Abstract
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.
Highlights
Turbulence is one of the greatest unsolved mysteries of physics
This study addresses a basic problem in turbulence analysis, namely, the number of unknowns in the Reynolds stress tensor, which is obviously a fundamental question in fluid mechanics
The Reynolds-averaged Navier– Stokes (RANS) equations were formulated more than 120 years ago, much of the existing literature and standard turbulence textbooks are incorrect regarding the number of unknowns in the RANS, and this has prevented the turbulence problem from being solved
Summary
This study addresses a basic problem in turbulence analysis, namely, the number of unknowns in the Reynolds stress tensor, which is obviously a fundamental question in fluid mechanics. The research reveals that the Reynolds stress tensor has only three independent unknowns instead of the six stated in the existing literature, including textbooks. The RANS equations were formulated more than 120 years ago, much of the existing literature and standard turbulence textbooks are incorrect regarding the number of unknowns in the RANS, and this has prevented the turbulence problem from being solved. The aims are to revisit Reynolds’ averaging formulation of turbulence and clarify the number of independent unknowns in the Reynolds stress tensor and/or RANS equations. We raise an important question about the number of independent unknowns in both the Reynolds stress tensor and RANS equations, and provide three mathematical. We conclude with perspectives on future developments in turbulence research
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