Nature of flow and turbulence structure around an in-stream vertical plate in a shallow channel and the implications for sediment erosion
Abstract
[1] Detailed knowledge of the dynamics of large-scale turbulence structures is needed to understand the geomorphodynamic processes around in-stream obstacles present in rivers. Detached Eddy Simulation is used to study the flow past a high-aspect-ratio rectangular cylinder (plate) mounted on a flat-bed relatively shallow channel at a channel Reynolds number of 2.4 × 105. Similar to other flows past surface-mounted bluff bodies, the large amplification of the turbulence inside the horseshoe vortex system is because the core of the main necklace vortex is subject to large-scale bimodal oscillations. The presence of a sharp edge at the flanks of the obstruction fixes the position of the flow separation at all depths and induces the formation and shedding of very strong wake rollers over the whole channel depth. Compared with the case of a circular cylinder where the intensity of the rollers decays significantly in the near-bed region because the incoming flow velocity is not sufficient to force the wake to transition from subcritical to supercritical regime, in the case of a high-aspect-ratio rectangular cylinder the passage of the rollers was found to induce high bed-shear stresses at large distances (6–8 D) behind the obstruction. Also, the nondimensional values of the pressure root-mean-square fluctuations at the bed were found to be about 1 order of magnitude higher than the ones predicted for circular cylinders. Overall, this shows that the shape of the in-stream obstruction can greatly modify the dynamics of the large-scale coherent structures, the nature of their interactions, and ultimately, their capability to entrain and transport sediment particles and the speed at which the scour process evolves during its initial stages.
1. Introduction
[2] Understanding the flow and transport phenomena around bed obstacles and large-scale bed roughness elements inserted in alluvial streams and shallow channels is a complex problem of great interest to water resources engineering and management [Roy et al., 2004; Keylock et al., 2005; Fael et al., 2006; Hardy et al., 2007]. Flow around large in-streamflow obstructions (dimension of the obstacle is comparable to channel depth) generates complex large-scale coherent structures that are much stronger than the eddies associated with the sweep and ejection events observed in turbulent channel flows with smooth or rough beds. Consequently, these large-scale turbulent structures drive the local scour process [Paola et al., 1986; Koken and Constantinescu, 2008a, 2008b] around the flow obstruction. Most of the research in this area was driven by the need to predict local scour around river training structures (e.g., groynes), bridge abutments and bridge piers because excessive scour can lead to the failure of the structure and, in the case of piers and abutments, to the collapse of bridges.
[3] A large number of experimental studies have been devoted to predicting the scour depth around in-streamflow obstructions of different shapes [e.g., see Melville and Coleman, 2000; Sumer and Fredsoe, 2002]. However, despite the fact that most of the sediment entrainment is due to the action of large-scale turbulent eddies, their role in the scour development has been only partially understood and is seldom accounted explicitly in existing scour prediction methods (e.g., see the discussion in the work of Ettema et al. [2006]). The problem is further complicated by the fact that the turbulent structures of focal interest (e.g., the horseshoe vortex system, the downflow parallel to the upstream face of the obstruction, the vortex tubes in the separated shear layers and the wake vortices) are not isolated from each other, thus the need to study and account for their interactions. Existing scour prediction methods do not sufficiently recognize all the processes at play during scour. This is one of the main reasons why the existing, frequently used, prediction methods yield scour depth estimates that substantially exceed observed depths at obstructions (e.g., piers) of width that is comparable or larger than the channel depth (wide piers). Overestimation of the scour depth as the pier width increases becomes an increasingly unacceptable economic proposition for large bridges where these piers are used.
[4] As a prerequisite to develop new scour prediction methods that can incorporate more physics, one has first to be able to understand the turbulence structure and the dynamics of the eddies controlling the sediment entrainment, transport and deposition during the scour process. With the development of experimental and numerical techniques that can provide detailed information on the flow and turbulence structure, there has been lots of interest in using these tools to understand the flow physics and, in particular, the role played by the large-scale unsteady coherent structures and macroturbulence events that control the scour mechanism, the transport processes around the in-stream obstruction and past it.
[5] Though several experimental investigations of the instantaneous and mean-flow fields and, in some cases, turbulence statistics at circular [e.g., Dargahi, 1989, 1990; Yeh, 1996; Ahmed and Rajaratnam, 1998; Graf and Yulistiyanto, 1998; Graf and Istiarto, 2002; Roulund et al., 2005; Unger and Hager, 2007; Dey and Raikar, 2007] and low-aspect-ratio rectangular [e.g., Yeh, 1996; Dey and Raikar, 2007; Raikar and Dey, 2008] piers with flat or scoured bed were conducted in recent years, there is very little known about the flow fields and associated sediment transport processes around high-aspect-ratio rectangular cylinders at high angle of attack.
[6] A primary motivation of the present investigation is that flow obstructions in the form of a high-aspect-ratio rectangular cylinder resemble wide-pier walls that are used more and more at large bridges. Even if the long piers are build such that the angle of attack is small at normal flow conditions in the river, the angle of attack can increase considerably at flood conditions for braided channels or, over time, in meandering channels. As a result of the flow deflection, the pier skew angle relative to the main-channel axis is altered. This is particularly the case when a pier is situated near an abutment in a compound channel which is, in many cases, subject to varying flow orientations as the flow stage rises (high flow conditions). The present investigation considers only the extreme case in which the approach flow is perpendicular to a thin-plate vertical obstruction (angle of attack is 90° corresponding to maximum skew) of rectangular shape with a width to depth ratio D/H = 1. Several experimental investigations [e.g., Laursen and Toch, 1956; Ettema et al., 1998] have documented the significant increase of the maximum scour depth with the skew angle for rectangular piers of varying aspect ratios.
[7] The scour hole developing around piers of intermediate widths (0.7 < D/H < 5, according to the classification in the work of Melville and Coleman [2000]) and the associated sediment transport processes have aspects that are similar to both scour at an abutment (groyne wall) and scour at piers whose shape is close to square or circular. In particular, as the ratio D/H increases the intensity of the HV system peaks in vertical sections situated close to the flanks of the pier and the maximum scour depth shifts from the symmetry plane to the flanks of the pier.
[8] The problem is also significant in the general context of understanding the physics of junction flows. The fact that the section of the in-stream obstruction has sharp edges means the position at which the boundary layer separates is fixed (the flanks of the obstruction) and, more importantly, is the same at all flow depths. This is not the case of the flow past cylinders of smooth shape (e.g., circular) at high Reynolds numbers where the angle at which the attached boundary layer separates decreases considerably with increase distance from the bed. This is because in the near-bed region the incoming flow velocity within the channel boundary layer is not high enough to force the drag crisis to occur and the separation point to move downstream. The separation line is fixed and determined by the geometry in the case of a rectangular cylinder, independently of the Reynolds number. This is the main reason for the observed differences in the large-scale shedding process between rectangular and circular cylinders where the position of the separation line is function of the Reynolds number. As will be shown by the present investigation, the characteristics of the large-scale shedding mechanism behind the surface-mounted high-aspect-ratio rectangular cylinder resemble those observed behind infinitely long smooth-shape cylinders past the drag crisis (supercritical regime). This strongly modifies the interactions among the large-scale coherent structures that play a significant role in the entrainment and transport of sediment in the near-bed region compared to the case of a surface-mounted obstruction of smooth shape. Consequently, this points toward the need to properly account for the different physics in scour prediction methods, as most of these formulae are calibrated on the basis of laboratory studies conducted primarily for circular cylinders and do not adequately consider factors relevant to wide piers, long skewed piers or in-stream obstacles of similar shape.
