I apologize to others, but all other answers are wrong.

The reason for the error is that in reality there are with the name Boussinesq two great definitions. The first, that everyone is talking about, is called "the Boussinesq Approximation." In the text written by Singh Janamejay it is presented clearly and better understanding I suggest reading: M. Lesieur, Turbulence p36: 38.

However there is also the Boussinesq coefficient, an analogy made ​​in the nineteenth century by Boussinesq between the coefficient of molecular viscosity with a coefficient of turbulent viscosity (called turbulent viscosity, Coefficient of Boussinesq - epslon). (J. Boussinesq, 1877. Essai sur la théorie des eaux courantes, Memoires present par divers savants à l'Académie des Sciences XXIII: 1 1-680). How is an ancient text, and how the researchers of today does not tell of things that have fallen into disuse, no one realizes the difference.

To better understand what is this coefficient, and the problems that he has read: 2007 SCHMITT (SCHMITT, François G. 2007 About Boussinesq's turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity) in: http://www .fg-schmitt.fr/Publications_files/2007 20Schmitt%% 20CRAS.pdf.

Dear Deewakar Sharma, the text read: "Mixing length models in Boussinesq aproximation need modifications to damp the eddy viscosity to zero the flow approaches a solid wall and Also to account for the outer intermittent region", there is an error of the author. For this instead of writing "Boussinesq aproximation" should have written "coefficient of turbulent viscosity of Boussinesq". If written correctly, would see this as "turbulent viscosity coefficient" is variable depending on the distance to the wall (as in the cases of mixing length Prandtl and Kármán), where the "coefficient of turbulent viscosity" varies according to distance wall.

This article is about the Boussinesq approximation in buoyancy-driven flows. For other uses, see Boussinesq approximation (disambiguation).

In fluid dynamics, the Boussinesq approximation (pronounced: [businɛsk], named for Joseph Valentin Boussinesq) is used in the field of buoyancy-driven flow (also known as natural convection). It states that density differences are sufficiently small to be neglected, except where they appear in terms multiplied by g, the acceleration due to gravity. The essence of the Boussinesq approximation is that the difference in inertia is negligible but gravity is sufficiently strong to make the specific weight appreciably different between the two fluids. Sound waves are impossible/neglected when the Boussinesq approximation is used since sound waves move via density variations.

Boussinesq flows are common in nature (such as atmospheric fronts, oceanic circulation, katabatic winds), industry (dense gas dispersion, fume cupboard ventilation), and the built environment (natural ventilation, central heating). The approximation is extremely accurate for many such flows, and makes the mathematics and physics simpler.

The approximation's advantage arises because when considering a flow of, say, warm and cold water of density \rho_1 and \rho_2 one needs only consider a single density \rho: the difference \Delta\rho= \rho_1-\rho_2 is negligible. Dimensional analysis shows that, under these circumstances, the only sensible way that acceleration due to gravity g should enter into the equations of motion is in the reduced gravity g' where

g' = g{\rho_1-\rho_2\over \rho}.

(Note that the denominator may be either density without affecting the result because the change would be of order g(\Delta\rho/\rho)^2). The most generally used dimensionless number would be the Richardson number and Rayleigh number.

The mathematics of the flow is therefore simpler because the density ratio (\rho_1/\rho_2, a dimensionless number) does not affect the flow; the Boussinesq approximation states that it may be assumed to be exactly one

....source Wikipedia,

Please read the book "A first course in turbulence" by Tennekis and Lumely to clarify your question.

i think is another good book :

Turbulent Flows

AUTHOR: Stephen : B. Pope

I apologize to others, but all other answers are wrong.

The reason for the error is that in reality there are with the name Boussinesq two great definitions. The first, that everyone is talking about, is called "the Boussinesq Approximation." In the text written by Singh Janamejay it is presented clearly and better understanding I suggest reading: M. Lesieur, Turbulence p36: 38.

However there is also the Boussinesq coefficient, an analogy made ​​in the nineteenth century by Boussinesq between the coefficient of molecular viscosity with a coefficient of turbulent viscosity (called turbulent viscosity, Coefficient of Boussinesq - epslon). (J. Boussinesq, 1877. Essai sur la théorie des eaux courantes, Memoires present par divers savants à l'Académie des Sciences XXIII: 1 1-680). How is an ancient text, and how the researchers of today does not tell of things that have fallen into disuse, no one realizes the difference.

