The structure of turbulent flow in an open channel bend of strong curvature with deformed bed: Insight provided by detached eddy simulation

Authors


Abstract

[1] Results of a detached eddy simulation (DES) are used to better understand the effects of the mean flow three-dimensionality and secondary currents on turbulence and boundary shear stresses and the mechanisms through which the momentum and Reynolds stresses are redistributed in a strongly curved 193° bend with fixed deformed bed corresponding to the later stages of the erosion and sedimentation process. The ratio between the radius of curvature of the curved reach and the channel width is close to 1.3. The large channel curvature and the point bar induce flow separation near the inner bank and the formation of several strong separated shear layers (SSLs), where production by mean shear dominates. DES shows that in addition to the main cell of cross-stream circulation developing in the deeper part of the bend, several streamwise-oriented vortices (SOV) form at the inner bank. DES satisfactorily captures the distribution of the streamwise velocity and streamwise vorticity in relevant cross sections compared to experiment. Comparison with a Reynolds-averaged Navier-Stokes (RANS) simulation shows that DES predicts more accurately the velocity redistribution and cross-stream motions in the channel. This is because RANS significantly underpredicts the circulation and turbulence amplification inside the cores of the SOV vortices. DES is then used to clarify the influence of the SOV vortices and SSLs on the boundary shear stress. DES reveals the presence of several regions of large amplification of the pressure RMS fluctuations near the inner and outer banks, which can locally increase the bed erosion and affect the bank stability in the case of a bend with erodible banks. The mean flow bed shear stress distribution predicted by DES is significantly different than that predicted by RANS, while DES predictions of the mean flow are more accurate. This means that use of eddy-resolving techniques like DES in mobile bed simulations of flow in curved alluvial channels should result in more accurate predictions of bathymetry.

1. Introduction

[2] Natural rivers generally contain meandering regions. As a result of the centripetal forces acting on the flow in regions of significant channel curvature, curvature-induced secondary currents are forming at all stages of the erosion and deposition processes. The study of the nonlinear interactions between the main flow and the cross flow and their effects on the turbulence (e.g., strong anisotropy and generation of large-scale energetic eddies) is an active area of open channel flow research [e.g., see Blanckaert and Graf, 2004; Blanckaert and de Vriend, 2005a, 2005b, 2010; Blanckaert, 2009]. This is particularly important for reaches of strong channel curvature (ratio between the mean channel curvature radius, R, and the channel width, B, smaller than 2).

[3] The role of secondary currents is particularly complex in alluvial channels where regions of sediment deposition (point bars) and sediment erosion (pools) are forming because of the interactions among the secondary currents, the streamwise flow, the turbulence, and the sediment. Several experimental studies considered the case of flow over naturally deformed bathymetry in curved open channels with vertical nonerodible banks [e.g., see Odgaard and Bergs, 1988; Blanckaert and Graf, 2001; Blanckaert and de Vriend, 2004; Abad and Garcia, 2009; Roca et al., 2009; Jamieson et al., 2010]. Another factor that adds to the flow complexity is that the flow can separate in regions situated close to the inner bank (e.g., see Bagnold [1960], Leeder and Bridges [1975], Ferguson et al. [2003], and Frothingham and Rhoads [2003] for field observations and Blanckaert [2010, 2011] for laboratory observations) and in the lee of the submerged point bars [Frothingham and Rhoads, 2003]. This is accompanied by the formation of strong separated shear layers (SSLs), which are regions of large, localized, shear-driven turbulence production. The presence of low-velocity recirculation regions has important implications for sedimentation [Ferguson et al., 2003] and can alter the development of the main cell of cross-stream circulation in the curved bend. In the case when the channel banks are erodible, strong secondary currents may endanger bank stability by increasing the flow depth near the outer bank. Leeder and Bridges [1975], Ferguson et al. [2003], and Blanckaert [2011] provide a detailed discussion of the possible effects of inner bank flow separation on the outer bank flow.

[4] The most common type of coherent structure associated with secondary flow in curved bends is the main cell of cross-stream circulation, or helix, within which the fluid particles follow a helicoidal path. If the bed is not strongly deformed, the main cell extends over a large part of the cross section [e.g., see Blanckaert, 2010]. Once the pool-point bar structure develops, this cell generally extends only over the outer, deeper part of the channel [Thorne and Rais, 1985; Blanckaert, 2010; Zeng et al., 2008a].

[5] Previous studies [e.g., Rozovskii, 1957; Blanckaert and de Vriend, 2004; Blanckaert, 2011] have shown that a weaker secondary counterrotating cell of cross-stream circulation can form close to the free surface and the outer bank in bends of mild and strong curvature. Such outer bank cells were also observed in natural channels [Bathurst et al., 1977, 1979; Dietrich and Smith, 1983; Thorne and Hey, 1979; de Vriend and Geldof, 1983; Thorne et al., 1985]. It is thought [e.g., see Blanckaert and Graf, 2004] that the formation of the outer bank cell results in a buffer layer that protects the outer bank from the influence of the main cell of cross-stream motion. This results in smaller sidewall shear stresses in the region adjacent to the cell, thus enhancing bank stability. On the other hand, Bathurst et al. [1979] postulated that the outer bank cell may endanger bank stability by advecting high-momentum fluid near the free surface, toward the lower part of the bank. Still, as the vertical extent of this cell is limited once a pool region has developed near the outer bank, the presence of a strong main cell is expected to significantly increase the outer wall shear stresses with respect to the case of a straight channel. This is why state-of-the-art methods to predict bank erosion incorporate the effects of the cross-stream flow [e.g., see Pizzuto, 2008].

[6] As the present study will show, in the case of strongly curved bends, in addition to the main cross-stream circulation cell situated in the deeper (outer) part of the bend, secondary currents in the form of localized streamwise-oriented vortical cells (SOVs) can form close to the inner bank. The effects of the SOV cells on the distributions of the inner bank and bed shear stresses in the neighboring regions are unknown.

[7] As the bathymetry evolves in time, the position and strength of the secondary currents and the way they contribute to the redistribution of the streamwise momentum change. For large bed deformations, the bathymetry contributes directly to the redistribution of the velocity in the cross section. This effect is called topographic steering of the flow [Whiting and Dietrich, 1993] (also see Blanckaert [2010] for a discussion of this effect for the strongly curved bend considered in the present study). Eventually, if the inflow conditions are relatively constant over a sufficiently large period of time, the large-scale features of the bathymetry shape will reach an approximate equilibrium state. Still, small-scale bed forms like ripples and dunes continue to form and move over the large-scale bathymetry features after equilibrium was reached.

