Optimal Reconstruction of Inviscid Vortices

TL;DR

This paper develops an optimization-based method to reconstruct inviscid vortex models from data, accurately capturing vortex structures and characteristics in 3D axisymmetric flows, including realistic vortex rings.

Contribution

It introduces a variational approach for continuous reconstruction of the vorticity function, improving vortex modeling accuracy over empirical fits and previous methods.

Findings

Successful reconstruction of Hill's vortex with high accuracy.

More precise vorticity functions for realistic vortex flows.

Enhanced representation of vortex structure and integral properties.

Abstract

We address the question of constructing simple inviscid vortex models which optimally approximate realistic flows as solutions of an inverse problem. Assuming the model to be incompressible, inviscid and stationary in the frame of reference moving with the vortex, the "structure" of the vortex is uniquely characterized by the functional relation between the streamfunction and vorticity. It is demonstrated how the inverse problem of reconstructing this functional relation from data can be framed as an optimization problem which can be efficiently solved using variational techniques. In contrast to earlier studies, the vorticity function defining the streamfunction-vorticity relation is reconstructed in the continuous setting subject to a minimum number of assumptions. To focus attention, we consider flows in 3D axisymmetric geometry with vortex rings. To validate our approach, a test case involving Hill's vortex is presented in which a very good reconstruction is obtained. In the second example we construct an optimal inviscid vortex model for a realistic flow in which a more accurate vorticity function is obtained than produced through an empirical fit. When compared to available theoretical vortex-ring models, our approach has the advantage of offering a good representation of both the vortex structure and its integral characteristics.