# One-Dimensional Fluid-Structure Interaction Models in Presurized Fluid-Filled Pipes: A Review
## 3 Numerical and Experimental Research
### One Degree-of-Freedom Models  
The classic water-hammer model (two-equation model) is a sophisticated version of the basic 1-DOF system (Equations (1) and (2)), where the right-hand side term of the continuity equation is adapted in order to account for the pipe-wall distensibility. Although the bulk modulus of compressibility and a finite acoustic wave speed are considered in the fluid, in terms of density variation, the fluid is assumed to be incompressible and pressure changes are related to velocity changes by embedding fluid compressibility and pipe-wall distensibility into the wave celerity value, which is regarded as a constant parameter and can be either experimentally or numerically determined. Research works such as [1; 42; 46; 75; 76; 77; 78] contributed to the development of wave celerity formulae. The latest presented correcting factors to account for axial FSI. The fundamental equations of classic water-hammer theory (i.e., mass and momentum conservation) can be derived from Navier-Stokes equations [79] or by directly applying the Reynolds Transport Theorem [80] to a control volume of the pipe. From an FSI standpoint, these fundamental equations can be also reached from the system of equations presented in Section 2, as the classical \begin{table}
\begin{tabular}{c l l} \hline \hline
**DOF** & **Description** & **Main Contributions** \\ \hline \multirow{2}{*}{1} & Only the fluid transient is described. & \multirow{2}{*}{[1; 41; 42; 43; 44; 45; 46]} \\  & Equations solved: (1), (2) & \\ \hline \multirow{2}{*}{1,3} & Solid surging is coupled with the fluid. & \multirow{2}{*}{[12; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57]} \\  & Equations solved: (1), (2), (5), (6) & \\ \hline \multirow{2}{*}{1,2,3} & Fluid, breathing and solid surging & \multirow{2}{*}{[5; 47; 58; 59; 60]} \\  & interact. Equations solved: & \\  & (1), (2), (3), (4), (5), (6) & \\ \hline \multirow{2}{*}{1,3 and 4,6 or 5,7} & Fluid and solid surging, and either & \multirow{2}{*}{[40; 53; 61; 62; 63; 64; 65; 66; 67; 68]} \\  & swaying and yawing or heaving & \multirow{2}{*}{[40; 53; 61; 62; 63; 64; 65; 66; 67; 68]} \\  & and pitching are taken into account. & \\ \hline \multirow{2}{*}{1,3} & Fluid and solid surging, swaying, & \multirow{2}{*}{[5; 47; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68]} \\  & and (7), (8), (11), (12) or (9), (10), (13), (14) & \\ \hline \multirow{2}{*}{1,3,4,5,6,7,8} & Fluid and solid surging, swaying, & \multirow{2}{*}{[51; 69; 60; 61; 62; 63; 64; 65; 66; 67; 68]} \\  & having, yawing, pitching and rolling & \\ \hline \multirow{2}{*}{1,3,4,5,6,7} & Fluid and solid surging, swaying, & \multirow{2}{*}{[5, 47; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68]} \\  & having, yawing, pitching and rolling & \\ \hline \multirow{2}{*}{1,3,4,5,6,7} & Fluid and solid surging, swaying, & \multirow{2}{*}{[5, 47; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68]} \\  & are coupled. Equations solved: & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary table of main 1D FSI models in hydraulic transients research. theory considers a combination of the first two degrees-of-freedom. The fundamental momentum conservation equation is directly the one presented in 1-DOF (Equation (1)). For mass conservation (continuity equation), the cross-sectional area of the control volume is assumed to vary and this variation is related to the fluid inner pressure by applying a quasi-static assumption in the 2-DOF. This derivation is described in Appendix B.