Localization of Hopf bifurcations in fluid flow problems

We are concerned with the precise localization of Hopf bifurcations in various fluid flow problems. This is when a stationary solution loses stability and often becomes periodic in time. The difficulty is to determine the critical Reynolds number where a pair of eigenvalues of the Jacobian matrix crosses the imaginary axis. This requires the computation of the eigenvalues (or at least some of them) of a large matrix resulting from the discretization of the incompressible Navier-Stokes equations. We thus present a method allowing the computation of the smallest eigenvalues, from which we can extract the one with the smallest real part. From the imaginary part of the critical eigenvalue we can deduce the fundamental frequency of the time-periodic solution. These computations are then confirmed by direct simulation of the time-dependent Navier-Stokes equations.

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