Generalized Contour Dynamics: A Review

Contour dynamics is a computational technique to solve for the motion of vortices in incompressible inviscid flow. It is a Lagrangian technique in which the motion of contours is followed, and the velocity field moving the contours can be computed as integrals along the contours. Its best-known examples are in two dimensions, for which the vorticity between contours is taken to be constant and the vortices are vortex patches, and in axisymmetric flow for which the vorticity varies linearly with distance from the axis of symmetry. This review discusses generalizations that incorporate additional physics, in particular, buoyancy effects and magnetic fields, that take specific forms inside the vortices and preserve the contour dynamics structure. The extra physics can lead to time-dependent vortex sheets on the boundaries, whose evolution must be computed as part of the problem. The non-Boussinesq case, in which density differences can be important, leads to a coupled system for the evolution of both mean interfacial velocity and vortex sheet strength. Helical geometry is also discussed, in which two quantities are materially conserved and whose evolution governs the flow.     

Self-similar point vortices and confinement of vorticity

This papers deals with the large time behavior of solutions of the incompressible Euler equations in dimension 2. We consider a self-similar configuration of point vortices which grows like the square root of the time. We study the confinement properties of a blob of vorticity initially located around the first point vortex and moving in the velocity field produced by itself and by the other point vortices. We find a su?cient condition on the point vortices such that the vorticity stays confined around the first point vortex at a rate better than the square root of the time. The relevance to the large time behavior of the Euler equations is discussed.     

Parametric instability of a many point-vortex system in a multi-layer flow under linear deformation

The paper deals with a dynamical system governing the motion of many point vortices located in different layers of a multi-layer flow under external deformation. The deformation consists of generally independent shear and rotational components. First, we examine the dynamics of the system’s vorticity center. We demonstrate that the vorticity center of such a multi-vortex multi-layer system behaves just like the one of two point vortices interacting in a homogeneous deformation flow. Given nonstationary shear and rotational components oscillating with different magnitudes, the vorticity center may experience parametric instability leading to its unbounded growth. However, we then show that one can shift to a moving reference frame with the origin coinciding with the position of the vorticity center. In this new reference frame, the new vorticity center always stays at the origin of coordinates, and the equations governing the vortex trajectories look exactly the same as if the vorticity center had never moved in the original reference frame. Second, we studied the relative motion of two point vortices located in different layers of a two-layer flow under linear deformation. We analyze their regular and chaotic dynamics identifying parameters resulting in effective and extensive destabilization of the vortex trajectories.     

Vortex design problem

In this investigation we propose a computational approach for the solution of optimal control problems for vortex systems with compactly supported vorticity. The problem is formulated as a PDE-constrained optimization in which the solutions are found using a gradient-based descent method. Recognizing such Euler flows as free-boundary problems, the proposed approach relies on shape differentiation combined with adjoint analysis to determine cost functional gradients. In explicit tracking of interfaces (vortex boundaries) this method offers an alternative to grid-based techniques, such as the level-set methods, and represents a natural optimization formulation for vortex problems computed using the contour dynamics technique. We develop and validate this approach using the design of 2D equilibrium Euler flows with finite-area vortices as a model problem. It is also discussed how the proposed methodology can be applied to Euler flows featuring other vorticity distributions, such as vortex sheets, and to time-dependent phenomena.     

Evolution of two concentrated vortices in a two-dimensional bounded domain

We prove that two initially concentrated vortices with opposite vorticity of an incompressible ideal fluid moving in a two-dimensional bounded domain, remain concentrated during the time. The motion of their centers converges to the solution of the point vortex model with the corresponding initial conditions.     

Stability and Structure of Stretched Vortices

The radially symmetric Burgers vortex is an example of a solution to the Navier-Stokes equations in which the intensification of vorticity due to vortex stretching is balanced by the diffusion of vorticity through viscosity. The linear stability of the symmetric Burgers vortex to a class of two-dimensional perturbations is demonstrated. Analytical solutions obtained from a perturbation analysis and numerical computations are presented of nonsymmetric Burgers vortices in which the radial flow field in a plane perpendicular to the vorticity is nonsymmetric.     

Convergence of the point vortex method for 2-D vortex sheet

We give an elementary proof of the convergence of the point vortex method (PVM) to a classical weak solution for the two-dimensional incompressible Euler equations with initial vorticity being a finite Radon measure of distinguished sign and the initial velocity of locally bounded energy. This includes the important example of vortex sheets, which exhibits the classical Kelvin-Helmholtz instability. A surprise fact is that although the velocity fields generated by the point vortex method do not have bounded local kinetic energy, the limiting velocity field is shown to have a bounded local kinetic energy.

