Turbulent Flow in Pipes and Channels as Cross-Stream “Inverse Cascades” of Vorticity

A commonplace view of pressure-driven turbulence in pipes and channels is as “cascades” of streamwise momentum toward the viscous layer at the wall. We present in this paper an alternative picture of these flows as “inverse cascades” of spanwise vorticity, in the cross-stream direction but away from the viscous sublayer. We show that there is a constant spatial flux of spanwise vorticity, due to vorticity conservation, and that this flux is necessary to produce pressure-drop and energy dissipation. The vorticity transport is shown to be dominated by viscous diffusion at distances closer to the wall than the peak Reynolds stress, well into the classical log-layer. The Perry-Chong model based on “representative” hairpin/horsehoe vortices predicts a single sign of the turbulent vorticity flux over the whole log-layer, whereas the actual flux must change sign at the location of the Reynolds-stress maximum. Sign-reversal may be achieved by assuming a slow power-law decay of the Townsend “eddy intensity function” for wall-normal distances greater than the hairpin length-scale. The vortex-cascade picture presented here has a close analogue in the theory of quantum superfluids and superconductors, the “phase slippage” of quantized vortex lines. Most of our results should therefore apply as well to superfluid turbulence in pipes and channels. We also discuss issues about drag-reduction from this perspective. 1 ar X iv :0 80 9. 11 65 v2 [ ph ys ic s. fl udy n] 1 6 Se p 20 08