### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Flume Experiment
- 3. Numerical Model and Computational Details
- 4. Flow Structure and Bed Scour Mechanisms
- 5. Summary and Conclusions
- Acknowledgments
- 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 *Re*_{D} = 2.4 × 10^{5}, 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 *Re*_{D} = 2.4 × 10^{5} with those from simulations performed at a much lower Reynolds number, *Re*_{D} = 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

- Top of page
- Abstract
- 1. Introduction
- 2. Flume Experiment
- 3. Numerical Model and Computational Details
- 4. Flow Structure and Bed Scour Mechanisms
- 5. Summary and Conclusions
- Acknowledgments
- 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 10^{4}. 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 10^{5} and 10^{6}, 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 (*Re*_{D} = 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 × 10^{5} were compared to those from LES at *Re*_{D} = 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 × 10^{5} 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 (*Re*_{D} = 2.4 × 10^{5}) 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 (*Re*_{D} = 2.4 × 10^{5}) 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 *Re*_{D} = 18,000 with an equilibrium scour bed (case SL) [*Koken and Constantinescu*, 2008b] and with a DES simulation conducted at *Re*_{D} = 18,000, with equilibrium scour bathymetry taken from the experiment conducted at *Re*_{D} = 2.4 × 10^{5} (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.

### 3. Numerical Model and Computational Details

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

[18] In the case of a rough surface, the value of is estimated by solving [*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, *k*_{s} and are set equal to zero. The model constants in the above equations are: , , , , , , , and .

[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 with a new length-scale *d*_{DES} = min(*d*, C_{DES}Δ), where the model parameter C_{DES} 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 (1)) are balanced, the length scale in the LES regions *d*_{DES} = C_{DES}Δ becomes proportional to the local grid size and yields an eddy viscosity proportional to the mean rate of strain and Δ^{2} 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 *Re*_{D} = 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 × 10^{5}), 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 u_{τ}/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 *Re*_{D} = 2.4 × 10^{5}. The turbulent fluctuations (zero mean velocity) were obtained from a preliminary LES simulation of fully developed turbulent flow in a periodic channel (*Re*_{D} = 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.

### 5. Summary and Conclusions

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