本文以大尺度渦流模擬法(LES)搭配dynamic Smagorinsky model(D.S.M.)之次格點模式計算三維背向階梯紊流場。模擬條件為18400之雷諾數，其以階梯高度(h)與自由流速度(U_c)來定義，擴張比則為1.66。將模擬結果與Fessler and Eaton (1999)之實驗數據做比較。並針對背向階梯流場之現象做綜合討論。
由於實驗值部分缺少第三維(w方向)之紊流擾動數據，因此利用二維模擬數值加上三分之二的次格點動能構成模擬之部分紊流動能來與實驗值做比較，發現在各個截面LES模擬結果與實驗結果皆有不錯的符合度。流場暫態行為方面，瞬時再接觸點之時序資料指出，再接觸長度在平均再接觸點7.39 h前後約略1 h的地方來回擺盪。而頻譜圖則顯示其暫態行為之頻率大約小於分離點後之渦流溢放頻率一個數量級，與實驗觀察到之結果吻合。本文也歸納出，為確保經次格點模式計算之渦流大小符合等向性值，根據Pope (2004)提出之看法，次格點紊流動能必須小於總紊流動能之20%；而在格點配置方面，因剪流層為背向階梯流場之重要機制且其影響會持續影響至相當下游，所以除了迴流區以外，往下游延伸之剪流層區亦需要做額外之加密。

Turbulent flow over backward-facing step was simulated by large eddy simulation (LES) model with the dynamic Smagorinsky sub-grid model (DSM). The computational case is conducted at a Reynolds number of 18400 based on the step height (h) and the inlet free-stream velocity (U_c) with the expansion ratio of 1.66. The the predictions are compared with available experimental data performed by Fessler and Eaton (1999). The turbulent characteristics the characteristic the backward-facing step flow are also discussed in this study.
For lack of w-direction fluctuation component of experimental data, the predicted partial turbulent kinetic energy, which is defined as two fluctuation components (u_rms^ , v_rms^ ) plus two-third of sub-grid turbulent kinetic energy, is adapted for the comparison with available measured data. It is found that both the predicted mean stream-wise velocities as well as partial turbulent kinetic energy are in good agreement with experimental data. The temporal simulation results indicate that the reattachment point is not fixed but oscillates forth and back over a range of 1 step-height distance, which is in consistent with the reported observation in Eaton and Johnston’s experiment (1980). Moreover, the fact that the moving frequency of “flapping motion” is approximately one order of magnitude less than that of “vortex shedding” can be easily observed in energy spectral distributions at the respective locations near the mean reattachment point and the separation point. It is also concluded that, according to the suggestion Pope (2004), at most 20% of total turbulent kinetic energy should be resolved in the sub-grid scale of eddies in the calculation with the sub-grid turbulence model to meet the isotropic characteristics of sub-grid eddies. The shear layer is a critical mechanism for development of the backward facing step flow. Therefore, the highly grid resolution is essential not only for recirculation zone but also for shear layer extending to the downstream region far from step.

[1].Keating, A., Piomelli U., Bremhorst K., and Nesic S., “Large eddy simulation of heat transfer downstream of a backward-facing step,” J. Turbulence. Vol. 5, 2004.
[2].Aider, J. L., and Danet A., “Large-eddy simulation study of upstream boundary conditions influence upon a backward-facing step flow,” C. R. Mecanique, Vol. 334, pp. 447-453, 2006.
[3].Armaly, B. F., Durst, F., Pereira, J. C. F., and Schonung, B., “Experimental and Theoretical Investigation of Backward-Facing Step Flow,” J. Fluid Mech., Vol. 127, pp. 473-496, 1983.
[4].Smirnov A., Shi S., and Celik, I., “Random Flow Generation Technique for Large Eddy Simulation and Particle Dynamics Modelings,” J. Fluids Engineering, Vol. 123, pp.359-371, 2001.
[5].Leonard B. P., “The ultimate conservative difference scheme applied to unsteady one-dimensional advection”, Computer Methods in applied Mechanics and Engineering, 88, pp. 17-74, 1991.
[6].Bradshaw, P., and Wong, F. Y. F., “The reattachment and relaxation of a turbulent shear layer,” J. Fluid Mech, Vol. 52, pp. 113-135, 1972.
[7].Brederode, V. de, and Bradshaw, P., “Three-dimensional flow in nominally two-dimensional separation bubbles,” Imperial College of Science and Technology, Technical Aero Report No. 72-19, 1972
[8].Chan, C. K., Zhang, H. Q., and Lau, K. S. “Numerical simulation of gas-particle flow behind a backward-facing step using an improved stochastic separated flow model.”, Computational Mechanics, Vol. 5, pp. 412-417, 2001.
[9].Chan, C. K., Lau, K. S., and Yu, K. F., “Numerical simulation of gas-particle flow in a single-side backward-facing step flow,” J. Computational and applied mathematics, Vol. 163, pp. 319-331, 2004.
[10].Driver, D. M., Seegmiller, H. L., “Features of a Reattaching Turbulent Shear Layer in Divergent Channel Flow,” AIAAJ, Vol. 25, No. 7, pp. 914-919, 1987.