[9] Finally, the problem is also relevant for the study of flow past in-stream natural or artificial islands of elongated shape present in shallow streams or coastal areas. Understanding the influences of the shape of the obstacle (e.g., island) and degree of flow shallowness on the characteristics and dynamics of the wake region as well as acquiring a better understanding of entrainment and deposition phenomena around this kind of obstacles is of interest for several major river restoration projects (e.g., at the mouth of the Mississippi River related to the flooding of New Orleans, at the Stoddard Bay part of the Upper Mississippi River Basin (UMRB) related to efforts to restore the river ecosystem). The main goals of these projects are to restore the barrier islands and wetlands to protect inland urban areas from the effects of storm surges and to restore and improve river habitat for aquatic plants, fish, mussels and birds that disappeared owing to the system of locks and dams that transformed the UMRB into a series of navigation pools which inundated vast amounts of river valley. A clear understanding of sediment transport mechanisms around these artificial islands is essential to minimize the costs of the restoration project and to ensure that a change in the river conditions (e.g., occurrence of a flood that changes the velocity and approach angle relative to the island) would not compromise the habitat restoration efforts.
[10] The present paper tries to produce a full description of the general features of the flow field and turbulence structure in the vicinity of a high-aspect-ratio rectangular obstruction at large angle of attack and discusses the roles of the dominant large-scale coherent structures and of their interactions in controlling the sediment entrainment phenomena around the in-stream obstruction at conditions close to initiation of scour (flat bed channel) on the basis of results of an eddy-resolving numerical simulation. The Reynolds number in the simulation (Re = 2.4 × 105) corresponds to the upper limit at which laboratory investigations are typically conducted. Beyond the Reynolds number of the simulation, scale effects are not expected to play an essential role as far as the turbulent flow fields are concerned.
[11] The potential of Large Eddy Simulation (LES) to study the physics of flows relevant for river engineering and fluvial sedimentology has been already demonstrated (e.g., see discussion in the work of Keylock et al. [2005], Zedler and Street [2001, 2006], McCoy et al. [2007, 2008], and Hinterberger et al. [2007]). In particular, the flow past a vertical groyne in a channel with flat and scour hole (equilibrium conditions) was studied by Koken and Constantinescu [2008a, 2008b] and the flow past a circular pier with scour hole was studied by Kirkil et al. [2008]. In these studies, conducted on sufficiently fine meshes to avoid the use of wall functions, the channel Reynolds number was around 16,000, the incoming flow contained realistic turbulent fluctuations and a dynamic subgrid-scale (SGS) Smagorinsky model was used. This allowed a detailed investigation of the flow physics and of the role of the coherent structures in the sediment entrainment and transport processes. However, one limitation of these studies is the relatively low Reynolds numbers used in the LES simulations.
[12] For massively separated flows, like the one of interest in the present study, the use of the wall function approach, which is based on the validity of the law of the wall close to solid surfaces, is questionable. If the governing equations are integrated through the viscous sublayer, then the mesh requirements needed to resolve the attached boundary layers in well-resolved LES increase very fast with the Reynolds number. This limits the use of LES without wall functions to relatively low Reynolds numbers, especially for complex flows containing multiple wall surfaces which do not allow the use of spectral methods. To be able to numerically study the physics of these flows at Reynolds numbers closer to field conditions and investigate scale effects, a hybrid RANS-LES method has to be used. For example, a grid with at least 107 grid points is required to performed a well resolved LES without wall functions of the flow studied in the present paper at a Reynolds number of about 2.4 × 105. This is more than one order of magnitude higher than the estimated grid size (5–10 million grid points) needed by a hybrid RANS-LES model. Additionally, in LES the time steps have to be smaller because one has to accurately captures the dynamics of all eddies within most of the inertial range, in particular the eddy content in the very thin wall-attached boundary layers. Compared to hybrid methods, the total CPU time required by LES without wall functions at Re = 105–106 is at least two order of magnitude higher. This strongly makes the case for the use of hybrid methods to simulate complex turbulent flows at high Reynolds numbers.
[13] A first type of hybrid models uses a zonal approach [see, e.g., Wang and Moin, 2002] in which a RANS model is used in the vicinity of solid surfaces and a SGS model, the same as those used in well resolved LES, is used away from the solid boundaries. The main advantage of this approach is that the state-of-the-art physics-based SGS models developed for LES can be used to resolve the LES regions. These models include improved versions of the dynamic Smagorinsky model [see, e.g., Bou-Zeid et al., 2005; Stoll and Porte-Agel, 2006] and subfilter-scale models that partially reconstruct the turbulent stresses from the resolved velocity field. The main disadvantage is that a special treatment is required at the interface between the RANS and LES regions to match the two solutions and to avoid the development of spurious oscillations.
[14] A second type of hybrid models uses the same base (RANS) turbulence model in the whole domain (RANS and LES regions). Thus, no special treatment is required to match the solutions in the regions where the model switches from RANS mode to LES mode. The most popular and successful nonzonal model is called Detached Eddy Simulation (DES). In the RANS regions, DES typically uses a RANS model with near-wall modeling capabilities, thus the use of wall functions is avoided provided the mesh is sufficiently fine in the wall normal direction. In DES [Spalart, 2000] the SGS model used in the LES region is a RANS model in which the turbulence length scale is modified to allow energy cascade to the small scales as in classical LES. One of the disadvantages of DES is that the SGS model employed in regions where DES is in LES mode incorporates less physics compared to the dynamic Smagorinsky model or to other state-of-the-art SGS models. For internal flows (e.g., flow past surface-mounted bodies in a channel) the predictions of DES in terms of the large-scale eddy content, their dynamics, and the prediction of the mean flow and turbulence statistics are dependent on the presence, or not, of resolved turbulence in the incoming flow. For instance, this is discussed on the basis of comparison with highly resolved LES by Chang et al. [2007] in their DES study of flow and contaminant ejection from a bottom river cavity. As the incoming flow in all applications of relevance to river engineering is turbulent and the flow predictions can be significantly affected by the characteristics of the incoming flow, it is important to specify inflow conditions that match as close as possible those present in experiments and in the field. The problem is that it is not easy to generate a channel flow with the correct turbulence content at a high Reynolds number.
