Linear Three Dimensional Stability of Two-Sided Non-Facing Lid Driven Cavity Flows

The two-sided non-facing lid driven cavity flow for which the upper wall is moved to the right and the left wall to the bottom with equal speeds is investigated numerically for its linear three dimensional stability. The two-dimensional basic steady-state is first obtained by the Taylor-Hood finite element method through Newton iteration process. The triangulation of finite element mesh is based on a transformed Chebyshev Gauss-Lobatto collocation nodes, which is also used for the spatial discretization of the linear stability equations with a high-order finite-difference scheme. The resulting generalized eigenvalue problem in a matrix form is then solved by the implicitly restarted Arnoldi method with the shift-and-invert algorithm. Through the eigenvalue computation of linear stability equations, the most unstable stationary mode for long wave instability and two pairs of symmetrical travelling modes for short wave instability are found. The critical Reynolds number of the most unstable stationary mode occurs at Reynolds number Re_c=261.5 which is far smaller than that of the two-dimensional instability Re_c^2d=1 061.7. Through the plotting of spatial iso-surface of streamwise disturbance velocity and streamwise distur-bance vorticity for all three types of eigenmodes, it is also found that the unstable disturbances are located within the region of the two large primary vortices of the basic steady-state, which are considered as the disturbance energy source for the onset of instability.