Linear stability of lid‐driven cavity flow

Previous experimental studies indicate that the steady two‐dimensional flow in a lid‐driven cavity becomes unstable and goes through a sequence of transitions before becoming turbulent. In this study, an analysis of this instability is undertaken. The two‐dimensional base flow is computed numerically over a range of Reynolds numbers and is perturbed with three‐dimensional disturbances. The partial differential equations governing the evolution of these perturbations are then obtained using linear stability analysis and normal mode analysis. Using a finite difference discretization, a generalized eigenvalue problem is formulated from these equations whose solution gives the dispersion relation between complex growth rate and wave number. An eigenvalue solver using simultaneous iteration is employed to identify the dominant eigenvalue which is indicative of the growth rate of these perturbations and the associated eigenfunction which characterizes the secondary state. This paper presents stability curves to...

[1]  Natarajan Ramanan,et al.  MULTIGRID SOLUTION OF NATURAL CONVECTION IN A VERTICAL SLOT , 1989 .

[2]  William H. Press,et al.  Numerical recipes , 1990 .

[3]  Robert L. Street,et al.  Three‐dimensional unsteady flow simulations: Alternative strategies for a volume‐averaged calculation , 1989 .

[4]  Cyrus K. Aidun,et al.  Transition to unsteady nonperiodic state in a through‐flow lid‐driven cavity , 1992 .

[5]  S. G. Rubin,et al.  A diagonally dominant second-order accurate implicit scheme , 1974 .

[6]  William J. Stewart,et al.  A Simultaneous Iteration Algorithm for Real Matrices , 1981, TOMS.

[7]  H. B. Keller,et al.  Driven cavity flows by efficient numerical techniques , 1983 .

[8]  R. M. Clever,et al.  Transition to time-dependent convection , 1974, Journal of Fluid Mechanics.

[9]  L. E. Scriven,et al.  Finding leading modes of a viscous free surface flow: An asymmetric generalized eigenproblem , 1988, J. Sci. Comput..

[10]  Michael A. Saunders,et al.  Algorithm 583: LSQR: Sparse Linear Equations and Least Squares Problems , 1982, TOMS.

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

[12]  O. Burggraf.  Analytical and numerical studies of the structure of steady separated flows , 1966, Journal of Fluid Mechanics.

[13]  A. Acrivos,et al.  Steady flows in rectangular cavities , 1967, Journal of Fluid Mechanics.

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

[15]  T. Taylor,et al.  A Pseudospectral method for solution of the three-dimensional incompressible Navier-Stokes equations , 1987 .

[16]  P. Moin,et al.  Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations , 1984 .

[17]  R. Kessler,et al.  Nonlinear transition in three-dimensional convection , 1987, Journal of Fluid Mechanics.

[18]  Cyrus K. Aidun,et al.  Global stability of a lid‐driven cavity with throughflow: Flow visualization studies , 1991 .

[19]  Robert L. Street,et al.  The Lid-Driven Cavity Flow: A Synthesis of Qualitative and Quantitative Observations , 1984 .

[20]  R G Storer,et al.  Numerical studies of toroidal resistive magnetohydrodynamic instabilities , 1986 .

[21]  F. Busse,et al.  Non-linear properties of thermal convection , 1978 .

[22]  Arthur Rizzi,et al.  Computer-aided analysis of the convergence to steady state of discrete approximations to the euler equations , 1985 .

[23]  Some observations on the influence of longitudinal vortices in a lid-driven cavity flow , 1988 .

[24]  Eckart Meiburg,et al.  High Marangoni number convection in a square cavity , 1984 .