Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 16.3.1 - A 3D Eulerian-Lagrangian Numerical Model for Sediment Transport
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 Material Information
Title: 16.3.1 - A 3D Eulerian-Lagrangian Numerical Model for Sediment Transport Granular Media
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: van Wachem, B.
Yu, X.
Hsu, T.-J.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: Lagrangian simulations
sedimentation transport
 Notes
Abstract: In this study, a new approach for sediment transport utilizing 3D turbulence resolving Eulerian-Lagrangian framework is presented. Themodel employs slightly modified Navier-Stokes equations to determine the motion of the fluid phase. The motion of the sediment phase is elucidated by a Lagrangian or Discrete Element Method (DEM), implying that the individual trajectory of each particle is determined by approximating Newtons second law of motion. The forces acting on each particle are gravity, the traction force of the fluid phase, and the force resulting from the interaction with other particles. The traction force of the fluid phase is determined by an empirical equation which has been validated for low particle Reynolds numbers. The total effect of the traction force summed over all particles is exactly opposite to the inter-phase momentum transfer added to the fluid-phase momentum equations. The force accounting for inter-particle collisions is based upon a so-called “soft-sphere” approach, following Tsuji (1994), which estimates the force based upon the deformation of each particle resulting from the interaction with other particles. We have performed simulation of sediment transport a 3 dimensional domain of approximately 50x30x50 particle diameters. The two directions perpendicular to gravity are treated as periodic; i.e. the domain is fictitiously extended to infinity in those directions. The flow is driven by an external pressure drop that corresponds to Shields parameter around 1.1. Next to the Eulerian-Lagrangian simulations, turbulence-averaged 1DV Eulerian-Eulerian simulations based upon the kinetic theory of granular flow are performed as well. The 3D model results are compared and the results from the Eulerian-Lagrangian simulations are analyzed to obtain flow statistics, which can be used as a starting point for Eulerian-Eulerian closure model validation and development.
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
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Resource Identifier: 1631-vanWachem-ICMF2010.pdf

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


A 3D Eulerian-Lagrangian Numerical Model for Sediment Transport


Berend van Wachem* Xiao Yut and Tian-Jian Hsut

Department of Mechanical Engineering, Imperial College London, Exhibition Road, London, SW7 2AZ
t Civil and Environmental Engineering, University of Delaware, Newark, DE 19716, USA
b.van-wachem @imperial.ac.uk and thsu @udel.edu
Keywords: Lagrangian simulations, Sedimentation Transport




Abstract

In this study, a new approach for sediment transport utilizing 3D turbulence resolving Eulerian-Lagrangian framework
is presented. The model employs slightly modified Navier-Stokes equations to determine the motion of the fluid phase.
The motion of the sediment phase is elucidated by a Lagrangian or Discrete Element Method (DEM), implying that
the individual trajectory of each particle is determined by approximating Newtons second law of motion. The forces
acting on each particle are gravity, the traction force of the fluid phase, and the force resulting from the interaction
with other particles. The traction force of the fluid phase is determined by an empirical equation which has been
validated for low particle Reynolds numbers. The total effect of the traction force summed over all particles is exactly
opposite to the inter-phase momentum transfer added to the fluid-phase momentum equations. The force accounting
for inter-particle collisions is based upon a so-called "soft-sphere" approach, following Tsuji (1994), which estimates
the force based upon the deformation of each particle resulting from the interaction with other particles. We have
performed simulation of sediment transport a 3 dimensional domain of approximately 50x30x50 particle diameters.
The two directions perpendicular to gravity are treated as periodic; i.e. the domain is fictitiously extended to infinity
in those directions. The flow is driven by an external pressure drop that corresponds to Shields parameter around 1.1.
Next to the Eulerian-Lagrangian simulations, turbulence-averaged 1DV Eulerian-Eulerian simulations based upon
the kinetic theory of granular flow are performed as well. The 3D model results are compared and the results from
the Eulerian-Lagrangian simulations are analyzed to obtain flow statistics, which can be used as a starting point for
Eulerian-Eulerian closure model validation and development.