[15] DES predictions of flow past a vertical-wall abutment and a wing-shaped cylinder mounted on a flat surface at high Reynolds numbers (1.0 × 105 to 4.5 × 105) were reported by Paik and Sotiropoulos [2005] and Paik et al. [2007] using steady inflow conditions (no resolved inflow turbulence). Though both the LES study of Koken and Constantinescu [2008a] and the DES study of Paik et al. [2007] successfully captured the bimodal oscillations of the main necklace vortex at the base of the obstruction [Devenport and Simpson, 1990], some of the characteristics of the flow dynamics in the HV region were found to be qualitatively different. For instance, the DES of Paik et al. [2007] showed the formation of a secondary instability in the form of hairpin-like vortices on the core of the main necklace vortex, a phenomenon absent in LES conducted at a lower Reynolds number. Whether or not this was due to scale effects or to the different conditions in the approach flow (steady versus unsteady containing resolved turbulence velocity fluctuations) is not entirely clear. We think the latter is the main explanation on the basis of the fact that such secondary instabilities (large-scale hairpin vortices) were also observed to develop on the core of the spanwise vortices in the separated shear layer on top of a bottom channel cavity at low Reynolds numbers with steady inflow conditions [Chang et al., 2006]. However, when in the same flow and at the same Reynolds number realistic turbulent fluctuations were added at the inflow, the hairpin-like eddies disappeared. Also relevant, the study of Shah and Ferziger [1997] found that the specification of time-accurate turbulent-like inflow boundary conditions is essential for successfully predicting the experimentally observed location of the HV system in the flow past a bottom-mounted obstruction using LES. Ideally, the turbulent like inflow conditions (e.g., time series of the velocity components) are obtained from a preliminary (LES or DES) simulation of the flow in a straight channel of identical section to the inflow section in the simulation containing the flow obstruction. The present DES study of flow past a vertical-plate obstruction at a relatively high Reynolds number (Re = 2.4 × 105) in which unsteady fluctuations are added at the inlet will help clarify whether scale effects or the incoming flow characteristics are responsible for the formation of a secondary instability in the form of hairpin-like eddies on the core of the main necklace vortex.
[16] In the following the numerical method, boundary conditions, and the DES model are briefly described. Then, the dynamics of the main large-scale coherent structures in the flow and their effect on the mean flow and turbulence statistics are analyzed. Particular attention is paid to the structure of the HV system in front of the vertical-plate obstruction, the structure of the near wake and the interaction of the legs of the necklace vortices with the separated shear layers and with the rollers shed in the near-wake region. The effect of the formation of these coherent structures and of their interactions on the distributions of the instantaneous and time-averaged bed friction velocity and of the pressure root-mean-square (rms) fluctuations is discussed. Based also on comparison with DES results obtained in our group for the flow past a circular cylinder under similar conditions (Reynolds number, width to channel depth ratio), the paper provides a discussion of the effects of the shape of the obstruction (relative bluntness) on the flow, turbulence structure and sediment entrainment mechanisms.
2. Numerical Method
[17] A general description of the DES code used in the present work is given in the work of Constantinescu and Squires [2004] and Constantinescu et al. [2003]. 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. To minimize the level of numerical dissipation away from solid boundaries (LES mode), the convective terms are discretized using a blend of fifth-order accurate upwind biased scheme and second-order central scheme using the blending function proposed by Travin et al. [2000]. The 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 the alternate direction implicit (ADI) approximate factorization scheme. Time integration is done using a double-time stepping algorithm. Local time stepping is used to accelerate the convergence at each physical time step. The source terms in the turbulence model equations are treated implicitly. The time discretization is second-order accurate.
:
is estimated by solving ∂/∂n = /d [Spalart, 2000; Zeng et al., 2008], where n is the wall normal direction. This makes the modified viscosity and the SGS viscosity to be formally nonzero at the rough surface. For smooth walls, ks and are set equal to zero. The model constants in the above equations are: Cb1 = 0.135, Cb2 = 0.622, σ = 0.67, κ = 0.41, Cv1 = 7.1, Cw2 = 0.3, Cw3 = 2.0 and Cw1 = Cb1/κ2 + (1 + Cb2)/σ.[19] The SA version of DES is obtained by replacing the turbulence length scale d with dDES which is defined as dDES = min(d, CDESΔ) where the model parameter CDES is equal to 0.65 and Δ is a measure of the local grid size. When the production and destruction terms of the model are balanced, the length scale in the LES regions dDES = CDESΔ becomes proportional to the local grid size and yields an SGS viscosity proportional to the mean rate of strain and Δ2 as in LES with a Smagorinsky model.
[20] Detailed grid sensitivity and validation studies for the flow past spheres and for flow and contaminant transport in a channel with a bottom cavity are discussed in the work of Constantinescu and Squires [2003, 2004], Constantinescu et al. [2003], and Chang et al. [2007]. An additional validation test case was conducted for the case of a relatively low-Reynolds-number flow (Re = 18,000) past a circular cylinder mounted on a flat-bed channel. For this test case data from a well-resolved LES simulation [Kirkil et al., 2006] obtained using a fully nondissipative code employing unstructured meshes and a dynamic Smagorinski model [Mahesh et al., 2004] were available. Despite the differences in the numerics, grid topologies (unstructured versus multiblock structured meshes), grid sizes and SGS model (dynamic Smagorinsky versus DES) the agreement for the mean flow and turbulence statistics between the two simulations was found to be very good. This was true not only in the wake region where DES is expected to perform well, but also in the critical HV region situated upstream of the obstacle which is tougher to predict using DES. Sample results are presented in Figure 1 which shows the distribution of the total production term in the transport equation for the turbulent kinetic energy (tke) in the symmetry plane (polar angle ϕ = 0°) upstream of the cylinder. For a circular cylinder, this section is the one in which the strength of the HV system is the largest. The DES predictions of the location of the region of high positive turbulence production and the levels of turbulence production inside this region are in very good agreement with LES. Moreover, DES captures the two-peak shape of this region induced by the bimodal oscillations of the main necklace vortex between the two preferred modes. The good agreement was in large part due to the fact that in both calculations the inflow conditions were similar. In both simulations the instantaneous velocity flow fields from a preliminary simulation of the turbulent flow in a channel of identical section were fed through the inflow section.
∂ui/∂xj)D/U3) in the transport equation for the tke in the symmetry plane (ϕ = 0°) of a circular cylinder at Re = 18,000. (a) DES. (b) LES. The origin of the system of coordinates is at the axis of the circular cylinder. In the ϕ = 0° plane the face of the cylinder is situated at ∣x/D∣ = 0.5D.3. Description of Computational Domain and Boundary Conditions
[21] The dimensions of the computational domain (Figure 2a) and the main parameters of the simulation were chosen to correspond to a preliminary physical model study performed in the environmental flume of the Iowa Institute of Hydraulic Research (Figure 2b). The main goal of the preliminary experiment was to determine what is the maximum Reynolds number at which the free surface deformations can be considered negligible (the numerical code treats the free surface as a nondeformable surface), to determine the shedding frequency of the large-scale rollers in the wake and the approximate location of the main necklace vortex as it wraps around the upstream base of the rectangular cylinder, and to confirm some previous qualitative observations that noticed the very large coherence of the HV system forming at the base of surface-mounted obstacles that resemble a vertical flat plate at high angle of attack.
[22] The flume has a test section which is 20 m long, 3 m wide and 2 m deep, permitting physical model studies at relatively large channel Reynolds numbers (105 < Re < 106). In the experiment and simulation the width of the in-stream obstruction, D, was equal to the channel depth, H, and its thickness was 0.07 D. The aspect ratio of the rectangular cylinder was close to 14.5. The cylinder was placed in the symmetry plane of the flume, at a streamwise position where the incoming flow was already fully developed. In the experiment, the average channel velocity was U = 0.45 m/s and D = 0.53 m. This corresponds to a channel Reynolds number of 2.4 × 105 and a Froude number Fr = U/
close to 0.2, where g is the gravitational acceleration. At this value of the Froude number the deformations of the free surface close to the cylinder are not significant, thus the use of a rigid lid boundary condition at the free surface in the numerical simulation is justified. The domain width in the simulation is 5.7 D, which corresponds to the width of the flume (see sketch in Figure 2a). The blockage ratio is 0.17. This will result in an acceleration of the flow as it passes the flanks of the cylinder compared to the case when the cylinder is situated far from other flow obstructions or the channel sides. The computational domain extends 5 D upstream of the cylinder, which is sufficient for the flow close to the inflow section not to be affected by the adverse pressure gradients induced by the presence of the cylinder, and 15 D downstream of it. The origin of the system of coordinates is located at the center of the cylinder on the bottom surface with the x axis in the streamwise direction.
