On the role of conditional averages in turbulence theory.

It is shown that conditional averages in the form of expected values of functions of the velocity at an arbitrary point given the velocities at a finite number of distinc points, appear naturally in certain types of turbulence theories and that the closure problems in such theories ultimately reduce to the approximation of these averages. Two examplary theories are considered. The first is characteristic of turbulence models formulated in terms of probability density functions whereas the second is related to the derivation of optimal algorithms for the numerical integration of the turbulent Navier-Stokes equations at large Reynolds numbers. Some mathematical properties of conditional expected values, including relations between conditional and unconditional second order tensor moments and results for the special case of isotropic turbulence are also presented.