LARGE EDDY SIMULATION OF SEPARATED FLOW IN A
STREAMWISE PERIODIC CHANNEL CONSTRICTION
Lionel Temmerman and Michael A. Leschziner
Department of Aeronautics
Imperial College of Science, Technology and Medicine
Prince Consort Rd.
London SW7 2BY, UK
ABSTRACT and time-averaged. At the high Reynolds num-
The ability of LES to resolve separation bers encountered in practical applications, the
from the leeward side of a curved "hill" in a pe- resource requirements of a wall-resolved sim-
riodically constricted channel is investigated. ulation quickly become prohibitive, and eco-
The emphasis is on the e ectiveness of di er- nomical compromises must be sought.
ent combinations of subgrid-scale models and One such compromise is to bridge the semi-
wall-functions used on relatively coarse grids. viscous near-wall layer by a "wall-function",
Accuracy is judged by reference to a highly- based on an assumed shape for the instanta-
resolved simulation on a 4:6 106 nodes grid, neous velocity between the nodes closest to
allowing Reynolds-stress budgets, realisability the wall and the wall itself. Variants in-
and structural features to be analysed. It is clude a power-law pro le (Werner and Wen-
demonstrated that even gross- ow parameters, gle, 1991) and the log-law pro le (Grotzbach,
such as separation-bubble length, are very sen- 1987). Both are designed to return an es-
sitive to modelling approximations and grid timate of the instantaneous wall shear stress
quality. for a given velocity at the wall-nearest node,
which then serves as a wall boundary condi-
tion for the outer LES domain. While wall-
INTRODUCTION functions have been used extensively in ows
LES is highly e ective for resolving ows which are largely una ected by boundary-layer
which are dominated by free shear layers sep- details, little is known about their e ectiveness
arating at sharp edges and governed, predom- in attached as well as separated ows in which
inantly, by large-scale structures. In contrast, near-wall processes are expected to be highly
ows for which the gross ow features are in uential. Other, more elaborate approaches
materially a ected by viscous near-wall pro- rest on matching RANS-type near-wall mod-
cesses pose a major challenge to LES, espe- els, based on conventional statistical closures,
cially at high Reynolds numbers. Near a wall, to the outer LES domain. Variants include the
LES is required, in e ect, to approach DNS, two-layer approach of Balaras et al (1996), re-
as the dynamically important scales dimin- cently investigated by Cabot and Moin (1999)
ish rapidly towards the dissipative ones, and for a separated backward-facing-step ow, and
turbulence approaches a two-component state. the "Detached Eddy Simulation" strategy of
This behaviour has important implications in Spalart et al (1997).
respect of grid density and quality, resource- This paper investigates the e ectiveness of
requirements and near-wall representation of di erent combinations of SGS models and
subgrid-scale transport and dissipation. wall-functions in simulating separation from
The above issues are especially pertinent a curved surface. The geometry, shown in
to ows in which separation is provoked on Fig. 1, is a streamwise-periodic, spanwise-
a gently curved or sloping wall by the sus- homogeneous channel segment with one wall
tained action of an adverse pressure gradient containing hill-shaped constrictions, 9 hill
on the decelerating boundary layer - for exam- heights apart. The Reynolds number is 10595.
ple, on a highly-loaded aerofoil or blade. The This is a modi cation of an experimental con-
structure of such a boundary layer will in u- guration examined by Almeida et al (1993),
ence the separation line, both instantaneous motivated by a combination of cost consid-
erations and the observation that the experi- u+ = A ln(y+) + B if 5 < y+ 30 (4)
mental ow was not fully periodic, that span-
wise con nement provokes spanwise variations, u+ = 1 ln(Ey+) if y+ > 30 (5)
and that the short distance between consec-
utive hills did not permit a signi cant post- with A = ( 1 ln(30E ) 5)=ln(6) and B = 5
reattachment recovery region to be established A ln(5).
