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A unified approach for inviscid compressible and nearly incompressible flow by least-squares finite element method. (English) Zbl 1019.76024

Summary: We present a unified approach to treat inviscid compressible and incompressible flows. The formulation is based on the primitive variables \(W=(p,u,v)\). In the case where the energy equation is present, we can either use the enthalpy \(h\) or temperature \(T\) in the primitive variables. The least-squares finite element method is applied in an unstructured framework. This method is well-defined for this set of variables, globally conservative and stable for any continuous interpolations, both for inviscid compressible and incompressible flows.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing

Software:

Modulef
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References:

[1] Baruzzi, G.; Habashi, W.G.; Guèvremont, W.G.; Hafez, M.M., A second order finite element method for the solution of the transonic Euler and and navier – stokes equations, Internat. J. numer. methods fluids, 20, 671-693, (1995) · Zbl 0840.76030
[2] Le Beau, G.J.; Ray, S.E.; Aliabadi, S.K.; Tezduyar, T.E., SUPG finite element computation of compressible flows with the entropy and conservation variables formulations, Comput. methods appl. mech. engrg., 104, 397-422, (1993) · Zbl 0772.76037
[3] M. Bernadou, et al., Modulef: Une bibliothèque modulaire d’éléments finis, Technical Report, INRIA, 1988
[4] Bochev, P.B.; Gunzburger, M.D., Analysis of least-sqaures finite element methods for the Stokes equations, Math. comp., 63, 479-506, (1994) · Zbl 0816.65082
[5] Briley, W.R.; McDonald, H.; Shamroth, S.J., A low Mach number Euler formulation and application to time iterative LBI schemes, Aiaa j., 21, 1467-1469, (1983)
[6] Bruneau, C.H.; Chattot, J.J.; Laminie, J., Computation of 3D vortex flows past a flat plate at incidence through a variational approach of the full steady Euler equations, Internat. J. numer. methods fluids, 9, 305-323, (1989) · Zbl 0665.76066
[7] Bruneau, C.H.; Chattot, J.J.; Laminie, J.; Guiu-Roux, J., Finite element least square method for solving full steady Euler equations in a plane nozzle, (), 161-166 · Zbl 0508.76094
[8] Bruneau, C.H.; Laminie, J., A method to compute 3D hypersonic flows with Euler model, Comput. fluid dyn. J., 1, 4, 347-360, (1993)
[9] Chattot, J.J.; Guiu-Roux, J.; Laminie, J., Finite element calculation of steady transonic flow in nozzles using primary variables, () · Zbl 0508.76094
[10] Chattot, J.J.; Guiu-Roux, J.; Laminie, J., Numerical solution of a first – order conservation equation by a least square method, Internat. J. numer. methods fluids, 2, 209-219, (1982) · Zbl 0484.76014
[11] Chattot, J.J.; Guiu-Roux, J.; Laminie, J., Numerical solution of a first-order conservation equation by a least square method, Internat. J. numer. methods fluids, 2, 209-219, (1982) · Zbl 0484.76014
[12] Choi, Y.H.; Merkle, C.L., The application of precontioning in viscous flows, J. comput. phys., 105, 293-311, (1988)
[13] Chorin, A.J., A numerical method for solving incompressible viscous flow problems, J. comput. phys., 2, (1967) · Zbl 0168.46501
[14] Fletcher, C.A.J., A primitive variable finite element formulation for inviscid compressible flow, J. comput. phys., 33, (1979) · Zbl 0451.73068
[15] M.M. Hafez, Variational approaches to CFD: Applications to potential, Euler and Navier-Stokes equations, in: 24th Computational Fluid Dynamics, VKI Lecture Series on Computational Fluid Dynamics. VKI LS 1993-04, VKI, Rhode-Saint-Genèse, March 1993
[16] Harlow, F.H.; Amsden, A.A., A numerical fluid dynamics calculation method for all speeds, J. comput. phys., 8, 197-213, (1971) · Zbl 0221.76011
[17] Hauke, G.; Hughes, T.J.R., A comparative study of different sets of variables for solving compressible and incompressible flows, Comput. methods appl. mech. engrg., 153, 1-44, (1998) · Zbl 0957.76028
[18] Huang, Y.; Lerat, A., Second-order upwinding through a characteristic time-step matrix for compressible flow calculations, J. comput. phys., 142, 2, 445-472, (1998) · Zbl 0932.76052
[19] 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
[20] Issa, R.I., Solution of the implicitly discetized fluid flow equations by operator-splitting, J. comput. phys., 62, 40-65, (1986) · Zbl 0619.76024
[21] Jiang, B.N., A stable least-squares finite element method for nonlinear hyperbolic equations, Internat. J. numer. methods fluids, 8, 933-942, (1988) · Zbl 0666.76087
