The Kármán–Howarth–Monin–Hill equation is employed to study the production and interscale energy transfer in a boundary layer undergoing bypass transition due to free-stream turbulence. The energy flux between different length scales is calculated at several streamwise locations covering the laminar, transitional and turbulent regimes. Maps of scale energy production and flux vectors are visualised on two-dimensional planes and three-dimensional hyper-planes 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. To provide deeper insight into the origin of the inverse cascade process, we decompose the energy flux vector into components arising from nonlinear interactions between velocity fluctuations, mean flow inhomogeneity, pressure and viscous effects. The inverse cascade is mainly due to the nonlinear interaction component, and in the earliest stages of transition this component competes with that 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. Once the transition process is complete, the maps become very similar to those observed in other fully developed turbulent flows, such as channel flow. Finally we characterise the nonlinear interaction term using probability density functions (PDFs) evaluated at different wall-normal heights. The PDFs are asymmetric and wide-skirted as in homogeneous isotropic turbulence, but are skewed towards positive values reflecting the inverse cascade.

Transient energy growth suppression is a common control objective for feedback flow control aimed at delaying transition to turbulence. A prevailing control approach in this context is observer-based feedback, in which a full-state feedback controller is applied to state estimates from an observer. The present study identifies a fundamental performance limitation of observer-based feedback control: whenever the uncontrolled system exhibits transient energy growth in response to optimal disturbances, control by observer-based feedback will necessarily lead to transient energy growth in response to optimal disturbances for the closed-loop system as well. Indeed, this result establishes that observerbased feedback can be a poor candidate for controller synthesis in the context of transient energy growth suppression and transition delay: the performance objective of transient energy growth suppression can never be achieved by means of observer-based feedback. Further, an illustrative example is used to show that alternative forms of output feedback are not necessarily subject to these same performance limitations, and should also be considered in the context of transient energy growth suppression and transition control.

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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, du 2 ( x, r), 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 (T T ) , γ (LL) and γ (T L) 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, du 2 (T T ) , du 2 (LL) and du 2 (T L) and decompose du 2 ( 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 du 2 (T T ) 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 du 2 (T T ) 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 (T T ) , at different locations. In particular, we study the growth of the volume enclosed within an iso-surface of γ (T T ) 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.

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