International Journal of Heat and Fluid Flow 24 (2003) 463–469 www.elsevier.com/locate/ijhﬀ Numerical simulation of the ﬂow around a circular cylinder at high Reynolds numbers Pietro Catalano a,* , Meng Wang b, Gianluca Iaccarino b, Parviz Moin b a b CIRA––Italian Aerospace Research Center, 81043 Capua (CE), Italy Center for Turbulence Research, Stanford University/NASA Ames Research Center, Stanford, CA 94305-3030, USA Received 30 November 2002; accepted 23 March 2003 Abstract The viability and accuracy of large-eddy simulation (LES) with wall modeling for high Reynolds number complex turbulent ﬂows is investigated by considering the ﬂow around a circular cylinder in the supercritical regime. A simple wall stress model is employed to provide approximate boundary conditions to the LES. The results are compared with those obtained from steady and unsteady Reynolds-averaged Navier–Stokes (RANS) solutions and the available experimental data. The LES solutions are shown to be considerably more accurate than the RANS results. They capture correctly the delayed boundary layer separation and reduced drag coeﬃcients consistent with experimental measurements after the drag crisis. The mean pressure distribution is predicted reasonably well at ReD ¼ 5 105 and 106 . However, the Reynolds number dependence is not captured, and the solution becomes less accurate at increased Reynolds numbers. Ó 2003 Elsevier Science Inc. All rights reserved. Keywords: Large-eddy simulation; Wall modeling; Unsteady RANS; High Reynolds number ﬂows; Circular cylinder; Navier–Stokes equations 1. Introduction The severe grid-resolution requirement in the nearwall region has been the major roadblock to the use of large-eddy simulation (LES) for practical applications. This arises because of the presence of small but dynamically important eddies in the near-wall region. To resolve these vortical structures, the number of grid points required scales as the square of the friction Reynolds number (Baggett et al., 1997), which is nearly the same as for direct numerical simulation. As a practical remedy, LES can be combined with a wall-layer model. In this approach, LES is conducted on a relatively coarse mesh which scales with the outer ﬂow scale, thus making the computational cost only weakly dependent on the Reynolds number. The dynamic eﬀects of energy-containing eddies in the wall layer (viscous and buﬀer regions) are determined from a wall model * Corresponding author. Tel.: +39-0823-623244; fax: +39-0823-623028. E-mail address: [email protected] (P. Catalano). which provides to the outer LES a set of approximate boundary conditions, often in the form of wall shear stresses. In recent years, wall models based on turbulent boundary layer (TBL) equations and their simpliﬁed forms have received much attention. These models, used with a Reynolds-averaged Navier–Stokes (RANS) type eddy viscosity, have shown good promise. The approach has been successfully tested by Balaras et al. (1996) in a plane channel, square duct, and rotating channel, and by Cabot and Moin (2000) in a plane channel and backward-facing step. More recently, Wang and Moin (2002) employed this approach to simulate the ﬂow past the 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 and Moin, 2000) at a small fraction of the computational cost. The main objectives of the present work are to further assess the viability and accuracy of LES with wall modeling for high Reynolds number complex turbulent ﬂows and to compare this approach with RANS models. To this end, the ﬂow around a circular cylinder at Reynolds numbers (based on the cylinder diameter D) 0142-727X/03/$ - see front matter Ó 2003 Elsevier Science Inc. All rights reserved. doi:10.1016/S0142-727X(03)00061-4 464 P. Catalano et al. / Int. J. Heat and Fluid Flow 24 (2003) 463–469 o oui 1 op oui o þ ðm þ mt Þ ¼ þ ui uj ; ox2 ox2 q oxi ot oxj i ¼ 1; 3 ð1Þ where x1 , x2 , and x3 denote the streamwise, wall-normal, and spanwise directions, respectively, and u1 , u2 and u3 the corresponding velocity components. Eq. (1) has been implemented in a body-ﬁtted, locally orthogonal coordinate system as shown in Fig. 1. To make a distinction, the ﬁxed Cartesian coordinates (see also Fig. 1) and velocities are denoted by x, y, z and u, v, w. In the present work a simpliﬁed version of the wall model is employed, X1 The numerical method for LES and wall model implementation are the same as in Wang and Moin (2002). The energy-conservative