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 Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings Title: 5.4.2 - A 3D Two-Phase Numerical Model for Sediment Transport
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 Material Information Title: 5.4.2 - A 3D Two-Phase Numerical Model for Sediment Transport Computational Techniques for Multiphase Flows Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings Physical Description: Conference Papers Creator: Chauchat, J.Ouriemi, M.Aussillous, P.Médale, M.Guazzelli, E. Publisher: International Conference on Multiphase Flow (ICMF) Publication Date: June 4, 2010
 Subjects Subject: sediment transporttwo-phase modelgranular rheologynumerical model
 Notes Abstract: We have developed a three dimensional numerical model based on the two-phase equations to study the bed-load transport Chauchat and Médale (2009). We have considered two formulations of the model based on a two fluid or a single mixed-fluid description. The governing equations are discretized by a finite element method and a penalisation method is introduced to cope with the incompressibility constraint whereas a regularisation technique is used to deal with the visco-plastic behaviour of the granular phase. The accuracy and efficiency of the numerical models have been compared with the analytical solution of Ouriemi et al. (2009a). It turns out that one must takes a smaller regularisation parameter (one order of magnitude) in the mixed-model than in the two-fluid one for a comparable accuracy. Using an Arc Length Continuation algorithm coupled with this model we have investigated the evolution of the solution in terms of the height of the flowing granular layer and the particle flux against the longitudinal pressure gradient. The results of these simulations are in good agreement with the analytical solution in the 2D case and three-dimensional computations have been carried out in a square and circular cross-section ducts showing that the effect of the geometry is non-trivial. General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
 Record Information Bibliographic ID: UF00102023 Volume ID: VID00129 Source Institution: University of Florida Holding Location: University of Florida Rights Management: All rights reserved by the source institution and holding location. Resource Identifier: 542-Chauchat-ICMF2010.pdf

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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010

A 3D Two-Phase Numerical Model for Sediment Transport

Julien Chauchatt,* Malika OuriemiP,* Pascale Aussillous*,

Marc M6dale* and Elisabeth Guazzelli*
IUSTI CNRS UMR 6595 Polytech'Marseille Aix-Marseille Universit6 (Ul), France
t Laboratoire des Ecoulements G6ophysiques et Industriels, Universit6 Joseph Fourier, INPG, CNRS, Grenoble, France
SIFP-Lyon, Solaize, France
julien.chauchat@polytech.univ-mrs.fr and elisabeth.guazzelli@polytech.univ-mrs.fr
Keywords: Sediment transport, two-phase model, granular rheology, numerical model

Abstract

We have developed a three dimensional numerical model based on the two-phase equations to study the bed-load
transport ( I.iiiIli.ii and M6dale (2009). We have considered two formulations of the model based on a two fluid or a
single mixed-fluid description. The governing equations are discretized by a finite element method and a penalisation
method is introduced to cope with the incompressibility constraint whereas a regularisation technique is used to deal
with the visco-plastic behaviour of the granular phase. The accuracy and efficiency of the numerical models have
been compared with the analytical solution of Ouriemi et al. (2009a). It turns out that one must takes a smaller
regularisation parameter (one order of magnitude) in the mixed-model than in the two-fluid one for a comparable
accuracy. Using an Arc Length Continuation algorithm coupled with this model we have investigated the evolution of
the solution in terms of the height of the flowing granular layer and the particle flux against the longitudinal pressure
gradient. The results of these simulations are in good agreement with the analytical solution in the 2D case and
three-dimensional computations have been carried out in a square and circular cross-section ducts showing that the
effect of the geometry is non-trivial.

Introduction

The transport of sediment or more generally the trans-
port of particles by a fluid flow is a problem of major
importance in geophysical flows such as coastal or river
morphodynamic or in industrial flows with the hydrate
or sand issues in oil production and granular transport in
food or pharmaceutical industries. This problem has been
extensively studied in the literature since the middle of the
twentieth century but poorly understood actually (Einstein
1942; Meyer-Peter and Muller 1948; Bagnold 1956; Yalin
1963).
Recently, Ouriemi et al. 12', a ,., have proposed a two-
phase model describing the bed-load transport in lami-
nar flows that allows to incorporate more physics than
in previous modelling based on particle flux or erosion
deposition approaches. This two-phase model is based
on a Newtonian rheology for the fluid phase and a fric-
tional rheology for the particulate phase (p(I) Forterre
and Pouliquen (2008) or Coulomb friction) while the fluid-
particle interaction is assumed to follow a Darcy law. This

