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Wang2001.pdf
Center for Turbulence Research
Annual Research Briefs 2001
45
Prediction of high Reynolds number flow over a
circular cylinder using LES with wall modeling
By Meng Wang, Pietro Catalano †,
AND
Gianluca Iaccarino
1. Motivation and objectives
The objective of this work is to assess the viability and accuracy of large-eddy simulation (LES) with wall modeling for high Reynolds number complex wall-bounded flows. It
is well known that the conventional LES is extremely expensive at high Reynolds numbers
due to the need to resolve the small but dynamically-important near-wall flow structures.
As a practical alternative, LES can be coupled with a wall model which models these
near-wall effects and provides the LES with a set of approximate boundary conditions,
often in the form of wall shear stress (Cabot & Moin 2000).
In recent years, wall models based on turbulent boundary layer (TBL) equations and
their simplified forms (Balaras, Benocci & Piomelli 1996; Cabot & Moin 2000) have
received much attention. These models, used with a Reynolds-averaged Navier-Stokes
(RANS) type of eddy viscosity, have shown promise for complex-flow predictions. For
instance, Wang & Moin (2001) employed this approach to simulate the flow past the
asymmetric trailing edge of an airfoil at chord Reynolds number of 2.15 × 106 , and
obtained very good agreement with solutions from the full LES (Wang & Moin 2000) at
a small fraction of the computational cost.
The flow around a circular cylinder represents a canonical problem for validating new
approaches in computational fluid dynamics. It is therefore reasonable or even necessary
to subject the hybrid LES/wall-modeling methodology to the same “grand challenge”.
To take the best advantage of wall modeling, we concentrate on the super-critical flow
regime in which the boundary layer on the cylinder becomes turbulent prior to separation.
This is, to our knowledge, the first such attempt using LES, although a related method
known as detached-eddy simulation (DES), in which the entire attached boundary layer is
modeled, has been tested in this type of flow (Travin et al. 1999). Breuer (2000) recently
conducted an LES study at a high sub-critical Reynolds number of ReD = 1.4 × 105 ,
and showed fairly good comparison with experimental data in the near wake. In the
present work, three simulations, at ReD = 5 × 105 , 1 × 106 , and 2 × 106 , have been
performed. Preliminary results and comparisons with experimental data are summarized
in this article.
2. Numerical method and procedure
The same LES code and wall model implementation as used by Wang & Moin (2001)
are used for the present calculations. The energy-conservative numerical scheme is of
hybrid finite-difference/spectral type, written for a C-mesh (Mittal & Moin 1997). The
time advancement is achieved by the fractional-step method, in combination with the
Crank-Nicolson method for viscous terms and third-order Runge-Kutta scheme for convective terms. A multi-grid iterative procedure is used to solve the Poisson equation for
† Centro Italiano Ricerche Aerospaziali (CIRA), Italy
46
M. Wang, P. Catalano & G. Iaccarino
pressure. The subgrid-scale stress (SGS) tensor is modeled using the dynamic SGS model
(Germano et al. 1991; Lilly 1992).
The computational domain has a spanwise size of 2D (D = cylinder diameter), over
which the flow is assumed periodic and 48 grid points are distributed uniformly. In the
planes perpendicular to the span, 401×120 grid points are used in the C-mesh, extending
approximately 22D upstream of the cylinder, 17D downstream of the cylinder, and 24D
into the far-field. Potential-flow solutions are imposed as boundary conditions in the farfield, and convective boundary conditions are used at the outflow boundary. Running at
a maximum CFL number of 1.5, the non-dimensional time step ∆tU∞ /D typically varies
between 0.0030 and 0.0045. To obtain the results presented here, the simulations have
advanced at least 150 dimensionless time units. The statistics are collected over the last
75 or so time units.
Approximate boundary conditions on the cylinder surface are imposed in terms of wall
shear stress estimated from a wall model of the form
∂
∂ui
1 ∂p
(ν + νt )
=
,
∂x2
∂x2
ρ ∂xi
i = 1, 3
(2.1)
This is a simpler variant of the TBL equation model (Cabot & Moin 2000) which allows
for easier implementation and lower computational cost. Although Wang & Moin (2001)
have shown that the full TBL equations (with dynamically adjusted νt ) give better results
in their trailing-edge flow, the discrepancy may be partly related to a surface curvature
discontinuity which is absent from the cylinder surface. Since the pressure is taken from
the LES at the edge of the wall layer, Eq. (2.1) can be integrated to the wall to obtain
an algebraic model for the wall shear stress components (Wang 1999)
ρ
1 ∂p δ ydy
uδi −
,
(2.2)
τwi = δ dy
ρ ∂xi 0 ν + νt
0 ν+νt
where uδi denotes the tangential velocity components from LES at the first off-wall
velocity nodes, at distance δ from the wall. In attached flows these nodes are generally
placed within the lower edge of the logarithmic layer. In the present flow, however, δ +
(in wall units) is found to vary from 0 to 100 depending on the local skin friction. The
2
+
+
1 − e−yw /A ,
eddy viscosity is modeled by a damped mixing-length model: νt /ν = κyw
+
= yw uτ /ν is the distance to the wall in wall units, κ = 0.4, and A = 19.
