Modification of Velocity Formulations in a Two-Layer Boussinesq-Type Model for Water Waves

Abstract

In order to improve the accuracy of velocity formulation in a Boussinesq-type wave model, with a two-layer Boussinesq-type model with the highest spatial derivative of 2 being chosen as the research object, a third-order term with constant coefficient is proposed to modify the velocity formulation. The coefficient is optimized by minimizing the error between the summation of the integration of horizontal and vertical velocities of the equation and that of the analytical linear Stokes wave velocity components in the range of 0<kh< 8 (where k is wave number, h is still water depth). At a 1% tolerance error, the applicable water depths of the modified formulations for horizontal and vertical velocities are up to kh=7.34 and kh=7.83, respectively, which are larger than those of the original formulations. The evolution of the steady-state wave and the focused wave is numerically simulated by using the numerical model. The horizontal velocity under the maximum surface elevation crest is in good agreements with the analytical solution of stream function and published experimental data, which verifies the effectiveness of the modified formulations. The studies show that the velocity accuracy of the improved equation is greatly improved. This method provides an important reference for the improvement of velocity field of other Boussinesq-type models.

Keywords

• boussinesq-type equation
• velocity formulation
• analytical solution
• numerical simulation