Interscale causality in near-wall turbulence

This work aims to map how information is transferred between different lengthscales through the momentum equations in near-wall turbulence using periodic channels, with a particular focus on streamwise (x) and spanwise (z) lengthscales. The flow can then be viewed as a dynamical system where the state space variables Φ= {u,w,v,p} are the three velocity components and the pressure, which depend on the x- and z-wavenumbers k = (k_x,k_z), y, and t. The evolution equations for Φ are the Navier-Stokes equations plus the continuity equation, which can then be written as F(Φ_k,y,t) = N(Φ_k′,y,t,Φ_k′′,y,t), where k, k′, k′′ and satisfy a triadic relationship, F represents the linear part of the equations and N the nonlinear terms, which are responsible for the transfer of information from other lengthscales into lengthscale k. The objective of this work is to characterize and quantify the relative importance of the different contributions to N. While intimately connected to triadic interactions, which are derived from energy considerations, this analysis, based instead on the momentum equations, aims to provide additional cause-effect information to the inter-scale interactions. For the near-wall cycle, for instance, we observe that the most significant interactions are of w structures of wavelengths λ_x⁺≈100, λ_z⁺≈1000 advecting u structures of wavelengths λ_x⁺≈1000, λ_z⁺≈100, to produce a signature in u at wavelengths λ_x⁺≈100, λ_z⁺≈100. This interaction is essentially the generation of meandering in streamwise streaks.