prior to the acceleration of the ow by the fol- In the above near-wall approximations, the
lowing hill. The ow o ers the important ad- velocity scale in y + is formed with the wall
vantage of not requiring boundary conditions, shear stress, restricting their validity, in a sta-
except for those at the two walls. The assess- tistical sense, to the state of turbulence-energy
ment of alternative wall-function implementa- equilibrium. In RANS computations, the ap-
tions is based on data derived from two highly- plicability of the log-law may be extended con-
resolved simulations, using the same grid of siderably by using the turbulent energy to
4:6 106 nodes, in which the SGS viscosity is scale y , a substitution based on the equivalence
of order of the uid viscosity. The two simu- u 2 = C 0:5k. This concept may also be ap-
lations were performed with two entirely dif- plied to LES ( Murakami et al, 1993), with k
ferent codes, one by Mellen et al (2000) and chosen to be the resolved turbulent energy at
the other by the authors. Here, the simulation the wall-nearest node. Thus, the universal wall
data are used, in e ect, in lieu of experimental distance becomes:
data to investigate sensitivity to SGS mod- 1
elling, grid density and approximate near-wall
y+ = yC 4 k 1
Otherwise, the wall-law (LLK) is identical to
the rst form, eqs. (1) and (2).
The fourth variant (WW) is a simpli ca-
tion of the two-layer log-law form proposed
by Werner and Wengle (1991). This is based
on an explicit power-law approximation to the
log-law outside the viscous sublayer, interfaced
with the linear pro le in the viscous sublayer:
u+ = y+ if y+ 11:8 (7)
Figure 1: Constricted channel geometrywith instantanaeous u+ = 8:3 (y+) 1 if y+ > 11:8
NEAR-WALL TREATMENT SGS models considered here are all based on
Four di erent formulations have been inves- the eddy-viscosity concept,
tigated, three utilising di erent forms of the
log-law and the fourth a power-law approxi- ij
mation. The simplest (LL2) is based on the ij ; 3 kk = ;2 t S ij (9)
assumption that the near-wall layer consists Simulations have been performed with the
(in an instantaneous sense) of a fully viscous SGS models given in Table 1. A detailed expo-
sublayer and a fully turbulent layer with the sition of the above models is not possible here,
interface at y + 11: but a few clarifying comments are given below.
u+ = y+ if y+ 11 (1) The basic Smagorisky model (SMAG) is
used in conjunction with the van Driest damp-
u+ = 1 ln(Ey+) if y+ > 11 (2) ing functions (WF1, WF2) which ensure that
the SGS viscosity vanishes as the wall is ap-
A three-layer generalisation (LL3) of the proached. The di erence between variants
above form accounts for the bu er region which WF1 and WF2 lies in the value of A+ .
is described by a modi ed log-function t The dynamic Smagorinsky model (DYN) is
(Dahlstrom and Davidson, 2000) as follows: that proposed by Germano et al (1991) and
modi ed by Lilly (1992). Test- ltering is per-
u+ = y+ if y+ 11 (3) formed in the streamwise-spanwise grid planes.
Model Designation Model Description Temmerman et al (2000).
