Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 6.1.2 - Enhanced settling due to particle-laden convective instability
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 Material Information
Title: 6.1.2 - Enhanced settling due to particle-laden convective instability Instabilities
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Yu, X.
Hsu, T.J.
Balachandar, S.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: convective instability
settling
particle-laden flow
Equilibrium Eulerian approximation
 Notes
Abstract: Convective instability may play an important role in determining both the location and timing of initial sediment deposition off river plume. It has been observed in laboratory experiments for concentration as large as few gl􀀀1, the convective instability occurs and lead to enhanced apparent settling velocity on the order of few cm/s which could not be explained by conventional Stokes settling law. In addition to the well-known double-diffusive convection mechanism, another mode of convective instability development is by heavier particles settling cross the density interface,i.e., settling-induced instability. The Eulerian-Eulerian two-phase equations is simplified with equilibrium Eulerian approximation appropriate for fine sediment of small particle response time. The resulting governing equations are similar to that of Rayleight-Taylor problem for density-driven flow except for the consideration of particle settling in the sediment conservation equation. A linear stability analysis for settling-induced convection is carried out to study the finger formation at the density interface. Relationships between growth rate and initial sediment concentration, mix layer thickness, grain diameter and water depth are obtained to help us further understand the dominant mechanisms causing the convective instability in a more realistic setting. Fully nonlinear analysis is further carried out through numerical solution of the complete equations computed by a 3D pseudo-spertral Navier-Stokes solver. Some discrepancies between results from the linear stability analysis and the fully nonlinear numerical solutions are discussed.
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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Enhanced settling due to particle-laden convective instability


Xiao Yu* Tian-Jian Hsu and S. Balachandart

Department of Civil and Environmental Engineering, University of Delaware, Newark, DE 19716, USA
t Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611, USA
yuxiao@udel.edu, thsu@udel.edu and balals@ufl.edu
Keywords: convective instability, ,sc'lli particle-laden flow, Equilibrium Eulerian approximation




Abstract

Convective instability may play an important role in determining both the location and timing of initial sediment
deposition off river plume. It has been observed in laboratory experiments for concentration as large as few gl 1, the
convective instability occurs and lead to enhanced apparent settling velocity on the order of few cm/s which could
not be explained by conventional Stokes settling law. In addition to the well-known double-diffusive convection
mechanism, another mode of convective instability development is by heavier particles settling cross the density
interface,i.e., settling-induced instability. The Eulerian-Eulerian two-phase equations is simplified with equilibrium
Eulerian approximation appropriate for fine sediment of small particle response time. The resulting governing
equations are similar to that of Rayleight-Taylor problem for density-driven flow except for the consideration of
particle settling in the sediment conservation equation. A linear stability analysis for settling-induced convection
is carried out to study the finger formation at the density interface. Relationships between growth rate and initial
sediment concentration, mix layer thickness, grain diameter and water depth are obtained to help us further understand
the dominant mechanisms causing the convective instability in a more realistic setting. Fully nonlinear analysis
is further carried out through numerical solution of the complete equations computed by a 3D pseudo-spertral
Navier-Stokes solver. Some discrepancies between results from the linear stability analysis and the fully nonlinear
numerical solutions are discussed.


Introduction


River water intruding into coastal ocean typically results
in a surface layer of sediment-laden water over relatively
dense saline seawater. To predict the location of ini-
tial sediment deposition off river plume, it is critical to
estimate the equivalent settling velocity of sediment in
salt-stratified sediment-laden river plume. Existing liter-
atures in coastal oceanography mostly consider Stokes
settling law for primary particle or floc aggregate, which
usually gives settling velocity of no more than few mm/s
(e.g., Hill et al. 2000). However, limited but valuable
field evidences suggest when sediment concentration in
the river plume exceed 0(10) g/1 during episodic river
flooding, equivalent settling velocity exceed few cm/s,
which cannot be explained by conventional Stokes set-
tling law (Warrick et al. 2008).
Convective instability across the density interface may
play an important role in such rapid sedimentation pro-
cesses (e.g., Thacker and Lavelle 1978; David et al.
1999; Parsons et al. 2000; McCool and Parsons 2004).


