## A transient solution method for the finite element incompressible Navier- Stokes equations.(English)Zbl 0619.76027

The authors present a numerical method for solving the unsteady incompressible Navier-Stokes equations for plane flows. The procedure is based on the spatial discretization of the Navier-Stokes equations and the continuity equation via the conventional Galerkin finite element method, employing the primitive variable formulation. The ordinary triangular element of linear interpolation is used for the velocity field and the pressure is assumed to be constant on each element. A direct time integration method, developed by the authors elsewhere, is applied to the integration of the finite element equations. The integration method has unique features in its formulation as well as in its evaluation of the contribution of extended functions. Particular processes concerning the continuity condition and the boundary conditions lead to a set of nonlinear recurrence equations representing the evolution of the velocities and pressures under the constraint of incompressibility. As for the nonlinear terms, an iterative process is performed until convergence is achieved in each integration step.
Numerical results are presented for flow past a rectangular cylinder of various width to height ratio at different Reynolds numbers, and the calculated flow fields are compared with experimentally observed ones reported in the literature. Agreement with the experimentally visualized flow fields turns out to be fairly well.
Reviewer: J.Siekmann

### MSC:

 76D05 Navier-Stokes equations for incompressible viscous fluids
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### References:

 [1] The Finite Element Methods, 3rd edn, McGraw-Hill, London, 1977. [2] and , ’A survey of the finite element applications in fluid mechanics’, in et al. (eds), Finite Elements in Fluids, Vol. 3, Wiley New York, 1978, Chap. 21, pp. 363-396. [3] and , ’Navier-Stokes equations using mixed interpolation finite element in flow problems’, in et al. (eds), Finite Element Methods in Flow Problems, University of Alabama, Huntsville Press, 1974, pp. 121-143. [4] and , ’Primitive variables versus stream function finite element solutions of the Navier-Stokes equations’, in et al. (eds), Finite Elements in Fluids, Vol. 3, Wiley, New York, 1978, Chap. 4, pp. 73-87. [5] Huyakorn, Compt. and Fluids 6 pp 25– (1978) [6] Gresho, Adv. Water Res. 4 pp 175– (1981) [7] and , ’Mixed finite element solution of fluid flow problems’, in et al. (ed.) Finite Elements in Fluids, Vol. 4, Wiley, New York, 1982, Chap. 1, pp. 1-20. [8] Ying, Compt. Meth. Appl. Mech. Engng. 36 pp 23– (1983) [9] Yang, Int. j. numer. methods fluids 3 pp 377– (1983) [10] Int. j. numer. methods fluids 4 pp 43– (1984) [11] Argyris, Compt. Meth. Appl. Mech. Engng. 45 pp 3– (1984) [12] Hughes, J. Comp. Phys. 30 pp 1– (1979) [13] Bercovier, J. Comp. Phys. 30 pp 181– (1979) [14] Heinrich, Comp. and Fluids 9 pp 73– (1981) [15] Reddy, Comp. Meth. Appl. Mech. Engng. 35 pp 87– (1982) [16] ’RIP-methods for stokesean flows’, in et al. (eds), Finite Elements in Fluids, Vol. 4, Wiley, New York, 1982, Chap. 15, pp. 305-318. [17] Christies, Int. j. numer. methods eng. 10 pp 1389– (1976) [18] Heinrich, Int. j. numer. methods eng. 11 pp 131– (1977) [19] Donea, Int. j. numer. methods fluids 1 pp 63– (1981) [20] Brooks, Com. Meth. Appl. Mech. Engng. 32 pp 199– (1982) [21] and , ’Advection-dominated flows, with emphasis on the consequences of mass lumping’, in et al. (eds), Finite Elements in Fluids, Vol. 3, Wiley, New York, 1978, Chap. 19, pp. 35-350. [22] Gresho, Int. j. numer. methods fluids 4 pp 557– (1984) [23] Donea, Comp. Meth. Appl. Mech. Engng. 30 pp 53– (1982) [24] Kawahara, Int. j. numer. methods Fluids 3 pp 137– (1983) [25] Taneda, J. Phys. Soc. Japan 30 pp 262– (1971) [26] Okajima, J. Fluid Mech. 23 pp 379– (1982) [27] and , ’Flow around rectangular prism–numerical calculations and experiments, No. 2’, Bull. of Research Inst. for Appl. Mech., (50), 67-80 (1979) (in Japanese). [28] Flow-Induced Vibration, Van Nostrand Reinhold Company, New York, 1977, Chap. 1. · Zbl 0385.73001 [29] Shimizu, Trans. JSME (2) 44 pp 2699– (1978) [30] and , ’A computational method for time integration of the heat conduction equations’, in et al. (eds), Interdisciplinary Finite Element Analysis, Cornell University, New York, 1981, 697-710. [31] and , ’Direct time integration method for the transient heat conduction equation’, Proc. JSCE, (313), 23-36 (1981) (in Japanese). [32] and , ’Application of the high accurate time integration to solution procedure of the Navier-Stokes equations’, Proc. JSCE, (326), 29-40 (1982) (in Japanese). [33] and , ’A solution procedure for the finite element equations of transient, incompressible, viscous flows’, Proc. JSCE, (351), 59-68 (1984) (in Japanese).
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