The Objective Deformation Component of a Velocity Field

TL;DR

This paper introduces a method to decompose any velocity field into a rigid-body component and a deformation component, where the deformation component is frame-indifferent and physically observable, ensuring consistent physical quantities across frames.

Contribution

The paper presents a novel approach to extract the deformation velocity component from an arbitrary velocity field, making key flow quantities frame-indifferent and physically meaningful.

Findings

Deformation velocity component is frame-indifferent.

Flow quantities like momentum and energy become frame-invariant.

Closest rigid-body velocity field minimizes the L^2 norm difference.

Abstract

For an arbitrary velocity field $\mathbf{v}$ defined on a finite, fixed spatial domain, we find the closest rigid-body velocity field $\mathbf{v}_{RB}$ to $\mathbf{v}$ in the $L^2$ norm. The resulting deformation velocity component, $\mathbf{v}_{d}=\mathbf{v-\mathbf{v}}_{RB}$, turns out to be frame-indifferent and physically observable. Specifically, if $\mathbf{Q}_{\text{RB}}(t)$ is the rotation tensor describing the motion of the closest rigid body frame, then $\mathbf{v}$ is seen as $\mathbf{Q}_{\text{RB}}^{T}\mathbf{v}_{d}$ by an observer in that frame. As a consequence, the momentum, energy, vorticity, enstrophy, and helicity of the flow all become frame-indifferent when computed from the deformation velocity component $\mathbf{v}_{d}$.