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Keywords:

  • large eddy simulation;
  • local scour;
  • turbulent flow

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Flume Experiment
  5. 3. Numerical Model and Computational Details
  6. 4. Flow Structure and Bed Scour Mechanisms
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References

[1] The flow and turbulence structure around a vertical-wall spur dike that extends over the whole depth of a straight channel are investigated using detached eddy simulation (DES). The channel Reynolds number in DES and the corresponding experiment is ReD = 2.4 × 105, which is typical of applications in small rivers and outside the range where well-resolved large eddy simulation (LES) can be conducted. The bathymetry at equilibrium scour conditions is obtained from a loose bed experiment. This paper discusses the main mechanisms which drive the growth of the scour hole upstream and downstream from the spur dike during the later stages of the scour process, and how these mechanisms change between the initial (flat bed conditions) and later stages of the scour process. Scale effects are investigated by comparing simulation results at ReD = 2.4 × 105 with those from simulations performed at a much lower Reynolds number, ReD = 18,000. Results show that while the structure of the horseshoe vortex (HV) system changes with respect to the case in which the bed is flat, the main necklace vortex of the HV system is still subject to large-scale aperiodic oscillations, similar to the ones observed in flows past in-stream bluff-body obstacles mounted on a flat surface. Present results show that the amplification of the horizontal vorticity within the lower part of the separated shear layer (SSL) and the associated formation of streaks of high-bed shear stress below the region where the SSL eddies are convected in the near-bed region is a general feature of high-Reynolds-number flow past a vertical-wall spur dike placed in a loose-bed channel at all stages of the scour process.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Flume Experiment
  5. 3. Numerical Model and Computational Details
  6. 4. Flow Structure and Bed Scour Mechanisms
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References

[2] The presence of man-made or natural obstructions in natural alluvial channels (e.g., river training structures, large-scale bed roughness, submerged or emerged islands, bank protrusions) induces important modifications in the flow which result in the formation of large-scale energetic coherent structures [e.g., see Lane et al., 2004; Hardy et al., 2007; Koken and Constantinescu, 2008a, 2008b; Kirkil and Constantinescu, 2010]. When these eddies are situated or convected close to the bed surface, the shear stress and the pressure fluctuations at the bed get amplified and sediment particles are entrained from the bed sediment layer. As a result, a scour hole starts developing around the flow obstruction.

[3] Experiments [e.g., Dargahi, 1989; Melville and Coleman, 2000; Fael et al., 2006; Unger and Hager, 2007; Dey and Barbhuiya, 2005, 2006a, 2006b] have shown that, past the initial stages of the scour process, the growth of the scour hole in front of surface-mounted bluff bodies (e.g., cylinders of different shapes, obstructions mounted at one of the channel's banks) is mainly driven by the horseshoe vortex (HV) system. The large-scale eddies shed in energetic shear layers penetrating up to the bed surface is another important mechanism for scour [Rhoads and Sukhodolov, 2001, 2004; Kirkil et al., 2008]. Finally, in cases when large-scale vortex shedding is present, the passage of the wake vortices affects bathymetry evolution downstream from the in-stream obstruction.

[4] The study of flow past in-stream obstacles mounted on a flat bed has been the object of numerous experimental and numerical investigations (e.g., for review see Koken and Constantinescu [2008a]). The main focus of most of these studies was to characterize the structure of the HV system forming at the upstream base of the bed-mounted obstacle. In a pioneering experimental study of the flow past a surface-mounted wing-body, Devenport and Simpson [1990] have showed that the large turbulence amplification in the HV region is due to the fact that the core of the main necklace vortex of the HV system is subject to large-scale aperiodic bimodal oscillations. The presence of bimodal oscillations was later confirmed in numerous numerical studies conducted at relatively low Reynolds numbers using direct numerical and large eddy simulations [e.g., Martinuzzi and Tropea, 1993; Hussein and Martinuzzi, 1996; Rodi, 1997; Shah and Ferziger, 1997; Koken and Constantinescu, 2008a; Kirkil et al., 2008]. In these investigations of the flow past bluff bodies (e.g., cylinders of different shapes, cubes) mounted on a flat bed, the Reynolds number defined with the width of the obstacle and the average incoming flow velocity was around 104. More recently, hybrid Reynolds-averaged Navier Stokes (RANS)-LES methods like detached eddy simulation (DES) were successfully used to study the dynamics of the coherent structures past bluff bodies mounted on a flat surface at Reynolds numbers between 105 and 106, which are closer or within the range encountered in most practical applications in river and coastal engineering [e.g., see Kirkil and Constantinescu, 2009; Koken and Constantinescu, 2009].

[5] In most applications of relevance to water resources, the in-stream obstacle is mounted on an erodible bed. The experimental study of Dargahi [1989] found that the structure of the HV system forming at the base of a circular cylinder is relatively unchanged past the initial stages of the scour process. Moreover, significant changes are observed in the coherent structures present in the flow and their dynamics between the conditions at the start of the scour process (flat bed) and the time when the maximum scour depth reaches about one-third of the maximum scour depth at equilibrium scour conditions. The changes are much less important as the scour hole continues to evolve toward equilibrium scour conditions. The formation of a sufficiently deep scour hole stabilizes the HV system and modifies the interaction between these necklace vortices with the bed and the eddies convected in the SSL. This is why the sediment entrainment mechanisms change not only in front of the obstacle, where the erosion is driven by the HV system, but also on its sides and behind it. Most of the scour takes place after the scour hole is large enough to limit the amplitude of the large-scale oscillations of the necklace vortices and to affect the dynamics of the other eddies forming around the obstacle. Thus, to explain sediment erosion processes around in-stream bluff-body obstructions one needs to understand the dynamics of the coherent structures for conditions in which the scour hole around the obstacle is relatively large.

[6] To understand the physics of flow past in-stream obstructions placed in loose-bed channels, the use of experimental and/or numerical techniques that can capture the dynamics of the large-scale eddies in the flow, in particular of the ones situated in the vicinity of the bed, is required. Compared to 2-D Particle Image Velocimetry (PIV) [e.g., see Unger and Hager, 2007] and other similar experimental techniques, LES-type simulations have the advantage that they provide the distributions of the flow variables over the whole three-dimensional flow field rather than just within two-dimensional sections. This allows visualizing the main coherent structures, characterizing their shapes and their positions relative to the bed. This type of information is critical for understanding the role played by the large-scale turbulence in the entrainment and transport of sediment particles from the loose bed.

[7] On the other hand, sufficiently well-resolved eddy-resolving simulations with a movable bed capability needed to simulate the development of the scour hole until equilibrium conditions are reached are computationally too expensive. Most simulations of this type are conducted using Reynolds-averaged Navier Stokes (RANS) codes [for review, see Roulund et al., 2005; Nagata et al., 2005]. In the case of RANS simulations, the uncertainties related to the modeling of the sediment entrainment and deposition are too high to have full confidence in the numerical predictions of the bed evolution. Although the eddy-resolving simulations conducted with fixed deformed bed do not capture the evolution of the bed in the vicinity of the spur dike, they provide a detailed description of the flow and turbulent structure at a certain stage of the scour process. As in most applications of interest, the time-scales needed to calculate converged statistics are much smaller than the ones associated with significant changes in the bathymetry; eddy-resolving simulations with a fixed deformed bed should be able to predict with reasonable accuracy the mean flow, the turbulence statistics, and the dynamics of the energetically important eddies in the flow at a certain stage of the scour process. This approach is particularly appropriate for cases when experiments are conducted with clear water scour conditions and relatively small particles. In such cases, one can neglect the effect of the entrained sediment on the coherent structures.

