3.1. Symmetric cases

The Q criterion (Dubief & Delcayre Reference Dubief and Delcayre2000) is used in figure 2 to visualize the SOV cells and the other helical cells of secondary flows developing downstream of the confluence apex in the α = 60° cases. The scalar Q corresponds to the second invariant of the velocity gradient tensor. No such SOV cells form in the α = 0° case. The SOV cells forming on the fast-speed side of the ML are denoted as FS, while the ones on the slow-speed side are denoted as SS. In some of the cases, more than one SOV cell is present on one side of the ML. The cells are identified using numbers with 1 corresponding to the primary SOV cell that is the closest to the ML centreline (e.g. SS1, FS1). Consistent with their formation mechanism, SS1 and FS1 are counter-rotating (e.g. see streamwise vorticity distributions in figure 2). Meanwhile, the SOV cells on each side of the ML are co-rotating. The secondary SOV cells can be more coherent (e.g. higher circulation) than the neighbouring primary SOV cell and/or extend further downstream in the main channel.

Figure 2. Visualization of SOV cells in the mean flow using the Q-criterion. (a) α = 60°, S-RF, VR = 1.5; (b) α = 60°, S-RF, VR = 2.2; (c) α = 60°, S-RF, VR = 2.2, H = D/2; (d) α = 60°, S-RF, VR = 2.2, H = 2D; (e) α = 60°, AS-RF, VR = 1.5; ( f) α = 60°, AS-RF, VR = 2.2; (g) α = 60°, AS-RS, VR = 1.5; (h) α = 60°, AS-RS, VR = 2.2. Also shown is the streamwise vorticity, ${\omega _x}D/U$, at several cross-sections. The vertical solid lines indicate the position of the ML boundaries.

The systems of SOV cells in the symmetric α = 60°, H = D, S-RF cases (figure 2a,b) are qualitatively similar, with two SOV cells forming on each side. In the VR = 1.5 simulation, away from the confluence apex, the coherence of FS1 and FS2 is larger than that of SS1 and SS2, respectively, as seen from comparing the streamwise variations of the circulation of these vortices in figure 3. This shows that the coherence of the SOV cells forming on the high-momentum side of the ML is higher, which is a general characteristic of SOV cells forming between convergent flows at least for cases with α ₁ ≈ α ₂. Interestingly, the core size and coherence of the secondary cell SS2 are larger compared to those of the primary cell SS1 (figures 2a and 3). The opposite is true at most river confluences with natural bathymetry, but that is because the ML is generally situated over the deepest part of the downstream channel (e.g. Constantinescu et al. Reference Constantinescu, Miyawaki, Rhoads, Sukhodolov and Kirkil2011, Reference Constantinescu, Miyawaki, Rhoads, Sukhodolov and Kirkil2012), so the flow is deeper in the region occupied by the primary SOV cells.

Figure 3. Streamwise variation of non-dimensional circulation of the SOV cells on the fast-speed (FS) and slow-speed (SS) sides of the ML for the H = D cases. (a) FS1 and SS1; (b) FS2 and SS2.

As VR increases from 1.5 to 2.2 in the S-RF cases with H = D, the coherence of FS1 increases, while that of FS2 and SS2 decreases (figure 3). As for the S-RF, VR = 1.5 case, the circulation of FS2 is larger than that of SS2 away from the confluence apex. In the VR = 2.2 simulation, the circulation of FS1 becomes larger than that of FS2 around the region where the cores of high velocity in the two streams collapse. In both symmetric cases, the circulation of the main SOV cells (FS2 and SS2 for VR = 1.5 and FS1, FS2, SS2 for VR = 2.2) peaks around the location where the cores of high velocity in the two streams collapse (e.g. x/D ≈ 30). This is expected, as the coherence of the SOV cells scales with the magnitude of the adverse pressure gradients generated on each side of the ML by the decelerating transverse component of the flow in the incoming stream relative to the ML. In both symmetric cases, the SOV cells do not penetrate inside the ML. Still, bottom-attached cells can form inside the ML (e.g. see figure 2b).

The main effects of decreasing the flow shallowness via increasing the flow depth in the S-RF, VR = 2.2 cases (figure 2c,d) are an increase of the core size, circulation and length of the SOV cells. For example, FS1 penetrates until x/D = 85 in the H = 2D simulation but only up to x/D = 30 in the H = D/2 simulation. A similar variation is observed for the main SOV cell on the low-speed side, SS2. Although when non-dimensionalized by the actual flow depth, the lengths of these vortices do not decay with increasing flow shallowness, their circulation does. This means that, for very high degrees of flow shallowness, the effects of the SOV cells on streamwise momentum redistribution will be important only close to the confluence apex. The other effect of increasing the flow depth is that the SOV cells on both sides of the ML move away from the centreline. In the H = 2D simulation (figure 2d) the SOV cells are situated far away from the ML boundaries while in the H = D/2 simulation FS1 moves inside the ML. This means that the interaction of the SOV cells with the KH vortices will be much lower for cases with low flow shallowness.

3.2. Asymmetric cases

In the H = D, AS-RF cases (figure 2e, f), the ML remains approximatively parallel to the centreline of the downstream channel even close to the apex. As a result, the angle between the core of high velocity in the higher-speed stream and the ML is close to zero and the adverse pressure gradients generated on that side of the ML are small. Only one SOV cell, FS1, forms on the high-speed side and its coherence is much lower than its counterpart on the low-speed side, SS1 (e.g. the peak circulation of SS1 is approximately 5 times larger than that of FS1 in the VR = 1.5 simulation, figure 3a). On the low-speed side, the angle between the core of high velocity and the ML is very high, close to 60°. This induces large adverse pressure gradients starting very close to the confluence apex. Three co-rotating SOV cells form together with several smaller, bottom-attached cells in the VR = 1.5 and VR = 2.2 AS-RF simulations. This is a good example of a case where the most coherent SOV cells do not form on the high-momentum side of the ML as is generally the case for symmetric, or nearly symmetric, planform geometries.

Another interesting feature of the system of SOV cells in the AS-RF simulations is that, although SS1 is situated outside of the ML in the formation region, its core moves gradually inside the ML (e.g. for x/D > 15). This strongly affects the dynamics of the KH vortices inside the ML. The fast turning of the flow on the low-speed side of the ML also generates helical cells of secondary flow. The SS3 eddy in figure 2(e, f) is not an SOV cell but rather a helical cell of secondary flow occupying a large part of the channel cross-section. Its width attains more than 20D at some streamwise locations.

