High performance computations of steady-state bifurcations in 3D incompressible fluid flows by Asymptotic Numerical Method

This paper presents a powerful numerical model that implements the Asymptotic Numerical Method to compute 3D steady-state incompressible fluid flow solutions. This continuation algorithm enables to explore branches of steady-state solutions, stable or unstable, to accurately determine any simple steady-state bifurcation points and their emanating bifurcated branches. The powerfulness of the model stands on an optimal step length continuation thanks to the combination of power series analysis in the framework of ANM along with an efficient parallel implementation of the resulting algorithm on high performance computers. The outcome of this approach is demonstrated throughout 3D incompressible fluid flows inside a sudden expansion channel (expansion ratio E=3, cross-section aspect ratio 10≤B≤20). We have computed for the first time up to four steady symmetry breaking (pitchfork) bifurcations together with their associated bifurcated branches. The main characteristic of this 3D symmetric expansion configuration is that for a given cross-section aspect ratio the first bifurcation mode induces a top-bottom asymmetry, as in the 2D case, whereas the subsequent ones modulate the former in the span-wise direction with increasing wave numbers.