Inertial similarity of velocity distributions in homogeneous isotropic turbulence

Abstract

The hypothesis of cross-independence of two turbulent velocities has been proposed by Tatsumi (2001) for closing the equations for the velocity distributions given by Lundgren (1967) and Monin (1967). Using this hypothesis, the equations for the one- and two-point velocity distributions in homogeneous isotropic turbulence are derived in closed forms, and the velocity distributions are obtained from these equations. One-point velocity distribution is obtained as an inertial normal distribution (N1) including the energy dissipation rate ε ̄ as only parameter. The normal distribution changes self-similarly in time t, starting from a uniform distribution (U) of zero probability density at t=0, growing up with t as a changing normal distribution, and tending to the delta distribution ( δ) around zero for t→∞. During this process, the kinetic energy Ē decays as t −1 and the energy dissipation rate ε ̄ as t −2. Two-point velocity distribution is expressed in terms of the velocity-sum and velocity-difference distributions. The both distributions are obtained for all distances r>0 as another inertial normal distribution (N2), having 1 2 ε ̄ instead of ε ̄ as parameter. These distributions are confirmed to satisfy the coincidence conditions for r→0, that is, N2→N1 for the velocity-sum distribution and N2→ δ for the velocity-difference distribution. Under the inertial similarity at which all viscous length-scales vanish, such changes of the distributions take place discontinuously for r→0. Such singular changes of the distributions for r→0 should be regularized by taking account of finite viscosity, but such a work will be dealt with in a separate paper. The inertial similarity of homogeneous isotropic turbulence associated with the normal one- and two-point velocity distributions seems to give good prospects for the extension of the present approach to inhomogeneous turbulent flows and more complex turbulent phenomena.