Transactions on Modelling and Simulation vol 10, 1995 WIT Press, ISSN X

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1 Hydrodynamic coefficients and motions due to a floating cylinder in waves D.D. Bhatta, M. Rahman Department of Applied Mathematics, Technical University of Nova Scotia, Halifax, Nova Scotia, Canada B3J 2X4 Abstract Hydrodynamic coefficients and responses due to surge, heave and pitch motions induced by wave excitation for a floating vertical circular cylinder in water of finite depth are formulated in this paper. Both the combined effects of scattering and radiation are considered in evaluating the total velocity potential. The total velocity potential is decomposed as the linear combination of four velocity potentials; one due to scattering in the presence of an incident wave on fixed structure (diffraction problem), and the other three due to radiation respectively by surge, heave and pitch motion on calm water (radiation problem). For each case, the velocity potential is derived by considering two regions, namely, interior region and exterior region. The complex matrix equations are solved numerically to determine the unknown coefficients which are used to compute the hydrodynamic quantities. Numerical results of hydrodynamic coefficients and the motions due to surge, heave and pitch are presented in graphical forms for different depth to radius and draft to radius ratios. The total forces due to the combined effects of diffraction and radiation are also derived and presented in graphical forms for different wave parameters. 1 Introduction The evaluation of the hydrodynamic coefficients and forces on offshore structures is one of the most important tasks for ocean engineers. The forces exerted by surface waves on offshore structures such as offshore drilling rigs or submerged oil storage tanks are of important considerations in the design of large submerged or semisubmerged structures. So accurate prediction of these loads is absolutely necessary to design safe offshore structures. When the structure is large compared to wavelength, the incident waves upon arriving at the structure undergo significant scattering, and hence the diffraction theory is to be used to account for the scattering. There are two types of offshore structures : floating and fixed. In this paper, we consider floating structures. A rigid floating structure may undergo six degrees of freedom : three translational and three rotational. Assuming a suitable coordinate system, OXYZ, the translational motions in the x, y, and z directions are referred as surge, sway, and heave respectively; and the rotational motions about x, y, and z axes are referred as roll, pitch, and yaw respectively. Here z axis is considered to be vertically

2 396 Computational Methods and Experimental Measurements upwards from its still water level. Often the structure is restrained to have fewer degrees of freedom due to the type of mechanical connection used to fasten it to the seafloor. Many scientific investigations have been performed in the past by many well-known scientists and engineers in this field. The problem of scattering of surface waves by a circular dock was carried out by Miles and Gilbert [1] and then by Garrett [2]. Garrett presented the results for the horizontal and vertical force and moment on the dock. Black, Mei and Bray [3] have calculated the wave forces on a truncated cylinder which either extends to the free surface or rests on the seabed. Isaacson [4] extended Garrett's method for a submerged truncated cylinder sitting on the sea-bed. Numerical results for the added mass and damping coefficients of semi-submerged two-dimensional heaving cylinders in water offinitedepth were presented by Bai [5]. Yeung [6] presented a set of theoretical added masses and damping coefficients for a floating circular cylinder in finite-depth water. Sabuncu and Calisal [7] obtained hydrodynamic coefficients for vertical cylinders at finite water depth. Rahman and Bhatta [8] have obtained closed form analytical solutions for the added mass and damping coefficients due to an oscillating circular cylinder in waves. It is worth mentioning here that all these researchers studied scattering and radiation problem independently, but did not investigate the combined effects of the radiation and diffraction to evaluate the total wave loads on a floating cylinder. This paper concerns with the investigation of the combined effect of diffraction and radiation by afloatingcircular cylinder. The analysis is made considering thefirstorder theory. Analytical solutions have been presented to evaluate the hydrodynamic coefficients and motions for a vertical circular cylinder due to surge, heave and pitch motion in water of finite depth in the presence of an incident wave. The results for total wave loadings due to the combined effects of diffraction and radiation by a floating cylinder are presented in graphical forms. 2 Mathematical formulation for a floating cylinder We assume that the fluid is incompressible, the fluid motion is irrotational and the waves are of small amplitude. Here we consider the coefficients related to the motion with three degrees of freedom, namely, two translational motions in the x and z direction, i.e. surge (back and forth) and heave (up and down) and one rotational motion about y direction,i.e. pitch motion. We consider a surface wave of amplitude A incident on a vertical circular cylinder of radius a in water offinitedepth h. The body is assumed to have motions with three degrees of freedom in the presence of incident wave with angular frequency a. The wave is parallel to x-axis at the time of incidence

