Dear Mr. Vatankhah,

I was intrigued by what did you mean when you said "real water surface slope".

The water surface on open channel flows could follow the bed slope or vary depending on energy curve of the flow, depending on the consideration that we had admited for the fluid: Newtonian fluid (water), thus leading to turbulent dissipation, or ideal. But you mentioned critical depth, leading to critical flow configuration where the Froude number is equal to 1 (gentle slope). Many things can arise from all these informations and many flow configurations like hydraulics jumps or large-time effects, such as gradually varied flows.

Actually I was not able to understand your question pretty well, and I'd be glad if you could explain better what is your problem and what are you intending on doing.

Best regards,

Guilherme FIOROT

Dear Mr Fiorot,

My mean is clear. I am talking about gradually varied flow governing equation in a prismatic open channel, that is:

dy/dx=(S0-Sf)/(1-F^2)

what is water surface slope (dy/dx) at critical flow depth when F=1?

Best regards,

Vatankhah

Good morning Mr Vatankhah,

Now I see your point!

In fact, I'm no expert on gradually varied flows but if you look at Graf & Altinakar (2000), at page 144, Chapter 4, you will find many examples on how the surface will behave depending on the bed slope, energy slope, mean flow height, critical height, etc... Theoretically, we see that dy/dx tends to infinity for F=1, and then atan(infty) = 90 degress to respect the bed, as you mentioned, this is not plausible.

I think that you will find chapter 4 from Graf & Altinakar very useful! Sorry I didn't help much.

best regards,

GHFiorot

http://books.google.fr/books/about/Hydraulique_fluviale.html?hl=fr&id=hR6RHZugNFgC

Re-thinking the problem, I really don't know what have been done to solve this problem, but I'm pretty sure it's solved. Maybe You could solve the problem at h=hc manipulating continuity equation... I have been studying shock waves on open channel, but there it's like an hydraulic jump, so we use Rankine-Hugoniot shocks conditions to calculate between h=hc and we do not know its value at precise height h=hc.

Just some thoughts, nothing conclusive...

Dear Mr. Fiorot,

Thanks for your help and your idea,

Best,

Vatankhah

Dear Dr. Vatankhah,

Thank you very much for your nice question. I believe that this is a fundamental hydraulic problem.

Indeed, applying the hopital's rule implicitly implies that the uniform flow condition will occur, i.e. S0=Sf. That is, at the critical section the water surface slope would be equal to the bed slope. Nice to mention that the critical flow regime is not a stable condition. We can usually observe a hydraulic jump or free overfall soon after the critical condition. Consequently, the hypothesis of considering the uniform flow condition is also questionable.

I think that such a flow condition is rather a Rapidly Varied Flow instead of a Gradually Varied Flow. In such a case the streamline curvature might be significant. Also the pressure distribution may not be hydro-static. In this regard referring to the studies of Castro Orgaz on the critical flow condition might be helpful.

Now let me a little change the original question:

What is the water surface slope at the near critical flow condition?

I hope it to be useful.

Thank you very much.

Bijankhan.

Dear Dr. Bijankhan,

Thank you very much for your nice comments.

However, this question is still an open one.

What is main difference between water surface slope which obtained theoretically at a critical flow depth (or as you mentioned near it) in a GVF and SVF?

Best,

Vatankhah

Dear Dr. Vatankha

I just joined Researchgate and I found one of your questions from a few months (Jan. 2013). I enclose below my contribution if it is not too late.

Under critical condition, water surface slope and bed slope are the same (Uniform flow)

Using Darcy-Weisbach relationship to critical flow in a rectangular channel, for example, we get a simple relation of the critical slope Ic as :

Ic=(f/8)(1+2Nc) where f is the friction factor and Nc is the critical aspect ratio equal to hc/b in which hc is the critical depth and b is the channel width. For two-dimensional flow corresponding to an infinitely large value for b (i.e Nc=0), one can write : Ic=f/8

Dear Dr Achour,

Thank you for your answer.

We discus about gradually varied flow profiles not uniform flows.

What is the slope of water surface profile at critical depth in a given open channel?

slope of water surface profile= dy/dx=(So-Sf)/(1-F^2)

Best regards,

Vatankhah

Dear Dr. Vatankhah

Obviously, I did not quite understand the question before. The problem remains and deserves considération.

So many thanks Dr. Achour.

You have a conceptual problem in question. When you use the equation gradually varied flow, is necessary have a gradually varied flow!

First, there is no way to assume in the same stretch of canal regime change through the critical height in this cases:

Mild slope M1=> M3, M2 => M1; steep slope S1 => S2, S1 => S3, horizontal slope H1=> H3 and adverse slope = A1> A3 (Bakhmeteff notation).

In the case of the flow passage supercritical flow to subcritical flow, you have the presence of an Hydraulic Jump (M3 => M1, M3 => M2, S3 => S1, critical slope = C2> C1; H3 => H1, A3 => A1) and, you don’t have Gradually Varied Flow, then the equation is simply not valid.

In the case of having a critical slope the boundary condition is asymptotic, i.e. the slope equals the bottom slope, resulting dy/dx = infinity.

Finally we have the presence of a critical height in channel M, H or A when there is a overfall or change of slope to steep slope. In both cases we have a singularity, the curvature of the water line are finite, not being construed the derivation of equation Gradually Varied Flow (more or less just 3 to 4 hc to the singular point).

Dear Dr Vatankhah

i remember that professor Kashefipour (Shahid chamran University) discussed about this subject when i was B.Sc.

i think we should seem the assumptions during the formulas development. the mention formula has been derived based on the GVF and can not be used for the Critical flow. 

thank you for your attention to the basic concepts and challenge in the open channel hydraulic

 best regard

Hello

What is the method for appointing nondimension terms in your papers about gradully varied flow?

Thanks