Comparison between Euler and Heun’s water surface profile: As shown in Figure 1 next page, it seems that the water surface profile from Heun’s method is exactly the same as that from Euler’s method. In fact there is a discrepancy between the two which is not observable from the graph because their difference is in an order of 1×10-6. The following table summarizes the super-critical flow’s water depth differences from the two numerical methods, it shows that the water depth from Heun’s method is only slightly higher than that from Euler’s method. This tiny discrepancy between the two methods is reasonable recalling what has been discussed before, the only difference between these two methods is that Heun’s method uses the average value of the previous two water surface slopes to calculate water depth while Euler’s method only uses one slope which has an magnitude of 1*1×10-3 (e.g. dY/dX1 = 0.0016826).
| Difference = YHeun - YEuler
Figure 2 – Comparison between measured and calculated water surface profile:
Comparison between measured and calculated water surface profile: As shown in Figure 2, there is a discrepancy between the measured and calculated water surface profile, the measured water surface profile appears to be higher than the calculated one in particular within the sub-critical flow region. Various factors could contribute this discrepancy. * Explanation from an energy point of view
The theoretical water surface profile is calculated based on the assumption that the channel wall is smoothed. Even though both Euler and Heun’s methods have modified the friction factor equation after considering the channel wall is made of Perspex, a relative roughness ε/D =0 may still overstates the fact. In reality during the lab experiment, the channel wall could be dirty, there could be dust sediments lying on the bottom of the channel, all these fact could cause a higher friction than that included within the theoretical equations. The key point to be made here is that the energy dissipation during the lab experiment could be slightly higher than the expected energy loss included within those numerical equations. If this is the case, it will have a direct effect on the water depth. To explain this discrepancy specific energy theory is used. The super-critical flow is studied first as shown in the channel flow diagram above. As water flows from the upstream control point A down to point B, the specific energy decreases gradually due to the friction factors mentioned, this means the specific energy at point B H0B should be smaller than the specific energy at point A H0A. This energy change between point A and B corresponds to a leftward movement on the specific energy curve within the super-critical flow part, which also corresponds to an increase in water depth Y. The key point here is more energy has been lost during the experiment than expected from the numerical equations, this indicates on the specific energy curve the leftward movement for the real experiment was bigger than the one for the numerical method and the further movement to the left the higher the water depth. As shown in the specific energy diagram above, the experimental specific energy at point Breal is located further left than the theoretical specific energy at point Btheory, Breal corresponds to a water depth Y1 while Btheory corresponds to a lower water depth Y2. This is the reason that the measured water...
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