Evolution of conditionally-averaged second order structure functions in a transitional boundary layer

Abstract

We consider the bypass transition in a flat plate boundary layer subject to free-stream turbulence and compute the evolution of the second-order structure function of the streamwise velocity, du2(,), from the laminar to the fully turbulent region using DNS. In order to separate the contributions from laminar and turbulent events at the two points used to define du(→x,→r), we apply conditional sampling based on the local instantaneous intermittency, τ (1 for turbulent and 0 for laminar events). Using τ(→x,t), we define two-point intermittencies, γ(TT), γ(LL) and γ(TL) which physically represent the probabilities that both points are in turbulent or laminar patches, or one in turbulent and the other in a laminar patch, respectively. Similarly, we also define the conditionally-averaged structure functions, ⟨du2⟩(TT), ⟨du2⟩(LL) and ⟨du2⟩(TL) and decompose ⟨du2⟩(→x,→r) in terms of these conditional averages. The derived expressions 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 space well inside the transition region, even after the initial break down to form turbulent spots. Analysis of the ⟨du2⟩(TT) fields reveal that the outer mode is the dominant secondary instability mechanism. Further analysis reveals how turbulence spots penetrate the boundary layer and approach the wall. The peaks of ⟨du2⟩(TT) in scale space appear in larger streamwise separations as transition progresses and this is explained by the strong growth of turbulent spots in this direction. On the other hand, the spanwise separation where the peak occurs remains relatively constant and is determined by the initial inception process. We also analyse the evolution of the two-point intermittency field, γ(TT), at different locations. In particular, we study the growth of the volume enclosed within an iso-surface of γ(TT) and notice that it increases in both directions, with the growth in the streamwise direction being especially large. The evolution of these conditional two-point statistics sheds light into the transition process from a different perspective and complements existing analyses using single-point statistics.