he mean velociy profile, however, beer undersanding of he physics is required. In his paper, a near-wall eddy-viscosiy formulaion for is presened. A RANS-like eddyviscosiy is imposed in he near-wall region and is correced wih he resolved urbulen sress. The 1

2 RANS eddy-viscosiy is obained from a resolved channel flow a = 39 and sored in a look-up able. This approach can be used wih eiher a no-slip boundary condiion or a wall model. Resuls are presened for channel flow a = 39 wih a wall-resolved grid, and up o = 1, using coarse grids. Near-wall reamen.1 Look-up ables Look-up ables were generaed wih a mehod similar o wha was used in [6] for RANS wall funcions for boh he wall shear sress, τ w = ρu τ, and eddy-viscosiy, ν. The ables are consruced using an averaged velociy profile obained from he resolved of channel flow a = 39, which is exrapolaed in he logarihmic region. The averaged velociy profile ( ) is ransformed o ( ) = (Re), where Re = yu/ν. The fricion velociy follows as u τ = u/. The look-up able for he eddy-viscosiy, ν + (y+ ), is obained from he non-dimensional RANS equaion for channel flow (1 + ν + )du+ /d = 1 /, where d /d is he gradien of he averaged velociy aken from he. When he ables are used only for values ha lie in he universal region, he eddy-viscosiy can be creaed wihou he las erm on he righ hand side.. Dynamic procedure for near-wall eddy-viscosiy correcion The basic idea consiss of comparing he averaged equaion for he sreamwise velociy componen in channel flow ( d ûˆv + (ν + ν les ) dû ) = d ˆp (1) dy dy dx wih he corresponding RANS equaion ( d (ν + ν rans ) du ) dy dy = d p dx where ˆ() denoes he filer and () a plane- and ime-average. If he same mean pressure gradien is imposed in boh equaions, hen () ûˆv + (ν + ν les ) dû dy = (ν + νrans ) du dy (3) Assuming ha ν les can be wrien as ν les and he wall-normal mean velociy gradien are independen, he subgrid sress dû d û. For an ideal and RANS, he velociy gradiens in equaion dy = νles dy (3) are equal: d û/dy = dū/dy. Wih hese assumpions, equaion (3) provides a relaion beween he and RANS eddy-viscosiy. ν les = ν rans + ûˆv / dū dy. (4) Since he urbulen sress, ûˆv, and he velociy gradien, dū/dy, have opposie signs, he second erm on he righ hand side of equaion (4) is negaive. Thus in he presence of urbulen flucuaions, he mean viscosiy is always less han he RANS viscosiy, wih he difference being a dynamic correcion for he resolved flucuaions. In our approach, equaion (4) is used o compue he insananeous eddy-viscosiy in he near-wall region. The velociy gradien is obained from he look-up able while he urbulen

3 sress comes from he. The averaging operaor used in equaion (4) can be plane- and/or ime-averaging. When using his approach, i is necessary o clip he eddy-viscosiy. As is sandard pracice when using he dynamic model, he eddy-viscosiy is clipped whenever is value drops below zero. An alernaive is o clip he eddy-viscosiy a he level of he dynamic model. 3 Numerical resuls This secion presens resuls for channel flows compued over a wide range of Reynolds numbers. All simulaions have been performed using a second-order cenered finie difference code [16] wih he dynamic Smagorinsky SGS model [4]. The sreamwise, wall-normal and spanwise dimensions of he channel are πh h πh. For a resolved a = 39, he grid is uniform in he sreamwise and spanwise direcions wih 18 and 96 cells, respecively. In he wall-normal direcion a hyperbolic angen disribuion is used wih 19 cells. A he wall, he firs cell cener is a y while in he channel cener he grid spacing is 1. This grid resoluion is abou wice as coarse in each direcion as he grid in [8] for a DNS using a pseudo-specral mehod. The firs es is o ensure ha he eddy-viscosiy from he look-up able, ν rans, provides an accurae RANS soluion for = 39 when used insead of ν les in he enire domain. When he compuaion is sared from he resolved, he flucuaions dissipae quickly and he soluion revers o a RANS soluion. Figure 1 compares he RANS soluion o he resolved. DNS, Moser (1999) ν rans in all cells 1 1 = Figure 1: Channel flow a = 39, vs. RANS The same grid has been used o evaluae he proposed correced eddy-viscosiy, ν corr, defined by equaion (4). ν corr and, for comparison, ν rans were applied near he wall up o various locaions ( = 3,69 and 11), above which he viscosiy from he dynamic model was used. The resuling eddy-viscosiy profiles are shown in Figure. Noe ha ν rans is an order of magniude greaer han ν corr. The corresponding mean and rms velociy profiles are shown in Figures 3 and 4. The resuls using ν corr agree significanly beer wih he resolved han hose using ν rans. The spanwise energy specra of he sreamwise velociy are shown in Figure. In he cener of he channel ( = 39), all resuls show similar behavior o he resolved. However, closer o he wall ( = 3), he resuls using ν rans wih he swich a = 11 have significanly more damping a 3

