Evolution of material surfaces in the temporal transition in channel flow

We report a Lagrangian study on the evolution of material surfaces in the Klebanoff-type temporal transitional channel flow. Based on the Eulerian velocity field from the direct numerical simulation, a backward-particle-tracking method is applied to solve the transport equation of the Lagrangian scalar field, and then the isosurfaces of the Lagrangian field can be extracted as material surfaces in the evolution. Three critical issues for Lagrangian investigations on the evolution of coherent structures using material surfaces are addressed. First, the initial scalar field is uniquely determined based on the proposed criteria, so that the initial material surfaces can be approximated as vortex surfaces, and remain invariant in the initial laminar state. Second, the evolution of typical material surfaces initially from different wall distances is presented, and then the influential material surface with the maximum deformation is identified. Large vorticity variations with the maximum curvature growth of vortex lines are also observed on this surface. Moreover, crucial events in the transition can be characterized in a Lagrangian approach by conditional statistics on the material surfaces. Finally, the influential material surface, which is initially a vortex surface, is demonstrated as a surrogate of the vortex surface before significant topological changes of vortical structures. Therefore, this material surface can he used to elucidate the continuous temporal evolution of vortical structures in transitional wall-hounded flows in a Lagrangian perspective. The evolution of the influential material surface is divided into three stages: the formation of a triangular bulge from an initially disturbed streamwise-spanwise sheet, rolling up of the vortex sheet near the bulge ridges with the vorticity intensification and the generation and evolution of signature hairpin-like structures with self-induced dynamics of vortex filaments.