7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Sediment Transport and Dunes in Pipe Flow
Malika Ouriemii,* Julien Chauchatt,* Pascale Aussillous*,
Marc M6dale* and Elisabeth Guazzelli*
IUSTI CNRS UMR 6595 Polytech'Marseille AixMarseille Universit6 (Ul), France
t Laboratoire des Ecoulements Gdophysiques et Industriels, Universit6 Joseph Fourier, INPG, CNRS, Grenoble, France
t IFPLyon, Solaize, France
julien.chauchat@polytech.univmrs.fr and elisabeth.guazzelli@polytech.univmrs.fr
Keywords: Sediment transport, twophase model, granular rheology
Abstract
We propose a twophase model having a Newtonian rheology for the fluid phase and friction for the particle phase
to describe bedload transport in laminar pipe flows. This model is able to provide a description of the flow of the
mobile granular layer. When the flow is not significantly perturbed, simple analytical results of the particle flux
varying cubically with the Shields number can be obtained. This prediction compares favorably with experimental
observations of the timeevolution of the bed height in conditions of laminar flow. A simple linear stability analysis
based on this particle flux also accounts reasonably well for the dune formation observed.
Introduction
Figure 1: Bedload transport.
When particle beds are submitted to shearing flows, the
particles at the surface of the bed can move as soon as
hydrodynamic forces acting on them exceed a fraction of
their apparent weight. Bedload refers to the sediment in
transport that is carried by intermittent contact with the
streambed by rolling, sliding, and bouncing, as shown in
Fig. 1. This situation occurs in a wide variety of natural
phenomena, such as sediment transport in rivers or by air,
and in industrial processes, such as hydrate or sand issues
in oil production and granular transport in food or phar
maceutical industries. A very common feature that arises
is the formation of ripples, i.e. small waves on the bed
surface, or of dunes, i.e. larger mounds or ridges.
The widely recognized mechanism for dune or ripple
formation is the fluid inertia or more precisely the phase
lag between the bottom shear stress and the bed waviness
generated by the fluid inertia, see e.g. Charm & Hinch
(2006) and references therein. In that case, the shear stress,
the maxima of which are slightly shifted upstream of the
crests, drags the particles from the troughs up to the crests.
However, a complete description of the bed instability is
still lacking as the coupling between the granular media
and the fluid is poorly understood. Empirical and/or phe
nomenological laws relating the particle flux to the bottom
shear stress have been used in the literature.
The situation addressed here is that in which the bed
load can be considered as a mobile granular medium where
the particles mainly interact through contact forces. In the
past decade, advances has been made in the understand
ing of granular flows. In particular, it has been shown that
a simple theological description in terms of a friction co
efficient may be sufficient to capture the major properties
of granular flows (GDR Midi 2004). This description has
been found to be also successful when an interstitial fluid
is present (Cassar, Nicolas & Pouliquen 2005). We have
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
chosen to use this rheology to describe the granular phase.
We have used a twophase model having a Newtonian
rheology for the fluid phase and friction for the particu
late phase. The model equations have been solved nu
merically in onedimension and analytically in asymptotic
cases. This continuum approach is able to provide an on
set of motion for the particle phase and a description of the
flow of the mobile granular layer. At some distance from
threshold, we obtain the simpler analytical result of the
particle flux varying cubically with the Shields number.
This algebraic formulation seems quite satisfactory for
describing experimental observations of bedload trans
port in pipe flows (Ouriemi et al. 2009 Part 1). Based
on this particle flux, a simple linear stability analysis has
been performed to predict the threshold for dune forma
tion. This basic analysis accounts reasonably well for the
experimental observations for dune formation (Ouriemi et
al. 2009 Part 2).
The twophase model
We have chosen to use the standard twophase equations,
see e.g. Jackson (2000), and to propose some closures
appropriate to the present problem.