The main effect of increasing VR in the AS-RF cases is to increase the coherence of FS1 (figure 3a) and to decrease the coherence and length of SS2. The reduction in the coherence of SS2 is consistent with the expected decrease in the magnitude of the adverse pressure gradients on the low-speed side of the ML due to the reduction of U ₂ and the fact that the ML position does not change much as VR is increased from 1.5 to 2.2 (figure 2e, f).

In the asymmetric AS-RS cases (figure 2g,h), the ML is pushed toward the sidewall on the low-speed side of the downstream channel. The angle between the core of high velocity from the right incoming channel and the ML is approximately 20°–25°. As a result, the transverse component of the momentum in this stream relative to the ML is significant, which explains the larger coherence of the main SOV cell forming on the right side of the ML in the AS-RS cases (SS1) compared to the corresponding cell (FS1) in the AS-RF cases (figure 3a). While the angle between the core of high velocity in the incoming left stream and the ML decreases slightly in the AS-RS cases compared to the AS-RF cases, the increase in the velocity magnitude in the same incoming stream (e.g. from 0.61U ₀ to 1.39U ₀ for VR = 2.2) is sufficiently large to increase the adverse pressure gradients generated on the left side of the ML. This explains why the coherence of the main SOV cell on the left side of the ML in the AS-RS cases (FS1) is larger compared to that of the corresponding cell (SS1) in the AS-RF cases (figures 2 and 3a). Similar to what was observed in the AS-RF cases, the core of the main SOV cell on the left side of the ML migrates inside the ML for x/D > 25 and a curvature-induced helical cell of secondary flow (FS2) is present next to FS1 in the AS-RS cases.

The lateral displacement of the ML toward the right sidewall of the downstream channel grows with increasing VR. This increases the adverse pressure gradients generated by the interaction of the core of high velocity in the slower stream and the ML. As a result, a secondary SOV cell, SS2, forms in the α = 60°, AS-RS, VR = 2.2 case (figure 2h).

3.3. Dynamics of SOV cells and their effect on the mean flow and turbulence

The SOV cells play a determinant role in the redistribution of the streamwise velocity downstream of the confluence apex and in locally enhancing mixing for confluent flows with a sufficiently high confluence angle. Moreover, advection of high streamwise momentum fluid from the free surface toward the channel bed by the rotating SOV cell can result in a significant increase of the bed shear stress and thus of sediment erosion in the case of a loose-bed channel. The capacity of an SOV cell to displace high streamwise velocity fluid toward the bed is proportional to its circulation. This is illustrated by the mean streamwise velocity plots in figure 4 that show results for the α = 60°, S-RF, VR = 1.5 and α = 60°, AS-RS, VR = 2.2 (H = D) cases in the region where the two streams collide. In the AS-RS simulation, the near-bed tongue of high streamwise velocity extends over the whole width of the FS2 cell.

Figure 4. Non-dimensional streamwise velocity, $u/{U_0}$ (top frame) and mean pressure fluctuations, $\overline {p^{\prime}p^{\prime}} /{(\rho U_0^2)^2}$ (bottom frame) in the x/D = 30 cross-section. (a) α = 60°, S-RF. VR = 1.5; (b) α = 60°, AS-RS, VR = 2.2. The vertical white lines show the boundaries of the ML.

As any coherent vortex forming in a surrounding turbulent flow, the core of each SOV cell is oscillating and its coherence is varying with time. The largest turbulence amplification occurs in the regions where the coherence of the SOV cell is the highest (e.g. see figure 4 for the FS1 cell). While in the S-RF case, the mean pressure fluctuations inside the core of FS1 are comparable, but slightly lower, than those inside the ML, in the AS-RS case the mean pressure fluctuation inside FS1 are about two times larger than those induced by the passage of the KH vortices. Moreover, two distinct patches of high mean pressure fluctuations are present inside the core of FS1. As explained bellow, this is a first indication that the core of FS1 is subject to large-scale, aperiodic bimodal oscillations in the x/D = 30 cross-section.

Under certain conditions, the cores of the most coherent SOV cells may undergo bimodal oscillations toward and away from the ML. This was shown to be the case at several concordant-bed river confluences for the main SOV cell forming on the ML side where the strongest adverse pressure gradients are generated (Constantinescu et al. Reference Constantinescu, Miyawaki, Rhoads, Sukhodolov and Kirkil2011, Reference Constantinescu, Miyawaki, Rhoads, Sukhodolov and Kirkil2012). The oscillations are called bimodal if the velocity histograms display a two-peak shape at locations situated inside the core of the vortex based on its mean flow position. Bimodal oscillations are generally observed only over a limited part of the SOV cell, around the region where the core of high velocity from the incoming stream reaches the ML. It is in this region that the capacity of the SOV cell to enhance mixing is the largest. Using the instantaneous flow fields, the two peaks in the velocity histograms can be associated with two preferred (extreme) states of the oscillating SOV cell. The first state is observed when the SOV cell is interacting with the ML eddies, while the second state corresponds to time instances when the SOV cell is situated the farthest from the ML. These two states were referred to by Constantinescu et al. (Reference Constantinescu, Miyawaki, Rhoads, Sukhodolov and Kirkil2012) as the mixing interface mode and the bank mode, respectively.

Among the cases with α = 60°, only the two AS-RS cases contain one SOV cell (FS1) that is subject to bimodal oscillations. This is somewhat expected as in these cases FS1 is situated on the high-speed side of the ML and the angle between the core of high velocity and the ML is large. For example, the region of bimodal oscillations of FS1 extends between x/D = 7 and x/D = 60 in the AS-RS, VR = 2.2 simulation. The presence of bimodal oscillations is confirmed by the velocity histogram at a point situated inside the core of FS1 in the x/D = 10 section (figure 5) that shows a two-peak distribution. The effect of the bimodal oscillations of FS1 on the flow structure is visualized in figure 6 that shows a full cycle lasting approximately 20D/U ₀ in which FS1 moves from the mixing interface mode to the bank mode and then back to the mixing interface mode. The range of non-dimensional frequencies associated with the bimodal oscillations is St = fU ₀/D = 0.04–0.055. Examination of the instantaneous streamwise vorticity fields in the x/D = 10 cross-section shows that, over the time intervals when v/U ₀ ≈ −0.2 (right peak in the velocity histogram in figure 5), the core of FS1 is situated inside the ML and strongly interacts with the KH vortices (mixing interface mode). During these time intervals the coherence of the bottom-attached vortex present in between FS1 and FS2 is enhanced (t = 0D/U ₀ and t = 20D/U ₀ frames in figure 6). This vortex is also present in the mean flow field (figure 2h). When FS1 moves away from the ML and enters the bank mode, it strongly interacts and exchanges mass and momentum with FS2 (t = 5.6D/U ₀ frame in figure 6). The size and circulation of the bottom-attached vortex are much smaller during the time FS1 is in the bank mode. Thus, bimodal oscillations are an important mechanism for enhancing mass and momentum exchange in the transverse direction.

