The Reynolds Navier-Stokes turbulence equations of incompressible flow are closed rather than unclosed

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.

The study of turbulence is not simple owing to its complex and forbidding mathematical descriptions, as well as the profound difficulties of inherent instabilities and even chaotic processes.People believe that turbulence prediction can be attained by understanding solutions to Navier-Stokes equations.However, understanding of the Navier-Stokes equations remains minimal, while there is still surprisingly little that can be predicted with relative certainty [15,17,18].In respect of the turbulence problem, a myriad of tentative theories have been proposed, each with its own doctrines and beliefs, whilst often focused on particular experiments; however, there is not much in the way of a coherent theoretical framework [1][2][3][4][5][6][7][8][10][11][12][13][14][15][16].Turbulence is a unique subject that engineers, mathematicians, and physicists tend to view in rather different ways.Many engineers promote the use of semi-empirical models of turbulence, while mathematicians advocate the use of purely statistical models [19][20][21][22][23][24], and the formulism of chaos theory and fractals [25][26][27].
In 1972 a new chapter was launched in turbulence theory: Orszag and Patterson demonstrated that it was possible to perform direct numerical simulation (DNS) of a fully turbulent flow [28].It is important to understand that DNS does not require any turbulence model to parameterize influence of the turbulent eddies.Rather, every eddy, from the largest to the smallest, is computed.Technically speaking, the turbulence can be solved by DNS if computers have infinite speed.However, a huge chasm remains between what the engineer needs to know, and what can be realized by DNS, using current computers.Even if DNS can assist to solve turbulence issues and problems, one still requires turbulence modelling to acquire a physical understanding of it.
In 1895 Reynolds published a seminal work on turbulence [29], in which he proposed that flow velocity u and pressure p are decomposed into its timeaveraged quantities, ū, t, p, and fluctuating quantities, u ′ , p ′ ; thus, the Reynolds decompositions are: u = ū(x, t) + u ′ (x, t) and p(x, t) = p(x, t) + p ′ (x, t), where coordinates and times are (x, t).With decomposition the Navier-Stokes equation is then transformed into Reynolds-averaged Navier-Stokes equations (or RAN-S equations), where the Reynolds stress tensor τ where T is the period of time over which the averaging takes place and must be sufficiently large to give meaningful averages.Reynolds stress is apparent stress owing to the fluctuating velocity field u ′ .Since introduction of the Reynolds stress tensor, the closure problem of turbulence, namely the Reynolds equations are unclosed, has eluded scientists and mathematicians for centuries.The Reynolds equations can not be solved unless some additional restrictions are somehow determined.
As we know, the Navier-Stokes momentum equation is ρu ,t + ∇ • Π = 0, continuity equation of incompressible flow is ∇ • u = 0, where the energy-moementum tensor given by Π = pI + ρu ⊗ u − µ(∇u + u∇), dynamic viscosity µ, gradient operator ∇ = e i ∂ i , base vector in the i-coordinate e i , and tensor product ⊗ .By introducing the Reynolds decomposition and averaging operation, we have the Reynolds equations and continuity equation of the mean velocity as follows, respectively: For a general three-dimensional flow, there are four independent equations governing the mean velocity field; namely three components of the Reynolds equations together with one mean continuity equation.However, these four equations contain more than four unknowns.In addition to ū and p (four quantities), there are also the Reynolds stresses.The Reynolds equations are unclosed.This is a manifestation of the closure problem.
In 1940, P.-Y.Chou presented another approach [30,31], who pointed out that because the Navier-Stokes equations are the basic dynamical equations of fluid motion, it is insufficient to consider only the mean turbulent motion.The turbulent fluctuations are as important as the mean motion and the equations for turbulent fluctuations also need to be considered.
Subtracting the mean motions from the Navier-Stokes equation and continuity equation, Chou [30,31] obtained the equations of the turbulence fluctuations ρ ū′ ,t + ρ∇ and ∇ • u ′ = 0.After having the above fluctuation equations, Chou [30,31] introduced hierarchy of equations for velocity fluctuation correlations, however, any velocity correlation equation of a given order always obtains an unknown velocity correlation of one higher order, i.e. all hierarchy are still not closed.
Regarding the closure issue, here we have a completely different perspectives, namely, the Reynolds mean and fluctuation turbulence equations are closed and no additional restrictions are needed.The so-called turbulence closure problem is not an issue at all.
To support this claim, let's re-writing the equations of mean and fluctuation motions as follows denoting kinematic viscosity ν = µ/ρ, the above integral-differential equations can be expressed equivalently as Amazingly the integral-differential equations in Eqs.(1,2,3,4) and or Eqs.(5,6,7,8) have exactly eight equations, which equal to the numbers of unknowns, namely, three components of mean velocity ū, three components of fluctuation velocity u ′ , one mean pressure p and one fluctuation pressure p ′ .That is why the integraldifferential equations system of Eqs.(1,2,3,4)and or Eqs.(5,6,7,8) are closed !All mistakes regarding the numbers of unknown quantities stem from misunderstandings of the formulation of Reynolds stress tensor, i.e., all literatures report the symmetric Reynolds stress tensor has six unknowns; however, the Reynolds stress tensor actually is not a general tensor but the product of fluctuation velocity, has only three unknowns, namely the three components of fluctuation velocity u ′ .
This can be proved easily as follows: the Reynolds stress tensor can be defined by and fluctuation velocity convective terms The formulations in Eq.(9,10) reveal that the Reynolds stress tensor τ = −ρu ′ ⊗ u ′ and fluctuation velocity con-vective terms can be calculated by the three components of fluctuation velocity u ′ 1 , u ′ 2 , u ′ 3 , hence the Reynolds stress tensor has only three unknowns.Therefore, our statement can be written as follows: The Reynolds Navier-Stokes turbulence equations of incompressible flow in Eqs.(1,2,3,4) are closed rather than unclosed.
It is my great pleasure to have shared and discussed some of the above with Michael Sun from Bishops Diocesan College, whose pure and direct scientific sense inspired me.