A vorticity-only formulation and a low-order asymptotic expansion solution near Hopf bifurcation
Résumé
A vorticity-only formulation is used in order to study the behavior of the solutions of the Navier-Stokes equations for two-dimensional incompressible flows as the Reynolds number is increased. This approach allows one to limit the numerical-solution domain to the vortical region of the flow, thereby reducing the number of the state variables of the system. The vorticity-only formulation is obtained from a vorticity-stream function formulation by inverting the Poisson equation relating the vorticity to the stream function and substituting the expression for the velocity in the vorticity-transport equation. The vorticity at the solid boundary is determined from the boundary conditions. The resulting dynamical system consists of a set of first-order ordinary differential equations having only quadratic nonlinearities. This system is then used to address the behavior of the solution beyond the stability boundary, within the context of the theory of dynamical systems. This part of the paper is general and is based on the use of a singular-perturbation technique, known as the method of multiple time scales; formulae for the nonlinear analysis near a Hopf bifurcation are given explicitly in terms of the coefficients of the dynamical system. Preliminary numerical results for the validation of the formulation are presented for the limited case of two-dimensional flows around a circular cylinder. These results include the steady-state solution with varying Reynolds number, the eigenvalues and the eigenvectors of the related stability matrix, and the characteristics of the corresponding limit cycle.