Standing wave in a rectangular basin

Purpose

The sloshing test case is useful for testing the model numerical dissipation for different temporal schemes.

Problem

Assuming no Coriolis, bottom friction, advection, diffusion, and that the water elevation is much smaller than the water depth, the governing equations are given by

${\frac {\partial \eta }{\partial t}}+h{\frac {\partial U}{\partial x}}=0$

(1)

${\frac {\partial U}{\partial t}}=-g{\frac {\partial \eta }{\partial x}}$

(2)

where

$t$ = time [s]

$x$ = horizontal coordinate [m]

$U$ = current velocity [m/s]

$h$ = water depth [m]

$\eta$ = water level with respect to the Still Water Level (SWL) [m]

Eliminating the velocity from both equations leads to the classical wave equation

${\frac {\partial \eta ^{2}}{\partial t^{2}}}=-gh{\frac {\partial \eta ^{2}}{\partial x^{2}}}$

(3)

Given a closed rectangular basin of length $L$ and width $W$, the following solution can be found for simple harmonic motion (Lamb 1945)

$\eta (x,t)=A\cos {(kx)}e^{i\omega t}$

(4)

where $k=m\pi /L$ is the wave number (eigenvalue), with $m=1,2,...$, $\omega$ is the wave frequency, and $T$ is the wave (eigen) period given by

$T={\frac {2}{\sqrt {gh{m/L}^{2}}}}$

(5)

When $m=1$ the solution is referred to as the fundamental mode. The fundamental mode is the solution used here.

Model Setup

Figure 1. Computational grid for the sloshing test case.

The model domain is 1000 m long and 300 m wide. The computational grid is shown in Figure 1. The grid has a constant grid spacing of 100 m.

The temporal tem is approximated by

${\frac {\partial \phi }{\partial t}}={\frac {(1+0.5\theta )\phi ^{n+1}-(1-\theta )\phi ^{n}+0.5\theta \phi ^{n-1}}{\Delta t}}$

(6)

where $\theta$ is a weighting factor between 0 and 1. For $\theta$ = 0, the scheme reduces to the first-order backward Euler scheme, and with $\theta$ = 1, the scheme reduces to the second-order backward scheme (Ferziger and Peric 1997).

The model setup is summarized in Table 1. Table 1. Summary of the model setup for the sloshing test case.

Model Parameter	Value
Water depth	10 m
Wave amplitude	1 cm
Time step	18 s
Simulation duration	10 hrs
Ramp period	0 s
Grid resolution	100 m
Advection	Off
Diffusion	Off
Bottom friction	Off
Temporal scheme	1st and 2nd order

Results and Discussion

Figure 2. Water level profiles at different times for the sloshing test case calculated with the first and second order temporal schemes.

A time series of the water level at for both the 1st and 2nd order temporal schemes is shown in Figure 2. The 1st order temporal scheme shows significant numerical dissipation which reduces the amplitude of the standing wave. The 2nd order temporal scheme shows much less numerical dissipation and follows closely the analytical solution.

Figure 3. Water level profiles at different times for the sloshing test case calculated with the second order temporal scheme.

In order to demonstrate the low dissipation of the second order scheme, a snap shot of the water levels after approximately 16 and 16.5 wave cycles is shown for the second order temporal scheme in Figure 3.

Figure 4. Water level time series calculated at x=L using the first and second order temporal schemes.

The difference in numerical dissipation between 1st and 2nd order temporal schemes can be appreciated in Figure 4 which shows a time series of the water levels calculated at $x=L$. In the case of the 1st order scheme, the wave amplitude is reduced by approximately half within 4 wave cycles, while the 2nd order schemes maintains the same wave amplitude for the duration of the simulation.

It is noted that there is a small inconsistency between the problem governing equations and the model governing equations. The numerical model solves the shallow water equations in conservative form and does not assume that the wave amplitude is small relative to the water depth. However, as long as the the wave amplitude is specified such that it is much small than the water depth, the model governing equations are approximately equivalent to equations (1) and (2).

References

• Ferziger, J. H., and Peric, M., 1997. Computational Methods for Fluid Dynamics, Springer-Verlag, Berlin/New York, 423 p.
• Lamb, H., 1945. Hydrodynamics, 6th ed., Dover, New York, 738 p.
• Mei, C.C., 1992. The Applied Dynamics of Ocean Surface Waves, World Scientific, London, 740 p.

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