Coherent structure dynamics and sediment erosion mechanisms around an in-stream rectangular cylinder at low and moderate angles of attack

Authors


Abstract

[1] Local scour around elongated in-stream structures (e.g., high–aspect ratio rectangular bridge piers) is mainly driven by the interactions between the erodible bed and the large-scale coherent structures generated by the presence of the flow obstruction. The present investigation uses eddy-resolving numerical simulations to study the mean flow and turbulence structure around a high–aspect ratio rectangular cylinder placed in a flat bed channel. Simulations are conducted for three angles of attack (α = 0°, 15°, and 30°) at a channel Reynolds number of 2.4 × 105. This paper focuses on the dynamics of the large-scale coherent structures forming around the rectangular cylinder and their role in controlling sediment entrainment at conditions corresponding to the start of the scour process. Simulation results show that most of the sediment is entrained from the bed by the eddies shed inside the separated shear layers (SSLs), by the legs of the necklace vortices, and by the strongly accelerated flow on the outer side of the SSLs. For α = 0° and 15°, the horseshoe vortex (HV) system plays a relatively minor role in the entrainment of sediment in front of the cylinder, and the passage of the wake vortices (rollers) results in a small amplification of the bed friction velocity. In contrast, for α = 30°, the unsteady dynamics of the main necklace vortices part of the HV system and of the roller vortices results in a significant amplification of the instantaneous bed friction velocity. The mean flux of sediment entrained from the bed calculated on the basis of the mean flow field is found to underestimate by 2–3 times the same quantity when estimated more correctly on the basis of the instantaneous flow fields. The primary reason for this underestimation is sediment entrainment in regions where the mean flow bed friction velocity is smaller than the critical value for entrainment. Quantitative information on the extent of the regions of high values of the bed friction velocity and its standard deviation and on the sediment entrainment flux as a function of the angle of attack is essential to guide scour protection measures around the in-stream noncircular obstructions.

1. Introduction

[2] The presence of large natural (e.g., large-scale bathymetry features) or man-made (e.g., groyne fields and other structures used in river restoration projects) flow obstructions in alluvial channels greatly modifies the local structure of the flow and controls the geomorphodynamic processes [Paola et al., 1986; Sumer and Fredsoe, 2002; Fael et al., 2006; Koken and Constantinescu, 2008a, 2008b; Constantinescu et al., 2010]. In these flow regions, the turbulence production and energy are concentrated in large-scale eddies in the form of necklace vortices at the base of the obstruction, wake rollers and vortical tubes inside the separated shear layers (SSLs). These eddies, rather than the classical sweep and ejection events, together with the near-bed regions of strong flow acceleration situated near the sides of the in-stream obstruction control the development of the bathymetry in the vicinity of the obstruction.

[3] The development of scour and deposition regions and the presence of regions of high turbulence and/or low velocity (e.g., recirculation regions behind in-stream structures or in between two consecutive structures) are generally beneficial if the goal is to enhance habitat for fish and vegetation. Even in these cases, it is sometimes desirable to be able to control the sediment transport and erosion processes. In the case of in-stream obstructions whose main role is to support a certain structure, the development of severe local scour is undesirable and dangerous. In particular, this is the case of bridge piers and abutments due to the possible structural failure of bridges if local scour is too severe.

[4] Most experimental studies of scour around in-stream structures nonattached to the channel banks considered the case of a circular cylinder [e.g., Dargahi, 1990; Dey et al., 1995; Unger and Hager, 2007; Graf and Yulistiyanto, 1998; Roulund et al., 2005]. Scour processes around bridge piers of relatively circular shape can be studied on the basis of experimental and numerical investigations conducted with surface-mounted circular cylinders. Because of the circular shape, scour is independent of cylinder orientation relative to the incoming flow direction. This is one of the main reasons why most scour prediction formulas are developed for circular cylinders. Then, correction factors with respect to scour predicted for a circular cylinder with a diameter equal to the projected width of the noncircular cylinder along the direction perpendicular to the incoming flow are applied to account for a different shape (e.g., rectangular) and orientation (angle of attack) of the incoming flow relative to the axis of the cylinder [e.g., see Melville and Coleman, 2000]. The problem is that changes in the shape of the cylinder (e.g., see Dey and Raikar [2007] and Raikar and Dey [2008], who studied low–aspect ratio rectangular cylinders) and in the angle of attack can result in a flow field whose large-scale eddy content is qualitatively and/or quantitatively very different from the one observed around circular cylinders. Consequently, the scour mechanisms, the shape and the extent of the scoured region around the noncircular in-stream obstruction will be different compared to those observed for circular cylinders.

[5] As the angle of attack increases, scour depth increases primarily because the effective frontal width of the noncircular cylinder becomes larger. Several experimental investigations [e.g., Laursen and Toch, 1956; Mostafa, 1994; Ettema et al., 1998] have documented the significant increase of the maximum scour depth with the angle of attack for rectangular cylinders of varying aspect ratios. For example, the local scour depth at a rectangular cylinder with a length to width ratio of 8 is nearly tripled at an angle of attack of 30° [Melville and Coleman, 2000]. Unfortunately these investigations do not provide a detailed analysis of the changes in the flow structure and sediment entrainment mechanisms with the angle of attack at different stages of the scour process. As discussed by Ettema et al. [2006], such information is needed to develop more physics-based scour prediction methods.

[6] The case of high–aspect ratio rectangular cylinders is a very relevant and difficult one. Currently, the use of high–aspect ratio rectangular piers that are aligned with the flow at normal flow (design) conditions is more and more common because of their lower cost, especially for large bridges. The problem is that pier orientation to approach flow can progressively change in time because of changes in the channel or thalweg curvature, subchannel alignment, and development of a channel bar. Flow around other features, notably an abutment situated relatively close to the pier, may also alter approach flow orientation and magnitude. Accurately accounting for this at the design stage is very difficult. Even more important, the angle of attack at bridge crossings may change significantly during floods.

[7] Changes of about 30% in the angle of attack at high flood conditions are not uncommon, especially for braided shallow streams. The occurrence of such large angles of attack over time periods of the order of days can dramatically change the sediment entrainment and transport around the elongated in-stream obstruction and result in development of large scour holes that can endanger the stability of the bridge pier or compromise habitat restoration efforts. This motivates the need to investigate the changes in the flow structure, sediment entrainment mechanisms, and extent of the regions where the bed will be scoured around high–aspect ratio rectangular cylinders between α = 0° when the flow is aligned with the cylinder (design conditions) and angles of attack that are typical of those observed at high flood conditions (α math formula30°).

[8] Field, laboratory and numerical investigations can be used to elucidate the changes in the flow structure and scour processes around rectangular cylinders with the angle of attack. As discussed by Kirkil and Constantinescu [2009, 2010], each of these approaches has its advantages and disadvantages. In the present paper we report a numerical investigation of these processes for conditions present at the initiation of scour (flat-bed channel) based on results of fully three-dimensional eddy-resolving simulations. Such simulations allow a detailed investigation of the flow physics and the mean flow predictions are generally more accurate than those obtained using classical RANS models [Rodi, 1997; Keylock et al., 2005].

