Physics-Based Turbulence Anisotropy Closure Including Nonlocal and Nonequilibrium Effects in Turbulent Flows.

Abstract

A new physics-based anisotropy closure including nonlocal and nonequilibrium effects in turbulent flows has been obtained. The new closure is motivated by fundamental studies of the vorticity alignment in turbulent flows, where decomposition of the total strain rate $S_{ij}$ into its local and nonlocal constituents, $S^R_{ij}$ and $S^B_{ij}$, respectively, reveals a substantial nonlocal, quasi-linear aspect to the vorticity dynamics. Through direct calculation of $S^R_{ij}$ and $S^B_{ij}$ from their exact integral relations, it is shown that the vorticity preferentially aligns with the most extensional eigenvector of $S^B_{ij}$. A similar result is obtained using an expansion formulation for $S^B_{ij}$, which allows the nonlocal strain to be calculated as a series of Laplacians of $S_{ij}$.

The fundamental vorticity alignment studies indicate that the anisotropy dynamics may be understood as a quasi-linear system. Nonlocal effects in this system are accounted for through a new nonlocal formulation for the rapid pressure-strain correlation. Using this formulation, a nonlocal transport equation for the anisotropy is obtained, and solution of a quasi-linear version of this equation gives a new closure for the anisotropy that includes nonlocal and nonequilibrium effects in turbulent flows. The new closure is written in an analogous form to the local equilibrium closure originally proposed by Boussinesq, except that the mean strain rate $overline{S}_{ij}$ is replaced with the nonlocal, nonequilibrium effective strain rate $widetilde{S}_{ij}$. The effective strain is naturally written as a convolution integral over the entire straining history of the flow, although a time-local formulation for $widetilde{S}_{ij}$ that can be implemented in computational fluid dynamics codes is also outlined.

Application of the new closure to a range of nonequilibrium and nonlocal tests provides significantly improved predictions of the anisotropy compared to standard approaches based on the local equilibrium closure. With respect to the nonequilibrium tests, particular focus is placed on periodically-sheared turbulence, where the degree of nonequilibrium is determined by the shearing frequency in the flow. The nonlocal tests include fully-developed turbulent channel flow and the zero pressure gradient turbulent boundary layer. Practical implementation of the new closure in existing computational frameworks is outlined, and computational results are presented for the boundary layer case.