The origin of skin-friction increase in laminar-to-turbulence transition: a stochastic Lagrangian analysis

A general feature of transition in wall-bounded flows is the significant increase in skin friction, whose origin has been speculated but never rigorously demonstrated. In this work, the skin friction is expressed in terms of wall vorticity and can thus be calculated as the expectation of a stochastic Cauchy invariant in backward time. Contributions arise from (i) wall vorticity flux (Lighthill source) and (ii) pre-existing interior vorticity evolved by nonlinear advection, viscous diffusion, vortex stretching and tilting. These contributions are quantified along backward stochastic Lagrangian trajectories, which are determined by the exact Navier-Stokes solutions. Our analysis is performed using the transitional boundary layer dataset of the Johns Hopkins Turbulence Databases (Z. Wu, J. Lee, C. Meneveau and T. Zaki, 2019, Phys. Rev. Fluids, 4(2), 023902), and examines an ensemble of wall-stress maxima in the transitional region. The analysis demonstrates that the dominant source of skin-friction increase in transition is spanwise stretching of pre-existing near-wall spanwise vorticity. Our formulation may assist more generally in understanding physical phenomena in transitional and turbulent wall-bounded flows, such as drag reduction, flow separation and extreme stress events.