^{1}, E. R. Johnson

^{2}and P. J. Morrison

^{3}

### Abstract

The deformation of two-dimensional vortex patches in the vicinity of fluid boundaries is investigated. The presence of a boundary causes an initially circular patch of uniform vorticity to deform. Sufficiently far away from the boundary, the deformed shape is well approximated by an ellipse. This leading order elliptical deformation is investigated via the elliptic moment model of Melander, Zabusky, and Styczek [J. Fluid Mech.167, 95 (Year: 1986)10.1017/S0022112086002744]. When the boundary is straight, the centre of the elliptic patch remains at a constant distance from the boundary, and the motion is integrable. Furthermore, since the straining flow acting on the patch is constant in time, the problem is that of an elliptic vortex patch in constant strain, which was analysed by Kida [J. Phys. Soc. Jpn.50, 3517 (Year: 1981)10.1143/JPSJ.50.3517]. For more complicated boundary shapes, such as a square corner, the motion is no longer integrable. Instead, there is an adiabatic invariant for the motion. This adiabatic invariant arises due to the separation in times scales between the relatively rapid time scale associated with the rotation of the patch and the slower time scale associated with the self-advection of the patch along the boundary. The interaction of a vortex patch with a circular island is also considered. Without a background flow, the conservation of angular impulse implies that the motion is again integrable. The addition of an irrotational flow past the island can drive the patch towards the boundary, leading to the possibility of large deformations and breakup.

The authors would like to thank the Woods Hole Oceanographic Institute whose Geophysical Fluid Dynamics program provided the inspiration and basis for the preceding research. The authors remain grateful to Professor David Dritschel for providing a copy of his contour dynamics code on which our code is based. A.C. is grateful for support from an EPSRC studentship. P.J.M. was supported by the U.S. Dept. of Energy (Contract No. DE-FG05-80ET-53088).

I. INTRODUCTION

II. METHODS

A. Elliptic model

B. Representing an ellipse

C. Model equations

D. Contour dynamics

III. STRAIGHT BOUNDARY

A. Model validation

IV. MOTION AROUND A CORNER

A. Adiabatic invariance

V. CHAOTIC BEHAVIOUR OF THE ELLIPTIC MODEL

VI. INTERACTION WITH AN ISLAND

A. No background flow

B. Background flow

C. Comparison with full solution

VII. CONCLUSION

### Key Topics

- Rotating flows
- 132.0
- Vortex dynamics
- 36.0
- Flow visualization
- 6.0
- Viscosity
- 5.0
- Collective models
- 4.0

## Figures

The motion of a vortex patch above a straight boundary is equivalent, via the method of images, to that of a vortex patch and an image vortex patch, of opposite vorticity, obtained by a reflection in the boundary.

The motion of a vortex patch above a straight boundary is equivalent, via the method of images, to that of a vortex patch and an image vortex patch, of opposite vorticity, obtained by a reflection in the boundary.

Possible motions in the r-θ plane as predicted by the elliptic model with e/ω = 1/18, corresponding to y c /R = 1.5. The contour that intersects r = 1, at which point θ is undefined, represents the motion of an initially circular patch (solid line). Any contour within this circular motion contour represents a nutating motion (dotted-dashed line). Contours outside the circular motion contour represent vortex patches undergoing rotation (dashed lines). For sufficiently large values of r, larger than those shown here, vortex patches are extended indefinitely.

Possible motions in the r-θ plane as predicted by the elliptic model with e/ω = 1/18, corresponding to y c /R = 1.5. The contour that intersects r = 1, at which point θ is undefined, represents the motion of an initially circular patch (solid line). Any contour within this circular motion contour represents a nutating motion (dotted-dashed line). Contours outside the circular motion contour represent vortex patches undergoing rotation (dashed lines). For sufficiently large values of r, larger than those shown here, vortex patches are extended indefinitely.

Maximum deformation r max of an initially circular patch of vorticity in terms of the distance of its centroid from the boundary, as given by the elliptic model. The model predicts that the patch is extended indefinitely when y c /R < 1.01.

Maximum deformation r max of an initially circular patch of vorticity in terms of the distance of its centroid from the boundary, as given by the elliptic model. The model predicts that the patch is extended indefinitely when y c /R < 1.01.

Motion of an initially circular patch with centroid at y c = 1.5R. Patch motion is from left to right at time intervals of 5/ω. Contour dynamics solution is given by solid lines, and the elliptic approximation by dashed lines.

Motion of an initially circular patch with centroid at y c = 1.5R. Patch motion is from left to right at time intervals of 5/ω. Contour dynamics solution is given by solid lines, and the elliptic approximation by dashed lines.

