Real-time nonlinear cylinder wave force reconstruction in stochastic wave field considering second-order wave effects
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.
Section snippets
Fundamentals
Fig. 1 shows the schematic of the wave-cylinder interaction problem. The cylinder is assumed to be a large-scale cylinder with a radius of . It is rigid and bottom-mounted at the horizontal tangent plane of the seabed. The origin of the coordinate system is set at the center of the cross-section. The axes of polar coordinates and Cartesian coordinates are located at the still water level (SWL). The z-axis is perpendicular to the SWL and positive upward. The depth water surrounding
Experimental setup
A hydrodynamic experiment with a bottom-mounted circular cylinder under stochastic wave actions was performed at the Wind Tunnel and Wave Flume Laboratory, Harbin Institute of Technology in China. The length of the wave flume is 50 m and the width of it is 5 m. In this experiment, a vertical circular cylinder with a radius of 0.3 m was installed in the center of the wave flume. JONSWAP spectrum was adopted for generating the wave field with a significant wave period of 0.8 s. The significant
Conclusion
This study presents a nonlinear method for wave force reconstruction by utilizing measured surface elevation data and considering the second-order wave effects. Two different algorithms, FFT and RLS, for real-time wave force reconstruction are implemented. Their computational accuracy and efficiency are discussed in the paper. The main findings of the investigation include the following.
(1) The use of wave elevation historical data to determine the approximate expressions of quadratic transfer
CRediT authorship contribution statement
Jiabin Liu: Conceptualization, Methodology, Validation, Formal analysis, Writing - original draft, Writing - review & editing. Qinghe Fang: Writing - review & editing. Anxin Guo: Conceptualization, Writing - review & editing, Project administration, Funding acquisition. Hui Li: Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The financial supports from the National Natural Science Foundation of China (51725801), China Postdoctoral Science Foundation funded project (2019M661285, BX20200109) and Supported by Fundamental Research Funds for the Central Universities are greatly appreciated by the authors.
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