[8] Detailed field studies of flow in natural river bends [e.g., Thorne and Rais, 1985; Ferguson et al., 2003; Frothingham and Rhoads, 2003; Blettler et al., 2010] are relatively scarce because of the high costs involved. Moreover, changing conditions in the river during the measurement period adds another level of complexity that has to be taken into account when analyzing the data. The resolution at which the measurements are performed within the area of interest in such field studies and the accuracy of these measurements are lower than those of state-of-the-art laboratory experiments. In this regard, though subject to scale effects, experiments conducted in controlled environments allow a more detailed investigation of the mean flow and turbulence structure. Still, in many cases the experimental investigations report detailed measurements only in one or few cross sections, which is not enough for a detailed characterization of the secondary flow over the whole bend and accurate identification of the position and distribution of the regions of concentrated vorticity associated with the cells of cross-stream circulation and SSLs in that section. Moreover, the distributions of the shear stresses and pressure RMS fluctuations at the deformed bed and the sidewalls cannot be accurately estimated.

[9] Besides providing a higher-resolution description of the mean flow and turbulence, eddy-resolving numerical simulations conducted on fine meshes can be used to examine the variation of the mean flow and turbulence over the entire flow depth, including the regions close to the bed and the two banks, where large-scale coherent structures interact with the channel boundaries. Large-eddy simulation (LES) was already successfully used by Stoesser et al. [2010], van Balen et al. [2009, 2010a], and Moncho-Esteve et al. [2010] to predict the mean flow and turbulence in curved open channels of mild and strong curvature at conditions corresponding to the start of the scour and erosion process (flat bed). The success of LES is not surprising because an advanced turbulence model, capable of accurately resolving turbulence anisotropy effects and the kinetic energy transfer between the mean flow and turbulence, is needed to capture the velocity redistribution in flat bed bends. In contrast, Reynolds-averaged Navier-Stokes (RANS) models with isotropic turbulence closures are much less successful in predicting the details of the secondary flow (e.g., the formation of the outer bank cell) in such channels and overestimate the friction losses [e.g., see van Balen et al., 2010a].

[10] However, past the initial stages of the scour and erosion process, when the bed develops large-scale deformations, the superiority of 3-D LES over 3-D RANS is not so clear [e.g., see van Balen et al., 2010b]. This is because the role of the anisotropy-induced secondary motions that are difficult to capture accurately with RANS is smaller. In fact, 3-D RANS [e.g., see Khosronejad et al., 2007; Zeng et al., 2008b, 2010] was shown to be relatively successful in predicting equilibrium bathymetry in loose bed open channels of medium curvature (4 < R/B < 8). As the redistribution of the velocity is driven and redirected to a larger degree by the action of the pressure gradients induced by the large-scale bed deformations, fully 3-D RANS should, in principle, successfully predict the flow in bends in which topographic steering effects are significant [Ferguson et al., 2003].

[11] Zeng et al. [2008a] were the first to use RANS to calculate the flow in the 193° bend of high curvature (R/B = 1.3) studied experimentally at École Polytechnique Fédérale de Lausanne by Blanckaert [2002, 2010]. RANS with a movable bed predicted the maximum scour depth with reasonable accuracy (within 20% of the mean channel depth). The main conclusion was that discrepancies between measurements and RANS predictions were due to the failure of the turbulent flow module of the morphodynamic code to predict the flow and turbulence structure in the bend and to deficiencies in the modeling of the sediment transport that determines the evolution of the bed. This means that to increase the accuracy of numerical models used to predict flow and to simulate sediment transport and morphodynamic changes in loose bed open channel bends and natural streams, more complex sediment transport (in particular, bed load transport) models and more advanced turbulence models have to be used.

[12] The main options available for increasing the accuracy of the predictions of the mean flow and turbulence structure in curved open channels and natural streams are to use anisotropic Reynolds stress RANS models, LES, or hybrid RANS-LES approaches. The last two approaches fall in the category of eddy-resolving techniques that directly resolve the large-scale coherent structures up to a certain size that depends on the local mesh size. Basically, this is achieved by making the eddy viscosity proportional to the local grid spacing either in the whole domain (LES) or in the regions situated away from the solid boundaries (hybrid RANS-LES). Recently, van Balen et al. [2010b] reported results of a LES simulation of the same test case considered in the present paper. The LES was conducted using a finite volume method in which an immersed boundary method was used to represent the deformed bathymetry. The classical Smagorinsky model with a constant coefficient was used as a subgrid-scale model. The smooth and rough (channel bottom) walls were treated using the wall function approach.

[13] For field-scale simulations, the main options are RANS, LES with wall functions, and hybrid RANS-LES approaches. As our long-term goal is to use the same numerical model to study flow and sediment transport in natural streams at field conditions, the use of a hybrid RANS-LES model that allows performing eddy-resolving simulations at field channel Reynolds numbers (e.g., around 106) appears to be the best option. This is because well-resolved (no wall functions) LES is too expensive to simulate the flow at such high Reynolds numbers, while a standard LES with wall functions uses a much more simplified approach to account for the flow processes in the near-bed region compared to an advanced hybrid RANS-LES model like detached eddy simulation (DES). For most complex turbulent flows, DES is significantly more accurate compared not only to RANS and unsteady RANS but also to LES with wall functions [Spalart, 2009]. In fact, one can think of such hybrid RANS-LES models as LES with a more sophisticated wall model.

[14] The good performance of DES in predicting massively separated flows (e.g., flow past bridge piers, bridge abutments, river groynes, etc.) over a wide range of Reynolds numbers was already proven [e.g., see Spalart, 2009; Kirkil and Constantinescu, 2009]. However, flow in curved bends does not fall into this category, especially if large emerged or submerged islands or hydraulic structures are not present in the channel. A careful assessment of the performances of DES for this type of flow in which strong adverse pressure gradients are present but the flow does not resemble a wake flow is needed.

[15] The main goals of this paper are to (1) assess the predictive capabilities of DES to capture the details of the mean flow developing over deformed bed at equilibrium in a curved bend of high curvature and compare with the predictive abilities of RANS and LES with wall functions and (2) use the data to examine details of the three-dimensional flow and turbulence structure and estimate relevant quantities (e.g., the boundary shear stresses and pressure RMS fluctuations) that cannot be accurately estimated from laboratory experiments.

2. Experimental Test Case

[16] Figure 1 shows a sketch of the flume in which the M89 clear-water scour mobile bed experiment was carried out by Blanckaert [2002, 2010]. Full descriptions of the flume, experiment, and measurement techniques are given by Blanckaert [2010]. The laboratory flume consisted of three sections: a 9 m long straight inflow channel reach followed by a 193° bend with constant centerline radius of curvature R = 1.7 m and a 5 m long straight outflow reach. The total length of the flume was 22.7 m along the centerline. The width of the flume was B = 1.3 m, and the sidewalls (banks) were vertical. The averaged flow depth in the M89 test case was H = 0.141 m, and the inflow discharge was 0.089 m3/s. The bed was covered with quasi-uniform sand with an average diameter of about d50 = 0.002 m. The bathymetry was measured after the flow and sediment transport reached equilibrium. The average height of the small-scale traveling dunes was 0.03–0.05 m. The Reynolds number and the Froude number calculated using H and the flume-averaged velocity were 68,400 and 0.41, respectively.

Figure 1.

Sketch of the flume in which the experiment was conducted.

