Journal of Geophysical Research: Earth Surface

Numerical analysis of the effect of momentum ratio on the dynamics and sediment-entrainment capacity of coherent flow structures at a stream confluence

Authors

  • George Constantinescu,

    Corresponding author
    1. Departments of Civil and Environmental Engineering and IIHR-Hydroscience and Engineering, University of Iowa, Iowa City, Iowa, USA
      Corresponding author: G. Constantinescu, Department of Civil and Environmental Engineering, University of Iowa, Iowa City, IA 52242 USA. (sconstan@engineering.uiowa.edu)
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  • Shinjiro Miyawaki,

    1. Departments of Civil and Environmental Engineering and IIHR-Hydroscience and Engineering, University of Iowa, Iowa City, Iowa, USA
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  • Bruce Rhoads,

    1. Department of Geography, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA
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  • Alexander Sukhodolov

    1. Department of Ecohydrology, Institute of Freshwater Ecology and Inland Fisheries, Berlin, Germany
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Corresponding author: G. Constantinescu, Department of Civil and Environmental Engineering, University of Iowa, Iowa City, IA 52242 USA. (sconstan@engineering.uiowa.edu)

Abstract

[1] The flow and turbulence structure at stream confluences are characterized by the formation of a mixing interface (MI) and, in some cases, of streamwise-oriented vortical (SOV) cells flanking the MI. Depending on the junction angle and planform symmetry, as well as the velocity ratio across the MI, the MI can be in the Kelvin-Helmholtz (KH) mode or in the wake mode. In the former case, the MI contains predominantly co-rotating large-scale quasi two-dimensional (2-D) eddies whose growth is driven by the KH instability and vortex pairing. In the latter case, the MI is populated by quasi 2-D eddies with opposing senses of rotation. This study uses eddy resolving simulations to predict details of flow structure for KH- and wake-mode conditions at a confluence for which field measurements are available. Results indicate that SOV cells at this confluence, which occur in both modes, redistribute momentum and mass, enhancing the potential for entrainment of bed material beneath the cells and for extraction of fluid and suspended sediment from the MI. The simulations predict that the cores of some of the primary SOV cells are subject to large-scale bimodal oscillations toward and away from the MI that contribute to amplification of the turbulence close to the MI and enhance the capacity of the SOV cells to entrain sediment. At this confluence, which has a concordant bed and a large angle between the incoming streams - conditions that generate strong adverse lateral pressure gradients adjacent to the MI - the oscillating SOV cells interact with MI eddies to generate large bed friction velocities in the zone of scour immediately downstream of the confluence.

1. Introduction

[2] Confluences are fundamental components of river networks [Paola, 1997] that play an important role in regulating the movement of sediment through these networks [Rhoads and Kenworthy, 1995; Rice, 1998]. The convergence of flow induced by the configuration, or planform, of the two conjoining channels results in highly three-dimensional patterns of fluid motion and the production of turbulence. FollowingKenworthy and Rhoads [1995], the region of complex flow in the immediate vicinity of the junction is referred to as the confluence hydrodynamic zone. This zone generally extends several channel widths downstream of the upstream junction corner, or confluence apex, where the two channels initially meet. Mixing effects can extend far beyond the confluence hydrodynamic zone [Bouchez et al., 2010].

[3] The structure of flow in the confluence hydrodynamic zone influences patterns of sediment transport, which, in turn, shape the channel bed. This zone typically is characterized by the formation of a large scour hole. Confluence scour constitutes an important environmental and river-management concern (e.g., for the design of bridges and structures anchored on the channel bottom), especially for large rivers. Scour holes can be deep, with depths up to five times greater than the mean depth of the upstream channels, and the orientation of these features can change considerably over time [Best and Ashworth, 1997; Paola, 1997; Rhoads et al., 2009]. The diverse morphologic and flow patterns at confluences also provide a variety of favorable habitat conditions for fish and other aquatic organisms [Rice et al., 2008].

[4] A prominent feature of flow within the confluence hydrodynamic zone (Figure 1) is the formation of a shear layer and associated mixing interface (MI). Field and laboratory experiments have examined in considerable detail the large-scale turbulent eddies within the MI and the relation of these eddies to momentum exchange and mixing between the two incoming streams [Best, 1988; Biron et al., 1993b, 1996; Paola, 1997; Rhoads and Kenworthy, 1999; Bradbrook et al., 2000a, 2000b; Sukhodolov and Rhoads, 2001; Rhoads and Sukhodolov, 2001, 2004, 2008]. The main parameters that determine the position and alignment of the MI are the momentum and velocity ratios between the two incoming streams, the magnitude of the angles between the incoming streams and the downstream channel, and whether or not changes in bathymetry at the entrance to the confluence are gradual (concordant bed) or abrupt (non-concordant bed). Flow structure at natural stream confluences is further complicated by topographic steering effects induced by bar forms on the channel bed, by large scale bed roughness [Hardy et al., 2009, 2010] and by large-scale irregularities of the channel banks [Rhoads and Sukhodolov, 2001].

Figure 1.

Sketch showing the main types of coherent structures present in the confluence hydrodynamic zone for the case the mixing interface (MI) is in (a) wake mode (momentum ratio is close to unity); (b) Kelvin-Helmholtz (KH) mode (high momentum ratio,Mr≫ 1). The coherence and transverse large-scale oscillations (red arrow) of the SOV cells are generally the strongest in the region where the axes of the two incoming channels reach the MI. For the KH mode, the higher momentum stream pushes the MI toward the bank situated on the other side of the main channel, the MI eddies are co-rotating, the recirculation region is relatively small and the coherence of the main SOV cell is much larger of the high-momentum side. The gray region visualizes the stagnation zone.

[5] Based on theoretical consideration on how vortical structures develop in mixing layers between two parallel streams of different mean velocities and how wake vortices develop behind a bluff body, Constantinescu et al. [2011a]proposed classifying confluence MIs into two types: Kelvin-Helmholtz (KH) and wake modes. Although past studies have generally characterized confluence hydrodynamics using the momentum ratio (Mr = [ρ1Q1U1]/[ρ2Q2U2] where ρi, Qi, and Ui are the density, discharge, and bulk velocity in the lateral [i = 1] and main [i = 2] tributaries), the occurrence of the KH and wake modes of MI dynamics is governed by transverse fluid shear, indicating that the velocity ratio (Vr = U1/U2), rather than Mr, is the most appropriate metric for characterizing these modes. It is possible to have different flows with different momentum flux ratios, but similar velocity ratios; however, for the cases analyzed in this paper Vr corresponds directly to the value of Mr.

[6] When Vrdiffers greatly from unity, the transverse shear across the MI and the Kelvin-Helmholtz (KH) instability play dominant roles in the formation and growth of the large-scale eddies [see alsoSukhodolov et al., 2010]. In the KH mode, the MI is populated by co-rotating eddies whose axes are close to vertical relative to the bed (Figure 1b). Similar to the classical case of a mixing layer developing between two streams of unequal velocities, the main mechanism responsible for the increase in the thickness of the MI with the distance from the upstream junction corner is vortex pairing [Winant and Browand, 1974].

[7] When the mean transverse shear across the MI is relatively small (Vr ≈ 1) and a zone of flow stagnation exists at the upstream junction corner, the shear layers along the two sides of the confluence apex can interact in a way similar to the separated shear layers (SSLs) forming on the two sides of a bluff body. These interacting shear layers generate large eddies (rollers) containing vorticity of opposite signs, i.e., the MI is populated by eddies with alternating senses of rotation (Figure 1a). In this wake mode, the eddy structure of the MI is similar to the von Karman vortex street that forms behind cylinders and other bluff bodies [Chen and Jirka, 1995]. Flow conditions and confluence planform geometries that favor the development of a large stagnation zone at the upstream junction corner should strengthen the wake mode and enhance the generation of high circulation rollers. As the MI eddies move away from the upstream junction corner, their size increases but no mechanism exists to increase their circulation. This condition contrasts with MIs in the Kelvin-Helmholtz mode where vortex pairing results in an increase in circulation, enhancing the capacity for MI eddies to erode the channel bed at large distances from the confluence apex. At natural stream confluences both modes can simultaneously influence the development of eddies within the MI, but often one mode will dominate over the other. Flow conditions, confluence geometry, and bathymetry will determine which mode is dominant.

[8] Besides the MI, streamwise-oriented vortical (SOV) cells flanking the MI (Figure 1) are another prominent type of coherent flow structure at concordant confluences, i.e., those with channel beds at nearly the same elevation upstream of the confluence [e.g., Rhoads and Kenworthy, 1995; Paola, 1997]. Such cells, or regions of strong helical motion, are expected to form and considerably affect momentum and mass exchange processes at concordant confluences where the angle between the two incoming streams is large. Under such conditions, the two converging streams have significant transverse momentum with respect to the orientation of the MI [Rhoads and Sukhodolov, 2008]. The convergence of the flows results in an increase in pressure, or the elevation of the water surface, within the center of the confluence. The rapid loss of transverse momentum in the region where the two streams collide leads to the development of strong adverse pressure gradients in the center of the confluence adjacent to the MI that steer or turn the flow into the downstream channel. The raised free surface induces strong downwelling of the fluid along the MI. The descending fluid moves laterally along the bed in a divergent pattern on the two sides of the MI and then rises toward the free surface. The result is the formation of a pair of counter-rotating vortices (primary SOV cells) parallel to the orientation of the MI. Depending on the spatial pattern of the pressure gradients in the transverse direction and bathymetry on the two sides of the MI, numerical simulations suggest that the formation of the primary SOV cells may be accompanied by the formation of secondary SOV cells that are co-rotating with the primary SOV cell on each side of the MI [Constantinescu et al., 2011a]. Evidence for the formation of primary SOV cells is quite strong for confluences with a concordant bed [e.g., Ashmore et al., 1992; Rhoads and Kenworthy, 1995, 1998; Rhoads, 1996; Paola, 1997; Rhoads and Sukhodolov, 2001; Sukhodolov et al., 2010]. Their presence and role at confluences with discordant beds are less clear because bed discordance may disrupt [De Serres et al., 1999, Ribeiro et al., 2012] the coherence of the SOV cells. In such cases, MI vortices distorted by the rapidly changing bed morphology are key drivers of flow dynamics and mixing. Even at confluences with strongly coherent SOVs, MI vortices can play an important role in transport and mixing, especially at confluences where the KH mode is dominant. The coherence of the SOV cells will diminish with the angle between the incoming streams and no SOV cells are expected to form at confluences when the two tributaries are nearly parallel – an outcome confirmed by Kirkil and Constantinescu [2008, 2009a], who applied the same numerical model used in the present work for the limiting case of a MI developing between two parallel streams of unequal velocities.

