Forces and moments exerted by incident internal waves on a plate-cylinder structure in a two-layer fluid

Abstract

Vertical cylindrical structures are widely used in offshore structures, for instance oil platforms, bridge foundations and so on. This paper presents incident internal waves interaction with a vertical circular cylinder under an elastic plate in a two-layer fluid in the framework of the potential flow theory. By means of an inner product and angular eigenfunction expansions, the effects of the characteristic parameters of structures and fluid on the shear pressures, horizontal forces and moments are investigated with numerical and graphical results. The conservation of energy as well as the relative error at the interface between the open water and plate-covered regions are discussed. Three cases of different boundary conditions of the cylinder are considered in order to investigate the the constraint forces and restraining moments of its ends. The results reveals that more energy can exchange between two wave modes owing to the presence of the elastic plate, which strengthens the horizontal forces on the cylinder. Incident waves of internal wave mode for a given frequency can cause complex polar distribution of the pressure around the cylinder. The values of constraint forces at ends of the cylinder are several times than those of the horizontal forces in some frequencies.

Appendix

$$\begin{aligned}&\sum _m \left[ B_{n,0_{m}}\text{J}_{n}^{\text{B}_1}(\kappa _{0_m} R) +C_{n,0_{m}}\text{Y}_{n}^{\text{B}_1}(\kappa _{0_m} R)\right] \frac{\partial {\tilde{Z}}_{0_{m}}}{\partial z}\nonumber \\&\qquad + \sum _{j}\left[ B_{n,j} {\text{I}}_{n}^{\text{B}_1}(\kappa _j R)+C_{n,j}\text{K}_{n}^{\text{B}_1}(\kappa _j R) \right] \frac{\partial {\tilde{Z}}_{j}}{\partial z}=0, \end{aligned}$$

(52)

$$\begin{aligned}&\sum _m \left[ B_{n,0_{m}}{\text{J}}_{n}^{\text{B}_2}(\kappa _{0_m}R)+C_{n,0_{m}}\text{Y}_{n}^{\text{B}_2}(\kappa _{0_m}R)\right] \frac{\partial {\tilde{Z}}_{0_{m}}}{\partial z}\nonumber \\&\qquad + \sum _{j}\left[ B_{n,j}{\text{I}}_{n}^{\text{B}_2}(\kappa _j R)+C_{n,j}\text{K}_{n}^{\text{B}_2}(\kappa _j R)\right] \frac{\partial {\tilde{Z}}_{j}}{\partial z}=0,\end{aligned}$$

(53)

$$\begin{aligned}&\sum _m \left[ B_{n,0_{m}}{\text{J}}_{n}(\kappa _{0_m}a)+C_{n,0_{m}}{\text{Y}}_{n}(\kappa _{0_m}a)\right] \frac{\partial {\tilde{Z}}_{0_{m}}}{\partial z} \nonumber \\&\qquad + \sum _{j}\left[ B_{n,j}{\text{I}}_{n}(\kappa _j a)+C_{n,j}\text{K}_{n}(\kappa _j a) \right] \frac{\partial {\tilde{Z}}_{j}}{\partial z}=0, \end{aligned}$$

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$$\begin{aligned}&\sum _m \left[ B_{n,0_{m}}\frac{\partial {\text{J}}_{n}(\kappa _{0_m}a)}{\partial r}+ C_{n,0_{m}}\frac{\partial {\text{Y}}_{n}(\kappa _{0_m}a)}{\partial r}\right] \frac{\partial {\tilde{Z}}_{0_{m}}}{\partial z} \nonumber \\&\qquad + \sum _{j}\left[ B_{n,j}\frac{\partial {\text{I}}_{n}(\kappa _ja)}{\partial r}+C_{n,j}\frac{\partial {\text{K}}_{n}(\kappa _ja)}{\partial r} \right] \frac{\partial {\tilde{Z}}_{j}}{\partial z}=0. \end{aligned}$$

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$$\begin{aligned}&A_{n,0_{1}}\text{H}_n(k_{0_1} R) P_{0_{1}0_\text{1}}\nonumber \\&\quad =\sum _m \left[ B_{n,0_{m}}{\text{J}}_n(\kappa _{0_m} R)+C_{n,0_{m}}{\text{Y}}_n(\kappa _{0_m} R)\right] Q_{0_m0_{1}}\nonumber \\&\qquad + \sum _{j}\left[ B_{n,j}{\text{I}}_n(\kappa _j R)+C_{n,j}\text{K}_n(\kappa _j R) \right] Q_{j 0_{1}},\quad (l=0_{1}), \end{aligned}$$

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$$\begin{aligned}&I_{0_{2}}{\text{i}}^n {\text{J}}_n(k_{0_2} R) P_{0_{2}0_{2}}+A_{n,0_{2}}{\text{H}}_n(k_{0_2} R) P_{0_{2}0_{2}}\nonumber \\&\quad =\sum _m \left[ B_{n,0_{m}}{\text{J}}_n(\kappa _{0_m} R)+C_{n,0_{m}}{\text{Y}}_n(\kappa _{0_m} R)\right] Q_{0_m0_{2}}\nonumber \\&\qquad + \sum _{j}\left[ B_{n,j}{\text{I}}_n(\kappa _j R)+C_{n,j}\text{K}_n(\kappa _j R)\right] Q_{j 0_{2}},\quad (l=0_{2}), \end{aligned}$$

