A fluid mechanic’s analysis of the teacup singularity

Abstract

The mechanism for singularity formation in an inviscid wall-bounded fluid flow is investigated. The incompressible Euler equations are numerically simulated in a cylindrical container. The flow is axisymmetric with the swirl. The simulations reproduce and corroborate aspects of prior studies reporting strong evidence for a finite-time singularity. The analysis here focuses on the interplay between inertia and pressure, rather than on vorticity. The linearity of the pressure Poisson equation is exploited to decompose the pressure field into independent contributions arising from the meridional flow and from the swirl, and enforcing incompressibility and enforcing flow confinement. The key pressure field driving the blowup of velocity gradients is that confining the fluid within the cylinder walls. A model is presented based on a primitive-variables formulation of the Euler equations on the cylinder wall, with closure coming from how pressure is determined from velocity. The model captures key features in the mechanics of the blowup scenario.

Anisotropic finite-time singularity in the three-dimensional axisymmetric Euler equation with a swirl

“…It should be remarked, that the approximation considered here (9) for the present model ( 16) and ( 17) is the opposite of Barkley's work [21].…”

“…The previous system was already numerically studied in the early 90s [30,31] and, more recently, but in a finite domain in Refs. [20][21][22].…”

“…To fix ideas, let us consider a situation in which one considers an anti-symmetric initial flow [20,21]. This flow is invariant under…”

“…The author refers to the review article by Gibbon [19] who carefully review the most notable progress including a summary table with expectations of existence, or not, of finite-time blow-up solutions of Euler, as well as Navier-Stokes, equations. More recently, Luo and Huo [20] provided numerical evidence of the existence of a finite time blow-up for the axi-symmetric Euler equations at the outer boundary of the domain and Barkley [21] analyses this singularity from an hydrodynamical aspect showing that the formation of the singularity is driven by the wall. Lastly, Elgindi and Jeong prove the existence of a finite-time singularity of axi-symmetric Euler equations in a "hour-glass" like domain excluding the axis of rotation [22].…”