[11].Durst, F. and Tropea, C., “Turbulent backward-facing step flows in two-dimensional ducts and channels,” In Proc. Third Intl Symp. On Turbulent Shear Flows, University of California, Davis, pp. 18.1-18.5, 1981.
[12].Eaton, J. K. and Johnston, J. P. “A Review of Research on Subsonic Turbulent Flow Reattachment,” AIAAJ, Vol. 19, No. 9, pp.1093-1100, 1981.
[13].Eaton, J. K. and Johnston, J. P. “Low Frequency Unsteadyness of a Reattaching Turbulent Shear Layer,” Turbulent Shear Flow 3, pp. 161-170, 1982.
[14].Fessler, J. R., and Eaton, J. K., “Turbulence modification by particles in a backward-facing step flow,” J. Fluid Mech., Vol. 394, pp. 97-117, 1999.
[15].FLUENT 6.3.26 User’s Guide. FLUENT Inc., 2006.
[16].Friderich,“Analysing turbulent backward-facing step flow with the lowpass-filtered Navier-Stokes equations”. J. Wind Engng Indust Aerodyn., Vol. 35, pp. 101-128, 1990.
[17].Germano, M., Piomelli, U., Moin, P., and Cabot, W.H., “A dynamic Subgrid-Scale Eddy Viscosity Model,” Phys. Fluid A, Vol. 3, pp. 1760-1765, 1999.
[18].Grigoriadis D.G.E., Bartzis J.G., Goulas A. “Efficient treatment of complex geometries for large eddy simulation of turbulent flows” J. Computer and Fluids, Vol.33, pp. 201-222, 2004.
[19].Choi H., and Moin P., “Effects of the computational time step on numerical solutions of turbulent flow”, J. Computational Physics, Vol. 113, 1994, pp. 1-4
[20].Jeon S., Choi H., Yoo J. Y., and Moin P., “Space-time characteristics of the wall shear stress fluctuations in a low Reynold number flow.”, Phys. Fluids, Vol. 11, pp. 3084-3094, 1999.
[21].J.L.Aider and A.Danet “Large eddy simulation study of upstream boundary conditions influence upon a backward-facing step flow,” Comptes Rendus Mecanique, Vol.334(7), pp. 447-453, 2006.
[22].J.R. Fessler., and J.K. Eaton, “Turbulence modification by particles in a backward-facing step flow,” J. Fluid Mech., Vol. 314, pp. 97-117, 1999.
[23].Kaiktsis L., Karniadakis GE., and Orszag SA.,“Onset of three-dimensionality, equilibria, and early transition in flow over a backward facing step,” J. Fluid Mech., Vol. 231, pp. 501-528, 1991.
[24].Kim, W. W., and Menon, S., “Application of the localized dynamic sub-grid-scale model to turbulent wall-bounded flows,” Technical Report AIAA-97-0210, American Institute of Aeronautics and Astronautics, 35th Aerospace Sciences Meeting, Reno, NV, 1997.
[25].Lee, H., and Moin, P., “Direct simulations of turbulent flow over backward-facing step.”, Rep. TF-58. Thermosciences Division, Dept. of Mechanical Engineering, Stanford University, 1997.
[26].Lilly, D. K., “A proposed modification of Germano subgrid scale closure method,” Phys. Fluid A., Vol. 4, pp-633-635.
[27].Mathey, F., and Cokljat, D., and Bertoglio, J.P. and Sergent, E. “Assessment of the vortex method for Large Eddy Simulation inlet conditions”, Progress Comp. Fluid Dynam., Vol. 6, pp.58-67, 2006.
[28].Panjwani B, Ertesvåg IS, Gruber A, Rian KE. “Large eddy simulation of backward facing step flow,” Fifth National Conference on Computational Mechanics, MekIT'09, Trondheim, Norway. (2009)
[29].Pope., S. B., “Ten questions concerning the large-eddy simulation of turbulent flow,” New J Phys., Vol. 6, pp. 1-24, 2004.
[30].SAGAUT, P., Large eddy simulation for incompressible flows. Springer,Berlin, 2002.
[31].Simpson, R. L., “Turbulent Boundary-Layer Separation,” Ann. Rev. Fluid Mech., Vol. 21, pp.205-234, 1989.
[32].Sinha, S. N., Gupta, A. K. and Oberai, M. M. “Laminar Separating Flow Over Backsteps and Cavities .1. Backsteps,” AIAAJ, Vol. 19, pp.1527-1530, 1981.
[33].Smagorinsky, J., “General Circulation Experiments with the Primitive Equations I, The Basic Experiment,” Mon. Wea. Rev., Vol. 91, 1963, pp. 99-164.
[34].Spazzini, P. G., Iuso, G., Onorato, M., Zurlo, N., and Cicca, G. M. D., “Unsteady Behavior of Back-Facing Step Flow,” Experiments in Fluids, Vol. 30, pp. 551-561, 2001.
[35].Tihon, J., Legrand, J., and Legentilhomme, P., “Near-wall investigation of backward-facing step flows,” Experiments in Fluids, Vol. 31, pp. 484-493, 2001.
[36].Westphal, R. V. and Johnston, J. P. “Effect of initial conditions on turbulent reattachment downstream of a backward-facing step,” AIAAJ, Vol. 22, pp. 1727-1731, 1984