Elliptically Desingularized Vortex Model for the Two-Dimensional Euler Equations

Abstract

A new self-consistent model of the incompressible Euler equations in two dimensions is presented. The vorticity is assumed to be distributed in well separated disjoint piecewiseconstant elliptical finite-area vortex regions (FAVORs) D k with area A k , The evolution equations for four variables that describe each FAVOR are derived by truncating a physicalspace moment description by omitting terms O ((A k /R ka ) 2 ). (R ka is the inter-FAVOR centroid distance.) The model is validated by comparing steady-state configurations and dynamical evolutions with contour dynamical results.PACS numbers: 47.10. + g Our new self-consistent low-order model of the two-dimensional (2D) incompressible Euler equations (EE) is a decisive improvement over the well-known singular (point) or invariant-core vortex models. 1 The model, derived by perturbation methods, is validated by comparing solutions with corresponding contour dynamical (CD) results. 2 We show that in first order, the elliptical shape is physically reasonable and mathematically cogent for the well-separated, disjoint finite-area vortex regions (FAVORs) which create the flow. The elliptical shape is used to approximate the vortical structures observed experimentally in a strain field 3 or after a "merger" or "pairing" event. 4,5 The mathematical justification is given below. Thus, each elliptical FAVOR is described by four dynamical variables: the centroid, 3c k = (x k ,y k )\ the aspect ratio, \ k \ and the orientation of the major axis, <f> k .The \ k and <f> k introduce sufficient "internal" degrees of freedom per FAVOR so that a pair of isolated regions can attract and "collapse" (i.e., centroid coalescence in a finite time)--a process analogous initially to merger or pairing. Note that when a collapse event is in progress, the low-order model begins to breakdown for it violates the assumption of the perturbation expansion. However, critical collapse distances are close to the corresponding critical merger distances for the full EE. 2 We derive the model from the EE in vorticity/stream-function form,where the velocity field u = (u, v) = (\ff y , -i// x ). We consider the flow created by constant vorticity (o k in Ndisjoint domains, D k (t), with constant area A k , where the diameter of each (ocA k^2 ) is assumed to be much less than the smallest inter-FAVOR distance, R ka , as shown in Fig.

Vortex structures with complex points singularities in two-dimensional Euler equations. New exact solutions

“…The third group of terms in (8) with first derivatives of the δ-function describes the vorticity which is generated by a group of dipole singularities M:…”

“…It is interesting that these classical solutions are still topical (see, for example, the recent works [5,6]). The generalizations of these solutions are the models of different coherent structures, vortex patches, and vortex crystals (see, for example, [7][8][9][10][11][12][13] and references therein), which are well observed in numerical and laboratory experiments (see, for example, [14][15][16][17][18][19][20][21]). Let us note that the Stuart solution is based on a Liouville-type equation for the stream function.…”

The Steady Motion of a Symmetric, Finite Core Size, Counterrotating Vortex Pair

“…Based on Saffman's results, Pierrehumbert [7] noted difficulties with the numerical procedure at the intersection and acknowledged the possibility of a corner. Wu, Overman, and Zabusky [11] [4] attempted such an analysis using a stream function formulation involving the integral of Biot-Savart contributions from differential elements of vorticity. Using the ratio of vortex area to the square of vortex spacing as an expansion parameter, they argued that the vortices were, to lowest order, ellipses.…”

Geostrophic vortex dynamics

“…More recently Styczek (1984 and) have introduced an elliptically desingularized model for the two-dimensional Euler equations (the "moment model") that they have applied to the symmetric merger problem with success. With the moment model, the merger process can be reduced to a four dimensional integrable Hamiltonian system that can be solved analytically, yielding a value of 3.2 for the critical merger distance.…”