Governing equation for transient pipe flow, part 2

In the previous part, the so called extended water hammer equations was discussed (FSI), and a reduction of the Navier-Stokes equation was started.

Imagine a valve being slam shut at the left end. Water streaming from the right. The valve causes a pressure wave, a shoch wave (water hammer) propagating to the right at about 1200 m/s. The widening of the pipe wall causes axial stress, which will shrink the diameter through the poisson coupling. This causes the precursor wave propagating at about 5000 m/s.

The equations were 2D. We want them 1D. This is done by averaging the pressure and flow across the cross section. Mathematically this is done by multiplying with 2𝜋R, integrating with respect to r from 0 to R and dividing by 𝜋R^2.

V and P is the average velocity and pressure, and expressed as:

This is just a fancy way of saying V and P are the same all over the cross section of the pipe. This is of course not 100% correct physically, particularly not for the velocity V, but this is the simplification we must do to make them 1D.

But we had one more equation from the last part:

The assumption is that the characteristic fluid length scale, L is large compared to the characteristic diameter, D. That is L/D >> 1 under the constrain that the wavelength is "long". With this assumption, it can be shown that the radial inertia is negligible compared to the axial inertia. We can simply neglect the first term in equation 3.7. This is called "the long wavelength approximation", and is found to be correct for frequencies up to 63% of the first radial liquid filled pipe frequency (which normally is much higher than the first axial frequency). Proof of this is found in my thesis. Then we are left with the last term in equation 3.7 to be equal to zero, which simply states that the pressure is constant in the cross section. Thus "the long wavelength approximation" will also "one dimensionalize" the pressure, and remove one equation from the set.

On the pipe wall, the radial fluid velocity must be equal to the radial pipe velocity:

Inserting this, and we are left with only these two equations:

We have removed one equation. We have transformed them from 2D to 1D, but we have added a radial wall velocity term. This looks like a complication, but is a key point for the end result in part 3.