Exploring unresolved inquiries regarding the meaning of Reynolds averaging and decomposition: A review

Abstract

In the late 19th century, Osborne Reynolds published two papers whose impact on atmospheric turbulence studies can hardly be overstated. The first, Reynolds (1883) established both his eponymous, dimensionless number and his reputation as the father of turbulence science, which is beyond doubt. However, his second famous paper (Reynolds, 1895) sowed seeds of confusion regarding the mathematical separation of average (mean) and fluctuating (turbulent) components of a fluid flow. Here, we revisit both the prehistory and aftereffects of Reynolds’s second famous article, which seems to have been published largely thanks to his already entrenched reputation. We show that successions of authors have misrepresented Reynolds’s innovations – now known as Reynolds averaging and decomposition (RAAD) –, putting his name to methodologies that he never intended. We attribute this, in part, to Reynolds’s predilection for long, inscrutable sentences, as well as his self-contradiction regarding the methodology for averaging the normal stress (or pressure). We examine two additional issues that are intimately related to using RAAD to define turbulent fluxes, namely its application to intensive versus extensive variables and the appearance of “Leonard terms” in the averaged equation of motion, neither of which is completely resolved. Throughout the manuscript, we identify a set of unanswered questions concerning RAAD and conclude that a complete mathematical description of turbulence is unlikely to emerge without addressing these issues, including the original inconsistency that was introduced by Osborne Reynolds himself.