ESURFEarth Surface DynamicsESURFEarth Surf. Dynam.2196-632XCopernicus GmbHGöttingen, Germany10.5194/esurf-3-105-2015The role of velocity, pressure, and bed stress fluctuations in bed load transport
over bed forms: numerical simulation downstream of a backward-facing stepSchmeeckleM. W.schmeeckle@asu.eduSchool of Geographical Sciences and Urban Planning, Arizona State University, Tempe, Arizona, USAM. W. Schmeeckle (schmeeckle@asu.edu)9February2015311051122July201417July201415January201526January2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://www.earth-surf-dynam.net/3/105/2015/esurf-3-105-2015.htmlThe full text article is available as a PDF file from https://www.earth-surf-dynam.net/3/105/2015/esurf-3-105-2015.pdf
Bed load transport over ripples and dunes in rivers exhibits strong
spatial and temporal variability due to the complex turbulence field
caused by flow separation at bedform crests. A turbulence-resolving
flow model downstream of a backward-facing step, coupled with
a model integrating the equations of motion of individual sand
grains, is used to investigate the physical interaction between bed
load motion and turbulence downstream of separated flow. Large bed
load transport events are found to correspond to low-frequency
positive pressure fluctuations. Episodic penetration of fluid into
the bed increases the bed stress and moves grains. Fluid penetration
events are larger in magnitude near the point of reattachment than
farther downstream. Models of bed load transport over ripples and
dunes must incorporate the effects of these penetration events of
high stress and sediment flux.
Introduction
The details of turbulent flow over dunes and ripples in
rivers and oceans have been described by field and laboratory
experiments seefor an extensive review, as well as
high-resolution, turbulence-resolving numerical simulations
. However
attempts to couple turbulence to the transport of sediment over
bedforms have usually relied on empirical formulas, wherein the sediment
flux is either a direct function of boundary shear stress or
indirectly through entrainment rate and deposition rate formulas
(although see , and ). Unlike suspended sediment fields, experiments detailing
the spatiotemporal pattern of bed load transport over ripples and
dunes have not been reported.
hypothesized that the mean bed load sediment
flux could be calculated as the integral of the
probability density of bed stress due to turbulence times the sediment flux as a function of stress.
They further hypothesized that the bed stress distribution could be determined
from the distribution of near-bed downstream velocity. The experiments of
simultaneously measuring sediment flux and near-bed fluid velocity
over a flat bed and downstream of a backward-facing step showed that
the relationship between near-bed fluid Reynolds stress and bed load
transport was not simple, and the spatially varying distribution of
velocity fluctuations relative to the shear velocity must be
considered in formulating transport relationships over ripples and
dunes. They also found that there was not a simple monotonic relationship
between instantaneous, downstream, near-bed velocity and sediment flux. Specifically,
longer duration positive fluctuations of near-bed velocity were found to transport
more sediment per unit time than shorter duration events. As such, the hypothesis
of needs significant modification to be useful downstream of separated flows.
Experimental measurement of turbulence is often limited to a time
series of fluid velocity components at a single point or, more rarely,
several points. In such instances, the detection of spatially and
temporally evolving turbulence structures is difficult. Quadrant
analysis has been used to detect certain types of turbulence
structures . Quadrant analysis involves joint
examination of the fluctuating components of fluid velocity in the
downstream, x, and bed-perpendicular, z, directions. u′
and w′ are the downstream and bed-perpendicular fluctuating
components of fluid velocity. With u′ and w′
measurements plotted on a two-dimensional graph, the first quadrant
(Q1) is a point with u′>0 and w′>0; this is also
known as an outward interaction. Quadrant 2 events (Q2, u′<0
and w′>0) are termed bursts or ejections, quadrant 3 events
(Q3, u′<0 and w′<0) are called inward
interactions, and quadrant 4 events (Q4, u′>0 and
w′<0) are known as sweeps. Q2 and Q4 events transport downstream
momentum toward the bed, and are thus positive contributions to the
Reynolds stress component, -ρu′w′‾,
whereas Q1 and Q3 events are negative contributions.
used a coupled turbulence-resolving numerical
model of flow and a particle model of sediment motion to simulate the
interaction between turbulence and sediment movement over a flat bed.
