The Approach of Turbulence to the Locally Homogeneous Asymptote as   Studied using Exact Structure-Function Equations
  Equations that follow from the Navier-Stokes equation and incompressibility but with no other approximations are called "exact" here. Exact equations relating 2nd and 3rd-order structure functions are obtained, as is an exact incompressibility condition on the 2nd-order velocity structure function. Ensemble, temporal, and spatial averages are all considered because they produce different statistical equations. Exact equations have particularly simple forms for the cases of an average over a sphere in r-space as well as DNS that has periodic boundary conditions. The trace mitigates the effect of anisotropy in the equations, thereby revealing that the trace of the third-order structure function is expected to be superior for quantifying asymptotic scaling laws. The midpoint and the difference of the two points at which the hydrodynamic quantities are obtained are X and r; t is time. Dependencies on X and on the orientation of r and on t fade as the asymptotic statistical states of local homogeneity, local isotropy, and local stationarity, respectively, are approached. The exact equations are thus applicable to study of the approach toward those asymptotic states. A new definition of local homogeneity is contrasted with previous definitions. The approach toward the asymptotic state of local homogeneity is studied by using scale analysis to determine the required approximations and the approximate equations pertaining to experiments and simulations of the small-scale structure of high-Reynolds-number turbulence, but without invoking local isotropy. Those equations differ from equations for homogeneous turbulence. 
Reginald J. Hill