Hydrodynamic processes and sediment erosion mechanisms in an open channel bend of strong curvature with deformed bathymetry

Authors


Abstract

[1] Most rivers exhibit regions of strong channel curvature that are characterized by more complex and variable flow and erosion patterns, compared to regions of lower curvature. Studies investigating high-curvature bends using eddy-resolving techniques have been limited, and the effect of bend angle on flow and erosion has rarely been investigated. This study investigates flow in a 135° nonerodible bank open channel bend of high curvature: ratio of radius of curvature, R, to channel width, B, is 1.5. The bathymetry is obtained during the final stages of a clear water scour experiment. Large Eddy Simulation is used to investigate the effect of secondary flow on the redistribution of streamwise momentum, the details of coherent structures, and mechanisms leading to erosion within the bend. Results are compared with those from a similar numerical study of a 193° sharply curved open channel bend with R/B = 1.35. The angle of the 135° bend is representative of typical regular meander geometry, while the larger angle of the 193° bend is representative of a tortuous meander geometry. The different bathymetries induced important quantitative and qualitative differences in the vortical and turbulence structure within the open channel for the two cases. Inner bank streamwise-oriented vortical (SOV) cells formed in both cases, but the position and extent of shear layers forming between regions of fast and slow moving fluid differed, and flow did not separate at the inner bank in the 135° bend. An outer-bank cell was observed in the 135° bend, but not in the 193° bend. Distributions of predicted boundary shear stresses indicated the capacity of the flow to erode the outer bank of a sharply curved bend under two representative regimes found in the field.

1 Introduction

[2] Straight rivers are relatively rare and tend to occur only in areas with very large slopes [Church, 1992; Rosgen, 1994]. The majority of meandering rivers contain regions in which channel curvature effects on flow, turbulence, and sediment transport are important. In curved river reaches, a main channel cell of secondary circulation develops which typically causes fluid closer to the surface to move towards the outer bank, and fluid near the bed to move towards the inner bank. This main channel secondary circulation has classically been explained as being mainly induced by centrifugal forces, which are directed towards the outer bank, and a transverse pressure gradient, which is directed towards the inner bank [Rozovskii, 1957]. The mechanisms, responsible for secondary circulation, however, are actually quite complex, particularly in river reaches which are considered to have strong curvature (as will be defined later) [Blanckaert and de Vriend, 2004].

[3] In meandering rivers, pools tend to form along the outer bank, while point bars form along the inner bank. The evolution of the bathymetry contributes directly to the redistribution of velocity. For example, as sediment deposits along the inner bank, flow starts moving away from the inner bank, toward deeper regions within the channel. In other words, the flow becomes “topographically steered” toward the channel thalweg [Dietrich and Smith, 1983]. Recently, Blanckaert [2010] showed that topographic steering is not dominated by advective momentum transport [Whiting and Dietrich, 1993], but mainly by the topography according to Chezy's law. Additional complications may occur due to interactions between large-scale moving bed forms (e.g., dunes) and secondary flow in bends with a movable bed and continuous sediment supply [Abad, 2008]. Topographic steering by progression of bed forms was discussed by Abad and García [2009b].

[4] Bend geometry is also important to consider, as it may substantially affect the structure of the flow, velocity distributions, erosion, and river migration. In a first approximation, the strength of the secondary flow (e.g., represented by the circulation of the main cell), which directly affects the rate of advection of high streamwise velocities toward the outer bank, appears to vary monotonically with the ratio of the local curvature radius, R, to the channel width, B, or with R/H, where H is the flow depth [Kashyap et al., 2012]. Though several nondimensional parameters may be used to classify curved channels, the ratio R/B is generally used to classify bends as being of low (R/B > 8), medium (8 > R/B > 2.5), or high (R/B < 2.5) curvature. The threshold value of 2.5 is in fact the middle of the interval (2 < R/B < 3) where transition between medium and high-curvature bends is generally observed. As discussed by Blanckaert and de Vriend [2010] and Blanckaert [2011], the scatter in the threshold value is largely due to the dependence on other factors, such as bend length, bed roughness, and flow depth. As R/B decreases, the degree of nonlinearity of the interactions between the secondary cross-streamflow and the streamwise momentum increases, which increases the anisotropy of the cross-stream turbulence and modifies sediment erosion and deposition patterns in loose bed channels [Blanckaert and de Vriend, 2003, 2004, 2010; Ottevanger et al., 2012].

[5] Many field and laboratory investigations of flow in channels of medium and high curvature have reported the presence of a weaker cell of cross-stream circulation close to the corner between the outer bank and the free surface [e.g., Rozovskii, 1957; Bathurst et al., 1977; Thorne and Hey, 1979; Blanckaert and de Vriend, 2004; Blanckaert, 2011; Blanckaert et al., 2012]. While there is agreement that the formation of this cell is driven by the anisotropy of the cross-stream turbulence, there is some debate as to whether this cell protects the outer bank from erosion by keeping the core of high streamwise velocity away from the bank [e.g., Blanckaert and Graf, 2001, 2004] or if it endangers bank stability by advecting high momentum fluid toward the base of the outer bank especially in regions where a large pool has already formed [Bathurst et al., 1979].

[6] Another example of large-scale flow structures induced by strong nonlinear interactions is the streamwise-oriented vortical (SOV) cells forming close to the inner bank of high-curvature bends [e.g., Constantinescu et al., 2011a]. Moreover, strong shear layers can form in between the core of high streamwise velocities and the region of slower flow moving downstream close to the inner bank. In some cases, the flow can also separate close to the inner bank of sharply curved bends [e.g., Leeder and Bridges, 1975; Ferguson et al., 2003; Frothingham and Rhoads, 2003; Blanckaert, 2010, 2011] and in the lee of submerged point bars [Frothingham and Rhoads, 2003]. Flow separation results in the formation of strongly energetic shear layers [Constantinescu et al., 2011a]. Constantinescu et al. [2011a] noted that these shear layers and their associated SOVs can impinge on the outer bank and thereby influence outer bank erosion, but the generality of this erosion mechanism remains unknown.

[7] There are relatively few in-depth studies that have investigated the effects of bend angle on erosion and/or migration over a wide range of bend angles [Kondratiev et al., 1982; Abad, 2008; Yeh et al., 2009]. An analysis conducted by Kondratiev et al. [1982], as reported by Yalin and da Silva [2001] on rivers from Europe and the United States showed that river migration rate varies with bend angle. The study showed that the downstream migration rate of a meander peaks at about a bend angle of 40° and the outward migration (expansion in the radial direction) rate peaks at about a bend angle of 110°. However, in the field, sinuosity, which is the ratio of curved channel length to the straight-line valley length [Knighton, 1998], is more commonly reported compared to bend angle. In this study, in order to determine a common bend angle for a natural river, a data set of 42 meandering rivers (minimum sinuosity > 1.2, average sinuosity = 1.6) from Leopold and Wolman [1960] were analyzed. The geometry of these natural meanders was approximated by fitting the meanders to sine-generated curves [Langbein and Leopold, 1966; Yalin and da Silva, 2001]. The sinuosity values given in Leopold and Wolman [1960] were used to calculate the Bessel value, and the initial deflection angle (θ°) was extracted from Figure 5.3 in Yalin and da Silva [2001], which in turn was used to determine the bend angle. This revealed an average bend angle of 134° for the data set, which we consider to represent a common bend angle.

