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A conservative adaptive projection method for the variable density incompressible Navier-Stokes equations. (English) Zbl 0933.76055

We present a method for solving the equations governing time-dependent, variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids. The method is based on a projection formulation in which we first solve advection-diffusion equations to predict intermediate velocities, and then project these velocities onto a space of approximately divergence-free vector fields. Our treatment of the first step uses a specialized second-order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high Reynolds number flow. Density and other scalars are advected in such a way as to maintain conservation, if appropriate, and free-stream preservation. Numerical examples demonstrate the algorithms’s accuracy and convergence properties. \(\copyright\) Academic Press.

MSC:

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

Software:

Wesseling
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References:

[1] Almgren, A. S.; Bell, J. B.; Colella, P.; Howell, L. H., An adaptive projection method for the incompressible Euler equations, 11th AIAA Computational Fluid Dynamics Conference (1993)
[2] Almgren, A. S.; Bell, J. B.; Colella, P.; Howell, L. H.; Welcome, M., A high-resolution adaptive projection method for regional atmospheric modeling, Proceedings of the U.S. EPA NGEMCOM Conference (1995)
[3] Almgren, A. S.; Bell, J. B.; Howell, L. H.; Colella, P., An adaptive projection method for the incompressible Navier-Stokes equations, Proceedings of the IMACS 14th World Conference (1994)
[4] Almgren, A. S.; Bell, J. B.; Szymczak, W. G., A numerical method for the incompressible Navier-Stokes equations based on an approximate projection, SIAM J. Sci. Comput., 17 (1996) · Zbl 0845.76055
[5] Bell, J. B.; Berger, M. J.; Saltzman, J. S.; Welcome, M., Three-Dimensional Adaptive Mesh Refinement for Hyperbolic Conservation Laws (1991)
[6] 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
[7] Bell, J. B.; Colella, P.; Howell, L. H., An efficient second-order projection method for viscous incompressible flow, 10th AIAA Computational Fluid Dynamics Conference (1991)
[8] Bell, J. B.; Marcus, D. L., A second-order projection method for variable-density flows, J. Comput. Phys., 101, 334 (1992) · Zbl 0759.76045
[9] Berger, M. J.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys., 82, 64 (1989) · Zbl 0665.76070
[10] Berger, M. J.; Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys., 53, 484 (1984) · Zbl 0536.65071
[11] Berger, M. J.; Rigoustsos, I., An Algorithm for Point Clustering and Grid Generation (1991)
[12] Brown, G. L.; Roshko, A., On density effects and large structure in turbulent mixing layers, J. Fluid Mech., 64, 775 (1974) · Zbl 1416.76061
[13] Chien, K.-Y.; Ferguson, R. E.; Kuhl, A. L.; Glaz, H. M.; Colella, P., Inviscid dynamics of two-dimensional shear layers, Comput. Fluid Dyn., 5, 59 (1995)
[14] Clark, Terry L.; Farley, R. D., Severe downslope windstorm calculations in two and three spatial dimensions using anelastic interactive grid nesting: A possible mechanism for gustiness, J. Atmos. Sci., 41, 329 (1984)
[15] Colella, P., A direct Eulerian MUSCL scheme for gas dynamics, SIAM J. Comput., 6, 104 (1985) · Zbl 0562.76072
[16] Colella, P., A multidimensional second order Godunov scheme for conservation laws, J. Comput. Phys., 87, 171 (1990)
[17] Howell, L. H.; Bell, J. B., An adaptive-mesh projection method for viscous incompressible flow, SIAM J. Sci. Comput., 18, 996 (1997) · Zbl 0901.76057
[18] Konrad, J. H., An Experimental Investigation of Mixing in Two-Dimensional Turbulent Shear Flows with Applications to Diffusion-Limited Chemical Reactions (1977)
[19] McCormick, S. F., Multilevel Adaptive Methods for Partial Differential Equations (1989) · Zbl 0707.65080
[20] Minion, M. L., On the stability of Godunov-projection methods for incompressible flow, J. Comput. Phys., 123, 435 (1996) · Zbl 0848.76050
[21] Minion, M. L., A projection method for locally refined grids, J. Comput. Phys., 127, 158 (1996) · Zbl 0859.76047
[22] Monkewitz, P. A.; Huerre, P., Influence of the velocity ratio on the spatial instability of mixing layers, J. Physics of Fluids, 25, 1137 (1982)
[23] Puckett, E. G.; Almgren, A. S.; Bell, J. B.; Marcus, D. L.; Rider, W. G., A higher-order projection method for tracking fluid interfaces in variable density incompressible flows, J. Comput. Phys., 130, 269 (1997) · Zbl 0872.76065
[24] Sandberg, W.; Ramamurti, R.; Löhner, R., Simulation of Torpedo Launch Using a 3-D Incompressible Finite Element Solver and Adaptive Remeshing (1995)
[25] Ramamurti, R.; Löhner, R.; Sandberg, W., Evaluation of a Three-Dimensonal Finite Element Incompressible Flow Solver (1994)
[26] Skamarock, W. C.; Klemp, J. B., Adaptive grid refinement for two-dimensional and three-dimensional nonhydrostatic atmospheric flow, Mon. Weather Rev., 121, 788 (1993)
[27] Steinthorsson, E.; Modiano, D.; Crutchfield, W. Y.; Bell, J. B.; Colella, P., An adaptive semi-implicit scheme for simulations of unsteady viscous compressible flow, 12th AIAA Computational Fluid Dynamics Conference (1995)
[28] Stevens, D. E., An Adaptive Multilevel Method for Boundary Layer Meteorology (1994)
[29] Stevens, D. E.; Bretherton, C. S., A forward-in-time advection scheme and adaptive multilevel flow solver for nearly incompressible atmospheric flow, J. Comput. Phys., 129, 284 (1996) · Zbl 0869.76059
[30] Wesseling, P., An Introduction to Multigrid Methods (1992) · Zbl 0760.65092
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