scheme, written in a staggered grid system in body-ﬁtted coordinates, is of hybrid ﬁnite diﬀerence/spectral type (Mittal and Moin, 1997); it employs second-order central diﬀerences in streamwise and wall-normal directions and Fourier collocation in the spanwise direction. The fractional step approach, in combination with the Crank–Nicholson method for viscous terms and third order Runge–Kutta scheme for the convective terms, is used for time advancement. The continuity constraint is imposed at each Runge–Kutta substep by solving a Poisson equation for pressure using a multigrid iterative procedure. The subgrid scale (SGS) stress tensor is modeled by the dynamic Smagorinsky model (Germano et al., 1991) in combination with a leastsquare contraction and spanwise averaging (Lilly, 1992). Approximate boundary conditions are imposed on the cylinder surface in terms of wall shear stress components swi (i ¼ 1,3) estimated from a TBL equationbased wall model (Balaras et al., 1996; Cabot and Moin, 2000; Wang and Moin, 2002), which has the following general form in Cartesian coordinates: 2 2. Numerical method for LES y X of 0.5 106 , 1 106 and 2 106 is considered, and the results are compared to those from steady and unsteady RANS and the available experimental data. The ﬂow around a circular cylinder, with its complex features, represents a canonical problem for validating new approaches in computational ﬂuid dynamics. To take the best advantage of wall modeling, we have concentrated on the super-critical ﬂow regime in which the boundary layer becomes turbulent prior to separation. This is, to the authorsÕ knowledge, the ﬁrst such attempt using LES. A related method, known as detached eddy simulation (DES), in which the entire attached boundary layer is modeled, has been tested for this type of ﬂow by Travin et al. (1999). Recently an LES study has been conducted by Breuer (2000) at a high sub-critical Reynolds number of 1.4 105 , and a good comparison with the experimental data, especially in the near wake, has been shown. θ 0 D/2 x Fig. 1. Coordinate systems used. z and x3 axes are out of the plane. in which the substantial derivative (last two terms) is dropped from the right-hand side of Eq. (1). Since the pressure is assumed x2 -independent in the thin wall layer and is imposed from the outer ﬂow LES solution, one can then integrate Eq. (1) to the wall to obtain a closed form expression for the wall stress components (Wang and Moin, 2002) Z oui q 1 op d dx2 swi ¼ l ¼ u dx R di 2 d dx2 q oxi 0 m þ mt ox2 x2 ¼0 0 mþmt ð2Þ where udi denotes the tangential velocity components from LES at the ﬁrst oﬀ-wall node, at distance d from the wall. In attached ﬂows these nodes are generally placed within the lower edge of the logarithmic layer. In the present ﬂow, however, dþ (in wall units) is found to vary from 0 to 130 depending on the local skin friction. The eddy viscosity mt is obtained from a mixing length model with a near-wall damping mt þ 2 ¼ jywþ ð1 eyw =A Þ ð3Þ m where ywþ is the distance to the wall in wall units, j ¼ 0:4 is the von Karman constant, and A ¼ 19. Simulations are conducted employing a C-mesh in the planes perpendicular to the span. The computational domain extends approximately 22D upstream of the cylinder, 17D downstream of the cylinder, and 24D into the far-ﬁeld. In the streamwise direction 401 grid points, 144 of which on the cylinder surface and 2 129 points in the wake, are employed. In the wall-normal direction 120 points are used. The spanwise domain size is 2D, over which the ﬂow is assumed periodic, and 48 grid points are distributed uniformly. Note that the spanwise domain size is shorter than the typical values used for lower Reynolds number ﬂows (e.g. pD for ReD ¼ 3900 in Beaudan and Moin (1994) and Kravchenko and Moin (2000)). This is justiﬁed because of the reduced spanwise P. Catalano et al. / Int. J. Heat and Fluid Flow 24 (2003) 463–469 3300 465 4. Results and discussion 300 3000 250 2700 2400 ∆x+1 ∆x+3 1800 150 1500 1200 ∆x+2 200 2100 100 900 600 50 300 0 0 30 60 90 θ 120 150 0 180 þ þ Fig. 2. Sizes of the wall-adjacent cells: (- - -) Dxþ 1 ; (—) Dx2 ; (––) Dx3 . correlation length for higher Reynolds number ﬂows. The same spanwise domain size of 2D was used by Breuer (2000) at ReD ¼ 1:4 105 . Potential-ﬂow solutions are imposed as boundary conditions in the farﬁeld, and convective boundary conditions are used at the outﬂow boundary. The sizes of the wall-adjacent grid cells for LES are