Numerical tools for the stability analysis of 2D flows. Application to the two and four-sided lid-driven cavity

This paper deals with the numerical study of bifurcations in the twodimensional lid-driven cavity. Two specific geometries are considered. The first geometry is the two-sided non facing (2SNF) cavity: the velocity is imposed on the upper and the left sides of the cavity. The second geometry is the four-sided (4S) cavity where all the sides have a prescribed motion. For the first time, the linear stability analysis is performed by coupling two specific algorithms. The first one is dedicated to the computation of the stationary bifurcations and the bifurcated branches. Then, a second algorithm is dedicated to the computations of Hopf bifurcations. In this study, for both problems, it is shown that the flow becomes asymmetric via a stationary bifurcation. The critical Reynolds numbers are close to 1070 and 130 respectively for the 2SNF and 4S cavity. Following the stationary bifurcated branches, supplementary results concerning the stability are found. Firstly, for both examples, a second stationary bifurcation appears on the unstable solution, for a Reynolds number equal to 1890 and 360, respectively, for the 2NSF and the 4S cavity. Secondly, a second stationary bifurcation is found on the stable solutions of the 4S lid-driven cavity for a critical Reynolds number close to 860. Nevertheless, no Hopf bifurcation has been found on this stable bifurcated branch for Reynolds numbers between 130 and 1000. Concerning the 2SNF lid-driven cavity, Hopf bifurcation points have been determined on these stable bifurcated solutions. The first bifurcation occurs for a Reynolds number close to 3000 and a Strouhal number equals to 0.47.