Mean and turbulent flow structure during the amalgamation process in fluvial bed forms

Authors

  • Christian E. Frias,

    1. Department of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania, USA
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  • Jorge D. Abad

    Corresponding author
    1. Department of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania, USA
    • Corresponding author: J. D. Abad, Department of Civil and Environmental Engineering, University of Pittsburgh, Benedum Hall 731, 3700 OHara St., Pittsburgh, PA 15261, USA. (jabad@pitt.edu)

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Abstract

[1] Fluvial channels present bed forms such as dunes and ripples that alter instantaneous hydrodynamics parameters such as flow velocities, water surface profiles, bed shear stresses, and Reynolds stresses and create turbulent coherent structures that are significantly different from those presented in flat bed conditions. It is known that LES-based models are more suitable than RANS models to reproduce the complex hydrodynamics around bed forms. Herein, a LES model is applied to describe the mean and turbulent flow structure under superimposed bed forms. Three cases were simulated: RUN I (train of ripples), RUN II (superimposed bed forms), and RUN III (amalgamated bed forms). The LES modeling was performed using a free surface condition to allow the model to develop undulations and boils on the water surface caused by effect of the bed forms. Some important conclusions from this study are: the division of high and low shear stresses on the stoss side of the dune, the progression of the flow field topology from RUN I and RUN III, and the type of turbulent coherent structures found in each stage. The region of high shear stresses was related to turbulence production, in which the streamwise velocity fluctuations (where strips structures are related to streaks) were associated to the modification of the bed morphology. The turbulence Horseshoes Vortices (THV) were more frequent in RUN I than in the other two cases (where streamwise rolls were more frequent). Finally, the frequency of the bursting events increased from RUN I to RUN II and decreased from RUN II to RUN III. Implications of detailed hydrodynamics into bed forms processes are also presented and discussed.

1. Introduction

[2] From past studies [Kennedy, 1963; Engelund and Fredsøe, 1974; Parsons et al., 2005; Best, 2005; Yalin, 1964; Yue et al., 2006], it has been observed that the shape of the river bed presents periodic irregularities that can be classified as dunes or ripples according to their size. These dunes and ripples, hereinafter called also bed forms, significantly influence the entrainment, transport, and deposition of sediments in river channels. A very interesting feature during the transport and deposition process is the superimposition and subsequent amalgamation of ripples over dunes. Best [2005], Jerolmack and Mohrig [2005], and Fernandez [2001] showed that bed forms that are evolving from the superimposed condition to amalgamated bed forms greatly modify the flow field structure. Thus, the knowledge of the dynamics of bed forms has a great importance in the geophysical processes of any natural river or stream [Abad and Garcia, 2009]. One of the important features that can be characterized by understanding such structures is the approximation of the mean water depth at different times from the sedimentary record [Paola and Borgman, 1991].

[3] Efforts to understand the influence of the bed forms on the flow field have been performed experimentally [e.g., Muller and Gyr, 1986; Fernandez, 2001; Hyun et al., 2003; Best et al., 2004; Fernandez et al., 2006] and numerically [e.g., Yue et al., 2005, 2006; Stoesser et al., 2008; Abad, 2008]. From these studies, important conclusions have been outlined. For example, Abad [2008] described that bed forms produce a modification on the natural secondary flow of a bend, and that the shear stresses exerted on the banks due to bed form progression are higher than conditions without bed forms, thus resulting in an increase of 50% on sediment transport due to an increased fluvial erosion rate.

[4] According to Gabel [1993], four types of creation and destruction of dunes are observed: dune splitting, dune combination, dune diminution, and dune creation. Herein, the flow field of the dune combination process called amalgamation is analyzed. It has been demonstrated that the mean and turbulent flow changes distinctly as bed forms amalgamate [Best, 2005]. Moreover, Jerolmack and Mohrig [2005] have described the connection between evolving topography, turbulence, and sediment transport by performing experiments with interaction of bed forms. Most of the past studies have focused in the development of the turbulent coherent structures over the dunes [Yue et al., 2006; Stoesser et al., 2008], and very little has been done to explain the effect of the amalgamation process into the hydrodynamics (except Fernandez [2001] and Fernandez et al. [2006]). An important conclusion from Fernandez [2001] and Fernandez et al. [2006]'s work is that “the presence of a smaller bed form superimposed on the stoss side of a ripple at the ripple-dune transition produces higher levels of turbulence intensity, turbulent kinetic energy and Reynolds stresses than either over ripples or the amalgamated dune bedstate.”