[9] The main advantage of a numerical approach based on eddy-resolving simulations is that the whole 3-D flow fields are available together with the distributions of the variables that control bed erosion. Analysis of the instantaneous and time-averaged solutions should allow understanding the dynamics of the main types of large-scale coherent structures in the flow and their effect on scour. The main limitation is that such computations cannot be used to also predict the evolution of the bed during the whole duration of the scour process. This is mainly because of the huge computational cost. Still, as the time scales over which the changes in the bathymetry significantly affect the flow field are much larger than those of the main coherent structures in the flow, such investigations can provide an accurate picture of the flow structure and sediment entrainment mechanism for a given bathymetry corresponding to a certain stage of the scour process. For example, such an approach was already used to understand scour mechanisms around spur dikes and circular cylinders by Koken and Constantinescu [2009], Kirkil et al. [2009], and Escauriaza and Sotiropoulos [2011]. In the later study, the Eulerian prediction of the turbulent flow fields by DES was coupled with a Lagrangian model that was used to predict entrainment of inertial particles initially placed on the (flat) bed upstream of a circular cylinder. The simulations of Escauriaza and Sotiropoulos [2011] showed that upstream of the cylinder, the ejection of the particles from the bottom boundary layer, their movement and trajectories were controlled to a large extent by the unsteady dynamics of the necklace vortices.

[10] In the present study, detached eddy simulation (DES) is used to account for the effect of the unresolved scales on the resolved flow. The simulations are conducted with a rectangular cylinder of a length to width ratio of 14.5 at a channel Reynolds number of 2.4 math formula 105, which is high enough such that the flow structure and dynamics of the coherent structures are qualitatively similar to those observed at field conditions, at least in small to medium size rivers. The channel Froude number is 0.2, so the flow is subcritical. The incoming flow is fully developed and contains resolved fluctuations from a precursor simulation. Three cases with different angles of attack are considered. The base case A00 is conducted with a zero angle of attack (α = 0°). The other two simulations, A15 and A30, are conducted with an angle of attack of 15° and 30°, respectively. The latter two simulations are representative of flow past high–aspect ratio cylinders with moderate (10° < α < 45°) angles of attack.

[11] In a related DES study, Kirkil and Constantinescu [2009, 2010] investigated the flow past the same rectangular cylinder at an angle of attack of 90°, representative of the high angle of attack regime (90° > α > 45°). For this orientation of the cylinder, the adverse pressure gradients in front of the cylinder are the largest. This is the reason why the maximum scour depth at equilibrium scour conditions is observed for α = 90°. Their investigations have also shown that important changes take place in the sediment entrainment mechanisms between piers of circular shape and rectangular piers at high angle of attack having the same projected width. However, the direct relevance of this case is rather limited for the present study, as very rarely the angle of attack at a rectangular in-stream obstruction designed to be aligned with the flow at normal flow conditions increases to more than 45°.

[12] The numerical method, turbulence modeling approach and simulation set up are discussed next. Sections 3 and 4 discuss the effect of the angle of attack on the structure of the mean flow and dynamics of the large-scale coherent structures present inside the SSLs, the wake and the horseshoe vortex (HV) system. The modification of sediment erosion mechanisms with the angle of attack is discussed. Section 5 focuses on the identification of regions where sediment will be entrained on the basis of the distributions of the friction velocity in the mean and instantaneous flow fields and the pressure RMS fluctuations at the bed. Section 5 also shows that estimating the mean sediment entrainment flux on the basis of the mean flow fields, similar to what is generally done in RANS-based numerical models, can lead to a substantial underprediction of entrainment. Section 6 revises the main findings and identifies directions for future work.

2. Numerical Method and Simulation Setup

[13] The eddy-resolving turbulence model, the numerical model and the boundary conditions are the same as those used by Kirkil and Constantinescu [2009] and are only briefly reviewed here. DES uses the same base turbulence model in the RANS and LES regions. Away from the solid boundaries, DES is similar to large eddy simulation (LES), in the sense that the subgrid scale viscosity is proportional to the square of the local grid spacing. Close to solid boundaries, DES reduces to a RANS model. No special treatment is required to match the solutions at the boundary between the LES and RANS regions. The model resolves the flow within the viscous sublayer, so the calculation of the bed shear stresses does not rely on the assumption of the presence of a logarithmic layer in the velocity profile. This is the main reason why DES is more accurate than LES with wall functions for prediction of high Reynolds number separated flows.

[14] The present study uses the Spalart-Allmaras (SA) RANS model as the base model in DES [see Spalart, 2009; Kirkil and Constantinescu, 2009]. The one-equation SA model solves a transport equation for the modified eddy viscosity, math formula. 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 math formula with a new length scale dDES = min(d,CDESΔ). The model parameter CDES is equal to 0.65 on the basis of calibration for isotropic turbulence and Δ is a measure of the local grid size. When the production and destruction terms in the transport equation for math formula 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. A modified version of DES called delayed DES [Spalart et al., 2006] is used. Delayed DES tries to always preserve the RANS mode inside the boundary layers, thus alleviating the grid-induced separation problems observed in standard DES conducted with steady inflow conditions. Moreover, the inflow fluctuations provide a spectral enrichment of the near-wall flow which further improves the performance of DES [Keating and Piomelli, 2006].

[15] The 3-D incompressible Navier-Stokes equations are integrated using a fully implicit fractional step method [Constantinescu et al., 2002; Constantinescu and Squires, 2004; Chang et al., 2007]. 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 fifth-order accurate upwind biased scheme and 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 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 time discretization is second order accurate.

[16] Validation of the DES code based on comparison with experimental data and/or results of well-resolved LES for flow in channels with bottom-mounted cavities, open channels with sidewall-mounted obstacles (e.g., spur dikes), flow past surface-mounted bluff bodies (in stream piers and rectangular plates at high angle of attack) with flat and scoured bed, sharp curved bends and river confluences is discussed by Chang et al. [2007], Kirkil [2008], Koken and Constantinescu [2009], Kirkil et al. [2009], Kirkil and Constantinescu [2009, 2010], Koken [2011], and Constantinescu et al. [2011a, 2011b].

[17] In particular, a DES simulation of the flow was a surface-mounted circular cylinder at Re = 106 [Kirkil, 2008] conducted using the same code, level of mesh refinement and type of boundary conditions showed that the turbulence statistics (e.g., the turbulence energy production) in the symmetry plane upstream of the cylinder were in good agreement with experiments [Devenport and Simpson, 1990] conducted for the flow past a wing-shaped cylinder with a thick incoming turbulent boundary layer. The comparison is justified as the upstream shape of the cylinder in the experiment was close to circular. The width/diameter D of the cylinder was used to nondimensionalize the experimental/numerical results. In particular, DES successfully predicted the shape and nondimensional levels of the turbulent energy production within the region of positive production where the core of the main necklace vortex is subject to bimodal oscillations and the relative location of this vortex in the mean flow relative to the cylinder compared to experiment (∼0.25D). Also similar to experiment, an elongated region of negative production was predicted in the bed vicinity close to the upstream face of the cylinder. Though the shape of the region of high turbulent energy production changed from mushroom-like to elliptical in DES and LES simulations conducted at a much smaller Reynolds number (Re = 18,000 [see Kirkil and Constantinescu, 2009, Figure 1]), DES and LES predicted a change in the mean position of the main necklace vortex of only 0.05D, from 0.25D to 0.3D, over a change of 2 orders of magnitude in the value of the Reynolds number. Similar to the Re = 106 simulations, analysis of the velocity histograms from the LES and DES at Re = 18,000 with an incoming turbulent flow containing resolved fluctuations confirmed that the main necklace vortex was subject to bimodal oscillations. Qualitatively, the distribution of the turbulent energy production in the symmetry plane upstream of the cylinder remained the same.

[18] One should also mention that DES with steady inflow of the flow past the wing-shaped cylinder [Paik et al., 2007], while successfully capturing the distribution of the turbulence statistics and the bimodal oscillations of the main necklace vortex (e.g., the velocity histograms displayed two peaks within the region of oscillation of the main necklace vortex), predicted the location of the core of this vortex to be situated somewhat upstream of the measured position.

