The infinite set of the Lundgren-Monin equations for the multi-point velocity distributions of fluid turbulence is closed by making use of the cross-independence closure hypothesis proposed by Tatsumi (Geometry and Statistics of Turbulence, 2001, p. 3), and the minimum deterministic set of equations is obtained as the equations for the one-point velocity distribution f, the two-point velocity distribution f(2) and the two-point local velocity distribution f(2)*. In practice, the two-point distributions f(2) and f(2)* are more conveniently expressed in terms of the velocity-sum and -difference distributions g+, g− and g+*, g−*, respectively.
As an outstanding result, the energy dissipation rate is expressed in terms of the distribution g− which is mainly contributed from small-scale turbulent fluctuations, making clear analogy with the ‘fluctuation-dissipation theorem’ in non-equilibrium statistical mechanics.
It is to be remarked that the integral moments of the equations for the distributions f and f(2) give the equations for the mean flow and the mean velocity procducts of various orders, which are identical with the corresponding equations derived directly from the Navier--Stokes equation. This results clearly shows the exact consistency of the cross-independence closure and gives an overall solution for the classical closure problem concerning the mean velocity products since they are derived from the known distributions.
Although the present work is confined to the two-point statistics of turbulence, the analysis can be extended to the higher-order statistics and even to turbulence in other fluids such as magneto and quantum fluids.