[8] Eddy-resolving simulations allow for obtaining the spatial and temporal distributions of the bed shear stress and other quantities (e.g., pressure root-mean-square (RMS) fluctuations in the vicinity of the bed) that control the entrainment of sediment. Such information is almost impossible to obtain from experiments, especially inside the scour hole where strong adverse pressure gradient, flow separation, and large nonuniform bed slopes are present. Moreover, the instantaneous distributions of the bed shear stress can be used to establish a direct link between the presence of an energetic eddy in a certain region near the bed and its capacity to entrain sediment. A large contribution to sediment entrainment in turbulent flows is because of the short-leaved peaks associated with the passage of coherent structures near the bed [Sterling et al., 2008]. For example, the data sets generated from such simulations should allow for explanations as to how scour occurs in regions where the mean bed shear stress is below the critical value for sediment entrainment. Obviously, in these regions the entrainment is driven energetic eddies convected, at times, at small distances from the bed surface.

[9] Reynolds-number-induced scale effects for flows past in-stream obstructions with a deformed bed were investigated recently by Kirkil et al. [2009] and Kirkil and Constantinescu [2010] for the case of circular and rectangular cylinders, respectively, with a scour hole corresponding to equilibrium conditions. The results of DES conducted at a channel Reynolds number (ReD = UD/ν, U is the mean streamwise velocity in the channel, D is the channel diameter, and ν is the molecular viscosity) of 2.4 × 105 were compared to those from LES at ReD = 18,000 for the case of a circular cylinder [Kirkil et al., 2008]. Directly relevant for the present study, Koken and Constantinescu [2009] investigated the effect of the Reynolds number for the flow past a vertical-wall spur dike of identical geometry to the one considered in this paper. The investigation of Koken and Constantinescu [2009] considered only the case of a flat bed corresponding to conditions present at the start of the scouring process. They found that an increase of the channel Reynolds number from 18,000 to 5 × 105 induced significant qualitative and quantitative differences in the structure of the flow inside the HV system region, the distribution of the nondimensional bed friction velocity, and the dynamics of the vortex tubes convected inside the SSL.

[10] The present study reports a similar DES investigation of the flow and turbulence structure around a vertical-wall spur dike, but considers the case of a large scour hole (equilibrium scour conditions). The obstacle is mounted at one of the vertical sidewalls of a straight channel. The incoming flow is fully developed. The channel Reynolds number (ReD = 2.4 × 105) is high enough to be representative of field conditions in small streams. The width of the obstacle, W, is 1.5 D.

[11] The experiment conducted to obtain the bathymetry, the numerical method, and the simulation set up are discussed in sections 2 and 3. The flow and the turbulence structure predicted by a high Reynolds number DES simulation (ReD = 2.4 × 105) of the flow past a vertical-wall spur dike with a large scour hole are then analyzed (case SH). Section 4 provides a detailed discussion of the dynamics of the main coherent structures on the basis of results of a DES simulation using the Spalart Allmaras (SA) model as the base RANS model. The influence of the turbulence model is investigated based on a comparison with the results of the DES using the k-ω shear stress transport (SST) model and of LES using the Smagorinsky model and a simplified modeling of the near-wall flow. The main changes in the flow structure and scour mechanisms between conditions present at the start and at the end of the scour process are discussed based on the comparison of high-Reynolds-number DES simulations conducted with a flat bed (case FH) [Koken and Constantinescu, 2009] and a scoured bed (case SH). Reynolds-number-induced scale effects are discussed based on the comparison of case SH with an LES simulation conducted at ReD = 18,000 with an equilibrium scour bed (case SL) [Koken and Constantinescu, 2008b] and with a DES simulation conducted at ReD = 18,000, with equilibrium scour bathymetry taken from the experiment conducted at ReD = 2.4 × 105 (case SL-HB). The LES simulation was performed using a dynamic Smagorinsky model and a fully nondissipative viscous solver using unstructured meshes [Mahesh et al., 2004]. An important observation is that the equilibrium bathymetry is not the same in cases SL and SH. This is because the changes in the flow induced by the increase in the Reynolds number resulted in different bed shear stress distributions, and thus in a modified bathymetry at equilibrium scour conditions.

2. Flume Experiment

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Flume Experiment
  5. 3. Numerical Model and Computational Details
  6. 4. Flow Structure and Bed Scour Mechanisms
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References

[12] Experiments were conducted in a flume 3 m wide and 2 m deep. The test section of the flume was 20 m long. The width (W = 0.8 m) and thickness (0.1 m) of the rectangular obstacle and the flow depth (D = 0.53 m) were the same as the ones in the flat-bed experiment simulated in case FH. The mean incoming channel velocity was U = 0.45 m s−1. The incoming flow was fully developed at the location of the obstacle. The main nondimensional flow parameters were ReD = 2.4 × 105 and equation image, where equation image is the gravitational acceleration.

[13] A uniform layer of sediment with a median diameter of 1.05 mm was placed on the bottom of the flume. The thickness of the sand layer was 1.0 m which was sufficient for the bottom of the flume to remain covered by sediment at all locations during the experiment. The critical friction velocity for sediment entrainment on a horizontal surface was uτc0/U = 0.056, where uτc0 was estimated on the basis of a Shields' diagram. The experiment was conducted under clear water scour conditions. The experiment was run until the bed reached equilibrium. This process took ∼5 days.

[14] The equilibrium bathymetry is shown in Figure 1. Because of the relatively large value of U/uτc0 and the small ratio between the width of the flume and the width of the obstacle (close to 4), the scour hole extended laterally until it was close to the sidewall opposite to the one to which the spur dike is mounted. Although some contraction scour was present, local scour remained the main mechanism responsible for the bed deformation. The largest flow depths occurred close to the flank of the spur dike. The maximum flow depth was 2.34 D, which is ∼20% larger than the one (1.9 D) recorded in the experiment conducted at Re = 18,000. This larger value of the maximum scour depth relative to the flow depth away from the obstacle is closer to the values generally observed around hydraulic structures of similar shapes in the field (e.g., bridge abutments). The bathymetry was measured in the horizontal directions on a grid with spacing of 0.1 m that was sufficient to capture the relevant large-scale features of the bed surface.

image

Figure 1. Computational domain and bathymetry contours for case SH (equilibrium scour conditions).

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3. Numerical Model and Computational Details

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Flume Experiment
  5. 3. Numerical Model and Computational Details
  6. 4. Flow Structure and Bed Scour Mechanisms
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References

[15] Koken and Constantinescu [2009] provide an in-depth discussion of the main types of numerical techniques that can be used to investigate the dynamics of the large-scale eddies in junction flows. In principle, a LES without a wall function is the best option to simulate these flows. However, at high Reynolds numbers this approach is computationally too expensive. LES with wall functions was shown to be much less successful in accurately predicting complex turbulent flows because of the very simplified way near-wall effects are accounted for. Hybrid RANS-LES approaches offer a better alternative and are more accurate because the near-wall flow is resolved with a more sophisticated model (e.g., a low-Reynolds-number version of a one- or two-equations RANS model is used in DES).

[16] The use of mesh densities that are high enough to resolve the flow in the critical regions, the specification of inflow boundary conditions that mimic as close as possible those present in the laboratory or in the field (e.g., containing resolved turbulence), and the use of viscous flow solvers that employ weakly dissipative discretization schemes, especially in the regions where the LES mode of DES is active, are critical to accurately predict flow past surface-mounted bluff bodies at high Reynolds numbers.