Figure 5. Probability density function (velocity histogram) of the spanwise component of the instantaneous velocity, v/U ₀, at a point situated inside the core of FS1 in the x/D = 10 cross-section (α = 60°, AS-RS, VR = 2.2 case). The two arrows point toward the peaks associated with the mixing interface mode and the bank mode.

Figure 6. Non-dimensional instantaneous streamwise vorticity (top) and concentration (bottom) in the x/D = 10 cross-section for the α = 60°, AS-RS, VR = 2.2 case illustrating the bimodal oscillations of FS1 between the mixing interface mode (t = 0D/U ₀ and t = 20D/U ₀) and the bank mode (t = 5.6D/U ₀) when FS1 exchanges mass with FS2.

4.1. Effect of SOV cells on scalar transport

Compared to the α = 0° case (figure 7a), the region occupied by the scalar introduced at the confluence apex is much wider at all elevations situated in between the bed and the free surface in the α = 60° cases with H = D (figure 7b–d). Besides the main region of higher scalar concentration due to scalar advected by the cores of the KH vortices, a second streak of high scalar concentration is sometimes present on the channel side/sides where strongly coherent (primary) SOV cells form (e.g. figure 7b, d, f). The SOV cells entrain scalar transported by the KH vortices into their cores starting very close to the confluence apex. The SOV cells do not penetrate to the top boundary. As a result, the width of the region containing scalar is smaller near the free surface compared to below mid-depth levels over the upstream part of the channel that contains SOV cells (x/D < 120). Downstream of this location, the width of the region containing scalar reaches a close to constant value (figure 7b–d). For some of the α = 60°, VR = 2.2 cases with H = D, the width of the region occupied by the scalar in the depth-averaged flow is more than two times larger than the one observed in the α = 0°, VR = 2.2 case (figure 7a). This is a first indication that for constant flow depth the mass exchange between the two streams increases strongly with the angle between the confluent flows.

Figure 7. Non-dimensional concentration field, C, in the instantaneous flow (z/D = 0.9) and in the mean flow (depth-averaged concentration). (a) α = 0°, S-RF, VR = 2.2; (b) α = 60°, S-RF, VR = 2.2; (c) α = 60°, AS-RF, VR = 2.2; (d) α = 60°, AS-RS, VR = 2.2; (e) α = 60°, S-RF, VR = 2.2, H = D/2; ( f) α = 60°, S-RF, VR = 2.2, H = 2D.

Cheng & Constantinescu (Reference Cheng and Constantinescu2020) have shown that for shallow MLs developing between parallel streams the scalar introduced at the confluence apex diffuses differently near the bed and near the free surface even though no SOV cells are present and the scalar is mainly advected by the KH vortices. Starting close to the confluence apex, the region containing scalar is larger near the free surface compared to near the bed as the ML boundary becomes inclined on the slower stream side. The maximum difference occurs at the end of the transition regime. For the α = 0°, S-RF, VR = 2.2 case, the maximum difference between the width of the region containing scalar near the free surface and near the bed is close to 10 % of the depth-averaged width (e.g. see x/D = 155 cross-section in figure 8a). Past the start of the quasi-equilibrium regime, the boundary of the region containing scalar on the slower stream side reduces its inclination and eventually becomes close to vertical (see x/D = 373 cross-section in figure 8a).

Figure 8. Distribution of the non-dimensional concentration, C, and two-dimensional streamline patterns in selected cross-sections. (a) α = 0°, S-RF, VR = 2.2; (b) α = 60°, S-RF, VR = 2.2; (c) α = 60°, AS-RF, VR = 2.2; (d) α = 60°, AS-RS, VR = 2.2. The vertical white lines show the boundaries of the ML.

In the α = 60° cases with H = D, the distribution of the mean scalar concentration is widely non-uniform in the vertical direction (figure 8b–d). The main reason is the presence of SOV cells that interact with the KH vortices during their oscillations or even move inside the ML region (e.g. FS1 in figure 2g,h). On both sides of the ML, the sense of rotation of the SOV cells is such that they advect scalar away from the ML centreline toward the channel sidewalls. Part of this scalar is entrained inside the cores of the primary SOV cells and part of it inside the bottom boundary layer. The secondary SOV cells play a similar role as they advect part of the scalar from the bottom boundary layer farther away from the ML, while the other part is entrained inside their cores. Moreover, if secondary SOV cells interact with the primary SOV cells, the secondary cell can extract some of the scalar present inside the core of the primary SOV cell (e.g. see figure 6). Eventually, the scalar advected inside the cores of the SOV cells is released to the surrounding fluid in the region where the SOV cells lose their coherence. This type of interaction is also observed in the symmetric S-RF, VR = 2.2 case. In the x/D = 30 and 60 sections (figure 8b), the amount of scalar extracted on the high-speed side of the ML is larger because this side contains the most coherent SOV cell (FS1). A reduction of VR does not automatically translate into a decrease of the amount of scalar extracted by the SOV cells from the ML region. For example, the amount of scalar extracted in the S-RF, H = D simulations with VR = 1.5 and 2.2 is approximately the same for x/D < 60.

As already commented on, larger differences in the coherence/circulation of the SOV cells on the two sides of the ML are observed at asymmetric confluences. In such cases, the extraction of scalar from the ML region takes place primarily on the side containing the more coherent SOV cells. For example, in the α = 60°, AS-RF, VR = 2.2 case (figure 2f), the circulation of FS1 is relatively low and its core remains outside of the ML region, which explains the small amount of scalar entrained on the high-speed side of the ML in the x/D = 15 cross-section in figure 8c. On the low-speed side, SS1 has moved inside the ML. The co-rotating SS2 and SS3 cells are also entraining scalar inside their cores and advecting scalar laterally near the channel bottom. This is the primary mechanism responsible for the dispersion of the scalar before the end of the transition regime (x/D = 200–220). Over this regime, the distribution of the scalar is highly non-uniform over the vertical (e.g. see figure 8c for x/D = 15 and x/D = 60) and is mainly due to the near-bed intrusion of scalar on the side containing the more coherent SOV cells.

The efficiency of the SOV cells to extract scalar from the ML region increases sharply if at least one of the primary SOV cells is subject to bimodal oscillations. For example, this is the case for the FS1 cell in the AS-RS, VR = 2.2 case (figure 2h). The FS1 cell moves inside the ML region during the mixing interface mode starting close to the confluence apex (figure 8d). Most of the scalar is drawn into the cores of FS1 and FS2 in the x/D = 15 section. Given the very high coherence of these two cells close to the confluence apex, there is little scalar advected laterally inside the bottom boundary layer. In the x/D = 45 cross-section, high scalar concentration fluid is present inside the ML region, but this is simply because the core of FS1 has migrated inside this region. As for the AS-RF cases, most of the lateral advection of the scalar in the AS-RS cases occurs beneath the free surface and takes place on the side containing the more coherent SOV cells.

