# Modelling Regime Changes of Dunes to Upper-Stage Plane Bed in Flumes and in Rivers
## 5 Results with Flume Conditions  
We start with testing whether the model is able to model a transition to upper-stage plane bed for flume conditions, that is, the computational scenario A4 as presented by [PERSON] et al. (2009) [PERSON] et al. (2009) (see also Section 3.3). With this scenario, [PERSON] et al. (2009) [PERSON] et al. (2009) show that their model is able to predict transitions to the upper-stage plane bed. The model also clearly shows hysteresis effects; the relation between discharge and water depths is as significantly different for the rising limb of the hydrograph as it is for the falling limb. The parameters of the scenario are typically equivalent to a flume scenario. With the model runs presented here, the suspension parameter \(u*/w_{s}\) varied between 0.9 and 1.8 during the scenario, so with this sediment and this flow regime the suspension regime is present for most of the time. Using scenario A4 from [PERSON] et al. (2009) [PERSON] et al. (2009), the following development of the dune field over time is found with the dune evolution model as developed in the present study. Here in Figure 6 the washing out and regrowth of the dunes can be clearly observed. 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 7. As can be seen in the beginning of the run, dunes start developing along with increasing discharge. At a certain high discharge, the shear stress and the step length become so high that the dunes are washed out. Due to the decrease in form drag and thereby _total shear stress_, the water level stabilizes temporarily, despite still rising discharge. The bed remains washed out until the discharge and step length become so low that dunes start developing again. Figure 5: The non-dimensional step length of the non-dimensional step length model of [PERSON] et al. (2009) [PERSON] et al. (2009), using \(\alpha_{S}\) for bed load, and the currently used step length model Equation (19), using \(\alpha_{S}\), which represents the grain-shear-stress-dependent part of non-dimensional step length in the new model. In this figure the depth is assumed to be equal to \(h_{\mathit{ref}}\) in order to more clearly show the relation with the [PERSON] step length model. [PERSON] et al. (2009) [14] found roughly the same moments of washing out (0.6 h) and re-emergence of dunes (1.7 h). Furthermore, the dune evolution model of [PERSON] et al. (2009) [14] predicted a dune with a maximum height of about 4 cm, which corresponds rather well with the present model result (3.75 cm). The relation between the step length parameter \(\alpha\) and the specific discharge for the rising and falling stages of the hydrograph can be seen in Figure 8.  
Figure 6: Dune field over time, using scenario A4 from [PERSON] et al. (2009). Figure 7: Dune crest and trough position (black lines) and water depth (blue line) over time, using scenario A4 from [PERSON] et al. (2009). Here a hysteresis effect is already observed, caused by the transition to upper-stage plane bed and then later returning to the dune regime. With the same discharge, dunes can be present in the rising stage and not in the falling stage. Because of the presence of dunes, a part of the flow power is lost due to form drag. This means the shear stress acting on the sand grains (effective bed shear stress) is relatively lower, and a lower step length is selected than when there are no dunes present. The effect of hysteresis on water depth can be seen in Figure 9.  
Figure 8: Specific discharge versus non-dimensional step length \(\alpha\) separated for the rising and falling stages of the hydrograph. Figure 9: Specific discharge versus water depth separated for the rising and falling stages of the hydrograph, using scenario A4 from [PERSON] et al. (2009). In the rising part of the hydrograph, dunes are able to develop firstly at medium discharges, while in the falling part, the dunes start developing at the end of the flood wave at lower discharges.