Vortical structures embedded within broader sweep structures were
found to bring fluid into and out of the bed, and were sites of
sediment entrainment and transport. In this article I extend the model
of and to the case of bed
load transport downstream of a backward-facing step, largely matching
the experiments of . Quadrant analysis is extended
to include sediment flux, bed stress, and fluid pressure. Flow over
a backward-facing step, like that over bed forms, causes flow
separation, but does not have the complicating effect of flow
acceleration by an upstream sloping bed.
Methodology
The fluid is modeled by the large eddy simulation (LES) technique in
which the spatially filtered Navier–Stokes equations are integrated
using the finite-volume method. The equations of motion of each
sediment grain are integrated over time using the distinct element
method (DEM). The sand grains are assumed to be spheres, and forces between
particles are calculated when grain boundaries overlap. The LES and
DEM models are coupled in momentum. The flow field is interpolated to
the particle centers and used to derive fluid forces on the
particles. In turn, each fluid force acting on the particles is given
as a resistance term to the fluid momentum equations at the fluid cell
containing the center of the particle. Only drag, pressure gradient,
and buoyancy forces are included as fluid–particle forces. The bed of
particles is about three to four grain diameters thick above the lower
fluid boundary, and non-moving particles of the particle bed rapidly
damp the fluid velocity. The details of the numerical model reported
here are the same as reported in .
The flow magnitude, step height (xstep), particle diameter
(D) and density (ρs), and flow depth of the numerical
simulations are specified to nearly match the experiments of
. The computational domain extends 0.2 m
upstream of the 0.04 m backward-facing step and 1.2 m
(30 step heights) downstream of the step, and 0.1 m (2.5) step
heights across stream. The fluid domain in the vertical dimension extends 0.16 m
above the step, and downstream of the step the vertical domain is 0.2025 m. At rest, the topmost
particles are roughly 0.0025 m above the lower wall. Thus, the
flow height downstream of the step is about 0.2 m from the bed
of particles to the top of the numerical domain. The grid used in
this study is a structured mesh of 4 655 000 hexagonal cells. The
grid is evenly spaced in the downstream and cross-stream
directions. The downstream and cross-stream grid lengths are 0.002
and 0.00143 m, respectively. The vertical grid spacing is
nonuniform, with smaller grid cells containing the particles and
near-bed flow. The vertical grid dimension is also significantly
reduced in a zone containing the separation bubble shear layer. The
vertical dimension of cells containing particles at the bottom of the
numerical domain is 0.00025 m. The downstream and cross-stream
grid dimensions are slightly larger than the diameter of the
particles, but the particle diameters are about 3.6 times larger than
the vertical grid dimension. As such there is rarely more than one particle
in a grid cell.
If fluid and particles are coupled in mass and momentum, the fluid solver becomes unstable when the particles are of the same size as the fluid grid cells. Each particle is treated
as a source of resistance in the cell where the particle(s) center is located.
It is possible to smooth the effect of a particle over a broader number of cells to achieve a
stable algorithm, but this smooths the sharp interface between the bed and the overlying relatively
sediment-free fluid. Smoothing of the bed interface was deemed not appropriate,
because most of the fluid momentum is damped within a grain diameter below the bed interface.
Here, fluid and particles are coupled in momentum but not in mass.
The flow around each particle is not directly modeled; only the damping of flow by
the integrated force of each particle is modeled. Thus, the flow separation and turbulence
generated by flow around particles and in the interstices of particles is not modeled.
Further, the calculated pressure field within the bed may be different than reality,
because the fluid continuity equation does not account for the reduced volume of fluid within the bed.
It is difficult to predict how these assumptions degrade the fidelity of the simulations.
However, the integrated momentum effect of each particle on each fluid cell
(and vice versa) is calculated in the model and should lead to relatively accurate fluid simulations at the scale of the grid.
The domain of the DEM model begins at the step in the x direction
but otherwise coincides with the fluid domain boundaries. The particle domain is
periodic in the downstream and cross-stream directions. The
diameters of the 415 000 particles in this simulation are randomly
drawn from a normal distribution with a median of 0.9 mm and
a standard deviation of 0.1 mm. The diameters are varied to
avoid close packing arrangement of the bed during the simulation. The
particle parameters for the DEM model are the same as in
except that the Young's modulus is increased to
5×106Pa.