[19] The other critical region for flow past surface-mounted bluff bodies is the separated shear layer (SSL) that controls sediment entrainment close to the extremity of the obstruction and the formation and shedding of roller vortices. On the basis of comparison with acoustic Doppler velocimetry (ADV) measurements, the same DES code was shown to correctly predict the amplification of the turbulence within the SSL forming as the flow is convected past the flank of a vertical wall spur dike placed in a channel with flat bed at a comparable Reynolds number [Koken, 2011].

[20] All results are presented in nondimensional form with the flow depth, D, as the length scale and the mean velocity in the channel, U, as the velocity scale. Figure 1 (top) shows the computational domain. The domain width is 5.7D. The computational domain extends 5D upstream of the centroid of the cylinder. The length of the pier, L, is equal to the channel depth, D. Its thickness is 0.07D. The aspect ratio of the cylinder is 14.5. These dimensions correspond to a set of experiments (D = 0.53, U = 0.45 m s−1, width of the flume = 3 m) performed by Kirkil and Constantinescu [2009] with the same rectangular cylinder for an angle of attack of 90°. The main axis of the cylinder is contained within the symmetry plane of the channel in case A00. The blockage ratio is less than 9% in all three simulations. Though the velocity and bed friction velocity will be somewhat larger in the flow acceleration regions close to the lateral edges of the rectangular cylinder compared to the case of a very wide channel, contraction scour effects are relatively small for this value of the blockage ratio. This was also confirmed by the loose bed experiment conducted by Kirkil and Constantinescu [2010] for the same cylinder with an angle of attack of 90° for which the blockage ratio was close to 15%.

Figure 1.

(top) General view of the computational domain and (bottom) visualization of the main coherent structures in the mean flow using the Q criterion in the simulation with α = 30°.

[21] Turbulent inflow conditions corresponding to fully developed turbulent channel flow with resolved turbulent fluctuations are applied at the inflow section. Chang et al. [2007] showed that the agreement between highly resolved LES and DES conducted on a much coarser mesh improved significantly when the inflow contained turbulent fluctuations. These fluctuations were obtained from a preliminary straight channel flow calculation at a lower Reynolds number and scaled on the basis of the ratio between the nondimensional turbulent kinetic energy in the outer flow at the two Reynolds numbers. At the outflow, a convective boundary condition is used. The free surface is treated as a shear-free rigid lid, which is justified as the channel Froude number is around 0.2. The cylinder's faces are treated as no-slip smooth surfaces. The nondimensional bed roughness was close to 40 wall units corresponding to a layer of sand with a mean diameter d50 = 1.05 mm. This sand size was used by Kirkil and Constantinescu [2010] in their loose-bed experiment with a rectangular cylinder at an angle of attack of 90°. The implementation of the rough-wall boundary condition in the base SA model is discussed by Spalart [2000] and Zeng et al. [2008, 2010]. Kirkil and Constantinescu [2009] provide a more detailed discussion of the implementation of the boundary conditions and justification (e.g., use of the rigid lid assumption) of the approaches used to specify them.

[22] The computational domain was meshed using about 10 million cells. A minimum grid spacing of one wall unit was used in the wall normal direction to avoid the use of wall functions. The grid spacing in the HV region was about 200 wall units. The grid spacing was around 20–100 wall units in the critical region situated around the two lateral sides of the rectangular cylinder where the SSLs are forming. The origin of the system of coordinates is located at the centroid of the cylinder on the bottom surface, with the x axis in the streamwise direction. The time step was 0.025D/U.

3. Vortical Content and Dynamics of the Separated Shear Layers and Wake Flow

[23] The recirculation regions and the SSLs are visualized in the mean flow using 3-D streamlines (Figure 2) and the distribution of the vertical vorticity, ωzD/U, at the free surface (Figure 3). Though some 3-D effects are present, the distributions of ωzD/U at other levels are similar to those shown in Figure 3. This is because, as explained by Kirkil and Constantinescu [2009], in the case of cylinders with sharp edges, the position of the separation point in a z = constant plane is fixed by the geometry and is the same at all depths, including within the near-bed region.

Figure 2.

Visualization of the (time-averaged) mean flow around the cylinder in the simulations with (left) α = 0°, (middle) α = 15°, and (right) α = 30° using 3-D streamlines. The streamlines originate in the near-bed region, upstream of the cylinder. The streamline color represents the elevation, z/D.

Figure 3.

Distribution of the nondimensional vertical vorticity, ωnD/U, at the free surface (z/D = 1) in the mean flow in the simulations with (top) α = 0° and (bottom) α = 15°. The regions with |ωnD/U| < 0.5 were blanked.

[24] For very small angles of attack (e.g., case A00), the SSLs generated by the interaction of the incoming flow with the upstream face of the cylinder reattach on the lateral faces and form two small recirculation bubbles. Another small recirculation region containing two bubbles forms behind the downstream face of the cylinder, as the boundary layers on the two lateral faces separate. For moderate angles of attack (e.g., cases A15 and A30), the incoming flow is deflected by the upstream edge of the cylinder and remains attached over the whole length of its left (relative to the incoming flow direction) lateral face. This is expected, as the attached boundary layer on the left face develops in a favorable pressure gradient induced by the flow blockage, whose magnitude increases with α. A relatively short SSL of high vorticity forms past the downstream edge of the left face.

[25] Meanwhile, the flow convected on the right side of the cylinder separates and forms a recirculation bubble that extends over the whole right lateral face of the cylinder and for some distance behind the cylinder. The bubble is laterally bordered by the two SSLs that originate at the upstream and downstream edges of the cylinder (Figure 3, bottom). Its mean length, Lb, and width, Wb, increase monotonically with α (e.g., Lb = 0.14D, 1.45D, and 1.88D and Wb = 0.15D, 0.6D, and 0.9D for α = 0°, 15°, and 30°, respectively). The mean pressure distribution around the cylinder is also strongly dependent on α, as shown by the distribution of the pressure coefficient at the free surface, math formula, in Figure 4 (p0 is the pressure away from the cylinder). The maximum pressure difference between the pressure at the stagnation point and the minimum pressure within the recirculation region expressed nondimensionally increases from 1.53 for α = 0°, to 2.55 for α = 15° and to 2.8 for α = 30°.

Figure 4.

Distribution of the nondimensional pressure, math formula, in the mean flow (z/D = 1) in the simulations with (top) α = 0°, (middle) α = 15°, and (bottom) α = 30°. The regions with |Cp| < 0.2 were blanked to better identify the regions of high and low pressure around the cylinder.

[26] The 3-D streamlines launched in the near-bed region upstream of the cylinder show that the flow within this tornado-like vortex is subject to upwelling motions that, in the mean, convect flow and sediment particles toward the free surface (Figure 2). The circulation of the core of this tornado like vortex measured in horizontal planes peaks toward the middle of the channel height (e.g., see case A30 in Figure 2) where the trajectories of the 3-D streamlines are converging toward the axis of the vortex before diverging again from it as the free surface is approached. For very large angles of attack (α > 60°) the flow behind the cylinder will become more symmetric and a second recirculation eddy will form close the left edge at the back of the cylinder. For α = 90°, the two recirculation eddies are symmetrical in the mean flow [see Kirkil and Constantinescu, 2009, Figure 7]. The flow within these recirculation bubbles is oriented toward the free surface at all angles of attack.

[27] The formation of the recirculation bubble, together with the acceleration of the flow convected on the right side of the cylinder, induces the formation of a long SSL of high-vorticity past the upstream edge of the cylinder (e.g., see Figure 3, bottom, for α = 15°). The slowly moving flow within the recirculation bubble in the vicinity of the lateral face of the cylinder does not create the conditions needed for a distinct SSL to form at the downstream edge of this face, as was the case for α = 0°. Thus, as opposed to case A00, where the rollers form because of the interaction between the two SSLs of equal strength (Figure 3, top), in cases A15 and A30 the rollers form because of the interaction between the higher-vorticity SSL forming close to the upstream face of the cylinder with the lower-vorticity SSL forming close to the back face of the cylinder. This is why the circulation and size of the core of the rollers forming on the two sides of the cylinder is quite different (see discussion of Figures 5 and 6). Only for very large angles of attack, the vorticity magnitude and spatial extent of the two SSLs and the circulation of the rollers of positive and negative circulation will become again comparable (e.g., see Kirkil and Constantinescu [2009, Figure 3b] for α = 90°).

