Pipe flow analytical solution
 

Hello everybody, this is my first post in this forum :)

I am studying the flow of water along a tiny pipe (capillary tube). The data of the problem is the following:

- Length of the pipe: 10 mm
- Diameter of the pipe: 1 mm
- Inlet velocity: 0.5 m/s
- Outlet pressure: ambient pressure
- Reynolds number: 500 (laminar)
- No-slip condition at the wall

I ran a simulation using OpenFoam (SimpleFoam solver) and I got the next results:

https://i.imgur.com/DvTdpTA.png

This is the horizontal velocity distribution of the flow (cut in the XZ plane). As it can be seen, the inlet velocity (at the center of the pipe) is 0.5 m/s and the outlet velocity is approximately 0.88 m/s.
What I want to do is to compare the numerical solution with the analytical solution, that is, calculate the outlet velocity by hand. Is there a formula that provides the outlet velocity as a function of the length, the diameter, etc?

I also obtained the gauge pressure distribution:

https://i.imgur.com/nvNxjXR.png

https://i.imgur.com/zCiZn8o.png


I know that this is a classical fluid mechanics problem (Poiseuille flow) but when I studied it in class we considered that the pipe was infinitely long and the derivatives on the x axis were 0. Here I cannot assume that.

Can somebody help me? Thank you very much,

Carlos.

To the best of my knowledge there is no such analytical solution, only for the fully developed flow. There are, instead, formulas to estimates the length necessary to achieve the fully developed condition.

However, from your pressure distribution, the last 10%-20% of the pipe might be close enough to the fully developed condition (note that, if an analytical solution existed, it would tell you that the fully developed condition, for this case, would be reached only for x->Inf).

What you could do to double check this, besides actually comparing the outlet profile with the analytical, fully developed one, is to set up the case to use periodicity in the streamwise direction, so that you fully replicate the conditions underlying the analytical solution.

It really depends from what you want to do, testing OF on a case as similar as possible, or obtaining that very specific analytical solution (which I don't think it actually exists).

In line of principle, if we start from the problem of two parallel flat plates, this 2D problem can be considered in the framework of two spatially evolving boundary layers (hence you know the analytical Blasius solution) that merge at a certain distance x=Xm producing the Poiseulle solution.
However, in the case of a pipe flow I think one could consider an extension of the theory for a boundary layer in (r,z) plane with axisymmetric condition. But I never tried to develop such a comparison.

I suppose that, given the Re number, you can also simply check you get a parabolic velocity profile after the merging lenght is reached.

While I agree with the recommendations of the previous posters, I would not worry about the velocity profile until you make sure that you are satisfying mass conservation. It may seem trivial, but it is one piece of the physics where you know the answer. I've seen lots of simulations over the years where the velocity profiles look reasonable but the user didn't conserve mass.

Hello, I need your help
I'm new to openfoam, I'm doing my graduation project, I did the same as charlos, I made a tube with diameter 2.5cm and length 10cm, the flow must be laminar with velocity 0.5http://pipe m / s so I use the simpleFoam solver but the results look wrong, the velocity is almost the same in all tube positions (I put a picture), I have to get the same result as carlos and the speedprofile must be parabolic, I didn't understand where the problem is, please help me http://pipe