where yw
3. Results and discussion
In Fig. 1, the contours of the vorticity magnitude at a given time instant and spanwise plane are plotted for ReD = 106 . Large coherent structures are visible in the wake,
but they are not as well organized and periodic as in typical Kármán streets at lower
(sub-critical) and higher (post-critical) Reynolds numbers. Compared to flows at lower
Reynolds number (e.g. Kravchenko & Moin 2000; Breuer 2000), the boundary-layer separation is much delayed and the wake is narrower, resulting in a much smaller drag coefficient. Note that the rather thick layer seen along the cylinder surface consists mostly of
vorticity contours of small magnitude. These levels are necessary for visualizing the wake
structure, but are not representative of the boundary-layer thickness. The true boundary
layer, with strong vorticity, is extremely thin in the attached region.
A comparison with two sets of experimental data of the mean pressure distribution
High Reynolds number flow over a cylinder
47
2
1
0
-1
-2
-2
0
2
4
6
8
Figure 1. Instantaneous vorticity magnitude at a given spanwise cut for flow over a circular
cylinder at ReD = 106 . 25 contour levels from ωD/U∞ = 1 to ωD/U∞ = 575 (exponential
distribution) are plotted.
1
Cp
0
-1
-2
-3
0
50
θ
100
150
Figure 2. Mean pressure distribution on the circular cylinder.
Present LES at ReD = 106 ;
6
◦ Experiment of Warschauer & Leene (1971) at ReD = 1.26 × 10 (spanwise averaged); Experiment of Flachsbart (in Zdravkovich 1997) at ReD = 6.7 × 105 .
on the cylinder surface is depicted in Fig. 2. Very good agreement is observed between
the LES at ReD = 106 and the experiment of Warschauer & Leene (1971) which was
performed at ReD = 1.26×106 . The original Cp data of Warschauer & Leene exhibit some
spanwise variations; for the purpose of comparison the average value is plotted. Relative
to the measurements of Flachsbart (see Zdravkovich 1997) at ReD = 6.7 × 105 , the LES
Cp shows smaller values in the base region. Note that Flachsbart’s data contain a kink
near θ = 110o , indicating the presence of a separation bubble. This type of separation
bubble is characteristic of the critical regime, and is difficult to reproduce experimentally
or numerically due to sensitivity to disturbances.
In Table 1, we compare the mean drag coefficient, the base pressure coefficient, and
the Strouhal number from the LES at ReD = 106 with the experimental values. The
agreement with the measurements of Shih et al. (1993) is reasonably good. The LES
somewhat overpredicts the drag coefficient compared with Shih et al. (1993), but underpredicts it relative to Achenbach (1968) (cf. Fig. 3). The Strouhal number of 0.22 from
48
M. Wang, P. Catalano & G. Iaccarino
LES
Exp. (Shih et al. 1993)
Exp. (Others, see Zdravkovich 1997)
CD
−Cp,base
St
0.31
0.24
0.17-0.40
0.32
0.33
–
0.28
0.22
0.18-0.50
Table 1. Drag, base pressure coefficient and Strouhal number for the flow around a circular
cylinder at a Reynolds number of 106 .
Shih et al. is for a rough-surface cylinder; no coherent vortex shedding was observed
for smooth cylinders at ReD larger than 4 × 105 . Indeed, it is generally accepted that
periodic vortex shedding does not exist in the super-critical regime of flow over a smooth
cylinder (Zdravkovich 1997). From our simulation, a broad spectral peak of the unsteady
lift centered at St ≈ 0.28 is found. It can be argued that although the LES is performed
for a smooth cylinder, the discretization of the cylinder surface and the numerical errors due to under-resolution may act as equivalent surface roughness, causing the flow
field to acquire some rough-cylinder characteristics. The flow at high Reynolds number
is very sensitive to surface roughness and to the level of free-stream turbulence, which
contribute to the wide scatter of CD and St among various experiments in the literature
(Zdravkovich 1997), listed at the bottom of Table 1. Other factors causing the data scatter include wind-tunnel blockage and end-plate effects. Our simulation results fall easily
within the experimental range. Generally speaking, there is a lack of detailed experimental data at super-critical Reynolds numbers. In particular, velocity and Reynolds-stress
profile measurements are non-existent, making a more detailed comparison impossible.
To assess the robustness of the computational method, we have performed simulations
at ReD = 5 × 105 and 2 × 106 , in addition to the initial attempt at ReD = 1 × 106 .
