Vortex Criteria can be Objectivized by Unsteadiness Minimization

TL;DR

This paper demonstrates that reference frame optimization for unsteady fluid flows is objective and independent of initial transformations, correcting previous misconceptions and establishing the validity of a variational formulation.

Contribution

It proves the objectivity of the variational formulation for reference frame optimization, clarifies misconceptions, and relates it to existing optimization approaches.

Findings

Optimized velocity fields are objective and transformation-independent.

The variational formulation of reference frame optimization is objective.

Clarifies the relationship between variational and other optimization methods.

Abstract

Reference frame optimization is a generic framework to calculate a spatially-varying observer field that views an unsteady fluid flow in a reference frame that is as-steady-as-possible. In this paper, we show that the optimized vector field is objective, i.e., it is independent of the initial Euclidean transformation of the observer. To check objectivity, the optimized velocity vectors and the coordinates in which they are defined must both be connected by an Euclidean transformation. In this paper we show that a recent publication [1] applied this definition incorrectly, falsely concluding that reference frame optimizations are not objective. Further, we prove the objectivity of the variational formulation of the reference frame optimization proposed in [1], and discuss how the variational formulation relates to recent local and global optimization approaches to unsteadiness minimization.