#### 3.1. Velocity Distributions

This study explores 3D flow structures in meandering channels under the three different conditions of sinuosity.

Figure 3 shows the mean velocity and turbulent kinetic energy (TKE) distributions simulated at the free surface with the SST

$k-\omega $ model. In this figure, the mean velocity distributions exhibit that the maximum velocity develops near the inner bank due to the centrifugal force, while the minimum velocity evolves along the outer bank. This trend is contradictory to the velocity distribution in natural rivers because the generated channels have the flat and smooth bed [

56,

59]. In nature, rivers generally present the highest velocity along the outer bank because of sediment erosion and deposition. The significant velocity gradients in the transverse direction around the inlet of the meander bend increase TKE near the bend outlet because of a strong momentum transfer between the inner bank and the outer bank. As sinuosity increases, the discrepancy between the maximum inner-bank velocity and the minimum outer-bank velocity increases. Compared to S136 and S157, consequently, S190 exhibits the stronger velocity gradients across the channel width, which drive the higher level of TKE adjacent to the bend apex, as shown in

Figure 3.

The momentum transfer within the meander bend is realized by the secondary flow. With the helical motion of the secondary flows, the high momentum fluid near the water surface transports toward the outer bank, whereas the low momentum fluid near the channel bed moves to the inner bank. The simulation results adequately reproduce these cross-sectional flow patterns, as depicted in

Figure 4, in which the spanwise velocity vectors are overlain with the mean velocity distributions at the bend apex, simulated with three turbulence models under the different sinuosity conditions. This figure clearly shows that the secondary flow intensity is enhanced with increasing sinuosity, which also leads to an increase and a decrease in the velocity magnitude near the inner bank and the outer bank, respectively. All turbulence models used in this study successfully simulate the aforementioned helical secondary flows at the bend apex. Even if the entire flow features are consistent over all turbulence models, the SST

$k-\omega $ model reproduces the stronger intensity of the secondary flow than that of the

$k-\epsilon $ and

$k-\omega $ models. Hence, the velocity near the water surface and channel bed, simulated with the SST

$k-\omega $ model, are higher than that of other turbulence models.

The influence of the turbulence models on the flow simulations is investigated more rigorously using the depth-averaged streamwise velocity and width-averaged spanwise velocity profiles, as shown in

Figure 5. From this figure, one can notice that the shear flows become stronger in both lateral and vertical directions as sinuosity increases. Although the topological characteristics exert a dominant control over the overall flow patterns in meandering channels, the detailed flow behaviors are governed by the turbulence models. In S136, the velocity profiles are barely sensitive to the turbulence models. In S190, in contrast, the SST

$k-\omega $ model simulates the larger deviations in both the streamwise and spanwise velocity profiles than those of other turbulence models. Here, the transverse gradient of the streamwise velocity for

$y/W<0.3$ near the outer bank increases more sharply with the SST

$k-\omega $ model. The vertical profile of the spanwise velocity obtained from the SST

$k-\omega $ model also shows the secondary flow to be stronger than that of other turbulence models. These results indicate that the SST

$k-\omega $ model produces the stronger shear flows in the sharply curved meander bend compared to the

$k-\epsilon $ and

$k-\omega $ models. Therefore, the performance of the turbulence models may directly impact solute transport mechanisms related to the bulk shear dispersion as well as the turbulent diffusion.

#### 3.2. Separated Recirculating Flows

The high sinuosity causes the transverse gradient of the streamwise velocity biased largely toward the inner bank. This strong velocity gradient in the transverse direction forms the low-velocity zones near the outer bank, where it is notable that the negative streamwise velocity (backflow) is simulated at the bend apex with the SST

$k-\omega $ model for S190, as shown in

Figure 5a. From the backflow, it is evident that horizontal recirculating flows take place in the vicinity of the outer bank owing to the adverse pressure gradients associated with the sharp bend curvature. At high sinuosity, the streamwise velocity structures are substantially influenced by the turbulence models (

Figure 5a). Thus, it can be inferred that the turbulence models may directly affect the recirculating flow characteristics.

Figure 6 shows that the separated recirculation zones emerge near the outer bank of the bend inlet in high-sinuosity cases of S157 and S190, as shown in

Figure 6. Hickin [

60] revealed that the onset of the horizontal flow separation occurs at a specific level of the curvature ratio to channel width,

${R}_{c}/W\le 2$, which is consistent with

${R}_{c}/W$ of S157 and S190, as presented in

Table 1. According to

Figure 6, however, the onset and size of the recirculation zones highly depend on the turbulence models. In S157, the

$k-\omega $ and SST

$k-\omega $ models adequately reproduce the onset of the recirculating flows, while the

$k-\epsilon $ model fails to simulate the flow separation. In S190, all turbulence models are successful in reproducing the separated recirculating flows. Nevertheless, the

$k-\epsilon $ and

$k-\omega $ models unpredict the reattachment length of the flow separation for both S157 and S190 compared to the SST

$k-\omega $ model (

Figure 6).

