# Modelling Regime Changes of Dunes to Upper-Stage Plane Bed in Flumes and in Rivers
## 7 Discussion  
In the present study, under flume conditions, differences have been observed in outcome between the new model and the k-\(\epsilon\) model of [PERSON] et al. (2009) ([PERSON] et al., 2009), that is, step lengths have to become significantly larger to reach the transition to an upper-stage plane bed, and there are differences of 25% in water depths. Of course, the models differ in significant ways. Firstly, [PERSON] et al. (2009) ([PERSON] et al., 2009) have used a non-hydrostatic flow model with a non-linear k-\(\epsilon\) model for turbulence closure, whereas in the idealized dune evolution model, a hydrostatic flow model with a constant eddy viscosity has been used. Secondly, their transport module also used a separate suspended sediment model, while in this research suspended sediment has been only modelled implicitly. Pinpointing the exact reason of the differences is hard because the A4 flume case under review is a'synthetic' case; there are no actually measured values as a reference to determine which model is closer to the truth. The [PERSON] et al. (2009) ([PERSON] et al., 2009) ([PERSON] et al., 2009) dune model is more physically complex, which suggest that its results should be better. However, it is still promising that the new relatively simple model presented in this research is able to represent similar bedform behaviour during a flood wave as the more complex model in a qualitative sense. Figure 21: Specific discharge versus water depth separated for the rising and falling stages of the hydrograph, using the more extreme river scenario. During the research for this paper, it was found that the timing of a transition to upper-stage plane bed is sensitive to the value of \(\alpha\). As an example, the results of the first 45 min for scenario A4 of [PERSON] et al. (2009) [14] with \(\alpha_{max}=300\) as well as \(\alpha_{max}=350\) are shown in Figure 22. Here it can be seen that the bed is washed out at about 37 min with \(\alpha_{max}=300\) instead of 35 min as with \(\alpha_{max}=350\), and that the maximum dune height is 3.75 cm and 2.8 cm, respectively. This shows that the results are sensitive to the settings of the step length model, though there is no extreme effect on general model behaviour. To ensure that the model still represents the dune regime well, the model was run again for the experimental conditions of [PERSON] et al. (2005a, 2005b) [52; 53] tested in the Section Results of [PERSON] et al. (2017) [26]. Compared to the previous results, the dune height is now generally slightly lower, though still represented well. Dune length is very similar to before, represented well except for the artificial defects of bedforms D and E. Migration rate and thereby transport rate are much higher than before, and are now generally overestimated instead of underestimated. This is in line with the results presented in Section 3.3 of [PERSON] et al. (2016) [26], where the migration rate and thereby transport rate increased strongly with step length. Overall the results are slightly worse than with the model version optimized for the dune regime, though dune dimensions are still represented well by the new model. The question arises whether the step length should be varied only due to the changing flow regime as is done now, or along the dune as well (because of local variation in shear stress). Explorative experiments of [PERSON] et al. (2012) [26] suggest that the latter is not necessary, but this warrants further research. This also applies for more realistic non-linear relations of the step length model, as the current one (Equation (19)) is still relatively simple. Although the model results show hysteresis effects with regard to discharge and water depth caused by the development of dunes, it should be realized that the present model does not describe the hysteresis effects due to the phase lead of discharge with respect to water depth, which may be relevant for field situations.