Possible origin of Prandt's mixing-length theory

Abstract

PRANDTL'S hypothesis^1,2 about turbulent motion in a simple shear layer proposes that the typical values of the fluctuating velocity components in the x and y directions, u and v, are each proportional to lδU/δy where l is the mixing length (Mischungsweg). Prandtl^2,3 says that l “may be considered as the diameter of the masses of fluid moving as a whole in each individual case; or again, as the distance traversed by a mass of this type before it becomes blended in with neighbouring masses….”; and also that l is “somewhat similar, as regards effect, to the mean free path in the kinetic theory of gases”. It follows that the Reynolds shear stress −ρuv is proportional to ρl²(δU/δy)² and l is defined so that the constant of proportionality is unity. Prandtl, however, described this expression as “only a rough approximation”.