## 4 Discussion
### Dependence of Particle Velocity on Particle and Flow Features  
Mean streamwise particle velocities can be modelled from fitting exponential distributions of different particle densities and channel flow rates. Below, their interdependence is quantified, and any trends are assessed. Figure 15 shows a comparison of the particle velocity exponential distributions fit to empirical data obtained experimentally from the instrumented particle, for different particle densities and flow Reynolds numbers. From Figure 15a,c, it is observed that with increasing flow Reynolds numbers, the probability of occurrence of higher streamwise particle velocity increases for both surfaces. For example, in bed surface A (Figure 15a), when flow Reynolds number increased by 15% from 43,353 to 50,789 the probability of occurrence increased from 3.6 to 3.95, which is about 15% for the same streamwise particle velocity of 0.12 m/s. In addition, as the particle density decreases, transport for the same flow becomes easier and the probability of occurrence of the higher streamwise particle velocity increases for both surfaces. For example, in bed surface B (Figure 15c), when particle density decreased by 15% from  
Figure 15: Instrumented particle velocity modelled by the exponential distribution: (**a**) for different flow Reynolds numbers, bed surface A and fixed instrumented particle density of 1132 kg/m\({}^{3}\), (**b**) for different instrumented particle densities, bed surface A and fixed Reynolds number of 50,789, (**c**) for different flow Reynolds numbers, bed surface B and fixed instrumented particle density of 1132 kg/m\({}^{3}\), and (**d**) for different instrumented particle densities, bed surface B and fixed Reynolds number of 51,488. 1224 to 1085 kg/m\({}^{3}\), the probability of occurrence increased from 3.81 to 4.25 (50%) for the same streamwise particle velocity of 0.09 m/s. The point where individual probability distributions meet (each having a different rate of change for particle velocities) indicates the streamwise particle velocity where the distributions are having the same probability of occurrence. This crossing is not clearly distinguishable for many cases, and it is prone to uncertainties in fitting the modelled probability distributions. More distinct are peaks (showing the most commonly occurring particle velocity) and mean of the individual distributions, as well as the extent of their tail (illustrating the magnitude of extreme particle velocities). The effects of different particle densities and flow Reynolds number on the fitted exponential distribution parameters, modelling instrumented particle velocities, are also tabulated in Tables 6 and 7, offered in Appendix A, along with further discussion, to allow for a clearer representation of the variations in the body and specifically the tail of the distribution (that may be hard to discern in Figure 15). Figure 16 shows the plot of the effect of different channel flow conditions (i.e., Reynolds and Shields numbers) and test particle densities kg/m\({}^{3}\), from the exponential distribution fittings on the streamwise particle velocity m/s, for bed surfaces A and B. The Shields numbers are used to generalize the results for different combinations of flow and particle conditions. For bed surface A and fixed particle density 1132 kg/m\({}^{3}\) (Figure 16a), with a \(\sim\)35% increase in flow Reynolds number (from 38,190 to 50,789, or an equivalent 50% increase in the Shields number, from 0.09 to 0.135--Figure 16c), the particle will interact with the bed surface less and move for longer hop lengths (consistent to what is shown in Figure 12a,c, and Figure 14a,c, respectively), resulting in a \(\sim\)140% increase in mean particle velocity (from 0.038 to 0.092 m/s).