[9] Laboratory and field studies have led to an improved understanding of confluence dynamics, mixing, and transport. These investigations have provided insight into bed morphology and distributions of mean velocity at cross sections within and downstream of confluences. Some of these studies also provide detailed information on velocity spectra, turbulent stresses, and sediment transport at discrete locations [e.g., Rhoads, 1996; Sukhodolov and Rhoads, 2001; Boyer et al., 2006]. Such studies are, however, limited if the dynamics of the large-scale turbulence and the details of the secondary flow within the entire confluence hydrodynamic zone are of critical interest. The most important limitation is the spatial resolution of the measurements, which typically is not sufficient to completely resolve flow within the SOV cells or turbulent structures generated within the MI. Determining the exact position and extent of these important flow features based on sparse measurements is challenging.

[10] Recent advances in the numerical simulation of complex turbulent flows at large (field) Reynolds numbers [e.g., Spalart, 2009] allow an alternative approach to investigate confluence hydrodynamics [Bradbrook et al., 2000b; Constantinescu et al., 2011a]. In the case of stream confluences, results of eddy-resolving numerical simulations can be used to visualize and examine the unsteady dynamics of the large-scale coherent structures (e.g., SOV cells, MI eddies) and to quantitatively assess the effects of these structures on instantaneous and mean bed shear stresses. Such an approach is an important step toward evaluating the capacity of these structures to entrain sediment and, ultimately, to understanding the main mechanisms driving morphological adjustments at confluences.

[11] This study uses 3-D eddy-resolving simulations for an asymmetrical (y-shaped) confluence with a concordant bed and a large angle between the two incoming streams to examine in detail the flow physics for two different cases in which the MI is primarily in the wake mode and, alternatively, mainly in the Kelvin-Helmholtz mode. Abundant data from past field investigations at this confluence confirm that flow is characterized by the development of SOV cells. The research builds on the modeling work ofConstantinescu et al. [2011a], which examined the effects of the SOV cells on mean flow and turbulence structure for a low momentum ratio test case at this confluence in which the wake mode was dominant. The model predicted that some SOV cells have a large capacity to generate high bed shear stresses, and thus entrain sediment, but did not offer an explanation for this prediction - an issue explored in this paper. In addition, the study presented here investigates numerically the flow physics for a high momentum ratio case in which the Kelvin-Helmholtz mode is dominant. Another novel contribution is an evaluation of the capacity of the MI eddies and SOV cells to transport mass in both cases based on simulations of scalar transport. The simulations address several important sets of questions related to the role of large-scale turbulence in momentum and mass exchange processes in the confluence hydrodynamic zone.

[12] 1) Do strongly coherent SOV cells form along the MI when the Kelvin-Helmholtz mode is dominant? If so, does the structure of the SOV cells differ from the structure of such cells when the wake mode is dominant? If differences in SOV cells do occur, how do these differences influence the capacity of the cells to entrain bed material compared to the transport capacity of the quasi 2-D eddies convected inside the MI? These questions are important because the coherence of MI eddies should be substantially enhanced in cases where the Kelvin-Helmholtz mode is dominant.

[13] 2) What are the spatial and temporal dynamics of SOV cells under conditions of steady mean flow and how are these dynamics related to turbulence amplification, to MI modes, to the flow inside the MI, to exchange of fluid and fine suspended sediment between the MI and adjacent freestream, and to spatial and temporal patterns of bed shear stress?

[14] The numerical method is discussed in section 2. The same section reviews validation of the numerical solver for several types of flows that are relevant to the present investigation of confluence dynamics. Section 3 provides details on the site at which the field experiments were conducted together with the flow conditions in the two simulated test cases. The effect of the momentum ratio on the instantaneous and mean flow structure is outlined in section 4. Section 5describes the predicted dynamics of the SOV cells. The simulated effect of large-scale coherent turbulence structures on the distribution of the depth-averaged turbulent kinetic energy is discussed insection 6. Section 7examines the influence of the velocity (momentum) ratio on the potential capacity of the large-scale coherent structures to entrain sediment from the bed andsection 8 explores the predicted effect of the SOV cells on mass transport based on the advection of a passive scalar injected close to the upstream junction corner. Section 9 summarizes the main findings and section 10 discusses the relevance of the present results in a more general context.

2. Numerical Model and Validation

[15] The simulations in the present study are performed using Detached Eddy Simulation (DES) with the Spalart-Allmaras (SA) Reynolds-Averaged Navier-Stokes (RANS) model as the base model [Spalart, 2000, 2009]. Based on a comparison with well-resolved Large Eddy Simulation (LES), the SA version of DES predicts more accurately mass transport compared to RANS and other versions of DES [Chang et al., 2007]. No special treatment is required to match the solutions at the boundary between the regions where DES is in LES mode and in RANS mode. In particular, the calculation of the bed friction velocity does not rely on the assumption of the presence of a logarithmic layer in the velocity profile, as is the case when wall functions are used. The one-equation SA model solves a transport equation for the modified eddy viscosity, inline image. The boundary condition implementation for rough walls is described in Spalart [2000]. The SA version of DES is obtained by replacing the turbulence length scale d (distance to the nearest wall) in the destruction term of the transport equation for inline image with a new length scale dDES = min(d, CDESΔ), where the model parameter CDES is equal to 0.65 and Δ is a measure of the local grid size. When the production and destruction terms in the transport equation for inline image are balanced, the length scale in the LES regions, dDES = CDESΔ, becomes proportional to the local grid size and yields an eddy viscosity proportional to the mean rate of strain and Δ2, as in LES. This approach allows for an energy cascade down to grid size.

[16] Detailed descriptions of the viscous solver, governing equations, and DES-SA model used to perform the present simulations are given inConstantinescu and Squires [2004], Chang et al. [2007] and Constantinescu et al. [2011a]and are not repeated here. The governing 3D incompressible Navier-Stokes equations formulated in generalized curvilinear coordinates are discretized on a non-staggered grid and then integrated using a fully implicit fractional-step method. The convective terms in the momentum and the advection-diffusion equations used to simulate scalar transport are discretized using the fifth-order accurate upwind biased scheme. All the other terms in the momentum, scalar transport and pressure-Poisson equations are discretized using second-order central differences. Source terms in the turbulence model equations are treated implicitly. The discrete momentum (predictor step), scalar transport and turbulence model equations are integrated in pseudo-time using the alternate direction implicit approximate factorization scheme. Time integration is done using a double time stepping algorithm with local time stepping. The time discretization is second order accurate. The simulations were performed using a parallel version of the code based on Multiple Parallel Interface.

[17] The capacity of present DES code to accurately simulate shallow mixing layers, flow in curved channels with naturally deformed bathymetry, and the unsteady dynamics of large-scale vortices induced by strong adverse pressure gradients is directly relevant for the confluence flow studied in the present work.Constantinescu et al. [2011b] reported a detailed validation study of flow in a sharply curved channel (ratio between mean radius of channel curvature and channel width close to 1.3) with equilibrium scour bathymetry at a channel Reynolds number of 68,400. DES successfully captured the redistribution of the streamwise velocity within the bend and the details of secondary flow revealed by the comprehensive experiments of Blanckaert [2010]. DES predictions were a better match to experimental data compared to both RANS and LES with wall functions.

[18] The same DES model [Kirkil and Constantinescu, 2008, 2009a] successfully captured the experimentally measured streamwise variation of the centerline displacement and width of a shallow mixing layer developing downstream of a splitter plate separating two parallel turbulent streams of unequal velocities in a flat-bed channel. Analysis of the velocity spectra confirmed the quasi-2D character of the large-scale KH billows within the mixing layer observed in experiment. The spanwise variation of the streamwise velocity profiles across the mixing layer was accurately captured by DES. The configuration considered byKirkil and Constantinescu [2008]is a limiting case for a river confluence with concordant bed in which the angles between the two tributaries and the downstream channel are close to zero and the MI is in the Kelvin-Helmholtz mode (Case 2).

[19] Finally, Constantinescu et al. [2011a] reported a detailed validation study for Case 1 based on the measurements collected as part of a field study by Rhoads and Sukhodolov [2001] and Sukhodolov and Rhoads [2001]. Very good agreement with the field data was obtained for the two-dimensional distributions of the streamwise velocity at the three cross sections (A1, A and C inFigure 2a) where measurements were available. Additionally, Figure 3 provides a direct comparison of the predicted vertical profiles of the streamwise velocity with field measurements at three stations for each of the three sections. The model predicted a narrow zone of high turbulent kinetic energy near the center of the confluence similar to that observed in the pattern of field data (see Figure 9 in Constantinescu et al. [2011a]). The exact locations of the predicted and measured zones of high TKE show some differences, but overall the DES predictions are greatly superior to RANS simulations, which fail to predict important features of the measured distributions of TKE and secondary flow (Figure 10 in Constantinescu et al. [2011a]). In particular, RANS cannot reproduce accurately the coherence and circulation of the SOV cells which results in underprediction of TKE within the MI and its vicinity by 1–2 orders of magnitude. Possible reasons for the discrepancy in the position of the zone of high TKE for field measurements and DES simulation include failure of the model to capture completely the influence of bathymetric and bank conditions on the flow field due to the rather coarse resolution of the topographic information on which the computational grid is based and/or failure of the field data to capture fully the effects of low-frequency oscillations of coherent structures in the vicinity of the MI during the measurement interval (≈90 s).

Figure 2.

