COMPUTATION OF ADDED MASS AND DAMPING COEFFICIENTS DUE TO A HEAVING CYLINDER

J. Appl. Math. & Computing Vol. 3(007), No. 1 -, pp. 17-140 Website: http://jamc.net COMPUTATION OF ADDED MASS AND DAMPING COEFFICIENTS DUE TO A HEAVING CYLINDER DAMBARU D BHATTA Abstract. We present the boundary value problem (BVP) for the heave motion due to a vertical circular cylinder in water of finite depth. The BVP is presented in terms of velocity potential function. The velocity potential is obtained by considering two regions, namely, interior region and exterior region. The solutions for these two regions are obtained by the method of separation of variables. The analytical expressions for the hydrodynamic coefficients are derived. Computational results are presented for various depth to radius and draft to radius ratios. AMS Mathematics Subject Classification: 76B15, 35Q35, 35J05. Key words and phrases : Water wave, heave motion, added mass, damping coefficients, circular cylinder. 1. Introduction The determination of wave loads on an offshore structure immersed in water in the presence of a free surface is one of the difficult tasks in the design of the structure. 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 semimerged structures. It is useful to calculate the hydrodynamic coefficients on the structure due to translational and rotational motions. Bai [1] presented numerical results for the added mass and damping coefficients of semi-submerged two-dimensional heaving cylinder in water of finite depth. He showed that the added mass is bounded for all frequrencies in water of finite depth. Dean and Dalrymple [3] presented a review of potential flow hydrodynamics. Solutions for standing and progressive small amplitude water waves provide the basis for application to numerous problems of engineering interest. Received August, 005. Revised September 6, 006. c 007 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 17

18 Dambaru Bhatta They discussed the formulation of the linear water ware theory and development of the simplest two-dimensional solution for standing and progressive waves. An introduction to various offshore structures are described by Chakrabarti []. He presented wave theories using stream and potential functions and wave force on structures of common geometrical shape such as cylinder, sphere. Floating structure dynamics were discussed by him in details. Numerical results for the added mass of bodies heaving at low frequency in water of finite depth were presented by McIver and Linton [6]. Rahman and Bhatta [8] presented the closed form solutions for the added mass and damping coefficient of a large surface-piercing bottom-mounted vertical circular cylinder undergoing horizontal oscillations. They studied the high-frequency and low-frequency behavior using asymptotic forms. Debnath [4] presented theoretical studies of nonlinear water waves over the last few decades. His work is primarily devoted to the mathematical theory of nonlinear water waves with applications. He studied the theory of nonlinear shallow water waves and solitons, with emphasis on methods and solutions of several evolution equations that are originated in the theory of water waves. Rahman [9] presented an introduction to the mathematical and physical aspects of the theory of water waves. He discussed the wave theory of Airy, nonlinear wave theory of Stokes, tidal dynamics in shallow water. He mentioned about the dynamics of floating offshore structures. Johnson [5] presented an introduction to mathematical ideas and techniques that are directly relevant to water wave theory. Beginning with the introduction of the appropriate equations of fluid mechanics, together with the relevant boundary conditions, the ideas of nondimensionalisation, scaling and asymptotic expansions are briefly explored. In the present work, we formulate the boundary value problem for a floating heaving circlular cylinder in water of arbitrary uniform depth. Here we consider the heave motion of the cylinder with the assumption that there is no incident wave. We present computational results for added mass and damping coefficients for the heaving cylinder for various depth to radius and draft to radius ratios.. Mathematical formulation A rigid floating structure undergoes six degrees of freedom : three translational and three rotational. Assuming a suitable coordinate system, OXY Z, 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 we consider the heave motion by a circular cylinder of radius a in water of finite depth h. The cylinder is assumed to be floating with a draft b in water. Cylindrical coordinate system (r, θ, z) is assumed with z-axis vertically upwards from the still water level (SWL), r measured radially from the z-axis and θ from the positive x-axis. Geometry is depicted in Figure 1. Then for an incompressible, irrotational and invisid fluid and for small amplitude wave, we can introduce a velocity potential φ(r, θ, z) such that

