Abstract

Laminar-to-turbulence transition in boundary layers is accompanied by concentration of the mean spanwise vorticity near and at the boundary.  The increase of wall vorticity, or equivalently the wall stress, is even more pronounced in the instantaneous fields when a nascent patch of turbulence first reaches the wall.  The fluid dynamical mechanism for this increase is examined rigorously using the stochastic Lagrangian formulation of the Navier-Stokes equations, which quantifies all possible contributors.  We express the instantaneous wall vorticity as the expectation of a stochastic Cauchy invariant in backward time, with terms due the (a) wall-vorticity flux (Lighthill source) and (b) interior vorticity evolved by nonlinear advection, viscous diffusion, vortex stretching and tilting.  We evaluate these terms for an ensemble of transition events from a simulation of bypass transition that is publicly available through the JHU database.  The results yield a clear picture of the origin of the enhanced wall vorticity in this flow, which is not the Lighthill source but instead a different mechanism suggested by Lighthill (1963).