SMAG Smagorinsky (Cs = 0:1)
SMAG + WF1 Smagorinsky (Cs = 0:1)
wall-damping function (A+ = 5) CHANNEL-FLOW SIMULATIONS
SMAG + WF2 Smagorinsky (Cs = 0:1)
wall-damping function (A+ = 25) As a precurser to the main study of sep-
MSM Mixed Scale Model arated ow, simulations were undertaken for
DYN Dynamic Smagorinsky model fully-developed channel ow, to investigate the
WALE WALE Model (Cw = 0:1)
LDYN Localised dynamic performance of alternative SGS models and ex-
Smagorinsky model amine the ability of the wall laws to return the
log-law behaviour with coarse near-wall grids
Table 1: Summary of SGS models used. for a Reynolds number that is comparable to
Stability is enhanced by spatial averaging over that of the hill ow. Simulations were per-
any statistically homogeneous direction and by formed for the case Re = 590 for which DNS
constraining the SGS viscosity to remain posi- data by Moser et al (1999) are available for
tive. comparison. Most wall-layer-resolving simula-
The localized dynamic Smagorinsky model tions were done on a 96 64 64 grid, covering
(LDYN) by Piomelli and Liu (1995) is a variant a box of 2 h 2h h and having cell-aspect
of the previous model, wherein the dynamic ratios ( x+ y + x+ ) = (38 2 42 29), the ;
coe cient is allowed to vary during the test- lowest y + = 2 being that at the wall. Statis-
ltering operation. This requires the use of a tical properties were assembled over a period
iterative solver to nd the dynamic coe cient of 12 ow-through times.
at each time-step. Alternatively, the value of <νt>/ν
this coe cient at the previous time-step can
be used as an approximation.
The mixed-scale model (Sagaut, 1996) is
based on a weighted geometric average of SGS
-3 SMA + wf1
viscosities in which the velocity scale is related,
SMA + wf2
respectively, to Sij (as done in the Smagorin- -5
sky model) and to the SGS turbulence energy 10
1 10 +
y 100 1000
k, obtained by the application of a test lter Figure 2: SGS viscosity for channel ow, wall-resolved sim-
analogous to that used for dynamic modelling. ulation.
Finally, the WALE model (Nicoud and In wall-layer-resolving simulations, the SGS
Ducros, 1999) is speci cally designed to re- viscosity level in the upper portion of the
turn the correct wall asymptotic y 3 variation bu er layer and its asymptotic near-wall be-
of the SGS viscosity. It does so by a particu- haviour were observed to be particularly in u-
lar manipulation of the strain tensors and its ential. Variations of this viscosity are shown
components within an expression of similar in in Fig. 2. The only models found to return
structure to the Smagorinsky model. the theoretical y +3 decay reasonably well are
the WALE and the dynamic formulations, al-
COMPUTATIONAL PROCEDURE though the latter gives substantially higher vis-
The LES equations are solved using a cosity values away from the wall. Table 2 com-
second-order fractional-step method with a pares errors in centre-line velocity and wall-
multiblock/multigrid non-orthogonal nite- shear Reynolds numbers, while Fig. 3 shows
volume approach with the variables stored at the velocity pro les, in wall co-ordinates, and
cell centroids. The time derivative is ap- turbulence-intensity pro les obtained with the
proximated by a second-order backward Euler WALE model, judged to give the best overall
scheme. Convection and di usion are approx- behaviour.
imated by second-order centred interpolation 30 0.2
DNS - u’rms/Ub
and are advanced in time using the Adams- 25
WALE - u’rms/Ub
DNS - v’rms/Ub
Bashforth scheme. Pressure is obtained by
DNS WALE - v’rms/Ub
DNS - w’rms/Ub
solving the pressure-Poisson equation using
+ 15 0.1
u WALE - w’rms/Ub
partial diagonalisation in conjunction with a
V-cycle multigrid scheme and LSOR relax- 0 0
ation. The code is fully parallelised. Details
1 100 1000 0 0.2 0.4 0.6 0.8 1
of the parallel implementation on various ar- Figure 3: Velocity and turbulence intensities for channel
chitectures and its e ciency can be found in ow, wall-resolved simulation.
SGS Model Re Error uc =Ub Error number is 10595. The mesh is close to orthog-
DNS 584 - 1.1418 - onal, of low aspect ratio over most of the ow
SMAG 617 +5.6 % 1.1533 +1.00 %
MSM 578 -1.0 % 1.1424 +0.05 % domain and mesh expansion ratio below 1:05.