Once the SOV cells are destroyed, the scalar distribution in the α = 60° cases continues to evolve such that the width of the region containing the scalar becomes close to constant in the vertical direction and the scalar concentration is close to uniform over the flow depth at large distances from the confluence apex (e.g. see x/D = 373 cross-section in figures 8b and 8d). This is qualitatively similar to what was observed in the α = 0° case (figure 8a).

For same planform geometry and VR, the capacity of the SOV cells to extract scalar from the ML region increases with decreasing flow shallowness. In the α = 60°, S-RF, VR = 2.2, H = 2D simulation most of the scalar in entrained into the cores of FS1 and FS2 starting close to the confluence apex, which explains the presence of the two streaks of high concentration in the mean flow in figure 7( f). Although FS1 also entrains some high-concentration scalar in the corresponding simulation with H = D (figure 7b), the effect of FS1 on scalar transport is much reduced as the streak of high concentration penetrates only up to x/D ≈ 30 compared to x/D ≈ 170 for H = 2D (figure 7f). In the simulation conducted with H = 2D, the capacity of the SS1 and SS2 cells to entrain scalar inside their cores is much more reduced compared to that of FS1 and FS2 because, as already discussed, the increase in the flow depth pushes the cores of the SS1 and SS2 vortices away from the ML (figures 2b and 2d).

4.2. Effects of planform geometry, velocity ratio and flow shallowness on ML structure

Cheng & Constantinescu (Reference Cheng and Constantinescu2020) have shown that for shallow MLs developing between parallel streams the ML develops an undulatory shape during the early stages of the quasi-equilibrium regime. In the α = 0°, S-RF, VR = 2.2 simulation, the transition regime ends around x/D = 155 and ML undulations are present until the end of the computational domain (figure 7a). In the mean, depth-averaged flow, the ML moves gradually toward the low-speed side as the velocity difference between the flows on the two sides of the ML decays exponentially with the distance from the ML origin and the ML centreline becomes close to parallel to the channel's sidewalls.

The instantaneous free-surface, scalar concentration fields show that shallow MLs developing between non-parallel streams (figure 7b–f) have a structure that is qualitatively similar to those developing between parallel streams. Following a region of rapid growth of the KH vortices mostly via vortex pairing, the KH vortices enter a regime where they rotate less fast and are severely stretched in the streamwise direction by the mean shear acting across the ML (figure 9). This is the main reason for the waviness of the ML past the end of the transition regime. This waviness is visualized in the α = 60° cases (instantaneous concentration fields in figures 7b–7e) as undulations of the region containing higher-concentration scalar near the ML centreline.

Figure 9. Non-dimensional concentration, C, in the instantaneous flow (z/D = 0.9) for the α = 60°, S-RF, VR = 1.5 case illustrating the evolution of the KH vortices. The red arrows illustrate the merging of two KH vortices that generates a larger KH vortex in the region where these vortices attain their maximum coherence (x/D ≈ 100). Downstream of this location, the cores of the large KH vortices rotate less, do not merge and start being severely stretched (blue arrows in the t = 0D/U ₀ frame). Their cores eventually assume a wave-like shape (blue arrows in the t = 50D/U ₀ frame) as they connect with the stretched cores of the neighbouring KH vortices.

For symmetric cases with same VR and constant flow depth (H = D), the increase of the confluence angle from 0° to 60° induces a large growth of the coherence of the KH billows immediately downstream of the region where the cores of high velocity in the two incoming streams finish their alignment with the centreline of the downstream channel and the curvature of the ML starts decreasing (e.g. around x/D = 30 in the α = 60°, S-RF, VR = 2.2 case, figure 7b). Over this region, the flow accelerates in the streamwise direction on both sides of the ML and the mean streamwise velocity difference between the two streams is increasing. This favours the formation of close to circular KH vortices (e.g. see instantaneous concentration plot in figure 7(b) for 30 < x/D < 80 and the KH vortices visualized in figure 16a). Such a region is not present for MLs developing between parallel streams (figure 7a) where the streamwise velocity difference decreases monotonically with the distance from the ML origin. The growth of the KH vortices is proportional to the mean shear across the ML. Thus, the coherence of the KH vortices over the upstream part of the ML decays with decaying VR for a constant confluence angle. In the α = 60° simulations with same VR and flow depth, the coherence of the KH vortices over the transition regime is larger in the symmetric case compared to the asymmetric cases. This is due to the primary SOV cell (SS1 for the AS-RF cases; FS1 for the AS-RS cases) moving inside the ML and strongly interacting with the KH vortices in the asymmetric cases. Still, for same VR, the coherence of the KH vortices in the asymmetric, α = 60° cases remains significantly larger compared to the one in the α = 0° case.

In the symmetric VR = 2.2 cases with H = D, the increase of the confluence angle delays the transition to the quasi-steady regime (e.g. from x/D ≈ 155D for α = 0° to x/D = 200-220 for α = 60°) and decreases the amplitude of the ML undulations (figure 7a,b). The start of the quasi-steady regime (x/D = 170-190 for VR = 1.5 and x/D = 200–220 for VR = 2.2) is fairly independent of planform geometry in the α = 60° simulations.

As expected, for same planform geometry and VR, decreasing the flow depth results in a reduction of the streamwise distance corresponding to the end of the transition regime (x/D ≈ 60 or x/H = 120 for H = D/2 compared to x/D = x/H ≈ 200 for H = D, see figures 7b and 7e). In the H = 2D simulation (figure 7f), the ML does not reach the quasi-equilibrium regime by the end of the computational domain. Moreover, similar to what was observed by Cheng & Constantinescu (Reference Cheng and Constantinescu2020) for MLs with α = 0°, in the α = 60°, S-RF, VR = 2.2, H = D/2 case the ML undulations are not present in the instantaneous flow fields over the later stages of the quasi-equilibrium regime (e.g. for x/D > 150 or x/H > 300 in figure 7e).

