4.3 Computational domain B
4.3.3 Investigation of the diffusivity
The influence of the turbulent viscosity on the hydrodynamics is negligible for such slow hydrodynamics. Generally, the turbulent diffusivity is assumed to be equal to the turbulent viscosity (as long as no tracer measurements are available, as it is the case here). However, the turbulence can have a strong influence on the transport as there is only advection and diffusion, while there are more important forces in the hydrodynamics.
Therefore, a sensitivity study has been carried out investigating two cases νT = 0.0001m2/sandνT = 0.01m2/sfor mean discharge condition and com-putational domain B. Further results have indicated that νT <0.0001m2/s do not lead to visible differences compared to the case νT = 0.0001m2/s.
Therefore, the results are not shown here.
The distribution of the tracer concentrations after 1, 2 and 3 hours are shown in figure 4.17. The maximum concentrations after 1, 2, 3 hours are: 40 mg/l, 30 mg/l and 12 mg/l for the case νT = 0.01m2/s and: 50 mg/l, 18 mg/l, 8 mg/l for the case νT = 0.0001m2/s.
A high impact of the turbulent diffusion can be clearly observed, when the results after 2 hours are compared for both cases (see fig. 4.18, left middle and right middle). The maximum concentrations between the tank and the upper river bank is still 30 mg/l after 2 hours.
Figure 4.18 (left: middle and bottom) shows the spatial variation of tracer concentrations at section B-B after 1, 2, and 3 hours, while figure 4.18 (right:
middle and bottom) shows the temporal variation of tracer concentration at point A. Figure 4.18 (left: middle and bottom) shows the influence of turbu-lent diffusion for the case ofνT = 0.01m2/s, where the tracer concentrations were almost constant at the cross section after 1, 2 and 3 hours with con-centration of 30, 23, 7 mg/l. The concon-centrations for νT = 0.0001m2/swere smaller [after 1 hour: 25 mg/l, after 2 hours: 11 mg/l and after 3 hours: 4 mg/l].
Figure 4.18 (right: middle and bottom) shows the temporal variation of tracer concentration at point A. For the case νT = 0.0001m2/s, the tracer reached point A after about 1200 seconds and had a peak after 1 hour with a maximum concentration of about 13 mg/l followed by a constant value of about 1.5 mg/l after two hours. If the results for νT = 0.0001m2/s are compared with the previous case, the influence of the turbulent diffusion can be clearly observed, as the tracer reached point A after 1800 seconds and the maximum concentration was only about 8 mg/l after about 1.5 hours.
Finally, from these results it can be observed that the tracer did not reach the Landwehr channel in all cases. The influence of the turbulent diffusion on the results is significant. However, as no tracer measurements were available, the value was set toνT = 0.0001m2/srepresenting a value from experiences for the further simulations.
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Figure4.17:2Dtracertransportafterthreetimesteps,left:νT=0.01m2/s,right:νT=0.0001m2/s
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viscosity=diffusivity = 0.01 m²/s viscosity=diffusivity=0.0001 m²/s Figure4.18:Left:spatialvariationoftracerconcentrationatsectionB-B[leftmiddle:νT=0.01m2/s,left bottom:νT=0.0001m2/s].Right:temporalvariationoftracerconcentrationatpointA:[right middle:νT=0.01m2/s,rightbottom:νT=0.0001m2/s]
4.3.4 3D simulation of hydrodynamics and transport
For the three-dimensional simulations only the computational domain B and the mean discharge condition have been considered.
3D grid: Generally, to mesh the 3D domain, we just need to take the 2D horizontal grid and then duplicate it along the vertical. There are two dif-ferent vertical discretizations “z” and “σv”. While the z-method divides the domain using equal spacing between each plane, the σv-method uses variable spacing. The latter one is generally recommended to better approximate the processes close to the vertical boundaries. The z-method has been chosen for this simple case as the bottom is plane, and no circulations (wind effect) or stratifications (density) are investigated, hence it is not necessary to refine the vertical grid at the free surface or the bottom. The vertical discretization and the 3D grid are discussed and presented in detail in section 5.5.1, where the bathymetry of the Unterhavel river is much more complicated compared to the Spree and wind influences are investigated. In such cases theσv-method is recommended as it enables to refine the vertical planes close to the bottom and free surface.
The space discretization of the 3D domain was carried out by taking the 2D grid with 6235 nodes and 11945 elements from section 4.3 (see fig. 4.11) and then duplicating it along the vertical. Thus, a 3D grid consisting of 10 planes, 62350 nodes and 54110 prisms was obtained. The same time step and the same initial and boundary conditions as for the 2D simulations have been chosen here. As in the 2D simulations, the inflow of tracers was imposed on three points simulating a damage of the tanks. In a vertically-averaged 2D simulation assigning a vertical depth to the points, where the damage occurs is not possible, however it is in 3D and this depth was chosen to be 1.5 m below the free surface.
Turbulence: The constant turbulent viscosity model has been chosen, so that the horizontal and vertical turbulent viscosity values are then constant
throughout the domain and time. In this work, the turbulent viscosities are equal to the turbulent diffusivities. For the horizontal directions, the same value as in the 2D simulations was set to νth = νT h = 0.0001m2/s and in the vertical direction a 10 times higher value was chosen to νtv = νT v = 0.001m2/s, being a common ratio of horizontal to vertical turbulence in shallow water flow.
