# One-Dimensional Fluid-Structure Interaction Models in Presurized Fluid-Filled Pipes: A Review
## 3 Numerical and Experimental Research
### Two Degree-of-Freedom Models  
The historical development of four-equation models can be traced back from [1] who already pointed out the need of considering axial stress waves. Pipe axial inertia and Poisson coupling was included qualitatively in the analyses of [82; 83]. [PERSON]'s work was extended in [3], who presented the four basic fundamental equations and introduced the concept of precursor waves. From an experimental standpoint [84] was the first to observe precursor waves, which are, at the same time, the evidence of the Poisson coupling effect. Finally, [PERSON]. [10; 85; 86] used the simplified version of [PERSON]'s equations that represent the well-known four-equation systems for axial FSI. [PERSON]'s work was revisited and analysed in [87]. For the description of pressure waves in pipe systems, two or four-equation models are sufficient [23]. Four-equation models consider the combination of classic theory with the 3-DOF equations. Hence, four fundamental equations, two for the fluid and two for the pipe axial movement, are to be solved. The right-hand side terms of the continuity equations of the 1-DOF and 3-DOF systems must be adapted in order to describe the Poisson coupling in terms of the dependent variables of the four-equation model (i.e., respectively, axial stress of the pipe-wall and fluid pressure). This derivation is explained in Appendix C from which Equations (25) and (27) are obtained:  
\[\frac{\partial V}{\partial t}+\frac{1}{\rho_{f}}\frac{\partial p}{\partial z }=0, \tag{24}\]  
\[\frac{\partial V}{\partial z}+\frac{1}{\rho_{f}a_{h}^{2}}\frac{\partial p}{ \partial t}=\frac{2\
u}{E}\frac{\partial\sigma_{z}}{\partial t}, \tag{25}\]  
\[\frac{\partial U_{z}}{\partial t}-\frac{1}{\rho_{p}}\frac{\partial\sigma_{z}} {\partial z}=0, \tag{26}\]  
\[\frac{\partial U_{z}}{\partial z}-\frac{1}{\rho_{p}a_{3}^{2}}\frac{\partial \sigma_{z}}{\partial t}=-\frac{r\
u}{eE}\frac{\partial p}{\partial t}. \tag{27}\]  
Equations (24)-(27) can be used to describe pipe systems like the one depicted in Figure 2, where the pipe axial movement is allowed and axial stress waves interact with the inner fluid pressure waves. Several numerical methods can be used to solve the above system of equations, either integrating both the fluid and the structure in the same numerical scheme (e.g., MOC-MOC) or by a combination between different schemes (e.g., MOC-FEM). The four-equation system was solved using MOC procedure for the first time in [10; 85]. A series of tests carried out on a vertical pipe line located in a subterranean salt cavern were presented by [50; 85; 88]. Measurements were shown from a water-main bridge in [89; 90; 91], and in [10; 92] from a loading line between tanks and ships. These measurements were used to develop and validate the four-equation model and to understand FSI mechanisms. The Method of Characteristics for both the fluid and the structure (i.e., MOC-MOC) was shown to have useful advantages in [93]. This approach was supported by experimental evidence from [94; 95; 96; 97], who carried out measurements in which FSI effects were particularly well isolated by means of suspended pipe rigs that were excited by the impact of a solid rod. In combination with their numerical developments, they showed how FSI coupling changes the natural vibrating frequencies, which cannot be predicted by uncoupled approaches. An FDM scheme in his four-equation model was used in [47] as a simplified version of a six-equation model which was solved by MOC. Fluid and axial stress waves in conduits were modelled by [98] who used a MOC approach taking into account only junction coupling but ignoring Poisson coupling. An FDM approach was used in [49], the implementation was validated by tests on a thin-walled straight pipe for Poisson coupling as well as junction coupling at a closed-free pipe end. An explicit Joukowsky-like expression was presented by [99]. The expression was derived from the four-equation system aiming at estimating maximum pressures during water-hammer with FSI.