# One-Dimensional Fluid-Structure Interaction Models in Presurized Fluid-Filled Pipes: A Review
## 3 Numerical and Experimental Research
### Four Degree-of-Freedom Models  
According to the classification proposed in Section 2.2, eight-equation models solve the system of equations for either 1,3,4,6-DOFs or 1,3,5,7-DOFs. These kind of models are used to describe in-plane axial, torsional and flexural pipe displacements, respectively, in the \(\hat{x}\hat{z}\) or \(\hat{y}\hat{z}\) planes. Radial deformation is nested in the celerity of the 1-DOF as in the classic water-hammer theory. Poisson coupling may be included such that the system of equations to be solved becomes composed of Equations (24)-(27) (i.e., the four-equation model) together with Equations (7), (8), (11) and (12) or Equations (9), (10), (13) and (14). The 4,5,6,7-DOFs are only coupled by means of junction coupling. Pipe systems like the one depicted in Figure 6 can be described by 4-DOF models. In this pipe scheme, a water-hammer wave generated by the valve manoeuvre would induce not only transient pressures but axial stress, shear stress and bending waves in the pipe wall, hence exciting the 1,3,4,6-DOFs. Figure 5: Comparison of transient pressures considering thick-wall theory (thick solid blue line), thin-wall theory (thin broken red line) and experimental data (thin solid black line) [60], \(e/r=0.15\). The eight-equation model for curved pipes for 1,3,4,6-DOFs was presented in [6], the transient fluid flow was coupled with the axial, shear and bending transient waves of the pipe-wall. The same equations using MOC were solved by [64], who validated the results against new experimental data. Radial inertia was included by [9] who solved a nine-equation model. A MOC-MOC scheme in combination with cavitation was used in [65; 66; 67], the cavitation effect was modelled by means of a lumped parameter model. An FVM method was used by [68] to solve the eight-equation model, which was tested for different set-ups (v.i. Figure 7). In this analysis [55], it was pointed out that a two-phase flow model is needed for simulations of more universal FSI problems occurring in pipelines. A compilation of sixteen experiments dedicated to systems with a single elbow (L-pipes) was presented in [117], eight experiments focused on the frequency-domain approach and eight on the time-domain. The experiments based on the time-domain approach are presented in Table 2.  
Figure 6: Reservoir-pipe-valve system with a 90\({}^{\circ}\) elbow in the middle section of the pipe. Figure 7: Numerical output from [68] considering a free moving valve (black dashed line), anchored (red solid line) and compared with the classic water–hammer model output (purple doted line). From an experimental standpoint, Ref. [118] tried to prove that a pressure wave reflects partially when passing through a rigidly supported elbow. This work generated in-depth discussion pointing out the importance of considering FSI even for rigid supports assuming that the movement of anchorages is nearly impossible to avoid. This idea was supported by [62] who stated that pipelines are never anchored sufficiently to eliminate motion due to a water-hammer surge. A complete pipe rig was used in [63]; nonetheless, they experienced difficulties in getting rid of undesired FSI effects, emphasizing the importance of properly testing experimental setups preventing such phenomena. Finally, [PERSON]. [120] verified that there is no pressure wave reflection from an immobile elbow but that there is due to the elbow movement. These findings were confirmed once more in [123]. A pipe system with multiple elbows, bends and junctions can be described by eight-equation models if these are located in the very same plane. This is the case for the experiment carried out in the University of Guanajuato, Mexico, in collaboration with the University of Lisbon, Portugal. Data from a pipe rig assembled by concentric elbows of 90\({}^{\circ}\) was collected by [40; 124]. The apparatus was equipped with pressure transducers and accelerometers. Water-hammer events were generated by a downstream valve manoeuvre.