[17] The bathymetry at equilibrium conditions is depicted in Figure 2. The measured equilibrium bathymetry shows that scour develops in the outer half of the cross section upon entering the curved reach. The flow depth reaches a maximum of about 3H in the cross section situated at 60° within the bend (section D60 in Figure 1) before reducing to a value of about 2.2H in sections D120 and D150. Scour increases again upon approaching the bend exit. The flow reaches a depth of about 2.6H in section D180. Then, the scour decays in the straight outflow reach, and the bed is close to flat in the outer part of section P2.9, situated 2.9 m downstream of the end of the curved reach. Two main regions of sediment deposition form in the inner half of the cross section. They are situated in between sections D30 and D100 and in between sections D180 and P1.0, respectively.

Figure 2.

Bathymetry at equilibrium conditions. The bed elevation (z/D) is measured with respect to the mean position of the free surface (z/D = 0) in the inlet section.

[18] Detailed velocity measurements of the flow at equilibrium conditions by means of an acoustic Doppler velocity profiler (ADVP) are available in representative sections for validation [Zeng et al., 2008a; Blanckaert, 2010]. The accuracy of the ADVP measurements is questionable at distances of less than 2 cm from the bed, the sidewalls, and the free surface. Blanckaert [2010] provides more details on the measurements and discusses the accuracy of the ADVP technique.

3. Numerical Model

[19] DES uses the same base turbulence model in the regions where DES is in unsteady RANS mode (close to solid walls) and in LES mode (away from solid walls). No special treatment is required to match the solutions at the boundary between the LES and RANS regions. The model resolves the near-wall flow. Thus, the calculation of the bed shear stresses does not rely on the assumption of the presence of a logarithmic layer in the velocity profile, as is the case in which wall functions are used. The present study uses the Spalart-Allmaras (SA) RANS model as the base model in DES [Spalart, 2000]. The one-equation SA model solves a transport equation for the modified eddy viscosity, equation image. The boundary condition implementation for rough walls is described by Spalart [2000]. The SA version of DES is obtained by replacing the turbulence length scale, d (distance to the nearest wall), in the destruction term of the transport equation for equation image with a new length scale, dDES = min(d, CDESΔ), where the model parameter CDES is equal to 0.65 and Δ is a measure of the local grid size. The value of CDES was obtained on the basis of calibration with data for isotropic turbulence [Spalart, 2009]. When the production and destruction terms in the transport equation for equation image are balanced, the length scale in the LES regions, dDES = CDESΔ, becomes proportional to the local grid size and yields an eddy viscosity proportional to the mean rate of strain and Δ2, as in LES with a Smagorinsky model. This allows the energy cascade down to grid size. Close to smooth or rough walls, DES reduces to a low-Reynolds-number version of a one-equation RANS model, which can resolve the near-wall flow significantly more accurately than using a wall function-based approach.

[20] One should also stress that even away from the walls, where DES is in LES mode, DES is not fully equivalent to the Smagorinsky model with a constant coefficient employed in most LES of engineering flows. This is the model used by van Balen et al. [2010b] to simulate the same test case. The equivalent Smagorinsky coefficient arising from assuming production and dissipation are nearly equal in the LES mode of DES is not a constant. Rather, its value adapts to the amount of energy in the smallest resolved scales. This behavior of the model parameter is qualitatively similar to that of the dynamic version of the Smagorinsky version. However, the relationship that determines the local value of the coefficient is significantly different than the one used in the dynamic version of the Smagorinsky model.

[21] Detailed descriptions of the DES code and DES-SA model equations are given by Constantinescu and Squires [2004] and are not repeated here. The RANS version of the same code was used by Zeng et al. [2008b, 2010] to calculate flow and sediment transport in open channels bends of medium curvature with movable bed and by Zeng et al. [2008a] to calculate the flow and sediment transport in the geometry considered in the present paper.

[22] The 3-D incompressible Navier-Stokes equations are integrated using a fully implicit fractional step method. The governing equations are transformed to generalized curvilinear coordinates on a nonstaggered grid. For DES calculations, the convective terms in the momentum equations are discretized using a blend of a fifth-order-accurate upwind biased scheme and a second-order central scheme. All other terms in the momentum and pressure-Poisson equations are approximated using second-order central differences. The discrete momentum (predictor step) and turbulence model equations are integrated in pseudotime using an alternate direction implicit approximate factorization scheme. Time integration is done using a double time stepping algorithm with local time stepping. The time discretization is second order accurate. Validation of the DES code for flow in channels with bottom-mounted cavities, open channels with sidewall-mounted obstacles, and flow past surface-mounted bluff bodies with a flat and scoured bed is discussed by Chang et al. [2007], Koken and Constantinescu [2009], Kirkil et al. [2009], and Kirkil and Constantinescu [2009, 2010].

4. Simulations Setup

[23] At the inflow section, turbulent inflow conditions corresponding to fully developed turbulent channel flow with resolved turbulent fluctuations are applied in DES. A steady fully developed precalculated RANS solution was used to specify the inflow conditions in the RANS simulations [Zeng et al., 2008a]. At the outflow, a convective boundary condition is used in DES. All the solid surfaces are treated as no-slip boundaries. In both RANS and DES, the equivalent total bed roughness estimated using the procedure described by Zeng et al. [2008a] was 0.037 m. This value is comparable to the measured height (0.03–0.05 m) of the small-scale traveling dunes present in the experiment.

[24] The free surface is treated as a rigid lid, which is an acceptable approximation if the local Froude number is smaller than 0.8 (e.g., see discussion by Kirkil and Constantinescu [2009, 2010] and Zeng et al. [2010]). The rigid lid assumption works well as long as the superelevation of the free surface is less than 10% of the channel depth [van Balen et al., 2010a]. The LES of van Balen et al. [2010b] confirmed the validity of this assumption for the present test case. To further minimize the effects of using an approximate method to represent the free surface, the free surface deformation measured in the experiment was taken into account when the computational domain and mesh were generated. This resulted in a much smaller level of nonuniformity of the pressure distribution on the free surface during the simulation compared to the case when the free surface is modeled as a horizontal surface. In the latter case, a large transverse mean pressure gradient, corresponding to the water superelevation at the outer bank, is predicted in regions of high bend curvature. The latter treatment was used in the LES of van Balen et al. [2010b].

[25] The computational domain in DES was meshed using about 12 million cells (1820 × 192 × 35 points in the streamwise, spanwise, and vertical directions, respectively). Away from the solid walls, the average edge of a cell was 5–8 mm in all three directions. The mesh in the RANS simulation was much coarser (0.5 million cells), but the RANS solution was checked to be grid independent. In both RANS and DES, a minimum grid spacing of one wall unit was used in the wall normal direction to avoid the use of wall functions. The location of the inflow section in DES and RANS was the same (9 m upstream of the bend entrance).

[26] The DES simulation required about 12,000 CPU hours of computer time. In comparison, a steady RANS simulation performed with a near-wall model requires only several hundreds hours of CPU time. The difference is due to the fact that DES has to be run as a time-accurate simulation with a sufficiently small time step, needed to capture the dynamics of the energetically eddies in the flow, even if the mean flow is steady. DES also requires a finer mesh in regions where small but energetically important eddies form. Compared to LES (Smagorinsky model) with wall functions, SA-based DES is about 30% more expensive because of the additional transport equation that has to be solved for the modified eddy viscosity and the finer mesh in the wall normal direction required by the use of a near-wall RANS model in the regions where DES is in RANS mode.

