a b s t r a c tThis paper deals with the computation of steady bifurcations in the framework of 2D incompressible Navier-Stokes flow. We first propose a numerical method to accurately detect the critical Reynolds number where this kind of bifurcation appears. From this singular value, we introduce a numerical tool to compute all the steady bifurcated branches. All these algorithms are based on the Asymptotic Numerical Method [1,2]. The critical values are determined by using a bifurcation indicator [3][4][5] and the bifurcated branches are computed by using an augmented system which was first introduced in solid mechanics [4,6]. Several numerical examples from 2D Navier-Stokes show the reliability and the efficiency of the proposed methods.

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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