STATISTICAL CHARACTERISTICS OF UNSTEADY REYNOLDS-AVERAGED NAVIER STOKES SIMULATIONS

Numerical Heat Transfer, Part B, 46: 1 18, 2005 Copyright # Taylor & Francis Inc. ISSN: 1040-7790 print/1521-0626 online DOI: 10.1080/10407790490515792 STATISTICAL CHARACTERISTICS OF UNSTEADY REYNOLDS-AVERAGED NAVIER STOKES SIMULATIONS Inanc Senocak and Wei Shyy Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, Florida, USA Stein Tore Johansen SINTEF Materials Technology, Trondheim, Norway The statistical characteristics of unsteady Reynolds-averaged Navier Stokes (RANS) turbulence models, utilizing the eddy-viscosity concept, are investigated. Detailed assessment is made to the original Launder-Spalding k «model as well as a nonlinear modification utilizing multiple scales from both turbulent variables and the local mean strain rate. The purpose of the study is to shed light on the issues of unsteady RANS computations, not to recommend a good model. The models are tested for flow past a square cylinder, employing progressively refined resolutions. The comparison between the length scale imposed by the turbulence model and the length scale of the spatial resolution helps understand the fidelity of the simulations. If the eddy viscosity is low, the flow field can be numerically underresolved in RANS computations, and consequently the solutions exhibit noticeable sensitivity to numerics such as discretization schemes and grid distribution. Under such situations, in practical computations, grid-independent solutions will be difficult to attain with conventional numerical schemes used in RANS simulations. It is demonstrated that in order to appreciate the statistical behavior of a turbulence model, both time-averaged and time-dependent results need be examined. 1. INTRODUCTION There are alternative modeling strategies for the simulation of turbulent flows, providing different levels of flow description. In direct numerical simulation (DNS), all length and time scales of the flow are resolved. Since the computational cost increases close to the third power of the Reynolds (Re) number, DNS of high- Re-number flows is beyond the capabilities of present-day computational resources [1]. In large-eddy simulation (LES), main turbulent motions are resolved numerically and subscale motions are modeled. Depending on the flow problem at hand, the Received 1 December 2003; accepted 18 June 2004. This study was supported by the NASA University Research, Engineering and Technology Institute (URETI) program and the CARPET program of the Norwegian Research Council. The present address of Inanc Senocak is Center for Turbulence Research, Stanford University, Building 500, Stanford, California 94305-3035, USA. Address correspondence to Wei Shyy, University of Florida, 231 Aerospace Building, PO Box 116250, Gainesville, FL 32611, USA. E-mail: wss@mae.ufl.edu 1

2 I. SENOCAK ET AL. NOMENCLATURE A, B turbulence model constants C e1, C e2, C m turbulence model constants D characteristic length scale of the square cylinder f frequency of the vortex shedding k turbulent kinetic energy L R recirculation length P pressure P t turbulent production Re Reynolds number S strain rate strain rate tensor S ij St Strouhal number t time u velocity vector U free-stream velocity d ij Kronecker delta function D length scale of spatial resolution e turbulent dissipation rate m, m t laminar viscosity, eddy viscosity r density s k, s e turbulence model constants t ij ; t R ij stress tensor, Reynolds stress tensor computational cost of LES can also be demanding and unpractical for engineering applications. Another strategy is the Reynolds-averaged Navier Stokes (RANS) simulations, which is a statistical approach because the governing equations corresponding to the ensemble-averaged flow field are solved. There are also studies blending the RANS and LES methods, which are referred to as hybrid RANS=LES methods, aimed at reducing the cost of LES. In this approach, the boundary layers are treated with the RANS method, and the separated flows are calculated with LES. An updated discussion on various methods to model turbulence can be found in Spalart [2]. Flow past a square cylinder involves complex features such as self-induced vortex shedding in the wake, separated shear layers, and large streamline curvatures. Detailed experimental data are available in Lyn and Rodi [3] and Lyn et al. [4]. This case has been studied extensively in the literature to evaluate unsteady RANS turbulence models. For example, Bosch and Rodi [5] have done simulations of such a flow problem with different turbulence models. They have shown that the choice of inflow conditions and inflow locations has noticeable influences on the predictions. They have also shown that the Kato-Launder [6] modification of the k e model produces better results than the standard (i.e., the Launder-Spalding) k e model [7]. They have found that the reduced turbulent kinetic energy levels at the front face of the cylinder enhance the strength of the vortex shedding and improve the predictions. Rodi [8] has presented a comparison of LES and unsteady RANS calculations for flow past bluff bodies with a variety of turbulence models. The comparison with experimental data has shown that the predictions are not entirely satisfactory, even though the main flow features can be better captured with some of the models considered. Kenjeres and Hanjalic [9] have studied the Rayleigh-Benard convection at high Rayleigh numbers with the unsteady RANS approach. They have used an algebraic low-re-number k e y 2 stress=flux model and reproduced the coherent structures and large-scale unsteadiness in accordance with the DNS and experimental results. Iaccarino et al. [10] have examined the accuracy of RANS models in predicting flow past blunt bodies using the v 2 f model of Durbin [11]. With unsteady RANS computations, they have achieved substantially better agreement with experiment

