Deformation of vortex patches by boundaries (original) (raw)
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The interaction of an elliptical patch with a point vortex
Fluid Dynamics Research, 2000
An integrable model is proposed to analyze the two-dimensional asymmetrical interaction of two vortices, either co-rotating or counter-rotating, in the absence of viscosity. To this purpose two assumptions are made: one vortex is uniform and elliptical and the other one is a point vortex. It follows a system with three degrees of freedom, for which ÿrst two integrals of the motion are known: the excess energy and the second-order moment of the vorticity ÿeld. By considering the latter as a parameter, the two remaining degrees of freedom are combined into a complex variable z, hence the isolines of the excess energy may be analyzed in the z-plane, to study the motion of the system. In particular, the number of the extremal points of the excess energy ÿeld, which identify the stationary conÿgurations of the system, is calculated in di erent regions of the parameter space. The excess energy ÿeld, associated to each of these regions, leads to the speciÿcation of the system dynamics for any possible initial condition. Depending on the values of the parameters and on the initial conditions, we ÿnd di erent types of motion, corresponding to periodic, merging (also for counter-rotating vortices) and diverging solutions. Diverging interactions lead to a kind of straining out of the patch and they are possible only for counter-rotating vortices, with the ratio between the circulation of the point vortex and the one of the patch equal to − 1 2. Particular attention is given to the interactions leading to merging, where the analysis in terms of an elliptical patch under rotating strain provides an useful physical interpretation.
Vortex Interactions Subjected to Deformation Flows: A Review
2018
Deformation flows are flows incorporating shear, strain and rotational components. These flows are ubiquitous in the geophysical flows, such as the ocean and atmosphere. They appear near almost any salience, such as isolated coherent structures (vortices and jets), various fixed obstacles (submerged obstacles, continental boundaries). Fluid structures subject to such deformation flows may exhibit drastic changes in motion. In this review paper, we focus on the motion of a small number of coherent vortices embedded in deformation flows. Problems involving isolated one and two vortices are addressed. When considering a single-vortex problem, the main focus is on the evolution of the vortex boundary and its influence on the passive scalar motion. Two vortex problems are addressed with the use of point vortex models, and the resulting stirring patterns of neighbouring scalars are studied by a combination of numerical and analytical methods from the dynamical system theory. Many dynamica...
Dynamics of a 2D vortex doublet under external deformation
2004
The influence of an external strain (or shear) field on the evolution of two identical vortices is investigated in a twodimensional incompressible fluid. Using point vortex modeling, two regimes of the vortex doublet (co-rotation and irreversible separation) are determined; the critical intensity of the large scale flow separating these two regimes for a given initial separation of vortices, is calculated. Finite-area effects are then considered for the vortices. The steady states of piecewise constant vortices are computed algebraically and numerically; positive strain (or shear) favors vortex deformation. This deformation has a dominant elliptical component. An elliptical model of two vortices confirms the point vortex model results for centroid trajectories, and the steady state model results concerning the influence of positive strain on vortex deformation. It also provides an estimate of critical merger distance in the presence of large scale flow. Finally, the finite-time, nonlinear evolution of the vortex doublet is simulated with a numerical code of the 2D vorticity equation. The various regimes (stationarity, merger, co-rotation, ejection) are classified in the plane of initial vortex separation and of external deformation. These regimes are analyzed, and the critical merger distance is evaluated for negative and positive external strain; the results are in agreement with the elliptical model prediction. Merger efficiency, defined as the ratio of final to initial vortex circulation, is computed; for the same initial distance, it is smaller for negative strain. It also depends in a more complex way of the initial vortex distance.
On the evolution of nearly circular vortex patches
Communications in Mathematical Physics, 1988
Recently, the classical problem of the evolution of patches of constant vorticity was reformulated as an evolution equation for the boundary of the patch. We study this equation in the neighborhood of the circular vortex patch and introduce a hierarchy of area-preserving nonlinear approximate equations. The first of these equations is shown to have a rich rigid structure: it possesses an exhaustive increasing sequence of linear invariant manifolds of arbitrarily large finite dimensions. On each of these manifolds the equation can be written as an explicit finite system of ordinary differential equations. Solutions of these ODEs, starting from arbitrarily small neighborhoods of the circular vortex patch, are shown to blow up.