Introduction


The phenomena of sediment transport is defined as the
movement of solid particles due to a combination of
gravity acting on the particles (i.e. the sediment), the
intra-particle forces, and the movement of the fluid in
which the sediment is entrained. Sediment transport
due to fluid motion occurs in rivers, the oceans, lakes,
due to currents, waves, and tides. Sediment transport
in water involves complicated fluid-particle interactions
and inter-granular interactions. A comprehensive
description of various mode of sediment transport,
namely, bedload and suspended load, requires a multi-
phase flow approach. Existing modeling approach for
sediment transport utilize Reynolds-averaged approach
in the Eulerian-Eulerian framework (Dong and Zhang
2002; Hsu et al. 2004; Amoudry and Liu 2009) or
Eulerian-Lagrangian framework (Drake and Calantoni
2001).


Modelling of sediment transport in rivers and coastal
areas is important in identifying erosion, changes
in fluvial and coastal morphology and the health of
ecosystem. Models have been successfully used to
study wave-induced sand transport under energetic
waves, i.e., sheet flow condition (Dong and Zhang
2002; Hsu and Hanes 2004; Yu et al. 2010). Through
Reynolds-averaging, these models parameterized all the
scales of turbulence and related turbulence-sediment
interaction. The coefficients involved in the turbulence
closure hence become highly empirical. Moreover, in
those models where sediment phase is modeled in an
Eulerian framework, kinetic theory of granular flow
is usually adopted for the closure of particle stresses.
Strictly speaking, kinetic theory becomes inappropriate
when sediment volume concentration becomes greater
than random-loss-packing ( -.' and hence addi-
tional bed treatment in the regime of enduring contact











in sediment transport is also necessary.

Simulations

Fluid-phase governing equations

The equations for the fluid-phase are derived analo-
gously to the Navier-Stokes equations. However, they
include a local volume fraction (Anderson and Jackson
1967) and additional source terms accounting for the
presence of particles and the driving force. The govern-
ing equations are the mass continuity and conservation
of momentum, which are given by equations (1) and (2)
respectively,


D(at f) d(aofpPfv )
at x1


D( 9fpfjV3) 9aftpfV3^) T9(af')
at + axi axi


Op
af O^


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


the Langrangian framework, where particle collisions
are modelled via the soft-sphere model proposed
by Tsuji et al. (1991b), which accounts for some
non-reversible deformation.

The interactions of particles with other particles
and walls are dynamic of nature. This is because
the particle movements are essentially defined by the
particle-particle interactions; particle-wall interactions;
particle-fluid interactions and/or body forces (Kuang
et al. 2008). The trajectories of individual particles are
considered (i.e. described in a Langrangian framework)
and Newton's 2nd law is solved for each individual par-
ticle, accounting for the fluid-particle, particle-particle,
and particle-wall interactions, and approximating the
integral with the Verlet algorithm.

Newton's 2nd law for the particles is written out


dvp P3 (Vf
dt ap -


Vp) + mpg


VpVPf


where af is the local volume fraction of the fluid phase,
,3 represents the local averaged reciprocal of the parti-
cle time-scale, or drag coefficient arising from the local
behaviour of the particles, Tf represents the source term
linear in the velocity field; S} represent the additional
source terms, which are used for driving the flow, and
T' the stress tensor of the fluid, given by


2 ) Ov'
3 O(3)


f (t vi vj \ (
a zi X3 01(h


where pf is the shear and Af represent the bulk viscos-
ity of the fluid. As a first approach, the sub-grid scale
stresses are not considered.

The simulations performed in this paper use a fi-
nite volume scheme, a second order backward Euler
time discretisation and a second order accurate central
scheme to approximate the advective terms in the conti-
nuity and momentum equations. The solving procedure
is made parallel with the MPI libraries. During the
simulations, the CFL number is kept constant at 0.4,
leading to a slight variation in time-step over time.

Particle-phase governing equations

A discrete element model (DEM) proposed by Cundall
and Strack (1979) is used to model the particles. The
individual trajectories of the particles are determined in


N
+VpSdrive + S [Fpw + Fpp]


and for the rotational momentum


dw -
dt


where mp is the mass of the particle; Ip is the momen-
tum of rotational inertia; Tp the torque of the particle;
w, is rotational velocity; vp is the translational veloc-
ity; and 3 is the drag function as proposed by Wen and
Yu (1966), where the reciprocal of the Eulerian fluid-
particle timescale is given by


3 apafPflvf
P d---
4 dp


Vp 2.65
af


and CD represents the coefficient of drag for an individ-
ual particle and af represents the fluid volume fraction.
The right hand side terms of equation (4) are outlined in
table 1.