Boundary conditions for fluid velocity are no shear at the upper
boundary, periodic conditions in the cross-stream direction, and zero
gradient at the outlet. The no-slip condition is applied at the lower
boundary, but the fluid velocity becomes negligible before reaching
this boundary because of the presence of a bed of particles above
it. The inlet boundary condition is specified as the velocity
0.15 m downstream of the inlet. This is similar to a periodic
boundary condition wherein the inlet and outlet are the same velocity,
but the recycled velocity is taken before the backward step, thus
ensuring fully developed boundary layer turbulence upstream of the
backward-facing step.
Results
Prior to recording simulation results, the flow reached dynamic
equilibrium after about 30 s of simulated time. Results reported here
are for 20 s of simulated time. This length of time provided adequate
statistics, but it was not so long that the bed of particles developed bedforms
and areas of the bed without sufficient numbers of particles.
However, bed elevation
changes of one to two particle diameters were apparent in response
to the passage of individual turbulent events. The position, velocity, and fluid
force of each particle of known diameter is recorded at
40 Hz. Similarly, the fluid velocity and pressure are saved
simultaneously at 40 Hz along two near-bed horizontal slices
at z=1mm and z=5mm and along a slice
perpendicular to the cross-stream at the center of the numerical
domain, y=0.05 m. The lower boundary of the fluid and particle
domain is at z=-0.0025m and the topmost particles of the
bed at rest are at approximately z=0. A local depth-integrated downstream
sediment flux, qsx, is calculated by summing the product of each
particle volume and velocity that is found in a 0.01 × 0.01 m
horizontal area of the bed and then dividing by the local bed area
(i.e., division by 0.0001 m2). Downstream bed
shear stress, τbx is calculated by summing the downstream
component of fluid force acting on all particles with centers
contained in the same 0.01 × 0.01 m horizontal grid, and
then dividing by the local bed area. The saved fluid velocity and
pressure at z=1mm are extracted at points in the center of
the grid of local bed areas used to calculate boundary stress and
sediment flux. In this manner, the data examined in this article
are of 20 s of simulated time at a rate of 40 Hz at 12 000 (1200 downstream
by 10 cross-stream) points of fluid velocity and pressure, boundary
shear stress, and depth-integrated downstream sediment flux.
Visualization of the downstream fluid velocity, u; vertical
velocity, w; particle velocity magnitude, |U|; and fluid pressure fluctuation, p′; at an instant
in time. (a) Downstream velocity on a vertical slice at the
middle of the cross-stream domain. (b) Downstream velocity
on a horizontal slice at z=1mm. (c) Particle
velocity magnitude. (d) Vertical velocity on a horizontal
slice at z=1mm. (e) Fluid pressure fluctuation
on a horizontal slice at z=1mm. (f) Fluid
pressure at the middle of the cross-stream domain. An animation associated with this figure can be found in the Supplement.
It should be noted that the bed stress, as defined here, is the sum
particle force per bed area. It is not the near-bed Reynolds stress
component, -ρu′w′‾. Averaged over
sufficient time, these two quantities are equivalent by a balance of
forces. However, at a particular instant in time, there are fluid
accelerations, and τbx≠-ρu′w′, except
by coincidence.
Simultaneous visualization of the fluid pressure fluctuation,
p′=p-p‾; downstream fluid velocity, u; and
particle velocity magnitude, |U|, reveals
a positive covariance of particle motion with near-bed,
downstream fluid velocity and fluid pressure (Fig. 1). Low-frequency,
cross-channel variations in pressure are apparent in Fig. 1, which
were also noted in the direct numerical simulations of
. Figure 1 shows that positive pressure fluctuations are
associated with both large near-bed downstream fluid and particle
velocity magnitude. There are small areas of the bed with large
negative vertical velocity (red areas of Fig. 1d). Fluid that
penetrates the bed leads to neighboring areas where fluid exits the
bed (blue areas of Fig. 1d). These large fluctuations in vertical
velocity are associated with significant sediment motion (Fig. 1c).