shown in Fig. 2 in wall units. They are extremely large, because the use of a wall model in principle bypasses the need to resolve the inner scales. In the staggered grid system used, the ﬁrst oﬀ-wall nodes for the tangential velocities are located at dþ ¼ Dxþ 2 =2, where the LES and wall-model velocities are required to match. Note that the wall units in Fig. 2 are aﬀected by the skin friction errors to be discussed later, and should thus be viewed as a crude estimate. To obtain the LES results presented here, the simulations have advanced more than 300 dimensionless time units. The statistics are collected over the last 200 time units. Running at a maximum CFL number of 1.5, the non-dimensional time step DtU1 =D typically varies between 0.0030 and 0.0045. Three Reynolds numbers, ReD ¼ 0:5 106 , 1 106 , and 2 106 , have been considered. The discussion will be mainly focused on the case of ReD ¼ 1 106 , with emphasis on important ﬂow parameters, such as the drag coeﬃcient, the base pressure coeﬃcient, the Strouhal number, and their dependence on the Reynolds number. The mean pressure distribution on the cylinder surface is compared to two set of experimental data in Fig. 3. A very good agreement is observed between the LES at ReD ¼ 1 106 and the experiment by Warschauer and Leene (1971) which was performed at ReD ¼ 1:2 106 . The original data of Warschauer and Leene (1971) exhibit some spanwise variations (see Zdravkovich, 1997), and for the purpose of comparison the average values are plotted. The unsteady RANS also provides a mean pressure coeﬃcient in satisfactory agreement with both LES and the experimental data, while, as expected, the steady RANS yields a poor result. Relative to the measurements of Falchsbart at ReD ¼ 6:7 105 , the numerical results show lower values in the base region. It is worth noting that FalchsbartÕs data contain a kink near h ¼ 110°, indicating the presence of a separation bubble. This type of separation bubble is characteristic of the critical regime, and is diﬃcult to reproduce 1 RANS simulations are carried out using a commercial CFD code, FLUENT. It is based on a second order ﬁnite volume discretization and the SIMPLE pressure correction technique for enforcing the divergence-free condition of the velocity ﬁeld; the time integration is three-level fully implicit. The eddy viscosity is obtained using the standard k–e model (Launder and Spalding, 1972) with wall functions. Although unstructured grids can be used in FLUENT, for the simulations presented here we employed the same C-mesh as used in the LES. Both steady and unsteady RANS calculations have been performed for comparison. Steady simulations are performed using only half of the computational domain ðy P 0Þ. Starting from this solution and its mirror image in the region y < 0, a disturbance was imposed to break the symmetry, and a time-accurate simulation was carried out. A non-dimensional time-step DtU1 =D of 0.01 was used and the simulation was run for 300 time units (D=U1 ). 0.5 0 -0.5 Cp 3. Numerical method for RANS -1 -1.5 -2 -2.5 -3 0 30 60 90 θ 120 150 180 Fig. 3. Mean pressure distribution on the cylinder: (—) LES at ReD ¼ 1 106 ; (- - -) RANS at ReD ¼ 1 106 ; (––) URANS at ReD ¼ 1 106 ; () experiment by Warschauer and Leene (1971) at ReD ¼ 1:2 106 (spanwise averaged); (M) experiment by Falchsbart (in Zdravkovich, 1997) at ReD ¼ 6:7 105 . 466 P. Catalano et al. / Int. J. Heat and Fluid Flow 24 (2003) 463–469 experimentally or numerically due to sensitivity to disturbances. The contours of the vorticity magnitude, as computed by LES and URANS for ReD ¼ 1 106 at a given time instant and spanwise plane, are plotted in Fig. 4. In the LES results, some coherent structures are visible in the wake, but they are not as well organized as in typical K arm an streets at sub-critical and post-critical Reynolds numbers. The rather thick layers along the cylinder surface consist 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 and mostly laminar in the attached ﬂow region. The shear layers are more coherent in the URANS than in the LES. A clear vortex-shedding pattern is exhibited in the URANS results. The mean streamwise velocity distribution (time and spanwise averaged) obtained by LES is presented in the lower half of Fig. 5. Compared to ﬂows at lower Reynolds numbers (Kravchenko and Moin, 2000; Breuer, 2000), the boundary layer separation is much delayed, and the wake is narrower, resulting in a smaller drag coeﬃcient. The time-averaged URANS velocity distribution is plotted in the upper half of Fig. 5, which shows a thicker wake, resulting in a higher drag coeﬃcient. A LES URANS Fig. 4. Instantaneous vorticity magnitude at a given spanwise cut for ﬂow over a circular cylinder at ReD ¼ 1 106 . 