Unsteady ship waves in shallow water of varying depth based on Green-Naghdi equation

摘要

Based on Green-Naghdi equation this work studies unsteady ship waves in shallow water of varying depth. A moving ship is regarded as a moving pressure disturbance on free surface. The moving pressure is incorporated into the Green-Naghdi equation to formulate forcing of ship waves in shallow water. The frequency dispersion term of the Green-Naghdi equation accounts for the effects of finite water depth on ship waves. A wave equation model and the finite element method (WE/FEM) are adopted to solve the Green-Naghdi equation. The numerical examples of a Series 60 (C_B = 0.6) ship moving in shallow water are presented. Three-dimensional ship wave profiles and wave resistance are given when the ship moves in shallow water with a bed bump (or a trench). The numerical results indicate that the wave resistance increases first, then decreases, and finally returns to normal value as the ship passes a bed bump. A comparison between the numerical results predicted by the Green-Naghdi equation and the shallow water equations is made. It is found that the wave resistance predicted by the Green-Naghdi equation is larger than that predicted by the shallow water equations in subcritical flow (F_r = gh~(1/2) < 1), and the Green-Naghdi equation and the shallow water equations predict almost the same wave resistance when F_r = gh~(1/2) > 1, the frequency dispersion can be neglected in supercritical flows.