Figure 5.

Distribution of (top) the nondimensional eddy viscosity at z/D = 0.9, (middle) vertical vorticity at z/D = 0.9, and (bottom) vertical vorticity z/D = 0.032 in an instantaneous flow field in the simulation with α = 0°.

Figure 6.

Distribution of (top) the nondimensional eddy viscosity at z/D = 0.9, (middle) vertical vorticity at z/D = 0.9, and (bottom) vertical vorticity z/D = 0.032 in an instantaneous flow field in the simulation with α = 30°.

[28] Figures 5 (case A00) and 6 (case A30) give more details on the structure and vortical content of the SSLs and near wake in the instantaneous flow fields. The results for case A15 were not included because they are qualitatively similar to those for case A30. As expected, at both angles of attack, the SSLs are populated by strong eddies in the form of vortex tubes that are convected away from the edge of the cylinder where the SSLs originated, while the near wake is populated by roller vortices. The dynamics of the rollers is important because they can transport sediment at large distances from the cylinder and their passage can induce large bed shear stresses and entrain sediment. Examination of the mean flow fields (e.g., see discussion of Figures 2 and 3) does not allow capturing, and even less analyzing quantitatively, these mechanisms because the rollers are not present in the mean flow. The large regions of concentrated vorticity, and sometimes of large bed friction velocity (see Figure 11), observed in the instantaneous flow fields that are induced by the rollers disappear in the mean flow because the passage of a roller at a certain location is followed by large periods of time when the flow above that location does not contain large-scale eddies. This shows the importance of analyzing not only the mean flow and turbulence structure, but also the vortical content of the instantaneous flow fields to be able to describe sediment entrainment and transport processes around a surface-mounted cylinder.

[29] Comparison of the nondimensional eddy viscosity, νt/ν (ν is the kinematic viscosity), and vertical vorticity distributions in Figures 5 and 6 shows that the wake structure in case A00 presents some differences with the one observed for moderate angles of attack (e.g., case A30). While the wake shape is undular at all angles of attack, as rollers containing patches of vertical vorticity of opposite signs are shed from the two SSLs, it is only for very small (α ∼ 0°) or very large (α ∼ 90°) angles of attack that the circulation and average size of the core of the rollers shed from the two sides of the cylinder are comparable.

[30] Some of these eddies originating in the SSLs forming at the upstream face of the cylinder in case A00 are convected close to the two lateral faces of the cylinder and perturb the vortex tubes convected inside the two SSLs forming at the back of the cylinder. The merging of vortex tubes originating in one or the other of the SSLs at the back of the cylinder results in the formation of the roller vortices in case A00. The perturbations of the vortex tubes in the SSLs forming at the back of the cylinder by vortical eddies convected close to the lateral faces of the cylinder are stronger close to the channel bed. This is because close to the bed the SSLs forming at the corners of the upstream face are, at most times, positioned very close to the two lateral faces. Away from the bed, most of the eddies originating in the SSLs forming at the upstream face are convected past the end of the cylinder without interacting with the vortex tubes convected inside the SSLs at the back of the cylinder. As a result, the rollers forming at the back of the cylinder in case A00 decrease significantly their coherence close to the bed. By this we mean that the circulation associated with the patch of high-vorticity eddies that define the core of a roller decreases. Thus, their capacity to entrain sediment beneath them is relatively small.

[31] For moderate angles of attack, the clockwise rotating rollers induced by the merging of vortex tubes from the SSL on the left side are more coherent (higher magnitude of the total circulation and smaller core size) than the counterclockwise rotating rollers forming from eddies originating in the SSL on the right side of the cylinder. However, the patches of concentrated vorticity associated with the roller vortices can still be clearly observed in the near-bed region at both moderate and large angles of attack [see also Kirkil and Constantinescu, 2009, Figure 14]. Thus, not only their capacity to transport suspended sediment but also their capacity to entrain sediment particles as they are convected away from the cylinder is important. The two SSLs interact in a region situated at a short distance behind the cylinder. The eddies inside the SSL on the right side are convected for a longer distance before they arrive in the region where the other SSL gets into their way and forces them to move leftward, toward the lateral face of the cylinder. Then, these eddies are convected upstream parallel to the lateral face of the cylinder. Some of them reach the upstream part of the SSL. Thus, their path is close to circular and they approximately follow the edge of the recirculation bubble.

[32] Two main reasons explain the lower coherence (e.g., as measured by circulation of the patch of vorticity associated with the roller at a given vertical location) of the rollers originating in the SSL on the right side of the cylinder. The first one is that the vortex tubes diffuse within the downstream part of the SSL. The second one is that the new roller that detaches from the cylinder contains lots of highly three-dimensional eddies from the recirculation bubble that were engulfed by the SSL eddies moving first away and then toward the region where the SSL originates. As these eddies do not contain predominantly vertical vorticity of one sign and do not resemble vortex tubes, the new roller forming as a result of the merging of all these eddies with the vortex tubes shed in the SSL will have a larger size and a lower magnitude of the total circulation.

[33] By contrast, the vortex tubes shed in the SSL on the left side of the cylinder start merging very close to the formation region and the roller does not contain a significant amount of other eddies convected into the roller from outside the SSL. This produces a more coherent roller with a larger capacity to entrain sediment. Thus, it is expected that the growth of the scour hole behind of the cylinder will be faster on the left side, at least during the initial stages of the scour process.

[34] Compared to case A00, the coherence of the clockwise and counterclockwise rotating rollers in cases A15 and A30 is much larger and their trajectories diverge faster away from the cylinder. This is why the lateral extent of the scour hole, away from the edges of the cylinder, will be larger on both sides in the cases with moderate angles of attack. Another important point is that the rate of growth of the scour hole behind the cylinder is affected not only by the size and circulation of the rollers in the near bed region but also by their frequency of passage.

[35] Analysis of velocity spectra in the wake and animations showing the vorticity fields show that the most energetic nondimensional frequency expressed as a Strouhal number (St = fD/U, f is the frequency associated with the shedding of roller vortices in the wake) has decreased from 0.75 for α = 0° to 0.46 for α = 15° and to 0.3 for α = 30°. Such a spectrum is illustrated in Figure 7 for α = 30°. One can see that most of the energy is concentrated at the frequency associated with the large scale vortex shedding. However, animations show that the time interval at which the rollers detach behind the cylinder is not constant. Variations of ±20% are very common. This is consistent with the fact that the decay of the energy away from the peak value in Figure 7 is not very sharp. The DES and experiment conducted by Kirkil and Constantinescu [2009] for α = 90° predicted St = 0.18 and 0.185, respectively. This is because the distance between the two SSLs increases with α. Consequently, the time needed for one shear layer to move and block the path of the eddies shed in the other SSL is longer. Thus, the frequency of passage will be lower. Though the passage of the rollers at a higher frequency is expected to allow for more sediment entrainment by the rollers per unit time in the simulations with a lower angle of attack, the circulation of the rollers at similar streamwise distances from the cylinder is decreasing significantly as α is decreased (see also discussion of Figure 11). The physical length scale that affects the shedding behind a cylindrical bluff body is the projected width, W, along the direction normal to the incoming flow. Using this length scale, the values of the Strouhal number St′ = fW/U are in fact monotonically increasing with the angle of attack (St′ = 0.05 for α = 0°, St′ = 0.12 for α = 15°, St′ = 0.15 for α = 30° and St′ = 0.18 for α = 90°). The fact that the nondimensional shedding frequency past high–aspect ratio rectangular cylinders placed in a relatively shallow channel, St′, is strongly dependent on α is another proof of the complexity of the flow and makes it even more difficult to propose simple models to account for the effect of the wake rollers on sediment entrainment. The dependence of St and St′ with both α and the nondimensional pressure difference between the pressure at the stagnation point and the minimum pressure within the recirculation bubble behind the cylinder is not linear.

