The paper describes the energy fluxes simultaneously occurring in the space of scales and in the physical space of wall-turbulent flows. The unexpected behaviour of the energy fluxes consists of spiral-like paths in the combined physical/scale space where the controversial reverse energy cascade plays a central role. Two dynamical processes are identified as driving mechanisms for the fluxes, one in the near-wall region and a second one further away from the wall. The former, stronger, one is related to the dynamics involved in the near-wall turbulence regeneration cycle. The second suggests an outer self-sustaining mechanism which is asymptotically expected to take place in the eventual log layer and could explain the debated mixed inner/outer scaling of the near-wall statistics. The observed behaviour may have strong repercussions on both theoretical and modelling approaches to wall turbulence, as anticipated by a simple equation which is shown able to capture most of the rich dynamics of the shear-dominated region of the flow.

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.

The separating and reattaching flows and the wake of a finite rectangular plate are studied by means of direct numerical simulation data. The large amount of information provided by the numerical approach is exploited here to address the multi-scale features of the flow and to assess the self-sustaining mechanisms that form the basis of the main unsteadinesses of the flows. We first analyse the statistically dominant flow structures by means of three-dimensional spatial correlation functions. The developed flow is found to be statistically dominated by quasi-streamwise vortices and streamwise velocity streaks as a result of flow motions induced by hairpin-like structures. On the other hand, the reverse flow within the separated region is found to be characterized by spanwise vortices. We then study the spectral properties of the flow. Given the strongly inhomogeneous nature of the flow, the spectral analysis has been conducted along two selected streamtraces of the mean velocity field. This approach allows us to study the spectral evolution of the flow along its paths. Two well-separated characteristic scales are identified in the near-wall reverse flow and in the leading-edge shear layer. The first is recognized to represent trains of small-scale structures triggering the leading-edge shear layer, whereas the second is found to be related to a very large-scale phenomenon that embraces the entire flow field. A picture of the self-sustaining mechanisms of the flow is then derived. It is shown that very-large-scale fluctuations of the pressure field alternate between promoting and suppressing the reverse flow within the separation region. Driven by these large-scale dynamics, packages of small-scale motions trigger the leading-edge shear layers, which in turn created them, alternating in the top and bottom sides of the rectangular plate with a relatively long period of inversion, thus closing the self-sustaining cycle.

Exact budget equations for the second-order structure function tensor $\langle \unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}u_{j}\rangle$ , where $\unicode[STIX]{x1D6FF}u_{i}$ is the difference of the $i$ th fluctuating velocity component between two points, are used to study the two-point statistics of velocity fluctuations in inhomogeneous turbulence. The anisotropic generalised Kolmogorov equations (AGKE) describe the production, transport, redistribution and dissipation of every Reynolds stress component occurring simultaneously among different scales and in space, i.e. along directions of statistical inhomogeneity. The AGKE are effective to study the inter-component and multi-scale processes of turbulence. In contrast to more classic approaches, such as those based on the spectral decomposition of the velocity field, the AGKE provide a natural definition of scales in the inhomogeneous directions, and describe fluxes across such scales too. Compared to the generalised Kolmogorov equation, which is recovered as their half-trace, the AGKE can describe inter-component energy transfers occurring via the pressure–strain term and contain also budget equations for the off-diagonal components of $\langle \unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}u_{j}\rangle$ . The non-trivial physical interpretation of the AGKE terms is demonstrated with three examples. First, the near-wall cycle of a turbulent channel flow at a friction Reynolds number of $Re_{\unicode[STIX]{x1D70F}}=200$ is considered. The off-diagonal component $\langle -\unicode[STIX]{x1D6FF}u\unicode[STIX]{x1D6FF}v\rangle$ , which cannot be interpreted in terms of scale energy, is discussed in detail. Wall-normal scales in the outer turbulence cycle are then discussed by applying the AGKE to channel flows at $Re_{\unicode[STIX]{x1D70F}}=500$ and $1000$ . In a third example, the AGKE are computed for a separating and reattaching flow. The process of spanwise-vortex formation in the reverse boundary layer within the separation bubble is discussed for the first time.

Starting from physical insight on the energy transfer phenomena in wall turbulent flows, it is shown how modeling of subgrid stresses in large-eddy simulations can be improved. Each model should aim at reproducing the double feature of energy sink and source of the small scales of wall flows which become relevant when large filter lengths are considered. Here we propose one possible choice where the main ingredient is the coupling of the classical linear formulation of eddy viscosity with the nonlinear anisotropic features of the velocity increments tensor. This approach, which actually presents most of the features of the mixed models, captures the near-wall dynamics for very large filter lengths reproducing the small scales source physics responsible for backward energy transfer. A posteriori tests show excellent agreement with direct numerical simulation of turbulent channel flows even when very coarse grids are considered. The capability of the balance of the filtered second order structure function as a post-processing tool to evaluate the physics of any model is also shown.