5. Description of the Mean Flow, Turbulence, and Main Vortical Structures

[27] Blanckaert [2009, 2010] provide a detailed presentation of the main flow characteristics in the flat and deformed bed cases and analyze the main hydrodynamic processes on the basis of the mean flow and turbulence measurements. In the following, we will focus our discussion on flow features (SOVs and SSLs) that cannot be easily visualized and/or characterized in a quantitative way on the basis of experimental measurements as well as on their effect on the mean flow and turbulence intensity.

[28] The mean flow structure predicted by DES is visualized in Figure 3 at the free surface and in a horizontal plane situated at 0.5D below the free surface. The results are presented in nondimensional form. The length scale is selected to be D = 0.115 m. The velocity scale is U = 0.61 m/s. Compared to the flat bed case [Zeng et al., 2008a], the main difference is the formation of two recirculation eddies over the shallower part of the bend. The center of the first eddy (RE1) is situated close to the D90 cross section. This eddy is induced by the combined effects of the high bank curvature and bed superelevation due to sediment deposition that force the flow to separate close to section D40. Downstream of section D120, the flow expands asymmetrically and reattaches to the inner bank. The second recirculation eddy (RE2) is mainly driven by the formation of a wake behind the second region of high sediment deposition centered around section P0.5. The flow separates in the lee of the submerged mount of sediment, which is part of the point bar forming at the inner bank. The separation is strong enough and the flow is sufficiently shallow such that RE2 extends up to the free surface. Such eddies were observed to form in natural channels by Frothingham and Rhoads [2003].

Figure 3.

Visualization of the mean flow (top) at the free surface and (bottom) in a horizontal section situated at z/D = −0.5. (a) Two-dimensional streamline patterns, (b) streamwise velocity, uξ/U, and (c) out-of-plane vorticity, ωzD/U.

[29] The main effect of the formation of the two recirculation eddies is to push the main current closer to the outer wall. Strong SSLs form in between these eddies and the main current (Figure 3c). The high shear-driven turbulence production within the shear layer region results in the formation and shedding of energetic turbulent eddies over the downstream part of the SSLs (see Figure 18).

[30] Close to the free surface, the shape of the SSL on the outer side of RE2 follows the direction of the boundary layer on the outer side of the submerged point bar. The SSL penetrates until close to the channel centerline. A similar SSL was observed in the LES with wall functions [van Balen et al., 2010b, Figure 5a].

[31] The evolution of the SSLs is much more complex for RE1. A strong SSL forms in between the core of high streamwise velocity fluid and the slowly moving fluid situated close to the inner bank. The formation of this SSL is due to the fact that the core of high streamwise momentum fluid tries to preserve its original direction, which is parallel to the direction of the inflow straight reach. The large bank curvature does not allow the core of high-velocity fluid to follow the inner bank. Simulations conducted in our group [e.g., see Kashyap et al., 2010] have shown that this shear layer is also present in curved channels with a flat bed, where the flow does not separate in horizontal planes at the inner wall, provided that the inner bank curvature is sufficiently large (R/B < 2 in our simulations) for the core of high-velocity incoming fluid to be deflected away from the inner bank. Close to the entrance into the bend, the shape of the outer face of the point bar is such that it starts deflecting the core of high streamwise velocity flow around section D30. Near the free surface, the SSL penetrates up to the outer wall. This explains the disappearance of the region of high incoming streamwise velocity around section D90 in Figure 3b. At lower levels, the SSL penetrates less. This allows the higher-velocity fluid to be convected downstream. For example, Figures 3b and 3c show that at z/D = −0.5 the downstream part of the SSL is strongly deflected close to the channel centerline, such that the SSL becomes parallel to the outer bank and is situated at a distance of about B/3 from it. The strong vertical nonuniformity of the shape of the SSL partially explains why the core of high streamwise velocity fluid is situated below the free surface in the central part of the bend (around section D90). The flow recovers gradually past the flow constriction imposed by the presence of the strong SSL close to the free surface. Downstream of section D120, the maximum streamwise velocity in the cross section is again observed close to or at the free surface (Figure 3b).

[32] Additionally, a second SSL forms on the outer side of the recirculation eddy RE1 (Figure 3c). Similar to the SSL forming on the outer side of RE2, this SSL originates in the attached boundary layer that separates in the region (around section D75) where the curvature of the lateral face of the mount of deposited sediment at the inner bank changes abruptly. The adverse pressure gradients are strong enough to allow the formation of a wake in the near-bed region that eventually extends up to the free surface. The shape of the second SSL is much less variable in the vertical direction. It is interesting to point out that LES [see van Balen et al., 2010b, Figure 5a] captured only the presence of a SSL similar to the second SSL observed around RE1 in DES.

[33] The Q criterion [Hunt et al., 1988] was used in Figures 4 and 5 to visualize the main large-scale eddies in the mean flow. The quantity Q used to visualize vertical structures is the second invariant of the velocity gradient tensor (Q = −0.5(∂ui/∂xj)(∂uj/∂xi)) and represents the balance between the rotation rate and the strain rate. The vortex V1 corresponds to the main cell of cross-stream circulation developing inside the outer half of the cross section. Its circulation peaks in between sections D60 and D90. As it passes RE1, the coherence loss of V1 is enhanced by vortex stretching, as the core of high streamwise velocity fluid starts reoccupying the whole width of the channel [see also Ferguson et al., 2003]. The value of the Q isosurface used to visualize the cores of the main vortices in Figure 5 was too high for the core of V1 to be present in section D120. The distribution of the streamwise vorticity in section D120 (Figure 11) will show that V1 is still present in section D120, but its coherence is significantly reduced because of the fast diffusion of V1 downstream of section D90. The eddy present in Figure 4 close to the channel centerline, in between sections D60 and D90, corresponds to the SSL on the outer side of RE1. The dominant vorticity component inside that eddy is vertically oriented rather than streamwise oriented.

Figure 4.

Three-dimensional visualization of the vortical structure of the mean flow predicted by detached eddy simulation (DES) using the Q criterion (DES).

Figure 5.

Visualization of streamwise-oriented vortices (SOVs) in representative cross sections using the Q criterion (DES). The sense of rotation of the SOVs is indicated in parentheses (C, clockwise; CC, counterclockwise).