[5] Turbulent coherent structures are vortex features that present different sizes and locations in time [Stoesser et al., 2008]. Since these coherent structures can change the morphology of bed forms, it is important to measure the effect quantitatively by means of laboratory measurements or numerical modeling. However, it is known that due to limitations to obtain spatially resolved instantaneous flow data on experimental and field measurements [Venditti and Church, 2005], this characterization is quite challenging. Numerical simulations can overcome this problem by providing instantaneous parameters that can be spatially or temporal resolved to have a better insight of the physical phenomenon. Venditti and Church [2005] described that conclusions based on experimental measurements cannot be made to explain the role that the turbulence plays in the initiation of bed forms. On the other hand, it has been demonstrated that the sediment transport is enhanced by the variation of the stresses produced by turbulent coherent structures near the bed. The bursting of these structures produce the change in the flow field that allow the resuspension of sediment and modification of the bed as well as Kelvin-Helmholtz instabilities [Venditti and Church, 2005; Escauriaza, 2008]. On the other hand, concerning the numerical approximations, it has been demonstrated that using large eddy simulations (LES), it is possible to model and describe the behavior of coherent turbulent structures [Keylock et al., 2005].

[6] There are known differences between bed forms observed in the field and in the lab; one important difference to mention is the lack of a recirculation zone on field bed forms in contrast to experimental fixed bed forms [Carling et al., 2000a, 2000b]. However, despite this difference it has been proven that hydrodynamic parameters such as Reynolds stresses and velocities remain similar between field and experimental cases [Sukhodolov et al., 2006]. In this paper, the analysis and discussion of the results of LES for the entire amalgamation process (train of ripples, superimposed bed form stage, and amalgamated bed forms stage) are presented. This can be considered as a complementary study to the work of Fernandez [2001] and Fernandez et al. [2006] as we further provide an analysis of the water surface interaction with the bed form (upwelling and downwelling structures) as observed by Nezu and Nakagawa [1993], streamlines' topological characterization, and bursting events quantification. Further, the characteristics of the turbulent structures are described by means of the swirling strength isosurfaces in contrast to the turbulence moments and quadrant analysis performed by Fernandez [2001].

2. Methodology

2.1. Hydrodynamic Governing Equations

[7] The LES method demands an intermediate amount of computational resources, more than the Reynolds Averaged Navier-Stokes (RANS) method and less than a Direct Numerical Simulation (DNS) approach. The LES methodology filters the Navier-Stokes governing equations in order to resolve the intermediate and large scales and to model the small ones [Li and Wang, 2002]. The hydrodynamics of the intermediate and large scales are considered local. Whereas, the behavior of the flow field for the small scales are considered universal and their effect on the large scales can be determined by some parameterization [Bradbrook et al., 2000]. To differentiate between the large and small scales, a box filter in three dimensions is applied to the velocity field [Pope, 2000]. However, according to Hartel and Kleiser [1998] the selection of the filter has little effect on the solution of the system.

[8] The filtered Navier-Stokes equations used herein are given by the equations shown below:

display math(1)
display math(2)

[9] Where inline image describing the Cartesian coordinates, inline image, and inline image are the Subgrid-scale (SGS) filtered velocities in the x1,x2, and x3 direction, respectively, ρ is the density of the fluid, inline image is the SGS filtered pressure, ν is the kinematic viscosity, and inline image is the SGS stress tensor.

[10] The difference between the Navier-Stokes equation and the Navier-Stokes filtered equation is the addition of the SGS stress tensor inline image. The SGS stress tensor contains the fluctuations for the subgrid scale and is given by the following equation.

display math(3)

[11] In order to solve the system shown by equations (1) and (2), a closure model was applied for the SGS stress tensor. These closure models can be categorized as algebraic models, differential subgrid models, scale similarity models, or dynamic models [de Villiers, 2006]. Among these models, the ones that better reproduce the physics of wall bounded flows, such as channels or pipes, are the differential subgrid models and the dynamic models [de Villiers, 2006]. Thus, herein a Smagorinsky dynamic model is used. The major advantage of this model is the adjustment of the coefficient in areas near the wall where it is lowered because of the higher shear stresses.

