Date of download: 11/7/2017 Copyright © ASME. All rights reserved. From: Nonlinear Wave Loads on a Submerged Deck by the Green–Naghdi Equations J. Offshore Mech. Arct. Eng. 2015;137(1):011102-011102-9. doi:10.1115/1.4028997 Figure Legend: Schematic of the numerical wave tank, showing different regions referred to in the text, and the domain boundaries. Also shown are the water depth in different regions and plate width (B). Not to scale.

Date of download: 11/7/2017 Copyright © ASME. All rights reserved. From: Nonlinear Wave Loads on a Submerged Deck by the Green–Naghdi Equations J. Offshore Mech. Arct. Eng. 2015;137(1):011102-011102-9. doi:10.1115/1.4028997 Figure Legend: Horizontal force amplitude against submergence depth calculated by the G–N equations versus laboratory measurements of Ref. [28] on a fully submerged deck due to incoming periodic waves of different wave heights (T¯ = 6.24,B/hI = 0.833,LP/hI = 2.5,tP/hI = 0.156). T¯ is the dimensionless wave period defined by Eq. (18).

Date of download: 11/7/2017 Copyright © ASME. All rights reserved. From: Nonlinear Wave Loads on a Submerged Deck by the Green–Naghdi Equations J. Offshore Mech. Arct. Eng. 2015;137(1):011102-011102-9. doi:10.1115/1.4028997 Figure Legend: Vertical force amplitude against wave period calculated by the G–N equations versus laboratory measurements of Ref. [28] on a fully submerged deck due incoming periodic waves (hII/hI = 0.422, B/hI = 0.833, LP/hI = 2.5, tP/hI = 0.156, and hI = 0.3048 m). Wave height of each of the data points are given in Table 1.

Date of download: 11/7/2017 Copyright © ASME. All rights reserved. From: Nonlinear Wave Loads on a Submerged Deck by the Green–Naghdi Equations J. Offshore Mech. Arct. Eng. 2015;137(1):011102-011102-9. doi:10.1115/1.4028997 Figure Legend: Amplitude of the horizontal (F¯x1) and vertical (F¯x3) forces versus wave height on a fully submerged deck calculated by the G–N equations and hydran, due to incoming periodic waves (T¯ = 8.58,B/hI = 3,hII/hI = 0.5)

Date of download: 11/7/2017 Copyright © ASME. All rights reserved. From: Nonlinear Wave Loads on a Submerged Deck by the Green–Naghdi Equations J. Offshore Mech. Arct. Eng. 2015;137(1):011102-011102-9. doi:10.1115/1.4028997 Figure Legend: Amplitude of the vertical force versus wave period on a fully submerged deck calculated by the G–N equations and hydran, due to incoming periodic waves (H/hI = 0.05, B/hI = 5, and hII/hI = 0.4)

Date of download: 11/7/2017 Copyright © ASME. All rights reserved. From: Nonlinear Wave Loads on a Submerged Deck by the Green–Naghdi Equations J. Offshore Mech. Arct. Eng. 2015;137(1):011102-011102-9. doi:10.1115/1.4028997 Figure Legend: Maximum horizontal positive and horizontal negative forces (F¯x1), and vertical uplift and vertical downward forces (F¯x3) on a fully submerged deck calculated by the G–N equations versus force amplitudes predicted by LWA, due to incoming periodic waves (λ/hI = 30, B/hI = 15, and hII/hI = 0.4). λ is the wave length.

Date of download: 11/7/2017 Copyright © ASME. All rights reserved. From: Nonlinear Wave Loads on a Submerged Deck by the Green–Naghdi Equations J. Offshore Mech. Arct. Eng. 2015;137(1):011102-011102-9. doi:10.1115/1.4028997 Figure Legend: Comparison of the time series of horizontal force on a fully submerged model due to cnoidal waves calculated by the G–N equations (flat plate model) versus the Euler equations (model of deck with girders) given in Ref. [26] (H/hI = 0.1, λ/hI = 32.39, hII/hI = 0.2, B/hI = 4.29, LP/hI = 2.10, tP/hI = 0.18, and tG/hI = 0.54). Water depth in the Euler solution is hI = 0.071 m, corresponding to shallow-water condition.

