Annual Journal of Hydraulic Engineering, JSCE, Vol.52, 2008, February
Annual Journal of Hydraulic Engineering, JSCE, Vol.52, 2008, February
NUMERICAL MODELLING OF BEDFORM
DEVELOPMENT; FORMATION OF
SAND-WAVELETS IN HYDRODYNAMICALLY
SMOOTH FLOW OVER AN ERODIBLE BED
Y Q. NGUYEN1, John C. WELLS2
1
Graduate student, 2Member of JSCE, Dr. of Eng., Associate Professor
Department of Civil & Environmental Engineering, Ritsumeikan University (1-1-1 Nojihigashi, Kusatsu, Shiga, Japan)
Formation of the initial bedforms, known as sand-wavelets, developing from a flat erodible bed in
hydrodynamically smooth flow has been studied numerically. The computational model consists of
hydrodynamic and sediment transport equations. The unsteady hydrodynamic equations are solved by
Large-Eddy-Simulation (LES) and Immersed-Boundary-Method (IBM) on a non-bodyfitted grid. Dynamics
of individual grains are neglected and only bedload is considered, as calculated from Van Rijn’s equation
(1984, Journal of Hydraulic Engineering, Vol. 110). Two-dimensional (2D) bed evolution is coupled with
the above 3-dimensional (3D) hydrodynamic equations. The computed wavelength agrees with linear
stability analysis and with experimental data in the literature. The length of the sand-wavelets is observed to
correspond to the disturbance region of bed shear stress caused by discontinuities on the bed surface, as
postulated by Yalin (1977, Mechanics of Sediment Transport, Pergamon press).
Keywords: numerical simulation, bedform, sand-wavelets, bedload transport.
1. INTRODUCTION
When an erodible riverbed or seabed is exposed to a
current, the bed usually becomes unstable and
bedforms appear1),2). Coleman & Melville3) and
Coleman et al4) reported that the first sand-waves
observed on a flat bed are instigated by random pileups
of sediment. When the height of these pileups is about
that of the bed roughness, small sand-waves of very
low height but finite wavelength are formed. They
defined these nascent sand-waves as sand-wavelets;
such structures grow further in time to mature ripples
or dunes depending on the flow characteristics.
Various trends for the length of sand-wavelets have
been reported. Coleman et al4) showed from their
experimental results and data in the literature, the
sand-wavelet’s wavelength is scaled only with the
grainsize, nearly independent of the flow conditions
for laminar, transitional and turbulent flows with
particle Reynolds number up to 50. By fitting all the
data, they proposed a relation between wavelet length
and grainsize as 175 .
, where λ and d are the
wavelength and mean grainsize, respectively, in
millimeters. In contrast, Langlois & Valance5) reported
that the wavelength is roughly independent of
grainsize for particle Reynolds number in their
experimental range of 1-14 in turbulent flows. Kuru et
al6) concluded that the wavelength increases with an
increase in the flow rate of the carrier fluid in the
transitional flows.
To re-examine these different conclusions for
turbulent flows, the test cases of Coleman & Melville3),
Coleman et al4), Langlois & Valance5), and Kuru et al6)
have been re-plotted (see Fig. 3). With this
arrangement,
the
wavelength
when
non-dimensionalized by the grainsize is found to be a
function of grain Reynolds number. For particle
Reynolds number above about 4, all data sets show a
wavelength around 200. By contrast, for the lower
range of the Reynolds number, there is a strong
discrepancy between the results of Coleman et al4) and
Langlois and Valance5) and a strong scatter of Kuru et
al’s6) results, and little trend is apparent.
Aiming at obtaining a clear trend of variation of
sand-wavelet’s wavelength for grain Reynolds number
smaller than 3, i.e. the regime of hydrodynamically
smooth flows, the formation of initial bedforms is
numerically studied in this paper. With numerical
simulation, we also hope to get more understanding of
the interaction between the bedform and the flow
during the initial stage of the bedform as described by
Yalin1),7) (see Section 3).
The computational method consists of a model for
fluid flow and a model for the sediment transport. The
fluid flow is solved with unsteady 3D
Large-Eddy-Simulation (LES) method while the
deformed
boundary
is
treated
with
the
Immersed-Boundary-Method (IBM) on a fixed
- 163 -
Cartesian non-bodyfitted grid. The model for sediment
transport includes equations of sediment flux and
equations of bed profile evolution in 2D which is
coupled with the 3D LES equations.
By employing the IBM, remeshing to fit the bed
surface when it evolves in time is not required.
Moreover, Cartesian solvers are used instead of
body-fitted solvers, yielding better cost-efficiency8).