The influence of the turbulence models on the flow separation is further examined using its width and velocity magnitude. The transverse distributions of the depth-averaged mean velocity at the cross-section across the center of the recirculation zones is plotted in

Figure 7. With this figure, the width of the recirculation zones is implicitly estimated by measuring the distance between the center of the recirculation zone and the wall. Please note that this study assumes the minimum-velocity point other than the near-wall boundary as the center of the recirculation zone, as illustrated in

Figure 7b. In S136, no flow recirculation is simulated with all turbulence models as the minimum velocity occurs very near the wall, as shown in

Figure 7a. For S157 and S190, the SST

$k-\omega $ model reproduces the faster and wider recirculating flows than those of other turbulence models, as depicted in

Figure 7b,c. Particularly in S157, the

$k-\epsilon $ model simulates the width of the recirculation zone close to zero, which indicates no occurrence of the flow separation (

Figure 7b). These trends are consistent with the findings from

Figure 6.

In the meandering channel studied in this work, the extent of the separated flow recirculation would be critical in quantifying the magnitude of the non-Fickian transport because the larger recirculation zones trap more solute particles and directly contribute to longer residence times of solutes. Additionally, the recirculating flow velocity may be significantly related to the trapping effects. The mass exchange between the main flow zone and the recirculation zone occurs via turbulent diffusion along the interface between the two different flow regimes, where the turbulent shear layer develops. Here, the fast recirculating flows suppress the solute diffusion from the recirculation zone into the main flow zone because of the strong inertial effects, thereby enhancing the trapping effects. As previously demonstrated, the onset, size, and velocity magnitude of the recirculating flows are considerably sensitive to the turbulence models. In consequence, it can be expected that the turbulence models control the non-Fickian transport behaviors simulated in the meandering channels.

#### 3.3. Solute Transport and Dispersion

For solute transport simulations, 2 × 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-\epsilon $,

$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 × 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-\epsilon $ 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 [

41,

61,

62]. 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-\epsilon $ model exhibits the steeper power-law slope and smaller truncation time than those of other turbulence 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-\epsilon $ 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). Thus, the difference in the BTC shape among the turbulence models is purely driven by the difference in the shear flow effects determined by the transverse gradients of the mean velocity, as shown in

Figure 10a. In S157, meanwhile, the

$k-\omega $ and SST

$k-\omega $ models reproduce the flow separation in the vicinity of the outer bank (

Figure 6a). This separated flow traps solute particles to the outer bank via the recirculating flows, then enhance the late-time tailing, as demonstrated in

Figure 10b. As a result, the remarkable change in the BTC parameters between S136 and S157 is observed with the

$k-\omega $ and SST

$k-\omega $ models due to the trapping effects of the flow separation. More specifically, the SST

$k-\omega $ model exhibits that the absolute value of the power-law slope, and truncation time of S157 are about half and twice the values of S136, respectively (

Table 2). On the other hand, the

$k-\epsilon $ model fails to predict the onset of the flow separation for S157. The absence of the trapping effects caused by the flow separation induces the minor difference in the BTC shape between S136 and S157 with the

$k-\epsilon $ model (

Figure 9a,b). In S190, the onset of the separated recirculating flows is successfully simulated with all turbulence models even though the size of the recirculation zone varies with the turbulence models (

Figure 6b). As a consequence, the

$k-\epsilon $ model shows the sudden change in the BTC shape between S157 and S190, wherein the absolute value of the power-law slope, and truncation time of S190 are about one-third and twice the values of S157, respectively (

Table 2). Moreover, the discrepancy in the power-law slope among the turbulence models is considerably reduced in S190.

According to

Table 2, the simulation cases involving the recirculating flows commonly have truncation times larger than 3. This result implies that this truncation time value can be suggested as the threshold for identifying the non-Fickian transport signature ascribed to the flow separation in the meandering channels of this study. Nevertheless, the

$k-\epsilon $ and

$k-\omega $ models exhibit the smaller truncation times and steeper power-law slopes than those of the SST

$k-\omega $ model. As previously discussed, this is because the reattachment length, width, and velocity magnitude of the recirculation zones differ by different turbulence models. Compared to the

$k-\epsilon $ and

$k-\omega $ models, the SST

$k-\omega $ reproduces the larger and faster recirculating flows (

Figure 6 and

Figure 7), and thus exacerbates the late-time tailing of the BTCs (

Figure 9) owing to the stronger storage (trapping) effects on solute transport, as visualized in

Figure 10c. Furthermore, the enhanced secondary flows simulated with the SST

$k-\omega $ model (

Figure 5b) may promote the trapping effects by vigorously conveying solute particles to the flow separation zones near the outer bank via the transverse dispersion. Therefore, the turbulence models significantly affect the solute transport simulations in meandering-open channel flows since these turbulence closure schemes govern the separated recirculating flow and secondary flow structures directly controlling the non-Fickian transport behaviors.