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Computational grid

A distinction is made between the definition of grids in horizontal and vertical directions. In the horizontal direction recti- or curvilinear coordinates are used. The grid definition in vertical direction is defined by means of a fixed number of layers in a such a way that both the bottom topography and the free surface can be accurately represented; see also Figure 5.1.

First we need to define the size and direction of the computational domain in the horizontal plane. The area of interest should be kept at least two wave lengths away from the boundaries. Note that if a sponge layer needs to be included, the computational domain needs to be extended with the size of that sponge layer. It is always wise to choose the grid axes being aligned as much as possible with the dominant wave direction. You may also generate a curvilinear grid.

An important aspect of specifying a computational grid is the spatial resolution. In principle, most energetic wave components need to be resolved accurately on the grid. Basically, we take sufficient number of grid points per wave length associated with the peak wave energy. For low waves, i.e. H/d $ \ll$ 1 with H a characteristic wave height (either significant or RMS) and d the (still) water depth, it is sufficient to take 50 grid cells (or 51 grid points) per peak wave length. However, for relatively high waves, it is better to take at least 100 grid cells per peak wave length. So, at least, we need to know something about a typical water depth and a typical peak period. Based on the linear dispersion relation the corresponding peak wave length can be found.

An example. We have a domain of 1500 m alongshore and 1200 m crosshore of which the deepest part is 20 m. We impose a wave spectrum at the entrance with a significant wave height of 2 m and a peak period of 10 s. According to the linear dispersion relation the peak wave length is about 120 m and Hs/d = 0.1. Hence, the waves are not too low but also not too high either. So, for safety, we choose 100 grid cells per peak wave length which implies a grid size of 1.2 m. Since the dominant wave direction is crosshore, we may relax the grid size alongshore by choosing 3 m. So, the total number of grid cells for the computational domain would be 500 x 1000.

Keep in mind, however, that waves become shorter when the water depth decreases. Hence, for a desired number of grid cells per wave length the grid resolution must therefore be higher locally. In this respect, a curvilinear grid may be useful. The desired number of grid cells for this case may be lesser than the one associated with the peak wave length. A rule of thumb is 15 to 20.

Another issue is the choice of the number of layers. This choice mainly depends on two types of application:

If one is interested in the vertical flow structures, e.g. undertow and stratified flows, then at least 10 or perhaps more vertical layers should be adopted. If necessary, a non-equidistant layer distribution can be specified; see Figure 5.1. The layer thickness hk, which is the distance between two consecutive layer interfaces, may be defined in a relative way, i.e. a percentage of the water depth similar to the $ \sigma$ -coordinates, or in an absolute way, i.e. a constant or fixed layer thickness as expressed in meters. To make sure that the sum of the layer thicknesses equals the water depth, at least one layer must be defined in a relative way.
Figure 5.1: Vertical grid definition with K layers and K+1 layer interfaces.
The choice for fixed layers may be useful in the case of a bathymetry exhibiting strong variation of the depth, such as a lake with some pits, in order to keep the vertical resolution relatively high along the bottom.

Concerning the wave transformation, the number of layers is determined by the linear frequency dispersion. In particular, the dimensionless depth, kd with k the wave number, decides the number of layers. The higher the value of kd, the more vertical layers needed. In addition, the accuracy with which the phase velocity of the wave components, c = $ \omega$/k with $ \omega$ the angular frequency, is obtained depends on the discretization of the vertical pressure gradient in the momentum equations. For wave transformation usually the Keller-box scheme is adopted; see Section 5.4.3.

The range of applicability of the SWASH model to values of kd indicating the relative importance of linear wave dispersion for primary waves is given in Table 5.1. This range is determined by requiring
Table 5.1: Range of dimensionless depth as function of number of layers K in SWASH.
K range error
1 kd $ \leq$ 0.5 1%
1 kd $ \leq$ 2.9 3%
2 kd $ \leq$ 7.7 1%
3 kd $ \leq$ 16.4 1%

a relative error in the normalized wave celerity (= c/$ \sqrt{{gd}}$) of at most 1%. An exception is the use of one vertical layer where the relative error is 3%, which is acceptable for many applications. Here, at most three layers may be considered enough for typical wave simulations. Moreover, the layers are assumed to have variable thicknesses and be equally distributed, which is the usual choice for wave simulations. So, do not use fixed layers!