As observed in the experiments3),4), the
sand-wavelets are instigated by very small
disturbances on the bed surface which grow in time,
interact with the flow, and cause the local shear stress
distribution to deviate from the standard one1),7).
Therefore, to model correct behavior of the bedform at
this initial stage, the flow solver must accurately
reproduce local turbulence, particularly bed shear
stress fluctuations around the disturbances.
Among available methods which explicitly represent
the turbulence, hence can satisfy the above
requirements, LES and RANS (Reynolds-Averaged
Navier-Stokes equations) can be considered as
practical candidates9). Between the two, Keylock et al9)
showed that LES is preferable for this kind of problem
since most RANS models are intended for accurate
representations of the mean flow field only. For
example, Chang & Scotti10) , comparing LES and with
the ω RANS model for separating flows over
ripples,
reported
that
RANS
substantially
underpredicted Reynolds stress and overpredicted
vertical velocity, while LES agreed very well with
DNS and experiment, even in the nonseparating
regions. The present flow is non-separating, and with
suitable tuning, it is possible that RANS could yield
performance that is similar to LES but at much lower
cost. If the present model were intended as a design
tool, the RANS model would then be clearly
preferable. However, our purpose here is to compute a
highly reliable turbulent flow field for the physical
(mechanistic) study of wavelet formation, and we
believe LES/IBM to be clearly more credible for such
a purpose. Furthermore, it is planned to proceed to
study later stages of bedform development in which
flow separation, and 3D beform formation, occurs.
Clearly, it will be simpler and more credible to use a
common CFD method, namely LES/IBM, that allows
comparative analysis of early versus late stages of
bedform formation.
In the LES method, the flow equations have to be
solved in 3D. A 2D model for the bed evolution
facilitates the computation considerably, and should be
sufficient because it is observed in experiments3)5) that
the sand-wavelets at low particle Reynolds number are
essentially two-dimensional.
To the authors’ knowledge, this is the first
numerical study on the initiation of sandwave from a
flat bed with a highly reliable solution of the turbulent
flow field.
2. NUMERICAL METHOD
(1) LES and IBM method for unsteady
hydrodynamic equations
The governing equations for LES are
non-dimensionalized by the total flow depth H and
mean friction velocity :
0
(1a)
% !"#$ &'
(1b)
,
where (̃ ' is the subgrid stress; *+ .- is
the flow Reynolds number based on the mean friction
velocity and fluid viscosity ν; &' is the artificial
body-force accounting for the presence of solid bodies.
/0 and /̃ indicate filtered and non-dimensionalized
quantities, respectively.
The subgrid stress model is the Shear-Improved
Smagorinsky model proposed by Leveque et al11):
0
(̃ ' 2' (̃ 33 256 7'
(2)
1
:, <̃= 8>? ∆=A 8B7089
:, <̃=B BC7089
:, <̃=DB=,
With 56 89
0 89
:, <̃= E 89
:, <̃= 89
:, <̃=F, and
7'
A
0 7'
0 =
:, <̃=B 827'
B7089
.A
.
where Cs=0.16 and ∆ G∆ ∆H ∆I J 1 . Here,
K, KH, KI are local grid spacings in the stream x,
vertical y, span z directions, respectively. C. D denotes
ensemble average which is evaluated along the
homogenous, z, direction. This model was selected
because of its simplicity, gentle restriction on time
step, moderate computational cost, with performance
that is comparable to the dynamic Smagorinsky
model11).
The artificial body-force is evaluated by the direct
forcing method12),13). Eq. (1b) is time-discretized as:
MN
M
'
' K<G*,7 &' J
(3)
where K< is the computational time step; RHS
contains the advection, pressure, and viscous term. To
impose the desired value OP ' within the solid body,
the artificial body-force should be:
QR S
&' *,7 (4)
T
inside the flow region occupied by the solid body and
zero elsewhere. Usually, the solid surface does not
coincide with the grid points. In this case, &' at the
grid point closest to the solid surface but inside the
solid body is linearly interpolated between the value
that would yield OP ' inside the solid body and the
value of zero inside the fluid portion. &' is evaluated
at every time step. This method is shown to have
2nd-order accuracy, and the condition for pressure is
satisfied automatically on the solid surface.
Eq.(1-2) are solved by a finite difference method on
a fixed Cartesian, staggered non-bodyfitted grid with
4th order central discretization in space and 2nd order
- 164 -
is approximately four times denser in x direction than
that of the DNS case (64×124×64). However, Balaras8)
showed that, even though the IBM method requires
higher grid resolution than the body-fitted method,
using the IBM with a Cartesian solver is still more
cost-efficient in computational time.
