  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.
