Reference map technique for Lagrangian exploration of coherent structures

Abstract: We explore the application of the reference map technique, originally developed for the Eulerian simulation of solid mechanics, in Lagrangian kinematics of fluid flows. Unlike traditional methods based on explicit particle tracking, the reference map facilitates the calculation of flow maps and gradients without the need for particles. This is achieved through an Eulerian update of the reference map, which records the take-off positions of fluid particles. Using the reference map, measures of Lagrangian fluid-element deformation can be computed solely from Eulerian data, such as displacement, deformation gradient, Cauchy-Green tensor, and Finite-Time Lyapunov Exponent (FTLE) fields. We first demonstrate the accuracy of FTLE calculations based on the reference map against the standard particle-based approach in a 2D Taylor-Green vortex. Then, we apply it to turbulent channel flow at $Re_\tau=180$ where Lagrangian coherent structures identified as ridges of the backward-time FTLE are found to bound vortical regions of flow, consistent with Eulerian coherent structures from the Q-criterion. The reference map also proves suitable for material surface tracking despite not explicitly tracking particles. This capability can provide valuable insights into the Lagrangian landscape of turbulent momentum transport, complementing Eulerian velocity-field analysis. The evolution of initially wall-normal material surfaces in the viscous sublayer, buffer layer, and log layer sheds light on the Reynolds stress-generating events from a Lagrangian perspective. Eliminating the need for tracking numerous particles, the reference map approach offers a convenient and promising avenue for future investigations into Lagrangian kinematics and dynamics of fluid flows.