Abstract
Large eddy simulation (LES) of the neutrally stratified turbulent Ekman layer is performed. In particular, we compare three LES models with direct numerical simulation (DNS), which was validated against existing DNS. The models considered are a standard nondynamic LES model, the Smagorinsky model (SM), a standard dynamic LES model, the stabilized dynamic Smagorinsky model (DSM), and a new linear dynamic model (LDM), which was derived from a realizable stochastic turbulence model. The following conclusions are obtained. The SM does not represent an appropriate model for the flow considered. Mean velocity and turbulence intensities are poorly predicted. With respect to instantaneous fields, the SM provides a tilting of turbulence structures in the opposite direction as seen in DNS. The stabilized DSM also suffers from significant shortcomings. First, its behavior depends on the wall distance. Close to the wall, it produces acceptable turbulence structures. Away from the wall, it suffers from the same shortcomings as the SM. Second, it incorrectly describes the effect of grid coarsening. The new LDM is free from the disadvantages of the SM and stabilized DSM. Its predictions of both mean and instantaneous velocity fields agree very well with DNS. The relevant conclusion is the following. The use of a dynamic LES method represents a mean for correctly simulating large-scale structures (means and stresses), but it does not ensure a correct simultaneous simulation of small-scale structures. Our results indicate that a dynamic method designed in consistency with a realizable stress model can correctly simulate both large-scale and small-scale structures.
Award Identifier / Grant number: NNX12AJ71A
Funding statement: This work was sponsored by the University of Wyoming School of Energy Resources via a graduate assistantship for E. Kazemi Foroushani. The authors are very thankful for additional support from the National Renewable Energy Laboratory (NREL) monitored by Dr. Mike Robinson. The authors would like to acknowledge the support for this work through a gift from BP Alternative Energy North America Inc. to the UW Wind Energy Research Center. S. Heinz would like to acknowledge the partial support through NASA’s NRA research opportunities in aeronautics program (Grant No. NNX12AJ71A) with Dr. P. Balakumar as the technical officer.
Acknowledgments
The authors are very thankful to Prof. M. Stöllinger for interesting discussions and helpful suggestions for improvements. The authors are also thankful to the referees for their very helpful comments. Computational resources provided through the University of Wyoming are gratefully acknowledged.
References
[1] P. A. Durbin and B. A. Petterson, Statistical theory and modeling for turbulent flows, John Wiley and Sons, Chichester, New York, Weinheim, Brisbane, Singapore, Toronto, 2001.Search in Google Scholar
[2] S. Heinz, On Fokker-Planck equations for turbulent reacting flows. Part 1. Probability density function for Reynolds-averaged Navier–Stokes equations, Flow Turbul. Combust. 70 (2003), 115–152.10.1023/B:APPL.0000004933.17800.46Search in Google Scholar
[3] S. Heinz, On equations for turbulent reacting flows. Part 2. Filter density function for large eddy simulation, Flow Turbul. Combust. 70 (2003), 153–181.10.1023/B:APPL.0000004934.22265.74Search in Google Scholar
[4] S. Heinz, Statistical mechanics of turbulent flows, Springer, Berlin, Heidelberg, New York, 2003.10.1007/978-3-662-10022-6Search in Google Scholar
[5] S. Heinz, Unified turbulence models for LES and RANS, FDF and PDF simulations, Theor. Comput. Fluid Dyn. 21 (2007), 99–118.10.1007/s00162-006-0036-8Search in Google Scholar
[6] M. Lesieur, O. Métais, and P. Comte, Large-eddy simulations of turbulence, Cambridge University Press, Cambridge, UK, 2005.10.1017/CBO9780511755507Search in Google Scholar
[7] S. B. Pope, Turbulent flows, Cambridge University Press, Cambridge, UK, 2000.10.1017/CBO9780511840531Search in Google Scholar
[8] P. Sagaut, Large eddy simulation for incompressible flows, Springer, Berlin, Heidelberg, New York, 2002.10.1007/978-3-662-04695-1Search in Google Scholar
[9] M. Germano, A dynamic subgrid-scale eddy viscosity model, J. Fluid Mech. 238 (1992), 325–336.10.1063/1.857955Search in Google Scholar
[10] M. Germano, U. Piomelli, P. Moin and W. H. Cabot, A dynamic subgrid-scale eddy viscosity model, Phys. Fluids A 3 (1991), 1760–1765.10.1063/1.857955Search in Google Scholar
[11] S. Ghosal, T. S. Lund, P. Moin and A. N. D. K. Akselvoll, A dynamic localization model for large-eddy simulation of turbulent flows, J. Fluid Mech. 286 (1995), 229–255.10.1017/S0022112095000711Search in Google Scholar
[12] D. K. Lilly, A proposed modification of the Germano subgrid-scale closure method, Phys. Fluids A 4 (1992), 633–635.10.1063/1.858280Search in Google Scholar
[13] C. Meneveau, T. S. Lund and W. H. Cabot, A Lagrangian dynamic subgrid-scale model for turbulence, J. Fluid Mech. 319 (1996), 353–385.10.1017/S0022112096007379Search in Google Scholar