UNSTEADY RANS SIMULATIONS 3 than a steady RANS computation. They have also shown that for flow past a cube on a surface, a three-dimensional RANS simulation is needed to capture the dominant flow structures. Nichols and Nelson [12] have done a comparison of RANS and hybrid RANS=LES models. The models have been tested separately for flow over a cylinder, an airfoil, and a cavity. They have found the performance of hybrid RANS=LES to be encouraging. However, they have also noted that hybrid RANS=LES models make sharp transition from RANS to LES and produce virtually no eddy viscosity outside the boundary layer, which have led to nonphysical solutions for the flow over a cavity. Smith and Yakhot [13] have discussed the short- and long-time behavior of eddyviscosity models. They have introduced an eddy-viscosity formula that couples the short- and long-time behavior of turbulence. Borue and Orszag [14] have done highresolution simulations of flow in a periodic box with large-scale forcing (Kolmogorov flow) and stated that the linear eddy-viscosity formula overpredicts the eddy viscosity in regions of high shear rate. They have shown that linear eddy-viscosity formula produces a space-varying eddy viscosity inconsistent with the experimentally observed constant eddy viscosity. Similar to Smith and Yakhot [13], they have proposed that C m should be a linearly decreasing function of the dimensionless shear and found that their results were consistent with that of Smith and Yakhot [13]. The standard k e model proposed by Launder and Spalding [7] is by far the most widely used engineering turbulence model, with well-known characteristics. Several modifications have been proposed to improve its prediction of nonstationary flows. In most studies it has been found that the standard k e model induces too high an eddy viscosity, which dampens the periodic unsteady motion. Most of the modifications have aimed at decreasing the level of eddy viscosity to offer improved predictions. An important issue that has not been adequately examined is the interplay between the time-dependent and time-averaged characteristics of the ensembleaveraged dependent variables in the unsteady RANS computation, especially when the level of eddy viscosity is low. Spalart [2] defines a method as RANS if the grid refinement does not override the empirical content of the turbulence model. In such cases, the role of grid refinement is numerical. However, in most LES studies, where the subgrid scale model is tied explicitly to the local grid size, the grid refinement increases the fidelity of the simulations by reducing the need for physical modeling. In this study, flow past a square cylinder is employed to assess two turbulence models, (1) the standard k e model and (2) the k e model with the modification of Smith and Yakhot [13]. A grid-refinement study is performed to examine the influence of the particular modification on the predictions and on the statistical characteristic of the RANS simulations. Both time-dependent and time-averaged aspects of the computed flow field are examined with the aid of experimental measurements. 2. GOVERNING EQUATIONS AND NUMERICAL TECHNIQUES The Reynolds-averaged Navier Stokes equations in their conservative form are employed. The equations are presented below in Cartesian coordinates. qr þ H ðruþ ¼0 qt ð1þ

