# Computational Evaluation of Shock Wave Interaction with a Cylindrical Water Column
## 3 Results
### Shock-Column Interaction
#### 3.2.2 Quantitative Validation  
The quantitative validation of the proposed CFD model was performed by examining some geometrical parameters of the deformed water column versus time, while making a comparison with reference numerical data [20], as well as experimental findings [11]. Here, the instantaneous shape of the distorted column was determined as that associated with the isoline at \(\alpha=0.5\) for the volume fraction of water. In order to demonstrate the effect of increasing the numerical resolution, the results corresponding to solutions CFD II and CFD III are presented. The time history of the decreasing centerline width of the column \(w\), normalized by the initial diameter \(d_{0}\), is shown on the left side of Figure 6. The two different lines that were reported for the reference numerical solution corresponded to \(\alpha=0.25\) (blue dashed line) and \(0.99\) (red dashed-dotted line), whereas green dotted and black solid lines were used for CFD II and CFD III, respectively. Apparently, a good agreement was achieved between the present solutions and both references. When making an analogous comparison for the increasing cross-stream breadth \(b\), which is depicted on the right side of the same figure, CFD analysis appeared to provide overestimated results with respect to the experimental findings. However, at the present relatively high Mach number, the reference numerical solution provided similar results. Furthermore, the deformation of the coherent water body was examined in terms of cross-section area \(A\), whose time history is given on the left side of Figure 7. Again, a good agreement with both references was obtained for the decreasing area, which is shown normalized by the initial circular section area. Figure 6: Time history of the normalized centerline width (**left**) and cross-stream breadth (**right**) of the water column: CFD II (green dotted line) and CFD III (black solid line), compared to reference numerical solutions (blue dashed and red dashed-dotted lines) and experimental data (symbols). It is worth stressing that the comparison with experimental measures, which were obtained using holographic interferograms, was affected by the inherent uncertainty associated with the definition of the boundary of the deforming water column. As observed in [20], while it was not possible to unambiguously identify the method used in the experiments [11] to define the boundary, the presence of a diffuse interface provided by the numerical methods further worsened the issue. The time evolution of the leading-edge drift \(\Delta x_{\text{le}}\) is examined on the right side of Figure 7, wherein CFD data appeared overestimated with respect to experimental visualizations, again according to what was found in [20]. However, as observed in [7], the leading-edge drift did not coincide with \(\Delta x_{\text{cm}}\), that is the center-of-mass (CM) drift. The time-dependent position of the CM is defined by:  
\[x_{\text{cm}}=\frac{\int_{\Omega}\alpha\rho_{l}x\ d\Omega}{\int_{\Omega}\alpha \rho_{l}\ d\Omega}\, \tag{9}\]  
which inherently involves only the flow region that is actually occupied by the liquid phase, and the CM velocity in the streamwise direction can be analogously evaluated, namely:  
\[u_{\text{cm}}=\frac{\int_{\Omega}\alpha\rho_{l}u\ d\Omega}{\int_{\Omega}\alpha \rho_{l}\ d\Omega}. \tag{10}\]  
By inspection of Figure 8, wherein the time histories of the CM drift and velocity are given, the present CFD data appeared to be slightly overestimated with respect to reference numerical results [20], with the positive effect of increasing the numerical resolution being evident. In fact, while the present resolution sufficed to practically achieve converged evolution for the leading-edge drift, a finer FV grid should be used to better capture the CM drift, according to what was found in [10].