Analysis of nonlocal effects in turbulence closures with application to wall-bounded flows
Keywords: Engineering.
Mechanical Engineering.

Turbulence modeling of wall-bounded flows is important to a wide variety of applications, including in engineering, e.g., aircraft and ships, and in geophysics, e.g., atmospheric boundary layers. In turbulent flows, direct numerical simulation (DNS) of the governing equations is often intractable due to the wide range of scales that must be resolved. Instead, turbulence modeling approaches, such as Reynolds-averaged Navier-Stokes (RANS) models, are widely-used in engineering applications. RANS models compute averaged quantities, such as the mean velocities needed for the prediction of drag or lift on an aircraft, that are sufficient for many engineering applications. Widely-used RANS models rely on a local and isotropic eddy viscosity approximation that is inadequate for complex flows. This work begins by revisiting an existing framework for incorporating nonlocal and anisotropic effects, namely Reynolds stress transport models. While we include recent experimental/DNS data from literature with the goal of improving model accuracy and develop an analytical near-wall model for capturing the leading-order behavior of all components of the Reynolds stresses near the wall, we encounter substantial challenges in making model improvements, particularly due to the distinct model requirements for both near-wall, low turbulent intensity regions, and away-from-wall, high turbulent intensity regions. This motivates both the development of an alternative modeling approach and a physical investigation of nonlocal and anisotropic effects in turbulent wall-bounded flows, which are the focuses of the rest of this work. The macroscopic forcing method (MFM) of Mani and Park (2021) allows computation of the exact nonlocal and anisotropic eddy viscosity. However, to compute the exact eddy viscosity, a brute force application of MFM requires as many DNSs as degrees of freedom in the averaged space. We develop a systematic and cost-effective approach for quantifying and modeling the nonlocal eddy viscosity that needs only information about a few of the low-order eddy viscosity moments, which can be computed efficiently using only one simulation per moment. The resulting model form, using what we call matched moment inverse operators, closely approximates the shape of the true nonlocal eddy viscosity kernel and is in the form of a partial differential equation rather than a difficult to solve integro-partial differential equation. We then focus on a physical investigation of the exact nonlocal and anisotropic eddy viscosity in turbulent wall-bounded flows. To alleviate the computational cost of a brute force MFM approach, we develop an adjoint-based macroscopic forcing method (adjoint MFM) for targeted computation of the nonlocal eddy viscosity, which relates the Reynolds stresses at a given location to the mean velocity gradient at all points in space, using only one simulation per desired location. While prior works have examined the wall-normal nonlocal eddy viscosity in turbulent channel flow, we use adjoint MFM to examine the streamwise and wall-normal nonlocal eddy viscosity in turbulent channel flow at select near-wall locations.