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Summary This paper deals with bifurcation analysis methods based on the asymptotic‐numerical method. It is used to investigate 3‐dimensional (3D) instabilities in a sudden expansion. To do so, high‐performance computing is implemented in ELMER, ie, an open‐source multiphysical software. In this work, velocity‐pressure mixed vectors are used with asymptotic‐numerical method–based methods, remarks are made for the branch‐switching method in the case of symmetry‐breaking bifurcation, and new 3D instability results are presented for the sudden expansion ratio, ie, E=3. Critical Reynolds numbers for primary bifurcations are studied with the evolution of a geometric parameter. New values are computed, which reveal new trends that complete a previous work. Several kinds of bifurcation are depicted and tracked with the evolution of the spanwise aspect ratio. One of these relies on a fully 3D effect as it breaks both spanwise and top‐bottom symmetries. This bifurcation is found for smaller aspect ratios than expected. Furthermore, a critical Reynolds number is found for the aspect ratio, ie, Ai=1, which was not previously reported. Finally, primary and secondary bifurcations are efficiently detected and all post‐bifurcated branches are followed. This makes it possible to plot a complete bifurcation diagram for this 3D case.

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This paper deals with the numerical study of bifurcations in the twodimensional (2D) lid-driven cavity (LDC). 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 side 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 computation 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 the 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 LDC 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 LDC, 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 equal to 0.47.