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Is the steady viscous incompressible two-dimensional flow over a backward-facing step at \(Re=800\) stable? (English) Zbl 0784.76050
Summary: A detailed case study is made of one particular solution of the 2D incompressible Navier-Stokes equations. Careful mesh refinement studies were made using four different methods (and computer codes): (1) a high- order finite-element method solving the unsteady equations by time- marching; (2) a high-order finite-element method solving both the steady equations and the associated linear-stability problem; (3) a second-order finite difference method solving the unsteady equations in streamfunction form by time-marching; and (4) a spectral-element method solving the unsteady equations by time-marching. The unanimous conclusion is that the correct solution for flow over the backward-facing step at \(Re=800\) is steady – and it is stable, to both small and large perturbations.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76M20 Finite difference methods applied to problems in fluid mechanics
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76D05 Navier-Stokes equations for incompressible viscous fluids
Software:
ENTWIFE; PITCON
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[1] ’A summary report on the 14 July minisymposium on outflow BCs for incompressible flow’, in Proceedings, 4th Int. Symp. on Computational Fluid Dynamics, University of California at Davis, 9-12 September 1991, p. 436.
[2] and , ’Summary of two minisymposia on outflow boundary conditions for incompressible flow’, Int. j. numer. methods fluids (in press).
[3] Gartling, Int. j. numer. methods. fluids 11 pp 953– (1990)
[4] Gresho, Comput. Methods Appl. Mech. Eng. 87 pp 201– (1991)
[5] Kaiktsis, J. Fluid Mech. 231 pp 501– (1991)
[6] and , ’Study of incompressible flow using an implicit time-dependent technique’, Proc. 6th Comput. Fluid Dyn. Conf., AIAA, New York, 1983, pp. 686-696.
[7] Kim, J. Comput. Phys. 59 pp 308– (1985)
[8] Sethian, J. Comput. Phys. 74 pp 283– (1988)
[9] Ghoniem, AIAA J. 25 pp 168– (1987)
[10] ’NACHOS II–A finite element code for incompressible flow problems’, Sandia National Laboratories Report, SAND86-1816 and SAND86-1817, Albuquerque, NM, 1987.
[11] and , ’On the time dependent solution of the incompressible Navier-Stokes equations in two and three dimensions’, in Recent Advances in Numerical Methods in Fluids, Vol. 1, Pineridge Press, Swansea, 1980, pp. 27-81.
[12] ENTWIFE User Manual (Release I), Harwell Report AERE-R 11577, 1985.
[13] and , Incompressible Flow and the Finite Element Method, (in preparation).
[14] Numerical Analysis of Parameterised Nonlinear Equations, Wiley-Interscience, New York, 1986.
[15] and , Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York, 1974.
[16] and , ’On a generalized eigenvalue problem arising from discretisations of the Navier-Stokes equations’, (in preparation).
[17] and , ’Iterative methods for the detection of Hopf bifurcations in finite element discretisations of incompressible flow problems’, SIAM J. Sci. Comput. (submitted).
[18] Malkus, Int. J. Eng. Sci. 19 pp 1299– (1981)
[19] Jennings, J. Inst. Math. Appl. 15 pp 351– (1975)
[20] Schreiber, J. Comput. Phys. 49 pp 310– (1983)
[21] ’A multigrid method for solving the biharmonic equation on rectangular domains’, Arbeitspapiere der GMD No. 143, Gesellschaft für Mathematik und Datenverarbeitung, St. Augustin, 1985.
[22] Gresho, Adv. Appl. Mech. 28 pp 45– (1992)
[23] Goodrich, J. Comput. Phys. 84 pp 207– (1989)
[24] ’An unsteady time asymptotic flow in the square driven cavity’, in K. Gustafson and W. Wyss (eds), Proc. IMACS 1st Int. Conf. on Comput. Phys., University of Colorado at Boulder, 1990; NASA TM 103141, 1990.
[25] ’An efficient and robust algorithm for two dimensional time dependent incompressible Navier-Stokes equations: high Reynolds number flows’, in , and (eds), Numerical Methods in Laminar and Turbulent Flow, Vol. VII: Proc. 7th Int. Conf. Stanford, 15-19 July, 1991; NASA TM 104424, 1991.
[26] (in preparation).
[27] and (in preparation).
[28] NEKTON User’s Guide, Version 2.7, Nektonics Inc., Cambridge, MA, 1991.
[29] Patera, J. Comput. Phys. 54 pp 468– (1984)
[30] ’Simulations of two-dimensional transient flow over a backward-facing step using a spectral-element method’, in (ed.), Fluids Engineering 1992 Abstracts, FED-Vol. 133. American Society of Mechanical Engineers, New York, 1992, pp. 185-186.
[31] ’Large-eddy simulation of turbulent flow using the finite element method’, Ph.D. Thesis, U. C. Davis, Department of Mechanical Engineering, (1993).
[32] Gresho, Int. j. numer. methods fluids 4 pp 557– (1984)
[33] Gresho, Int. j. numer. methods fluids 4 pp 619– (1984)
[34] B. Blackwell and D. W. Pepper (eds), ’Benchmark problem for heat transfer codes’, Proc. Winter Annual Meeting of the ASME, Anaheim, HTD-Vol. 222, CA, 8-13 November, 1992.
[35] and , ’Numerical calculations of two dimensional laminar flow and heat transfer for a backward facing step’, in B. Blackwell and D. W. Pepper (eds), Proc. Winter Annual Meeting of the ASME, Anaheim, 1992, p. 1.
[36] , and , ’Numerical simulation of laminar flow and heat transfer over a backward facing step’, In B. Blackwell and D. W. Pepper (eds), Proc. Winter Annual Meeting of the ASME, Anaheim, 1992, p. 21.
[37] Gresho, Bull. Faculty Sci. Eng. (Chuo Univ.) 30 pp 45– (1987)
[38] and , ’Step-wall diffuser benchmark prediction/boundary condition implementation in an AKCESS’, in B. Blackwell and D. W. Pepper (eds), Proc. Winter Annual Meeting of the ASME, Anaheim, 1992, p. 45.
[39] and , ’Flow over a backward facing step: a benchmark problem for laminar flow with heat transfer’, in B. Blackwell and D. W. Pepper (eds), Proc. Winter Annual Meeting of the ASME, Anaheim, 1992, p. 73.
[40] ’ANSWER: A benchmark study for backward facing step’, in B. Blackwell and D. W. Pepper (eds), Proc. Winter Annual Meeting of the ASME, Anaheim, 1992, p. 13.
[41] ’A finite-volume CFD code’, in B. Blackwell and D. W. Pepper (eds), Proc. Winter Annual Meeting of the ASME, Anaheim, 1992, p. 53.
[42] and , ’A spectral-element solution to the backward-facing step problem using NEKTON’, in B. Blackwell and D. W. Pepper (eds), Proc. Winter Annual Meeting of the ASME, Anaheim, 1992, p. 59.
[43] Yee, J. Comput. Phys. 97 pp 249– (1991)
[44] Schumack, J. Comput. Phys. 94 pp 30– (1991)
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