[8] The present study is the first to use Large Eddy Simulation (LES) without wall functions to investigate flow with natural-like bathymetry. It simulated flow in a 135° bend during the final stages of clear water scour, and compares results to those obtained in the numerical study of Constantinescu et al. [2011a] conducted in a 193° high-curvature bend during the final stages of live-bed scour in order to get beyond single case studies and to draw some more general results from a comparative perspective. Based on the previous discussion, the 135° bend is representative of typical regular meander geometry (see classification in Table 1 of Schumm [1963]), whereas the 193° bend is representative of a tortuous meander geometry. It should be noted that although each case represents different final scour conditions (i.e., clear water versus live-bed), previous experiments have suggested that equilibrium topography for clear water and live-bed scour are similar, and showed the same dominant bed features [Roca et al., 2007; Fazli et al., 2009]. We expect the main difference between the two scour conditions to be the presence of small dunes for the live-bed condition.

[9] The main goals of the present paper are to investigate flow processes that lead to bed and bank erosion processes and to show how these processes are affected by large-scale turbulence based on high-resolution LES without wall functions. The main research questions we try to answer in the present study are the following:

  1. [10] Are strongly coherent SOV cells at the inner bank and the associated shear layer a general characteristic of flow in sharply curved bends and do the coherent eddies shed in the shear layer always penetrate up to the outer bank of sharply curved bends?

  2. [11] Can the outer bank cell endanger bank stability by advecting high momentum fluid towards the bank?

  3. [12] Can erosion at the inner bank be a concern for sharply curved bends?

  4. [13] Do the energetically important curvature-induced coherent structures present near the bed and the banks of sharply curved bends contribute significantly to the increase of the boundary stress in regions where the erosion potential is high?

[14] In the present paper, we will try to answer these questions based on results of simulations performed with bathymetry reflecting the final stages of scour. The present findings will strictly apply to the later stages of the scour and erosion processes in the alluvial channel bend modeled in the laboratory experiment. However, most of the findings are expected to apply also to meandering natural channels with a developed pool-point bar structure containing high-curvature reaches.

[15] It is also important to mention in regard to the third research question that the two cases help to understand how flow separation at the inner bank (which occurs for the 193° bend but not the 135° bend) and the presence of a highly coherent outer bank cell (which is present for the 135° bend but not the 193° bend) may affect erosion. Not much is known about the effect of flow separation at the inner bank on sediment transport despite its importance for the development of the point bar and the river planform evolution [Ferguson et al., 2003; Blanckaert, 2010]. In regard to the fourth research question, one should also point out that information on the boundary shear stress distribution is very valuable in calibrating models for outer-bank migration [Ikeda et al., 1981; Odgaard, 1989; Blanckaert and de Vriend, 2010]. Such information is readily available from 3-D LES.

2 Role of Eddy Resolving Techniques to Investigate Flow in Open Channels

[16] Much existing knowledge of flow and turbulence structure in open channel bends comes from laboratory [e.g., Blanckaert and Graf, 2001; Blanckaert and de Vriend, 2004; Blanckaert and de Vriend, 2005a, 2005b; Abad, 2008; Abad and García, 2009a; Blanckaert, 2010; Jamieson et al., 2010] and field studies [e.g., Bathurst et al., 1977, 1979; Thorne et al., 1985; Ferguson et al., 2003; Frothingham and Rhoads, 2003; Sukhodolov, 2012]. Though field investigations are not subject to scale effects, they have some limitations. The most important one is the spatial resolution at which the measurements are performed. Though subject to scale effects, laboratory experimental studies conducted under controlled conditions allow conducting more detailed measurements of the main flow variables. Most of these studies, however, have concentrated on detailed measurements of the mean flow and turbulence statistics only in one or two cross sections. In this regard, the detailed measurements of the flow in a 193° sharply curved bend (R/B = 1.3) reported by Blanckaert [2009, 2010] and Blanckaert et al. [2012] constitute a unique set of data that can be used to understand the physics of high-curvature open channel bend flow. However, even the high-resolution data collected in this laboratory experiment did not allow a quantitative characterization of some of the important flow structures forming in the immediate vicinity of the inner and outer walls and did not allow for accurate estimation of the distributions of the pressure fluctuations, friction velocity, and its standard deviation.

[17] The 3-D Reynolds Averaged Navier-Stokes (RANS) simulations were shown to be fairly successful in capturing the redistribution of the streamwise momentum in bends [e.g., Kashyap et al., 2009, 2012]. Despite some success in predicting the extent of the pool region and maximum scour depth in curved alluvial channels of medium and high curvature [e.g., Rüther and Olsen, 2005; Khosronejad et al., 2007; Zeng et al., 2008a, 2008b, 2010], 3-D RANS models with sediment transport and movable bed capabilities have failed to predict all the relevant details of the bathymetry at equilibrium conditions. One of the main reasons is the inaccurate description of the mean flow and turbulence structure, which points towards the need to use more advanced turbulence models. By contrast, numerical simulations using LES [Keylock et al., 2005] or hybrid RANS-LES approaches [Spalart, 2009] conducted on sufficiently fine meshes were shown to accurately predict mean flow and turbulence statistics in open channels bends of medium and large curvature and river reaches with realistic bathymetry [e.g., van Balen et al., 2010b; Constantinescu et al., 2011a, 2011b, 2012; Keylock et al., 2012]. Equally important, such numerical simulations were able to capture the dynamics of the energetically important large-scale eddies in the flow at both laboratory and field scales, and to clarify their role in entraining sediment in the case of a loose bed.

[18] Directly relevant for the present study, van Balen et al. [2010a, 2010b] used LES with wall functions and the classical Smagorinsky subgrid scale model to predict the mean flow and turbulence in the 193° high-curvature (R/B = 1.3) bend studied experimentally by Blanckaert [2009, 2010] at conditions corresponding to the start (flat bed) and end (equilibrium bathymetry) of the erosion and deposition process. The same test case with equilibrium bathymetry was simulated using a hybrid RANS-LES method called Detached Eddy Simulation by Constantinescu et al. [2011a].

[19] The present LES simulation of flow in a 135° bend with vertical smooth sidewalls does not explicitly consider sediment transport and evolution of the bathymetry. As the laboratory experiment was conducted under clear water conditions (negligible amount of suspended sediment in the flow) and the movable bed forms observed at close to equilibrium conditions were not significant (small-scale ripples), the conditions in LES were sufficiently close to the laboratory experiment such that the flow and turbulence structure revealed by LES were thought to be representative of those in the experiment.

3 Physical Experiment in the 135° Bend

[20] The experiment for the 135° bend case was conducted at the University of Ottawa. The 135° bend flume had a 12.2 m straight entrance section, a 3.6 m curved section, and a 2.4 m straight exit section. The channel sidewalls were smooth and vertical, and R = 1.5 m, and B = 1 m (see Post [2007] for further flume details). The bottom of the flume was filled to a depth of approximately 30 cm with a sediment having a mean particle size d50 = 0.689 mm.

[21] The experiment was run under steady clear water scour conditions with a discharge of 0.0464 m3/s until close to equilibrium scour was reached. This was considered to occur when scour along the inner and outer flume walls appeared negligible (< 1 mm/day), which in this experiment occurred after 105 h. The experiment, however, was run for a total of 167 h after which time small amounts of scour could still be seen along the deepest area of the thalweg, although the rate appeared low. No sediment was supplied through the inlet.

[22] Discharges were measured using a V-notch weir installed in the flume exit tank and were validated by also calculating discharge from the acoustic Doppler velocimetry measurements in the straight inlet section. The initial bed slope was zero, and the initial water depth (H) in the upstream part of the straight inlet section of the flume was 0.15 m. The average bed friction velocity in the incoming flow within the inlet straight reach was 0.016 m/s, which was smaller than the critical friction velocity for sediment entrainment on a flat bed given by Shields' diagram (0.0197 m/s), and the experimental conditions were considered to be close to hydraulically smooth. Bathymetry measurements were taken with a Leica DistoTM pro4a laser altimeter which had an accuracy of ±0.0015 m.

