^{1}, Eva Kanso

^{1}and Paul K. Newton

^{1}

### Abstract

We describe the viscous evolution of a collinear three-vortex structure that corresponds initially to an inviscid point vortex fixed equilibrium, with the goal of elucidating some of the main transient dynamical features of the flow. Using a multi-Gaussian “core-growth” type of model, we show that the system immediately begins to rotate unsteadily, a mechanism we attribute to a “viscously induced” instability. We then examine in detail the qualitative and quantitative evolution of the system as it evolves toward the long-time asymptotic Lamb–Oseen state, showing the sequence of topological bifurcations that occur both in a fixed reference frame and in an appropriately chosen rotating reference frame. The evolution of passive particles in this viscously evolving flow is shown and interpreted in relation to these evolving streamline patterns.

This work is partially supported by the National Science Foundation through the CAREER Award No. CMMI 06-44925 (F.J. and E.K.) and Grant No. NSF-DMS-0804629 (P.K.N.).

I. INTRODUCTION

II. PROBLEM SETTING

III. THE MULTI-GAUSSIAN MODEL

IV. COMPARISON TO NAVIER–STOKES

V. EVOLUTION OF VORTICITY IN THE MULTI-GAUSSIAN MODEL

VI. CONCLUSIONS

### Key Topics

- Rotating flows
- 87.0
- Vortex dynamics
- 48.0
- Bifurcations
- 20.0
- Numerical modeling
- 14.0
- Viscosity
- 12.0

## Figures

Fixed point vortex equilibrium: three collinear and equally spaced point vortices with the outer vortices of strength and the middle vortex of strength .

Fixed point vortex equilibrium: three collinear and equally spaced point vortices with the outer vortices of strength and the middle vortex of strength .

Vorticity contours (top row) and streamlines (bottom row) of Navier–Stokes simulation for at , 2.8, 43.7, and 47.4. The vortex configuration rotates unsteadily for . The center vortex stretches and diffuses out first, then the outer two vortices begin to merge. Eventually the vortex configuration approaches a single Gaussian vortex.

Vorticity contours (top row) and streamlines (bottom row) of Navier–Stokes simulation for at , 2.8, 43.7, and 47.4. The vortex configuration rotates unsteadily for . The center vortex stretches and diffuses out first, then the outer two vortices begin to merge. Eventually the vortex configuration approaches a single Gaussian vortex.

Vorticity contours (top row) and streamlines (bottom row) of multi-Gaussian model for and at four instants , 25, 60, and 300. Similar to Navier–Stokes simulation, the vortex configuration rotates unsteadily for , the center vortex stretches and diffuses out first, then the outer vortices merge, eventually the vortex configuration approaches a single Gaussian vortex.

Vorticity contours (top row) and streamlines (bottom row) of multi-Gaussian model for and at four instants , 25, 60, and 300. Similar to Navier–Stokes simulation, the vortex configuration rotates unsteadily for , the center vortex stretches and diffuses out first, then the outer vortices merge, eventually the vortex configuration approaches a single Gaussian vortex.

Rotation rate and rotation angle as functions of time of multi-Gaussian model for , , and .

Rotation rate and rotation angle as functions of time of multi-Gaussian model for , , and .

Evolution of the streamlines of the multi-Gaussian model. The separatrices are depicted in thick lines with arrows showing the direction of the flow. Instantaneous hyperbolic points are at intersections of separatrices while elliptic points are represented by circles.

Evolution of the streamlines of the multi-Gaussian model. The separatrices are depicted in thick lines with arrows showing the direction of the flow. Instantaneous hyperbolic points are at intersections of separatrices while elliptic points are represented by circles.

Instantaneous stagnation points: (a) a pair of hyperbolic stagnation point located at for , , and . This pair collides at (0,0) at bifurcation time . (b) A pair of elliptic stagnation points located at . This pair collides with the now hyperbolic origin at bifurcation time .

Instantaneous stagnation points: (a) a pair of hyperbolic stagnation point located at for , , and . This pair collides at (0,0) at bifurcation time . (b) A pair of elliptic stagnation points located at . This pair collides with the now hyperbolic origin at bifurcation time .

Multi-Gaussian model: (a) norm of residual vs for and , (b) norm of the difference in the vorticity field of the multi-Gaussian model and the single-peaked Lamb–Oseen vortex with circulation , and (c) shows the difference in velocity field. Clearly, for long time, the model approaches the single-peaked Gaussian but in short time, the multi-Gaussian, while not numerically accurate in comparison to the Navier–Stokes model as indicated in (a), its dynamics is richer than the single Gaussian as indicated in (b) and (c).

Multi-Gaussian model: (a) norm of residual vs for and , (b) norm of the difference in the vorticity field of the multi-Gaussian model and the single-peaked Lamb–Oseen vortex with circulation , and (c) shows the difference in velocity field. Clearly, for long time, the model approaches the single-peaked Gaussian but in short time, the multi-Gaussian, while not numerically accurate in comparison to the Navier–Stokes model as indicated in (a), its dynamics is richer than the single Gaussian as indicated in (b) and (c).