It is noted that SWASH uses its own dispersion relation, which is an approximate one of the exact linear dispersion relation, given by

$\displaystyle \omega^{{2}}_{{}}$ = gk tanh(kd )

This approximate relation is derived using the Keller-box scheme (see Section 5.4.3) and depends on the number of equidistant layers employed in the model. The linear dispersion relation of SWASH using one vertical layer, i.e. depth-averaged, is given by

$\displaystyle \omega^{{2}}_{{}}$ = gk $\displaystyle {\frac{{kd}}{{1+\frac{1}{4}k^2d^2}}}$

while for two and three equidistant layers, it is given by

$\displaystyle \omega^{{2}}_{{}}$ = gk $\displaystyle {\frac{{kd+\frac{1}{16}k^3d^3}}{{1+\frac{3}{8}k^2d^2+\frac{1}{256}k^4d^4}}}$


$\displaystyle \omega^{{2}}_{{}}$ = gk $\displaystyle {\frac{{kd+\frac{5}{54}k^3d^3+\frac{1}{1296}k^5d^5}}{{1+\frac{5}{12}k^2d^2+\frac{5}{432}k^4d^4+\frac{1}{46656}k^6d^6}}}$

respectively. Thus the approximate dispersion relation is consistent with the model, particularly for relatively high wave frequencies. This will lead to more accurate results. The approximate dispersion relation in SWASH is only available for one, two and three equidistant layers with variable thickness (i.e. sigma planes, not fixed layers). SWASH shall indicate this in the PRINT file.

So, for primary waves with kd $ \leq$ 2.9, use of one layer is sufficient, while at least two equidistant layers need to be chosen if kd $ \leq$ 7.7. For most typical nearshore wave simulations, two layers may be enough. In the example above, kd = 1.04, so one vertical layer would be enough.

However, one must realized that, for a given number of layers, relatively high harmonics may propagate too slow at a given depth. For example, using one layer, SWASH is accurate up to a kd value of 2.9 for primary waves. For a depth of 20 m (deepest part in the example above), the shortest wave that can accurately modelled at this depth has a frequency of 0.18 Hz (minimum wave period of 5.6 s; derived from the approximate dispersion relation above). In other words, for a given number of layers and a water depth, there is a maximum frequency above which a wave component has an incorrect celerity; see Table 5.2. This means, for instance, that the phase differences between the concerning harmonics are wrong.
Table 5.2: Maximum frequency (in Hz) as function of still water depth (in m) and number of layers.
d (m) K=1 K=2 K=3
1 0.82 1.37 2.00
5 0.37 0.61 0.89
10 0.26 0.43 0.63
15 0.21 0.35 0.52
20 0.18 0.31 0.45
25 0.16 0.27 0.40
30 0.15 0.25 0.36
35 0.14 0.23 0.34
40 0.13 0.22 0.32
45 0.12 0.20 0.30
50 0.12 0.19 0.28
100 0.08 0.14 0.20

This is particularly important when nonlinear effects are dominant5.1.

Ideally, the maximum frequency is about 1.5 to 2 times the peak frequency at a given depth. It is then assumed that all components above this maximum frequency have a little bit amount of energy (here the presuming spectrum shape is a Jonswap one). As a consequence, the phase differences between the representative wave components, including the relatively short waves, are thus well controlled in the model.

Referred to the example above, the peak frequency at the wavemaker boundary is 0.1 Hz, while the depth is 20 m. The required maximum frequency should be at least 0.15 Hz or preferably higher. So, the use of one layer would be critical when propagation of short waves with frequencies higher than 0.18 Hz needs to be modelled accurately.

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