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Analysis of incompressible massively separated viscous flows using unsteady Navier-Stokes equations. (English) Zbl 0683.76027
Summary: The unsteady incompressible Navier-Stokes equations are formulated in terms of vorticity and stream-function in generalized curvilinear orthogonal coordinates to facilitate analysis of flow configurations with general geometries. The numerical method developed solves the conservative form of the vorticity transport equation using the alternating direction implicit method, whereas the streamfunction equation is solved by direct block Gaussian elimination. The method is applied to a model problem of flow over a backstep in a doubly infinite channel, using clustered conformal coordinates. One-dimensional stretching functions, dependent on the Reynolds number and the asymptotic behaviour of the flow, are used to provide suitable grid distribution in the separation and reattachment regions, as well as in the inflow and outflow regions. The optimum grid distribution selected attempts to honour the multiple length scales of the separated flow model problem. The asymptotic behaviour of the finite differenced transport equation near infinity is examined and the numerical method is carefully developed so as to lead to spatially second-order-accurate wiggle-free solutions, i.e. with minimum dispersive error. Results have been obtained in the entire laminar range for the backstep channel and are in good agreement with the available experimental data for this flow problem, prior to the onset of three-dimensionality in the experiment.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
76M99 Basic methods in fluid mechanics
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[1] and , ’Progress on interacting boundary-layer computations at high Reynolds number’, Proc. First Symp. on Numerical and Physical Aspects of Aerodynamic Flows, Long Beach, CA, 1981.
[2] ’Multistructured boundary layers on flat plates and related bodies’. Advances in Applied Mechanics, Vol. 14, Academic Press, 1974, pp. 145-239.
[3] Sychev, Mehanika Zhidkosti i Gaza 3 pp 47– (1972)
[4] ’Laminar separation: a local asymptotic flow description for constant pressure downstream’, in Flow Separation, AGARD Conf. Proc. on Flow Separation, CP 168, 1975.
[5] Smith, Proc. Roy. Soc. A 356 pp 443– (1977)
[6] Smith, J. Fluid Mech. 92 pp 171– (1979)
[7] Steger, AIAA J. 16 pp 679– (1978)
[8] ’Incompressible Navier-Stokes and parabolized Navier-Stokes solution procedures and computational techniques’, VKI Lecture Notes for Series on Computational Fluid Dynamics, 1982.
[9] Ghia, Comput. Fluids 9 pp 123– (1981)
[10] and , ’Semi-elliptic globally-iterative analysis for two-dimensional subsonic internal viscous flows’, presented at NASA-Lewis Workshop on Computational Fluid Mechanics, Cleveland, OH, 20-21 October 1982.
[11] Briley, J. Fluid Mech. 47 pp 713– (1971)
[12] Ghia, AIAA J. 12 pp 1659– (1974)
[13] and , ’Transonic flows with viscous effects’, in Transonic, Shock and Multi-Dimensional Flows: Advances in Scientific Computing, Academic Press, 1982.
[14] and , ’Study of unsteady incompressible flow using nonuniform curvilinear grids, time marching and a direct method’, Multigrid Methods, NASA CP-2202, October 1981.
[15] ’On the construction of fast solvers for elliptic equations’, VKI Lecture Notes, Brussels, 29 March-2 April, 1982.
[16] Buzbee, SIAM J. Numer. Anal. 7 pp 627– (1970)
[17] Dorr, SIAM Rev. 12 pp 248– (1970)
[18] Sweet, J. Comput. Phys. 12 pp 422– (1973)
[19] Sweet, SIAM J. Numer. Anal. 11 pp 506– (1974)
[20] Schumann, J. Comput. Phys. 20 pp 171– (1976)
[21] Schwarztrauber, SIAM J. Numer. Anal. 11 pp 1136– (1974)
[22] Nand, Trans. Inst. Chem. Eng. 52 pp 361– (1974)
[23] and , ’Reattachment length and circulation regions downstream of a two-dimensional single backward facing step’. Momentum and Heat Transfer Processes in Recirculating Flows, HTD-, Vol. 13 ASME, New York, 1980, pp. 1-8.
[24] Roache, AIAA J. 8 pp 530– (1970)
[25] and , ’Lecture notes for course entitled: Calculation of recirculating flow’. Heat Transfer Report HTS/74/2, Imperial College, London, 1973.
[26] ’A direct numerical method for the solution of unsteady Navier-Stokes equations in generalized orthogonal coordinates’, Ph.D. Dissertation, University of Cincinnati, Cincinnati, OH, 1983.
[27] Ghia, AIAA J. 17 pp 298– (1979)
[28] Kumar, J. Fluid Mech. 97 pp 27– (1980)
[29] and , ’Implicit numerical methods for the compressible Navier-Stokes and Euler equations’, VKI Lecture Notes, Brussels, 29 March-2 April, 1982.
[30] Armaly, J. Fluid Mech. 127 pp 473– (1983)
[31] Smith, Q. Appl. Math. 13 pp 233– (1955)
[32] ’Generation of streamwise vortices in separating nominally two-dimensional boundary layer flow’, 18th Midwestern Mechanics Conf., University of Iowa, 1983, pp. 193-196.
[33] Rubin, Comput. Fluids 9 pp 163– (1981)
[34] and , ’A direct method for the solution of unsteady two-dimensional incompressible Navier-Stokes equations’, Proc. Second Symp. on Numerical and Physical Aspects of Aerodynamic Flows, Long Beach, CA, 1983.
[35] and , ’Steady of incompressible separated flow using an implicit time-dependent technique’, AIAA-CP 834, 1983, pp. 686-696.
[36] Kim, J. Comput. Phys. 59 pp 308– (1985)
[37] and , ’A direct algorithm for solution of incompressible three-dimensional unsteady Navier-Stokes equations’, AIAA CP-874, 1987, pp. 408-421.
[38] Ghoniem, AIAA J. 25 pp 168– (1987)
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