A two-layer viscous Boussinesq-type model for surface waves: Derivation, analysis, numerical implementation, and verification

Abstract

A two-layer viscous Boussinesq-type model is developed to simulate the wave energy dissipation during wave propagation in deep water. The viscous terms are incorporated into both the dynamic and kinematic boundary conditions at the free surface, and the corresponding analytical solution of the second-order amplitude has been derived for the first time. The linear and nonlinear properties of the model are analyzed with different viscosity coefficients. When the viscosity coefficient is 1 × 10−4 m2/s, the linear phase velocity, decay rate, second-order amplitude, and velocity profiles of the viscous model are accurate for up to h/L0 (h is water depth, L0 is characteristic wavelength) ≈ 8.66, 5.86, 3.60, 3.60, and 7.51 within 1% error, respectively. The finite difference method is adopted for the numerical implementation of the model. To verify the linear and nonlinear properties of the model, computed results for linear waves and focused wave group in deep water are compared with linear analytical solutions and experimental data, respectively.