However, the mechanisms causing such instability, such
as gravitational settling and double-diffusion, remains
unclear and hence more detailed studies are necessary.
Using fine sediment approximation (Ferry and Bal-
achandar 2001) to the Eulerian-Eulerian two-phase
equations for fluid-particle system, we investigate con-
vective instability and its implication to the occurrence
of rapid sedimentation of fine sediments off river plume
in the field. The idealized particle-laden two-layer
flow problem is first solved by linear stability analy-
sis in order to understand critical parameters affecting
the growth rate and characteristic size of the instabil-
ity. Fully nonlinear 3D analysis is further investigated
by adopting a Navier-Stokes solver based on pseudo-
spectral scheme previously demonstrated to be capable
of carrying out direct numerical simulations (DNS) for
turbulent flow (Mariano et al. 2008).


Conditions for settling-driven fingering

Green (1987) presents a criterion to determine whether











double-diffusive convection or gravitational settling
dominates particle transport across the interface based
on the ratio (F) of the flux by double diffusion (Fdd) to
the flux by settling (Fs):

Fdd
F,

For F > 1, double diffusion is the dominant mecha-
nism causing the instability, while for F < 1 gravi-
tational settling dominates. Green (1987) modified the
double-diffusive flux presented by Schmitt (1079) to ac-
commodate particles rather than a dissolved substance in
the upper layer. Assume initially there is no particles in
the lower layer and no dissolved substance in the upper
layer, the double-diffusive convective flux is
1 4
Fdd =pb(gi)3(C .)3

where b is a nondimensional constant determined by ex-
periment (1/20), g is the acceleration of gravity, K is the
diffusion coefficient of the fastest diffusing substance
and C, is the sediment concentration in the upper layer
(expressed in units of mass/mass). For a dilute particle
suspension. The volumetric expansion coefficient is de-
fined as


0 OP
(p18m) eo
\p09C,^


pS pf
P


where p8 is the density of particles.
The flux by settling could be simply calculated as the
product between settling velocity and concentration

Fs = WC

where C is the sediment mass concentration. For small
particles, the settling velocity can be estimated using
Stokes Law
W, 19(p pf)
18p y
where d is the particle diameter and v is the kinematic
viscosity of fluid.
Although in natural system, both double-diffusive and
settling-driven fingering may be present at the same
time, we can always estimate the relative importance of
these two different mechanisms using parameter F de-
fined above. For typical sediment properties and flow
conditions in natural river, i.e., d = 40m, C
2 1'.. /1, we have Fdd = 3.85 x 10 4kg/(m2 s) and
F, 3.80 x 10 3kg/(m2 s), which gives F z 0.1
and gravitational settling is the dominant mechanism for
instability. Hence, as a preliminary step, we focus on
studying instability induced by gravitational settling in
this paper.


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


Model formulation

It will be useful and instructive to study the main
characteristics of a complex flow with the help of a
somewhat simple flow configuration that possesses
the real physical and numerical features of the full
problem. For our problem, theoretically the sharp
planar interface might continuously descend until the
sand settles at the bottom and form a dense package.
However, the interface instability grows through time
and develops a finger like patten. Here, the system is
modeled as a mixture with variable density. In the initial
stage, the fluid is horizontally stratified, with denser
water-sand mixture on top of clear fluid. Therefore, we
make contact to the classic Rayleigh-Taylor instability
according to which the destabilization of the interface
between the layers is inevitable.

The flow is assumed to be consist of two layers with
heavier sediment-laden water on top of clear water. In
the initial state, the sediment concentration is given by
specifying the mixing thickness 6, which represents the
finite mixing layer thickness typically exists at the bot-
tom of the river plume. The density interface is located
at y = 0 at t = 0 as shown in Fig.1. Cartesian coor-
dinates (x, y) and velocities (u, v) are equivalently re-
ferred to as the streamwise and normal component, re-
spectively. The mixture density can be written as a func-
tion of sediment concentration, p pf +c(p8 -p ) with
c representing volumetric concentration of sediment,
and for typical sand, p8 given as 2.65 x 103kg m3.
Initially the sediment concentration profile is given as
co(z) 0.5Cmax(tanh(ay) + 1) with a defined as
a = 2,where the mixing thickness 6 defined as


I00 (
-f"0 -4


(c Cmax/2)2>
maCa


which is analogy to the definition of momentum thick-
ness of the mixing layer.
Equilibrium Eulerian approximation is adopted to de-
scribe the two-phase fluid-particle system here. For
small particles, whose particle response time T is suf-
ficiently smaller than the flow time scale, the particle
velocity can be expanded in terms of T and local car-
rier flow field, see Jim Ferry & S. Balachandar (2000).
Here, only the first order term is kept, which gives
u" = u + Ws, suggesting that the particles are mov-
ing with the carrier fluid flow except gravitational set-
tling. The sediment concentration is considered to be
low, hence the particle interactions will be ignored. Be-
sides, the density variations is considered to be small
such that Boussinesq approximation could be used. The







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


H








Figure 1: Sketch of the flow configuration



governing equation can be written as


Oui 0
=0
dOxi
bui bui 1 9p D2Ui
- +Uj +v
Dt dxj p dxi dxj dxj
dc dc D2c
+ (ui + WA,3) = K
at oxi Oxoxii


form are


Oui
=0
Oxi
Oui Oui
a+ u


Dp 1 D2ui
D + Re dxjdj
Oxi Re Oxj8xj


'R,. ..


