Flow Structures Around Non-Hyperbolic Singularities on the Wall with One Vanishing Eigenvalue: A Discussion of the Flow Structures and the Unfoldings

摘要

The report investigates flow structures near a plane wall. The flow is assumed to be steady, incompressible and it satisfies the no-slip boundary conditions on the wall. Local solutions of the continuity equation and the Navier-Stokes equations are obtained by performing series expansions near a point at the wall and these solutions are used to classify the topology of the three-dimensional, steady streamline patterns near the wall. These streamline patterns are described by the trajectories of a third-order system of which the singular points play a special role. They have been distinguished in hyperbolic and non-hyperbolic singularities. In the report only the non-hyperbolic singularities with one eigenvalue having a vanishing real part will be considered. It appears that the pressure gradient normal to the wall is of crucial importance. After determining the flow structures, bifurcation theory is used to find the physical unfoldings of the degenerate singularities. These unfoldings will be used to discuss some separated flow structures. Some critical remarks will be presented on the definitions of open separation.