## 4 Discussion  
### Dependence of Hop Travel Distance on Particle and Flow Features  
The research herein is the first to use an instrumented particle to assess hop distances. Particle hop lengths can be obtained from the Gamma distribution fittings of different particle densities and channel flow rates to assess and quantify any interesting trends. Greater magnitudes of hop lengths imply that the particle will transverse a distance along the channel with a smaller number of interactions with the bed surface. Figure 9 shows the comparison of the modelled hop lengths distributions (using the Gamma PDF) for the range of assessed particle densities and flows (defined with flow Reynolds number). From Figure 11a,c, the extent and extrema of particle hop length distributions are seen to increase with increasing flow Reynolds number for both surfaces (A and B). For example,  
Figure 10: Empirical and modeled (exponential) distribution of instantaneous instrumented particle streamwise velocities for the flow rate of 49 (L/s), particle density of 1132 (kg/m\({}^{3}\)), and bed surface A.  
as seen in Figure 11c (also Table 4, offered in Appendix A), for the bed surface B, the instrumented particle with density of 1132 kg/m\({}^{3}\) will travel a maximum hop distance 160% greater (from 0.0009 to 0.00235 m), for a 15% increase in flow Reynolds number (from 41,841 to 47,379). In addition, a similar trend is observed as the particle density decreases (Figure 11b,d), with the range of hop lengths increasing with solid particle densities, for both surfaces. For example, as seen in Figure 11b (bed surface A and Reynolds number of 50,789--also Table 5, offered in Appendix A), when particle density decreased by 10% from 1180 to 1085, the peak hop length increased about 70% (from 0.0019 to 0.0032 m). The effects of different particle densities and flow Reynolds number on the Gamma distribution parameters modelling hop lengths are also tabulated in Tables 4 and 5, 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 11). Figure 12 shows the effect of flow Reynolds numbers and Shields numbers on its average hop length (modelled from the Gamma distributions fittings), for bed surfaces A and B, and the assessed particle densities. Figure 11: Hop length modelled by the Gamma 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. Figure 12 displays that the average hop length in general increases with lower particle density. For the example of bed surface A, with fixed flow Reynolds number of 50,789 (Figure 12c), the average hop length increases by 80% from 0.018 to 0.032 m, when particle density decreases by about 15% (from 1224 to 1085 kg/m\({}^{3}\)). The average hop length increases with higher flow Reynolds (Figure 12a,b) and Shields (Figure 12c,d) numbers. For the example shown in Figure 12a, mean hop travel distance increases from 0.015 to 0.028 m (90%) with only about 15% increase in flow Reynolds number (from 43,353 to 50,789). The same increase is seen in Figure 12c, for increasing the Shields number from 0.074 to 0.099 (35%), for fixed particle density of 1180 kg/m\({}^{3}\) and bed surface A. With a higher Shields number and lower density, there is a higher hopping (saltation) length on average, as the particle interacts with the bed fewer times, while covering the same distance downstream.