# A Numerical Study of the Influence of Channel-Scale Secondary Circulation on Mixing Processes Downstream of River Junctions

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## Abstract

**:**

## 1. Introduction

#### 1.1. Near Field Processes—Secondary Circulation and Shear-Driven Turbulence

#### 1.2. Implications for Mixing

## 2. Case Study and Methods

#### 2.1. Case-Study

^{2}when it joins the Kama river about 250 river km north of the city of Perm. It has an average annual flow rate of 508 m

^{3}·s

^{−1}and an average summer (June to August) flow rate of 385 m

^{3}·s

^{−1}. The Kama, with a larger upstream contributing area of 51,300 km

^{2}, becomes the name of the post-confluence river, even though it has a lower mean annual flow (385 m

^{3}·s

^{−1}) and mean summer flow (144 m

^{3}·s

^{−1}). Lower Kama flows reflect the fact that the Vishera drains the Ural foothills which receive greater rainfall. However, given the difference in climate forcing, there is also an important intra-annual variation in the discharge ratio of the two channels illustrated here for 2018 (Figure 1). In 2018, from January through to early April, the Vishera has a marginally higher discharge than the Kama. The Vishera has a higher altitude such that as snowmelt begins in April, the Kama rises first with Q

_{r}> 1 until late May. Flow in the Vishera increases in May, and Q

_{r}declines to <1. The Vishera snowmelt season is complete by the end of July, but the Vishera discharge remains higher than the Kama discharge until early October because of the effect of the Urals on summer precipitation. From early October, the two basins are affected by autumn storms that may be as snow in the Vishera, but where snow does not always accumulate for more than a few weeks. Thus, the Q

_{r}can fluctuate, as a function of both precipitation events and snowmelt release. In our experiments, we drive the numerical simulations (see below) using three different measured values of Q

_{r}(Table 1). The first two, Q

_{r}= 0.62 and Q

_{r}= 0.76, are characteristic of the period June through November when the Vishera generally has a higher discharge than the Kama. The third, Q

_{r}= 1.99, is characteristic of the period in April-May when the Kama has a higher discharge than the Vishera.

^{−4}and the mean width is 500 m, but this varies substantially because of the presence of mid-channel islands. The junction angle between the two rivers is about 50°. Just upstream of the junction the w:d ratio of the Kama is circa 100 and the Vishera is circa 110, which are values much higher than those typical in numerical studies of river junctions. There is no clear discordance at the junction (Figure 2b) and the w:d ratio increases markedly immediately downstream to circa 220. Figure 2b shows that the downstream, post-junction channel initially comprises two laterally attached point bars that can be partially exposed at certain flows to create islands. The channel then curves toward the true left, with two exposed mid-channel bars. The presence of bars and curvature downstream makes this junction an ideal case for evaluating how far-field-induced flow processes might modify the mixing initiated at the junction. Downstream of the junction there is a major hydropower plant that has flooded the Kama valley, but the junction is well upstream of this influence.

^{−1}and of the Vishera 140–180 mgL

^{−1}). The Kama river appears different in color before it merges with the Vishera river because upstream of the junction it crosses a marshland, and so has a higher content of organic matter and iron, and so a darker color. This does not impact the densities significantly.

#### 2.2. Numerical Simulation

_{i}

_{,j,k}are the velocity components in the x, y, and z directions, μ is the molecular viscosity, μ

_{t}is the dynamic viscosity, k is the turbulent kinetic energy, δ is the Kronecker delta function, and g is gravity. In order to be able to represent the effects of turbulence anisotropy on secondary circulation (i.e., Prandtl type 2 secondary circulation) a Reynolds Stress Model (RSM) was used to close Equation (1). Although the magnitude of secondary circulation associated with turbulence anisotropy may be small in rivers with irregular boundaries as compared with that associated with channel scale effects such as pressure gradients [59,60], it may be significant for more regular geometries or in far field situations with fewer channel scale effects [61,62,63,64,65]. Representing such effects require a Reynolds Stress Model [66]. In an RSM model explicit transport, production, and dissipation terms are introduced through a momentum equation for the Reynolds stresses:

_{l}= 1.8.

_{μ}= 0.09.

_{m}is the coefficient of molecular diffusion, D

_{t}is the effective coefficient of turbulent diffusion associated with the turbulent viscosity μ

_{t}through the relation D

_{t}= (μ

_{t}/ρ)/Sc

_{t}, where $S{c}_{t}$ is the turbulent Schmidt number [67], that was set equal to 0.7.

