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...

Vortex blob methods are typically characterized by a regularization length scale, below which the dynamics are trivial for isolated blobs. In this article, we observe that the dynamics need not be trivial if one is willing to consider distributional derivatives of Dirac delta functionals as valid vorticity distributions. More specifically, a new singular vortex theory is presented for regulariz...

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...

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...

Vortex methods for inviscid incompressible two-dimensional uid ow are usually based on blob approximations. This paper presents a vortex method in which the vorticity is approximated by a piecewise polynomial interpolant on a Delaunay triangulation of the vortices. An e cient reconstruction of the Delaunay triangulation at each step makes the method accurate for long times. The vertices of the ...

In this paper, numerical simulation for a two-dimensional viscous and incompressible flow past the elliptical airfoil is presented by Random Vortex Blob (RVB). RVB is a numerical technique to solve the incompressible, two-dimensional and unsteady Navier-Stocks equations by converting them to rotational non-primitive formulations. In this method, the velocity vector at a certain point can be cal...

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...

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...

Sufficient conditions for the existence of quasi-pe~odic solutions of two different desingularized vortex models for 2-dimensional Euler flows are derived. One of these modds is the vortex blob model for the evolution of a periodic vortex sheet and the other is a second order elliptic moment model (DEMM) for the evolution of widely separated vortex regions. The method involves the identificatio...