[17] The Spalart Allmaras (SA) one-equation model is used as the base RANS model in the main DES simulations discussed in the present paper. The SA RANS model solves a transport equation for the modified eddy viscosity equation image

  • equation image

where S is the magnitude of the vorticity, ν is the molecular viscosity, u j is the contravariant resolved velocity, t is the time, d is the turbulence length scale, and ξ j is the curvilinear coordinate in the j direction. The other variables and parameters are equation image and equation image. The eddy (SGS) viscosity νt is obtained from

  • equation image

where equation image, equation image, equation image, equation image, and equation image.

[18] In the case of a rough surface, the value of equation image is estimated by solving equation image [Spalart, 2000; Zeng et al., 2008, 2010], where n is the wall-normal direction. This yields nonzero values of the modified viscosity and SGS viscosity at the rough surface. For smooth walls, ks and equation image are set equal to zero. The model constants in the above equations are: equation image, equation image, equation image, equation image, equation image, equation image, equation image, and equation image.

[19] The SA version of DES is obtained by replacing the turbulence length scale d (distance to the nearest wall) in the destruction (dissipation-like) term of the transport equation for equation image with a new length-scale dDES = min(d, CDESΔ), where the model parameter CDES is equal to 0.65 and Δ is a measure of the local grid size. When the production and destruction terms in the transport equation for equation image (equation (1)) 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 [Keylock et al., 2005]. The governing equations are integrated through the viscous sublayer and no wall functions are used.

[20] The main features of the numerical method, which is second-order accurate in both space and time, are discussed by Koken and Constantinescu [2009]. The governing equations are formulated in generalized curvilinear coordinates on a nonstaggered grid. The time integration is done using a double-time-stepping algorithm. The discrete momentum and turbulence model equations are integrated in pseudotime using the alternate direction implicit approximate factorization scheme. More details on the numerical method and the DES model are given by Constantinescu and Squires [2004] and Chang et al. [2007]. The code and the SA version of the DES model used to perform the simulation of case SH are the same as the ones used to perform the SA DES simulation of case FH. The sensitivity of the predictions to the base RANS model is analyzed based on comparison with the results obtained using the SST version of DES, as described by Chang et al. [2007]. Finally, the effect of the turbulence model and near-wall treatment are investigated by comparing the DES predictions to those of the LES with a classical Smagorinsky model (wall functions are used) on a mesh that has the same level of grid refinement away from the solid surfaces. The LES simulation is carried out using the same viscous solver used to perform the DES simulations.

[21] Constantinescu and Squires [2003, 2004] and Chang et al. [2007] discuss the results of grid sensitivity and validation studies for DES simulations of the flow past spheres and channel flow over a bottom cavity conducted using the same code. 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 obtained from a preliminary straight-channel flow calculation. This is the approach adopted in the present study. It is consistent with the flow conditions present in the experiment in which the incoming flow was fully developed.

[22] Koken and Constantinescu [2008a, 2009] used both LES and DES to calculate the flow at ReD = 18,000 past the same vertical-wall spur dike considered in the present study. In these low Reynolds-number simulations used to assess the predictive abilities of DES, the channel bed was flat. The mean flow and turbulence statistics predicted by DES were very close to those from the corresponding well-resolved LES.

[23] For higher Reynolds numbers, on the basis of comparison with acoustic-Doppler-velocimetry (ADV) and laser-Doppler-velocimetry (LDV) 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 spur dike placed in a channel with a flat bed [Koken, 2011] and the turbulence statistics within the HV system forming around the upstream base of a circular cylinder mounted on a flat-bed channel [Kirkil, 2008]. Also relevant for the present test cases, in which the bed is highly deformed as a result of local scour and deposition, the same DES code was shown to accurately predict the mean flow and turbulence structure in a sharp bend of high curvature with equilibrium bathymetry [Constantinescu et al., 2011a] and around a river confluence (the Reynolds number in the main channel was ∼1.6 × 105), for which detailed velocity and turbulence measurements were available from a field study [Constantinescu et al., 2011b]. Moreover, for the test cases with a highly deformed bed, DES predicted more accurately the flow and the turbulence statistics compared to RANS. In all of these simulations, the level of mesh refinement in the critical flow regions was similar to the one used in the present investigation.

[24] As in the flat-bed simulation (case FH), the domain width is 5.6 D. The computational domain extends 8 D upstream of the axis of the obstacle and 30 D downstream from it. The mesh contains 7.4 million cells (480 × 192 × 80 in the streamwise, spanwise, and vertical direction, respectively). The first grid point off the solid surfaces is situated at ∼0.25 wall units (10−5 D), assuming the average bed friction velocity is uτ/U = 0.04. The nondimensional width and thickness of the obstruction are identical to those in case SL [Koken and Constantinescu, 2008b]. The computational domain was meshed using close to 4 million cells in the LES simulation of case SL. The LES and DES simulations of cases SL and SH, respectively, were performed on meshes that were sufficiently fine in the wall-normal direction to avoid the use of wall functions. In the DES simulation of case SH, the mesh spacing in the critical regions containing dynamically important eddies (e.g., HV region, SSL) was similar to the one used in the DES simulation of case FH [Koken and Constantinescu, 2009]. The time step was 0.025 D/U in the simulation of cases SH and SL-HB.

[25] The boundary conditions in the DES simulations of case SH are identical to the ones used in case FH. The channel sidewalls, the surfaces of the obstacle, and the deformed channel bottom were treated as no-slip boundaries. The free surface was treated as a shear-free rigid lid. This is justified because the free surface deformations in the experiment were not significant and the channel Froude number was smaller than 0.5. For these flow conditions, the inherent assumption of the rigid-lid approximation that the pressure field is close to hydrostatic in the vicinity of the free surface is acceptable. A convective boundary condition was used at the outflow. The position of the outflow section in the computational domain roughly matched that of the end of the test section of the flume. The boundary conditions in the LES simulation of case SH were the same as in the DES, but the first grid point off the walls was situated at ∼30 wall units. Except for the inflow velocity, the boundary conditions in cases SH and SL-HB were identical.

[26] The mean streamwise velocity profile at the inlet was obtained from a preliminary RANS simulation of fully developed turbulent flow in a periodic channel at ReD = 2.4 × 105. The turbulent fluctuations (zero mean velocity) were obtained from a preliminary LES simulation of fully developed turbulent flow in a periodic channel (ReD = 18,000), whose section was identical to that used to investigate the flow past the vertical-wall obstacle. The nondimensional fluctuations from the lower Reynolds-number simulation were added to the nondimensional RANS mean streamwise velocity profile and the total velocity fields provided the inlet condition in a time-accurate manner. Koken and Constantinescu [2009] provide a more detailed discussion of inflow boundary conditions.

4. Flow Structure and Bed Scour Mechanisms

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Flume Experiment
  5. 3. Numerical Model and Computational Details
  6. 4. Flow Structure and Bed Scour Mechanisms
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References

4.1. Horseshoe Vortex System

4.1.1. Mean Flow

[27] The Q criterion [Hunt et al., 1988] is used in Figure 2a to visualize the mean flow coherent structures within the recirculation region situated upstream of the spur dike and inside the scour hole in case SH. The quantity Q is the second invariant of the (resolved in LES) velocity gradient tensor, Q = 0.5 (∂ui/∂uj) × (∂uj/∂xi), and represents the balance between the rotation rate and the strain rate. A large corner vortex, CV1, is present within the recirculation region. The role of CV1 is to convect fluid and momentum from the free surface into the core of the main necklace vortex, HV1 (Figure 2b).

image

Figure 2. Visualization of the vortical structure of the mean flow in the vicinity of the spur dike for case SH (SA-DES) using: (a) Q criterion; (b) 3-D streamlines. The necklace vortices part of the HV system are denoted HV1–HV4. CV1 is the main vortex in the recirculation region situated upstream of the spur dike.

