Adjoint macroscopic forcing method for computing the nonlocal eddy viscosity in a turbulent channel flow

The nonlocal eddy viscosity relates the Reynolds stress at a spatial point to the mean velocity gradient at all points. With this, one can inform models, e.g., RANS models, about the sensitivity to the mean velocity gradient and the suitability of local approximations. Previous brute force approaches (Mani and Park, Phys. Rev. Fluids, 2021; Hamba, Phys. Fluids, 2005) compute the nonlocal eddy viscosity by forcing the mean velocity gradient at each point in the averaged space and examining the Reynolds stress response. They require a separate simulation for each mean velocity gradient point. So, obtaining the nonlocal eddy viscosity requires as many simulations as degrees of freedom in the averaged space. For large problems, the number of simulations required becomes cost prohibitive. We present the adjoint macroscopic forcing method (MFM) as a strategy to obtain the nonlocal eddy viscosity at a given Reynolds stress location using a single simulation. This method recovers the Reynolds stress dependence at a point of interest, such as a separation point or near the wall, on the mean velocity gradient at all points. We demonstrate the adjoint MFM on a canonical turbulent channel flow.