Abstract
The present work describes the multidimensional behaviour of scale-energy production, transfer and dissipation in wall-bounded turbulent flows. This approach allows us to understand the cascade mechanisms by which scale energy is transmitted scale-by-scale among different regions of the flow. Two driving mechanisms are identified. A strong scale-energy source in the buffer layer related to the near-wall cycle and an outer scale-energy source associated with an outer turbulent cycle in the overlap layer. These two sourcing mechanisms lead to a complex redistribution of scale energy where spatially evolving reverse and forward cascades coexist. From a hierarchy of spanwise scales in the near-wall region generated through a reverse cascade and local turbulent generation processes, scale energy is transferred towards the bulk, flowing through the attached scales of motion, while among the detached scales it converges towards small scales, still ascending towards the channel centre. The attached scales of wall-bounded turbulence are then recognized to sustain a spatial reverse cascade process towards the bulk flow. On the other hand, the detached scales are involved in a direct forward cascade process that links the scale-energy excess at large attached scales with dissipation at the smaller scales of motion located further away from the wall. The unexpected behaviour of the fluxes and of the turbulent generation mechanisms may have strong repercussions on both theoretical and modelling approaches to wall turbulence. Indeed, actual turbulent flows are shown here to have a much richer physics with respect to the classical notion of turbulent cascade, where anisotropic production and inhomogeneous fluxes lead to a complex redistribution of energy where a spatial reverse cascade plays a central role.
Highlights
The multiscale feature of turbulent flows has always drawn the attention of scientists since Richardson’s work (Richardson 1922) on the turbulent energy cascade
Much of the current understanding of fully developed turbulence is based on this result and relies on a general equation which is known as the Kolmogorov equation, δu3
This conjecture is consistent with the behaviour of the single-point turbulent kinetic energy source, s = − uv −, which is displayed for three Reynolds numbers, Reτ = 550, Reτ = 950 and Reτ = 2000, in figure 4
Summary
The multiscale feature of turbulent flows has always drawn the attention of scientists since Richardson’s work (Richardson 1922) on the turbulent energy cascade. To provide a more complete view, the four-fifths law in the form of a balance equation for second-order structure function, originally proposed by Hill (2002), was used by Marati, Casciola & Piva (2004) to address the energy transfer in both spatial and scale spaces for a turbulent channel flow. The second-order structure function can be written as δu2 = 2 k (Yc + ry/2) + 2 k (Yc − ry/2) − 2 ui(Yc + ry/2)ui(Yc − ry/2) where k = uiui/2 and increments are considered only in the wall-normal direction for simplicity This expression allows us to highlight how spatial inhomogeneity enters the space of wall-normal scales ry. The mean unbalance of the terms of (2.1) is found to be less than 1.5 % of the local dissipation
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