Implementation of particle collisions

The particle-particle and particle-wall interactions as
taken into account in this work are assumed to be
local; i.e. no long-range forces are included. For
establishing the nature of the interaction, each particle
pair could be interrogated. However, this would lead
to a scaling of the computational effort with N2, N
being the number of particles. Instead, a particle-mesh
algorithm is adopted, in which each of the particles
is assigned a cell in the particle mesh based upon its


phases J f
+Tffv3+S + 1 p)-p1]












Table 1: Terms of right hand side of equation(4)

Term Force type

a3Vf Vp. Drag force
mpg Body force due to gravity
VpVPf Force due to the pressure gradient
VpSdrive External driving force
Fp, Particle-wall contact force
Fpp Particle-particle contact force



location. Using this particle-mesh, each particle is
only tested for interaction against particles located in
the same or directly neighboring particle mesh cells.
Although there is some additional computational effort
and a slightly more complex algorithm required for
this approach, it leads to a scaling of N logN with the
number of particles, making it a lot more favourable for
large numbers of particles.

The particle collisions are modelled by the soft-
sphere model as described by Cundall and Strack
(1979). In brief, this model uses a spring-dashpot-slider
arrangement to describe the particle behaviour before,
during and after a collision. The damping coefficient
as introduced by Tsuji et al. (1991a) accounts for the
non-ideal behaviour during the collision of the particles
due to irreversible plastic or visco-elastic deformation.

The normal and tangential contact forces are given
by the sum of forces due to the spring and dashpot.
From Hertzian contact theory the normal and tangential
contact forces are,

Fij = (-kn6 i. G n)n (7)
Ftij = -kt6t G (8)
where G is the velocity vector of particle i relative to
particle j, Get is the slit velocity vector at the contact
point. The subscripts n and t represent the normal and
tangential components respectively and 6 is the displace-
ment, or overlap, of particles i and j during collision.
Three parameters are required by the soft-sphere model;
stiffness (k), damping coefficient (rI) and the coefficient
of friction (p). p is a well known empirical quantity,
stiffness and the damping coefficient must be estimated
using equations

/n = ac Mk6 (9)


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


mp. The relationship between a and the coefficient of
restitution is well defined by Tsuji et al. (1991b). The
spring constants are, based upon elastic deformation,


4 1 1
S- 3 E +
h6g \ A


,, (2 -i 2-,j ri rj)_2 I
kt 8 G + rir 6 (12)
Gi G \ rirj
where r is the radius of the particles, a the Poisson's ra-
tio; E the Young's modulus, G the shear modulus (given
by G = (iE ) and 6, the magnitude of the deforma-
tion in the normal direction. Note as above, when a par-
ticle reaches a wall, r, -+ oc, hence, r' + -1.

Simulation Parameters

The fluid properties used in the simulations are those
of water, with density p = 998 and viscosity v
1.00 x 10 3Pa s. The driving force of the fluid is deter-
mined by the shields parameter. The shields parameter is
a dimensionless parameter that characterizes the type of
sediment transport regime. The Shields parameter, i.e.,
the nondimensionalized bed shear stress, is given by
T
O (13)
(ps pf) gds
where T is the bed shear stress, g is the gravitational ac-
celereation, d, represents the particle diameter, and p,
and pf the solids and fluid density respectively. For sta-
tistical steady flow considered in this study, the given
bed shear stress can be further related to pressure drop
in the streamwise direction:
9p T
(14)
Ox h
where h is the height of the domain. The driving force
is made equal to the pressure drop, i.e.


Drive


And this driving force is distributed over the local
fluid and particle phase weighted by the local volume
fraction.

For the simulations, sand (silicon dioxide) particles
were taken and assumed to have constant properties, see
Table 2


Results and Discussion


a/t = a 'kt (10)
where M -= m and m is the mass of the particle.
Note that for wall collisions m, -- oc, hence M


Eulerian-Lagrangian Simulation Results

The simulations carried out with the Shields parameter
of 1.1. The resulting magnitude of the shear stress is


E r
ri ( rirj







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


Table 2: Summary of the sand particle properties used
in computations.
Variable Value Units
d, 200 .10-6 m
ps 2650 kg/m3
Youngs Modulus 75 .107 Pa
Poisson ratio 0.17
Friction Coefficient 0.45
Restitution Coefficient 0.95


T = 3.89 Pa and the pressure gradient = 55.6 Pa.

