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Comparison between Prandtl, Navier-Stokes and Euler solutions for a vortex dipole impinging on a wall

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We argue that d’Alembert’s paradox (1749) is still unresolved for very large Reynolds number flows. Prandtl (1904) assumed that there exists a viscous boundary layer attached to the wall and predicted that the drag force dissipates energy there at a rate proportional to Re^(−1/2). Kato (1984) proved that, in the limit of infinite Reynolds number, the energy dissipation rate tends to zero if and only if the solution of the Navier-Stokes equation converges towards the solution of the Euler equations (with the same initial data) and then occurs in a very thin boundary layer of thickness proportional to Re^(−1). By performing direct numerical simulations of a dipole crashing into a wall we show that Kato’s scaling is more appropriate than Prandtl’s scaling as soon as the boundary layer detaches from the wall.

Author(s):

Romain Nguyen van yen    
Freie Universität Berlin
Germany

Matthias Waidmann    
Freie Universität Berlin
Germany

Marie Farge    
LMD-CNRS, ENS, Paris
France

Kai Schneider    
M2P2 - CNRS
France

Rupert Klein    
Freie Universität
Germany