Surface gravity water waves 149
4. Change the time step to Δt = 0.285 s and Δt = 0.286 s. Conduct the
simulations and explain the results.
5. Change the wave period to T = 2 s and 32 s. Conduct the simulations
and compare the effects of the submerged breakwater on the two different period waves.
Example 6.3
This exercise demonstrates the creation of a one-dimensional linear
long wave (i.e. tsunami) due to a sudden oscillation of the sea floor (i.e.
earthquake) (Koutitas, Gousidou-Koutita and Papazachos 1983). The
data used for the simulation are as follows:
Water depth = 4 m
Bed friction = 0.001 s –1
Longitudinal length of the solution domain = 400 m
Parameter for the seismic vertical displacement = 2 m
Parameter for the seismic time occurrence = 4 s
The discretization steps for the simulation were taken as Δx = 2 m
and Δt = 0.1 s. The governing equation was given by Equation 6.23,
but the turbulence energy losses were neglected. The initial condition
was a horizontal water surface, while both boundaries were described
by a free radiation condition. The seismic oscillation was induced by
a gradual 2 m rising of the seabed within a period of 4 s. The movement occurred at a seafloor segment of 20 m located at the middle of
the domain.
The propagation of the resulted tsunami wave after times 12 s and
24 s is illustrated in Figure 6.5. Considering that the celerity for shallow waves is c
gh
o =
, then for a water depth of 4.0 m, the tsunami
front is expected to move with celerity of 6.26 m/s, a fact that is consistent with the simulation results (Figure 6.5).
Computer code 6.3
% Example 6.3 1-D Linear Long Wave Generated from Sudden Sea
Floor Movement
% ho = Water depth [m];
% fbo = bed friction;
% nd = Parameter of time variation of sea floor movement;
% zbmax = Parameter for maximum sea floor movement [m];
% Dx = Spatial step [m];
% Dt = Time step [s];
% nx = Number of computational steps;
% nt = Number of time steps;
clc; clear all; close all;