×

zbMATH — the first resource for mathematics

Prediction of the critical Reynolds number for flow past a circular cylinder. (English) Zbl 1119.76031
Summary: We attempt to estimate the critical Reynolds number, \(Re_{c}\), for the first wake instability of the wake associated with a flow past a circular cylinder. Linear stability analysis and direct time integration of the governing equations for incompressible flows are carried out via a stabilized finite element formulation. The generalized eigenvalue problem resulting from the finite element discretization of the equations from linearized stability analysis is solved using a subspace iteration method to get the most unstable eigenmode. The results from the two methods are in good agreement. We investigate the effect of spatial resolution and location of computational boundaries. It is found that, for high blockage (ratio of the diameter of cylinder to the lateral width of domain), \(Re_{c}\) first decreases and then increases with increase in blockage. It is also observed that the Strouhal number at \(Re_{c}\) is quite sensitive to the blockage. This might possibly explain the scatter in the data from various researchers in the past.

MSC:
76E99 Hydrodynamic stability
76M10 Finite element methods applied to problems in fluid mechanics
76D25 Wakes and jets
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Behr, M.; Hastreiter, D.; Mittal, S.; Tezduyar, T.E., Incompressible flow past a circular cylinder: dependence of the computed flow field on the location of the lateral boundaries, Comput. meth. appl. mech. engrg., 123, 309-316, (1995)
[2] Berger, E.; Wille, R., Periodic flow phenomena, Ann. rev. fluid mech., 4, 313-340, (1972)
[3] Coutanceau, M.; Bouard, R., Experimental determination of the main features of the viscous flow in the wake of a circular cylinder in uniform translation. part 1. steady flow, J. fluid mech., 79, 231, (1977)
[4] Coutanceau, M.; Bouard, R., Experimental determination of the main features of the viscous flow in the wake of a circular cylinder in uniform translation. part 2. unsteady flow, J. fluid mech., 79, 257, (1977)
[5] Ding, Y.; Kawahara, M., Three dimensional linear stability analysis of incompressible viscous flows using the finite element method, Int. J. numer meth. fluids, 31, 451-479, (1999) · Zbl 0965.76041
[6] Fornberg, B., A numerical study of steady viscous flow past a circular cylinder, J fluid mech., 98, 819-855, (1980) · Zbl 0428.76032
[7] Gresho, P.M.; Chan, S.T.; Lee, R.L.; Upson, C.D., A modified finite element method for solving the time-dependent incompressible Navier-Stokes equations. part 2 applications, Int. J. numer. meth. fluids, 4, 619, (1984) · Zbl 0559.76031
[8] Jackson, C.P., A finite-element study of the onset of vortex shedding in flow past variously shaped bodies, J. fluid mech., 182, 23-45, (1987) · Zbl 0639.76041
[9] Meyer, A., Modern algorithms for large sparse eigenvalue problems, (1987), Akademie-Verlag Berlin · Zbl 0613.65032
[10] Mittal, S., On the performance of high aspect-ratio elements for incompressible flows, Comput. meth. appl. mech. engrg., 188, 269-287, (2000) · Zbl 0981.76056
[11] Mittal, S.; Kumar, B., Flow past a rotating cylinder, J. fluid mech., 476, 303-334, (2003) · Zbl 1163.76442
[12] Morzynski, M.; Afanasiev, K.; Thiele, F., Solution of the eigenvalue problems resulting from global non-parallel flow stability analysis, Comput. meth. appl. mech. engrg., 169, 161-176, (1999) · Zbl 0959.76045
[13] Morzynski, M.; Thiele, F., Numerical stability analysis of a flow about a cylinder, Z. angew. math. mech., 71, T424-T428, (1991)
[14] Norberg, C., An experimental investigation of the flow around a circular cylinder: influence of aspect ratio, J. fluid mech., 258, 287-316, (1994)
[15] Norberg, C., Flow around a circular cylinder: aspects of fluctuating lift, J. fluids struct., 15, 459-469, (2001)
[16] Posdziech, O.; Grundmann, R., Numerical simulation of the flow around an infinitely long circular cylinder in the transition regime, Theoret. comput. fluid mech., 15, 121-141, (2001) · Zbl 1067.76072
[17] A. Roshko, On the development of turbulent wakes from vortex streets, Technical Report 1191, NACA, 1954.
[18] Roshko, A., Perspectives on bluff body aerodynamics, J. wind engrg. ind. aerodyn., 49, 79-100, (1993)
[19] Saad, Y.; Schultz, M., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. scientific statist. comput., 7, 856-869, (1986) · Zbl 0599.65018
[20] Sahin, M.; Owens, R.G., A numerical investigation of wall effects up to high blockage ratios on two-dimensional flow past a confined circular cylinder, Phys. fluids, 16, 5, 1305-1320, (2004) · Zbl 1186.76455
[21] Shair, F.H.; Grove, A.S.; Petersen, E.E.; Archivos, A., The effect of confining walls on the stability of the steady wake behind a circular cylinder, J. fluid mech., 17, 546-550, (1963) · Zbl 0113.19503
[22] Stewart, G.W., Methods of simultaneous iteration for calculating eigenvectors of matrices, (), 169-185 · Zbl 0353.65020
[23] Stewart, G.W., Simultaneous iteration for computing invariant subspaces of Nonhermitian matrices, Numer. math., 25, 123-136, (1976) · Zbl 0328.65025
[24] Tezduyar, T.E.; Mittal, S.; Ray, S.E.; Shih, R., Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements, Comput. meth. appl. mech. engrg., 95, 221-242, (1992) · Zbl 0756.76048
[25] Wilkinson, J.H., The algebraic eigenvalue problem, (1965), Clarendon Press Oxford · Zbl 0258.65037
[26] Williamson, C.H.K., Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers, J. fluid mech., 206, 579-627, (1989)
[27] Williamson, C.H.K., Vortex dynamics in the cylinder wake, Ann. rev. fluid mech., 28, 477-539, (1996) · Zbl 0899.76129
[28] Zebib, A., Stability of viscous flow past a circular cylinder, J. engrg. math., 21, 155-165, (1987) · Zbl 0632.76063
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.