c 2005 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 73 .... surface from the fluid, and (n4,n5,n6)=(x, y, z) Ã (nx,ny,nz).
J. Appl. Math. & Computing Vol. 17(2005), N0. 1 - 2, pp. 73 - 91
NUMERICAL METHOD IN WAVE-BODY INTERACTIONS S. H. MOUSAVIZADEGAN AND M. RAHMAN∗
Abstract. The application of Green’s function in calculation of flow characteristics around submerged and floating bodies due to a regular wave is presented. It is assumed that the fluid is homogeneous, inviscid and incompressible, the flow is irrotational and all body motions are small. Two methods based on the boundary integral equation method (BIEM) are applied to solve associated problems. The first is a low order panel method with triangular flat patches and uniform distribution of velocity potential on each panel. The second method is a high order panel method in which the kernels of the integral equations are modified to make it nonsingular and amenable to solution by the Gaussian quadrature formula. The calculations are performed on a submerged sphere and some floating spheroids of different aspect ratios. The excellent level of agreement with the analytical solutions shows that the second method is more accurate and reliable. AMS Mathematics Subject Classification : 76B07, 76M15. Key words and phrases : Potential flow, Green’s function, singularity, panel method, radiation, diffraction.
1. Introduction A combination of two independent classical problems is considered in order to find the hydrodynamic characteristics of motions of a body in time harmonic waves. One of them is the radiation problem where the body undergoes prescribed oscillatory motions in otherwise calm fluid, and the other is the diffraction problem where the body is held fixed in the incident wave field and determines the influence of it over the incident wave. The boundary integral equation methods (BIEM) or boundary element methods (BEM) are well known as the effective tools in computation of the boundary value problems associated with the wave body interactions. The main advantages of these methods are that they Received December 2, 2003. ∗ Corresponding author. c 2005 Korean Society for Computational & Applied Mathematics and Korean SIGCAM.
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reduce dimensionality of the problem by one and transform an infinite domain of interest to finite boundaries in which the far-field condition is automatically satisfied. There are two approaches which lead to different types of integral formulation. The so-called direct boundary integral formulation is derived through the application of Green’s second identity. Another one is referred to as the indirect boundary integral formulation, which represents an unknown function with the aid of a surface distribution of Green’s function. For more detailed description see Brebbia [1]. One of the most widely used boundary integral equation method (BIEM) is that of Hess and Smith [4], in which the indirect method is used to solve the problem of potential flow without free surface effect. Hess and Smith subdivided the body surface into N quadrilateral flat panels over which the source strength distribution was assumed to be uniform. Based on this low order method, several notable works were developed to calculate the wave body interactions, such as Faltinsen and Michelson [2], Garrison [3] and Inglis and Price [6]. Different modifications were applied to this low order method. These modifications are concerned with the geometrical description of the body surface, density distribution of the singularities on each panel, and how and where the body surface boundary condition are applied. A modification to the Hess and Smith method was presented by Landweber and Macagno [7]. The method mainly differs in the treatment of the singularity of the kernel of the integral equations and applying the Gaussian quadrature to obtain numerical solutions. Webster [12] developed a method by using triangular patches over which the source strength is chosen to vary linearly and the panels are submerged somewhat below the actual surface of the body. Some higher order panel methods in which the body surface and the velocity potential have been approximated with higher order polynomials have been developed during the past few years. Recently, Newman and Lee [10] have presented a high order panel method in which the surface of the body and the velocity potential are modeled by B-spline. It gives the ability to describe the geometry exactly. The direct boundary integral formulation is applied to obtain the solution of the radiation problem of a submerged sphere and the radiation and diffraction problems of several floating semi-ellipsoids of different aspect ratios in time harmonic waves. The radiation problem of the submerged sphere was solved by two methods. The first method is a low order method in which the original form of the free surface Green’s function is used and the body is modeled as a faceted form of triangular patches. The second metod is an arbitrary higher order method in which the non-singular form of the free surface Green’s function is applied to calculate the solutions. The radiation and diffraction problems of the semi-ellipsoids were solved by the second method which we call non-singular method (NSM). The Gaussian quadrature is applied in the NSM to discretize
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the integral equations. The Gaussian quadrature points are calculated exactly from the mathematical description of the body surface. The Gaussian points are also used as collocation points where the body surface boundary conditions are applied. The body can be considered as a single panel or multiple panels, depending on the geometry of the body. The number of Gaussian points which correspond to the order of the Gaussian quadrature can be adjusted with respect to the panel geometry. The comparison of the results with analytical solutions demonstrate the accuracy of the non-singular method. 2. Mathematical formulation We assume that the fluid is homogeneous, inviscid and incompressible and the fluid motion is irrotational. It is also assumed that the motions are small. We consider a body that is submerged or floating on the surface of the fluid. Two sets of coordinate systems are considered. One set is a right handed coordinate system, o − xyz, fixed in the fluid with oz opposing the direction of gravity and o − xy lying in the undisturbed free surface. The other set is attached to the body and is defined with respect to the geometry of the body. The body is subjected to an incoming waves of amplitude A and frequency ω propagating in positive x-direction. The total velocity potential may be written in general case as Φ(x, y, z, t) =
