We consider nonlinear, directionally spread irregular wave fields in deep water and study the statistical properties of the total inline force that would be induced by the waves on a vertical circular cylinder. Starting from the two-dimensional Morison equation, we specifically investigate the effect of wave steepness and directionality on the probability density functions (PDFs) of the inertia and drag forces. To do so, we derive new analytical expressions for the PDFs of these forces based on first-order theory and compare them with the results of fully nonlinear numerical simulations. We show that the inertia force for the main direction (x) of the wave field is in general unaffected by nonlinear effects, while the inertia force for the direction perpendicular to the main direction (y) is subject to substantial third-order effects when the steepness is appreciable and the wave field becomes relatively long crested. Moreover, we show that the drag force for the x-direction is in general subject to substantial second-order effects. The drag force for the y-direction is also affected by second-order effects, but to a much smaller degree than the x-direction. It is, however, strongly affected by third-order effects under the same conditions as the inertia force in the y-direction. We conclude that the total force can be accurately approximated by first-order theory when the ratio kpD/ε is large (with kp the peak wavenumber, D the cylinder's diameter and ε the wave steepness), while first-order theory underestimates the probability of large forces considerably when this ratio is small.
- Surface gravity waves