[23] The wall surfaces were treated as no-slip boundaries. The nondimensional bed roughness was close to 40 wall units corresponding to a layer of sand with a mean diameter of d50 = 1 mm which is typical for physical model studies with loose bed performed in the environmental flume. At the inflow section, flow conditions corresponding to fully developed turbulent channel flow were specified. The mean streamwise velocity profile was obtained from a preliminary RANS simulation in a periodic channel at the same Reynolds number. The turbulent fluctuations (zero mean velocity) were obtained from a preliminary simulation of the flow in a periodic channel (straight channel reach simulation in which periodic boundary conditions are used for the velocity and turbulence variables at the inflow and outflow sections) whose section was identical to the one used to investigate the flow past the rectangular cylinder. The fluctuations were then added to the RANS mean streamwise velocity profile and the total velocity fields were fed in a time-accurate manner through the inflow section. A convective boundary condition was used at the outflow. The free surface was modeled as a shear-free rigid lid.
[24] The mesh contains 8.1 million cells (432 × 288 × 65 in the streamwise, spanwise and vertical direction, respectively). This large number of mesh points was needed to capture the dynamically important eddies forming around the in-stream obstruction, to resolve the thin attached boundary layers and to have a sufficient resolution to avoid the use of wall functions in DES at Re = 2.4 × 105. The grid spacing in the wall-normal direction was close to one wall unit (assuming uτ/U∼0.045). The corresponding physical height of the first row of cells off the wall was 5 × 10−5 m. The mesh was refined near all the walls, and in the regions containing the HV system and the upstream part of the separated shear layers (SSLs) at the flanks of the rectangular cylinder.
[25] The time step in the DES simulation was 0.02 D/U which is sufficiently low to accurately capture the range of energetic frequencies associated with the dynamics of the energetically dominant structures in the flow (e.g., necklace vortices in the HV region and rollers in the wake). After the flow reached the statistically steady state, flow statistics were computed over 60 D/U. Simulations were run on 24 processors of a PC Cluster.
4. Mean Flow and Turbulence Statistics in the Horseshoe Vortex System Region
[26] The positions of the main necklace vortex HV1 and of the secondary vortex HV2 in the mean flow are visualized in Figures 3a and 3b using a Q isosurface [Dubief and Delcayre, 2000]. The core of HV1 is situated at a distance of 0.45–0.55 D from the upstream face of the cylinder and its two flanks. This distance is close to twice the one predicted by a DES simulation of the flow past a surface-mounted circular cylinder of diameter D (H/D∼1) at a comparable Reynolds number (Figure 3c). Close to the sides at the cylinder, the legs of the small junction vortex are interacting with the upstream part of the separated shear layers (SSLs) in the near-bed region. The position of HV1 relative to the rectangular cylinder was investigated using dye visualizations (e.g., see Figure 2b). The dye visualizations confirmed the core of HV1 was located, in average, at a distance of 0.5 D from the upstream face of the rectangular cylinder and its coherence was much stronger than that observed for a circular cylinder.
[27] The nondimensional vorticity magnitude (ωtD/U) distributions inside the HV region in representative vertical sections (Figure 4) show the circulation of HV1 does not peak in the symmetry plane (y/D = 0.0), as was the case for the circular cylinder. Rather, the maximum circulation is observed in a plane (Figure 4b) cutting through the extremity of the rectangular cylinder. This is very similar to the results obtained by Koken and Constantinescu [2008a] for the case of the flow past a vertical-wall spur dike. The increase of the circulation of HV1 in Figure 4b compared with the symmetry plane (Figure 4a) is 30%. Past the section shown in Figure 4b, the circulation decays monotonically in vertical sections that are perpendicular to the core of HV1. The decay is close to 20% in Figure 4c. The vorticity decay is much sharper in the legs of HV1.
[28] Another interesting feature of the vorticity distributions in Figure 4 is the presence of a near-bed patch (see arrow in Figures 4a and 4b; see inset) of relatively high vorticity in between the face of the cylinder and HV1. The high values of the vorticity inside this region are induced by the passage of eddies convected first with the downflow and then away from the cylinder toward HV1. The vorticity amplification in this region is the largest at sections cutting through the flanks of the cylinder (e.g., in Figure 4b). Qualitatively, the vorticity distribution does not change between Figure 4b and Figure 4c where the vorticity sheet associated with the SSL plays the role of a curved surface that forces some of the eddies convected with the incoming flow to change direction and move toward the bed as a result of their interaction with the SSL. In average, the eddies convected with the downflow are more energetic at these sections compared to sections close to the symmetry plane. This provides one reason for the fact that the circulation of HV1 does not peak in the symmetry plane.
[29] Though a well defined patch of vorticity associated with the main secondary vortex HV2 is present in Figure 4a and 4b, the ratio between the circulation of HV2 and that of HV1 is less than 0.25 at all sections which shows that HV1 is by far the dominant necklace vortex in the HV region.
[30] The distributions of the resolved tke and pressure RMS fluctuations, 
, in Figure 4 show that the turbulence intensity is greatly amplified inside the HV region. This is similar to what was observed for circular cylinders (e.g., the distributions in the symmetry plane are shown in Figures 6b and 6c). Similar to the flow past circular cylinders and spur dikes, the main reason for the turbulence amplification is the bimodal switching of the core of HV1 between the back-flow and the zero-flow modes. The mechanism through which the primary necklace vortex switches between the two modes and the structure of the flow inside the HV region was first described by Devenport and Simpson [1990] for a wing-shaped body mounted on a flat surface. Koken and Constantinescu [2008a] provide a similar discussion for the case of a vertical spur dike in a flat-bed channel. Evidence of the presence of the bimodal oscillations is given in Figure 5 where the histograms of the streamwise velocity are bimodal inside the region where the core of HV1 oscillates (e.g., see the histogram at point 1). As expected, the histograms recover their usual one-peak shape outside this region (e.g., see the histogram at point 2 situated in the incoming boundary layer).
[31] The estimated averaged time interval associated with the bimodal switching is 2 D/U which is comparable to the one (4.14 D/U, D is the maximum width of the wing-shaped body) estimated experimentally by Devenport and Simpson [1990] at a comparable Reynolds number but for a body of different shape. The power spectrum of the streamwise velocity at point 1 (Figure 5d) shows the most energetic frequency in the HV region corresponds to St = 0.27. As expected, this value is close to the estimated average time the main necklace eddy spends in one mode (0.5(1/St)D/U∼2 D/U). The range of energetic frequencies associated with the bimodal oscillations was estimated to be 0.07 < St < 0.3. The power spectrum of the streamwise velocity in log-log scale (Figure 5c) shows an inertial range is present until St∼5–7. This is consistent with the estimated cut-off frequency given the time step in the computation, the average velocity magnitude and the average cell size in the HV region.
[32] The shape of the patch of high values changes from being relatively circular in the symmetry plane (Figure 4a) to having an elliptical shape in the sections cutting close to the flanks of the cylinder, in which the circulation of HV1 is the strongest. All the three vertical sections in Figure 4 are locally perpendicular to the axis of HV1 in the mean flow. The elliptical shape of the patch of high values indicates the amplitude of the large-scale bimodal oscillations is also the largest at these sections.
