A Taylor-Galerkin-based algorithm for viscous incompressible flow. (English) Zbl 0686.76019
Summary: The development and behaviour of a new finite element algorithm for viscous incompressible flow is presented. The stability and background theory are discussed and the numerical performance is considered for some benchmark problems. The Taylor-Galerkin approach naturally leads to a time-stepping algorithm which is shown to perform well for a wide range of Reynolds numbers (1\(\leq Re\leq 400)\). Various modifications to the algorithm are investigated, particularly with respect to their effects on stability and accuracy.
MSC:
| 76D10 | Boundary-layer theory, separation and reattachment, higher-order effects |
| 76M99 | Basic methods in fluid mechanics |
Keywords:
fractional step method; finite element algorithm; viscous incompressible flow; Taylor-Galerkin approachReferences:
| [1] | and , ’An algorithm for the three-dimensional transient simulation of non-Newtonian fluid flows’, in G. N. Pande and J. Middleton (eds), Transient/Dynamic Analysis and Constitutive Laws for Engineering Materials, Vol. II (Proc. Int. Conf. on Numerical Methods in Engineering: Theory and Applications, NUMETA 1987, Swansea), Nijhoff, Dordrecht, 1987. |
| [2] | , and , ’Splitting up techniques for computations of industrial flows’, in and (eds), Numerical Methods for Fluid Dynamics, Vol. II, Clarendon Press, Oxford, 1986, pp. 103-128. |
| [3] | , , and , ’On the numerical solution of 3D-incompressible flow problems’, in and (eds), Numerical Methods for Fluid Dyanamics, Vol. II, Clarendon Press, Oxford, 1986, pp. 499-512. |
| [4] | and , Computational Methods for Fluid Flow, Springer-Verlag, New York, 1983. · Zbl 0514.76001 · doi:10.1007/978-3-642-85952-6 |
| [5] | van Kan, SIAM J. Sci. Stat. Comput. 7 pp 870– (1986) |
| [6] | Donea, Int. j. numer. methods eng. 20 pp 101– (1984) |
| [7] | Löhner, Int. j. numer. methods fluids 4 pp 1043– (1984) |
| [8] | and , Difference Methods for Initial Value Problems, 2nd edn, Wiley, New York, 1967. |
| [9] | ’Initial-value problems by finite difference and other methods’, in (ed.), Proc. Conf. IMA ’State of the Art in Numerical Analysis’, York, April 1976, Academic Press, New York, 1977. |
| [10] | Brooks, Comput. Methods Appl. Mech. Eng. 32 pp 199– (1982) |
| [11] | Quartapelle, J. Comput. Phys. 40 pp 453– (1981) |
| [12] | and , ’A new semi-implicit method for solving the time-dependent conservation equations for incompressible flow’, in Numerical Methods in Laminar and Turbulent Flow, Pineridge Press, Swansea, 1985, pp. 3-21. |
| [13] | Donea, Comput. Methods Appl. Mech. Eng. 30 pp 53– (1982) |
| [14] | Chorin, Math. Comput. 22 pp 745– (1968) |
| [15] | Temam, Arch. Rat. Mech. Anal. 32 pp 135– (1969) |
| [16] | Issa, J. Comput. Phys. 62 pp 40– (1986) |
| [17] | and , Finite Element Methods and Navier-Stokes Equations, D. Reidel, Dordrecht, 1986. · Zbl 0649.65059 · doi:10.1007/978-94-010-9333-0 |
| [18] | and , ’Advection-dominated flows with emphasis on the consequences of mass lumping’, in Finite Elements in Fluids, Vol. 3, Wiley, New York, 1978, pp. 745-756. |
| [19] | ’Galerkin finite element methods and their generalisations’, Oxford University Computer Laboratory, N.A. Report 6, 1986 (invited lecture at Conf. IMA ’State of the Art in Numerical Analysis’, Birmingham, April 1986). |
| [20] | and , ’A stability analysis of the Lagrange-Galerkin method with non-exact integration’, Oxford University Computer Laboratory, N.A. Report 14, 1986. |
| [21] | and , An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973. |
| [22] | Löhner, Comput. Methods Appl. Mech. Eng. 45 pp 313– (1984) |
| [23] | and , Handbook for Automatic Computation, Vol. II, Linear Algebra, Springer-Verlag, New York, 1971, pp. 50-56. · doi:10.1007/978-3-642-86940-2 |
| [24] | and , The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, MA, 1974. |
| [25] | Wathen, IMA J. Numer. Anal. 7 pp 449– (1987) |
| [26] | , , and , ’Finite elements for compressible gas flow and similar systems’, paper presented at 7th Int. Conf. on Computer Methods and Applications in Science and Engineering, Versailles, December 1985. |
| [27] | A Handbook of Numerical Matrix Inversion and Solution of Linear Equations, Wiley, New York, 1968. |
| [28] | and , Applied Iterative Methods, Academic Press, London, 1981. · Zbl 0459.65014 |
| [29] | Donea, Comput. Methods Appl. Mech. Eng. 45 pp 123– (1984) |
| [30] | ’Superconvergent estimation of the gradient from linear finite element approximations on triangular elements’, University of Reading, N.A. Report 3, 1985. |
| [31] | Bourcier, Recherche Aerospatiale 131 pp 23– (1969) |
| [32] | ’Contribution a I’étude des écoulements visqueux en milieu confiné’, Thèse Doctorat d’Etat, IMFM, Université d’Aix-Marseille, 1978. |
| [33] | Burggraf, J. Fluid. Mech. 24 pp 113– (1966) |
| [34] | Pepper, Comput. Fluids 8 pp 213– (1980) |
| [35] | Nallasamy, J. Fluid Mech. 79 pp 391– (1977) |
| [36] | Ku, Comput. Fluids 13 pp 99– (1985) |
| [37] | Schreiber, J. Comput. Phys. 49 pp 310– (1983) |
| [38] | Sivaloganathan, Int. j. numer. methods fluids 8 pp 417– (1988) |
| [39] | Gupta, J. Comput. Phys. 31 pp 265– (1979) |
| [40] | Pan, J. Fluid Mech. 28 pp 643– (1967) |
| [41] | Kawaguti, J. Phys. Soc. Japan 16 pp 2307– (1961) |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
