# One-Dimensional Fluid-Structure Interaction Models in Presurized Fluid-Filled Pipes: A Review
## Appendix B Two-Equation Model  
In the classic water-hammer theory, only 1-DOF is described and the distensibility of the pipe in the radial direction is taken into account neglecting the radial inertia of the pipe-wall and the fluid, and assuming a quasi-steady linear-elastic circumferential deformation of the pipe-wall.  
On the one side, if inertial terms (proportional to \(\partial W/\partial t\)) are neglected in the momentum equation of the 2-DOF, Equation (3) becomes the well-known hoop stress formula:  
\[\sigma_{\theta}=\frac{rp}{e}\ . \tag{10}\]  
Applying time partial derivative to both sides of Equation (10) and expanding differential terms, one obtains:  
\[\frac{\partial\sigma_{\theta}}{\partial t}=\frac{p}{e}\frac{\partial r}{ \partial t}+\frac{r}{e}\frac{\partial p}{\partial t}\ \ . \tag{11}\]  
The left-hand side of Equation (11) can be written in terms of circumferential strain:  
\[E\frac{\partial e_{\theta}}{\partial t}=\frac{p}{e}\frac{\partial r}{ \partial t}+\frac{r}{e}\frac{\partial p}{\partial t}\ \ \, \tag{12}\]  
and knowing that \(\epsilon_{\theta}=\partial r/r\), one gets:  
\[E\frac{\partial\epsilon_{\theta}}{\partial t}=\frac{pr}{e}\frac{\partial \epsilon_{\theta}}{\partial t}+\frac{r}{e}\frac{\partial p}{\partial t}\ \ . \tag{13}\]  
Rearranging Equation (13) and assuming \(\frac{pr}{e}<<E\), one obtains:  
\[\frac{\partial\epsilon_{\theta}}{\partial t}=\frac{r}{eE}\frac{\partial p}{ \partial t}\ \ . \tag{14}\]  
On the other side, the classic water-hammer theory does not consider any axial movement of the pipe. Hence, in Equation (4), \(\partial U_{z}/\partial z=0\) and becomes:  
\[\frac{\partial\sigma_{\theta}}{\partial t}=E\frac{W}{r}\ \ \, \tag{15}\]which in terms of circumferential strain is:  
\[\frac{\partial e_{\theta}}{\partial t}=\frac{W}{r}\ \ . \tag{10}\]  
Combining Equation (11) with Equation (10), an expression for the radial velocity of the pipe-wall, in a function of the inner pressure, is obtained:  
\[W=\frac{r^{2}}{eE}\frac{\partial p}{\partial t}\ . \tag{12}\]  
Substituting Equation (12) into the right-hand side of the continuity equation of the 1-DOF:  
\[\frac{1}{K}\frac{\partial p}{\partial t}+\frac{\partial V}{\partial z}=-\frac {2r}{eE}\frac{\partial p}{\partial t} \tag{13}\]  
rearranging Equation (13):  
\[\frac{\partial V}{\partial z}+\left(\frac{1}{K}+\frac{D}{eE}\right)\frac{ \partial p}{\partial t}=0\ \ . \tag{14}\]  
Finally, defining the elastic wave celerity as:  
\[a_{h}=\sqrt{\frac{K}{\rho_{f}\left(1+\frac{DK}{eE}\right)}}\ \, \tag{15}\]  
the continuity equation Equation (13) for classic water-hammer theory is obtained:  
\[\frac{\partial V}{\partial z}+\frac{1}{\rho_{f}a_{h}^{2}}\frac{\partial p}{ \partial t}=0\ \ . \tag{16}\]  
The fundamental system of equations of the classic water-hammer theory, neglecting damping mechanisms, is therefore composed by Equations (1) and (13), forming the following system of Equations (14):  
\[\begin{array}{c}\includegraphics[width=142.26378 pt]{figures/a1.eps}\\ \end{array} \tag{17}\]  
If the acoustic wave celerity in the unconfined fluid is considered (\(a_{1}=\sqrt{\frac{K}{\rho_{f}}}\)), Equations (1) and (13) are equivalent. Hence, the only difference between 1-DOF wave propagation and classic water-hammer theory is determined by how the elastic wave celerity is defined. The first assumes an entirely rigid pipe, while the second takes into account the hoop distensibility of the pipe-wall.