W


s s


(s 1)gci3


where c is the concentration of sediment, ,u are fluid
velocities. The initial mixing thickness 6 is chosen as the
characteristic length scale for convenience, although it
should not be the correct scale for the convective fingers.
The velocity scale is chosen as U = ^/g ,where g'
g(s 1)Cma, so that


Xi
Xi*
8'

tU
t* 6
X^=^


ui p
ui* U' pP* U2

c WW
Cmax W U


The resulting governing equations in the dimensionless


Du' Dv'
Dx Dy
Du' Dp' 1 (D2
Dt Dx + ReSc 0e
Dv' Dp' 1 (D2
t D9y ReScc D
Dc' Dc' ,dco
W, + v
9t Dy dy


ui D2u'2

v' 2 v'U,
/+ D ) Ric'
R2 +y2


ReSe d2c' D 'y2 c
ReSc 8x2 Dy)


The incompressibility for 2D flow can be formulated in
terms of a streamfunction q'. Since the perturbations
satisfies V u' 0, it is permissible to introduce a per-
turbation streamfunction 7' defined by


y '


The disturbance field 0'(x, y, t) now can be taken of the
following traveling-wave form

(x, y, t) = (y)ei(kx-wt)


dc dc 1 d2c
+ (ui + W, 6,3)
t D3xi ReSc OxiDxi

where the nondimensional groups are defined as

8U
Re -

(s 1)g6Ac
Ri =
U2

with H representing the half depth of the whole water
column and U is the characteristic velocity scale. For
simplicity, the superscripts have been neglected in the
above equations.


Linear stability analysis

For simplicity, a two-dimensional flow system is stud-
ies here by stability analysis. For field 0, we take the
decomposition in the form of =< 0 > +9', where
< 4 > is the mean of quantity 0, which could be veloci-
ties, pressure or concentration, and 0' is the perturbation
from the mean state. We are interested in the stability of
the system, so introducing small perturbations into the
problem the resulting linear stability equations govern-
ing the disturbance now becomes







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


where 0 denotes the complex amplitude, and k is the
real wavenumber of the perturbation in the streamwise
direction. Upon substituting the above ansatz into the
linearized governing equation and eliminating the pres-
sure terms in the usual way, we will obtain an Orr-
Sommerfeld type fourth order ordinary differential equa-
tion.

dco dc 1 d2
-iw + +W = -k c+
dy dy ReSc y

(k -- k k2Ric = 0
(k dy2 i Re( dy2 )

The initial concentration profile is given by co(y)
0.5(tanh(ay) + 1) in the dimensionless form.
The temporal long-term evolution of this type of distur-
bance is characterized by w whose real part describes the
frequency w, and whose imaginary part the correspond-
ing growth rate wi. The one-dimensional eigenvalue
problem A0- w9 with matrix A obtained by Cheby-
shev expansion with Chebyshev-Guass-Lobatto colloca-
tion points can be easily solved using existing eigenvalue
package.


Linear stability result

In the absence of sediment, the flow becomes clear
fluid at rest, which should be always stable. By setting
Ri 0 excluding the concentration equation and
solving the Orr-Sommerfeld equation, we obtain wi < 0
for all modes, suggesting that the initial perturbation is
always attenuated and the system is stable.
For Ri / 0, model solution gives wi > 0 for small
wavenumber k and the system is unstable. Fig.2 shows
the growth rate as a function of wavenumber k for
d 40pm,Cmax 1 x 10-4 and 6/H 1/30.
Evidently, the growth rate is not a monotonic function
of wavenumber k. The growth rate increases sharply
first then decreases smoothly towards large wavenum-
ber. The peak corresponds to the most unstable mode,
which grows most rapidly and shall become dominate
mode soon after the onset of instability. Hence, the
most unstable mode can be used to estimate the size of
sediment finger.
Our principal interest is to study the growth-rate as
a function of different flow parameters and sediment
properties, such as grain diameter d, mixing layer
thickness 6, initial concentration C,,,, and flow depth
H.