2.1.1. Dynamic Smagorinky Model

[12] A major problem that has been discovered in the plain Smagorinky SGS model is the disadvantage of considering the SGS viscosity in regions very close to the wall where the flow is no longer turbulent [Keylock et al., 2005]. To address this problem, a dynamic model that allows the Smagorinsky constant Cs to vary in time and space was needed. A dynamic model could be considered as a procedure instead of a model since it can be applied to any of the algebraic models, scale similarity, or differential SGS models [de Villiers, 2006]. It assumes that the behavior of the resolved scales is similar to the SGS scales [de Villiers, 2006]. In order to differentiate the zones close to walls (or zones of high shear stress) and the rest of the domain, a test filter is defined. The test filter is denoted as inline image and it is equal to double size of the original filter width ( inline image). If this test filter is applied to the Filtered Navier-Stokes equations (2), then a new subgrid stress T can be modeled as shown by equation (4).

display math(4)

[13] At this point, the Germano identity shown in the following equation is used to calculate the model coefficients needed for the method.

display math(5)

[14] Where Lik is the Leonard stress ( inline image) and inline image is the test filter stress. A detailed explanation for the use of the Germano identity to obtain the model coefficients can be found in de Villiers [2006]. In summary, the model coefficient is given by the following equation.

display math(6)

[15] Where inline image.

2.2. Computational Setup

[16] Three cases were considered, a periodic train of simulated ripples (RUN I), a bed form superimposed on one of the simulated ripples (RUN II), and a complete amalgamated bed form over the simulated ripples representing a dune-size larger bed form (RUN III). For all the cases, a water depth, inline image, of 0.10 m was used. The geometry of the cases was based on the work of Fernandez et al. [2006]. Thus, for RUN I each ripple had a leeside slope angle of 25°, a wavelength, inline image, of 0.30 m, and a crest height, hs of 0.015 m. For RUN II, a smaller bed form was superimposed with a wavelength, inline image, of 0.10 m and a height, hs, of 0.01 m (see Figure 1). For RUN III, an amalgamated bed form with a bed form ratio, inline image, of 4.00 was used. All the cases considered a flume with a width of 300 mm, a height of 200 mm, and a length of 900 mm. For all of these cases, three complete bed forms were located in the computational grid, and the analysis was performed in the middle bed form. The simulations were carried out using structured computational grids. The grid spacing was selected following Stoesser et al.'s [2008] configuration. This spacing is determined using the wall units shown in the following equation [Pope, 2000].

display math(7)

where inline image are the wall units, ν is the kinematic viscosity, and inline image is the shear velocity.

Figure 1.

Computational grid for RUN II. The mesh has been refined at the water-air interface and near the wall. The dashed rectangles show the refined regions for the interface and near the wall.

[17] As Pope [2000] states on wall bounded flows, the domain can be divided into a viscous wall region ( inline image) and an outer layer ( inline image) with the viscous sublayer located at inline image. Then, the grid spacing is calculated from the wall units distance expressed as inline image.

[18] Thus, inline image in streamwise direction, inline image in spanwise direction, and inline image near the bed form surface. The grid spacing in the y and z direction presents the finest value at the walls and at the water-air interface ( inline image). An example of the computational grid (around 11 million of nodes) for the RUN II (superimposed bed forms) is shown in Figure 1. The CFD (Computational Fluid Dynamics) model used for the simulations was OpenFOAM [OpenFOAM-Foundation, 2012], which is an open source code that treats the two fluid stages (water and air) with a Volume of Fluid (VOF) approach. The algorithm used to solve the Navier-Stokes equations was the Pressure Implicit with Splitting of Operators (PISO) algorithm. The simulations were run in parallel using 64 cores with a total computing time of 27,000 h per case approximately. The computing time is distributed among the 64 cores, which results in about 18 days of computing time per case.

