A Comparison of RANS-Based Turbulence Modeling for Flow over a Wall-Mounted Square Cylinder

Transcription

1 A Comparison of RANS-Based Turbulence Modeling for Flow over a Wall-Mounted Square Cylinder P. L. Davis 1, A. T. Rinehimer 2, and M.Uddin 3 N C Motorsports and Automotive Research Center, Department of Mechanical Engineering and Engineering Science, The University of North Carolina at Charlotte, Charlotte, NC 28223, USA 1 pdavis24@uncc.edu 2 arinehim@uncc.edu 3 muddin@uncc.edu s ABSTRACT Using experimental data from a study of flow over a wall-mounted square cylinder (h=4 as a baseline, three Reynolds-Averaged Navier-Stokes turbulence models are used in a commercial CFD code Star- CCM+ to compare the relative accuracy of the tested models. In virtually every standard of comparison applied in this study, the Realizable k ε with a twolayer treatment proved to be far superior to both the k ε V2F (All y + hybrid wall treatment) and the k ω models. To observe mesh dependence for each of the models, all three were run on three different polyhedral mesh cases. Resulting first prism-layer heights of y + 12, y + 5, and y + 1 allowed comparison of results with the mesh resolved to the buffer layer, the buffer layer/viscous sublayer transformation, and into the viscous sublayer, respectively. In all cases, the Realizable k ε proved superior. The mesh study also suggests that applying the twolayer treatment to Realizable k ε allows it to operate well into the viscous sublayer, an area in which k ε models are traditionally expected to suffer degradation in accuracy. The k ω scheme, however, does not show improvements with increased mesh quality. In contradiction to the expected results, mesh independence is reached by y + 5 for k ω in this study. 1. INTRODUCTION To perform a veracity check of popular RANS turbulence models in predicting separated flows, experimental data reported by Bourgeois et al is used as a basis of comparison. The flow being examined consists of a single wall-mounted square cylinder of dimensions (h/d =4) sitting within a steady, wellprofiled air flow of Reynolds number Re=12, Time and phase-averaged flow data sets as well as velocity profiles for a three-dimensional grid are made available by the organizers of the 2012 CFDSC Challenge COMPUTATIONAL EQUIPMENT All CFD simulations were performed using the commercial CFD code Star-CCM+ version 6.04 by CD-Adapco. All simulations were performed on the UNC-Charlotte cluster with 32 cores. Additional postprocessing was performed using EnSight by CEI. 3. SURFACE & VOLUME MESHING All CAD data were drawn and surface meshed using ANSA by Beta CAE. All volume meshing was performed within Star-CCM+ using a polyhedral meshing scheme. A polyhedral scheme was chosen due to the omni-directional nature of wall-bounded turbulent flows (Timothy Yen, CD-ADAPCO, personal communication, February 23, 2012). For the best computational accuracy, an ideal mesh would have each cell oriented with one face normal to the flow. When compared to a tetrahedral or hexahedral cell, a polyhedral cell has more faces, and therefore it has more optimal flow directions (normal to a face) than a tetrahedral or hexahedral cell. With more potential optimal flow directions, the polyhedral cell has an increased likelihood of a particular flow direction being at or near the optimal flow direction of one of the polyhedral cell faces. Also, polyhedral cells have more neighbors which allows for better gradient approximations, especially near boundaries and corners. 3 As a starting point, a mesh with ~16x10 6 cells was generated with polyhedral cells, to be called the

