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Research Papers: Ocean Engineering

Nonlinear Wave Loads on a Submerged Deck by the Green–Naghdi Equations

[+] Author and Article Information
Masoud Hayatdavoodi

Department of Maritime Systems Engineering,
Texas A&M University at Galveston,
Galveston, TX 77554
e-mail: masoud@tamu.edu

R. Cengiz Ertekin

Fellow ASME
Ocean and Resources Engineering Department,
SOEST, University of Hawaii at Manoa,
2540 Dole Street, Holmes 402,
Honolulu, HI 96822
e-mail: ertekin@hawaii.edu

1Corresponding author.

Contributed by the Ocean, Offshore, and Arctic Engineering Division of ASME for publication in the JOURNAL OF OFFSHORE MECHANICS AND ARCTIC ENGINEERING. Manuscript received February 24, 2014; final manuscript received October 16, 2014; published online December 3, 2014. Assoc. Editor: Celso P. Pesce.

J. Offshore Mech. Arct. Eng 137(1), 011102 (Feb 01, 2015) (9 pages) Paper No: OMAE-14-1014; doi: 10.1115/1.4028997 History: Received February 24, 2014; Revised October 16, 2014; Online December 03, 2014

We present a comparative study of the two-dimensional linear and nonlinear vertical and horizontal wave forces and overturning moment due to the unsteady flow of an inviscid, incompressible fluid over a fully submerged horizontal, fixed deck in shallow-water. The problem is approached on the basis of the level I Green–Naghdi (G–N) theory of shallow-water waves. The main objective of this paper is to present a comparison of the solitary and periodic wave loads calculated by use of the G–N equations, with those computed by the Euler equations and the existing laboratory measurements, and also with linear solutions of the problem for small-amplitude waves. The results show a remarkable similarity between the G–N and the Euler solutions and the laboratory measurements. In particular, the calculations predict that the thickness of the deck, if it is not “too thick,” has no effect on the vertical forces and has only a slight influence on the two-dimensional horizontal positive force. The calculations also predict that viscosity of the fluid has a small effect on these loads. The results have applications to various physical problems such as wave forces on submerged coastal bridges and submerged breakwaters.

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References

Hayatdavoodi, M., and Ertekin, R. C., 2012, “Nonlinear Forces on a Submerged, Horizontal Plate: The GN Theory,” The 27th International Workshop on Water Waves and Floating Bodies, Copenhagen, Denmark, Apr. 22–25, pp. 69–72.
Hayatdavoodi, M., and Ertekin, R. C., 2014, “Wave Forces on a Submerged Horizontal Plate. Part I: Theory and Modeling,” J. Fluids Struct (in press).
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Zhao, B., Duan, W., and Ertekin, R., 2014, “Application of Higher-Level GN Theory to Some Wave Transformation Problems,” Coastal Eng., 83, pp. 177–189. [CrossRef]
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Seiffert, B., Hayatdavoodi, M., and Ertekin, R. C., 2014, “Experiments and Computations of Solitary-Wave Forces on a Coastal-Bridge Deck. Part I: Flat Plate,” Coastal Eng., 88, pp. 194–209. [CrossRef]
Hayatdavoodi, M., Seiffert, B., and Ertekin, R. C., 2014, “Experiments and Computations of Solitary-Wave Forces on a Coastal-Bridge Deck. Part II: Deck With Girders,” Coastal Eng., 88, pp. 210–228. [CrossRef]
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Riggs, H. R., Suzuki, H., Ertekin, R. C., Kim, J. W., and Iijima, K., 2008, “Comparison of Hydroelastic Computer Codes Based on the ISSC VLFS Benchmark,” Ocean Eng., 35(7), pp. 589–597. [CrossRef]
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Ertekin, R. C., 1984, “Soliton Generation by Moving Disturbances in Shallow Water: Theory, Computation and Experiment,” Ph. D. dissertation, University of California at Berkeley, Berkeley, CA.
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Figures

Grahic Jump Location
Fig. 1

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.

Grahic Jump Location
Fig. 2

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).

Grahic Jump Location
Fig. 3

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.

Grahic Jump Location
Fig. 4

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)

Grahic Jump Location
Fig. 5

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)

Grahic Jump Location
Fig. 6

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.

Grahic Jump Location
Fig. 7

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.

Grahic Jump Location
Fig. 8

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.

Grahic Jump Location
Fig. 9

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.

Grahic Jump Location
Fig. 10

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.

Grahic Jump Location
Fig. 14

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.

Grahic Jump Location
Fig. 13

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.

Grahic Jump Location
Fig. 12

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.

Grahic Jump Location
Fig. 11

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.

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