## 3 Results and Discussion
### Solute Transport and Dispersion  
For solute transport simulations, \(2\times 10^{5}\) particles are instantaneously injected as a point source introduced from the center of the inlet cross-section, and the simulated mean velocity and turbulence fields at the steady state are used as input variables to govern advection and diffusion of the solute particles, respectively. The time step is set to retain \(Cr\) to be unity [7]. The effects of sinuosity on solute dispersion are first investigated. Figure 8 presents one-dimensional (1D) breakthrough curves (BTCs) at the channel outlet, simulated with the \(k-\varepsilon,k-\omega\) and SST \(k-\omega\) models. This figure indicates probability density functions of particle residence times until they pass the outlet, which can conventionally be considered as particle residence time distributions. Herein, the probability density is calculated as the ratio of the number of particles, which pass the outlet at each measurement interval of 20 s (bin width of Figure 8), to the total particles of \(2\times 10^{5}\) divided by the bin width. As shown in Figure 8, the longitudinal dispersion increases as sinuosity increases because the higher sinuosity leads to the larger elongation of the solute spreading in the streamwise direction, honoring the larger velocity gradients in the transverse direction. This trend is commonly observed in the BTCs simulated with all turbulence models. Yet, the tail scaling of the BTCs differ according to the turbulence models, as depicted in Figure 8. The scaling (slope) of the BTC tails is conventionally considered as a vital factor in quantifying the non-Fickian dispersion [61; 62]. Figure 9 shows the impact of the turbulence models on the BTC tailing with a variation of channel sinuosity and demonstrates that the \(k-\omega\) and SST \(k-\omega\) models exhibit the heavier-tailed BTCs than those of the \(k-\varepsilon\) model. To quantify this disparity among the turbulence models, this study estimates the tail power-law slope and the truncation time, which are measures of the non-Fickian dispersion [61; 62; 41]. Here, the truncation time is defined as the ratio of the maximum residence time to the peak concentration time [57], and the power-law slope is calculated using the least-squares regression. Please note that the flatter power-law slope (smaller its absolute value) and larger truncation time characterize the stronger non-Fickian dispersion with the heavier BTC tailing. The power-law slope and truncation time for all simulation cases are summarized in Table 2. In S136, the \(k-\varepsilon\) model exhibits the steeper power-law slope and smaller truncation time than those of other turbulence  
Figure 8: 1D breakthrough curves of S136, S157, and S190, in which particle residence times are measured at the channel outlet, simulated with (**a**) the \(k-\varepsilon\) model, (**b**) the \(k-\omega\) model, and (**c**) the SST \(k-\omega\) model. models since this turbulence model simulates the relatively smaller transverse gradient of the mean velocity (weaker shear-flow effect), as shown in Figure 7a. According to Figure 9, this tendency is more pronounced in higher-sinuosity cases of S157 and S190. As sinuosity increases to 1.57 from 1.36, the power-law slope and truncation time ratios of the \(k-\varepsilon\) model to the SST \(k-\omega\) model increase and decrease to 2.70 and 0.42 from 1.96 and 0.65, respectively, as presented in Table 2. Besides, S157 shows that the SST \(k-\omega\) model yields the larger truncation time and flatter power-law slope than those of the \(k-\omega\) model even though no significant difference in the BTC parameters between these two turbulence models is found in S136. The occurrence of the separated flow recirculation can explain the increasing discrepancy in the BTC parameters between the turbulence models, corresponding to increasing sinuosity. In S136, no flow separation is simulated with all turbulence models (Figure 7a).