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Numerical Convergence Study of Nearly Incompressible, Inviscid Taylor–Green Vortex Flow

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Abstract

A spectral method and a fifth-order weighted essentially non-oscillatory method were used to examine the consequences of filtering in the numerical simulation of the three-dimensional evolution of nearly-incompressible, inviscid Taylor–Green vortex flow. It was found that numerical filtering using the high-order exponential filter and low-pass filter with sharp high mode cutoff applied in the spectral simulations significantly affects the convergence of the numerical solution. While the conservation property of the spectral method is highly desirable for fluid flows described by a system of hyperbolic conservation laws, spectral methods can yield erroneous results and conclusions at late evolution times when the flow eventually becomes under-resolved. In particular, it is demonstrated that the enstrophy and kinetic energy, which are two integral quantities often used to evaluate the quality of numerical schemes, can be misleading and should not be used unless one can assure that the solution is sufficiently well-resolved. In addition, it is shown that for the Taylor–Green vortex (for example) it is useful to compare the predictions of at least two numerical methods with different algorithmic foundations (such as a spectral and finite-difference method) in order to corroborate the conclusions from the numerical solutions when the analytical solution is not known.

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References

  1. Gottlieb D., Orszag S.A. (1977). Numerical Analysis of Spectral Methods: Theory and Application, CBMS-NSF Regional Conference Series in Applied Mathematics Vol. 26, SIAM, Philadelphia

  2. D. Gottlieb M.Y. Hussaini S.A. Orszag (1984) Spectral Methods for Partial Differential Equations SIAM Philadelphia

    Google Scholar 

  3. C. Canuto M.Y. Hussaini A. Quarteroni T.A. Zang (1990) Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics Springer-Verlag New York

    Google Scholar 

  4. Canuto C. (1996). Spectral methods. In: Lesieur M., Comte P., Zinn-Justin J. (ed). Computational Fluid Dynamics, Proceedings of the Les Houches Summer School on Theoretical Physics 1993 Session LVIX, North-Holland, Amsterdam

  5. C. Bernardi Y. Maday (1997) Spectral methods P.G. Ciarlet J.L. Lions (Eds) Handbook of Numerical Analysis Vol. 5. North-Holland Amsterdam

    Google Scholar 

  6. Y. Morinishi T.S. Lund O.V. Vasilyev P. Moin (1998) ArticleTitleFully conservative higher order finite difference schemes for incompressible flow J. Comp. Phys. 143 90–124 Occurrence Handle10.1006/jcph.1998.5962

    Article  Google Scholar 

  7. F.E. Ham F.S. Lien A.B. Strong (2002) ArticleTitleA fully conservative second-order finite difference scheme for incompressible flow on nonuniform grids J. Comp. Phys. 177 117–133 Occurrence Handle10.1006/jcph.2002.7006

    Article  Google Scholar 

  8. J.G. Wissink (2003) ArticleTitleOn unconditional conservation of kinetic energy by finite- difference discretizations of the linear and non-linear convection equation Comp. and Fluids 33 315–343 Occurrence Handle10.1016/S0045-7930(03)00057-4

    Article  Google Scholar 

  9. F. Nicoud (2000) ArticleTitleConservative high-order finite-difference schemes for Low-Mach number flows J. Comp. Phys. 158 71–97 Occurrence Handle10.1006/jcph.1999.6408

    Article  Google Scholar 

  10. A.J. Majda A.L. Bertozzi (2002) Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics Vol. 27 Cambridge University Press Cambridge

    Google Scholar 

  11. D.L. Brown M.L. Minion (1995) ArticleTitlePerformance of under-resolved two-dimensional incompressible flow simulations J. Comp. Phys. 122 165–183 Occurrence Handle10.1006/jcph.1995.1205

    Article  Google Scholar 

  12. M.L. Minion D.L. Brown (1997) ArticleTitlePerformance of under-resolved two-dimensional incompressible flow simulations, II J. Comp. Phys. 138 734–765 Occurrence Handle10.1006/jcph.1997.5843

    Article  Google Scholar 

  13. D. Drikakis P.K. Smolarkiewicz (2001) ArticleTitleOn spurious vortical structures J. Comp. Phys. 172 309–325 Occurrence Handle10.1006/jcph.2001.6825

    Article  Google Scholar 

  14. G.K. Batchelor (2000) An Introduction to Fluid Dynamics Cambridge Mathematical Library Cambridge University Press Cambridge