Date of download: 11/7/2017 Copyright © ASME. All rights reserved. From: Nonlinear Wave Loads on a Submerged Deck by the Green–Naghdi Equations J. Offshore Mech. Arct. Eng. 2015;137(1):011102-011102-9. doi:10.1115/1.4028997 Figure Legend: Comparison of the time series of (a) vertical force and (b) horizontal force on a fully submerged deck due to a solitary wave calculated by the G–N equations versus the Euler equations (A/hI = 0.2, hII/hI = 0.6, B/hI = 2.67, LP/hI = 1.3, and tP/hI = 0.11) given in Ref. [13]. Water depth in the Euler solution is hI = 0.114 m.

Date of download: 11/7/2017 Copyright © ASME. All rights reserved. From: Nonlinear Wave Loads on a Submerged Deck by the Green–Naghdi Equations J. Offshore Mech. Arct. Eng. 2015;137(1):011102-011102-9. doi:10.1115/1.4028997 Figure Legend: Comparison of (a) vertical uplift force and (b) vertical downward force on a fully submerged deck due to a solitary wave calculated by the G–N equations versus the Euler equations and laboratory experiments of Ref. [13] (hII/hI = 0.6, B/hI = 2.67, LP/hI = 1.3, and tP/hI = 0.11). Water depth in the laboratory experiments is hI = 0.114 m.

Date of download: 11/7/2017 Copyright © ASME. All rights reserved. From: Nonlinear Wave Loads on a Submerged Deck by the Green–Naghdi Equations J. Offshore Mech. Arct. Eng. 2015;137(1):011102-011102-9. doi:10.1115/1.4028997 Figure Legend: Comparison of (a) horizontal positive force and (b) horizontal negative force on a fully submerged deck due to a solitary wave calculated by the G–N equations versus the Euler equations and laboratory experiments of Ref. [13]. Plate geometry and wave conditions are given in Fig. 9.

Date of download: 11/7/2017 Copyright © ASME. All rights reserved. From: Nonlinear Wave Loads on a Submerged Deck by the Green–Naghdi Equations J. Offshore Mech. Arct. Eng. 2015;137(1):011102-011102-9. doi:10.1115/1.4028997 Figure Legend: Comparison of (a)vertical uplift force and (b)vertical downward force on a fully submerged deck due to a solitary wave calculated by the G–N equations versus calculations and measurements of Ref. [13] (hII/hI = 0.4, B/hI = 2.67, LP/hI = 1.3, and tP/hI = 0.11). Water depth in the laboratory experiments is hI = 0.114 m.

Date of download: 11/7/2017 Copyright © ASME. All rights reserved. From: Nonlinear Wave Loads on a Submerged Deck by the Green–Naghdi Equations J. Offshore Mech. Arct. Eng. 2015;137(1):011102-011102-9. doi:10.1115/1.4028997 Figure Legend: Comparison of (a) horizontal positive force and (b) horizontal negative force on a fully submerged deck due to a solitary wave calculated by the G–N equations versus calculations and measurements of Ref. [13]. Plate geometry and wave conditions are given in Fig. 11.

Date of download: 11/7/2017 Copyright © ASME. All rights reserved. From: Nonlinear Wave Loads on a Submerged Deck by the Green–Naghdi Equations J. Offshore Mech. Arct. Eng. 2015;137(1):011102-011102-9. doi:10.1115/1.4028997 Figure Legend: Two-dimensional (a) vertical uplift force and (b) vertical downward force of cnoidal and solitary waves propagating over a submerged flat plate (B/hI = 2.67 and hII/hI = 0.6) calculated by the G–N equations. Starting from the left data point, the cnoidal wave lengths are λ/hI = 12.3, 15, 16.7, 18.5, and 20.2. Plate width is kept constant.

Date of download: 11/7/2017 Copyright © ASME. All rights reserved. From: Nonlinear Wave Loads on a Submerged Deck by the Green–Naghdi Equations J. Offshore Mech. Arct. Eng. 2015;137(1):011102-011102-9. doi:10.1115/1.4028997 Figure Legend: Two-dimensional (a) horizontal positive force and (b) horizontal negative force of cnoidal and solitary waves propagating over a submerged flat plate calculated by the G–N equations. Plate geometry and wave conditions are given in Fig. 13.