Symmetry breaking and preliminary results about a Hopf bifurcation for incompressible viscous flow in an expansion channel

This computational study shows, for the first time, a clear transition to two-dimensional Hopf bifurcation for laminar incompressible flows in symmetric plane expansion channels. Due to the well-known extreme sensitivity of this study on computational mesh, the critical Reynolds numbers for both the known symmetry-breaking (pitchfork) bifurcation and Hopf bifurcation were investigated for several layers of mesh refinement. It is found that under-refined meshes lead to an overestimation of the critical Reynolds number for the symmetry breaking and an underestimation of the critical Reynolds number for the Hopf bifurcation.

[1]  T. Mullin,et al.  Nonlinear flow phenomena in a symmetric sudden expansion , 1990, Journal of Fluid Mechanics.

[2]  D. Drikakis.  Bifurcation phenomena in incompressible sudden expansion flows , 1997 .

[3]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[4]  R. Codina.  Stabilized finite element approximation of transient incompressible flows using orthogonal subscales , 2002 .

[5]  F. Durst,et al.  Low Reynolds number flow over a plane symmetric sudden expansion , 1974, Journal of Fluid Mechanics.

[6]  Zvi Rusak,et al.  The dynamics of a laminar flow in a symmetric channel with a sudden expansion , 2001, Journal of Fluid Mechanics.

[7]  Steven A. Orszag,et al.  Instability mechanisms in shear-flow transition , 1988 .

[8]  H. Kuhlmann,et al.  Global stability of multiple solutions in plane sudden-expansion flow , 2012, Journal of Fluid Mechanics.

[9]  James Serrin,et al.  Mathematical Principles of Classical Fluid Mechanics , 1959 .

[10]  Fred Wubs,et al.  Numerical Bifurcation Methods and their Application to Fluid Dynamics: Analysis beyond Simulation , 2012 .

[11]  F. Battaglia,et al.  Bifurcation of Low Reynolds Number Flows in Symmetric Channels , 1996 .

[12]  Roger Pierre,et al.  Localization of Hopf bifurcations in fluid flow problems , 1997 .

[13]  Nicola Parolini,et al.  Numerical investigation on the stability of singular driven cavity flow , 2002 .

[14]  高等学校計算数学学報編輯委員会編.  高等学校計算数学学報 = Numerical mathematics , 1979 .

[15]  H. K. Moffatt.  Viscous and resistive eddies near a sharp corner , 1964, Journal of Fluid Mechanics.

[16]  H. H. Fernholz,et al.  Report on the first European Mechanics Colloquium, on the Coanda effect , 1965, Journal of Fluid Mechanics.

[17]  Krishnamurthy Jayaraman,et al.  Asymmetric flows in planar symmetric channels with large expansion ratio , 2002 .

[18]  Paul-Louis George,et al.  A Numerical Study of Some Hessian Recovery Techniques on Isotropic and Anisotropic Meshes , 2011, SIAM J. Sci. Comput..

[19]  F. Durst,et al.  Asymmetric flows and instabilities in symmetric ducts with sudden expansions , 1978, Journal of Fluid Mechanics.

[20]  Timothy A. Davis,et al.  An Unsymmetric-pattern Multifrontal Method for Sparse Lu Factorization , 1993 .

[21]  Jie Shen,et al.  Hopf bifurcation of the unsteady regularized driven cavity flow , 1991 .

[22]  G. McKinley,et al.  Simulations of extensional flow in microrheometric devices , 2008 .

[23]  Karl Gustafson,et al.  Hopf bifurcation in the driven cavity , 1990 .

[24]  Milton S. Plesset,et al.  Viscous Effects in Rayleigh-Taylor Instability. , 1974 .

[25]  D. Degani,et al.  Stability and existence of multiple solutions for viscous flow in suddenly enlarged channels , 1990 .

[26]  A. Revuelta.  On the two-dimensional flow in a sudden expansion with large expansion ratios , 2005 .

[27]  J. Svendsen,et al.  Valves in the Heart of the Big Apple VI: Evaluation and Management of Valvular Heart Diseases 2010 , 2010, Cardiology.

[28]  Carmen Ginghină.  The Coandă effect in cardiology , 2007, Journal of cardiovascular medicine.

[29]  P. Drazin,et al.  Bifurcations of two-dimensional channel flows , 1986, Journal of Fluid Mechanics.