    Google Scholar 

  15. J.T. Beale T. Kato A. Majda (1984) ArticleTitleRemarks on the breakdown of smooth solutions for the 3-D Euler equations Comm. Math. Phys. 94 61–66 Occurrence Handle10.1007/BF01212349

    Article  Google Scholar 

  16. Ponce G. (1985). Remarks on a paper by J.T. Beale, T. Kato, and A. Majda. Comm. Math. Phys. 98, 349–353

  17. A. Majda (1986) ArticleTitleVorticity and the mathematical theory of incompressible fluid flow Comm. Pure Appl. Math. 39 S187–S220

    Google Scholar 

  18. Orszag S.A. (1974). In Glowinski, R., Lions, J. L. (eds.), Proceedings of the Symposium on Computer Methods in Applied Sciences and Engineering, part II, Springer-Verlag, New York

  19. M.E. Brachet D.I. Meiron S.A. Orszag B.G. Nickel R.H. Morf U. Frisch (1983) ArticleTitleSmall-scale structure of the Taylor–Green vortex J. Fluid Mech. 130 411–452

    Google Scholar 

  20. Mossi M. (1999). Simulation of Benchmark and Industrial Unsteady Compressible Turbulent Fluid Flows, Ph.D. thesis, É cole Polytechnique Fé dé rale de Lausanne

  21. S. Benhamadouche D. Laurence (2002) ArticleTitleGlobal kinetic energy conservation with unstructured meshes Int. J. Num. Meth. Fluids 40 561–571 Occurrence Handle10.1002/fld.303 Occurrence HandleMR1932997

    Article  MathSciNet  Google Scholar 

  22. M.E. Brachet M. Meneguzzi H. Politano P.L. Sulem (1988) ArticleTitleThe dynamics of freely decaying two-dimensional turbulence J. Fluid Mech. 194 333–349

    Google Scholar 

  23. M.E. Brachet D.I. Meiron S.A. Orszag B.G. Nickel R.H. Morf U. Frisch (1984) ArticleTitleThe Taylor–Green vortex and fully-developed turbulence J. Stat. Phys. 34 1049–1063 Occurrence Handle10.1007/BF01009458

    Article  Google Scholar 

  24. M.E. Brachet (1991) ArticleTitleDirect simulation of three-dimensional turbulence in the Taylor–Green vortex Fluid Dyn. Res. 8 1–8

    Google Scholar 

  25. M.E. Brachet M. Meneguzzi A. Vincent H. Politano P.L. Sulem (1992) ArticleTitleNumerical evidence of smooth self-similar dynamics and possibility of subsequent collapse for three-dimensional Euler flows Phys. Fluids A 4 2845–2854 Occurrence Handle10.1063/1.858513

    Article  Google Scholar 

  26. R.H. Morf S.A. Orszag U. Frisch (1980) ArticleTitleSpontaneous singularity in three-dimensional inviscid incompressible flow Phys. Rev. Lett. 44 572–575 Occurrence Handle10.1103/PhysRevLett.44.572

    Article  Google Scholar 

  27. K. Ohkitani J.D. Gibbon (2000) ArticleTitleNumerical study of singularity formation in a class of Euler and Navier-Stokes flows Phys. Fluids 12 3181–3194 Occurrence Handle10.1063/1.1321256

    Article  Google Scholar 

  28. P.G. Saffman (1981) ArticleTitleDynamics of vorticity J. Fluid Mech. 106 49–58

    Google Scholar 

  29. C.-W. Shu S. Osher (1988) ArticleTitleEfficient implementation of essentially non-oscillatory shock-capturing schemes J. Comp. Phys. 77 439–471 Occurrence Handle10.1016/0021-9991(88)90177-5

    Article  Google Scholar 

  30. D. Gottlieb J.S. Hesthaven (2001) ArticleTitleSpectral methods for hyperbolic problems J. Comp. Appl. Math. 128 83–131 Occurrence Handle10.1016/S0377-0427(00)00510-0

    Article  Google Scholar 

  31. H. Vandeven (1991) ArticleTitleFamily of spectral filters for discontinuous problems J. Sci. Comp. 6 159–192 Occurrence Handle10.1007/BF01062118

    Article  Google Scholar 

  32. H.O. Kreiss J. Oliger (1972) ArticleTitleComparison of accurate methods for the integration of hyperbolic equations Tellus 24 199–215

