Solving Water Hammer Equations: Joukousky vs Wave Equations

[rppearso]

Hello all,

So I found this article on water hammer and I have since beefed up my math skills.

In the attached PDF for some reason they felt the need to convert the Joukousky equation to a wave equation, can't the joukousky equation be solved analytically for pressure vs velocity and then stepped through time?

Why would they need to convert this to a wave equation? Couldn't the first order PDE just be solved directly?

 

[rppearso]

the initial equation is in terms of V and P so the equation has to be reduced to a single dependant term in terms of x and t otherwise its not solvable. I just need to rationalize how they get from the water hammer equation to the wave equation. Could a valve Cv equation then be used simultaneously to solve for valve closure/opening time?

 

[rppearso]

What would be the damping term for frictional losses on water pressure waves in a pipe? What about the forcing term of the valve slowly opening/closing?

 

[256bits]

Hello all,

So I found this article on water hammer and I have since beefed up my math skills.

In the attached PDF for some reason they felt the need to convert the Joukousky equation to a wave equation, can't the joukousky equation be solved analytically for pressure vs velocity and then stepped through time?

Why would they need to convert this to a wave equation? Couldn't the first order PDE just be solved directly?

( Surprising we don't have someone more knowledge here to help you out, but I will give it a try. )

It can and it gives the pressure peak from the "first wave".

Other transient effects, such as,
other important transient phenomena such as line packing, instantaneous wall shear stress values and the Richardson annular effect
show up with the 2D model.
In fact, for long pipelines, and highly viscous flow, the 2D gives a more realistic view of how high the "local" pressure can become. Joukousky seems to underperform in those situations.

Google some more on hammer to get a better insight.
I did and found and now know much more than I did before.
http://www.sciencedirect.com/science/article/pii/S0307904X07002569
http://www.ksb.com/linkableblob/ksb-pk/80892-193671/data/Druckstoss_Know-how_Band_1_en-data.pdf

 

[rppearso]

Awesome, thank you for the help. I found the second link and I think the only way to do a simplified PDE solution is to assume frictionless and a constant forcing term. The moment anything more complicated is introduced it become a finite element problem (which I am just beginning to learn). I know how to solve the 2D wave equation PDE pretty well but that is only the most simplified solution. So it will be another year of reading lol.

Also since we are at it does anyone know how to solve the full electromagntic wave equation which includes current density (for some reason in antenna theory we introduced the concept of "A" in order to solve the problem but I was told it could be solved with Greens functions. Are greens functions similar to Bessel functions in where you are just trying to solve for more complex eigen values?

Seems like a water hammer problem with a simplifed damping term could be solved the same way without going to finite element for simple cases?

 

[Nidum]

http://www.fluidmechanics.co.uk/hydraulic-calculations/transient-flow/

http://www2.ce.metu.edu.tr/~ce534/docs/CE534-WebFiles/intro-transient-flow.pdf