General
A thing that was done when deducing the governing equations was to neglect all convective terms. The details of this is rather elaborate, and involves lots of equations (different forms of Navier-Stokes equations). A full treatment is not done here, only the key points are included. The full analysis can be found in Appendix A of my thesis.
The basic principle is to make the Navier-Stokes equations from Part one non-dimensional in the same manner as was done in the Long wavelength approximation in the previous post.
Why would we neglect the convective terms in the first place? The reason is that they add considerable complexity in solving these equations. At the same time, if they happen to be unimportant for the solution nonetheless, it's only silly to include them.
We know that liquids are compressible. It's what makes transient analysis of liquid filled pipe interesting and important. However, we also know that liquids are only slightly compressible. Gas on the other hand is highly compressible. Can this compressibility be related in some way to some characteristic property, like for instance fluid velocity? Yes it can.
The key property describing the compressibility in relation to transient piping flow analysis is the Mach number. The Mach number M is defined as:
M = V/a
Where V is the fluid transport velocity and a is the wave propagation velocity (commonly known as the speed of sound). The speed of sound in water is typically around 1400 m/s, while the speed of sound in air is about 340 m/s at room temperature and atmospheric pressure. The (transport) velocity of the fluid in a pipe is for liquids around 1-10 m/s at max, while for gas it can be 100 m/s or more in some cases depending on application. For choked gas flow the Mach number is reached. It's safe to say that for liquids in pipes, the Mach number is very small, while this is certainly not the case in general for gas flow (although it could be the case in some specific circumstances as shown later).
Continuity
We make the NS equations non-dimensional carefully choosing appropriate characteristic properties. The continuity equation (in 2D cylindrical coordinates) pops out to become:
Here we see that the order of smallness for all convective terms is related to the Mach number squared. This is beneficial. If the Mach number already is small, a typical value is 0.004 for water in a pipe, then this squared becomes 0.000016, which is even smaller. A table for all the convective and space derivative terms is set up:
We see that the order of smallness for all the convective terms is very small, as long as M << 1, and they can be neglected. This means that when M << 1, then the convective terms can be neglected both for 1D and 2D equations in the continuity equation. Even if the Mach number is not super small, it has very little effect in the continuity equation. For typical flow in (large) gas lines, the Mach number is around 0.1 -ish, which means the order of smallness becomes 0.005 -ish (0.5 %) which is well inside the accuracy we can hope to achieve in a calculation.
Momentum
Doing the same for the momentum equation, the following non-dimensional equations in axial and radial direction pops out.
Now, this is unfortunate. The momentum equation is not related to Mach number at all. No matter what Mach number we use, the order of smallness of the convective terms becomes O(1) and cannot simply be neglected. There's nothing we can do about it, it's just the way things are.
But, there are a few other things we can do. One thing that can be done, is in combination with the continuity equation to force the convective terms to be small. We say that the transport velocity is small and negligible. This is called the acoustic approximation, and is used for acoustic analysis (typically in 3D). In 1D this will simply produce the ordinary water hammer equations, but without the friction terms. A kind of side effect is that the ordinary water hammer equations can be used to calculate acoustics in gas pipes, and very accurately so, as long as the long wavelength approximation is taken into account. This was actually the first application of LVTrans, calculation of acoustics in gas risers.
What is done in my thesis is to apply the long wavelength approximation, L/D >> 1 to make the whole thing 1D. The continuity and momentum equations then become:
A.13 and A.14 is combined to obtain:
Now the Mach number is back, and we can compare the time derivative terms for smallness:
This shows that the in the
combined set of equations for continuity and momentum, the residing convective term becomes insignificant when M << 1. This means that M << 1 does not in general remove convective terms in the momentum equation at all. It's only when applying the long wavelength approximation (1D flow) that the combined set combined will show the convective terms to be insignificant, and only when M << 1. the final non-dimensional momentum equation becomes:
Equations for (slightly compressible) gas flow
For the sake of completeness, it is worth considering the more general form of the non-dimensional equations where only the long wavelength approximation have been introduced (1D flow). We simply remove all the radial terms to obtain (removing all the subscripts for clarity):
These are two equations governed by two constants: The Mach number and the Reynolds number. It must be noted that the variation of density with respect to pressure is constant for these equations, the equation of state is over-simplified. The following non-dimensional variables are used in this post.