The origin of vorticity in viscous incompressible flows

TL;DR

This paper introduces an Eulerian, adjoint-based method to trace the origin of vorticity in viscous incompressible flows, providing a new perspective that complements stochastic Lagrangian approaches and enhances understanding of turbulence dynamics.

Contribution

It presents the first Eulerian, adjoint-based framework for quantifying vorticity origins in viscous flows, establishing mathematical equivalence with stochastic Lagrangian methods.

Findings

The adjoint approach accurately captures vorticity origins in turbulent flows.

The method can incorporate vorticity injection at boundaries.

Application to turbulent channel flow reveals insights into high-stress event mechanisms.

Abstract

In inviscid, incompressible flows, the evolution of vorticity is exactly equivalent to that of an infinitesimal material line-element, and hence vorticity can be traced forward or backward in time in a Lagrangian fashion. This elegant and powerful description is not possible in viscous flows due to the action of diffusion. Instead, a stochastic Lagrangian interpretation is required and was recently introduced, where the origin of vorticity at a point is traced back in time as an expectation over the contribution from stochastic trajectories. We herein introduce for the first time an Eulerian, adjoint-based approach to quantify the back-in-time origin of vorticity in viscous, incompressible flows. The adjoint variable encodes the advection, tilting and stretching of the earlier-in-time vorticity that ultimately leads to the target value. Precisely, the adjoint vorticity is the volume-density of the mean Lagrangian deformation of the earlier vorticity. The formulation can also account for the injection of vorticity into the domain at solid boundaries. We demonstrate the mathematical equivalence of the adjoint approach and the stochastic Lagrangian approach. We then provide an example from turbulent channel flow, where we analyze the origin of high-stress events and relate them to Lighthill's mechanism of stretching of near-wall vorticity.