scholarly journals Reynolds' Turbulence Solution

2021 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Stress Tensor ◽  
Reynolds Stress ◽  
Velocity Fluctuation ◽  
Turbulence Modeling ◽  
Stokes Equations ◽  
Symmetric Tensor ◽  
Navier Stokes ◽  
Difficult Situation ◽  
Velocity Fluctuations ◽  
Reynolds Stress Tensor

This study revisits the Reynolds-averaged Navier--Stokes equations (RANS) 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 on 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.

scholarly journals Revisiting the Reynolds-averaged Navier–Stokes equations

Open Physics ◽  
2021 ◽  
Vol 19 (1) ◽  
pp. 853-862
Author(s):  
Bohua Sun
Keyword(s):  
Stress Tensor ◽  
Reynolds Stress ◽  
Velocity Fluctuation ◽  
Stokes Equations ◽  
Symmetric Tensor ◽  
Navier Stokes ◽  
Difficult Situation ◽  
Velocity Fluctuations ◽  
Reynolds Stress Tensor ◽  
Reynolds Averaged Navier Stokes

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.

scholarly journals Reynolds' Turbulence Solution

2019 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Stress Tensor ◽  
Reynolds Stress ◽  
Current Literature ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Closure Problem ◽  
Navier Stokes Equations ◽  
Difficult Situation ◽  
Reynolds Stress Tensor ◽  
Shed Light

The study found an error in current literature, including numerous textbooks, about the number of independent 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 independent three unknowns, which are functions of the three components of fluctuation velocity. This research discovers that the misconception about the number of independent unknowns in the RANS could stem from misinterpreting the Reynolds stress tensor. The misconception 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. 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 more than a century.

scholarly journals Reynolds Turbulence Solution

2019 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Stress Tensor ◽  
Reynolds Stress ◽  
Current Literature ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Closure Problem ◽  
Difficult Situation ◽  
Reynolds Stress Tensor ◽  
Boundary Layer Equations

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.

scholarly journals On the Reynolds-Averaged Navier-Stokes Equations

2019 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Stress Tensor ◽  
Reynolds Stress ◽  
Current Literature ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Closure Problem ◽  
Navier Stokes Equations ◽  
Fluctuation Velocity ◽  
Reynolds Stress Tensor ◽  
Reynolds Averaged Navier Stokes

This paper attempts to clarify an long-standing issue about the number of unknowns in the Reynolds-Averaged Navier-Stokes equations (RANS). This study shows that all perspectives regarding the numbers of unknowns in the RANS stem from the misinterpretation of the Reynolds stress tensor. The current literature consider that the Reynolds stress tensor has six unknown components; however, this study shows that the Reynolds stress tensor actually has only three unknown components, namely the three components of fluctuation velocity. This understanding might shed a light to understand the well-known closure problem of turbulence.

scholarly journals A Novel Simplification of the Reynolds-Chou-Navier-Stokes Turbulence Equations of Incompressible Flow

2018 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Velocity Field ◽  
Stress Tensor ◽  
Reynolds Stress ◽  
Incompressible Flow ◽  
Velocity Fluctuation ◽  
Explicit Representation ◽  
Navier Stokes ◽  
Mean Velocity ◽  
The Mean ◽  
Turbulence Formulations

Based on author's previous work [Sun, B. The Reynolds Navier-Stokes Turbulence Equations of Incompressible Flow Are Closed Rather Than Unclosed. Preprints 2018, 2018060461 (doi: 10.20944/preprints201806.0461.v1)], this paper proposed an explicit representation of velocity fluctuation and formulated the Reynolds stress tensor in terms of the mean velocity field. The proposed closed Reynolds Navier-Stokes turbulence formulations reveal that the mean vorticity is the key source of producing turbulence.

scholarly journals An Intrinsic Formulation of Incompressible Navier-Stokes Turbulent Flow

2019 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Velocity Field ◽  
Turbulent Flow ◽  
Stress Tensor ◽  
Reynolds Stress ◽  
Velocity Fluctuation ◽  
Navier Stokes ◽  
Natural Consequence ◽  
Mean Velocity ◽  
Turbulence Transition ◽  
The Mean

This paper proposed an explicit and simple representation of velocity fluctuation and the Reynolds stress tensor in terms of the mean velocity field. The proposed turbulence equations are closed. The proposed formulations reveal that the mean vorticity is the key source of producing turbulence. It is found that there are no velocity fluctuation and turbulence if there were no vorticity. As a natural consequence, the laminar- turbulence transition condition was obtained in a rational way.