SMAG & wf1 595 +1.8 % 1.1444 +0.23 % The y + -value at the nodes closest to the lower
SMAG & wf2 538
-1.05 % wall is around 0:5. SGS e ects are represented
WALE 542 -7.1 % 1.1440 +0.20 % by the WALE model, giving viscosity values
LDYN 520 -10.9 % 1.1221 -1.73 % below that of the uid viscosity over most of
the ow domain. A similar simulation was also
Table 2: Wall shear stress and centreline velocity for channel performed by Mellen et al (2000), and this gave
ow, wall-resolved simulation.
SGS Model Re Error uc =Ub Error
results in close agreement to the present ones.
DNS 584 - 1.1418 - A few statistical results, derived by integra-
WALE + LL2 558.5 -4.4 % 1.12 -1.91 % tion over 55 through- ow periods, are shown
WALE + LL3 557.8 -4.5 % 1.118 -2.08 %
WALE + LLK 537.6 -7.9 % 1.13 -1.03 %
in Figs. 5-8.
WALE + WW 598.4 2.5 % 1.133 -0.08 %
Table 3: Wall shear stress and centreline velocity for channel
ow, wall-function simulation.
Simulations were then performed with the
four wall-laws and the WALE model for Re =
590 over a deliberately coarser grid of 64 Figure 5: Streamlines for the highly-resolved simulation.
32 32 nodes giving cell-aspect ratios ratios Fig. 5 gives the the time-averaged stream-
( x+ y + x+ ) (58 37 58). Table 3 com- function eld. The ow separates at x = 0:22h
pares errors in centre-line velocity and wall- and reattaches at x = 4:72h. Velocity and
shear Reynolds numbers, while Fig. 4 gives ve- normal-stress pro les are included in compar-
locity and turbulence-intensity pro les for the isons to follow in the next section. Adherence
four wall-law formulations. The results illus- to realisability constraints is demonstrated in
trate that, despite the evidently serious resolu- Fig. 6 which gives 3 cross- ow pro les relating
tion limitations which arise when coarse grids the second and third stress invariants on Lum-
are used, the simulations are able to resolve the ley's realisability map. The simulation was
essential features of the statistical elds. also used to extract stress budgets, currently
employed to examine second-moment closures.
Fig. 7 gives the turbulence-energy budget at
WALE + LL2
the location x = 2h, midway along the sepa-
20 20 WALE + WW
WALE + LL3
+ 15 WALE + LLK + 15
ration bubble. Dissipation was obtained from
the balance of other processes. The behaviour
across the shear-layer is observed to be qualita-
tively close to that reported by Le et al (1997)
in their DNS of a backward-facing-step ow,
while the near-wall behaviour is quite similar
to that in an ordinary channel ow (Mansour
WALE + WW
WALE + LL3 0.04
WALE + LL2
et al, 1988).
WALE + LLK DNS
WALE + WW
WALE + LL3
0.02 WALE + LL2
WALE + LLK
0.2 Axisymmetric contraction
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1 2D Turbulence
y/h y/h x/h = 0.05
0.15 x/h = 2.0
x/h = 8.0
Figure 4: Velocity and turbulence intensities for channel
ow, wall-function simulations. -II 0.1
HIGHLY-RESOLVED SIMULATION OF
SEPARATED CHANNEL FLOW
-0.01 0 0.01 0.02
To enable the accuracy of coarse-grid/wall- Figure 6: Realisability map for 3 cross- ow traverses.
