Wind and Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Stabilized finite element technique and its application for turbulent flow with high Reynolds number

  • Huang, Cheng (School of Naval Architecture, Ocean and Civil engineering, Shanghai Jiao Tong University) ;
  • Yan, Bao (School of Naval Architecture, Ocean and Civil engineering, Shanghai Jiao Tong University) ;
  • Zhou, Dai (School of Naval Architecture, Ocean and Civil engineering, Shanghai Jiao Tong University) ;
  • Xu, Jinquan (School of Naval Architecture, Ocean and Civil engineering, Shanghai Jiao Tong University)
  • Received : 2009.12.11
  • Accepted : 2011.05.01
  • Published : 2011.09.25
⟨ Previous Next ⟩

Abstract

In this paper, a stabilized large eddy simulation technique is developed to predict turbulent flow with high Reynolds number. Streamline Upwind Petrov-Galerkin (SUPG) stabilized method and three-step technique are both implemented for the finite element formulation of Smagorinsky sub-grid scale (SGS) model. Temporal discretization is performed using three-step technique with viscous term treated implicitly. And the pressure is computed from Poisson equation derived from the incompressible condition. Then two numerical examples of turbulent flow with high Reynolds number are discussed. One is lid driven flow at Re = $10^5$ in a triangular cavity, the other is turbulent flow past a square cylinder at Re = 22000. Results show that the present technique can effectively suppress the instabilities of turbulent flow caused by traditional FEM and well predict the unsteady flow even with coarse mesh.

