Linear and nonlinear models of anisotropic turbulence

Because of mean distortion, most turbulent flows are anisotropic. Two-point descriptions, forming the heart of this review of anisotropic models, capture the continuum of anisotropically structured turbulent scales and, moreover, allow exact treatment of the linear terms representing mean distortion, only needing closure assumptions for the nonlinear part of the model. The rapid-distortion limit, in which nonlinear terms are neglected, is the main subject of Section 2, while Section 3 introduces nonlinearity. It is shown that, even with significant nonlinearity, many features of turbulence can, at least qualitatively, be understood using linear theory alone, e.g, the directionality of velocity fluctuations and correlation lengths induced by strong mean shear near a wall or straining by duct flow, whereas some, e.g. wave resonances in rotating turbulence, involve a subtle combination of linear and nonlinear terms. The importance of linear effects is reflected in the triadic models of Section 3, which contain no approximations of the linear terms and whose anisotropic nonlinear closures are heavily dependent on linear theory. Despite being fundamentally less satisfactory (because they involve additional ad hoc hypotheses to compensate for the lack of two-point information), one-point models dominate industrial calculations because they are robust, well-established, and computationally relatively cheap. Although there are too many spectral degrees of freedom for a one-point model to reproduce two-point results in all circumstances, two-point theories-in particular RDT-have been exploited to develop new one-point models, as discussed in Section 4. Given the significant limitation of classical two-point models to homogeneous turbulence, some inhomogeneous extensions are described in Section 5.