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

Abstract

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.

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

“…The validity of this mechanism for a transitional boundary-layer flow has been verified recently by Wang et al. (2022). One would expect on the basis of this argument to find spanwise-extended vortex structures as principal elements of wall-bounded turbulent flows.…”

“…Quoting again from Lighthill (1963, p. 96), 'it concentrates most of the vorticity much closer to the wall than before, although at the same time allowing some straggling vorticity to wander away from it farther'. The validity of this mechanism for a transitional boundary-layer flow has been verified recently by Wang et al (2022). One would expect on the basis of this argument to find spanwise-extended vortex structures as principal elements of wall-bounded turbulent flows.…”

“…The conclusion of Wang et al. (2022) from their analysis of the numerical data was that, despite strong viscous effects in the near-wall region, the theory of Lighthill (1963) explained well the origin of high wall-stress events observed in transitional flow.…”

“…The viscous modifications of ideal vortex dynamics can be incorporated by means of a stochastic Lagrangian formulation of incompressible Navier-Stokes in vorticity-velocity representation (Constantin & Iyer 2011;Eyink, Gupta & Zaki 2020a), which represents viscous diffusion of vorticity by an average over stochastic Brownian perturbations of Lagrangian particle motions. This approach was exploited by Wang et al (2022) to investigate the origin of the enhanced wall vorticity and skin friction in a transitional zero-pressure-gradient boundary layer, as discussed in the passage from Lighthill (1963) quoted in the Introduction. This study used the Lighthill vorticity source σ in (1.1) as Neumann boundary conditions, so that the wall vorticity at points of local maximum amplitude could be expressed in terms of two contributions: (i) the Lighthill source integrated over earlier times and (ii) the initial conditions for the vorticity as modified by subsequent advection, stretching and viscous diffusion.…”

“…This approach was exploited by Wang et al. (2022) to investigate the origin of the enhanced wall vorticity and skin friction in a transitional zero-pressure-gradient boundary layer, as discussed in the passage from Lighthill (1963) quoted in the Introduction. This study used the Lighthill vorticity source in (1.1) as Neumann boundary conditions, so that the wall vorticity at points of local maximum amplitude could be expressed in terms of two contributions: (i) the Lighthill source integrated over earlier times and (ii) the initial conditions for the vorticity as modified by subsequent advection, stretching and viscous diffusion.…”