(57)

$$\begin{aligned}&A_{n,i} {\text{K}}_n(k_i R) P_{i i}\nonumber \\&\quad =\sum _m \left[ B_{n,0_{m}}{\text{J}}_n(\kappa _{0_m} R)+C_{n,0_{m}}{\text{Y}}_n(\kappa _{0_m} R)\right] Q_{0_mi}\nonumber \\&\qquad + \sum _{j}\left[ B_{n,j}{\text{I}}_n(\kappa _j R)+C_{n,j}\text{K}_n(\kappa _j R) \right] Q_{j i}, \quad (l=i=1,2,\ldots ,M). \end{aligned}$$

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$$\begin{aligned}&A_{n,0_{1}}\frac{\partial {\text{H}}_n(k_{0_1} R) }{\partial r} P_{0_{1}0_\text{1}}\nonumber \\&\quad =\sum _m \left[ B_{n,0_{m}}\frac{\partial {\text{J}}_n(\kappa _{0_m} R)}{\partial r} +C_{n,0_{m}}\frac{\partial {\text{Y}}_n(\kappa _{0_m} R) }{\partial r}\right] Q_{0_m0_{1}}\nonumber \\&\qquad + \sum _{j}\left[ B_{n,j}\frac{\partial {\text{I}}_n(\kappa _j R) }{\partial r} +C_{n,j}\frac{\partial {\text{K}}_n(\kappa _j R)}{\partial r} \right] Q_{j 0_{1}},\quad (l=0_{1}), \end{aligned}$$

(59)

$$\begin{aligned}&I_{0_{2}}{\text{i}}^n \frac{\partial {\text{J}}_{n}(k_{0_2} R)}{\partial r} P_{0_{2}0_{2}}+A_{n,0_{2}}\frac{\partial {\text{H}}_n(k_{0_2} R) }{\partial r} P_{0_{2}0_{2}}\nonumber \\&\quad =\sum _m \left[ B_{n,0_{m}}\frac{\partial {\text{J}}_n(\kappa _{0_m} R)}{\partial r} +C_{n,0_{m}}\frac{\partial {\text{Y}}_n(\kappa _{0_m} R) }{\partial r}\right] Q_{0_m0_{2}}\nonumber \\&\qquad + \sum _{j}\left[ B_{n,j}\frac{\partial {\text{I}}_n(\kappa _j R) }{\partial r} +C_{n,j}\frac{\partial {\text{K}}_n(\kappa _j R)}{\partial r} \right] Q_{j 0_{2}},\quad (l=0_{2}), \end{aligned}$$

(60)

$$\begin{aligned}&A_{n,i} \frac{\partial {\text{K}}_n(k_i R)}{\partial r} P_{i i}\nonumber \\&\quad =\sum _m \left[ B_{n,0_{m}}\frac{\partial {\text{J}}_n(\kappa _{0_m} R)}{\partial r} +C_{n,0_{m}}\frac{\partial {\text{Y}}_n(\kappa _{0_m} R) }{\partial r}\right] Q_{0_mi}\nonumber \\&\qquad + \sum _{j}\left[ B_{n,j}\frac{\partial {\text{I}}_n(\kappa _j R) }{\partial r} +C_{n,j}\frac{\partial {\text{K}}_n(\kappa _j R)}{\partial r} \right] Q_{j i}, \quad (l=i=1,2,\ldots ,M). \end{aligned}$$

(61)

$$\begin{aligned}&\sum _m \left[ B_{n,0_{m}}\frac{\partial {\text{J}}_n(\kappa _{0_m} a)}{\partial r} + C_{n,0_{m}}\frac{\partial {\text{Y}}_n(\kappa _{0_m} a) }{\partial r}\right] Q_{0_m0_{1}}\nonumber \\&\qquad + \sum _{j}\left[ B_{n,j}\frac{\partial {\text{I}}_n(\kappa _j a) }{\partial r} +C_{n,j}\frac{\partial {\text{K}}_n(\kappa _j a)}{\partial r} \right] Q_{j 0_{1}}=0,\quad (l=0_{1}), \end{aligned}$$

(62)

$$\begin{aligned}&\sum _m \left[ B_{n,0_{m}}\frac{\partial {\text{J}}_n(\kappa _{0_m} a)}{\partial r} +C_{n,0_{m}}\frac{\partial {\text{Y}}_n(\kappa _{0_m} a) }{\partial r}\right] Q_{0_m0_{2}}\nonumber \\&\qquad + \sum _{j}\left[ B_{n,j}\frac{\partial {\text{I}}_n(\kappa _j a) }{\partial r} +C_{n,j}{\tilde{K}}'_{n,j}(a) \right] Q_{j 0_{2}}=0,\quad (l=0_{2}), \end{aligned}$$

(63)

$$\begin{aligned}&\sum _m \left[ B_{n,0_{m}}\frac{\partial {\text{J}}_n(\kappa _{0_m} a)}{\partial r} +C_{n,0_{m}}\frac{\partial {\text{Y}}_n(\kappa _{0_m} a) }{\partial r}\right] Q_{0_mi}\nonumber \\&\qquad + \sum _{j}\left[ B_{n,j}\frac{\partial {\text{I}}_n(\kappa _j a) }{\partial r} +C_{n,j}\frac{\partial {\text{K}}_n(\kappa _j a)}{\partial r} \right] Q_{j i}=0, \quad (l=i=1,2,\ldots ,M), \end{aligned}$$

(64)

where

$$\begin{aligned}&Q_{cl}= \int _{-H}^{-h_1}{\tilde{Z}}_{c}Z_{l} {\text{d}}z + \gamma \int _{-h_1}^{0}{\tilde{Z}}_{c}Z_{l} {\text{d}}z,\nonumber \\&(c=0_{1},0_{2},{\text{I}},{\text{II}},1,2,\ldots ). \end{aligned}$$

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