Computation of added mass 19 y r a x z x b h Figure 1. Cylinder heaving in finite depth water. φ r + 1 φ r r + 1 φ r θ + φ =0, z (1) g φ z σ φ =0 at z =0, () φ =0 z at z =, (3) φ = w z on r a, z = b, (4) φ =0 r at r = a, b z 0 (5) where σ represents the angular frequency and w is the heaving velocity. Due to symmetry in θ direction, we assume φ and w take the forms φ(r, θ, z) = ψ m (r, z) cos mθ, (6) w = m=0 w m cos mθ, (7) m=0 Now we consider the whole fluid region as two regions, interior region and exterior region. Interior region is the region below the cylinder, i.e., the region where r a, z and z b. Exterior region is the region where r a and z 0. We seek the solution for interior and exterior regions now.

130 Dambaru Bhatta 3. Interior solution Substituting (6) and (7) into (1)-(5), we have ψ (i) m m r ψ(i) m =0, (8) ψ m (i) =0 z at z =, (9) ψ m (i) = w m z on z = b. (10) Superscript (i) is used to denote the solution for the interior region and = r + r r +. We decompose ψ(i) z m into homogeneous and nonhomogeneous part as ψ m (i) = ψ(i) m h + ψ m (i) p. (11) For homogeneous part using separation of variables method, we can write ψ (i) m h = α ( m0 r m α mn I m (k n r) + cos k n (z + h) (1) a) I m (k n a) where α mn s are constants, k n = nπ h b,,,... and I m(k n r) is the modified Bessel function of first kind and order m ( McLachlan [9] ). The particular solution is ψ m (i) p (r, z) = w ] m [(z + h) r. (13) (h b) Hence interior solution is At r = a, we have ψ (i) m (r, z) =α m0 ( r m α mn I m (k n r) + cos k n (z + h) a) I m (k n a) [(z + h) r + w m (h b) ]. (14) ψ m (i) (a, z) = α m0 + α mn cos k n (z + h) + w ] m [(z + h) a. (15) (h b) So α mn s can be obtained from α mn = h b b ψ (i) m (a, z) cos k n (z + h)dz I np. (16)

I 0p = I np = w m (h b) = w m(h b) 6 w m (h b) Computation of added mass 131 ] [(z + h) a dz [ ( ) ] a 3. (17) h b b b ] [(z + h) a cos k n (z + h)dz = ( 1)n w m (h b) n π. (18) 4. Exterior solution In this case the boundary value problem is ψ (e) m g ψ(e) m z m r ψ(e) m =0, (19) σ ψ m (e) =0 at z =0, (0) ψ m (e) =0 z at z =, (1) ψ m (e) =0 r at r = a, b z 0. () Superscript (e) is used to denote the solution for the exterior region. We use separation of variables method to obtain ψ m (e). Thus the exterior solution can be written as m = β m0h m (1) (s 0 r) Z H m (1) s0 (z)+ (s 0 a) ψ (e) j=1 β mj K m (s j r) Z sj (z) (3) K m (s j a) where β mj s are constants, H (1) m (s 0 r) is the Hankel function of first kind of order m and K m (s j r) is the modified Bessel function of second kind and order m. Here s 0 and s j satisfy the relations σ = gs 0 tanh s 0 h, (4) σ = gs j tan s j h j =1,,... (5) In deriving the solution (3), we use the following notations : Z s0 (z) =N 1 s 0 cosh s 0 (z + h), (6) Z sj (z) =N 1 s j cos s j (z + h), (7)

