Lagrangian-History Closure Approximation for Turbulence

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Abstract
The direct-interaction approximation for turbulence is extended to predict the covariance and average Green's function of a generalized velocity u(x, t|r), defined as the velocity measured at time r in the fluid element which passes through the point x at time t. The resulting formulas for triple moments involve integrals over the Eulerian time-history of the fluid. The approximation is then altered so that the integrals are instead over Lagrangian histories, measured along the particle paths. The alteration is determined by requiring simultaneously that energy be conserved; that there exist formal inviscid equipartition solutions; and that the Lagrangian dynamics of homogeneous flows exhibit invariance under random Galilean transformations. In the altered equations, the relaxation times associated with energy transfer are Lagrangian memory times determined by the viscous and pressure forces. As a result, the approximation implies the Kolmogorov inertial- and dissipation-range laws. The corresponding approximation for convection of a passive scalar field yields some exact results of Taylor and implies Richardson's law for the relative diffusion of two particles.