4.3. Transverse shift of ML centreline

The transverse shift of the ML centreline with respect to the apex point (y/D = 0) is denoted l_cy. For the α = 0°, S-RF, VR = 2.2 case, the increase of l_cy/D with x/D in figure 10 is close to linear only near the ML origin (x/D < 25). As the ML enters the transition regime (x/D ≈ 25), the rate of increase of l_cy/D with x/D starts decaying because the dynamics of the KH vortices is significantly affected by bed friction. With respect to the linear growth regime observed very close to the confluence apex, the rate of increase of l_cy/D decreases by approximately 70 % once pairing of KH billows ceases and the ML enters the quasi-equilibrium regime (x/D ≈ 155) and continues to decrease as the ML approaches equilibrium. The centreline shift toward the low-speed side is a function of the non-dimensional streamwise velocity difference between the two sides of the ML and approaches a constant value as this velocity difference goes to zero (Cushman-Roisin & Constantinescu Reference Cushman-Roisin and Constantinescu2020).

Figure 10. Shift of the ML centreline, l_cy/D, near the free surface as a function of the distance from the origin.

Similar to the case of parallel streams, the ML centreline moves toward the low-speed side in the symmetric (α = 60°, S-RF) cases and l_cy increases with increasing VR (figures 7b and 10). Given that the widths of the two incoming channels are equal and the widths of the incoming and downstream channels are kept constant in all simulations, the equilibrium value of |l_cy| should mainly be a function of VR assuming that lateral mixing between the two streams can be neglected (Cushman-Roisin & Constantinescu Reference Cushman-Roisin and Constantinescu2020). Consistent with this hypothesis, l_cv reaches a close to constant value (l_cy ≈ 3.6D) near the end of the computational domain (x/D > 330) in the S-RF and AS-RF, VR = 1.5 simulations with H = D. The magnitude of the transverse shift is much larger (|l_cy| ≈ 5.6D) near the end of the domain (x/D = 375) in the AS-RS, VR = 1.5 simulation because the deflection of the ML by the faster flow is larger over the region where two incoming streams collapse and so is the readjustment length before the incoming flows lose most of their transverse momentum. Moreover, the magnitude of l_cy continues to decrease at the end of the computational domain, so |l_cy| may approach 3.6D at sufficiently large distances from the ML origin.

Results in figure 10 suggest the equilibrium value of |l_cy| is close to 9.0D in the VR = 2.2 simulations with H = D. This value is reached in the AS-RF case where l_cy is close to constant for x/D > 320. The symmetric cases with α = 0° and α = 60° have non-zero rates of growth near the end of the computational domain. In the α = 0° simulation, |l_cy| ≈ 7.5D at x/D = 375, but the rate of increase of |l_cy| for x/D > 250 is approximately two times larger than that of the α = 60°, S-RF, VR = 2.2 case for which l_cy ≈ 8.5D at x/D = 375. Even in the AS-RS, VR = 2.2 case where l_cy reaches −14D at x/D = 85, downstream of the region where the two streams collapse, the recovery is fairly fast such that |l_cy| ≈ 8.0D at the end of the domain.

For same planform geometry and VR, comparison of the α = 60°, S-RF, VR = 2.2 simulations conducted with H = D/2 and H = D suggests that the nearly constant value of |l_cy| past the end of the transition regime is fairly independent of the flow depth. Even in the H = 2D simulation, where the ML does not reach the quasi-equilibrium regime, the values of l_cy for x/D > 300 are close to 8.5D.

4.4. ML width

For all cases, the streamwise growth of the ML width, δ(x), near the free surface (z/D = 0.85) is nonlinear away from the ML origin (figure 11). In most simulations, the length of the downstream channel was sufficiently large for the ML width to become close to constant (e.g. for x/D > 190 for the α = 60°, S-RF, VR = 1.5 case) or to reach a regime where its rate of change is relatively small compared to values observed during the transition regime. As opposed to the α = 0° case where the rate of growth of the ML width decays monotonically with increasing streamwise distance, the variation of dδ/dx is highly non-monotonic in most of the α = 60° cases. In many of the asymmetric cases, a region characterized by a rapid growth of the ML width is present during the later stages of the transition regime (e.g. starting around x/D = 100 in the AS-RF, VR = 2.2 case and x/D = 160 in the AS-RS, VR = 2.2 case). Inside this region, the trajectories of the KH vortices become less regular as some of these eddies penetrate more on the low-speed side of the channel. As a result, the ML thickness increases rapidly.

Figure 11. ML width, δ/D, near the free surface as a function of the distance from the origin. The legend is the same as in figure 10.

For all α = 60° cases with H = D, the effect of increasing VR is to increase the ML width. For example, the ML width toward the end of the domain is approximately 3 times larger in the S-RF, VR = 2.2 case compared to the S-RF, VR = 1.5 case. The increase is of the order of 100 % for the AS-RF cases and 70 % for the AS-RS cases. The same trend was observed by Cheng & Constantinescu (Reference Cheng and Constantinescu2020) for MLs developing between parallel streams. As VR increases, so does the mean shear acting across the ML, which results in the formation of more-coherent KH vortices. The increase in VR also increases the average number of vortex-pairing events occurring during the transition regime which is the main mechanism responsible for the increase of the ML width.

For constant VR and constant flow depth, the asymmetric cases are characterized by a larger ML width compared to the symmetric cases beginning some distance after the start of the transition regime. For the symmetric cases, the main effect of increasing α from 0° to 60° is a reduction of the ML width of around 10 % past the end of the transition regime. In the α = 60°, S-RF, VR = 2.2 case, the coherence of the KH vortices and δ(x) increase rapidly in the region (x/D < 40) where the two incoming streams collapse and the mean streamwise velocity difference between the two streams increases in the streamwise direction. At the end of this region, the ML width is larger compared to the corresponding α = 0° case. No vortex pairing is observed between x/D = 40 and x/D = 90, which explains the low increase of δ over this region in the α = 60°, S-RF, VR = 2.2 case. Overall, results in figure 11 show that VR is the leading factor controlling the ML width past the end of the transition regime.

For constant planform geometry and VR, the effect of increasing the flow depth is to increase the ML width. In the α = 60°, S-RF, VR = 2.2, H = D/2 simulation δ/D ≈ 4.7 past the initial stages of the quasi-equilibrium regime. The corresponding H = D simulation shows δ/D ≈ 9.5 toward the end of the computational domain where the ML is still in the initial stages of the quasi-equilibrium regime. In the H = 2D simulation, the ML width is still characterized by a significant rate of growth at the end of the computational domain where δ/D ≈ 17. In terms of the flow depth, which is the physical length scale, the non-dimensional ML width during the later stages of the quasi-regular regime is around 9.5 in the α = 60°, S-RF, VR = 2.2, H = D/2 simulation. This value is expected to slightly increase with decreasing flow shallowness.