Even when the elliptic model is not valid, it still gives a good approximation to the region occupied by the vortex patch at short times. Here, we show the contour dynamics and elliptic model solutions with y c = R at time intervals of 2/ω.

Even when the elliptic model is not valid, it still gives a good approximation to the region occupied by the vortex patch at short times. Here, we show the contour dynamics and elliptic model solutions with y c = R at time intervals of 2/ω.

The motion of a vortex patch around a corner is equivalent, via the method of images, to that of a vortex patch and its image vortex patches obtained by reflection in the boundaries x = 0 and y = 0. The two image vortex patches obtained from a single reflection have the opposite vorticity to the original patch, the third image patch has the same vorticity.

The motion of a vortex patch around a corner is equivalent, via the method of images, to that of a vortex patch and its image vortex patches obtained by reflection in the boundaries x = 0 and y = 0. The two image vortex patches obtained from a single reflection have the opposite vorticity to the original patch, the third image patch has the same vorticity.

Evolution of an initially circular vortex patch in a quarter plane with initial centroid (1.5R, 100R). The time origin t = 0 is defined to occur when x c = y c . (a) Path of the patch centroid. (b) Variation of elliptic shape with time. Far away from the corner, the patch undergoes a periodic motion associated with the evolution of a patch above a straight boundary. The periodic evolution of the patch shape before and after the corner is almost exactly the same.

Evolution of an initially circular vortex patch in a quarter plane with initial centroid (1.5R, 100R). The time origin t = 0 is defined to occur when x c = y c . (a) Path of the patch centroid. (b) Variation of elliptic shape with time. Far away from the corner, the patch undergoes a periodic motion associated with the evolution of a patch above a straight boundary. The periodic evolution of the patch shape before and after the corner is almost exactly the same.

(a) Evolution of the action I for the motion shown in Figure 7 . The values of I before and after the corner interaction are almost exactly the same. (b) The difference in action ΔI before and after the corner is shown for patches whose centroid speed is modified by a factor of α−1. These values of ΔI are calculated from a root mean square average over q of the difference in action for all vortex patches with the same initial action and energy (and consequently separation from the wall) as the patch shown in Figure 7 . The difference ΔI decreases exponentially as α is increased (corresponding to ε being decreased), which is consistent with adiabatic behaviour.

(a) Evolution of the action I for the motion shown in Figure 7 . The values of I before and after the corner interaction are almost exactly the same. (b) The difference in action ΔI before and after the corner is shown for patches whose centroid speed is modified by a factor of α−1. These values of ΔI are calculated from a root mean square average over q of the difference in action for all vortex patches with the same initial action and energy (and consequently separation from the wall) as the patch shown in Figure 7 . The difference ΔI decreases exponentially as α is increased (corresponding to ε being decreased), which is consistent with adiabatic behaviour.

(a) Streamlines for the irrotational background flow Ψ(x, y) = βxy. (b) Paths of patch centroids under the point vortex approximation for the case β = −0.028ω. Whenever β/ω < 0, a stagnation point exists, and vortex patches are trapped in the corner region.

(a) Streamlines for the irrotational background flow Ψ(x, y) = βxy. (b) Paths of patch centroids under the point vortex approximation for the case β = −0.028ω. Whenever β/ω < 0, a stagnation point exists, and vortex patches are trapped in the corner region.

Poincaré sections in the x c -y c plane of forward intersections with θ = 0. The strain rate is β = −0.028ω and two different values of H are considered. (a) H/(ω2 R 4) = −0.44. The Poincaré section closely matches the earlier point vortex paths. (b) H/(ω2 R 4) = −0.27. The Poincaré section reveals chaotic behaviour. Resonances have opened up in some of the previously circular contours, and a “sea of chaos” has formed around the edge of the region.

Poincaré sections in the x c -y c plane of forward intersections with θ = 0. The strain rate is β = −0.028ω and two different values of H are considered. (a) H/(ω2 R 4) = −0.44. The Poincaré section closely matches the earlier point vortex paths. (b) H/(ω2 R 4) = −0.27. The Poincaré section reveals chaotic behaviour. Resonances have opened up in some of the previously circular contours, and a “sea of chaos” has formed around the edge of the region.

Motion of a vortex patch, initially circular with centroid ( − 2.5R, 0), around an island of radius a = R. The motion is visualised by snapshots of the patch location at time intervals of 40/ω for the elliptic model (dashed line) and the full contour dynamics solution (solid line). The path of the centroid under the elliptic model is also shown (grey line).

Motion of a vortex patch, initially circular with centroid ( − 2.5R, 0), around an island of radius a = R. The motion is visualised by snapshots of the patch location at time intervals of 40/ω for the elliptic model (dashed line) and the full contour dynamics solution (solid line). The path of the centroid under the elliptic model is also shown (grey line).

