## 5 Scale Effect
### Scale Effect of Slump-Type Landslide Tsunamis  
With the good agreement between the Splash3D result and experimental data, the scale effect is investigated numerically under the criteria of Froude number similarity, Bingham number, and normalized yield stress. Table 1 shows the parameters enlarged from the laboratory scale to the field scales. Two kinematics are taken into consideration for the scaling analysis. The first one is the slump deformation, and the second one is the wave induced by the slump deformation. Figure 11 shows the calculated results of the slump-type landslide tsunamis at \(T_{1}=\ 1.00\) s, \(T_{2}=\ 3.16\) s, \(T_{3}=\ 10.00\) s, and \(T_{4}=\ 31.60\) s. In Figure 11, with the different geometric scales, the dimensionless wave profiles including the onshore breaking waves and the offshore propagated waves coincide with each other. This indicates that the water waves are dominated by the Froude number. In Figure 11, the dimensionless geometries of the slump profiles are similar to one another. However, the differences in the slumpshapes are larger than that in the water free-surface profiles. It is due to the generation of large eddies and strong turbulence in the head area of the slumps, and the behaviors of eddies and turbulence are controlled by the Reynolds number. The conditions of Reynolds number similarity, Froude number similarity, and Bingham number similarity cannot be established simultaneously unless the model scale is the same as the prototype scale. However, satisfying both Froude and Bingham number similarities is a difficult task. The difficulty comes from the scaling down or scaling up the slump material. In a general situation, the slump sample is collected in the field and used in the laboratory experiments, or vice versa. Therefore, the geometry is scaled down to the laboratory scale and the Froude number similarity is kept. However, using the same slump sample from the field will twist the Bingham number similarity. To investigate this problem, a scenario is designed shown in Figure 12, red lines. In this scenario, the setup is scale \(H_{4}\). However, the slump parameters are adopted from scale \(H_{1}\). The inverse way (setup: \(H_{1}\); slump: \(H_{4}\)) can easily be done numerically. However, the result shows a stationary slump in the simulation. This is due to that the \(H_{4}\) slump characteristic behaviors are too sturdy in scale \(H_{1}\). To avoid this problem, setup \(H_{4}\) and slump \(H_{1}\) are adopted. The red-line slump (setup: \(H_{4}\); slump: \(H_{1}\)) moves much faster than the black-line slump (setup \(H_{4}\); slump \(H_{4}\)). A large red-line eddy can be seen at t = 30 s and 40 s. This phenomenon indicates that the \(H_{1}\) slump behaviors are friable in the \(H_{4}\) scale. The wave height, however, remains similar in these two cases. This result indicates that the water wave is dominated by \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Laboratory**[16] & **Scale \(H_{2}\)** & **Scale \(H_{3}\)** & **Scale \(H_{4}\)** \\ \hline Slump (m) & \(0.65\times 0.65\) & \(6.5\times 6.5\) & \(65\times 65\) & \(650\times 650\) \\ Water depth (m) & \(1.6\) & \(16\) & \(160\) & \(1600\) \\ Cell size (m) & \(4.35\times 10^{-3}\) & \(4.35\times 10^{-2}\) & \(4.35\times 10^{-1}\) & \(4.35\) \\ Mx = dz & \(4.35\times 10^{-3}\) & \(4.35\times 10^{-2}\) & \(4.35\times 10^{-1}\) & \(4.35\) \\ Viscosity (Pa s) & \(50\) & \(1.6\times 10^{3}\) & \(5\times 10^{4}\) & \(1.6\times 10^{6}\) \\ Yield stress (Pa) & \(10^{3}\) & \(10^{4}\) & \(10^{5}\) & \(10^{6}\) \\ Model time (s) & \(1\) & \(3.16\) & \(10\) & \(31.6\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters for laboratory scales and field scales.