4 higher wave-numbers. The sreamwise specra of he sreamwise velociy a he corresponding locaions are presened in Figure 8. Flow srucures are visualized using he sreamwise voriciy, ω x. In Figure 6, he vorices compued wih ν corr used up o = 3 appear similar o he resolved. If he swiching occurs a = 11, fewer urbulen srucures are reained when ν rans is used, as shown in Figure 7. Larger srucures are visualized using a lower value of ω x and are presened in Figures 9 and 1. Compuaions have also been performed for = 39 on a coarse grid using cells wih he firs cell cener a y 1 + = 3. νcorr has been applied a up o hree wall-adjacen cells wih he corresponding = 3,69 and 11. As shown in Figure 11, he mean velociy is prediced more accuraely compared o he wall-model compuaion ha uses only he viscosiy from he dynamic model. A comparison o Figure reveals ha he eddy-viscosiy ν corr adjuss o he coarseness of he grid. xamples of he spanwise energy specra of he sreamwise velociy a = 3 and = 39 are presened in Figure 1. Flow srucures are presened in Figure 13. Noe ha he isosurfaces of sreamwise voriciy correspond o lower values han he ones presened for he fine grid. Addiional simulaions were performed for = 1,,4, 1, and 1, using grids consising of cells uniformly disribued in each direcion. The mean velociy profiles are shown in Figure 14, which summarizes he resuls wih he correced eddy-viscosiy mehod (wih ν corr applied in cell 1) for hese Reynolds numbers. A comparison of he proposed correced eddy-viscosiy mehod o previous work is presened in Figure 1 for = 4. The shifed wall model of Piomelli e al. [1] produces a resul similar o our compuaion wih he wall model only. The sub-opimal conrol based wall model resuls of Nicoud e al. [9] are comparable o our correced eddy-viscosiy mehod considering ha hey argeed a logarihmic profile wih κ =.41 insead of κ =.4. However, heir approach is significanly more compuaionally expensive and requires a priori knowledge of he mean velociy profile hroughou he channel. Figure 1 also compares he roo-mean square (rms) velociy profiles o he resuls of Kravchenko e al. [7] obained using a zonally embedded mesh wih B-splines and no-slip boundary condiions. The compuaions on he coarse grids are no able o capure he seep near-wall gradiens observed by [7]. Finally, a grid dependency sudy for = 4 is presened in Figure 16. Three grids were used: 1) cells wih y 1 + = 3; ) cells wih y+ 1 = 1; 3) cells wih y 1 + = 11. Consisenly, wih higher grid resoluion he resuls improve. 4 Conclusions A near-wall eddy-viscosiy formulaion for a high Reynolds numbers is proposed. A RANSlike eddy-viscosiy is imposed in he near-wall region and is correced wih he resolved urbulen sress. This approach assures ha flucuaions are presen in he near-wall region while he correc soluion for he mean velociy is recovered. The use of a wall model on coarse grids allows he mehod o be applied o channel flow over a wide range of Reynolds numbers. The resuls indicae ha he performance of his model is no Reynolds number dependen. In addiion, he proposed approach is compuaionally inexpensive, simple o implemen and is poenially applicable o complex flows, as shown for RANS in [6]. Acknowledgemens This research was sponsored by he DO hrough he ASC program a CITS, he AFOSR Gran # F and by G Aircraf ngines hrough he USA program. 4