Bathymetry around the confluence between the Kaskaskia River (KR) and Copper Slough (CS). (a) Case 1; (b) Case 2. The bathymetry is visualized using nondimensional bed elevation values, z′/D. The datum corresponds to the position of the free surface in Case 1. The free surface in Case 2 is situated at z′/D = −0.58. (c) Also shown is the full computational domain used in simulation of Case 1.

Figure 3.

Comparison of predicted mean streamwise velocity (solid line) with the measurements (symbols) for Case 1. The profiles are compared at several stations within cross sections (top) A1, (middle) A and (bottom) C. For a given cross section, the nondimensional transverse distance from the west bank line is indicated at the top of each frame. Each vertical profile obtained from the numerical simulation (solid line) starts at the bed and ends at the free surface (z/D = 0).

3. Test Cases and Simulations Setup

[20] The flow at the confluence of Kaskaskia River (KR, west [W] side of the confluence) and Copper Slough (CS, east [E] side of the confluence) in east central Illinois, U.S.A. is chosen as a representative test case of an asymmetrical confluence with a concordant bed. The cross sections of both tributaries upstream of the confluence are trapezoidal. The upstream channel for the Kaskaskia River is fairly well aligned with the downstream channel. The Copper Slough joins the Kaskaskia River at an angle of about 60°. The inner and the outer banks correspond to the east (E) and the west (W) banks, respectively. The radius of curvature of the flow around the east bank divided by the mean width of the downstream channel is close to 3.5. Results of field research at this confluence are reported by Rhoads [1996], Rhoads and Kenworthy [1995, 1998], Rhoads and Sukhodolov [2001, 2004, 2008], Rhoads et al. [2009], and Sukhodolov and Rhoads [2001]. The present paper describes the results of simulations for two test cases corresponding to confluence bathymetry (Figure 2) and flow conditions in the two incoming streams (Table 1) as documented on May 27, 1998 [Rhoads and Sukhodolov, 2001; Sukhodolov and Rhoads, 2001] and on May 26, 1999 [Rhoads and Sukhodolov, 2008]. Little or no bed material was transported by the flow during these events and no bed forms were present on the bed. Typical of flows at this stage [Kenworthy and Rhoads, 1995], suspended sediment consisted entirely of wash load (silt, clay, very fine sand). Detailed information on the textural characteristics of the bed material is given in Rhoads et al. [2009]. The main difference between the two test cases is the magnitude of Mr, which is about one (1.03) for Case 1 and about 5.4 for Case 2. In the latter test case, Copper Slough has much more momentum than does Kaskaskia River. Thus, the wake mode is dominant in Case 1, while the Kelvin-Helmholtz mode is dominant in Case 2. The physical Reynolds number defined using average values of flow velocity and depth in the downstream channel is 166,000 for Case 1 and 77,000 for Case 2. For both test cases, all quantities are non-dimensionalized using the mean velocity (U) and the mean flow depth in the downstream channel (D) for Case 1, i.e., U = 0.45 m/s and D = 0.36 m.

Table 1. Main Flow Parameters of the Confluence Flow in Case 1 and Case 2
CaseStreamQi (m3/s)Ui (m/s)ρQiUi (kg m/s)Di (m)ReFrMr
  • a

    Ave. row contains variables calculated using the average value of the bulk velocity and mean depth in the channel downstream of the junction between the two incoming streams Kaskaskia River (KR) and Copper Slough (CS).

1Ave.a  0.45  0.36166,0000.241.0
KR1.410.425970.48
CR1.340.466150.32
 
2Ave.a  0.34  0.2377,0000.225.4
KR0.350.19650.28
CR0.750.453370.19

[21] Bathymetry data were collected within the confluence hydrodynamic zone with an average resolution of 2 m in the streamwise direction and 1 m in the transverse direction. This resolution includes all major bathymetric features (bars, scour hole, failed blocks of bank material) within the confluence. The bathymetry transects available within the region of interest from the field experiment were used to define the computational domain. No smoothing was applied on the measured bathymetry profiles except near the banks. As can be seen from Figure 2, the minimum bed elevations of the scour holes in the two test cases measured with respect to the position of the free surface in Case 1 (z′ = 0.0D) were about the same (≅−3.0D), but the volume of the scour hole in Case 2 was smaller than the volume in Case 1. The mean flow depth in Case 2 was about two thirds of that in Case 1. The mean width of the downstream channel was close to 25D in both cases. A submerged block of failed bank material was present in Case 1 along the west bank of the confluence, close to where the Kaskaskia River enters the junction (see bathymetry in Figure 2a around Section A3). In Case 2, the bank line at this same location was much different. Furthermore, between 1998 and 1999 a submerged bar developed close to the east bank between Sections A and C; thus, the flow in Case 2 was much shallower in this region compared to Case 1.

[22] The computational domain (e.g., see Figure 2cfor Case 1) is meshed with close to 5 million cells. The number of grid points in the vertical direction is close to 30. This domain adequately represented individual coherent structures in the flow. A typical computational cell has dimensions of 0.05D–0.1D in the spanwise direction within and close to the MI, 0.1D–0.25 D in the streamwise direction and less than 0.04D in the vertical direction. The wall-normal grid spacing of the first row of cells off the bed and the banks is less than two wall units based on the average value of the bed friction velocity in the incoming channels. Using information from the field studies, the channel bed and banks are treated as rough surfaces with a mean value of the roughness height equal to 0.01 m – a value consistent with the 84th percentile (d84) of bed material in the confluence [Rhoads et al., 2009], a common metric of roughness scale. The 50th percentile (d50) of bed material in the confluence was 4 mm.

[23] For both test cases, inflow conditions corresponding to fully developed turbulent channel flow with resolved turbulent fluctuations are applied. These conditions conform with those in the field experiment where the channel reaches upstream of the confluence are long and straight. The inlet section is situated about two channel widths upstream of the confluence apex in both tributaries (Figure 2c). For each inflow section, the turbulent fluctuations (zero mean velocity) are obtained from a precursor well-resolved LES simulation of fully developed turbulent flow in a straight channel conducted at a physical channel Reynolds number of 20,000. This low Reynolds number was used because LES without wall functions is computationally expensive at the field Reynolds number. The LES fluctuations in the two incoming streams were scaled such that average value of the turbulent intensity away from the wall surfaces matched the value measured in the field experiment (5–7%). The scaled fluctuations were then added to the mean velocity profile obtained from a RANS simulation of fully developed turbulent flow for the Reynolds number and geometrical conditions in the corresponding incoming stream (Table 1). The total velocity fields (RANS mean plus LES fluctuations) were stored in a file and then fed in a time-accurate manner through the two inflow sections. The presence of a minimum amount of resolved turbulence in the incoming flow is critical because of the perturbations this turbulence induces in the shear layers on the two sides of the confluence apex [see alsoChang et al., 2006, 2007].

[24] A convective boundary condition is used at the outflow of the domain. The outflow section is situated at about two channel widths downstream of Section C (Figure 2c) to minimize effects on flow within the confluence hydrodynamic zone. The free surface is modeled as a shear-free rigid lid. Given that the Froude numbers defined with the average depth and velocity in the incoming and downstream channels is less than 0.3 in both test cases (Table 1), this approximation is acceptable. Though the model does not directly simulate variations in water surface elevation (e.g., superelevation within the confluence), the model does account for such effects in an approximate way by predicting non-uniform pressure distributions on the rigid-lid surface.

[25] Transport of mass in the vicinity of the MI is investigated by solving an advection-diffusion equation for a passive (conserved) tracer similar to the approach used byBradbrook et al. [2000b] and Biron and Lane [2008]. A tracer with a normalized concentration CSS = 1 is introduced continuously at all flow depths in two small regions situated about 4D upstream of the junction corner and close to the two banks that form the junction corner. The concentration of the tracer in the two incoming streams is CSS = 0. The flux of tracer is set equal to zero at the channel bottom, free surface, and the banks. The time step is 0.1 D/U. Statistics were collected over an interval of 900D/U after the flow and tracer transport reached a statistically steady state (500D/U) and were checked for convergence.

[26] Estimates of the eddy (vortex) diameter are obtained by first calculating the area of high (positive or negative depending on the sense of rotation of the eddy) along-the-axis vorticity component within a plane perpendicular to the axis of the vortex. The value of this area, along with the assumption that each eddy has nearly circular shape, is then used to calculate the vortex diameter.

4. Effect of Momentum Ratio on Instantaneous and Mean Flow Structure

[27] Values of the momentum and velocity ratios of the two tributaries close to unity as well as the formation of a zone of flow stagnation in the vicinity of the upstream junction corner in Case 1 favor the development of a strong wake mode that allows: 1) the Kelvin Helmholtz instability inside the separated shear layers to grow fast and develop into strongly coherent tube-like structures; and 2) the vortex tubes inside the two separated shear layers to interact and shed large eddies into the MI. Generally this shedding happens when the boundary layers on one or both sides of the confluence apex separate upstream of the junction corner and the size of the stagnation zone is large. The streamlined shape of the junction corner region in Case 2 (Figure 2b) reduces the size of the stagnation zone forming between the two separated shear layers and the confluence apex and constrains the development of large-scale eddies inside the separated shear layer on the Kaskaskia River side, which detaches from the channel bank upstream of the confluence apex (Figure 4b). By contrast, in Case 1 both separated shear layers contain strongly coherent eddies and the size of the stagnation zone is larger than in Case 2 at all flow depths. The larger size of the stagnation zone in Case 1 is primarily due to upstream separation of the boundary layer on the CS side and its movement away from the adjacent bank.

Figure 4.

Distribution of the vertical vorticity, ωzD/U, in the instantaneous flow in a horizontal surface situated 0.1 D below the free surface. (a) Case 1; (b) Case 2. The black dash-dot line visualizes the shear layer forming at the east bank due to the strong curvature of the inner bank and the presence of a submerged bar of deposited sediment in Case 2. The green dashed line visualizes the SSL forming in Case 1 at the west bank due to the presence of a submerged bar of failed bank material near Section A3 (see alsoFigure 2a). The black dashed line follows the centerline of the Copper Slough (CS) stream until it intersects the MI. Figure is redrawn from Miyawaki et al. [2010].