[34] Three more SOVs, denoted V2, V3, and V4, form in the vicinity of the inner bank and then slowly diverge from it. These three vortices are not connected. The circulation of the SOVs increases quickly close to the position where they originate and then decays slowly in the streamwise direction. The formation of these vortices, in particular that of V3 (e.g., section D120 in Figure 5) and V4 (e.g., section P2.0 in Figure 5), is favored by the presence of a deeper region close to the inner bank in some parts of the flume. One can also argue that the formation of the SOVs triggers the changes in the bathymetry observed at equilibrium conditions close to the inner bank. As will be discussed in section 7, this appears to be the case of V4, which induces a significant cross-stream boundary shear stress even when the bathymetry is at equilibrium. This is possible because the bed slope beneath V4 is relatively high. As the discussion of the mean flow structure in a representative cross section will show, the coherence of the SOVs (e.g., V3 and V4) as measured by their circulation can be quite strong. This can also be inferred from the fact that the areas of the cores of V1 in sections D60 and D90 and of V4 in section P2.0 are comparable in Figure 5. Coupled with the fact that a large amplification of the turbulent kinetic energy (TKE) is observed inside the cores of some of these SOVs (see Figure 6), their presence may substantially amplify the bed friction velocity and pressure RMS fluctuations on the solid surfaces (banks and bed) situated in their vicinity, at various stages of the erosion and deposition process.

Figure 6.

Turbulent kinetic energy, 100k/U2, in sections (top) D120 and (bottom) P2.9 predicted by DES.

[35] The main features of the distribution of the depth-averaged nondimensional TKE in Figure 7a can be explained on the basis of the vortical structure of the flow and the position of the regions in which strong energetic turbulent eddies are produced. The patch of high TKE close to the outer bank, situated in between sections D90 and D120, is due to the large TKE amplification inside the core of V1 (e.g., see Figure 6). Interestingly, the TKE amplification is not very high inside the upstream part of V1, where the position of its core is relatively stable in the instantaneous flow fields. Once V1 starts growing, its core becomes more unstable, and phenomena such as vortex breakdown are observed. They explain the large amplification of the TKE inside the region occupied by V1 past section D90. The patch of high TKE close to the channel centerline, situated in between sections D90 and D120, is mainly due to the passage of the highly energetic eddies detaching from the downstream part of the SSLs forming in the vicinity of RE1. The depth-averaged TKE is lower over the deeper region where most of the incoming high-velocity fluid is pushed past the bend apex. The core of high velocity is situated below the free surface (Figures 3b and 11). The turbulence levels inside the core of high-velocity fluid are comparable to those inside the straight inflow reach. It is only close to the free surface, where the SSL forming because the core of high-streamwise momentum fluid tries to preserves its original direction, that the TKE is strongly amplified. The downstream part of the SSL that is populated by strongly energetic eddies penetrates until close to the outer bank (Figure 18) and increases the TKE.

Figure 7.

Distribution of the depth-averaged turbulent kinetic energy, 100〈k〉/U2, in the channel. (a) DES and (b) Reynolds-averaged Navier-Stokes (RANS) simulation.

[36] The region of relatively high TKE close to the inner wall of the straight outflow reach in Figure 7a is induced by V4 (see also TKE distribution in section P2.9 in Figure 6). Additionally, the TKE is amplified in the wake of RE2. Finally, a region of smaller but noticeable amplification of TKE compared to the background levels corresponding to fully developed flow in a straight channel is present in the vicinity of the inner bank, in between section D90 and the bend exit. It is due to the amplification of the TKE inside the core of V3 (e.g., see TKE distribution in section D120 in Figure 6).

[37] The main differences between the TKE distributions predicted by DES and RANS are the lower TKE predicted by RANS inside the core of V1 and the SSL forming in the vicinity of RE1. The latter is directly related to the known difficulties of RANS to predict the correct amplification of the TKE due to the passage of large-scale energetic eddies in a SSL. Also, RANS predicts larger TKE levels at the boundary between RE2 and the core of high-speed fluid. This is mainly because the size of RE2 is larger in RANS. This results in a larger streamwise velocity gradient in the spanwise direction and an increased turbulence production by shear in RANS.

[38] As discussed by Zeng et al. [2008a], the rapid development of the transverse bed slope past section D30 induces a rapid spanwise redistribution of the streamwise discharge over the section width. The spanwise distribution of streamwise discharge per unit width qs is modulated by the bed topography. The unit discharge in Figure 8 was calculated using the local flow depth and the depth-averaged streamwise velocity. Experimental [Blanckaert, 2010] and simulation results show that in section D60 more than 90% of the discharge flows through the outer half of the cross section and the maximum unit discharge is about 3 times higher than in the straight approach flow. The shallow inner half of the cross section over the point bar developing between sections D30 and P1.5, which contains the two recirculation regions, transports very little of the flow discharge. Downstream of section D60, about 75% of the flow discharge is conveyed through the outer half of the cross section.

Figure 8.

Distribution of the normalized streamwise unit discharge, qs, in the channel. (a) Experiment, (b) RANS, and (c) DES.

6. Analysis of Curvature-Induced Effects on the Streamwise Velocity and Secondary Flow

[39] Though the comparison with experiment is not very easy, as velocity measurements are not available close to the two banks (Figure 8a), it is clear that RANS overpredicts the size of the two recirculation regions that correspond to the regions with low or negative values of qs at the inner bank. One can argue that the same is true for DES, but the error is smaller, by about 40% for RE1, compared to RANS. Moreover, DES captures the formation of a finger-like region of high qs away from the outer wall around section D120, a feature not reproduced by RANS.

[40] The RANS and DES predictions of the nondimensional streamwise vorticity in section D60 are compared in Figure 9 with the experiment. This section cuts through the main cell of cross-stream circulation, V1. Similar to the experiment, DES captures the formation of a relatively circular patch of high streamwise vorticity at the end of the bottom-attached boundary layer induced by the cross-stream motions in the vicinity of the bed, over the inner half of the section. This compact patch of high streamwise vorticity coincides with the position of the core of V1, as visualized by the Q isosurface in Figure 5. Meanwhile, RANS does not capture the presence of a distinct region of high streamwise vorticity that can be associated with V1. In fact, additional comparison of the strength of the cross flow motions in relevant cross sections showed the circulation of V1 was consistently and significantly underpredicted by RANS compared to DES. The underprediction of the circulation and coherence of V1 by RANS explains the severe underestimation (by more than 50% at some locations) of the transverse velocity close to the free surface at stations located over the deeper part of section D60 (e.g., see profiles at η/D = 2.25 and η/D = 4.5 in Figure 10, where η is the transverse coordinate in the section measured from the centerline). At η/D = 4.5, DES shows a closer level of agreement with the experiment not only when compared to RANS but also when compared to LES with wall functions.

Figure 9.

(left) Mean flow velocity, uξ/U, (middle) streamwise vorticity, ωξD/U, and (right) 2-D streamlines in section D60. (a) Experiment, (b) RANS, and (c) DES.

Figure 10.

Comparison of measured (black circles), RANS (dashed blue line), DES (solid line with red squares), and large-eddy simulation (LES) (solid black line) vertical profiles of the nondimensional transverse velocity in section D60.

[41] The distributions of the streamwise velocity in section D60 (Figure 9) are also affected by how well the simulations capture V1. In the experiment and DES, the core of V1 corresponds to a region where the streamwise velocity is smaller than that in the surrounding flow. The role of V1 is to convect higher streamwise velocity fluid from the free surface first downward, parallel to the outer bank, and then upward, parallel to the deformed bed. As RANS fails to predict the formation of a high-circulation compact vortex core, the redistribution of the streamwise velocity in section D60 is captured less accurately compared to DES.