4 Numerical Model and Simulation Setup

[23] A collocated finite-volume scheme is used to solve the filtered 3-D Navier-Stokes equations. We refer to the papers by Mahesh et al. [2004], Mahesh et al. [2006], and McCoy et al. [2007] for a detailed description of the numerical method and the subgrid scale Smagorinsky model. The collocated fractional-step scheme is nondissipative yet robust at high Reynolds numbers on highly skewed meshes without the use of numerical dissipation. In the predictor-corrector formulation, the Cartesian velocity components defined at the center of the cell and the face-normal velocities defined at the center of the face are essentially treated as independent variables. The fractional step algorithm is second-order accurate in both space and time. All the operators, including the convective terms, are discretized using central schemes. Time discretization is achieved using a Crank-Nicholson scheme for the convective and viscous operators in the momentum (predictor step) equations. After discretization in time, the governing equations are solved using the Successive Over-Relaxation method. No wall functions are used, and the governing equations are integrated through the viscous sublayer. The code has the capability to use unstructured hybrid meshes which allows clustering of the cells in regions where the dynamics of the flow requires a fine mesh while maintaining a high mesh quality throughout the domain.

[24] The LES code was successfully validated for various complex flows of interest in hydraulics and river engineering where large-scale coherent structures play an important role in explaining momentum and mass transport [e.g., Tokyay and Constantinescu, 2006; Kirkil et al., 2008; Koken and Constantinescu, 2008a, 2008b; McCoy et al., 2008; Constantinescu et al., 2009; Keylock et al., 2012]. Several of the aforementioned studies considered flow over a naturally scoured bed. Kashyap [2012] also found good agreement between LES predictions and experimental measurements for a high-curvature bend study, particularly in its predictions of SOVs.

[25] The numerical simulation was modeled after a flume experiment with a mobile bed conducted at the University of Ottawa using the main experimental parameters given in Table 1. A sketch of the flume and bathymetry used in the numerical model are shown in Figure 1. The computational domain contained a 135° curved high-curvature reach (R/B = 1.5) connecting two straight inlet and outlet reaches. The mean inlet channel water depth, H, in the experiment was 0.15 m and was used as the length scale for nondimensionalizing the flow and geometrical variables in the simulation. To save computational time, the inlet in the simulation was shortened to 14.5H (see discussion below for the inlet boundary condition), and to minimize the effect of the outflow boundary on the flow, the outlet reach was made longer (27H).

Table 1. Flow Parameters for the 135° and 193° Bend Flume Experiments
CaseUHu0*u*crd50Re*ReFr
(m/s)(m)(m/s)(m/s)(mm)(u*d50/v)(UH/v)U/(gH)1/2
  1. U is the mean inlet streamwise velocity, H is the water depth at the upstream boundary of the flume, u0* is the mean straight channel bed friction velocity, u*cr is the Shields critical bed friction velocity for the d50 size, Re* is the particle Reynolds number, Re is the straight channel Reynolds number, and Fr is the Froude number.
135°0.3100.1500.01520.01970.68913.860,0000.255
193°0.6100.115--2.000-83,0120.735
Figure 1.

Sketch of flume in which the 135° bend experiment with a mobile bed was conducted. The dashed line shows the flume centerline. The right frame shows the bathymetry at close to equilibrium scour conditions. The bed elevation (z/H) is measured with respect to the mean position of the free surface (z/H = 0) in the inlet section (far upstream boundary).

[26] Figure 1 also shows the locations of several cross sections within the computational domain. The cross sections within the curved reach are denoted by D followed by the value of the polar angle (0° < θ < 135°). Cross sections in the straight outlet reach are denoted D135 + αH, where α is the nondimensional centerline distance (H is used for nondimensionalization) between the end of the curved reach (section D135) and the specified cross section. The mean velocity, U = 0.31 m/s, of the incoming flow at the inlet is used as the velocity scale. The channel Reynolds and Froude numbers were Re = UH/ν = 60,000 and Fr = (U/gH)1/2 = 0.26, where g is the gravitational acceleration and ν is the kinematic viscosity.

[27] Inflow conditions corresponding to fully developed turbulent channel flow containing resolved turbulence were applied at the channel entrance. The mean streamwise velocity distribution was obtained from a precursor Reynolds-Stress-Model RANS simulation at Re = 60,000 of fully developed turbulent flow in a straight channel. A second LES precursor simulation was conducted to obtain the (zero-mean) velocity fluctuations corresponding to a fully developed turbulent channel flow. The total (mean plus fluctuations) velocity fields were then fed through the inflow section of the computational domain containing the curved reach. This method to specify inflow conditions is fairly similar to the conditions present in the experiment where the incoming flow in the inlet straight reach was fully turbulent and contained turbulent eddies. The same method was successfully used to model a variety of flows in open channels using eddy-resolving techniques [e.g., Chang et al., 2006; McCoy et al., 2008; Constantinescu et al., 2011a, 2011b]. The constant flow discharge was 6.67UH2. The free surface was modeled using a symmetry boundary condition (rigid lid approximation). This is acceptable (see also discussion in Kirkil and Constantinescu [2010], van Balen et al. [2010a, 2010b], Constantinescu et al. [2011a]) because of the low value (<0.3) of the Froude number. A convective boundary condition was used at the outflow. All the solid boundaries of the channel were treated as no slip surfaces.

[28] The unstructured mesh contained only hexahedral cells. The mesh consisted of a total of 3.4 million grid points. The grid cell closest to the bed and sidewalls had a nondimensional size Δn1/H = 0.001 in the wall normal direction, where Δn1 is the dimensional grid spacing. This corresponds to less than three nondimensional wall units (y+ = 3), where y+ = (Re)(u*/U)(Δn1/H). In this equation, Re is the channel Reynolds number, and u* is the main channel bed shear velocity. The time step in the simulation was 0.01H/U. Statistics were collected after the flow reached a statistically steady state, a process that took about 200H/U.

5 Validation of the Numerical Model

[29] The LES code was also used to simulate one of the test cases (H89) studied experimentally by Blanckaert [2009] and Blanckaert et al. [2012]. The test case corresponded to the flow in a 193° open channel bend with flat bed for which R/B = 1.3, B/H = 8.2, Re = 68,400, and Fr = 0.34 (Figure 2a) [Zeng et al., 2008a]. The channel sidewalls were vertical and smooth. The boundary conditions used in this LES simulation with flat bed were identical to the ones used to simulate the flow in the 135° bend with a deformed scour bed.

Figure 2.

Sketch of flume in which the 193° bend experiment with a mobile bed [Blanckaert, 2010] was conducted [Constantinescu et al., 2011a]. The dashed line shows the flume centerline. The right frame shows the bathymetry at equilibrium scour [Constantinescu et al., 2011a]. The bed elevation (z/H) is measured with respect to the mean position of the free surface (z/H = 0) in the inlet section (far upstream boundary).

[30] To better put in perspective the performance of the present LES code, the comparison of the distributions of the streamwise vorticity, ωξD/U, and 2-D streamline patterns in representative cross sections in Figures 3 and 4 includes not only LES and experimental measurements but also the results of a RANS simulation [Zeng et al., 2008a]. For example, RANS does not predict the strong amplification of ωξ near the inner wall in sections D90 and D120 (Figure 3). This is because RANS does not capture the secondary SOV cell forming in between the inner wall and the shear layer bordering the core of high streamwise velocity present past section D40. This SOV cell is also captured by the 2-D streamline patterns inferred from velocity measurements in section D120 (Figure 4). LES more accurately predicts the vorticity distribution within the main cell of cross-stream circulation which, as expected for flat bed bends, occupies most of the cross section [Abad and García, 2009a; Blanckaert et al., 2012]. In all three cross sections, RANS severely underpredicts the magnitude of the streamwise vorticity and the thickness of the patch of high vorticity magnitude associated with the secondary flow, especially over the inner half and middle part of the cross section.