[28] The proximity of the two separated shear layers and the small angle between these separated shear layers explain why the shapes of the billows shed downstream of the stagnation zone in Case 1 are elongated compared to the form of billows observed in the wake of bluff bodies (e.g., circular cylinders). The mean velocity distributions show that, close to the free surface, the differential shear across the MI is around 0.05–0.1 U downstream of Section A1. In other words, a weak Kelvin-Helmholtz mode is present over part of the MI in Case 1. By contrast, the most upstream part of the MI in Case 2 resembles a vorticity sheet (Figure 4b). Under these conditions, the wake mode is negligible and the perturbations of the vorticity sheet by the incoming turbulence on the two sides of the MI are not sufficient to generate large-scale vortices (Kelvin-Helmholtz billows) even though the differential shear across the vorticity sheet is large. Large-scale billows are only first observed around Section A1 for Case 2 (see alsoFigure 19a).

[29] Compared to Case 1 where the MI is located within the central part of the downstream channel, the large momentum ratio in Case 2 and deflection of shallow flow near the east bank (flow depth less than 10% of the mean flow depth in the downstream channel) by a submerged bar shift the MI toward the west bank. The position of the MI in both cases is in good agreement with the location inferred from field data [Rhoads and Sukhodolov, 2001, 2008; Sukhodolov and Rhoads, 2001]. Near Section C, the trajectories of the MI eddies in Case 2 are deflected by the west bank. Around Section C, these eddies attain a diameter of about 10 times the mean flow depth in the downstream channel. Analysis of the temporal evolution of the instantaneous flow fields shows that, on average, two vortex-pairing events occur among neighboring Kelvin-Helmholtz billows between Section A2, where the width of the MI is less than two times the mean flow depth, and Section C. The width of the MI is estimated in an approximate way by determining the width of the region where MI eddies visualized using both the out-of-plane vorticity and a passive scalar introduced at the confluence apex (seesection 8) are convected in the instantaneous flow fields. Although the average diameter of the MI eddies around Section C is considerably smaller in Case 1 than in Case 2, shallow flow effects become significant in Case 1 as the average diameter of the eddies is close to four times the mean flow depth.

[30] The dynamics and coherence of the MI eddies at and downstream of Section C in Case 1 are strongly affected by their interaction with: 1) the large-scale turbulence originating in the region where the incoming flow in Kaskaskia River is advected over a submerged block of failed bank material along the west bank near Section A3 (Figures 2a and 4a); and 2) the narrow shear layer that develops due to an abrupt lateral decrease in streamwise velocity toward the east bank downstream of Section A1 induced by sharp curvature of this bank (Figure 2a). The development of inner bank shear layers is a general characteristic of flow in sharply curved open channels with flat or deformed beds [Constantinescu et al., 2011b]. The core of high momentum fluid parallel to the banks of the Copper Slough upstream of the region of high bank curvature does not change direction fast enough for this core to follow the bank line. A shear layer forms between the core of high momentum fluid and fluid moving slowly in the same direction near the inner bank.

[31] The water level in Case 2 is sufficiently low such that the flow does not submerge the block of failed material around Section A3 (Figures 2b and 4b). This is why large-scale energetic eddies are absent on the Kaskaskia River side of the downstream channel in Case 2. The flow separates as it is deflected laterally by the submerged bar along the inner bank between Sections A and C. A shear layer of positive vorticity forms at about the same location as the shear layer induced by abrupt bank curvature in Case 1. However, its effect on the structure of the flow within the MI is negligible because the MI is displaced toward the west bank in Case 2.

[32] The formation of strong SOV cells at the KRCS confluence is expected due to the large angle between the Copper Slough and the downstream channel. The system of SOV cells predicted in the two cases is depicted in Figure 5 using the Q criterion [Hunt et al., 1988]. The quantity Q is the second invariant of the (resolved in DES) velocity gradient tensor (Q = −0.5 ∂ui/∂xj⋅∂uj/∂xiD2/U2) and represents the balance between the rotation rate and the strain rate. The same value of Q (=10) was used in all analyses to allow relative estimation of the size and coherence of the coherent structures in the two cases. The primary SOV cells on the two sides of the MI are denoted SVE1 (East side) and SVW1 (West side). Their axes follow the boundaries of the MI. The secondary SOV cells are denoted SVE2 and SVW2. The cores of the SOV cells in the mean flow are fairly circular and are regions of high streamwise vorticity (see Figure 6for Section A1 in Case 1). The high-vorticity fluid inside their cores is transported downstream, but the streamwise velocity in the core of the SOV cells is smaller that that of the surrounding flow by about 20–30%. This difference is especially evident in the streamwise velocity distributions for the instantaneous flow fields (Figure 6). The same is true of the streamwise velocity distributions for the mean flow (Figure 6), but the difference between streamwise velocities within and outside of the vorticity cores is smaller because of large-scale temporal oscillations of the SOV cells.

Figure 5.

Visualization of the main vortical structures in the mean flow using a Q iso-surface (Q = 10). (a) Case 1; (b) Case 2. The 3-D ribbons visualize the helical motion of the particles inside the SOV cells. The red segments visualize the approximate position of the MI close to the free surface in Sections A1, A and C. Figure is redrawn fromMiyawaki et al. [2010] and Constantinescu et al. [2011a].

Figure 6.

Flow structure at Section A1 for Case 1 as visualized by: (a) streamwise-oriented vorticity,ωsD/U, mean flow; (b) streamwise velocity, us/U, mean flow; (c) streamwise velocity, us/U, instantaneous flow. A Q isosurface (Q = 10) is used to visualize the cores of the clockwise rotating (dash line) and counter-clockwise rotating (solid line) SOV cells (see alsoFigure 5a). MW denotes the local width of the MI. The dotted rectangle in frame a shows the extent of region visualized in Figure 10. Mean streamwise flow is toward the viewer. Frame a is taken from Constantinescu et al. [2011a].

[33] Important differences are observed in the system of SOV cells in the two cases. In Case 1, the flow in the Copper Slough approaches the MI at a large angle close to the upstream junction corner and requires a certain distance to deflect the MI toward the west bank. Meanwhile, flow in the Kaskaskia River is initially aligned with the downstream channel. The adverse pressure gradients needed to decelerate the transverse component (relative to the MI) of the flow on the Copper Slough side of the MI close to the upstream junction corner are much larger than the requisite gradients on the Kaskaskia River side of the MI. Thus, the circulation of SVE1 is 2–3 times larger than that of SVW1 close to Section A3. The difference between the circulations of SVE1 and SVW1 decreases until Section A1 where the circulations of the two SOV cells are close to equal. This change in relative circulation strength is not surprising because the momentum of the two incoming streams and the angles between the MI and the two incoming streams downstream of Section A2 are about the same. The coherence of the secondary SOV cells on both sides of the MI is considerably smaller than that of the primary SOV cells.

[34] In Case 2, the coherence of the SOV cells is strongly dependent on the velocity and momentum of the two incoming streams. The core of high velocities on the Copper Slough side rapidly loses transverse momentum as it approaches the MI. Strong adverse pressure gradients are generated in the transverse direction relative to the MI and induce the formation of strongly coherent SOV cells on the high-momentum side. The SOV cells on the Kaskaskia River side are quite weak even in the region between Sections A3 and A where the core of large velocities within the Kaskaskia River approaches the MI at a high angle. The ratio between the circulations of SVE1 and SVW1 is close to 3.5 in this region. Downstream of Section A1, the core of high velocity flow on the Kaskaskia River side becomes relatively well aligned with the MI. The low transverse momentum of flow on the Kaskaskia River side of the MI induces a rapid decay of the circulation and coherence of SVW1. Thus, the circulation of SVE1 is about eight times larger than that of SVW1 at Section A. Another important difference with the structure of the SOV system observed in Case 1 is that the circulation of SVE1 is larger than that of SVE2 only until Section A1. Downstream of Section A, the coherence of SVE1 decays faster than that of SVE2. Meanwhile, SVE2 moves into the deepest part of the section.

[35] Thus, the differences in the value of the momentum ratio and, to a certain extent, the bathymetry between the two cases are responsible for the change in the primary mechanism responsible for the formation and growth of the quasi 2-D eddies within the mixing interface, the changes in the position of the interface and the coherence of the SOV cells on the two sides of the interface.

[36] Analysis of the distributions of the turbulent kinetic energy, k, and mean pressure fluctuations, inline image, in the two cases shows that the turbulence is strongly amplified inside the cores of the most coherent SOV cells compared to the levels observed in the surrounding turbulent flow, especially over the region where the two incoming streams collide. In particular, for Case 1 the largest values of k and inline image are observed within and adjacent to the core of SVE1. The turbulence intensity in this region is 4–5 times larger than that predicted inside the MI and at least two times larger than values in the region occupied by the primary SOV cell on the other side of the MI (Figure 7). The levels of turbulence amplification inside the secondary SOV cells (e.g., SVE2) with respect to the background levels are negligible. As discussed in the next section, the reason for the large amplification of the turbulence within and near the core of SVE1 is directly related to the oscillatory motion of SVE1.

Figure 7.

Distribution of the mean pressure fluctuations, inline image, at Sections A1, A and C for Case 1. A Q isosurface (Q = 10) is used to visualize the cores of the clockwise rotating (dash line) and counter-clockwise rotating (solid line) SOV cells (see alsoFigure 5a). MW denotes the local width of the MI. The arrows point toward the location of the pressure peaks. Two distinct peaks are observed in the region where the core of SVE1 is subject to bimodal oscillations at Sections A1 and C. Even at Section A, where only one peak is observed, the region of high inline image induced by the oscillations of SVE1 is very elongated in the spanwise direction. Mean streamwise flow is toward the viewer.