approach allows to predict the threshold of motion for the
particle phase and give a description of the flow of inside
the flowing granular layer. Away from the threshold of
motion, a simpler analytical model for the particle flux
is obtained which gives a quite satisfactory description of
experimental observations of bed-load transport in pipe
flows Ouriemi et al. (2009a).
Based on this theoretical model we have developed a
3D numerical model that allows to simulate bed-load trans-
port in 2D or 3D configurations (( Ii.iiili.ii and M6dale
2009). It is restricted to the cases where the granular bed
does not change its shape in the course of time, conse-
quently ripples and dunes formation are beyond the scope
of this paper. We consider a mixed-fluid and a two-fluid
formulation, the models equations are discretized by a fi-
nite element method and a penalisation method is used to
impose the incompressibility constraint. The granular rhe-
ology is analoguous to a viscoplastic behaviour especially
by the existence of a yield stress with the particularity that
this yield stress depends on the depth inside the granular
layer. A regularisation technique is used to deal with the

viscoplastic rheology and is shown to give satisfactory re-
sults.

Two-phase model

The present model is based on Jackson (2000, 1997) aver-
aged equations using the closures developed by Ouriemi
et al. (2009a). These equations are summarized hereafter
in dimensionless form using the following scaling: the
length is scaled by H, the channel height (see figure 1),
and the stresses is scaled by ApgH, and therefore the time
is scaled by l/ApgH where Ap pp pf. The problem
is expressed in terms of the solid volume fraction y, the
mixture velocity u" and the particulate velocity up.

Two-fluid model

V. (u"

V. up

H3 DD

H3 DuD
Ga RpD
d3 DL

Vp + V ||
VP V.

H 2 Up
-K +A 1Ap1 7|

-Vp Vp7f

+V. 7 + v. )

(1 _- )H7-2 t \$
K V II |
(1)
In these equations, k Vu + (V.. F with k = m
or k p, R, pf/pp represents the density ratio and
Ga = d3pfApg/r2 is the Galileo number where d is the
particle diameter. The Galileo number is a Reynolds num-
ber based on the settling velocity of particles.

Mixed-fluid model

1 R3 )D uL
C., H 1+ R) D
D

-Vp P V pp + PM
Ap II ( II

+V. T I +

7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010

with ps the internal friction coefficient and the permeabil-
ity K -(1 )3d2
ity K = with kOK w 180 (Happel and Brenner
1973).

Numerical model

The Finite Element Method (FEM) is based on the dis-
cretisation of the variational formulation associated with
equations (1) for the two-fluid model and on equation (2)
for the mixed-fluid model.
The two-fluid formulation is based on the solution of
the system (1) in a weakly coupled way. The algorithm
associated with the mixed-fluid formulation is based on
the solution of the system (2). The non-linearities in the
governing equations are solved by a Newton-Raphson al-
gorithm by taking the first variation of the variational for-
mulations.
The specific issue raised by the previous two-phase model
lie in the calculation of the frictional stress. Following
(Jop et al. 2006), the frictional stress can be written as
TP = lP

with I. = i'/I P |. The particulate viscosity di-
verges as the particulate shear rate tends toward zero (i.e.:
in the static zone) raises obvious numerical problems. The
basic idea to overcome this issue consists in regularizing
the viscosity by adding a small quantity (A) to the denom-
inator of the particulate viscosity ,. -' 1'/( +A)
then the divergence is controlled by this parameter and the
viscosity is kept finite. In other words, the static zone in
the frictional rheology is replaced by a very viscous zone.
In the next section of this paper we will discuss the influ-
ence of the value of the parameter A on the model solution.
In our implementation, we use piecewise quadratic poly-
nomial approximation for the velocity and piecewise lin-
ear discontinuous approximation for the pressure. In the
computations, we have employed a 27-nodes hexahedra
element (H27) for the velocities. The incompressibility
constraint is solved by a penalisation method. The code
is developed with the PETSc library Balay et al. (2001,
2004, 1997) which provides several parallel iterative and
direct solvers. As we use a penalisation method to cope
with the incompressibility constraint, all the algebraic sys-
tems have been solved by the MUMPS direct solver Amestoy
et al. (2000, 2001, 2006) with a penalty parameter set to
109 for all the simulations presented in this paper.

Results

To close these equations we need to prescribe the effec-
tive fluid viscosity rie = (1 + 5/2y) (Einstein 1906), the
particulate viscosity = /i 1-'/ I|| Ij | (Jop et al. 2006)

We present the results of the previous two-phase numeri-
cal model applied to the flow of a Newtonian fluid over a
granular bed. First we compare the results of the numer-
ical model with the analytical solution of Ouriemi et al.

