Description implementation weirs in FINEL2D
1. Introduction
Basically a weir is simply a structure in the water flow, causing contraction of the flow and energy loss. This refinement is not always necessary for the required accuracy, in those cases the computation is needlessly slowed down by the extra elements. Another drawback of representing a weir in the bathymetry is the inaccurate calculated solution of the flow over the sill. In reality conservation of energy holds upstream of the weir, and conservation of momentum downstream. |
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Therefore in FINEL2D weirs are represented as a sub-grid feature at the edge of an element. On a weir-edge an empirical discharge formula is used as internal boundary condition. The empirical formula calculates the discharge corresponding to the energy head over the weir, taking into account the geometry of the weir. In such way the grid size is independent from the weir width, and no inaccurate solution is obtained as the empirical formula takes into account both the upstream conservation of energy and the downstream conservation of momentum. |
Section 2 gives technical details about the implementation of weirs in FINEL2D and test results are shown in Section 3.
2. Technical documentation weir formulation
A weir is implemented as an internal boundary condition in FINEL. The idea is taken from a journal paper [ref. 2], although the implementation in FINEL is different in some aspects. The boundary condition consist of the weir discharge as function of the energy level upstream and downstream of the weir and taking into account the weir geometry.
At the element side where a weir is situated, two extra "dummy" boundary elements are placed inside the mesh. The water level and current velocity on these dummy elements are prescribed to satisfy the weir discharge boundary condition. See the sketch below for clarification.
On the element side between element e1 and e2 a weir is situated. Therefore two dummy elements are created (d1 and d2). The interaction between element e1 and e2 is replaced by an interaction between e1<-> d1 and an interaction between d2 <-> e2. The conditions in d1 and d2 are calculated using the weir discharge taking into account the energy level in e1 and e2; qweir = f(E1, E2). Both d1 and d2 get prescribed the same discharge qweir. Of course the user does not see these dummy elements, and only the flow contraction and energy loss caused by the weir is visible in the results of the real elements e1 and e2.
Figure 2: Weir on element side between e1 and e2 (left) -> internally two dummy elements d1 and d2 are created; on these two dummy elements qweir is prescribed (right).
The Dutch ministry of public works has carried out experimental analysis of weir flow. In an internal memo by A. Sieben [ref. 1], the results are described. The memo gives an empirical relation to determine the weir discharge knowing the upstream and downstream energy level and taking into account the influence of the width of the weir crest, the weir slope upstream and downstream and the weir sill height. This empirical relation has been implemented to calculate the weir discharge in FINEL. It is given by:
On both dummy elements a normal velocity and corresponding water level is prescribed resulting in a discharge equal to qweir. The Riemann invariant (R=u+2c upstream and R=u-2c downstream) is used to find values for un and H for both dummy elements. The system of equations is solved with a fast Newton-Rhapson iteration. For the test simulations maximal 2 or 3 iterations were needed. The velocity and water level of the dummy element upstream is not equal to the values of the downstream dummy element, but both result in the same discharge qweir. The tangent velocity of d1 is equal to e2 and d2 is equal to e1.
To keep the weir flow stable relaxation is used. Standard a relaxation factor 0.1 is used (Un+1=0.1*Un+1+0.9*Un), but when the downstream velocity <0.1 m/s or upstream or downstream the velocity is against the energy head over the weir then a stronger relaxation is used with a factor 0.01. This relaxation may seem large, but because of the small time steps in FINEL the relaxation doesn’t cause a large time gap in reaction of the weir on changing flow conditions. And smaller relaxation does lead to oscillating weir behavior.
3. Test results
3.1 Test subcritical, critical and stagnating weir flow
First a test is carried out with a rectangular mesh, and a weir halfway in y-direction of 1 m high (Lk=4m, mu=md=3). Water level boundary condition upstream 2 m, downstream dropping from 2 m to 1 m in 3 hours, then increasing from 1 m to 2 m in the next 3 hours. Last 3 hours both boundary conditions are 2 m. This test involves the start of flow over a weir, the transition from sub critical to critical weir flow and the transition to flow stagnation.