[37] Although a well-defined scour hole exists for both cases, the model did not predict separation of flow from the channel bed as it moves through the zone of scour. This prediction is consistent with the field data, which also did not document flow separation from the bed within the scour hole [Kenworthy and Rhoads, 1995; Rhoads, 1996; Rhoads and Kenworthy, 1998; Rhoads and Sukhodolov, 2001]. The lack of separation is attributable to the small adverse pressure gradients along paths of the flow streamlines, which are nearly parallel to the side slopes of the scour hole. Thus, the rate of change in bed elevation along these flow paths is too small to produce flow separation.

5. Dynamics of the SOV Cells

[38] The simulations show that in the region where the transverse (relative to the MI) pressure gradients are strongest, the cores of the primary SOVs are often subject to large-scale aperiodic bimodal oscillations, similar to the ones observed for necklace vortices in junction flows. To determine the bimodal nature of these oscillations, the approach ofDevenport and Simpson [1990]is used to analyze histograms of the velocity components in the directions (vertical or transverse in the case of SOV cells) that do not coincide with the axis of the vortex inside the region where the core of that vortex oscillates. If the histograms show a double-peaked shape, then the oscillations are bimodal, meaning the core of the vortex oscillates between two extreme modes associated with the two peaks in the velocity histograms. The switch from one mode to the other does not take place at a definite frequency. Analysis here focuses on the structure of the flow around the SOV vortices subject to bimodal oscillations and the mechanisms that are responsible for transition from one mode to the other.

[39] For Case 1, a peculiar characteristic of the region of high amplification of inline image within the region where the core of SVE1 oscillates is that it contains two distinct zones of maximum inline image (e.g., see Sections A1 and C in Figure 7). Even in Section A where the region contains only one maximum, the shape of this zone is fairly elliptical – a result expected when the distance between the centers of two separate circular patches of high inline image becomes smaller than the mean radii of the patches. The reason for this variation of inline imageinside the horseshoe vortex region is that the core of SVE1, rather than randomly moving around its mean flow (time-averaged) position, is subject to large-scale bimodal oscillations toward and away the MI, similar to those observed for the necklace vortices in turbulent junction flows [e.g.,Devenport and Simpson, 1990; Simpson, 2001]. Eddy resolving simulations conducted for bluff bodies mounted on a flat bed [e.g., Koken and Constantinescu, 2009; Kirkil and Constantinescu, 2009b] and on scoured deformed bed [e.g., Kirkil et al., 2009; Koken and Constantinescu, 2008; Kirkil and Constantinescu, 2010] have shown that the twin maxima in the spatial distribution of inline image in the region where the primary necklace vortex oscillates are a strong indication that these oscillations are bimodal. Thus, the validity of a similar explanation for the dynamics of SVE1 has to be explored further through analysis of the velocity histograms.

[40] In a turbulent flow without large-scale coherent vortices, the histograms of the velocity fluctuations in a certain direction contain only one sharp peak around the mean value of the velocity component and their shapes are fairly symmetric. The velocity histogram calculated at a point within the Kaskaskia River upstream of the junction where the open channel flow is fully developed and the mean value of the vertical velocity is close to zero exhibits such a shape (Figure 8). Even when the flow contains large scale eddies that pass quasi-regularly through a certain region (e.g., MI quasi 2-D eddies), the histograms at points situated within this region contain only one peak, but the decay away from the mean value is not as sharp as that for flows without large eddies.

Figure 8.

Probability-density function of the vertical component of the instantaneous velocity,v/U, for Case 1 at a point situated in the Kaskaskia River upstream of the confluence (mid-depth and mid span).

[41] The histograms at points situated inside the core of SVE1 are bimodal at all sections within the confluence hydrodynamic zone (e.g., see Figure 9). The separation between the two peaks associated with the predominant states, or modes, decays past Section A1 in the downstream direction, suggesting that the average amplitude of the bimodal oscillations decays gradually past the region where the high velocity cores of the two incoming streams converge. For Case 1, this region is situated around the location (between Sections A2 and A1) where the centerlines of the two incoming tributary channels intersect the MI. Within this region, the two converging streams lose the largest amount of transverse momentum as flow approaches the MI and rapidly changes its direction from the orientation of the tributary to the orientation of the downstream channel. To produce this change in direction, large adverse pressure gradients are generated in the transverse direction in the vicinity of the two sides of the MI, which results in formation of strong SOV cells and the generation of bimodal oscillations for the most coherent of these vortices. Histograms at points situated inside the core of SVW1 display two closely spaced peaks, which suggest SVW1 is subject to weak bimodal oscillations. This weak bimodality is consistent with the fairly circular shape of the region of high inline image values present around the location of SVW1 in Section A (Figure 7). At all sections, the values of inline image inside the region where SVW1 oscillates are smaller than those inside the corresponding region for SVE1.

Figure 9.

Probability-density functions of the vertical component of the instantaneous velocity,v/U, for Case 1 at a point situated close to the axis of SVE1 in the mean flow. (a) Section A1; (b) Section A; (c) Section C. The peaks associated with the bank mode (BM) and the interface mode (IM) are shown with arrows. The double-peaked histograms of the instantaneous velocity confirm the bimodal nature of the large-scale oscillations of SVE1 for Case 1.

[42] The mechanisms responsible for the formation of the primary and secondary SOV cells and the generation of bimodal oscillations in regions of strong adverse pressure gradients are, in many respects, similar to those responsible for the formation and unsteady dynamics of necklace vortices generated within the horseshoe-vortex system that develops at the base of a surface-mounted bluff body. In both cases the bimodal oscillations of the vortex is the main reason for the large amplification of the turbulence (e.g., up to one order of magnitude for inline imageand Reynolds stresses) compared to the background levels and for the oscillating vortex to induce high bed shear stresses beneath it. The oscillations involve movement of the vortices toward and away from a boundary and are driven by the adverse pressure gradients in the direction perpendicular to that boundary. In the case of junction flows, the relevant boundary is the upstream no-slip surface of a bed-mounted obstacle. This boundary is fixed in time and space and is also independent of the flow. In the case of a river confluence, the boundary is the MI, the position of which is determined by the confluence geometry and flow conditions. The boundary is dynamic and contains large-scale eddies that can interact with the core of SVE1 when the vortex gets close to the MI. In the case of junction flows, the primary and secondary necklace vortices in front of the obstacle follow approximately the junction line between the bed and the obstacle, while the legs of the vortices tend to align with the incoming flow direction. In the case of river confluences, the SOV cells are parallel to the internal flow boundary, but as the MI extends past the region where the two streams converge, the SOV cells remain parallel to the MI over their entire length. While for junction flows the main flow direction is, in most cases, perpendicular to the boundary, or upstream face of the obstacle, at stream confluences the main flow direction is oriented obliquely in relation to the boundary, in this case the MI. Thus, at confluences only the transverse component (with respect to the local orientation of the MI) of the incoming flow contributes to the generation of SOV cells.

[43] The bimodal character of the oscillation of SOV cells can be discerned by examining the structure of the flow at Section A1 at two different times (Figure 10). In the first instance, which is representative of times when SVE1 is in what is referred to here as the interface mode (IM), SVE1 is situated close to the MI and does not exchange vorticity with SVE2. The core of SVE1 is fairly circular and its interaction with the bottom-attached boundary layer induces the formation of a bottom-attached cell called BAW that separates SVE1 from SVE2 and rotates in opposite direction. During the transition to the other preferred state, which is referred to as the bank mode (BM), the circulation of SVE1 increases. The core of SVE1 moves toward BAW and starts extracting vorticity of the same sign from the core of SVE2. At Section A1, where the energy of the bimodal oscillations is the largest, the transition to the bank mode generally results in the complete merging of SVE2 and SVE1. The core of SVE1 increases in size and assumes a more elliptical shape. A strong jet-like flow oriented parallel to the channel bottom develops beneath SVE1. As the intensity of the jet-like flow decreases, SVE1 moves closer to the MI, initiating the transition to the interface mode.

Figure 10.

Velocity vectors and streamwise-oriented vorticity,ωsD/U, in Section A1 (Case 1) at two time instances when: (a) SVE1 is in the interface mode; (b) SVE1 is in the bank mode. The dotted rectangle in Figure 6ashows the extent of region visualized in Figure 10. The shaded patch shows the change in position and shape of the core of SVE1 as it switches between the two modes associated with the bimodal oscillations. The clockwise-rotating bottom attached vortex, BAW, is shown using a dash line. Mean streamwise flow is toward the viewer.

[44] The two modes are somewhat similar to two modes observed in junction flows: the zero-flow mode, in which the main necklace vortex is fairly circular and close to the upstream face of the obstacle and the back flow mode, in which the core of the necklace vortex acquires an elliptical shape and is situated away from the obstacle.Devenport and Simpson [1990]have shown that the transition to the zero-flow mode is generally triggered by the injection of a patch of high-vorticity and low-momentum fluid moving toward the obstacle first into the downflow parallel to the face of the obstacle and from there into the necklace vortex. The transition to the back flow mode generally occurs when a patch of low-vorticity and high-momentum fluid from the upper part of the channel is injected into the downflow.

[45] The injection of patches of high and low vorticity from the two incoming streams moving toward the MI modulates the coherence of the SOV cells in a manner similar to that observed for junction flows. However, in the case of a stream confluence the mechanisms triggering the transition to the two modes are more complex than those for junction flows. Because secondary flow within the MI involves movement of fluid toward the bed and away from the bed toward the two banks, the presence of a strongly coherent SOV cell along the MI results in the injection of near-bed diverging flow into the cell core along with MI eddies advected laterally by this flow. This mechanism appears to be equally, if not more, effective in starting the transition to the bank mode than the mechanism related to injection of eddies from the incoming channel. Moreover, the interactions between quasi-vertical MI eddies and the streamwise oriented SOV cell results in the generation of 3D turbulent eddies, which contribute strongly to the increase of the turbulent kinetic energy. Analysis of the instantaneous flow fields shows that SVE1 spends significantly more time in the interface mode compared to the bank mode. Similar to the case of junction flows, the transition from one mode to the other does not happen quasi periodically. Rather, it happens at unevenly spaced time intervals. The average frequency associated with the switching in Case 1 (0.035Hz) is several times larger than the average frequency corresponding to passage of the MI eddies in Case 1 (0.16Hz and 0.23Hz). It is also is consistent with the frequency of a prominent peak in the transverse velocity spectra (≈0.04Hz) based on velocity time series measurements (300 s time series available at selected locations) obtained within or on the margins of the MI in the field experiment corresponding to Case 1 [Rhoads and Sukhodolov, 2004].

