# How to calculate the centerline velocity inside a pipe?

Well, I was reading about fluid-solid/structure interaction models for make a simulation in COMSOL, and in few papers about it [1] I find an expression about the centerline velocity. I understand the concept: it's the velocity of the central part of a fully developed velocity profile for a cylindrical pipe. I found an interesting expression for its calculation, but I don't know how to get it, the expression is:

$$U_{inlet} = U t^2 \sqrt{(0.04-t^2)^2 + (0.1t)^2}$$

where I've got a value for $U$ in $[m/s]$ and $t$ is in seconds.

But today reading an example made in Comsol [2], from part of its own documentation I found almost the same equation but different:

$$u_{inlet}= \dfrac{Ut^2}{\sqrt{(0.04-t^2)^2 + (0.1t)^2}}$$

The coefficient values are the same, and the expression is very similar, but in the first paper I read, the term inside the square root was multiplying, and here is dividing, but I can't see clear why.

I tried to find out for sure checking the books available that I have: White (6th ed.), Streeter (8th ed.) and one of computational fluid dynamics, but I have not found anything to help me. Someone here will have an idea?

• It looks like it's just an empirical correlation. Could you add references to the papers? Commented Oct 18, 2014 at 3:04
• @user3823992 I think so, I edited the question for add the references you asked. Commented Oct 18, 2014 at 6:48
• I'd guess that it's the first paper that has a typo. I've tried plotting the two cases and the division case makes a lot more sense. It produces a rise-overshoot-decay shape. Multiplying just causes the curve to increase constantly (over 100 m/s at 5 seconds!). Commented Oct 18, 2014 at 16:20
• Well I was trying the Wolfram Alpha site with the function I asked. I plotted bot functions using an interval from -pi to pi and the first equation, without division case looks more as a parabolic profile meanwhile the second isn't continue at zero. This is because the division, you can't dive between zero. So, for me makes more sense the first one, but I want to know how you plotted them? Commented Oct 20, 2014 at 22:40
• I used wolfram too (because lazy). Parabolic makes sense a a spatial profile, not temporal (t stands for time here). I should think that it would start at t=0. As I noted before, the parabolic profile increases without limit. Commented Oct 21, 2014 at 15:44

The limit as $$t$$ goes to $$\infty$$ for:
$$u_{inlet}= \dfrac{Ut^2}{\sqrt{(0.04-t^2)^2 + (0.1t)^2}}$$
is $$U$$, which is more reasonable than $$u_{inlet}$$ going to $$\infty$$ for the first equation.