The hydrodynamic models that have been used in this research all compute with the depth-averaged velocity. In reality the flow velocity is not constant over the depth and the velocity at the surface is usually greater than the depth-averaged velocity.
For instance in the case of the closed basin, a stationary situation is reached in depth-averaged models when all velocities are zero; therefore the new formulation has no influence on the final wind setup. When, on the other hand, the same test is performed in 3D (or in 1DV or 2DV with the use of multiple layers) then a constant circulation will remain in the final situation. At the surface the water will flow in the same direction as the wind, while there is a return flow at the bottom, see figure 5.16.
usurface
0 u z
Figure 5.16: Velocity profile in a 3D-computation
So, in the case of a 3D computation a constant flow velocity at the surface will remain. Therefore the wind stress in a stationary situation becomes smaller with the use of the new wind formulation. This results in a lower wind setup. In conclusion it might be said that the effects of the new wind formulation will become stronger when a 3D-model is used. This requires further study.
5.10 Correcting the wind speed to the surface
5.10.1 Introduction
As stated in section 4.3, the wind velocity is defined at a height of 10 m, while the velocity at the surface should actually be used to subtract the flow velocity from the wind. A parameter α has been introduced, which is the relation between the wind at the surface and the wind at 10 m height. With the use of this factor the influence of the flow velocity becomes stronger. In this experiment the effects of this adjustment in 1D and 2D are tested.
5.10.2 Effects in 1D
A semi-closed basin with a constant depth of 5 m and a length of 10 km is modelled.
On the left side there is an open boundary with ζ = 0; on the right side the boundary is closed. The wind velocity is 30 m/s. This model has been run with different values of α.
uwind
ζ=0
Figure 5.17: Setup of the experiment: 1D closed basin - side view
The results are presented in figure 5.18. It is obvious that the introduction of the α correction factor has a positive effect on the damping and the time needed for the model to reach a steady state.
0 1 2 3 4 5 6 7 8
(b) Time series - detailed
Figure 5.18: Surface elevation at end of basin
5.10.3 Effects in 2D
To find out what the effects of the α-factor in a 2D-model are, the Ketelmeer-schematization (section 5.7) is used, see figure 5.19. The wind velocity is 30 m/s; all other parameters are the same as in section 5.7.
depth= 0.5m depth= 10.0m
depth= 10.0m station 1 station 2
Figure 5.19: Tidal flat model
It is clear that the influence of the flow velocity in the new formulation becomes stronger due to the use of the factor α. The results in table 5.2 and figure 5.20 show that the water level in the 2-dimensional Ketelmeer-schematization is further reduced when the α is applied.
wind-formulation alpha water level in station 1 reduction
wind 1 n/a 0.9138 m
-wind 3 1.0 0.8907 m 2.5%
wind 3 0.5 0.8703 m 4.8%
wind 3 0.2 0.8209 m 10.2%
Table 5.2: Results for different values of α
0 0.5 1 1.5
Surface elevation in station 1 (100,25)
time in hrs
Surface elevation in m
wind 1 − traditional formulation wind 3 − alpha = 1.0 wind 3 − alpha = 0.5 wind 3 − alpha = 0.2
(a) Water level in station 1
0 0.5 1 1.5
Horizontal velocity ux at station 2 (50,25)
time in hrs
Velocity in m/s
wind 1 − traditional formulation wind 3 − alpha = 1.0 wind 3 − alpha = 0.5 wind 3 − alpha = 0.2
(b) Velocity in station 2
Figure 5.20: 2D results for different values of α
Chapter 6
Grid size
6.1 Introduction
Apart from the wind-formulation, there may be other factors which influence the results during the modelling of very shallow areas. Capel (2001) states in his MSc-thesis on Tropical Cyclone-induced Storm Surges that the grid size might have an important effect on the results. He did a few theoretical experiments at which the results show under-prediction of the storm surge height if a coarse grid is chosen.
Blain, Westerink and Luettich (1998) also did grid convergence studies for the pre-diction of hurricane storm surges, but they found that under-resolution (i.e. the use of a coarse grid) on the continental shelf leads to significant over-prediction of the primary storm surge.
These two studies have both been tested on large scale models, though Capel used a theoretical 1d-shore with stationary wind input while Blain et al. (1998) performed the tests on existent cyclone-models, which included more (large scale) physical effects.
In order to find out what the effects of the grid size on small domain models like the IJsselmeer-model are, a few convergence tests will be performed with smaller 1-dimensional models.
6.2 Setup of the experiment
A few experiments will be carried out to test the influence of the grid size on shallow water areas for a constant wind in a 1D-model. First, three tests are carried out with a sloping bottom, testing three different slopes. Consequently, two experiments are done with a closed basin with variable depths.
All tests have a closed boundary on the right side. The shores (tests 1,2 and 3) have ζ = 0 as boundary condition on the left side, while the other tests (closed basin) have a closed boundary on the left side as well.