The equations of continuity for the fluid and the particle
phases are respectively
at xf)
a0t a('..
at Oxi
where ..f is the local mean fluid velocity, .., the local
mean particle velocity, p the particle volume fraction, and
e 1 p the void fraction or fraction of space occupied
by the fluid.
The momentum equations for the fluid and particle phases
are respectively
D (., 'f)
Pf Dt
P Dt
Oxj
i).ri
nfi + ., ,
+nf, + Oppgi,
Dxj
where gi is the specific gravity force vector, pf the fluid
density, pp the particle density, n the number density (num
ber of particles per unit volume). The force fi represents
the average value of the resultant force exerted by the fluid
on a particle. The stress tensors at and o may be re
garded as effective stress tensors associated with the fluid
and particle phases, respectively. We need some closure
for the interphase force and stresses of the two phases.
The interphase force can be decomposed into a gener
alized buoyancy force and a force which gathers all the
Figure 2: Sketch of a particle bed submitted to a
Poiseuille flow in a two dimensional channel.
remaining contributions (Jackson 2000). In the present
bedload case, the remaining contribution reduces to the
dominant viscous Darcy drag. The effective stress ten
sor associated with the fluid phase is supposed to be of
Newtonian form with an effective viscosity r, (that for
simplicity we can take as the Einstein viscosity) while
the stress tensor of the particle phase comes only from
direct particleparticle interactions and is described by a
Coulomb friction model where:
the tangential stress is proportional to the load (i.e.
the particle pressure pP) when the granular shear
rate is positive (i.e. is equal to ypp with a friction
coefficient p which mostly depends upon the parti
cle geometry and which is given by the tangent of
the angle of repose),
the tangential stress is indeterminate when the gran
ular shear rate is zero.
Bedload transport in pipe flow
In the calculation, we have considered a flat bed of thick
ness hp consisting of particles having a diameter d. This
bed is submitted to a stationary and uniform Poiseuille
flow in a two dimensional channel of thickness D, as de
picted in Fig. 2. The twophase equations presented above
are shown to reduce to the momentum equation for the
mixture (particles + fluid) and the Brinkman equation for
the fluid velocity. Calculations of bedload transport have
been performed numerically but also analytically at some
distance from threshold where the mobile layer is larger
than a grain size. The details of these calculations can be
found in Ouriemi et al. (2009 Part 1).
Here we present a much simpler calculation which gives
the flavor of the physical mechanisms involved. The mo
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
2 3 4 5 6
/(0oAP9 + ) xio3
Figure 3: Typical parabolic profile predicted inside the
bed.
mentum equation of the mixture can be written as
T(y) + Tf(y) = T(hp) (h y),
8 r
where Opf /Ox is the constant fluid pressure gradient driv
ing the Poiseuille flow. This equation shows that the mix
ture shear stress Tf +TP increases linearly with depth from
the surface value due to the horizontal pressure gradient
Opf /dx. It also describes the exchange between the shear
stress of the fluid phase Tf and that of the solid phase TP.
On the one hand, at the top of the granular bed hp, the
particle shear stress is zero and increases inside the bed
until it reaches ppP where the granular medium starts to
be sheared. The particle pressure is proportional to the
apparent weight of the solid phase
j"' =4oApg(hp Y),
and increases inside the bed. The particle shear stress can
keep the value ppP until it reaches y = he inside the
bed. On the other hand, at the top of the granular bed
hp, the fluid shear stress is equal to Tf(hp) and goes to
zero at he below which the granular medium is immobile
and behaves as a porous medium. Moreover, the Darcy
drag term is dominant in the Brinkman equation for the
fluid velocity. Therefore, inside the bed, there is very lit
tle slip between the two phases and both particle and fluid
phases move at the velocity of the mixture with the fol
lowing parabolic profile, see Fig. 3,
S (pp, g (y he)2
0,0 uAP + (  + h) (7)
TIe 2
0.01 L
0.1
1 10
Figure 4: Dimensionless kinematic velocity as a function
of the initial Shields number for different com
binations of particles and fluids. The solid line
shows the prediction c* (0) oc i oc 07/2. The
vertical dotted line shows the limit of this pre
diction for large 0.
