The origin of vorticity in viscous incompressible flows

In inviscid incompressible flows, vortex lines are material lines that can be traced forward and backward in time without ambiguity. For viscous Navier-Stokes, this symmetry is no longer present. A stochastic interpretation of vorticity was recently introduced to account for the influence of viscosity, where particles are tracked back-in-time along stochastic trajectories. The terminal vorticity at the target location is then evaluated as the expectation of the initial vorticity being stretched, tilted/twisted along the particle paths. We present a new Eulerian approach to determine the origin of vorticity in backward time, and we prove mathematically that it is exactly equivalent to the stochastic interpretation. With our new equation, we can dissect the contributions to the target vorticity in terms of stretching and tilting of the interior vorticity and a wall contribution. We apply our approach to turbulent channel flow to study the back-in-time origin of high-stress events on the wall.