Structures and Dynamics of Lone Schur Flows with Vorticity but no Swirls

TL;DR

This paper investigates the dynamics of flows with purely real eigenvalues of the velocity gradient tensor, revealing complex vortical structures without swirls, and introduces a semi-analytical method for their computation.

Contribution

It establishes a governing equation and a computational algorithm for lone Schur flows, highlighting their unique vortical dynamics and connections to turbulence near critical dimensions.

Findings

Rich vortical structures without swirls in simulated flows

Discovery of flux loop scenario and turbulence analogy at critical dimension

No homoclinic loops observed in the studied flows

Abstract

We study the dynamics and indications of the flows with all the eigenvalues of the velocity gradients being real, thus `lone', \textit{i.e.}, without forming the complex conjugate pairs associated to the swirls. A generic prototype is the `lone Schur flow (LSF)' whose velocity gradient tensor is uniformly of Schur form but free of complex eigenvalues. A (partial) integral-differential equation governing such LSF is established, and a semi-analytical algorithm is accordingly designed for computation. Simulated evolutions of example LSFs in 2- and 3-spaces show rich dynamics and vortical structures, but no obvious swirls (nor even the homoclinic loops in whatever distorted forms) could be found. We discovered the flux loop scenario and the anisotropic analogy of the incompressible turbulence at or close to the critical dimension $D_c =4/3$ decimated from 2-space.