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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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Figures

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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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