scholarly journals An Intrinsic Formulation of Incompressible Navier-Stokes Turbulent Flow

2018 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Velocity Field ◽  
Turbulent Flow ◽  
Stress Tensor ◽  
Reynolds Stress ◽  
Velocity Fluctuation ◽  
Navier Stokes ◽  
Natural Consequence ◽  
Mean Velocity ◽  
Turbulence Transition ◽  
The Mean

This paper proposed an explicit and simple representation of velocity fluctuation and the Reynolds stress tensor in terms of the mean velocity field. The proposed turbulence equations are closed. The proposed formulations reveal that the mean vorticity is the key source of producing turbulence. It is found that there are no velocity fluctuation and turbulence if there were no vorticity. As a natural consequence, the laminar- turbulence transition condition was obtained in a rational way.

scholarly journals Closed integral-differential equations of incompressible Navier-Stokes turbulent flow

2018 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Stress Tensor ◽  
Reynolds Stress ◽  
Incompressible Flows ◽  
Navier Stokes ◽  
Mean Velocity ◽  
Fluctuation Velocity ◽  
Reynolds Stress Tensor ◽  
Mean Pressure ◽  
Integral Differential Equations ◽  
Three Components

This paper showed that turbulence closure problem is not an issue at all. All mistakes in theliterature regarding the numbers of unknown quantities in the Reynolds turbulence equations stemfrom the misunderstandings of physics of the Reynolds stress tensor, i.e., all literature has statedthat the symmetric Reynolds stress tensor has six unknowns; however, it actually has only threeunknowns, i.e., the three components of fluctuation velocity. We showed the integral-differentialequations of the Reynolds mean and fluctuation equations have exactly eight equations, which equalto the numbers of quantities in total, namely, three components of mean velocity, three componentsof fluctuation velocity, one mean pressure and one fluctuation pressure. With this understanding,the closed Reynolds Navier-Stokes turbulence equations of incompressible flows were formulated.This study may help to solve the puzzle that has eluded scientists and mathematicians for centuries.

scholarly journals The Reynolds Navier-Stokes Turbulence Equations of Incompressible Flow Are Closed Rather Than Unclosed

2018 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Stress Tensor ◽  
Reynolds Stress ◽  
Incompressible Flow ◽  
Navier Stokes ◽  
Mean Velocity ◽  
Fluctuation Velocity ◽  
Reynolds Stress Tensor ◽  
Mean Pressure ◽  
Integral Differential Equations ◽  
Three Components

This paper shown that turbulence closure problem is not an issue at all. All mistakes in the literature regarding the numbers of unknown quantities in the Reynolds turbulence equations stem from the misunderstandings of physics of the Reynolds stress tensor, i.e., all literatures have stated that the symmetric Reynolds stress tensor has six unknowns; however, it actually has only three unknowns, i.e., the three components of fluctuation velocity. We shown the integral-differential equations of the Reynolds mean and fluctuation equations have exactly eight equations, which equal to the numbers of quantities in total, namely, three components of mean velocity, three components of fluctuation velocity, one mean pressure and one fluctuation pressure. That is why we claim in this paper, that the Reynolds Navier-Stokes turbulence equations of incompressible flow are closed rather than unclosed. This study may help to solve the puzzle that has eluded scientists and mathematicians for centuries.

scholarly journals The Reynolds Navier-Stokes Turbulence Equations of Incompressible Flow Are Closed Rather Than Unclosed

2018 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Stress Tensor ◽  
Reynolds Stress ◽  
Incompressible Flow ◽  
Navier Stokes ◽  
Mean Velocity ◽  
Fluctuation Velocity ◽  
Reynolds Stress Tensor ◽  
Mean Pressure ◽  
Integral Differential Equations ◽  
Three Components

This paper shown that turbulence closure problem is not an issue at all. All mistakes in the literature regarding the numbers of unknown quantities in the Reynolds turbulence equations stem from the misunderstandings of physics of the Reynolds stress tensor, i.e., all literatures have stated that the symmetric Reynolds stress tensor has six unknowns; however, it actually has only three unknowns, i.e., the three components of fluctuation velocity. We shown the integral-differential equations of the Reynolds mean and fluctuation equations have exactly eight equations, which equal to the numbers of quantities in total, namely, three components of mean velocity, three components of fluctuation velocity, one mean pressure and one fluctuation pressure. That is why we claim in this paper, that the Reynolds Navier-Stokes turbulence equations of incompressible flow are closed rather than unclosed. This study may help to solve the puzzle that has eluded scientists and mathematicians for centuries.

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