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Reynolds-averaged Navier--Stokes (RANS) closure must be sensitive to the flowphysics, including nonlocality and anisotropy of the effective eddy viscosity.Recent approaches used forced direct numerical simulations to probe theseeffects, including the macroscopic forcing method (MFM) of Mani and Park($\textit{Phys. Rev. Fluids}$ $\textbf{6}$, 054607 (2021)) and the Green'sfunction approach of Hamba ($\textit{Phys. Fluids}$ $\textbf{17}$, 115102(2005)). The resulting nonlocal and anisotropic eddy viscosities are exact andrelate Reynolds stresses to mean velocity gradients at all locations. They canbe used to inform RANS models of the sensitivity to the mean velocity gradientand the suitability of local and isotropic approximations. However, thesebrute-force approaches are expensive. They force the mean velocity gradient ateach point in the averaged space and measure the Reynolds stress response,requiring a separate simulation for each mean velocity gradient location. Thus,computing the eddy viscosity requires as many simulations as degrees of freedomin the averaged space, which can be cost-prohibitive for problems with manydegrees of freedom. In this work, we develop an adjoint-based MFM to obtain theeddy viscosity at a given Reynolds stress location using a single simulation.This approach recovers the Reynolds stress dependence at a location ofinterest, such as a separation point or near a wall, on the mean velocitygradient at all locations. We demonstrate using adjoint MFM to compute the eddyviscosity for a specified wall-normal location in an incompressible turbulentchannel flow using one simulation. In contrast, a brute-force approach for thesame problem requires $N=144$ simulations (the number of grid points in thenon-averaged coordinate direction). We show that a local approximation for theeddy viscosity would have been inappropriate.

Targeted computation of nonlocal closure operators via an adjoint-based macroscopic forcing method