Analytical Theories and Stochastic Models

Abstract

Our objective in this chapter is to give to the reader a good understanding of the analytical theories and stochastic models of turbulence sometimes referred to as “two-point closures” since, as will be seen, they deal with correlations in two different points of the space (or two different wave numbers k and k¹ such that k+k¹ = 0 in the Fourier space). A whole book would not be sufficient to contain all the details of the algebra which is involved, and the reader will be referred to the quoted references for further details: of particular interest for that purpose are refs [99], [11], [10], [74]. We will mainly use the so-called E.D.Q.N.M. approximation (Eddy-Damped Quasi-Normal Markovian approximation), and will situate it among other theories of the same type. These theories can generally be presented from two different points of view, the stochastic model point of view, and the closure point of view. Some of these theories, as will be seen, do not exactly correspond to these points of view, but they lead to spectral equations of the same family, which can be solved with the same methods. We will not spend too much energy deriving the “best” analytical theory, for it seems that they all have qualitatively the same defects and qualities, and differ essentially in the values of the inertial-range exponents. We will concentrate principally on the E.D.Q.N.M. (which in the case of isotropy can be solved numerically at a much cheaper cost than the direct simulations, even at high Reynolds number), and will discuss to what extent the results can be relied upon for “real” turbulence (that is turbulence governed by the Navier-Stokes equation).