2.3. Boundary and Initial Conditions

[19] Boundary and initial conditions were given for the velocity, pressure, SGS dynamic viscosity, turbulent kinetic energy (TKE), and the fluid fraction function (α). The fluid fraction function is used in the VOF method to track the fluid contained in each cell ( inline image: water, inline image: air, inline image: interface). The equation to be solved for the fraction function is a scalar transport equation given by the following equation.

display math(8)

[20] The inlet boundary condition for the velocity was a logarithmic profile with initial random fluctuations following the procedure described by Keylock et al. [2005]. The logarithmic profile is given by the following equation.

display math(9)

where inline image is the shear velocity (m/s), z is the vertical direction coordinate (m), ν is the viscosity of the water ( inline image), and κ is the von Kármán constant (0.41).

[21] In order to have a flow rate equal to the flow rate reported in each case according to Fernandez [2001] and Fernandez et al. [2006], the shear velocity to be used in equation (9) was calculated. This shear velocity is different from the one reported by Fernandez et al. [2006] since it is not a spatial average on the entire domain but a value used to get the same inlet flow rate of the experimental cases using equation (9). The parameters for equation (9) are presented in Table 1.

Table 1. Logarithmic Inlet Profile Parameters, Where inline image Is the Streamwise Depth Averaged Velocity (m/s), inline image Is the Shear Velocity (m/s), and Q Is the Flow Discharge ( inline image)
RUNIIIIII
u*0.01820.01820.0177
inline image0.3720.3740.363
Q0.01110.01120.0108

[22] The walls were simulated as a no-slip boundary condition for velocity. At the outlet, different patches were created for the outlet of the air and the outlet of the water. Additionally, a wall simulating a weir was placed in order to preserve the water surface elevation for average flow and the pressure distribution. The pressure for the water patch at the outlet was a zero Dirichlet boundary condition (p = 0) while, for the rest of the variables, a zero gradient was considered for the water and air patches. Thus, the outlet boundary condition works as a Poisson equation, in which the pressure is calculated at the outlet. In order to let the flow exit at the top region, a boundary condition that switches the zero gradient ( inline image) and zero value ( inline image) according to the direction of the velocity was used. For the pressure, a Dirichlet boundary condition that represented the total pressure ( inline image) was used. The initial conditions were obtained from a previous simulation in a coarser mesh. The coarser mesh simulation was run for 100 s in only one of the cases (RUN I). Next, the results were mapped to be used as input for the three cases.

3. Results, Analysis, and Discussion

3.1. Validation of the LES Modeling

[23] For the validation, only comparisons for RUN II (superimposed bed forms) are described, since this case presented a more complex geometry and flow structure. Thus, Figure 2 shows the modeled average streamwise velocity for RUN II. The contours show a region of high velocity and one where the flow recirculates (negative values).

Figure 2.

Average streamwise velocity inline image (m/s) for the three cases. The contours show only the water region portion. Six cross sections are used for comparison against experimental measurements.

[24] The computed streamwise average velocity and the computed Reynolds stresses were compared with the experimental data as shown by Figures 3 and 4, respectively. Overall, the streamwise average velocity values are in agreement with the experimental data at the zone near the bed (inner layer). However, it can be seen that in the numerical results there is a trend to exceed the average velocity obtained experimentally at zones above the outer layer (z >30mm) and between cross sections 280 and 364 mm, which is associated with the effect of the air-water interface modeled using the VOF method. On the other hand, the results for Reynolds stresses showed similar patterns as those from the measurements. Some differences are observed near the inner layer between cross sections 334 and 364 mm.

Figure 3.

Experimental and numerical results for the average streamwise velocity (m/s) for selected cross sections in RUN II. Experimental results were obtained from a vertical profile on case RUN II in Fernandez et al. [2006].

Figure 4.

Experimental and numerical results for the Reynolds stresses inline image for selected cross sections in RUN II. Experimental results were obtained from a vertical profile on case RUN II in Fernandez et al. [2006].

3.2. Flow Field Structure

[25] In contrast to previous studies such as Stoesser et al. [2008], Yue et al. [2005], or Fröhlich et al. [2005], the RUN II involves two crests. Figure 5 shows the average streamlines for the three cases. It can be seen that the nodes, saddle, and half-saddle topology signatures described by Gyr and Hoyer [2006] are presented in RUN I and RUN II, while RUN III shows only nodes and saddles. The reattachment length increases by about 75% once the bed forms are amalgamated as shown in Figures 5a–5c.

Figure 5.

Average flow streamlines colored by average velocity magnitude (m/s) for (a) RUN I, (b) RUN II, and (c) RUN III. The saddles, nodes, and half-saddles structures are shown for RUN I, RUN II, and RUN III, respectively. The reattachment point is located at a half saddle for all the cases.

