Three-dimensional linear stability analysis of lid-driven magnetohydrodynamic cavity flow

We present numerical results of a linear three-dimensional (3D) stability analysis of a square lid-driven cavity flow under the influence of an external magnetic field which is directed parallel to the lid. The Lorentz force has a strong influence on the two-dimensional (2D) flow structure, thereby changing number, shape and strength of the eddies. The resulting 3D stability behavior is rather complex since it depends on the 2D flow structure. Although increasing magnetic fields are able to damp 3D instability, in a parameter region around Re=3100, several branches of the neutral stability curve do exist. For a fixed Reynolds number in this range, an increase of the magnetic field strength may lead to a transition from a stable flow to an oscillatory unstable one.

[1]  U. Ghia,et al.  High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method , 1982 .

[2]  Pinhas Z. Bar-Yoseph,et al.  The effect of an external magnetic field on oscillatory instability of convective flows in a rectangular cavity , 2001 .

[3]  Hans J. Rath,et al.  Three-dimensional centrifugal-flow instabilities in the lid-driven-cavity problem , 2001 .

[4]  T. Alboussière,et al.  On the stability of the Hartmann layer , 1999 .

[5]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[6]  T. Lundgren,et al.  Transition to turbulence in an elliptic vortex , 1996, Journal of Fluid Mechanics.

[7]  S. Balachandar,et al.  Direct Numerical Simulation of Flow Past Elliptic Cylinders , 1996 .

[8]  P. N. Shankar,et al.  The eddy structure in Stokes flow in a cavity , 1993, Journal of Fluid Mechanics.

[9]  Cyrus K. Aidun,et al.  A direct method for computation of simple bifurcations , 1995 .

[10]  Kurt Liffman.  Comments on a Collocation Spectral Solver for the Helmholtz Equation , 1996 .

[11]  M. D. Deshpande,et al.  FLUID MECHANICS IN THE DRIVEN CAVITY , 2000 .

[12]  Gunter Gerbeth,et al.  The scenario of three-dimensional instabilities of the cylinder wake in an external magnetic field: A linear stability analysis , 2001 .

[13]  S. Grossmann.  The onset of shear flow turbulence , 2000 .

[14]  D. Gérard-Varet.  Amplification of small perturbations in a Hartmann layer , 2002 .

[15]  Bayly.  Three-dimensional instability of elliptical flow. , 1986, Physical review letters.

[16]  Mutsuto Kawahara,et al.  Linear stability of incompressible fluid flow in a cavity using finite element method , 1998 .