We consider the inter-scale energy transfer process in a boundary layer undergoing bypass
transition using a generalised Karman-Howarth equation. The analysis is performed using
a high-resolution direct numerical simulation (DNS) database. The bypass transition is
triggered by homogeneous and isotropic free-stream turbulence satisfying a Von Karman
spectrum.
We first consider the evolution of the second-order structure function (defined as the square of the difference of the streamwise velocity fluctuations between two points) at different locations in the transition region. We apply conditional sampling based on the local
instantaneous intermittency and derived new analytic expressions that generalise existing
decompositions of single-point statistics to two-point statistics. It is found that in the
transition region, laminar streaky structures maintain their geometrical characteristics in
the physical and scale spaces well inside the transition region, even after the initial break
down to form turbulent spots. Further conditional analysis reveals that the outer mode is the dominant secondary instability mechanism and shows how turbulence spots penetrate the boundary layer and approach the wall. We also analyse the evolution of the two-point intermittency field, and find that the volume enclosed by an isosurface of a given value grows in both directions, with the growth in the streamwise direction being especially large.
In order to study the dynamics of the structure function and explore the production and
interscale energy transfer in bypass transition, the Karman-Howarth-Monin-Hill (KHMH)
equation is then employed. This is the evolution equation of the scale energy, i.e. the energy contained within eddies of a specific length scale. This equation is very general and can be applied to inhomogeneous and anisotropic flows, like bypass transition. Maps of scale energy production and flux vectors are visualised on two-dimensional planes and three-dimensional hyperplanes that comprise both physical and separation spaces. In the transitional region, the maps show strong inverse cascade in the streamwise direction near the wall. The energy flux vectors emanate from a region of strong production and transfer energy to larger streamwise scales. The inverse cascade is mainly due to the non-linear interaction flux component, and this component competes with the one due to mean flow inhomogeneity. By superposing the instantaneous velocity fields and the energy flux vectors, we relate the inverse cascade
process to the growth of turbulent spots.
We also apply the conditional averaging framework that we developed for the structure
function to the KHMH equation and derive a conditionally-averaged KHMH equation as well
as expressions for the decomposition of energy flux vectors. Because conditional averaging
does not commute with the spatial differentiation operation, the derivatives of conditionallyaveraged two-point variables are calculated directly in the scale space. The conditionally-averaged KHMH analysis shows that the strong inverse cascade in the streamwise direction is mainly due to the non-linear interaction terms across the upstream and downstream laminar/turbulent interfaces of the turbulent spots; we find that the contribution from the downstream one dominates. Many of the observed features are explained by a schematic cartoon that considers a propagating diamond-shaped turbulent spot. Finally, we compare
the 3-dimensional energy flux map within the turbulent spots and within the fully turbulent
region. A similar shape of the flux map is found, proving the dynamical similarities between
turbulent spots and fully turbulent region in terms of interscale transfer.