13 Dambaru Bhatta with Now at r = a, we have N s0 = 1 ( 1+ sinh s ) 0h, (8) s 0 h N sj = 1 ( 1+ sin s ) jh. (9) s j h ψ (e) m (a, z) = β mj Z sj (z). (30) The functions Z sj (z) are orthonormal over [, 0] as shown below 0 0 1 Zs h 0 (z) dz = 1 cosh s 0 (z + h) dz h N s0 = 1 ( h + sinh s ) 0h hn s0 s 0 =1. (31) For j =1,, 3..., we have 1 0 Zs h j (z) dz = 1 0 cos s j (z + h) dz h N s0 = 1 ( h + sin s ) jh hn s0 s j = 1 (3) and 1 0 Z s0 (z) Z sj (z) dz = 1 0 cosh s 0 (z + h) cos s j (z + h) dz. h h Ns0 Nsj Considering the following integral so that we have I = 1 h = 1 I = 0 0 h 0 cosh s 0 (z + h) cos s j (z + h) dz [ e s0(z+h) + e s0(z+h)] cos s j (z + h) dz [ e s 0u + e s0u] cos s j udu = es0h s 0 + (s 0 cos s j h + s j sin s j h) s 0 s j s 0 + s j

Computation of added mass 133 + e s0h s 0 + ( s 0 cos s j h + s j sin s j h)+ s 0 s j s 0 + s j = s 0 + (s 0 sinh s 0 h cos s j h + s j cosh s 0 h sin s j h) s j = g ( s 0 + ) cosh s 0 h cos s j h (gs 0 tanh s 0 h + gs j tan s j h) s j = 0 (33) because of σ = gs 0 tanh s 0 h = gs j tan s j h [equations (4), (5)]. Also for l =1,,.., m =1,,..and l m, it can shown that 1 h 0 Z sl (z) Z sm (z) dz = 1 h 0 cos s l (z + h) cos s m (z + h) Nsl Nsm dz =0. (34) Hence Z sj (z) are orthonormal over [, 0]. Thus due to the orthonormality of the functions Z sj (z), β mj can be obtained from (30) as β mj = 1 h 0 ψ (e) m (a, z)z sj (z)dz. (35) Matching conditions at r = a are for z b. Body surface condition is 5. Matching condition φ (i) (a,θ,z)=φ (e) (a,θ,z) (36) φ (i) ] ] = φ(e) (37) r r r=a for b z 0. From the equation (16) and condition (36) α mn = h b r=a φ (e) ] = 0 (38) r r=a b b ψ (i) m (a, z) cos k n(z + h)dz I np = ψ m (e) (a, z) cos k n (z + h)dz I np h b = β mj L nsj I np (39)

134 Dambaru Bhatta where L 0sj = 1 h b L nsj = 1 h b b b For j = 0 and j =1,,..respectively, we have Z sj dz, (40) Z sj cos k n (z + h)dz. (41) L ns0 = ( 1)n N 1 s 0 (h b)s 0 sinh s 0 (h b) (h b) s 0 + n π, (4) L nsj = ( 1)n N 1 s j (h b)s j sin s j (h b) (h b) s j n π. (43) Now from conditions (37) and (38), we have G m0 + mα m0 = α mn G mn cos k n (z + h)+ β mj G mj Z sj (z) (44) for z b, β mj G mj Z sj (z) = 0 for b z 0 (45) where G m0 = w ma (h b) G mn = k nai m(k n a) I m (k n a) G m0 = s 0aH (1) m (s 0 a) H m (1) (s 0 a) (46) (47) (48) G mn = s nak m (s na) (49) K m (s n a) Now multiplying the equations (44) and (45) by Z sl (z), l=0, 1,... and integrating in the regions of validity and adding them we get ( G m0 + mα m0 ) L 0sl = α mn G mn L nsl + h h b β mjg mj δ sj s l. (50) Now substituting the values of α mn from equation (39), we get a system of equations E lj βmj = X ml (51)

