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Initial and boundary conditions

To solve the continuity and momentum equations, appropriate boundary conditions need to be imposed at the boundaries of the computational grid.

In general, initial conditions are more important for relative short simulations (e.g. a few minutes in case of short waves or a few days in case of tidal waves). Boundary conditions are by far more important for longer simulations. Moreover, often no information is available in order to start the simulation. Therefore, the simulation will usually start with zero velocities and a spatially constant water level, and the simulation will be long enough to get a steady-state solution; see also Section 5.4.1.

Waves may be generated along one or two boundaries. These are called wavemaker boundaries. First, it is assumed that the variation of the depth along these wavemaker boundaries is slowly. Second, it is advised to place these wavemaker boundaries away from the area of interest, and from steep topography. At the wavemaker boundary, we may imposed either regular or irregular waves. For one-dimensional cases they are by definition long-crested or uni-directional. For a two-dimensional case short-crested or multi-directional waves can also be specified. Usually, a time series need to be given for incident waves. This may either be synthesized from parametric information (wave height, period, etc.) or derived from a surface elevation time series.

For regular waves, at least the wave height and the wave period must be specified. Optionally, the wave direction can be specified as well. Alternatively, a time series can be imposed.

For irregular waves, either a spectrum or time series can be enforced. In the case of a spectrum, both the shape and wave characteristics need to be specified. The usual shape is either Jonswap or Pierson-Moskowitz. Sometimes a TMA shape is desired. The wave characteristics are determined by the following parameters: the significant or RMS wave height, peak or first order mean period, peak wave direction, and directional spreading (only in case of short-crested waves). The frequency range [fmin, fmax] is such that the highest frequency, fmax, equals 3 times the peak frequency (or mean frequency), while the lowest one, fmin, equals half of the peak/mean frequency.

Alternatively, a spectrum file may be given. There are two types of files:

Using a spectrum a time series of surface elevation will be synthesized. At least, the length of this series should correspond to the time period over which surface elevation and velocities are outputted after steady-state condition has been established. This time period should be long enough to provide statistically reliable wave data. After this time period the time series repeats itself. This duration of the time series is called the cycle period (see command BOUnd ... SPECTrum ... [cycle]). The recommended range is from 100 to 300 wave periods. If the cycle period is denoted as Tcycle, then the frequency step $ \Delta$f to be used for the evaluation of the parametric spectrum (e.g. Jonswap) equals

$\displaystyle \Delta$f = $\displaystyle {\frac{{1}}{{T_{\rm cycle}}}}$

Thus the spectrum is divided into N frequency bins with uniform spacing $ \Delta$f,

N = $\displaystyle {\frac{{f_{\rm max} - f_{\rm min}}}{{\Delta f}}}$

Referring to the above example, we impose a Jonswap spectrum at the wavemaker boundary. The peak period is 10 s, so that the frequencies are in between 0.05 Hz and 0.3 Hz. The duration of the time series of surface elevation to be synthesized is set to 30 minutes, which is supposed to be accurate enough to get sufficient statistics like wave height and mean period. Hence, in total, 450 wave components will be generated at the entrance of the computational domain.

Please be careful in choosing the cycle period. The larger this period the more wave components will be involved at the wavemaker boundary. Based on these components, SWASH will synthesize time series for the orbital velocities in each grid point and each vertical layer along the boundary. That would enhance the turn-around time of the simulation a lot.

When imposing a spectrum at the boundary, one has to realize that some so-called evanescent modes might be included as well. These modes show exponential decay with distance from the boundary at which the spectrum is imposed. As such, they can not be "seen" by the model. Evanescent waves are a general property of the underlying model equations. The frequency at which the evanescent modes are generated is the cut-off frequency and is determined by the dispersive properties of the model equations. It is given by

$\displaystyle \omega_{{\rm cf}}^{}$ = 2 K $\displaystyle \sqrt{{\frac{g}{d}}}$

with K the number of layers used in the model. Hence, the lowest wave period to be considered in the model simulation equals 2$ \pi$/$ \omega_{{\rm cf}}^{}$. So, given the depth at the boundary and the number of layers used, the cut-off frequency is determined above which the evanescent waves are generated at the wavemaker boundary where the spectrum is imposed; see Table 5.3.
Table 5.3: Cut-off frequency (in Hz) as function of still water depth (in m) and number of layers.
d (m) K=1 K=2 K=3
1 1.00 1.99 2.99
5 0.45 0.89 1.34
10 0.32 0.63 0.95
15 0.26 0.51 0.77
20 0.22 0.45 0.67
25 0.20 0.40 0.60
30 0.18 0.36 0.55
35 0.17 0.34 0.51
40 0.16 0.32 0.47
45 0.15 0.30 0.45
50 0.14 0.28 0.42
100 0.10 0.20 0.30

These evanescent modes will be removed by SWASH. Note that these modes carry a little bit energy and thus negligible. SWASH will give a warning when at least 10% of the total wave components are the evanescent modes that have been removed. If there are too much evanescent modes on the boundary, i.e. these modes together contain a significant amount of energy of the wave spectrum, the user is advised either to enlarge the number of layers (see also Table 5.2) or to truncate the imposed spectrum (e.g. SWAN spectrum), i.e. the highest frequency of the spectrum is not larger than the given cut-off frequency.

In the above example, one layer (K = 1) has been chosen. We assume that along the wavemaker boundary we have a uniform depth of 20 m. So the cut-off frequency is 0.22 Hz (see Table 5.3; the lowest wave period is thus 4.5 s). However, the highest frequency is 0.3 Hz. So there are 144 evanescent modes on the boundary, which is about 30%, thus reasonably. They will be removed from the boundary.

For high waves, sub and super harmonics are generated due to nonlinearity. These waves are called bound waves as they are attached to the primary wave and travel at its phase speed instead of that of a free wave at the same frequency. If linear wave conditions are enforced at the boundaries, the model will generate free components with the same magnitude but 180o out of phase with the bound waves at the wavemaker in order to satisfy the linear wave boundary condition. The presence of bound and free waves that travel at different speeds will lead to a spatially nonhomogeneous wave field with the wave height changing continuously over the domain. For such a case, it is recommended to add bound waves at the wavemaker boundaries (see command BOUnd ... ADDBoundwave ...).

To simulate entering waves without some reflections at this boundary, a weakly reflective condition allowing outgoing waves must be adopted (command BTYPE WEAK). This type of radiation condition has been shown to lead to good results within the surf zone.

Waves propagating out of the computational domain are absorbed by means of a sponge layer placed behind an output boundary. It is recommended to take the width of the sponge layer at least 3 times the typical wave length. However, for long waves a Sommerfeld radiation condition might be a good alternative.

When no boundary conditions are specified at a boundary, this boundary is considered to be a close one. This boundary is fully reflective. Alternatively, periodic boundary conditions can be applied at two opposite boundaries. This means that wave energy leaving at one end of the domain enters at the other side. In this case no reflections at these boundaries occur. This is recommended in the case of a simulation of a field case where longshore bottom variations are negligible. In such a case the computational domain is made repeated in a representative direction (see command CGRID ... REPeating X|Y).

next up previous index
Next: Numerical parameters Up: Setting up your own Previous: Input grids   Index
The SWASH team 2017-04-06