10 .
100 1000 104
t"
Figure 5: Temporal evolution of the bed height for dif
ferent combinations of particles and fluids.
The solid lines represent the numerical inte
gration of equation (11). The timescale is
(6Re/Ga)1/2 (D/d)(re/Apgd) where Re is
the Reynolds number and Ga the Galileo num
ber Ga =pf A,,; ,il/q?2.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
The bedload thickness has a very simple linear vari
ation in Shield number 0 = 7f(hp)/Apgd (ratio of the
shear stress at the top of the bed to the apparent weight of
a single particle)
hp h,
Figure 6: Profile of the small dunes.
Od
Poo + f /Apg
Assuming that the critical Shields number for the onset of
grain motion corresponds to the thickness of the mobile
layer being half a particle, one has
0' z yo/2.
Finally, the particle flux can be expressed as
Figure 7: Side and top views of the vortex dunes.
qp/A,,,i
TIE
~24 O
where re is the effective viscosity of the fluid phase.
These predictions have been tested against experimen
tal measurements of bed profile evolution in a pipe flow.
The principle of the experiment is (i) to fill the pipe with
fluid and particles and to build an uniform flat bed, and
(ii) to record the evolution of the bed height as a function
of time using a laser sheet illumination for a given flow
rate. If we choose to be in the condition of bedload trans
port, i.e. above the critical Shields number, the bed height
is always seen to decrease with increasing time as the test
section is not fed in with particles. The indirect method
for measuring the flux of particles is to use the equation
for the conservation of particles
dOh dqp i~l.,
o + = 0, (11)
at I Fx
which can be written as a kinematic wave equation in
dimensionless form as a function of the nondimensional
Shields number, see details in Ouriemi et al. (2009 Part 1).
There is good agreement between the experimental ob
servations and the theoretical prediction based on the flux
of particles found earlier in the new twophase model.
First, the prediction 0' p=/o/2 with o = 0.55 in
the bulk of the bed and p 0.43 (Cassar, Nicolas &
Pouliquen 2005) agrees well with the experimental value
0.12 0.03 (Ouriemi et al. 2007). Second, the cubic
law for the particle flux seems satisfactory for describ
ing the velocity of the kinematic waves which are trig
gered at the entrance of the tube, see Fig. 4, as well as the
timeevolution of the bed height in conditions of bedload
transport for pipe flows, see Fig. 5.
Dune formation in pipe flow
Different dune patterns are observed as the flow rate is in
creased from the laminar to the turbulent regimes. Small
Figure 8: Top and side views of the sinuous dunes.
Figure 9: Destabilizing mechanism: The shear stress,
the maxima of which are slightly shifted up
stream of the crests, drags the particles from the
troughs up to the crests
hp(x)
/ > x
Figure 10: Stabilizing mechanism: Gravity force favors
particle downhill motion
dunes present small amplitudes and only exist in laminar
flow, see Fig. 6. Vortex dunes are characterized by the
existence of vortices at their front and are found either
in laminar or turbulent flow, see Fig. 7. Sinuous dunes,
showing a double periodicity, appear in turbulent flow,
see Fig. 8.
104
Re
1000 A
100
/ Rec
0 ,' I~X A Sinuous dunes
910 '* A Vortex dunes
I n, x 0 Small dunes
S/ Flatbed
1 ..... Model validity
/ 0 Flat bed
/ X No motion
/Oe
0.1
10 100 1000 104 105 106 107
G.11,!1 l, 2
Figure 11: Phase diagram of the dune patterns.
To predict the small dune formation, we have performed
a simple linear stability analysis where inertia in the fluid
produces a phaselag in the shear stress which is destabi
lizing, see Fig. 9, while the component of gravity down
an incline stabilizes the perturbations, see Fig. 10. Fol
lowing Charm & Hinch (2000), we first calculated the
perturbed fluid flow over a wavy bottom considered as if
fixed. Then, we used the particle flux found by Ouriemi et
al. (2009 Part 1) to relate the bed height evolution to the
shear stress at the top of the bed through the particle mass
conservation. The threshold for dune formation is found
to be controlled by the Reynolds number, see further de
tails in Ouriemi et al. (2009 Part 2).
