Research Papers: CFD and VIV

Numerical Analysis of Second-Order Mean Wave Forces by a Stabilized Higher-Order Boundary Element Method

[+] Author and Article Information
Yan-Lin Shao

Department of Mechanical Engineering,
Technical University of Denmark,
Nils Koppels Allé,
Kgs. Lyngby 2800, Denmark;
Shipbuilding Engineering Institute,
Harbin Engineering University,
Harbin 150001, China
e-mail: yshao@mek.dtu.dk

Contributed by the Ocean, Offshore, and Arctic Engineering Division of ASME for publication in the JOURNAL OF OFFSHORE MECHANICS AND ARCTIC ENGINEERING. Manuscript received June 26, 2018; final manuscript received December 3, 2018; published online January 22, 2019. Assoc. Editor: Yi-Hsiang Yu.

J. Offshore Mech. Arct. Eng 141(5), 051801 (Jan 22, 2019) (9 pages) Paper No: OMAE-18-1081; doi: 10.1115/1.4042197 History: Received June 26, 2018; Revised December 03, 2018

A stabilized higher-order boundary element method (HOBEM) based on cubic shape functions is presented to solve the linear wave-structure interaction with the presence of steady or slowly varying velocities. The m-terms which involve second derivatives of local steady flow are difficult to calculate accurately on structure surfaces with large curvatures. They are also not integrable at the sharp corners. A formulation of the boundary value problem in a body-fixed coordinate system is thus adopted, which avoids the calculation of the m-terms. The use of body-fixed coordinate system also avoids the inconsistency in the traditional perturbation method when the second-order slowly varying motions are larger than the first-order motions. A stabilized numerical method based on streamline integration and biased differencing scheme along the streamlines will be presented. An implicit scheme is used for the convective terms in the free surface conditions for the time integration of the free surface conditions. In an implicit scheme, solution of an additional matrix equation is normally required because the convective terms are discretized by using the variables at current time-step rather than that from the previous time steps. A novel method that avoids solving such matrix equation is presented, which reduces the computational efforts significantly in the implicit method. The methodology is applicable on both structured and unstructured meshes. It can also be used in general second-order wave-structure interaction analysis with the presence of steady or slowly varying velocities.