25 contour levels from xD=U1 ¼ 1 to xD=U1 ¼ 575 (exponential distribution) are plotted. Fig. 5. Mean streamwise velocity distribution predicted by LES and URANS. 45 contour levels from U=U1 ¼ 0:2 to U=U1 ¼ 1:7 are plotted. quantitative comparison between LES and URANS in terms of the mean streamwise and vertical velocity proﬁles in the cylinder wake is shown in Fig. 6. The upper and lower parts correspond to two streamwise locations inside and outside of the mean recirculation region, respectively. The LES predicts a wide range of ﬂow scales and hence more mixing of the ﬂow. The velocity deﬁcit from LES is larger at x=D ¼ 0:75, which is in the middle of the mean recirculation bubble as predicted by LES, but it is soon surpassed by the URANS prediction further downstream in the wake. Unfortunately, there is a general lack of detailed experimental data at super-critical Reynolds numbers. In particular, velocity and Reynolds-stress proﬁle measurements are non-existent, making an experimental validation of the wake proﬁles impossible. Another comparison between the LES and URANS is made in Fig. 7 in terms of lift and drag time histories. It is again clear that the URANS predicts a very well organized and periodic ﬂow at this Reynolds number, whereas the LES results have broadband turbulence characteristics. The overly dissipative nature of the URANS calculations is also evident by observing the small amplitudes of the lift and, especially, drag oscillations. The drag coeﬃcient, base pressure coeﬃcient, Strouhal number, and mean recirculation length for the ﬂow at Reynolds number of 1 106 are summarized in Table 1. The agreement with the measurements of Shih et al. (1993) is reasonably good. The LES overpredicts the drag coeﬃcient compared to Shih et al. (1993), but underpredicts the CD relative to Achenbach (1968) (cf. Fig. 9). The Strouhal number of 0.22 from Shih et al. (1993) is for a rough cylinder. It is generally accepted that periodic vortex shedding does not exist in the supercritical regime for smooth cylinders (Shih et al., 1993; Zdravkovich, 1997). From our simulations, a distinct spectral peak is observed at St 0:35, as shown clearly in the Evv spectra in Fig. 8. This ﬁgure depicts the frequency spectra of the streamwise and vertical velocities at x=D ¼ 0:70 and 1.50, y=D ¼ 0:15. It can be argued that the discretization of the cylinder surface and the numerical errors due to the under-resolution may act as equivalent surface roughness, causing the ﬂow ﬁeld to acquire some rough cylinder characteristics. The wide scatter of CD and St among various experiments in the literature (Zdravkovich, 1997), listed at the bottom of Table 1, suggests high sensitivity of the ﬂow to perturbations due to surface roughness and free-stream turbulence in the super-critical regime. Our simulation results fall easily within the experimental range. To assess the robustness of the computational method, LESs at ReD ¼ 5 105 and 2 106 have also been performed. The predicted mean drag coeﬃcients are plotted in Fig. 9 along with the drag curve of Achenbach (1968). The CD at the two lower Reynolds numbers is 1.2 0.3 0.8 0.15 V/U∞ U/U∞ P. Catalano et al. / Int. J. Heat and Fluid Flow 24 (2003) 463–469 0.4 0 467 0 -0.15 -0.4 -1 -0.5 0 0.5 1 -0.3 -2 -1.5 -1 -0.5 y/D 0 0.5 1 1.5 2 0.5 1 1.5 2 y/D 1.2 0.3 1 0.15 V/U∞ U/U∞ 0.8 0 0.6 -0.15 0.4 0.2 -1 -0.5 0 0.5 1 -0.3 -2 -1.5 -1 y/D -0.5 0 y/D Fig. 6. Mean streamwise and vertical velocities at x=D ¼ 0:75 (upper ﬁgures) and x=D ¼ 1:50 (lower ﬁgures): (—) LES; (––) URANS. 