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TECHNICAL PAPERS
Oct 15, 2009

Detached Eddy Simulation Investigation of Turbulence at a Circular Pier with Scour Hole

Publication: Journal of Hydraulic Engineering
Volume 135, Issue 11

Abstract

This paper uses results from detached eddy simulation to reveal the dynamics of large-scale coherent eddies in the flow around a circular pier with an equilibrium scour hole. This is important for the sediment transport because the local scour process is controlled to a large extent by the large-scale coherent structures present in the near-bed region. The present paper investigates the dynamics of these coherent structures, their interactions and their role in entraining sediment in the later stages of the scour process when the horseshoe vortex system is stabilized by the presence of a large scour hole. The pier Reynolds number was 2.06×105 , outside the range of well-resolved large-eddy simulation (LES). Additionally, scale effects are investigated based on comparison with LES results obtained at a much lower Reynolds number of 16,000 in a previous investigation. The paper provides a detailed study of the dynamics of the main necklace vortices of the horseshoe vortex system, including an investigation of the bimodal oscillations, their effect on the amplification of the turbulence within the scour hole and the interactions of the necklace vortices with the downflow. Several mechanisms for the growth of the downstream part of the scour hole in the later stages of the scour process are discussed. Similar to the low-Reynolds-number simulation, and consistent with experimental observations, the presence of strong upwelling motions near the symmetry plane resulted in the suppression of the large-scale vortex shedding in the wake. The fact that the nondimensional values of the turbulent kinetic energy and pressure RMS fluctuations in the higher Reynolds number simulation were consistently lower inside the regions of high turbulence amplification associated with the main necklace vortex, the separated shear layers and the near-wake shows that changes in the flow and turbulence due to the Reynolds number and scour hole geometry can be quantitatively significant over Reynolds numbers between 104 and 105 .

Introduction

Local scour at a pier is attributable to the forces exerted on the bed by the complex, highly three-dimensional (3D) and unsteady flow field generated by the pier. The flow field is marked by turbulence structures with a wide range of scales, and by a pronounced downflow at the pier’s leading edge. The present paper presents insights into the flow field at a pier with a developed scour hole formed in a sand bed at a relatively high pier Reynolds number, and discusses the linkages between flow field and sediment entrainment and transport from the scour hole. These linkages are relevant for piers with relatively large scour holes corresponding to the later stages of the scour process. The complexity of the flow field and the presence of a large scour hole hamper the use of existing flow instrumentation to obtain detailed insights into the flow and turbulence structure.
To date, most of the focus on the scour flow field has been on the horseshoe vortex (HV) system forming around the upstream base of the pier. It is thought to be the main mechanism responsible for the growth of the scour hole past the initial stages of the scour process (e.g., Dargahi 1990; Melville and Coleman 2000). However, the HV is part of a system of turbulence structures that together with the downflow and the flow acceleration around the flanks of the pier are the erosive flow mechanisms of primary importance. The turbulence structures [e.g., necklace vortices of HV system, eddies shed in the separated shear layers (SSLs), large-scale wake rollers] are not isolated from each other. They intrinsically connect within the flow field. Thus, it is not enough to focus on one turbulence structure, notably the well-known HV system to understand the physics of scour processes around the pier.
To understand pier scour requires quantitative and qualitative knowledge of the flow field formed at a pier during the scouring process and of the dynamics of the large-scale structures that are responsible for entrainment and transport of most sediment particles. This need is reflected by the numerous experimental studies of flow fields around bridge piers conducted especially over the last couple of decades; many of these studies are summarized in Melville and Coleman (2000) and Sumer and Fredsoe (2002). Dargahi (1990) documented the changes in the scour mechanism from flat to equilibrium scour conditions at a circular pier ( RD=UD/ν=4×104 , U =mean incoming flow velocity; D =diameter or equivalent width of the pier; and ν =molecular kinematic viscosity) placed in a loose bed channel. In particular, he observed no significant change in the structure of the HV system past the initial stages of the formation of the scour hole. This is a significant result for the relevance of the present investigation that focuses on the dynamics of coherent structures for the case of a scour hole at equilibrium scour conditions. 3D velocity distributions inside an equilibrium scour hole at a circular pier were measured by Dey et al. (1995) using a five-hole pitot tube. Measurements of the mean-flow fields and turbulence around a circular pier with equilibrium scour bed at RD105 were reported by Graf and Istiarto (2002) in several polar planes (0°<ϕ<180°) . The bed shear stress distribution was inferred from the velocity measurements (extrapolation toward the bed of the estimated Reynolds stresses at stations situated close to the bed surface). The study by Roulund et al. (2005) provides velocity measurements in the symmetry plane upstream and downstream of the pier using a laser Doppler anemometer and estimations of the bed shear stress around the pier using a hot film probe for a circular bridge pier on a flat bed at RD105 . Dey and Raikar (2007) investigated the turbulent flow fields around circular piers (RD4×104) within a developing scour hole having depths of 0.25, 0.5, 0.75, and 1.0 times the depth at equilibrium scour conditions based on acoustic Doppler velocimeter measurements. The characteristics of the HV system with the development of the scour hole were discussed based on the observed similarity with the velocity and turbulence characteristic scales. Unger and Hager (2007) performed a particle image velocimetry (PIV) study of the flow around a circular pier over a wide range of Reynolds numbers (5×104<RD<3.5×106) . The interest was in the visualization of the flow patterns in the vertical symmetry plane and in several horizontal planes as the scour process evolves toward equilibrium conditions. The downflow and the HV system were identified as the main reasons for the growth of the scour hole. They also confirmed the fact that the main scour activity is concentrated in the region situated in front of the pier where the coherence of the HV system is highest. Their experimental observations confirmed the variation in the position of the boundary layer separation point on the pier from ϕ90° away from the bed to ϕ180° close to the bed, consistent with the large eddy simulation (LES) results of Kirkil et al. (2008). Several of these investigations found that the structure of the HV system varies little past the initial stages of the scour process.

