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 are 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 .