The purpose of this study was  to explore  the central core  region  of a  plane  turbulent Cou-
ette flow  by means of large-eddy  simulations.  First  it  was  demonstrated  how  accurately  a 
low  Reynolds  number flow  could  be simulated.  After  having verified  the  reliability  of the 
LES  approach. simulations were  performed  at a  substantially  higher  Re.  It was  observed 
that  the  mean  velocity  exhibited  a  practically  linear  variation  in  the  core  region.  The 
extent  of the  core  increased  with  Re,  whereas  the  slope  of the mean  velocity  profile  was 
significantly  reduced

Andersson, H. I.

Kalmár-Nagy, Tamás

Kristoffersen, R.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

PERJODICA POLyTECHNIC .... SER. ME CH. ENG. VOL. 42, NO. I, PP. 47-56 (1998) LARGE-EDDY SIMULATION OF TURBULENT PLANE COUETTE FLOW H. L ANDERSSON*, R. KRISTOFFERSEN* and Tamas KALM . \R-NAGY** • Applied Mechanics Group The Norwegian University of Science and Technology N-7034 Trondheim, Norway "Department of Applied Mechanics Technical University. of Budapest H-1.521 Budapest, Hungary E-mail: nagykhan.tam.comell.edu Received: April 29, 1997 Abstract The purpose of this study was to explore the central core region of a plane turbulent Cou-ette flow by means of large-eddy simulations. First it was demonstrated how accurately a low Reynolds number flow could be simulated. After having verified the reliability of the LES approach. simulations were performed at a substantially higher Re. It was observed that the mean velocity exhibited a practically linear variation in the core region. The extent of the core increased with Re, whereas the slope of the mean velocity profile was significantly reduced. Keywords: large-eddy simulation. Couette-folw, turbulence. 1. Introduction The shear-driven motion of an incompressible fluid bounded by two parallel walls in relative motion, known as plane Couette flow, is still a problem of practical interest in contexts varying from lubrication theory to thin-film coating. This Bmv has therefore been studied experimentally for four decades. However, due to difficulties in establishing a reliable and fully developed flow field, the scatter in experimental data sets is disturbing. On the other hand, fully developed flows, in \vhich the streamlines of the mean field are parallel, are particularly amenable to exploration by means of direct large-eddy simulations. To this end a series of D:.'\S at Reynolds number Re = L~~h equal to 1300 was performed by BECH & .-\!\,DERSSO!\' [1]. Here, Ut(. is the velocity difference between the walls, which are separated at a distance '2h. An extensive comparison between the most convincing computer simulation and new laboratory measurements was provided by BECH et al. [3]. It was found that the near·\vall region of the turbulent Couette flow closely resembled those of pressure-driven Poiseuille Bow and turbulent boundary layers. The central core region, which typically occupied 50 per cent of the channel 48 H.I. ANDERSSON et al. cross section, exhibited a nearly uniform mean shear rate. Unfortunately. firm conclusions cannot be drawn from the DNS due to the relatively limited extent of the nearly uniform the central region (core). In Couette flow at a higher Re, however, the relative extent of the cen-tral core, as compared to the near-wall region, is greater, thereby enabling a more thorough exploration of this part of the flow. Since DN S is not feasible at higher Re, the large-eddy simulation approach is adopted to achieve this goal. First, however, a series of LESs have been performed at the lower Re = 1300 in order to demonstrate the reliability of LES. 2. Analytical Background Fig. 1 shows the configuration for