4 I. SENOCAK ET AL. q qt ðruþþh ðru uþ ¼ HP þ H ðt ij þ t R ij Þ t ij þ t R qu i ij ¼ðmþm t Þ þ qu j 2 qx j qx i 3 d qu k ij qx k ð2þ ð3þ where r is the density, u is the velocity, t is the time, P is the pressure, m t is the turbulent viscosity, and m is the laminar viscosity. The baseline Navier Stokes solver, documented by Shyy [15] and Shyy et al. [16], employs a pressure-based algorithm and a finite-volume approach to solve the fluid flow and energy equations, written in body-fitted curvilinear coordinates, on collocated, multiblock grids in 2-D and 3-D domains. The convection terms are discretized using the second-order upwind scheme, and the second-order central difference scheme is used for the discretization of the diffusion terms. The time derivative terms are discretized using a fully implicit first-order backward scheme. To offer a contrast, the first-order upwind scheme is also employed for selected cases. These cases are explicitly described. 2.1. The k «Turbulence Model In this study, the k e model with the wall function is adopted as the turbulence closure. The transport equations for the turbulent kinetic energy (k) and its dissipation rate (e) are written as follows, respectively: qrk qt þ H ðrkuþ ¼ðtR ij HÞu re þ H m þ m t Hk s k qre qt þ H ðreuþ ¼C e e1 k P t C e2 r e2 k þ H m þ m t He s e The turbulence production is defined as ð4þ ð5þ P t ¼ðt R ij HÞu ð6þ The empirical constants have the following values: C e1 ¼ 1.44, C e2 ¼ 1.92, s k ¼ 1.0, s e ¼ 1.3. The eddy viscosity is defined as m T ¼ C m r k2 e In the standard k e model, C m is constant and has the following value: C m ¼ 0:09 ð7þ ð8þ In addition to the standard k e model, we also consider the modification of Smith and Yakhot [13] and Borue and Orszag [14], which has the following form: C m ¼ A 1 exp Be ð9þ ks

UNSTEADY RANS SIMULATIONS 5 where S ¼ p ffiffiffiffiffiffiffiffiffiffiffiffi 2S ij S ij S ij ¼ 1 qu i þ qu j 2 qx j qx i ð10þ Borue and Orszag [14] have suggested A ¼ 0.085 and B ¼ 5.0. However, these values are not consistent with the C m ¼ 0.09 value in the wall-function formulation. To be consistent, we use A ¼ 0.646 and B ¼ 0.5 along with C m ¼ 0.09 in the wall-function formulation, which can be computed based on the equilibrium assumption as described in the Appendix. We refer to this hereafter as the nonlinear model. In essence, the nonlinear model introduces an additional time scale based on the local mean (in the ensemble sense) strain rate of the time-dependent flow field. The combined effect of the turbulent scale, reflected by the turbulent kinetic energy and dissipation rate, and the mean velocity field, reflected by the strain rate, determines the level of the eddy viscosity. 3. RESULTS AND DISCUSSION The influence of the computational domain size and inlet turbulence data on unsteady RANS simulations has been discussed by Bosch and Rodi [5]. In Figure 1, the outline of the computational domain used in the present study is shown. In the present study, all computations are two-dimensional. Three different grid densities are employed, ranging from 13,600 nodes for the coarse grid to 27,800 nodes for the medium grid, to 54,500 nodes for the fine grid. The inlet and outlet boundaries are placed far away from the square cylinder, giving a domain of size (53D614D) based on the length of the square cylinder edge (D). The Reynolds number based on the characteristic length of the square cylinder is 22,000. We have studied the sensitivity of the predicted Strouhal number, St ¼ fd=u, to assign proper inlet values of turbulence model variables (k and e), using the nonlinear model. Lyn et al. [4] report a Strouhal number of 0.132 0.004 and (2%) turbulence intensity at the inlet. In our simulations, the turbulent kinetic energy (k) is defined based on the same turbulence intensity, and three different values of the turbulent kinetic energy dissipation rate (e) are tested. Table 1 shows the findings on the coarse grid. We have investigated the Figure 1. Computational domain. The center of the square is located at x ¼ 0.0 and y ¼ 0.0.

