Dear Nevena.

I think the text sent by Brian is an excellent start to your work. However if you wish to follow up with more provocative circumstances and boundary conditions more complex, I suggest you look in traditional books that work with "surge tanks" . There are introductory texts, such as Pickford, texts or even slightly broader as Wylie & Streeter (1978) and Chaudhary (1987) or an older Jaeger (particularly like the first one more).

I look at these important texts for two reasons: first, they present real cases in which such problems are solved, and second, these books present several problems solved to manage to compare with your simulations, in this books have models and results shalt find them very simple solution of "engineers".

They are old books that have satisfactory solutions to your problems.

Pickford, J., 1969 "Analysis of Surges", Macmillan and Co. Ltd., bath, U.K.

Wylie, E. B., Streeter, L., 1978 “Fluid Transients”, Mc Graw Hill Inc., New York, U.S.A.,

Chaudhary, M. H., 1987 "Applied Hydraulic Transients", Van Nostrand Reinhold Co., New York, U.S.A.,

Jaeger, C. 1977, “Fluid transients in hydro-electric engineering practice”, Blackie, London

(in http://pt.scribd.com/doc/11432816/Fluid-Transients-Jaeger)

1. You probably mean Navier Stokes equations.

2. The question is not very clear. 1D flow between two reservoirs in steady or unsteady cases is something that is solved typically using control volume approach and using Navier Stokes, simplified down to 1D case of Bernoulli equation or similar.

Dear Mr. Liberzon, thank you very much for the correction. That was only typing mistake and it is already changed.

Here are additional details:

- two reservoirs connected with a pipe, with a quite amount of water in one and just a small amount of water in the second one (this is only, due to necessity to stick to one formulation of flow-in equation - in other words, I don't have any transitional flow at the inlet into the pipe and outlet out of the pipe and into the second reservoir, but already developed behavior )

- the case is simple as it can be - 1D flow, incompressible fluid, where I did two cases:

(I) Bernoulli equation (simplified NS, dV/dt = 0) + Continuity equation (control volume) and

(II) NS equation (velocity calculation over time) +Poisson eq. (in order to define pressure distribution as a consequence of given boundary conditions)

-I have got similar results in both cases, but I need a published paper as a benchmark, to prove my results and to test just built models with given different input parameters.

An approximate solution to this problem involves macroscopic balances for the conservation of mass in the two tanks ( expressed as a change in differential height -Delta h - of the liquid levels between the two tanks), and quasi steady rectilinear flow in the connecting pipe. The time rate of change of Delta h is related the local pressure drop in the pipe. To approximate entrance and exits effects you can add a discharge coefficient factor. I doubt that this problem has been published in a scientific journal- more likely to be found in fluid mechanic textbooks ( or process control textbooks) as a worked example or a chapter problem. Understanding the limitations of the model is crucial!

Dear Mr. Higgins, thank you very much for the additional note. Yes, I am aware of these details, but my guess is that local pressure drop in the pipe as parameter to determine dt becomes important, when one has inertia component in the equations (which I don't have at the moment because of all applied simplifications). Therefore, I picked enough small dt step, in order to avoid numeric instability. Concerning effects at the inlet and outlet point, the behavior is defined by different equations, and I can use it without any problem.

The results, that I have got are pretty much what I would expect as hydraulic engineer, only at this stage I need to see if someone did describe it through some published papers.

Sincere regards,

NP

I do not understand something, that might be hindering the search for a benchmark

When you make a citation from a resolution "1D", implicitly're not considering the development of Bondary Layer at beginning of movement in the conduit.

For me it was not clear the problem The boundary conditions are not well defined

Dear Mr. Maestri, I didn't even mention type of boundary conditions, as I wonder if someone else has something similar, with which I can compare my results, without particular wish to bother all of you with some additional details. Nevertheless, as I have two reservoirs at both ends of the pipe, and for this simple simulation I took boundary conditions as a constant value of pressure (for every time step, defining it through the Dirichlet boundary formulation). Also, as I have friction at the walls of pipe, this is so-called "no-slip" boundary condition.

I hope, it is a bit more clear now.

Sincere regards,

NP

Here is a quasi steady analysis of your flow problem, assuming that flow in the connecting pipe is laminar.

Thank you very much Mr. Higgins! This is exactly, what you have mentioned in previous post.

I did exactly the same in one of my cases, but I wonder, why the slope of Head Decrease is a bit steeper in the second case, as I use the same geometry, the same input data and the same parameters.

Sincere regards,

Nevena

Your test case shows two calculations; Bernoulli-continuity Eqn and Navier-Stokes Calculation. Each of these calculations has implied approximations and need not be the same. For example, the Bernoulli equation is unlikely to be valid in the connecting pipe unless the flow there is irrotational or inviscid. If the Reynolds number is low, so that the flow is laminar in the connecting pipe, then Bernoulli's equation does not apply as the flow is neither irrotational or inviscid. One could attempt to invoke a Bernoulli type equation ( inviscid vs irrotational) to relate pressure, velocity and gravitational head in the tank, but you have to be very careful what type of approximation you are making, (inviscid flow, irrotational flow, steady? etc). Different approximations different results. A full CFD calculation using the Navier-Sokes equations is probably the best but is the effort warranted. If you have experiments then evaluating key dimensional groups for your problem should allow you to determine what approximations to make and hence what type of analysis is necessary.

Dear Mr. Higgins, thank you very much for given hints and short description with formulas. It is always good to think out of the box, especially when doing aproximations while building mathematical model and when working in the group, where the final decision is not only mine.

All the best,

Sincere regards,

N.P.

Your are welcome!

Dear Nevena.

I think the text sent by Brian is an excellent start to your work. However if you wish to follow up with more provocative circumstances and boundary conditions more complex, I suggest you look in traditional books that work with "surge tanks" . There are introductory texts, such as Pickford, texts or even slightly broader as Wylie & Streeter (1978) and Chaudhary (1987) or an older Jaeger (particularly like the first one more).

I look at these important texts for two reasons: first, they present real cases in which such problems are solved, and second, these books present several problems solved to manage to compare with your simulations, in this books have models and results shalt find them very simple solution of "engineers".

They are old books that have satisfactory solutions to your problems.

Pickford, J., 1969 "Analysis of Surges", Macmillan and Co. Ltd., bath, U.K.

Wylie, E. B., Streeter, L., 1978 “Fluid Transients”, Mc Graw Hill Inc., New York, U.S.A.,

Chaudhary, M. H., 1987 "Applied Hydraulic Transients", Van Nostrand Reinhold Co., New York, U.S.A.,

Jaeger, C. 1977, “Fluid transients in hydro-electric engineering practice”, Blackie, London

(in http://pt.scribd.com/doc/11432816/Fluid-Transients-Jaeger)

Dear Rogerio, thank you so much for suggested books! I appreciate any kind of help, and this one was not of "any kind". Originally I am hydraulic engineer and therefore quite familiar with the behaviour of cases. Yet when it comes to programming these hydraulic phenomena, many different questions arise.

I am looking forward to reading in details all the material!

Sincere regards,

Nevena

Dear neneva

Any further information with more details you can send an email to [email protected], which you can help it I will.