Justification of the Approach and Relevant Numerical Studies

The highly unsteady 3D flow around a pier around is hard to visualize, let alone measure, in their entirety. Presently, it is not possible, by means of laboratory experiment, to visualize the entire instantaneous flow field around a bridge pier or to obtain accurate information on the flow and turbulence statistics very close to the bed (e.g., distributions of the mean and instantaneous bed shear stress and pressure RMS fluctuations). Simulations using eddy-resolving techniques and sufficiently fine meshes to integrate the equations through the viscous sublayer allow us to obtain this information. Numerical models using eddy-resolving techniques like LES and hybrid RANS-LES techniques (e.g., Rodi 1997; Spalart 2000; Tokyay and Constantinescu 2006; McCoy et al. 2008) have been shown to be much more successful to predict massively separated flows and flows dominated by unsteady large-scale coherent structures compared to the classical Reynolds averaged Navier-Stokes (RANS) approach. Also, the knowledge of the mean-flow and turbulence statistics only is not enough to allow investigating the role played by the energetically important coherent structures in the flow. Even if the evolution of the scoured bed was not part of the solution, as the time scales of the large-scale coherent structures are much smaller than the time scales associated with bathymetry evolution, information from these simulations can help to better understand the scouring mechanism at different stages of the erosion and deposition process around the pier. Together with PIV based studies (e.g., Unger and Hager 2007), high-resolution eddy-resolving numerical simulations are the best approach to elucidate fundamental aspects of momentum and sediment transport mechanisms around bridge piers.
Ideally, sufficiently resolved LES simulations without wall functions and employing Smagorinsky subgrid-scale (SGS) models based on the flow physics like the dynamic Smagorinsky model (e.g., Bou-Zeid et al. 2005) should be used to investigate such complex flows. Kirkil et al. (2008) reported such a study for a pier with scour hole at a relatively low Reynolds number (RD=16,000) . A similar LES investigation was conducted by Koken and Constantinescu (2008a,b) for abutments and by McCoy et al. (2007) for emerged and submerged groynes exposed to the flow. One important question is to what extent the findings from these LES studies and, for that matter, the findings from detailed flume studies conducted at relatively low Reynolds numbers (RD104) are relevant for practical applications. At very large Reynolds numbers (RD>105) , the mesh and time-step requirements make the use of LES without wall functions not feasible. To investigate Reynolds-number-induced scale effects using eddy-resolving techniques, LES with wall functions or hybrid RANS-LES methods are required. Hybrid zonal (e.g., Wang and Moin 2002) and nonzonal RANS-LES methods can resolve the dynamically most important eddies in the flow and can be used to study flow past bridge piers at RD closer to those encountered in rivers.
Nonzonal RANS-LES methods like detached eddy simulation (DES) use the same base turbulence model in the RANS and LES regions and no special treatment is required to match the solutions at the boundary between the LES and RANS regions. The model used in the LES region is a RANS model in which the turbulence length scale is modified to allow the energy cascade to the small scales like in LES. For internal flows (e.g., flow past surface-mounted bodies or bottom cavities in a channel) the predictions of DES, including the dynamics of the large-scale coherent structures, are dependent on the presence, or not, of resolved turbulence in the incoming flow (e.g., Chang et al. 2007). The problem is that it is not easy to generate a flow with the proper turbulence content in a channel at a high Reynolds number. Basically all experimental investigations of local scour at bridge piers are conducted with an incoming turbulent channel flow. Thus, it is important that the inflow conditions in the numerical simulations are as close as possible to those present in experiments or in the field.
Kirkil et al. (2008) reviewed recent RANS investigations of the flow past circular piers with fixed or evolving bed. Their study also summarizes prior studies using LES and hybrid RANS-LES methods for related flows (e.g., flow past abutments, piers of complex shape, and other obstacles mounted on flat beds). LES of the flow past a circular pier with flat and deformed bed was performed by Tseng et al. (2000) and Choi and Yang (2002) using relatively coarse meshes (largest simulations used 250,000 cells), wall functions and a constant coefficient SGS model. The incoming flow did not contain any resolved turbulence. Kirkil et al. (2008) were the first to use sufficiently well-resolved LES (mesh contained close to 4 million cells and RD=16,000 ) to study, besides the mean-flow and turbulence statistics, the dynamics of the large-scale coherent structures in the flow past a circular pier with equilibrium scour hole in a relatively shallow channel. The simulations were conducted using a nondissipative energy-conserving code (Mahesh et al. 2004) and the dynamic Smagorinsky model was used to estimate the SGS viscosity. The incoming flow was fully turbulent and contained realistic fluctuations obtained from a preliminary channel LES simulation. The simulation captured the presence of bimodal aperiodic oscillations inside the HV system region, as well as the associated increase in the resolved turbulent kinetic energy (k) and pressure RMS fluctuations (p2¯) . The study showed, for the first time, that bimodal oscillations of the necklace vortices are present in the later stages of the scouring process and, thus, are a general characteristic of bridge pier flows. Dye visualizations and large-scale PIV were used to confirm some of the flow phenomena identified based on analysis of the LES flow fields. Several mechanisms that can explain the growth of the scour hole laterally and behind the pier were identified. DES predictions of flow past wing-shaped cylinders mounted on a flat surface at RD=1.15×105 were reported by Paik et al. (2007) using steady inflow conditions (no resolved inflow turbulence). DES predictions of flow past a vertical spur dike (bridge abutment) in a flat-bed channel at RD=5×105 were reported Koken and Constantinescu (2008c, 2009) using inflow conditions that contained resolved fluctuations. A similar investigation was conducted by Kirkil and Constantinescu (2009) for a high aspect ratio rectangular cylinder at high angle of attack in a flat-bed channel at RD=2.5×105 .
Development of practical design approaches for predicting scour depths at bridges has been hampered by inadequate knowledge about, and formulation of, important component processes at play during scour. A better understanding of the turbulence characteristics, and in particular of the role of the large-scale coherent structures, should lead to scour prediction methods that incorporate more physics and predict more accurately the scour depth for a broader range of flow characteristics (Ettema et al. 2006). Though not always the case [e.g., scour prediction methods proposed by Melville and Coleman (2000) and by Oliveto and Hager (2002a,b)], some scour prediction methods are calibrated using mostly data from relatively low-Reynolds-number flume studies. Thus, information on scale effects associated with an increase in the Reynolds number obtained from either simulations or experiments should be useful toward developing expressions for correction parameters in these methods. At a more fundamental level, there is a need to understand if the turbulence structure, and thus the sediment entrainment and transport mechanisms, change significantly between the Reynolds numbers at which many flume studies are conducted and Reynolds numbers (RD>105) above the range that is known to be subject to significant scale effects in other massively separated turbulent flows [e.g., see Williamson (1996), for flow past infinitely long cylinders].
The DES reported here investigates flow past a circular pier with scour hole corresponding to conditions close to equilibrium scour for RD=2.06×105 . The scour hole dimensions were determined from a flume experiment. Following a brief description of the numerical method, DES model and simulation setup, the dynamics of the large-scale coherent structures, the mean-flow and turbulence statistics are analyzed. The main goal of the present paper is to discuss the dynamics of these structures for the case of a cylinder with a developed scour hole, representative of the later stages of the scour process. The dynamics of these structures is expected to change significantly between the initial stages when the HV system is much more unstable (e.g., Kirkil and Constantinescu 2007) and the advanced stages when the scour hole stabilizes the HV system and modifies its interactions with the SSLs. On the other hand, the dynamics is expected to be similar once the scour hole is large enough to stabilize the HV system. Scale effects are discussed based on comparison of DES with results from an LES simulation at RD=16,000 (Kirkil et al. 2008). In both simulations the incoming flow contained resolved fluctuations and the ratio between the channel depth away from the pier (H) and the diameter of the pier (D) was the same (H/D=1.12) . One should point out that, as opposed to the flat-bed case, differences in the two solutions are not due solely to the difference in the Reynolds number but also to differences in the geometry of the scour hole, which is deeper in the high Reynolds number simulation. In the ensuing sections, the DES solution at RD=2.06×105 is termed the HR solution while the LES solution at RD=16,000 (Kirkil et al. 2008) is termed the LR solution.

