Local similarity of velocity distributions in homogeneous isotropic turbulence

Succeeding to the previous paper [Tatsumi and Yoshimura, 2004. Inertial similarity of velocity distributions in homogeneous isotropic turbulence. Fluid Dyn. Res. 35, 123-158] which dealt with the inertial similarity of the velocity distributions of homogeneous isotropic turbulence, the local similarity of the velocity distributions is investigated. The equations for the one- and two-point velocity distributions are expressed in the local dimensionless variables based on the mean energy-dissipation rate ε¯ and the kinetic viscosity ν and solved by making use of the cross-independence closure hypothesis [Tatsumi, 2001. Mathematical physics of turbulence. In: Kambe, T., et al. (Eds.), Geometry and Statistics of Turbulence. Kluwer Academic Publishers, Dordrecht, pp. 3-12]. The velocity distributions are obtained as continuous solutions in the local variables which coincide with the inertial normal distributions out of the local similarity range. The one-point velocity distribution, which was given by the normal distribution N1 with the parameter α0 under the inertial similarity, is expressed in the local variables as the distribution N1 but with the local parameter α0*(=α0/ν). The two-point velocity distribution is expressed in terms of the velocity-sum distribution and the velocity-difference distribution as before. The velocity-sum distribution, which was given by the normal distribution N2 with the parameter α0/2 under the inertial similarity, is obtained as the normal distribution N3 with the local parameter α+*(r*) which changes with the local distance r*(=|r*|) between the two points. Since α+*(r*) tends to α0*/2, corresponding to α0/2 of N2, for r*→∞ and to α0*, corresponding to α0 of N1, for r*→0, the distribution N3 satisfies the boundary conditions at both ends of the local similarity range. The velocity-difference distribution, which was given by the isotropic distribution N2 under the inertial similarity, becomes axi-symmetric with respect to r* in the local range. The lateral distribution is obtained as the one-dimensional normal distribution N4 which satisfies the boundary conditions at both ends of the local range. The longitudinal distribution is obtained in three different similarity forms in the local range. It is expressed as the intermediate normal distribution N5 with the parameter α-0*(r*) in the intermediate subrange, as the algebraic distribution A1 with the same α-0*(r*) in the inertial subrange, and as the slightly asymmetric algebraic distribution A2 in the viscous subrange. The physical concepts of these results are discussed in comparison with existing experimental and numerical results.