6 I. SENOCAK ET AL. Table 1. Sensitivity of Strouhal number predictions to the inlet turbulence data (k and e) a Case no. Convection scheme Inlet value of m T =m Strouhal no. 1 First-order upwind 350 0.127 2 Second-order upwind 350 0.141 3 Second-order upwind 250 0.138 4 Second-order upwind 891 0.145 a The experimental value of the Strouhal number is 0.132 0.004. influence of convection schemes and inlet conditions on the Strouhal number predictions. As seen in the comparison of case 1 and case 2 in Table 1, based on the same inlet turbulence data, a lower Strouhal number is predicted with the first-order upwind scheme because of its dissipative nature. It appears that both numerical and eddy viscosity influences the Strouhal number predictions. In the following, we choose to use the second-order upwind scheme with a ratio of eddy to laminar viscosity of 250, i.e., case 3 in Table 1. Table 2 documents the grid sizes and the corresponding Strouhal number and time-averaged recirculation length predictions of the turbulence models employed. As the computational grid is refined, the nonlinear model predicts higher values of the Strouhal number, whereas the standard k e model predicts lower values and they are relatively insensitive to grid refinement. The standard k e model predictions observed in our study are also consistent with the results of Bosch and Rodi [5]. A high Strouhal number indicates stronger unsteady motion, which is the vortex shedding in the wake of the cylinder, as highlighted in Figure 2. In the nonlinear model predictions, the vortices form on the front corners of the square cylinder and shed in the wake at x=d 1.0, which is in agreement with the experiments of Lyn et al. [4]. For the standard k e model case, the vortices are shed at x=d 2.0. The time-averaged recirculation lengths, given in Table 2, show that the length of the separation zone is significantly overpredicted by the standard model, whereas the nonlinear model provides better predictions as the grid is refined. It appears that there is a discrepancy with regard to the fidelity of the Strouhal number and recirculation length predictions. Apparently, neither model can predict both quantities equally well. In Figure 3 the streamwise velocity along the wake centerline is shown. The results of the nonlinear model are highly grid-dependent for the grid resolutions Table 2. Grid sizes and corresponding Strouhal number and time-averaged recirculation length of the simulations a No. of grid nodes (edge total) Strouhal no. L r =D Experiment 0.132 0.004 1.38 Coarse: 10 13,600 0.138 (0.129) 4.28 (3.71) Medium: 20 27,800 0.159 (0.126) 1.73 (3.37) Fine: 30 54,500 0.155 (0.122) 1.25 (3.05) a Based on the inlet condition m T =m ¼ 250. The values in parentheses are from standard k e model results.

UNSTEADY RANS SIMULATIONS 7 Figure 2. Illustration of instantaneous spanwise vorticity distribution. adopted in the present study. Indeed, there is no sign of grid convergence and the predictions are largely varying with each grid. The results in the region of x=d > 1.0 improve as the grid is refined, but the results of the near-wake region do not seem to improve at all. For instance, the magnitude of u velocity in the recirculation zone is significantly lower than the experimental value. On the other hand the standard k e model, although not in good agreement with the experiments, has a modest grid dependency and exhibits typical grid convergence behavior. One can notice that a nearly grid-independent solution is obtained in the near-wake of the square cylinder. This unusually high grid dependency of the nonlinear model, as compared to the grid dependency of the standard k e model, has led us to investigate the possible reasons. Figure 4 helps shed light on the reasons for high grid dependency of the nonlinear model. A grid-refinement study is carried out for the normalized turbulent eddy viscosity m T =m. In the wake of the square cylinder (70.5 < y=d < 0.5), the turbulent eddy viscosity obtained from the nonlinear model is highly dependent on the grid, which is different from the standard k e model. As was shown in Figure 2, the wake is populated with both positive and negative vortices, which makes velocity a rapidly varying quantity in this region. Since the magnitude of the strain rate (S),

