Origin of enhanced skin friction at the onset of boundary-layer transition

Boundary-layer transition is accompanied by a significant increase in skin friction whose origin is rigorously explained using the stochastic Lagrangian formulation of the Navier-Stokes equations. This formulation permits the exact analysis of vorticity dynamics in individual realizations of a viscous incompressible fluid flow. The Lagrangian reconstruction formula for vorticity is here extended for the first time to Neumann boundary conditions (Lighthill source). We can thus express the wall vorticity, and, therefore, the wall stress, as the expectation of a stochastic Cauchy invariant in backward time, with contributions from (a) wall vorticity flux (Lighthill source) and (b) interior vorticity that has been evolved by nonlinear advection, viscous diffusion, vortex stretching and tilting. We consider the origin of stress maxima in the transitional region, examining a sufficient number of events to represent the increased skin friction. The stochastic Cauchy analysis is applied to each event to trace the origin of the wall vorticity. We find that the Lighthill source, vortex tilting, diffusion and advection of the outer vorticity make minor contributions. They are less important than spanwise stretching of near-wall spanwise vorticity, which is the dominant source of skin-friction increase during laminar-to-turbulent transition. Our analysis should assist more generally in understanding drag generation and reduction strategies and flow separation in terms of near-wall vorticity dynamics.