**Production, transport and dissipation of turbulent stresses in channels **

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**Alessandro Chiarini**

The present work describes the production, dissipation and transport phenomena of every Reynolds stress component in wall-bounded turbulent flows, considering the multidimensional space of scales and positions. This has required the derivation of the exact budget equation for the second-order structure function ⟨δu_{i}δu_{k}⟩ — where δu_{i} is the increment of the i - th velocity component at position X_{c} and separation r, i.e. δu_{i} = u_{i}(X_{c} + r∕2,t) - u_{i}(X_{c} - r∕2,t) — that we name Anisotropic Generalized Kolmogorov Equation (AGKE). This set of equations allows us to understand the mechanism by which each ⟨δu_{i}δu_{k}⟩ is transmitted scale-by-scale across different regions of the flow.

The terms of the AGKE have been computed by analysing a DNS-generated database for a turbulent channel flow at Reynolds number Re_{τ} = 200, by means of a solver specifically created for the present work. The analysis carried out for the first time in the present work, brings to light important differences among the transfers of the several second-order structure functions, highlighting the anisotropic character of the flow. When studying the transfers of the diagonal components ⟨δu_{i}δu_{i}⟩ (with i = 1,2,3), two main classes have been identified and associated with two particular redistribution mechanisms of turbulent energy. One denotes transfers occurring in the buffer layer and involving the structures related to the near-wall cycle, whereas the other highlights transfers towards the bulk of the flow through both attached and detached scales of motion. Regarding the off-diagonal term ⟨-δuδv⟩, similarly, two main transfers have been identified. On one side, ⟨-δuδv⟩ is transferred from the bulk of the flow towards the wall through both incoherent motions and stream-wise coherent vortices placed at different wall-normal distances. On the other side, a transfer of ⟨-δuδv⟩ occurs in the near-wall region involving the so-called uv-structures placed in the viscous sublayer and those associated with the near-wall cycle.

The newly derived AGKE, describing the Reynolds stresses in both the space of scales and the physical space, provides an insightful description of the interactions that occur among turbulent fluctuations at different scales in presence of spatial inhomogeneities, progressing beyond the simple concept of “energy”, which — after all — provides a satisfactory description of the second-order statistics in the homogeneous and isotropic case only. As a result, the AGKE may benefit both theoretical and modelling approaches to wall turbulence. For example, it may be used to improve existing models used in large-eddies simulations, where the effect of the small unresolved scales on the resolved motion should be accurately reproduced.