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Zhao, R. , and Faltinsen, O. M. , 1989, “A Discussion of the mj-Terms in the Wave-Current-Body Interaction Problem,” Third International Workshop on Water Waves and Floating Bodies, Oystese, Norway, May 7–10. http://www.iwwwfb.org/Abstracts/iwwwfb04/iwwwfb04_53.pdf
Nestegård, A. , and Shao, Y. L. , 2014, “Wave Forces on Floating Units in Extreme Waves and Current,” DNVGL, Høvik, Norway, DNV Report No. 2013-1640.
Stansberg, C. T. , Kaasen, K. E. , Abrahamsen, B. C. , Nestegård, A. , Shao, Y. L. , and Larsen, K. , 2015, “Challenges in Wave Force Modelling for Mooring Design in High Seas,” Offshore Technology Conference, Houston, TX, May 4–7, Paper No. OTC-25944-MS.
ITTC, 2011, “The Specialist Committee on Computational Fluid Dynamics—Final Report and Recommendation to the 26th ITTC,” 26th International Towing Tank Conference, Rio de Janeiro, Brazil, Aug. 28–Sept. 3. https://ittc.info/media/5528/09.pdf
Yan, H. , and Liu, Y. , 2011, “An Efficient High-Order Boundary Element for Nonlinear Wave-Wave and Wave-Body Interactions,” J. Comput. Phys., 230(2), pp. 402–424. [CrossRef]
Ducrozet, G. , Bingham, H. B. , Engsig-Karup, A. P. , and Ferrant, P. , 2010, “High-Order Finite Difference Solution for 3D Nonlinear Wave-Structure Interaction,” J. Hydrodyn., 22(Suppl. 5), pp. 225–230. [CrossRef]
Shao, Y. L. , and Faltinsen, O. M. , 2014, “A Harmonic Polynomial Cell (HPC) Method for 3D Laplace Equation With Application in Marine Hydrodynamics,” J. Comput. Phys., 274, pp. 312–332. [CrossRef]
Shao, Y. L. , and Faltinsen, O. M. , 2014, “Fully-Nonlinear Wave-Current-Body Interaction Analysis by a Harmonic Polynomial Cell (HPC) Method,” ASME J. Offshore Mech. Arct. Eng., 136(3), p. 031301. [CrossRef]
Sclavounos, P. D. , 1992, “The Quadratic Effect of Random Gravity Waves on a Vertical Boundary,” J. Fluid. Mech., 242(1), pp. 475–489. [CrossRef]
Shao, Y. L. , 2016, “Higher-Order Effects on the Mean Wave Drift Forces,” Sevan Marine, Arendal, Norway, Report No. 59340-SM-U-RA-18001.
Faltinsen, O. M. , and Loken, A. , 1978, “Drift Forces and Slowly Varying Horizontal Forces on a Ship in Waves,” Symposium on Applied Mathematics Dedicated to the Late Prof. Dr. R. Timman, Delft, The Netherlands, Jan. 11–13, pp. 22–41.
Molin, B. , 1979, “Second-Order Diffraction Loads Upon Three-Dimensional Bodies,” Appl. Ocean Res., 1(4), pp. 197–202. [CrossRef]
Chen, X. B. , 2007, “Middle-Field Formulation for the Computation of Wave-Drift Loads,” J. Eng. Math., 59(1), pp. 61–82. [CrossRef]
Liang, H. , and Chen, X. B. , 2017, “A New Multi-Domain Method Based on an Analytical Control Surface for Linear and Second-Order Mean Drift Wave Loads on Floating Bodies,” J. Comput. Phys., 347, pp. 506–532. [CrossRef]
Lee, C. H. , 2007, “On the Evaluation of Quadratic Forces on Stationary Bodies,” J. Eng. Math., 58(1–4), pp. 141–148. [CrossRef]
Xiang, X. , and Faltinsen, O. M. , 2010, “Maneuvering of Two Interacting Ships in Calm Water,” 11th Practical Design of Ships and Other Floating Structures, Rio de Janeiro, Brazil, Sept. 19–24, Paper No. 2010-2019. https://www.researchgate.net/publication/266472289_Maneuvering_of_Two_Interacting_Ships_in_Calm_Water
Shao, Y. L. , 2010, “Numerical Potential-Flow Studies on Weakly-Nonlinear Wave-Body Interactions With/Without Small Forward Speeds,” Ph.D. thesis, Department of Marine Technology, Norwegian University of Science and Technology, Trondheim, Norway. https://brage.bibsys.no/xmlui/handle/11250/237758
Shao, Y. L. , and Faltinsen, O. M. , 2010, “Use of Body-Fixed Coordinate System in Analysis of Weakly Nonlinear Wave-Body Problems,” Appl. Ocean Res., 32(1), pp. 20–33. [CrossRef]
Shao, Y. L. , and Faltinsen, O. M. , 2012, “Second-Order Diffraction and Radiation of a Floating Body With Small Forward Speed,” ASME J. Offshore Mech. Arct. Eng., 135(1), p. 011301.
Zienkiewicz, O. C. , Taylor, R. L. , and Nithiarasu, P. , 2014, The Finite Element Method for Fluid Dynamics, 7th ed., Butterworth-Heinemann, Oxford, MA.
Servan-Camas, B. , 2016, “A Time-Domain Finite Element Method for Seakeeping and Wave Resistance Problems,” Ph.D. thesis, School of Naval Architecture and Ocean Engineering, Technical University of Madrid, Madrid, Spain. http://oa.upm.es/39794/1/BORJA_SERVAN_CAMAS.pdf
Faltinsen, O. M. , and Timokha, A. N. , 2009, Sloshing, Cambridge University Press, New York.
Geuzaine, C. , and Remacle, J.-F. , 2009, “Gmsh: A Three-Dimensional Finite Element Mesh Generator With Built-In Pre- and Post-Processing Facilities,” Int. J. Numer. Methods Eng., 79(11), pp. 1309–1331. [CrossRef]
Shao, Y. L. , and Faltinsen, O. M. , 2014, “A Numerical Study of the Second-Order Wave Excitation of Ship Springing by a Higher-Order Boundary Element Method,” Int. J. Nav. Archit. Ocean Eng., 4(4), pp. 1000–1013. https://www.sciencedirect.com/science/article/pii/S2092678216302680
Shao, Y. L. , and Helmers, J. B. , 2014, “Numerical Analysis of Second-Order Wave Loads on Large-Volume Marine Structures in a Current,” ASME Paper No. OMAE2014-24586.
MacCamy, R. C. , and Fuchs, R. A. , 1954, “Wave Forces on Piles: A Diffraction Theory,” Beach Erosion Board, Corps of Engineers, Washington, DC, Report No. TM-69. http://acwc.sdp.sirsi.net/client/en_US/default/index.assetbox.assetactionicon.view/1007800?rm=BEACH+EROSION+0%7C%7C%7C1%7C%7C%7C2%7C%7C%7Ctrue
Eatock Taylor, R. , and Hung, S. M. , 1987, “Second-Order Diffraction Forces on a Vertical Cylinder in Regular Waves,” Appl. Ocean Res., 9(1), pp. 19–30. [CrossRef]
Cheung, K. F. , Isaacson, M. , and Lee, J. W. , 1996, “Wave Diffraction Around a Three-Dimensional Body in a Current,” ASME J. Offshore Mech. Arct. Eng., 118(4), pp. 247–252. [CrossRef]
Kinoshita, T. , and Bao, W. , 1996, “Hydrodynamic Forces Acting on a Circular Cylinder Oscillating in Waves and a Small Current,” J. Mar. Sci. Technol, 1(3), pp. 155–173. [CrossRef]
Kinoshita, T. , Bao, W. , and Zhu, R. , 1997, “Higher-Order Boundary Element Method for the Interaction of a Floating Body With Both Waves and Slow Current,” J. Mar. Sci. Technol., 2(4), pp. 268–279. [CrossRef]
Zhao, R. , and Faltinsen, O. M. , 1989, “Interaction Between Current, Waves and Marine Structures,” Fifth International Conference on Numerical Ship Hydrodynamics, Hiroshima, Japan, Sept. 24–28, pp. 513–528. https://www.nap.edu/read/1604/chapter/43#514
Liu, Y. H. , Kim, C. H. , and Kim, M. H. , 1993, “The Computation of Mean Drift Forces and Wave Run-Up by Higher-Order Boundary Element Method,” Int. J. Offshore Polar Eng., 3(2), pp. 101–106. https://www.onepetro.org/download/journal-paper/ISOPE-93-03-2-101?id=journal-paper%2FISOPE-93-03-2-101
Kim, C. H. , 2008, Nonlinear Waves and Offshore Structures, World Scientific Publishing, Singapore.
Newman, J. N. , 1977, Marine Hydrodynamics, MIT Press, Cambridge, MA.
Mavrakos, 2018, private communication.
Mavrakos, S. A. , 1988, “The Vertical Drift Force and Pitch Moment on Axisymmetric Bodies in Regular Waves,” Appl. Ocean Res., 10(4), pp. 207–218. [CrossRef]
Liang, 2018, private communication.