Computation of added mass 135 where E lj = ml 0sl L 0sj + X ml = G m0 L 0sl mi 0pL 0sl h h b G mjδ sj s l + + G mn L nsl L nsj, (5) I np G mn L nsl. (53) 6. Added mass and damping coefficients The radiated force can be written as F = iσρ π a θ=0 r=0 φ (i) (r, θ, b)r dr dθ. (54) Decomposing the radiated force into components in phase with the acceleration and the velocity, the added mass, µ and damping coefficient, λ are obtained as µ + i λ σ = ρ w π a θ=0 r=0 φ (i) (r, θ, b)r dr dθ. (55) Since θ integration is from 0 to π, and the integrand contains cos mθ, there will be contribution for m = 0 only. Thus [ { 1 w 0 (h b) 1 ( ) } a + α 00 8 h b ] + h b ( 1) n α 0n I 1 (k n a) π a ni 0 (k n a) { = πa ρ 1 w 0 (h b) w 0 1 ( ) } a + β 0j L 0sj I 0p 8 h b + h b ( 1) n I 1 (k n a) π a ni 0 (k n a) β 0j L nsj I np { = πa (h b)ρ 1 3 + 1 ( ) } a + γ 0j L 0sj 8 h b + 4 h b ( 1) n I 1 (k n a) γ 0j L nsj ( 1)n π a ni 0 (k n a) n π (56) µ + i λ σ = πa ρ w 0 where γ 0j = β0j w. 0(h b)

136 Dambaru Bhatta 7. Computational results We write complex matrix equation (51) as N p E lj γ 0j = X 0l (57) where E lj and X 0l s are given by E lj = h h b G N n 0jδ sj s l + G 0n L nsl L nsj, (58) N n X 0l = g 00 L 0sl + I np G 0n L nsl (59) where l =0, 1,..., N p and j =0, 1,..., N p and a g 00 = (h b) and I np = ( 1)n n π. (60) Thus E is a square matrix of order (N p + 1) and X is a vector of length (N p + 1). The equation (57) is solved by using methods available in JMSL. We compute s j a, j =1,,... for given s 0 a from the equations (4) and (5) for h/a =5.0. The various values of s j a, j =1,,..., 15 are presented in the columns of the Table 1 for s 0 a =0.5, 1.0, 1.5,.0,.5 respectively. Table 1. Solutions of equation (5) for various s 0 a for h/a =5.0. n s 0 a =0.5 s 0 a =1.0 s 0 a =1.5 s 0 a =.0 s 0 a =.5 1 0.4653 0.388 0.3614 0.3486 0.3410 1.1773 1.1099 1.0660 1.038 1.0199 3 1.833 1.786 1.746 1.711 1.6896 4.4739.4353.4014.3730.3500 5 3.1101 3.0788 3.050 3.047 3.005 6 3.7437 3.7173 3.696 3.6701 3.6498 7 4.3757 4.3530 4.3314 4.311 4.96 8 5.0069 4.9869 4.9679 4.9497 4.936 9 5.6374 5.6196 5.605 5.5861 5.5703 10 6.675 6.514 6.359 6.08 6.066 11 6.897 6.886 6.8685 6.8547 6.8413 1 7.567 7.5133 7.500 7.4876 7.475 13 8.1560 8.1437 8.1316 8.1198 8.1083 14 8.785 8.7737 8.765 8.7514 8.7406 15 9.4143 9.4036 9.3931 9.387 9.376

Computation of added mass 137 To compute added mass and damping, we need to solve the equation (57) first. We verify the convergence of the solution by the following way. Figure Figure 3 Fig. : Real part of the solution γ 01 of equation (57) as function of N p. Fig. 3: Imaginary part of the solution γ 01 of equation (57) as function of N p. Then we vary N n keeping N p fixed (N p = 0). We consider s 0 a = 1.0, h/a =5.00 and b/a =1.00. Then fixing N n (= 0), we vary N p and observe the real and imaginary parts of the solution of (57) as a function of N p. The real and imaginary parts of the solution γ 01 for this case are presented in Figure and Figure 3 respectively. Figure 4 Figure 5 Fig. 4: Real part of the solution γ 01 of equation (57) as function of N n. Fig. 5: Imaginary part of the solution γ 01 of equation (57) as function of N n.