Some characteristic results are shown in the Figure 3, a 2d top view of the flow and a side view of the water level along the mesh, clearly showing the drop in water level at the weir. As the total influence of the weir is sub-mesh, only the resulting flow contraction from upstream to downstream is visible, not the intermediate flow condition above the weir crest itself.
Figure 3: Characteristic weir flow results top view and side view (Note red line gives an indication of the weir, it is not implemented in the bathymetry).
Figure 4 shows the results are continuous and smooth without large wiggles. The oscillations in V velocity tangent to the weir are very small (order 10-4 m/s). The discharge in FINEL is very close to the discharge calculated without relaxation in MATLAB from the FINEL flow results. The transitions from zero discharge to subcritical discharge to critical weir discharge to subcritical and zero discharge again all go smooth. Inertia makes slowing down to zero discharge takes some time after the driving force is zero.
Due to the flow contraction the velocity downstream is larger than upstream. Just after 400 minutes the velocity gets under 0.1 m/s and a stronger relaxation (0.01) is used, this is visible in the jump of the black line showing the difference between the discharge in FINEL and the MATLAB calculated discharge from the flow field results. Minor oscillations in the discharge (<0.2 m3/s) are visible near the stagnant point, but this does not impact the flow conditions significantly.
Figure 4: Test subcritical, critical and stagnating weir flow results.
3.2 Test drying and flooding of weir
The weir flow 2 test simulation is almost identical to weir flow 1, only the boundary conditions are different. Upstream the water level drops from 2 to 0.5 m in the first 3 hours of the simulation. Downstream the water level increases from 0.5 to 2 m in the last 3 hours of the simulation. So the middle 3 hours the water level is at 0.5 m (0.5 m under the weir crest level) and no flow should occur.This test involves drying of the weir and then flooding again.
The results in Figure 5 show a smooth transition during drying of the weir at 180 minutes and again a smooth transition during wetting at 360 minutes.
Figure 5: Test drying and wetting of weir results.
3.3 Groyne field
Finally a series of groynes is placed in a long 20 km long channel. Boundary conditions consist of a discharge of 5847 m3/s upstream and a water level of 5 m downstream. The bottom slope is 1:10.000. Manning is 0.025, leading to Chezy=52,3 m0.5/s. The bottom slope is in equilibrium with the energy dissipation caused by friction.
Figure 6: 20 km channel with 2x 7 groynes on each side of the river: the influence on the water level is clear.
A close up at the section with groynes in Figure 7 shows the influence of the groynes on the water level and velocity in the river.
Figure 7: Close up water level (top) and current velocities (down) in groyne field.
3.4 Arbitrary weirs
Also a test simulation with some arbitrary placed weirs (diagonal, along X direction and along Y direction) is carried out.It shows the weirs act as a kind of semi-permeable streamliner. When a weir is not perpendicular to the flow it does not impact the flow.
Figure 8: Current velocity with arbitrary weirs.
3.5 WAQUA test simulations
In a previous project (1459) the empirical weir formulas from A. Sieben were implemented in a test version of WAQUA. Now the same test simulations are executed with FINEL, to compare the results. Figure 9 and 10 show the simulation and the used boundary conditions. A prismatic channel with weirs halfway is used, the upstream discharge increases step wise from zero to 6.25 m2/s in 5 steps and then back to zero again. The water level downstream is 2.5 m at the start of the simulation and increases to 3 m after 1260 minutes. All flow conditions from no flow to subcritical and critical weir flow are captured in this way with two different downstream water levels. Because the changes in boundary conditions are stepwise and then constant for 3 hours, it is possible to see whether stable constant solutions are obtained.
Figure 9: Sketch of WAQUA test simulations.
Figure 10: Time series boundary conditions WAQUA test simulations, discharge upstream (red line) and water level downstream (blue line).
Some simulations are carried out to demonstrate the influence of the 3 weir parameters mu, md en Lk. Every time 1 of 3 parameters is varied and the other 2 stay constant. An overview of the parameters in the simulation is given in Table 1 below.
