Flow development and eddy structure in an L‐shaped cavity with lids moving in the same directions have been investigated using both tools from topological and numerical methods. In particular, structural bifurcation near a nonsimple degenerate point is investigated by making a local analysis of the velocity field based on a Taylor series expansion. The streamlines of a Hamiltonian vector field system are simplified by using the homotopy invariance of the index theory. A series of bifurcation curves are constructed to determine the sequence of flow structures by which eddies are generated in the L‐shaped cavity.
The structural bifurcation of a 2D divergence free vector field u(·, t) when u(·, t 0 ) has an interior isolated singular point x 0 of zero index has been studied by Ma and Wang [16]. Although in the class of divergence free fields which undergo a local bifurcation around a singular point, the ones with index zero singular points are generic, this class excludes some important families of symmetric flows. In particular, when u(·, t 0 ) is anti-symmetric with respect to x 0 , or symmetric with respect to the axis located on x 0 and normal to the unique eigendirection of the Jacobian Du(·, t 0 ), the vector field must have index 1 or -1 at the singular point. Thus we study the structural bifurcation when u(·, t 0 ) has an interior isolated singular point x 0 with index -1, 1. In particular we show that if such a vector field with its acceleration at t 0 both satisfy aforementioned symmetries then generically the flow will undergo a local bifurcation. Under these generic conditions, we rigorously prove the existence of flow patterns such as pairs of corotating vortices and double saddle connections. We also present numerical evidence of Stokes flow in a rectangular and cylindrical cavity showing that the bifurcation scenarios we present are indeed realizable.
The aim of this paper is to obtain streamline patterns of axisymmetric flow and their bifurcations for 2-D incompressible flows close to non-simple singular point. The streamlines of a Hamiltonian vector field system are simplified by using the homotopy invariance of the index theory. Using a homotopy invariance of the index, we develop a theory for the sufficient and necessary conditions for structural bifurcation of axisymmetric flow near non-simple degenerate critical points. The variation of parameters in the flow field can cause structural bifurcations. The bifurcation of the degenerate flow structure is obtained when it is perturbed slightly.
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