functions simulations to be assessed for the The simulation allowed a-priory testing of
separated ow shown in Fig. 1, a highly- the wall-functions to be undertaken. Fig. 8
resolved reference simulation was performed gives an example in which the Werner-Wengle
over a grid of 4:6 106 (196 128 186) cells. approximation was used to extract the instan-
The channel is 9h long, 3:036h high and 4:5h taneous wall-shear stress from the velocity re-
deep, h being the hill height. The Reynolds solved at di erent distances (grid lines) from
x/h = 2.0 Balance
Viscous diffusion Grid SGS & ( h )sep:
( x )reat:
Nx Ny Nz Wall Model
196 128 186 WALE + NS 0.22 4.72
176 64 92 WALE + NS 0.38 3.45
Turbulence energy ( * 2/3)
176 64 92 WALE + WW 0.32 4.56
176 64 92 WALE + LL3 0.34 4.32
112 64 92 WALE + NS 1.12 2.17
112 64 92 WALE + WW 0.46 4.00
112 64 92 WALE + LL2 0.54 2.95
112 64 92 WALE + LL3 0.53 2.98
112 64 92 WALE + LLK 0.49 3.38
x/h = 2.0 112 64 92 SMA & WF2 0.50 3.59
Viscous diffusion + WW
112 64 92 MSM + WW 0.45 4.18
112 64 92 DYN + WW 0.46 3.56
0.02 Pressure diffusion
112 64 92 LDYN + WW 0.47 3.56
Turbulence energy ( * 2/3)
Table 4: Separation and reattachement locations for
SGS modelling and wall-law approximations -
to permit an appreciation of the relative im-
0 0.1 0.2 0.3 0.4 0.5
portance of the three issues in relation to pre-
Figure 7: Turbulent-energy budget at x=h = 2:0 left: across
the shear layer right: zoom on the lower wall region, highly-
resolved simulation. Table 4 provides a comparison of pre-
the wall in the mid-span plane. This test dicted separation and reattachment positions
illustrates the smoothing e ect of the wall- obtained with di erent SGS models, wall-
function treatment and its tendency to seri- functions and grid densities (NS indicates "no-
ously underestimate the peak wall-shear stress. slip" conditions). As seen, both positions are
These defects re ect , in parts, the fact that materialy sensitive to all three parameters. Es-
the near-wall velocity pro les, shown in Fig. 9 pecially poor results are obtained when the
at 3 streamwise locations, do not adher, ei- streamwise grid density is low in the vicinity of
ther statistically or instantaneously, to the the separation point (112 64 92 grid) and
velocity-pro le assumptions underpinning the when the no-slip condition is applied. Reason-
wall laws. ably good results are returned by the combina-
tion of WALE model and the Werner-Wengle
or 3-layer wall law approximation, provided
Distance to the wall = 0.0151 h adequate resolution around the separation lo-
Distance to the wall = 0.0511 h
Distance to the wall = 0.0854 h
0.01 x/h = 2.0 x/h = 6.0
0 LL Dense grid
2 LLK 2 LL3
0 2 4 6 8 WW LLK
x/h y/h NS y/h WW
Figure 8: Instantaneous wall-shear stress derived from the
Werner-Wengle wall-treatment at 3 grid lines progressively
removed from the wall.
-0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Figure 10: Distribution of streamwise velocity with the
log-law WALE model and 4 wall-treatments on the coarsest grid.
x/h = 0.5
Comparisons of streamwise velocity and
x/h = 2.0
x/h = 6.0
streamwise stress at two locations, one in
the recirculation zone and the other in the
post-reattachment recovery region, are given
in Figs. 10-12. The velocity pro les (Fig. 10
1 + 100
Figure 9: Near-wall velocity pro les at 3 streamwise loca- and 11) were obtained using the coarse and
tions derived from highly-resolved simulation. medium grids and the WALE model. Fig. 12
shows the streamwise stresses obtained with
the medium grid (176 64 92), the WALE
WALL-FUNCTION SIMULATIONS OF model and three wall-treatments.
SEPARATED CHANNEL FLOW The results reinforce the observation that
Coarse-grid simulations were performed substantial errors can arise especially from
along three parametric 'axes' - grid density, insu cient resolution around the separation
x/h = 2.0 x/h = 6.0
This work is part of the "LESFOIL" EU-
project (No. BRPR-CT97-0565), nanced
LL3 Dense grid
2 NS 2 LL3
through the Brite-Euram programme.
0 0.2 0.4
0.6 0.8 1 REFERENCES
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