[33] In the case of the tke, besides the large amplification of the tke inside the region where the core of HV1 oscillates, a second small patch of high tke values is present in the near-bed region where the downstream part of the jet-like flow developing beneath the core of HV1 changes direction as HV1 switches between the zero-flow mode (the core is close to circular and is situated closer to the face of the cylinder) and the back-flow mode (the core is more elliptical and is situated far from the face of the cylinder). The presence of this second patch of high tke values was also observed in DES simulations of the flow past a circular cylinder at 105 < Re < 106 and in the measurements of Devenport and Simpson [1990] for the flow past a wing-shaped body at Re∼125,000 [see Paik et al., 2007, Figure 7b]. For comparison, the tke distribution in the symmetry plane predicted for the circular cylinder using DES is presented in Figure 6a. Qualitatively, the tke and distributions are similar in the simulations past the circular (Figures 6a and 6b) and rectangular (Figure 4a) cylinders. The main difference is the relative level of turbulence amplification inside the HV region on the basis of the nondimensional values of the tke and which is 3–5 times higher in the case of the high-aspect-ratio rectangular cylinder. Also, the size of the core of HV1 and the distance between its axis and the bed are close to three times larger in the simulation of the flow past the rectangular cylinder compared to the circular cylinder case. As already mentioned, dye visualizations confirmed the very high coherence and intensity of the main necklace vortex for the high-aspect-ratio rectangular cylinder.
5. Mean Flow and Turbulence Statistics in the Separated Shear Layers and Near-Wake Regions
[34] The flow inside the two recirculation bubbles behind the cylinder is visualized using 3-D streamlines in Figure 7. The flow inside the cores of these eddies, that resemble tornado-like vortices, is oriented toward the free surface. The streamlines entrained into these two vortices are originating in the lower half of the channel (z/D < 0.5), upstream of the cylinder (x/D < −3), in a region situated relatively close to the symmetry plane, ∣y/D∣ < 0.3. As they approach the upstream face of the cylinder, the streamlines are deflected laterally and toward the bed. The mean-flow streamlines are then moving relatively parallel to the SSL on the side of the cylinder. Then, while remaining relatively close to the bed, the streamlines are moving inward toward the symmetry plane and from there toward the back of the cylinder where they are entrained into one of the two tornado-like vortices. Close to the free surface, the streamlines escape the core of these eddies and are ejected into the wake flow downstream of the recirculation bubbles.
[35] Figure 8 provides more information on the turbulence structure in the near-wake region. In a spanwise vertical plane situated just behind the cylinder (x/D = 0.1; see Figure 8a) the largest tke values are observed inside the SSL. The other region of high tke values is situated between the two sides of the cylinder. Interestingly, the largest values are observed on the interior side of the SSL. The levels of amplification of the tke and in these two regions are relatively independent of the distance from the bed. This is because the SSL detaches at the flanks of the cylinder at all vertical locations, as the position of the separation is dictated by the geometry of the flow obstruction. This is in contrast to cylinders of circular shape where the position of the separation is dictated by the flow (e.g., is function of the Reynolds number) and thus can change with the vertical location. As will be discussed later, this difference affects significantly the characteristics of the large-scale vortex shedding behind the cylinders. There is a third region situated near the bed at ∣y/D∣∼1.05 in which the tke and values are above the background turbulence levels associated with the fully developed turbulent channel flow. It corresponds to one of the legs of the main necklace vortex HV1 (see Figure 8a).
values were blanked.[36] The section at x/D = 1 cuts through the downstream part of the recirculation bubble. The largest tke values occur inside the recirculation bubble. Interestingly, the width of the region of high tke values decreases with the distance from the free surface. This is the opposite of the tke behavior in the DES simulation past a circular cylinder at Re > 2 × 105 (Figure 9) in which the wake flow is supercritical away from the bed region (SSLs detach at a polar angle larger than 100°) and subcritical in the near-bed region (SSLs detach at a polar angle close to 88°). The largest values occur around ∣y/D∣ = 0.5 (sides of the rectangular cylinder, see vertical dashed lines in Figure 8b). Compared to the levels predicted around the middle of the channel, decays by about 20–30% as the bed is approached inside the region centered around ∣y/D∣ = 0.5. By comparison, the decay predicted in the same region in the flow past a circular cylinder is close to 100% (cf. Figures 8b and 9).
values were blanked.[37] The section at x/D = 2 (Figure 8c) is situated downstream of the recirculation bubble. The tke and distributions are qualitatively similar to those in the section at x/D = 1.0. The main difference is the faster decay of compared to that of the tke between these two sections. This decay takes place over the whole depth of the channel.
[38] Comparison of the nondimensional values of the tke and at equivalent sections shows that the tke is larger by a factor of about two and by a factor of about eight for the high-aspect-ratio rectangular cylinder compared to the circular cylinder. Thus, for the rectangular cylinder the turbulence intensity is considerably larger not only in the HV region but also in the near-wake region compared to the circular cylinder.
6. Dynamics of Coherent Structures
[39] Figure 10 visualizes the vortical structure of one of the instantaneous flow fields using a Q isosurface. Besides the main necklace vortex HV1, several other eddies are present upstream of the rectangular cylinder. Though some of these eddies resemble hairpin vortices their coherence is much smaller compared to that observed by Paik et al. [2007] in a simulation with steady inflow (no fluctuations in the incoming flow). This means the secondary instability responsible for the formation of hairpin-like vortices wrapping around the core of the main necklace vortex observed by Paik et al. [2007] does not play a major role in the case of an incoming turbulent flow containing turbulent eddies which is the case of interest for river engineering applications. A small junction vortex is also present at the base of the rectangular cylinder.
[40] Examination of the instantaneous flow fields show there are two main types of events that result in the growth of the coherence of HV1 in front of the rectangular cylinder.
[41] The first one is due to the interactions between HV1 and secondary necklace vortices. As a result of these interactions, vorticity is fed into the core of HV1 which increases its strength at the locations along its core where the two vortices are close enough to exchange vorticity. These interactions take place when the downstream part of the jet-like flow beneath HV1 is oriented along the bed and no ejection of vorticity of opposite sign to that of HV1 is taking place. Such an event is captured in Figures 11a and 11b which show the distribution of the out-of-plane vorticity in the symmetry plane (y/D = 0). At t = t1, the secondary necklace vortex HV2 is relatively strong but does not exchange vorticity with HV1. At t = t1 + 0.4 D/U, HV2 feeds vorticity into the core of HV1. The process stops by t = t1 + 0.7 D/U when the downstream part of the jet-like flow starts changing direction. As a result, at t = t1 + 1.1 D/U a tongue of vorticity whose sign is opposite to that of the vorticity in the core of the necklace vortices is being ejected from the channel bottom, in between the cores of HV1 and HV2. The downstream part of the jet-like flow is oriented away from the channel bed and its intensity peaks. The tongue of vorticity engirdles part of the core of HV1. As a result, at t = t1 + 1.6 D/U the intensity of the jet-like flow and the circulation of HV1 decrease, and the component of the jet-like flow oriented away from the channel bed is very weak. Though subject to large temporal variations, the average timescale associated with this type of interactions is around 1.8 D/U which is close to the average time needed for HV1 to switch from one mode to the other.