Figure 7.

Spanwise velocity energy spectrum at a point situated at (x/D, y/D, z/D) = (3.0, 0.0, 0.9) in the wake of the cylinder in the simulation with α = 30°.

[36] The Q criterion [Dubief and Delcayre, 2000] is used in Figure 8 to visualize the coherent structures in the instantaneous flow. The Q criterion shows that the SSLs, the recirculation and near-wake regions are populated by a wide range of eddies. Interestingly, not all the dynamically important eddies have their axes oriented vertically, as will be expected in the case of a very deep flow where the vertically oriented vortex tubes and the wake rollers will account for most of the large-scale eddy content of the flow. As the angle of attack increases (see results for case A15 and especially for case A30 in Figure 8), the relatively high shallowness of the flow induces the formation of high-vorticity eddies whose axes are close to parallel to the bed. These eddies are predominantly present in the regions connecting two successively shed rollers. Such eddies are also present in the range of high angles of attack and play a similar role [Kirkil and Constantinescu, 2009]. The presence of such high-vorticity eddies in the vicinity of the bed provides an additional mechanism for the entrainment of the sediment in the near-wake region.

Figure 8.

Visualization of the main vortical structures around the cylinder in the instantaneous flow in the simulations with (left) α = 0°, (middle) α = 15°, and (right) α = 30° using the Q criterion. The color of the Q isosurface represents the elevation, z/D.

4. Horseshoe Vortex System

[37] The adverse pressure gradients and the downflow induced by the deceleration of the incoming flow as it reaches the cylinder induce the formation of a system of unsteady necklace vortices at all angles of attack. These vortices are visualized using the Q criterion in Figures 8 and 9 in the instantaneous and mean flows, respectively. The number of the necklace vortices and the spatial extent of the legs of these vortices are highly variable in the instantaneous flow fields. Smaller necklace vortices are shed from the region where the incoming boundary layer on the channel bed separates. These smaller secondary vortices disturb the cores of the larger, more stable, necklace vortices. This is one of the main mechanisms responsible for the observed temporal variation in the circulation of the main necklace vortices in vertical sections which are close to perpendicular to the axis of the vortex in the mean flow. The circulation is calculated on the basis of integration of the out-of-plane vorticity inside the patch of high-magnitude out-of-plane vorticity associated with the vortex. A threshold value of math formula0.1D/U is used to separate the vorticity associated with the background turbulence from the one part of the core of the vortex in the mean and/or instantaneous flow fields.

Figure 9.

Visualization of the necklace vortices and separated shear layers in the mean flow in the simulations with (left) α = 0°, (middle) α = 15°, and (right) α = 30° using the Q criterion. The view is from above.

[38] Comparison of the large-scale eddies present in the HV region in Figure 8 shows the cores of the main necklace vortices and the length of their legs on both sides of the cylinder increase with the angle of attack. For example, at most times the legs of the main necklace vortices can still be visualized in the instantaneous flow fields until about 0.2D from the most upstream edge of the cylinder in case A00. This distance increases to about 0.4D in case A15 and to 0.8–1.0D in case A30. Even though the legs of the main necklace vortices are much shorter in the mean flow (Figure 9), the presence of such coherent vortices oriented parallel to the bed for a sufficiently long time in the instantaneous flow fields is sufficient to entrain sediment. Thus, for moderate angles of attack the main necklace vortices may play a significant role in the growth of the scour hole during the initial stages of the scour process not only in front of the cylinder but also on its two sides.

[39] Figure 9 allows examining the effect of the angle of attack on the HV system in the mean flow. In case A00, the HV system contains only one necklace vortex whose relative position with respect to the cylinder and spatial extent are the same on both sides of the cylinder. This is expected because of the symmetry of the flow with respect to the main axis of the cylinder for α = 0°. The legs of the primary necklace vortex are approximately parallel to the outer face of the SSLs forming at the two upstream edges of the cylinder.

[40] At moderate angles of attack, the HV system in the mean flow contains two main necklace vortices (e.g., see cases A15 and A30 in Figure 9). Besides the primary vortex that is positioned close to the upstream tip of the cylinder and extends on its both sides, a second necklace vortex forms on the side where the flow separates. This second vortex is situated upstream of the primary necklace vortex. Its circulation is in average lower but, at times, comparable to that of the primary vortex. The cores of the two necklace vortices are approximately parallel to the outer face of the SSL in front of them. The formation of the recirculation bubble increases the region over which the incoming flow is diverted on the right side of the cylinder. The SSL bordering the recirculation region acts as a shield that decreases the streamwise momentum of the incoming flow, diverts the flow laterally, and induces strong adverse pressure gradients in front of it. These pressure gradients are larger than the ones induced on the other side of the cylinder where the flow is deflected with a lesser amount of frontal obstruction. This is why for moderate angles of attack the coherence of the HV system is, in average, stronger in front of the front face of the cylinder and on the side of the recirculation bubble. The presence of a HV system containing two main necklace vortices is observed up to α = 90°. As α approaches 90°, the circulation of the two necklace vortices on the two sides of the cylinder becomes again comparable [e.g., see Kirkil and Constantinescu, 2009, Figure 3].

[41] The cores of the two main necklace vortices are subject to oscillations in time. Analysis of the velocity histograms at locations situated inside the cores of the two main necklace vortices show the histograms contain two peaks rather than one, similar to other numerical investigations of the turbulent HV system in front of bluff bodies with flat and deformed bed [e.g., see Paik et al., 2007; Kirkil and Constantinescu, 2009, 2010; Kirkil et al., 2009; Koken and Constantinescu, 2009]. This means that the primary necklace vortex is subject to large-scale bimodal oscillations toward and away the base of the cylinder [see also Devenport and Simpson, 1990]. Such bimodal oscillations were also present in the simulation with α = 90° [Kirkil and Constantinescu, 2009]. Thus, one can conclude that the primary necklace vortex forming around a rectangular cylinder is subject to large-scale low-frequency oscillations at all angles of attack. However, the amplitude of these oscillations, which affects the sediment entrainment capacity of the necklace vortex, is strongly dependent on the magnitude of α.

[42] Another difference between the cases of low and moderate angles of attack and the case with α = 90° is the absence of large-scale hairpin vortices developing over part of the core of the main necklace vortex during part of the bimodal oscillation cycle. This instability is very strong in cases in which the approaching boundary layer does not contain resolved turbulence, as shown by the DES of Paik et al. [2007]. However, especially for cases with moderate degree of bluntness of the obstacle this instability is not observed (e.g., see Kirkil [2008] for the case of a circular cylinder and present results for α < 30°). For obstacles with a high degree of bluntness and for relatively high Reynolds numbers, large-scale hairpin vortices wrapping around the main necklace vortex are observed locally on the core of the main necklace vortex in DES with incoming turbulent flow containing resolved fluctuations (e.g., see Kirkil and Constantinescu [2009, Figure 10] for α = 90°). However, the degree of organization of these vortices is much more reduced compared to the one observed in DES with steady inflow [e.g., see Paik et al., 2007, Figure 15]. These results are fully consistent with those observed by Chang et al. [2006] who studied the effect of inflow (steady versus fully turbulent containing resolved fluctuations) on the structure of a separated shear layer over a bottom channel cavity. In the case the incoming flow was steady, large-scale spanwise vortices were shed regularly close to the edge of the cavity. As these vortices were convected downstream, the disturbances along the cores of these vortices grew into arrays of hairpin vortices wrapping around the cores of the main spanwise vortices, very similar to what was observed by Paik et al. [2007] for the main necklace vortices. However, when the incoming flow contained resolved turbulence, the interactions of the eddies convected by the overflow with the vortices forming in the upstream part of the SSL strongly increased the three dimensionality of the flow and the range of energetic eddies present over the mouth of the cavity and did not allow the formation of arrays of hairpin vortices.

