Production, transport and dissipation of turbulent stresses across scales and space

We present a theoretical framework for describing production, transport, redistribution and dissipation of every Reynolds stress component occurring among different scales and along directions of statistical inhomogeneity. It is based on the exact budget equations for the second-order structure function tensor ⟨δuiδuj⟩. This set of equations, that we name Anisotropic Generalized Kolmogorov Equations or AGKEs adds the scale information to the classic analysis of the single-point budget of the Reynolds stresses, while it allows the consistent definition of scales in directions of statistical inhomogeneity compared to a spectral analysis of the two-point Reynolds stress budgets. Fluxes of Reynolds stresses in space and across scales can be defined and their properties analysed.