The LES and DNS have the same periodic boundary
conditions in the stream x and span z directions, and
no-slip condition on the lower surface while they have
different boundary condition on the upper surface: free
slip in the LES+IBM case, to approximate the free
surface, and rigid in the DNS case. The bulk Reynolds
number, Re=Ub*(H-α)/ν, where Ub is the bulk
velocity; H is the total flow depth; α is the wave
amplitude, in both cases are matched to about 4300.
All the compared quantities have good agreements
on the lower boundary, and because of different
boundary conditions on the upper surface, they differ
slightly far away from the lower wall. Grid refinement
tests of the LES code have been done with two
different grid resolutions and excellent agreements
have been observed. The agreements between the
LES+IBM and the DNS results show the accuracy of
the present hydrodynamic model.
Fig.1 Comparison between the present LES results with DNS data
by Ohta et al15). From top to bottom: mean streamwise
velocity, mean vertical velocity, Reynolds stress, turbulent
kinetic energy scaled by the bulk velocity Ub; (○) DNS, (---)
LES (128×72×64), (━)LES (256×128×64). Flow from left to
right.
Wave
amplitude
α=0.05*(H-α).
Wave
length=3.84*(H-α).
Adams-Bashforth method for time marching14). The
grid is uniform in the stream x and span z directions.
On the vertical y direction, grid is equal-spaced until
the highest point of the solid surface and then a
hypertangential distribution is used up to the upper
boundary. The boundary conditions are periodic in the
x and z directions, and non-slip and free-slip on the
lower and upper surface, respectively.
To validate the LES+IBM code, a test case of flow
over a sinusoidal bed has been conducted and the
results compared to DNS data by Ohta et al15) as
shown in Fig. 1. The computational domain in the
present simulation is one wavelength in the stream
direction, while it is four wavelengths in the DNS case,
and half a wavelength in the span direction, which is
the same as the DNS case. The grid resolution, over
one wavelength, in the present simulation (256×128×64)
(2) Model for bedload transport
For the present range of particle Reynolds number,
Rep<3, the viscosity influence is large, dynamics of
individual particles can be neglected and the granular
medium can be treated as a continuum1). Therefore,
change of bed profile is corresponding to shear stress
distribution on the bed surface, which determines the
sediment flux. Moreover, Coleman & Melville3),
Coleman et al4), and Langlois & Valance5) reported that
formation of sand-wavelets is mainly due to bedload
transport. Therefore, only bedload transport is included
in the model. The bedload equation by Van Rijn16) has
been selected because it takes particle Reynolds
number into account and covers the range of Rep≤5.
For consistency with the hydrodynamic equations,
equations in this section are also non-dimensionalized
with total flow depth H and mean friction velocity .
Van Rijn16) employed two non-dimensional
parameters. The particle parameter is
U V *+XA Y
W$
.1
\ A
W$ ]
A
Z [ ^ *+
_
.1
(5a)
with *+X ⁄- the particle Reynolds number,
and the transport stage parameter is
b
a 1.0
(5b)
bc
with de the critical Shields parameters, and
# A
^ d
#$
(5c)
.A
\ A
6 %.j
Z [ ^ _ 0.053 $.k
\U
W$ ]
(5d)
%
d 8?S #=f\ [
is the local Shields parameter obtained from local
friction velocity uτ. Then his equilibrium bedload flux
equation becomes:
- 165 -
g"h Gravity effects are accounted as2),19):
ye ye E1 25
λ
15
10
5
0
*+ 1.5 0.5 -0.5 -1.5 -2.5
Fig.2 Sand-wavelet developing from plane bed with an initial
perturbation of one sinusoidal wave for the test case of
Rep=1.0; Flow from left to right. From bottom: t̃=0.0, 4.5,
8.5, 12.5… the sand-wavelet is marked with the dash
rectangular at t̃=16.5 with length, indicated by the arrows, of
λ/d≃420. The dashed line highlights the development path
of the sandwave. Vertical dimensions are stretched for
legibility. At left, time development of y+max and y+min around
the sandwave is shown.
%
#$
8?S =f\
is the mean Shields number, s is
the ratio of grain density to fluid density, g is the
gravitational acceleration.
It is assumed that the local bedload flux does not
adapt instantaneously to the local bed shear stress17),
hence there is a lag time, or equivalently, a lag distance
between the two. This lag distance is shown to play a
central role on formations of sandwaves on the bed17).
