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Radiation and Diffraction of Water Waves by a Submerged Body with Ice Cover in Finite Depth

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Abstract

This paper presents the three-dimensional Green-function method to predict the radiation and diffraction of water waves by a submerged body in water of uniform finite depth with an ice cover. The fluid is assumed to be perfect and irrotational, the ice is modelled as an elastic plate. The zero-speed Green function of finite depth satisfying the linearized covered-surface condition is derived in three dimensions, the numerical results for the Green function and its derivatives are given. The integral equations are established by distributing the source strength on the body surface, the radiation and diffraction problems are solved. A submerged sphere is taken as an example, the effects of the water depth and the flexural rigidity of ice cover on hydrodynamics are analysed, and the good agreement with the analytical solutions reveals that the present method is correct and reliable.

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Funding

This work is supported by the Fund of Key Laboratory of Hydrodynamics (grant no. 350324100K3012AA00).

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Authors and Affiliations

  1. State Key Laboratory of Hydrodynamics, China Ship Scientific Research Center, Wuxi, China

    L. Z. Hao & Z. Y. Pan

Authors
  1. L. Z. Hao
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  2. Z. Y. Pan
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Correspondence to L. Z. Hao or Z. Y. Pan.

Appendix A

Appendix A

The expression for EPGF has been obtained as shown in Eq. (2.13), the third term in EPGF is denoted as \({{G}_{3}}\left( {P,Q} \right) = {\text{P}}{\text{.V}}.\int_0^\infty g (k)dk\). The appropriate technique for numerically solving \({{G}_{3}}\left( {P,Q} \right)\) is given in this section as there is a singular point in the integrand.

Based on the asymptotic analysis method, the singularity of \(g\left( k \right)\) at k = k1 is approximated with formula \({{c}_{1}}{\text{/}}\left( {k - {{k}_{1}}} \right)\), c1 is a constant and can be calculated as follow

$${{c}_{1}} = \mathop {\lim }\limits_{k \to {{k}_{1}}} g\left( k \right) \times \left( {k - {{k}_{1}}} \right) = \frac{{2{{S}_{1}}\left( {{{k}_{1}}} \right)}}{{S_{2}^{'}\left( {{{k}_{1}}} \right)}}\mathcal{F}\left( {{{k}_{1}}} \right){{J}_{0}}\left( {{{k}_{1}}R} \right).$$
(A.1)

Then

$${{G}_{3}}\left( {P,Q} \right) = \mathop \smallint \limits_0^{2{{k}_{1}}} \left[ {g\left( k \right) - {{c}_{1}}{\text{/}}\left( {k - {{k}_{1}}} \right)} \right]dk + \mathop \smallint \limits_{2{{k}_{1}}}^{ + \infty } g\left( k \right)dk$$
(A.2)

the integrand in the first term in the above equation is bounded, \(g\left( k \right)\) approaches to 0 rapidly for large k and no special measures is taken for the oscillation in the integrand of the second term. Equation (A.2) can be successfully solved with numerical method.

The calculation of the derivatives of EPGF has a similar process and we won’t tell the details here.

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Hao, L.Z., Pan, Z.Y. Radiation and Diffraction of Water Waves by a Submerged Body with Ice Cover in Finite Depth. Fluid Dyn 56 (Suppl 1), S70–S87 (2021). https://doi.org/10.1134/S0015462822020045

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  • DOI: https://doi.org/10.1134/S0015462822020045

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