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[28] The structure of the HV system in the mean flow is significantly affected by scale effects. While only one main necklace vortex, HV1, is observed in case SL (Figure 3), a second large necklace vortex, denoted HV2 in Figure 2a, is present in case SH. The formation of HV2 is not a direct consequence of the difference in the Reynolds number between cases SL and SH, in the sense of a change in the flow structure observed for identical bathymetries as a result of the increase in the Reynolds number. Rather, the formation of HV2 is mainly a result of the difference in the shape of the scour hole and larger relative scour depths (z/W) in case SH. This is confirmed by the fact that HV2 is also present in the SL-HB simulation. The streamlines launched inside the core of HV2 (Figure 2b) follow a helicoidal trajectory. Similar to HV1, HV2 draws into its core some fluid from the upper part of the channel. In this sense, HV2 does not play the usual role of a secondary necklace vortex. HV2 is situated inside the scour hole and above HV1. The axes of HV1 and HV2 are not parallel. Near the sidewall, HV2 is situated farther away from the face of the obstacle compared to HV1. As the obstacle is approached, HV2 gets closer to the flank (extremity) of the spur dike.

image

Figure 3. Visualization of the vortical structure of the mean flow in the vicinity of the spur dike in case SL using the Q criterion. The main necklace vortex is denoted HV1. CV1 is the main vortex in the recirculation region situated upstream of the spur dike. WAV and CV2 are small junction vortices.

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[29] Close to the channel sidewall, where HV1 occupies the deeper part of the scour hole, the circulation within the core of HV1 in vertical sections oriented perpendicular to the axis of the vortex is ∼50% larger than the one within the core of HV2. The region of high vorticity within the core of HV1 is situated away from the bed, while that within the core of HV2 is situated closer to the bed. So, close to the sidewall, HV2 has a larger potential to entrain sediment despite having a lower circulation. As HV2 gets away from the scoured bed surface and moves over HV1, its coherence decays significantly. Meanwhile, the patch of high vorticity associated with HV1 moves very close to the bed surface, while the core of HV2 is situated within the top part of the scour hole region. Moreover, the shape of the patch of high vorticity associated with HV1 is very elongated and follows the bed surface. The vorticity distribution in a vertical section cutting through the flank of the obstacle in Figure 4a is representative of the vorticity distribution within the scour hole at sections situated away from the sidewall at which the obstacle is mounted. In this section, the circulation of HV1 is more than two times larger than that of HV2. Away from the channel sidewall, HV1 is responsible for most of the sediment entrained from the upstream part of the scour hole.

image

Figure 4. Out-of-plane vorticity contours in a vertical section cutting through the HV system (case SH, mean flow: (a) SA DES, (b) SST DES, and (c) LES). The position of the section relative to the spur dike is shown in the inset.

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[30] Besides HV1 and HV2, two smaller necklace vortices, HV3 and HV4, are present in the mean flow inside the scoured region (Figures 2a and 4a). They are induced by the sudden change in the bed slope. Similar to case SL, in the instantaneous flow fields, smaller necklace vortices detach from the upstream region of the scour hole and interact with the main necklace vortex. Ejections of patches of vorticity of opposite sign to that present inside HV1 are observed regularly close to the bed. HV4 is a counter-rotating bottom-attached vortex induced by HV1. The large-scale variations in the coherence of HV4 are directly related to the occurrence of these vorticity ejection events. These events result in a temporary decrease of the circulation of HV1.

4.1.2. Turbulence Statistics and Bimodal Oscillations

[31] Similar to case SL, the turbulent kinetic energy (TKE) and pressure RMS fluctuations, equation image (Figure 5) increase significantly above the levels corresponding to the surrounding turbulent flow in the region where the core of HV1 is subject to large-scale aperiodic oscillations. Still, the nondimensional values of these two variables within the region of strong turbulence amplification are at least 50% lower in case SH compared to case SL (Figure 6) [see also Koken and Constantinescu, 2008b]. Meanwhile, the overall levels of the TKE in cases SH (Figure 5) and SL-HB (Figure 7) are comparable. This means that most of the differences between cases SH and SL are due to the different shape of the scour hole and its larger relative depth in case SH, which reduce the mean amplitude of the large-scale oscillations of HV1.

image

Figure 5. (a–c) Three representative vertical sections illustrating (left) two-dimensional mean-flow streamline patterns; (middle) nondimensional TKE, k/U2; and (right) pressure RMS fluctuations, equation image for case SH (SA DES). The position of the sections is shown in the inset in Figure 5a. The vertical dashed line in Figure 5c marks the position of the spur dike.

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image

Figure 6. TKE, k/U2, in representative vertical sections for case SL. The position of the sections is shown in the inset in Figure 6b.

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image

Figure 7. TKE, k/U2, in representative vertical sections for case SL-HB. The position of the sections is shown in the inset in Figure 7b.

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[32] The relative reduction of the turbulence intensity in case SH, with respect to case SL, increases at sections situated farther away from the sidewall. For example, the reduction of the nondimensional values of TKE and equation image is ∼50% at section a, which is parallel to the sidewall and intersects the obstacle at ∼0.5 D from its flank, and >100% the center of this region at sections b and c which are situated around the flank of the obstacle and downstream of it (see Figures 5 and 6). In case SH (Figure 5), the region of high TKE amplification is relatively circular and the TKE peaks at the center of this region at sections b and c in the inset in Figure 5a). By contrast, the region of high TKE values at section a has a more elliptical shape and contain two peaks. This is similar to the TKE distribution observed in case SL not only at sections situated close to the sidewall, but also at sections situated close to the flank of the obstacle (Figure 6). In cases SH (Figure 5) and SL-HB (Figure 7), the intensity of the large-scale oscillations of HV1 peaks at sections situated between the sidewall and the flank of the obstacle. By contrast, in case SL and in the high-Reynolds-number flat-bed simulation (case FH), the TKE amplification is the highest at sections cutting through the flank of the obstacle. The different geometry of the scour hole is the main reason for this qualitative change in the distribution of the TKE inside the HV region between cases SL and SH.

[33] The position of the region of very large equation image values is not always similar in cases SH (Figure 5) and SL [see Koken and Constantinescu, 2008b]. Similar to the TKE, the distributions of equation image are qualitatively very similar in vertical sections situated between the sidewall and 0.3 D from the flank of the spur dike. At sections cutting through the flank of the dike and through the leg of HV1 (e.g., sections b and c in the inset in Figure 5a), equation image is strongly amplified inside a small region situated close to the bed in case SH. This region is part of the larger area of relatively high amplification of the TKE and equation image induced by the large-scale oscillations of HV1. Also, this region is distinct from the small patch of high equation image values corresponding to the corner junction vortex. The high levels of equation image inside these two near-bed regions enhances the entrainment of sediment from the deeper parts of the scour hole. By contrast, in case SL, the patch of high equation image values inside the scour bed was situated at a significantly larger distance from the bed and occupied about the same region as the patch of high TKE values in the same section.

