×

A stabilized finite element method based on SGS models for compressible flows. (English) Zbl 1120.76331

Summary: We present an appropriate extension of the stabilized finite element formulation, introduced in [A. Corsini, F. Rispoli, A. Santoriello, A variational multiscale high-order finite element formulation for turbomachinery flow computations, Comput. Methods Appl. Mech. Engrg. 194 (2005) 4797-4823] for the prediction of incompressible flows, aimed at compressible flows. The stabilized formulation is the so-called variable subgrid scale method (V-SGS) based on an approximation of the class of subgrid scale models (SGS) derived from the Hughes variational multiscale method (Hughes-VMS) introduced in [T.J.R. Hughes, Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg. 127 (1995) 387-401]. It is characterized by a variable stabilization parameter within the domain of element.
We also propose an innovative procedure for computing the stabilization parameter by using the one-dimensional element Green’s functions corresponding to an advective-diffusive differential operator. The stabilization parameter is defined as the sum of two components, the first providing its mean value and the second its element-wise space dependency. On this basis the present work proposes the stabilization matrix for the multidimensional advective-diffusive system of equations governing compressible flows. The procedure developed for computing the stabilization parameter can be applied to other differential operators as it has been shown in [A. Corsini, F. Rispoli, A. Santoriello, A variational multiscale high-order finite element formulation for turbomachinery flow computations, Comput. Methods Appl. Mech. Engrg. 194 (2005) 4797-4823]. The proposed stabilization device is validated by simulation of inviscid and viscous supersonic flow configurations.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Corsini, A.; Rispoli, F.; Santoriello, A., A variational multiscale high-order finite element formulation for turbomachinery flow computations, Comput. methods appl. mech. engrg., 194, 4797-4823, (2005) · Zbl 1093.76032
[2] Hughes, T.J.R., Multiscale phenomena: green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. methods appl. mech. engrg., 127, 387-401, (1995) · Zbl 0866.76044
[3] 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
[4] 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
[5] Hughes, T.J.R.; Mallet, M.; Franca, L.P., New finite element methods for the compressible Euler and Navier Stokes equations, Comput. methods appl. sci. engrg., VII, 339-360, (1986) · Zbl 0678.76069
[6] M. Mallet, A finite element method for computational fluid dynamics, Ph.D. Thesis, Department of Civil Engineering, Stanford University, 1985.
[7] Hughes, T.J.R., Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and navier – stokes equations, Int. J. numer. methods fluids, 7, 1261-1275, (1987) · Zbl 0638.76080
[8] 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
[9] Aliabadi, S.K.; Tezduyar, T.E., Space – time finite element computation of compressible flows involving moving boundaries and interfaces, Comput. methods appl. mech. engrg., 107, 209-223, (1993) · Zbl 0798.76037
[10] Aliabadi, S.K.; Tezduyar, T.E., Parallel fluid dynamics computations in aerospace applications, Int. J. numer. methods fluids, 21, 783-805, (1995) · Zbl 0862.76033
[11] Tezduyar, T.E.; Senga, M., Stabilization and shock-capturing parameters in SUPG formulation of compressible flows, Comput. methods appl. mech. engrg., 195, 1621-1632, (2006) · Zbl 1122.76061
[12] Hughes, T.J.R.; Franca, L.P.; Hulbert, G., A new finite element formulation for computational fluid dynamics: VIII the Galerkin/least-square method for advective – diffusive equations, Comput. methods appl. mech. engrg., 73, 173-189, (1989) · Zbl 0697.76100
[13] F. Shakib, Finite element analysis of the compressible Euler and Navier-Stokes equations, Ph.D. Thesis, Department of Mechanical Engineering, Stanford University, 1988.
[14] Shakib, F.; Hughes, T.J.R.; Johan, Z., A new finite element formulation for computational fluid dynamics: X. the compressible Euler and navier – stokes equations, Comput. methods appl. mech. engrg., 89, 141-219, (1991) · Zbl 0838.76040
[15] Hughes, T.J.R.; Stewart, J., A space – time formulation for multiscale phenomena, J. comput. appl. math., 74, 217-229, (1996) · Zbl 0869.65061
[16] Oberai, A.A.; Pinsky, P.M., A multiscale finite element method for the Helmholtz equation, Comput. methods appl. mech. engrg., 154, 281-297, (1998) · Zbl 0937.65119
[17] Hughes, T.J.R.; R Feijóo, G.; Mazzei, L.; Quincy, J.B., The variational multiscale method – a paradigm for computational mechanics, Comput. methods appl. mech. engrg., 166, 3-24, (1998) · Zbl 1017.65525
[18] Hauke, G.; Garcia-Olivares, A., Variational subgrid scale formulations for the advection – diffusion – reaction equation, Comput. methods appl. mech. engrg., 190, 6847-6865, (2001) · Zbl 0996.76074
[19] Hauke, G., A simple subgrid scale stabilized method for the advection – diffusion – reaction equation, Comput. methods appl. mech. engrg., 191, 2925-2947, (2002) · Zbl 1005.76057
[20] Masud, A.; Khurram, R.A., A multiscale/stabilized finite element method for the advection – diffusion equation, Comput. methods appl. mech. engrg., 193, 1997-2018, (2004) · Zbl 1067.76570
[21] 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
[22] Roach, G.F., Green’s functions-introductory theory with applications, (1970), VNR London · Zbl 0186.47104
[23] Tezduyar, T.E.; Osawa, Y., Finite element stabilization parameters computed from element matrices and vectors, Comput. methods appl. mech. engrg., 190, 411-430, (2000) · Zbl 0973.76057
[24] Tezduyar, T.E., Stabilized finite element formulations for incompressible flow computations, Adv. appl. mech., 28, 1-44, (1992) · Zbl 0747.76069
[25] Carette, J.C.; Deconinck, H.; Paillere, H.; Roe, P.L., Multidimensional upwinding: its relation to finite elements, Int. J. numer. methods fluids, 20, 935-955, (1995) · Zbl 0840.76032
[26] 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
[27] Nigro, N.; Storti, M.; Idelsohn, S., Gmres physics-based preconditioner for all Reynolds and Mach numbers: numerical examples, Int. J. numer. methods fluids, 25, 1347-1371, (1997) · Zbl 0910.76036
[28] Liepmann, H.W.; Roshko, A., Elements of gasdynamics, (1957), John Wiley & Sons New York · Zbl 0078.39901
[29] Mittal, S., Finite element computation of unsteady viscous compressible flows, Comput. methods appl. mech. engrg., 157, 151-175, (1998) · Zbl 0953.76051
[30] 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.