Hydrodynamic processes and sediment erosion mechanisms ... - Wiley

JOURNAL OF GEOPHYSICAL RESEARCH: EARTH SURFACE, VOL. 118, 480–496, doi:10.1002/jgrf.20042, 2013

Hydrodynamic processes and sediment erosion mechanisms in an open channel bend of strong curvature with deformed bathymetry G. Constantinescu,1 S. Kashyap,2 T. Tokyay,1 C. D. Rennie,2 and R. D. Townsend2 Received 7 March 2012; revised 31 January 2013; accepted 5 February 2013; published 24 April 2013.

[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.

Citation: Constantinescu, G., S. Kashyap, T. Tokyay, C. D. Rennie, and R. D. Townsend (2013), Hydrodynamic processes and sediment erosion mechanisms in an open channel bend of strong curvature with deformed bathymetry, J. Geophys. Res. Earth Surf., 118, 480–496, doi:10.1002/jgrf.20042.

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

1 Department of Civil and Environmental Engineering and IIHRHydroscience and Engineering, The University of Iowa, Iowa City, Iowa, USA. 2 Department of Civil Engineering, University of Ottawa, Ottawa, Ontario, Canada.

Corresponding author: S. Kashyap, Department of Civil Engineering, University of Ottawa, Ottawa, ON, Canada, K1N 6 N5. ([email protected]) ©2013. American Geophysical Union. All Rights Reserved. 2169-9003/13/10.1002/jgrf.20042

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,

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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 highcurvature 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 straightline 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 highcurvature 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 largescale turbulence based on high-resolution LES without wall functions. The main research questions we try to answer in the present study are the following: [10] 1. 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? [11] 2. Can the outer bank cell endanger bank stability by advecting high momentum fluid towards the bank? [12] 3. Can erosion at the inner bank be a concern for sharply curved bends? [13] 4. 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?

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[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 highcurvature 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,

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CONSTANTINESCU ET AL.: EROSION MECHANISMS IN A RIVER BEND Table 1. Flow Parameters for the 135 and 193 Bend Flume Experiments U

H

u0*

u*cr

d50

Re*

Re

Fr

Case

(m/s)

(m)

(m/s)

(m/s)

(mm)

(u*d50/v)

(UH/v)

U/(gH)1/2

135 193

0.310 0.610

0.150 0.115

0.0152 -

0.0197 -

0.689 2.000

13.8 -

60,000 83,012

0.255 0.735

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.

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 predictorcorrector 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). [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 + aH, where a 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

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).

483


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). numbers were Re = UH/n = 60,000 and Fr = (U/gH)1/2 = 0.26, where g is the gravitational acceleration and n 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-StressModel 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 ( 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 t/t0 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 t/t0 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 t/t0 confirms that the transverse component is the primary contributor (ffi80%) 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 t/t0 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 down-

Figure 13. (left) Streamwise and (right) transverse components of the nondimensional bed shear stress.

491


Figure 14. Nondimensional shear stress magnitude, t/t0, at the channel sidewalls. (a) inner bank and (b) outer bank. Lx is the streamwise distance measured along the channel sidewall. stream. Its cross-stream circulation is large enough to induce a substantial transverse component of t/t0 (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 t/t0 > 1.25 between sections D15 and D70. Comparison of the distributions of the vertical component of t/t0 (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 t/t0 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 t/t0 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. [62] The main region of high shear stress (t/t0 > 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 t/t0 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. [63] In the 135 case, the peak values of t/t0 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

Figure 15. Vertical component of the nondimensional shear stress at the inner bank. Lx 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.

Figure 16. Distribution of the mean pressure fluctuations, p’2 =r2 U 4 , 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.

492


Figure 17. Distribution of the mean pressure fluctuations, p’2 =r2 U 4, at the channel sidewalls. (a) inner bank and (b) outer bank. values of t/t0 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 pres sure fluctuations, p’2 =r2 U 4 , 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. [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 p’2 =r2 U 4 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 p’2 =r2 U 4 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 p’2 =r2 U 4 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 mecha nism is responsible for the streak of high p’2 =r2 U 4 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 1. 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

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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.

[73] Acknowledgments. 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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