## 3 Computational Modeling Method
### Fluid Flow Modeling  
The Fluent module of Ansys is computationally used with a k-\(\epsilon\) turbulence model to solve the energy-momentum equations for the fluid domain containing water where a high pressure of 16.6 MPa is applied instantaneously at the inlet. This pressure is constantly supplied at the inlet for 20 microseconds and the change in velocity and pressure of the pipe featuring the elbow is monitored. The fluid domain is solved in the three-dimensional volume of the 3 m model pipe where the inlet is kept at the straight end of the pipe, as depicted in the CEA tests. The walls of the fluid domain are partitioned (Figure 5) such that each partition represents the input surface of the simplified models (550 mm and 150 mm). The fluid domain is solved twice for the two input walls for the simplified models, and the corresponding wall pressure is imported to the structural models to perform the FSI simulations. This research incorporates a partitioned FSI approach where the fluid domain and the structural domain are solved independently and joined together with a coupling algorithm. This independent solving capability uses unique solver codes for both the flow and structural equations. The fluid equations are solved independently through the governing equations for large displacement problems for incompressible Newtonian fluids with Navier-Stokes criterion from the following Equation (1) [33; 34]:  
\[\rho^{F}\frac{u}{t}\Big{|}_{x}+\rho^{F}\Big{(}u-u^{G}\Big{)}\cdot\
abla u-2\mu \
abla\varepsilon(u)+\
abla\overline{p}=\rho^{F}b^{F} \tag{1}\]  
\[\
abla u=0 \tag{2}\]  
where \(\rho^{F}\) and \(b^{F}\) represent the density and body forces of the fluid with \(\overline{p}\) as the physical pressure, viscosity \(\mu\), and fluid velocity \(u\). Since the mass is conserved within the fluid, we obtain Equation (2). The strain rate tensor is represented by \(\varepsilon(u)\), and it is derived from the following Equation (3):  
\[\varepsilon(u)=\frac{1}{2}\Big{(}\
abla u+(\
abla u)^{T}\Big{)} \tag{3}\]  
The instantaneous high-pressure inlet can be depicted as the water hammer phenomenon expected to cause a large deformation at the free end of the pipe which produces  
Figure 5: Wall partition for the CFD simulations for the corresponding simplified structural models. the pipe whip effect. The fluid flow is solved in the three-dimensional region of the pipe using the finite volume method (FVM) where the elements are modelled with uniform hexahedral elements (Figures 2 and 4) with inflated layers of mesh at the fluid domain wall to create accurate flow modeling. The flow is kept under transient conditions, and the longitudinal high velocity motion of the incompressible fluid flow inside the pipe can be given by the following Equation (4):  
\[\rho_{f}A_{f}\mathrm{d}x\frac{DV}{Dt}A_{f}\bigg{(}P+\frac{\partial P}{ \partial x}-P\bigg{)}\mathrm{d}x=0 \tag{4}\]  
where \(\rho_{f}\) and \(A_{f}\) are the density and area of the fluid inside the pipe, with \(x\) being the flow direction of the fluid with respect to time \(t\), and \(P\) is the pressure of the fluid in the center [35]. The computational fluid dynamics (CFD) analysis in the Ansys Fluent module is solved for three-dimensional flow model under the action of the inlet pressure surge at the initial instance. The walls of the pipe are partitioned based on the length of the corresponding simplified models, as shown in Figure 5, such the resultant fluid pressure distribution inside the pipe wall with respect to time is then transferred to the FSI study. Wall 1 is the entire fluid domain which is used for the fully 3D computational study. Walls 2, 3, 4, and 5 are used as the imported pressure input for the 550 mm model, and walls 3 and 4 are used for the 150 mm model.