According to field observation and laboratory ex-
periments, Shields parameter greater than 1.0 represents
typical "sheet flow" condition where intense sediment
transport occurs, the bed is statistically flat (no bedforms
or ripples) with a concentrated layer of sediment moving
near the bed. A snapshot of the numerical simulation is
shown in Figure 1, where the particles are coloured by
their velocity.
The simulations predict the basic characteristics of sheet
flow. Sediment concentration decreases rapidly away
from the bed and all the transport occurs within 2 mm
near the bed. In this moving layer, the fluid velocity
also increases rapidly from 2 cm/s to about 20 cm/s.
Numerical model predicts the statistically-averaged
fluid velocity is almost identical to that of particle
velocity for relatively fine sediment in water considered
here.

Eulerian-Eulerian Model Results

A Eulerian-Eulerian two-phase model for sediment
transport is adopted in this study. Turbulence-
averaged two-phase equations are simplified into one-
dimensional-vertial (1DV) to model fully-developed
sheet flow sand transport driven by a steady or oscil-
latory flow forcing (Hsu et al. 2004; Yu et al. 2010).
The eddy viscosity and k-e closure for two-phase flow
are adopted for fluid Reynolds stress and turbulent
suspension of sediment. Kinetic theory of granular
flow of Jenkins and Savage (1983) and empirically
modified closure for stress due to enduring contact are
utilized for the closure of particle stress. More detailed
model formulations and model applications to sediment
transport can be found in Hsu et al. (2"' 14); Yu et al.
(2010).

The Eulerian-Eulerian model results compared to
the the Eulerian-Lagrangian model results for the same
case are shown in Figure 2 for vertical profiles of
fluid velocity (solid curve in the left panel), sediment


Figure 1: A snapshot of the Eulerian-Lagrangian sim-
ulations, showing the individual particles. The particles
are coloured by their velocity.



concentration (solid curve in the middle panel) and fluid
turbulent intensity (2TKE), where TKE represents
the fluid turbulent kinetic energy (right panel). The
fluid stress consists of turbulent Reynolds stress and
viscous stress with the the viscous stress contribution
negligible in this case. Eulerian-Eulerian model predicts
sediment suspended much higher in the water column
due to strong flow turbulence. Strong turbulence can
be observed from large turbulent intensity near the
bed. Some discrepancies with the Eulerian-Lagrangian
model results can be observed. Most evidently,
Eulerian-Eulerian model predicts more intense sediment
suspension throughout the water column and higher
mobility of sediment. Although it is likely that the
flow turbulence and granular temperature may be
over-predicted in the Eulerian-Eulerian model due to
uncertainties in the closures, it is also possible that in
the Eulerian-Lagrangian model, the flow turbulence
is under-predicted due to low resolution used in this
simulation. The under-prediction of flow turbulence
can be seen from the almost linear fluid velocity profile


























0 50 100 150 0 20 40 60 0 5 10 15
u (cm/s) vol cone (/) sqrt(2*TKE) (cm/s)

Figure 2: Eulerian-Eulerian model results(line)
compared to the the Eulerian-Lagrangian model re-
sults(symbols) for the same case, for vertical profiles of
fluid velocity (solid curve in the left panel), sediment
concentration (solid curve in the middle panel) and fluid
stress (right panel, solid curve) and particle stress (right
panel, dotted curve).


away from the bed. On the other hand, Eulerian-
Eulerian model also predicts almost identical fluid and
sediment velocities (not shown here), consistent with
Eulerian-Lagrangian model results.


Conclusions

A new approach for sediment transport utilizing 3D
turbulence resolving Eulerian-Lagrangian framework is
presented. Preliminary simulation results suggest the
main characteristics of sheet flow in the concentrated
regime of sediment transport is captured. However,
higher numerical resolution for fluid phase is required,
or subgrid turbulence closure need to be implemented in
order to capture flow turbulence and the resulting turbu-
lent suspension of sediment.


Acknowledgements

Financial support for Dr. Berend van Wachem's travel
to ICMF-2010 is partly provided by the Royal Academy
of Engineering. Financial support to Hsu and Yu is
provided by U.S. National Science Fundation (OCE-
0644497).


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


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