2 coarse mesh case hereafter. After preliminary simulations and subsequent examinations of the y + values, two additional meshes were created. The medium mesh with ~26x10 6 cells had further refinements in the floor, behind the cylinder, and around the cylinder surfaces in order to attain a maximum y + 5. In the fine mesh case, additional refinements were made to the cells around the cylinder edges and the prism-layer meshes on the tunnel floor. This increased the total cell count to ~29x10 6 and reduced the first prism layer heights down to produce a y + 1. Even though some simulations may have suggested further refinements could yield slight improvements, for comparative purposes all simulations were run on these three meshes. Coarse Medium Fine Total Volume Cells ~16x10 6 ~26x10 6 ~29x10 6 Floor Tot. Prism Hgt. 4mm 9mm 9mm Floor Prism Levels Floor Prism 1 st Hgt mm mm mm Cyl. Total Prism Hgt. 4mm 9mm 9mm Cyl. Prism Levels Cyl. Prism 1 st Hgt mm mm mm Cyl. Corners 0.25mm 0.125mm 0.125mm Table 1-Volume Mesh Control Values 4. TURBULENCE MODELS In order to examine a range of RANS modeling, three versions of k ω and k ε turbulence models are selected from the available Star CCM+ physics models for detailed examination. The three models are used with all default settings given in version The first model, the Standard k ω, is a wellestablished model capable of resolving through the boundary layer. 4 The second model is Realizable k ε, an improvement over the standard k ε model. 5 k ε schemes model turbulence based on turbulent core flows, making them traditionally unsuitable for applications in the viscous sublayer. 6 And finally the third model examined is the V2F variant of k ε. This model is expected to offer the possibilities of capturing near-wall turbulence effects more accurately k ω Setup Since k ω is capable of resolving flow through the boundary layer, accuracy is expected to improve as the mesh size is reduced to move y + into the viscous sublayer (y + 5). Salim and Cheah 6 suggested this approach and, accordingly, the fine mesh was tuned to result in a near-wall prism layer of y K-Epsilon Realizable Setup As mentioned, the traditional k ε model is not well-suited in resolving flows in near-wall regions. At or near the viscous sublayer, the viscous forces are dominant over the turbulent forces, effectively damping any effects of turbulence. Due to this damping, traditional k ε can t resolve flows in this region, and thus the medium mesh is designed with a near-wall prism layer that yields y + 5. It could be expected that resolving k ε at a lower value, such as the fine mesh s value of y + 1, could potentially show no improvement. There is likelihood that this may even be detrimental to the accuracy of the solution, a reason why a two-layer wall treatment is also applied. The two-layer approach 8 applies a modified model in the viscous sublayer region, allowing k ε to be applied in meshes that could otherwise be unsuitable for such a model. Because three common meshes are being used rather than an optimized mesh for each individual model, the two-layer approach is expected to produce better results with a mesh that is otherwise not optimized for a k ε scheme. One drawback of the two-layer approach in this study is the fact that, as mentioned, more traditional k ε schemes could be expected to perform poorly with y + < 5. The twolayer approach will likely mitigate these effects, so a full understanding of the effects of y + may not be possible in this study. 4.3 K-Epsilon V2F Setup The V2F variant of k ε model solves two additional transport equations, and is expected to more accurately predict the effects of turbulence near walls. 9 It is to be expected that this scheme will require more computational time, but with such complex wall-bounded flows as this problem, the additional time could be a solution cost worth paying. After running several simulations with this model, the actual difference in computational time was significant, requiring approximately six-fold more time to perform the same iteration count with k ε V2F. Star-CCM+ has two possible wall treatments for the k ε V2F model: All y + and Low y +. The Low y + treatment is intended for situations where the mesh resolves the viscous sublayer, and would likely be suited for the fine and possibly even the medium mesh. But the resolution of the coarse mesh is insufficient for the Low y +. Since all three meshes were to be run on the identical modeling setup, the All y + was the necessary choice.

3 5. COMPARISON WITH EXPERIMENTAL DATA Viewing the various flow visualizations comparing the three turbulence models shows distinct differences in the accuracy of the simulations. Observations from all the figures show that k ω and k ε V2F produced nearly identical results, in spite of using significantly different modeling techniques. However, despite their close agreement, neither k ω nor k ε V2F produces results as close to the experimental data as the Realizable k ε scheme. Figs. 1 and 2 show that Realizable k ε produces a wake shape in the X-Z plane that most closely resembles the experimental data s profile. While k ω and k ε V2F show significantly more downward motion (Fig. 2b and 2 than the experiment, Realizable k ε shows very similar characteristics at the mid-plane. A look at the mid-plane vector plot sheds some light as to why this difference exists. As can be seen in Fig. 3a, the experimental data shows a recirculation region immediately behind the rear top edge of the cylinder, something which also exists in the Realizable k ε case. Also visible in both the experiment and the Realizable k ε case is a distinct saddle region as described by Perry and Chong 10, although it is located slightly higher and farther back in the Realizable k ε. Looking at Fig. 3b and 3d, neither the recirculation at the top nor the saddle region is visible in the k ω and the k ε V2F cases. A further inspection of Fig. 3b and 3d shows that in the k ω and the k ε V2F cases the streamlines closest to the top of the cylinder are already slightly downturned by the time they reach the rear edge, and continue to turn until they reach an approximately 45 approach angle with the ground. They continue with this direction until they interact with the ground, FIG.1 x-velocity on X-Z Plane for Experimental k-ω k-ε Real. k-ε V2F FIG.2 z-velocity on X-Z Plane for Experimental k-ω k-ε Real. k-ε V2F

