Real-time nonlinear cylinder wave force reconstruction in stochastic wave field considering second-order wave effects

Highlights

• A nonlinear method for real-time wave force reconstruction is presented.

• Approximate expressions of QTFs are built with undetermined coefficients.

• The nonlinear method implemented by FFT and RLS algorithms is investigated.

Abstract

This study provides a novel method for reconstructing real-time nonlinear wave forces on a large-scale circular cylinder by considering second-order wave effects. Potential theory is utilized for deriving the expression of wave forces with the measured data of wave elevation. Approximate expressions of quadratic transfer functions are built with undetermined coefficients, which are resolved by using the historical data of measured wave elevation. Two different algorithms, including fast Fourier transform (FFT) and recursive least squares (RLS), are adopted for real-time reconstruction. Hydrodynamic tests are conducted in the wave flume on a circular cylinder to examine the effectiveness of the nonlinear reconstruction method. Comparative results demonstrate that the accuracy of real-time reconstructed wave forces is significantly enhanced by the present method. The over-prediction errors at force crests and the under-prediction errors at force troughs have been reduced. Furthermore, comparative results show that the nonlinear method implemented by the FFT algorithm provides more accurate results, whereas the RLS algorithm is more time cost efficient.

Introduction

In the field tests of offshore structures, how to reconstruct the wave forces on their foundations in a stochastic wave field has aroused great interest among engineers. The reconstructed results can provide valuable references for similar structures, which will be constructed in the same ocean region in the future. A real-time wave force reconstruction can further contribute to the active control of structural dynamic responses. According to the incident waves known or unknown, the wave force reconstruction problem can be classified into two categories. When the incident waves are known, the wave forces acting on the cylinder can be evaluated by theoretical methods (MacCamy and Fuchs, 1954) or numerical methods (Aristodemo et al., 2017, Chow et al., 2019, Mohseni et al., 2018, Paulsen et al., 2014). When the incident waves are unknown, researchers need to reconstruct the wave forces indirectly. For a small-scale cylinder, researchers attempt to reconstruct the wave forces by using the Morison equation with measured data of wave elevation (Boccotti et al., 2012, Lotfollahi-Yaghin et al., 2012). It indicates a shortcut for determining the wave forces on the cylinder by using water surface elevation data. The wave fields, however, are assumed to be undisturbed by the structures in the aforementioned studies. For a larger-scale structure, Liu et al. (2018) established a prediction method to reconstruct the wave forces by using the measured data of wave elevation around the cylinder. For circular cylinders, a linear method that affords an excellent reconstruction of the linear wave forces is provided. However, it is also found that the high- and low-frequency wave forces are under-predicted and over-predicted, respectively, because the nonlinear wave forces are incorrectly considered.

The nonlinear wave problem has always been a hot topic in hydrodynamic research. In the framework of potential theory, a so-called Stokes expansion by the powers of wave steepness is widely applied in the analytical study of wave interactions. The wave characteristics obtained through a simple superposition of linear waves become unreliable when second-order effects are significant. Motivated by previous studies (Longuet-Higgins, 1962, Longuet-Higgins, 1963), a second-order solution for wave elevations in the case of bidirectional bichromatic waves is provided by Sharma and Dean (1981) and reviewed by Dalzell (1999) and Forristall (2000). A third-order solution for multidirectional irregular waves can be found in the studies of Zhang and Chen (1999) and Madsen and Fuhrman (2012). The abovementioned studies are conducted under the condition of excluding the effects of existing structures. When the size of the structures is large compared with the wavelength, diffraction waves significantly change the wave field, and nonlinear wave effects become more complex. For a vertical axisymmetric body that interacts with unidirectional bichromatic incident waves, Kim and Yue (1990) derived an explicit second-order diffraction and radiation solution by using the ring source integral equation method. With the aid of radiation potential, Cong et al. (2018) provided a semi-analytic solution for second-order wave loads on a bottom-mounted cylinder without explicitly obtaining the second-order diffraction potential when the incident waves are bichromatic and bidirectional. Although Chen (1994) and Cong et al. (2012) made the integration process more efficient, time cost remains to be the main problem for the real-time wave force reconstruction in engineering practice. Furthermore, most of the previous studies have focused on the problem of regular wave actions. Actually, stochastic waves are more frequent in real ocean scenarios. In this case, the wave force reconstruction for offshore structures becomes more difficult.

The main objective of this study is to build a nonlinear method to reconstruct the wave forces on a large-scale cylinder by considering second-order wave effects. Recognizing the difficulties in determining the properties of incident waves, the water surface elevation around the cylinder is utilized to reconstruct the wave forces acting on the cylinder. The content of this paper is organized as follows. Section 2 first reviews the mathematical models for wave force reconstruction and the linear method provided by Liu et al. (2018). Then, the nonlinear method for wave force reconstruction is introduced with two different algorithms, fast Fourier transform (FFT) and recursive least squares (RLS). Section 3 describes the experimental validation for the nonlinear method. Time domain, frequency domain, and time cost analysis are performed to evaluate the nonlinear method. Section 4 lists the main conclusions of present work.