A One-Dimensional Integral Equation for a Porous Horizontal Disc under Water Waves

Technical Summary: A One-Dimensional Integral Equation for a Porous Horizontal Disc under Water Waves

Problem Statement
This study investigates the wave scattering and radiation phenomena generated by a thin, porous, circular plate submerged in deep water. The primary objective is to formulate and solve the hydrodynamic problem to determine the added mass and damping coefficients acting on the structure. The analysis focuses on the influence of the complex porous effect parameter, $G$, which characterizes the plate's resistance and inertial properties. The problem is mathematically modeled as a second-kind hypersingular Fredholm integral equation defined over the unit disk.

Methodology
The research employs a potential flow theory framework, assuming the fluid motion is small-amplitude, irrotational, incompressible, and inviscid. The governing boundary value problem is reduced to a hypersingular integral equation for the velocity potential jump across the disk.

Key methodological steps include:

1. Dimensional Reduction: Following the mathematical framework established by Farina & Martin (1997), the three-dimensional problem is reduced to a one-dimensional integral equation under the assumption of axisymmetric motion. This is achieved by expanding physical quantities into Fourier series and introducing an unknown auxiliary function, $\psi$.
2. Governing Equation: The reduction yields a one-dimensional Fredholm integral equation of the second kind (Equation 21) featuring singular kernels. This equation incorporates the wavenumber $K$, the porosity parameter $G$, and the submergence depth $d$.
3. Numerical Solution: The hypersingular nature of the equation precludes the direct application of standard Nyström methods with Gauss-Legendre quadrature. Instead, the authors utilize the Boundary Element Method (BEM) coupled with the D01GCF routine from the NAG library. This routine employs the Korobov-Conroy Number-Theoretic Method (NTM) to rigorously evaluate potential integrals. The singularity is managed by decomposing the kernel and partitioning the integration domain into subintervals adjacent to the singular point.
4. Validation: The numerical implementation is validated through a grid independence test and by comparing results against established literature, specifically the work of De Freitas et al. (2021) and Farina & Martin (1997).

Key Contributions and Results
The paper presents a validated numerical formulation for calculating hydrodynamic forces on submerged porous discs. The results are analyzed across five specific scenarios varying the porosity parameter $G$ (real, imaginary, and complex) and the submergence depth $d$:

• Porosity Influence: The study demonstrates distinct behaviors based on the nature of $G$:
  • Real $G$ (Resistive): Represents high-resistance porous structures. As the real component increases, the disk behaves more like a solid plate, hindering fluid penetration. This results in reduced added mass and damping effects compared to less resistive configurations.
  • Imaginary $G$ (Inertial): Represents structures dominated by inertial effects within the pores. In this regime, the fluid tends to follow the disk's motion, acting as an effective added mass. This leads to significantly high peaks in both added mass and damping coefficients, particularly in near-free-surface configurations where resonance in the fluid layer above the disk is induced.
• Numerical Agreement: The proposed formulation shows excellent agreement with the reference data from De Freitas et al. (2021). Quantitative validation using Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) confirms the consistency of the derived added mass and damping coefficients across various wavenumbers ($Ka$) and depths.
• Connection to Classical Theory: The study identifies that when the wavenumber $K=0$, the governing equation simplifies to a form resembling the classical Love-Lieb equation. This establishes a theoretical link between the hydrodynamic problem of a submerged porous disc and the electrostatic problem of coaxial circular plates, suggesting the proposed formulation is a generalization of the Love equation for fluid-structure interaction contexts.

Significance and Claims
The authors claim that the research successfully adapts the analytical strategy of Farina & Martin (1997), originally developed for rigid plates, to the context of porous bodies. The primary significance lies in the derivation of a streamlined one-dimensional integral formulation that preserves the essential mathematical structure of the original model while accommodating the complexity of porous media.

The paper modestly notes that while the numerical consistency validates the approach, the treatment of singularities in such equations remains a challenge, necessitating the development of alternative numerical strategies beyond standard quadrature methods. The study is limited to small submergence depths ($d=0.1$ and $d=0.2$), focusing on resonance phenomena intensified by the proximity of the free surface. The authors conclude that the complex porosity parameter $G$ serves as a critical tuning mechanism for controlling extreme values of incident hydrodynamic forces, with the real component acting dissipatively and the imaginary component amplifying resonance.