Word representation of streamline topologies for structurally stable vortex flows in multiply connected domains

Abstract

Let us consider incompressible and inviscid flows in two-dimensional domains with multiple obstacles. The instantaneous velocity field becomes a Hamiltonian vector field defined from the stream function, and it is topologically characterized by the streamline pattern that corresponds to the contour plot of the stream function. The present paper provides us with a procedure to construct structurally stable streamline patterns generated by finitely many point vortices in the presence of the uniform flow. Starting from some basic structurally stable streamline patterns in a disc of low genus, i.e. a disc with a small number of holes, we repeat some fundamental operations that append a streamline pattern by increasing one genus to them. Owing to the inductive procedure, one can assign a sequence of operations as a representing word to each structurally stable streamline pattern. We also give the canonical expression for the word representation, which allows us to make a catalogue of all possible structurally stable streamline patterns in a combinatorial manner. As an example, we show all streamline patterns in the discs of genus 1 and 2.

Transitions between streamline topologies of structurally stable Hamiltonian flows in multiply connected domains

“…Here we may set U = 1, φ = 0 and a = 1 without loss of generality, since we are interested in the topological streamline structure of the uniform flow in the neighborhood of the origin. We call the singular point at the origin of D ζ (M) the 1-source-sink point whose definition is given as follows [10].…”

“…The global transition analysis given in this paper is based on the theory of word representations for streamline topologies of structurally stable Hamiltonian vector fields satisfying the slip boundary condition in two-dimensional multiply connected domains with a dipole singularity [10], which is reviewed as follows. In order to characterize multiply connected domains topologically, we use the term genus element instead of genus usually used in mathematical studies.…”

“…The construction of this paper is as follows. In the next section, we review the theory of word representations for structurally stable Hamiltonian vector fields in multiply connected domains developed in [9,10]. We then make some preparations for describing transitions between structurally stable Hamiltonian vector fields.…”

“…For example, Aref and Brøns [8] have considered a classification of streamline topologies generated by potential flows with point-vortex singularities, called vortex flows, in the 2D unbounded plane. This study has been extended to the Hamiltonian vector fields, which are a generalization of the vortex flows, in 2D multiply connected domains [9,10], in which the classification of structurally stable streamline topologies that are unchanged under small perturbations are considered. In these studies, the Hamiltonian vector fields are assumed to satisfy the slip boundary condition.…”

“…Here, we are concerned with topological structures of the level sets of the Hamiltonian, which we call streamlines by analogy from incompressible fluid flows. Classification of structurally stable streamline patterns has been considered in Yokoyama and Sakajo (2013), where a procedure to assign a unique sequence of words, called the maximal word, to these patterns is proposed. Thanks to this procedure, we can identify every streamline pattern with its representing sequence of words up to topological equivalence.…”