Water wave interaction with an array of submerged circular plates: Hankel transform approach

In this paper, water wave interaction with an array of thin submerged horizontal circular plates is investigated within the framework of linear potential flow theory. To consider a more general case, the circular plates studied in this paper are not limited to be rigid and impermeable and, instead, they can be perforated and/or elastic. A Hankel transform approach is employed to formulate integral equations in terms of unknown functions related to the jump in velocity potential across each plate. A Galerkin method is adopted to the solution of these integral equations, and the velocity potential jump across the plate is expressed in terms of Fourier-Gegenbauer series, incorporating the known square-root behavior at the edge of the plate in a rapidly convergent numerical scheme. For elastic plates, the plate motion is expanded in modes of free vibration with the edge constraint conditions accounted for intrinsically. The unknown coefficients of the plate motion are further expressed in terms of the unknown coefficients related to the velocity potential jump. The Hankel transform–based model is found to be valid for multiple plates distributed arbitrarily, including the staggered arrangement, for which the traditional eigenfunction matching method would not work. In-depth discussions have been made to the hydrodynamic responses of staggered arrangement of plates. It is found that the staggered arrangement of plates can result in notable wave focusing but with less energy dissipation. The largest principal strain is observed on the front region of the plate submerged at a shallower depth.

Figure 1

Sketch of three arrangements of a pair of submerged circular plates: (a) coaxial arrangement, (b) side-by-side arrangement, and (c) staggered arrangement.

Figure 2

Definition of the coordinate systems (top view) (after [12]). $R_{n,j}$=0, $R_{n,j}\ge R_n+R_j$ and $0<R_{n,j}<R_n+R_j$ correspond to the coaxial arrangement, side-by-side arrangement, and staggered arrangement, respectively (see Fig. 1).

Figure 3

Impact of the angular cutoffs (i.e., in terms of $M$) on dimensionless wave excitation force or moment acting on a pair of submerged rigid and impermeable circular plates in a staggered arrangement for $L=6, R_1/h=0.5, d_1/h=0.2, R_2/h=0.3, d_2/h=0.1, x_1=y_1=y_2=0, x_2/h=0.5, \beta =\pi /6, kh=2.0$ and 5.0: (a) and (d) heave excitation forces; (b) and (e) roll excitation moments; (c) and (f) pitch excitation moments. (a)–(c) plate 1; (d)–(f) plate 2.

Figure 4

Impact of the radial cutoffs (i.e., in terms of $L$) on dimensionless wave excitation force or moment acting on a pair of submerged rigid and impermeable circular plates in a staggered arrangement for $M=10, R_1/h=0.5, d_1/h=0.2, R_2/h=0.3, d_2/h=0.1, x_1=y_1=y_2=0, x_2/h=0.5, \beta =\pi /6, kh=2.0$ and 5.0: (a) and (d) heave excitation forces; (b) and (e) roll excitation moments; (c) and (f) pitch excitation moments. (a)–(c) plate 1; (d)–(f) plate 2.

Figure 5

Variation of dimensionless wave excitation force or moment acting on a pair of submerged rigid and impermeable circular plates in a staggered arrangement vs wave number in terms of $kh$ for $R_1/h=0.5, d_1/h=0.2, R_2/h=0.3, d_2/h=0.1, x_1=y_1=y_2=0, x_2/h=0.5$, and $\beta =\pi /6$: (a) heave excitation forces, (b) roll excitation moments, and (c) pitch excitation moments. Lines: numerical results [33]; symbols: present semianalytical results.

Figure 6

Contour of the deflection of a single submerged elastic perforated circular plate $|{\eta }_1|/A$ for $R_1/h=2.0, d_1/h=0.2, \overline{\chi }=\overline{\gamma }=0.01, \overline{p}=1.0, \beta =0, kh=\pi /2$ and simply supported edge conditions: (a) published result based on the eigenfunction matching method [12], and (b) present semianalytical result.

Figure 7

Contour of the wave amplitude above a single submerged elastic perforated circular plate $\left|\mathcal{E}\right|/A$ for $R_1/h=2.0, d_1/h=0.2, \overline{\chi }=\overline{\gamma }=0.01, \overline{p}=1.0, \beta =0, kh=\pi /2$ and simply supported edge conditions: (a) published result based on the eigenfunction matching method [12], and (b) present semianalytical result.

