# One-Dimensional Fluid-Structure Interaction Models in Presurized Fluid-Filled Pipes: A Review
## 3 Numerical and Experimental Research
### Seven Degree-of-Freedom Models  
The fourteen-equation model includes all the degrees-of-freedom presented in Section 2 except the 2-DOF corresponding to the radial inertia of the pipe-wall, which is nested in the celerity of the 1-DOF like in the classic water-hammer theory. Hence, the system to be solved is composed of Equations (24)-(27) (i.e., the four-equation model) together with Equations (7)-(16). Pipe systems like the one depicted in Figure 8 can be described by 7-DOF models, where all the related DOFs would be excited by a water-hammer wave generated at the downstream valve. \begin{table}
\begin{tabular}{c l c} \hline \hline
**Reference** & **Experimental Setup** & **Transient Test** \\ \hline
[118] & 45\({}^{\circ}\)–180\({}^{\circ}\), hor. mtire, & \\  & hor. curved bends & valve closure: 2–5 ms \\  & 0.85 \(<R_{c}/D<\) 5.0, & initial flow vel.: 0.6–2.4 m/s \\  & rigid (2 jacks) & \\ \hline
[62] & 30\({}^{\circ}\)–150\({}^{\circ}\), hor. mtire, & valve closure: 2 ms \\  & rigid and free & initial flow vel.: 2–3 m/s \\ \hline
[63] & hor. 114.3 mm, & \\  & \(D=70.6\) mm, & gun: 150 bar \\  & \(R_{c}/D=1.6\), rigid & pulse 3 \\ \hline
[64] & \(R_{c}/D=6\) & pellet impact 0.2 m/s \\ \hline
[119] & hor. \(R_{c}/D=0.8\) & valve closure: 4 ms \\
[120] & initial flow vel.: 1.2 m/s \\ \hline
[66] & hor. 0.88 kg & rod impact 0.15 m/s \\ \hline
[122] & hor. \(R_{c}/D=2.2\) & impact hammer pulse 1–2 ms \\ \hline
[123] & vert. elbow \(R_{c}/D=1.5\) & valve opening: 20–200 ms \\  & initial flow vel.: 2–17 m/s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Main time-domain experiments carried out for single-elbow pipes. The fourteen-equation model in the time-domain was introduced by [4]. The model was finally implemented by [7; 8; 120] with MOC approach, both in the fluid and in the structure. Experimental fluid pressure and structural velocity measurements from [7], corresponding to a similar set-up as the one depicted in Figure 8, are shown in Figure 9. A good fitting with measurements was obtained, but the analysis concluded that further model developments were necessary. The work was extended by [125] using an experimental set-up consisting of a copper pipe containing a U-bend free to move in an in-plane fashion. This method was used also by [126], who simulated an accident. Experiments were carried out in a PVC pipe containing a U-shaped section at the laboratory of Delft Hydraulics, the Netherlands, in [69]. The author concluded that classic water-hammer theory was not accurate enough to describe the behaviour of the pipe-rig and, consequently, the FLUSTRIN project was launched. A complex and large-scale apparatus (Figure 10) held by suspension wires and specially designed for FSI tests was assembled at Delft Hydraulics laboratory and used for the development  
Figure 8: Reservoir-pipe-valve system with two out-of-plane 90\({}^{\circ}\) elbows. Figure 9: Experimental pressure measurements next to the downstream valve and at a bend [7]. and verification of the FLUSTRIN code, which is based on a MOC-FEM approach [70; 71]. In this framework, [PERSON]. [127; 128] carried out a series of numerical benchmark tests. Coupled and uncoupled Poisson effect solutions were compared for the Delft Hydraulics Benchmark Problem F [129], which is a good approach for verifying fourteen-equation model implementations (v.i. Figure 11). Experimental measurements were used in this comparison and a guideline was provided suggesting when FSI is important. The same computer code was used by [51; 72; 74] with similar purposes of comparing with other modelling assumptions and using experimental tests for validation. [PERSON]. [74] suggested that, for four-equation modelling, an MOC-MOC approach is more convenient, while, for higher degrees-of-freedom, an MOC-FEM scheme is preferable as higher grid resolution is required. A similar MOC-FEM code with differences in the implementation of the Poisson coupling mechanism was presented in [73].