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Performance of under-resolved two-dimensional incompressible flow simulations. II. (English) Zbl 0914.76063
Summary: This paper presents a study of the behavior of several difference approximations for the incompressible Navier-Stokes equations as a function of the computational mesh resolution. In particular, the under-resolved case is considered. The methods considered include a Godunov projection method, a primitive variable ENO method, an upwind vorticity stream-function method, centered difference methods of both a pressure-Poisson and vorticity stream-function formulation, and a pseudospectral method. It is demonstrated that all these methods produce spurious, nonphysical vortices of the type described by the authors in part I [ibid. 122, No. 1, 165-183 (1995; Zbl 0849.76043)] when the flow is sufficiently under-resolved. The occurrence of these artifacts appears to be due to a nonlinear effect in which the truncation error of the difference method initiates a vortex instability in the computed flow. The implications of this study for adaptive mesh refinement strategies are also discussed. \(\copyright\) Academic Press.

76M20 Finite difference methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI
[1] Almgren, A.S.; Bell, J.B.; Colella, P.; Howell, L.H.; Welcome, M.L., A conservative adaptive projection method for the variable density incompressible navier – stokes equations, Preprint, LBNL-39075/UC-405, (1996)
[2] Almgren, A.S.; Bell, J.B.; Szymczak, W.G., A numerical method for the incompressible navier – stokes equations based on an approximate projection, LLNL-unclassified report, UCRL-JC-112842, (1993)
[3] Baker, G.R.; Shelley, M.J., On the connection between thin vortex layers and vortex sheets, J. fluid mech., 215, 161, (1990) · Zbl 0698.76029
[4] Bell, J.B.; Colella, P.; Glaz, H.M., A second order projection method for the incompressible navier – stokes equations, J. comput. phys., 85, 257, (1989) · Zbl 0681.76030
[5] J. B. Bell, P. Colella, L. H. Howell, June 1991, An efficient second-order projection method for viscous incompressible flow, Proceedings of the Tenth AIAA Computational Fluid Dynamics Conference, 360, 367, AIAA
[6] M. J. Berger, M. Aftosmis, J. Melton, 1996, Accuracy, adaptive methods and complex geometry, Proceedings of the First AFOSR Conference on Dynamic Motion CFD, June 1996
[7] Berger, M.J.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. comput. phys., 82, 64, (1989) · Zbl 0665.76070
[8] Berger, M.J.; Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equations, J. comput. phys., 53, 484, (1984) · Zbl 0536.65071
[9] Brown, D.L.; Minion, M., Performance of underresolved two-dimensional incompressible flow simulations, J. comput. phys., 121, (1995)
[10] Chorin, A.J., Numerical solution of the navier – stokes equations, Math. comp., 22, 742, (1968)
[11] Drazin, P.G.; Reid, W.H., Hydrodynamic stability, (1981), Cambridge Univ. Press Cambridge · Zbl 0449.76027
[12] Shu, W.E.; Shu, C.-W., A numerical resolution study of high order essentially non-oscillatory schemes applied to incompressible flow, ICASE rept., 92-39, (1992)
[13] Shu, W.E.; Shu, C.-W., A numerical resolution study of high order essentially non-oscillatory schemes applied to incompressible flow, J. comput. phys., 110, 39, (1994) · Zbl 0790.76055
[14] Gustafsson, B.; Kreiss, H.-O.; Oliger, J., Time dependent problems and difference methods, (1995), Wiley New York
[15] Harlow, F.H.; Welch, J.E., Numerical calculation of time-dependent viscous incompressible flow of fluids with free surfaces, Phys. fluids, 8, (1965) · Zbl 1180.76043
[16] Henshaw, W.D., A fourth-order accurate method for the incompressible navier – stokes equations on overlapping grids, J. comput. phys., 113, 13, (1994) · Zbl 0808.76059
[17] Henshaw, W.D.; Kreiss, H.-O.; Reyna, L., On the smallest scale for the incompressible navier – stokes equations, Theoret. comput. fluid dynam., 1, 65, (1989) · Zbl 0708.76043
[18] Kopriva, D.A., A practical assessment of spectral accuracy for hyperbolic problems with discontinuities, J. sci. comput., 2, 249, (1987) · Zbl 0666.65080
[19] M. F. Lai, J. Bell, P. Colella, 1993, A projection method for combustion in the zero Mach number limit, Proceedings of the Eleventh AIAA Computational Fluid Dynamics Conference, AIAA, June 1993, 776
[20] Levy, D.; Tadmor, E., Non-oscillatory central schemes for the incompressible navier – stokes equations, Tech. rep., 96-37, (1996)
[21] Liu, X.-D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J. comput. phys., 115, 200, (1994) · Zbl 0811.65076
[22] Majda, A.; McDonough, J.; Osher, S., The Fourier method for nonsmooth initial data, Math. comput., 77, 439, (1988)
[23] M. L. Minion, May1994, Two Methods for the Study of Vortex Patch Evolution on Locally Refined Grids, University of California, Berkeley
[24] Minion, M.L., A note on the stability of Godunov-projection methods, J. comput. phys., 123, (1996) · Zbl 0848.76050
[25] Minion, M.L., A projection method for locally refined grids, J. comput. phys., 127, (1996) · Zbl 0859.76047
[26] Jiang, G.Shan; Shu, C.Wang, Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 202, (1996) · Zbl 0877.65065
[27] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. comput. phys., 77, 439, (1988) · Zbl 0653.65072
[28] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, J. comput. phys., 83, 32, (1989) · Zbl 0674.65061
[29] D. E. Stevens, 1994, An Adaptive Multilevel Method for Boundary Layer Meteorology, University of Washington
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