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.
Highlights
In 1926, Einstein published a short paper explaining the meandering of rivers [1]
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
This secondary local maximum in the pressure field first appears at time t 0.002 and accounts for the stress required to accelerate the flow around the saddle point—bending incoming axial velocity near the cylinder wall to radially inward velocity near the midplane
Summary
In 1926, Einstein published a short paper explaining the meandering of rivers [1]. He famously began the paper by discussing the secondary flow generated in a stirred teacup—the flow widely known to be responsible for the collection of tea leaves at the centre of a stirred cup of tea. A pressure field is instantaneously generated to provide the radially inward force necessary to keep fluid parcels moving along circular paths This results in high pressure at the cylinder wall where the circulation is largest (z = ±L/4) and low pressure where there is no azimuthal flow (z = 0 and z = ±L/2). The pressure is the only stress acting within an inviscid fluid and it is the only means to provide force to, and thereby accelerate, this flow. It is at the heart of the teacup effect and it is natural to investigate its role in the singularity. An important focus of this work will be disentangling the contributions to the stress associated with incompressibility from those associated with flow confinement
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