Temporal mean and the 10th and 90th percentile flow and transport
parameters are plotted against distance downstream relative to step
height, x/hstep. (a) Downstream fluid velocity
at z=1mm, (b) vertical fluid velocity at z=1mm, (c) fluid pressure at z=1mm,
(d) bed shear stress, and (e) depth-integrated
downstream sediment flux. N95 is the measured sediment flux of
. A smoothed line of a moving average of
q∗‾ of all points within 0.025 m upstream
and downstream is also shown.
Figure 2 shows the temporal statistics (mean, 10th, and 90th percentile) vs. downstream distance for u, w, p, τbx,
and qsx. The position of the point of reattachment is plotted at
a vertical dotted line in the five plots of Fig. 2. Figure 2a shows
that the mean near-bed velocity (at z=1mm) increases
rapidly near the point of reattachment, and it increases, albeit much
less rapidly, all the way to the downstream outlet. Interestingly, the
difference between the 90th and 10th percentile of velocity is larger
near the outlet than in the reattachment zone. However, the difference
between the 90th and 10th percentile of bed stress (Fig. 2d) is
smaller near the outlet. The 10th percentile transport rates are essentially zero
(or slightly negative) downstream of flow reattachment (Fig. 2e).
The mean sediment transport rate (Fig. 2e) increases rapidly downstream
of reattachment and does not show a peak in transport at 20 step heights
as do the results of . However, they suggested that the
peak could be the result of sampling error.
The fluid pressure (Fig. 2c) rises rapidly from the recirculation
region through the zone of reattachment (from about
x/hstep=3 to 7). The largest magnitude of this
upstream-directed pressure gradient is about 400 Pam-1,
which leads to a stress of about -0.5 Pa at
x/hstep=5. This “pressure gradient stress” is about one-third to one-quarter of the negative bed stress in the recirculation
and reattachment zone. However, this stress is distributed throughout
the bed of particles, and the resulting pressure force on individual
grains is more than an order of magnitude smaller than is required to
entrain the topmost grains that are able to move.
While Fig. 1 qualitatively shows the spatial covariance of some of the
fluid and particle variables, Fig. 3 shows some of the significant
temporal correlation pairs of variables
u, |w′|, τbx, and p. The absolute magnitude of the
vertical velocity fluctuation, |w′| is used rather than w
because transport was found to peak when the fluctuations of vertical
velocity were high. It is perhaps unsurprising that u′ is
positively correlated with τbx and qsx, but Fig. 3 also
shows the positive correlation with fluid pressure, p.
Correlation coefficients vs. time lag for various pairs of
flow and transport variables as indicated in the legend in
(c). The lag is of the second variable relative to the
first variable in the legend shown in (c).
DiscussionPermeable splat events
Given that the force on bed grains results primarily from fluid drag,
it is somewhat paradoxical that the temporal variance in the bed
stress is much larger near reattachment, despite the smaller variance
in downstream velocity at reattachment, relative to farther
downstream. This apparent paradox is due to the fluctuations in
vertical velocity being much larger near the point of reattachment
than farther downstream (Fig. 2b). A large negative vertical velocity
brings high downstream fluid velocity into the bed, thus creating peak
bed stresses. Consider the plots in Fig. 2 between about
x/hstep=8 and x/hstep=12. Figure 2a shows that the
90th percentile of the near bed velocity continues to increase
downstream. However, Fig. 2d shows that the 90th percentile in boundary
shear stress is as high or higher than farther downstream. Figure 2e
shows that the transport, similarly, is as large as farther
downstream, despite having a lower near-bed downstream
velocity. Figure 2b shows that the magnitude of the 10th percentile of
vertical velocity is larger in this zone than farther
downstream. These large negative vertical velocities bring high-momentum fluid into the bed, increasing the force on grains and
causing transport.
When a localized volume of fluid approaches and impinges on an
impermeable boundary, the boundary-normal velocity must stagnate, and
the fluid gets redirected to move parallel to the
wall. refer to these wall impingements as “splat
events”, and note the occurrence of splats near
flow reattachment in their simulations of turbulence over dunes. In
the simulations reported here, the bed is a permeable boundary, and
splats can penetrate the bed. To satisfy fluid continuity,
infiltration of the bed by a splat must be accompanied by
exfiltration of the bed surrounding the splat. Permeable splat events
are apparent near reattachment in Fig. 1d (areas of intense red and
blue) and the dynamics of the splat events are apparent in the
Supplement animation of Fig. 1. remarked that
significant entrainment of bed load grains occurs on the boundaries
between areas of bed infiltration and exfiltration.