[42] The nature of the interactions of the secondary necklace vortices with the primary necklace vortex, both in front of the obstruction and past it, was also studied in previous investigations of junction flows [e.g., Thomas, 1987]. Compared to the laminar unsteady regime [see, e.g., Thomas, 1987; Lin et al., 2003] in which the nature of these interactions is regular, and the shedding of secondary necklace vortices and their merging with the primary vortex take place at a definite frequency, which increases with the Reynolds number, in the turbulent case studied here the nature of these interactions is much more random. This is the case for the interactions taking place in the region situated in front of the cylinder as well as for the interactions among the legs of the necklace vortices on the sides and downstream of the cylinder. The average nondimensional shedding frequency of the secondary necklace vortices in the present simulation (Re = 2.4 × 105) was estimated to be St∼0.5 which is comparable to the frequency (St∼0.35) measured by Thomas [1987] for Re∼13,000 at which the laminar HV system transitions to a turbulent HV system.
[43] The second one is due to the convection of highly energetic eddies with the downflow toward the channel bed (e.g., the eddy denoted SV in Figure 11c) and from there away from the face of the cylinder (Figure 11d). When the vorticity of these eddies has the same sign as that of HV1, the merging event (Figure 11d) results in an increase of the coherence of HV1 at locations along its core where this interaction took place.
[44] The interactions between HV1 and the secondary necklace eddies do not take place uniformly along the core of HV1. For example, at the time instance at which the relative position of the cores of the necklace vortices is visualized in Figure 12a, HV2 extends mostly on the left side of the cylinder. While the legs of HV1 and HV2 start merging and exchange vorticity, the degree of interaction between the cores of the two vortices is much lower in front of the cylinder (−0.5 < y/D < 0.5). At subsequent times, the core of HV1 is moving away from the cylinder over most of its length while the core of HV2 remains approximately at the same location. The result is an amplification of the coherence of HV1 over most of the left side of the cylinder.
[45] At the time instance shown in Figure 12b, the coherence of HV1 on both sides of the cylinder peaks at locations situated close to the two flanks of the rectangular cylinder rather than at locations close to the symmetry plane, as expected to be the case for a circular cylinder. This behavior is characteristic of groynes where the intensity of the HV system peaks at locations situated close to the tip of the groyne [Koken and Constantinescu, 2008a]. Observe also that the left leg of HV1 extends up to the large billow of vorticity behind the cylinder. The billow corresponds to the large-scale roller that formed on the left side of the cylinder and is being convected away from it. Animations show these interactions, which result in the downstream part of the leg of HV1 being entrained into the roller, happen regularly each time a large-scale roller is being shed away from the cylinder. The coherence of HV1 is always larger on the side on which the large-scale roller is being shed.
[46] The process through which one large-scale roller forms and is then shed from each side of the rectangular cylinder during one shedding cycle is visualized in Figure 13. The process is quasi-regular. The time period between the successive shedding of a roller from the same side of the cylinder is 5.5 D/U, which corresponds to a Strouhal number of 0.18. The velocity power spectra in Figure 14c clearly show the fact that most of the large-scale energy in the wake is associated with the shedding of the rollers.
[47] Chen and Jirka [1995] studied the influence of flow shallowness on the turbulent wakes behind circular cylinders and high aspect ratio rectangular cylinders. They found that the shallow near-wake characteristics of plane wakes behind cylinders extending over the whole depth of the channel are function of a shallow wake parameter S = cfD/H, where cf is the friction coefficient, D is the width of the cylinder, and H is the channel depth. For small values of the parameter S, corresponding to a relatively low influence of the bed friction on the wake flow, the wake contains rollers that are shed regularly behind the cylinder. This vortex street wake type is similar to the one observed in the flow past infinitely long cylinders. Past a certain threshold value Sca the wake transitioned to an unsteady bubble wake type in which flow instabilities grow downstream of a recirculation bubble attached to the body. Finally, as S in further increased past a second threshold value Scc, a steady bubble wake type is observed in which the attached bubble is followed by a turbulent wake that does not contain growing instabilities. The values of Sca and Scc were found to be somewhat dependent on the shape of the cylinders. For plates or high aspect ratio rectangular cylinders Chen and Jirka [1995] estimated Sca = 0.16 and Scc = 0.4. In the present simulation H = D and cf ∼ 0.0035, so S = 0.0035 ≪ 0.16, which is consistent with the observed shedding of large-scale rollers in the wake. As in the present simulation the local stability parameter Sb = cf(2b/H) defined with the width of the wake (2b) remains smaller than 0.16 over the whole length of the computational domain, the rollers are expected to maintain their coherence and continue to increase their size until they exit the domain. This is consistent with the numerical results. So, in the test case considered in the present paper the wall friction effects are not strong enough to suppress the transverse growth of the disturbances that are responsible for the formation and grow of the rollers. Finally, the predicted value of the Strouhal number (St = 0.18) is very close to the value (St = 0.17–0.18) determined by Chen and Jirka [1995] for wakes behind plates with S < 0.04. In fact, Chen and Jirka [1995] showed that for nonporous plate wakes the value of the Strouhal number remains relatively constant until S ∼ 0.32.
[48] In Figure 13a the roller (A) on the right side has detached and its core moves toward the left side as it is convected away from the cylinder. The trajectories of the eddies shed inside the SSL on the left side of the cylinder curve inward toward the back of the cylinder. This allows eddies containing vorticity of the same sign (negative in Figure 13) to accumulate behind the cylinder on its left side. The eddies inside this patch start merging. As a result, this patch gradually forms the new roller (B) that is going to be shed from the left side of the cylinder. In Figure 13b these eddies form a relatively compact zone of negative vorticity whose mean size has increased compared to Figure 13a. As the strength of this patch of negative vorticity associated with roller B increases and roller A moves further away from the cylinder, the size of roller B increases too (Figure 13c). As roller B starts to detach from the back of the cylinder, its size continues to grow owing to the influx of eddies from the SSL (Figures 13c and 13d). At a certain point the roller gets into the way of the eddies shed in the SSL on the right side of the cylinder (Figure 13d). Meanwhile, a new roller (C) containing positive vorticity starts forming on the right side of the cylinder. As the size and the strength of roller C increase, roller B moves away from the cylinder (Figure 13e). Then, roller C starts detaching from the back of the cylinder and gets into the way of the eddies shed on the SSL on the left side of the cylinder (Figure 13f). This is the end of a full shedding cycle.
[49] Many of the individual eddies that merge to form the rollers do not lose their coherence during the merging process. The vorticity distribution within each roller never resembles that of a regular vortex. It is rather a collection of smaller eddies containing vorticity of the same sign that maintain their coherence while being advected together. The alternate shedding of large-scale rollers induces the undular shape of the wake region.
[50] The large-scale vortex shedding behind the rectangular cylinder at Re = 2.4 × 105 has many similarities to the one observed in the flow past a circular cylinder at Reynolds numbers that are high enough for the drag crisis to occur away from the near-bed region. The main resemblance is the formation and shedding of very strong rollers from the two sides of the cylinder. The mechanism through which these rollers form is very similar.