Numerical Method

The present study used the Spalart-Allmaras (SA) RANS model as the base model in DES (e.g., Spalart 2000). The SA model is based on a transport equation for the modified eddy viscosity, ν̃ . The transport equation for ν̃ is
ν̃t+ujν̃ξj=cb1S̃ν̃+1σ{[(ν+ν̃)ν̃]+cb2(ν̃)2}cw1fw[ν̃d]2
(1)
where S=magnitude of the vorticity; uj=contravariant resolved velocity; t=time ; d=distance to the closest wall; and ξj=curvilinear coordinate in the j direction. The other variables and parameters are
S̃S+(ν̃/κ2d2)fv2
fv2=1ν̃/(ν+ν̃fv1)
The eddy (SGS) viscosity νt is obtained from
νt=ν̃fv1
(3)
where
fv1=χ3/(χ3+Cv13)
(4)
χ=ν̃/ν+0.5ksd
(5)
fw=ĝ[1+Cw36ĝ6+Cw36]1/6
(6)
ĝ=r+Cw2(r6r)
(7)
rν̃S̃κ2d2
(8)
To account for roughness effects the distance to the (rough) wall, d , is redefined [see also Spalart (2000)] as
d=dmin+0.03ks
(9)
where dmin=distance to the closest wall and ks=equivalent roughness height (e.g., this term can include the form roughness due to presence of ripples or small dunes that are not resolved by the grid). For smooth walls, ks is taken equal to zero. The model constants in the aforementioned equations are: Cb1=0.135 , Cb2=0.622 , σ=0.67 , κ=0.41 , Cv1=7.1 , Cw2=0.3 , Cw3=2.0 , and Cw1=Cb1/κ2+(1+Cb2)/σ .
The SA version of DES is obtained by replacing the turbulence length scale d with lDES which is defined as lDES=min(d,CDESΔ) where the model parameter CDES is equal to 0.65 and Δ is a measure of the local grid size. When the production and destruction terms of the model are balanced, the length scale in the LES regions lDES=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, which allows the energy cascade down to grid size. The eddy viscosity predicted by DES in the LES regions goes to zero if the local grid size decreases to zero as in classical LES.
A general description of the DES code used in the present work is given in Constantinescu and Squires (2004). The 3D incompressible Navier-Stokes equations are integrated using a fully implicit fractional-step method. The governing equations are transformed to generalized curvilinear coordinates on a nonstaggered grid. The discrete momentum (predictor step) and turbulence model equations are integrated in pseudotime using alternate direction implicit approximate factorization scheme. Time integration is done using a double time stepping algorithm and local time stepping is used to accelerate the convergence at each physical time step. Source terms in the turbulence model equations are treated implicitly. The time discretization is second-order accurate. Detailed grid sensitivity and validation studies for the flow past spheres and for flow and contaminant transport in a channel with a bottom cavity are discussed in Constantinescu and Squires (2003,2004), Constantinescu et al. (2002,2003) and in Chang et al. (2007). In particular, the study of Chang et al. (2007) showed that the agreement between highly resolved LES (Chang et al. 2006) and DES on a much coarser mesh improved significantly when the inflow contained turbulence fluctuations obtained from a preliminary channel flow calculation. This is the approach adopted in the present study.

Test Case Conditions, Flume Experiments, and Simulation Setup

Experiments were conducted in a flume at Iowa Institute of Hydraulic Research (IIHR). Its test section is 20 m long, W=3m wide and 2 m deep. The diameter of the circular pier was D=0.46m , the flow depth was H=0.53m and the mean incoming channel velocity was U=0.45m/s . The main nondimensional flow parameters were RD=2.06×105 (RH=2.4×105) and FH=U/gH0.2 .
A uniform layer of sediment with a median diameter d50=1.05mm was placed on the bottom of the flume. The thickness of the sand layer was ts=1.0m . The critical friction velocity was uτc0/U=0.056 , where uτc0 was estimated based on the Shields diagram. The experiment was conducted under clearwater scour conditions and was run until the scour hole growth was negligible. This process took close to 5 days. The discharge was slowly reduced such that the bathymetry changes during the reduction of the flow discharge were small. The measured decrease of the maximum scour depth during the reduction of discharge and dewatering of the flume was less than 1.5%.
The equilibrium bathymetry is shown in Fig. 1(a). The maximum flow depth in the scour hole was HS=2.09D corresponding to a scour depth of 0.97D . The scour depth is significantly larger than that (0.68D) present in the experiment conducted at RD=16,000 (Kirkil et al. 2008). This larger value of Hs relative to H is closer to the values generally observed at circular bridge piers. Table 1 compares the main hydraulic and geometric parameters in the experiments performed to obtain the bathymetry in simulations LR and HR.
Fig. 1. (a) View of flume with scoured bed; (b) computational domain corresponding to experiment
Table 1. Main Parameters in the Experiments Conducted to Obtain Bathymetry Used in the LR and HR Simulation
  H (m) D (m) U (m/s) RH RD d50 (mm) ts (m) HS/D W (m)
LR0.10.090.1818,00016,0000.680.231.680.91
HR0.530.460.45240,000206,0001.051.002.093.0
Bathymetry measurements around the pier were taken at 0.1-m intervals, both in the longitudinal and transversal directions. The bathymetry data were then digitized for the purpose of generating the scoured bed surface used in the numerical simulation [Fig. 1(b)]. The measured bathymetry is essentially symmetric with respect to the ϕ=0° (the polar angle ϕ is measured starting at the symmetry plane upstream of the pier) plane. The scour and deposition patterns are similar to those observed to form in previous experimental studies of scour around piers of circular section (e.g., Dargahi 1990; Graf and Istiarto 2002; Kirkil et al. 2008). The slope of the bed inside this part of the scour hole was around 32°. The scour hole is not as deep behind the pier but it extends to larger distances away from the pier. Several small-scale ripples are observed inside the scoured region downstream of the pier. In the central region behind the pier (x/D5) , a deposition region in the form of a dune is present. The axis of the deposition dune is streamwise oriented. The deposition dune is situated farther away from the pier compared to the lower RD experiment of Kirkil et al. (2008).
The computational domain [Fig. 1(b)] extends 7D upstream of the pier axis and 20D downstream of it. The domain width is 6.67D , the same as that of the flume used to perform the experiment. The depth of the channel in the flat-bed region is H=1.12D . Though this ratio seems small compared to the usual ratios of H/D encountered in practice at circular piers, the dynamics of the HV system within the scour hole is not expected to be significantly affected by the flow shallowness if the flow structures that may form in the upstream stagnation region close to the free surface (e.g., surface roller) do not interact directly with the HV system. Our experiments showed that this is the case once H>0.7D . Still, the relatively shallow flow conditions considered in the present investigation may have a strong influence on some aspects of the flow (e.g., near-wake region). Reducing the flow shallowness in our test case would have required an increase of the total number of computational mesh points needed to sufficiently resolve the flow proportional to the increase in the ratio H/D .
The computational domain contains about 9.2 million cells (480×240×80) . The mesh in several representative sections is shown in Fig. 2. The larger number of grid points compared to that used in the LES simulation at RD=16,000 (Kirkil et al. 2008) was needed to resolve the attached boundary layers at RD=2.06×105 and to capture the dynamically important eddies forming around the pier. The increase in the mesh size in DES is much smaller than the increase required by a sufficiently well-resolved LES simulation at the same RD because in DES the thin attached boundary layers are resolved using RANS. The grid spacing in the wall-normal direction was close to 0.25 wall units (assuming the nondimensional friction velocity uτ/U is 0.05) on all the wall surfaces.
Fig. 2. Computational mesh in a horizontal and a vertical plane
The lateral boundaries corresponding to the position of the flume sidewalls, the pier surface and the deformed channel bottom were treated as no-slip boundaries. To account for the small unresolved roughness in the bed-surface mesh, the modified eddy viscosity at the wall was calculated such that the nondimensional bed roughness was close to 100. The boundary condition implementation for rough walls is described in Spalart (2000) and Zeng et al. (2008). The free surface was treated as a shear-free rigid lid, which is justified as the free surface deformations in the experiment were negligible (Δz/H<0.04) . The maximum deformations (Δz0.02m) in the experiment were recorded at the upstream stagnation point. The use of a rigid lid approximation accounts to a certain degree for the effect of the increase of the free surface level near the stagnation point on the flow, as the pressure distribution at the free surface is not uniform. The inferred free surface superelevation at the stagnation point was within 20% of the experimental value. The rigid lid approximation is widely used for simulating flows with FH<0.5 (e.g., see Paik et al. 2007). At the outflow, a convective boundary condition was used. The inflow conditions corresponded to fully developed turbulent channel flow and contained resolved turbulence fluctuations. The time step was 0.025D/U .
One issue is the capability of DES to accurately capture the dynamics of the unsteady coherent structures in the flow past surface-mounted bodies. The related DES-LES study of Koken and Constantinescu (2008a, 2009) for vertical wall abutments with incoming fully turbulent flow showed that the mean-flow and turbulence statistics predicted by a DES simulation at a channel Reynolds number of 18,000 were very close to LES results at the same Reynolds number, despite the different numerics in the two codes and, of course, use of a DES SGS model instead of the dynamic Smagorinsky model.

