5.2 Grid generator JANET and pre-processing stage
5.3.1 Investigation of mean discharge condition
After the grid generation has been completed, initial and boundary condition must be set. Initial conditions cover the whole computational domain and boundary conditions cover all boundaries of the computational domain for the whole simulation period.
Initial conditions: The simulations started with a prescribed initial water level of 29.5 m (SenGUV, 2006) equal in all points of the computational mesh and initial flow velocities were zero.
Boundary conditions: The following boundary conditions have been se-lected for the hydrodynamic simulations:
Neumann-boundary conditions: At two inflow boundaries, discharges were imposed. They were set as a constant values representing mean dis-charge conditions (SenGUV, 2006). A value of 30.7 m3/s was imposed as inflow boundary at Pichelssee and 8.3 m3/s as inflow boundary at Teltow channel, see fig. 5.5.
Dirichlet-boundary conditions: Water levels were prescribed at outflow boundaries at Jungfernsee and Glienicker See. The prescribed constant water level is equal 29.5 m. As the water level variations are small, choosing a constant value is justified.
All other boundaries are closed or no flow boundaries. Figure 5.5 shows the respective liquid and solid boundaries.
After all required inputs, for example: initial and boundary conditions as well as physical and numerical parameters, information about geometry and boundary condition files have been given in the steering file, the 2D simula-tions were carried using the program TELEMAC-2D.
The difference between low discharge (e.g. inflow at Pichelssee: 12 m3/s) and mean discharge (e.g. inflow at Pichelsee: 30.7 m3/s) is small compared
to the high discharge (e.g. inflow at Pichelssee 117 m3/s). Hence the low discharge condition has not been considered here as no big difference can be expected.
Pichelssee Q=30.7 m³/s
Teltow channel Q=8.3 m³/s Jungfernsee
H=29.5 m
Glienicker See H=29.5 m
Figure 5.5: Computational domain with boundary conditions for the mean discharge conditions
Discretisation and solver: In order to avoid long simulation times and to ensure stability the methods of characteristics (with fractional step) and the conservative scheme with SUPG method have been chosen to solve the advection step for the velocity components and the water depth respectively.
The BiCGSTAB (Biconjugate Gradient Stabilized Method) was chosen as the solver. Finally, the mass balance is checked over the entire domain every time step. This option balances inflow and outflow with the mass changes in the domain and thus determines the relative error on mass conservation for each time step (HINKELMANN, 2005; HERVOUET, 2007). The relative error was about 0.1528543.10−6 in these simulations.
Time step: The time step has been chosen considering the Courant con-dition to avoid poor quality of results due to too high values. The Courant number influences the quality of the results, for example, numbers much higher than 10 may produce much dumping and thus poor quality results (HINKELMANN, 2005). After variations of the time step, it was chosen to 3 seconds corresponding to a Courant number in the smallest element about 6:
Cr= vmax+√ g.hmax
4x/4t = 0.38 +√
9,81.9,5
5/3 w6
This was a suitable compromise between CPU time and solution accuracy.
Bottom friction: Unfortunately, no data about friction coefficient were available for the Unterhavel domain. SCHUMACHER (2001) presented Manning-Stickler friction coefficients of the Spree till the Spandau lock (the upper boundary of our computational domain close to Pichelssee) and the Kleinmachnow lock (the other boundary of computational domain close to Teltow Channel input). These values range between 30m13/sand 50m13/s.
Once the model was set up, it has been tested with different Manning-Stickler friction coefficients. Three scenarios for the bottom friction have been carried
out with coefficient values of 30, 40 and 50m13/s to check this influence.
The velocity output obtained from this analysis in a section A-A (shown in fig. 5.11) is illustrated in the figure 5.6. It can be concluded from the figure that there is no significant effect of Manning-Stickler’s coefficient on the model results. Similar observations have been made in the Spree (see sec. 4) and other slow flowing systems (OMAR, 2009; HINKELMANN, 2005). Therefore, the friction coefficient was set to 40m13/sin the following simulations.
0 0.025 0.05
0 100 200 300 400 500
0 0.025 0.05
0 100 200 300 400 500
Maning-Strickler 30
0 0.025 0.05
0 100 200 300 400 500
0 0.025 0.05
0 100 200 300 400 500
Maning-Strickler 40
0 0.025 0.05
0 100 200 300 400 500
Maning-Strickler 50
[m]
scalar velocity [m/s]
Figure 5.6: Comparison of scalar velocity for three friction coefficients at a section A-A (fig. 5.11)
The constant viscosity option, which assumes the turbulent viscosity is con-stant throughout the domain and time has been chosen with a value ofν= 0.0001 m2/s. The flow velocities were very small (in the range of mm/s, see fig. 5.9) in large parts of the computational domain except small passages.
