Comparison with 1D Analytical Solution

The governing equations for simplified 1D tidal motion over a constant depth channel can be expressed as (Wu et al., 1994):

$$\begin{split} &\frac{\partial \eta }{\partial t} + H \frac... ...ial t} + g \frac{\partial \eta } {\partial x} = 0 \end{split}$$ (9.1)

Where H is the constant depth below the mean level as in figure 3.1.

  

Figure 6.1: Points of level measurement.

The open boundary conditions are specified by a sinusoidal wave (6.2) forcing the West border of the channel of figure 6.1 and a non reflecting boundary following Sommerfeld radiation equation on the East wall of the channel to simulate an infinite length channel and a wave that travels along it. The 1D form the Sommerfeld equation is rewritten as (6.3).

  

$$\eta(0,t) = A \sin ( \omega t)$$ (9.2)

$$\eta _t+c\eta _x$$ = 0 (9.3)

Where A is the imposed wave amplitude, $\omega$ is the wave radiation frequency, $\omega=2\pi/T$, where T is the tidal period, (12.4206 hours for the case of the mean semi diurnal component M₂) and c is the wave celerity.

The exact solution of equations (6.1) is of the form:

 

$$\eta(x,t)=A \sin ( \omega t - x / \lambda)$$ (9.4)

$$u(x,t)=\frac aH \sqrt{gH} \sin ( \omega t - x / \lambda)$$ (9.5)

From that solution it arises that, for the constant depth channel, both the celerity and wave length are constant, being their values $c=\sqrt{gH}$, $\lambda=c \; T$.

As initial conditions (t=0) the water level is set to 1 meter and the velocity to 0 m/s. Amplitude A in the simulation is 5 cm (micro-tide regime). After a short transient, the channel reaches a stationary regime.

Figure 6.2 represents the level output, at points $\alpha_0$ $\beta_0$ and $\gamma_0$ of figure 6.1, for a 72-hour simulation, shown besides the analytical solution.

  

Figure 6.2: Numerical and analytical level comparison.

As it's shown in figure 6.2, there is no significant difference between model and analytical solution. Figure 6.3 is a zoomed version of this comparison showing an error of less than 0.5 mm in the amplitude (1%). That error could be attributed to numerical viscosity (Hirsch, 1991; Rizzi & Engquist, 1987; Roache, 1972) and to small differences between the real position of the points in the grid and computed position in the analytical solution (Augenbaum et al., 1991). An enlightening animation of the wave traveling along the channel could be seen in Kaplan (1998a).

The numerical code attains a very good approximation to the analytical solution. The developed low reflecting boundary condition has proven to be a very good implementation for the free border simulation and very useful in the infinite length channel modeling.

  

Figure 6.3: Zoomed view of the numerical and analytical level comparison.