Development of one-point statistical closure from a spectral description of the turbulence

The statistical two-point description of turbulence has been regarded as a convenient mathematical framework for better understanding and modelling of the one-point quantities. K'arm'an and Howarth (1938) used this technique to study the evolution of the turbulent length scales in decaying isotropic turbulence. It was also used later by Crow (1968) to express the rapid distortion of isotropic turbulence and by Naot et al. (1973) to derive a transport equation for the Reynolds stress tensor. The two-point description of turbulence provides indeed a convenient framework to study the length scales equations. Rotta (1951) applied the integral operator of the two-point correlation equations to derive the equation for the product of the Reynolds stresses and a scalar integral length-scale. The two-point description served also to derive transport equations for the length-scale tensor (Lin and Wolfshtein (1980), Donaldson and Sandri (1981) or Oberlack and Peters (1993)), and to provide a better insight in the closure of the classical one-point dissipation equation (Jovanovi'c et al. (1995), Oberlack (1994)). However, most of these studies were confined to homogeneous turbulent flows. The two-point spatial approach has also been used to develop multiple-scales models (Schiestel, 1987), though this can also be achieved by starting from a description of the two-point equations in the spectral space through a Fourier transformation. The latter approach is adopted in this study.