The predicted mean drag coefficients are plotted in Fig. 3 along with the drag curve
of Achenbach (1968). While the simulations predict CD rather well at the two lower
Reynolds numbers, the discrepancy becomes large at ReD = 2 × 106 . More significantly,
the LES solutions show relative insensitivity to the Reynolds number, in contrast to the
experimental data which exhibit an increase in CD with Reynolds number after the drag
crisis. Similar Reynolds-number insensitivity has been observed for the other quantities
shown previously. Poor grid resolution, which becomes increasingly severe as the Reynolds
number increases, is the primary suspect.
Finally, the skin-friction coefficients predicted by the wall model in the LES calculations
are plotted in Fig. 4 against the experimental data of Achenbach (1968) at ReD = 3.6 ×
106 . The levels are very different on the front half of the cylinder, but are in reasonable
agreement on the back half. The boundary-layer separation and the recirculation region
are captured rather well by the LES, indicating that they are not strongly affected by
the upstream errors. The different Reynolds numbers in the LES and the experiment can
account for only a small fraction of the discrepancy. Note that our computed Cf values are
comparable to those reported by Travin et al. (2000) using DES. Travin et al. attribute
the overprediction of Cf before separation to the largely-laminar boundary layer in the
experiment, which has not been modeled adequately in either simulation. Grid resolution
is another potential culprit in the present work. In addition, an overprediction of the skin
High Reynolds number flow over a cylinder
49
2
1.6
CD
1.2
0.8
0.4
0
101
102
103
104
ReD
105
106
Figure 3. Drag coefficient as a function of Reynolds number.
• Present LES.
107
Achenbach (1968);
0.015
0.01
Cf
0.005
0
-0.005
-0.01
-0.015
0
60
120
180
θ
240
300
360
ReD = 5 × 105 ;
Figure 4. Skin friction distribution on the cylinder from LES:
6
6
ReD = 1 × 10 ;
ReD = 2 × 10 . ◦ Experiment of Achenbach (1968) at
ReD = 3.6 × 106 .
friction by the present wall model has also been observed by Wang & Moin (2001) in
the acceleration region of the trailing-edge flow, suggesting that this simplified model
may have difficulty with strong favorable pressure gradients. If this proves to be a major
factor, the more general TBL equation model should provide a better alternative.
4. Concluding remarks
A bold numerical experiment has been carried out to compute the flow around a circular cylinder at supercritical Reynolds numbers using LES. The simulation is made
possible by the use of a wall-layer model which alleviates the near-wall grid resolution
requirements. Preliminary results are promising in the sense that they correctly predict
the delayed boundary-layer separation and reduced drag coefficients consistent with measurements after the drag crisis. In quantitative terms, the mean pressure distributions
50
M. Wang, P. Catalano & G. Iaccarino
and overall drag coefficients are predicted reasonably well at ReD = 5 × 105 and 106 .
However, the computational solutions are inaccurate at higher Reynolds numbers, and
the Reynolds-number dependence of the drag coefficient is not captured.
It must be emphasized that the results presented here are very preliminary. The grid
used near the cylinder surface, particularly before separation, is quite coarse judged by
the need to resolve the outer boundary-layer scales. The effect of the wall model under
coarse grid resolution and in the laminar boundary layer is not clear. Evidently, a more
systematic investigation is needed to separate the grid resolution and wall modeling
effects, and to fully validate the numerical methodology in this challenging flow.
REFERENCES
Achenbach, E. 1968 Distribution of local pressure and skin friction around a circular
cylinder in cross-flow up to Re = 5 × 106 . J. Fluid Mech. 34, 625-639.
Balaras, E., Benocci, C. & Piomelli, U. 1996 Two-layer approximate boundary
conditions for large-eddy simulation. AIAA J. 34, 1111-1119.
Breuer, M. 2000 A challenging test case for large eddy simulation: high Reynolds
number circular cylinder flow. Int. J. Heat Fluid Flow 21, 648-654.
Cabot, W. & Moin, P. 2000 Approximate wall boundary conditions in the large-eddy
simulation of high Reynolds number flow. Flow Turb. Combust. 63, 269-291.
Germano, M., Piomelli, U., Moin, P. & Cabot, W. H. 1991 A dynamic subgridscale eddy viscosity model. Phys. Fluids A 3, 1760-1765.
Kravchenko, A.G. & Moin P. 2000 Numerical studies of flow over a circular cylinder
at ReD = 3900. Phys. Fluids 12, 403-417.
Lilly D. K. 1992 A proposed modification of the Germano subgrid scale closure method.
Phys. Fluids A 4, 633-635.
Mittal, R. & Moin, P. 1997 Suitability of upwind-biased finite difference schemes for
large-eddy simulation of turbulence flows. AIAA J. 35, 1415-1417.
Travin, A., Shur, M., Strelets, M. & Spalart, P. 1999 Detached-eddy simulations
past a circular cylinder. Flow Turb. Combust. 63, 269-291.
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