Abstract
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 regularized Euler fluid equations of ideal incompressible flow in the plane. We determine the conditions under which such regularized Euler fluid equations may admit vorticity singularities which are stronger than delta functions, e.g., derivatives of delta functions. We also describe the symplectic geometry associated with these augmented vortex structures, and we characterize the dynamics as Hamiltonian. Applications to the design of numerical methods similar to vortex blob methods are also discussed. Such findings illuminate the rich dynamics which occur below the regularization length scale and enlighten our perspective on the potential for regularized fluid models to capture multiscale phenomena.
Highlights
Vortices are important in hydrodynamics because they are the sources for the incompressible flow field
The point vortex approach did not produce a competitive numerical method until the 1970s, when the problems related to singularities were overcome by regularizing the singular vortex kernel to form a vortex blob
The convergence rate of the mth kernel was found to be of order hmq for any q ∈ (0, 1) where h = δq is a grid-spacing parameter and δ > 0 is a length scale associated with the smoothing kernel (Beale and Majda 1982, 1985)
Summary
Vortices are important in hydrodynamics because they are the sources for the incompressible flow field. The vorticity distribution at any instant of time determines both the current state of the flow and its future evolution, for given boundary conditions. This property holds for any Hamiltonian system, and it can be shown that the dynamics of vortices can be usefully expressed in Hamiltonian form. In the vorticity and stream function formulation of an ideal incompressible planar fluid, the evolution of the vorticity distribution ω(x, y, t) is given by. Stochastic perturbations were further included to model viscosity (Chorin 1973) These adjustments to the classical point vortex method yielded the vortex blob method, which quickly became of practical use for realistic fluid flow modeling.
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