  We report a Lagrangian study on the evolution of material surfaces in the
K-type temporal transitional channel flow. Based on the Eulerian velocity field
from the DNS, a backward-particle-tracking method is applied to solve the
transport equation of the Lagrangian scalar field, and then the iso-surfaces 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 proposed criteria, so that the
initial material surfaces can be approximated as vortex surfaces, and keep
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 be used to elucidate the continuous
evolution of vortical structures in transitional wall-bounded 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.
