Some current CFD issues relevant to the incompressible Navier-Stokes equations. (English) Zbl 0760.76018

Gresho, P. M.

Comput. Methods Appl. Mech. Eng. 87, No. 2-3, 201-252 (1991).

Summary: The goals of this paper are to carefully define a particular class of well-set incompressible Navier-Stokes problems in the continuum (partial differential equation/PDE) setting and to discuss some relevant and sometimes poorly understood issues related to these well-posed PDE problems, both in the continuum world and in its computer counterpart. Emphasized are three common formulations used to generate computer codes: the primitive variable (velocity-pressure) formulation and two derived variable formulations (pressure Poisson equation and vorticity transport equation).

MSC:

76D05	Navier-Stokes equations for incompressible viscous fluids
76M99	Basic methods in fluid mechanics

Keywords:

primitive variable (velocity-pressure) formulation; pressure Poisson equation; vorticity transport equation

References:

[1]	Gresho, P. M., Some interesting issues in incompressible fluid dynamics, both in the continuum and in numerical simulation, Adv. in Appl. Mech., 28 (1991), to appear. · Zbl 0741.76011
[2]	Gresho, P. M., Incompressible fluid dynamics: Some fundamental formulation issues, (Ann. Rev. Fluid Mech., 23 (1991), Annual Reviews: Annual Reviews Palo Alto, CA), 413-453 · Zbl 0717.76006
[3]	Engleman, M. S.; Sani, R. L.; Gresho, P. M., The implementation of normal and/or tangential boundary conditions in finite element codes for incompressible fluid flow, Internat. J. Numer. Methods Fluids, 2, 225 (1982) · Zbl 0501.76001
[4]	Orszag, S. A., Numerical simulation of incompressible flows within simple boundaries. 1. Galerkin (spectral) representations, Stud. Appl. Math., 50, 4, 293-327 (1971) · Zbl 0237.76012
[5]	Zang, T. A.; Krist, S. E., Numerical experiment on stability and transition in plane channel flow, Theoret. and Comput. Fluid Dyn., 1, 41 (1989) · Zbl 0712.76041
[6]	Pironneau, O., Finite Element Methods for Fluids (1989), Wiley: Wiley Chichester · Zbl 0665.73059
[7]	Gunzburger, M. D., Finite Element Methods for Viscous Incompressible Flows, (A Guide to Theory, Practice, and Algorithms (1989), Academic Press: Academic Press Boston) · Zbl 0601.76019
[8]	Orszag, S. A., Numerical simulation of incompressible flows within simple boundaries: Accuracy, J. Fluid Mech., 49, 75-112 (1971) · Zbl 0229.76029
[9]	Hussaini, M. Y.; Zang, T. A., Spectral methods in fluid dynamics, Ann. Rev. Fluid Mech., 19, 339 (1987) · Zbl 0636.76009
[10]	Ferziger, J. H., Review: Simulation of incompressible turbulent flows, J. Comput. Phys., 69, 1 (1987) · Zbl 0638.76070
[11]	Zang, T. A., On the rotation and skew-symmetric forms for incompressible flow simulations, Appl. Numer. Math., 7, 1, 27-40 (1991) · Zbl 0708.76071
[12]	Jiang, B.-N.; Chang, C. L., Least-squares finite elements for the Stokes problem, Comput. Methods Appl. Mech. Engrg., 78, 297-311 (1990) · Zbl 0706.76033
[13]	Jiang, B.-N., A least-squares finite element method for incompressible Navier Stokes problems, Internat. J. Numer. Methods Fluids (1991), in press.
[14]	Gresho, P. M.; Sani, R. L., On pressure boundary conditions for the incompressible Navier Stokes equations, Internat. J. Numer. Methods Fluids, 7, 1111 (1987) · Zbl 0644.76025
[15]	Glowinski, R.; Pironneau, O., On a mixed finite element approximation of the Stokes problem (I): Convergence of the approximate solution, Numer. Math., 33, 397 (1979), (Part II of this reference is in [16].) · Zbl 0423.65059
[16]	Glowinski, R., Numerical Methods for Non-linear Variational Problems (1984), Springer: Springer New York · Zbl 0575.65123