Figure 3.

Distribution of streamwise vorticity, ωξH/U, in sections (top row) D90, (middle row) D120, and (bottom row) D180 for the 193° bend case with a flat bed. (a) LES (left), (b) experiment (middle), and (c) RANS (right).

Figure 4.

Distribution of 2-D streamline patterns in section D120 for the 193° bend case with a flat bed. (a) LES (left), (b) experiment (middle), and (c) RANS (right). Results show that RANS fails to capture the secondary cell at the outer bank observed in experiment and LES.

[31] As shown by the 2-D streamline patterns for the flat bed case in Figure 4, RANS fails to capture the presence of the secondary cell at the outer bank. By comparison, LES shows that an outer-bank cell is present between sections D30 and P1.0. The LES with wall functions of van Balen et al. [2010a] also captured the presence of the outer-bank cell, but the cell extended only over the upstream part of the bend. The fact that the present LES predicts an outer-bank cell extending over the whole length of the curved reach is consistent with experimental measurements. We attribute the improved accuracy of the present LES to the fact that the model does not use wall functions.

6 Results and Discussion

6.1 Bathymetry Corresponding to Final Stages of Scour

[32] The main features of the final bathymetries in the two cases are discussed next. This is important because once the pool and point bar develop inside the curved open channel, topographic steering effects play an important role in the redistribution of the streamwise velocity, affect the formation, position and coherence of the SOV cells, and the other regions of high turbulence production (e.g., shear layers) and, in some cases, can force separation of the flow at the inner bank.

[33] In the 135° case (Figure 1), scour starts close to the entrance in the curved reach. The flow depth reaches a peak value of about 1.8H in section D100 at about 1/3B from the outer bank. However, scour at the outer bank remains small until section D90. Then, severe scour occurs in the immediate vicinity of the outer bank in between sections D120 and D135 + 10H. The maximum flow depth in this region reaches 2.3H. The point bar starts around section D30 and remains attached to the inner bank until the end of the curved reach. The minimum flow depth is around 0.3H in the immediate vicinity of the inner bank. Downstream of the exit from the curved reach, the minimum flow depth increases.

[34] In the 193° case (Figure 2), scour develops in the outer half of the cross-section downstream of section D00. The flow depth reaches a maximum of about 3.2H in section D60. The scour close to the outer bank reduces gradually in the downstream part of the curved reach. Scour increases again upon approaching the bend exit. The flow reaches a depth of about 2.6H in section D180. Downstream of section P2.0, situated 2.0 m (18H) from the end of the curved reach, scour decays in the outer part of the cross section. Two main regions of sediment deposition form in the inner half of the cross section. The minimum flow depth is about 0.1H–0.2H around sections D60 and P0.5.

[35] In summary, the main differences between the two cases are that: (1) The most severe scour at the outer bank occurs in the upstream half of the curved reach for the 193° case (3.2H) and close to the exit from the curved reach in the 135° case (2.3H); (2) The bed deformations close to the outer bank remain small until section D90 in the135° case. By contrast, in the 193° case, the largest scour developed close to the outer bank around section D60. (3) The point bar much is much wider in the 193° case. The amount of sediment deposited close to the inner bank past the end of curved reach in the 135° case is relatively small compared to the 193° case; (4) Overall, both scour and deposition are more severe in the 193° case.

6.2 3-D Flow and Large-Scale Coherent Structures

[36] The Q criterion [Hunt et al., 1988] is used to visualize the large-scale eddies in the flow and their relative position with respect to the channel bed and banks in Figure 5. The information provided by the Q criterion is mainly qualitative and serves to understand the position in space of the core of the main vortices. In the 135° case, the main cell of cross-stream circulation, V1, extends as a well-defined vortex from the entrance into the curved reach until close to its exit (section D110). Its centerline is situated within the inner half of the section. The cell occupies the central part of the cross section inside the downstream part of the curved reach. This is different from what was observed experimentally and numerically in the 193° case (Figure 4) [Constantinescu et al., 2011a where the main cell was located close to the outer bank at all streamwise locations within the curved reach. In both cases, the main cell approximately follows the deepest part of the bathymetry within the strongly curved reach. Thus, in sharply curved bends, topographic steering controls the position of the main cell in the later stages of the scour and deposition process after a well-defined pool and bar structure has developed. This result is expected to hold for natural meanders containing high-curvature regions.

Figure 5.

3-D visualization of the vortical structure of the mean flow predicted by LES using the Q criterion. C and CC indicate a clockwise-rotating and a counterclockwise rotating vortex when looking downstream in the bend, respectively.

[37] The flow structure close to the inner bank is significantly more complex in the 135° case where two (clockwise rotating when looking downstream in the bend) SOV cells (V2 and V3) and a large counterclockwise rotating SOV cell, V4, are present (Figure 5). Only one SOV cell was present in the 193° case. Important qualitative differences are also observed in the straight outflow reach. The larger extent and relative depth of the pool within this reach in the 135° case appears to enhance and/or lead to an increase in the size of the large clockwise-rotating SOV cell, V6, within the deeper part of the bathymetry, starting around section D100 (see also 2-D streamline patterns in Figure 9). The vertical extent of this SOV is limited by the presence of a large counterclockwise rotating secondary cell at the outer bank, V5, close to the free surface (Figure 9).

[38] Experiment and simulations showed that a relatively strong secondary outer-bank cell similar to V5 did not form near the free surface for the 193° case. Thus, a comparison of these two cases may help to better understand the changes in the flow structure and their effects on the capacity of the flow to induce bank erosion between a sharply curved bend in which a large outer-bank secondary cell extends over most of the curved reach and continues well into the straight outflow reach and a similar bend in which no such secondary cell forms. Previous experimental and numerical studies that provide a detailed discussion of the effect of this secondary cell on the flow structure in bends of high curvature exist for cases with a flat bed [e.g., van Balen et al., 2010a; Blanckaert et al., 2012] and deformed bed [Blanckaert and de Vriend, 2004; van Balen, 2010, chapter 9]. The formation of this cell is dependent on many factors, and Blanckaert [2011] and Blanckaert et al. [2012] have also shown that it will be enhanced with increasing bank steepness and roughness, and also with increasing curvature, H/R. The effect of bend angle on this outer bank cell is not fully understood. While it did not appear in the 193° bend, it has been found in bends of greater angles such as the 240° experimental bend of Termini and Piraino [2011] which had developed topography. Such cells have also been commonly observed in field studies (Sukhodolov [2012] [bend angle = 150°], Bathurst et al. [1979] [bend angle = 38° to 105°], and Thorne et al. [1985] [bend angle ≈ 180° to 190°]) in regions where a large pool has formed at the outer bank. Unfortunately, field studies do not allow a detailed characterization of the flow structure and estimation of the bank shear stresses at all flow depths, needed to understand the effect of the outer-bank cell on the flow forcing on the outer bank.

[39] The effects of the presence of these large-scale coherent structures on the mean flow and turbulence statistics is analyzed next based on the distributions of the 2-D streamline patterns, nondimensional streamwise velocity, vertical vorticity and turbulent kinetic energy (TKE) at the free surface, and in a horizontal section situated at 0.5H below the free surface (Figure 6). The most important qualitative difference between the two cases is the absence of flow separation in horizontal planes in the 135° case. By contrast, large recirculation regions were present in the 193° case behind the shallowest parts of the inner bank bar (Figure 3) [Constantinescu et al., 2011a].

Figure 6.