7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010

z have performed simulations on the half of the domain in
u= v= = o the transverse direction for obvious symmetry reasons.
H Figure 4 shows the velocity profile in a cross section
,f = o 0 = wf = o of the ducts. The contour colors represent the x-velocity
OfS a s of the mixture (um). The horizontal thick solid line at
ax d P ax z 0.5 represents the position of the granular bed. The
fluid and the mixture are sheared in both z and y direc-
Stions inducing an increase in the friction compared with
vip =- o vf wP= 0 the two-dimensional case. Due to this shear increase the
oup Hp OuP velocity is lower than in the two-dimensional case. Figure
ax ax 5 shows the velocity profiles of the fluid phase velocity in
__ blue and the particulate phase velocity in red (an offset of
ad af 10-3 has been added to make the particulate phase veloc-
v1 = w = v = uv = wp = 0 and =0
az ity visible) obtained with the two-fluid model. These re-
Figure 1: Sketch of the flow of a Newtonian fluid over a sults illustrate the good behaviour of the numerical model
granular bed. for three-dimensional flow configurations.

12i i 1.11 for the bed-load transport in laminar shearing flows
to validate quantitatively the two formulations of the two-
phase flow model. We have also looked at the computa- Analytic
tional efficiency of the numerical model associated with UP
both formulations.
The sketch of the problem and boundary conditions are 0.8
given in figure 1. The lower half of the domain is filled
with particles at = 0.55 immersed in a fluid and the 0.6
upper part is filled with pure fluid ( 0). Therefore N
in this problem the values of the dimensionless numbers 0.4
are: Re 2 102, Ga 11, R = 0.4 and d/H
30. The regularisation parameter is set to A 10 6 s 1 0.2
when its value is not mentioned in the figure captions.
We solved by FEM the two formulations of the two-phase 0 0.001 0.002 0.003 0.004 0.005
flow model for a 6x1x40 mesh with a requested absolute U
residual lower than 1011 per degree of freedom. (a) Two-fluid model
Figures 2a and 2b present the comparison of the hori-
zontal velocity profiles obtained for the two formulations 1 Analytic
of the two-phase flow model (two-fluid and mixed-fluid) UM
by numerical simulations compared with the analytical
solution proposed by Ouriemi et al. (2009a). In both fig- 0
ures the black solid line represents the analytical solution.
The good agreement between the numerical solution and 0.6
the analytical one gives a first qualitaitve validation of the N
FEM model for the bed-load transport. 0.4
The spatial convergence analysis for both formulations
(see figure 3) shows that the regularisation of the granular 0.2
rheology reduces the spatial order of convergence to order
one whereas a second order is theoretically hopped. Also, O ' 0 0 0
0.001 0.002 0.003 0.004 0.005
to reach the same accuracy with both formulations we U
observe that the regularisation for the mixed-fluid model (b) Mixed-fluid model
must be decreased by one order of magnitude compared
ith te d easo- d bone Figure 2: Longitudinal velocity profiles for the flow of
with the two-fluid one.
wie ow l model t a Newtonian fluid over a granular bed between two in-
We now apply the model to three-dimensional configu-
finite parallel planes obtained by numerical simulations
rations: a square and a circular cross-section ducts, with
the same values of the dimensionless numbers Re 2 102 for a) the two-fluid model and b) the mixed-fluid model
Ga = 11, R 0.4, d/H 30 and Bn 2 104. We 'compared with the analytical solution of Ouriemi et al.
(2009a).

7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010

Error e = f(h)

le-08

le-09

x' O
0

S .....-- e_p (le-6) ---
S(.. e_f (le-6) ---..... ---
e_m(le-6) El
e_m(le-7) 0
0.01 0.1

Figure 3: RMS Error against analytical solution for the flow of a Newtonian fluid over a moving granular bed between
two infinite parallel planes: ep and ef stands for the particulate phase and the fluid phase error respectively for the
two-fluid model whereas em designates the mixture velocity error for the mixed-fluid model. The value in brackets is
the value of the regularisation parameter A. The RMS error is defined as: e 7= ( (U U ana)2)1/2 where N is
the number of nodes in the mesh.

-025

(a) Square duct

U
S00040
S 00035
0 0030
0 0025
00020
0 0015
S 00010
0 0005
0 0000

----

0 00025 0005 -0 5

(b) Cylindrical duct

Figure 4: Velocity profile obtained by numerical simulations with the mixed-fluid model for a) the square cross-section
duct (6x20x40) and b) the circular cross-section duct (6x896).

