Non-stationary boundary layers and energy dissipation in incompressible flows
Seminar Room 1, Newton Institute
In fully turbulent incompressible flows, the two presumed culprits for energy dissipation are thin boundary layers on the one hand, and three-dimensional stretching of vortex tubes on the other hand. Although both suspects seem equally important, much more theoretical effort has been invested to study the latter, due to the emphasis that has been put on homogeneous turbulence since the 1930s. We argue that the same effort should be dedicated to interrogate the former, especially since it abides the simpler two-dimensional (2D) setting.
Indeed, while the shortcomings of the perfect fluid model, which lead to the d'Alembert paradox, are well understood, the failure of the Prandtl viscous boundary layer theory to predict the right scaling of energy dissipation still lacks a complete explanation. Several possibilities have been explored in recent years, like finite time singularities in the Prandtl equations, ill-posedness of the Prandtl equations, or even earlier breakdown of the asymptotic expansion itself. What happens beyond the Prandtl regime is even more mysterious. According to a criterion proven by Kato, energy should then be dissipated at scales as fine as the inverse of the Reynolds number, but the process by which this could happen remains elusive.
We review the main issues at hand, using a 2D dipole-wall interaction as illustrative test case. First we re-derive the Euler-Prandtl equations in the vorticity formulation, solve them numerically, and pinpoint the origin of the discrepancy with a reference Navier-Stokes solution. We then proceed to explore alternative asymptotic expansions, allowing for a localized collapse of the boundary layer to finer scales, emphasizing in particular the importance of the topology of the vorticity field, as well as the consequences regarding the scaling of energy dissipation.