Therefore, the influence of turbulence is minor here and the value for the viscosity is reasonable. The Reynolds number was:
Re= vmean∗dmean
νt = 0.2∗5.5
10−4 '1.104
Re= vmax∗dmax
νt = 0.38∗9.5
10−4 '3.104
Further, a sensitivity study were carried out for different conditions of vis-cosities (see sec. 5.4.2).
All information about mesh geometry and calculated values at each printout time step are contained in the result file using a special TELEMAC format called SELAFIN. The numerical results have been visualized using the soft-ware RUBENS, which allows the graphical post processing of the simulation results (see sec. 3.1.4).
As mentioned, at the beginning of the simulations the water elevation was constant. Figure 5.7 shows the spatial variation of water depth after 3 days at three locations: Wannsee, Jungfernsee and Schwanenwerder island, while figure 5.8 shows the temporal variation of water depth and free surface at three points in the domain in the first 3 days. As observed in this figure, the variations of water depth and free surface are very small (in the range of a few millimeters) in the Unterhavel domain for mean discharge condition.
Further steady state conditions have been reached after 3 days.
Figure 5.9 shows the flow field in some parts of the Unterhavel. The horizon-tal variability can clearly be seen in many parts of the domain, this cannot be modeled with a one-dimensional or even a simpler model. The range of the velocities in the Unterhavel domain mainly varies between 0.001m/s and 0.015m/s depending on the location, i.e. the depth and the width. In the small elements it increases to several cm/s with a maximum value of
∼ 0.38m/s (see fig. 5.13). The flow velocities in the wider areas of the Unterhavel are very small (really stagnant) and increase up to several cm/s around the narrow channels and the entrance of the domain resulting from the inflow boundary conditions.
The average velocity in the Wannsee is between 0.005m/s and 0.01m/s.
Figure 5.9 (right) shows the incoming velocity in the Wannsee from the southern part. A large eddy and circulation is occurring as there is also inflow from the northern part. Further, dead zones with stagnant water are observed here. Similar conditions occur in other parts of the Unterhavel (large eddies, dead zones).
In figure 5.9 we can also see that parts of water entering through the Teltow channel flow to the Wannsee, i.e. treated wastewater which is transported in the Teltow channel can enter the Wannsee.
Figure 5.10 illustrates the temporal variation of scalar velocity and water depth at three points X, Y and Z in the first three days. Scalar velocity is very small at point X and no peak appears. At point Y the velocity reaches the maximum value after 12 hours then stays constant with about 1.3cm/s. At point Z the velocity has not reached a steady state after three days. This occurs later. Nevertheless, the assumption of steady state after 3 days is reasonable as the remaining variation in small parts of the domain are smaller than 1mm/s. The water level variations are the same as in figure 5.8.
Spatial variations of scalar velocity and water depth at four sections after three days are illustrated in figure 5.11, which confirm the previous results.
Variations of the flowrate (i.e, discharge per meter width) in four cross sec-tions of the domain are visualized in figure 5.12. In section A-A, C-C and D-D steady state conditions occur already after two days. However, in section B they are not reached after three days as there is inflow from the southern and northern part leading to a large eddy. It is referred to the comment to figure 5.10.
The relationships between the scalar velocity and the water depth has been illustrated in figure 5.13 in order to get a better understanding of the flow variability. In large areas with water depth between (0.5 m) and (-9.5 m) the flow velocities are very small with values less than 10 cm/s or 5 cm/s.
Some large flow velocities with values higher than 20 cm/s occur in areas with water depth between 3 and 3.5 m. From this figure it can be concluded that the velocities of about 57000 nodes (∼ 80% from the total number of nodes) were smaller than 0.1 m/s, while only about 5700 nodes (<10% from the total number of nodes) had the velocities higher than 0.2 m/s.
