Analysis of interscale energy transfer in a boundary layer undergoing bypass transition

Abstract

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.

On the role of the laminar/turbulent interface in energy transfer between scales in bypass transition

“…This pattern was observed in Yao et al. (2022).

Figure 5.Nonlinear interscale energy flux vector (denoted LHS) and the constituent components , where , in the plane at location .

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“…This plot is very similar to that of figure 8( b ) that shows the nonlinear component , the latter being the dominant component (see Yao et al. (2022) for a detailed discussion on the other components). Both plots depict a dense cluster of stream-tubes with energy flowing to larger scales (inverse cascade) with energy originating at and .…”

“…It can be easily proved (see Yao et al. 2022) that for this component is equal to . The corresponding production component due to inhomogeneity of the mean flow in physical space, , is much smaller.…”

“…It is therefore difficult to explain the origin of the inverse cascade found in Yao et al. (2022). For example, does it arise from the turbulent conditions within the spot?…”

“…The most general form of the KHMH equation, applicable to inhomogeneous and anisotropic flows, was derived directly from Navier–Stokes equations by Hill (2002). The equation was applied recently to transitional boundary layers and demonstrated strong inverse cascade in the transition region, especially in the streamwise direction (see Yao, Mollicone & Papadakis 2022). Analysis of instantaneous velocity fields and flux vectors revealed that the inverse cascade was related to the growth of turbulent spots.…”