Visualization of the mean flow at (top) the free surface and (bottom) in a horizontal section situated at z/H = −0.5. (a) 2-D streamline patterns, (b) streamwise velocity, uξ/U, (c) vertical vorticity, ωzH/U, and (d) TKE, k/U2.

[40] Despite the absence of flow separation, several regions of high magnitude vertical vorticity corresponding to energetic shear layers are observed in the 135° case. The main vorticity sheet starts around section D30 and ends inside the straight outflow reach. Its formation is due to the fact that the core of high streamwise momentum fluid within the straight inflow reach does not lose its streamwise momentum fast enough and does not acquire transverse momentum to follow the surface of the high-curvature inner bank. As a result, a region of high mean shear develops at the boundary between the core of high streamwise velocity and the much slower fluid moving over the point bar. Though the mechanism responsible for the formation of this shear layer is the same in the two cases, the position of this shear layer is quite different. In the 135° case, the shear layer is well aligned with the outer bank past section D90 and does not reach the vicinity of the outer bank. In the 193° case, the shear layer penetrated until close to the outer bank close to the free surface. The strongly energetic eddies convected within the shear layer were shown by Constantinescu et al. [2011a] to locally amplify the capacity of flow to erode the outer bank close to the free surface. As shown by the distribution of the instantaneous vertical vorticity in Figure 7a, though such energetic eddies are also convected within the main shear layer in the 135° case and, at times, the eddies approach the outer bank (e.g., around section D100), they never really reach the bank surface. The main reason is the presence of the outer-bank secondary cell (V5), visible as a region containing mostly negative vertical vorticity in Figure 7a. This cell protects the outer bank against the possible penetration of the shear layer eddies close to the free surface. Thus, the inner-bank shear layer is expected to play a role in outer-bank erosion only for long and relatively narrow high-curvature bends.

Figure 7.

Distribution of the (left) vertical vorticity, ωzH/U, in an instantaneous flow field and (right) mean pressure, p/ρU2, at the free surface. The left frame visualizes the eddies shed in the shear layer developing between the core of high streamwise velocity fluid and the slowly moving fluid close to the inner bank and in the outer-bank boundary layer which thickens rapidly as it develops into a region of strong adverse pressure gradients (see right frame). The right frame shows the pressure computed at the horizontal rigid lid, which can easily be converted to a hydrostatic flow depth that is indicative of the real elevations of the free surface.

[41] Another characteristic of flow in sharply curved bends is the formation of a thick region of low streamwise velocity close to the outer bank within the upstream half of the curved reach (Figure 6b). For the 135o case, the thick boundary layer like region is present between sections D00 and D90 at the free surface and in between sections D00 and D60 at z = −0.5H. Its main characteristic is that fairly energetic eddies are convected inside it (Figure 7a). Its presence is due to several reasons. The first is that the core of high streamwise velocities moves toward the inner bank at the entrance into the curved reach, thus reducing the transverse gradient of the streamwise velocity close to the outer bank. The second is the presence of a strongly coherent secondary cell. Finally, the high curvature of the outer bank modifies the distribution of the mean pressure in horizontal planes such that the outer-bank boundary layer inside the upstream part of the curved reach develops within a region of strong adverse pressure gradients (Figure 7b). The presence of energetic eddies also explains the large increase of the turbulent kinetic energy (TKE) within this layer (Figure 6d). The main region of high TKE corresponds to the downstream part of the main shear layer between the inner bank and the main flow, where the Kelvin-Helmholtz instability induces the formation of strongly energetic eddies (Figure 7a). The coherence of these eddies is larger close to the free surface. Such a layer transporting large-scale turbulence is also expected to be observed in meandering rivers in regions where the outer-bank curvature is high and interaction of flow with geometry generates large adverse pressure gradients.

[42] The other region of high TKE in the 135° case is due to production by mean shear associated with a high streamwise velocity gradient in the transverse direction between the cores of the SOV cells (V2, V3, and V4) and the surrounding flow. Meanwhile, the core of high streamwise velocities within the upstream part of the curved reach contains mostly incoming flow without large-scale turbulence from the shear layers forming on its two sides. This explains the low TKE levels within this region. In meandering rivers, the core of high streamwise velocities entering a new high-curvature region may be more energetic than that in the isolated bend considered here because of large-scale turbulence generated in the upstream meanders that is convected with the mean flow.

6.3 Depth-Averaged Flow

[43] The distribution of the streamwise unit discharge, qs, in a curved channel is very important for several reasons. It demonstrates the role of the bathymetry in the redistribution of the streamwise flow within the curved channel. Additionally, qs is the main variable that determines the patterns of bed entrainment in the case of depth-averaged 2-D numerical models that have the capability to predict sediment transport and changes in the bed morphology. The main differences between the distributions of qs in the two cases (Figure 8) are primarily due to the different bathymetries developing during the final stages of scour.

Figure 8.

Distribution of the normalized streamwise unit discharge, qs, in the channel for (left) the 135° case and (right) the 193° case (adapted from Constantinescu et al. [2011a]). In both cases, most of the discharge is convected over the deeper regions of the bathymetry.

[44] In the 193° case, the transverse slope within the outer half side of the cross section increases rapidly, and severe scour penetrates to the outer bank starting close to the entrance into the curved reach. Consequently, most of the incoming flow is pushed toward the deepest parts of the bathymetry extending for about B/3 away from the outer bank. The main features of the transverse distribution of qs in a cross section do not change until some distance downstream of the second region of separated flow close to the inner bank. The presence of two regions of separated flow decreases the width of the region of positive qs and increases the amount of flow convected over the deepest parts of the bathymetry.

[45] In the 135° case, the regions of scour develop much more gradually into the curved reach. The milder topographic steering effects over the upstream half of the curved reach and the absence of regions of separated flow over the point bar in the 135° case are two reasons why the regions of high qs correlate better with the position of the regions of high flow depth in the channel. The largest values of qs are first observed close to the inner bank around the entrance into the curved reach. Gradually, curvature effects push the core of high qs values towards the outer bank. In a fairly good approximation, the line defined by the peak values of qs in a cross section follows the line of maximum flow depth between sections D00 and D120. The maximum values of qs are only about two times larger than the mean value in the straight inflow reach. The severe scour region at the outer bank extending for about 1.5B past the end of the curved reach, and the very large length of the point bar in the straight outflow reach partly explain why the flow recovers very slowly in the 135° case. Topographic steering plays a major role in delaying the return to a uniform distribution of qs past the end of the curved reach.

6.4 Curvature and Topographic Steering Effects on Streamwise Velocity, Secondary Flow, and Turbulence

[46] Analysis of the distributions of the streamwise velocity (uξ), streamwise vorticity (ωξ), and TKE at representative cross sections in Figures 9 and 11 allows for better understanding of the effect of curvature and topographic steering on the mean flow and turbulence within the channel for the 135° bend. The streamwise variation of the circulation (Γ) of the cells of cross-stream circulation shown in Figure 10 is a good indicator of the regions where the effect of the SOV cells on the mean flow and boundary stresses is the largest. Of course, the proximity of the core of the SOV cell to the boundary also affects the erosion potential of the cell. The circulation of an SOV cell in a cross section is calculated by integrating the streamwise vorticity within the region occupied by the core of that SOV. The extent of the core of a SOV cell is determined based on the approximate position of the SOV cell given by the Q criterion and 2-D streamline patterns. The integration over the patch of high streamwise vorticity values associated with the core of a SOV cell is performed until the vorticity away from the axis of the cell decays below a certain threshold level associated with small-scale turbulence. As in some sections, it is difficult to separate between the cores of V2 and V3 and that of the main cell, V1, only the total circulation, Γ+ associated with the secondary clockwise rotating flow is shown in Figure 10. Though V6 is distinct from V1 based on the analysis of the regions of high streamwise vorticity, its effect is combined with that of the other clockwise rotating cells.