[26] According to the streamlines shown in Figure 5 and the topology description given by Gyr and Hoyer [2006], it is observed that for RUN I, a single saddle is formed. On the contrary, RUN II and RUN III presented two saddles. For all the cases, the reattachment point is defined by a half-saddle structure. The saddles separate the recirculation zone. The center of each recirculation zone is described by a node for all the stages. Thus, for the train of ripples stage (RUN I) we have three recirculation zones of different sizes. The largest one is located closer to the reattachment point. For the superimposed bed forms stage (RUN II), four recirculation zones are observed. The largest one is located again closer to the reattachment point. For the amalgamated bed forms stage (RUN III), three recirculation zones are observed. However, two of these recirculation zones are very close to each other and can be considered as one. Therefore, it can be said that RUN III is composed by two main recirculation zones. For RUN II, one of the saddles is located above the lower crest separating the largest recirculation from the two other recirculation zones in between the crests. Although, these results are for a fixed bed model, some preliminary conclusions related to the sediment transport can be done based on the flow field. Thus, it is due to this separation in the recirculation zone that suspended sediments might not be trapped in the vortices and thus deposited on the stoss side of the downstream bed form. Additionally, this mechanism produces the migration of the superimposed ripple by transporting sediment grains from the ripple's stoss side onto the dune's stoss side. At the same time, deposition processes are occurring at the leeside of the larger and smaller bed form until a complete amalgamation can be reached. From a different point of view, the amalgamation process starts with two recirculation zones that are stretched upstream when the bed forms are superimposing. That is why the resulting topology at RUN II is two small recirculation zones between crests and one main recirculation zone downstream of the largest bed form crest. Next, the migration of the small bed form on the larger bed form (amalgamation) pushes the stretched recirculation zone back into a similar topology as in RUN I but with larger dimensions. In other words, the amalgamation process will produce an oscillating topology in which the sizes of the recirculation zones will be determined by the location of the superimposed bed form crest.

3.3. Interaction Ripple-Dune Structure and Turbulence

[27] To understand the interaction of the ripple-dune structure with turbulence, a visualization of the velocity fluctuations and invariants of the velocity gradient polynomial [Haimes and Kenwright, 1999; Gaston, 2005] was performed. Figures 6 and 7 depict the turbulent coherent structures using the swirling strength parameter ( inline image) for the three cases at a selected time step. The swirling strength is defined as the imaginary part of the eigenvalue of the velocity gradient ( inline image) [Adrian et al., 2000; Adrian, 2007]. For more information regarding to the definition of this parameter, the reader can refer to Adrian et al. [2000] and Adrian [2007].

Figure 6.

Swirling strength isosurfaces with elevation contours (mm) for (a) RUN I, (b) RUN II, and (c) RUN III.

Figure 7.

Swirling strength isosurfaces with velocity contours (m/s) for (a) RUN I, (b) RUN II, and (c) RUN III.

[28] These turbulent coherent structures, due to their similarities at different scales, are given by fractal laws as stated by Gyr and Hoyer [2006]. The contours in Figure 6 are colored by value of elevation to show the three-dimensional shape of the structures and by velocity magnitude in Figure 7. Based on the swirling strength isosurface visualization, it seems that the turbulence Horseshoes Vortices (THV) [Grigoriadis et al., 2009; Escauriaza and Sotiropoulos, 2011] are created downstream of the bed form's leeside but not observed at the vertical plane located at coordinate y = 0. Instead, the THV tend to appear toward the side walls in periodical bursts. The turbulent structures present a more chaotic behavior along the amalgamation process (from RUN I to RUN III). The THV are labeled as TH1-1, TH1–2, TH1–3 for RUN I, TH2-1, TH2-2 for RUN II, and TH3-1 for RUN III. On the other hand, the streamwise roll vortex structures are labeled as SV2-1, SV2-2 for RUN II and SV3-1 and SV3-2 for RUN III. The distribution of these structures presents a more chaotic behavior along the amalgamation process. It is observed in RUN I that the streamwise rolls are not as noticeable as those in RUN II and RUN III. By contrasting the contours of elevation and velocity from Figures 6 and 7, it is clear that the majority of the low elevation structures coincide with the low-speed structures, also known as low-speed streaks [Smith and Meltzer, 1983; Adrian et al., 2000; Asai et al., 2002]. For the case of RUN II, the range of velocity for the streamwise rolls SV2-1 and SV2-2 (0.2–03 m/s) is lower than the upper part of the THV such as TH2-1 and TH2-2 (0.4–0.5 m/s). However, for RUN III, the values of velocities for TH3-1 and SV3-1 (0.2–0.45 m/s) are almost in the same range in contrast to the SV3-2 structure, which is between 0.1 and 0.3 m/s. The latter is again related to the height of the structures, thus showing that the higher the coherent structures, the faster they are. This plays an important role on the rate of sediment transported to the outer layer. The streamwise rolls are a consequence of broken THV, thus, the sequence is characterized starting from low-speed streaks to high-velocity THV, then to low-speed streamwise rolls. The streamwise rolls reduce the velocity compared to the THV due to the energy that is lost when a THV breaks. Finally, the height of the structures for the three cases is approximately 15%–25% of the average water depth of 100 mm. This height is considered to be related to the bursts that transport the sediment from the viscous and inner layer to the outer layer.