Keywords

References

  1. Bardina, J., Ferzuger, J.H. and Reynolds, W.C. (1980), "Improved subgrid scale models for large eddy simulation", AIAA paper, Fluid and Plasma Dynamics Conference.
  2. Bao, Y., Zhou, D. and Zhao Y.J. (2010a), "A two-step Taylor-characteristic-based Galerkin method for incompressible flows and its application to flow over triangular cylinder with different incidence angles", Int.J. Numer. Meth. Fl., 62(11), 1181-1208.
  3. Bao, Y., Zhou, D. and Huang, C. (2010b), "Numerical simulation of flow over three circular cylinders in equilateral arrangements at low Reynolds number by a second-order characteristic-based split finite element method", Comput. Fluids., 39(5), 882-899. https://doi.org/10.1016/j.compfluid.2010.01.002
  4. Becker, R. and Vexler, B. (2007), "Optimal control of the convection-diffusion equation using stabilized finite element methods" , Numer. Math., 106(3), 349-367. https://doi.org/10.1007/s00211-007-0067-0
  5. Brooks, A.N. and Hughes, T.J.R. (1982), "Streamline Upwind/Petrov Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations", Comput. Method. Appl. M., 32(1), 199-259. https://doi.org/10.1016/0045-7825(82)90071-8
  6. Buchan, A.G., Candy, A.S., Merton, S.R. and Pain, C.C. (2008), "The inner element sub-grid scale finite element method for the Boltzmann transport equation" , Preprint submitted to Elsevier.
  7. Collis, S.S. and Heinkenschloss, M. (2002), "Analysis of the streamline Upwind/Petrov Galerkin method applied to the solution of optimal control problems", CAAM TR02-01, March.
  8. Deardorff, J.W. (1970), "A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers", J. Fluid Mech., 41(2), 453-480. https://doi.org/10.1017/S0022112070000691
  9. Dettmer, W. and Peric, D. (2006), "A computational framework for fluid-structure interaction: Finite element formulation and applications", Comput. Appl. M., 195(41), 5754-5779.
  10. Donea, J., Giuliani, S., Laval, H. and Quartapelle, L. (1984), "Time-accurate solution of advection-diffusion problems", Comput. Method. Appl. M., 45(1-3), 123-146. https://doi.org/10.1016/0045-7825(84)90153-1
  11. Durao, D.F.G., Heitor, M.V. and Pereira, J.C.F. (1998), "Measurements of turbulent and periodic flows around a square cross-section cylinder", Exp. Fluids, 6(5), 298-304.
  12. Franca, L.P. and Frey, S.L. (1992), "Stabilized finite element methods: II. The incompressible Navier-Stokes equations", Comput. Method. Appl. M., 99(2), 209-233. https://doi.org/10.1016/0045-7825(92)90041-H
  13. Franke, R. and Rodi, W. (1993), "Calculation of vortex shedding past a square cylinder with various turbulent models", Turbulent Shear Flows VIII, 189-204.
  14. Germano, M., Piomelli, U., Moin, P. and Cabot, W.H. (1991), "A dynamic subgrid-scale eddy viecosity model", Phys. Fluid., 3(7), 1760-1765. https://doi.org/10.1063/1.857955
  15. Hasebe, H. and Nomura, T. (2009), "Finite element analysis of 2D turbulent flows using the logarithmic form of the k-å model", Wind Struct., 12(1), 21-47. https://doi.org/10.12989/was.2009.12.1.021
  16. Heinkenschloss, M. and Leykekhman, D. (2008), Local error estimates for supg solutions of advectiondominated elliptic linear-quadratic optimal control problems , Tech. Rep. TR08-30, Department of Computational and Applied Mathematics, Rice University.
  17. Hughes, T.J.R., Franca, L.P. and Balestra, M (1986), "A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations", Comput. Method. Appl. M., 59(1), 85-99. https://doi.org/10.1016/0045-7825(86)90025-3
  18. Hughes, T.J.R., Franca, L.P. and Hulbert, G.M. (1989), "A new finite element formulation for computational fluid dynamics: VIII, The Galerkin least-squares method for advective diffusive equations", Comput. Method. Appl. M., 73(2), 173-189. https://doi.org/10.1016/0045-7825(89)90111-4
  19. Itoh, Y. and Tamura, T. (2008), "Large eddy simulation of turbulent flows around bluff bodies in overlaid grid systems", J. Wind Eng. Ind. Aerod., 96(10), 1938-1946. https://doi.org/10.1016/j.jweia.2008.02.065
  20. Jeong, U.Y. and Koh, H.M. (2002), "Finite element formulation for the analysis of turbulent wind flow passing bluff structures using the RNG k-a model", J. Wind Eng. Ind. Aerd., 90(3), 159-169.
  21. Jiang, C.B. and Kawahara, M. (1993), "A three step finite element method for unsteady incompressible flows", Comput. Mech., 11(5), 355-370. https://doi.org/10.1007/BF00350093
  22. Jimenez, A., Crespo, A., Migoya, E. and Garcia, J. (2008), "Large-eddy simulation of spectral coherence in a wind turbine wake" , Environ. Res. Lett., 3(1).
  23. Lee, S. and Bienkiewicz, B. (1998), "Finite element implementation of large eddy simulation for separated flows", J. Wind Eng. Ind. Aerod., 77-78(2), 603-617. https://doi.org/10.1016/S0167-6105(98)00176-7
  24. Lyn, D.A., Einav, S., Rodi, W. and Park, J.H. (1995), "A laser-Doppler velocimeter study of ensemble-averaged characteristics of the turbulent near wake of a square cylinder", J. Fluid Mech., 304, 285-319. https://doi.org/10.1017/S0022112095004435
  25. Lyn, D.A. and Rodi, W. (1989), "Phase-averaged turbulence measurements in the separated shear region of flow around a square cylinder", Proceedings of the 23th cong. Ind. Ass. Hydraulic Research, Ottawa, August.
  26. Ma, S.W., Song. L.F., Ong, S.L. and Ng, W.J. (2004), "A 2-D streamline upwind Petrov/Galerkin finite element model for concentration polarization in spiral wound reverse osmosis modules" , J. Membrane Sci., 244(1-2), 129-139. https://doi.org/10.1016/j.memsci.2004.06.048
  27. Murakami, S. and Mochida, A. (1995), "On turbulent vortex-shedding flow past 2D square cylinder predicted by CFD", J. Wind Eng. Ind. Aerd., 54-55, 191-211. https://doi.org/10.1016/0167-6105(94)00043-D
  28. Onate, E. (1998), "Derivation of stabilized equations for advective-diffusive transport and fluid flow problems", Comput. Method. Appl. M., 151(1), 233-267. https://doi.org/10.1016/S0045-7825(97)00119-9
  29. Pain, C.C., Eaton, M.D., Smedley-Stevenson, R.P. and Piggott, M.D. (2006), "Space-time streamline upwind Petrov-Galerkin method for the Boltzmann transport equation" , Comput. Method. Appl. M.,195(33-36), 4334- 4357. https://doi.org/10.1016/j.cma.2005.09.005
  30. Ramakrishnan, S. and Collis, S.S., (2006), "Partition selection in multiscale turbulence modeling" , Phys.Fluid, 18(7).
  31. Selmin, V., Donea, J. and Quartapelle, L. (1985), "Finite element methods for nonlinear advection", Comput. Method. Appl. M., 52(1-3), 817- 845. https://doi.org/10.1016/0045-7825(85)90016-7
  32. Smagorinsky, T.S. (1963), "General circulation experiment with primitive equations: Part I, Basic experiments", Mon. Weather Rev., 91, 99-164. https://doi.org/10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2
  33. Tezduyar, T.E. (2007), "Finite elements in fluids: Stabilized formulations and moving boundaries and interfaces", Comput. Fluids, 36(2), 191-206. https://doi.org/10.1016/j.compfluid.2005.02.011
  34. Tutar, M. and Celik, I. (2007), "Large eddy simulation of a square cylinder flow: Modelling of inflow turbulence.", Wind Struct., 10(6), 511-532. https://doi.org/10.12989/was.2007.10.6.511
  35. Uchida, T. and Ohya, Y. (2003), "Large-eddy simulation of turbulent airflow over complex terrain" , J. Wind Eng. Ind. Aerod., 91(1-2), 219-229. https://doi.org/10.1016/S0167-6105(02)00347-1
  36. Yu, D.H. and Kareem, A. (1997), "Numerical simulation of flow around a rectangular prsim", J. Wind Eng. Ind. Aerod., 67-68, 195-208. https://doi.org/10.1016/S0167-6105(97)00073-1
  37. Zienkiewicz, O.C. and Codina, R. (1995), "A general algorithm for compressible and incompressible flow. Part I: The split characteristic based scheme", Int. J. Numer. Meth. Flu., 20(8), 869-885. https://doi.org/10.1002/fld.1650200812