_{c}and $S{c}_{t}$ are molecular and turbulent Schmidt numbers, J

_{w}is the diffusion flux of concentration at the wall, and P

_{c}is defined by:

_{avg}is the mean flow velocity. The turbulence intensity was defined by

#### 2.3. Analysis of Model Outputs

_{y}and secondary flow as V

_{x}. However, the situation for the P and the N simulations was more complicated as the primary flow direction did not follow mesh lines. So, in practice, and to be consistent, we applied the no next cross-stream discharge condition [69] to all sections for the R, the P and the N simulations, that is the direction of primary velocity was chosen to be that which produced no net secondary flux in the horizontal.

_{x}, V

_{y}, and V

_{z}component. As the mesh lines in the z direction were vertical, no further correction was required. However, we needed to determine primary velocity (V

_{p}) and the horizontal component of secondary velocity (V

_{s}) from V

_{x}and V

_{y}. The primary and secondary velocities were defined as:

_{x}and Q

_{y}are the discharges defined in the x and y directions respectively. As the mesh was irregular, we interpolated V

_{x}and V

_{y}onto a regular mesh (i.e., of known cell size) of commensurate resolution to the average mesh spacing (10 m horizontal and 0.08 m vertical) and then used the mesh spacing with V

_{x}and V

_{y}to determine Q

_{x}and Q

_{y}.

_{i}is the concentration at location i for the n predictions within a cross-section. In Equation (22), with perfect mixing, ε becomes 0. If there is any inhomogeneity in concentration, then ε > 0. The maximum inhomogeneity will maximize ε, but this value of ε will be dependent upon the relative number of cells in the two confluent channels, and hence the discharge ratio. Thus, this statistic allows us to quantify the patterns of decline (and influences upon them) and the point at which perfect mixing is reached (when ε = 0). We also calculate Equation (22) for velocity. In operational terms, we make two modifications. First, the mean concentration is defined for all sections as the mean concentrations of the two inlets, such that Equation (22) is always calculated with reference to the original (perfectly) unmixed case. Second, J

_{i}values equal to 0 cannot be used with Equation (22). Instead we use an imperceptibly small value of J

_{i}(10

^{−5}), noting that replacing values J

_{i}= 0 with J

_{i}< 10

^{−4}produces stable values of ε to 3 decimal places.

_{s}in each section:

_{si}is the secondary flow in cell i with vertical dimension $d{z}_{i}$ and horizontal dimension $d{s}_{i}$, s being the secondary flow direction.

#### 2.4. Mesh Sensitivity and Validation

_{r}= 0.62. The focus of these tests was assessment of the extent to which mixing and secondary circulation predictions were mesh sensitive. Meshes were coarsened by 6.9% and then 20.5% of the mesh used in the simulations. Table 2 shows the result and confirms that for the entropy statistics, at both 0 km and 5 km downstream the predictions were stable. This was also the case for the secondary circulation intensity at 5 km downstream but not so for secondary circulation intensity at 0 km downstream. We attribute this to the difficulty that the model has in getting the shear predictions correct right at the junction corner and where rapid changes in geometry amplify the effects of mesh changes. However, given that the concentration and velocity entropies are stable throughout, and that a particular focus on this paper is on the far-field secondary circulation induced by the post junction channel, we believe Mesh 2 to be sufficient for the simulations.

_{Kama}was 1010 m

^{3}·s

^{−1}and the Q

_{Vishera}was 1170 m

^{3}·s

^{−1}giving a Q

_{r}= 0.86. The image was analyzed using a supervised classification (Figure 3a) and the mixing interface was digitized and superimposed on an additional numerical simulation (Figure 3b). This comparison shows that the model is capable of reproducing the position of the mixing interface very effectively.

## 3. Results

#### 3.1. R Simulations: Regular Planform and Bathymetry

_{r}= 0.62, this is the Vishera), that is toward the true right. The primary velocities reflect the discharge ratio at 0 km (i.e., primary velocities are higher in the Vishera, Figure 5) such that there is some shear between the two tributaries at the junction but these differences decrease progressively with distance downstream as Vishera water slows and Kama water accelerates. The stream function (Figure 5) shows the expected formation of two counter rotating vortices at 0 km downstream, that is in the near-field. The contours show that the intensity of this circulation declines rapidly in magnitude, to less than 5% of the 0 m values by only two multiples of the post-junction width (1000 m) downstream. The Kama cell declines in cross-section extent and the Vishera increases, as the Vishera water migrates to the true right. In the absence of any kind of asymmetry in the near-field (Figure 5, 0 km) the cells remained confined within their respective waters in terms of concentration. The stream function also reveals the presence of spatially restricted Prandtl Type 2 secondary circulation attached to the banks and there is perhaps some evidence that these interact with the channel scale secondary circulation reflected in the splitting of the true right vortex on the Kama side of the channel.