The local, non-equilibrium, or lagged, bedload flux,g
is obtained via the lag distance, or the non-equilibrium
adaption-length m"h , through a relaxation law18),19):
h
n
hShop
qop
(6a)
Following Philips & Sutherland18) and Bui et al19),
the non-equilibrium adaption-length m"h is taken as
the saltation length proposed by Van Rijn16):
m"h 3[.,^U.r a .s
(6b)
Variation of bed profile is calculated by
Exner-Polya equation1):
81 t=
u
vh
vn
(8)
u
g"h g"h [1.0 ^
(9)
ye : critical Shields parameter for zero-slope bed;
determined from White’s experiment20), and }? :
friction angle, average value of tan}? 0.63 as
quoted in Richards21).
Eq. (6a) and (7) are also solved by the finite
difference method on the same grid as Eq.(1-2) with
2nd order accuracy in space and time.
It is assumed that the time scale of the flow
development is much smaller than that of the bedform
development21). Moreover, for Rep≤2.5, the bed can be
treated as smooth, fixed one when the hydrodynamic
equations, Eq.(1a,1b) are solved1),7)22). The local friction
velocity is then calculated from the shear stress, as
averaged in the span z direction during an interval
∆a of 50 or 100 LES time steps.
20
where d .
u
F
zM{|
(7)
Where w is the bed elevation; n is the bed porosity,
taken to be 0 in this model, since the granular medium
is treated as a continuum one; and x is the bed surface
tangential direction.
# ]

‚*+
ƒ
:
„
(10)
where n is the velocity component along the bed
surface; … is the surface-normal direction. Eq. (10) is
:
„
averaged from …N #$ 2 † 4

After the Eq.(1-2) are solved and the bed shear stress
is obtained by Eq.(10), the bedload flux is computed
by Eq. (5,6,8,9) and bed profile is updated by Eq.(7).
At the next step, the Eq.(1-2) are solved over the new
bed profile, and the process is repeated until the
desired bedform (sand-wavelet) has been observed.
The interval ∆a is necessary to assure that the flow
have enough time to adapt to the new bed profile.
3. RESULTS AND DISCUSSIONS
Fig. 2 shows an example of the evolution of the
initial bed profile with one sinusoidal wave occupying
half of the domain, and the maximum and minimum
bed elevation. From the crest of the initial wave, a
small bump appears and grows in height, while its two
ends are not clear until the height saturates at a certain
value. The wave crest becomes longer in time until the
downstream end of the wave starts being eroded
strongly, with the scoured sediment deposited further
downstream to form a new wave. At this instant, two
troughs fore and aft of the wave can be specified, and
the wave is identified as sand-wavelet, consistent with
the wave-detection method by Coleman et al4). The
trough-to-trough length, i.e. wavelength λ indicated by
arrows in Fig.2, of this sand-wavelet is found to be
independent of the wavelength and height of the initial
sinusoidal wave on the initial bed profile provided that
the disturbance produced by the initial bed profile is
strong enough to help the bedform evolve.
Due to the high computational cost of the 3D
hydrodynamic equations, the computational domain is
limited. Accordingly, further development of the
- 166 -
Fig.3 Computed wavelets’ wavelengths vs. reported experimental
data for turbulent flow.
bedform has not been computed.
The top of the wave is seen to be quite symmetric
fore-and-aft, which is similar to the bed profile in
Figure 4 of Coleman et al4) for runs with Rep=1.5→
2.1.
Tests with different values of the mean Shields
number d show that the wavelength at a specific
value of Rep is independent of d ; Higher d only
shortens the time for the sand-wavelet to appear. In
addition, as in the case of ripples1), formation of
sand-wavelets should not be dependent on the flow
depth which, however, appears in the bedload
equations through the non-dimensionalizing process.
However, changing H does not change the results
significantly. For example, at Rep=1.0, tests with
H/d=300 and 500 yield λ/d=419 and 428, respectively.
The simulations were run with values of Rep from
0.5 to 2.5 and the wavelengths of the sand-wavelets
obtained at each Rep are plotted in Fig. 3, together with
the reported experimental ones. That the computed
points are distributing on the lower limit of the
experimental ones shows the agreement between them.
In the simulation results, the sand-wavelets are caught
when they just appear, hence their wavelength should
be corresponding to the smallest observed in the
experiments. In experiments, the wavelength is
averaged in the whole observed field. Therefore, it is
possible that second-generation waves (induced by the
first waves downstream) are also taken into account,
and the averaged wavelength may be greater than that
of the first-generation sand-wavelets that are obtained
by simulation.
