Streamline topologies near a fixed wall using normal forms

Abstract

Bakker's work is revisited in a more general setting allowing a curvature of the fixed wall and a time dependence of the streamlines. The velocity field is expanded at a point on the wall, and the expansion coefficients are considered as bifurcation parameters. A series of nonlinear coordinate changes results in a much simplified system of differential equations for the streamlines (a normal form) encapsulating all the features of the original system. From this, a complete description of bifurcations up to codimension three close to a simple linear degeneracy is obtained. Further, the case of a non-simple degeneracy is considered. Finally the effect of the Navier-Stokes equations on the local topology is investigated.

Codimension three bifurcation of streamline patterns close to a no-slip wall: A topological description of boundary layer eruption

“…The number of bifurcation parameters which remain in the normal form is the co-dimension of the degenerate critical point. Under the non-degeneracy condition a 0,3 = 0 normal forms and bifurcation diagrams of co-dimension up to three has been obtained in [28,31,32]. This is known as the simple case, while the case a 0,3 = 0 is non-simple.…”

“…see [28,31]. Two different bifurcation diagrams result from this normal form, depending on the sign σ ofã 0,4 /a 2,2 .…”

“…It is interesting to note that the stream function (9) cannot occur in a steady flow. From the steady Navier-Stokes equations further conditions on the coefficients appear which ensure [31] thatã 0,4 < 0. Hence, the analysis of this paper is relevant only for unsteady flows.…”

“…Following Hartnack [31] we introduce the new variable ξ = x + a 1,3 2a 2,2 y to eliminate the term a 1,3 xy 3 . As a consequence of the assumptionã 0,4 = 0, the streamfunction then reads …”

“…For unsteady flows time can also be considered a parameter in the system (1). A general bifurcation theory for streamline patterns has been developed by several authors [28,29,30,31,32,33] and many applications to specific flow problems such as vortex breakdown [34,35], driven cavities [36,37,38], the cylinder wake [39,40] and peristaltic flows [41] are available. The analysis of topological bifurcations consists in identifying degenerate streamline patterns and their unfoldings, that is, parametrized families of velocity fields which can represent all possible perturbations of the given degenerate pattern.…”