_{r}= 0.62 for all discharge ratios; the entropy declines only very slowly throughout the simulation domain such that Kama and Vishera water remains unmixed by 8 km downstream. There is a slightly higher rate of decay to 1000 m, and a very slightly higher level of entropy for Q

_{r}= 1.99 than Q

_{r}= 0.76 and for Q

_{r}= 0.76 than for Q

_{r}= 0.62; that is a very weak discharge ratio effect. Mirroring these changes, the entropy in velocity tends rapidly to zero, suggesting a tendency to velocity homogeneity in the near-field, reflected also in Figure 5. It never falls perfectly to zero because there is always shear on the bed and the banks. The secondary flow intensity drops to almost zero within 500 m (i.e., one post junction channel width) distance downstream. These results point to only a weak potential for advective processes to cause mixing in the near field, with the secondary flow in the two rivers serving to keep their waters separate (Figure 5).

#### 3.2. P Simulations: Natural Planform, Regular Bathymetry

_{r}. Figure 7 also shows that the two rivers remain poorly mixed for Q

_{r}= 0.62. The primary velocities are more asymmetric than the R case, and this asymmetry remains throughout, albeit decreasing in intensity. With the Q

_{r}used here, the initial secondary circulation intensity in the near field (as represented by secondary fluxes in Figure 7, 0 km) is greater in the Kama but lower in the Vishera than with the R case (Figure 5, 0 km). This reflects the direction of planform curvature of the post-junction channel which when compared with the R case is in the same sense in the Vishera but in the opposite sense in the Kama. Thus, the angle of turn for the Vishera is reduced and that for the Kama is increased, explaining this difference. The decline in secondary circulation in moving from the near-field to the far-field is also slower, with more intense secondary circulation for the R case for both Kama originating and Vishera originating water by 1 km downstream. The secondary circulation cells for the first 1000 km (Figure 7) suggest that there is sufficient asymmetry for the Vishera cell to extend into Kama water and so it is perhaps surprising that the mixing rate is not more rapid. However, this only occurs for a short distance (100 s of m). With the flux values in Figure 7 at 1000 m downstream, complete cross-channel mixing would take 1000 s of m distance downstream and it is clear that by 2000 m downstream the Vishera cell no longer crosses the mixing interface. After 2 km downstream, the secondary flux becomes more complex to interpret as it is influenced by mid-channel bars, which impart flow curvature and also major changes in the w:d ratio. For instance, the bar at 4 km causes the curvature of both Kama- and Vishera-derived waters to reverse and the result is a reversal in the direction of secondary circulation (Figure 7).

_{r}. As with the R simulations the velocity entropy declines rapidly in the near-field to 1000 km. However, there is also evidence that the downstream mid-channel bars lead to increases and decreases in velocity entropy. This is most notable for the first mid-channel bar (Figure 2b). However, this does not result in any real increase in secondary circulation (Figure 8), suggesting that the entropy comes from an increase in the total wetted perimeter (and so the number of boundary cells with low velocity) and flow acceleration into the true right branch of the mid-channel bar (Figure 7). This may explain why the far-field, and mid-channel bars in particular, has little impact on mixing in the P case (Figure 4, Figure 7 and Figure 8).

#### 3.3. N Simulations: Natural Planform and Bathymetry

_{r}= 1.99) shows that when the Kama has a greater discharge than the Vishera, the Kama remains unmixed for a much further distance downstream. Comparison of Figure 4 with Figure 3 is also interesting. Figure 3 is the validation scenario with the Q

_{r}closest to 1, but also substantially higher discharges in both the Kama and the Vishera reflecting an unusually cold and wet summer. In this situation, the rivers were observed and modelled as mixing much less significantly (Figure 3) than in 2018, the focus of the R, P, and N simulations in this study.

