# Extreme Wave Loading on a Vertical Circular Cylinder
## 4 Extreme Wave Loads on the Vertical Cylinder
### On the Scaling of Peak Horizontal Impact Forces  
Assumes that the horizontal momentum carried by the impacting water is transferred completely into the impact force on the structure, then the instantaneous peak horizontal force applied to the structure per unit width can be calculated by [22]  
\[f_{\rm{xmax}}=\Delta m\cdot u \tag{8}\]  
where \(\Delta m\) is the mass flow rate of the impacting water per unit width, and \(u\) is the corresponding fluid velocity. Referring to Figure 2, \(\Delta m=\rho uh_{d}\), assuming that the fluid velocity is uniform over the depth of inundation.  
For non-breaking waves in shallow waters, we assume that the linear horizontal velocity at the wave crest can be approximated by [40]  
\[u=\frac{A_{c}}{h}\sqrt{g\hbar} \tag{9}\]  
And thus,  
\[f_{\rm{xmax}}=\rho u^{2}h_{d}=\rho A_{c}^{2}\frac{g}{h}h_{d} \tag{10}\]  
Assuming that the horizontal force is uniform across the structure width, then the peak horizontal force \(F_{xmax}\) scales with the square of the linear wave amplitude and the inundation level. This is consistent with the scaling proposed by [22] for a wave into a solid deck. While for a circular cylinder, there may be a factor (smaller than 1) considering the change across the structure width. Nevertheless, we note that this treatment/assumption is adequate at this stage as only the scaling is of concern here.  
Using the scaling above, the amplitude/peak of the horizontal impact force is now non-dimensionalized by dividing by \(\rho g(A_{L})^{2}h_{d}\). Here the wave amplitude is selected as \(A_{L}\), which is the maximum value of the linearized wave elevation envelope at the position of the cylinder, i.e., \(x=9.3\) m, as discussed in Section 3.2 and Figures 8 and 9. It can be seen that for all four wave conditions investigated, the non-dimensional horizontal impact force is approximately constant for the whole range of inundation levels and increases slightly with the increasing wave steepness. This suggests that the impact force is independent of the inundation level or the air gap if [PERSON]'s scaling is applied.  
It is worth noting that [PERSON]'s scaling discussed above is introduced based on a 'destruction of momentum' argument, which assumes that the horizontal momentum of the impacting water (below the undisturbed free surface but above the deck bottom) is transferred completely into the impact on the vertical edge of a structure. Also, the linear  
Figure 11: Variation in the impact force on the truncated cylinder with the inundation level. (**Left**) horizontal forces. (**Right**) vertical forces.  
Stokes shallow-water wave model is used for approximating the wave crest kinematics. These assumptions may explain the observed scatter in the results in Figure 12. A certain amount of water is deflected away by the curved face of the cylinder (see, e.g., Figure 2 top left photo), i.e., some horizontal momentum is deflected rather than transferred to the impact. And the linear Stokes water wave is certainly not a good approximation for waves in intermediate, nearly shallow waters, as investigated in this work.  
Nevertheless, this work, as shown in Figure 12, provides further support to the momentum block model proposed by [22] and extends its applicability to structures with curved surfaces, although more systematic analysis is required. We leave this additional complication for future work. The momentum block model is useful and could be applied for practical engineering analysis for estimating the violent impact introduced by, e.g., breaking waves.