Figure 9.

(left) Mean-flow velocity, uξ/U, (center) streamwise vorticity, ωξH/U, and (right) 2-D streamline patterns in representative cross-sections. z/H = 0 is the mean free surface position.

Figure 10.

Streamwise variation of nondimensional circulation magnitude, Γ, of the vortical structures associated with the main cell of cross-stream circulation (Γ + = ΓV1 + ΓV2 + ΓV3 + ΓV6), the main inner bank SOV cell (ΓV4), the (counterclockwise rotating) secondary outer-bank cell (ΓV5) and the SOV cell associated with the main cell of cross-stream circulation in the straigth outflow reach (ΓV6). The circulation is nondimensionalized by UH, where U is the inlet mean streamwise velocity, and H is the inlet water depth.

[47] The total circulation induced by curvature increases rapidly past the entrance to the curved reach and peaks around section D60 (Figure 10). By comparison, the peak value of the circulation associated with the main cell in the 193° case occurred around section D90 [van Balen et al., 2010a, 2010b]. In both test cases, the circulation decreases monotonically within the remaining part of the curved reach, and the rate of decrease of the circulation is smaller in the straight outflow reach.

[48] In the 135° case, bed deformations with respect to the initial flat bed conditions start close to the entrance into the curved reach. By section D30 (Figures 9a), the center of the core of the main cell, V1, is situated over the deepest part of the section. Downstream of section D30, the core of high streamwise velocity starts moving towards the outer bank, close to the free surface (e.g., see section D60 in Figures 9). In agreement with previous experimental and numerical studies of flow in medium- and high-curvature curved channels, the maximum of streamwise velocity is not observed at or just beneath the free surface, but rather close to the deepest part of the cross section (between sections D40 and D120 for the 135° case). This is due to the redistribution of streamwise velocity by the main cell, V1. The large coherence of V1 in this section induces strong cross-stream velocities close to the bed. However, the region of high ωξ is not limited to the near-bed region beneath V1 but extends up to the free surface and then continues toward the outer wall. The later part of this region is due to the presence of V3 whose sense of rotation is the same as that of V1.

[49] The patch of concentrated high negative values of streamwise vorticity and low streamwise velocity close to the inner bank is associated with V4 that occupies the whole channel depth. The slight dip in the bathymetry levels beneath the centerline of V4 is due to the large coherence of V4. This is also confirmed by results in Figure 10 that show that the circulation of V4 is the largest at this streamwise location.

[50] The largest TKE values in Figure 11 (section D60) are observed within the shear layer forming in between V4 and V2 that are rotating in opposite directions. Relatively large TKE values are also observed within the main shear layer close to the free surface that borders the core of highest streamwise velocity in section D60. The TKE values within the core of V4 are only slightly larger than the background TKE. This means that the core of V4 is not subject to large-scale oscillations in the instantaneous flow fields toward and away from the inner bank, and does not transport large-scale turbulence similar to the shear layers. In the same section, the main cell, V1, convects the energetic eddies within the outer-bank boundary layer away from the outer bank within the lower half of the cross section. This explains why the region of high TKE is much thicker close to the bed compared to close to the free surface.

Figure 11.

Turbulent kinetic energy, 100 k/U2, in sections D30, D60, and D120. SL denotes shear layer regions where production of turbulence by mean horizontal shear is large. The black solid lines show the position of the main eddies as visualized by the Q = 0.10 isosurface.

[51] The coherence of V4 has significantly decayed by section D120. The region of highest TKE in section D120 (Figure 11, section D120) corresponds to the main shear layer, which is prevented from reaching the outer bank by V5. Downstream of section D120 along the outer bank, the core of high streamwise velocity fluid extends over the whole channel depth. Still, the vertical distribution of the streamwise velocity is very nonuniform due to V6 and V5. One should also notice that the combined effect of V6 and V5 in the cross section is to convect fluid toward the outer bank at about its half depth.

[52] The increase in TKE in the attached boundary layer at the outer-bank beginning at the bend entrance (e.g., see sections D30 and D60 in Figure 11a) is related to the generation of large-scale turbulence associated with a boundary layer developing into a strong adverse pressure gradient (Figure 7). Though turbulence production by shear plays the major role in the amplification of the TKE within the channel [Constantinescu et al., 2011a, 2011b], it does not mean that turbulence anisotropy effects are not important. For example, these effects contribute to the formation of the secondary outer-bank eddy, V5, and the changes in the momentum transfer mechanisms close to the outer bank. However, the TKE amplification within V5 is not substantial. Present results show an amplification of the TKE at the outer edge (the edge away from the outer bank) of the outer-bank cell close to the free surface, which is consistent with results shown by Blanckaert et al. [2012].

[53] As mentioned, between sections D40 and D120, the region of maximum of uξ is not observed at or just beneath the free surface but is closer to the channel bed (Figure 9). While the region of high uξ is confined to the lower part of the flow depth, near the channel bed, it is not centered at the deepest part of the channel thalweg. This is mainly due to the cores of the secondary SOV cells that are regions of high circulation and low streamwise velocity. Results for the two bends analyzed here suggest that the formation of strongly coherent SOV cells at the inner bank is a characteristic of flow in sharply curved bends. This explains the complex shape and nonuniform distribution of streamwise velocity within the region that convects most of the flow within the channel in the 135° case. It is only downstream of the exit from the curved reach (section D135) that the core of high streamwise velocity extends from the bed to the free surface and the distribution of uξ is fairly uniform in the vertical direction close to the outer bank.

[54] A careful examination of the lines of constant streamwise velocity in sections D135 and D135 + 2.8H in Figure 9 shows that these lines diverge away from the outer bank as the free surface is approached due to the presence of the secondary outer-bank cell, V5. This means that close to the free surface V5 protects the bank against possible erosion induced by high streamwise velocities. However, at the same time, the peak values within the core of high streamwise velocity fluid have moved very close to the outer bank in section D135 + 2.8H. This is because of the convection of high streamwise velocity fluid by the combined action of V5 and V6 towards the outer bank at about half bank depth level. Thus, the observation of Bathurst et al. [1979] that the advection of high momentum fluid below the outer-bank cell can endanger bank stability is also confirmed by the present results which are restricted for the case of vertical banks. More precisely, the amplification of the bank shear stress is expected to be the largest towards the mid-depth levels where the transverse gradient of the streamwise velocity is the largest. Future research is needed to confirm the validity of this result for bends with lower bank inclination and how it is affected by other geometrical factors.

[55] While in both cases the cores of the SOV cells near the inner bank contain low streamwise velocity fluid, no significant amplification of the TKE was observed within these SOV cells (V2 and V4) in the 135° case. By contrast, the TKE amplification inside the cores of the inner bank SOV cells was comparable to that observed inside the main shear layer in the 193° case (Figure 6) [Constantinescu et al., 2011a]. Thus, the turbulence structure within the inner-bank SOV cells is strongly dependent on the channel bathymetry and geometry that can favor or impede the large-scale oscillations of the cores of these eddies. One should also point out that the inner bank SOV cells in which the turbulence was strongly amplified formed in regions situated immediately downstream of recirculating flow. Such regions were not present in the 135° case where the flow did not separate at any streamwise location over the point bar. The presence of inner bank SOVs has also been observed in high-curvature natural channels (Figure 3b) [Bathurst et al., 1979].