3 Computational Methods and Experimental Measurements 397 on the cylinder and propagating along positive direction. The draft of the cylinder in water is b. The geometry is depicted in Figure 1. We consider the cylindrical coordinate system (r, 0, z) with z vertically upwards from the still water level ( SWL ), r measured radially from the z- axis and 0 from the positive x -axis. For an incompressible and inviscidfluid,and for small amplitude wave theory with irrotational motion, we can introduce a velocity potential $ which can be written as $(r, 0,z,t) = #e[<^(r, 0,z)e~'**] where Re stands for the real part. From Bernoulli's equation we get pressure, P(r, 0, z,t), as P - /?ff- where p is thefluiddensity. The force components F^.Fy.F^ along x, y, z directions are given by Fx = - I * I P(a,0,z,t)acosOdzdO Je=o J-b rl* yo Fy = - I \ P(a,e,z,t)asinOdzde J0=0 J-b F, = f [* P(r,0,-b,t)rdrdO Je=o Jo respectively. Since the incident wave is parallel to x-axis at the time of incidence, the nonzero horizontal component is F^. Because of the linearity of the situation, the velocity potential <^>(r, 0, z) can be decomposed into four velocity potentials fa, <^i, fa and fa where fa is the velocity potential due to the diffraction of an incident wave acting on the fixed cylinder, and <^>i, fa and fa are velocity potentials due to the radiation of surge, heave and pitch respectively. Thus (f) can be written as <j> = ^ + 6^1+6^3 + 06^5 where 6, 6, 6 are displacements for surge, heave and pitch respectively. Now by dividing the entire fluid domain into two domains, (a) interior domain : region below the cylinder i.e. r < a, h < z < 6; (b) exterior domain : region for r > a and h < z < 0, we write the velocity potential for the interior domain as 0* and the velocity potential for the exterior domain as <^. Then F^ and FZ can be written as the real part of /*e~*** and fze~* * respectively where fx and fz are given by e=o J-b -f 06^5(0, 0, z)} cos OdzdO (1) e=o Jo +ais</>t(r,0,-b)}rdrm. (2) To obtain the velocity potential fa the following boundary value problem is to be solved VV = 0 (3)

4 398 Computational Methods and Experimental Measurements cr^s g = 0 on z 0 r > a (4) oz 86 -I- = 0 on z = -h (5) with other boundary conditions as stated below : a) for surge motion -- = 0 on z = -b 0 < r < a (6) c\ I -? = -iaii cos 9 on r = a -6< z < 0 (7) b) for heave motion o / -f = -2<7&3 on z = -6 0 < r < a (8) <9z c\ i -I- = 0 on r = a -b< z < 0 (9) c) for pitch motion f\ I = icr $rcoso on z b 0 < r < a (10) oz 9<^ = ia^zcoso on r a b<z<0. (11) First we consider this problem by separating it into two problems, diffraction problem and radiation problem. We then superpose the solutions to obtain the total solution. The diffraction problem will give us the exciting force and from the radiation problem we will get the hydrodynamic coefficients, added mass or added moment of inertia and damping. 3 Evaluation of the forces This section is concerned with the evaluation of wave forces due to the combined effects of diffraction and radiation. The component of the horizontal force /p can be computed from where j^d is the diffraction force, f^\ the force due to the surge motion and /rs the force due to the pitch motion. The vertical force component /^ can be written as /, = /* + /,3 (13) where /^ is the diffraction force, and /^ the force due to heave motion. Due to lengthy mathematical expressions involved in the force calculation, we are unable to show them here but the interested reader is referred to the work of Rahman and Bhatta [9].

