Statistical mechanics of turbulence based on cross-independence closure hypothesis

A new approach to statistical mechanic of turbulence based on the cross-independence closure hypothesis is presented and its relationship with Kolmogorov's theory of locally isotropic turbulence is discussed. For homogeneous isotropic turbulence, the one-point velocity distribution is obtained as the inertial normal distribution N1 with the parameter alpha = epsilon/3, epsilon being the energy-dissipation, and no viscosity nu. The energy-dissipation epsilon satisfies the fluctuation-dissipation theorem and causes the inviscid energy catastrophe E greater than 0 in the limit of nu approaches 0. Since no energy supply is assumed for homogeneous turbulence, the energy E decays in time t as E is proportional to t(exp -1) and hence epsilon is proportional to t(exp -2). Two-point velocity distribution is expressed in terms of the velocity-sum distribution and the velocity-difference distribution, and the latter distributions are expressed as another inertial normal distribution N2 with the parameter alpha/2 for r greater than 0, r being the distance of the two points. Although these distributions change discontinuously at r = 0 for satisfying the boundary conditions, they are continuous functions of the local coordinate r* = r/eta. eta = (nu(exp 3)/epsilon)(exp 1/4) being Kolmogorov's length. In the local range, the velocity-sum distribution is expressed as the local normal distribution N3 with the self-energy-dissipation alpha* + (r*) for the velocity-sum as the parameter. The velocity-difference distribution in the local range is axisymmetric with respect to the vector r*, and the lateral component is expressed as the (one-dimensional) local normal distribution N4 with the self-energy-dissipation alpha* - (r*) for the velocity-difference as the parameter. The longitudinal velocity-difference distribution in the local range is obtained as algebraic non-normal distributions A1 and A2 for the inertial and viscous subranges respectively. For inhomogeneous turbulence, the velocity is decomposed into the mean velocity and the fluctuation velocity around it, and the equations for the mean velocity and the distributions of the one-and two-point fluctuation velocities are derived. The general characters of the equations are discussed with systematic application to inhomogeneous turbulence in scope.
資料番号: AA0063908002
レポート番号: JAXA-SP-07-026E