4 Saddle Region Saddle Region FIG.3 Vector Profiles on X-Z Plane for Experimental k-ω k-ε Real. k-ε V2F then turn to a horizontal direction by about 1h behind the rear of the cube. This strong 45 -downward flow explains the significant negative flow to the near-floor region in Fig. 2b and 2d. Both the experimental data and the Realizable k ε show that the flow close to the top of the cylinder is still upwards at the rear edge of the cylinder, but then quickly curves downward and becomes the recirculation zone. The next layer above this passes over the recirculation zone, and curves more slowly towards the ground at somewhat less than 45. But, unlike in the k ω and the k ε V2F cases, the downward flow is deflected by the saddle region and thus approaches the ground at a much lower angle of approach. While Fig. 3b and 3d show the downward flow interacting with the floor at ~1h, Fig. 3b and 3d show a flow with such a shallow approach that they have not reached the floor even at the end of the visible region (2.5h). This difference in approach angle is visible in other views as well. FIG.4 x-velocity on X-Y Plane at z=0.5h for Experimental k-ω k-ε Real. k-ε V2F Viewing from above, Fig. 4 again shows the closest simulation to be the Realizable k ε. At 0.5h, the flow-direction wake is nearly horizontal and passing through the 0.5h plane all the way to the 2.5h end of the view in both Fig. 4a and 4c. But the downward angle of approach in the k ω and the k ε V2F may be seen as a significantly shorter region of influence by the wake in the 0.5h plane in Fig. 4b and 4d.

5 Viewed from above, the streamlines in all four views of Fig. 6 show twin counter-rotating regions behind the cylinder. The flow past these regions curves first inward then back outward in a bottleneck fashion in both the experiment and the Realizable k ε case. The k ω and k ε V2F cases do not show this bottlenecking, and instead continue to converge towards the mid-plane. FIG.5 y-velocity on X-Y Plane for Experimental k-ω k-ε Real. k-ε V2F Cross-flow velocities in Fig. 5 are also very telling as to the difference in structures generated by the three turbulence models. Again, both the experiment and Realizable k ε are in close agreement. Both produce three pairs of evenly-spaced counter-rotating regions in the twin wake trails. By comparison, k ω and k ε V2F show a single pair directly behind the cubes (as does the experiment). But instead of the two additional pairs of counter-rotating confined regions, a single long region continuing well past the viewing area are visible. FIG.6 Vector Profiles on X-Y Plane for Experimental k-ω k-ε Real. k-ε V2F

6 One area in which the Realizable k ε case does not produce the best results (and in fact the only observable structure in any manner throughout this entire study where Realizable k ε was not best) is in the size and intensity of the recirculating regions in Fig. 6. All three turbulence models demonstrate a larger and more organized rotation region than the experiment, with Realizable k ε being the largest. The flow visualization in Fig. 6 may, however, be somewhat misleading when it comes to recirculation strength due to the large differences in vector seeding density by EnSight between the experimental and CFD data. Looking at the cylinder from the rear, Fig. 7 shows the experimental data as having a pair of large regions of cross-flow, with two more pairs of counter-flowing areas below this. While all three turbulence models show this structure to some extent, the Realizable k ε most closely matches the experimental structure in size, location, and intensity. Fig. 8 also shows that the vertical velocity profile of Realizable k ε most closely matches the experimental data. The experiment shows a confined core of downward flow at the top half of the cylinder, while k ω and the k ε V2F show a core nearly the full height of the cylinder. FIG.7 y-velocity on Y-Z Plane for Experimental k-ω k-ε Real. k-ε V2F FIG.8 z-velocity on Y-Z Plane for Experimental k-ω k-ε Real. k-ε V2F

7 Velocity (m/s) Velocity (m/s) Position (mm) -10 Position (mm) Coarse Mesh X Velocity Medium Mesh X Velocity Fine Mesh X Velocity Coarse Mesh Z Velocity Medium Mesh Z Velocity Fine Mesh Z Velocity Coarse Mesh X Velocity Medium Mesh X Velocity Fine Mesh X Velocity Coarse Mesh Z Velocity Medium Mesh Z Velocity Fine Mesh Z Velocity FIG.9 Selected Velocity Profile for k-ω with Three Mesh Cases 6. MESH INDEPENDENCE In order to examine mesh independence, all simulations were monitored with a set of velocity profiles in various regions of the flow path, as well as with visual comparisons of flow structure and scalar plots. Two samples of these velocity profiles may be seen in Fig. 9 and 10 for k ω and Realizable k ε. Due to computational time, a mesh-dependence study was not performed on k ε V2F. As previously discussed, it could be expected that k- ω would improve with an increasing resolution in the boundary layer. With the medium mesh (y+ 5) resolving to the edge of the viscous sublayer and the fine mesh (y+ 1) resolving well into the viscous sublayer, it was expected that some improvement in accuracy would be yielded by the fine mesh. Salim and Cheah also predicted such behavior. 6 However, Fig. 9 shows that, at least for velocity profiles, this was not the case. As can be seen, there is no significant difference between the medium and fine mesh cases in velocity profiles at the selected regions, suggesting that the k ε scheme used in this study reached mesh independence by y+ 5. This unexpected result was also seen by Salim and Cheah. In their study, their fine mesh (y+ 2) did improve the accuracy of skin friction predictions, but FIG.10 Selected Velocity Profile for Realizable k-ε with Three Mesh Cases their fine mesh showed no improvements in predicting velocity profiles compared to either their coarse (y+ 32.5) or medium mesh (y+ 12.5). Also as discussed, unlike k ω, k ε was expected to not see improvement with a mesh resolved into the viscous sublayer. But again, this study did not support the initial assumptions for k ε, at least not with the two-layer treatment applied. Instead, the fine mesh case produced the best results for the Realizable k ε, as may be seen in Fig. 10. Salim and Cheah had similar results. In their study at the finest mesh (y+ 2) Realizable k ε, with a Standard Wall Function (SWF), did perform the worst of any model tested. But by adding an Enhanced Wall Function (EWF), a two-layer treatment similar to that used here in Star-CCM+, Salim and Cheah found that Realizable k ε was among the best model tested with the fine mesh. It is likely that, had this current study compared a Realizable k ε without the twolayer treatment, poor results similar to Salim and Cheah would have been seen with the fine mesh. 7. CONCLUSION When comparing the three models used in this study, one model has shown a clear and distinct advantage in predicting the flow of a wall-mounted square

