Transient inviscid incompressible pipe flow

[bob_confused2]

Hello
I am trying to better understand transient fluid dynamics in pipes. First, I am attempting what I believe should be relatively simple problem. I have a constant area horizontal pipe partially filled with a stationary incompressible inviscid fluid. The part of the pipe that is filled is upstream of a burst disc which separates the rest of the pipe. Upstream of the burst disc is at pressure P1 and downstream is P2. At t=0sec, the disc bursts. What is the velocity v2 of the fluid as it flows as a function of time? Here, I am assuming that I am looking only at the velocity at the location of the burst disc and that the flow is uniform. Once I understand this problem, I hope to add in friction and varying location.

I've started with F=ma=m(dv/dt)
--> rho(A)dx(dv/dt)=-AdP
rho(dv/dt) = -dP/dx

And I want to see how long it takes for the velocity to reach the velocity that would be calculated from Bernoulli's equation: P1-P2=rho(v2)^2/2
Any assistance/guidance would be greatly appreciated.
Thank you

 

[Andy Resnick]

This is not a simple problem, although with some care it could be simplified. The origin of the problem is the 'partially' filled part- there is a fluid-fluid interface that can *deform* as the fluid flows. This type of problem, free surface flows, is a class of problem that has resisted clean solutions since forever.

The problem can perhaps be simplied by instead of partially filling the pipe, the pipe is full and subjected, at t = 0, to a pressure spike at one end. The pressure gradient will induce fluid flow, and the pressure wave will propagate at the speed of sound. Even now this problem is very difficult to solve, but significantly easier than before. I don't know if anyone has published a solution to this simplied problem.

 

[bob_confused2]

Thanks. I will try to take a step back even further to try the simplified problem you suggested and then try to add complexity. No wonder my head has been hurting so much as I attacked this problem.

 

[bob_confused2]

For people checking out this post and are interested in learning more about transient 1D flow, I found this online resource:
http://www.scribd.com/doc/8717066/Fluid-Mechanics-Hydraulics-of-Pipeline-Systems

 

[rcgldr]

constant area horizontal pipe partially filled with a stationary incompressible inviscid fluid. pressure is P2.

How does a incompressable fluid have any pressure if it partially fills a pipe? What is supplying the force? Is it gravity?

Incompressable fluids introduce all sorts of problems. For one thing the speed of information propagation (speed of sound) is infinite. If you had a partially filled container, composed of a vacuum and the incompressable fluid, then how would the fluid distribute itself in a zero g environment (assuming the fluid doesn't vaporize)? dP/dx can be infinite in an incompressable fluid such as a incompressable fluid traveling through a pipe that varies in diameter via a vertical wall (transition distance is zero).

 

[reasonableman]

Hmmm, I'm not sure but from your description it sounds like a 'dam break' problem. I've worked through one of these problems but that was a while ago. One difference might be that the dam break surface is completely free. Generally it's hardcore mathematicians that work on these so the maths is formidable and I seem to remember most solutions are implicit. I notice your link has characteristics and I think those are required...

Jeff, I recognise quite a lot of those assumptions, the effects you mention are generally ignored.