CMX Lunch Seminar
My interests lie at the interface between high-performance computation, pattern formation and nonlinear phenomena. I am interested in instabilities in fluid, chemical and biological systems and how best to understand these through efficient direct numerical computation. I am currently devoting a lot of attention to a singularity in the Euler equations and to the onset of turbulence in shear flows. Other key words describing my research are: large-scale computation, mathematical modeling, hydrodynamic stability, transition to turbulence, reaction-diffusion equations, excitable media, symmetry-breaking bifurcations, and spatio-temporal chaos.
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 swirl. The simulations reproduce and corroborate aspects of prior studies by Luo and Hou reporting strong evidence for a finite-time singularity. The analysis here focuses on the interplay between inertia and pressure, rather than on vorticity. 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.