Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Reynolds Turbulence Solution

Version 1 : Received: 10 August 2019 / Approved: 12 August 2019 / Online: 12 August 2019 (05:39:11 CEST)
Version 2 : Received: 29 October 2019 / Approved: 30 October 2019 / Online: 30 October 2019 (04:49:17 CET)
Version 3 : Received: 1 December 2021 / Approved: 3 December 2021 / Online: 3 December 2021 (10:16:34 CET)

A peer-reviewed article of this Preprint also exists.

Sun, Bohua. "Revisiting the Reynolds-averaged Navier–Stokes equations" Open Physics, vol. 19, no. 1, 2021, pp. 853-862. Sun, Bohua. "Revisiting the Reynolds-averaged Navier–Stokes equations" Open Physics, vol. 19, no. 1, 2021, pp. 853-862.


The study found an error in current literature, including textbooks, about the number of unknowns in the Reynolds stress tensor and/or in Reynolds-averaged Navier-Stokes equations (RANS). Current literature claims that the Reynolds stress tensor has six unknowns; however, this article shows that the Reynolds stress tensor only has three unknowns, namely the three components of fluctuation velocity. This research discovers that the misconception about the number of unknowns in the RANS could stem from misinterpreting the Reynolds stress tensor. The misconception might be one of the biggest scientific mistake in classical physics and has hampered the development of turbulence for longtime. In order to find a way out of this difficult situation, we return to the time of Reynolds in 1895 and revisit Reynolds' averaging formulation of turbulence. In light of Reynolds' deterministic view on turbulence, this paper proposes a general algorithm for three dimensional turbulence flows. The study found that the magnitude of velocity fluctuations or turbulence is proportional to the flow pressure, which is a remarkable discovery. As applications, the Reynolds turbulence solution of the turbulent Burgers equation and the Prandtl boundary layer equations have been obtained, the beauty of these relevant solutions is that there is no adjustable parameters. The present investigation can be considered as a renaissance of Reynolds' work in 1895, which might shed light on the well-known closure problem of turbulence, and help to understand the puzzle of the turbulence closure problem that has eluded scientists and mathematicians for centuries.


turbulence; number of unknowns; the Reynolds stress tensor; RANS; turbulence closure problem


Physical Sciences, Fluids and Plasmas Physics

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