Maximum deformation r max of an initially circular patch of vorticity in terms of the initial separation of its centroid from the boundary, s c (0) − a, as given by the elliptic model. Larger islands lead to more deformation, and, in the limit a → ∞ the result for deformation above a straight boundary is recovered. Solutions with s c (0) − a < R are unphysical since the vortex patch overlaps with the island.

Maximum deformation r max of an initially circular patch of vorticity in terms of the initial separation of its centroid from the boundary, s c (0) − a, as given by the elliptic model. Larger islands lead to more deformation, and, in the limit a → ∞ the result for deformation above a straight boundary is recovered. Solutions with s c (0) − a < R are unphysical since the vortex patch overlaps with the island.

(a) Streamlines for the irrotational background flow Ψ(x, y) = Uy(1 − a 2(x 2 + y 2)−1) around an island of size a = R. (b) Paths of patch centroid under point vortex approximation with a = R and U/(ωR) = 0.1. The separatrix (heavy line) splits the motion into three regions: vortices passing below the island, vortices passing above the island, and vortices orbiting the island. This separation into three regions is generic and does not depend on the strength of the vortex.

(a) Streamlines for the irrotational background flow Ψ(x, y) = Uy(1 − a 2(x 2 + y 2)−1) around an island of size a = R. (b) Paths of patch centroid under point vortex approximation with a = R and U/(ωR) = 0.1. The separatrix (heavy line) splits the motion into three regions: vortices passing below the island, vortices passing above the island, and vortices orbiting the island. This separation into three regions is generic and does not depend on the strength of the vortex.

Motion of initially circular vortex patches past an island of the same size (R = a) calculated using the elliptic model. The irrotational background flow has strength U/(ωR) = 0.1. The motion of vortex patches is shown for three different values of the release height y 0. Each motion is visualised by the path of the ellipse centroid alongside snapshots of the elliptic shape at time intervals of 10/ω.

Motion of initially circular vortex patches past an island of the same size (R = a) calculated using the elliptic model. The irrotational background flow has strength U/(ωR) = 0.1. The motion of vortex patches is shown for three different values of the release height y 0. Each motion is visualised by the path of the ellipse centroid alongside snapshots of the elliptic shape at time intervals of 10/ω.

Maximum deformation r max of elliptic vortex patches with the same size as the island (R = a) over a range of background flow strengths U and release heights y 0. The region of parameter space for which vortex patches are indefinitely extended is shaded in dark grey. Elsewhere, contours of r max are shown. For further information, the location of the separatrix for point vortices is shown (heavy line) as is the location of the island (light grey shaded region).

Maximum deformation r max of elliptic vortex patches with the same size as the island (R = a) over a range of background flow strengths U and release heights y 0. The region of parameter space for which vortex patches are indefinitely extended is shaded in dark grey. Elsewhere, contours of r max are shown. For further information, the location of the separatrix for point vortices is shown (heavy line) as is the location of the island (light grey shaded region).

Motion of vortex patches with varying release heights driven towards an island by background flow; each panel shows a lower release height than the one before. The island and the patch have the same area, and the background flow has strength U/(ωR) = 0.1. Solutions were calculated using the elliptic model (dashed lines) and using contour dynamics (solid lines). The patch motion is visualised by snapshots of the patch position at intervals of 10/ω. In (a), the patch passes above the island and the elliptic model (r max = 1.53) is in good agreement with the full solution; there is similar agreement in (f) when the patch passes below the island (r max = 1.30). For intermediate release heights (b)–(e), the elliptic model predicts indefinite extension, as shown in (b). For these intermediate heights, the patch is either deformed into a non-elliptical shape while passing the island before returning to an approximately circular shape downstream (b), (c), (e); or the patch is extended around both sides of the island (d).

Motion of vortex patches with varying release heights driven towards an island by background flow; each panel shows a lower release height than the one before. The island and the patch have the same area, and the background flow has strength U/(ωR) = 0.1. Solutions were calculated using the elliptic model (dashed lines) and using contour dynamics (solid lines). The patch motion is visualised by snapshots of the patch position at intervals of 10/ω. In (a), the patch passes above the island and the elliptic model (r max = 1.53) is in good agreement with the full solution; there is similar agreement in (f) when the patch passes below the island (r max = 1.30). For intermediate release heights (b)–(e), the elliptic model predicts indefinite extension, as shown in (b). For these intermediate heights, the patch is either deformed into a non-elliptical shape while passing the island before returning to an approximately circular shape downstream (b), (c), (e); or the patch is extended around both sides of the island (d).

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