Comparison of rotation angle between (a) the Navier–Stokes simulations and (b) the multi-Gaussian model. Navier–Stokes simulations are conducted with the same initial vorticity field for Reynolds numbers , 2000, 3000, and 4000. The results of the multi-Gaussian model are obtained for and , 1/200, 1/300, and 1/400. The trend of both models is qualitatively similar.

Comparison of rotation angle between (a) the Navier–Stokes simulations and (b) the multi-Gaussian model. Navier–Stokes simulations are conducted with the same initial vorticity field for Reynolds numbers , 2000, 3000, and 4000. The results of the multi-Gaussian model are obtained for and , 1/200, 1/300, and 1/400. The trend of both models is qualitatively similar.

Comparison of the times of the first bifurcation given by the Navier–Stokes simulations and the multi-Gaussian model in log-log plot. Simulation results are plotted as squares for , 500, 1000, 2000, 3000, 4000, 5000 and the dashed line is a fitted linear line obtained by least squares distance rule. The solid line is the result from the multi-Gaussian model.

Comparison of the times of the first bifurcation given by the Navier–Stokes simulations and the multi-Gaussian model in log-log plot. Simulation results are plotted as squares for , 500, 1000, 2000, 3000, 4000, 5000 and the dashed line is a fitted linear line obtained by least squares distance rule. The solid line is the result from the multi-Gaussian model.

The velocity field induced by the collinear vortex structure becomes analogous to that of a Rankine vortex. In particular, the component of velocity along the -axis is depicted. In (a), we show the velocity profiles for , 0.1, 0.15, 0.2, 0.3, and 0.5. The maximum velocity decreases as increases. When is small, e.g., , is negative close to the origin. This is because the vorticity is still relatively concentrated at the vortex centers. In (b), we superimpose on the plots of vs (solid lines) the velocity of a Rankine vortex (dashed lines) with vorticity and time-dependent core. Clearly, the velocity field is similar to that induced by a rigid rotation close to the origin and it is similar to an inverse decay at large distance from the origin.

The velocity field induced by the collinear vortex structure becomes analogous to that of a Rankine vortex. In particular, the component of velocity along the -axis is depicted. In (a), we show the velocity profiles for , 0.1, 0.15, 0.2, 0.3, and 0.5. The maximum velocity decreases as increases. When is small, e.g., , is negative close to the origin. This is because the vorticity is still relatively concentrated at the vortex centers. In (b), we superimpose on the plots of vs (solid lines) the velocity of a Rankine vortex (dashed lines) with vorticity and time-dependent core. Clearly, the velocity field is similar to that induced by a rigid rotation close to the origin and it is similar to an inverse decay at large distance from the origin.

Relative velocity field: two new pairs of stagnation points appear from infinity as time . (a) -component of the pair of stagnation points . This pair eventually converge to as , respectively. (b) -component of the pair of stagnation points . This pair reach at bifurcation time and collapse at (0,0) at bifurcation time . Parameters are and .

Relative velocity field: two new pairs of stagnation points appear from infinity as time . (a) -component of the pair of stagnation points . This pair eventually converge to as , respectively. (b) -component of the pair of stagnation points . This pair reach at bifurcation time and collapse at (0,0) at bifurcation time . Parameters are and .

Evolution of the separatrices of the relative velocity field . Instantaneous hyperbolic points are at intersections of separatrices and elliptic points are represented by circles. The outside separatrices and elliptic points in (b) and (c) are plotted out of scale.

Evolution of the separatrices of the relative velocity field . Instantaneous hyperbolic points are at intersections of separatrices and elliptic points are represented by circles. The outside separatrices and elliptic points in (b) and (c) are plotted out of scale.

Colored passive tracers advected by the velocity field given in Eq. (11) and depicted in the frame rotating with the vortex structure. As time evolves, the passive tracers stretch and mix forming large lobes at a finite distance from the initial location of the vortex structure. The separatices of the relative velocity field are superimposed in black at various instants in time.

Colored passive tracers advected by the velocity field given in Eq. (11) and depicted in the frame rotating with the vortex structure. As time evolves, the passive tracers stretch and mix forming large lobes at a finite distance from the initial location of the vortex structure. The separatices of the relative velocity field are superimposed in black at various instants in time.

Homotopic equivalences of the separatrices of the fluid velocity field . Bifurcation topologies are placed in boxes.

Homotopic equivalences of the separatrices of the fluid velocity field . Bifurcation topologies are placed in boxes.

Homotopic equivalences of the separatrices of the relative velocity field . Bifurcation topologies are placed in boxes.

Homotopic equivalences of the separatrices of the relative velocity field . Bifurcation topologies are placed in boxes.

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