5 Two-dimensional surface water model
5.4 Salinity transport
5.4.1 Test cases
Before applying it to the Nile Estuary case, the ability of the TELEMAC2D modeling system to model and quantify the effect of horizontal density variation was first investigated using two theoretical cases of a rectangular channel and a trapezoidal channel. For the two theoretical cases, four scenarios were modeled which are:
1) Stagnant water with horizontal density variation only and ignoring diffusion 2) Stagnant water with diffusion only and ignoring horizontal density variation 3) Stagnant water with both diffusion and horizontal density variation
4) Flowing water with both diffusion and horizontal density variation
The aim of the previous scenarios was to check how the horizontal variation of density can affect the results and to compare its impact with the one of diffusion. In addition, the shape impact was also analyzed.
5.4.1.1 Rectangular channel
The channel dimensions are 5 m, 200 m and 1000 m for the depth, the width and the length respectively, and with zero bottom slope. A triangular grid of 10 m
Freshwater
35 25 15 5
River mouth
Saltwater
River Sea
Contour lines of salinity concentration (%0)
5.4 Salinity transport discretization length was generated by MATISSE; the grid was refined in the middle part of the channel where a 4 m discretization length was used due to the choice of the initial conditions (see Figure 5.21). The total number of nodes was 3,303 and the total number of elements was 6,344 (Figure 5.20).
The channel has two open boundaries which are an U.S. boundary where the flow was given (zero in case of stagnant water and 80 m3/s in case of flowing water) and zero salinity, and a D.S. boundary where a water level of 5 m and a salinity concentration of 35 mg/l were imposed. The discharge and salinity values chosen in case of flowing water were very similar to the conditions of the Nile Estuary. The same Manning friction coefficient as for the Nile case of 0.022 m1/3/s and a constant turbulent viscosity (ν = 0.01m2/s) were employed here. The time step of the simulation was 5 seconds.
Figure 5.20: Grid of the rectangular channel (Mahgoub and Hinkelmann, 2012)
The Positive Streamline Invariant (PSI) distributive scheme was chosen after testing the other available schemes (SUPG and Method of Characteristics) which caused either stability problems or gave high negative salinity. Thereof, PSI scheme is very suitable to salinity transport simulations. The solver used in this case was GMRES. A hydrodynamic simulation was first done for the case of flowing water until reaching a steady state condition, but for the other cases it was not necessary because the water was stagnant.
0 250 500 750 1000 Bottom(m)
0 200
150 100 50 [m] 0
The initial condition for salinity was that half of the channel (starting from the D.S.
boundary) was saline water and the other half was fresh water as shown in Figure 5.21.
The diffusion (molecular and turbulent) was assumed to be constant with a value of 0.001 m²/s.
Figure 5.21: Initial conditions for salinity transport (Mahgoub and Hinkelmann, 2012) The simulation of the rectangular channel showed smaller impact for the diffusion than of density variation as shown in Figures 5.22A and 5.22B for 100 days of simulation time. For longer simulation time the impact of density variation will be bigger as it still shows slow changes while for the case of diffusion no further change was noticed after 10 days of simulation.
As the impact in case of turbulent diffusion only was quite small, combining both diffusion and density variation showed very similar results as the case of density variation only (Figure 5.22C).
When introducing a flow from the U.S. boundary (80 m3/s), the saline water moved towards the D.S. boundary very quickly (Figure 5.22D shows the result after about 80 minutes only) due to the momentum produced from the flow velocity which was higher than the one due to the spatial variation of density. No doubt that changing the discharge will affect the movement speed of the saline water.
0 250 500 750
36 30 24 18 12 6 0 Salinity (kg/m3)
200 150 100 50 0 [m]
5.4 Salinity transport
Figure 5.22: Salinity transport for the rectangular channel for A) stagnant water with horizontal density variation only, B) stagnant water with diffusion only, C) stagnant water with horizontal density variation and diffusion and D) flowing water with horizontal density variation and diffusion (Mahgoub and Hinkelmann, 2012)
5.4.1.2 Trapezoidal channel
The channel has a total length of 1000 m; the cross section is shown in Figure 5.23.