8 I. SENOCAK ET AL. Figure 3. Time-averaged streamwise velocity along the wake centerline: (s) experimental data; ( ) fine grid; (---) medium grid; (- -) coarse grid. which appears in the eddy viscosity formula of the nonlinear model, is computed from the time-dependent velocity field, the eddy viscosity is much more affected from the resolution as compared to the standard k e model. The instantaneous turbulent eddy-viscosity distribution of the nonlinear model, given in Figure 5, clearly shows the rapid variation in the wake. Furthermore, the nonlinear model computes very low time-averaged turbulent viscosity levels, about m T =m 20, in the free-stream region, as compared to the standard k e model, where m T =m 150. With such low levels of physical dissipation, m T =m 20, the flow field is highly underresolved with the conventional numerical schemes used in RANS computations. In other words, the numerical dissipation inherent in these schemes becomes comparable to the physical dissipation induced by the turbulence model. Consequently, the solutions do not progress in an orderly manner as the grid is refined. In Figure 5 the distribution of instantaneous and time-averaged turbulent eddy viscosity in the computational domain are shown for both models. Unlike the standard k e model, the location of the inlet boundary becomes important for the

UNSTEADY RANS SIMULATIONS 9 Figure 4. Time-averaged normalized turbulent eddy viscosity ðm T =mþ profiles at x=d ¼ 1.5: ( ) fine grid; (---) medium grid; (- -) coarse grid. nonlinear model, which is observed from the strong variation of turbulent eddy viscosity at the inlet region. A turbulent length scale, normalized by the characteristic length scale D, can be computed based on ðk 3=2 =edþ, which characterizes the structures of the turbulent flow field. In Figure 6 the length-scale predictions obtained from each model are shown. We also plot the length scale of the spatial resolution, normalized by the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi characteristic length scale D, D ¼ ðdxþ 2 þðdyþ 2 =D. As the grid is refined, the standard k e model predicts almost the same length scales, which are much larger than the length scale of the spatial resolution. This observation is consistent with the fact that the turbulent eddy viscosity yielded by the standard k e model is high, indicating that the entire flow computation is heavily influenced by the turbulence closure, accounting for the full turbulence statistics. In other words, the numerical resolution requirement is less severe with the standard k e model. For example, in the wake, the length scale is comparable to the characteristic length of the square cylinder (D) and in the free-stream region it is comparable to the distance between the square cylinder and the test section wall (5D). The turbulent length scales predicted by the nonlinear model are more sensitive to the numerical resolution.

10 I. SENOCAK ET AL. Figure 5. Time-averaged and instantaneous turbulent eddy-viscosity distribution in the computational domain. Figure 6. Influence of grid refinement on time-averaged turbulent length-scale ðk 3=2 =eþ predictions at x=d ¼ 1.5: ( ) fine grid; (---) medium grid; (- -) coarse grid.

UNSTEADY RANS SIMULATIONS 11 In the present computations, the turbulent length scale of the nonlinear model is comparable to the length scale of the spatial resolution (D), noticeably in the nearwake region. This again shows that the numerical resolution interacts with the modeled turbulent length scales, thus creating difficulties in finding grid-independent solutions. The nonlinear model s performance does not meet the suggestion made by Spalart [2] that in RANS computations, grid refinement should not override the empirical content of the turbulence model. Figure 7 shows turbulent kinetic energy profiles at the front face of the square cylinder. As discussed by Bosch and Rodi [5], the standard k e model predicts high levels of turbulent kinetic energy at stagnation points. Bosch and Rodi used the modification of Kato and Launder [6] to lower the production of the turbulent kinetic energy. Essentially, in the Kato-Launder modification the turbulence production term is modified to include a rotation parameter that is proportional to the magnitude of the local mean vorticity, which vanishes on the stagnation streamline. The nonlinear model accomplishes similar goals via a nonlinear form for the C m term of the eddy-viscosity formula. Unlike the wake region, the nonlinear model does not show any grid dependency at the front face of the square cylinder. This is Figure 7. Time-averaged turbulent kinetic energy at the front face of the square cylinder (x=d ¼70.65): ( ) fine grid; (---) medium grid; (- -) coarse grid.

