Governing equation for transient pipe flow, part 3

 To get further from part 2, Hooke's law in cylindrical coordinates is introduced.

Then by combining Equation 3.13, 3.15 and 3.16 we get:

We have already discussed that the radial inertia term is unimportant (Long wavelength approximation). We can therefore substitute the tangential stress term for the quasi-stationary relation for a thin walled cylinder found in most text books, and assuming the pipe wall is much thinner than the pipe diameter.

A bit reorganizing, and Equation 3.17 becomes:

This is now fully 1D, and the stress term can be substituted using Hooke's law in 1D:

We now get the final fluid equations:

 

This is the same equations for transient pipe flow as can be found in any text book (when re-organized a bit), except the last term in Equation 3.19. This is the so called Poisson coupling of the fluid to axial motion of the pipe. The fluid wave propagation velocity is approximated by parenthesis before the second term (when multiplied by the fluid density), and is also essentially the same as found in most text books.

An interesting point is that when deducing the Water-hammer equations from Navier-Stokes equations, the FSI coupling pops out automatically, even for fully 1D equations. The only way to remove that coupling, is to force the axial movement of the pipe to be small compared with the pressure forces. We simply say that axial displacement of the pipe wall, u, is "small". In other words, it is assumed that the piping system is in practice 100% rigid. The other way is to set the Poisson ratio to be zero, which it isn't. The Poisson ratio of steel is a constant, and about 0.3.

For normal engineering purposes we simply say that the piping system is "rigid enough" for the transient fluid simulations to be correct. This is usually the case for the vast majority of systems. The only way the piping system is normally included, is in the calculation of the pressure wave velocity, see for instance Equation 3.18 and 3.19 where it can change the wave propagation velocity substantially. This will be correct also for thick walled pipes, except in cases where the pipe wall material becomes very "soft".

For the pipe itself, the axial motion ends up being described by:

The pressure is coupled to axial displacement by the Poisson ratio. When we say that the axial displacements are small compared with pressure forces, we can flip it around and say that the pressure forces are large compared with the axial displacements. Thus, vibrational analysis of the mechanical piping system without including the fluid transients, is therefore fundamentally wrong from the start. We see that the pressure forces are proportional to R/e, the radius of the pipe over the wall thickness. By increasing the wall thickness we will also reduce FSI effects, which fits well with intuition in this case.

For a full FSI analysis, junction coupling, bending and torsion also must be included, as well as all the clamping structures. It is a bit funny however, that transient simulations of the fluid inside the pipe can be done in most cases by disregarding the piping structure elasticity, but doing a vibrational analysis of the piping structure cannot normally be done correctly without including the fluid transients.