[51] However, there are several important differences between the characteristics of the large-scale vortex shedding behind the high-aspect ratio rectangular cylinder and the one behind a circular cylinder, even if the wake flow is supercritical. The first one is the fact that away from the near-bed region the relative strength of the rollers (e.g., their circulation at a certain level) is higher in the case of the rectangular cylinder and the rollers maintain their compactness for much larger distances as they are shed away from the cylinder. Though the large-scale shedding is close to regular in both cases and the width to channel depth ratio is similar, the Strouhal number associated with the formation and convection of the rollers is different. The Strouhal number is 0.18 in the rectangular cylinder simulation (Figure 14c) and 0.27 in the circular cylinder simulation. The values inferred from experiment were 0.185 and 0.26, respectively. We think these differences are due to the fact that the process that results into the eddies from one of the separated shear layer coalescing into a roller vortex is more efficient in the case of rectangular cylinders. The separated shear layer from one side penetrates deeper on the other side during the shedding cycle and, as a result, more of the eddies convected inside the separated shear layer from the other side of the cylinder are trapped into the eddy that will develop into a new roller. The fact that the separation point has the same position at all depths should further increase the coherence of the rollers in the near bed region in the case of a cylinder with sharp edges.
[52] The main difference occurs in the near-bed region. In the circular cylinder case, the flow remains subcritical (the mean polar angle at separation is close to 88°) in the near-bed region despite being supercritical away from the bed (the mean polar angle at separation is close to 103°). The intensity of the large-scale rollers and their compactness are substantially diminished as the bed is approached. By contrast, in the case of a rectangular cylinder, the separation takes place at the edge of the cylinder at all depths and the large-scale vortex shedding is much more uniform over the whole depth of the flow (e.g., compare the out-of-plane vorticity distributions in Figure 14). Consequently, the strength of the rollers varies very little as the bed is approached, and the characteristics of the large-scale vortex shedding are largely unaffected by the proximity of the bed. Of course, the near-bed turbulence produces some additional jittering and accelerates the breaking of the larger eddies forming the rollers into smaller highly 3-D eddies, but the strongly coherent rollers are clearly observable even at very small distances from the bed (e.g., at z/D = 0.032 in Figure 14b). This was not the case for the circular cylinder. Still, with respect to the case of a rectangular cylinder mounted on a slip surface, the energy of these rollers is expected to be lower owing to the stabilizing effect of the bed friction. The effect of the bed friction is expected to increase with the degree of flow shallowness (larger values of the D/H ratio).
[53] The vorticity distributions in the instantaneous flow (Figure 15) show interesting differences between the eddy content in the near bed region and away from it. The presence of a large roller (A) that starts detaching from the right side of the cylinder is observed at both levels (z/D = 0.5 and z/D = 0.1). A less coherent roller (B) that was shed from the other side of the cylinder can also be observed at a larger distance from the cylinder. However, not all the large-scale coherent structures have their axes oriented along the vertical direction. At z/D = 0.5 two elongated regions of high vorticity magnitude are present. They are denoted C1 and C2. The vertical vorticity component inside these regions is low. These regions correspond to the formation in the near-wake region of elongated coherent structures whose axes make a relatively small angle with the horizontal direction. These eddies can be very coherent as also shown from the visualization of the vortical content of the near-wake flow using a Q isosurface in Figure 15c. In this figure the eddies similar to C1 and C2 are denoted D1 and D2. Over the depth of the channel more than one of these elongated, relatively horizontal, eddies can be present. They appear to be similar to the finger vortices that form as a secondary instability in between two successively shed rollers in the flow past an infinitely long cylinder. In the latter case the finger vortices are significantly weaker than the roller vortices in the near wake. This does not appear to be the case in the present simulation, where the relative shallowness of the flow is one of the reasons the coherence of these eddies is larger than the one of the other eddies in the near-wake region.
7. Analysis of the Shear Stress and Pressure Root-Mean-Square Fluctuations at the Bed
[54] The largest values of at the bed (Figure 16a) are present inside the SSLs and over the upstream and middle parts of the recirculation bubbles behind the rectangular cylinder. The nondimensional values of are about ten times higher than the ones predicted for the circular cylinder case (Figure 16b). This means that the pressure fluctuations play a much more important role in the entrainment of sediment particles at the bed in the case of a high-aspect-ratio rectangular cylinder.
in the two frames is different. The regions of relatively low bed friction velocity and values, close to those recorded in a fully turbulent channel flow, were blanked.[55] For a mean sediment size of 1 mm, which is typical of local scour experiments performed in the environmental flume of IIHR, the (nondimensional) critical value of the Shields bed-friction velocity is uτc0 = 0.056 U. The entrainment regions in the mean flow predicted based only on the critical value of the bed friction velocity are also shown in Figure 16 (the regions where uτ < uτc0 were blanked). The presence of large pressure fluctuations in the case of a rectangular cylinder will allow entrainment of sediment particles at values of uτ that are considerably lower than uτc0.
[56] In Figure 16a, the largest uτ values are present in the strong flow acceleration region near and downstream of the two flanks of the rectangular cylinder. This is consistent with the distribution of uτ in the flow past the circular cylinder for which the largest values are observed in a region centered around ∣ϕ∣ = 45° where the horizontal velocity in the near-bed region around the cylinder is the highest. A secondary region of relatively high uτ values is present beneath the core of HV1. The amplification rate inside this region peaks close to the two extremities of the rectangular cylinder where, as was previously discussed, the intensity of HV1 is the highest. Beneath HV1, the values of the bed friction velocity close to the two sides of the cylinder (∣y/D∣ = 0.5) are about 30% higher than those in the symmetry plane (y/D = 0). The regions of high uτ values induced by the strong flow acceleration as it passes the cylinder and by the presence of HV1 merge together.
[57] Compared to the circular cylinder case, the values of uτ in the mean flow remain larger than uτc0 for a much longer distance (x/D ∼ 3 in Figure 16a compared to x/D ∼ 1.5 in Figure 16b) in the region where the large-scale rollers detach and are convected away from the cylinder. The strong coherence of the rollers in the near-bed region (see discussion of Figure 14) is responsible for the large uτ values induced at the bed by their passage. The distribution of uτ in one of the instantaneous flow fields (Figure 17a) shows that uτ > uτc0 beneath the last 5–6 rollers shed in the near-wake region. The instantaneous values of uτ beneath these rollers are comparable to the ones recorded in the acceleration region around the sides of the cylinder and beneath HV1. This means that in the initial stages of the scour process, the sediment entrainment behind the rectangular cylinder, in particular along the trajectory followed by the rollers detaching from the two sides of the cylinder, will be comparable to that occurring close to the flanks of the cylinder and probably higher than the one occurring in front of the cylinder. The presence of regions where uτ > uτc0 at large distances behind the rectangular cylinder in the instantaneous flow fields (Figure 17a) means that lots of sediment particles will be entrained from the near-wake region in the initial stages of the scour process. The actual entrainment region will be much larger than the one predicted on the basis of the bed friction velocity distribution in the mean flow (Figure 16a). This is because sediment particles can be entrained locally by the passage of a strong roller at locations where the value of uτ in the mean flow is significantly smaller than uτc0. By comparison, the instantaneous bed friction velocity distribution for the circular cylinder in Figure 17b does not show the presence of regions of relatively high uτ values at large distances behind the cylinder, consistent with the observed decrease in the coherence of the rollers as the bed is approached.
[58] More quantitative evidence of the large amplification of uτ caused by the quasi-regular passage of the rollers in the wake is given in Figure 18a that shows the time series of uτ at a point situated on the trajectory of the rollers (point 5 in Figure 16a). The instantaneous values can be up to three times larger than uτc0. When the rollers are not present, the bed friction velocity values are only 30% of the critical value. Though large-scale variations associated with changes in the coherence of HV1 are also observed in the time series of uτ (Figure 18b) at a point situated beneath HV1 (point 6 in Figure 16a), the amplitude of these oscillations is smaller and the large-scale oscillations are more irregular.