Figure 8

Contours of wave power dissipation, ${\kappa }_{\mathrm{diss}}$ [(a) and (d)], heave excitation forces acting on plate 1, $|F_3^{\left(1\right)}|/\left(\rho g\pi h^{2}A\right)$ [(b) and (e)] and 2, $|F_3^{\left(2\right)}|/\left(\rho g\pi h^{2}A\right)$ [(c) and (f)] with the incident direction $\beta$ and the distance between the two plates $R_{1,2}=x_2-x_1$ for $y_1=y_2=0, R/h=2.0, d_1/h=0.1, d_2/h=0.2, \overline{\chi }=\overline{\gamma }=0.01, \overline{p}=1.0$, and $kh=\pi /2$: (a)-(c) simply supported edge conditions; (d)-(f) clamped edge conditions.

Figure 9

Contours of wave power dissipation, ${\kappa }_{\mathrm{diss}}$ [(a) and (d)], heave excitation forces acting on plate 1, $|F_3^{\left(1\right)}|/\left(\rho g\pi h^{2}A\right)$ [(b) and (e)] and 2, $|F_3^{\left(2\right)}|/\left(\rho g\pi h^{2}A\right)$ [(c) and (f)] with the porosity parameter $\overline{p}$ and the distance between the two plates $R_{1,2}=x_2-x_1$ for $y_1=y_2=0, R/h=2.0, d_1/h=0.1, d_2/h=0.2, \overline{\chi }=\overline{\gamma }=0.01, \beta =\pi /2$, and $kh=\pi /2$: (a)–(c) simply supported edge conditions, and (d)–(f) clamped edge conditions.

Figure 10

Contours of wave power dissipation, ${\kappa }_{\mathrm{diss}}$ [(a) and (d)], heave excitation forces acting on plate 1, $|F_3^{\left(1\right)}|/\left(\rho g\pi h^{2}A\right)$ [(b) and (e)] and 2, $|F_3^{\left(2\right)}|/\left(\rho g\pi h^{2}A\right)$ [(c) and (f)] with the wave frequency $kh$ and the distance between the two plates $R_{1,2}=x_2-x_1$ for $y_1=y_2=0, R/h=2.0, d_1/h=0.1, d_2/h=0.2, \overline{\chi }=\overline{\gamma }=0.01, \beta =\pi /2$, and $\overline{p}=1.0$: (a)–(c) simply supported edge conditions, and (d)–(f) clamped edge conditions.

Figure 11

Contour of the wave amplitude above a pair of elastic perforated circular plates, simply supported edge conditions, $R_{1,2}=x_2-x_1$ for $y_1=y_2=0, R/h=2.0, d_1/h=0.1, d_2/h=0.2, \overline{\chi }=\overline{\gamma }=0.01, \overline{p}=1.0, \beta =\pi /2$, and $kh=\pi /2$: (a) $R_{1,2}=0$, i.e., coaxial; (b) $R_{1,2}/h=2.0$, i.e., staggered; (c) $R_{1,2}/h=4.0$; (d) $R_{1,2}/h=6.0$, i.e., side by side.

Figure 12

Contour of the deflection of a pair of submerged elastic perforated circular plates, simply supported edge conditions, $R_{1,2}=x_2-x_1$ for $y_1=y_2=0, R/h=2.0, d_1/h=0.1, d_2/h=0.2, \overline{\chi }=\overline{\gamma }=0.01, \overline{p}=1.0, \beta =\pi /2$, and $kh=\pi /2$: (a) $R_{1,2}=0$, i.e., coaxial; (b) $R_{1,2}/h=2.0$, i.e., staggered; (c) $R_{1,2}/h=4.0$; (d) $R_{1,2}/h=6.0$, i.e., side by side.

Figure 13

The instantaneous velocity vector field around a pair of submerged elastic perforated circular plates with simply supported edge conditions at $x/h=0,\pm 1$ and $y/h=0$ for $t=0, y_1=y_2=0, R/h=2.0, d_1/h=0.1, d_2/h=0.2, \overline{\chi }=\overline{\gamma }=0.01, \overline{p}=1.0, \beta =\pi /2, kh=\pi /2$, and $R_{1,2}/h=2.0$, i.e., the staggered arrangement corresponding to Figs. 11 and 12.

Figure 14

Distribution of the maximum principal strain of a pair of submerged elastic perforated circular plates with simply supported edge conditions, $R_{1,2}=x_2-x_1$ for $y_1=y_2=0, R/h=2.0, d_1/h=0.1, d_2/h=0.2, \overline{\chi }=\overline{\gamma }=0.01, \overline{p}=1.0, \beta =\pi /2$ and $kh=\pi /2, R_{1,2}/h=2.0$, i.e., the staggered arrangement corresponding to Figs. 11, 12, and 13.

Figure 15

Three integration paths across the pole.