Hydrodynamic interactions of water waves with a group of independently oscillating truncated circular cylinders

Abstract

In this study, we examine the water wave radiation by arrays of truncated circular cylinders. Each cylinder can oscillate independently in any rigid oscillation mode with a prescribed amplitude, including translational and rotational modes such as surge, sway, heave, pitch, roll, and their combinations. Based on the eigenfunction expansion and Graf’s addition theorem for Bessel functions, we developed an analytical method that includes the effects of evanescent modes in order to analyze such arrays of cylinders. To investigate the effects of several influential factors on convergence, our objective is to dramatically reduce the number of tests required and determine the influencing relationships between truncation number and convergence behavior for different factor combinations. We use the orthogonal test method to fulfill the objective. Lastly, we present our results regarding the effects of evanescent modes on hydrodynamic coefficients.

Appendix

The unknown radiation coefficients $C_{pm}^s ,D_{nm}^{s}$ can be solved from the equation set

$$\begin{aligned}&C_{pm}^{s}+\sum \limits _{n=0}^{\infty } {F_{pn}^m D_{nm}^s } =R_s^{pm}, \end{aligned}$$

(A1a)

$$\begin{aligned}&D_{nm}^s -\sum \limits _{p=0}^{\infty } {G_{np}^m C_{pm}^s } =S_s^{nm}. \end{aligned}$$

(A1b)

The expressions of the known coefficients in Eq. (A1a, A1b) are

$$\begin{aligned} F_{p0}^{m}= & {} -\frac{2}{d-h}\int _{-d}^{-h}\frac{\mathrm{H}_m \left( {k_0 a} \right) }{N_0^{1/2} \mathrm{H}_m^{\prime }\left( {k_0 a} \right) }\cosh [k_0 \left( {z+d} \right) ]\nonumber \\&\cos \left[ {\frac{p\uppi \left( {z+d} \right) }{d-h}} \right] \mathrm{d}z \nonumber \\= & {} -\frac{2\mathrm{H}_m \left( {k_0 a} \right) k_0 (d-h)\left( {-1} \right) ^{p}\sinh [k_0 (d-h)]}{\mathrm{H}_m^{\prime }\left( {k_0 a} \right) N_0^{1/2} \left[ {k_0^2 (d-h)^{2}+\left( {p\uppi } \right) ^{2}} \right] }, \end{aligned}$$

(A2a)

$$\begin{aligned} F_{pn}^{m}= & {} -\frac{2}{d-h}\int _{-d}^{-h}\frac{\mathrm{K}_m \left( {k_{n} a} \right) }{N_{n}^{1/2} \mathrm{K}_m^{\prime }\left( {k_{n} a} \right) }\cos [k_{n} \left( {z+d} \right) ]\nonumber \\&\cos \left[ {\frac{p\uppi \left( {z+d} \right) }{d-h}} \right] \mathrm{d}z \nonumber \\= & {} -\frac{2\mathrm{K}_m \left( {k_{n} a} \right) k_{n} (d-h)\left( {-1} \right) ^{p}\sin [k_{n} (d-h)]}{\mathrm{K}_m^{\prime }\left( {k_{n} a} \right) N_{n}^{1/2} \left[ {k_{n}^2 (d-h)^{2}-\left( {p\uppi } \right) ^{2}} \right] }, \end{aligned}$$

(A2b)

$$\begin{aligned} G_{00}^{(m)}= & {} \frac{1}{k_0 d}\int _{-d}^{-h} {\frac{\left| m \right| }{2a}Z_0 (z)\mathrm{d}z}\nonumber \\= & {} \frac{m\cdot \sinh k_0 (d-h)}{\sqrt{2}\cdot a\cdot d\cdot k_0 ^{2}\left[ {1+\frac{\sinh (2k_0 d)}{2k_0 d}} \right] ^{1/2}},\,\,\,n=0,p=0, \end{aligned}$$

(A3a)

$$\begin{aligned} G_{n0}^{(m)}= & {} \frac{1}{k_{n} d}\int _{-d}^{-h} {\frac{\left| m \right| }{2a}Z_{n} (z)\mathrm{d}z}\nonumber \\= & {} \frac{m\cdot \sin [k_{n} (d-h)]}{\sqrt{2}\cdot a\cdot d\cdot k_{n} ^{2}\left[ {1+\frac{\sin (2k_{n} d)}{2k_{n} d}} \right] ^{1/2}},\,\,\,n\geqslant 1,p=0, \end{aligned}$$