Sumer et al22) used linear stability analysis,
consisting of a linearized model for hydrodynamic
equations and Bagnold’s bedload formula for sediment
transport, in the hydrodynamically smooth regime
(Rep<4.0) and found that the fastest growing
wavelength is dependent on the friction angle }? :
220 ˆ ⁄- ˆ 610 for tan}? 0.32 † 0.75 and
for the average value of tan}? 0.63,
‰#$
‰
A1
273 Š‹ (11)

\
!"Œ
As shown in Fig.3, the
wavelength can be fitted as:
Ž
‰

\
!"Œ
present
computed
(12)
Obviously, from Eq. (11) and (12), the results from
Fig.4 Distribution of friction velocity (above) around small bumps
(below). Bulk Reynolds number Re=5000.
the two methods differ only by the prefactor. Since we
are at present short of factors that determine }? , such
as the particle properties, the flow properties, tests with
other values of }? have not been done.
Yalin7) gives an empirical formula for developed
ripple for Rep ≤2.5:
‰
AA

(13)
\
!"Œ
Since Eq.(13)= constant×Eq.(12), it seems likely
that the formation of the sand-wavelets (initial
bedform) and the ripples (mature bedform) should be
related to similar origins in this range of Rep.
According to Yalin1),7), one possible origin is the
instability of the bed surface. Incidental discontinuities
on the bed surface can cause the derivative of bed
shear stress (⁄ , hence g ⁄ to deviate from
zero and the bed starts growing in time following Eq.
(7). This hypothesis is consistent with the observations
by Coleman & Melville3) and Coleman et al4) that the
sand-wavelets are instigated by random pileups of
sediment (see Section 1).
In the present simulations, the height of the first
waves when its top stops growing is wN w ⁄- 1.5 † 2.0 . This wave can be considered as the
‘discontinuity’ mentioned by Yalin1),7). To understand
effects of such a discontinuity on the bed shear stress
distribution, flows over bumps on a flat bed with
heights wN 1.0 2.0 and with different bump
lengths were solved with the LES+IBM method,
Eq.(1-2), and are shown in Fig. 4.
As expected, the peak of the friction velocity
distributions are always before the peak of the bumps,
i.e. there is a phase lag between the two. Although the
bump does not cause flow separation, i.e. the friction
velocity is always positive, there is a region, or
‘shadow’ with length Lx+, where the bed shear stress
deviates from the standard value. Interestingly, Lx+ is
almost independent of the bump size, and is about 300
which is similar to the sand-wavelet’s wavelength, Eq.
(12). Therefore, the sand-wavelets appear to be formed
adapting to the length of this ‘shadow’, and hence,
- 167 -
Yalin’s1),7) postulation is confirmed. Note that the
‘shadow’ length is from the top of the bump while the
wavelength is measured on the whole wave.
In Fig. 4, the longest bump has the gentlest gradient
of the friction velocity on the top but the highest
gradient at the downstream-end point. This may
explain for observations in the simulations as
presented in Fig. 2. Initially, disturbances caused by
the initial bed profile are responsible for the
appearance of the first small wave. This wave
continues growing in both height and length because of
sharp shear stress gradient on its top and downstream
side, similar to the smallest bump in Fig. 4. When its
length becomes larger, the shear stress gradient on its
top becomes gentler until is not strong enough to help
the wave gain in height while it can still gain in length
because the downstream shear stress gradient is
becoming stronger. When the wavelength is equivalent
to that of the ‘shadow’, the shear stress gradient at the
downstream end of the wave is strongest, and causes
the strong erosion which help to identify the
sand-wavelet.
4. SUMMARY
Formation of sand-wavelets in hydrodynamically
smooth flow over an erodible bed has been studied
numerically. The non-dimensional wavelengths vary as
a function of particle Reynolds number as ⁄ 400/*+X up to Rep=2.5. Agreement with
experimental and analytical results in the literature has
been obtained. Similar to the case of ripples, the
sand-wavelets are seen to develop adapting to the
disturbance of the bed shear stress caused by the
discontinuities on the bed surface, which is consistent
with the postulation by Yalin1),7) and the observations
by Coleman & Melville3) and Coleman et al4).
ACKNOWLEDGMENT: We are grateful for support
from the Monbukagakusho Ministry for a scholarship
for the first author. Dr. Dinh Xuan Thien at CFD
laboratory, Ritsumeikan University is appreciated for
giving the initial LES code and for helpful discussions.
Professor Y. Shimizu and Dr. S. Giri at Hokkaido
University are also thanked for useful discussions. The
anonymous reviewers are grateful for helping to
improve the quality of this paper.
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(Received September 30, 2007)