[42] In DES, V1 is still present at section D120 (Figure 11). The streamwise vorticity levels within its core are much lower compared to section D60, while the core is more diffused. This is consistent with the streamwise expansion of V1 in Figure 4. The experiment and DES show the presence of a relatively large patch of higher-magnitude streamwise vorticity compared to that in the surrounding flow. This patch is situated close to the free surface and the outer wall. In contrast, no patch of vorticity corresponding to V1 can be observed in section D120 in the RANS results. This is consistent with the underestimation by RANS of the strength and compactness of the vorticity distribution inside the core of V1 at section D60.

Figure 11.

(left) Mean flow streamwise velocity, uξ/U, (middle) streamwise vorticity, ωξD/U, and (right) 2-D streamlines in section D120. (a) Experiment, (b) RANS, and (c) DES.

[43] A strong SOV, V3, is present close to the inner bank in DES. Unfortunately, no velocity measurements are available inside that region. The presence of a strong vortex at the inner bank is fully consistent with the local scour observed close to the inner bank (η/D < −4; see also Figure 5). Moreover, the vorticity levels between η/D = −2 and η/D = −4.5 are in fairly good agreement in experiment and DES. In contrast, away from the bed, RANS predicts significantly lower vorticity values over this region. Two-dimensional streamline patterns in Figure 11 show the presence of an eddy similar to V3 in the RANS simulation. However, its circulation is a couple of times lower compared to that of V3 in DES. As already pointed out, the TKE levels predicted by DES are very high inside the core of V3 (Figure 6). The presence of a strongly coherent vortex that can induce relatively high transverse bed shear stresses and large turbulent fluctuations offers a good explanation for the local scour observed close to the inner wall in section D120.

[44] Comparison of the streamwise velocity distributions in Figure 11 shows other interesting differences between the RANS and DES predictions. The submerged core of large streamwise velocities has started moving toward the inner bank (see also Figure 3b). Experimental measurements show that the center of this core is situated close to the centerline. The region of high streamwise velocity (uξ/U > 0.8) penetrates up to η/D ∼ −3.5 on the inner side and up to η/D ∼ 3 on the outer side of the section. Though some differences are observed between experiment and DES, the overall agreement is quite satisfactory. RANS shows several qualitative and quantitative differences with both experiment and DES. The peak values inside the core of large streamwise velocities are smaller. Meanwhile, the area occupied by the core is larger. In particular, rather than extending up to η/D ∼ 3, the core penetrates in the lower part of the section until close to the outer bank. This is possible because of the severe underprediction of the coherence of V1 in the same section.

[45] Comparison of the streamwise velocity profiles in section D180 (Figure 12a) shows that the LES and DES predictions are remarkably close and better capture the velocity gradient over the deeper parts of the flow compared to RANS. The better accuracy of DES compared to RANS is not limited to cross sections in which strong SOVs are present. For example, comparison of the vertical profiles of streamwise velocity in section P0.5 (Figure 12b) shows that DES predicts larger streamwise velocities over the central part of the cross section (e.g., see station situated at η/D = 0), which is in quite good agreement with the experiment. Even closer to the inner bank (η/D = −2.25), DES predictions are slightly closer to the experiment compared to RANS. The main reason for the larger errors given by RANS in section P0.5 is that the streamwise velocity is substantially overestimated close to the outer bank.

Figure 12.

Comparison of measured (black circles), RANS (dashed blue line), DES (solid line with red squares), and LES (solid black line) vertical profiles of the nondimensional streamwise velocity in (a) section D180 and (b) section P0.5.

[46] Farther downstream in the straight outflow reach, RANS predictions of the streamwise velocity are fairly close to DES in terms of the overall level of agreement with experiment. Some differences can still be pointed out. For example, at most sections, RANS overestimates more than DES the peak streamwise velocity within the core of high uξ values near the outer bank. This is exemplified in Figure 13 for section P2.9. However, most of the important differences in the flow structure between DES and RANS in the straight outflow reach are observed in the distributions of the cross-flow motions.

Figure 13.

(left) Mean flow streamwise velocity, uξ/U, (middle) streamwise vorticity, ωξD/U, and (right) 2-D streamlines in section P2.9. (a) Experiment, (b) RANS, and (c) DES.

[47] For example, RANS fails to predict the circular region of negative vorticity associated with V4 at η/D ∼ −4.5 and the tongue of negative vorticity starting at the crest of the bathymetry hump centered around η/D = 0 in section P2.9. Both flow features are present in the streamwise vorticity field calculated on the basis of velocity measurements. The vorticity values in the two regions of negative vorticity are comparable in DES and experiment (∼−1.5U/D). In contrast, the streamwise vorticity values predicted by RANS are very low over the whole section. Comparison of the 2-D streamline patterns in Figure 13 shows that RANS predicts the formation of a recirculation eddy close to the bed, in between the inner wall and η/D = −0.5. However, the circulation of this eddy, which occupies the region where the strongly coherent vortex V4 and the weaker cell centered at η/D = −2.5 are present in DES, is very small. The 2-D streamline patterns in experiment and DES are similar in this region. Examination of the instantaneous flow fields from DES shows that the circulation of V4 is sufficiently large to induce the formation of a bottom-attached vortex that, at times, ejects patches of positive vorticity into the outer flow. This is the main mechanism that modulates the intensity of V4 in time. This detaching boundary layer is situated too close to the bed to be accurately captured by the measured velocity field in section P2.9.

7. Coherent Structures and Predicted Patterns of Boundary Shear Stresses and Pressure RMS Fluctuations

[48] Bend scour and bank erosion can be affected by the presence of large-scale energetic eddies in their vicinity, such as the SOVs and the eddies forming via the growth of the Kelvin-Helmholtz instabilities in the downstream part of SSLs [Leeder and Bridges, 1975]. Such turbulent structures can significantly amplify the boundary shear stresses above their time-averaged values and increase the pressure RMS fluctuations.

[49] Figure 14 compares the bed shear stresses predicted by DES and RANS in the mean flow. Figure 15 shows the distributions of the streamwise and transverse components of the bed shear stress. The boundary stresses, τ, are nondimensionalized by the average value of the boundary shear stress in the inlet section, τ0, where the flow is fully developed. Both simulations predict a strong amplification of the bed shear stresses within the curved reach, in between the channel centerline and the outer bank. The main qualitative difference is that the region of large bed shear stress (τ/τ0 > 3) within the bend is situated in between sections D30 and D120 in DES and in between D60 and D193 in RANS. The peak value of τ/τ0 is larger by about 50% in DES. Figure 15 shows that the high transverse bed slope region near the outer bank, situated between sections D30 and D120, is the only one where the transverse bed shear stress is comparable to the peak values of the streamwise bed shear stress. It roughly corresponds to the region where the coherence of V1 is the highest. Downstream of section D120, the compactness of the core of V1 decreases strongly. Thus, the capacity of V1 to induce large transverse bed shear stresses decreases. Meanwhile, the core of high streamwise velocity starts moving again toward the free surface, reducing the velocity gradients near the bed. This decreases the streamwise component of τ. As expected, the magnitude of τ is small inside the two recirculation eddies, RE1 and RE2, forming near the inner bank.