To better understand what is this coefficient, and the problems that he has read: 2007 SCHMITT (SCHMITT, François G. 2007 About Boussinesq's turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity) in: http://www .fg-schmitt.fr/Publications_files/2007 20Schmitt%% 20CRAS.pdf.

Dear Deewakar Sharma, the text read: "Mixing length models in Boussinesq aproximation need modifications to damp the eddy viscosity to zero the flow approaches a solid wall and Also to account for the outer intermittent region", there is an error of the author. For this instead of writing "Boussinesq aproximation" should have written "coefficient of turbulent viscosity of Boussinesq". If written correctly, would see this as "turbulent viscosity coefficient" is variable depending on the distance to the wall (as in the cases of mixing length Prandtl and Kármán), where the "coefficient of turbulent viscosity" varies according to distance wall.

Note : None of the books mentioned above employ the notation coefficient Boussinesq turbulent viscosity , but if you look in books of the decades of 20 and 30 will find this notation , mainly French and German books .

It is not to damp "viscosity" to zero. It is to damp "eddy viscosity" to zero. This is because on the wall, there is no turbulence instability and very close to the wall, aka viscous sublayer, the fluid viscosity effect must dominate over the turbulent instability.

Dear Shiuh-Hwa Shyu".

In the work of "Boussinesq turbulent viscosity coefficient" (1877), which was defined as "eddy viscosity" correspodia roughly speaking with the "eddy viscosity"". The concept of viscous sublayer is postulated for the first time in 1974 by Eckelmann (approximately 100 years after the work )". Therefore it is not possible to compare ratings. At that time _ "Boussinesq turbulent viscosity coefficient" did not exist the prior setting _ "laminar sublayer".

What I explained was the confusion between "Boussinesq's Hypothesis" and "Boussinesq turbulent viscosity coefficient" ". I know perfectly well as define the currently turbulent boundary layer, "Boussinesq turbulent viscosity coefficient".He is who did not know there 136 years ago!

Many people attribute different phenomena to Boussinesq. In particular there are three common used definitions. These are the Boussinesq approximation (1), the Boussinesq assumption (2) and the Boussinesq hypothesis (3).

1) In the Boussinesq Approximation the buoyancy force is taken only into account as an external body force, which is a function of temperature, in the momentum equation. In other terms the density is taken as a constant.

2) The Boussinesq Assumption usually refers to the model of eddy viscosity to model turbulent Reynols stresses as Newtonian viscosity is a model for laminar stresses.

3) The Boussinesq Hypothesis is usually the same as the 2nd phenomenon.

We clearly talk here about the 2nd and 3rd definition.

In this case, we talk about numerical approximations. Now at a wall we have the fact that the wall will not be penetrated in the normal direction and that the tangential gradient should be converging to a value associated to the laminar viscosity. Therefore the eddy viscosity should vanish approaching the wall.

Perhaps Sharma has to define what he means by the "book" he came across at the first time.

It is not viscosity that goes to zero, it is the "fictitious" Boussinesq Turbulence Viscosity ( NOT turbulent viscosity) that goes to zero as walls dampen turbulence.

Just to be clearer definition read the work of Boussinesq where he tries to make an analogy of the "turbulent viscosity coefficient" with the coefficient of viscosity, it is clear in your own text from the variability of the same, such that the coefficients epslon :

“doit dépendre em chaque point, non-seulement de la température e peut-ètre de la pression p, mais encore et surtout de l’intensité de l’agitation moyenne qui s’y trouve produite.”

The variability due to the pressure part of his hypothesis, he later described to the time that there was no correlation with that according to Darcy's experiments and others.

The full text of Boussinesqu can be found at: http://gallica.bnf.fr/ark:/12148/bpt6k56673076.r=.langEN and the highlighted text on page 46.

I can give two justifications. The first is that the EV term (nu_{turb} grad^{s} u ) models the Reynolds' stresses. Since the velocity vanishes at fixed solid walls the Reynolds stresses does as well and its replacement should also. This means turbulent viscosity vanishes at walls. Otherwise a modelling error occurs. If small it may be negligible. But since gradients are large at walls the modelling error would be huge. The second: If you want a mathematical justification from the Navier-Stokes equations see Article Damping Functions correct over-dissipation of the Smagorinsky Model