Inter-scale Causality Relations in Wall Turbulence

We investigate the inter-scale energy transfer in the buffer layer between different streamwise and spanwise lengthscales. In a periodic channel, inter-scale transfer of momentum in the periodic directions is only possible through wavenumber interactions in the non-linear advection term. The aim of this study is to map the inter-scale interactions which transfer energy through advection to different lengthscales of interest. The non-linear advection term has nine contributions, which correspond to three different advecting and advected velocity components. Because the flow is anisotropic, each of these contributions can be viewed as portraying a different physical mechanism. Maps of inter-scale transfer are therefore constructed for each of the nine contributions to the advection term in order to understand the relative importance of advecting and advected velocity components in wall turbulence. As a preliminary exploration, two maps of the inter-scale energy transfer in the buffer layer are presented for Re-tau = 180, one for transfer to a 'target' lengthscale dynamically relevant in the near-wall cycle, lambda(+)(x) = 188, lambda(+)(z) = 94, and one for transfer to a 'target' lengthscale in the dissipative range, lambda(+)(x) = 42, lambda(+)(z) = 21. In the first case, we observe that the transfer of energy is dominated by transfer into streamwise-velocity (u) by spanwise-elongated structures of spanwise-velocity (w) that advect streamwise-elongated structures of streamwise-velocity (u). This is nothing but the process of streak meandering induced by spanwise-elongated (w) structures. In the second case, the inter-scale transfer can be split into three dominant mechanisms. The first and second mechanisms are observed across many of the nine contributions to the advection term. Velocity structures with size similar or larger than streamwise streaks either advect (mechanism 1) or are advected by (mechanism 2) velocity structures with lengthscales close to the target one. The third mechanism is only significant for streamwise (u) velocity structures advecting streamwise (u) and spanwise (w) velocity, and involves structures with an intermediate scale which is in-between the dissipation scale and the near-wall cycle dynamic lengthscale, but still in the dissipation range. In this mechanism, intermediate-scale streamwise-velocity structures advect intermediate-scale streamwise- and spanwise-velocity structures.