Parametric analysis of steady bifurcations in 2D incompressible viscous flow with high order algorithm

• A new algorithm for parametric studies with evolving geometric parameter is proposed.
• The algorithm associates Asymptotic Numerical Method, Padé Approximants and Homotopy.
• 2 examples, in fluid mechanics with incompressible Newtonian fluid, are solved.
• This algorithm is compared to Newton’s algorithm and ANM based algorithm.
• Results show the efficiency of the solution with lower computational times.

This work deals with the computation of steady bifurcation points in 2D incompressible Newtonian fluid flows. The problem is modeled with the Navier–Stokes equations with an evolving geometric parameter. The aim of the present study is to propose a reliable and efficient numerical method for parametric steady bifurcation calculations. The numerical algorithm is based on the coupling of a continuation method with a homotopy technique. The continuation method lies on the asymptotic numerical method with Padé approximants for an initial linear stability analysis with an initial geometric configuration. The homotopy technique completes the calculation with the computation of critical Reynolds numbers for different discrete values of the geometric parameter. Two classical numerical problems are approached. The first one is the flow in sudden expansion. The geometric parameter is the height of the expansion inlet. The second problem is the flow in a divergent/convergent channel. In this case, the geometric parameter is the length of the channel. Comparisons of results with those obtained from the literature are performed, showing the efficiency of the proposed algorithm. The aim of this study is to determine the critical Reynolds numbers of the flow using few computations for each geometric parameter.