A Fourier Series Approximation for Deep-water Waves

Abstract

The Brazilian multinational petroleum corporation Petrobras and partners are developing the Buzios oil field, which is approximately 210 km offshore of Brazil. The Floating production storage and offloading (FPSO) is spread-moored in a maximum water depth of 2,030 m and has facilities to receive oil from sub-sea wells. It also has production plant facilities to process fluids, stabilize them, and separate produced water and natural gas, which is re-injected into a dedicated reservoir. Processed liquids are metered, stored in the FPSO cargo storage tanks, and offloaded to export tankers. The design life of the FPSO is 30 years. The relative water depth is defined as   , where    is the wave number,    is the angular frequency,  is the wavelength,  is the wave period,  is water depth, and  is gravity. According to DNV (2010a), it is normally not necessary to investigate wave periods longer than 18 s. Therefore, the relative water depth is greater than 25 for the project. When the relative water depth is greater than 2, deep-water wave theories are applicable (Chakrabarti, 1987; DNV, 2010b; Shin, 2019). Well-known wave theories include Airy theory, Stokes theory, Dean’s stream function theory, Fenton’s theory, and trochodial (Gerstner) theory for deep-water waves in offshore structure design. Trochodial theory is an exact solution of the Euler equation with vorticity. The first rotational solution was described by Gerstner in 1802 and was independently rediscovered later by Rankine (1863). A mathematical analysis of trochodial theory was performed by Constantin (Henry, 2008). The wavelength is independent of the trochoidal wave’s height, unlike in Stokes’ wave theory and observations. The trajectories of a water particle are closed circles, in contrast with the usual experimental observation of Stokes drift associated with wave motion. Therefore, trochodial theory is of limited use for offshore structure design. Airy theory, Stokes theory, Dean’s stream function theory, and Fenton’s theory are irrotational wave theories, unlike Trochodial theory. The wavelength is also independent of Airy wave’s height, and Airy theory is applicable for  ≤ (Chakrabarti, 1987), where  is the wave height. Therefore, Airy theory is unsuitable for describing waves near the Miche limit (DNV, 2010b); i.e.,   , where  is the wavelength calculated by Airy theory. Journal of Ocean Engineering and Technology [ARTICLE IN PRESS] https://doi.org/10.26748/KSOE.2021.092 pISSN 1225-0767 eISSN 2287-6715