Energy Balance Relations for Flow Through Thick Porous Structures

In wave–structure interaction problems, energy balance relations are often derived and used to check the accuracy of the computational results obtained using numerical methods. These energy identities are also used to get qualitative information about various physical quantities of interest. It is well known that for rigid structures, the energy identity is K_r²+K_t²= 1, where K_r and K_t are the reflection and transmission coefficients, respectively. Even if we take flexible barriers, then also the aforementioned energy identity will hold. Now, for wave past a thick porous structure, often a major portion of the incoming wave energy is dissipated due to the structural porosity. So, the aforementioned energy identity will be modified into K_r²+K_t²+K_D= 1, where K_D takes into account the amount of dissipative wave energy. These energy identities are available in the literature for thin porous barriers. But derivation of the energy identity is complicated for thick porous structures due to complex momentum equation and boundary conditions. In the present paper, an appropriate energy identity will be derived for water waves past a thick rectangular porous structure. In this regard, Green’s second identity is used in multi-domain regions with the arguments velocity potential and its complex conjugate. With the help of complex function theory, the final form of the same is written in a compact form. Now, to compute each quantity associated with the energy identity, the associated boundary value problem is converted into a system of Fredholm integral equations. Finally, using the boundary element method, the components present in the energy identity are obtained and checked for validation.