Table 1: WAQUA simulation weir parameters.
First a direct comparison between WAQUA and FINEL with the standard weir from Table 1 is carried out. The results show a weir flow diagram with the ratio S=E2/E1 on the horizontal axis and the ratio between the actual weir flow and the theoretic maximal weir flow 
on the vertical axis.
Also a time line of the energy head over the weir is given. The results from WAQUA and FINEL are very close to each other. Both show a smooth continuous weir flow diagram (Figure 11 top), in FINEL at almost zero flow a small negligible wiggle is visible.
Figure 11: Comparison FINEL and WAQUA weir 1.
Test simulations 1 to 9 are shown in Figure 12-15. They show the influence of the weir parameters mu, md, Lk and Δzd. Both crest width and height have similar influence, a longer or higher weir does cause more energy loss. The influence of the upstream slope is not too large, but a steeper upstream slope does result in a slightly larger energy loss. The influence of the downstream slope is also relatively small, but a bit more complicated than the influence of the upstream slope. With subcritical flow a steeper downstream slope results in larger energy loss, but with critical flow a steeper downstream slope results in smaller energy loss! This does agree with the experiments where qweir is based on [ref. 1], but it can be unexpected at first sight.
FINEL can carry out all these test simulations, only at small discharge some small oscillations are apparent. For practical applications this will probably not cause problems, because the currents are already too low to have major impact and most oscillations are found in hypothetically simulations (Lk=0, or Δzd=0).
Figure 12: Influence upstream slope WAQUA test simulations.
Figure 13: Influence downstream slope WAQUA test simulations.
Figure 14: Influence weir crest level WAQUA test simulations.
Figure 15: Influence weir crest width WAQUA test simulations.
3.6 Ladder of cascades
In the journal paper [ref. 2] describing the implementation of a weir as internal boundary in a Roe-like solver a test simulation with a ladder of cascades is carried out. This test is also carried out with FINEL, although it is not too important for practical application in estuary environment, because it is about discontinuous stationary flow with super critical flow and the transition to sub critical flow in a bore. The test consists of a 900 m long channel with steep bottom slope of 0.003, a little friction with Manning=0.009 and three identical weirs of 0.25 m high. Upstream a discharge of 3.33 m2/s is imposed and downstream a water level of 2 m. To keep the upstream boundary non supercritical an extra weir is imposed near the entrance.
In figure 16 the final stationary result is shown. Downstream of the two middle weirs super critical flow occurs, with a bore there is a transition to sub critical flow in front of the next weir. The weirs are imposed as sharp crested weirs with Lk=0, mu=0 and md=0. This results in a Cd0 factor of 1.16 which means 1.16 times the perfect weir discharge. As can be seen the super critical flow condition downstream of the weir does not cause any trouble. A direct comparison with the results in the paper is impossible, because no information is given about the exact weir discharge formula and which contraction coefficient is used in the paper. But the results are close to the results in the paper, giving confidence in the way the weirs are implemented in FINEL.
Figure 16: Final stationary results ladder of cascades, location of weirs shown with red lines.
3.7 Weir together with silt and sand transport
When calculations are performed with silt and/or sand transport simply the silt and/or sand flux is transported over the weir by applying the upstream concentration at the downstream dummy element. Some simple test simulations have been executed (not shown here) and this does not lead to erroneous results. Because of the flow contraction and current velocity increase downstream of the weir the bottom has a tendency to deepen downstream of the weir.
Another test involves the evolution of an equilibrium bottom profile. At the start of simulation a uniform bottom elevation of -10m is used. Upstream a discharge of 20.000 m3/s and downstream a water level of 0m is prescribed as boundary conditions. Halfway a weir with zweir=9 m is situated. Engelund Hansen is used to calculate the morphological changes.
Figure 17: Equilibrium bottom profile with weir after 100 days .
After 100 days morphological calculations a bottom slope with an equilibrium depth of 11.54m everywhere (upstream and downstream of the weir) is found. At the weir the flow contraction has caused a jump of 13 cm in the bottom elevation.
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4. references
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