Hydrodynamic coefficients for a floating semi-porous compound cylinder in finite ocean depth

Abstract

Employing the methods of separation of variables and matched eigenfunction expansions for velocity potential, analytical solutions are proposed for a water wave radiation problem of a floating semi-porous compound cylinder in finite ocean depth. The configuration of the semi-porous compound cylinder is such that it consists of an impermeable inner cylinder rising above the free surface and a coaxial truncated porous cylinder around the lower part of the inner cylinder with the top of the porous cylinder being impermeable. The condition on the porous boundary is defined by applying Darcy’s law as in Williams et al. (Ocean Eng 27:1–28, 2000) . The translational motions in the x- and z-directions, i.e., surge and heave motions, are investigated. A mathematical model is developed which can be considered as an extension of a number of the earlier works, e.g., Kokkinowrachos et al. (Ocean Eng 13:505–538, 1986) and Calisal and Subancu (Ocean Eng 11(5):529–542, 1984), in which significance of porosity of the structure was neglected. Numerical investigation is taken up here in order to examine the influence of submerged depth, radii, porous coefficient, and water depth on added mass and radiation damping, two most important entities in radiation problems, with respect to surge and heave motions. It is found that the variation of porous coefficient, radii, and depth has a significant influence on the added mass and damping coefficients for the semi-porous compound cylinder. The added mass is found not sufficiently affected by lower values of porous coefficient G, but exhibits significant variation corresponding to higher values of G. Another important observation is that the damping coefficients oscillate alternately between negative and positive values which can be attributed to coupled behavior between different motions. The results establish that an appropriate optimal ratio of various parameters may be considered in designing ocean structures with minimum adverse hydrodynamic effect. The effectiveness of the present model is validated by comparing it with an available result which shows an excellent agreement.

Appendix: Porous coefficient G

The dimensionless porous parameter G can be defined as $\displaystyle G=\displaystyle \frac{L \rho \sigma }{\varepsilon k_0}$, as was used by Chwang [7]. In general, being complex, it can be expressed as ${G_r +\mathrm{i} G_i}$ (Yu [35]), where $G_r$ and ${G_i}$, respectively, denote the real and the imaginary parts. In practice, G always possesses positive real and imaginary parts except when the resistance effect against the flow dominates the inertial effect of the fluid inside the porous material in which case G becomes real. Similarly, when the inertial effect dominates the resistance effect, G becomes purely imaginary. The parameter G may be viewed as a Reynolds number for the flow passing through the fine pores of the wall (Chwang [7]). It is also a measure of the porous effect of the wall. ${G = 0}$ implies that the porous wall is impermeable. On the other hand, as G approaches infinity, the porous wall is completely permeable to fluid, that is, there would be no porous wall at all. Basically here we work with Darcy’s law, which is applicable for low Reynolds number flow through porous wall. Also Sankarbabu et al. [21] proposed to choose G as

$$\begin{aligned} G= & {} \frac{L\rho \sigma }{\varepsilon k_0}\\= & {} \frac{L\rho \sqrt{gk_0 \tanh k_0h}}{\varepsilon k_0} \quad \quad \text{(From } \text{ dispersion } \text{ relation) }\\= & {} \frac{\rho L \sqrt{gh}}{\varepsilon }\sqrt{\frac{\tanh k_0h}{k_0h}}, \end{aligned}$$

with h as the draft of the structure.

Initially, the value of G for a given porosity is obtained by trial and error method. On substitution of the obtained G and other known values in the above expression, the unknown material constant (L) for the given porosity of the outer cylinder is calculated. Based on linear extrapolation, the material constant values for other porosities are computed resulting in values of G.

Conforming to the assumptions and conditions used in this work, G is taken to be real since the flow dominates the inertial effect of the fluid inside the porous cylinder.