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Fig. 1

Definition of different coordinate systems

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Fig. 2

The relationship between the calm-water surface and the xy-plane

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Fig. 3

Definition of the wave elevations observed in the body-fixed coordinate system oxyz and the inertial coordinate system OXYZ

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Fig. 4

12-node cubic order boundary elements on mean free surface and a bottom mounted vertical cylinder

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Fig. 5

The 12-node cubic element in the physical plane

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Fig. 6

The 12-node cubic element in the ξ−η plane. The numbers are indices for the transformed coordinates (ξj,ηj,0) corresponding to the coordinates (xj,yj,zj) in the physical plane. The lengths of the four sides are identical and equal to 2.

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Fig. 7

Streamlines close to a circular cylinder. Only half of the cylinder and streamlines are shown.

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Fig. 8

Upwind points along a streamline. Filled circle is the free point where streamline-differentiation will be performed. Filled triangles are the upstream points next to the point of interest.

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Fig. 9

Comparison of the nondimensional amplitude of first-order in-line diffraction force with the analytical results based on MacCamy and Fuchs's [26] theory. A is the incident wave amplitude. h = R.

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Fig. 10

Comparison of the nondimensional mean-drift force on a bottom-mounted circular cylinder. A is the incident wave amplitude. h = R.

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Fig. 11

Nondimensional horizontal mean drift force on a vertical circular cylinder versus kR. A is the wave amplitude. d = h=R, Fr = −0.1. k is the incident wavenumber.

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Fig. 12

The horizontal wave drift force on a fixed truncated circular cylinder. h=2R, d=R. k is the wavenumber of the incident waves.

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Fig. 13

The numerical results of vertical mean wave force. The quadratic part of the second-order force is calculated based on the reformulation in Eq. (24). Comparisons with the near-field and far-field results of Zhao and Faltinsen [31] are made. ω0 is the frequency of the incident wave, R is the radius of the cylinder, and g is the gravitational acceleration.



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