^{1,a)}, Eva Kanso

^{2}and Paul K. Newton

^{2}

### Abstract

We revisit the two vortex merger problem (symmetric and asymmetric) for the Navier-Stokes equations using the core growth model for vorticity evolution coupled with the passive particle field and an appropriately chosen time-dependent rotating reference frame. Using the combined tools of analyzing the topology of the streamline patterns along with the careful tracking of passive fields, we highlight the key features of the stages of evolution of vortex merger, pinpointing deficiencies in the low-dimensional model with respect to similar experimental/numerical studies. The model, however, reveals a far richer and delicate sequence of topological bifurcations than has previously been discussed in the literature for this problem, and, at the same time, points the way towards a method of improving the model.

The work of F.J. and E.K. was partially supported by the National Science Foundation through the CAREER award (Award No. CMMI 06-44925) and the grant (Grant No. CMMI 07-57092). Part of the work carried out was supported by the National Science Foundation Grant to P.K.N. (Grant No. NSF-DMS0804629).

I. INTRODUCTION

A. Background

B. Some relevant literature on vortex merger

II. VISCOUS EVOLUTION OF SYMMETRIC AND ASYMMETRIC MERGERS

A. Vorticity and absolute velocity fields

1. Symmetric case

2. Asymmetric case

B. Relative velocity field

1. Symmetric case

2. Asymmetric case

C. Passive tracer evolution

1. Symmetric case

2. Asymmetric case

III. DISCUSSIONS AND CONCLUSIONS

### Key Topics

- Rotating flows
- 83.0
- Vortex dynamics
- 71.0
- Bifurcations
- 29.0
- Viscosity
- 14.0
- Navier Stokes equations
- 10.0

## Figures

Schematics of (a) symmetric and (b) asymmetric co-rotating vortex pairs.

Schematics of (a) symmetric and (b) asymmetric co-rotating vortex pairs.

(a) Rotation rate and (b) rotation angle θ for the co-rotating vortex pair, *Re* = 1000.

(a) Rotation rate and (b) rotation angle θ for the co-rotating vortex pair, *Re* = 1000.

Symmetric vortex pair with *Re* = 1000: (a) Vorticity along the connecting line. The two vorticity peaks eventually decay and merge to a single peak at the origin. (b) Distance between the vorticity peaks and origin as a function of time. The peaks merge at time τ = 0.125. (c) Vorticity evolution at the origin. The maximum vorticity occurs at τ = 0.0625.

Symmetric vortex pair with *Re* = 1000: (a) Vorticity along the connecting line. The two vorticity peaks eventually decay and merge to a single peak at the origin. (b) Distance between the vorticity peaks and origin as a function of time. The peaks merge at time τ = 0.125. (c) Vorticity evolution at the origin. The maximum vorticity occurs at τ = 0.0625.

Symmetric vortex pair with *Re* = 1000: Streamlines of the absolute velocity field in inertial frame at τ = 0, 0.025, 0.050 and 0.075. Each plot shows a [−1, 1] × [−1, 1] window in (*x*, *y*) plane.

Symmetric vortex pair with *Re* = 1000: Streamlines of the absolute velocity field in inertial frame at τ = 0, 0.025, 0.050 and 0.075. Each plot shows a [−1, 1] × [−1, 1] window in (*x*, *y*) plane.

Asymmetric vortex pair with *Re* = 1000: (a) Vorticity along the connecting line. One vorticity peak eventually decay and merge with the valley. (b) Distance between the vorticity peaks (and valley) and origin as function of time. One peak and the valley merge at time τ ≈ 0.0725. (c) Vorticity evolution at the origin. The maximum vorticity occurs at τ ≈ 0.0304.

Asymmetric vortex pair with *Re* = 1000: (a) Vorticity along the connecting line. One vorticity peak eventually decay and merge with the valley. (b) Distance between the vorticity peaks (and valley) and origin as function of time. One peak and the valley merge at time τ ≈ 0.0725. (c) Vorticity evolution at the origin. The maximum vorticity occurs at τ ≈ 0.0304.

