Vertical alignment of stagnation points in pseudo-plane ideal flows

TL;DR

This paper investigates the vertical alignment of stagnation points in pseudo-plane ideal flows, revealing topological and dynamical conditions that govern vortex structure and alignment in ideal and viscous fluids.

Contribution

It provides a topological explanation for vortex alignment in pseudo-plane flows and extends the understanding to viscous fluids under certain conditions.

Findings

Stagnation points tend to be vertically aligned in steady pseudo-plane flows.

Topological properties of pressure Hessian explain vortex alignment.

Alignment phenomena are influenced by flow rotation and inertial periods.

Abstract

Recent studies of pseudo-plane ideal flow (PIF) reveal a ubiquitous presence of vortex alignment in both homogeneous and stratified fluids, and in both inertial and rotating reference frames as well. The exact solutions of a steady-state PIF model suggest that stagnation points tend to be vertically aligned and the concentric structure represents a fixed-point phenomenon of the Euler equations. Exception occurs in the rotating frame when a flow holds inertial period and skew center becomes possible. Properties of stagnation points based on Morse theory are obtained, leading to a topological explanation of vertical alignment via pressure Hessian. The study thus uncovers a new aspect of vortex behavior in ideal fluid that requires vortex center to align with the direction of gravity when vortex evolution reaches a laminar end state characterized by steady pseudo-plane velocities. Though the phenomenon arises from the constraint of the Euler equations, under specific conditions the topological theory is applicable to viscous fluid and explains the curvilinear tilting of von K\'arm\'an swirling vortex.