The grid configurations, the boundary conditions, the physical and the numerical parameters are the same as in the case of the rectangular channel. The MATISSE grid
0 250 500 750 1000 0 250 500 750 1000 0 250 500 750 1000
0 250 500 750 1000
36 30 24 18 12 6 0 Salinity (kg/m3)
200 150 100 50 0
A
B
C
D 200
150 100 50 0 200 150 100 50 0 200 150 100 50 0 [m]
generator was used in this case also to generate the grid. The total number of nodes was 3,325 and the total number of elements was 6,388 (Figure 5.24). The initial conditions as shown in Figure 5.21 are applied in this case also.
Figure 5.23: Cross section of the trapezoidal channel (Mahgoub and Hinkelmann, 2012)
Figure 5.24: Grid of the trapezoidal channel (Mahgoub and Hinkelmann, 2012) The changes were faster in this case than in the case of the rectangular channel. After 10 days of simulation, an obvious difference can be noticed between the four modeled scenarios. In the case of horizontal density variation only, most of the channel was turned to be saline water, which means that the impact of the salinity difference forced the fresh water to leave the system (Figure 5.25A).
0 250 500 750 1000 200
150 100 50 0
Bottom (m) 5 4 3 2 1 0 (0.00)
(5.00 m)
120 m
40 m 40 m
[m]
5.4 Salinity transport
Figure 5.25: Salinity transport for the trapezoidal channel for A) stagnant water with horizontal density variation only, B) stagnant water with diffusion only, C) stagnant water with horizontal density variation and diffusion and D) flowing water with horizontal density variation and diffusion (Mahgoub and Hinkelmann, 2012)
Changes in the salinity concentration with respect to the depth variation were also noticed, where higher concentrations were in the deeper parts, this is consistent with the results of Hansen and Ratrray (1965), where they concluded that lateral variations of the depth cause a lateral variation of the turbulence and bottom friction which results in a tilt of the gravitational flow responsible for the density-driven flow in 2D model. In addition, the horizontal pressure force due to horizontal density gradient is
0 250 500 750 1000 0 250 500 750 1000
0 250 500 750 1000 0 250 500 750 1000
36 30 24 18 12 6 0 Salinity (kg/m3)
200 150 100 50 0
A
B
C
D 200
150 100 50 0 200 150 100 50 0 200 150 100 50 [m] 0
proportional to the depth; therefore the tendency of the heavier salty water to replace the lighter fresh water landward is stronger when increasing the depth (Li et al., 1998).
The impact of the turbulent diffusion was much smaller than the impact of density variation; it was limited in a small part in the middle of the channel (Figure 5.25B).
The influence of turbulent diffusion was quite similar in both the trapezoidal channel and the rectangular channel (Figures 5.22B and 5.25B) which is not the case for the impact of the density variation.
When combining diffusion and density variation together (Figure 5.25C), only a slight difference compared to the case of the density variation only can be seen; instead of the uniform variation with respect to the horizontal axes, the variation of salinity was only in the lower lateral boundary.
Like the case of the rectangular channel mentioned earlier, introducing a slow flow to the system (80 m3/s) caused that the saltwater moved quickly towards the D.S.
boundary (Figure 5.25D). However the shape of the saltwater wedge is completely different compared to the case of the rectangular channel. That also emphasizes the strong impact of the shape on the salinity transport.
5.4.1.3 Comments on the test cases
The simulation of the test cases revealed that the impact of density variation was higher than the turbulent diffusion. The value used for turbulent diffusion was a high value (0.001 m²/s), so the impact of diffusion could be smaller if smaller value for turbulent diffusion was used. The shape considerably influenced the results, as a much faster change was noticed in the trapezoidal channel (approximately the whole domain turned to be saltwater in 10 days simulation time), while the change was quite slow in the rectangular channel. However, the impact of the shape was recorded only for the case of horizontal variation of density and it was not observed for the case of diffusion, thereof the diffusion is independent of the shape. The effect of the water depth was also noticed; higher salt concentration occurred with higher water depth (as described in the case of trapezoidal channel).
5.4 Salinity transport According to the plausible results of the test cases, the TELEMAC2D modeling system was applied to simulate horizontal density-driven flow in the Nile in the following. In addition, the test cases emphasized the importance of including the spatial variation of density when simulating tideless estuaries as in the case of the Nile River. However, 2D simulations cannot account for the stratification, they can consider only the spatial density variations due to barotropic and baroclinic gradients;
so the following simulations for the Nile can be considered as a first estimation, and 3D simulations are required for more realistic results.