    Google Scholar 

  33. L. Jameson (2000) ArticleTitleHigh order schemes for resolving waves: number of points per wavelength J. Sci. Comp. 15 417–439 Occurrence Handle10.1023/A:1011180613990

    Article  Google Scholar 

  34. X.-D. Liu S. Osher T. Chan (1994) ArticleTitleWeighted essentially non-oscillatory schemes J. Comp. Phys. 115 200–212 Occurrence Handle10.1006/jcph.1994.1187 Occurrence HandleMR1300340

    Article  MathSciNet  Google Scholar 

  35. G.-S. Jiang C.-W. Shu (1996) ArticleTitleEfficient implementation of weighted ENO schemes J. Comp. Phys. 126 202–228 Occurrence Handle10.1006/jcph.1996.0130

    Article  Google Scholar 

  36. D. Balsara C.-W. Shu (2000) ArticleTitleMonotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy J. Comp. Phys. 160 405–452 Occurrence Handle10.1006/jcph.2000.6443 Occurrence HandleMR1763821

    Article  MathSciNet  Google Scholar 

  37. Shu, C.-W. (1998). Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation Laws. In Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics Vol. 1697, Springer-Verlag, New York

  38. C.-W. Shu (1999) High order ENO and WENO schemes for computational fluid dynamics T.J. Barth H. Deconinck (Eds) High-Order Methods for Computational Physics. Springer-Verlag New York

    Google Scholar 

  39. A. Harten S. Osher B. Engquist S.R. Chakravarthy (1986) ArticleTitleSome results on uniformly high-order accurate essentially nonoscillatory schemes Appl. Num. Math. 2 347–377 Occurrence Handle10.1016/0168-9274(86)90039-5

    Article  Google Scholar 

  40. A. Harten S. Osher (1987) ArticleTitleUniformly high-order accurate non-oscillatory schemes SIAM J. Num. Anal. 24 279–309 Occurrence Handle10.1137/0724022

    Article  Google Scholar 

  41. A. Harten B. Engquist S. Osher R. Chakravarthy S. (1987) ArticleTitleUniformly high order essentially non-oscillatory systems, III J. Comp. Phys. 71 231–303 Occurrence Handle10.1016/0021-9991(87)90031-3

    Article  Google Scholar 

  42. C.-W. Shu S. Osher (1989) ArticleTitleEfficient implementation of essentially non-oscillatory shock-capturing schemes, II J. Comp. Phys. 83 32–78 Occurrence Handle10.1016/0021-9991(89)90222-2

    Article  Google Scholar 

  43. E. Tadmor (1989) ArticleTitleConvergence of spectral methods for nonlinear conservation laws SIAM J. Num. Anal. 26 30–44 Occurrence Handle10.1137/0726003

    Article  Google Scholar 

  44. Tadmor, E. (1989). Shock Capturing by the Spectral Viscosity Method. In Proceedings of ICOSAHOM 89, IMACS, North-Holland, Amsterdam

  45. Y. Maday S. Ould Kaber E. Tadmor (1993) ArticleTitleLegendre pseudospectral viscosity method for nonlinear conservation laws SIAM J. Num. Anal. 30 321–342 Occurrence Handle10.1137/0730016

    Article  Google Scholar 

  46. H. Ma (1998) ArticleTitleChebyshev–Legendre super spectral viscosity method for nonlinear conservation laws SIAM J. Num. Anal. 35 869–892 Occurrence Handle10.1137/S0036142995293900

    Article  Google Scholar 

  47. M. Carpenter D. Gottlieb C.-W. Shu (2003) ArticleTitleOn the conservation and convergence to weak solutions of global schemes J. Sci. Comp. 18 111–132 Occurrence Handle10.1023/A:1020390212806

    Article  Google Scholar 

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Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI, 02912, USA

    Chi-Wang Shu, Wai-Sun Don & David Gottlieb

  2. Lawrence Livermore National Laboratory, University of California, P.O. Box 808, Livermore, CA, 94551, USA

    Oleg Schilling & Leland Jameson

Authors
  1. Chi-Wang Shu
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  2. Wai-Sun Don
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  3. David Gottlieb
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  4. Oleg Schilling
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  5. Leland Jameson
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Correspondence to Chi-Wang Shu.

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Shu, CW., Don, WS., Gottlieb, D. et al. Numerical Convergence Study of Nearly Incompressible, Inviscid Taylor–Green Vortex Flow. J Sci Comput 24, 1–27 (2005). https://doi.org/10.1007/s10915-004-5407-y

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