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Abstract
The present paper is concerned with the “closure problem of turbulence,” which constituted the main subject of the Session on the “Energy transfer in homogeneous turbulence” in the last Turbulence Colloquium Marseille (TCM) 1961. It is usual in statistical theory of turbulence to express the random velocity field of turbulence in terms of the velocity correlations of various orders and deal with the set of dynamical equations governing these velocity correlations. This way is, however, associated with the difficulty of unclosedness since each equation involves two velocity correlations of different orders as unknowns according to the nonlinearity of the Navier–Stokes equation, and hence we have to introduce some relationship between the velocity correlations of different orders in order to make the set of equations closed. Among various closuretheories proposed so far, the quasi-normal closure by Proudman and Reid (Proudman and W.H. Reid, Phil. Trans. R. Soc. Lond. A 247 (Citation1954), pp. 163–189) and Tatsumi (T. Tatsumi, Proc. Roy. Soc. Lond. A 239 (Citation1957), p. 16.) and the direct-interaction closure by Kraichnan (R.H. Kraichnan, J. Fluid Mech. 5 (Citation1959), pp. 497–543.) have been taken up in the Session and critical survey and evaluation have been made on these theories. Apart from particular merits and demerits of the theories, it has been recognized that they still remain in approximate levels with either partial or no fulfillment of Kolmogorov’s nonzero inertial energy-dissipation hypothesis. More recently, Tatsumi (T. Tatsumi, in Geometry and Statistics of Turbulence, T. Kambe ed., Kluwer, Citation2001, pp. 3–12.) formulated statistical mechanics of fluid turbulence in terms of the Lundgren–Monin equations (T.S. Lundgren, Phys. Fluids, 10 (Citation1967), pp. 969–975; A.S. Monin, PMM J. Appl. Math. Mech. 31 (Citation1967), pp. 1057–1068.) for the multipoint velocity distributions of turbulence and the cross-independence closure of these equations. It has been shown by Tatsumi (T. Tatsumi, J. Fluid Mech. 670 (Citation2011), pp. 365–403.) that this closure provides us with exactclosure of the Lundgren–Monin equations and thus exact solution to any problem concerning the mean velocity products.