_{r}= 0.62, but these are displaced toward the Kama (the true right) suggesting that the real bathymetry modifies near field processes in a way that is likely to enhance mixing further upstream in the near-field zone. There is a second vortex on the Vishera side of the channel coincident with its high primary velocity core. By 500 m downstream, there is a single large vortex aligned directly over the mixing interface and two sub-vortices, one also aligned over the mixing interface and one entirely within the Vishera. The presence of a submerged mid-channel bar suggests the possibility of significant local topographic forcing and there is consistent flux of water from the true left to the true right over the bar because of the larger-scale vortex. The magnitude of this vortex is lower on the Vishera side and higher on the Kama side than with the R and P simulations at 500 m downstream; but what is more important is the development of a single vortex that would be capable of fluxing Vishera water toward the Kama at altitude, and vice versa at depth. The presence of a significant vortex on the Vishera side of the channel is consistent with the direction of curvature of the Vishera. The secondary circulation flux becomes more intense by 1000 m downstream and has evolved into a single vortex at the scale of the entire post-confluence channel by 2000 m downstream. At 4000 m, the channel is divided either side of an emergent mid-channel bar. The mixing interface has taken the true left (Vishera) side of the channel but the channel is more mixed than with either the R or the P cases.

_{r}, the entropy in concentration reduces to almost zero by 8000 m downstream (Figure 10). The entropy in velocity falls to 2000 m, then varies systematically because of the presence of islands, which increase the flow inhomogeneities and hence entropy as in the P case. The secondary flow intensity is much higher in magnitude. It declines to 2000 m downstream, more slowly than in the R and P cases (Figure 6 and Figure 8), but then there is evidence of high magnitude secondary flow intensity notably in relation to variations in the far-field bathymetry and channel planform. This is also reflected in Figure 9. The entropy in concentration shows a much stronger dependence on Q

_{r}than with the regular geometry. It is maintained at much higher levels to 4000 km downstream (see also Figure 4) when the Kama has a bigger discharge than the Vishera (Q

_{r}= 1.99) in this case. The concentration entropy when the Vishera is dominant (Q

_{r}< 1) declines more rapidly. We argue that this Q

_{r}effect is related to the direction of curvature in the downstream post-confluence channel, that is the far field. The curvature-driven component of secondary circulation in a curved channel has a linear dependence on the square of the streamwise velocity and an inverse dependence on the radius of curvature. When the Vishera has a relatively higher discharge, and so a relatively higher streamwise velocity, this acts in the same direction as the radius of curvature (Figure 4) aiding the development of a stronger streamwise vortex than the Kama’s, hence encouraging mixing. When the Kama has a bigger discharge, while the initial curvature into the confluence is likely to produce a stronger streamwise vortex, the downstream curvature of the post-confluence channel is in the opposite direction. The latter would serve to maintain two more equal counter rotating vortices for longer and so reducing the mixing.

## 4. Discussion

_{r}closer to 1, complete mixing was also delayed. The effect of the higher discharges will be higher longitudinal mean velocity which theory suggests should increase the longitudinal distance required for complete mixing (e.g., [72,73,74]).

^{3}·s

^{−1}and Vishera of 2940 m

^{3}·s

^{−1}. Figure 11 shows the simulated mixing at the surface (Figure 11a,b). In qualitative terms, it suggests that the inclusion of turbulence anisotropy (Figure 11b) makes very little difference to the mixing process as compared with bathymetric effects (Figure 4). Simulations for the full set of R, P, and N simulations may be of value to test this further and these conclusions may also depend upon the kind of model used to represent the effects of turbulent anisotropy on secondary circulation; but these initial tests suggest that the effect of Prandtl Type 2 secondary circulation on the mixing process is relatively small.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Time-series of discharges in the Kama and Vishera rivers and the associated discharge ratios.

**Figure 2.**The junction of the Vishera and Kama rivers in the region of Perm, western Urals, Russia, ©GoogleEarth (

**a**) showing the altitude of the river bed measured using a SyQwest HydroBox sonar to get flow depths and dGPS position to convert these into altitudes (

**b**) and the locations of cross-sections used in the analysis (

**c**).

**Figure 3.**The Google Earth image for the Kama-Vishera junction on the 19 July 2019, showing the classified Kama and Vishera water (

**a**). The numerical simulation (

**b**) for the same day showing (black line) the digitized mixing interface from 3a superimposed.

**Figure 4.**Ned bed concentrations for the three discharge ratios for R simulations (regular planform, regular bathymetry), P simulations (natural planform, regular bathymetry) and the N simulations (natural planform, natural bathymetry).