4.5. Spectral content of the ML

Figure 12 provides information on the streamwise evolution of the spectral content of the ML by comparing power spectra of the spanwise velocity fluctuations at points situated close to the ML centreline. This is supplemented in figure 13 by the autocorrelation function of the spanwise (v) velocity fluctuations,

(4.1)\begin{equation}{R_{vv}}(\Delta t) = \frac{{\langle v^{\prime}({t_0}) \cdot v^{\prime}({t_0} + \Delta t)\rangle }}{{{{(\langle {{v^{\prime}}^2}({t_0})\rangle \langle {{v^{\prime}}^2}({t_0} + \Delta t)\rangle )}^{0.5}}}},\end{equation}

at x/D = 30 where the ML is in the transition regime. The brackets 〈 〉 denote averaging over all values of the time ${t_0}$ and Δt is the time difference. The time interval between the origin, where R_vv(0) = 1, and the first maximum of R_vv can be interpreted as the passage time of the KH vortices at that location. During the transition regime, a −3 energy-decay subrange is often present on the high-frequency side of the peak frequency associated with the advection of KH vortices (figure 12). Such a −3 energy-decay subrange was observed experimentally by Uijttewaal & Booij (Reference Uijttewaal and Booij2000) for shallow MLs forming between parallel streams and by Vowinckel, Schnauder & Sukhodolov (Reference Vowinckel, Schnauder and Sukhodolov2007) for a shallow ML between two streams with a small confluence angle. Its presence indicates that the KH vortices are quasi-two-dimensional (Batchelor Reference Batchelor1969). Once the ML transitions to the quasi-equilibrium regime, the peak frequency in the spectra (e.g. at x/D = 224 and 373 in figure 12) generally corresponds to the undulatory motions of the ML.

Figure 12. Non-dimensional power spectra of spanwise velocity fluctuations calculated at a point situated close to the ML centerline and the free surface (z/D = 0.85). (a) α = 0°, S-RF, VR = 2.2; (b) α = 60°, S-RF, VR = 2.2; (c) α = 60°, S-RF, VR = 1.5; (d) α = 60°, AS-RF, VR = 2.2; (e) α = 60°, AS-RF, VR = 1.5; ( f) α = 60°, AS-RS, VR = 2.2; (g) α = 60°, AS-RS, VR = 1.5. The green line shows the presence of a –3 subrange. The arrows point toward peak frequencies in the velocity spectra.

Figure 13. Autocorrelation function of the spanwise velocity fluctuations, R_vv at x/D = 30 (x = 2 m). The autocorrelation function is evaluated close to the ML centreline near the free surface (z/D = 0.85).

4.6. TKE inside the ML region

For shallow MLs developing between parallel streams, the region of highest (depth-averaged) TKE corresponds to that where coherent KH vortices are present (e.g. see figure 14(a) for the α = 0°, VR = 2.2 case). This is because no other large-scale vortices are present in the flow. In the streamwise direction, the region of high TKE $(k/U_0^2 \gt 0.004)$ extends slightly past the location where the ML transitions to the quasi-equilibrium regime (e.g. x/D = 155). Turbulence production by mean shear is the main mechanism for the TKE amplification inside the ML. The ML undulations do not induce a noticeable increase of the TKE. As the confluence angle is increased to 60° (figure 14b), the length of the region of high TKE is approximately the same but its width increases much faster away from the ML origin and then remains close to constant. The TKE values near the ML centreline are higher until x/D = 40. This is due to the larger coherence of the KH billows in the α = 60°, VR = 2.2 case (figure 7b) compared to the α = 0°, VR = 2.2 case (figure 7a). The other main reason for the TKE increase is the temporal oscillations of the primary SOV cells SS1 and FS1 (figure 2b).

Figure 14. Non-dimensional, depth-averaged TKE, $\; k/U_0^2$. (a) α = 0°, S-RF, VR = 2.2; (b) α = 60°, S-RF, VR = 2.2; (c) α = 60°, AS-RF, VR = 2.2; (d) α = 60°, AS-RS, VR = 2.2. Values smaller than 0.0039 are blanked. The dashed line visualizes the approximate position of the ML centreline near the free surface.

Although for constant flow depth the length of the region of high TKE $(k/U_0^2 \gt 0.004)$ does not increase in the asymmetric α = 60° cases (figure 14c,d) compared to the α = 60° symmetric case, there is a significant increase of the subregion of very high TKE $(k/U_0^2 \gt 0.01)$ in the asymmetric cases. This subregion is skewed toward the confluence side where the most coherent SOV cells form (e.g. right side in the AS-RF cases and left side in the AS-RS cases). As the cores of some of these cells (e.g. SS1 in the AS-RF, VR = 2.2 case and FS1 in the AS-RS, VR = 2.2 case) moves inside the ML away from the confluence apex, it is hard to separate between the two mechanisms that lead to the strong TKE amplification near the ML centreline in these cases. It is only in the AS-RS cases, where FS1 is subject to bimodal oscillations, that the region of very high TKE splits into two elongated streaks associated with the advection of the KH billows and the oscillations of FS1 (figure 14d).

6.1. Role of SOV cells and KH vortices in sediment entrainment

The presence of energetic coherent structures in the vicinity of the channel bed may result in a large amplification of τ. Cheng & Constantinescu (Reference Cheng and Constantinescu2020) have shown that for shallow MLs forming between parallel streams, the deformed cores of the KH vortices can induce elongated patches of high τ close to the ML centreline and these values were comparable to the mean value of the bed shear stress in the incoming channel containing the faster flow.

Regardless of the planform geometry, the highest bed shear stresses in the α = 60° cases are induced by the most coherent SOV cells and helical cells of secondary flow. As illustrated in figure 16 for a symmetric case and for an asymmetric case, the values of τ induced by these cells on the high-momentum side (e.g. FS1 and FS2 for the S-RF, VR = 1.5 case and FS2 for the AS-RS, VR = 2.2 case) are more than two times larger than both $\bar{\tau }$ in the incoming channel containing the faster flow and the peak values of τ in the region where strongly coherent KH vortices interact with the channel bed. Meanwhile, the values of $\bar{\tau }$ are relatively low in the region where KH vortices are advected over the channel bed. By contrast, the values of τ and $\bar{\tau }$ are fairly comparable and high beneath the SOV cells and the helical cells of secondary flow, as these cells are present at all times in the vicinity of the bed. Thus, these cells have a much higher capacity to entrain sediment from the bed. The entrained sediment is then advected downstream inside their cores before being released to the surrounding flow. This is the primary mechanism for the formation of the confluence scour hole at natural river confluences. The largest values of τ and $\bar{\tau }$ do not occur exactly beneath the cores of the cells, but rather on the side where their rotation pulls larger streamwise velocity fluid from higher levels toward the bed (figure 16). Next to such cells, the transverse component of the bed shear stress is generally much lower than its streamwise component. This is also the reason why the cells on the slower stream side of the ML (e.g. SS2 in the S-RF, VR = 1.5 case, see figures 16a) have a much lower capacity to increase τ and $\bar{\tau }$ even when their coherence is comparable to that of the cells forming on the faster stream side (e.g. FS2 in the S-RF, VR = 1.5 case, see figure 16a).

