TRR 181 Seminar "Production of dissipative vortices by no-slip walls in incompressible flows in the vanishing viscosity limit"

We revisit the problem posed by Euler in 1748 that lead d'Alembert to formulate his paradox and address the following question: does energy dissipate when boundary layers detach from solid body in the vanishing viscosity limit, or equivalently in the limit of very large Reynolds number Re? To trigger detachment we consider a vortex-dipole impinging onto a wall. We compare numerical solutions of two-dimensional Euler, Prandtl, and Navier-Stokes equations [5]. We observe the formation of two opposite-sign boundary layers whose thickness scales like $Re^{-1/2}$, as predicted by Prandtl's theory in 1904. After a certain time when the boundary layers detach from the wall Prandtl's solution becomes singular, while the Navier-Stokes solution collapses down to a much finer thickness for the boundary layers in both directions (parallel but also perpendicular to the wall), that scales as $Re^{-1}$ in accordance with Kato's 1984 theorem [1]. The boundary layers then roll up and form vortices that dissipate a finite amount of energy, even in the vanishing viscosity limit [2,5]. These numerical results suggest that a new Reynolds independent description of the flow beyond the breakdown of Prandtl's solution might be possible. This lead to the following questions: does the solution converge to a weak dissipative solution of the Euler equation, analog to the dissipative shocks one get with the inviscid Burgers equation, and how would it be possible to approximate it numerically [3,4]? This is joint work with Natacha Nguyen van yen and Marie Farge.

[5] N. Nguyen van yen, M. Weidmann, R. Klein, M. Farge and K. Schneider, 2017. Energy dissipation caused by boundary layer instability at vanishing viscosity. arXiv:1706.00942, J. Fluid Mech., under revision.