# Modelling Regime Changes of Dunes to Upper-Stage Plane Bed in Flumes and in Rivers
## 6 Results with River Conditions
### Results for the River Scenario  
The development of the dune field, starting from an already developed dune field, can be observed in Figure 11.  
The dunes grow and shrink during the flood wave, though in general the dune height variation is low. From the same results, the development of the dune trough and crest positions (in the vertical) and water depth over time are shown in Figure 12. As can be seen in the beginning of the run, dunes start developing slowly along with increasing discharge. This continues past the moment of peak discharge until about 155 h, after which the dune height decreases. Due to the increase in discharge, the water level still increases and decreases, with seemingly little influence of the dune height.  
Figure 11: Dune field over time, using the river scenario.  
During the flood wave of 1998, dune heights in the Waal varied between 0.2 and 0.6 m on the centreline [47], while the model predicts values between 1.6 and 2.1 m. This is still a considerable difference, which indicates that further model improvements are necessary. Increasing the step length likely has a dampening effect on the dune height, but it also causes the dunes to wash out sooner. Therefore, care should be taken in adjusting the model settings. In Figure 13 the response of the water depth to the varying discharge can be seen.  
Figure 12: Dune crest and trough position (black lines) and water depth (blue line) over time, using the river scenario.  
Figure 13: Discharge versus dune height for the rising and falling stages of the hydrograph, using the river scenario. The arrow signifies the direction of development over time.  
The hysteresis loop observed here is in the same direction as for the flood wave of 1998 as reported by [PERSON] et al. (2002) [PERSON] et al. (2002), with dune heights in the falling limb being higher than in the rising limb (Figure 14). Here the dune height also keeps growing for a period of time after the discharge is already decreasing. The relation between the non-dimensional shear stress and the dune height can be seen in the following figure. The dunes do not wash out; however, they do decrease in height about 5 h after the maximum discharge has been reached.  
The relation between \(\alpha\) and the specific discharge for the rising and falling stages of the hydrograph can be seen in Figure 15. Here again a hysteresis effect is observed, caused by the lag between dune development and discharge. It should be noted that now, much higher step lengths are reached than for the flume scenario, up until \(\alpha=4750\), which is a dimensional step length of about 6 m. This is about 2.6 times higher than that of the flume scenario.  
Figure 14: Non-dimensional shear stress \(\theta\) versus dune height for the rising and falling stages of the hydrograph, using the river scenario. The arrow signifies the direction of development over time.  
Figure 15: Specific discharge versus non-dimensional step length \(\alpha\) separated for the rising and falling stages of the hydrograph, using the river scenario. The arrow signifies the direction of development over time.  
In the rising part of the hydrograph, dunes increase in height, while in the falling part the dune height first remains constant before decreasing later on. Because of the delay in dune height development compared to discharge, the water depth in the falling limb of the hydrograph is larger than in the rising limb, which agrees again in a qualitative sense with what is reported by [PERSON] et al. (2002) [PERSON] et al. (2002) for the flood wave of 1998.