Stochastic Lagrangian Dynamics of Vorticity. II. Channel-Flow Turbulence

Abstract

We here exploit a rigorous mathematical theory of vorticity dynamics for Navier-Stokes solutions in terms of stochastic Lagrangian flows and their stochastic Cauchy invariants, that are conserved on average backward in time. This theory yields exact expressions for the vorticity inside the flow domain in terms of the vorticity at the wall, as it is transported by viscous diffusion and by nonlinear advection, stretching and rotation. As a concrete application, we exploit an online database of a turbulent channel-flow simulation at $Re_τ = 1000$ (Graham et al. 2016) to determine the origin of the vorticity in the near-wall buffer layer. Following an experimental study of Sheng et al. (2009), we identify typical "ejection" and "sweep" events in the buffer layer by local minima/maxima of the wall-stress. In contrast to their conjecture, however, we find that vortex-lifting from the wall is not a discrete event requiring only ~1 viscous time and ~10 wall units, but is instead a distributed process taking place over a space-time region at least 1~2 orders of magnitude larger in extent. We show that Lagrangian chaos observed in the buffer layer can be reconciled with saturated vorticity magnitude by a process of "virtual reconnection": although the Eulerian vorticity field in the viscous sublayer has only a single sign of spanwise component, opposite signs of Lagrangian vorticity evolve by rotation and cancel by viscous destruction. Our analysis reveals many unifying features of classical fluids and quantum superfluids. We argue that "bundles" of quantized vortices in superfluid turbulence will also exhibit stochastic Lagrangian dynamics and will satisfy stochastic conservation laws resulting from particle relabelling symmetry.