On the evolution of nearly circular vortex patches

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.     

Motions of vortex patches

An evolution equation describing the motion of vortrex patches is established. The existence of steady solutions of this equation is proved. These solutions arem-fold symmetric regions of constant vorticity ω₀ and are uniformly rotating with angular velocity Ω in the range $$\tilde \Omega _{m - 1}< \tilde \Omega \leqslant \tilde \Omega _m (\tilde \Omega = \Omega /\omega _0 ,m \geqslant 2)$$ where \(\tilde \Omega _m = (m - 1)/2m\) . We call this class, ofm-fold symmetric rotating regionsD, the class of them-waves of Kelvin. Any may be regarded as a simply connected region which is a stationary configuration of the Euler equations in two dimensions. If then any magnification, rotation or reflection is also in with the same angular velocity Ω ofD. The angular velocity \(\Omega _m = \tilde \Omega _m \omega _0 \) corresponds only to the circle solution, which is a trivial member of every class ,m?2. The class corresponds to the rotating ellipses of Kirchoff. Other properties of the class are established.     

Global regularity for vortex patches

We present a proof of Chemin's [4] result which states that the boundary of a vortex patch remains smooth for all time if it is initially smooth.Partially supported by a National Science Foundation Postdoctoral FellowshipPartially supported by the National Science Foundation     

The stability of rotating vortex patches

In this paper we examine the nonlinear and linear stability of variousrotating vortex patches. These patches include the Kirchhoff ellipse, the Kelvin waves, and the co-rotating uniformm vortices. These are achieved by using relative variational methods and spectral analysis. Thus, we extend Arnol'd's idea for stability problems in [1965, 1969] to a non-smooth symmetric setting and also relate that to the usual linear stability analysis.Supported by National Science Foundation under grant DMS-8501746Dedicated to the memory of my grandmother, Lang-Chang Lee Wan     

Nonlinear stability of circular vortex patches

This paper proves that circular vortex patches in the plane are stable for the nonlinear dynamical system generated by the Euler equations of incompressible fluids. This is achieved by establishing a relative variational principle in terms of either energy or angular momentum. Thus, we exploit and extend Arnold's idea in (1965, 1969) to a nonsmooth setting as well.Research partially supported by DOE contract DE-AT03-82ER12097 to the University of California, Berkeley     

Basic sound generation mechanisms in inviscid vortex interactions at low Mach number

The sound generation mechanisms during finite core vortex interactions at low Mach number are investigated in the present study. The theoretical deductions show clearly that the basic sound generation mechanisms are associated with the vortex core deformation and the vorticity centroid dynamics, independent of the vortex system. Such deductions are substantiated by numerical experiments with the interactions of two-dimensional vortices, vortex pairs and vortex rings. Detailed discussions on the similarities and differences between the sound generation processes of the two-dimensional and axisymmetric vortex systems are given. The relative importance of the two sound generation mechanisms in these vortex systems, their characteristics and interactions, which are hardly found in existing literature, are also examined. The present findings have also generalized and substantiated the previous results of the authors on the topic.     

On the viscous and inviscid stability of magnetohydrodynamic shock waves

We study the viscous and inviscid stability of shock waves in barotropic and full magnetohydrodynamics. We show that there are magnetohydrodynamic shock waves that are one-dimensionally stable as viscous shock profiles while they are multidimensionally strongly unstable as planar shock discontinuities.     

On vortex streets behind Taylor columns

Computer experiments were performed to explore the flow in the vicinity of a truncated normal flat plate in a rapidly rotating fluid. A Taylor column formed above the flat plate and the vortex shedding in the wake of the Taylor column closely resembled the vortex street behind the solid plate. This is probably the first observation of a Kármán vortex street behind a Taylor column in a computational study.     

On the Lie point symmetry analysis and solutions of the inviscid Burgers equation

Lie point symmetries of the first-order inviscid Burgers equation in a general setting are studied. Some new and interesting solutions are presented.     

On the vortex flow in bounded domains

We consider the motion ofN vortices in bounded domains in IR². We prove that the set of initial positions which lead to a collapse of two or more vortices has Lebesgue measure zero. We extend this result to the stochastic motion of the vortices, where the stochasticity comes from a Wiener-noise term, which is added to the deterministic equation of motion.On leave of the Fachbereich Mathematik, RUB, 4630 Bochum, Federal Republic of Germany. Supported by a DFG-fellowshipPartially supported by Italian CNR     

On vortex shedding from a hexagonal cylinder

The unsteady wake behind a hexagonal cylinder in cross-flow is investigated numerically. The time-dependent three-dimensional Navier-Stokes equations are solved for three different Reynolds numbers Re and for two different cylinder orientations. The topology of the vortex shedding depends on the orientation and the Strouhal frequency is generally higher in the wake of a face-oriented cylinder than behind a corner-oriented cylinder. For both orientations a higher Strouhal number St is observed when Re is increased from 100 to 500 whereas St is unaffected by a further increase up to Re=1000. The distinct variation of St with the orientation of the hexagonal cylinder relative to the oncoming flow is opposite of earlier findings for square cylinder wakes which exhibited a higher St with corner orientation than with face orientation.     

On the scattering of sound by a vortex ring

An analytical treatment is given of the scattering of sound by a circular vortex ring translating at low Mach number in an unbounded stationary medium. The theoretical results complement the recent numerical investigation of this problem by Kambe and Mya Oo [1].     

Satellites in the inviscid breakup of bubbles

In this Letter, we stress the essential role played by gas inertia in the breakup of gas bubbles. Our results reveal that, whenever the gas to liquid density ratio Lambda=rhog/rhol is different from zero, tiny satellite bubbles may be formed as a result of the large gas velocities that are reached close to pinch-off. Moreover, we provide a closed expression for the characteristic satellite diameter, which decreases when decreasing Lambda and which shows order of magnitude agreement with the micron-sized satellite bubbles observed experimentally.     

On a graphics hardware-based vortex detection and visualization system

We present a graphics hardware-based system for interactive denoising, vortex detection, and visualization of vector data. No intermediate results need to be read back by the application once the vector field has been loaded onto the graphics adapter. Due to modern GPUs’ parallel processing capabilities, our system significantly outperforms software implementations and thus provides a valuable tool for the interactive exploration of vector fields and the understanding of flow structures.