Fully developed turbulence and its modeling

Statistical and numerical approaches to the modeling of homogeneous turbulence allowed to develop without external constraints limiting the effects of nonlinearities are presented. The statistical approach is based on both a phenomenological analysis, and on analytical two-point closure theories, including those of Kolmogorov, Obukhov and the quasi-normal theories derived by Millionchikov (1941) and Proudman and Reid (1954). The dynamics of large coherent two-dimensional turbulence structures may also be understood in terms of statistical theories if the predictability of freely evolving flows is taken into account. The numerical approach involves the numerical integration, by the aid of a computer, of the equations governing all the flows of interest; one of the major problems in this approach is the parameterization of the small scales. Of the two approaches, no factor allows one to be preferred over the other as a means of making progress towards the understanding of the problem, and further theoretical, experimental and numerical investigations of turbulence phenomena are called for.