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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.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
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[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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