Low-dimensional model for vortex merging in the two-dimensional temporal mixing layer

The two-dimensional temporal mixing layer shows spiraling and merging vortices and is an example of a flow problem in which, despite the complexity, the vortices as individual coherent structures can be clearly visualized. In this paper we present a method for the analysis of the data that describes the spiraling and merging of vortices. To that end we define a parameterized set of structures, the ‘phenomenological model manifold’, which approximates the apparent spatial structures. Then we let the parameters of the manifold vary in such a way that the succession of states resembles the evolving flow as well as possible. Two different model manifolds were designed, one model for which the vortices are described with Gaussian profiles, and another in which a more optimal spatial structure is used. Projection of the numerical data on these manifolds results in information about the strength, ellipticity and trajectories of the vortices. The method is also used to study the successive merging of vortices; differing from scaling arguments for an inviscid flow, the results show that the first pairwise merging evolves approximately 2.11 times faster than the second merging. Efficient procedures are described for the required extensive optimisation problems.