Functional Modeling (Isotropic Case)

Abstract

It would be illusory to try to describe the structure of the scales of motion and the interactions in all imaginable configurations, in light of the very large disparity of physical phenomena encountered. So we have to restrict this description to cases which by nature include scales that are too small for today's computer facilities to solve them entirely, and which are at the same time accessible to theoretical analysis. This description will therefore be centered on the inter-scale interactions in the case of fully developed isotropic homogeneous turbulence 1 , which is moreover the only case accessible by theoretical analysis and is consequently the only theoretical framework used today for developing subgrid models. Attempts to extend this theory to anisotropic and/or inhomogeneous cases are discussed in Chap. 6. The text will mainly be oriented toward the large-eddy simulation aspects. For a detailed description of the isotropic homogeneous turbulence properties, which are reviewed in Appendix A, the reader may refer to the works of Lesieur [439] and Batch-elor [45]. 5.1 Phenomenology of Inter-Scale Interactions It is important to note here the framework of restrictions that apply to the results we will be presenting. These results concern three-dimensional flows and thus do not cover the physics of two-dimensional flows (in the sense of flows with two directions 2 , and not two-component 3 flows), which have a totally different dynamics [403, 404, 405, 438, 481]. The modeling in the two-dimensional case leads to specific models [42, 624, 625] which will not 1 That is, whose statistical properties are invariant by translation, rotation, or symmetry. 2 These are flows such that there exists a direction x for which we have the property: ∂u ∂x ≡ 0. 3 These are flows such that there exists a framework in which the velocity field has an identically zero component.