0.5 CD 0.4 0.3 0.2 150 2 00 2 00 t U∞ / D 250 3 00 0.6 CL 0.3 0 -0.3 -0.6 150 t U∞ / D 250 300 Fig. 7. Time histories of lift and drag coeﬃcients. (—) LES; (- - -) URANS. predicted rather well, but the discrepancy becomes large at ReD ¼ 2 106 . More signiﬁcantly the LES solutions show relative insensitivity to the Reynolds number, in contrast to the experimental data that exhibit an increase in CD after the drag crisis. Poor grid resolution, which becomes increasingly severe as the Reynolds 468 P. Catalano et al. / Int. J. Heat and Fluid Flow 24 (2003) 463–469 Table 1 Drag, base pressure coeﬃcient, Strouhal number, and recirculation length for the ﬂow around a circular cylinder at ReD ¼ 1 106 CD CPbase 0.31 0.39 0.40 0.24 0.17–0.40 0.32 0.33 0.41 0.33 – St 0.35 – 0.31 0.22 0.18–0.50 Lr =D 1.6 1.04 1.37 – – 1.2 CD LES RANS URANS Exp. (Shih et al., 1993) Exp. (Others, see Zdravkovich, 1997) 2 0.8 number increases, is the primary suspect. Similar Reynolds number insensitivity is shown by the URANS results. The skin friction coeﬃcients predicted by the wall model employed in the LES computations are presented in Fig. 10 together with the experimental data of Achenbach (1968) at ReD ¼ 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, indicating that they are not strongly aﬀected by the upstream errors. The diﬀerent Reynolds numbers between the LES and the experiments can account for only a small fraction of the discrepancy. Note that the computed Cf values are comparable to those reported by Travin et al. (1999) using DES at ReD ¼ 3 106 . Travin et al. (1999) attribute the overprediction of the Cf before the separation to the largely laminar boundary layer that has not been adequately modeled. This is also the case in the present simulation. Both experiments and numerical simulations suggest that even at these super-critical Reynolds numbers, the boundary layers remain laminar in most of the favorable pressure gradient region. In our simulations no eﬀort was made to trigger transition, nor was the wall model 0 4 10 E vv /U∞ D 10 10-4 10-4 10-5 10-6 10-6 f D /U∞ 10 7 -3 10-5 0 10 A bold numerical experiment has been performed to compute the ﬂow around a circular cylinder at super- -3 10 6 5. Concluding remarks 10-2 -1 10 modiﬁed for laminar ﬂow application. Grid resolution is another potential culprit in the present work. In addition, an overprediction of the skin friction by the wall model adopted in the present LES computations has also been observed by Wang and Moin (2002) in the acceleration region of the trailing-edge ﬂow, suggesting that this simpliﬁed model may have diﬃculty with strong favorable pressure gradients. 10-2 10 5 Fig. 9. Drag coeﬃcient as a function of the Reynolds number. (—) Achenbach (1968); (d) LES; (j) URANS. 10-1 10-7 10 ReD 10-1 10 Euu /U∞ D 0.4 1 10-7 10 -1 10 0 10 1 f D /U∞ Fig. 8. Frequency spectra of streamwise (left ﬁgure) and vertical (right ﬁgure) velocities at two wake stations: (—) x=D ¼ 0:70, y=D ¼ 0:15; (- - -) x=D ¼ 1:50, y=D ¼ 0:15. P. Catalano et al. / Int. J. Heat and Fluid Flow 24 (2003) 463–469 facilities at NASA Ames Research Center and on facilities at the Center for Turbulence Research. We would like to thank Professor Javier Jimenez for valuable discussions. 0.015 0.01 Cf 0.005 References 0 -0.005 -0.01 -0.015 469 0 60 120 180 θ 240 300 360 Fig. 10. Skin friction distribution on the cylinder: (- - -) LES at ReD ¼ 0:5 106 ; (—) LES at ReD ¼ 1 106 ; (––) LES at ReD ¼ 2 106 ; ( ) experiments by Achenbach (1968) at ReD ¼ 3:62 106 . critical Reynolds number using LES. The simulations have been made possible by the use of a wall model that 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. The mean pressure distributions and overall drag coeﬃcients are predicted reasonably well at ReD ¼ 0:5 106 and 1 106 . However the computational solutions are inaccurate at higher Reynolds numbers, and the Reynolds number dependence is not captured. It should be noted that the grid used near the 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 section of the boundary layer is not clear. A more systematic investigation is needed to separate the grid resolution and the wall-modeling eﬀects, and to fully validate the numerical methodology for this challenging ﬂow. Acknowledgements This work was supported in part by the US Air Force Oﬃce of Scientiﬁc Research Grant No. F49620-00-10111. Computations were carried out on the NAS Achenbach, E., 1968. Distribution of local pressure and skin friction around a circular cylinder in cross-ﬂow up to Re ¼ 5 106 . J. Fluid Mech. 34, 625–639. Baggett, J.S., Jimenez, J., Kravchenko, A.G., 1997. Resolution requirements in large eddy simulations of shear ﬂows. Annual Research Briefs––1997. 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Experiments on mean and ﬂuctuating pressures of circular cylinders at cross ﬂow at very high Reynolds numbers. In: Proc. Int. Conf. on Wind Eﬀects on Buildings and Structures, Tokyo, Japan (see also Zdravkovich 1997), pp. 305–315. Zdravkovich, M.M., 1997. Flow Around Circular Cylinders. Fundamentals, vol. 1. Oxford University Press, 1997 (Chapter 6).