3.4. Interaction Bed Form and Water Surface Elevation

[29] In Figure 8, the spanwise instantaneous velocity fluctuations are shown. The vertical inclined flow structures reported by Stoesser et al. [2008] were encountered for the three cases. These inclined structures alternate between high and low spanwise instantaneous velocities in the longitudinal direction. The angle increases slightly with respect to the horizontal as the amalgamation process progresses (RUN I to RUN III).

Figure 8.

Spanwise velocity fluctuations contours for a plane located at y = 0 (a) RUN I, (b) RUN II, and (c) RUN III.

[30] These structures are related to the location and scope of the so-called “ejection rollers” due to a THV instability. These “ejection rollers” reach the water surface creating the well-known kolk-boil structure. These kolk-boil structure generate the upwelling and downwelling water surface shape that is shown in Figure 9 for the three cases (undulations) [Ashworth et al., 1996]. The deformation of the instantaneous water surface (Figures 9) is strongly associated to the bursting events. The instantaneous water surface presents small perturbations with respect to the average water surface, where the downwelling and upwelling structures are well defined. The difference in elevation between the upwelling and downwelling structures for RUN I is in the order of 1–2 mm, whereas for RUN II and RUN III these fluctuations are approximately greater than 5 mm. For RUN I, the downwelling structure is located exactly above the single crest, while for RUN II, it is located at the lowest crest. For RUN II, the highest crest influences the rise of the water surface creating an upwelling structure above it. For RUN III, the downwelling structure is located downstream of the crest of the bed form. Thus, it is shown that the amalgamation process modulates the perturbations in the water surface elevations and therefore the upwelling and downwelling structures.

Figure 9.

Instantaneous water surface at selected time steps for (a) RUN I, (b) RUN II, and (c) RUN III. The upwelling structures on the water surface are denoted as UW1-1, UW2-1, and UW3-1 for RUN I, RUN II, and RUN III, respectively. The downwelling structures on the water surface are denoted as DW1-1, DW2-1, and DW3-1 for RUN I, RUN II, and RUN III, respectively.

3.5. Shear Stress, Bed Sediment Transport, and Bursting Events

[31] Several researchers [Fedele and Garcia, 2001; Best, 2005; Parsons et al., 2005; Cantero et al., 2008; Shugar et al., 2010] stated that the bed shear stresses play an important role in the bed sediment transport and the bed evolution. The amalgamation process is strongly related to the sediment transport processes, which is related to the bed shear stress, thus a characterization of the spatial variability of the bed shear stresses is required. Figure 10 shows the bed shear stresses for all cases. A region of high shear stress denoted as HSS and a region of low shear stress denoted as LSS are observed. The LSS region is located between the first crest (highest crest for RUN II) and the reattachment point. Whereas, the HSS region is located after the reattachment point. It is well known that the HSS regions are related to erosion processes whereas the LSS regions are related to depositional processes [Best, 2005]. As explained previously, the distance from the crest to the reattachment point increases with the amalgamation process (from RUN I to RUN III). Therefore, the area of the HSS region decreases with the amalgamation process as observed in Figure 10. This behavior influences the bed sediment transport rate, manifesting that along the amalgamation process, the sediment flux might be reduced. In the HSS region, the values of shear stress alternates between low and high shear stress strips. This alternation will produce “trails” on the stoss side of the bed form located downstream. The evolution of these “trails” will be modulated by the streak coherent structures (alternation of low and high velocity fluctuations regions [Stoesser et al., 2008]).