8 cylinder. In virtually every measure of comparison, Realizable k ε demonstrates a superior ability to capture the mean flow of the complex structures measured in the experimental data. Neither k ω nor k ε V2F is able to predict the complex saddle structure behind the cylinder or the trail-edge recirculation zone at the top of the cylinder as seen in Fig.3. Without accurately predicting these structures, k ω and k ε V2F are also unable to predict the complex pattern of counter-rotating and alternately shedding vortices captured by the experiment. Realizable k ε, on the other hand, shows very good ability in predicting these complex patterns. While not an exact representation, the Realizable k ε cases are able to very closely predict the three pairs of counter-rotating regions behind the cylinder seen in Fig. 5. The results of the study of any correlation between mesh quality and turbulence modeling are not nearly as straight-forward. The k ω model, which is capable of modeling flows throughout the boundary layer, was expected to see improvements with a mesh resolved into the boundary layer. But in the case observed in this study, k ω reached mesh independence with the medium mesh despite being resolved only to y+ 5, a distance at the edge of the viscous sublayer. To the contrary, Realizable k-ε did not demonstrate mesh independence at y+ 5, and instead showed further improvements all the way to y+ 1. This contradicts the traditional assumption that k ε models perform poorly when resolved into the viscous sublayer. In the case of this study, just as in the literature, the likely reason for the improving results with a viscous sublayer-refined mesh is the application of the two-layer treatment. This treatment applies a different model to the k ε scheme in the viscous sublayer region, allowing for much finer meshes to be used without sacrificing accuracy. cylinder with a thin boundary layer. Physics of Fluids, 23(9):095101, 2011 [2] R. J. Martinuzzi. CFDSC Challenge Data. Retrieved : March 16, [3] M. Peric and S. Ferguson. The Advantage of Polyhedral Meshes. Technical Report, CD Adapco Group, [4] D. C. Wilcox. Turbulence Modeling for CFD, 2 nd ed. DCW Industries, 2006 [5] T. H. Shih, W. W. Liou, A. Shabbir, Z. Yang, and J. Zhu. A New k-ε Eddy Viscosity Model for High Reynolds Number Turbulent Flows Model Development and Validation. Computers Fluids. 24(3): , 1995 [6] Salim M. Salim and S. C. Cheah. Wall y+ Strategy for Dealing with Wall-Bounded Turbulent Flows. Physics of Fluids, 23(9):095101, 2011 [7] L. Davidson and P. V. Nielsen. Modifications of the v 2 -f Model for Computing the Flow in a 3D Wall Jet. Turbulence, Heat and mass Transfer, 23: , 2003 [8] W. Rodi. Experience with Two-Layer Models Combining the k-e Model with a One-Equasion Model Near the Wall. 29 th Aerospace Sciences Meeting, January 7-10, Reno, NV , 1991 [9] Star CCM+ Help Manual. What is the V2F Low- Reynolds Number K-Epsilon Model. CD- Adapco Group. [10] A.E. Perry and M. S. Chong. A Description of Eddying Motions and Flow Patterns Using Critical-Point Concepts. Ann. Rev. Fluid Mech., 19: , 1987 ACKNOWLEDGEMENTS This material is based upon work supported by the national Science Foundation Graduate Research Fellowship under Grant No Also, the authors would also like to thank Timothy Yen of CD-Adapco and Bill Dunn of CEI for their assistance with this project. REFERENCES [1] J.A. Bourgeois, P. Sattari, and R. J. Martinuzzi. Alternating half-loop shedding in the turbulent wake of a finite surface-mounted square