The very large negative stresses in the recirculation region which
peak at about x/hstep=4 in Fig. 2d are due to a negative
mean vertical velocity, w‾, at the particle bed (Fig. 2b)
and the large negative vertical velocity fluctuations, w10. There
is a mean penetration of fluid into the bed, and there are permeable
splat events. The downstream fluid velocity is also negative in the
bed of particles due to the adverse pressure gradient. However, once
again, the drag forces produced on the grains in this region are more
broadly distributed through the bed, in contrast to the bed well
downstream of flow reattachment, where the boundary shear stress is
concentrated on only the topmost particles. This set of conditions
also explains why the mean transport rate and near-bed downstream velocity are negligible even though Fig. 2d shows that the mean boundary
shear stress is negative at the point of reattachment.
Flow and transport data binned by downstream and vertical
velocity fluctuation pairs, u′–w′. Data are
aggregated for all points downstream of x/hstep=12.5. (a) Bin counts normalized by the largest bin. The
total percentage of counts for each quadrant are
shown. (b) The sum of downstream sediment transport in
each bin, Σqsx, normalized by the bin with the largest
transport sum. The percentage of transport for each quadrant is
shown. (c) The sum of the downstream bed stress
fluctuation for each bin, normalized by the magnitude of the bin
with the largest magnitude. Percentages shown in each quadrant are
for the sum of the stress fluctuation, Στbx′,
divided by the total absolute deviation, Σ|τbx′|. (d) The sum of the fluid
pressure fluctuation for each bin, Σp′, normalized
by the magnitude of the bin with the largest magnitude. Percentages
shown in each graph are for the sum of the pressure fluctuation
divided by the total absolute pressure deviation, Σ|p′|.
Flow, velocity quadrant, and transport variables at a time
instant. (a) Downstream velocity at z=1mm as
shown in Fig. 1. (b) Downstream force, fx, on
particles. Particles with |fx|<2×10-6 N are not
shown. (c) Downstream particle velocity. Particles with
|U|< 0.007 ms-1 are not
shown. (d) Sweeps and outward interactions at z=5mm. Areas with |u′w′|<
0.0004 m2s-2 are not shown. (e) Bursts and
inward interactions at z=5mm. Areas with |u′w′|<0.0004m2s-2 are not
shown. (f) Near-bed pressure fluctuation at z=1mm as shown in Fig. 1. An animation
associated with this figure can be found in the Supplement.
Quadrant analysis
Recall that the simulation data were collected for u, w, qsx,
p, and τbx simultaneously at a horizontal grid of points. In
Fig. 4 all of the data were aggregated from all points downstream of
x/hstep=12.5. Figure 4a shows the frequency of
u′–w′ paired bins, and the predominance of burst
and sweep events is apparent. In Fig. 4b, qsx is summed for each
u′–w′ bin. The bins are then normalized by
dividing all bins by the bin with the maximum sum of
qsx. Figure 4b shows that most of the transport (about 80 %)
takes place during sweeps and outward interactions. This result is
consistent with . In Fig. 4c, τbx′ is
summed for each u′–w′ bin, and each bin is
normalized by the largest magnitude bin. Percentages for each quadrant
in Fig. 4c are given by the sum Στbx′ and divided
by the total deviation, Σ|τbx′|. Sweeps are associated with high bed stress, and bursts are
associated with low bed stress. The pressure deviation is summed in
each u′–w′ bin in Fig. 4d and normalized by the
magnitude of the bin with the largest magnitude. Percentages for each
quadrant are given by the sum Σp′ and divided by the
total deviation, Σ|p′|. Sweeps and outward
interactions are associated with high-pressure events and bursts and
inward interactions are associated with low-pressure events.
