The Lagrangian Deformation Structure of Three-Dimensional Steady Flow

TL;DR

This paper introduces a Protean coordinate transform for steady 3D flows that simplifies deformation analysis, revealing underlying structures and enabling stochastic modeling of fluid deformation.

Contribution

The paper develops a novel streamline coordinate transform that simplifies deformation metrics and explicitly incorporates flow constraints, advancing understanding of 3D steady flow deformation.

Findings

Flow exhibits Gaussian deformation structure in random cases.

Transform recovers kinematic and topological deformation constraints.

Flow deformation can be characterized by few parameters.

Abstract

Fluid deformation and strain history are central to wide range of fluid mechanical phenomena ranging from fluid mixing and particle transport to stress development in complex fluids and the formation of Lagrangian coherent structures (LCSs). To understand and model these processes it is necessary to quantify Lagrangian deformation in terms of Eulerian flow properties, currently an open problem. To elucidate this link we develop a Protean (streamline) coordinate transform for steady three-dimensional (3D) flows which renders both the velocity gradient and deformation gradient upper triangular. This frame not only simplifies computation of fluid deformation metrics such as finite-time Lyapunov exponents (FTLEs) and elucidates the deformation structure of the flow, but moreover explicitly recovers kinematic and topological constraints upon deformation such as those related to helicity density and the Poincar\'{e}-Bendixson theorem. We apply this transform to several classes of steady 3D flow, including helical and non-helical, compressible and incompressible flows, and find random flows exhibit remarkably simple (Gaussian) deformation structure, and so are fully characterised in terms of a small number of parameters. As such this technique provides the basis for the development of stochastic models of fluid deformation in random flows which adhere to the kinematic constraints inherent to various flow classes.