[59] Comparison of the instantaneous bed-friction velocity distributions in Figure 19 and of the out-of-plane vorticity distributions in Figures 13d–13f demonstrates the high correlation between the position of the rollers and the regions of high uτ values behind the rectangular cylinder. The three frames in Figure 19 span half of a full shedding cycle (T = 5.5 D/U).
[60] In Figures 13d and 19a, the roller (B) on the left side of the cylinder has just detached. Strong eddies from the SSL are convected inside the core of roller B. The passage of these eddies over the bed and the acceleration of the flow on the outward side of the SSL induces large uτ values around the left flank of the cylinder. A new roller C starts forming on the right side. However, the speed of the eddies accumulating on the right side of the cylinder is relatively small so the amplification of uτ beneath roller C is smaller than the one beneath roller B.
[61] In Figures 13e and 19b, as vortex B moves away and slightly outward from the cylinder the coherence of the leg of HV1 increases on the left side of the cylinder. An elongated patch of high uτ values is observed beneath the core of HV1 as it wraps around the left side of the cylinder. The leg of HV1 interacts with the SSL on the left side and with roller B. This explains the large thickness of the region of high uτ values on the left side of the cylinder between x/D = 0 and x/D = 1.0. On the right side of the cylinder roller C continued to grow while its center moved closer to the symmetry plane.
[62] In Figures 13f and 19c, roller C has detached from the cylinder and its core moved even closer to the symmetry plane, such that to create the conditions for the formation of a new roller on the left side of the cylinder. The interaction between the leg of HV1, whose coherence remains strong, and roller B is at its peak. The passage of roller B induces a very concentrated patch of high uτ values at the bed. Animations of the out-of-plane vorticity in a horizontal plane situated close to the bed show that the downstream end of the leg of HV1 curves toward roller B and is drawn into it. As a result of these phenomena, the region of relatively high uτ values on the left side of the cylinder attains its largest size. At following times, the size of the core of roller B increases and the uτ values beneath it decrease, similar to the evolution of the patch of high uτ values associated with roller A on the right side of the cylinder in Figure 19. Meanwhile, vortex C starts moving outward as it is convected downstream such that it can start to interact with the right leg of HV1.
8. Summary and Conclusions
[63] The flow past a surface-mounted rectangular cylinder of high aspect ratio positioned perpendicular to the incoming flow, placed in a relatively shallow flat-bed channel (H = D), was investigated on the basis of results from a high-resolution DES simulation at a channel Reynolds number of 2.4 × 105. This configuration is an idealized prototype for several practical flow configurations present in rivers and coastal areas. Because of the proximity of alluvial beds, the large-scale coherent structures generated as a result of the flow obstruction induced by the obstacle cause the scouring and erosion of the bed.
[64] The present investigation showed the critical role played by the large-scale coherent structures and their interactions on the sediment entrainment and transport phenomena at conditions corresponding to the start of the scour process (flat bed). This information was used together with that provided by a detailed analysis of the mean flow and turbulence statistics to better understand the associated geomorphodynamic processes. The paper also discussed the similarities and differences with the case of a circular cylinder (effect of bluntness of the in-stream obstruction). A main difference with the flow past surface-mounted circular cylinders at high Reynolds numbers is that the position at which the boundary layer separates is fixed (the flanks of the rectangular cylinder) and is the same at all flow depths.
[65] Similar to the case of the flow past a circular cylinder and other surface-mounted bluff bodies, the large amplification of the turbulence intensity in the HV region was due to the fact that the core of the main necklace vortex was subject to large-scale bimodal oscillations. The distributions of the pressure RMS fluctuations and of the tke within the region where the core of the main necklace vortex oscillates between the zero-flow and the back-flow modes were similar to the ones observed for the circular cylinder. On the other hand, in the case of a high-aspect-ratio rectangular cylinder the average strength (e.g., nondimensional values of the circulation, tke and pressure RMS fluctuations at sections cutting through the core of the main necklace vortex) of the HV system was significantly higher and the core of the necklace vortex was situated at larger distances relative to the upstream face of the cylinder. While in the case of a circular cylinder the circulation of the main necklace vortex was the highest in the symmetry plane, in the case of the rectangular cylinder the circulation of the main necklace vortex peaked at sections located close to the two flanks of the cylinder, similar to the case of an isolated groyne or abutment where the intensity of the HV system peaks at locations close to the extremity of the groyne.
[66] Consistent with results from stability analysis [Chen and Jirka, 1995] and experiment, a vortex street type wake was present. The characteristics of the large-scale vortex shedding presented several important similarities and differences to the one observed for circular cylinders at high Reynolds numbers. The formation of very strong rollers behind the cylinder on its two sides followed by their advection in the near wake was similar to that observed in simulations of flow past circular cylinders for which the flow away from the near-bed region was supercritical. In the case of circular cylinders the intensity of the rollers decayed significantly close to the bed mainly because the separation point on the cylinder moved upstream and the flow remained subcritical in the near-bed region. In contrast to that, in the case of a rectangular cylinder, where the position of the separation is the same at all depths, the intensity of the rollers remained very high in the near bed region. The Strouhal number associated with the large-scale vortex shedding was 0.27 in the case of the circular cylinder and 0.18 in the case of the rectangular cylinder. Large-scale coherent structures resembling the fingers eddies developing between the rollers in the flow past infinitely long cylinders were observed in the near wake. The coherence of these eddies was much larger than the one expected for a secondary instability. The strong amplification of the strength of these eddies is probably due to the relatively shallow flow conditions considered in the present simulation. The interaction between the rollers and the legs of the main necklace vortex was much more regular in the case of the rectangular cylinder, as each time a roller would detach from the back of the cylinder the leg will be pushed outward while the downstream part of the leg will merge with the roller detaching from behind the cylinder.
[67] The distributions of the pressure RMS fluctuations and mean-flow bed-friction velocity at the bed were in many respects similar to those for a circular cylinder at high Reynolds numbers. However, the nondimensional values inside the regions of high amplification of the pressure fluctuations were about one order of magnitude higher than the ones predicted for circular cylinders, and the width and length of the region where the bed-friction velocity distribution was significantly amplified on the two sides of the cylinder were significantly larger. Also significant for the development of the scour hole in the initial stages of the scour process, the rollers were found to induce large bed friction velocity values at large distances behind the cylinder. Even if in the mean flow the values of the bed friction velocity are relatively low away from the cylinder, sediment particles will be entrained by the passage of the rollers.
[68] The present study shows that the shape of the obstruction can have significant qualitative and quantitative effects on the dynamics of the large-scale coherent eddies than control sediment entrainment and on the turbulence structure around an in-stream obstruction. One of the main challenges of developing new physics-based scour prediction methods is to account in a simple way for the changes in the flow physics triggered by the shape of the obstruction and the Reynolds number. To be able to do that a clear understanding of the flow, turbulence and sediment entrainment mechanisms at the different stages of the scour process at in-stream obstacles of common shapes is needed.
Acknowledgments
[69] The authors would like to thank Robert Ettema for his advice on various aspects of this research and the National Center for High Performance Computing (NCHC) in Taiwan, in particular W. H. Tsai, for providing the computational resources needed to perform some of the simulations as part of the collaboration program between NCHC and IIHR Hydroscience and Engineering.
