Evolutionary geometry of Lagrangian structures in a transitional boundary layer

Abstract

We report a geometric study of Lagrangian structures in a weakly compressible, spatially evolving transitional boundary layer at the Mach number 0.7. The Lagrangian structures in the transition process are extracted from the Lagrangian scalar field by a sliding window filter at a sequence of reference times. The multi-scale and multi-directional geometric analysis is applied to characterize the geometry of spatially evolving Lagrangian structures, including the averaged inclination and sweep angles at different scales ranging from one half of the boundary layer thickness to several viscous length scales. Here, the inclination angle is on the plane of the streamwise and wall-normal directions, and the sweep angle is on the plane of the streamwise and spanwise directions. In general, the averaged inclination angle is increased and the sweep angle is decreased with the reference time. The variation of the angles for large-scale structures is smaller than that for small-scale structures. Before the transition, the averaged inclination and sweep angles are only slightly altered for all the scales. As the transition occurs, averaged inclination angles increase and sweep angles decrease rapidly for small-scale structures. In the late transitional stage, the averaged inclination angle of small-scale structures with 30 viscous length scales is approximately 42°, and the averaged sweep angle in the logarithm law region is approximately 30°. Additionally, the geometry of Lagrangian structures in transitional boundary layer flow is compared with that in the fully developed turbulent channel flow.