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, periodic in the axial coordinate. The flow is axisymmetric with swirl. The simulations reproduce and corroborate aspects of the seminal work of Luo and Hou [Luo G, Hou T (2014) Potentially singular solutions of the 3D axisymmetric Euler equations. Proc Natl Acad Sci USA 111:12968-12973]. Contrary to standard practice, the analysis here focuses on the pressure and not the vorticity. Linearity of the pressure Poisson equation is exploited to decompose the pressure field into a superposition of independent contributions arising from the meridional flow and from the swirl. The swirl pressure is further decomposed into axial mean and fluctuating components, where the fluctuating component itself has distinct contributions maintaining incompressibility and confining the flow. The key pressure field driving singularity formation is shown to be that confining the fluid within the cylinder walls. This pressure is directly connected to the Luo and Hou model via the Hilbert transformation.