Gravity is disregarded not only for the BL flow model but for many flow problems characterized by a specific analysis of the magnitude order of each term in the momentum equations. For example in aerodynamics problem where the characteristic driven velocity (and the convective flux of the momentum) is much more relevant.

Gravity can act by the action of buoyancy but that requires the the non-dimensional numbers (Grashof, Rayleigh) show that it really has effect.

When there is a forced convection you can see that the gravity effects are disregardable as they act according to Gr/Re^2.

Gravity is disregarded not only for the BL flow model but for many flow problems characterized by a specific analysis of the magnitude order of each term in the momentum equations. For example in aerodynamics problem where the characteristic driven velocity (and the convective flux of the momentum) is much more relevant.

Gravity can act by the action of buoyancy but that requires the the non-dimensional numbers (Grashof, Rayleigh) show that it really has effect.

When there is a forced convection you can see that the gravity effects are disregardable as they act according to Gr/Re^2.

As noted by Flippo Maria Denaro, if buoyancy effects are negligible, then gravity effects are unimportant on the development of a boundary layer on a flat plate when gravity acts perpindicular to the flow direction. One might ask what happens when gravity acts in the direction of the flow, as would occur for the boundary layer development on a vertical plate. Well this problem has been solved by Ackerberg in 1968 (see Physics of Fluids Vol 11, p1278-1291, 1968). The solution is developed as an asymptotic expansion in terms of the small parameter \epsilon=(Fr/Re)^1/2. The analysis shows that the solution becomes indistinguishable from the solution when \epsilon =0, a short distance downstream. At this point the external inviscid flow is absorbed into the boundary layer.

As noted by Flippo Maria Denaro, if buoyancy effects are negligible, then gravity effects are unimportant on the development of a boundary layer on a flat plate when gravity acts perpindicular to the flow direction. One might ask what happens when gravity acts in the direction of the flow, as would occur for the boundary layer development on a vertical plate. Well this problem has been solved by Ackerberg in 1968 (see Physics of Fluids Vol 11, p1278-1291, 1968). The solution is developed as an asymptotic expansion in terms of the small parameter epsilon=(Fr/Re)^1/2. The analysis shows that the solution becomes indistinguishable from the solution when epsilon =0, a short distance downstream. At this point the external inviscid flow is absorbed into the boundary layer.

Actually, gravity is not neglected. In CFD, usually the gravitational term is not included and the pressure we get in the solution is subtracted from a reference state of hydrostatic equilibrium. This allows for additional accuracy.

https://www.sciencedirect.com/search?qs=gravity%20neglected%20for%20boundary%20layer%20over%20flat%20plate&show=25&sortBy=relevance

Depending on the application, the magnitude of the gravity effect can be negligible. As noted in the other comments, sometimes it cannot be neglected. In typical pipe flow applications, the effect is insignificant in relation to to other factors, not to mention other the overall uncertainty of the measurements. You'll note that in most papers that do address the gravity effect focus on flat plates, where it can make a difference; also, the limitations of CFD make the calculations much simpler for flat plates than for conduits with circular cross sections.

In modern fluid mechanics, based on a system of fundamental equations, the flow around obstacles for four types of fluids - strong and weakly stratified and homogeneous (potentially and actual) is investigated (for more details, see Zagumennyi Ya. V., Chashechkin Yu. D. Pattern of unsteady vortex flow around plate under a zero angle of attack (two-dimensional problem) // Fluid Dynamics, 2016, Vol. 51, No. 3, pp. 53-70. DOI: 10.7868 / S0568528116030). When studying the flows of an inhomogeneous fluid (stratified strong or weakly and potentially homogeneous), the equations contain term take into account the action of the gravitational force

( Chashechkin Yu. D., Zagumennyi Ya. V., Dimitrieva N.F. Dynamics of the formation and fine structure of the flow pattern in the laboratory and computational experiment // Supercomputing: Second Russian Supercomputing Days (RuSCDays 2016). International conference. Moscow, Russia, September 26-27, 2016. Revised Selected Papers. V. Voevodin and S. Sobolev (Eds.). Communications in computer and information science 687. Springer International Publishing AG 2016. P. 41-56. V. Voevodin and S. Sobolev (Eds.): RuSCDays 2016, CCIS 687, pp. 41-56, 2016. DOI: 10.1007 / 978-3-319-55669-7_4

Chashechkin Yu. D., Zagumennyi I. V., Dimitrieva N. F. Unsteady Vortex Dynamics Past a Uniformly Moving Tilted Plate // Topical Problems of Fluid Mechanics 2018, Prague, February 21-23, 2018. Proceedings. P. 47 - 56. Edited by Šimurda D., Bodnar T .: 2017. 350 p. DOI: https://doi.org/10.14311/TPFM.2018.007

ISBN 978-80-87012-65-9, ISSN 2336-5781.)

The results of numerical simulation of two-dimensional problems for different liquids are consistent between themselves, with analytical calculations of internal waves and data of laboratory experiments.

Yu.D.Chashechkin

Hi Saeb

Notice, the pdf text [edit] in the original message gives the clue: “Using an order of magnitude analysis …”.

Body forces, like the gravitational, are proportional to the volume (or mass) of material involved. In general boundary layers are thin, so that the volume (or mass) of fluid in it is small compared to the volume (or mass) of fluid in motion outside the boundary layer, where inertial forces, that is associated with fluid accelerations, predominates (e. g., flow around blunt bodies). For this reason, body forces within boundary layers are disregarded as compared to the other forces present. However, for a large wall surface (where the boundary layer is attached) and for very viscous liquids flowing at low Reynolds number, may be the boundary layer will grow enough in thickness so that body forces could, perhaps, become comparable to other forces.