Figure 11 presents the phase diagram of the dune pat
terns in the plane Repipe, Ga(hf/d)2. We choose this
plane to exhibit both the threshold for incipient particle
motion controlled by the Shields number and that for dune
instability predicted to be controlled by the Reynolds num
ber in the linear stability analysis. In this plane, the pre
dicted threshold for particle motion (0 0.12) is the
dashed line. The predicted instability threshold is the hor
izontal solid line. The dotted line indicates the domain
of validity of the algebraic law relating the dimensionless
particle flux to the Shields number found by Ouriemi et
al. (2009 Part 1) and thus indicates the domain of valid
ity of the instability threshold prediction of Ouriemi et al.
(2009 Part 2).
The three regimes of 'no motion', 'flat bed in motion',
and 'small dunes' are well delineated by these boundaries
in the given limit of validity. Clearly, the threshold for
incipient particle motion and that for small dune instabil
ity are observed to differ as there is a large region of 'flat
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
bed in motion' without any dune formation. The thresh
old prediction of the simple linear stability with a single
adjustable parameter that we have taken to be realistic,
Rec oc 1/p z 37.5 with p 0.43 (Cassar, Nicolas
& Pouliquen 2005), provides a correct boundary for the
'small dune' instability. Furthermore, the regimes of 'vor
tex dunes' and 'sinuous dunes' seem separated and their
thresholds also well described by the Reynolds number of
the pipe as a control parameter.
Conclusions
In summary, we have proposed a twophase model to de
scribe bedload transport in the laminar viscous regime,
i.e. the flux of particles in a flat mobile bed submitted
to laminar flows. The fluid phase has been assumed to
be a Newtonian viscous liquid with Einstein dilute vis
cosity formula applied to the concentrated situation while
the particle phase to have Coulomb solid friction with the
shear stress proportional to the pressure.
We have applied this continuum model to bedload trans
port in pipe flows. The relevant equations have been found
to be the Brinkman equation for the fluid phase and the
momentum balance equation for the mixture. They have
been solved numerically but also analytically. A very sim
ple analytical model, valid when the Poiseuille flow is not
significantly perturbed, finds that the bedload thickness
varies linearly with the Shields number whereas the parti
cle flux cubically with it.
We have compared these predictions with experimental
observations in pipe flow. The cubic law for the particle
flux seems satisfactory for describing the velocity of the
kinematic waves which are triggered at the entrance of
the tube as well as the timeevolution of the bed height in
conditions of bedload transport for pipe flows.
We have also examined the different dune patterns which
are observed when a bed composed of spherical particles
is submitted to a shearing flow in a pipe. 'Small dunes'
present small amplitudes and only exist in laminar flow.
'Vortex dunes' are characterized by the existence of vor
tices at their front and are found either in laminar or turbu
lent flow. 'Sinuous dunes', showing a double periodicity,
appear in turbulent flow. While the threshold for incipient
motion is determined by the Shields number, that for dune
formation seems to be described by the Reynolds number.
To predict the small dune formation, we have performed
a simple linear stability analysis. This analysis containing
the basic ingredient of the destabilizing fluid inertia and
stabilizing gravity is found sufficient to provide realistic
predictions.
The twophase nature of the problem has been only ac
counted for in the particle conservation equation in the
present study and thus a full twophase analysis should be
performed in the future, in particular through full three
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
dimensional computations as undertaken in Chauchat &
M6dale (2010) and in the accompanying paper (Chauchat
etal. 2010).
Acknowledgement
Funding from the Institut Francais du P6trole and Agence
National de la Recherche (Project Dunes ANR073_18
3892) are gratefully acknowledged.
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