Vorticity cascade and turbulent drag in wall-bounded flows: plane Poiseuille flow

Drag for wall-bounded flows is directly related to flux of spanwise vorticity outward from the wall. In turbulent flows a key contribution arises from cross-stream "vorticity cascade" by nonlinear advection and stretching of vorticity. We study this process using numerical simulation data of turbulent channel flow at $Re_\tau=1000$. The net transfer from the wall of fresh spanwise vorticity created by downstream pressure drop is due to two large opposing fluxes, one which is "down-gradient" or outward from the wall, where most vorticity concentrates, and the other which is "up-gradient" or toward the wall and acting against strong viscous diffusion in the near-wall region. We present evidence that the up-gradient transport occurs by a mechanism of correlated inflow and spanwise vortex stretching that was proposed by Lighthill. This mechanism is essentially Lagrangian, but we explicate its relation to the Eulerian anti-symmetric vorticity flux tensor. As evidence for the mechanism we study (i) statistical correlations of the wall-normal velocity and of wall-normal flux of spanwise vorticity, (ii) vorticity flux cospectra that identify eddies involved in nonlinear vorticity transport in the two opposing directions, and (iii) visualizations of coherent vortex structures which contribute dominantly to the transport. The "D-type" vortices contributing dominantly to down-gradient transport in the log-layer are found to be attached, hairpin-type vortices. However, the "U-type" vortices contributing dominantly to up-gradient transport are detached, wall-parallel, pancake-shaped vortices with strong spanwise vorticity, as expected by Lighthill's mechanism. We discuss modifications to the attached eddy model and implications for turbulent drag reduction.