[46] Bimodal oscillations of the core of SVE1 are also observed in Case 2. The centerline of the Copper Slough channel containing high momentum fluid intersects the MI between Sections A1 and A, where the coherence of SVE1 is the largest. Because the MI is in the Kelvin-Helmholtz mode, the largest values of inline image are observed within the MI rather than the region occupied by the most coherent SOV cells, as was the case for Case 1. Still, a clear amplification of inline image is observed around the region where the core of SVE1 oscillates. The regions of amplification of inline image induced by these oscillations between Sections A1 and A contains two peaks (e.g., see Figure 11 for Section A1), which suggests that the large scale oscillations have a bimodal character – a conclusion confirmed by the velocity histograms, which are double peaked between Sections A1 and A (e.g., see Figure 12a). In contrast to Case 1, bimodal oscillations of SVE1 are not observed downstream of Section B. For example, the histogram at Section C in Figure 12b contains only one distinct peak. In terms of the flow structure during the interface mode and the bank mode, the largest differences with Case 1 are observed during the bank mode. Though SVE1 extracts vorticity from the core of SVE2 during the time SVE1 is in the bank mode, no local merging of the two vortices is evident. The lateral displacement of the core of SVE1 as it switches between the interface mode and the bank mode is, on average, smaller than for Case 1. The overall lower energy and amplitude of the bimodal oscillations in Case 2 may also be due to the different structure of the MI. Because the coherence of the Kelvin Helmholtz billows is much greater than that of the eddies advected within a MI in which the wake mode is dominant, the SOV cells on the two sides of the MI are less efficient in extracting turbulent eddies from the MI region – a mechanism that can trigger the transition to bank mode.

Figure 11.

Flow structure in Section A1 for Case 2. (a) streamwise-oriented vorticity,ωsD/U, mean flow; (b) mean pressure fluctuations, inline image. The solid lines show the cores of the SOV cells as visualized by a Q isosurface (Q = 10, see also Figure 5b). MW denotes the local width of the MI. The black arrows point toward the two regions of amplification of the mean pressure fluctuations induced by the bimodal oscillations of SVE1. Mean streamwise flow is toward the viewer.

Figure 12.

Probability-density functions of the vertical component of the instantaneous velocity,v/U, for Case 2 at a point situated close to the axis of SVE1 in the mean flow in a given cross section. (a) Section A; (b) Section C. The peaks associated with the bank mode (BM) and the interface mode (IM) are shown with arrows. The histograms of the instantaneous velocity show that the large-scale oscillations of SVE1 are bimodal at Section A but not at Section C.

6. Effect of Large-Scale Coherent Structures on Turbulence Statistics

[47] Variation in the structure of the MI, the coherence of the SOV cells, and the spatial extent of bimodal oscillations of the SOV cells explain important qualitative and quantitative differences in the patterns of the nondimensional depth-averaged turbulent kinetic energy, inline image for the two test cases (Figure 13). In both cases, the amplification of inline imageis primarily due to the passage of coherent turbulent structures in a certain flow region and oscillations of large-scale coherent structures that are much lower in frequency than velocity fluctuations associated with small-scale turbulence. In Case 1, the convection of strong vortical eddies inside the separated shear layers and their interaction that results in the formation of the large-scale vortices for a MI in wake mode account for the large values of inline image around the upstream junction corner and within the recirculation zone. Outside of this region, large values of inline image occur along the entire length of SVE1 and, in particular, between Sections A2 and A1 where the amplitude of the bimodal oscillations is greatest. The distribution of the TKE in these cross sections (Figure 9 in Constantinescu et al. [2011a]) show the peak values occur in the region that corresponds to the position of SVE1 in the mean flow. However, amplification of inline image in the region situated between the centerline of the MI and SVE1 is due primarily to the passage of the MI eddies and their interactions with the cores of the SOV cells (see discussion in section 5). The levels of inline image within the regions occupied by SVW1 and SVE2 are comparable and are about 30–100% smaller than the magnitude of inline image values inside the region occupied by SVE1. The same is true for the separated shear layer induced by the submerged bar of failed bank material at the west bank over which strong turbulent eddies are convected (see also Figure 14). Past Section B, the region of amplified depth-averaged turbulent kinetic energy ( inline image > 0.01 U2) relative to levels in the incoming streams occupies more than half of the channel section. The cores of the SOV cells are subject to large-scale instabilities that result in the break up of the downstream part of the core into a succession of streamwise-oriented streaks of highly vortical fluid (Figure 14). As they are convected downstream, these streaks are stretched and change orientation due to interactions with the MI eddies and with the 3-D eddies originating in the separated shear layer along the west bank.

Figure 13.

Distribution of the depth-averaged turbulent kinetic energy,104 × inline image/U2. (a) Case 1; (b) Case 2. The dashed lines visualize the boundaries of the MI in the mean flow.

Figure 14.

Visualization of the 3 D vortical structure of the instantaneous flow for Case 1 using a Q isosurface (Q = 10). The two black solid lines indicate the approximate position of the MI. The red arrows point toward vertically oriented quasi-2D MI eddies. The dash-dot lines delimit the shear layer forming at the east bank due to the strong curvature of the inner bank and the separated shear layer at the west bank produced by flow moving over a submerged bar of failed bank material near Section A3 (see alsoFigure 2a).

[48] Comparison of the eddy content of the instantaneous flow in the two cases shows that the coherence of the MI eddies is significantly greater in Case 2 (Figure 15) than in Case 1 (Figure 14), not only close to the upstream junction corner, but also farther downstream due to merging of co-rotating billows. This enhanced coherence is the main reason why in Case 2 the largest values of inline imageare observed toward the inner-bank side of the MI rather than the region occupied by the SOV cells (Figure 13b), a finding consistent with field experiments [Rhoads and Sukhodolov, 2008]. Such a pattern is expected for confluences where the Kelvin-Helmholtz mode is strong. In contrast to Case 1 where large values of inline image were observed over the whole length of SVE1, the levels of inline imageinside SVE1 in Case 2 are larger than those in the surrounding flow only within the region (situated approximately between Sections A2 and B) where the cell is subject to bimodal oscillations. Still, in a given cross-section the values of inline image in the region occupied by SVE1 are less than half those inside the MI. The levels of amplification of inline imageinside SVE2 and SVW1 are negligible. Past Section B, the growth of instabilities propagating along the cores of the main SOV cells and vortex breakdown result in the formation of a region populated by highly 3-D energetic eddies whose orientation is variable in time (e.g., seeFigure 15). These eddies interact and stretch the billows within the MI.

Figure 15.

Visualization of the 3D vortical structure of the instantaneous flow for Case 2 using a Q isosurface (Q = 10). The red arrows point toward vertically oriented quasi-2D eddies within the MI. The two black solid lines indicate the approximate position of the MI. Some of the neighboring MI eddies are in the process of merging together (e.g., see second and fifth red arrow). The blue and green ribbons visualize the paths of particles drawn into the core of SVE2.

7. Implications of Large-Scale Coherent Structures for Bed Friction Velocity and Sediment Entrainment

[49] The flows investigated in this study were not transport effective, at least for bed material. The extant bed morphology was produced by flows much larger than those for which the velocity data were obtained, but which had similar momentum ratios to the measured flows. Extensive field work at this confluence over the years has documented the influence of high momentum and low momentum ratio transport-effective events on bed morphology at the confluence [seeKenworthy and Rhoads, 1995; Rhoads, 1996; Rhoads et al., 2009]. The location of the scour hole in the center of the confluence for Case 1 (Mr ≈ 1) corresponds to morphologic conditions produced by transport effective flows with Mr ≈ 1, whereas the shift in position of the scour hole toward the outer (west) bank and the development of a pronounced bar along the inner (east bank) for Case 2 (Mr > 1) conforms to the pattern of bed morphology produced by events with Mr > 1 [Rhoads et al., 2009] (Figure 2). Thus, the changes morphology at the site between Cases 1 and 2 can be linked to changes in momentum ratio, or equivalently velocity ratio, and the Mr values of the low flows examined here correspond to Mr values of formative flows that produced the extant morphologic conditions. Moreover, field data for transport effective stages, when bed forms are present, indicate that flow structure for such events is characterized by pronounced SOVs on each side of a mixing interface [Rhoads, 1996] – the same pattern documented at low flow stages and predicted by the simulations. As discussed below, simulations for the two test cases provide insight into predicted patterns of bed friction velocity for two different momentum ratios and suggest that mechanisms responsible for sediment entrainment within the confluence may differ for these two cases.

[50] The most striking feature of the nondimensional distributions of the bed friction velocity is that for both cases the largest predicted values of uτ/U occur beneath the SOV cells (Figure 16). While this result is not surprising for Case 1 as the MI is in wake mode and the highest predicted levels of turbulence are generated by SVE1, the results for Case 2 suggest that the SOV cells may be a primary mechanism for entraining sediment within the confluence hydrodynamic zone at high-angle asymmetrical confluences, regardless of the structure of the MI. In Case 2, however, the predicted instantaneous values of uτ/U beneath the strongly coherent billows (Figure 15) are at many times comparable to those beneath the strongly coherent SOV cells, which is not the case for Case 1 [Constantinescu et al., 2011a]. The time of passage of a large Kelvin-Helmholtz billow over a certain location inside the MI is of the order of seconds downstream of Section A1. This duration is much longer than the minimum time interval required for sediment particles with d84 < 1 cm to be entrained in a region where the bed friction velocity is larger than the threshold value for sediment entrainment [e.g., Celik et al., 2010]. Thus, if the Kelvin-Helmholtz mode is strong inside the MI, the quasi 2-D MI eddies may play a major role in sediment entrainment and transport. Comparison of the predicted distributions of uτ/U in Figures 16a and 16b show that the size of the region characterized by high values of uτ/U beneath the MI and thus the capacity of the mixing interface eddies to entrain sediment increases as the Kelvin-Helmholtz mode becomes dominant (e.g., for high values of the momentum ratio).

