Radiation and diffraction of a submerged sphere

Transcription

1 Radiation and diffraction of a submerged sphere S. H. Mousavizadeganl, M. Rahman2 & M. G. Satish3 'Dept. of Mechanical Engineering, Dalhousie University, Canada 2Dept. of Engineering Mathematics, Dalhousie University, Canada 'Dept. of Civil Engineering, Dalhousie University, Canada Abstract This paper presents an analytical investigation and numerical solutions to the problems of wave radiation and wave diffraction by a submerged sphere. We assume that the fluid is homogeneous, inviscid, and incompressible and the fluid motion is irrotational. The surge and the heave motions are studied and the hydrodynamic coefficients of these motions are calculated. The analytical solutions are obtained by expanding the velocity potential into a series of associated legendre functions in a spherical coordinate system. The problem is also solved numerically based on the boundary integral equation method (BIEM) using non-singular form of the Green's formula. The major advantages of this modified fomulation are that the exact geometry of the body can be applied to solve of the problem and the integral equations are amenable to solution by the quadrature formulas. The results of the computations are presented graphically in the form of non-dimensional parameters and good agreement between analytical and numerical solutions is observed. It is proposed that this modified formulation can be extended to the problem of motion of submerged or floating bodies in waves at forward speed. 1 Introduction Radiation and diffraction of time harmonic waves by submerged or floating bodies is of great importance for ocean engineers and naval architects. Diffraction by fixed structures such as offshore terminals and drilling rigs and radiation-diffraction by moored bodies such as tankers, floating offshore platforms are of practical instances in ocean engineering. In predicting the ship motions in waves, the forward-speeddependent terms in the free surface boundary condition can perhaps be treated as

2 14 Fluid Structure Interaction 11 nonhomogenous terms in the free-surface condition for radiation and diffraction at zero mean forward speed. Thus radiation and diffraction at zero mean forward speed is also of importance in naval architecture. The linearized problem associated with most practical marine structures can be solved by numerical methods. It is necessary to validate the numerical solutions before applying them into the real structures. Therefore, an analytical solution for the linearized potential flow for a small amplitude harmonic incoming wave is derived by using multipole expansion method. An analysis of the radiation and diffraction of a fully submerged body in an infinite water depth was investigated by Wang [10]. By employing special series solution, Wang solved the governing equation satisfied by the velocity potential. Wu [l21 considered a submerged spheroid and obtained analytical solutions for the linear forces. Rahman [5] obtained the fluid velocity potential in the form of double series of the associated Legendre functions for a completely submerged sphere in water of finite depth by using multipole expansions method. Rahman [6] studied the fields of the hydrodynamic diffraction pressure and fluid velocity around a submerged sphere in finite depth by using multipole expansion method. The same procedure is applied to find the fluid velocity potential and other fluid flow characteristics analytically. The importance of boundary integrallelements (BI/BE) methods in the solution of exterior Laplace's problem is universally recognized. The main advantage of (BIIBE) methods is that they 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 B1 formulation is derived through the application of Green's second identity, which represents an unknown function as the superposition of a single- and a double-layer potential. Another one is referred to as the indirect B1 formulation, which represents an unknown function with the aid of a surface distribution of Green's function with fictitious singularities of adjustable strengths. One of the most widely used BIEM is that of Hess [l], in which the indirect method is used to solve the problem of potential flow without free surface effect. Hess [l] subdivided the body surface into N quadrilateral flat panels over which the source strength distribution was assumed to be uniform. Landweber [3] devised a method which mainly differs from that of Hess [l] in the treatment of the singularity of the kernel of the integral equation and applying a quadrature to obtain numerical solutions, Noblesse [4]. The direct boundary integral formulation is applied to solve the radiation and diffraction problems. The term in the potential function is treated by the non-singalar method proposed by Landweber [3] and the harmonic terms of the potential function due to the free surfce effect are solved by using the routine submitted by Telste [g]. 2 Mathematical formulation Two sets of coordinate systems were considered. One is a right handed coordinate system fixed in the fluid with oz opposing the direction of gravity and o - xy lying in the undisturbed free surface. The other set is the spherical coordinate system

3 Fluid Structure Interaction (r, 8, $) with the origin at the center of sphere. The total velocity potential may be written as where ql and are the displacements for surge and heave motions. $1 and 43 are the velocity potentials corresponding to surge and heave oscillations of the body of unit amplitude. 4I is the incoming wave velocity potential with unit amplitude and is the wave diffraction velocity potential of unit amplitude. Figure 1: Sketch of the problem geometry and coordinates definition The velocity potentials for radiation and diffraction must satisfy the Laplace's equation V24 = 0 in the whole fluid domain, the free surface condition the bottom boundary condition d+/dx = 0 for x + -m and the radiation condition. Where K = w2lg is called wave-number and R = Jw. Also, the components of velocity potentials should satisfy the body surface condition d4j - -iwnj j = 1,3 for radiation problem dn d = 0 for diffraction problem. (5) (,)(m' + 4 ~ )