plane Couette flow, where the incom-pressible fluid is bounded by two infinite plates 2h distance apart, and the upper plate moves with velocity Uw , while the lower one is kept fixed. The streamwise length of the computational domain is Lx, while the spanwise length is L y . ~ U ~ ~ ~ Y~ Lx Fig. 1. Plane Couette flow The dimensionless governing equations are V·u au _..L u· Vu at' = 0, -Vp + 2...c,u. Re ' 2h (1) (2) where u = (It, V, w) and Re is the Reynolds number, written m terms of the characteristic scales as Re LARGE-EDDY SIMUL.4TION OF TURBULENT PL.4NE COUETTE FLOW The first boundary condition is the no-slip condition at the walls u(x,y,-h) = 0, U(x,y,+h) = (UW,O,O). In the homogeneous directions \ve impose periodic boundary conditions u(O,y,z) = u(Lx,y,z), u(x,O,z) = u(x,Lx,z). 3. Computational Approach 49 (3) (4) (5) (6) The three-dimensional unsteady N avier-Stokes equations were solved nu-merically using the Explicit Channel Code for Large-Eddy Simulation de-veloped by GAVRILAKIS et al. [5J. ECCLES is a finite-difference code with second-order central-diffrence approximations in space and a second-order Adams-B asforth scheme in time. The primitive variables are positioned in a staggered grid arrangement. The computational domain was 47ih x 27ih in the streamwise (x) and spanwise (z) directions, respectively, while the height was 2h. The initial velocity field was constructed of the laminar mean velocity profile UjUw = (z +h)/2h with random velocity fluctuations superimposed. From this unphysical velocity field the program was running until the flow reached a statistically steady state. This statistically steady fully developed turbulent Couette flow was identified by its property that du --J1 dz - pu'w' = Twal/, (7) I.e. the sum of the viscous shear stress and the Reynolds shear stress is con-stant across the entire channel (this can easily be derived from the Reynolds averaged N avier-Stokes equations). Then the ensemble of statistically independent samples was gathered with further simulations. Different computational grids and three sub-grid scale (SGS) models were considered, i. e. those due to Smagorinsky, Schumann and IvIoin & Kim. The gridpoints were uniformly distributed in the x- and y-directions, while uniform or tanh distribution was used in the inhomogeneous z-direc-tion (the latter to improve the resolution in the near-wall regions). 50 H.I. .4.NDERSSON et al. 4. The Effect of Grid Resolution and Subgrid-Scale Closures Fig. 2 shows mean velocity profiles from direct simulations (i.e. without SGS model) at Reynolds number 1300 and 7050 (left arid right figures, respec-tively) . 000 l -0.20 32x32x32 TANH grid • 32x32x32 UNIFORM grid 0 32x16x16 TANH grid + 32x16x16 UNIFORM grid -0.40 .s:: -.... N -0.60 -0.80 + 0.00 0.10 0.20 0.30 0.40 0.50 <u>/u wall Fig. 2a The effect of grid resolution on the mean velocity profile In the central region the mean velocity gradient is fairly the same for all the used grids (but different for different Reynolds numbers), however, at the significantly higher Reynolds number Re = 7050, the simulated flow field was substantially affected by the grid resolution, especially in the near-wall region. At the lower Reynolds number the results from simulations without SGS model were quite similar to those obtained with models. LARGE-EDDY SUlULATION OF TURBULENT PLANE COUETTE FLOW 51 0.00 -0.20 64x32x64 T ANH grid 6, 32x32x32 T ANH grid • 32x32x32 UNIFORM grid -0040 0 32x16x16 T ANH grid .J:: + 32x16x16 UNIFORM grid ~ N J+ -0.60 • F J+ -0.80 0.00 0.10 0.20 0.30 0040 0.50 Fig. 2b The effect of grid resolution on the mean velocity profile It was shown by ::VIIYAKE et al. [7] that the Smagorinsky model is not adequate for modelling Couette flow, because