6.5 Mechanisms Controlling Erosion During the Later Stages of Scour and Deposition

[56] The distribution of the nondimensional mean bed shear stress, τ/τ0, where τ0 is the mean value of the bed shear stress in the straight inflow reach (fully developed incoming turbulent flow), is shown in Figure 12a for the 135° bend. Overall, a good correlation exists between regions with a large unit discharge (Figure 8a) and regions of large values of τ/τ0. In particular, the streamwise component of τ/τ0 is amplified in regions where the peak values of the streamwise velocities are situated at small distances from the bed rather than near the free surface. This is the case for sections D60 to D120 (Figure 9), where large bed shear values are observed on the slope face of the inner-bank point bar, which is consistent with previous field observations for the location of maximum nondimensional shear stress [Dietrich and Whiting, 1989] and bedload transport [Mulhoffer, 1933, reported in Church, 1987]. The transverse component is, in turn, affected by the capacity of a cell of cross-stream motion to induce a large velocity gradient in the direction normal to the bed surface. The regions where a patch of high (positive) values of the streamwise vorticity is present at the bed surface beneath the cores of V1 and V6 (Figure 9) are also regions where the transverse component of τ/τ0 is high (Figure 13b).

Figure 12.

Nondimensional shear stress magnitude, τ/τ0, at the bed in (left) the 135° case and (right) the 193° case (adapted from Constantinescu et al. [2011a]). τ0 is the mean bed shear stress in the straight inlet channel.

Figure 13.

(left) Streamwise and (right) transverse components of the nondimensional bed shear stress.

[57] The presence of regions of high bed shear stress magnitude inside the scour hole may look surprising given the fact that the bathymetry is close to equilibrium and the scour developed under clear water scour conditions in the 135° case. However, on sloped surfaces, the local critical value of the bed shear stress is not the same as the value inferred from Shields diagram for flat bed conditions because of gravitational bed slope effects. If the secondary flow pushes sediment particles against the local bed slope, then the local critical value is larger than the one for a flat bed [e.g., Kirkil et al., 2008]. It is possible that even after correcting for gravitational effects, the bed shear stress could slightly exceed critical levels along the thalweg downstream of the bend apex. If the experiment had been run longer, then this may have lead to a slight deepening of the thalweg but should not have affected the overall scour patterns. The experiments were run for 7 days and did not show any noticeable changes in inner and outer bank levels after 4.4 days.

[58] The peak values of τ/τ0 are around four in the 135° case (Figure 12a) compared to around six in the 193° case (Figure 12b). The main reason for this difference is the very deep scour occurring close to the outer bank in the 193° case which induces significantly larger values of qs in the same region and a more coherent main cell of cross stream motion. This is a main reason why both the streamwise and the transverse components of τ/τ0 are larger in the 193° case (Figure 15) [Constantinescu et al., 2011a]. However, the extent of the region where τ/τ0 > 2 within the curved reach is comparatively larger in the 135° case where it covers most of the central part of the curved reach past section D30 rather than being confined to the upstream part of the curved reach (e.g., between sections D30 and D100 in the 193° case). The decrease of τ/τ0 in the downstream part of the curved reach in the 193° case is related to the movement of the region of high streamwise velocities away from the bed and the large increase in the size of the core of the main cell (Figures 9 and 11) [Constantinescu et al., 2011a]. Meanwhile, the contribution of the transverse component to the total value of the local bed shear stress inside the curved reach is up to around 50% in both cases. In fact, this contribution is larger than 50% just downstream of the bend entrance in the 135° case (Figure 13).

[59] Present results show that strongly coherent SOV cells generated by high curvature and turbulence anisotropy effects can be a main contributor to the shear stress when their core is situated close to a channel boundary. The elongated streak of relatively high τ/τ0 situated close to the inner bank between sections D30 and D90 is induced by one of the SOV cells (V4). As already discussed, V4 transports low streamwise momentum fluid and its cross-stream circulation is relatively high especially between sections D45 and D90 (Figure 10). Examination of the two components of τ/τ0 confirms that the transverse component is the primary contributor (≅80%) to the total shear stress at the bed in the region situated beneath V4.

[60] For the 135° bend, the other elongated streak of high τ/τ0 situated inside the straight outflow reach is induced by V6. This SOV cell is not confined to the deeper part of the pool. Rather, its core moves away from the outer bank (e.g., see Figure 5 and 2-D streamline patterns in Figure 9). The main role of this SOV cell, as far as morphodynamics is concerned, is to entrain sediment particles from the deeper parts of the pool and to push them against the transverse slope of the scour hole, while these particles move downstream. Its cross-stream circulation is large enough to induce a substantial transverse component of τ/τ0 (around 30% of the total stress). The large erosion capability of V6 is also confirmed by the large transverse slope of the bed observed in the bathymetry beneath its core. The presence of a large secondary cell (V5) in the upper part of the pool in the 135° case favors the confinement of V6 toward the deeper regions and limits the growth of its core. As a result, its sediment entrainment potential is larger compared to bends where the main cell of cross-stream circulation occupies most of the flow depth close to the outer bank.

[61] The large erosion potential of some of the SOV cells is also confirmed by the shear stress distribution at the inner bank in Figure 14 that shows the presence of a region with τ/τ0 > 1.25 between sections D15 and D70. Comparison of the distributions of the vertical component of τ/τ0 (Figure 15) and total shear stress (Figure 14a) at the inner bank indicates that the vertical component provides the main contribution to the total boundary shear stress. This finding is consistent with the fact that the SOV cells forming at the inner bank contain low streamwise velocity and high circulation fluid. The vertical component is oriented toward the free surface between sections D15 and D45 and toward the bed between sections D45 and D70 (Figure 15). This is because the amplification of τ/τ0 at the inner bank is first due to V2 and then due to V4 that rotates in opposite direction. Thus, the flow forcing on the inner bank is driven by the SOV cells rather than being induced by the presence of a core of high streamwise velocities in its vicinity. Moreover, the values of τ/τ0 within the regions of high shear stress at the inner bank are comparable to the peak values observed at the outer bank (Figures 14a and 14b). In natural channels, variations in curvature along the curved reach can affect the coherence of the SOVs close to the inner bank and their capacity to erode the bank. Hodskinson and Ferguson [1998] discuss the work of Page and Nanson [1982], Lewin [1978], and Andrle [1994] who demonstrated that inner-bank erosion of high-curvature bends does occur and can lead to channel widening.

Figure 14.

Nondimensional shear stress magnitude, τ/τ0, at the channel sidewalls. (a) inner bank and (b) outer bank. Lξ is the streamwise distance measured along the channel sidewall.

Figure 15.

Vertical component of the nondimensional shear stress at the inner bank. Lξ is the streamwise distance measured along the channel sidewall. Large values of the shear stress at the inner bank are primarily induced by the SOV cells in its vicinity.

[62] The main region of high shear stress (τ/τ0 > 1.25) at the outer bank is situated between sections D100 and D135 + 3H for the 135° case (Figure 14b) and between sections D120 and P2.0 in the 193° case (Figure 16) [Constantinescu et al., 2011a]. In both cases, the amplification of τ/τ0 is due to the movement of the core of high streamwise velocity values close to the outer bank. This is confirmed by the fact that in both cases, the vertical component of the total shear stress (not shown) is much smaller than the streamwise component at all streamwise locations. Thus, high erosion at the outer bank of natural channels containing high-curvature reaches is expected to occur in regions where the outer bank curvature decays rapidly.

Figure 16.

Distribution of the mean pressure fluctuations, inline image, at the channel bed. This quantity serves to identify regions of large near bed turbulence where the temporal variations of the instantaneous bed friction velocity around its mean value are also large.

