Adjoint-based computation of nonlocal eddy viscosity in turbulent channel flow

Reynolds-averaged Navier-Stokes (RANS) closure operators are generally nonlocal and anisotropic for many flows, including wall-bounded turbulence. The macroscopic forcing method (MFM) and Green's function-based approaches examine these effects using forced direct numerical simulations. For these approaches, the number of simulations required to compute the nonlocal and anisotropic eddy viscosity is equal to the number of degrees of freedom in the averaged space. We reduce the computational cost by introducing an adjoint-based formulation of MFM to obtain the nonlocal and anisotropic eddy viscosity at any Reynolds stress location using a single simulation. We then quantify the streamwise and wall-normal nonlocal eddy viscosity at near-wall locations in turbulent channel flow at Re-tau = 180. We demonstrate that the upstream nonlocality is not fully described by Lagrangian transport. We also quantify the significant differences in the upstream nonlocality between various components of the eddy viscosity tensor. Our results can be used to guide closure modeling, including the lengthscales and timescales used in Reynolds stress models.