[17]	Batchelor, G. K., An Introduction to Fluid Dynamics (1967), Univ. Press: Univ. Press Cambridge · Zbl 0152.44402
[18]	Strang, G., Introduction to Applied Mathematics (1986), Wellesley-Cambridge Press: Wellesley-Cambridge Press Wellesley · Zbl 0618.00015
[19]	Strang, G., A framework for equilibrium equations, SIAM Rev., 30, 2, 283 (1988) · Zbl 0900.49028
[20]	Gresho, P. M.; Sani, R. L., (Announcement — open boundary conditions (OBC) minisymposium. Announcement — open boundary conditions (OBC) minisymposium, Internat J. Numer. Methods Fluids, 11 (1990)), 952, (7)
[21]	Christodoulou, K. N.; Scriven, L. E., The fluid mechanics of slide coating, J. Fluid Mech., 208, 321 (1989) · Zbl 0681.76040
[22]	Kistler, S. F.; Scriven, L. E., Coating flow theory by finite element and asymptotic analysis of the Navier Stokes system, Internat. J. Numer. Methods Fluids, 4, 3, 207 (1984) · Zbl 0555.76026
[23]	Sackinger, P. A.; Brown, R. A.; Derby, J. J., A finite element method for analysis of fluid flow, heat transfer, and free interfaces in Czochralski crystal growth, Internat. J. Numer. Methods Fluids, 9, 453 (1989)
[24]	Leone, J. L.; Gresho, P. M., Finite element simulations of two-dimensional, viscous incompressible flow over a step, J. Comput. Phys., 41, 1, 167 (1981) · Zbl 0464.76038
[25]	Van Kan, J., A second-order accurate pressure-correction scheme for viscous incompressible flow, SIAM J. Sci. Statist. Comput., 7, 870 (1986) · Zbl 0594.76023
[26]	Lowery, P. S.; Reynolds, W. C., Numerical simulation of a spatially-developing, forced, plane mixing layer, (Report TF-26 (1986), Thermosciences Division, Department of Mechanical Engineering, Stanford University: Thermosciences Division, Department of Mechanical Engineering, Stanford University Stanford, CA)
[27]	Gresho, P. M.; Sani, R. L., (Minisymposium on Outflow Boundary Conditions, Univ. College of Swansea. Minisymposium on Outflow Boundary Conditions, Univ. College of Swansea, UK (10 July 1989)), (No Proceedings, but the ‘radiation BC’ was prominent among those utilized at this informal meeting).
[28]	Orszag, S. A.; Israeli, M., Numerical simulation of viscous incompressible flows, Ann. Rev. Fluid Mech., 6, 281 (1974)
[29]	Peyret, R.; Taylor, T. D., Computational Methods for Fluid Flow (1983), Springer: Springer New York · Zbl 0514.76001
[30]	Quartapelle, L.; Napolitano, M., Integral conditions for the pressure in the computation of incompressible viscous flows, J. Comput. Phys., 62, 340 (1986) · Zbl 0604.76021
[31]	Sani, R.; Eaton, B.; Gresho, P.; Upson, C., On outflow boundary conditions for stratified and/or rotating flows, (Proc. 5th Internat. Conf. Finite Elements in Flow Problems (1989), The University of Texas: The University of Texas Austin), 85-91
[32]	Roache, P. J., Computational Fluid Dynamics (1982), Hermosa Publishers: Hermosa Publishers New Mexico
[33]	Schüller, A., A multigrid algotithm for the incompressible Navier-Stokes equations, (Hackbusch, W.; Rannacher, R., Numerical Treatment of the Navier-Stokes Equation, Notes on Numerical Fluid Mechanics, Vol. 30 (1990), Vieweg: Vieweg Braunschweig), 124-133
[34]	Stevens, W. N.R., Finite element stream function vorticity solution of steady laminar natural convection, Internat. J. Numer. Methods Fluids, 2, 349 (1982) · Zbl 0497.76078
[35]	Glowinski, R.; Pironneau, O., Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem, SIAM Rev., 21, 2, 167 (1979) · Zbl 0427.65073
[36]	Campion-Renson, A.; Crochet, M. J., On the stream function-vorticity finite element solutions of Navier Stokes equations, Internat. J. Numer. Methods Engrg., 12, 1809 (1978) · Zbl 0394.76032
[37]	Thomasset, F., Implementation of Finite Element Methods for Navier Stokes Equations (1981), Springer: Springer New York · Zbl 0475.76036