12 I. SENOCAK ET AL. because at this location the length scale predicted by the turbulence model is much larger than the spatial resolution, as shown in Figure 8. Hence, the computations are well resolved in this region. It should be mentioned that the noticeable difference of the coarse-grid results is because the location of the coarse-grid results (x=d ¼ 70.675) does not match the location of the medium- and fine-grid results (x=d ¼ 70.650). The computations of the nonlinear model with the fine grid have produced an interesting behavior, which is demonstrated in the evolution of pressure in time, given in Figure 9. Flow disperses in time, and a regular vortex street formation is lost. The resulting wake structure does not seem to be physical, as shown in Figure 10. For instance, two large-scale vortices split and accelerate into the surrounding region, causing a dispersive vortex pattern to emerge. As a result, a typical periodic wake structure does not appear. The standard k e model does not exhibit such a dispersive nature, as can be seen in Figure 2. The regular wake structure is well preserved, but the locations of the vortex cores are aligned along the wake symmetry line. Instead, the vortex cores should follow a zigzag pattern, as observed in the experiments of O Neil and Meneveau [17]. Figure 8. Time-averaged turbulent length-scale predictions at the front face of the square cylinder (x=d ¼70.65): ( ) fine grid; (---) medium grid; (- -) coarse grid.

UNSTEADY RANS SIMULATIONS 13 Figure 9. Evolution of pressure in time: (top) standard k e model; (bottom) nonlinear model. Fine grid is used in both computations. Such unusual numerical solutions have been reported in the literature; they arise possibly because of the limitation imposed by 2-D simulations. Sohankar et al. [18] have done DNS of flow past square cylinders for a range of moderate Reynolds numbers (Re ¼ 150 500). They have reported that there is a transition from 2-D shedding to 3-D shedding for Re ¼ 150 200. Their 2-D simulations for Re ¼ 400 exhibited nonphysical solutions such as nonzero mean lift coefficient, dispersed wake structure, and asymmetrical profiles of mean quantities, which were absent in the case of 3-D simulations. Breuer [19] also experienced similar problems with 2-D LES simulations of flow past a circular cylinder at Re ¼ 3,900 that were also absent in 3-D simulations. The nonlinear model solution has an effective Reynolds number as high as 2,700 in the free-stream region of the wake, whereas the value is about 150 with the standard k e model. It appears that the effective Reynolds number based on the eddy viscosity is higher with the nonlinear model, necessitating 3-D simulations. We have not observed the dispersion problem for the coarse and medium grids, indicating that the numerical dissipation is sufficient to compensate for the high effective Reynolds number in those cases. In Figures 11 and 12, the time-averaged streamwise and cross-streamwise velocity profiles at various wake locations are compared. The nonlinear model does a better job at certain locations as compared to the standard k e model. Especially the

14 I. SENOCAK ET AL. Figure 10. Dispersed vortex shedding in the wake of square cylinder produced by the nonlinear model with the fine-grid computations. cross-streamwise velocity results are noticeably improved with the nonlinear model. This is because the vortex shedding is stronger with the nonlinear model and vortices alternately roll up farther from the wake centerline. Although improved timeaveraged results can be obtained with the nonlinear model, considering the high grid dependency of the results in the wake region, low levels of turbulent eddy viscosity, and the dispersion characteristics suggest that the model encounters difficulties in practical 2-D computations. Similar grid-dependency problems have also been noted in high-reynolds-number large-eddy simulation by Breuer [20], where it was found that the results based on 2-D simulations departed more from the experiments with further grid refinement. The reason is that 3-D flow structures in shedding flows comprises a systematic set of co- and counterrotating vortices aligned in the flow direction, which are absent in 2-D simulations [20]. On the other hand, the standard k e model performs in a more orderly manner in terms of grid refinement. This is a result of the higher eddy-viscosity levels, which are applied to the full wavelength spectrum. 4. CONCLUSIONS We have assessed the statistical characteristics of unsteady RANS computations, utilizing progressively refined spatial resolutions. The standard k e model has