[63] In the 135° case, the peak values of τ/τ0 at the outer bank are recorded far below the free surface at all streamwise locations, at a depth generally situated close to the boundary between V6 and V5. By contrast, the peak values of τ/τ0 within the corresponding region of high bank shear stress were always situated at or very close to the free surface in the 193° case (Figure 16) [Constantinescu et al. 2011a]. This is because the secondary outer bank cell present in the 135° case protects the outer bank against erosion close to the free surface. Meanwhile, the effect of the presence of two counter-rotating vortices (V5 and V6) in the vicinity of the outer bank downstream of section D100 is to gradually push the core of high streamwise velocities closer to the outer bank (see distributions of the streamwise velocity in sections D120 to D135 + 2.8H in Figure 9) and thus to increase the shear stress around the mid-depth level, even though the direct erosion potential of both V5 and V6 is small. This testifies to the complex effect of curvature on the flow structure near the outer bank and its potential to induce bank erosion in sharply curved bends.

[64] The passage of energetic eddies near the channel boundaries and/or the random oscillations of the cores of the large-scale vortices situated near these boundaries can significantly increase the magnitude of the shear stress above the mean values. Thus, the bed and bank erosion potentials will be higher than those estimated solely based on the mean value of the shear stress magnitude in regions of high turbulence intensity. The two bend cases analyzed here suggest that such regions appear to be a general characteristic of flow in sharply curved bends. The distributions of the mean pressure fluctuations, inline image, at the bed shown in Figure 16 and at the two banks shown in Figure 17 allow to identify these regions for conditions corresponding to the later stages of the erosion and deposition process in the 135° bend.

Figure 17.

Distribution of the mean pressure fluctuations, inline image, at the channel sidewalls. (a) inner bank and (b) outer bank.

[65] For the 135° bend, the boundary layer at the outer bank of the curved reach develops into a region subject to a large adverse pressure gradient within its upstream half (Figure 7b) which induces the formation of highly energetic turbulent eddies and a large increase in the boundary layer width (Figure 7a). The peak values of inline image within this region are recorded just downstream of section D00 (Figures 16 and 17b) and are due to the sudden change in curvature between the straight inflow reach and the curved reach. One expects the amplification of the turbulence around the entrance into the region of high channel curvature to be smaller in natural channels where the change in curvature takes place gradually. Some of the eddies generated in the attached boundary layer advect away from the outer-bank surface past the formation region but, because of the high curvature of the outer bank, approach again its surface between sections D70 and D100. This explains the variation of inline image in Figure 17b. A similar region of severe amplification of the turbulence intensity was observed around section D90 in the 193° case [Constantinescu et al., 2011a].

[66] For the 135° bend, the large amplification of inline image at the inner bank between sections D60 and D100 observed in Figure 17a is due to the presence of V2 and V4 in the immediate vicinity of the bank surface. The random temporal variations in the coherence of these vortices and in the distance between their cores and the bank surface generate ejection of patches of vorticity from the attached boundary layer. The end effect is an amplification of the pressure and the boundary shear stress fluctuations at the inner bank. The same mechanism is responsible for the streak of high inline image forming between sections D30 and D90 on the bed surface below the core of V4. Thus, in sharply curved bends, the SOV cells can not only significantly amplify the mean bed shear stress but also the mean pressure fluctuations and thus the local bed shear stress variance. This effect should be taken into consideration in design formulas used for bank erosion and when deciding on the extent of regions that should be protected against erosion.

7 Summary

[67] Comparison of results of eddy-resolving simulations of flow in a 135° bend with R/B = 1.5 and in a 193° bend with R/B = 1.3 revealed some common features of flow in sharply curved bends with natural bathymetry corresponding to the later stages of scour after a pool-point bar structure has developed. It also helped in at least partially answering the research questions defined in section 'Introduction'. LES gave insight into the distributions of boundary shear stresses and the potential of flow to erode the boundaries on the basis of the secondary flow patterns and associated large-scale coherent structures. The validation for the 193° flat bed case also showed that LES without wall functions was better able capture the characteristics of the flow structures compared to both LES with wall functions, and RANS.

[68] The presence of strongly coherent SOV cells at the inner bank and the associated shear layer(s) appears to be a general characteristic of open channel bend flows, provided that the ratio R/B is sufficiently low. The presence of the inner bank SOV cells is independent of flow separation over part of the point bar. Constantinescu et al. [2011b] found a similar flow structure near the high-curvature inner bank (R/B ~ 3) near the confluence of two natural streams. This suggests that the formation of the SOV cells and the associated shear layer near the inner bank of natural streams is a fairly local phenomenon that is mainly controlled by the local curvature of the inner bank. On the other hand, the ability of the eddies shed into the separated shear layer to penetrate close to the outer bank and enhance the flow attach on the outer banks appears to be case dependent. Present results also showed that the SOV cell situated in the immediate vicinity of the inner bank can locally amplify the boundary shear stresses to values that are comparable to peak levels recorded near the outer bank.

[69] Several field studies in natural river reaches [Bathurst et al., 1977; Thorne and Hey, 1979; de Vriend and Geldof, 1983; Sukhodolov, 2012] observed the formation of an outer-bank counterclockwise rotating SOV cell near the free surface. Present results agree with the observation of Bathurst et al. [1979] that the outer bank cell may endanger bank stability by advecting high-momentum fluid toward the outer bank. Even though this cell does seem to protect the outer bank close to the free surface, the prospect of undermining may present a threat to bank stability all the way to the free surface if a collapse were to occur. The largest shear stresses on the outer bank of the 135° bend occur around the exit of the curved reach, precisely in the region of high transverse velocity oriented toward the outer bank, situated in between the outer-bank cell and the main cell. This mechanism pushes higher streamwise velocity fluid toward the outer bank and thus locally increases wall shear stresses. However, wall shear stresses close to the free surface remain relatively low, as the bank in this area is protected by the presence of the outer bank cell.

[70] Present results showed that the secondary flow is directly responsible for a large percentage of the capacity of the flow to erode the bed inside sharply curved narrow bends. For the sharply curved bends (R/B ~ 1.5) analyzed in the present study, the transverse component accounted for up to 50% of the value of the bed shear stress magnitude. Even more relevant, the largest transverse bed shear percentages were generally observed in regions where the potential for bed erosion was the largest. It should be noted, however, that an analysis conducted by Blanckaert and de Vriend [2010] found that velocity redistribution by secondary circulation tends to be of leading order in narrow rivers with B/H < 10, but may be negligible in shallow ones with B/H > 50 [Blanckaert, 2011].

[71] In the cross sections where the circulation of the main cell of cross-stream circulation was high and its core was situated close to the bed, a substantial amount of high streamwise velocity fluid was convected against the bed slope toward the inner bank. This explains why regions of high bed shear stress were present over the shallower parts of the cross section.

[72] Flow in alluvial curved channels is more complex than that analyzed in the present test cases which assumed the boundaries to be fixed. For example, sediment transported as bed load and suspended load in the channel may dampen or amplify turbulence in a certain region. In rivers, the channel banks are generally erodible and can have a large inclination with respect to the vertical. Moreover, large-scale moving bed forms can be present in the channel and significantly modify the secondary flow [Abad and García, 2009b]. This will obviously affect the coherence and position of some of the large-scale coherent structures in the flow. Still, as the timescales associated with the large-scale coherent structures and their unsteady dynamics are generally much smaller than the timescales over which the large-scale features of the bathymetry change significantly, LES simulations with fixed bathymetry should provide relevant information on the flow structure and its capacity to entrain sediment.

Acknowledgments

[73] We gratefully acknowledge the Transportation Research and Analysis Computing Center (TRACC) at the Argonne National Laboratory for providing substantial amounts of computer time. Also, funding grants provided by the Canada Foundation for Innovation and Natural Sciences and Engineering Research Council of Canada are greatly appreciated. The second author would also like to acknowledge scholarship funds provided for this research by the University of Ottawa, the Natural Sciences and Engineering Research Council of Canada, and the Ontario Graduate Scholarship Program.

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