of the high anisotropy of the flmv near the walls. This ,yas also confirmed during the simulations. The Schumann's model underpredicted the velocities at the higher Re. while lvIoin & Kim's model turned out be the most appropriate. From here on ,ye only consider the results from the simulations with 32 x 32 x 32 and 64 x 32 x 64 streched grids (for Re = 1300 and Re = 1"050, respectively). In Fig. 3 we compare the mean profile and intensities (root mean square value of the turbulent velocity in each direction) for Re = 1300 from our simulation with the one measured by AYDI!\ & LEUTHEUSSER [1] and the most reliable DNS available [3]. 52 0.00 -0.20 -DAD ..c. N -0.60 -0.80 0.00 0.10 6. • o H.!. ANDERSSON et al. Bech et al., 1995 Aydin & Leutheusser, 1987 No model Moin & Kim model 0.20 0.30 U/Uwall DAD 0.50 Fig. Sa Comparison of simulations and experiment (mean profile and intensities) Re = 1300 The agreement is good, however, the experimental velocity profile has a generally steeper profile. It is surprising that the profile of our simulation matches the curve from the fully resolved simulation of Bech (\vith 256 x 70 x 256 gridpoints) very well. This means that the resolved scales are responsible for most of the turbulent transport. So, the underlying idea of LES is thus confirmed by these results. The intensity of the velocity fluctuation is dominant in the streamwise direction. The model slightly overpredicts the streamwise fluctuation. The higher Reynolds number is sufficiently high for the unresolved scales to con-tain important contribution. The effect of these scales is accounted for by the SGS model. U. l,nns U .. 3.00 2.00 1.00 0.00 LARGE-EDDY SIMULATION OF TURBULENT PLANE COUETTE FLOW 53 0 0 0 • • 0 • 0 • • ---/. • ~ ... 0 ° .. 0 /0 ~ • / • 0 / • 0 0 • <> ~ .' $ -1.00 -0.80 0 0 • 0 • • 0 0 0 o • o • o • ------- ------a e e e ~ ___ ~_ _ _ e ___ ~ __ e ~ ~ t ~ ~ • • 0 • 0 • 0 0 Bech et al.,1995 No model Moin & Kim model u • u 0 u v • v 0 v w • w 0 w -0.60 -0.40 -0.20 0.00 z/h Fig. Bb Comparison of simulations and experiment (mean profile and intensities) Re = 1300 The mean profile and turbulence intensities for this higher Re case from 'direct' and model simulations are shown in Fig. 4. In their experiment ROBERTSON & JOHNSON [9J found a peak level of 11 of about '2.7, i.e. below the peak level obtained in the LES. However, scaling of experimental data with the friction velocity may cause errors. since this quantity is difficult to accurately determine experimentally. 5. Discussion REICHARDT [8J was the first to observe that the plane Couette fio\'; exhibited a core region \vith nearly linear variation of the mean velocity U and that the slop e S = }h dd~ of the mean ve 10 ci ty profile U ( z) decreased with Vll'all ... increasing Reynolds number. In the limit of infinitely high Re BUSSE [4J found a lower bound S =0.'25 for this slope. Rather surprisingly, the DNS 54 H.!. Ai\'DERSSON et .1. of BECH et al. [3] showed that S = 0.22 at Re = 1300, i.e. slightly below the theoretically established lo\ver limit. It is therefore noteworthy that the LES for Re = 1300 gives an S-value of 0.20, i.e. in good agreement \vith the directly simulated Couette flow. When the Reynolds number is increased from 1300 to 7050, the extent of the core region. defined as the range in which the mean velocity U does not deviate more than 0.01 Uw from the tangent line at z = 0, increased from 51 % to 64% of the channel \vidth 2h. Even more interesting is the observation that the dimensionless slope reduced to S = 0.17, i.e. some 30 per cent below the theoretically determined lower bound. 0.00 -0.20 • Robertson & Johnson. 