Figure 16. Effect of the SOV cells and KH vortices on the bed shear stress. (a) α = 60°, S-RF, VR = 1.5; (b) α = 60°, AS-RS, VR = 2.2. The coherent structures in the mean and instantaneous flow fields are visualized using the Q criterion. The red arrows point toward the KH vortices inside the ML. Also shown is the corresponding distribution of the non-dimensional bed shear stress in the mean flow, $\bar{\boldsymbol{\tau }}/{\boldsymbol{\tau }_0}$, or in an instantaneous flow field, $\tau /{\boldsymbol{\tau }_0}$, where ${\boldsymbol{\tau }_0} = 0.0023\boldsymbol{\rho U}_0^2$ is the average bed shear stress in the two incoming streams. The left and middle frames show the coherent structures and bed shear stress in the instantaneous flow while the right frame shows the mean flow variables.

Figure 16 also allows us to describe the interactions between the cores of the KH vortices and the primary SOV cells. In the symmetric cases (e.g. see middle panel of figure 16a for x > 15D), the result of a KH vortex hitting the core of an SOV cell is to induce local disturbances along its core. The core of the SOV cell appears to locally wrap up around the core of the KH vortex. As a result, the cores of the primary SOV cells assume a helicoidal shape in the instantaneous flow fields in cases where strongly coherent KH vortices are generated inside the upstream part of the ML (e.g. SS1 and FS1 for the S-RF, VR = 1.5 case and FS1 for the AS-RS, VR = 2.2 case in the downstream part of the middle panels in figure 16). Another consequence is that the levels of τ are very non-uniform beneath the cores of the primary SOV cells. If one of the primary SOV cells is subject to bimodal oscillations (e.g. FS1 in the AS-RS, VR = 2.2 case, see middle panels in figures 16b and 6), then this cell strongly interacts with the KH vortices during the mixing interface mode.

6.2. Effect of planform geometry, velocity ratio and flow shallowness on the bed shear stress

To characterize the sediment entrainment capacity in flows where unsteady coherent structures are present, one needs to consider not only the distributions of $\bar{\tau }$ but also information on the local variability of $\tau $ with respect to its mean value. For example, no sediment entrainment will occur at a certain location based on the value of the mean bed shear stress if $\bar{\tau } \lt {\tau _c}$. However, sediment can still be entrained at that location if $\tau \gt {\tau _c}$ during the times an eddy passes over that location. Cheng, Koken & Constantinescu (Reference Cheng, Koken and Constantinescu2018) have shown that for this type of flows, one should analyse the distributions of $\bar{\tau }$ and ${\tau _{rms}}$ to characterize the sediment entrainment capacity of the flow. This is also why many codes used to perform simulations with a deformable loose bed employ an ‘enhanced’ value of the bed shear stress $({\tau _e} = \bar{\tau } + C{\tau _{rms}})$ in the sediment entrainment flux formula (Sumer et al. Reference Sumer, Chua, Cheng and Fredsøe2003; Cheng et al. Reference Cheng, Koken and Constantinescu2018). As the system of SOV cells and the distributions of $\bar{\tau }$ and ${\tau _{rms}}$ are qualitatively similar in the corresponding VR = 2.2 and VR = 1.5 simulations, only results for the VR = 2.2 cases are presented in figure 17.

Figure 17. Normalized bed shear stress in the mean flow, $\bar{\tau }/{\tau _0}$, (top frames) and root mean square of the bed shear stress fluctuations, ${\tau ^{rms}}/{\tau _0}$ (bottom frames), where ${\tau _0} = 0.0023\rho U_0^2$. (a) α = 0°, S-RF, VR = 2.2; (b) α = 60°, S-RF, VR = 2.2; (c) α = 60°, AS-RF, VR = 2.2; e; (d) α = 60°, AS-RS, VR = 2.2; (e) α = 60°, S-RF, VR = 2.2, H = 2D. The dotted line visualizes the approximate position of the ML centreline near free surface.

The mean bed shear stress is smaller beneath the ML than beneath the faster stream in the α = 0°, S-RF, VR = 2.2 case (figure 17a) and the streamwise variation of $\bar{\tau }$ in the faster and slower streams reflects the gradual evolution toward velocity equalization in the two streams (Cushman-Roisin & Constantinescu Reference Cushman-Roisin and Constantinescu2020). Still, the presence of coherent KH vortices results in relatively large values of ${\tau _{rms}}$ beneath the ML until the end of the transition regime. The peak values are close to $0.2{\tau _0}$ (${\tau _0}$ is the average bed shear stress in the two incoming streams) which should enhance the capacity of the flow to entrain sediment beneath the upstream part of the ML.

In the α = 60° cases with H = D, the largest values of $\bar{\tau }$ are induced by the SOV cells on the high-speed side of the ML. For example, in the S-RF, VR = 2.2 simulation (figure 17b) the long streak of high $\bar{\tau }$ extending up to x/D = 50 is induced by FS1 (figure 2b). The peak values of $\bar{\tau }$ inside this region are approximately 50 % larger than the largest $\bar{\tau }$ values in the high-speed stream downstream of the confluence apex. The passage of KH vortices is a main mechanism for the amplification of ${\tau _{rms}}$. The region with ${\tau _{rms}}/{\tau _0} \gt 0.1$ extents up to the location where the ML reaches the quasi-equilibrium regime. The lateral oscillations and changes in the coherence of FS1 and SS1 also contribute to the increase of ${\tau _{rms}}$ closer to the confluence apex. Still, despite the presence of strongly coherent SOV cells, the size of the region with ${\tau _{rms}}/{\tau _0} \gt 0.1$ and the peak values of ${\tau _{rms}}$ within this region are comparable to those predicted in the α = 0°, S-RF, VR = 2.2 case (figure 17a).

In the α = 60°, AS-RF cases, the largest $\bar{\tau }$ are not induced by the most coherent SOV cells that are situated on the side of the incoming channel making a large angle with the downstream channel (e.g. see figure 17(c) for VR = 2.2 where the highest $\bar{\tau }$ values are induced by FS1 rather than SS1). This is because the main reason for the amplification of $\bar{\tau }$ is not the increase of the transverse component, which is proportional to the circulation of the cell, but the increase of the streamwise component due to advection of higher-momentum fluid toward the bed. Although the KH billows are less coherent compared to the corresponding symmetric case, ${\tau _{rms}}$ is 20 %–30 % larger near the ML centreline. This is due to the larger unsteadiness of the SOV cells as they interact with the KH vortices.