[43] The nondimensional turbulent kinetic energy (TKE) distributions in a vertical plane cutting through the upstream face of the cylinder and parallel to the incoming flow confirm the fact that the second necklace vortex is not subject to bimodal oscillations (Figure 10). In cases A15 and A30, the region where the TKE values are a couple of times larger than those in the surrounding turbulent flow is limited to the region occupied by the core of the primary necklace vortex. As shown by Koken and Constantinescu [2009] and Kirkil and Constantinescu [2009], the large amplification of the TKE in the region where a necklace vortex is situated in the mean flow is a direct consequence of the bimodal oscillations of the vortex in the instantaneous flow fields.

Figure 10.

Distribution of the nondimensional turbulent kinetic energy in a vertical plane cutting perpendicular to the axis of the main necklace vortex close to the front face of the cylinder (see inset) in the simulations with (a) α = 0°, (b) α = 15°, and (c) α = 30°.

[44] Comparison of the TKE distributions in Figure 10 shows that the TKE amplification in case A00 is larger than that predicted in case A15. This may look a little bit surprising as one expects the size of the region of high TKE associated with the HV system to increase monotonically with α. However, the result can be easily explained because of the symmetry of the mean flow with respect to the axis of the rectangular cylinder in case A00. This induces a very strong downflow along the front face of the cylinder (observe also the very large amplification of the TKE within the downflow that convects eddies from the incoming flow toward the toe of the rectangular cylinder in case A00 compared to cases A15 and A30) and results in a very stable, high-circulation, necklace vortex whose large-scale oscillations are controlled by the injection of the eddies within the downflow. As already discussed, the coherence of this vortex decreases very fast in the spanwise direction compared to cases when the angle of attack is nonnegligible (Figures 8 and 9). As the angle of attack increases, the symmetry of the flow is lost and the incoming flow is diverted in a gentler way on the two sides of the cylinder, once it reaches the most upstream edge of the cylinder. This decreases the relative degree of bluntness and the strength of the downflow with respect to the case when the incoming flow encounters the front face of the cylinder (α = 0°). In the case of high–aspect ratio cylinders, an angle of attack of only couple of degrees is sufficient for this to happen. One can also argue that in practical applications (e.g., bridge piers in rivers) the rectangular cylinder will never be exactly aligned with the flow.

[45] For larger values of α, in a given vertical section cutting perpendicular to the axis of the main necklace vortex in the mean flow, the circulation of the vortex and the levels of the TKE within its core increase strongly with α, at least for moderate angles of attack. For example, comparison of the TKE distributions in Figure 10 shows that the size of the region where the TKE is larger than 0.025U2 is about 5 times larger in case A30 compared to case A15, while the peak TKE value is about 2 times larger. The size of the region of high TKE values and the peak TKE value continue to increase with α at large angles of attack. For example, the peak TKE value and the area where the TKE is larger than 0.025U2 within the HV region are more than 4 times larger for α = 90° compared to α = 30°. The strong variation of the circulation and level of amplification of the TKE of the main necklace vortex with the angle of attack is a direct consequence of the fact that the adverse pressure gradients increase with the flow blockage, which is proportional to the projected width of the cylinder. One should point out that, for high–aspect ratio rectangular cylinders at high angles of attack, the largest amplification of the TKE may occur close to the two lateral sides of the cylinder rather than close to the middle distance in between the two sides (e.g., see discussion by Kirkil and Constantinescu [2009]). This is similar to what happens in the case of vertical wall spur dikes where the largest circulation and amplification of turbulence within the main necklace vortices always occur close to the side of the obstruction [Koken and Constantinescu, 2008a, 2009].

5. Friction Velocity and Pressure RMS Fluctuations at the Bed

[46] The distributions of the magnitude of the bed friction velocity vector in the mean ( math formula) and instantaneous (uτ/U) flow fields are compared in Figure 11. As the first grid point off the solid walls is situated within the viscous sublayer, uτ is calculated using the definition ( math formula, where n is the wall normal direction and Umag is the magnitude of the projection of the velocity vector on a plane that is locally tangent to the wall surface). In the following discussion, we will arbitrarily define the regions of high bed friction velocity as those where uτ/U > 0.056. This value corresponds to the critical (Shields) friction velocity, uτc/U, for sediment with a mean diameter d50 = 1.05 mm. This is the sediment size used by Kirkil and Constantinescu [2010] in their loose-bed experiment conducted for the same rectangular cylinder oriented at an angle of attack of 90° and is representative of local scour studies conducted in large laboratory flumes.

Figure 11.

Distribution of the nondimensional bed friction velocity magnitude (top) in the mean flow and (bottom) in the instantaneous flow in the simulations with (left) α = 0°, (middle) α = 15°, and (right) α = 30°. The regions with a bed friction velocity magnitude smaller than 0.056U were blanked.

[47] Comparison of Figures 3 and 11 shows that the regions of high math formula are not limited to the upstream part of the strong SSLs within which vortex tubes are shed. The small regions of local amplification of the bed friction velocity induced by the individual vortex tubes are clearly distinguishable in the instantaneous contours of uτ/U in Figure 11. For moderate angles of attack (e.g., cases A15 and A30), the regions of high vorticity magnitude associated with the SSLs are thinner and are situated on the inner side (toward the back of the cylinder) of the regions of high uτ/U. Recall that the SSLs act as a shield that diverts the incoming flow. As a result, a region of high horizontal velocity flow and, thus, of high bed friction velocity is present on the outer side of the SSLs. This is consistent with experimental observation of scour occurring during its initial stages, when the bed is close to flat. For the reasons discussed in section 4, the size of the region of high uτ/U is larger on the right side in cases A15 and A30. Thus, during the initial stages of the scour process one expects the scour hole to grow at a faster rate on this side. The size of both regions, and thus the capacity of the flow to entrain sediment close to the cylinder, increases with α mainly because of the increase in the projected width of the cylinder (flow blockage). For example, the increase of α from 0° to 15° resulted in an increase by about 6 times of the total area of the regions where math formula > 0.056 in the mean flow. A further increase of α to 30° resulted in an increase by about 2 times of the same regions in the mean flow.

[48] At all angles of attack the strong amplification of the bed friction velocity at the sides of the cylinder is due to the combined effect of the legs of the main necklace vortices, contraction of the streamlines at the sides of the cylinder that increases velocity magnitude and shedding of energetic vortex tubes in the SSLs. Meanwhile, the capacity of the main necklace vortices to entrain sediment at the start of the scouring process in front of the upstream face of the cylinder is small not only in case A00 but also in case A15. In both cases the values of the bed friction velocity beneath the HV region in front of the cylinder are low not only in the mean flow but also in the instantaneous flow fields. It is only for larger angles of attack (e.g., case A30) that the main necklace vortices can, at times, induce values of uτ/U in front of the cylinder comparable to those observed within the flow acceleration regions at the two sides of the cylinder. This happens more often beneath the primary necklace vortex which is subject to large-scale low-frequency oscillations. During those times, sediment entrainment will occur despite the fact that the values of math formula are significantly lower. Thus, even at conditions corresponding to the start of the scour process, scour is expected to occur in front of the cylinder beneath the main necklace vortices for sufficiently high angles of attack. This shows the necessity to analyze the sediment entrainment mechanisms not only on the basis of the mean flow quantities but also on the basis of the distribution of these quantities in the instantaneous flow fields.

