scholarly journals On closure problem of incompressible turbulent flow

2018 ◽  
Author(s):  
BOHUA SUN
Keyword(s):  
Turbulent Flow ◽  
Stress Tensor ◽  
Reynolds Stress ◽  
Unsolved Problem ◽  
Incompressible Flows ◽  
Closure Problem ◽  
Fluctuation Velocity ◽  
Reynolds Stress Tensor ◽  
Three Components

This paper attempts to clarify an issue regarding the lasting unsolved problem of turbulence, namely the closure problem. This study shows that all perspectives regarding the numbers of un- known quantities in the Reynolds turbulence equations stem from the misunderstandings of physics of the Reynolds stress tensor. The current literatures have a consensus that the Reynolds stress tensor has six unknowns; however, this study shows that the Reynolds stress tensor actually has only three ones, namely the three components of fluctuation velocity. With this new understanding, the closed turbulence equations for incompressible flows are proposed.

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 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 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.

Experimental study of particle-driven secondary flow in turbulent pipe flows

2012 ◽  
Vol 709 ◽  
pp. 1-36 ◽  
Author(s):  
R. J. Belt ◽  
A. C. L. M. Daalmans ◽  
L. M. Portela
Keyword(s):  
Turbulent Flow ◽  
Secondary Flow ◽  
Stress Tensor ◽  
Reynolds Stress ◽  
Cross Section ◽  
Circular Pipe ◽  
Horizontal Pipe ◽  
Pipe Cross Section ◽  
Reynolds Stress Tensor ◽  
Pipe Flows

AbstractIn fully developed single-phase turbulent flow in straight pipes, it is known that mean motions can occur in the plane of the pipe cross-section, when the cross-section is non-circular, or when the wall roughness is non-uniform around the circumference of a circular pipe. This phenomenon is known as secondary flow of the second kind and is associated with the anisotropy in the Reynolds stress tensor in the pipe cross-section. In this work, we show, using careful laser Doppler anemometry experiments, that secondary flow of the second kind can also be promoted by a non-uniform non-axisymmetric particle-forcing, in a fully developed turbulent flow in a smooth circular pipe. In order to isolate the particle-forcing from other phenomena, and to prevent the occurrence of mean particle-forcing in the pipe cross-section, which could promote a different type of secondary flow (secondary flow of the first kind), we consider a simplified well-defined situation: a non-uniform distribution of particles, kept at fixed positions in the ‘bottom’ part of the pipe, mimicking, in a way, the particle or droplet distribution in horizontal pipe flows. Our results show that the particles modify the turbulence through ‘direct’ effects (associated with the wake of the particles) and ‘indirect’ effects (associated with the global balance of momentum and the turbulence dynamics). The resulting anisotropy in the Reynolds stress tensor is shown to promote four secondary flow cells in the pipe cross-section. We show that the secondary flow is determined by the projection of the Reynolds stress tensor onto the pipe cross-section. In particular, we show that the direction of the secondary flow is dictated by the gradients of the normal Reynolds stresses in the pipe cross-section, $\partial {\tau }_{rr} / \partial r$ and $\partial {\tau }_{\theta \theta } / \partial \theta $. Finally, a scaling law is proposed, showing that the particle-driven secondary flow scales with the root of the mean particle-forcing in the axial direction, allowing us to estimate the magnitude of the secondary flow.

Anisotropy Invariants of Reynolds Stress Tensor in a Duct Flow and Turbulent Boundary Layer

1998 ◽  
Vol 120 (2) ◽  
pp. 280-284 ◽  
Author(s):  
A. Mazouz ◽  
L. Labraga ◽  
C. Tournier
Keyword(s):  
Surface Roughness ◽  
Turbulent Flow ◽  
Stress Tensor ◽  
Reynolds Stress ◽  
Pipe Flow ◽  
Duct Flow ◽  
Reynolds Stress Tensor ◽  
Entire Thickness ◽  
Anisotropy Tensor ◽  
Flow Turbulence

The present study shows that the Reynolds stress anisotropy tensor for turbulent flow depends both on the nature of the surface and the boundary conditions of the flow. Contrary to the case of turbulent boundary layers with k-type surface roughness, the measured anisotropy invariants of the Reynolds stress tensor over a series of spanwise square bars separated by rectangular cavities (k-type) in duct flows show that roughness increases the anisotropy. There is a similarity between the effect of roughness on channel flow turbulence and that on pipe flow turbulence. The present data show that the effect of introducing a surface roughness significantly perturbs the entire thickness of the turbulent flow.

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.

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