Horseshoe Vortex System

Instantaneous Flow Structure

Fig. 3 shows the instantaneous-flow out-of-plane (ωn) vorticity contours in several polar planes for case HR. In the LR case [see Fig. 3 in Kirkil et al. (2008)] the largest amplification of the instantaneous vorticity was observed within the core of the main necklace vortex (MV) for all sections with |ϕ|<90° . In the HR case this is true only for the sections close to the symmetry plane |ϕ|<30° . At higher ϕ , there is a sharp decay of the vorticity amplification associated with the eddies that are part of the necklace vortices.
Fig. 3. Visualization of the instantaneous structure of the HV system in polar planes using out-of-plane vorticity contours: (a) ϕ=0° ; (b) ϕ=45° ; (c) ϕ=60° ; and (d) ϕ=90°
An elongated patch of high vorticity is present in the vicinity of the bed for |ϕ|<30° . As ϕ increases, this patch moves closer to the pier and thus to the deepest part of the scour hole in the section. By comparison, the vorticity amplification inside the core of MV is much smaller. Animations show the formation of the elongated patch of vorticity to be due to the convection of strong eddies from the downflow away from the pier, after they reach the bed. Though eddies originating inside the downflow are also convected away from the pier in the LR case, these eddies are much less energetic compared to the necklace vortices.
Fig. 4 gives more details on the dynamics of the MV in the ϕ=0° section. At small ϕ , the separated incoming boundary layer is the main source of vorticity for the MV. Compared to the LR case in which the incoming boundary later separates at the mouth of the scour hole, in the HR case the incoming boundary layer remains attached for about 0.5D downstream of the mouth of the scour hole. Due to its proximity to the core of the MV, most of the tongue of vorticity detaching from the incoming boundary layer merges with the MV. When this happens, the intensity and coherence of the MV increase because the tongue of vorticity detaching from the boundary layer has the same sign as the vorticity inside the MV.
Fig. 4. Visualization of temporal evolution of the instantaneous structure of HV system in the ϕ=0° plane using 2D streamline patterns and out-of-plane vorticity contours: (a) t=t1 ; (b) t=t1+0.2D/U ; and (c) t=t1+1.0D/U
The transfer of vorticity from the incoming boundary layer to the MV is not continuous. At times, a strong ejection of vorticity of opposite sign (see arrows in Fig. 4) to that of the vorticity inside the MV takes place from the bed region situated immediately upstream of the MV. Such an event is captured in the three frames of Fig. 4. As a result, the incoming boundary layer and the MV do not exchange vorticity for a certain time interval and the coherence of the MV decreases, provided that no strong eddy originating from the downflow is feeding a high amount of vorticity into the MV.

Bimodal Oscillations

Similar to the LR case and to the flat-bed case (Kirkil et al. 2008), the core of the MV undergoes bimodal oscillations. This means that bimodal oscillations of the HV system forming at the base of a circular pier are present during all the stages of the scouring process, provided the Reynolds number is high enough to generate a turbulent HV system. The two-peak shape of the streamwise velocity histograms in Fig. 5 confirms this. As shown in Fig. 6, during the time interval the tongue of vorticity of opposite (negative in Fig. 6) sign ejected from the bed separates the MV from the tongue of vorticity detaching from the incoming boundary layer, the MV is in the zero-flow mode. At the start of the zero-flow mode the size of the core of MV is large and the coherence of the vortex is high. Then, as a result of the interaction with the negative vorticity ejected from the bed and due to fact that the separated boundary layer cannot feed positive vorticity to the MV, the coherence of the MV starts to decrease and its shape becomes more elliptical (transition to the back-flow mode). This continues until an eddy from the downflow containing positive vorticity is convected in the near-bed region and this eddy is strong enough to dissipate most of the patch of negative vorticity. As this happens, the separated incoming boundary layer reconnects with the core of the MV that is now in the back-flow mode.
Fig. 5. Histograms of the probability density function of the streamwise velocity at points situated inside the HV system in the ϕ=0° plane: (a) pt. 1; (b) pt. 2. The position of the two points is shown in Fig. 6(a).
Fig. 6. Visualization of the instantaneous flow structure in the ϕ=0° plane using velocity vectors and out-of-plane vorticity contours: (a) HV system is in the zero-flow mode; (b) HV system is in the back-flow mode
Compared to case LR, the effect of the deeper scour hole is to impede even more the movement of the MV. While in the LR case the core of the MV was moving away from the pier for close to 0.25D , during the transition from the zero-flow mode to the back-flow mode (see Fig. 7 in Kirkil et al. 2008), in the HR case the change in its position is close to zero during most transitions between the two modes [compare axes of the MV in Figs. 6(a, b)]. The main difference in the flow structure associated with the two modes in the HR case is the fact that the downstream part of the jetlike flow is oriented parallel to the scoured bed in the back-flow mode [see curved solid line in Fig. 6(b)] and away from the bed in the zero-flow mode [see curved solid line in Fig. 6(a)]. Still, the bimodal oscillations induce significant changes in the intensity and direction of the jetlike flow, and in the size and shape of the core of the MV. Compared to the transition to the back-flow mode described by Devenport and Simpson (1990) for bluff bodies mounted on a flat surface in which a patch of high-momentum low-vorticity fluid is merging with the MV, in the present simulation with a large scoured bed the transition to the back-flow mode appears to be mostly associated with the convection of a patch of high-momentum and high-vorticity fluid first with the downflow and then away from the pier in the near-bed region. The convection of strong eddies from the incoming separating boundary layer can also play an important role in forcing the transition to the back-flow mode. Finally, the temporal variations in the coherence of the MV are larger and its size is smaller in the flat-bed case (Kirkil et al. 2008).
Fig. 7. Visualization of the eddy structure in a deformed surface situated at 0.09D from the bed: (a) instantaneous vorticity magnitude; (b) instantaneous vertical velocity

