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. 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