(a) Square duct

025 -

N 0

-0 25 F

W F- -- ------ ------

0 0 002 0 004 5
u u
(b) Cylindrical duct

Figure 5: Velocity profile obtained by numerical simulations with the two-fluid model for a) the square cross-section
duct (6x20x40) and b) the circular cross-section duct (6x896). The fluid phase velocity is in blue and the particulate
phase velocity is in red. An offset of 10 3 has been added to the velocity of the particulate phase (up) to make it
visible.

025

Figure 6 shows the application of the mixed-fluid model
coupled with an Arc Length Continuation algorithm that
allows us to explore the parameter space in terms of the
longitudinal pressure gradient dp/Ox. The agreement be-
tween the analytical solution of Ouriemi et al. 12'1 1'.1 and
the numerical model for the height of the flowing granular
layer Hp He and the particle flux Qp for a wide range
of longitudinal pressure gradient gives another validation
of our three dimensional numerical model. As for the ve-
locity profiles we have explored the parameter space for
three dimensional configurations that allows us to obtain
the height evolution of the flowing granular layer Hp He
and the particle flux Qp in rectangular and circular cross
section ducts. These results are presented in Figure 7 and
illustrate the non-trivial effect of the geometry on the be-
haviour of the flowing granular layer. This method ap-
plied to complex geometry is a powerful tool to compare
the prediction of the initial two-phase model with experi-
ments in real geometry such as rectangular or cylindrical
ducts. This is a strong argument for the development of a
three dimensional numerical model.

0.05 0.1
dpdx

0.15 0.2

7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010

Conclusions

In summary, we have developed a three dimensional nu-
merical model to simulate incompressible particulate two-
phase flows for the flow of a Newtonian fluid over a granu-
lar bed. The incompresibility is imposed by a penalisation
method and the viscoplastic characteristic of the granular
rheology is dealt with a regularisation technique. We have
studied the accuracy of our numerical solution by compar-
ison with the analytical solution derived by Ouriemi et al.
'2'i "1.,, in function of the spatial discretisation and the
value of the regularisation parameter for the mixed-fluid
and the two-fluid model. We have concluded that one
must take a regularisation parameter one order of mag-
nitude lower for the mixed-fluid model than for the two-
fluid one to reach the same accuracy.
The Finite Element method allows us to deal with ar-
bitrary geometries and we have shown numerical results

0.1 0.2 0.3
dpdx
(a) Height of the flowing layer

(a) Height of the flowing layer

3e-04

2e-04

le-04

Oe+00

0 0.05
dpdx
(b) Particle flux

6e-04

4e-04

2e-04

Oe+00

0.1 0.15

Figure 6: Comparison of the numerical results for the
2D (mesh size 4x1x640) configurations with the analyti-
cal solution of Ouriemi et al. 12'1, '.ii in terms of a) the
height of the flowing granular layer Hp He and b) the
particle flux Q, versus the longitudinal pressure gradient.

2D
Rectangular -.....
Cylindrical

0 0.1 0.2
dpdx
(b) Particle flux

0.3

Figure 7: Comparison of the numerical results for the 2D
and 3D configurations (rectangular and circular cross sec-
tion ducts) in terms of a) the height of the flowing granular
layer Hp He and b) the particle flux Qp versus the lon-
gitudinal pressure gradient. The mesh sizes for the 2D
case is 4x1x640, for the rectangular duct (aspect ratio of
W/H 0.2692) is 1x160x320 and for the cylindrical
duct is 1x179998.

- I -------------.............

of three-dimensional simulation of the bed-load transport
in a square and a circular cross-section ducts. This model
had also been coupled with an Arc Length Continuation
algorithm to describe the evolution of the height of the
flowing granular layer and the particle flux with the lon-
gitudinal pressure gradient in both 2D and 3D configura-
tions. This method is particularly interesting in the per-
spective of comparing the prediction of this model with
experiments that are always carried out in 3D configura-
tions.
Ouriemi et al. (2009b, 2010) have performed a sim-
ple linear stability analysis which provides realistic pre-
dictions for the formation of small dunes. The aim of
the present numerical model is to perform a full three-
dimensional stability analysis that accounts for the two-
phase nature of the problem.

Acknowledgement

Funding from the Institut Franqais du P6trole and Agence
National de la Recherche (Project Dunes ANR-07-3_18-
3892) are gratefully acknowledged.

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