4 16 Fluid Structure Interaction II 3 Analytical treatment A combination of two independent classical problems, the radiation problem, where the body undergoes prescribed oscillatory motions in otherwise calm fluid, and the diffraction problem, where the body is held fixed in the incident wave field and determined the influence of it over the incident wave, can be considered. The multipole expansion method is applied to solve these problems to find the hydrdynamic characteristcs. 3.1 Incident potential The incoming waves of amplitude A and frequency W propagating in positive X- direction can be described by OI = (Ag/w)e e ikrcos+. Using Rahman[S], the incident wave velocity potential can be expressed in terms of the associated Legendre functions as where EO = 1 and E, = 2 for m Diffraction potential The diffraction velocity potential satisfies the linear free surface condition, bottom condition, radiation condition and body surface condition. It can be expressed by making it $ independent as, Rahman[6] m=o where the +-independent velocity potential is n=m G: are the multipole potantials and can be expressed as, Thorne[9] P," (cos 6') (- G: = Tn+l + (n - m)! k-k + (-'Im+" 2TiKn+leK('+h) Jm (KR) (9) (n- m)! The second term can be expanded in the region near the body surface into a series of the associated Legendre functions and the potential can be expressed as m=0 n=m 00 P," (cos 6') + C.(n, m)r8+m~zm (cos%) - ZaiKn+lex("f") J,(KR) COS m$ ((0) (n- m)! l

5 where Fluid Structure Interaction I1 17 The AT coefficients are calculated by imposing the body surface condition for the diffraction problem. From the expression for QD and QI and the body surface condition we get - (-1)mCn,iKn+2e~(z+h) sin (n- m)! (KR)- J ~+I (KR)] 3.3 Exciting force The hydrodynamic pressure in the linearized potential theory related to the diffraction problem is p = -iwp(41 + +D). The exciting force on the sphere can be obtained by integration of the hydrodynamic pressure associated with the incident and diffraction velocity potentials over its surface, as 25T F, = %[ ls - iwpa2 db l (QI + l Q,) l,=,n sin 8d+ e-iut (13) where F, = (F,, F,, F,) and n = (n,, n,, n,). The outward normals on the body surface can be written as, n = (-l)jp: (cos 8) cos j+, where j = 1 for n, and j = 0 for n,. Total velocity potential at the surface of the sphere can be obtained by, Rahmanr n+ 1 &D (8, +) = (41 + $D) I,=, = a E E 7 AT PC (cos 8) cos m+ I" m=o n=a(m) (14) where A(m) = 1 for m = 0 and A(m) = 0 for m # 0. Considering Fej = %[feje-iwt] where fej are called the complex exciting force, and using (6), (10) and othogonality properties of cosine functions and associated Legendre polynomials, the complex exciting forces can be written as 4ipma3 (l + f e3 - ~j (l- j)! Considering j = 1 for exciting force in X-direction and j = 0 for exciting force in z-direction.

6 18 Fluid Structure Interaction Radiation problem We solve the radiation problem for a surging and a heaving sphere. The surge potential and heave potantial satisfy the same set of equations and boundary conditions as the diffraction problem except for the body surface condition. The radiation potantial can be expressed by a series of associated Legendre functions in the region near the body as, Thorne[9] where 00 + [C,- id,]r8p,"(cos 8) cos m$ s=m l 2~(-1)~+~ D, = p+s+le-2kh (m + s)!(n - m)! The complex coefficients BF are calculated by imposing the body surface condition for radiation problem. They can be found by solving the folowing infinite system of linear simultanous equations for an infinite number of unknown, Srokosz [l. The components of force and moment for the radiation problem can be written in the form of Fk = %{C:=, vje-iwt fkj). The complex force coeficients, fkj, are 2 fkj = W pkj - iwhkj = -iwplo 4jnxds (21) where pkj are the added mass coefficients and hkj are the damping coefficients. Using (17 ) and (20 ) and also orthogonality properties of associated Legendre functions, the added mass and damping coeficients for surge and heave can be written as where 8 and R indicate the imaginary and the real part of the coefficients.

