Introduction to two-point closures

An overview is given of nonlocal theories and models, ranging from linear to nonlinear. The background principles are presented and illustrated mainly for incompressible, homogeneous, anisotropic turbulence. In this case, which includes effects of mean gradients and body forces and related structural effects, the complete rapid distortion theory (RDT) solution is shown to be a building block for constructing a full nonlinear closure theory. Firstly, a general overview of the closure problem is presented, which accounts for both the nonlinear problem and the nonlocal problem. Then, some limitations of single-point closures are illustrated by simple examples, and we discriminate flows dominated by production effects and flows dominated by wave effects. A classical spectral description is introduced for the fluctuating flow and its multi-point correlations. Applications to stably-stratified and rotating turbulence are discussed. Linear effects captured by RDT include dispersivity of gravity waves, whereas the irreversible collapse of vertical motion and subsequent layering of the velocity field is only captured by nonlinear theories or high resolution DNS/LES computations. Applications to weak turbulence in compressible flows are touched upon at the very end.