5 References [1] J.S. Bagge, J. Jimenez, and A.G. Kravchenko. Resoluion requiremens in large-eddy simulaion of shear flows. CTR Annual Research Briefs, pages 1 66, [] W. Cabo. Wall models in large eddy simuaion of separaed flow. CTR Annual Research Briefs, pages 97 16, [3] W. Cabo and P. Moin. Approximae wall boundary condiions in he large-eddy simulaion of high Reynolds number flow. Flow, Turbulence and Combusion, 63:69 91,. [4] M. Germano, U. Piomelli, P. Moin, and W. Cabo. A dynamic subgrid-scale eddy-viscosiy model. Physics of Fluids, 3: , [] J. Jimenez and R.D. Moser. : where we are and wha we can expec. AIAA J., 38(4):6 61,. [6] G. Kalizin, G. Medic, G. Iaccarino, and P.A. Durbin. Near-wall behavior of RANS urbulence models and implicaions for wall funcions. J. Comp Phys., 4(1):6 91,. [7] A.G. Kravchenko, P. Moin, and R. Moser. Zonal embedded grids for numerical simulaions of wall-bounded urbulen flows. J. Comp. Phys., 17:41 43, [8] R.D. Moser, J. Kim, and N.N. Mansour. Direc numerical simulaion of urbulen channel flow up o = 9. Phys. Fluids, 11(4):943 94, [9] F. Nicoud, J.S. Bagge, P. Moin, and W. Cabo. wall-modeling based on subopimal conrol heory and linear sochasic esimaion. Physics of Fluids, 13(1): , 1. [1] N.V. Nikiin, F. Nicoud, B. Wasisho, K.D. Squires, and P.R. Spalar. An approach o wall modelling in large-eddy simulaions. Phys. Fluids. Leers, 1(7):169,. [11] U. Piomelli and. Balaras. Wall-layer models for large-eddy simulaions. Ann. Rev. Fluid Mech., 34: ,. [1] U. Piomelli, J. Ferziger, P. Moin, and J. Kim. New approximae boundary condiions for large eddy simulaions of wall bounded flows. Phys. Fluids A, (1): , [13] P. Sagau. Large ddy Simulaion for Incompressible Flows. Springer, Berlin,. [14] P.R. Spalar, W.-H. Jou, M. Sreles, and S.R. Allmaras. Commens on he feasibiliy of for wings, and on a hybrid RANS/ approach, pages 4 8. Firs AFOSR Inernaional Conference on DNS/, Ruson, LA, [1] M. Sreles. Deached eddy simulaion of massively separaed flows. AIAA Paper 1-879, 1. [16] J.A. Templeon. Wall Models for Large-ddy Simulaion based on Opimal Conrol Theory. PhD hesis, Sanford Universiy,.

6 4 swich a =3 swich a =69 swich a = ν + ν using ν rans Figure : Fine grid, = 39, eddy-viscosiy (ν les ) + using ν corr swich a =3 swich a =69 swich a =11 3 rms 1 1 w + rms 1 v + rms Figure 3: Fine grid, = 39, using RANS-like eddy-viscosiy ν rans swich a =3 swich a =69 swich a =11 3 rms 1 1 w + rms 1 v + rms Figure 4: Fine grid, = 39, using correced eddy-viscosiy ν corr 6

7 = 3 1 = DNS, Moser (1999) ν rans, swich a =3 ν rans, swich a =11 ν corr, swich a =3 ν corr, swich a = k z Figure : Fine grid, = 39, spanwise energy specra of sreamwise velociy a wo locaions k z Figure 6: Fine grid, = 39, sreamwise voriciy, isosurfaces of ω x = ±4 Figure 7: Fine grid, = 39, sreamwise voriciy, isosurfaces of ω x = ±4 7

8 = 3 1 = DNS, Moser (1999) ν rans, swich a =3 ν rans, swich a =11 ν corr, swich a =3 ν corr, swich a = k x Figure 8: Fine grid, = 39, sreamwise energy specra of sreamwise velociy k x Figure 9: Fine grid, = 39, sreamwise voriciy, isosurfaces of ω x = ±1 Figure 1: Fine grid, = 39, sreamwise voriciy, isosurfaces of ω x = ±1 8

9 1 swich a =3 swich a =69 swich a =11 wall model only ν Figure 11: Coarse grid, = 39, using correced eddy-viscosiy ν corr 1 1 = 3 1 = DNS, Moser (1999) wall model only ν corr, swich a =3 ν corr, swich a = k z Figure 1: Coarse grid, = 39, spanwise energy specra of sreamwise velociy, using ν corr k z Figure 13: Coarse grid, = 39, sreamwise voriciy, using ν corr, isosurfaces of ω x = ± 9

10 3 3. log( ) , = 39 = 39 = 1 = = 4 = 1 = 1 Figure 14: Coarse grid, various, using correced eddy-viscosiy ν corr, = 39 ν corr in cell 1 Piomelli (1989) Nicoud (1) rms Kravchenko (1996) ν corr in cell 1 wall model only Nicoud (1) 1. log( ) w + rms 1. v + rms Figure 1: Coarse grid, = 4, comparison o previous work, = x 17 x 16 3 x 33 x 3 48 x 49 x log( ) Figure 16: Coarse grid, = 4, using correced eddy-viscosiy ν corr for various grids 1