Enhanced stability of flows through contraction channels: combining shape optimisation and linear stability analysis

The first flow bifurcation, in channels with a sudden geometry contraction, is controlled through shape optimization to delay the onset of asymmetry. First, we confirm the existence of a pitchfork type bifurcation instability, already reported in similar geometries. The global mode responsible for this bifurcation leads to asymmetric flow for Reynolds numbers beyond a critical value. Second, we propose a global shape optimization methodology to introduce small modifications in the channel geometry that lead to flows with enhanced stability. Our results include three contraction ratios C = 2, 4, and 8, where C is the ratio of upstream to downstream channel widths. The shape optimization aims at minimizing the growth rate of the unstable mode responsible for asymmetry. Sensitivity analysis is used to find an appropriate geometry parametrization, which is defined through super-elliptic curves, and limited to small deformations. Additionally, a dynamically updated surrogate model (based on radial basis functions) is developed to drive the optimization. This substitutes expensive function evaluations, each requiring the solution of a steady Navier-Stokes base flow computation and a solution of an eigenvalue problem (linear stability analysis). Finally, a mode tracking algorithm identifies the eigenmode responsible for the onset of asymmetry during the optimization. The optimized geometries show rounded corners and are stable for Reynolds numbers well beyond the original values. For all contraction ratios, the critical Reynolds number increases by at least 7.9 times with respect to the original values. Three-dimensional simulations confirm that the optimized geometry is more stable than the original when periodic boundaries are used on the side walls. When comparing the drag of the optimized geometry to the original, we obtain a reduction of at least 64%