Figure 10.

Bed shear stress contours for (a) RUN I, (b) RUN II, and (c) RUN III. It is observed that the bed is divided in a high shear stress region (from 0.03 to 0.04 Pa) and a low shear stress region (from 0 to 0.01 Pa). The high shear stress region is located after the reattachment point while the low shear stress region is located between the highest crest and the reattachment point.

[32] To get a better insight into the influence of bed shear stresses and near-bed turbulence on sediment transport processes, a visualization of the streamwise velocity fluctuation near the bed (plane parallel to the bed at inline image) was obtained (Figure 11). It is observed that a line of negative velocity fluctuations divides the zone of HSS and LSS. As pointed out by Stoesser et al. [2008], in the zone of HSS there is an alternation of low-speed and high-speed streaks. The high-speed streaks are less spaced at the wall than at the middle zone. For RUN I, the streaks are more defined than for RUN II and RUN III, due to a more defined THV encountered for RUN I.

Figure 11.

Streamwise velocity fluctuations contours (m/s) for a plane parallel to the bed at inline image for (a) RUN I, (b) RUN II, and (c) RUN III.

[33] To quantify the bursting events, techniques such as quadrant analysis [Fernandez, 2001; Fernandez et al., 2006] and threshold crossing schemes such as U-level or modified U-level methods have been applied in previous studies [Binder et al., 1991; Gyr and Hoyer, 2006; Metzger et al., 2010; Sakai et al., 2011]. Another threshold technique is the Variable Interval Time Average or VITA method [Gyr and Hoyer, 2006]. Herein, the modified U-level method is used to determine the duration of the bursting events and the time between their occurrences. The results of this method are shown in Figure 12. The burst event is related to a strong negative streamwise fluctuation ( inline image) [Metzger et al., 2010] in a specific point of the flow domain, the location of which is detailed below. The method consists of detecting the moment at which the signal crosses a lower threshold (beginning of the event or leading edge) and a upper threshold (end of the event or trailing edge) [Luchik and Tiederman, 1987]. The value of the lower threshold is typically one standard deviation of the signal ( inline image) and the value of the upper threshold is set to inline image. Once the events are detected it is possible to average the duration of each event and the time between events. For all cases, a time series of 10 s was selected. The point was located at x = 330 mm, y = 0 mm, and z = 15 mm. The location of the point was selected in order to consider a point that falls inside the shear layer for the three cases. The average burst event duration is 0.15, 0.33, and 0.28 s for RUN I, RUN II, and RUN III, respectively. On the other hand, the time between events is 0.77, 1.28, and 1.20 s for RUN I, RUN II, and RUN III, respectively. These results indicate that there is a trend to decrease the rate of the bursts with the amalgamation process. However, once the bed reaches to the amalgamated state (RUN III) the bursting rate increases by 15% approximately. The time between events follow the same pattern as the duration of the even; it increases during the amalgamation process but once it is amalgamated it decreases by approximately 7%. This is related to the migration rate of the ripple formed at the stoss side of the downstream dune as previously described.

Figure 12.

Burst frequency for (a) RUN I, (b) RUN II, and (c) RUN III. Where inline image is the duration of a single event, inline image is the average event duration for each case, Te is the time between single events, and inline image is the average time between events for each case.