5 Computational Methods and Experimental Measurements Numerical results The numerical results for the exciting forces and the total forces are presented in graphical forms. Figure 2 shows the results of dimensionless horizontal exciting force as a function of AQG. This force increases from zero rapidly to a maximum value around AQG = 1, and then goes to an asymptotic value of 0.15 as XQO, becomes larger. It is interesting to note that the force increases steadily with the increase of depth to radius ratio. This is a classical behavior of the exciting force reported in literatures. Vertical exciting force is depicted in Figure 3 for the same set of parameters showing the usual trend. This force is unity at Aoa = 0 and then goes to zero as AQO increases. Total horizontal force is shown in Figure 4 for the set of parameters considered earlier. Although from this figure it appears that there exists a resonant diffraction parameter Ago at which the total force becomes maximum, it can be shown by taking more number of points in the computation that these are smooth curves. As Aoa increases and in practice after Aoa = 2 the total horizontal force decreases very smoothly showing an asymptotic behavior for large Aoa. 5 Conclusions A systematic mathematical technique has been presented in this paper to compute the total wave forces on a vertical circular cylinder due to surge, heave and pitch motion infinitedepth water in the presence of an incident wave. Analytical solution for the total velocity potential is obtained by dividing the whole boundary value problem into two problems, namely, diffraction problem of an incident wave acting on the fixed cylinder and radiation problem of the cylinder forced to move in otherwise still water. Mathematical solutions for the boundary value problems are obtained in two physical regions, namely, interior region and exterior region. The exciting force components are obtained by solving the diffraction problem and the added mass, added moment of inertia and damping coefficients are obtained by solving the radiation problem. Then the responses due to surge, heave and pitch induced by the wave excitation are determined from the equation of motion of the floating cylinder. Using Bernoulli's equation, pressure is computed which is used to determine the wave forces. Results for different depth to radius and draft to radius ratios are presented in various figures. Acknowledgement The authors are very grateful to the Natural Sciences and Engineering Research Council of Canada for financial support leading to this paper.

6 400 Computational Methods and Experimental Measurements References 1. Miles, J. W. and Gilbert, J. F. 1968; Scattering of gravity waves by a circular dock, Journal of Fluid Mechanics, 34, Garrett, C. J. R. 1971; Wave forces on a circular dock, Journal of Fluid Mechanics, 46, Black, J. L., Mei, C. C. and Bray, C. G. 1971; Radiation and scattering of water waves by rigid bodies, Journal of Fluid Mechanics, 46, Isaacson, M. 1979; Wave forces on compound cylinders, Proceedings of the Civil Engineering in the Oceans IV, ASCE, San Francisco, I, Bai, K. J. 1977; The added mass of two-dimensional cylinders heaving in water offinitedepth, Journal of Fluid Mechanics, 81, Yeung, R. W. 1981; Added mass and damping of a vertical cylinder in finite-depth water, Applied Ocean Research, 3, 3, Sabuncu, T. and Calisal, S. 1981; Hydrodynamic coefficients for a vertical circular cylinders at finite depth, Ocean Engng., 8, Rahman, M. and Bhatta, D. D. 1993; Evaluation of added mass and damping coefficient of an oscillating circular cylinder, Applied Mathematical Modelling, 17, Bhatta, D. D. and Rahman, M. 1993; Hydrodynamic coefficients and motions due to floating structures in waves, Department of Applied Mathematics, TUNS, Technical Report, 48 pages.

7 Computational Methods and Experimental Measurements 401 Figure 1 : Definition Sketch to-2.0, to-1.0 to.2.0,to.04 J I «I Figure 2 : Dimensionless horizontal exciting force.

8 402 Computational Methods and Experimental Measurements 1.0 o ' Figure 3 : Dimensionless vertical exciting force. 1.0 A - * o l\ Nl Wt-2.0.b/t-1.0 \J I I Figure 4 ' Dimensionless total horizontal force.