Statistical theory of turbulenc

Since the time of Osborne Reynolds it has been known that turbulence produces virtual mean stresses which are proportional to the coefficient of correlation between the components of turbulent velocity at a fixed point in two perpendicular directions. The significance of correlation between the velocity of a particle at one time and that of the same particle at a later time, or between simultaneous velocities at two fixed points was discussed in 1921 by the present writer in a theory of “Diffusion by Continuous Movements.” The recent improvements in the technique of measuring turbulence have made it possible actually to measure some of the quantities envisaged in the theory and thus to verify some of the relationships then put forward. The theory has also been developed in several directions which were not originally contemplated. The theory, as originally put forward, provided a method for defining the scale of turbulence when the motion is defined in the Lagrangian manner, and showed how this scale is related to diffusion. It is now shown that it can be applied either to the Lagrangian or to the Eulerian conceptions of fluid flow.