[34] A visualization of the bursting processes for all cases is shown by Figure 13. The time steps selected for this visualization were taken from two time steps before the bursting event, as detected by the modified U-level algorithm. On these time steps, a decomposition of the flow in two stages was observed. These stages can be summarized as (1) a flow within a structure given by the vortex (Figures 13a, 13c, and 13e), (2) a transient evolution of the vortex skeleton from its formation to its decay (Figures 13b, 13d, and 13f), and (3) a downstream convection of the whole structure. Once the flow reaches the last stage the instantaneous velocity vectors do not show any relevant recirculation pattern and, therefore none are shown herein. The time evolution is as follow, the vortex skeleton begins with a core of negative velocities (recirculation) and starts to travel downstream (for Figures 13a, 13c, and 13e) this vortex has already traveled up to a distance of x= 350 mm), next, this vortex “explodes,” or bursts, forming a structure of positive velocity flow (between 0.1 and 0.2 m/s). The vectors showed that this structure pushes the flow upward and thus boils appear at the water surface. The height of the vortex skeleton bursting structure is greater for RUN II than for the other two cases (approximately twice the height for RUN I and RUN III, 40 mm in contrast to 20 mm). It is assumed that this behavior is due to the presence of two shear layers caused by the ripple superimposed over the dune. In general, the bursting process is associated with the time scale of the ejections and sweeps, which are independent of viscosity [Grass, 1971; Ashworth et al., 1996]. Hence, the bursting process is associated with the sediment transport and bed morphology evolution as well as the shear stresses as explained previously by Best [2005].

Figure 13.

Instantaneous streamwise velocity contours and instantaneous velocity vectors for different times. The figure shows the evolution of the bursting process from one time step before the burst occurs and one time step at the onset of the burst. (a) RUN I, Time = 10.95 s, (b) RUN I, Time = 11.3 s; (c) RUN II, Time = 12.45 s, (d) RUN II, Time = 12.75 s; (e) RUN III, Time = 12.15 s, (f) RUN III, Time = 12.65 s. The point located at x = 330 mm, y = 0 mm, and z = 15 mm is denoted as a white circle.

4. Conclusions

[35] The flow field in a bed form amalgamation process was characterized by means of large eddy simulations over fixed laboratory conditions [Fernandez, 2001; Fernandez et al., 2006].

[36] The recirculation area of the three stages (train of ripples, superimposed bed forms and amalgamated bed forms) is divided by saddles. These structures (topological signature) are located at different locations along the stoss side of the bed form for RUN I and RUN III. It was observed that the location of the saddles tends to oscillate. At the train of ripples stage there are three recirculation zones, two of them are later at the superimposed bed stage stretched downstream. At this stage, four recirculation zones are defined to return to a two recirculation zones stage at the amalgamation stage. Moreover, there is an increase in the reattachment length as amalgamation process progresses, as observed by Fernandez et al. [2006].

[37] Turbulent coherent structures denoted as THV, streamwise rolls, and streaks were described for the amalgamation process. Contrasting the velocity field and the elevation of the THV for the three cases, more chaotic behavior was observed once the ripple is amalgamated (RUN III). From the velocity contours (isosurface), it has been concluded that the structures starts from low-speed streaks to high-velocity THV that burst and form low-speed streamwise rolls. Additionally, the structures tend to move toward the walls during the bursting events. This movement, indicates that these structures might influence the reshaping of alluvial banks. These turbulent coherent structures are directly related to both bedload and suspended sediment transport processes.

[38] The inclined structures caused by plotting the spanwise velocity fluctuations and the alternation of low-velocity and high-velocity streaks together with the alternation of bed shear stresses can cause complex bed morphology (e.g. trails or strips of sediments on the zone of erosion due to high shear stresses).

[39] Using a U-level method, the bursting events were described. The bursting events and the time between them increase in frequency as the amalgamation process evolves. However, this frequency is decreased once the ripples are superimposing (RUN II). These bursting events are associated with the shear stresses, sediment transport, the deformation of the water surface and the bed morphology evolution.

[40] The use of LES has tremendously aided to understanding of the instantaneous and mean flow structures, both temporarily and spatially, overcoming some of the limitations of experimental studies. Furthermore, with the detailed understanding of the hydrodynamics around superimposed bed forms and the amalgamation stage, more insight into sediment transport and fluvial geomorphology is obtained.

Acknowledgments

[41] The authors would like to thank R. Fernandez and J. Best for sharing the postprocessed experimental data used in this study and discussing the present version of the manuscript. Special thanks to L. Yilmaz for assisting on facilitating computational resources at the Center for Simulation and Modeling (SAM) of the University of Pittsburgh. Initial runs were performed in the cluster from the Environmental Fluid Mechanics Laboratory (EFM). Thanks to the Department of Civil and Environmental Engineering and the College of Engineering for providing the financial support based on J. Abad's start up funds. Thanks to Kristin Dauer for proofreading the manuscript.

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