Dynamic Response of Marine Risers to Wave-Crest Loads
Authors: Li Yousun 
Published: October 2008

Abstract
Intermittent wave-crest loads may induce significant response of a water-piercing riser system mainly for riser bending moments near the water level. This presentation addresses the dynamic behaviors and statistics of riser local response in the wave-crest zone and introduces efficient numerical solution methods. Through a typical example, it is demonstrated that riser bending moments can be much larger than those obtained from the frequency-domain method in which wave-crest effects are neglected. The riser response under wave-crest loads should be an important issue in the riser analyses.

Introduction
Most analyses of the dynamic response of an offshore structure to wave loads are based on the linear wave theory, in which the effects of the water surface elevation on the offshore structure are neglected. However, in reality the waves are splashed over the portion of the riser between the wave crests and troughs, as shown in Figure 1. The wave loads are applied to a level along the riser only when this level is submerged under the water free surface. Similarly, following the water free surface fluctuation, current loads are also intermittently applied to the riser. The wave-crest loads consist of the wave and current forces on the varying wetted length of the riser.
The wave-crest loads may induce a significant level of bending moments in the riser close to the mean water level. The extreme stresses aud fatigue damage rates in this portion of the riser should be correctly predicted.
The riser response within the wave-crest zone is strongly nonlinear to the waves. The solution of a dynamic system under stochastic field can be solved by spectral analyses or by time simulations. The spectral analyses relate the inputs and outputs in terms of spectral density fictions through linear (sometimes quadratic) transformations. Some efforts have been made to simplify the wave-crest loads to adapt frequency-domain analyses. For example, Ref 1 described the wave-crest load as a second-order force solved by bi-spectral analyses. However, this and other similar methods are based on many assumptions which may be far from reality. At this point we believe that simulation of long-duration time histories of the riser response is the best option, from which the extreme stress level can be directly read and the fatigue life can be estimated by a rain-flow counting algorithm. Main efforts in the time simulation are to reduce the computational time for shorter analysis cycles.

Review of Wave-Crest Mechanics
The Airy linear wave kinematics is developed by imposing the mean water level as the boundary condition. Full inclusion of the free water surface may result in complicated mathematical equations. For engineering purposes, we only need to modify the Airy linear wave kinematics to include wave kinematics in the wave-crest zone. Figure 2 shows various approaches for engineering purposes. The traditional form (Ref. 2) is to extend the exponential water velocity profie up to the water free surface. By this approach, the water velocities have an unrealistic exponential amplification within the wave-crest zone. A moderate approach is the linear extrapolation of the water velocity profde from the mean water level to the water free surface or, even further, assuming uniform distributed water velocities in the wave-crest zone (Ref 3). Opposite to the extrapolation is stretching (Ref. 4), which maps the water column up to the mean water level into the water column to the instant water free surface. The water velocities in the wave-crest zone by this approach are much smaller than those from the extrapolation approach. A compromise between linear extrapolation and stretching is delta st