Revisiting the Reynolds-averaged Navier–Stokes equations

Abstract This study revisits the Reynolds-averaged Navier–Stokes (RANS) equations and finds that the existing literature is erroneous regarding the primary unknowns and the number of independent unknowns in the RANS. The literature claims that the Reynolds stress tensor has six independent unknowns, but in fact the six unknowns can be reduced to three that are functions of the three velocity fluctuation components, because the Reynolds stress tensor is simply an integration of a second-order dyadic tensor of flow velocity fluctuations rather than a general symmetric tensor. This difficult situation is resolved by returning to the time of Reynolds in 1895 and revisiting Reynolds’ averaging formulation of turbulence. The study of turbulence modeling could focus on the velocity fluctuations instead of the Reynolds stress. An advantage of modeling the velocity fluctuations is, from both physical and experimental perspectives, that the velocity fluctuation components are observable whereas the Reynolds stress tensor is not.