[34] Analysis of the simulation results of case SH shows that some of the characteristics of the large-scale aperiodic oscillations of HV1 change as one moves away from the sidewall. The structure of the flow inside the scour hole at sections situated in between the sidewall and 0.3 D–0.7 D from the flank of the obstacle (Figure 8) is similar to that observed in case SL at sections situated close to the sidewall and cutting through the flank of the obstacle. The flow structure remains similar when the vortex is in the zero-flow mode (weak jet-like flow: the core of HV1 is more compact and fairly circular) and in the backflow mode (strong jet-like flow: the core of HV1 is larger and has a more elliptical shape). Similar to other junction flows with a flat or deformed bed (e.g., cylinders of different shapes), the transition to the zero-flow mode is generally triggered when a patch of low- and high-vorticity fluid is injected into the downflow. The transition to the backflow mode starts when a patch of high- and low-vorticity fluid is injected into the downflow. In case SH, the axis of HV1 moves by ∼0.5 D, as the core of HV1 oscillates between these two preferred states (e.g., see Figures 8a and 8b). This explains the two-peak distribution of the TKE at the same section (Figure 8c). A relatively strong eddy is present upstream of HV1 in the zero-flow mode. This eddy corresponds to HV2 in the mean flow.

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Figure 8. Two-dimensional (a) streamline patterns and (b) velocity vectors in the instantaneous flow showing HV1 (left) in the zero-flow mode and (right) in the backflow mode for case SH (SA-DES). The vertical section is parallel to the channel sidewall (see inset in Figure 8a). Figure 8c shows the distribution of the TKE, k/U2.

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[35] At sections cutting through the flank of the obstacle in case SH (Figure 9), the main necklace vortices occupy a much larger part of the scour hole at most times. In the backflow mode, the vorticity inside the core of HV1 is quite diffused and the vortex looses most of its coherence. Even in the zero flow mode the vorticity distribution inside HV1 is less compact compared to sections situated closer to the sidewall. This is why the resolved TKE in Figure 9c does not show a two-peak distribution, which is normally associated with regions where bimodal oscillations are present, and the region of high TKE values corresponds roughly to the core of HV1 in the zero-flow mode. Thus, the main difference with case SL is that the intensity of the bimodal oscillations does not peak anymore at sections situated close to the flank of the spur dike, but rather at the sections situated closer to the sidewall at which the spur dike is mounted.

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Figure 9. Two-dimensional (a) streamline patterns and (b) velocity vectors in the instantaneous flow showing HV1 (left) in the zero-flow mode and (right) in the back-flow mode for case SH (SA DES). The vertical section cuts through the flank of the spur dike (see inset in Figure 9a). Figure 9c shows the distribution of the TKE, k/U2.

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[36] To assess the influence of the turbulence model on the structure of the HV system, the vorticity distributions (Figure 4) and the turbulence statistics (Figures 5a and 10) are compared among the SA DES, SST DES, and LES simulations. All of the simulations show that a main necklace vortex HV1 and a secondary vortex near the edge of the scour hole (HV3) are present upstream of the obstacle. Analysis of the results shows that both SA DES and SST DES predict HV1 and is subject to strong bimodal oscillations, which explains the double-peak structure observed in the distributions of the TKE and equation image. Moreover, the level of amplification of the turbulence inside the scour hole is very similar (e.g., compare Figure 5a with Figure 10a). By contrast, LES (Figure 10b) predicts a main necklace vortex of a smaller size. HV1 occupies only the deeper part of the scour hole. Its circulation is smaller by 30%–50% than that predicted by the two DES simulations. Although the turbulence is amplified significantly inside the HV region, the distributions of the TKE and equation image do not display a double-peak structure and the peak levels of these two variables are ∼50% lower compared to DES. This means that the strength of the bimodal oscillations is greatly reduced compared to DES. The experiments clearly show that the core of the main necklace vortex is subject to large-scale oscillations at high Reynolds numbers. Thus, LES simulations with a simplified near-wall treatment are not able to predict the correct amplification of the turbulence in front of the bluff body, where the flow is subject to large, adverse pressure gradients and the main necklace vortex is subject to strong aperiodic oscillations. It is also relevant to mention that well-resolved LES and SA DES were able to capture strong bimodal oscillations and predicted a similar level of turbulence amplification at low Reynolds numbers.

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Figure 10. (left) Two-dimensional mean-flow streamline patterns; (middle) nondimensional TKE, k/U2; and (right) pressure RMS fluctuations, equation image, in a vertical section (section a in the inset of Figure 5a) for case SH: (a) SST DES; (b) LES. The contour scales for the two variables are the same as the ones used in Figure 5.

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4.2. Separated Shear Layer and Wake Regions

4.2.1. Mean and Instantaneous Flow Fields

[37] The coherent structures that populate the SSL originating at the flank of the spur dike and the near-wake region are visualized in Figures 11a and 11b. Similar to case FH (Figure 11d), the vortex tubes in the instantaneous flow fields are strongly distorted starting inside the upstream part of the SSL. The deformations are significantly larger than those present in case SL. Thus, Reynolds-number-induced scale effects on the dynamics of the vortex tubes convected inside the SSL are significant during all stages of the scour process. Compared to case FH (Figure 11c), the size of the main recirculation region downstream from the spur dike (Figure 11a) is smaller by ∼40% close to the free surface.

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Figure 11. Visualization of (a) the mean-flow pattern in a horizontal plane situated close to the free surface and (b) the vortical structure of the instantaneous flow downstream of the spur dike using the Q criterion for case SH. Figures 11c and 11d show the same variables for case FH.

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[38] The distributions of the vorticity magnitude in the mean and instantaneous flow fields are compared in Figure 12 at the free surface and over two curved surfaces situated at a constant distance of 0.25 and 0.05 D from the bed, respectively. The distribution of the instantaneous vorticity shows a wide range of energetic eddies are present in between the SSL and the sidewall, and inside the scour hole.

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Figure 12. Distribution of (left) the vorticity magnitude in the mean flow and (right) instantaneous flow in case SH (SA DES): (a) z/D = 1 (free surface); (b) Δh/D = 0.25; (c) Δh/D = 0.05. Δh is the distance from the deformed bed surface.

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[39] Analysis of the instantaneous vertical vorticity fields in the horizontal planes situated at small distances from the free surface (e.g., see Figure 12a) show that merging between successively shed vortex tubes takes place fairly regularly. The merging generally results in the formation of a more energetic vortical structure. The merging does not necessarily take place over the whole length of the vortex tube. On average, every third vortex tube shed from the flank of the obstacle merges with the one in front of it. A similar phenomenon was observed in case SL. Analysis of the animations of the vorticity fields show that the main mechanism responsible for perturbing the SSL near the free surface is the convection of small, energetic near-wake eddies parallel to the downstream face of the obstacle, toward its flank. A less important mechanism is the interaction between the 3-D eddies recirculated inside the near-wake region and the SSL eddies.

[40] As one moves toward deeper levels (below 0.9 D from the free surface), the energy of the 3-D eddies around the obstacle increases considerably because of the strong turbulence amplification inside the scour hole. Close to the bed (e.g., see Figure 12b), the eddies convected from the scoured region situated upstream of the obstacle play the most important role in perturbing the vortex tubes. These perturbations are driven by the eddies convected on the outer side of the SSL.