The spatial correlation between sweeps and outward interactions and
between bursts and inward interactions is apparent in Fig. 5. Areas of
the bed occupied predominately by sweeps and outward interactions
are also areas with high fluid pressure, large particle forces, and
large sediment fluxes. Conversely, bursts and inward interactions are
associated low pressure, small particle forces, and small sediment
fluxes. Sweeps and inward interactions occur together when a broad
volume of fluid moves toward the bed, bringing with it high downstream
velocity. Such a situation is apparent in Figs. 1 and 5 at
x/hstep≈17. When a broad sweep impinges on the
permeable bed, there is infiltration and exfiltration at spatial
scales smaller than the broader sweep structure. Areas of
exfiltration are apparent as outward interactions.
Downstream-elongated structures of high- and low-speed fluid begin to
emerge downstream of flow reattachment (Fig. 5a) (as also noted by
). These emerging streaks also produce streaks of high
particle forces (Fig. 4b) and particle motion (Fig. 4c).
Conclusions
Temporally averaged bed stress is not sufficient to
specify the rate of bed load transport downstream of separated flow
(compare Fig. 2d and e). Most of the transport takes place at high-stress events that are associated with both high downstream velocity
and high-magnitude vertical velocity events (Fig. 4b). The temporal
distribution of bed stress is broader near flow reattachment than
farther downstream (Fig. 4d), even though the temporal distribution of
near-bed downstream velocity is less broad near flow reattachment than
downstream. “Near-bed” and “in the bed” fluid velocities are
different. In this study near-bed was specified at z=1mm,
which is about one sand grain diameter above the top of the
bed. Negative vertical velocity events (splats) bring high downstream
momentum fluid into the bed, and those bed penetration events are
stronger near flow reattachment (Fig. 2b). Consequently, the 90th percentile of stress and the mean sediment flux reaches a peak in
a relatively short distance downstream of reattachment. This provides a probable explanation of the findings of
that instantaneous, near-bed downstream velocity was not sufficient to specify
the instantaneous sediment flux; the actual force on bed particles is also dependent
on the penetration of turbulence structures into the bed.
The upstream inclination of the stoss of bed forms, relative to the flat bed
considered here, is expected to increase the intensity of fluid
penetration events near flow reattachment.
The Supplement related to this article is available online at doi:10.5194/esurf-3-105-2015-supplement.
Acknowledgements
This material is based upon work supported by the National Science
Foundation under grant no. 1226288. All simulations were performed
at the Arizona State University Advanced Computing Center
(A2C2). Data produced in making this article can be obtained upon
email request to the author.
Edited by: D. Parsons
ReferencesBest, J.: The fluid dynamics of river
dunes: a review and some future research directions,
J. Geophys. Res.-Earth, 110, F04502, 10.1029/2004JF000218, 2005. Bogard, D. and
Tiederman, W.: Burst detection with single-point velocity
measurements, J. Fluid Mech., 162, 389–413, 1986.Chang, K. and
Constantinescu, G.: Coherent structures in flow over two-dimensional
dunes, Water Resour. Res., 49, 2446–2460, 10.1002/wrcr.20239, 2013.Chou, Y.-J. and
Fringer, O. B.: A model for the simulation of coupled flow-bed form
evolution in turbulent flows, J. Geophys. Res.-Oceans, 115, C10041,
10.1029/2010JC006103, 2010.Furbish, D. J. and Schmeeckle, M. W.: A probabilistic derivation of
the exponential-like distribution of bed load particle velocities,
Water Resour. Res., 49, 1537–1551, 10.1002/wrcr.20074, 2013.Giri, S. and Shimizu, Y.: Numerical computation of sand dune
migration with free surface flow, Water Resour. Res., 42, W10422,
10.1029/2005WR004588, 2006.