Discretization and solvers: For the advection of water depth, the conser-vative scheme without decentring was chosen for the sake of mass conserva-tion and numerical stability. The velocity advecconserva-tion is carried out with the method of characteristics (which has good physical properties: monotonicity, preserves sharp fronts, very fast, however problems with conservation) for a single processor, and is replaced by the distributive scheme N in parallelism.
For the advection of the tracer the explicit schemes MURD and PSI were used for the sake of conservation. These schemes are monotonic, and thus do not generate any tracer values less than a defined minimum. Moreover, these schemes are not limited to the Courant number and are better compared to, for example the method of characteristics which is the least diffusive but cannot ensure tracer conservation. However, the MURD scheme generally requires more CPU time (HINKELMANN, 2005; GALLAND et al., 1991;
HERVOUET, 2007). For the diffusion step the PCG (Preconditioned Con-jugate Gradiendt Method) has been chosen as for the 2D simulations (see sec. 3.1.1).
The programs RUBENS, POSTEL-3D and FUDAA-PREPRO have been chosen for the visualizing and analysing the 3D results (see sec. 3.1.4).
Results: Figure 4.19 (top) shows the 3D grid at the longitudinal section X-X (fig. 4.19, middle), where ten planes have been chosen using the z-method.
Two scenarios have been considered here (see fig. 4.19, scenario A: bottom left, scenario B: bottom right), where only the location of the source was changed. In scenario B, the source is in a ’dead zone’ while it is in the flow field of scenario A (see fig. 4.19) leading to faster transport in early stages.
source point was shifted in order to consider most leakages may be occur.
Figure 4.20 shows the 3D flow field at section Y-Y. We can clearly see that the vertical velocities are higher close to the tank when compared to the horizontal ones.
Figure 4.21 shows the vertical distribution of tracer concentration at section Y-Y (see fig. 4.20) for scenario A after 120, 360 and 1080 seconds. After 120 seconds (fig. 4.21, left), the maximum concentration was still close to the source point with a maximum value of 30 mg/l. The concentration increased after 360 seconds to the maximum value of 120 mg/l (fig. 4.21, middle right).
Figure (4.21, right) also shows that the tracer concentration was constant in the vertical direction due to advection and vertical turbulent diffusion.
Similar results can be seen in figure 4.22, which shows the tracer concen-tration after four time steps. After 30 seconds the tracer is still spreading from source point in all directions. Figure 4.22 (top right) shows that after 390 seconds the tracer started to spread vertically and it was constant in the vertical direction after about 2220 seconds (fig. 4.22, bottom right). In this shallow water river, where the water depth is small as in the Spree, the tracer spreads relatively fast in the vertical direction compared to the horizontal one due to the high vertical diffusion.
Finally, a 2D profile of the velocity and tracer occurs in a short distance (about 100 m) from the inlet justifying 2D simulations for further investiga-tions.
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Simulation of Hydrodynamics and Transport in the Unterhavel
This chapter presents a natural urban water system with a complex geome-try which is stressed by treated waste water. The computational domain is a part of the Unterhavel water system in Berlin. This chapter explains the 2D and 3D model of hydrodynamics and transport in detail starting from the grid generation step, setting boundary conditions, carrying out the simula-tions and visualizing the results. Various condisimula-tions are investigated: low, mean and high discharge, contaminations. Finally, the effect of wind is also investigated.
5.1 Project area and computational domain
The study area is the Unterhavel river in Berlin, Germany (fig. 5.1). The river Unterhavel flows from north to south along Berlin’s western boundary (fig. 1.3, 5.1, and sec. 1.3). This system consists of shallow lakes (e.g.
Wannsee, Glieniker See and Jungfernsee), small islands (e.g. Pfaueninsel).
Further, the river Spree joins the Unterhavel in Spandau (north), while the Teltow channel (which is characterized as channel-like river) joins it in the south east. The lakes are shallow, the rivers are slow-flowing and carry little water. Especially in summer, the system may become ecologically very sensitive.
It was the first time that the Unterhavel system has been investigated with
high resolution bathymetry and geometry as well as with the two- and three-dimensional numerical simulations for different conditions (e.g. discharges, wind) (JOURIEH and HINKELMANN, 2012).
The computational domain is located between Pichelssee in the north, where the Spree joins the river Havel, the junction point with the Teltow channel in the south east, the Jungfernsee and Glieniker See in the south west (fig.
5.1, 5.5). The whole domain has a mean depth of 5.5 m, a maximum depth of 9.5 m (fig. 5.3 and 5.4), an area of ∼30km2 km and a volume of ∼155 million m3 (SenGUV, 2006).
As already mentioned in section 1.3, the Teltow channel is a shallow channel located in the southern area of Berlin. It carries the highest load of sewage discharges of all Berlin rivers and channels coming from Berlin´s largest sewage treatment plant in Ruhleben (only from April to October) and from sewage treatment plant in Stahnsdorf (fig. 1.3 and 5.1).
One important question is, for example whether and how much treated wastewater can be transported from Teltow channel into regions such as the bathing area Wannsee where it is unwanted, due to unfavorable hydraulic conditions.
For this purpose, a 2D and a 3D numerical model were set up and hy-drodynamics as well as transport studies were carried out and analysed in the following, investigating different hydrodynamic and transport conditions (mean and extreme discharge conditions, bottom friction, turbulent viscosity and diffusivity, influence of wind).
Berlin Germany
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Figure 5.1: Study area, computational domain and model concepts