[49] The rollers offer another example where analysis of the instantaneous flow fields is essential to understand sediment entrainment mechanisms around the cylinder. In all three simulations, no sediment with d50 > 1.05 mm is entrained in the near wake on the basis of the distributions of math formula (Figure 11). However, examination of the instantaneous distributions of uτ/U in Figure 11 shows that rollers are entraining sediment in cases A15 and A30, at least close to the regions where they form. As the angle of attack increases, the area of the regions beneath the rollers where uτ/U > 0.056 increases (e.g., compare the instantaneous distributions of uτ/U for cases A15 and A30; moreover, the peak values of uτ/U beneath a roller as it detaches from the cylinder are comparable with the peak values observed beneath the SSLs in case A30). The same is true for the number of rollers present at a given time instant in the flow field who can entrain sediment (e.g., compare the distributions of uτ/U for cases A15 and A30 in Figure 11 with that in Figure 17a from Kirkil and Constantinescu [2009]). Experiments confirmed that scour past cylinders occurs not only in front and on the sides of the cylinders but also behind them [e.g., see Kirkil et al., 2009; Kirkil and Constantinescu, 2010]. The convection of the rollers is the main mechanism responsible for the growth of the scour hole behind the cylinder at moderate and large angles of attack.

[50] The values of uτ/U and math formula are relatively small in the region situated beneath the recirculation bubble in cases A15 and A30. This means that sediment particles can easily deposit in these regions. However, this is not likely to happen because of the weak upwelling motion of the mean flow within the recirculation bubble (see discussion of Figure 2).

[51] A more quantitative way to estimate the differences between uτ and math formula is to analyze the distributions of the standard deviation of the magnitude of the bed friction velocity magnitude, math formula (Figure 12). This quantity is estimated using the distributions of the velocity RMS fluctuations at the first grid point off the bed. Relatively large regions where math formula is comparable to uτc are predicted for cases A15 and A30 (e.g., downstream of the left edge of the cylinder relative to the flow direction where the highly energetic vortex tubes agglomerate to form a new roller). This means that in such regions sediment can be entrained, at times, even at locations where math formula is negligible. The region where math formula behind the cylinder extends up to x/D = 2 in case A15 and x/D = 4 in case A30. The width of this region increases approximately proportional to sin(α). The high values of math formula in this region are mainly due to the passage of strongly coherent wake rollers over the bed. The other large region where math formula is situated in cases A15 and A30 beneath the HV system, in front of the cylinder. Given also the relatively large values of math formula in the same region, where at most locations math formula, this shows that significant sediment entrainment occurs beneath the HV region in front of the cylinder for flat bed conditions. Further analysis, will show that the contribution of regions where math formula to the average sediment flux entrained from the bed cannot be neglected for moderate values of α and its relative contribution to the total flux increases monotonically with α.

Figure 12.

Distribution of the standard deviation of the bed friction velocity magnitude, math formula/U, in the simulations with (left) α = 0°, (middle) α = 15°, and (right) α = 30°. The regions with math formula/U < 0.02 were blanked.

[52] Besides the bed friction velocity, the pressure RMS fluctuations at the bed, math formula, shown in Figure 13, is the other variable that determines the capacity of the flow to entrain sediment. One should also point out that the distribution of math formula at the bed is qualitatively similar to that of math formula. This should not come as a surprise, as both variables characterize the turbulence intensity at the bed.

Figure 13.

Distribution of the nondimensional pressure RMS fluctuations at the bed in the simulations with (left) α = 0°, (middle) α = 15°, and (right) α = 30°. The regions with math formula < 0.015 were blanked.

[53] In the simulations with moderate angles of attack (cases A15 and A30), the largest values of math formula occur on the inner side (toward the recirculation region) of the SSL originating at the downstream face of the cylinder. This is the region where new rollers are forming. The large amplification of math formula is mainly triggered by the passage of the vortex tubes inside this SSL and by formation and shedding of new rollers at times intervals that are close to constant. During the time a roller forms, the pressure inside its core decays. Then, the pressure increases again once the roller gets away from the cylinder. This results in a strong amplification of math formula. Meanwhile, the levels of math formula are smaller by a factor of two or more in the SSL originating at the upstream face of the cylinder. As already discussed, this is due to the formation of lower circulation, larger size, rollers behind the upstream edge of the cylinder. Their shedding away from the cylinder results in smaller large-scale variations of the pressure in their formation region.

[54] At moderate angles of attack, large math formula values are also observed immediately downstream of the recirculation bubble. The levels decay fairly monotonically with the distance from the cylinder. This trend continues at high angles of attack [e.g., see Kirkil and Constantinescu, 2009]. The size of this region and the peak value of math formula within this region increase monotonically with α. As the flow blockage increases, the velocity gradient across the SSLs increases and the vortex tubes shed in the two SSLs increase their circulation. This results in the formation of rollers whose circulation remains significant at larger distances from the cylinder. Thus, at a given distance from the cylinder, the periodic reduction of the pressure at the bed induced by the passage of a roller increases with the angle of attack. Ultimately, this results in larger values of math formula inside the near wake.

[55] Figure 14 compares the probability distributions functions (pdf's) of math formula at three representative locations in the flow situated beneath the main necklace vortex, the SSL and inside the near wake region for case A15. The pdf's in the other two simulations are qualitatively similar in the three regions. Consistent with the analysis conducted by Sterling et al. [2008] for turbulent open channel flow in straight prismatic channels, a Gaussian distribution would not be the most appropriate distribution to describe the fluctuating nature of math formula. A gamma or lognormal distribution appears more appropriate, especially in regions of the flow where the coherence and/or presence of large-scale eddies varies greatly and values of math formula that are significantly larger than the mean value can be induced from time to time (e.g., beneath the SSLs in the flow considered here).

Figure 14.

Probability density function (pdf) of the normalized bed friction velocity magnitude at a point located (left) beneath the main necklace vortex in front of the cylinder, (middle) beneath the separated shear layer on the left side of the cylinder, and (right) in the near wake behind the cylinder at x/D = 3. Results are shown for α = 30°. The values of math formula on the x axis are nondimensionalized by the mean (time-averaged) value at that location, math formula.

[56] The importance of accounting for the effect of the large-scale flow unsteadiness on estimating the mean (time-averaged) flux of sediment entrained from the bed in a channel flow with a loose bed was proven by Sumer et al. [2003]. In the present work we use ideas proposed by Sumer et al. [2003] to estimate the effect of large-scale coherent structures on sediment entrainment in flow past surface-mounted obstructions on the basis of data available from eddy-resolving simulations. As a starting point, we estimated the mean flux based on the distributions of math formula and math formula separately. Semiempirical approaches for estimating sediment entrainment fluxes under clear water scour conditions typically assume that the (bed load) sediment flux is proportional to math formula (e.g., math formula = 4.2 in van Rijn's formula [van Rijn, 1984a]). Contributions to the sediment flux occur only at locations where math formula. Similarly, van Rijn's formula for predicting sediment entrainment over a loose bed of noncohesive sediment [van Rijn, 1984b] assumes that the local pick up rate P is proportional to math formula with math formula = 3, where P is expressed in units of volumetric flux per unit area and time. Directly relevant for the present approach in which In has the generic form of P given by van Rijn [1984b] and entrainment occurs when math formula in the fact that the same approach was successfully used using LES to predict formation and development of ripples in an oscillatory channel flow [Chou and Fringer, 2008]. This approach is particularly justified for cases where in the case of a movable bed scour develops under clear water scour conditions. Thus, for a given channel depth and mean flow velocity away from the obstacle and a fixed sediment diameter, the flux of sediment entrained from the bed at a certain time (t = nΔt) can be described by the following nondimensional quantity:

