Surface water waves propagating over a submerged forest

Abstract

This paper reports a semi-analytical theory of water waves obliquely propagating over a submerged vegetated belt area. A mathematical model is developed for small-amplitude periodic waves, scattered by an array of submerged vertical cylinders within an infinitely long strip. Assuming a strong contrast between the cylinder spacing and the typical wavelength, the multi-scale perturbation theory of homogenization is employed to derive the governing equations for the macro-scale wave dynamics and the boundary-value problem of micro-scale flows within a unit cell of the cylinder array. The constitutive coefficients in the macro-scale governing equations are computed using the solutions of the micro-scale boundary-value problem, being driven by the macro-scale pressure gradients. Flow turbulences in the vicinity of cylinders are represented by the eddy viscosity model in which the bulk eddy viscosity is determined by balancing the time-averaged rate of energy dissipation and the rate of work done by wave forces on the cylinders, integrated over the entire submerged forest. The wave forces are calculated by the Morison-type formula, in which a new drag formula as a function of Reynolds number is constructed based on existing and newly conducted experimental data. The potential decomposition method is employed in solving the waves/vegetation interaction problem on the macro-scale, which well captures the effects of wave scattering. The theory was checked with several sets of experimental data for normally incident waves. The agreement between the theory and experiments is very good for cases where the submerged forest heights vary from shallow to near water depth. Results for obliquely incident waves are also presented and discussed.