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SUPG finite element computation of compressible flows with the entropy and conservation variables formulations. (English) Zbl 0772.76037

SUPG-stabilized finite element formulations of compressible Euler equations based on the conservation and entropy variables are investigated and compared. The formulation based on the conservation variables consists of the formulation introduced by T. E. Tezduyar and T. J. R. Hughes [“Development of time-accurate finite element techniques for first-order hyperbolic systems …”, Rep. NASA-Ames Univ. Consort. Interchange, No. NCA 2-OR745-104 (1982)] plus a shock capturing term. The formulation based on the entropy variables is the same as the one by T. J. R. Hughes, L. P. Franca and M. Mallet [Comput. Methods Appl. Mech. Eng. 54, 223-234 (1986; Zbl 0581.76077)], which has a shock capturing term built in. These formulations are tested on several subsonic, transonic and supersonic compressible flow problems.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics

Citations:

Zbl 0581.76077
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References:

[1] Tezduyar, T. E.; Hughes, T. J.R., Development of time-accurate finite element techniques for first-order hyperbolic systems with particular emphasis on the compressible Euler equations, Report prepared under NASA-Ames University Consortium Interchange, No. NCA2-OR745-104 (1982)
[2] Hughes, T. J.R.; Franca, L. P.; Mallet, M., A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics, Comput. Methods Appl. Mech. Engrg., 54, 223-234 (1986) · Zbl 0572.76068
[3] Hughes, T. J.R.; Mallet, M., A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg., 58, 305-328 (1986) · Zbl 0622.76075
[4] Hughes, T. J.R.; Mallet, M., A new finite element formulation for computational fluid dynamics: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg., 58, 329-336 (1986) · Zbl 0587.76120
[6] Hughes, T. J.R.; Brooks, A. N., A multi-dimensional upwind scheme with no crosswind diffusion, (Hughes, T. J.R., Finite Element Methods for Convection Dominated Flows. Finite Element Methods for Convection Dominated Flows, AMD, Vol. 34 (1979), ASME: ASME New York), 19-35 · Zbl 0423.76067
[7] Brooks, A. N.; Hughes, T. J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32, 199-259 (1982) · Zbl 0497.76041
[8] Tezduyar, T. E.; Hughes, T. J.R., Finite element formulations for convection dominated flows with particular emphasis on the compressible Euler equations, (Proc. AIAA 21st Aerospace Sciences Meeting. Proc. AIAA 21st Aerospace Sciences Meeting, Reno, Nevada. Proc. AIAA 21st Aerospace Sciences Meeting. Proc. AIAA 21st Aerospace Sciences Meeting, Reno, Nevada, AIAA Paper 83-0125 (1983)) · Zbl 0535.76074
[9] Hughes, T. J.R.; Tezduyar, T. E., Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations, Comput. Methods Appl. Mech. Engrg., 45, 217-284 (1984) · Zbl 0542.76093
[10] Donea, J., A Taylor-Galerkin method for convective transport problems, Internat. J. Numer. Methods Engrg., 20, 101-120 (1984) · Zbl 0524.65071
[11] Le Beau, G. J.; Tezduyar, T. E., Finite element computation of compressible flows with the SUPG formulation, (Dhaubhadel, M. N.; Engelman, M. S.; Reddy, J. N., Advances in Finite Element Analysis in Fluid Dynamics, FED Vol. 123 (1991), ASME: ASME New York), 21-27
[12] Mallet, M., A finite element method for computational fluid dynamics, (Ph.D. Thesis (1985), Department of Civil Engineering, Stanford University)
[13] Shakib, F., Finite element analysis of the compressible Euler and Navier-Stokes equations, (Ph.D. Thesis (1988), Department of Mechanical Engineering, Standford University)
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