Extreme value prediction of inundation drag force with and without current Zhen Gao à , Torgeir Moan Centrefor Ships and Ocean Structures and Department of Marine Technology, Norwegian University of Science and Technology, Otto Nielsens vei 10, NO-7491 Trondheim, Norway article info Article history: Received 4 September 2008 Accepted 21 July 2009 Available online 28 July 2009 Keywords: Inundation drag force Extreme value prediction Mean up-crossing rate The Rice formula The Laplace method abstract This paper deals with drag forces due to irregular waves on a vertical slender structure in the splash zone, i.e. in the vicinity of still-water free surface, by considering the inundation effect due to instantaneous wave elevation. The force turns out to be a third-order quantity with respect to wave elevation. The focus of this paper is however limited to extreme value prediction of this force in stochastic waves. Based upon a transformation of random variables and use of the Rice formula, the mean up-crossing rate of inundation drag force is obtained in the frequency domain both by direct numerical integration and asymptotic evaluation for high levels using the Laplace method. The extreme value distribution of this force is then established by the Poisson probability law assuming independent up-crossing events. The proposed method agrees very well with time-domain simulations both for the mean up-crossing rate and the extreme value prediction. The effect of correlation between wave elevation and horizontal water particle velocity and the presence of current have been studied. & 2009 Elsevier Ltd. All rights reserved. 1. Introduction Offshore structures with slender components are widely used in the offshore oil and gas industry. Typical examples are fixed platforms like jackets and jack-ups. Floating platforms like semi- submersibles may also involve slender structural components. Mooring lines and risers are normally long and flexible slender members. Wave and current forces acting on a submerged slender structure can be evaluated by the well-known Morison formula (Morison et al., 1950), which generally includes a drag term and an inertia term. The horizontal force per unit length on a vertical slender component of a fixed structure can be written as F M ðtÞ¼ K d vðtÞjvðtÞj þ K m aðtÞ ð1Þ where v(t) and a(t) are the horizontal water particle velocity and acceleration, respectively, t is the time variable, K d ¼ 1/2rC d D, K m ¼ p/4rC m D 2 , r is the water density and D is the diameter of the slender component, C d and C m are the empirical drag and inertia coefficients, respectively. The horizontal velocity v(t) may also include current velocity which is normally modeled as a constant value in time. Moreover, if the motion of the slender component is taken into account, both relative velocity and relative acceleration should be considered. The Morison equation represents a nonlinear wave force formulation, but it is frequently associated with linear potential wave theory by which the velocity and acceleration can be well defined as Gaussian random processes. Many researchers have shown great interest in analyzing statistical properties of the Morison force, including the spectrum and probability density function of the force, probability distributions of the maxima and extreme values in a certain period (or the expected maximum value), where possible analytical solutions can be achieved. See for example, Borgman (1972), Tung (1975), Vinje (1980), Sarpkaya and Isaacson (1981), Ochi (1982), as well as the good review paper by Isaacson (1991). Many efforts have also been focused on the linearization, cubicization and even quinticization of the drag term of the Morison formula, such as Borgman (1967), Gudmestad and Connor (1983), Bruce (1985), Winterstein (1988), etc. The effect of current may also be included in these polynomial approxima- tions. The least-square approximation method and the moment- based approximation method are commonly adopted. A recent paper by Liaw and Zheng (2004) summarized these two major approximations. The linearization makes it much easy for spectral analysis of the drag force in the frequency domain and in general it can represent the force spectrum very well. However, it also underestimates extreme value of the drag force. The cubicization and quinticization might be necessary for this prediction. Structural response induced by a Morison-type force will be nonlinear even if structural system is linear and the total force needs to include all of the contributions from the force acting on each segment of the slender structure. Many studies have been carried out on this issue and most of them were based on linearization of the force. A statistical quadratization method can also be applied as shown by Quek et al. (1994). Naess and Yim (1996) proposed a new method for representing the drag force with the absence of current by defining a quadratic transfer function, which allowed them to study the statistical properties of both the drag force and the corresponding structural response. ARTICLE IN PRESS Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/oceaneng Ocean Engineering 0029-8018/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2009.07.010 à Corresponding author. Tel.: +4773551458; fax: +4773595528. E-mail address: zhen.gao@marin.ntnu.no (Z. Gao). Ocean Engineering 36 (2009) 1244–1250