(A3b)

$$\begin{aligned} G_{0p}^{(m)}= & {} \frac{1}{k_0 d}\int _{-d}^{-h} {\frac{p\uppi }{d-h}\frac{\mathrm{I}_m^{\prime }\left( {\frac{p\uppi a}{d-h}} \right) }{\mathrm{I}_m \left( {\frac{p\uppi a}{d-h}} \right) }\cos \left[ {\frac{p\uppi \left( {z+d} \right) }{d-h}} \right] Z_0 (z)\mathrm{d}z} \nonumber \\= & {} \frac{\mathrm{I}'_m \left( \frac{p\uppi a}{d-h}\right) \sqrt{2}p\uppi (d-h)(-1)^{p}\cdot \sinh [k_0 (d-h)]}{\mathrm{I}_m \left( \frac{p\uppi a}{d-h}\right) \cdot d\cdot \left[ {1+\frac{\sinh (2k_0 d)}{2k_0 d}} \right] ^{1/2}\cdot [k_0 ^{2}(d-h)^{2}+p^{2}\uppi ^{2}]},\nonumber \\&n=0,p\geqslant 1, \end{aligned}$$

(A3c)

$$\begin{aligned} G_{np}^{(m)}= & {} \frac{1}{k_{n} d}\int _{-d}^{-h} {\frac{p\uppi }{d-h}\frac{\mathrm{I}_m^{\prime }\left( {\frac{p\uppi a}{d-h}} \right) }{\mathrm{I}_m \left( {\frac{p\uppi a}{d-h}} \right) }\cos \left[ {\frac{p\uppi \left( {z+d} \right) }{d-h}} \right] Z_{n} (z)\mathrm{d}z}\nonumber \\= & {} \frac{\mathrm{I}'_m \left( \frac{p\uppi a}{d-h}\right) \sqrt{2}p\uppi (d-h)(-1)^{p}\cdot \sin [k_{n} (d-h)]}{\mathrm{I}_m \left( \frac{p\uppi a}{d-h}\right) \cdot d\cdot \left[ {1+\frac{\sin (2k_{n} d)}{2k_{n} d}} \right] ^{1/2}\cdot [k_{n} ^{2}(d-h)^{2}-p^{2}\uppi ^{2}]},\nonumber \\&n\geqslant 1,p\geqslant 1, \end{aligned}$$

(A3d)

(A4a)

(A4b)

The diffraction transfer matrices of the truncated single cylinders $\varvec{B}_{j}^\mathrm{E}$ and $\varvec{B}_{j}^\mathrm{C}$are obtained following the procedure given in Kagemoto and Yue [12], the elements of $\varvec{B}_{j}^\mathrm{E}$ (or $\varvec{B}_{j}^\mathrm{C} )$ are the amplitude of the q-th (or p-th) partial wave of the diffraction potential due to a single unit-amplitude incidence of mode n on cylinder j, which are listed below

$$\begin{aligned} {B}_{j}^\mathrm{E} (0,0,m)= & {} -\frac{\mathrm{J}'_m (k_0 a)}{\mathrm{H}'_m (k_0 a)}+\frac{D_0^m \cosh (k_0 d)}{\mathrm{H}'_m (k_0 a)N_0^{1/2} \hbox {e}^{\mathrm{i}m(\uppi /2-\beta )}},\nonumber \\&\quad n=0,q=0, \end{aligned}$$

(A5a)

$$\begin{aligned} {B}_{j}^\mathrm{E} (q,0,m)= & {} \frac{D_q^m }{\mathrm{K}'_m (k_q a)N_q^{1/2} \hbox {e}^{\mathrm{i}m(\uppi /2-\beta )}},\nonumber \\&\quad n=0,q\geqslant 1, \end{aligned}$$

(A5b)

$$\begin{aligned} {B}_{j}^\mathrm{E} (0,n,m)= & {} \frac{D[n]_0^m \cosh (k_0 d)}{\mathrm{H}'_m (k_0 a)N_0^{1/2}},\quad n\geqslant 1,q=0\end{aligned}$$

(A5c)

$$\begin{aligned} {B}_{j}^\mathrm{E} (q,n,m)= & {} \left\{ {\begin{array}{l} \frac{D[n]_q^m }{\mathrm{K}'_l (k_q a)N_q^{1/2}},\\ \quad n\geqslant 1,q\geqslant 1,q\ne n, \\ -\frac{\mathrm{I}'_m (k_q a)}{\mathrm{K}'_m (k_q a)}+\frac{D[n]_q^m }{\mathrm{K}'_m (k_q a)N_q^{1/2}}\\ \quad n\geqslant 1,q\geqslant 1,q=n, \\ \end{array}} \right. \end{aligned}$$

(A5d)

$$\begin{aligned} {B}_{j}^{\mathrm{C}} (0,0,m)= & {} \frac{C_0^m }{2a^{\left| m \right| }\mathrm{i}^{m}}, \quad n=0,p=0, \end{aligned}$$

(A6a)

$$\begin{aligned} {B}_{j}^{\mathrm{C}} (p,0,m)= & {} \frac{C_p^m }{\mathrm{I}_m \left[ {p\uppi a/(d-h)} \right] \mathrm{i}^{m}}\cdot \cos \left[ {\frac{p\uppi (z+d)}{d-h}} \right] ,\nonumber \\&\quad n=0,p\geqslant 1, \end{aligned}$$