UNSTEADY RANS SIMULATIONS 15 Figure 11. Comparison of time-averaged u-velocity profiles at different locations: (s) experimental data; ( ) nonlinear model; (---) standard k e model. been compared to the k e model with the modification of Smith and Yakhot [13], referred to as the nonlinear model in this study. Both models have been tested for flow past a square cylinder, and time-averaged results have been compared with the experimental data of Lyn and Rodi [3] and Lyn et al. [4] For the range of spatial resolutions considered in this study, the standard k e model exhibited a modest grid dependency with a typical grid convergence trend. On the other hand, the nonlinear model showed no sign of grid convergence, and it produced results that were highly influenced by the grid refinement, especially in the wake of the square cylinder. Grid dependency of each turbulence model was further investigated with the aid of turbulent eddy viscosity, length scales of the turbulence, and the spatial resolution. The comparison between the length scale imposed by the turbulence model and the length scale of the spatial resolution helps us to understand the fidelity of the simulations. The eddy viscosity of the nonlinear model is determined by the multiple scales, based on both mean turbulent quantities and mean velocity field; for the present case, it yields very low levels of turbulent eddy viscosity in some regions as compared to the standard k e model. Accordingly, the flow field can be underresolved and the computations do not progress in an orderly manner as the grid is refined. Furthermore, because of 2-D computations and high effective Reynolds numbers attained by the nonlinear model, dispersive behavior appeared

16 I. SENOCAK ET AL. Figure 12. Comparison of time-averaged cross-stream velocity at different locations: (s) experimental data; ( ) nonlinear model; (---) standard k e model. when employing sufficiently refined resolution with lower numerical dissipation. The issue needs to be further addressed by 3-D simulations. The nonlinear model can be useful for other complex flows and for different approaches such as in hybrid RANS=LES models [12]. On the other hand, the standard k e model performs in a more orderly manner with respect to grid refinement by imposing a higher level of eddy viscosity on a full-wavelength spectrum. To summarize, we have illustrated that the interplay between the physical (modeled) and numerical length scales can substantially affect the outcome of turbulent flow simulations. As a result, the time-dependent and time-averaged characteristics can also exhibit highly different patterns. REFERENCES 1. S. B. Pope, Turbulent Flows. Cambridge University Press, Cambridge, UK, 2000. 2. P. R. Spalart, Strategies for Turbulence Modeling and Simulations, Int. J. Heat Fluid Flow, vol. 24, pp. 147 156, 2000. 3. D. A. Lyn and W. Rodi, The Flapping Shear Layer Formed by Flow Separation from the Forward Corner of a Square Cylinder, J. Fluid Mech., vol. 267, pp. 353 376, 1994.