Dynamics of the Legs of the Main Necklace Vortices

Fig. 7 visualizes the instantaneous horizontal (ωxy) vorticity and vertical (w) velocity in a deformed surface situated everywhere at 0.09D from the channel bed and cutting through the core of the MV. As expected, ωxy is amplified in the region where the MV is present ( 0.7<r/D<1.3 , r is the radial distance from the center of the pier) and in the upstream part of the scour hole where a secondary necklace vortex is present at that moment in time. For |ϕ|<30° the legs of the MV curve toward the back of the pier rather than being parallel to the streamwise direction. Examination of the temporal evolution of the vorticity field shows that, at most of the time instances when the coherence of the legs of the MV is high, one of the legs of the MV extends close to the symmetry (ϕ=180°) plane. Then, the downstream end of the leg on one side of the pier detaches and is convected away from the pier in the form of a streak of vorticity. The streak of vorticity moves mostly in the streamwise direction and, similar to the LR case, orients itself at an angle such that the streak is moving against the local bed slope.
Due to the random nature of the oscillations of the HV system and the interaction between the legs of the necklace vortices and the surrounding eddies in the turbulent flow, the time period associated with successive detachments of patches of vorticity from the leg of the MV on one side of the pier can vary significantly. The same is true for the strength of the detached streak of vorticity and for the distance over which this eddy travels before dissipating. The dashed lines in Fig. 7 correspond to the axes of these elongated streaks of vorticity. Such eddies are present in Fig. 7 close to the pier close on both sides. They correspond to the downstream part of the legs of the MV that just starts to be convected away from the pier. Another long streak of vorticity can be observed on the left side (1.5<x/D<3) . Concomitantly, two shorter streaks are convected in the region with 1<x/D<2 on the right side of the pier. It appears these streaks are confined to one or the other sides of the flow with respect to the symmetry (ϕ=180°) plane.
In the LR case the detachment of streaks of vorticity from the downstream end of the legs of the MV was identified to be one of the main mechanisms explaining the lateral and downstream growth of the scour hole [see discussion of Fig. 5 in Kirkil et al. (2008)]. Most of the eddies detaching from the legs were convected laterally away from the pier similar to what is observed for the flat-bed case. Meanwhile, only a few eddies were convected behind the pier and could contribute to the downstream growth of the scour hole in the later stages of the scouring process. By contrast, in the HR case the events that result in the formation and convection of a streak of vorticity in the region situated behind the pier are the most frequent. This is also confirmed by dye-visualization experiments. The successive frames in Fig. 8 illustrate such an event in which the coherence of the leg of the necklace vortex increases while the leg moves behind the pier [Frame (b)]. Then, in Frames (c) and (d) the part of the leg situated behind the pier starts being convected downstream while remaining close to the bed.
Fig. 8. Dye visualizations showing dynamics of the legs of the MV

Mean-Flow and Turbulence Statistics

The nondimensional distributions of k and p2¯ in Fig. 9 provide additional information on the turbulence amplification inside the scour hole. Two regions of high amplification of k are observed in the ϕ=0° and ϕ=30° sections. One of these regions corresponds to the core of the MV that is subject to large-scale oscillations. For |ϕ|>30° , the region of high k amplification associated with the MV gets farther away from the bed. That was not the case in the LR simulation (see Fig. 6 in Kirkil et al. 2008). At a given polar angle, the nondimensional values of k inside the MV region are more than two times lower in the HR case compared to the LR case. In both cases the largest level of turbulence amplification in the MV region occurs in the symmetry (ϕ=0°) plane.
Fig. 9. Mean-flow 2D streamline patterns, resolved turbulent kinetic energy, pressure RMS fluctuations, and resolved turbulence production: (a) ϕ=0° ; (b) ϕ=30° ; and (c) ϕ=90°
The other region of large amplification of k is present in the region where patches of vorticity of opposite sign to the one inside the core of the MV are ejected from the bed. As already discussed, these ejections are associated with the switching of the HV system between the back-flow mode and the zero-flow mode. The amplification of the k levels in the region where these ejections of vorticity take place was also observed in the distribution of k measured in the symmetry plane of a wing-shaped body mounted on a flat surface by Devenport and Simpson (1990) and in a flat-bed simulation of flow past a circular cylinder by Kirkil and Constantinescu (2009). By contrast, the amplification of k in the same region was insignificant in the LR case. This strongly suggests the strong amplification of k due to the ejection of patches of vorticity of opposite sign to the vorticity inside the core of the MV is a Reynolds number effect.
At most of the locations where the resolved turbulence production term, Pk , in the transport equation for k (see plots of Pk in Fig. 9) is high, the values of k are also large compared to the background levels. This suggests that the local turbulence production associated with changes in the flow structure between the two modes is the main factor in the amplification of k inside the MV region and over the downstream end of the jetlike flow. The maximum values of Pk(D/U3) in the symmetry plane (0.4) are comparable to those measured by Devenport and Simpson (1990) at RD=1.25×105 (0.55) .
The other important difference with the LR case is the fact that the k and p2¯ distributions inside the scour hole are no longer similar. As shown in Fig. 9, in the HR case the largest amplification of p2¯ occurs in the region where the eddies convected with the downflow reach the bed. Then, p2¯ decays as the eddies are convected with the jetlike flow away from the pier. The p2¯ levels inside the MV region are couple of times smaller than those observed in the upstream part of the jetlike flow. In contrast to this, in the LR case no large amplification of p2¯ outside the region of bimodal oscillations of the core of the MV was observed for |ϕ|<90° . The same observation holds for the flat-bed results obtained in our group, which points toward the fact that the effect of the downflow on the pressure fluctuations inside the HV region increases as the scour hole grows and is dependent on the Reynolds number.

Separated Shear Layers and Near-Wake Region

Flow Structure behind the Pier

A Q isosurface (Dubief and Delcayre 2000) was used in Fig. 10(a) to visualize the large-scale vortices around the pier. The quantity Q is the second invariant of the resolved velocity gradient tensor [Q=0.5(ui/xj)(uj/xi)] . Positive Q isosurfaces indicate areas where the strength of rotation overcomes the strain, thus making those surfaces eligible as vortex envelopes. Besides the MV which is situated inside the scour hole, a pair of streamwise-oriented vortices is present behind the pier. Fig. 10(b) visualizes one of these streamwise-oriented vortices using 3D streamtraces. The presence of a pair of streamwise-oriented vortices originating on the back face of the pier was also observed in case LR. As one moves away from the pier, the axes of these two vortices move toward the free surface. Their main role is to supply vertical momentum to the flow near the symmetry plane behind the pier. This is consistent with the mean-flow two-dimensional (2D) streamline patterns in the same plane [Fig. 10(c)], which show the flow is convected away from the deformed channel bottom. The presence of strong upwelling phenomena behind the pier in the vicinity of the symmetry plane was confirmed by dye visualizations and inspection of the instantaneous 3D flow fields.
Fig. 10. Visualization of the flow structure behind the pier: (a) Q isosurface; (b) 3D streamtraces; and (c) mean-flow 2D streamline patterns in the symmetry (ϕ=180°) plane
The streamlines in Fig. 10(c) are originating from a small region at the bed, situated at around 0.15D behind the pier. The fluid is convected into that region from the two sides of the scour hole in the near-bed region. This is possible because near the bottom of the scour hole the attached boundary layer on the pier does not separate until close to ϕ=180° [see vorticity magnitude contour plot in a deformed surface situated at 0.1D from the bed in Fig. 11(a) and also discussion in the PIV study of Unger and Hager (2007)]. Similar to case LR, the separation point moves upstream as the distance from the bed increases [e.g., compare Frames (a)–(c) in Fig. 11]. In the plane situated at z/D=0.7 , the SSLs in the HR simulation detach close to ϕ=90° , which is the value expected for the flow past an infinitely long cylinder. Above this level, the SSLs are approximately vertical.
Fig. 11. Visualization of the SSLs in the mean flow: (a) vorticity magnitude contours in a deformed surface situated at 0.1D from the bed; (b) vorticity magnitude contours in a deformed surface situated at 0.5D from the bed; (c) vorticity magnitude contours in a horizontal plane (z/D=0.7) ; and (d) turbulent kinetic energy in a vertical plane behind the pier (x/D=0.51) . Also shown in Frame (d) are the relative positions of the surfaces in which the vorticity is plotted in Frames (a)–(c).

