We show that a certain class of vortex blob approximations for ideal hydrodynamics in two dimensions can be rigorously understood as solutions to the equations of second-grade non-Newtonian fluids with zero viscosity, and initial data in the space of Radon measures M(R2). The solutions of this regularized PDE, also known as the averaged Euler or Euler-α equations, are geodesics on the volume pr...

In this thesis we investigate the locomotion of fish and birds by applying both new and well known mathematical techniques. The two-dimensional model is first studied using Krasny’s vortex blob method, and then a new numerical method based on Wu’s theory1 is developed. To begin with, we will implement Krasny’s ideas for a couple of examples and then switch to the numerical implementation of the...

Expanding upon an observation of Hou [Math. Comp., submitted], exact desingularizations are presented of the Euler equations in two and three dimensions for which the singularity within the Biot-Savart integrand is reduced by one order. The reformulated equations are then solved numerically using either the point vortex method or the vortex blob method. The increased smoothness of the Biot-Sava...

This review paper offers both an overview of the recent developments in vortex statistics, and a detailed critical review of some of the advances as well as open problems in the area. A fruitful new direction in vortex dynamics concerns vortical systems with very large numbers of vortices in which statistical methods play a pivotal role. After a review of some of the older equilibrium statistic...

Turbulence is hard to control. Many experimental methods have been developed generate this elusive state of matter, leading fundamental insights into its statistical and structural features as well onset. In all cases, however, the material boundaries apparatus pose a challenge for understanding what turbulence has fed how it would freely evolve. Here we build control confined using elemental b...

In this paper, we prove the convergence of the vortex blob method for a family of nonlinear evolutionary partial differential equations (PDEs), the so-called b-equation. This kind of PDEs, including the Camassa-Holm equation and the Degasperis-Procesi equation, has many applications in diverse scientific fields. Our convergence analysis also provides a proof for the existence of the global weak...

this paper presents the results of numerical simulations of the laminar and turbulent flows inside diverging channels using random vortex methods (rvm). random vortex method is a mesh-free method which solves unsteady vorticity equation instead of solving navier-stokes equations directly to determine the velocity field. in this paper, the velocity potential, which is used as the initial conditi...