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 layer detaches from solid body in the vanishing viscosity limit, or in the limit of very large Reynolds number $Re$? To trigger detachment we consider a vortex dipole impinging onto a wall and we compare the 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 1904 theory. But after a certain time when 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 Euler's 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, 1-4
[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, 1-8
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