Abstract
In the history of turbulence research, the randomness of turbulent motions has mostly been represented by the statistical averages of the turbulent variables such as the mean velocity and the mean velocity products of various orders. Dynamical equations governing these statistical averages are derived from the Navier-Stokes equation, and statistical theory of turbulence is constructed upon these dynamical equations representing turbulence. In such theory of turbulence, we encounter the unclosedness of the equations, since the equation for the statistical average of a certain-order always includes those of the higher-order according to the nonlinearity of the Navier-Stokes equation, and hence we have to introduce some relationship between the statistical averages of different-orders for closing the set of equations. Actually, this closure problem constitutes the central difficulty of the theory of turbulence.
Another type of statistical approach to turbulence has been given by Lundgren1) and Monin2) using the equations for the multi-point probability distributions of turbulent velocity. Then, statistical mechanics of turbulence is formulated in terms of these statistical equations. Again, the set of these equations is found to be unclosed since the equation for the velocity distribution of a certain-number of points always includes the distributions of the largernumber of points. This time, however, the closure problem is much easier to deal with than before and actually an exact closure has been attained by the cross-independence closure proposed by Tatsumi3). The exactness is confirmed by Tatsumi4) showing that the moment equations derived from the closed equations are in complete agreement with those derived from the Navier-Stokes equation directly and also that the energy-dissipation rate ε due to this closure satisfies the ’fluctuation-dissipation theorem’ of non-equilibrium statistical mechanics.
