On the Reynolds number scaling of vorticity production at no-slip walls during vortex-wall collisions

Recently, numerical studies revealed two different scaling regimes of the peak enstrophy Z and palinstrophy P during the collision of a dipole with a no-slip wall [Clercx and van Heijst, Phys. Rev. E 65, 066305, 2002]: Z proportional to Re(0.8) and P proportional to Re(2.25) for 5 x 10(2) <= Re <= 2 x 10(4) and Z proportional to Re(0.5) and P proportional to Re(1.5) for Re >= 2 x 10(4) (with Re based on the velocity and size of the dipole). A critical Reynolds number Re(c) (here, Rec approximate to 2 x 10(4)) is identified below which the interaction time of the dipole with the boundary layer depends on the kinematic viscosity nu. The oscillating plate as a boundary-layer problem can then be used to mimick the vortex-wall interaction and the following scaling relations are obtained: Z proportional to Re(3/4), P proportional to Re(9/4), and dP/dt proportional to Re(11/4) in agreement with the numerically obtained scaling laws. For Re >= Re(c) the interaction time of the dipole with the boundary layer becomes independent of the kinematic viscosity and, applying flat-plate boundary-layer theory, this yields: Z proportional to Re(1/2) and P proportional to Re(3/2).