**Figure 5.**Predictions of concentration (

**left**) and primary velocity (

**right**, m·s

^{−1}) with the stream function for Q

_{r}= 0.62 and the R simulation geometry. Contours of flux are shown (units of m

^{2}·s

^{−1}) as calculated by the stream function. Plots are viewed looking upstream, such that the Kama is on the left of each plot (i.e., the true right) and the Vishera on the right of each plot (i.e., true left). The contour values are only shown on the primary velocity plots.

**Figure 6.**Longitudinal distribution of entropy of concentration, entropy of velocity, and secondary flow intensity for different values of the discharge ratio for the R simulations.

**Figure 7.**Predictions of concentration (

**left**) and primary velocity (

**right**, m·s

^{−1}) with the stream function for Q

_{r}= 0.62 and the P simulation geometry. Contours of flux are shown (units of m

^{2}·s

^{−1}) as calculated by the stream function. Plots are viewed looking upstream, such that the Kama is on the left of each plot (i.e., the true right) and the Vishera on the right of each plot (i.e., true left). The contour values are only shown on the primary velocity plots.

**Figure 8.**Longitudinal distribution of entropy of concentration, entropy of velocity and secondary flow intensity for different values of the discharge ratio for the P simulations.

**Figure 9.**Predictions of concentration (

**left**) and primary velocity (

**right**, m·s

^{−1}) with the stream function for Q

_{r}= 0.62 and the N simulation geometry. Contours of flux are shown (units of m

^{2}·s

^{−1}) as calculated by the stream function. Plots are viewed looking upstream, such that the Kama is on the left of each plot (i.e., the true right) and the Vishera on the right of each plot (i.e., true left). The contour values are only shown on the primary velocity plots.

**Figure 10.**Longitudinal distribution of entropy of concentration, entropy of velocity, and secondary flow intensity for different values of the discharge ratio for the N simulations.

**Figure 11.**Comparison of the standard k-ε model (

**a**) and the Reynolds Stress model (

**b**) predictions of mixing for a Kama discharge of 1480 m

^{3}·s

^{−1}and a Vishera discharge of 2940 m

^{3}·s

^{−1}.

Sample | Discharge Kama (Q_{Kama}) m^{3}·s^{−1} | Discharge Vishera (Q_{Vishera}) m^{3}·s^{−1} | Discharge Ratio (Q_{Kama}/Q_{Vishera}) |
---|---|---|---|

30 July 2018 | 260 | 418 | 0.62 |

22 July 2018 | 385 | 508 | 0.76 |

10 September 2018 | 542 | 273 | 1.99 |

Validation data, 19 July 2019 | 1010 | 1170 | 0.86 |

**Table 2.**Tests of sensitivity of the entropy statistics and secondary circulation (see Equations (22) and (23)) to mesh resolution for two cross-sections using the R simulations.

Parameter | 0 km −20.5% | 0 km −6.9% | 0 km Base Mesh | 0.5 km −20.5% | 0.5 km −6.9% | 0.5 km Base Mesh |
---|---|---|---|---|---|---|

Entropy in concentration | 0.596 | 0.601 | 0.601 | 0.539 | 0.572 | 0.572 |

Entropy in velocity | 0.032 | 0.031 | 0.031 | 0.010 | 0.016 | 0.016 |

Secondary circulation intensity (m^{3}·s^{−1}) | 1.390 | 1.502 | 1.761 | 0.051 | 0.055 | 0.055 |

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## Share and Cite

**MDPI and ACS Style**

Lyubimova, T.P.; Lepikhin, A.P.; Parshakova, Y.N.; Kolchanov, V.Y.; Gualtieri, C.; Roux, B.; Lane, S.N.
A Numerical Study of the Influence of Channel-Scale Secondary Circulation on Mixing Processes Downstream of River Junctions. *Water* **2020**, *12*, 2969.
https://doi.org/10.3390/w12112969

**AMA Style**

Lyubimova TP, Lepikhin AP, Parshakova YN, Kolchanov VY, Gualtieri C, Roux B, Lane SN.
A Numerical Study of the Influence of Channel-Scale Secondary Circulation on Mixing Processes Downstream of River Junctions. *Water*. 2020; 12(11):2969.
https://doi.org/10.3390/w12112969

**Chicago/Turabian Style**

Lyubimova, Tatyana P., Anatoly P. Lepikhin, Yanina N. Parshakova, Vadim Y. Kolchanov, Carlo Gualtieri, Bernard Roux, and Stuart N. Lane.
2020. "A Numerical Study of the Influence of Channel-Scale Secondary Circulation on Mixing Processes Downstream of River Junctions" *Water* 12, no. 11: 2969.
https://doi.org/10.3390/w12112969