The Geometry of Axisymmetric Ideal Fluid Flows with Swirl

TL;DR

This paper investigates the geometric properties of axisymmetric ideal fluid flows with swirl on a solid flat torus, revealing conditions for positive curvature and conjugate points that relate to flow stability.

Contribution

It characterizes the sectional curvature of the volumorphism group for axisymmetric flows with swirl, highlighting differences from non-swirl flows and linking curvature to flow stability.

Findings

Positive sectional curvature occurs under specific conditions involving flow velocity derivatives.

The criterion for positive curvature implies the existence of conjugate points along geodesics.

Results contrast with non-swirl flows where only Killing fields have nonnegative curvature.

Abstract

The sectional curvature of the volume preserving diffeomorphism group of a Riemannian manifold $M$ can give information about the stability of inviscid, incompressible fluid flows on $M$. We demonstrate that the submanifold of the volumorphism group of the solid flat torus generated by axisymmetric fluid flows with swirl, denoted by $\mathcal{D}_{\mu,E}(M)$, has positive sectional curvature in every section containing the field $X = u(r)\partial_\theta$ iff $\partial_r(ru^2)>0$. This is in sharp contrast to the situation on $\mathcal{D}_{\mu}(M)$, where only Killing fields $X$ have nonnegative sectional curvature in all sections containing it. We also show that this criterion guarantees the existence of conjugate points on $\mathcal{D}_{\mu,E}(M)$ along the geodesic defined by $X$.