On the three‐dimensional instability of elliptical vortex subjected to stretching
Physics of Fluids, 1996
It is known that two-dimensional vortices are subject to generic three-dimensional instabilities. This phenomenon is located near the core of vortices and depends on the eccentricity of their streamlines. In this paper we are concerned with the modification of this instability when stretching is applied to such vortices. We describe this instability by linearizing the Navier-Stokes equations around a basic state, which is an exact time-dependent solution. The complete system for the perturbations is reduced to a single equation for the perturbed velocity along the vortex span. In the limit of weak stretching, a perturbation theory can be performed and leads to a WKBJ approximation for the solution. This procedure demonstrates that a small amount of stretching is able to prevent the appearance of three-dimensional instabilities for vortices with a low enough eccentricity. Since most vortices are slightly elliptical in turbulent flows, the above computations are expected to cover a wide range of experimental cases. In particular, it is tentatively argued that this mechanism may explain recent experimental observations ͓Phys. Fluids 7, 630 ͑1995͔͒.
On shape derivative and free-boundary problems in vortex dynamics
2017
A new shape derivative formula for singular contour integrals with logarithmic kernels is proposed. This formula yields a simple numerical scheme to compute vortex patch equilibria. Owing to its simplicity, any steady configuration of point vortices can be extended to that of vortex patches. As a test problem, a doubly periodic array of vortex patches is considered to show the efficiency of the new formula. Many non-trivial families of stationary vortex patch lattices are presented.
Global Bifurcation of Rotating Vortex Patches
Communications on Pure and Applied Mathematics, 2019
We rigorously construct continuous curves of rotating vortex patch solutions to the twodimensional Euler equations. The curves are large in that, as the parameter tends to infinity, the minimum along the interface of the angular fluid velocity in the rotating frame becomes arbitrarily small. This is consistent with the conjectured existence [WOZ84, Ove86] of singular limiting patches with 90 • corners at which the relative fluid velocity vanishes. For solutions close to the disk, we prove that there are "Cat's eyes"-type structures in the flow, and provide numerical evidence that these structures persist along the entire solution curves and are related to the formation of corners. We also show, for any rotating vortex patch, that the boundary is analytic as soon as it is sufficiently regular.
The erosion of a distributed two-dimensional vortex in a background straining flow
Journal of Fluid Mechanics, 2001
Herein we present a simplified theory for the behaviour of a vortex embedded in a growing external straining flow. Such a flow arises naturally as a vortex moves relative to other vortices. While the strain may generally exhibit a complex time dependence, the salient features of the vortex evolution can be understood in the simpler context, studied here, of a linearly growing strain. Then, all of the typical stages of evolution can be seen, from linear deformation, to the stripping or erosion of low-lying peripheral vorticity, and finally to the breaking or rapid elongation of the vortex into a thin filament.When, as is often the case in practice, the strain growth is slow, the vortex adjusts itself to be in approximate equilibrium with the background flow. Then, the vortex passes through, or near, a sequence of equilibrium states until, at a critical value of the strain, it suddenly breaks. In the intermediate period before breaking, the vortex continuously sheds peripheral vortici...
Elliptic instability of a co-rotating vortex pair
Journal of Fluid Mechanics, 2005
In this paper, we report experimental results concerning a three-dimensional shortwave instability observed in a pair of equal co-rotating vortices. The pair is generated in water by impulsively started plates, and is analysed through dye visualizations and detailed quantitative measurements using particle image velocimetry. The instability mode, which is found to be stationary in the rotating frame of reference of the two-vortex system, consists of internal deformations of the vortex cores, which are characteristic of the elliptic instability occurring in strained vortical flows. Measurements of the spatial structure, wavelengths and growth rates are presented, as functions of Reynolds number and non-dimensional core size. The self-induced rotation of the vortex pair, which is not a background rotation of the entire flow, is found to lead to a shift of the unstable wavelength band to higher values, as well as to higher growth rates. In addition, a dramatic increase in the width of the unstable bands for large values of the rescaled core radius is found. Comparisons with recent theoretical results by Le concerning elliptic instability of co-rotating vortices show very good agreement.
This part introduces the reader to the understanding of fl uid motion in terms of vortex dynamics. First the conceptual background on vortex dynamics is drawn, developing an intuition why most fl uid phenomena involve vortices and why they are so relevant. Then the reader is accompanied along the life of vortices: where they come from, why they form as vortices, their interaction with other vortices, up to the fi nal dissipation. During this, vortices reveal to have a major infl uence on the wall shear stress along nearby tissues, and the process of vortex formation is associated to the development of forces on surrounding boundaries. Finally, an account is worthy of turbulence in terms of vorticity.