     

Numerical analysis of the dynamics of distributed vortex configurations

A numerical algorithm is proposed for analyzing the dynamics of distributed plane vortex configurations in an inviscid incompressible fluid. At every time step, the algorithm involves the computation of unsteady vortex flows, an analysis of the configuration structure with the help of heuristic criteria, the visualization of the distribution of marked particles and vorticity, the construction of streamlines of fluid particles, and the computation of the field of local Lyapunov exponents. The inviscid incompressible fluid dynamic equations are solved by applying a meshless vortex method. The algorithm is used to investigate the interaction of two and three identical distributed vortices with various initial positions in the flow region with and without the Coriolis force.     

Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations

We study the interplay between the local geometric properties and the non-blowup of the 3D incompressible Euler equations. We consider the interaction of two perturbed antiparallel vortex tubes using Kerr's initial condition . We use a pseudo-spectral method with resolution up to 1536 × 1024 × 3072 to resolve the nearly singular behavior of the Euler equations. Our numerical results demonstrate that the maximum vorticity does not grow faster than doubly exponential in time, up to t = 19, beyond the singularity time t = 18.7 predicted by Kerr's computations , . The velocity, the enstrophy, and the enstrophy production rate remain bounded throughout the computations. As the flow evolves, the vortex tubes are flattened severely and turned into thin vortex sheets, which roll up subsequently. The vortex lines near the region of the maximum vorticity are relatively straight. This local geometric regularity of vortex lines seems to be responsible for the dynamic depletion of vortex stretching.     

Local Instability in Curved Flow

Expressions are obtained for the rates of change of the vorticitycomponents about the tangent, principal normal, and binormalof inviscid flow of arbitrary configuration with curvature andtorsion of the particle paths, and non-uniform density. If a steady flow pattern exists the vorticity changes are exactlythose required to carry the fluid particles through it. A smallrotational displacement of a fluid element about each of thesedirections is then considered separately and if the additionalrate of change of vorticity has the same sign and order of magnitudeas the displacement the motion is unstable locally regardlessof the motion elsewhere. The equations depend on four quantities: the curvature, torsion,velocity and vorticity at the point where stability is beinginvestigated. These four quantities define a helical vortex,with the same stability properties, in which the flow is equivalent.There is therefore special merit in studying this case. It is found that a vortex with helical particle paths is unstablefor rotation of the fluid elements about directions which liebetween the vorticity vector and the direction of the axis ofthe motion, when the density is uniform. More generally, themotion is unstable for these disturbances if the stagnationpressure decreases radially outwards. A gradient of axial velocitycomponent always causes some local instability, and the mostlikely (fastest growing) disturbance is one in which rotationsoccur around a line everywhere bisecting the angle between thevorticity vector and the direction of the axis. The analysis shows that in two dimensional circular flow themost unstable disturbances are toroidal and in general curvedflow in two dimensions (without torsion) the most likely disturbancesare rotations around the tangent, i.e. longitudinal rolls. The criteria obtained for local instability do not agree witha Richardson-type criterion for local stability, but both showthe destabilizing effect of a radiating gradient of axial velocity.     

Modeling of local particle accumulation in disperse flows with kinematic singularities

The aim of the study is to model the formation of local particle accumulation zones near several typical kinematic singularities. The flows considered are: (i) a steady two–dimensional flow with localized vorticity of the Kelvin cat's eye type (vortex in a mixing layer), (ii) a steady axisymmetric flow formed by a vortex filament normal to a plane in viscous fluid (simple model of tornado), (iii) a neighbourhood of a zero acceleration point in two–dimensional unsteady (harmonic) flow. From parametric numerical calculations, we investigated the inertial mechanisms of forming local particle accumulation zones and found the threshold values of governing parameters separating qualitatively different particle velocity and density patterns. In particular, it is shown that the zero–acceleration point can either “attract” or “scatter” the particles. Zones of concentrated vorticity are typically devoid of particles. In the tornado–like flow, an axisymmetric “cup-shaped” particle accumulation region is formed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homostrophic vortex interaction under external strain,in a coupled QG-SQG model