7 4 Numerical treatment Fluid Structure Interaction The boundary-value problems of submerged sphere in time harmonic waves are also solved numerically based on boundary integral equation method (BIEM). It is well known in the potential theory, these boundary-value problems may be re-cast as integral equations via Green's theorem, Korsmeyer[2]. for radiation problem (24) P E SB for diffraction problem (25) where G is the free-surfce Green's function and Sg is the surface of the sphere. G satisfies all boundary conditions except the body boundary conditions and is defined,wehausen [l l] Where The Green's function may be written in the form of G($, q) = $ + H(p, q), where the term $ is the singular part and H(p, q) is the harmonic part of it. The method proposed by Landweber [3] is adopted to solve these integral equations. Based on that procedure, the integral equations yield the form 4, W j b ) 4P) an,

8 20 Fluid Structure Interaction 11 where ~ (q) is the source distribution over the body surface to make it equipotantial of potential 4,. It can be calculated through the iterative formula, Yang [l31 d 1 nk-(-1 dn, T - ffkz(l)) T d ~, Since 4, is constant in the interior of an equipotantial surface, its value may conveniently be computed by locating point p at the origin as The potentails and the hydrodynamic coefficients can be found by applying the Gaussian quadrature in solution of these integral equations and other relevant formulas. Figure 2: Surge Added-mass Coefficient as a function of wave-number for various values of submegence depth, (*, e, 0,... : calculated results) 5 Results and discussion The integral equations can be discretized along the surface of the body by applying the quadrature formulas. The discritized equations are linear systems with N unknown for the radiation and diffraction problems. They can be represented in the simple form of [A][x] = [B]. where [A] is the coefficient matrix formed by the derivative of Green's function, [B] is the coefficient matrix formed by multiplication of the Green's function and the free surface boundary condition for radiation problem or the incident wave potential for diffraction problem at the integration points and [X] the unknown matrix of velocity potentials around the body.the coefficient matrices and also the body surface boundary conditions are dependent on

9 Fluid Structure Interaction I1 21 the geometry of the body. If the exact expression for the geometry can be used in the numerical computation the accuracy of the result will be highly improved. This is one of the most important features, using the exact geometry, of the adopted method in treating the integral equations. Figure 3: Heave added-mass coefficient as a function of Ka and hla Figure 4: Heave damping coefficient as a function of Ka and hla Calculation were performed for a submerged sphere of radius one at different depths of submergence and various wave frequencies. Sixteen and 32 Gauss- Lagendre integration points were distributed along the 8 and $ directions. The results of the computation for non-dimensional surge added-mass, pl~/p($).rra3 and

10 22 Fluid Structure Interaction 11 heave added-mass p33/p($)m3 and heave damping coefficient are presented in figures 2, 3 and 4 as a function of wave-number for different depth of submergence of the sphere. The results were compared with the works by Wang [10]. These figures show that the proposed modified integral equations can yield satisfactory results. The absolute errors of the numerical results at differnt wave numbers are greatly improved in comparison with the other methods such as panel method. It is intended to extend these modified integral equation to the calculation of hydrodynamic charateristcs of submerged or floating bodies at forward speed. Acknowledgements We are very grateful to Natural Sciences and Enginnering Research Council ( NSERC ) of Canada for financial support leading to this paper. References [l] Hess, J. L., Smith, A. M. O., Calculation of nonlifting potenatial flow about arbitrary three-dimensional bodies, Journal of Ship Research, 8, pp , Sept [2]Korsmeyer, F. T., Bingham, H. B., The forward speed diffraction problem, Journal of Ship Research, Vol. 42, No. 2, pp , June 1998 [3] Landweber, L., Macagno, M., Irrotational flow about ship forms, IIHR Report No. 123, Iowa Institute of Hydraulic Research, The University of Iowa, Iowa City, Iowa, [4] Noblesse, F., Triantafyllou, G., Explicit approximation for calculating potential flow about a body, Journal of Ship Research, Vol. 27, No. 1, pp. 1-12, March [5] Rahman, M., Simulation of diffraction of ocean waves by a submerged sphere in finite depth, Applied Ocean Research, 23, pp ,2001. [6] Rahman, M., Effect of diffraction and radiation on a submerged sphere, IJMMS, 28:9, pp ,2001. [7] Srokosz, M. A., The submerged sphere as an absorber of wave power, J. Fluid Mech., Vol. 95, Part 4, pp , [8]Telste, J. G., Noblesse, F., Numerical evauation of the Green function of waterwave radiation and diffraction, Journal of Ship Research, Vol. 30, No. 2: pp , June [9] Thorne, R. C., Multipole expansions in the theory of surface waves, Proc. Camb. Phil. Soc., 49, , [10]Wang, S., Motions of spherical submarine in waves, Ocean Engng. Vol. 13. NO. 3, pp , [l liwehausen, J. V., Laitone, E. V., Surface Waves, Encyclopedia of phisics, Vol. 4, Fluid dynamics 3, Springer-Verlac, Berlin, [l21 Wu, G. X., Eatock Taylor, R., The exciting force on a submerged spheroid in regular waves, J. Fluid Mech. Vol. 182, pp , [l31 Yang, S. A., On the singularities of Green's formula and its normal derivatives with an application to surface-wave-body interaction problems, Int. J. Numer. Math. Engng. 47: pp ,2000.