Effect of concentration
Considering the classic Rayleight-Taylor instability, it is
known that the key controlling parameter is the Atwood


dp=4p m
dp=40 I m
d= 100 p m


05

3 04

S03
5


05 1 15 2 25 3 35
Wavenumber k (cm-1)


4 45 5


Figure 2: The growth ratewi as a function of wave num-
ber k. The concentration Ac 1 x 10 4,
and mixing thickness 6/H 1/30. The most
unstable mode for different sediment diam-
eter differs. It reveals that the finger width
will be a function of sediment diameter and
it decreases when the sediment diameter in-
creases.


number defined as
SP P2 (s )Cmax
P1 + P2 2 + (s 1)Cma
with s the specific density of sediment, which is given
as 2.65 for natural sand particles. For small Atwood
number A < 1, the Boussinesq approximation is valid,
which is the case studied here. For increasing concentra-
tion, the density differences also increase, which leads
to larger Atwood number and larger growth rate is ex-
pected. For very small concentration, we can expand the
above equation using Taylor Series, which gives us

A z 0.5(s 1)Cmax[1 0.5(s 1)Cmx]
S0.5(s 1)Cma + O(C,,x)

The classic Rayleigh-Taylor instability gives the growth
rate proportional to /A. Fig. 3 shows how the growth
rate varies with concentration for two choices of grain
sizes. By fitting the power law we acb using least
square method, we obtain b 0.60 for small particle
settling velocity of grain size dp 4/tm. If we set set-
tling velocity to zero, which gives the miscible two fluid
system, the curve overlaps with the one for dp 4pm.
b 0.60 obtained here is different from the classic
Rayleigh-Taylor result of b = 0.5 for two-immiscible in-
viscid fluid system. The presence of viscous effect and
sediment settling through the interface may also intro-
duce perturbations to this system. Therefore, the growth







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


Scm =001


Figure 3: The growth rate cw as a function of sedi-
ment concentration Cax for dp = 4pm and
dp = 40pm. The mixing layer thickness is
6/H = 1/30. In log-log plot, the curve close
to a straight line, which implies a power law
relationship between the growth rate wi and
volumetric concentration C,,x.


rate is deviated from the value of 0.6. Fig.3 also shows
the growth rate as a function of concentration for lager
settling velocity of grain diameter dp 40/m. For
larger sediment settling velocity of dp = 40pm, b in-
creases to 0.64.
Effect of initial mixing thickness
We can also change the initial mixing thickness to study
its effect on growth rate. In reality, the mixing layer
thickness at the bottom of a river plume is controlled by
fresh-saline water density-driven flow and the associated
ambient mixing processes by K-H waves and Hombole
waves. These complicated processes shall be studied in
details however they are highly idealized here by a mix-
ing layer thickness through initial condition.
Fig. 4 shows how the growth rate varies as a function
of 6/H. It can be observed from Fig. 4 that if the ini-
tial sediment concentration is strongly mixed, i.e. larger
6/H value, the growth rate decreases. Since the instabil-
ity occurs at the interface, which means we should con-
sider a local Atwood number instead of using Ac in the
above equation for A. For larger 6/H value, the local
concentration at certain level c is smaller, which gives
a smaller Atwood number, therefore, smaller growth
rate. This may imply that in the field condition, stronger
frontal mixing at the bottom of the river plume may de-
lay the development of sediment finger.

Effect of settling velocity and domain height
The settling velocity for fine particle can be given by
Stokes law. According to linear stability analysis the


Figure 4: The growth rate cw as a function of dimen-
sionless mixing high 6/H. Sediment diam-
eter dp 40pm and Cmax 1 x 10 4,
Cmax 1 x 10 2 respectively.


growth rate is not very sensitive to grain size for small
particle diameter. Fig. 5 shows that for dp < 15pm, the
growth rate does not change with sediment diameter. For
dp > 15pm, the growth rate decreases with increasing
sediment diameter. Our results suggesting the growth
rate is insensitive or decreasing with increasing settling
velocity appears to be inconsistent with limited obser-
vation in the laboratory experiment (Hoyal et al. 1999).
This may be due to the limitation of a linear stability
analysis. More future work on fully nonlinear analysis
is necessary.
The domain height also plays minor role in determin-
ing the growth rate. The boundary condition writes
u' v' 0 at both the top and bottom boundary, which
means the presence of boundary tends to attenuate the
perturbation and hence suppress instability. For larger
domain height, the growth rate only increases slightly
(comparing dashed curve to solid curve in Fig. 5). For
large enough domain height H, the boundary effect can
be neglected.