@R. P. Peçanha, where can I find the pdf file?!

Hi Saeb

I referred to the text highlighted in blue (order of magnitude analysis) on the pdf bit included in your own message (Boundary layer equations [Edit]), right below the flat plate drawing !

Thanks.

Sad to say, if you ask studenst to define Reynold's Number most will answer density x velocityx length over viscosity. It should be the ratio of the inertial force over the viscous force...so that explain why in flow problem the gravitional force is ignored. My advice is remember the basics

In the problem of flow without any heat transfer, the BL problem is simplified by ignoring gravity. There are forces of fluid weight acting on the fluid particles or control volume but at the same time the other forces of convective forces and viscosity forces are much greater than the weight forces. So in laminar flow there is no mixing between the upper layers and lower layers of fluid all of them moving together. But in turbulent flow gravity effect may overcome the viscosity effect, so turbulence model (e.g. k-epsilon Model) should be included in the solution which is a composition of many factors one of them is gravity, but still gravity in turbulent flow couldn't be solved individually.

Not sure to understand why gravity should be included in turbulent regime..

The Navier-Stokes equations are the same, the BCs. are the same, so I don't see the issue.

Gravity term is not included in the equation of Navier-Stock for turbulent flow. I want to say that the effect of gravity however is not included in the Navier-Stock equation but its effect is combined in the turbulence model.

If the gravity does not enter into the Navier-Stokes equations (is neglected in the momentum) how do you think it enters into its derived turbulence model?

Please, show with a clear demonstration.

Mr Akram,

You have provided three links in reply to the original question. However, I do not understand how these links help in answering that question. Would you please help me to understand? All that I can see is that the various authors state that gravity has been neglected but there is no statement about why it has been neglected.

@Mohamad Afifi. You are right. These are basics. Unfortunately during years of education, I've been exposed to a wide range of topics, from missile guidance to composite plate vibration and fluid mechanics and I've had some contributions in these separate fields thus far. As this is the case, I am kidding to tell, "I've been jack of all trades!"

As far as I know, including gravity doesn't not depend on flow being laminar or turbulent. It depends on the type of flow. NS equations consists of different terms like convective, viscous and body forces (gravity, electrostatics etc). Depending on the order of magnitude analysis, different terms can be dropped for different types of flow, as mentioned by Fillipo earlier. For example, convective term can be dropped for creeping flow, gravity term can be dropped for boundary layer flows etc. One such flow where gravity terms are included is simulation of Rayleigh-Taylor instability.

Hope this helps.

Anant, you are right and coming back to the BL question, the gravity term has a non-dimensional number that is in the form Gr/Re^2. Therefore, since the flow over a flat plate has a Reynolds number depending on the position x along the plate, one can estimate the extension Xs for which, given a proper value Gr, the term Gr/(Re_x)^2 is meaningful when compared to the O(1) coefficient in the other terms of the momentum. An estimation is

Re_x (Xs)= sqrt(Gr) - > Xs = sqrt(Gr)*ni/U_inf

Than you could consider also the coupling with the thermal BL in this region.

But at a first look that means a very small region if you consider air and U_inf>1. For example, for a velocity of 10m/s you get Xs=10^-6*sqrt(Gr).

I didn't understand why some researcher present here, talks about turbulent and laminar flow !? (It's just a question maybe someone will help me to understand !)

my little knowledge in this field is that the gravity is part of the boussinesq approximation which is used in buoyancy-driven flow, this approximation is usually used in natural and/or mixed convection flow whatever the flow being laminar or turbulent. but in forced convection, the buoyancy effects are neglected according to Richardson law which is the proportion between the forced convection flow represented by the Reynolds number and the natural flow (Grashof number). so in you case, Mr. Saeb, I think that you are confronted to the forced convection case.

Dear researchers;

Here we are talking about flow only without any relation to heat transfer; no convection no boussinesq .. etc.

....and over vertical lengthscales which are so small that variations in g are negligible.

It is a sort of approximation to solve NS equations that we neglect body forces including gravitational forces. Also for boundary layer flow, gravity forces could be an order of magnitude less compared to viscous and pressure forces.

When we are talking about turbulent flow, the turbulence model is very complex model and it depends on many constants have been predicted by using wide range of data from experiments. I estimate that all types of forces (including body forces) affect the flow and they are included in these constants, and there is no idea about the order of magnitude of body forces compared with other forces. Because we are using experimental data, we could't say that body forces are not effecting the flow without proof for that.

Interesting

Interesting

The Boundary layer theory assumes that there is a strong attraction between the solid surface and the fluid. But this is not always true. We all know about hydrophobic surfaces or the Non stick cookwares. They do not allow some types of fluids to stick to their surface .

Now as far as the BL Theory is considered, that is highly dependent of order of magnitude and there is a possibility that a liquid, like mercury, who have very high density and have non stick property the boundary layer theory will fail.

for low density materials (which is a general condition for fluids) boundary layer can be neglected to some order. However it affect the flow.

And finally.... the solutions of boundary layer is itself is a approximation.

before going for computation modelling the experimental results must be studied first.

It depends on the inertia force on the boundary. In some cases (low inertia force), you can not ignore gravity.

Gravitational forces are embedded into the pressure forces, so no negligence.

It is clear that the inclusion of gravitational forces depend on the applications and boundary conditions however in some cases the effect of gravitational forces are not relevant(less value compared with other quantities) and can be disregarded.

Dear researchers,

Have a nice weekend with your family....