UNSTEADY RANS SIMULATIONS 17 4. D. A. Lyn, S. Einav, W. Rodi, and J. H. Park, A Laser-Doppler Velocimetry Study of Ensemble-Averaged Characteristics of the Turbulent Near Wake of a Square Cylinder, J. Fluid Mech., vol. 304, pp. 285 319, 1995. 5. G. Bosch and W. Rodi, Simulation of Vortex Shedding past a Square Cylinder with Different Turbulence Models, Int. J. Numer. Meth. Fluids, vol. 28, pp. 601 616, 1998. 6. M. Kato and B. E. Launder, The Modeling of Turbulent Flow around Stationary and Vibrating Square Cylinders, Proc. 9th Symp. on Turbulent Shear Flows, Kyoto, Japan, 1993, pp. 10.4.1 10.4.6. 7. B. E. Launder and D. B. Spalding, The Numerical Computation of Turbulent Flows, Comput. Meth. Appl. Mech. Eng., vol. 3, pp. 269 289, 1974. 8. W. Rodi, Comparison of LES and RANS Calculations of the Flow around Bluff Bodies, J. Wind Eng. Ind. Aerodyn., vol. 69 71, pp. 55 75, 1997. 9. S. Kenjeres and K. Hanjalic, Transient Analysis of Rayleigh-Benard Convection with a RANS Model, Int. J. Heat Fluid Flow, vol. 20, pp. 329 340, 1999. 10. G. Iaccarino, A. Ooi, P. A. Durbin, and M. Behnia, Reynolds Averaged Simulation of Unsteady Separated Flow, Int. J. Heat Fluid Flow, vol. 24, pp. 147 156, 2003. 11. P. A. Durbin, Separated Flow Computations with the k e v 2 Model, AIAA J., vol. 33, pp. 659 664, 1995. 12. R. H. Nichols and C. C. Nelson, Application of Hybrid RANS=LES Turbulence Models, AIAA Paper 2003 0083. 13. L. M. Smith and V. Yakhot, Short- and Long-time Behavior of Eddy-Viscosity Models, Theory, Comput. Fluid Dynam., vol. 4, pp. 197 217, 1993. 14. V. Borue and S. A. Orszag, Numerical Study of Three-Dimensional Kolmogorov Flow at High Reynolds Numbers, J. Fluid Mech., vol. 306, pp. 293 323, 1996. 15. W. Shyy, Computational Modeling for Fluid Flow and Interfacial Transport. Elsevier, Amsterdam, The Netherlands, 1997. 16. W. Shyy, S. S. Thakur, H. Ouyang, J. Liu, and E. Blosch, Computational Techniques for Complex Transport Phenomena, Cambridge University Press, Cambridge, UK, 1997. 17. J. O Neil and C. Meneveau, Subgrid-Scale Stresses and Their Modelling in a Turbulent Plane Wake, J. Fluid Mech., vol. 349, pp. 253 293, 1997. 18. A. Sohankar, C. Norberg, and L. Davidson, Simulation of Three-Dimensional Flow around a Square Cylinder at Moderate Reynolds Numbers, Phys. Fluids, vol. 11, pp. 288 306, 1999. 19. M. Breuer, Numerical and Modeling Influences on Large Eddy Simulations for the Flow past a Circular Cylinder, Int. J. Heat Fluid Flow, vol. 19, pp. 512 521, 1998. 20. M. Breuer, A Challenging Test Case for the Large Eddy Simulation: High Reynolds Number Circular Cylinder Flows, Int. J. Heat Fluid Flow, vol. 21, pp. 648 654, 2000. APPENDIX In a simple turbulent shear flow, the strain rate is given by S ¼ qu ða1þ qy When the turbulence production equals the dissipation, the empirical observation suggests the following: huvi 0:30k ða2þ

18 I. SENOCAK ET AL. The turbulence production in a simple shear flow is written as P t ¼ huvis ða3þ The shear stress can be computed from the eddy-viscosity model as shown below: k 2 huvi ¼ C m e S ða4þ The assumption of equilibrium turbulence (P t ¼ e) along with the empirical observation given in (A2) yields e=ks ¼ 0:3. The wall function formulation is based on the equilibrium turbulence assumption; hence C m of the nonlinear model should satisfy C m ¼ A 1 exp Be! A ½1 expð 0:30BÞŠ ¼ 0:09 ða5þ ks The model parameters A ¼ 0.646 and B ¼ 0.5 are chosen to satisfy the above equation. Note that the model parameters suggested by Borue and Orszag (1996) are A ¼ 0.085 and B ¼ 5.0. In equilibrium flows these constants give huvi 0:245k, not huvi 0:30k.