In order to test the grid size sensitivity, five different grid sizes are chosen between 1 and 1000 m. The time step changes with the same factor to keep the Courant-number (σ = u∆x∆t) the same, so it varies between 0.01 and 10 seconds. Other parameters that are used:
C = 60 m1/2/s Cd = 0.0043
uw = 20 m/s (for tests 1,2 and 3) or 30 m/s (tests 4 and 5)
6.3 Results
6.3.1 Test 1: Shore with 2 different slopes Figure 6.1 shows setup for the first experiment.
uwind
20.0 m
station 1
5 km 5 km
2.0 m
Figure 6.1: Setup of experiment 1
The results, presented in figure 6.2 and table 6.1, show that the use of a coarse grid leads to severe under-estimation of the storm surge level in station 1. They also illustrate that decreasing the grid size leads to convergence. In this (small domain) case a grid size of about 10 m would be necessary to be able to neglect the errors due to the use of a coarse grid.
0 1 2 3 4 5 6
0 0.05 0.1 0.15 0.2 0.25
time (hrs)
surface elevation (m)
Water level in station 1
dx=1000 m dx=500 m dx=100 m dx=10 m dx=1 m
Figure 6.2: Water level at station 1 using different grid sizes, test 1
∆x mean water level under-estimation
1000 0.08383 12.8 cm 60%
500 0.11965 9.1 cm 46%
100 0.18424 2.7 cm 13%
10 0.20986 0.1 cm .5%
1 0.21101 -
-Table 6.1: Results at station 1 for test 1
6.3.2 Test 2: Shore with a steep slope
In test 2 again a hypothetical shore is simulated, this time with a constant slope that goes from 10 m below to 10 m above reference. The results are quite similar to test 1, as can be seen in figure 6.4 and 6.2.
Since the first test showed that in a similar case there is convergence for ∆x = 10m, the test with a grid size of 1 m - which takes several hours of computational time - is now omitted.
uwind
10.0 m
10.0 m
station 1
5 km 5 km
Figure 6.3: Setup of experiment 2
0 2 4 6 8 10 12
−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
time (hrs)
surface elevation (m)
Water level in station 1
dx=1000 m dx=500 m dx=100 m dx=10 m
Figure 6.4: Water level at station 1 using different grid sizes, test 2
∆x mean water level under-estimation
1000 0.16470 21.1 cm 56%
500 0.22999 14.6 cm 39%
100 0.33538 4.1 cm 11%
10 0.37605 -
-Table 6.2: Results at station 1 for test 2
6.3.3 Test 3: Shore with a gentle slope
In test 3 a gentle slope is modelled, as opposed to the steep slope in test 2. Results show about the same absolute under-estimation (20 cm for ∆x = 1000m), but the relative under-estimation is significantly smaller. Figure 6.6 shows a bump after about nine hours for the finer grids. This is due to the returning wave that arises when the water reaches the closed right boundary. In the tests that use the coarser grid, the water is dragged onto the slope more slowly, so the reflecting wave hasn’t reached station 1 after twelve hours.
uwind
station 1
5 km 5 km
2.0 m 2.0 m
Figure 6.5: Setup of experiment 3
0 2 4 6 8 10 12
−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
time (hrs)
surface elevation (m)
Water level in station 1
dx=1000 m dx=500 m dx=100 m dx=10 m
Figure 6.6: Water level at station 1 using different grid sizes, test 3
∆x water level after 6h under-estimation
1000 0.5951 19.8 cm 25%
500 0.6983 9.5 cm 12%
100 0.7751 1.8 cm 2%
10 0.7929 -
-Table 6.3: Results at station 1 after 6 hours for test 3
6.3.4 Test 4: Closed basin - 10 m deep
This test models a 10 km long basin with a constant depth of 10 m. The water level is monitored at station 3, which is located at the end of the basin, see figure 6.7.
The results, figure 6.8 and table 6.5, show that a coarse grid not only leads to an under-estimation of the mean water level, but also makes the oscillations damp out much faster.
uwind
station 3
10 km
10 m
Figure 6.7: Setup of experiment 4
0 2 4 6 8 10 12
0 0.05 0.1 0.15 0.2 0.25
time (hrs)
surface elevation (m)
Water level in station 3
dx=1000 m dx=500 m dx=100 m dx=10 m dx=1 m
Figure 6.8: Water level at station 3 using different grid sizes, test 4
∆x mean water level under-estimation
1000 0.09619 1.0 cm 10 %
500 0.10159 0.5 cm 5 %
100 0.10579 0.1 cm 0.6 %
10 0.10642 0.0 cm 0.05 %
1 0.10647 -
-Table 6.4: Results at station 3 for test 4
6.3.5 Test 5: Closed basin - 0.5 m deep
Finally, the same basin as in test 4 is modelled, but this time with a depth of 0.5 m. A coarse grid leads to a greater absolute under-estimation than for the deep basin, but the relative under-estimation is much smaller, only 5 % for ∆x = 1000m (see table 6.5).
uwind
station 3
10 km
2.0 m
Figure 6.9: Setup of experiment 5
0 2 4 6 8 10 12
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
time (hrs)
surface elevation (m)
Water level in station 3
dx=1000 m dx=500 m dx=100 m dx=10 m
Figure 6.10: Water level at station 3 using different grid sizes, test 5
∆x water level after 12h under-estimation
1000 1.530 10.0 cm 6%
500 1.580 5.0 cm 3%
100 1.621 0.9 cm .6%
10 1.630 -
-Table 6.5: Results at station 3 after 12 hours for test 5