Figure 14.

Distribution of the nondimensional shear stress, τ/τ0, at the bed. (a) DES and (b) RANS.

Figure 15.

DES predictions of the (a) streamwise and (b) transverse components of the nondimensional shear stress at the bed.

[50] The differences in the distributions of τ are less pronounced in the straight outflow reach. Both RANS and DES predict a strong amplification of the bed shear stress near the outer wall, where the core of high streamwise velocities is present. The coherence of V4 in DES is sufficiently large to induce a noticeable amplification of the bed shear stress near the inner bank. Figure 15 confirms that the transverse component of τ is significant below V4. No noticeable amplification of τ is observed in RANS near the inner bank. The bed shear stresses are also strongly amplified over the crest of the diagonal bar present in the downstream part of the straight outflow reach (Figure 2).

[51] Overall, RANS and DES predictions of τ show significant differences that would translate into even larger differences in sediment transport. As DES predicts more accurately the mean flow, DES simulations with a movable bed should predict more accurately the bathymetry evolution and equilibrium bathymetry compared to similar RANS simulations. Still, on the basis of only the τ distributions obtained over the fixed equilibrium bathymetry measured in the experiment, one cannot quantitatively estimate how the equilibrium bathymetry will differ in a RANS and a DES simulation with movable bed. One should also mention that significant differences between the bed shear stress patterns predicted by RANS and DES were also observed in other complex open channel flows (e.g., see Constantinescu et al. [2011] for river confluences) for which DES predictions of the mean flow were shown to be more accurate than those of RANS.

[52] The distribution of τ predicted by DES on the inner bank (Figure 16a) shows that at most streamwise locations, the magnitude of the shear stress is comparable or lower than the mean value corresponding to a fully developed flow in a straight channel. Though the streamwise component is the dominant one, the vertical component induced by V2 and V4 at locations where their circulation is relatively high and the core of the SOV is situated close to the inner bank can account for 20%–30% of the magnitude of the local sidewall shear stress (compare Figures 16a and 17).

Figure 16.

DES predictions of the nondimensional shear stress, τ/τ0, at the channel sidewalls. (a) Inner bank and (b) outer bank. Lξ is the streamwise distance measured along the channel sidewall.

Figure 17.

DES predictions of the vertical component of the nondimensional shear stress at the inner bank. Lξ is the streamwise distance measured along the channel sidewall.

[53] As expected, the magnitude of the shear stress in the mean flow is larger than τ0 at several locations along the outer bank. These regions are centered around sections D30 and P0.5 and correspond to locations where the core of high streamwise velocity fluid is situated close to the free surface and the outer bank. The small values of τ in between these two regions are explained by the fact that the core of high streamwise velocities is situated below the free surface and away from the outer bank in between sections D60 and D193. The vertical component of τ along the outer bank is everywhere much less than the streamwise component and is not shown. Despite being the strongest SOV, the core of V1 is situated too far from the outer wall to induce a large vertical component of the shear stress.

[54] A region of milder amplification of τ is observed close to the free surface, around section D90. The pressure RMS fluctuations are strongly amplified at the same location (Figure 20b). Recall that a SSL forms because the core of high streamwise momentum fluid entering the bend tries to preserves its original direction. Close to the free surface, this SSL penetrates until close to the outer wall (Figure 3c). The distributions of the vorticity close to the free surface in one of the instantaneous flow fields and in the mean flow are shown in Figures 18 and 3c, respectively. Animations of the vorticity fields show that the eddies separating from the downstream part of the SSL approach, at times, the attached boundary layer on the outer bank and induce ejection of patches of high-vorticity fluid. This results in large pressure fluctuations, the amplification of the shear stress on the outer wall, and the formation of highly energetic, relatively large scale eddies that are then convected downstream. Thus, despite the fact that in this region the mean value of the shear stress is relatively low, the potential for bank erosion is high. Figure 18 shows that close to the free surface, the core of incoming high streamwise velocity fluid contains less energetic eddies than the flow downstream of the SSL.

Figure 18.

Distribution of the vertical vorticity, ωzD/U, at the free surface in one of the instantaneous flow fields predicted by DES.

[55] The distribution of the pressure RMS fluctuations at the bed in Figure 19 shows that regions of high turbulent fluctuations do not generally correspond to regions of high bed shear stress in the mean flow. The large amplification of equation image/ρ2U4 close to the inner bank, in between sections D130 and P0.8, is somewhat expected because of the complex interactions of the energetic eddies originating in the SSLs forming around RE1 with the deformed bathymetry. Figure 20a shows that in between these two sections, the values of equation image/ρ2U4 over the whole height of the inner bank are comparable to those recorded on the bed, in the vicinity of the inner bank.

Figure 19.

Distribution of the pressure RMS fluctuations, equation image/ρ2U4, at the channel bed (DES).

Figure 20.

Distribution of the pressure RMS fluctuations, equation image/ρ2U4, at the channel sidewalls (DES). (a) Inner bank and (b) outer bank.

[56] The large amplification of equation image/ρ2U4 near the outer bank in the vicinity of section D90 has much more important consequences for scour. This is because high values of equation image/ρ2U4 are recorded within the region of peak values of the mean bed shear stress. The region of high equation image/ρ2U4 is not limited to the bed but also covers the deeper part of the outer bank in between sections D80 and D105 (see Figure 20b). At these locations, the sidewall shear stress in the mean flow is relatively low. The large turbulence fluctuations observed around section D90 in the deeper part of the flow, close to the outer bank, are driven by pockets of high vorticity and high streamwise velocity fluid that, at times, are convected laterally within the near bed layer, until they reach the boundary layer on the outer bank. Thus, the estimation of bank stability and bed erosion based only on the mean values of the bed shear stress can be misleading in this critical region of the curved channel, where large pressure RMS fluctuations and large values of the instantaneous bed shear stress can substantially increase the erosion potential of the turbulent flow.

8. Summary and Concluding Remarks

[57] The present study considered the flow in an open channel bend of strong curvature (R/B = 1.3) over realistic topography corresponding to equilibrium scour conditions. The flow and turbulent structure are controlled by the nonlinear interaction between the downstream velocity and the cross-stream circulation. This interaction is further complicated by the interaction with the pronounced riffle-pool bathymetry, which leaves a strong fingerprint on all characteristics of the flow field, including the formation of two recirculation eddies near the shallowest regions at the inner bank. The data generated from an eddy-resolving simulation conducted using DES on a fine mesh were used to enhance the insight into the physics of flow in bends of strong curvature over equilibrium bathymetry.

[58] DES revealed that in addition to the main cell of cross-stream circulation present in the deeper outer side of the channel bend, which is a general characteristic of flow in curved open channels, several strong streamwise-oriented vortices (SOVs) formed near the high-curvature inner bank. The circulation, levels of turbulence amplification, and exact extent of some of these vortices could not be estimated from experiment because of the lack of accurate measurements near the solid boundaries. Furthermore, these eddies play an important role. The increased erosion potential around the SOVs was used to explain some of the features observed in the bathymetry at equilibrium.