Separated Shear Layers

The curving of the two SSLs toward each other as the bed is approached is visualized in the contour plot of k in Fig. 11(d) at x/D=0.51 . The presence of the scour hole induces a clear amplification of k inside the SSLs compared to the levels observed in the upper part of the section (Δzd>1.5D) . The largest amplification of k is observed inside the SSLs at a distance of 0.30.7D from the deformed bed. Relatively high values of k are also present in between the two SSLs, a region which is characterized by strong momentum transfer in the vertical direction.
Fig. 12 visualizes the distribution of the vorticity magnitude at the free surface in the mean and the instantaneous flow fields. Despite the presence of a noticeable upwelling flow close to ϕ=180° , the free surface behind the cylinder remained flat in the experiment. Thus, the simulation results in this region are not expected to be subject to large errors due to the free surface treatment (rigid lid approximation). Near the free surface, the SSLs are diverging away from the symmetry plane not only in the mean flow but also, at most times, in the instantaneous flow. At most times [e.g., the vorticity distribution in Fig. 12(b) is representative] there is no interaction between the eddies convected in one of the SSLs with the SSL on the opposite side. This means the main mechanism responsible for the formation of large-scale wake rollers is very weak. A flat-bed simulation conducted with identical conditions (H/D=1.12) showed the formation and shedding of very strong rollers at a nondimensional frequency that is close to that expected for infinitely long cylinders. In the deformed bed case, most of the SSL eddies are moving away from the symmetry plane as they are convected downstream. This is partially due to the flow being convected upward near the symmetry plane, including near the free surface [Fig. 10(c)]. As the upward flow approaches the free surface it starts moving away from the symmetry plane on its two sides to ensure continuity. This creates a flow component oriented away from the symmetry plane in the top layer. As a result, the SSLs are pushed away from the symmetry plane in the downstream direction.
Fig. 12. Visualization of the SSLs and the wake at the free surface: (a) vorticity magnitude, mean flow; (b) vorticity magnitude, instantaneous flow; and (c) SGS viscosity, instantaneous flow
Compared to case LR [see Fig. 9 in Kirkil et al. (2008)], the eddies shed in the SSLs maintain their coherence for longer distances. Many of these eddies are still coherent at 45D behind the pier compared to an average distance of only 1D in case LR. Also, compared to case LR it is very rare that two successively shed eddies convected inside the same SSL will merge with each other, or that a strongly coherent SSL eddy will be entrained in the region between the two SSLs behind the cylinder.
The presence of the bed surface and the shallowness of the approach flow (H/D1) relative to scour hole depth strongly modifies the characteristics of the flow behind the pier over the whole depth of the channel compared to the flow features expected for infinitively long cylinders. Fig. 10(c) has shown the presence of a strong upward flow movement close to the symmetry plane. The combined effect of the presence of the upward flow, the scour hole downstream of the pier and the deformed shape of the SSLs is to significantly modify the characteristics of the flow in the near-wake region. The instantaneous wake at the free surface is visualized using eddy viscosity contours in Fig. 12(c). As opposed to the flat-bed simulations, e.g., see Kirkil and Constantinescu (2009), in which large-scale rollers formed behind the pier and the shape of the wake was undular, in the deformed bed cases (LR and HR), at most times, the shape of the wake is not undular though large eddies are present inside the wake.

Near-Wake Region

More details on the variation of the turbulence intensity in the region behind the pier are obtained from Fig. 13. Similar to case LR [see Fig. 13 in Kirkil et al. (2008)], k and p2¯ are strongly amplified in the near-wake region. In case LR the turbulence intensity behind the pier was high over the whole depth. In contrast to this, in case HR, between the pier and x/D1 , the levels of the turbulence intensity are much smaller in the upper layer (z/D>0.7) compared to those in the lower layer (z/D<0.7) . The nondimensional values of k in the highly turbulent region behind the pier are consistently lower (by up to 50%) in case HR.
Fig. 13. Resolved turbulent kinetic energy (left) and pressure RMS fluctuations (right) in vertical spanwise planes: (a) x/D=0.51 ; (b) x/D=0.7 ; and (c) x/D=1.3
In both cases k is amplified above the mean levels characterizing the surrounding turbulent flow inside the legs of the MV. Also, up to x/D=1.0 the p2¯ levels are relatively high in between the leg of the necklace vortex and the bed. However, in the HR case the leg of the MV curves strongly toward the symmetry plane. As a result, for x/D>0.7 the patch of high k values corresponding to the leg starts merging with the region of high turbulence intensity between the SSLs. This did not happen in the LR case, in which the legs (mean flow) remained relatively parallel to the streamwise direction. Similar to k , the nondimensional values of p2¯ in the region of strong turbulence intensity behind the pier (0.5<x/D<1.5) are, on average, lower by 30–50% compared to those observed in case LR.

Shear Stress and Pressure Fluctuations at the Bed

The largest values of p2¯ at the bed [Fig. 14(a)] are observed at the base of the pier (r/D<0.7) . They are induced by (1) the strong temporal variations in the intensity of the downflow; (2) the movement of the legs of the necklace vortices; (3) the wrapping of the legs around the sides and the downstream face of the pier followed by detachment of patches of vorticity from the downstream part of the legs; and (4) the eddies shed inside the SSL in the vicinity of the bed. As the attached boundary layer on the pier separates at ϕ180° , most of the SSL eddies in the near-bed region remain close to the symmetry plane. While some of these eddies are convected away from the pier, others are entrained in the region between the two SSLs and are pushed back toward the pier. The p2¯ levels inside most of the scour hole around the pier are also significantly higher (by at least two times) than those observed on the channel bed both upstream and downstream of the scoured region.
Fig. 14. (a) Resolved pressure RMS fluctuations; (b) mean-flow bed-friction velocity, at the bed
Fig. 14(b) shows the nondimensional bed-friction velocity, uτ/U , distribution in the mean flow. As the governing equations were integrated through the viscous sublayer, uτ was calculated using the definition ( uτ=(τ/ρ)1/2 , where the bed shear stress is τ=ρ(ν+νt)U1/Δn1 , U1 is the magnitude of the velocity component in the direction parallel to the bed at the first point off the bed and Δn1 is the wall-normal distance of the first point off the bed situated within the viscous sublayer). The bed-friction velocity is amplified beneath the region where the core of the MV and its legs oscillate. The largest values are observed close to the junction line between the pier and the bed for 50°<|ϕ|<160° The main reasons for the high values of uτ close to the base of the pier are similar to those responsible for the amplification of p2¯ in the same region. Overall, the distribution of uτ in the mean flow is similar to that predicted in case LR. By contrast, in the case of a flat bed, we observed that the maximum uτ in the mean flow occurred at ϕ45° on both sides of the pier. The amplification of uτ in the region of strong flow acceleration is consistent with experiments that reported scour is initiated in this region. In the flat-bed case, uτ is also amplified beneath the MV but the amplification is smaller than that observed in the acceleration region centered around ϕ=|45°| .
Animations of the instantaneous distributions of the bed shear stress, τ , for the HR case show a high temporal variability of this quantity inside the scour hole. Though, at most times, the largest values of τ are observed in the regions where the bed shear stress is high in the mean flow, there are time instances when the largest values occur at a short distance from the back of the pier. Such an event is visualized in Fig. 15(a) (see arrow pointing toward the strong elongated patch of high τ values forming on the right side of the pier). Then, the streak of high bed shear stress starts being convected away from the pier while remaining on the side where it originated [Fig. 15(b)]. The main reason for the formation of these streaks of high bed shear stress is the movement of one of the legs of the necklace vortices toward the back of the pier followed by the detachment of a streak of vorticity from the leg (see discussion of Figs. 7 and 8). As this streak is convected away from the pier, its vorticity is high enough to produce a significant amplification of τ beneath it. Such a large amplification of τ behind the pier was not common in the LR case.
Fig. 15. Distribution of the instantaneous bed shear stress: (a) t=t10 ; (b) t=t10+0.3D/U