[38]	Mizukami, A., A stream function-vorticity finite element formulation for Navier Stokes equations in multi-connected domain, Internat. J. Numer. Methods Engrg., 19, 1403 (1983) · Zbl 0519.76029
[39]	Girault, V.; Raviart, P.-A., Finite Elements Methods for Navier Stokes equations. Theory and algorithms (1986), Springer: Springer Berlin
[40]	Gunzburger, M. D.; Nicolaides, R. A.; Liu, C. H., Algorithmic and theoretical results on computation of incompressible viscous flows by finite element methods, Comput. & Fluids, 13, 3, 361 (1985) · Zbl 0572.76030
[41]	Gunzburger, M. D.; Peterson, J. S., On finite element approximations of the stream function-vorticity and velocity-vorticity equations, Internat. J. Numer. Methods Fluids, 8, 1229 (1988) · Zbl 0667.76044
[42]	Gunzburger, M. D.; Peterson, J. S., Finite element methods for the stream function-vorticity equations: Boundary condition treatments and multiply-connected domains, SIAM J. Sci. Statist. Comput., 4, 650 (1988) · Zbl 0652.76020
[43]	Tezduyar, T. E.; Glowinski, R.; Liou, J., Petrov-Galerkin methods on multiply-connected domains for the vorticity-stream function formulation of the incompressible Navier Stokes equations, Internat. J. Numer. Methods Fluids, 8, 1269 (1988) · Zbl 0667.76046
[44]	Bristeau, M. O.; Glowinski, R.; Periaux, J., Numerical methods for the Navier Stokes equations. Applications to the simulation of compressible and incompressible viscous flows, (Computational Physics Reports, 6 (1987), North-Holland: North-Holland Amsterdam), 73
[45]	Tezduyar, T. E., Finite element formulation for the vorticity-steam function form of the incompressible Euler equations on multiply-connected domains, Comput. Methods Appl. Mech. Engrg., 73, 3, 331-339 (1989) · Zbl 0687.76021
[46]	Tezduyar, T. E., Solution techniques for the vorticity-stream function formulation of two-dimensional unsteady incompressible flows, Internat. J. Numer. Methods Fluids (1989), in press. · Zbl 0687.76021
[47]	Heywood, J. D., The Navier Stokes equations: On the existence, regularity, and decay of solutions, Indiana Univ. Math. J., 25, 4, 639 (1980) · Zbl 0494.35077
[48]	Heywood, J. D.; Rannacher, R., Finite element approximation of the non-stationary Navier Stokes problem. Part 1: Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal., 19, 2, 275 (1982) · Zbl 0487.76035
[49]	Morino, L., Helmholtz decomposition revisited: Vorticity generation and trailing edge condition. Part 1: Incompressible flows, J. Comput. Mech., 1, 65 (1986) · Zbl 0625.76027
[50]	Orszag, S. A.; Israeli, M.; Deville, M. O., Boundary conditions for incompressible flows, J. Sci. Comput., 1, 1, 75 (1986) · Zbl 0648.76023
[51]	Tezduyar, T. E.; Liou, J., On the downstream boundary conditions for the vorticity-stream function formulation of two-dimensional incompressible flows, (Report UMSI 89/145 (September 1989), Univ. of Minnesota Supercomputer Institute) · Zbl 0825.76129
[52]	Sedov, L. I., Impulsive motion in incompressible fluids, (Two-dimensional Problems in Hydrodynamics and Aerodynamics (1965), Interscience: Interscience New York) · Zbl 0063.06852
[53]	Telionis, D. P., Unsteady Viscous Flows (1981), Springer: Springer New York · Zbl 0484.76054
[54]	Weinbaum, S., On the singular points in the laminar two-dimensional near wake flow field, J. Fluid Mech., 33, 38 (1968) · Zbl 0159.27804
[55]	Brenan, K. E.; Campbell, S. L.; Petzold, L. R., Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations (1989), North-Holland: North-Holland New York · Zbl 0699.65057
[56]	Gresho, P. M.; Sani, R. L.; Engleman, M. S., Incompressible Flow and the Finite Element Method (1992), Wiley: Wiley New York