7500 11000 14500 18000 21500
27004700670087001070012700 water depth [m]
0.5 1.4 2.3 3.2 4.1 5 5.9 6.8 7.7 8.6 9.5
[m] [m]
Figure5.7:Spatialvariationofwaterdepthforthemeandischargeconditionafter3days;leftJungfernsee,middlewholedomain,righttopSchwanenwerderisland,rightbottomWannsee
7500
11000
14500
18000
21500 27004700670087001070012700
water depth [m] 0.51.42.33.24.155.96.87.78.69.5 [m]
[m] 7.775
7.7875
7.8 086400172800259200
water depth at point X 0.495
0.4985
0.502 086400172800259200
free surface at point X 2.64
2.6425
2.645 086400172800259200
water depth at point Y [s]
[s] [s]
[m]
[m] 0.495
0.4985
0.502 086400172800259200
free surface at point Y
[m]
[s] 0.495
0.4985
0.502 086400172800259200
free surface at point Z 6.38
6.39
6.4 086400172800259200
water depth at point Z
[s] [s]
[m]
[m] [m]
X Y
Z Figure5.8:TemporalvariationofwaterdepthandfreesurfaceatconsideredpointsX,YandZformeandischarge condition
scalar velocity [m/s]
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
velocity [m/s]
0.1 velocity [m/s]
0.01
scalar velocity (m/s)
0 0.02 0.04 0.06 0.08 0.1
velocity [m/s]
0.01 velocity [m/s]
0.01
velocity [m/s]
0.01
Figure5.9:Velocityfieldforthemeandischargeconditioninthewholedomainandseveralsubdomains
7500
11000
14500
18000
21500 27004700670087001070012700
water depth [m] 0.51.42.33.24.155.96.87.78.69.5 [m]
[m/s] 7.775
7.7875
7.8 086400172800259200
water depth at point X 0
0.0125
0.025 086400172800259200
scalar velocity at point X 2.64
2.6425
2.645 086400172800259200
water depth at point Y [s]
[s] [s]
[m]
[m] 0
0.0125
0.025 086400172800259200
scalar velocity at point Y
[m/s]
[s] 0
0.0125
0.025 086400172800259200
scalar velocity at point Z 6.38
6.39
6.4 086400172800259200
water depth at point Z
[s] [s]
[m/s]
[m] [m]
; <
Z Figure5.10:Temporalvariationofscalarvelocity(left)andwaterdepth(right)attheconsideredpointsX,Yand Zinthefirst3daysformeandischargecondition
0 0.01 0.02 0.03 0.04 0.05
0100200300400500600700800900 scalar velocity (m/s) 0 0.01 0.02 0.03 0.04 0.05
0100200300400500 scalar velocity (m/s)
0 0.01 0.02 0.03 0.04 0.05
050100150200250300 scalar velocity (m/s)
0 0.01 0.02 0.03 0.04 0.05
0255075100125150175 scalar velocity (m/s) bottom [m]
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
AA
BB
C C D D section A-A
section B-B section D-D
section C-C [ m†/s ] 0 2.5 5 7.5 10
0100200300400500 water depth
0 2.5 5 7.5 10
0100200300400500600700800900 water depth 0 2.5 5 7.5 10
050100150200250300 water depth (m)
0 1 2 3 4 5
0255075100125150175 water depth (m) [m]
[m] m m [m] [m] [m/s] [m/s] [m]
[m] [m] m [m]
[m/s] [m/s]
[m]
Figure5.11:Spatialvariationofwaterdepthandscalarvelocityatdifferentsectionsafter3daysformeandischargecondition
00.01
0.02
0.03
0.04
0.05 02505007501000
one day two days three days
00.05
0.10.15
0.2 0100200300400500600
one day two days three days 0
0.1
0.2
0.3
0.4
0.5 050100150200250300
one day two days three days 00.010.020.030.040.05 0255075100125150175
one day two days three days
bottom [m] -10-9-8-7-6-5-4-3-2-10 AA B B C
C
D
D
section A-A section B-B
section D-D section C-C
[ m²/s ]
[ m²/s ]
[ m²/s ] [ m²/s ]
[m] [m] [m] Figure5.12:Spatialvariationofflowrate(dischargepermeterwidth)atdifferentsectionsinthefirst3daysfor meandischargecondition
0.5 5.25 10
0 0.1 0.2 0.3 0.4
water depth (m) 7 8.5 10
0 0.005 0.01 0.015 0.02 0.025 0.03
0.5 2.25 4
0 0.005 0.01 0.015 0.02 0.025 0.03
scalar velocity (m/s)
Figure 5.13: Correlations between scalar velocity and water depth for mean discharge condition