[22] Jiang, B.N., A least-squares finite element method for incompressible navier – stokes problems, Internat. J. numer. methods fluids, 14, 843-859, (1992) · Zbl 0753.76097
[23] Jiang, B.N., The least-squares finite element method. theory and applications in CFD and electromagnetics, (1998), Springer-Verlag Berlin
[24] Jiang, B.N.; Povinelli, L.A., Large scale compuation of incompressible viscous flow by least-squares finite element method, Comput. methods appl. mech. engrg., 114, 213-231, (1990)
[25] Jiang, B.N.; Povinelli, L.A., Least-squares finite element method for fluid dynamics, Comput. methods appl. mech. engrg., 81, 13-37, (1990) · Zbl 0714.76058
[26] Kariman, S.M.H.; Shneider, G.E., A pressure-based control volume finite element method for flow at all speeds, Aiaa j., 33, 1611-1618, (1995) · Zbl 0851.76040
[27] Klainerman; Majda, Compressible and incompressible fluids, Commun. pure appl. math., 25, 629-651, (1982) · Zbl 0478.76091
[28] Kreiss, H.O.; Lorenz, J.; Naughton, M.J., Convergence of the solutions of the compressible to the solutions of the incompressible navier – stokes equations, Adv. appl. math., 12, 187-214, (1991) · Zbl 0728.76084
[29] Lefebvre, D.; Peraire, J.; Morgan, K., Finite element least-squares solution of the Euler equations using linear and quadratic approximations, Internat. J. comput. fluid dynam., 2, 209-219, (1992)
[30] Majda, A., Compressible fluid flow and systems of conservation laws in several spaces variables, (1984), Springer-Verlag Berlin · Zbl 0537.76001
[31] Merkle, C.L.; Choi, Y.H., Computation of low speed flow with heat addition, Aiaa j., 25, 831-838, (1987)
[32] Merkle, C.L.; Choi, Y.H., Computation of low speed flow with heat addition, Internat. J. numer. methods engrg., 25, 293-311, (1988) · Zbl 0668.76077
[33] Mittal, S.; Tezduyar, T., A unified finite element formulation for compressible and incompressible flows using augmented conservation variables, Comput. methods appl. mech. engrg., 161, 229-243, (1998) · Zbl 0943.76050
[34] Morgan, K.; Löhner, R.; Zienkiewicz, O.C., An adaptive finite element procedure for compressible high speed flows, Comput. methods appl. mech. engrg., 51, 441-465, (1985) · Zbl 0568.76074
[35] Nigro, N.; Storti, M.; Idelsohn, S., GMRES physics-based preconditioner for all Reynolds and Mach numbers: numerical examples, Internat. J. numer. methods fluids, 25, 1347-1371, (1997) · Zbl 0910.76036
[36] Nigro, N.; Storti, M.; Idelsohn, S.; Tezduyar, T., Physics based GMRES preconditioner for compressible and incompressible navier – stokes equations, Comput. methods appl. mech. engrg., 154, 203-228, (1998) · Zbl 0957.76032
[37] Reddy, J.N., An introduction to the finite element method, (1993), McGraw-Hill New York · Zbl 0561.65079
[38] F. Shakib, Finite element analysis of the compressible Euler and Navier-Stokes equations, Ph.D. Thesis, Department of Mechanical Engineering, Stanford University, Stanford, CA, 1988
[39] D. Sidilkover, T.W. Roberts, R.C. Swanson, Textbook multigrid efficiency for the steady euler equations, Technical Report AIAA 97-1949, AIAA Paper, 1997 · Zbl 0976.76056
[40] Turkel, E., Preconditioning methods for solving the incompressible and low speed compressible equations, J. comput. phys., 72, 277-298, (1987) · Zbl 0633.76069
[41] Turkel, E., Review of preconditioning method for fluid dynamics, Appl. numer. math., 12, 257-284, (1993) · Zbl 0770.76048
[42] E. Turkel, Review of preconditioning techniqes for fluid dynamics, Technical Report 93-42, ICASE, 1993
[43] Turkel, E.; Fiterman, A.; van Leer, B., Preconditioning and the limit to the incompressible flow equations, (), 215-234, (see also ICASE Report 93-42)
[44] B. van Leer, W.T. Lee, P.L. Roe, Characteristic time-stepping or local preconditioning of the Euler equations, Technical Report 91-1552-CP, AIAA Paper, June 1991
[45] C. Viozat, Implicit upwind schemes for low Mach number compressible flow, Technical Report RR-2084, INRIA, 1997
[46] Volpe, G., Performance of compressible codes at low Mach number, Aiaa j., 31, 1, 49-56, (1993) · Zbl 0775.76140
[47] T Yu, S.; Jiang, B.N.; Wu, J., A div-curl-Grad formulation for three-dimensional low-Mach flows solved by the least-squares finite element method, Comput. methods appl. mechanics engrg., (1997)
[48] Yu, S.T.; Jiang, B.N.; Wu, J.; Liu, N.S., The least-squares finite element method for low Mach number compressible viscous flows, Internat. J. numer. methods engrg., 38, 3591-3610, (1995) · Zbl 0867.76047
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