138 Dambaru Bhatta The real and imaginary parts of the solution γ 01 are shown in Figure 4 and Figure 5 respectively. From the results computed, we see excellent truncation properties for N p and N n. From this observation, we choose N p = 0 and N n = 0 for further computational purpose. Truncating the equation (56), added mass and damping coefficient can be written as µ + i λ σ S = [{ 1 3 + 1 ( ) } a 8 h b + 4 π h b a where S = πa (h b)ρ N n and N p ( 1) n I 1 (k n a) ni 0 (k n a) + γ 0j L 0sj { Np }] γ 0j L nsj ( 1)n n π (61) L ns0 = ( 1)n N 1 s 0 (h b)s 0 sinh s 0 (h b) (h b) s 0 + n π, (6) L nsj = ( 1)n N 1 s j (h b)s j sin s j (h b) (h b) s j n π. (63) Computational results of (61) are presented next. Non-dimensional added mass and damping coefficients are presented in Figure 6 through Figure 9 as functions of s 0 a. Figure 6 shows the computational results of non-dimensional added mass for depth to radius ratio 3 (h/a = 3) and draft to radius ratios, b/a = 1.5, 1.0, 0.75. Figure 6 Figure 7 Fig. 6: Added mass for various b/a and h/a =3.0. Fig. 7: Added mass for various b/a and h/a =5.0

Computation of added mass 139 Results of added mass for h/a = 5 and b/a =1.5, 1.0, 0.75 are presented in Figure 7. For h/a =3, 5 and b/a =1.0, added mass is shown in Figure 8. Figure 8 Figure 9 Fig. 8: Added mass for various h/a and b/a =1.0. Fig. 9: Damping coefficient for various b/a and h/a =5.0 Figure 9 presents the computational results of non-dimensional damping coefficients for h/a = 3 and b/a = 1.5, 1.0, 0.75. Fig. 10: Damping coefficient for various b/a and h/a = 5.0 Results of damping for h/a = 5 and b/a =1.5, 1.0, 0.75 are presented in Figure 10. In these calculations, we take N p = 0 and N n = 0. At the same depth of water (for h/a fixed), heave added mass is higher for higher draft of the cylinder

140 Dambaru Bhatta (for larger b/a). For the same draft of the cylinder (for b/a fixed), added mass is higher for lower depth of water (smaller h/a). For all cases variation in added mass becomes smaller and smaller for wave numbers larger than 3 (s 0 a>3). Damping coefficients for all the cases are approaching to zero for s 0 a>3. 8. Acknowledgement Author would like to thank the referee for his encouraging remarks. References 1. K. J. Bai, The added mass of two-dimensional cylinders heaving in water of finite depth, Journal of Fluid Mechanics 81 (1977), 85-105.. S. K. Chakrabarti, Hydrodynamics of Offshore Structures, Computational Mechanics Publications, Southampton-Boston, 1986. 3. R. G. Dean and R. A. Dalrymple, Water Wave Mechanics for Engineers and Scientists, Prentice-Hall Inc, New Jersey, 1984. 4. L. Debnath, Nonlinear Water Waves, Academic Press, London, England, 1994. 5. R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge, 1997. 6. P. McIver and C. M. Linton, The added mass of bodies heaving at low frequency in water of finitre depth, Applied Ocean Research 13 (1991), 1-17. 7. N. W. McLachlan, Bessel functions for Engineers, Clarendon Press, Oxford, 1961. 8. M. Rahman and D. D. Bhatta, Evaluation of added mass and damping coefficient of an oscillating circular cylinder, Applied Mathematical Modelling 17 (199), 70-79. 9. M. Rahman, WATER WAVES: Relating Modern Theory to Advanced Engineering Practice, Oxford University Press, Oxford, 1994. Dambaru D. Bhatta received his PhD from Dalhousie University, Halifax, Canada. Currently he is a faculty in the Department of Mathematics, University of Texas-Pan American, TX, USA. His research interests include applied mathematics, partial differential equations, water wave, wave-structure interactions, fractional differential equations, digital signal processing. Department of Mathematics, The University of Texas-Pan American, 101 W. University Drive, Edinburg, TX 78541, USA e-mail : bhattad@utpa.edu