1970 No model o Moin & Kim model -0.40 .c. N -0.60 -0.80 0.00 0.10 0.20 0.30 0.40 0.50 u/u wall Fig. 4a Comparison of simulations and experiment (mean profile and intensities) Re = 7050 ........................................... _-_ .. -LARGE-EDDY SDIULATION OF TURBULEl'iT PL4NE COUETTE FLOW 4.00 3.00 2.00 • • • • m • • • • • • I , ........... --------------I. ~~~·~~-.-~-.-O-.-.- .... -.-.-.-.-.-. 0 1.00 I ........ • • • • • • • 6 • •••• le ' ••• • , . I .- • No model Moin & Kim model Robertson & Johnson , . u u '. ,~ . 0.00 ~~.-·--.-----r_--_.----_r-'-v--,_--~--r_V-'--,_----r_--_.----~ -1.00 -0.80 -0.60 z/h -DAD -0.20 I 0.00 55 Fig. 4b Comparison of simulations and experiment (mean profile and intensities) Re = 7050 :\loreover, a significant part of the mean velocity profile in Fig. 4 ex-hibits a nearly increasing velocity [- in contrast \\'ith the inverse cosine variation deduced by VON K.A.R\I.A.:\ [6]. Eqllally interesting is the observation that the slope of the mean veloc-ity profile is about 20% below the theoretically established lower bound for the limit of infinite Re [4]. It is moreover note\yorth~' that the region with linearly increasing [.- is associated with a constant turbulent shear stress -pu'w' (sum of resoh'ed and SGS-modelled part). The local turbulence intensity at the centreline is much higher than the turbulence intensity at the centre line in plane Poiseuille flow. This is probably because of the fact that for plane Couette flow the turbulence production is not zero at the centreline. 56 H.!. Al\'DERSSON et al. 6. Conclusions This study has shown that the large-eddy simulation provides a useful ap-proach to plane Couette flow at intermediate Reynolds numbers which can-not be directly simulated due to the capacity of present days supercomput-ers. The reliability of the present LES seems to compare favourably with most laboratory investigations. The simulations confirmed the conjecture that the mean velocity gradient in the core region decreases with the in-creasing Reynolds number. 7. Acknowledgement T. Kalmar-Nagy would like to acknowledge the grant from the Norwegian University of Science and Technology for his stay in Trondheim and the computer time on a CRAY-YMP generously provided by SINTEF. References [1] AYDIN, E. M. - LEUTHEUSSER, H. J. (1987): Experimental Investigation of Turbu-lent Plane Couette Flow, ASME Forum on Turbulent Flows FED Vo!. .51, Cincin-natti. [2] BECH, K. H. - ANDERSSON, H. I. (1994): In Direct and Large-eddy Simulation I, p. 13. [3] BECH, K. H. - TILLMARK, N - ALFREDSSON, P. H. - ANDERSSOK, H. 1.(199.5): An Investigation of Turbulent Plane Couette Flow at Low Reynolds Numbers, J. Fluid Mech., Mar. 10, 199.5, pp. 286, 291-32.5. [4] BUSSE, F. H. (1970): J. Fluid Mech., Vo1.41, p. 219. [.5] GAVRILAKIS, S. -TSAI, H. M. - VOKE, P. R. - LESLIE. D. C. (1986): Large-eddy Simulation of a Low Reynolds l\umber Channel Flow by Spectral and Finite Difference Methods, Notes on Numerical Fluid Mechanics, Vo!. 1.5, pp. 10.5-118. [6] VON K4.RM.4.N, T. (1937): J. Aeron. Sci. 4, p. 131. [7] MIYAKE, Y. KAJISHI:VIA, T. OBANA, S.(1987): Direct Numerical Simulation of Plane Couette Flow at a Transitional Reynolds Number. JSME International Jour-nal, Vo!. 30(2.59) pp . .57-6.5. [8] REICHARDT, H.(19.56): Uber die Geschwindigkeitsverteilung in einer geradigen tur-bulenten Couettestromung, Zeit. Angew. Math. Mech. Vo!':36, pp. 26-29. [9] ROBERTSON, J. M. - .JoHNSON, H. F.(1970): Turbulence Structure in Plane Couette Flow. ASCE J. Eng. Mech. Div. Vo!. 96, pp. 1171-1182. 

LARGE-EDDY SIMULATION OF TURBULENT PLANE COUETTE FLOW

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LARGE-EDDY SIMULATION OF TURBULENT PLANE COUETTE FLOW

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Periodica Polytechnica (Budapest University of Technology and Economics)