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Production of dissipative vortices by solid walls in incompressible fluid flows at vanishing viscosity

Banff International Research Station for Mathematical Innovation and Discovery

This is joint work with Romain Nguyen van yen and Marie Farge. 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. We observe the formation of two opposite-sign boundary layers whose thickness scales in $Re^{-1/2}$, 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]. 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]? References: [1] T. Kato, 1984 Remarks on zero viscosity limit for non stationary Navier-Stokes flows with boundary. Seminar on nonlinear PDEs, MSRI, Berkeley, 85-98. [2] R. Nguyen van yen, M. Farge and K. Schneider, 2011 Energy dissipative structures in the vanishing viscosity limit of two-dimensional incompressible flow with boundaries. Phys. Rev.Lett., 106(8), 184502. [3] R. Pereira, R. Nguyen van yen, M. Farge and K. Schneider, 2013 Wavelet methods to eliminate resonances in the Galerkin-truncated Burgers and Euler equations. Phys. Rev. E, 87, 033017. [4] R. Nguyen van yen et al., 2016 Energy dissipation caused by boundary layer instability at vanishing viscosity. Preprint.

Mathematics; Fluid mechanics; Partial differential equations

video/mp4

Author affiliation: Aix Marseille Université

2017-01-27

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59888

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/