Finite-Reynolds-number effects in turbulence using logarithmic expansions

Abstract

Experimental or numerical data in turbulence are invariably obtained at finite Reynolds numbers whereas theories of turbulence correspond to infinitely large Reynolds numbers. A proper merger of the two approaches is possible only if corrections for finite Reynolds numbers can be quantified. This paper heuristically considers examples in two classes of finite-Reynolds-number effects. Expansions in terms of logarithms of appropriate variables are shown to yield results in agreement with experimental and numerical data in the following instances: the third-order structure function in isotropic turbulence, the mixed-order structure function for the passive scalar and the Reynolds shear stress around its maximum point. Results suggestive of expansions in terms of the inverse logarithm of the Reynolds number, also motivated by experimental data, concern the tendency for turbulent structures to cluster along a line of observation and (more speculatively) for the longitudinal velocity derivative to become singular at some finite Reynolds number. We suggest an elementary hydrodynamical process that may provide a physical basis for the expansions considered here, but note that the formal justification remains tantalizingly unclear.