The interaction between two co-rotating vortices, embedded in a steady external strain field, is studied in a coupled Quasi-Geostrophic — Surface Quasi-Geostrophic (hereafter referred to as QG-SQG) model. One vortex is an anomaly of surface density, and the other is an anomaly of internal potential vorticity. The equilibria of singular point vortices and their stability are presented first. The number and form of the equilibria are determined as a function of two parameters: the external strain rate and the vertical separation between the vortices. A curve is determined analytically which separates the domain of existence of one saddle-point, and that of one neutral point and two saddle-points. Then, a Contour-Advective Semi-Lagrangian (hereafter referred to as CASL) numerical model of the coupled QG-SQG equations is used to simulate the time-evolution of a sphere of uniform potential vorticity, with radius R at depth −2H interacting with a disk of uniform density anomaly, with radius R, at the surface. In the absence of external strain, distant vortices co-rotate, while closer vortices align vertically, either completely or partially (depending on their initial distance). With strain, a fourth regime appears in which vortices are strongly elongated and drift away from their common center, irreversibly. An analysis of the vertical tilt and of the horizontal deformation of the internal vortex in the regimes of partial or complete alignment is used to quantify the three-dimensional deformation of the internal vortex in time. A similar analysis is performed to understand the deformation of the surface vortex.     

Wavelet filtering to study mixing in 2D isotropic turbulence

This paper presents the application of coherent vortex simulation (CVS) filtering, based on an orthogonal wavelet decomposition of vorticity, to study mixing in 2D homogeneous isotropic turbulent flows. The Eulerian and Lagrangian dynamics of the flow are studied by comparing the evolution of a passive scalar and of particles advected by the coherent and incoherent velocity fields, respectively. The former is responsible for strong mixing and produces the same anomalous diffusion as the total flow, due to transport by the coherent vortices, while mixing in the latter is much weaker and corresponds to classical diffusion.     

Trajectory statistics and turbulence evolution

The aim of this paper is to understand the tendency to organization of the turbulence in two-dimensional ideal fluids. A different perspective on vorticity separation and on the inverse cascade of energy yields from this study. Trajectory trapping or eddying appears to be strongly connected to these nonlinear processes. The statistics of the trajectories of the vorticity elements in a turbulent state is studied using a semi-analytic method. We show that the separation of the positive and negative vorticities is due to the attraction produced by a large scale vortex on the small scale vortices of the same sign. More precisely, a large scale velocity is shown to determine average transverse drifts, which have opposite orientations for positive and negative vorticity. They appear only in the presence of trapping and lead to energy flow to large scales due to the increase of the circulation of the large vortex. Recent results on drift turbulence evolution in magnetically confined plasmas are discussed in order to underline the idea that there is a link between the inverse cascade and trajectory trapping. The physical mechanisms are different in fluids and plasmas due to the different types of nonlinearities of the two systems, but trajectory trapping has the main role in both cases.     

Vortex Perturbation Dynamics

An initial value approach is used to examine the dynamics of perturbations introduced into a vortex under strain. Both the basic vortex considered and the perturbations are taken as three-dimensional. An explicit solution for the time evolution of the vorticity perturbations is given for arbitrary initial vorticity. Analytical solutions for the resulting velocity components are found when the initial vorticity is assumed to be localized. For more general initial vorticity distributions, the velocity components are determined numerically. It is found that the variation in the radial direction of the initial vorticity disturbance is the most important factor influencing the qualitative behavior of the solutions. Transient growth in the magnitude of the velocity components is found to be directly attributable to the compactness of the initial vorticity.     

Vortex damping of sloshing in tanks with baffles

The problem of damping the sloshing in tanks with sharp-edged baffles (thin inserts which partially span a longitudinal or transverse cross-section) is considered. Separation of the boundary layer and the formation of vortices occur at these sharp edges. It is assumed that the domains where there is significant vortex motion of the fluid are localized in small neighbourhoods of the sharp edges of the baffles. The non-linear vortex damping is determined from the distribution of the velocity intensity factors at these sharp edges in the same way as the linear damping, caused by the dissipation of energy in a boundary layer close to a wall, is determined from the fluid velocity distribution on the walls of a cavity. Both of the above-mentioned distributions are calculated by solving the same boundary-value problem on the oscillations of an ideal fluid. The second of the distributions characterizes the singular properties of the solutions of this problem on particular lines. A method based on the variation of the area of the baffles, which simplifies the calculation of the velocity intensity factors is described. The distinctive features arising when the method of finite elements is used are considered. The results of numerical calculations of the damping of sloshing in a cylindrical tank with a ring baffle are compared with experimental data.     

Numerical exploration of new features on three-dimensional transition of a cylinder wake

The three-dimensional transition of the wake flow behind a circular cylinder is studied in detail by direct numerical simulations using 3D incompressible N-S equations for Reynolds number ranging from 200 to 300. New features and vortex dynamics of the 3D transition of the wake are found and investigated. At Re = 200, the flow pattern is characterized by mode A instability. However, the spanwise characteristic length of the cylinder determines the transition features. Particularly for the specific spanwise charac-