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Quinn Lewis
Department of Geography and Environmental Management
Introduction
River confluences can create visual patterns of mixing when incoming flows differ in colour or turbidity. The extent to which flows mix at confluences is important for determining spatial patterns of water quality, channel morphology, benthic fauna and fish habitat. Because mixing processes are complex, predicting rates and characteristic scales of mixing is difficult.
This paper introduces a theoretical framework for mixing dynamics of shallow flows at river confluences where the mixing process is controlled by two modes of behaviour: one similar to a wake behind an obstacle and the other similar to a mixing layer between two parallel flows (Figure 1). The paper shows that lateral advective fluxes of momentum induced by helical motion driven by the planar geometry of confluences and by turbulence associated with lateral shear can greatly exceed the inhibiting effect of riverbed friction and thereby substantially enhance mixing at river confluences.
Figure 1: Patterns of mixing at the confluence of the rivers Danube and Save at Belgrade, Serbia.
a, In the mixing-layer mode with higher momentum water from the Danube. b, In the wake mode, showing mixing of waters with almost equal momentum and turbulent vortices arranged into an unsteady wake pattern.
Methodology
The theoretical framework is based on the mixing-length approach and focuses on the dynamics of mixing interfaces simultaneously controlled by multiple factors. The model is derived by integrating the two-dimensional shallow-flow equations of momentum in a steady river flow with a realistic schematization of confluence geometry. A mode-switching hypothesis proposes that the dynamics of mixing interfaces can change to a wake mode when the velocity difference between tributaries is small and the stagnation zone induces an internal velocity deficit that acts like a wake behind an obstacle.
Model predictions were tested against data obtained from field-based experiments that included detailed measurements of instantaneous flow velocities, topography of free surfaces and visualization of flow patterns using dyes. Experiments conducted for shallow mixing layers between parallel flows and shallow wakes in the lee of an obstacle allowed for rigorous testing of model predictions for interfaces with asymptotic behaviour, pure mixing-layer mode and pure wake mode (Figure 2a,b). In experiments with angled tributary channels, the dynamics of mixing interfaces in the intermodal state for a wide range of momentum flux ratio values was explored (Figure 2c).
Figure 2: Patterns of mixing in experiments.
a, Shallow mixing layer b, Shallow wake downstream of a circular obstruction (rectangle is enlarged in d). c, Shallow mixing interface at 40° confluence (rectangle is enlarged in e). d, Patterns of large-scale lateral turbulence interacting with smaller-scale bed-friction-induced turbulence. e, Pattern of mixing induced by streamwise-oriented vortical; dashed lines are mixing-interface boundaries.
Outcomes
Comparison of theoretical predictions with the results of field-based experiments showed that dynamics of the mixing interface modelled using only bed friction systematically underestimate the stabilization effects related to lateral advective and turbulent fluxes of momentum (Figure 3). Predicted lateral profiles of depth-averaged streamwise velocity closely match experimental data for the shallow mixing layer and shallow wake cases (Figure 3a,b). At short distances, the mixing interface for experiments behaved as a free lateral shear flow, while at long distances the data systematically deviated from the predictions. The model, which includes the effects of lateral fluxes of momentum by advection and turbulence, yielded predictions that correspond closely to measured experimental data on mixing-interface growth (Figures 3 c,d).
Fig. 3: Mixing interfaces with asymptotic behaviour: scales relations and comparison of predictions and measurements.
a, Mixing layer scaled by hyperbolic tangent function. b, Wake scaled by exponential function. c, Mixing in mixing-layer mode (dashed line is free mixing layer; solid lines are shallow mixing layer (blue, equation), and accounting for lateral advection and turbulence (red, equation). d, Mixing in wake mode (circular cylinder; dashed line is a free wake; solid lines are shallow wake, equation with (red) and without (blue) the effects of lateral advection and turbulence).
Intermodal behaviour, including mixing-layer and wake modes, is expected when scales contribute to the dynamics of the mixing interfaces. Comparison of experimental data with theoretical predictions showed that accounting for intermodal behaviour provided an accurate prediction of mixing with fast and slow rates (Figure 4). In angled shallow confluences, the stagnation zone near the confluence apex imposed a wake-like effect on the flow. To incorporate this effect into a multi-scale theoretical model, a lateral velocity profile was introduced in which the velocity scale is a sum of mixing-layer and wake contributions. The shape of the composite velocity profile is determined by the ratio of external and internal velocity scales, which provided a metric to characterize intermodal cases (Figure 4 a,b).
The theoretical framework highlighted the value of near-field processes in the vicinity of the confluence apex on the mixing in the far field because at short distances the wake sets the initial width of the interface in a weak mixing-layer mode (Figure 4c,d). The study suggests that understanding of stagnation-zone hydrodynamics can substantially improve predictions of mixing. The predictions and patterns of experimental data show how initial growth rates over distance associated with wake effects are much greater than those for the asymptotic mixing-layer case (Figure 3c) and similar to those for the asymptotic wake case (Figure 3d).
Figure 4: Mixing interface with intermodal behaviour: scale relations and comparison of predictions and measurements. a, In mixing-layer mode with a small wake effect (solid blue line is equation; dashed line is hyperbolic tangent function). b, In wake mode with a small mixing-layer effect. c, Mixing with a weak wake effect, solid red line is equation, red dashed line is free mixing layer, solid blue line is equation and dashed blue line is free wake). d, Mixing with a weak mixing-layer effect, the same definitions for lines as in c).
Conclusions
The study provides insight into the importance of different modalities of flow structure in controlling mixing at river confluences, thereby contributing to practical knowledge on the role of confluences in dispersal of contaminants in river systems. Specifically, the study shows how the mixing process can be understood theoretically and experimentally as consisting of two intrinsic modes of behaviour: wake mode and mixing-layer mode, and demonstrates that intermodal behaviour is the generic mechanism governing dynamics of the mixing interfaces. In addition, it demonstrates that advective effects, particularly those associated with streamline curvature caused by the angled planform geometry of confluences and the need for incoming flows to reorient in direction, play an important role in momentum exchange that drives these dynamics.
The theoretical framework proposed provides promising opportunities for predicting rates of mixing from remotely sensed images and should enhance capabilities to analyze mixing on the basis of geographical information systems. Although the framework captures many relevant aspects of mixing, additional factors, such as jet-like and cross-flow effects, require further experimental and theoretical assessment and requires an understanding of how advective and diffusive processes affect mixing under less complicated conditions.
Sukhodolov, A.N., Shumilova, O.O., Constantinescu, G.S., Lewis, Q. W., Rhoads, B. L.. Mixing dynamics at river confluences governed by intermodal behaviour. Nature Geoscience. 16, 89–93 (2023). https://doi.org/10.1038/s41561-022-01091-1
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