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 µ Re0.8ZRe08 and P µ Re2.25PRe225 for 5 × 102 = Re = 2 × 104 and Z µ Re0.5ZRe05 and P µ Re1.5PRe15 for Re = 2 × 104 (with Re based on the velocity and size of the dipole). A critical Reynolds number Re c (here, Rec » 2×104Rec2104) is identified below which the interaction time of the dipole with the boundary layer depends on the kinematic viscosity ¿. 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 µ Re3/4, P µ Re9/4ZRe34PRe94 , and dP/dt µ Re11/4Re114 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 µ Re1/2ZRe12 and P µ Re3/2PRe32.