The physical mechanism for vortex merging

Abstract

In this paper, we study the interaction of two co-rotating trailing vortices. It is well-known that vortices of like-sign ultimately merge to form a single vortex, and there has been much work on measuring and predicting the initial conditions for the onset of merger, especially concerning the critical vortex core radius. However, the physical mechanism causing this merger has received little attention. In this work, we directly measure the structure of the antisymmetric vorticity field that causes the co-rotating vortices to be pushed towards each other during merger. We discover that the form of the antisymmetric vorticity comprises two counter-rotating vortex pairs, whose induced velocity field readily pushes the two centroids together. The merging velocity computed from the antisymmetric vorticity field agrees closely with the merging velocity measured directly from the total (original) flow field.The co-rotating vortex pair evolves through four distinct phases. The initial stage comprises a diffusive growth, which can be either viscous or turbulent. In either case, the number of turns that they rotate around one another until the vortices start to merge increases with Reynolds number (Re). If one observes the streamlines in a rotating reference frame (moving with the vortices), then one finds an inner and outer recirculating region of the flow bounded by a separatrix streamline. When the vortices grow large enough in the first stage, diffusion across the separatrix places vorticity into the outer recirculating region of the flow, and this leads to the generation of the antisymmetric vorticity, causing convective merger. This second (convective) stage corresponds to the motion of the vortex centroids towards each other, and is a process which is almost independent of viscosity. During the late part of this stage, the antisymmetric vorticity is diminished by a symmetrization process, and the final merging into one vorticity structure is achieved by a second diffusive stage. The fourth and ultimate phase is one where the merged vortex core grows by diffusion.

Asymmetric vortex merger: mechanism and criterion

“…It is also noted that this time is comparable to the time at which the core size, a 2 (t), deviates from viscous (linear) growth [2]. In the final diffusive/axisymmetrization phase, the flow evolves towards axisymmetry by diffusion [4].…”

“…The determination of (a/b) cr has been the focus of a number of studies, e.g., [11,10,9]. The physical mechanisms of symmetric merger have also been considered, e.g., [7,8,4,13,1,2]. In the more general interaction of two unequal size and/or strength (asymmetric) vortices, there is a greater range of flow behavior and the interaction may result in the destruction of the smaller/weaker vortex.…”

Vortex filamentation: its onset and its role on axisymmetrization and merger

“…Melander et al (1988) proposed this hypothesis simply as an heuristic tool to explain the vortex interaction. Although it has been shown to be inaccurate for various vorticity distributions (Velasco Fuentes, 2001), the same hypothesis has been recently recovered and it has been proposed as the physical mechanism for merger (Cerretelli and Williamson, 2003). Hence we will analyze below the relation of the Eulerian and Lagrangian geometries with the different regimes and, especially, the role of filamentation on merger.…”

“…These regimes are the following: (a) vortices located far apart circle around each other endlessly without undergoing large deformations, (b) vortices located at an intermediate distance circle around each other while exchanging some vorticity, and (c) vortices located at a distance smaller than a certain critical value merge into a single vortex (e.g., Melander et al, 1988). The existence of these regimes has been explained invoking the Eulerian flow geometry (Melander et al, 1988;Cerretelli and Williamson, 2003). According to this hypothesis the evolution depends on how the initial vorticity field is located with respect to the separatrices of the co-rotating stream function (e.g., Fig.…”

“…It is well known that in the course of these processes the vortices expel filaments in amounts that depend on the initial conditions (vorticity distribution, aspect ratio of the elongated vortex, intercentroid distance of the vortex pair, etc.). Vortex filamentation is generally considered to occur because a stagnation point of saddle type lies within the vortex (Melander et al, 1987(Melander et al, , 1988Polvani et al, 1989;Mariotti et al, 1994;Cerretelli and Williamson, 2003) and it has been proposed as the cause of axisymmetrization (Melander et al, 1987) and merger (Melander et al, 1988;Cerretelli and Williamson, 2003). These hypotheses are based on comparing the vorticity distribution with the Eulerian flow geometry, which may be defined as the set of stagnation points of saddle type and the associated streamlines as observed in the instantaneous velocity field.…”