Linear Stability of Incompressible Flow Using a Mixed Finite Element Method

Three-dimensional flow instability in a lid-driven isosceles triangular cavity

“…Still, the máximum difference for the fine meshes is about 2.25 % which occurs for the máximum of the uo in table 1. Note that our meshes are substantially finer than those used by Ding & Kawahara (1998), who also used a finite-element method to solve the related problem of the (singular) lid-driven square-cavity problem. For basic state calculations by time-stepping, we used the time-step Ai = 0.01.…”

“…To date, the range of linear critical Reynolds numbers predicted for two-dimensional (spanwise wavenumber parameter k = 0) global flow instability, Re 2 d, c rit e [7400,8375], is wide enough to warrant further work. As far as three-dimensional global instability of spanwise homogeneous square lid-driven cavity flows is concerned, the analysis of Ding & Kawahara (1998) as well as the independently obtained complete parametric studies of three-dimensional instability in the square cavity by Theofilis (2000) and Albensoeder, Kuhlmann & Rath (2001 b) are mentioned. The first work provided an accurate description of the third unstable mode of the flow, while the second and third works completed the instability map of the flow by discovering independently the leading eigenmode of the flow, stationary mode SI with critical parameters (Re^= 0 «783, k^= 0 « 15.4) and the three additional travelling neutral modes, TI,T2 and T3,at (Re^° « 845,k^° « 15.8), (/?e^°«922,^°«7.4) and (Re*f° «961, k*f° « 14.3), respectively.…”

The linearized pressure Poisson equation for global instability analysis of incompressible flows

“…Linear instability of spanwise homogeneous flow in the square lid-driven cavity has been addressed by a number of authors [14,45]. Theofilis [58] and Albensoeder et al [4] were the first to independently discover unstable eigenvalue branches subcritical to those known up to that time and present the critical conditions of the square [4,58] and rectangular [63] cavity.…”

“…The FreeFEM++ based code used for the present instability analysis has been presented by Tammisola et al [53] and was further validated by Lashgari et al [32]. It uses an unstructured mesh comprising a total of 29,132 triangles and 14,787 vertices has been used to spatially discretize both the base flow of the regularized cavity (14) and the corresponding global eigenvalue problem. This implies a total number of degrees of freedom (and leading dimension of the matrix discretizing the EVP) of close to 2 × 10 5 .…”

“…In the present work, the domain is defined in = [0, 1] 3 and flow is set up by horizontal motion of the wall y = 1 along the positive Ox axis direction. In order to preserve spectral accuracy, in a manner analogous to (14), a regularization of the lid motion using p = 16 has been implemented, [33] …”