Concluding Comments

The dynamics of the large-scale coherent structures in the flow past a circular pier with a deep scour hole in a relatively shallow channel was investigated using DES for a flow condition typical of the upper range of Reynolds numbers at which flume studies of local scour around bridge piers are conducted under clearwater scour conditions. The flow mechanisms responsible for sediment entrainment and transport during the later stages of the scour process at a circular pier for RD=104(2×105) include bimodal, large-scale oscillations of the MV. Additionally, the curving of the legs of the necklace vortices toward the back of the pier followed by the detachment of a streak of vorticity and its convection away from the pier at a small distance from the bed (as depicted in Figs. 7 and 8) provide for a mechanism for sediment removal behind the pier. The topology of the flow in the near wake was similar in the low and high Reynolds number cases. In particular, the presence of strong upwelling motions near the symmetry plane behind the pier (ϕ=180°) and the relative shallowness of the channel impeded the interaction between the SSLs and suppressed the formation and regular shedding of large-scale roller vortices in the near wake. Finally, in both cases the angle at which the attached boundary layers on the sides of the pier separate was found to increase from 90° in the upper part of the channel to close to 180° in the vicinity of the bed.
The DES and LES simulations revealed that some scale effects are present over the range of Reynolds numbers simulated. For example, differences were observed between the two cases in terms of: (1) the distributions of k and p2¯ in the region where the large-scale bimodal oscillations of the MV were present; (2) the preferred mechanism associated with the transition between the two modes of oscillation; and (3) the probability the streak of vorticity detaching from the leg of the necklace vortex will be convected laterally or behind the pier. In particular, in the high Reynolds number case k was strongly amplified in the region where patches of vorticity of opposite sign to that of the vorticity inside the core of the MV are ejected away from the bed, and the coherence of the streaks of vorticity detaching behind the pier was higher. In the high Reynolds number simulation the largest values of p2¯ in the HV region were observed in the region where the eddies convected with the downflow reached the scour hole. This means that at high Reynolds numbers the pressure fluctuations play a critical role in the entrainment of sediment near the junction line between the upstream face of the pier and the bed. Another important finding of this study is that the nondimensional values of k and p2¯ in the regions of high turbulence amplification associated with the MV, the SSLs and the near wake, were consistently lower in the high-Reynolds-number simulation. This scale effect should be accounted for when results from experiments and numerical simulations at lower Reynolds numbers are used to calibrate scour prediction formulas. Scour prediction formulas calibrated using data mostly from low Reynolds number experiments tend to overpredict scour when used to predict scour at field conditions. The fact that the amplification of the turbulence in nondimensional terms is smaller at high Reynolds numbers suggests that Reynolds number effects are one of the reasons for the observed overprediction of scour in the field. Results of the present investigation do not allow a clear estimation of the importance of this effect compared to other factors responsible for the overprediction of scour in the field.
The present study considered a circular cylinder as pier shape. More complex shapes induce additional complexities of the flow field. The present simulation approach lends itself to investigating variations in flow structure and turbulence, as well as scour mechanisms with differences in such important variables as angle of attack, flow shallowness, pier width, and pier substructure. The presence of sharp edges (e.g., rectangular pier form), for instance, can change dramatically the angle at which the attached shear layers separate in the near-bed region (Kirkil and Constantinescu 2009). In turn, changes occur in the topology of the flow and the dynamics of the large-scale coherent structures in the near-wake region.

Notation

The following symbols are used in this paper:
CDES
=
model constant in DES;
D
=
diameter of the cylinder;
d
=
distance to the wall corrected for wall roughness;
dmin
=
distance to the closest wall;
d50
=
median size of bed material;
FH
=
channel Froude number;
g
=
gravitational acceleration;
H
=
channel depth;
HS
=
maximum scour depth;
lDES
=
turbulence length scale in DES;
k
=
turbulent kinetic energy;
ks
=
equivalent roughness height;
P
=
pressure;
Pk
=
resolved turbulent production term in the transport equation for the turbulence kinetic energy;
p
=
pressure fluctuation;
Q
=
second invariant of the resolved velocity gradient tensor {[Q=0.5(ui/xj)(uj/xi)]} ;
RD
=
Reynolds number defined with cylinder diameter (UD/ν) ;
RH
=
Reynolds number defined with channel depth (UH/ν) ;
r
=
radial distance measured from the axis of the cylinder;
S
=
vorticity magnitude;
S̃
=
modified vorticity in SA model;
t
=
time;
ts
=
initial depth of sand layer;
U
=
mean incoming channel flow velocity;
U1
=
magnitude of the velocity component in the direction parallel to the bed at the first point off the bed;
uj
=
Cartesian resolved velocity in j -direction;
uj
=
contravariant resolved velocity in j -direction;
uτ
=
bed-friction velocity;
uτc0
=
critical friction velocity for sediment entrainment on flat bed;
W
=
width of the flume;
w
=
vertical velocity;
x , y , z
=
coordinate axes;
Δ
=
measure of local grid size;
Δn1
=
wall-normal distance of the first point off the bed situated within the viscous sublayer;
Δz
=
free surface deformation in vertical direction;
Δzd
=
vertical distance measured from deformed bed;
κ
=
von Karman constant;
ν
=
molecular kinematic viscosity;
νt
=
eddy viscosity;
ν̃
=
modified eddy viscosity;
ξj
=
curvilinear coordinate in j -direction;
τ
=
bed shear stress;
ϕ
=
polar angle measured from symmetry plane upstream of the pier in counterclockwise direction;
ω
=
vorticity;
ωt
=
vorticity magnitude;
ωn
=
out-of-plane vorticity component; and
ωxy
=
in-plane horizontal vorticity magnitude.

Acknowledgments

The writers thank the National Center for High Performance Computing in Taiwan and Dr. W. H. Tsai for providing the computational resources needed to perform some of the simulations as part of a collaboration program between the two institutions.