Grass, A. and Ayoub, R.: Bed load transport of fine sand by laminar
andturbulent flow, Coastal Eng. Proc., 1, 1589–1599, 1982.Grigoriadis, D. G. E., Balaras, E., and
Dimas, A. A.: Large-eddy simulations of unidirectional water flow
over dunes, J. Geophys. Res.-Earth, 114, F02022, 10.1029/2008JF001014, 2009.Kraft, S., Wang, Y., and Oberlack, M.: Large eddy simulation of
sediment deformation in a turbulent flow by means of level-set
method, J. Hydraul. Eng., 137, 1394–1405, 10.1061/(ASCE)HY.1943-7900.0000439, 2011. Le, H., Moin, P.,
and Kim, J.: Direct numerical simulation of turbulent flow over
a backward-facing step, J. Fluid Mech., 330, 349–374, 1997. Lu, S. and Willmarth, W.:
Measurements of the structure of the Reynolds stress in a turbulent
boundary layer, J. Fluid Mech., 60, 481–511, 1973.Nabi, M., de Vriend, H. J., Mosselman, E.,
Sloff, C. J., and Shimizu, Y.: Detailed simulation of
morphodynamics: 3. Ripples and dunes, Water Resour. Res., 49,
5930–5943, 10.1002/wrcr.20457, 2013.Nelson, J. M., Shreve, R. L., McLean, S. R.,
and Drake, T. G.: Role of near-bed turbulence structure in bed load
transport and bed form mechanics, Water Resour. Res., 31,
2071–2086, 10.1029/95WR00976, 1995. Nelson, J. M., Burman, A. R.,
Shimizu, Y., McLean, S. R., Shreve, R. L., and Schmeeckle, M.:
Computing flow and sediment transport over bedforms, in:
Proceedings of the 4th IAHR Symposium on River, Coastal and
Estuarine Morphodynamics, 4–7 October 2005, Urbana, Illinois, USA,
vol. 2, edited by: Parker, G. and Garcia, M., Taylor & Francis
Group, London, UK, 861–872, 2006. Nguyen, Q. and
Wells, J. C.: A numerical model to study bedform development in
hydraulically smooth turbulent flows, J. Hydraul. Eng., 53, 157–162, 2009.Niemann, S., Fredsøe, J., and
Jacobsen, N.: Sand dunes in steady flow at low froude numbers: dune
height evolution and flow resistance, J. Hydraul. Eng., 137, 5–14,
10.1061/(ASCE)HY.1943-7900.0000255, 2011.
Omidyeganeh, M. and Piomelli, U.: Large-eddy simulation of
two-dimensional dunes in a steady, unidirectional flow, J. Turbul., 12, 1–31, 2011.Paarlberg, A. J.,
Dohmen-Janssen, C. M., Hulscher, S. J. M. H., and Termes, P.:
Modeling river dune evolution using a parameterization of flow
separation, J. Geophys. Res.-Earth, 114, F01014, 10.1029/2007JF000910, 2009.Penko, A., Calantoni, J., Rodriguez-Abudo, S., Foster, D., and Slinn, D.:
Three-dimensional mixture simulations of flow over dynamic rippled beds,
J. Geophys. Res.-Oceans, 118, 1543–1555,
10.1002/jgrc.20120, 2013. Perot, B. and Moin, P.:
Shear-free turbulent boundary layers, Part 1. Physical insights into
near-wall turbulence, J. Fluid Mech., 295, 199–227, 1995.Schmeeckle, M. W.:
Numerical simulation of turbulence and sediment transport of medium
sand, J. Geophys. Res.-Earth, 119, 1240–1262, 10.1002/2013JF002911, 2014.
Shimizu, Y., Schmeeckle, M. W., Hoshi, K.,
and Tateya, K.: Numerical simulation of turbulence over
two-dimensional dunes, river, coastal and estuarine
morphodynamics, in: Proceedings International Association for
Hydraulic Research Symposium, Genova, Italy, 251–260, 1999. Shimizu, Y., Schmeeckle, M. W., and
Nelson, J. M.: Direct numerical simulation of turbulence over
two-dimensional dunes using CIP method, J. Hydrosci.
Hydraul. Eng., 19, 85–92, 2001.Stoesser, T., Braun, C.,
García-Villalba, M., and Rodi, W.: Turbulence structures in
flow over two-dimensional dunes, J. Hydraul. Eng., 134, 42–55,
10.1061/(ASCE)0733-9429(2008)134:1(42), 2008.Zedler, E. A. and
Street, R. L.: Large-eddy simulation of sediment transport:
currents over ripples, J. Hydraul. Eng., 127, 444–452,
10.1061/(ASCE)0733-9429(2001)127:6(444), 2001.