display math

where we assumed math formula = 3 and A′ is the bed area where math formula. Then, one can estimate the mean flux, math formula, as the time average of math formula. In a RANS simulation, the distributions of math formula are not available. So, the mean flux is estimated on the basis of math formula as

display math

where math formula is the bed area where math formula. To get an idea about the error associated with estimating the mean entrainment flux on the basis of the mean flow fields as opposed to the instantaneous velocity fields, one can estimate both math formula and math formula from DES. The values of math formula and math formula are 2.8 × 10−6 and 5.1 × 10−6 for α = 0°, 10.1 ×10−6 and 20.1 × 10−6 for α = 15°, and 21.5 × 10−6 and 71.1 × 10−6 for α = 30°, respectively. Thus, an estimation of the mean entrainment flux based on the mean flow fields would underestimate this quantity by close to two times for α = 0° and by more than 3 times for α = 30°. An obvious question is whether these differences are due to neglecting the unsteady component of the bed friction velocity in regions where entrainment occurs on the basis of the mean flow predictions of math formula or because of entrainment in regions where math formula. To answer this question, one has to calculate

display math

and then the corresponding mean (time averaged) flux over the region where entrainment occurs in the mean flow only, math formula. The values of math formula are 4.3 × 10−6 for α = 0°, 14.1 × 10−6 for α = 15° and 34.9 × 10−6 for α = 30°. These values are in all three cases of the order of 50% larger than the corresponding value of math formula. This means, that most of the underestimation of the mean flux of sediment is due to neglecting the entrainment that takes place in regions where math formula. For α = 30°, this component accounts to close to 50% of the total mean flux calculated for flat bed conditions. For the flow considered in this study, these regions correspond to the near wake and the front of the cylinder and are associated with the action of the roller and necklace vortices. This type of analysis is basically impossible to conduct on the basis of data measured in experimental investigations or obtained from RANS simulations.

[57] The distribution of math formula can be used to approximately account for the effect of the unsteady large-scale coherent structures on the sediment entrainment capacity of the flow in the near wake region and in front of the cylinder, something that is not possible to do on the basis of the distributions of the mean flow variables (e.g., pressure, bed friction velocity). This is particularly important for RANS-based approaches with a movable bed where, in most models, these variables cannot be estimated directly. For example, to be able to calculate math formula accurately, one should use a Reynolds stress closure or, at least, a nonlinear RANS closure that provides estimations of the velocity RMS fluctuations. Once information from eddy-resolving simulations is available on the pdf of math formula in the main regions where entrainment occurs, one can use math formula (C is a random weight obtained from the assumed pdf) rather than math formula to estimate entrainment at a certain time. This stochastic way of estimating entrainment should account, albeit in an approximate way, for entrainment occurring in regions where math formula as well as for unsteadiness effects on the bed friction velocity in regions where math formula. Ultimately, this should result in an improvement of scour predictions in RANS simulations with a movable bed.

6. Summary and Concluding Remarks

[58] The present paper provided a description of the main features of the flow field and turbulence structure in the vicinity of a high–aspect ratio rectangular cylinder at small and moderate angles of attack, and a discussion of the roles of the energetically important coherent structures and of their interactions in controlling sediment entrainment phenomena around the surface-mounted cylinder at conditions close to initiation of scour (flat bed channel) on the basis of results of DES simulations. The angle of attack was varied between 0° and 30°. This study is directly relevant for understanding erosion mechanisms around rectangular bridge piers as the flow conditions in the stream change from normal conditions, where piers are designed to be aligned with the flow, to high flood conditions, where the angle of attack can increase up to 30°–35°. Thus, the present investigations considers the most relevant range of angles of attack for the design of rectangular piers that is one of the most common shapes of piers used to support bridges. Comparison is also provided with results obtained by Kirkil and Constantinescu [2009] for the same cylinder at an angle of attack of 90°, which is an extreme case in terms of the extent of the local scour around the pier, but less common for river engineering applications. Results from the present study can be used to determine region where rip-rap protection has to be used on the basis of the expected maximum angle of attack which is expected to occur at high flood conditions.

[59] The instantaneous and mean flow bed friction velocity distributions showed that the main necklace vortices induce large bed friction velocity values in front of the cylinder only over limited time periods. This explains why the amplification of the bed friction velocity in front of the cylinder in the mean flow is significantly lower than that observed at the sides of the cylinder. However, even for flat bed conditions the legs of the main necklace vortices contribute significantly to the amplification of the bed friction velocity around the sides of the obstruction. The other major contributions are due to the contraction of the streamlines and the associated increase in the velocity and to the shedding of vortex tubes in the SSLs. These findings are consistent with laboratory investigations of local scour around in-stream structures that consistently report that scour is initiated at the sides of the obstructions rather than in front of it.

[60] Though the shedding of large-scale rollers in the wake was observed at all angles of attack, their capacity to entrain sediment became significant only for α > 15°. At moderate angles of attack, the circulation and size of the rollers shed from the two sides of the cylinder was significantly different. Similar to the necklace vortices, the capacity of the rollers to entrain sediment in the formation region increased monotonically with α. Moreover, simulation results showed that the bed region situated beneath the paths followed by the rollers is characterized by large values of the pressure and bed friction velocity RMS fluctuations. This further increases the capacity of the flow to entrain sediment behind the cylinder (e.g., sediment can be entrained even if the bed shear stress is below the critical value if the turbulence intensity is high).

[61] The fact that the distributions of the bed friction velocity in the instantaneous flow fields were available from DES allowed quantifying the error in estimating the mean (time-averaged) total flux of sediment entrained from the bed on the basis of the mean flow field (simplified approach), similar to what is done to estimate entrainment in simulations conducted using RANS models. Present results show that estimating the total flux on the basis of the mean flow distributions of the bed friction velocity can lead to an underestimation of this quantity by about three times for the larger angle of attack. Moreover, analysis of entrainment in the instantaneous flow fields showed that most of this underestimation is due to neglecting entrainment that occurs in regions where large-scale coherent structures are situated at times close to the bed but the friction velocity in the mean flow is below the critical value for entrainment (e.g., beneath the necklace vortices in front of the cylinder and beneath the wake rollers). This result shows the important role of the unsteady dynamics of the large-scale coherent structures for local scour around in-flow structures. This important finding is consistent with the work of Sumer et al. [2003] that proved the significant effect of mean level of turbulence on bed load sediment transport in a channel flow with a loose bed. Moreover, using information from present DES (e.g., shape of the pdf's of the bed friction velocity), the present study proposes an approximate method to account for the effect of the turbulent fluctuations on entrainment in simulations that do not resolve most of the large-scale turbulence in the flow (e.g., RANS-based approach). This is important because RANS simulations are known to have problems predicting scour behind the cylinders. In these regions scour is driven by unsteady rollers shed in the wake. The novel method will allow accounting in an approximate way for the passage of the unsteady rollers on sediment entrainment in such RANS simulations with a movable bed.

[62] We intend to extend the present study of scour mechanisms around high–aspect ratio cylinders with low and moderate angles of attack to cases with a deformed scoured bed corresponding to all relevant stages of the scour process. This will allow us to explain the changes in the scour mechanism and quantifying the effect of the unsteady dynamics of large-scale coherent structures on sediment entrainment as the scour hole evolves toward equilibrium scour conditions.

Acknowledgments

[63] Financial support from the National Science Council, Taiwan, under grant NSC 98-2625-M-492-002 is highly appreciated. We are also grateful to the National Center for High-Performance Computing for computer time and facilities. The authors would like to thank S. Miyawaki for his help in setting up the model, generating the grids, and performing the simulations.

Ancillary