Governing equation for transient pipe flow, part 1

There are several ways to deduce the governing equations for transient pipe flow. These equations are the heart of any transient flow simulations software. The final equations are often called the Allievi equations, or the water-hammer equations.

This will be a short series of how I did it in my PhD thesis, directly from the Navier-Stokes equations. Going from the NS equations, instead of any other method, we can see all the simplifications that is done to achieve the wanted result. These equations are the so called extended water-hammer equations. They also take into account the elasticity of the pipe itself, which causes the not so well known precursor wave. 

When a valve is shut very fast, this will cause a water-hammer wave in the fluid and a similar "steel hammer" wave in pipe wall, an axial shock wave due to the elasticity of the wall.

When a valve is shut very fast, the velocity of the fluid close to the valve is changed from the steady state velocity to zero. If this change of velocity is fast enough, as it often is, this causes a chock wave in the fluid, starting at the valve, propagating upstream with a velocity of about 1200 m/s, the approximate speed of sound in a water-filled regular steel pipe. This is the normal water-hammer. However, the sudden pressure rise at the valve, also causes the pipe to expand radially. This radial expansion of the pipe wall will in turn cause an axial tension in the pipe through the poisson coupling.

The poisson coupling of a homogeneous solid material is the effect that if it's stretched, it will also become thinner in the perpendicular direction. Stretching a rubber band, causes it to become thinner, proportional to the poisson number, or poisson ratio of that material.

This axial tension will also propagate as a shock wave, but much faster than the water hammer wave. The coupling goes both ways however. Axial tension in the pipe wall causes the pipe to decrease in diameter, like a rubber band. This decrease, propagating roughly at the speed of sound in the pipe wall, about 5000 m/s in steel, will in turn cause a smaller pressure wave, also propagating at 5000 m/s. This is the so called precursor wave, that actually can be measured to foresee the coming water hammer wave some small time step later. This could be a millisecond, or several seconds, depending on the length of the pipe.

This precursor wave and the coupling is usually of little practical interest when dealing with most fluid systems (the fluid transients), but can be of outmost importance when dealing with the pipe structure and particularly vibrations in this structure. This is called fluid structure interaction, FSI, and is the cause of most failures in piping systems. FSI in general is a wide area encompassing many different phenomena.

OK, back to the equations, but it is important to keep in mind that every single transient fluid pipe phenomenon has some level of FSI in it, it's impossible to look at fluid transients without also keeping the piping system in mind.

The starting point is the 2D Navier-Stokes equations in cylindrical coordinates.

by observing this fluid element in axis-cylindrical coordinates:

No simplifications are made except 2D flow, symmetry around the x axis and isothermal flow.

The next simplification is to assume very low compressibility. A typical gas has high compressibility, while a liquid has very low compressibility. Many people don't even consider liquids to be compressible, but the very fact that it has some compressibility is the crucial point in all transient piping flow analysis. For very low compressible fluids, the variation of density with respect to pressure can be approximated by a constant value, that is:

K is the bulk modulus, the elasticity of the liquid and c is the wave propagation velocity. 𝜌 is the density of the fluid, and p is the pressure.

For the sake of clarity, all viscous terms are neglected at this point. Friction will be added later in the form of Moody friction. By assuming very low Mach numbers, M << 1, all the convective terms are neglected. For clarity we also neglect all external body forces. These can also easily be added later. For more elaborate proof, especially this M << 1, one can look in my thesis. We then end up with a set of much simpler equations.

These equations are still not easily solvable, particularly due to the variation in radial direction, they are still 2D. This will be handled in the next post.