Asymmetric vortex pair with *Re* = 1000: Streamlines of absolute velocity field in inertial frame at τ = 0, 0.008, 0.0146, and 0.05. Each plot shows a [ − 1, 1] × [ − 1, 1] window in (*x*, *y*) plane.

Asymmetric vortex pair with *Re* = 1000: Streamlines of absolute velocity field in inertial frame at τ = 0, 0.008, 0.0146, and 0.05. Each plot shows a [ − 1, 1] × [ − 1, 1] window in (*x*, *y*) plane.

The velocity field induced by the co-rotating vortex pair becomes analogous to that of a Rankine vortex. The component of velocity *v* _{η} along the ζ axis is depicted at various instants (solid). The velocity of a Rankine vortex with vorticity 2Γ and time-dependent core are superimposed (dashed). The velocity field is similar to a rigid rotation close to the origin and an inverse decay at larger distance from the origin.

The velocity field induced by the co-rotating vortex pair becomes analogous to that of a Rankine vortex. The component of velocity *v* _{η} along the ζ axis is depicted at various instants (solid). The velocity of a Rankine vortex with vorticity 2Γ and time-dependent core are superimposed (dashed). The velocity field is similar to a rigid rotation close to the origin and an inverse decay at larger distance from the origin.

Evolution of separatrices associated with the relative velocity field of the non-symmetric co-rotating pair in rotating frame. The arrows on separatrices show velocity direction. Hyperbolic points are represented by cross-sections of separatrices. Elliptic points are small circles. *Re* = 1000.

Evolution of separatrices associated with the relative velocity field of the non-symmetric co-rotating pair in rotating frame. The arrows on separatrices show velocity direction. Hyperbolic points are represented by cross-sections of separatrices. Elliptic points are small circles. *Re* = 1000.

Evolution of fixed points and separatrices associated with the relative velocity field for the symmetric co-rotating pair in rotating frame. The arrows on separatrices show velocity directions. Hyperbolic points are represented by cross-sections of separatrices. Elliptic points are small circles. *Re* = 1000.

Evolution of fixed points and separatrices associated with the relative velocity field for the symmetric co-rotating pair in rotating frame. The arrows on separatrices show velocity directions. Hyperbolic points are represented by cross-sections of separatrices. Elliptic points are small circles. *Re* = 1000.

Passive tracer evolution for the symmetric case.

Passive tracer evolution for the symmetric case.

Trajectories of passive tracers for the symmetric case. Their initial positions at τ = 0 are represented by ◯: (0 , 0.1), (0 , 0.5), (0 , − 0.7), (− 0.15 , 0), (− 0.4 , 1) and (1 , 0); and their final position at τ = 0.2 are marked with △.

Trajectories of passive tracers for the symmetric case. Their initial positions at τ = 0 are represented by ◯: (0 , 0.1), (0 , 0.5), (0 , − 0.7), (− 0.15 , 0), (− 0.4 , 1) and (1 , 0); and their final position at τ = 0.2 are marked with △.

Passive tracer evolution for the asymmetric case.

Passive tracer evolution for the asymmetric case.

Homotopic equivalences of the separatrices for symmetric co-rotating pair. Bifurcation states are depicted in boxes.

Homotopic equivalences of the separatrices for symmetric co-rotating pair. Bifurcation states are depicted in boxes.

Timeline of important events for the symmetric case.

Timeline of important events for the symmetric case.

Homotopic equivalences of the separatrices for asymmetric co-rotating pair. Bifurcation states are depicted in boxes.

Homotopic equivalences of the separatrices for asymmetric co-rotating pair. Bifurcation states are depicted in boxes.

Timeline of important events for the asymmetric case.

Timeline of important events for the asymmetric case.

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