[57]	Lötstedt, P.; Petzold, L., Numerical solution of nonlinear differential equations with constraints. I: Convergence results for backward differentiation formulas, Math. Comp., 46, 174, 491 (1986) · Zbl 0601.65060
[58]	Petzold, L.; Lötstedt, P., Numerical solution of nonlinear differential equations with contraints. II: Practical implications, SIAM J. Sci. Statist. Comput., 7, 3, 720 (1986) · Zbl 0632.65086
[59]	Gresho, P. M.; Chan, S. T., On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly-consistent mass matrix. Part II: Implementation, Internat. J. Numer. Methods Fluid, 11, 5, 621 (1990) · Zbl 0712.76036
[60]	Ghia, K.; Hanky, W.; Hodge, J., Use of primitive variables in the solution of incompressible Navier Stokes equations, AIAA J., 17, 3, 298 (1979) · Zbl 0403.76038
[61]	Alfrink, B., On the Neumann problem for the pressure in a Navier Stokes model, (Proc. 2nd Internat. Conf. Numerical Methods in Laminar and Turbulent Flow (1985), Pineridge Press: Pineridge Press Swansea), 389 · Zbl 0483.76037
[62]	Kawahara, M.; Ohmiya, K., Finite element analysis of density flow using the velocity correction method, Internat. J. Numer. Methods Fluids, 5, 981 (1985) · Zbl 0575.76005
[63]	Gresho, P. M., An analysis of the velocity correction method of M. Kawahara et al, (Bull. Faculty of Science and Engrg., 30 (1987), Chuo University: Chuo University Tokyo, Japan), 43
[64]	Anderson, C. R., Vorticity boundary conditions and boundary vorticity generation for two-dimensional viscous incompressible flows, J. Comput. Phys., 80, 72 (1989) · Zbl 0656.76034
[65]	Gresho, P. M.; Chan, S. T.; Lee, R. L.; Upson, C. D., A modified finite element method for solving the time-dependent, incompressible Navier-Stokes equations, Part I: Theory, Internat. J. Numer. Methods Fluids, 4, 6, 557 (1984) · Zbl 0559.76030
[66]	Gresho, P. M.; Lee, R. L., Don’t suppress the wiggles — They’re telling you something!, J. Comput. Fluids, 9, 2, 223 (1981) · Zbl 0436.76065
[67]	Gresho, P. M.; Lee, R. L.; Sani, R. L., On the time dependent solution of the incompressible Navier Stokes equations in two and three dimensions, (Recent advances in numerical Methods in Fluids (1980), Pineridge Press: Pineridge Press Swansea), 27 · Zbl 0446.76034
[68]	Gresho, P. M., On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly-consistent mass matrix. Part I: Theory, Internat. J. Numer. Methods. Fluids, 11, 5, 587 (1990) · Zbl 0712.76035
[69]	Chorin, A. J., The numerical solution of the Navier Stokes equations for an incompressible fluid, Bull. Amer. Math. Soc., 73, 6, 928 (1967) · Zbl 0168.46501
[70]	Chorin, A. J., Numerical solution of the Navier Stokes equations, Math. Comp., 22, 745 (1968) · Zbl 0198.50103
[71]	Chorin, A. J., On the convergence of discrete approximations to the Navier Stokes equations, Math. Comp., 23, 106, 341 (1969) · Zbl 0184.20103
[72]	Teman, R., Sur l’approximation de la solution des équations de Navier-Stokes par la methode des pas fractionaires I, Arch. Rat. Mech. Anal., 32, 2, 135 (1969) · Zbl 0195.46001
[73]	Teman, R., Sur l’approximation de la solution des équations de Navier-Stokes par la methode des pas fractionaires II, Arch. Rat. Mech. Anal., 33, 377 (1969) · Zbl 0207.16904
[74]	Bell, J. B.; Colella, P.; Glaz, H., A second-order projection method for the incompressible Navier Stokes equations, J. Comput. Phys., 85, 2, 257 (1989) · Zbl 0681.76030
[75]	Canute, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods in Fluid Dynamics (1988), Springer: Springer New York
[76]	Hung, S. C.; Kinney, R. B., Unsteady viscous flow over a grooved wall: A comparison of two numerical methods, Internat. J. Numer. Methods Fluids, 8, 11, 1403 (1988) · Zbl 0664.76042