In the AS-RS, VR = 2.2 case (figures 2h and 17d), where the most coherent SOV cells form on the high-speed side of the ML, the elongated region of high $\bar{\tau }$ induced by FS1 and FS2 is approximately 2.5 times longer and 2 times wider compared to the corresponding region of high $\bar{\tau }$ in the AS-RF, VR = 2.2 case. The peak values of $\bar{\tau }$ inside this region are about 50 % larger compared with the AS-RF, VR = 2.2 case. In the AS-RS simulations, the main reasons for the formation of a long region of high ${\tau _{rms}}$ on the high-speed side of the ML centreline are the bimodal oscillations of FS1 and the interactions between the KH vortices and the disturbed core of FS1. Finally, the regions of high ${\tau _{rms}}$ and $\bar{\tau }$ observed in the AS-RS cases and, to a lesser degree, in the AS-RF cases near the left sidewall of the confluence are due to the separated shear layer forming as the flow in the inclined channel enters the confluence.

In the α = 60°, S-RF, VR = 2.2, H = D/2 case the effect of the SOV cells on the distributions of $\bar{\tau }$ and ${\tau _{rms}}$ is negligible and no clear amplification of $\bar{\tau }$ is observed beneath the ML even close to the confluence apex where KH vortices are present in figure 7(e). However, as the flow depth becomes sufficiently large the SOV cells can significantly increase the bed shear stress beneath them. This is evident by comparing the distributions of $\bar{\tau }$ and ${\tau _{rms}}$ for the α = 60°, S-RF, VR = 2.2 simulations with H = D (figure 17b) and H = 2D (figure 17e). Although, in the H = 2D simulation the values of $\bar{\tau }$ beneath FS1 are slightly smaller compared to the H = D simulation, most of the sediment inside the confluence is likely to be entrained by FS2 whose sediment entrainment capacity is negligible in the H = D simulation. Moreover, the levels of amplification of ${\tau _{rms}}$ are very large not only beneath FS2 but also beneath FS1 and SS1. So, for deeper cases, the SOV cells are expected to drive the growth of the confluence scour hole, which is consistent with observations and simulations of flow dynamics at natural river confluences (e.g. see Constantinescu et al. Reference Constantinescu, Miyawaki, Rhoads, Sukhodolov and Kirkil2012).

6.3. Sediment entrainment capacity

To get a more quantitative idea about the effect of planform geometry and VR on the sediment entrainment capacity of confluent flows, one can estimate the mean (time-averaged) flux of entrained sediment based on the instantaneous flow fields, $\bar{E}$. Sediment entrainment formulas for non-cohesive sediment generally assume the sediment flux is proportional to ${(\tau - {\tau _c})^\gamma }\; $with γ > 1, where ${\tau _c}$ is the critical value for sediment entrainment for a given mean particle diameter, d ₅₀ (Chou & Fringer Reference Chou and Fringer2010). For example, γ = 1.5 was used in the van Rijn (Reference Van Rijn1984) expression to estimate the volumetric flux of sediment entrainment per unit time and area. In non-dimensional form, the volumetric flux given by van Rijn entrainment formula can be written as

(6.2)\begin{equation}E(t) = \frac{1}{{U_0^2{D^2}}}\int_{A^{\prime}} {{{(\tau - {\tau _c})}^{1.5}}\,\textrm{d}A^{\prime},}\end{equation}

where A’(t) is the downstream channel bed area where $\tau (t,x,y) \gt {\tau _c}$. The mean flux $\bar{E}$ is the time average of E(t). Using Shields’ diagram, ${\tau _c}/{\tau _0} = 1.5$ for sand particles with d ₅₀ = 300 μm if U ₀ = 0.23 m s⁻¹, D = 0.067 m.

Consistent with figure 17(a) that shows that $\bar{\tau }/{\tau _0} \lt 1.5$ over the entire region with x/D > 0, the total volumetric flux $\bar{E} \approx 0$ for the α = 0°, S-RF, VR = 2.2 case (figure 18). Based on the mean flow, the size of the region with $\bar{\tau }/{\tau _0} \gt 1.5$ is comparable in the S-RF and the AS-RF simulations with VR = 2.2 (figure 17b,c). The size of this region increases significantly in the corresponding AS-RS simulation (figure 17d). The predicted values of $\bar{E}$ in figure 18 are consistent with these trends. For VR = 2.2, $\bar{E}$ is approximately 30 % larger in the AS-RF simulation compared to the S-RF simulation, while the value of $\bar{E}$ in the AS-RS simulation is 5–6 times larger compared to the other α = 60° cases. This means that for relatively large angles between the incoming channels, planform geometry can have a very large impact on the capacity of the flow to entrain sediment and to form a confluence scour hole downstream of the location where the two streams come into contact.

Figure 18. Non-dimensional mean time-averaged total volumetric flux of entrained sediment, $\bar{E}$. Results are plotted for ${\tau _c}/{\tau _0} = 1.5$.

For fixed planform geometry, constant flow depth and total discharge inside the main channel, the influence of VR is more subtle. As VR increases, there is an automatic increase in the sediment entrainment potential on the faster stream side and a corresponding decrease on the slower stream side of the ML. However, given also the nonlinearity between E and $\tau - {\tau _c}$, it is not possible to a priori predict if $\bar{E}$ increases or decreases with VR or even if this relationship is monotonic. Results in figure 18 for the α = 60° cases show that increasing VR from 1.5 to 2.2 decreases $\bar{E}$ by approximately 6 % in the AS-RS simulations and by approximately 25 % in the S-RF and AS-RF simulations with H = D.

For fixed planform geometry, the effect of increasing the flow depth while maintaining the section-averaged streamwise flow velocity in the main channel constant is to increase the erosive capacity of the flow. For example, $\bar{E}$ increases by approximately 300 % as the flow depth is increased from H = D to H = 2D in the α = 60°, S-RF, VR = 2.2 simulations (figure 18).

As for mixing, it is of interest to quantify the contribution of the SOV cells to the sediment entrainment potential. For fixed platform geometry and flow depth, the influence of increasing VR from 1.5 to 2.2 on the relative contribution of the SOV cells to $\bar{E}$ is less than 5 % for α = 60° simulations. Consistent with results presented in figures 2 and 17, for constant flow depth and VR, the contribution of the SOV cells to $\bar{E}$ is the largest in the AS-RS cases where it attains 75 % compared to about 40 % in the AS-RF and S-RF cases. For constant planform geometry and VR, increasing the flow depth from H = D to H = 2D in the α = 60°, S-RF simulations enhances the relative contribution of the SOV cells to $\bar{E}$ from 40 % to 86 %.