Fully nonlinear numerical simulations

Full numerical solution of the nondimensionalized gov-
erning equations is computed by extending a 3D Navier-
Stokes solver based on pseudo-spectral scheme (Mari-
ano et al. 2008) in order to carry out fully nonlinear
analysis of convective instability and resulting sediment
finger development. An operator splitting method is im-
plemented to solve the momentum equation along with
the incompressibility condition. A low-storage mixed







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


H=O1m
- - H=O 2m


Figure 6: Concentration contour at t = 153s


Figure 5: The growth rate cw as a function of sediment
diameter dp. For different domain height, the
growth rate only differs slightly.





third-order Runge-Kutta and Crank-Nicolson schemes
are used for the temporal discritization of the advection
and diffusion terms respectively. More details on the
implementation of this numerical scheme can be found
in Cortese and Balachandar (1995). Since the interface
descends with the settling of sediment phase, a vertical
coordinate transform is further adopted into the pseudo-
spetral code. The coordinate transform is done by set-
ting znew = Zold Wst in the vertical direction only.
Fourier expansions for the flow variables in the stream-
wise and spanwise direction remain unchanged. In the
inhomogeneous vertical direction, a rational collocation
method with adaptively transformed Chebyshev points
Tee & Trefethen (2006) is also applied.
The computational domain is a box of size L, = 2 x Lz
and L, 0.5 x Lz (with L = 306), which extends from
x -2.0 to x 2.0, from y -0.5 to y 0.5 and
from z = -1.0 to z = 1.0. The flow is initialized from
rest with co = Cmax(tanh(az) + 1)/2 where heavier
water-sediment mixture on top of nearly clear fluid (see
Fig. 1). The simulations are performed using a resolution
of N, 256, N = 128 and N, 129.
Periodic boundary conditions are enforced for all the
variables in both streamwise and spanwise directions.
This is due to the characteristics of the spectral method
used. However, the computational domain is taken to
be large enough in these directions to allow the full de-
velopment of sediment fingers for a sufficient amount of
time. At the top and bottom walls, no-slip and no pene-
tration conditions are enforced for the continuous phase


Figure 7: Concentration contour at t = 170s



velocity. For the normalized concentration, we apply


and the mass of sediment should be conserved in the sys-
tem.
Fig. 6 and Fig. 7 shows preliminary results of two
snapshots of concentration contour at two different time
t 153sec and t 170sec. When these fingers are
macroscopically observable, as shown in Fig. 6 and Fig.
7, the number of fingers is fewer than that predicted by
the linear stability analysis. In this case, 3D simulation
under-predicts the number of finger exactly by one-half.
The discrepancy between the 2D linear stability analysis
and full 3D solutions could due to the action of pair-
ing mechanism upon the 3D microscopic fingers caus-
ing them to merge. Tan and Homsy (1987) explain this
as a secondary modulatory spanwise instability, which
causes the fingers to couple and coalesce. More detailed
numerical study shall be carry out in the future to in-
vestigate the 3D fully nonlinear dynamics of sediment
fingers.











Conclusions

The linear stability analysis for the initial evolution of a
water-sand interfaces is investigated. Equilibrium Eu-
lerian method is used here to describe the two-phase
particle-laden flow appropriate for fine particles. The
linear stability analysis provides us the growth rate,
which can be used to provide critical condition for the
occurrence of sediment finger during initial deposition
of sediment off river plume.
The results of stability analysis suggest that the growth
rate is a strong function of sediment concentration and
mixing thickness. However, the growth rate is less sen-
sitive to variation of grain size and flow depth (i.e., the
height of computational domain). The growth rate is a
power function of sediment concentration. For finer sed-
iment, this power law gives ci acb, where b is in the
range of 0.5 0.7 depending on sediment diameter and
mixing thickness 6.
Examination of for the most unstable waves has allowed
for the identification of the characteristic size of the sed-
iment fingers, such as finger width. Full numerical so-
lutions of the 3D nondimensional governing equations
show pairing mechanism that cannot be captured by lin-
ear stability analysis.


Acknowledgements

This study is supported by the U.S. Office of Naval
Research (N00014-09-1-0134) and National Science
Foundation (OCE-0913283) to University of Delaware
and by U.S. Office of Naval Research grant (N00014-
07-1-0494) to University of Florida. This work is
also partially supported by the National Center for
Supercomputing Applications under OCE70005N and
OCE80005P utilizing the NCSA Cobalt and PSC Pople.


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