Figure 16.

Distribution of the magnitude of the bed friction velocity, uτ/U, in the mean flow. (a) Case 1; (b) Case 2. The solid lines visualize the boundaries of the MI in the mean flow. The dash-dotted lines indicate the positions of the axes of the main SOV cells. The approximate position of the scour hole is visualized using two isobath (red dashed) lines. The two isobath lines correspond to levels situated 0.5D and 1D above point where the maximum scour depth is reached in Case 1 and to levels situated 0.35D and 0.7D above the point where the maximum scour depth is reached in Case 2. Figure is redrawn fromMiyawaki et al. [2010].

[51] The predicted pattern of uτ/U varies between the two cases in relation to the strength of the SOV cells. For a momentum ratio close to one, large uτ/U are induced beneath the SOV cells on both sides of the MI, whereas for a high momentum ratio, large uτ/U occur only beneath the SOV cells forming on the high-momentum side of the confluence (Figure 16). The streamwise locations where the strongly coherent SOV cells induce the largest uτ/U values are centered on the region where the high speed cores of the two incoming streams collide (between Sections A2 and A1 for Case 1 and between Sections A1 and A for Case 2) and where, if present, the bimodal oscillations are most energetic. Despite the fact that the mean velocity in the Copper Slough is comparable in the two cases and the circulation of SVE1 is largest in Case 1, the peak values of uτ/U beneath SVE1 are larger by 20–30% in Case 2 because the flow is much shallower and the cores of the SOV cells are situated closer to the bed than in Case 1.

[52] Details of the bed morphology in the two cases account for other regions of high amplification of uτ/U. For example, pronounced acceleration of the flow past the failed bar of bank material near the west bank explains the patch of high uτ/U values upstream of Section A3 in Case 1. The patch of high uτ/U values observed in Case 2 downstream of Section B within the region of greatest confluence scour is induced by high streamwise velocity fluid in the vicinity of the bed (Figure 17). This submerged core of high streamwise velocity fluid likely forms through redistribution of high, near-surface streamwise momentum by the strong SOV cells on the high momentum side of the MI. The overall pattern of bed friction velocity for Case 2 is generally consistent with the pattern of near-bed turbulent kinetic energy, which shifts from a narrow vertical band along the MI upstream to a horizontal pattern extending across the bed downstream as secondary flow advects streamwise momentum downward [Rhoads and Sukhodolov, 2008]. The downstream increase in the width of the zone of high bed friction velocity predicted by the numerical model is consistent with this pattern of the turbulent kinetic energy near the bed, which is related to bed shear stress [Biron et al., 2004].

Figure 17.

Distribution of the mean streamwise velocity, us/U, in Section C for Case 2. Mean streamwise flow is toward the viewer. The core of SVE2 is visualized using a Q isosurface (Q = 10).

[53] The spatial patterns of bed friction velocity for Cases 1 and 2 suggest that, if these patterns are sustained during transport-effective events, scour should extend close to the junction apex (Figure 16), whereas in both cases the zone of maximum scour is located downstream of the apex (Figure 2). Evaluation of the reason for this discrepancy requires velocity measurements during high-stage, channel-shaping events, which are difficult to obtain at the KRCS confluence with ADVs due to dangerous flow conditions and the short duration (a few hours) of such events. Velocity measurements obtained at this confluence with an electromagnetic current meter during a transport-effective event with Mr > 1 indicate that the core of high velocity shifts toward the bed immediately downstream of the confluence as the strength of secondary circulation increases with increasing flow velocities and momentum is advected from the surface toward the bed [Rhoads, 1996]. More pronounced acceleration of flow through the confluence, enhanced helical motion, enlargement of the zone of flow stagnation, and a spatial lag in helical cell development and oscillation may shift the zone of maximum bed friction velocities farther downstream during high flows. Additional modeling and field studies are required to examine this problem in detail.

8. Effect of SOVs on Mass Transport

[54] To examine the capacity of the SOVs to entrain mass from the surrounding regions, including from inside the MI, a numerical experiment was conducted in which a passive tracer (scalar) is continuously introduced within the attached boundary layers developing on the two converging banks of the junction, just upstream of the junction corner (Figures 18 and 19). The main purpose of this numerical experiment is to examine the possible roles of the large-scale MI eddies and SOV cells in the advection of fluid and mass downstream of the confluence apex for MIs in the wake mode and in the KH mode.

Figure 18.

Distribution of the time-averaged concentration of a passive scalar used to visualize how mass transport around the mixing interface in Case 1 is affected by the quasi 2-D eddies and the SOV cells. (a) horizontal surface situated 0.1 D below the free surface; (b) cross-sections A1, A and C. A scalar with a non-dimensional concentration CSS = 1 is continuously introduced inside the attached boundary layers developing on the two converging banks of the junction, just upstream of the junction corner. The incoming flow in the two tributaries has a concentration CSS = 0. The arrows point toward the accumulation of scalar in the shallow region close to the east bank around and downstream of Section C. The cores of the main SOV cells are visualized using a Q isosurface (Q = 10).

Figure 19.

Distribution of the concentration of a passive scalar used to visualize how mass transport around the MI in Case 2 is affected by the quasi 2-D eddies and the SOV cells. (a) instantaneous flow, horizontal surface situated 0.1 D below the free surface; (b) mean flow, horizontal surface situated 0.1D below the free surface; (c) mean flow, cross-sections A3, A1, A, B and C. The dotted lines visualize the boundaries of the mixing interface in the mean flow. The cores of the main SOV cells are visualized using a Q isosurface (Q = 10).

[55] Time-averaged distributions of the nondimensional concentration, CSS, in a horizontal plane situated close to the free surface differ greatly within the MI for the two cases. In Case 2 a substantial amount of the injected tracer is advected within the MI by the quasi 2-D billows (Figure 19a), resulting in high concentrations within the MI (Figures 19b and 19c). In striking contrast, the levels of CSS within the MI downstream of Section A2 are several times smaller in Case 1. These contrasting patterns of concentration are associated with differences in the relative roles of MI eddies and SOV cells in advection of the tracer within the confluence hydrodynamic zone.

[56] The coherence of the MI eddies in Case 1 is relatively low, especially at large distances from the upstream junction corner, while strongly coherent SOV cells are present on both sides of the interface. As the downwelling flow within the MI approaches the bed, it is diverted into the bottom boundary layer on the two sides of the MI. When the cores of the SOVs on both sides of the interface approach the MI and interact with the large-scale eddies within this interface, tracer is advected strongly away from the MI. The capacity of SVE1 to extract tracer from the MI is larger than that of SVE2, mainly because the core of SVE1 is subject to bimodal oscillations (Figure 18). A substantial amount of tracer is extracted from the MI directly into the core of SVE1 during the time when SVE1 is in the interface mode (Figure 10). The other mechanism that results in the entrainment of substantial amounts of tracer into the core of SVE1 occurs during transition to the bank mode when SVE1 strongly interacts with the bottom-attached vortex, which is a locus of highly concentrated tracer, and extracts fluid from the core of this vortex. Conversely, tracer is advected into SVE2 predominantly when SVE1 is in the interface mode. At these times, SVE2 is extracting some of the high-concentration scalar from the bottom-attached vortex (Figure 10).

[57] Given the sense of rotation of the SOV cells on both sides of the MI, the effect of these cells is to transport high-concentration fluid near the bed away from the MI and toward the two banks. This fluid accelerates beneath the cores of the primary and secondary SOV cells. As a result, tracer is advected against the slope of bed, toward the banks (Figure 18b). The convergence of near-bed flow toward the inner bank produces a region of high-concentration tracer in the shallow region downstream of Section B (Figure 18). Although the model uses a passive scalar, the transport processes at confluences where strongly coherent SOV cells are present may be qualitatively similar for fine suspended sediment with small settling velocities that are readily influenced by patterns of secondary flow within the confluence [Kenworthy and Rhoads, 1995]. The numerical simulation results imply that fine suspended sediment will accumulate in the shallow region close to the east bank. This prediction is consistent with field observations of fine-grained bars forming in this part of the confluence when the momentum ratio is much larger than one [Rhoads and Kenworthy, 1995].

[58] The concentration distributions at Sections A1, A and C show that once tracer from the MI is injected inside the cores of the SOV cells in Case 1, it is transported downstream with relatively low diffusion over the streamwise distance that the cells remain strongly coherent. For Case 1, the primary SOV cells on each side of the MI advect a substantial amount of the tracer entrained inside their cores over large distances. The transport mechanisms are expected to be qualitatively similar in the case of suspended sediment, at least for fine sediment. The main difference is that the source of suspended sediment is not limited to the entrainment of particle-laden fluid from the MI, as the SOV cells induce large bed friction velocities (Figure 16) and thus can entrain sediment from the bed. Present results support the validity of the model proposed by Paola [1997]for the growth of the confluence scour hole in which the SOV cells act as a mechanism for advecting high-momentum fluid near the bed, resulting in high friction velocities and entrainment of sediment along the path of the MI.

[59] The mechanisms by which the SOV cells extract tracer from the MI are similar in Case 2. The effect of these cells on the concentration distributions grows with the cell coherence and decays with an increase in the coherence of the MI eddies. For example, the low coherence of SVW1 (Figure 5b) explains why the amount of tracer extracted from the MI along the west side of this interface is negligible downstream of Section A1. By contrast, SVE1 extracts a large amount of tracer from the MI, especially between Sections A1 and A where its core is subject to bimodal oscillations. Still, compared to Case 1, the concentration within the MI is significantly larger because of the strong coherence of the MI eddies, which reduces the efficiency of SVE1 to extract tracer from the MI. Some of the tracer extracted from the MI is injected into the core of SVE1, which explains the elongated streak of high concentration fluid in the vicinity of the MI (Figures 19a and 19b). The remaining tracer is convected toward the east bank within the bottom layer, similar to the mechanism described in Case 1. A relatively small amount of tracer is drawn into the core of SVE2 (e.g., in Sections A1 and A). As already discussed, SVE2 plays an important role in advecting high concentration near-bed fluid toward the east bank through Section B. In Case 2, however, the coherence of SVE1 decreases substantially around Section B (see discussion ofFigure 15). Downstream of this section, the flow shoals over the submerged bar along the inner bank. The strength of the secondary flow toward the east bank diminishes as the core of high speed fluid on the Copper Slough side of the MI extends along the margin of the bar, diminishing the flow curvature compared to Case 1. As a result, inward-moving high concentration fluid near the bed does not penetrate the shallow region over the submerged bar between Sections B and C (Figure 19c). Near Section B, the decaying SVE1 approaches the MI, which is still populated by highly coherent KH billows. As a result, some of the tracer within the core of SVE1 is re-entrained into the MI (Figures 19a and 19b).