[59] Besides the SOVs, analysis of the mean and instantaneous flow fields revealed the important role played by the separated shear layers (SSLs), which are concentrated regions of high vorticity and high turbulence production by shear. DES revealed that in addition to the SSL forming as a result of the separation of the attached boundary layers on the lateral faces of the mounts of sediment that are part of the point bar forming at the inner bank, an energetic SSL forms because the core of high streamwise momentum fluid entering the high curvature bend region tries to preserves its original direction and cannot increase its transverse momentum sufficiently quickly, such that the core of high-velocity fluid continues to follow the bank line. We observed the formation of similar SSLs in other simulations conducted over flat bed and topography in curved open channels with a high curvature of the inner bank [Kashyap et al., 2010; Constantinescu et al., 2011]. Thus, this SSL appears to be a general characteristic of flow in open channels of strong inner bank curvature, and its formation is independent of the flow separating in the lee side of the large-scale deposition regions, which are part of the point bar forming at the inner bank.

[60] The exact position of the SSLs is very hard to estimate from experimental measurements because of their small thickness. The present results showed that the eddies convected inside the SSLs play an important role, as they can modify the turbulence structure of the flow and interact with the channel boundaries. Thus, the SSL eddies can locally increase the erosion potential. Ultimately, DES allowed us to explain the main features of the distributions of boundary shear stresses and pressure RMS fluctuations on the basis of the large-scale vortical structure of the mean and instantaneous flow fields.

[61] Results demonstrate that compared to RANS, DES is able to better capture the redistribution of the mean flow streamwise velocity and streamwise vorticity in relevant cross sections due to curvature- and bathymetry-induced effects. A main question is whether the differences in the distributions of the streamwise velocity and streamwise vorticity result in significant differences in the mean flow bed shear stress distributions predicted by RANS and DES. The present results showed that the answer is positive (Figure 14). The most important difference was that while DES predicted the largest bed shear stresses to occur between sections D30 and D120 in the curved reach of the channel, RANS predicted the largest bed shear stresses to occur much farther downstream, in between sections D70 and D180. It is also interesting to point out that the study by Ferguson et al. [2003] of a channel with natural pool-riffle topography and strong inner bend curvature (R/B < 1.4) also found that the region of maximum boundary shear stress at the outer bank was situated mostly upstream of the bend apex, where a recirculation region similar to RE1 formed, rather than downstream of it. This is consistent with present DES results.

[62] Consequently, it is reasonable to expect that simulations with a mobile bed and sediment transport conducted using RANS and DES will predict a different temporal evolution of the bathymetry and, ultimately, a different equilibrium bathymetry. On the basis of their success in predicting the mean flow structure in the curved channel, the use of DES should result in more accurate predictions of the bathymetry compared to RANS. Moreover, as DES provides quantitative information on the pressure RMS fluctuations at these solid boundaries in addition to the boundary shear stresses, we think that a major direction for future research is to incorporate this information into formulas and methods used to estimate the local erosion. This is important because high turbulence fluctuations can negatively affect bank stability and induce erosion even in regions where the mean boundary shear stress is below the critical values required for sediment entrainment at the bed or for bank failure in the case of channels with erodible banks.

[63] Comparison of results from the present DES and the LES conducted using wall functions by van Balen et al. [2010b] shows that, overall, DES does a better job in predicting the details of the mean flow structure in the high-curvature channel. Probably, the main reason is the more sophisticated level of modeling of the near-wall flow in DES, which uses a one-equation RANS model rather than wall functions, which are known to have serious deficiencies in predicting flows in strong adverse pressure gradients. van Balen et al. [2010b] identified the insufficient resolution of the bathymetry, which did not describe the moving dunes, as the main reason for the differences between the experimental data and LES. One should also point out that LES and the RANS predictions of Zeng et al. [2008a] were very close. There is no doubt that a better characterization of the geometry and roughness (e.g., resolving the dunes in the simulation) will result in more accurate predictions of the hydrodynamics. However, the present results show that other factors may significantly affect the overall predictive ability of a certain numerical model. The subgrid-scale model used away from the walls in DES more closely resembles a dynamic Smagorinsky model than the constant coefficient Smagorinsky model used by van Balen et al. [2010b]. Depending on the particular flow that is calculated, the latter model can be too dissipative in some of the regions situated away from the walls. As the LES and DES were obtained using codes with different numerics, some of the differences may be related to the viscous flow solver used to perform these simulations, even if the formal order of accuracy of the two numerical schemes is the same. The numerics is not a factor when comparing DES with RANS, as the same base solver was used by Zeng et al. [2008a]. Despite the fact that both simulations used the rigid lid assumption, the fact that the measured mean flow free surface deformations were taken into account in the definition of the computational domain in DES may have resulted in a better approximation of the flow processes near the free surface in DES.

[64] DES was already used to study separated flows past hydraulic structures such as bridge piers and abutments at field conditions and related scale effects [e.g., see Kirkil and Constantinescu, 2009, 2010; Koken and Constantinescu, 2009] and flow past bottom river cavities [Chang et al., 2007]. Recently, Kirkil and Constantinescu [2008] have shown that DES can be successfully used to predict mean flow, turbulence, and dynamics of the quasi-2-D eddies of shallow mixing layers at the laboratory scale. On the basis of a detailed comparison with field data, DES was also shown to predict the mean flow and turbulence structure (e.g., the formation of strong SOVs on both sides of the mixing layer interface) at a natural river confluence significantly more accurately than RANS [Constantinescu et al., 2011]. This test case is particularly relevant for the present investigation because of the relatively high curvature (R/B ∼ 3) of the natural river reach downstream of the confluence apex. DES of flow in natural channels of a total length of several kilometers is feasible on today's supercomputers.

[65] The success of DES in predicting flow in curved channels at moderate Reynolds numbers is particularly important, as this technique can be used to conduct numerical simulations at very high Reynolds numbers, thus allowing the study of scale effects between laboratory and field conditions. Most of the detailed experimental investigations of flow and sediment transport in curved open channels were conducted at channel Reynolds numbers less than, or around, 105. A main question that DES can help answer is to what extent the findings from these laboratory investigations apply in the field, where the Reynolds numbers can be much larger than 106. Such simulations will allow enlarging the range of geometrical and flow parameters over which detailed information on the flow physics, streamwise variation of the strength of the secondary flow, and integral parameters characterizing the flow in curved bends are available. This information should be helpful for developing more accurate parametrizations of the main processes controlling flow in curved bends.

Acknowledgments

[66] We gratefully acknowledge the National Center for High Performance Computing (NCHC) in Taiwan and the Transportation Research and Analysis Computing Center (TRACC) at the Argonne National Laboratory for providing substantial amounts of computer time. This paper was written while G. Constantinescu was on sabbatical leave at the Laboratory of Hydraulics, Hydrology and Glaciology (VAW) at ETH Zurich. George Constantinescu would like to thank W. Hager and the other researchers at VAW for their support.

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