(A6b)

$$\begin{aligned} {B}_{j}^\mathrm{C} (0,n,m)= & {} \frac{C[n]_0^m }{2a^{\left| m \right| }},\quad n\geqslant 1,p=0, \end{aligned}$$

(A6c)

$$\begin{aligned} {B}_{j}^\mathrm{C} (p,n,m)= & {} \frac{C[n]_p^m }{\mathrm{I}_m \left[ {p\uppi a/(d-h)} \right] }\cdot \cos \left[ {\frac{p\uppi (z+d)}{d-h}} \right] ,\nonumber \\&\quad n\geqslant 1,p\geqslant 1. \end{aligned}$$

(A6d)

Coefficients such as $C[n]_p^m$, $D[n]_q^m$ are determined by linear algebraic equations having the same form as Eq. (A1a, A7b), which can be found in Ref. [27] and are not detailed here.

The details of $\varvec{A}_{{\mathrm{R}j}}$, $\varvec{B}_{j}^\mathrm{E}$, $\varvec{T}_{{ij}}$, $\varvec{R}_{{is}}$ are given in Eq. (A7a-d). For computation, the infinite terms are truncated to $m_0, l_0, q_0, n_0$ terms, where $m_0, l_0, q_0, n_0$ are the upper limit values of the truncation numbers for m, l, q, n. For clarity, parentheses are inserted in the subscript of element of column vector shown in Eq. (A7a, A7b), which are omitted in Eqs. (20) and (13) in the main body of the manuscript.

The element in Eq. (A7c) can be obtained using Eq. (A5), as follows

The element ${T}_{{ij}} (n,m,l)$ in Eq. (A7d) is as follows

• if $n=0$, then $T_{ij} (n,m,l)=\mathrm{H}_{m-l} (k_0 L_{ij} )\hbox {e}^{\mathrm{i}\alpha _{ij} (m-l)}\;$;

• if $n\geqslant 1$, then $T_{ij} (n,m,l)=\mathrm{K}_{m-l} (k_{n} L_{ij} )\hbox {e}^{\mathrm{i}\alpha _{ij} (m-l)}(-1)^{l}$.

$$\begin{aligned} \varvec{A}_{\mathrm{R}j}^\mathrm{T}= & {} \left[ A_{\mathrm{R}(0,-m_0 )}^{(j)} A_{\mathrm{R}(0,-m_0 +1)}^{(j)} A_{\mathrm{R}(0,-m_0 +2)}^{(j)} \ldots A_{\mathrm{R}(0,m_0 -2)}^{(j)} \right. \nonumber \\&\left. A_{\mathrm{R}(0,m_0 -1)}^{(j)} A_{\mathrm{R}(0,m_0 )}^{(j)}A_{\mathrm{R}(1,-m_0 )}^{(j)} A_{\mathrm{R}(1,-m_0 +1)}^{(j)} \cdots \right. \nonumber \\&\left. A_{\mathrm{R}(1,m_0 -1)}^{(j)} A_{\mathrm{R}(1,m_0 )}^{(j)} \cdots A_{\mathrm{R}(n_0 -1,-m_0 )}^{(j)}\right. \nonumber \\&\left. A_{\mathrm{R}(n_0 -1,-m_0 +1)}^{(j)} \cdots A_{\mathrm{R}(n_0 -1,m_0 -1)}^{(j)} A_{\mathrm{R}(n_0 -1,m_0 )}^{(j)}\right] , \end{aligned}$$

(A7a)

$$\begin{aligned} \varvec{R}_{{i}s}^\mathrm{T}= & {} \left[ R_{(0,-m_0 ,s)}^{(i)} R_{(0,-m_0 +1,s)}^{(i)} R_{(0,-m_0 +2,s)}^{(i)} \cdots \right. \nonumber \\&\left. R_{(0,m_0 -2,s)}^{(i)} R_{(0,m_0 -1,s)}^{(i)} R_{(0,m_0 ,s)}^{(i)} R_{(1,-m_0 ,s)}^{(i)} R_{(1,-m_0 +1,s)}^{(i)} \cdots \right. \nonumber \\&\left. R_{(1,m_0 -1,s)}^{(j)} R_{(1,m_0 ,s)}^{(j)} \cdots R_{(n_0 -1,-m_0 ,s)}^{(i)} R_{(n_0 -1,-m_0 +1,s)}^{(i)}\right. \cdots \nonumber \\&\left. R_{(n_0 -1,m_0 -1,s)}^{(i)} R_{(n_0 -1,m_0 ,s)}^{(i)}\right] . \end{aligned}$$

(A7b)

(A7c)

(A7d)