[41] Similar to case SL, the shape of the SSL changes with the distance from the bed. The mean and instantaneous vorticity fields show that, away from the scour hole, the SSL is oriented at a small angle (20°–30°) with the streamwise direction, away from the sidewall at which the obstacle is attached (Figure 12a). The shape of the SSL is similar to the one expected for a vertical-wall spur dike placed in a channel of infinite depth. As the bed is approached, the angle at which the SSL eddies are shed changes. Most of the eddies are shed toward the sidewall at which the obstacle is attached. For instance, in Figure 12b (Δh = 0.25 D) the angle with the streamwise direction varies between 0° and −45°. The average angle is close to −30°. Very close to the bed (e.g., at Δh = 0.05 D in Figure 9c), most of the SSL eddies are shed at an angle which varies between −30° and −45°.

[42] One interesting difference between cases SH (Figure 12c) and SL is the fact that in case SL, two preferential directions corresponding to −40° and 15°, respectively, can be identified for the shedding of the vortex tubes close to the bed [Koken and Constantinescu, 2008b]. Such a phenomenon is also present in case SL-HB where the shedding angles associated with the two preferred directions are close to −30° and 0°, respectively (Figure 13b). This means that the flapping motion of the SSL close to the bed is a pure low-Reynolds-number effect and is only mildly affected by the shape of the scour hole.

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Figure 13. Distribution of the vorticity magnitude in the mean flow for case SL-HB (SA DES): (a) z/D = 1 (free surface); (b) Δh/D = 0.05. Δh is the distance from the deformed bed surface. The two arrows in Figure 13b indicate the preferential directions along which vortex tubes are shed close to the bed.

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[43] Close to the bed, the SSL eddies are convected toward the large submerged deposition hill. Some of these eddies can entrain sediment and transport it away from the flank of the obstacle and the deeper parts of the scour hole. This is one of the main mechanisms that explain the growth of the scour hole downstream from the spur dike in the later stages of the scour process and the formation of the submerged deposition hill. In particular, these eddies are responsible for the relatively large slope of the surface hill at the side situated on the path of the SSL eddies (see Figure 1).

[44] More details on the interaction between these eddies and the upslope face of the deposition hill can be inferred from the mean and instantaneous vorticity magnitude plots in Figure 14. The z/D = 0.3 plane cuts through the top part of the hill. The plane also cuts through the core of a streamwise-oriented junction vortex (vortex A in Figure 15) situated in between the face of the deposition hill and the sidewall at which the spur dike is attached. Some of the energetic eddies that are convected in the near-bed region from the SSL toward the hill become fairly elongated, and their axes tend to orient parallel to the deformed bed surface. This increases their efficiency in entraining sediment particles from the deeper regions and carrying the particles toward the crest of the deposition hill, until equilibrium conditions are reached.

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Figure 14. Distribution of (left) the vorticity magnitude in the mean flow and (right) instantaneous flow in case SH in a horizontal plane (z/D = 0.3) cutting through the core of vortex A.

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Figure 15. Visualization of the bed-sidewall junction vortex A and of the flow pattern inside the recirculation region downstream of the spur dike using 3-D mean-flow streamlines (case SH, SA DES). The arrows show the direction of the flow along the streamlines.

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4.2.2. Mean Flow Structure Inside the Recirculation Region

[45] The 3-D mean-flow streamlines in Figure 15 enable visualization of the structure of the flow within the main recirculation region and how the main recirculation bubble exchanges flow with the surrounding fluid. In particular, Figure 15 shows the presence of an elongated vortex (vortex A) near the junction line between the bed and the sidewall. Dye visualization experiments showed that this vortex is present at all stages of the scour process, (e.g., see also Koken and Constantinescu [2009] for case FH) and is, to a large degree, responsible for the formation of the elongated deposition hill. Vortex A entrains sediment from the deeper levels of the scour hole, close to the junction line, and carries the sediment against the slope of the deposition hill. At equilibrium conditions, the slope of the hill is large enough such that the shear stress induced by the vortex is not sufficient to entrain sediment particles uphill, against gravity. The length of vortex A is close to 11 D in the mean flow. In most of the instantaneous flow fields (e.g., see Figure 14b), the core of vortex A is quite distorted and the coherence of the vortex varies substantially along its axis.

[46] The flow inside the core of vortex A moves in the upstream direction for x/D < 3.5 and in the opposite direction for x/D > 3.5. This phenomenon was also observed in case SL, where the threshold value for the change in the flow direction was ∼3.1 D. Dye visualizations performed in the experiment conducted for case SH confirmed the change of direction of the flow within the core of vortex A. The threshold value was close to 3.2 D. The change of direction was also observed in case FH, but the change happened further downstream at x/D ≅ 13.5 D. The difference is explained by the change in the bed topography during the evolution of the scour process. Vortex A is fed by fluid convected in the bed vicinity, inside the main recirculation eddy. The flow moving in the upstream direction inside the core of vortex A is first re-entrained into the recirculation eddy forming downstream from the obstacle. Then the fluid particles follow a complex trajectory before being ejected back into the high-velocity region on the outward side of the SSL. The flow moving in the downstream direction inside the core of vortex A is convected directly into the mainstream of fluid passing the spur dike.

4.2.3. Turbulence Statistics

[47] Figure 16 shows the distributions of the TKE and equation image at the free surface, at midchannel depth (z/D = 0.5), and on a deformed surface situated at 0.3 D from the bed. Close to the free surface, the highest TKE levels are observed within the SSL. Over most of the region situated in between the SSL and the sidewall, at which the obstacle is attached, the TKE levels are several times larger than the TKE in the stream of flow convected past the spur dike. The pressure RMS fluctuations are also strongly amplified over the upstream part of the SSL, but then they decay fast for x > 3 D (x is measured from the upstream face of the obstacle). Also, equation image remains low in the wake region, outside the SSL. The TKE and equation image are strongly amplified in the upstream recirculating region in which the main corner vortex is present. A comparison of Figures 16a and 16b shows that the distributions of equation image remain largely similar between the free-surface and the mid-depth level. This is also true for the TKE in the upstream recirculation region. However, at z/D = 0.5, the TKE is strongly amplified on the inner side of the SSL because of the larger variation in the trajectories followed by the SSL eddies. Meanwhile, the equation image values remain relatively low. A likely explanation is that the circulation and rotational velocity of the vortex tubes entrained toward the sidewall are relatively small compared to the ones of the vortex tubes whose trajectories follows the axis of the SSL in the mean flow. This means that the decay of the pressure inside their cores is fairly small. Meanwhile, the convective velocity of all of the vortex tubes is similar. This explains the different effect on TKE and equation image. The distributions of TKE and equation image change significantly as the bed surface is approached. The deformed surface in Figure 16c (Δh = 0.3 D) cuts through the region where the core of HV1 is subject to bimodal oscillations. This explains the large amplification of the turbulence in the region situated upstream of the obstacle.

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Figure 16. Distribution of (left) the TKE, k/U2, and (right) pressure RMS fluctuations, equation image, in case SH (SA-DES): (a) z/D = 1 (free surface); (b) z/D = 0.5; (c) Δh/D = 0.3. Δh is the distance from the deformed bed surface.

