The objective deformation component of a velocity field

Abstract

For an arbitrary velocity field v defined on a finite, fixed spatial domain, we find the closest rigid-body velocity field vRB to v in the L2 norm. The resulting deformation velocity component, vd=v−vRB, turns out to be frame-indifferent and physically observable. Specifically, if QRB(t) is the rotation tensor describing the motion of the closest rigid body frame, then v is seen as QRBTvd 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 vd.