[14] H. Gopalan, S. Heinz and M. Stollinger, A unified RANS-LES model: computational development, accuracy and cost, J. Comput. Phys. 249 (2013), 249–279.10.1016/j.jcp.2013.03.066Search in Google Scholar
[15] S. Heinz, Realizability of dynamic subgrid-scale stress models via stochastic analysis, Monte Carlo Method. Appl. 14 (2008), 311–329.10.1515/MCMA.2008.014Search in Google Scholar
[16] S. Heinz and H. Gopalan, Realizable versus non-realizable dynamic sub-grid scale stress models, Phys. Fluids 24 (2012), 115105/1–23.10.1063/1.4767538Search in Google Scholar
[17] G. N. Coleman, Similarity statistics from a direct numerical simulation of the neutrally stratified planetary boundary layer, J. Atmos. Sci. 56 (1999), 891–900.10.1175/1520-0469(1999)056<0891:SSFADN>2.0.CO;2Search in Google Scholar
[18] G. N. Coleman, J. H. Ferziger and P. R. Spalart, A numerical study of the Ekman layer, J. Fluid Mech. 213 (1990), 313–348.10.1017/S0022112090002348Search in Google Scholar
[19] S. Marlatt, S. Waggy and S. Biringen, Direct numerical simulation of the turbulent Ekman layer: turbulent energy budgets, J. Thermophys. Heat Transfer 24 (2010), 544–555.10.2514/6.2010-1254Search in Google Scholar
[20] W. Marlatt, Direct numerical simulation of Ekman layer transition and turbulence, PhD Thesis, University of Colorado, Boulder, 1994.Search in Google Scholar
[21] K. Miyashita, K. Iwamoto and H. Kawamura, Direct numerical simulation of the neutrally stratified turbulent Ekman boundary layer, J. Earth Simul. 6 (2006), 3–15.Search in Google Scholar
[22] K. Shingai and H. Kawamura, Direct numerical simulation of turbulent heat transfer in the stably stratified Ekman layer, Therm. Sci. Eng. 10 (2002), 25–33.Search in Google Scholar
[23] K. Shingai and H. Kawamura, A study of turbulent structure and large-scale motion in the Ekman layer through direct numerical simulations, J. Turbul. 5 (2004), 1–18.10.1088/1468-5248/5/1/013Search in Google Scholar
[24] J. R. Taylor and S. Sarkar, Direct and large eddy simulations of a bottom Ekman layer under an external stratification, Int. J. Heat Fluid Flow 29 (2008), 721–732.10.1016/j.ijheatfluidflow.2008.01.017Search in Google Scholar
[25] S. Waggy, S. Marlatt, and S. Biringen, Direct numerical simulation of the turbulent EKMAN layer: instantaneous flow structures, J. Thermophys. Heat Transfer 25 (2011), 309–318.10.2514/1.50858Search in Google Scholar
[26] D. K. Lilly, The representation of small-scale turbulence in numerical simulation of experiments, in: Proceedings of the IBM Scientific Computing Symposium on Environmental Sciences. H. H. Goldstine, ed., pp. 195–210, IBM, Yorktown Heights, NY, 1967.Search in Google Scholar
[27] E. D. Villiers, The potential of large eddy simulation for the modelling of wall bounded flows, PhD Thesis, Imperial College, London, 2006.Search in Google Scholar
[28] P. J. Mason, Large-eddy simulation of the convective atmospheric boundary layer, J. Atmos. Sci. 46 (1989), 1492–1516.10.1175/1520-0469(1989)046<1492:LESOTC>2.0.CO;2Search in Google Scholar
[29] P. J. Mason and D. J. Thomson, Stochastic backscatter in large eddy simulation of boundary layers, J. Fluid Mech. 242 (1992), 51–78.10.1017/S0022112092002271Search in Google Scholar
[30] P. A. Durbin and C. G. Speziale, Realizability of second-moment closure via stochastic analysis, J. Fluid Mech. 280 (1994), 395–407.10.1017/S0022112094002983Search in Google Scholar
[31] S. B. Pope, On the relationship between stochastic Lagrangian models of turbulence and second-moment closures, Phys. Fluids 6 (1994), 973–985.10.1063/1.868329Search in Google Scholar
[32] U. Schumann, Realizability of Reynolds stress turbulence models, Phys. Fluids 20 (1977), 721–725.10.1063/1.861942Search in Google Scholar
[33] C. G. Speziale, R. Abid and P. A. Durbin, New results on the realizability of Reynolds stress turbulence closures, ICASE Report, 93–76 (1993), 1–47.Search in Google Scholar
[34] http://murasun.me.noda.tus.ac.jp/turbulence/.Search in Google Scholar
[35] E. Kazemi Foroushani, Direct and large eddy simulation of the turbulent Ekman layer, PhD Thesis, University of Wyoming, Laramie, 2014.Search in Google Scholar
[36] Openfoam, the open source cfd tool box, user guide, see www.openfoam.org, tech. report, April 2014.Search in Google Scholar
[37] S. Heinz, Mathematical modeling, 1st ed. Springer-Verlag, Heidelberg, Dordrecht, London, New York, 2011.Search in Google Scholar
[38] P. Saha, S. Heinz, E. Kazemi Foroushani and M. Stollinger, Turbulence structure characteristics of LES methods implied by stochastic turbulence models, in 52nd AIAA Aerospace Sciences Meeting and Exhibit, AIAA Paper 14–0592, National Harbor, MD, January 2014.10.2514/6.2014-0592Search in Google Scholar
[39] M. Bohnert and J. H. Ferziger, The dynamic subgrid scale model in large eddy simulation of the stratified Ekman layer, in: Engineering Turbulence Modelling and Experiments 2, W. Rodi and F. Martelli, eds, pp. 315–324, Elsevier, New York, 1993.10.1016/B978-0-444-89802-9.50034-4Search in Google Scholar
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