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[48] A comparison of the distributions of the mean flow vorticity at the free surface (Figures 12a, 17a, and 17b) and in a deformed surface situated 0.3 D from the bed (Figures 16c, 18a, and 18b) allows assessing the effect of the turbulence model on the structure of the SSL and of the flow around the spur dike. The position, shape, and level of vorticity amplification within the SSL are similar in the three simulations. At the free surface, the two DES simulations predict a more elongated main recirculation eddy upstream of the spur dike compared to the LES. However, the circulation of this eddy is relatively close in the three simulations. The effect of the turbulence model is more important in the deeper part of the flow. Consistent with the underprediction by LES of the strength of the bimodal oscillations in section a of Figure 5, the turbulence amplification in the region cutting through the core of the main necklace vortices situated upstream of the spur dike is uniformly lower compared to the DES. On average, the values of the TKE and equation image predicted by LES are ∼50% lower than the ones predicted by the two DES simulations. The SA DES and SST DES distributions of the TKE and equation image remain in very good qualitative and quantitative agreement in the wake of the spur dike. Over this region, LES predicts similar distributions of the turbulence statistics compared to DES. Still, LES predicts slightly larger values and a more uniform distribution of the turbulence variables inside the main recirculation region situated downstream from the spur dike compared to DES.

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Figure 17. Distribution of the vorticity magnitude in the mean flow in case SH at z/D = 1 (free surface): (a) SST DES; (b) LES; (c) the bathymetry contours. The contour scale for the vorticity is the same as the one used in Figure 12. Figure 17c shows the bathymetry contours.

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Figure 18. Distribution of (left) the TKE, k/U2, and (right) pressure RMS fluctuations, equation image, in case SH at Δh/D = 0.3: (a) SST DES; (b) LES. The contour scales for the two variables are the same as the ones used in Figure 16.

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[49] Comparison of the nondimensional distributions of the TKE and equation image in Figures 16 and 19 helps in understanding the changes in the turbulence structure around the spur dike between the start of the scour process (flat bed) and its end (equilibrium scour conditions). The values of TKE inside the SSL are comparable at z/D = 1 and z/D = 0.5 in cases FH and SH. However, because of the scour hole, the recirculation flow patterns in the wake region are different.

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Figure 19. Distribution of the (left) TKE, k/U2, and (right) pressure RMS fluctuations, equation image, in case FH (flat bed): (a) z/D = 1 (free surface); (b) z/D = 0.5. SA DES results are shown.

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[50] At the free surface, the decay of the TKE inside the downstream part of the SSL and the wake region is faster in case SH. The differences in the TKE distributions between case FH and SH increase at z/D = 0.5. In case FH, the TKE amplification inside the downstream part of the SSL is significantly larger at z/D = 0.5 compared to z/D = 1. TKE values that are a couple of times larger than the background turbulence levels are observed over a significantly larger distance from the spur dike at z/D = 0.5 (≅ 8 D) compared to the free surface (≅ 5 D). The changes in the TKE distributions with the distance from the free surface are qualitatively different in case SH. At z/D = 0.5, the TKE is amplified mostly inside of the upstream part of the SSL and around it, especially on its inner side. As previously discussed, this amplification is mostly because of the SSL eddies that are shed at varying angles toward the deposition hill. As these eddies interact with the face of the hill, their coherence decays. This explains why in case SH the TKE decays much faster in the streamwise direction inside the SSL and the wake region, especially away from the free surface.

[51] In both cases, the distributions of equation image do not vary significantly between z/D = 1 and z/D = 0.5. The location and size of the regions of high equation image are quite different in the two cases. In case FH, equation image peaks inside the upstream part of the SSL and in the region situated around x/D = 13, where the flow reattaches to the sidewall at which the spur dike is attached. With the exception of the region situated immediately behind the obstacle, equation image is significantly larger than the background levels over most of the wake region until x/D ≅ 20. In case SH, equation image is strongly amplified over the upstream part of the SSL. In the wake, no significant amplification of equation image is observed for x/D > 7. Similar to the TKE, equation image is several times larger inside the upstream recirculation region in case SH compared to case FH.

4.3. Friction Velocity and Pressure Root-Mean-Square Fluctuations at the Bed

[52] The distributions of equation image and of the mean-flow bed friction velocity, uτ/U, in case SH are shown in Figures 20a and 21b, respectively. The regions of high equation image are also regions of high values of the RMS of uτ. Thus, the distribution of equation image at the bed can be used to identify regions where the presence of large-scale coherent structures results in a relatively large local increase of the bed friction velocity over finite periods of time. This, in turn, can significantly affect sediment entrainment [Sterling et al., 2008]. The largest values of equation image are observed along the upstream face of the spur dike and are induced by the strong temporal variations in the intensity of the downflow parallel to the face of the obstacle. Similar levels of amplification are observed beneath the upstream part of the SSL and on the face of the deposition hill over which SSL eddies are convected in the near-bed region. Finally, equation image is significantly amplified over the other face of the hill because of vortex A.

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Figure 20. Distribution of the pressure RMS fluctuations at the bed, equation image: (a) case SH; (b) case SL; (c) case SL-HB. The two arrows in Figures 20b and 20c indicate the preferential directions along which vortex tubes are shed in the SSL close to the bed.

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Figure 21. Distribution of bed-friction velocity in case SH (SA DES): (a) uτ/U, instantaneous flow; (b) uτ/U, mean flow; (c) uτ/uτc0, mean flow (the regions where uτ/uτc0 < 1 were blanked); (d) uτ/uτc, mean flow (the regions where uτ/uτc < 1 were blanked). Figure 21e shows the distribution of uτ/U in the mean flow for case SL.

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[53] The positions of the regions of high-nondimensional values of equation image in case SH (Figure 20a) are similar to the ones predicted in case SL (Figure 20b). Still, some differences are observed. For example, in case SH, the downflow is very strong. This explains the presence of a region of very high values of equation image in front of the spur dike. The levels of equation image beneath HV1 are ∼50% lower than those in the region where the downflow reaches the bed. By contrast, the levels of equation image beneath HV1 and the downflow are comparable in case SL. The distributions of equation image upstream of the spur dike are qualitatively and quantitatively similar in cases SH and SL-HB. This means that equation image is determined mainly by the shape and relative depth of the scour hole in front of the spur dike that, in turn, control the strength of the downflow and of the bimodal oscillations. Downstream from the dike, the distribution of equation image in case SL-HB is qualitatively similar to the one observed in case SL. The effect of the shedding of the vortex tubes over a wide range of angles close to the bed (e.g., see discussion of Figure 13b in section 4.2.1) is clearly observed in the distributions of equation image in Figures 20b (case SL) and 20c (case SL-HB). As previously discussed, the flapping motion of the SSL close to the bed is a purely low-Reynolds-number effect.

[54] Similar to case SL (Figure 21e), the regions of high values of uτ/U in the mean flow are situated beneath the main necklace vortex, HV1, beneath the upstream part of the SSL, and in the region where the incoming flow is strongly accelerated on the outward side of the SSL. Compared to case SL, the values of uτ/U in these regions are lower in case SH (Figure 21b), on average, by ∼30%. This is mainly because of the difference in the Reynolds number, which produces a similar decrease of the nondimensional bed friction velocity in the incoming fully developed flow.

[55] The threshold value of the bed-friction velocity for sediment entrainment over a flat surface for the conditions considered in case SH (d50 = 1.05 mm) is uτc0 = 0.056 U. The regions where uτ/uτc0 > 1 are shown in Figure 21c. However, as discussed by Koken and Constantinescu [2008b], the fact that uτ/uτc0 > 1 does not necessarily indicate that sediment will be entrained in the case where the bed is deformed. In Figure 21d, uτc0 was adjusted to account for gravitational slope effects using the formulas given by Brooks and Shukry [1963], and the distribution of the ratio between the friction velocity, uτ, and the local critical bed-friction velocity, uτc, was plotted. The regions where uτ/uτc > 1 correspond to regions where sediment particles will be entrained. The simulation results predict that, where >98% of the region where the bed is deformed, no entrainment will occur for the flow conditions and bathymetry considered in case SH.

