# One-Dimensional Fluid-Structure Interaction Models in Presurized Fluid-Filled Pipes: A Review
## 3 Numerical and Experimental Research
### Two Degree-of-Freedom Models
A number of research contributions e.g., [100; 101; 102; 7; 103]) explained how to solve the four-equation system considering Poisson coupling. They presented the characteristic equations after MOC transformation and how to integrate them within the same characteristic grid using time-line interpolations as explained by [104]. The MOC transformation that allows hyperbolic partial differential equation systems to be converted to a set of ordinary differential equations was based on [105]. A FEM scheme was used by [52] for both the fluid and the structure. Time interpolation and wave adjustment methods are compared for MOC-MOC solutions in [106; 107]. A hybrid MOC-FEM approach, MOC for the fluid and FEM for the structure, was used in [48], experimental data was used for model verification. An FVM approach was presented in [55] to solve the four-equation model, which was successfully verified using the Delft Hydraulics Benchmark Problem A [51; 70]. Both approaches MOC-MOC and MOC-FEM are compared in [51], concluding that for straight pipe problems the MOC procedure is more accurate and efficient. In addition, several approaches were compared in [108] suggesting that MOC methods become difficult to implement for complex pipe geometries and restrictions, for which alternative methods to describe the structure behaviour like FEM or FVM become attractive. A MOC-MOC coupling was used in [109] to simulate a kind of FSI which was experimentally observed in pipe coils by [99; 110]. The MOC-MOC approach is also used in [111], who combined pipe-wall viscoelasticity, column separation and unsteady friction with fluid-structure interaction. The Delft Hydraulics Benchmark Problem A (20 m long, steel pipe, 0.4 m diameter) is a good test case for the verification of four-equation numerical codes (v.i. Figure 3). A theoretical development of an exact solution of the four-equation system by means of a recursion was presented  
Figure 2: Reservoir-pipe-valve system with a straight pipe which is free to move in the axial direction. in [12; 54; 112]. The drawback of the method is its exponential computational effort for longer simulation periods. Recently, in [56], the computation for the exact solution was upgraded in order to increase computational efficiency and applicability. The analysis suggested to keep the scope of exact solutions to generate test cases and to benchmark solutions for more conventional numerical methods. In addition, in [113], the efficiency was improved by using a hybrid cubic time-line interpolation scheme. It is important to highlight that numerical outputs, such as the one presented in v.i. Figure 3, show how FSI phenomena can cause pressure surges higher than the ones expected from classical theory. The Poisson coupling beat, which is a phenomenon that arises from resonance between 1-DOF and 3-DOF, was demonstrated in [115]. Poisson coupling beat was already numerically observed by [8]. So far, there is no experimental evidence about it, as damping mechanisms tend to hide the oscillating resonance between the pipe-wall and the fluid vibrations. More recently, [PERSON]. [57] numerically observed the Liebau effect in pipe flow using a four-equation model which was adapted to describe the inertia of thrust blocks. The Liebau effect is rather an object of study in the field of physiological flows and is defined as the occurrence of valveless pumping through the application of a periodic force at a place which lies asymmetric with respect to the system configuration [116]. The Liebau effect in pipe flow may be induced by Poisson coupling and should be object of further research (v.i. Figure 4). Figure 4: Numerical evidence of Liebau effect depicted in [57]. Figure 3: Four-equation code verified by means of the Delft Hydraulics Benchmark Problem A [114].