References

Bou-Zeid, E., Meneveau, C., and Parlange, M. (2005). “A scale dependent Lagrangian dynamic model for LES of complex turbulent flows.” Phys. Fluids, 17, 025105.
Chang, K., Constantinescu, G., and Park, S. O. (2006). “Analysis of the flow and mass transfer process for the incompressible flow past an open cavity with a laminar and a fully turbulent incoming boundary layer.” J. Fluid Mech., 561, 113–145.
Chang, K., Constantinescu, G., and Park, S. O. (2007). “Assessment of predictive capabilities of detached eddy simulation to simulate flow and mass transport past open cavities.” ASME J. Fluids Eng., 129(11), 1372–1383.
Choi, S. U., and Yang, W. (2002). “Numerical simulation of 3-D flows around bridge piers.” Proc., 1st Int. Conf. on Scour of Foundations, Texas A&M Univ., College Station, Tex., 206–213.
Constantinescu, G., Chapelet, M. C., and Squires, K. D. (2003). “Turbulence modeling applied to flow over a sphere.” AIAA J., 41(9), 1733–1743.
Constantinescu, G., and Squires, K. D. (2004). “Numerical investigation of the flow over a sphere in the subcritical and supercritical regimes.” Phys. Fluids, 16(5), 1449–1466.
Constantinescu, G. S., Pasinato, H., Wang, Y. Q., Forsythe, J. R., and Squires, K. D. (2002). “Numerical investigations of flow past a prolate spheroid.” ASME J. Fluids Eng., 124(4), 904–910.
Constantinescu, G. S., and Squires, K. D. (2003). “LES and DES investigations of turbulent flow over a sphere at Re=10,000 .” Flow, Turbul. Combust., 70, 267–298.
Dargahi, B. (1990). “Controlling mechanism of local scouring.” J. Hydraul. Eng., 116(10), 1197–1214.
Devenport, W. J., and Simpson, R. L. (1990). “Time-dependent and time-averaged turbulence structure near the nose of a wing-body junction.” J. Fluid Mech., 210, 23–55.
Dey, S., Bose, S. K., and Sastry, G. (1995). “Clearwater scour at circular piers: A model.” J. Hydraul. Eng., 121(12), 869–876.
Dey, S., and Raikar, R. V. (2007). “Characteristics of horseshoe vortex in developing scour holes at piers.” J. Hydraul. Eng., 133(4), 399–413.
Dubief, Y., and Delcayre, F. (2000). “On coherent vortex identification in turbulence.” J. Turbul., 1, 011.
Ettema, R., Kirkil, G., and Muste, M. (2006). “Similitude of large-scale turbulence in experiments on local scour at cylinders.” J. Hydraul. Eng., 132(1), 33–40.
Graf, W. H., and Istiarto, I. (2002). “Flow pattern in the scour hole around a cylinder.” J. Hydraul. Res., 40(1), 13–19.
Kirkil, G., and Constantinescu, S. G. (2007). “A comparison of the horseshoe vortex system at a circular bridge pier between initial and final stages of scour.” Proc., 5th Int. Symp. on Environmental Hydraulics, D. Boyer and O. Alexandrova, eds., Arizona State Univ., Tempe, Ariz.
Kirkil, G., and Constantinescu, S. G. (2009). “Flow and turbulence structure in the flow past an in-stream vertical plate obstruction placed in a relatively shallow open channel.” Water Resour. Res., 45, W06412.
Kirkil, G., Constantinescu, S. G., and Ettema, R. (2008). “Coherent structures in the flow field around a circular cylinder with scour hole.” J. Hydraul. Eng., 134(5), 572–587.
Koken, M., and Constantinescu, G. (2008a). “An investigation of the flow and scour mechanisms around isolated spur dikes in a shallow open channel. Part I. Conditions corresponding to the initiation of the erosion and deposition process.” Water Resour. Res., 44, W08406.
Koken, M., and Constantinescu, G. (2008b). “An investigation of the flow and scour mechanisms around isolated spur dikes in a shallow open channel. Part II. Conditions corresponding to the final stages of the erosion and deposition process.” Water Resour. Res., 44, W08407.
Koken, M., and Constantinescu, S. G. (2008c). “A combined numerical and experimental study of flow past an emerged groyne in a flat bed channel.” Proc., Int. Conf. on Fluvial Hydraulics, River Flow 2008, M. S. Altinakar and I. Kokpinar, eds., Kubaba Congress Department and Travel Services, Turkey.
Koken, M., and Constantinescu, G. (2009). “An investigation of the dynamics of coherent structures in a turbulent channel flow with a vertical sidewall obstruction.” Phys. Fluids, 21(1), 085104.
Mahesh, K., Constantinescu, S. G., and Moin, P. (2004). “A numerical method for large eddy simulation in complex geometries.” J. Comput. Phys., 197, 215–240.
McCoy, A., Constantinescu, G., and Weber, L. (2007). “A numerical investigation of coherent structures and mass exchange processes in a channel flow with two lateral submerged groynes.” Water Resour. Res., 43, W05445.
McCoy, A., Constantinescu, S. G., and Weber, L. (2008). “Numerical investigation of flow hydrodynamics in a channel with a series of groynes.” J. Hydraul. Eng., 134(2), 157–172.
Melville, B. W., and Coleman, S. E. (2000). Bridge scour, Water Resources Publications, Littleton, Colo.
Oliveto, G., and Hager, W. H. (2002a). “Temporal evolution of clear-water pier and abutment scour.” J. Hydraul. Eng., 128(9), 811–820.
Oliveto, G., and Hager, W. H. (2002b). “Further results to time development local scour at bridge elements.” J. Hydraul. Eng., 131(2), 97–105.
Paik, J., Escauriaza, C., and Sotiropoulos, F. (2007). “On the bimodal dynamics of the turbulent horseshoe vortex system in a wing body junction.” Phys. Fluids, 19(3), 045107.
Rodi, W. (1997). “Comparison of LES and RANS calculations of the flow around bluff bodies.” J. Wind Eng. Ind. Aerodyn., 69–71, 55–75.
Roulund, A., Sumer, B. M., Fredsoe, J., and Michelsen, J. (2005). “Numerical and experimental investigation of flow and scour around a circular pile.” J. Fluid Mech., 534, 351–401.
Spalart, P. R. (2000). “Trends in turbulence treatments.” AIAA Rep. No. 2000-2306.
Sumer, B. M., and Fredsoe, J. (2002). The mechanics of scour in the marine environment, World Scientific, River Edge, N.J.
Tokyay, T., and Constantinescu, S. G. (2006). “Validation of a large eddy simulation model to simulate flow in pump intakes of realistic geometry.” J. Hydraul. Eng., 132(12), 1303–1315.
Tseng, M. H., Yen, C. L., and Song, C. C. S. (2000). “Computation of three-dimensional flow around square and circular piers.” Int. J. Numer. Methods Fluids, 34, 207–227.
Unger, J., and Hager, W. H. (2007). “Downflow and horseshoe vortex characteristics of sediment embedded bridge piers.” Exp. Fluids, 42(1), 1–19.
Wang, M., and Moin, P. (2002). “Dynamic wall modeling for large eddy simulation of complex turbulent flows.” Phys. Fluids, 14, 2043–2056.
Williamson, C. H. K. (1996). “Vortex dynamics in the cylinder wake.” Annu. Rev. Fluid Mech., 28, 477–539.
Zeng, J., Constantinescu, S. G., and Weber, L. (2008). “A 3D non-hydrostatic model to predict flow and sediment transport in loose-bed channel bends.” J. Hydraul. Res., 46(3), 356–372.

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Published In

Go to Journal of Hydraulic Engineering
Journal of Hydraulic Engineering
Volume 135Issue 11November 2009
Pages: 888 - 901

History

Received: Aug 17, 2008
Accepted: May 22, 2009
Published online: Oct 15, 2009
Published in print: Nov 2009

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G. Kirkil, M.ASCE [email protected]
Postdoctoral Research Staff Member, Atmospheric, Earth, and Energy Div., Lawrence Livermore National Laboratory, P.O. Box 808, L-103, Livermore, CA 94551; formerly, Graduate Research Assistant, Dept. of Civil and Environmental Engineering, IIHR-Hydroscience and Engineering, Stanley Hydraulics Laboratory, The Univ. of Iowa, Iowa City, IA 52242. E-mail: [email protected]
G. Constantinescu, M.ASCE [email protected]
Associate Professor, Civil and Environmental Engineering, IIHR-Hydroscience and Engineering, Stanley Hydraulics Laboratory, The Univ. of Iowa, Iowa City, IA 52242 (corresponding author). E-mail: [email protected]
R. Ettema, M.ASCE [email protected]
Professor, College of Engineering and Applied Science, The Univ. of Wyoming, Laramie, WY 82071. E-mail: [email protected]

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