[56] Several small regions where uτ/uτc are present and slightly larger than one (by ∼10%–20%), are primarily within the scour hole, upstream of the spur dike. In addition to errors in the numerical model predictions, one suspects the bathymetry did not reach equilibrium in those regions or the resolution used to measure the bathymetry in that region was insufficient. The numerical model predicts in a very good approximation that the sediment entrainment around the spur dike is negligible. This serves as an indirect validation of the model, as the bathymetry in case SH corresponds to the equilibrium scour conditions. The model of Brooks and Shukry [1963] determines the occurrence of entrainment for particles of a certain size only on the basis of the bed shear stress value and the mean-flow pattern close to the bed. Thus, some sediment entrainment may occur in regions where high-pressure fluctuations are present, even if uτ/uτc < 1.

[57] Consistent with the distribution of uτ in the mean flow (Figure 21b), the regions in which the largest values of the instantaneous bed-friction velocity occur (e.g., see Figure 21a) are situated beneath the region where the core of HV1 oscillates, beneath the upstream part of the SSL, and in the strong flow acceleration region around the flank of the spur dike. An additional region containing pockets of high-bed friction velocity is present at all times in the instantaneous flow fields. This region corresponds to the area over which SSL eddies are convected at a small distance from the bed. It covers a large part of the upslope face of the deposition hill. Although, when averaged over time, the values of uτ/U in this region are not very high, the occurrence of large values in the instantaneous flow fields (up to two times the ones in the mean flow) can induce sediment entrainment locally if uτ/uτc > 1 for a sufficiently large amount of time. These near-bed eddies form primarily because of the interaction among the deformed vortex tubes, the eddies inside the downstream recirculation region, and the eddies convected upstream of the spur dike, close to its flank. The convection of these eddies plays an important role in the evolution of the scour process downstream from the obstacle in the later stages of the scour process.

[58] The simulation of case FH has already shown that the eddy content of the SSL developing over a flat bed at Re > 105 is much more complex than the one observed at Re ≅ 104. As a result of the large-scale eddies present in the SSL away from the flank of the spur dike, streaks of high bed-friction velocity were induced beneath the SSL region [Koken and Constantinescu, 2009]. Streaks of high uτ/U also form in case SH (Figure 21a). Figure 22 visualizes the temporal evolution of these streaks that are generally observed starting at a distance of ∼1.5 D from the flank of the spur dike. Most of these streaks tend to align with the lines of constant bathymetry elevation as they are convected over the upslope face of the deposition hill. The time of passage of the streaks over a certain location of the bed surface can be long enough to generate or significantly amplify local sediment entrainment with respect to the one predicted, based on the distribution of uτ in the mean flow.

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Figure 22. Distribution of bed shear stress, equation image, in the instantaneous flow for case SH (SA DES), when (a) t = t1; (b) t = t1 + 0.6 D/U (SA DES). The dashed lines correspond to the lines of constant bathymetry elevation. The solid black lines mark the streaks of high-bed shear stress.

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[59] Analysis of the time evolution of the bed shear stress distributions upstream of the spur dike shows that the region situated beneath the core of HV1 is characterized by large temporal variations. These variations are induced by the large-scale oscillations of the core of HV1. When the coherence HV1 is strong (generally, this happens when HV1 is in the zero-flow mode), the instantaneous bed friction velocity beneath HV1 is as high as the values observed over the upstream part of the SSL (e.g., see Figure 22b).

5. Summary and Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Flume Experiment
  5. 3. Numerical Model and Computational Details
  6. 4. Flow Structure and Bed Scour Mechanisms
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References

[60] Eddy-resolving simulations were used to investigate: (1) the flow past a vertical-wall spur dike placed in a channel with a deformed bed corresponding to equilibrium scour conditions; (2) the role of the large-scale turbulence in explaining erosion and deposition phenomena during the later stages of the scour process; and (3) the role of the turbulence model. The channel Reynolds number in the DES simulation (case SH) was 240,000. The scale effects were discussed on the basis of a comparison with results of a simulation (case SL) conducted at a much lower Reynolds number (Re = 18,000). In the case the bathymetry corresponds to equilibrium scour conditions, scale effects are felt directly, as the increase in the Reynolds number modifies the rate of growth and the development of the instabilities in the flow, and indirectly, as the shape of the bed surface at equilibrium conditions changes with the Reynolds number. The results showed that the changes in the equilibrium bathymetry with the Reynolds number can induce significant differences in the flow patterns and affect the dynamics of the coherent structures around the spur dike over the range of Reynolds numbers considered in the present study. For example, the presence of a second main necklace vortex (HV2) inside the scour hole in case SH is a unique feature of the flow field that was not observed in case SL, in which the shape of the scour hole was different. By contrast, the flapping of the SSL close to the bed is primarily a Reynolds number effect.

[61] Despite the fact that the development of the scour hole had a stabilizing effect on the HV system, the present study showed that the large-scale oscillations of the core of the main necklace vortex is the main reason for the turbulence amplification inside the HV region, even after a large scour hole forms around the base of the spur dike. The presence of these oscillations is independent of the Reynolds number, provided that the incoming flow is fully turbulent. The turbulence amplification associated with the aperiodic oscillations in case SH was the largest in the vertical sections parallel to the streamwise direction situated in between the sidewall and a location close to the flank of the in-stream obstacle. This is different from case SL and the high-Reynolds-number flat-bed case (case FH), as well as from the case of wide rectangular cylinders [Kirkil and Constantinescu, 2009], where the intensity of the aperiodic oscillations was the largest in the vertical sections cutting through the flank of the obstacle.

[62] Similar to case FH, the bed-shear-stress distributions in the instantaneous flow fields displayed a streaky structure beneath the downstream part of the SSL. These streaks are induced by the convection of fairly horizontal energetic eddies within the SSL at a small distance from the bed, first against the slope of the scour hole and then over the upslope face of the deposition hill. Present results strongly suggests this is one of the main mechanisms responsible for the growth of the scour hole downstream from the spur dike, and the development of the deposition hill in the later stages of the scour process for high-Reynolds-number flow conditions.

[63] Simulation results showed that the role of the base RANS model used in DES is relatively minor. The mean flow and turbulence structure predicted by SA DES and SST DES were very close both inside the upstream part of the scour hole as well as downstream from the spur dike. By contrast, LES with a simplified wall model strongly underpredicted the strength of the bimodal oscillations in the upstream part of the scour hole. LES and DES gave much closer predictions of the flow inside the SSL and wake regions. Experiments clearly show the flow at high Reynolds numbers is subject to strong, large-scale oscillations of the main necklace vortex. Moreover, well-resolved LES at low Reynolds numbers predicted such strong bimodal oscillations. Thus, the use for LES-type techniques with complex wall modeling capabilities appears to be a requirement to correctly capture the turbulence structure of flow past surface-mounted in-stream obstructions with a high degree of bluntness at large Reynolds numbers.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Flume Experiment
  5. 3. Numerical Model and Computational Details
  6. 4. Flow Structure and Bed Scour Mechanisms
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References

[64] The authors would like to thank Taiwan's National Center for High Performance Computing (NCHC) and the Transportation Research and Analysis Computer Center (TRACC) at the Argonne National Laboratory for providing the computational resources needed to perform most of the simulations.

References

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  2. Abstract
  3. 1. Introduction
  4. 2. Flume Experiment
  5. 3. Numerical Model and Computational Details
  6. 4. Flow Structure and Bed Scour Mechanisms
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References
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