[60] Numerical results suggest that regardless of the momentum ratio between the two streams the strongest SOV cells extract significant amounts of mass from the mixing interface and advect it downstream and laterally toward shallow low-velocity regions. Thus, the simulations predict that MI eddies and the SOV cells can both play important roles in mass transport at stream confluences.

9. Summary

[61] This study has used eddy-resolving simulations of two different flow conditions (Case 1 and Case 2) to enhance insight into the flow and turbulent structure at an asymmetric confluence with a concordant bed, a curved downstream junction corner, and a large angle between the two incoming streams. In Case 1 the momentum ratio between the two incoming streams,Mr, is close to one, while in the second case Mr = 5.4. These two cases illustrate changes in the structure and dynamics of the mixing interface (MI) for conditions in which the wake mode is dominant (Case 1) and the Kelvin-Helmholtz (KH) mode is dominant (Case 2). Simulation results show that although the position of the mixing interface with respect to the two banks changes considerably, strongly coherent streamwise-oriented vortical (SOV) cells develop on the two sides of the mixing interface in both cases. The streamwise cells are regions of concentrated streamwise (relative to the local shape of the mixing interface) vorticity, while the streamwise velocity inside the cores of the cells is less than that of the surrounding flow. The simulation results support inferences from past field studies at this research site and from other field and experimental investigations suggesting that strongly coherent cells are a general characteristic of flow at confluences with concordant beds provided that the angle between one or both of the incoming tributaries and the downstream channel is large and confluence geometry and bathymetry are such that strong adverse pressure gradients form away from the MI in the region where the two streams collide.

[62] The mechanisms responsible for the formation of the vertical eddies convected inside the mixing interface differ for the two cases. In Case 1, the quasi 2-D eddies shed from the junction corner region have opposing senses of rotation and their advection is similar to that of wake roller vortices past bluff bodies. Their growth in size is not accompanied by a noticeable increase in the magnitude of their circulation. Thus, the capacity of these eddies to increase bed friction velocity and potentially entrain sediment from the bed at large distances from the upstream junction corner is limited. In Case 2, the formation and dynamics of the quasi 2-D eddies is mainly driven by the Kelvin Helmholtz instability induced by the strong horizontal shear across the mixing interface. The increase in the mean size of the co-rotating quasi 2-D eddies as a result of vortex pairing is accompanied by a significant increase in their circulation, which can enhance bed friction velocities beneath MI eddies.

[63] Results show that when the momentum of the lateral tributary is much larger than the tributary aligned with the downstream channel, the coherence and streamwise extent of the primary and secondary cells on the lateral-tributary side of the confluence are significantly larger than the coherence and circulation of cells on the opposite side of the interface. This difference in cell development, in turn, affects the spatial distribution of shear velocities within the confluence, which has potential implications for sediment entrainment and patterns of erosion and deposition.

[64] According to model predictions, the core of the primary cell on the side of the lateral tributary undergoes large-scale bimodal oscillations toward (interface mode) and away (bank mode) from the mixing interface. These oscillations may explain why transverse velocity fluctuations in the field experiment with a momentum ratio close to unity persist well downstream of the upstream junction corner [Rhoads and Sukhodolov, 2004]. The bimodal oscillations generate strong local amplification of the turbulent kinetic energy and pressure fluctuations. Thus, the MI eddies, the SOV cells, and interactions between these two types of coherent flow structures, which reach peak intensity when the core of one of the SOV cells shifts closest to the centerline of the MI, are primary mechanisms for amplification of turbulence within center of the confluence and immediately downstream. This amplification of turbulence has strong effects on bed friction velocities. In both cases, the largest bed friction velocities occur beneath the region where the core of the cell is subject to strong bimodal oscillations. While in the case with a momentum ratio Mr ≅ 1, the bimodal oscillations are strongest and extend far from the upstream junction corner (e.g., they persist at a distance more than two times the mean width of the downstream channel), in the case with Mr = 5.4, the bimodal oscillations are limited to the central part of the confluence hydrodynamic zone.

10. Final Discussion

[65] Several investigations of mixing interface dynamics at confluences, including studies at the confluence investigated in the present study, have noted unsteadiness, or “flapping,” of the mixing interface position over time, which manifests itself as low-frequency peaks in power spectra of velocity data [Biron et al., 1993a; Rhoads and Sukhodolov, 2004, 2008; Bradbrook et al., 2000b; Keylock et al., 2005]. Variations in the path of the MI eddies extend over a certain streamwise distance and occur even if upstreamflow conditions are steady. The cause of these variations has yet to be adequately explained. Rhoads and Sukhodolov [2008]found a connection between the periodicity of dominant oscillations in the stagnation zone and those of the interface position, suggesting that the dynamics of these two hydrodynamic features may be connected, perhaps through backwater effects of the converging flows. The simulations in the current study suggest that flapping motion of the mixing interface may be related to bimodal oscillations of the streamwise-oriented vortical cells. As a cell subject to bimodal oscillations transitions to the interface mode, enhanced exchange of momentum with the mixing-interface eddies occurs. Below the free surface, some of these eddies are displaced laterally toward the cell over the streamwise distance corresponding to enhanced momentum exchange. As that cell transitions to the bank mode, interaction between the oscillating cell and the mixing-interface eddies becomes negligible. Consequently, the trajectories of these eddies generally remain close to the (mean-flow) centerline of the mixing interface. This change in the trajectories of the mixing-interface eddies induced by their interaction with cells below the free surface is manifested as a flapping motion in the free-surface dynamics of these eddies. Consistent with the change of the cell from one mode to the other, the frequency of flapping is much less than the frequency of passage of the mixing-interface eddies.

[66] Simulations of vortical flow structure, bed friction velocity patterns, and transport of a passive tracer suggest that the streamwise-oriented vortical cells contribute substantially to high values of bed friction velocity within the zone of scour at the confluence analyzed in the present study. Downwelling fluid on the cell margins advects momentum toward the bed, inducing large bed friction velocities. These cells also extract mass and momentum from the mixing interface. The capacity of the cells to extract fluid from the mixing interface is a strong function of the momentum ratio, which determines what mode is dominant inside the mixing interface. This capacity decreases as the coherence of the mixing-interface eddies increases (Kelvin-Helmholtz mode is dominant). Still, even in Case 2 (Mr = 5.4) the primary cell on the high momentum side of the interface extracts more than 50% of the mass rate of passive tracer introduced at the upstream junction corner. In both cases, strong secondary flow over the inner part of the downstream channel advects this tracer laterally toward the inner bank. Assuming that the dynamics of the passive tracer does not differ greatly from that of fine suspended sediment, the lateral movement of tracer injected into the mixing interface can partially explain the formation in Case 2 of fine-grained depositional bars along the inner bank of the downstream channel – a phenomenon documented in the field under Case 2 conditions.

[67] The development of a zone of scour is a prominent feature at confluences with concordant beds. Scour at the confluence investigated in the present study occurs under virtually all conditions with the position of scour shifting dynamically in relation to variations in momentum ratio of the incoming flows [Rhoads et al., 2009]. Ultimately the development of scour is caused by net removal of sediment and thus is a product of the spatial pattern of sediment transport. The cause of scour at confluences has been attributed to several factors, including high levels of turbulence along the shear layer/mixing interface, acceleration of flow through the confluence, and the development of strong helical motion that sweeps sediment laterally away from the center of the confluence toward the channel banks. All of these factors should contribute to high bed shear stresses and enhanced rates of sediment mobilization. Unraveling the individual effects of these factors is difficult, but the results presented here suggest that at confluences with concordant bed the helical motion associated with the streamwise oriented vortical cells can in some cases enhance local bed friction to a greater extent than the passage of the mixing interface eddies. Though not considered in the present study, bed load transport and active bed forms may complicate flow structure near the channel bed. However, even under such conditions, large-scale structure of the flow is still dominated by the development of streamwise helical cells on each side of a well-defined mixing interface and patterns of sediment transport are strongly influenced by these cells [Rhoads, 1996]. A direction for future study will be to investigate mean flow and turbulence characteristics at confluences with discordant beds, where the flow physics is even more complex than at concordant confluences (e.g., flow can separate within the scour hole) and SOV cells are less likely to develop.

[68] The predictions of this numerical experiment are suggestive of possible real-world processes, including interactions between streamwise oriented helical cells and Kelvin-Helmholtz vortices as well as the potential influence of such interactions on transport and mixing of fine sediments, on bed friction velocities, on bed-material transport, and on patterns of scour and deposition. Although the DES model predicts reasonably accurately measured patterns of mean flow and time-averaged turbulence, particularly compared to RANS predictions [Constantinescu et al., 2011a], some discrepancies exist and have yet to be completely resolved. Moreover, at present time the capacity to predict details of the flow, such as unsteady components related to turbulence and oscillation of vortices, exceeds the availability of field data at appropriate spatial and temporal resolutions needed to rigorously evaluate these predictions. Thus, many of the predictions here require further evaluation based on field measurements that capture the oscillatory and unsteady dynamics of coherent flow structures.

Acknowledgments

[69] We gratefully acknowledge the National Center for High Performance Computing (NCHC) in Taiwan and the Transportation Research and Analysis Computing Center (TRACC) at the Argonne National Laboratory for providing substantial amounts of computer time.