Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 9.4.4 - Understanding and Prediction of Hydrodynamics of Multiphase Flow Using CFD
ALL VOLUMES CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00102023/00229
 Material Information
Title: 9.4.4 - Understanding and Prediction of Hydrodynamics of Multiphase Flow Using CFD Computational Techniques for Multiphase Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Hendrickson, K.
Campbell, B.K.
Liu, Y.
Roberts, R.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: DNS
nonlinear interfacial instability
slug flow initiation and evolution
 Notes
Abstract: This work describes the development of two computational capabilities designed to investigate the hydrodynamics of multiphase (gas/liquid) flow in support of the design of deepwater pipelines and the development of advanced flow-monitoring devices. Of particular interest is understanding the transition and nonlinear evolution of stratified flows into liquid slugs in horizontal or incline pipes. Two complementary computational capabilities are developed. The first scheme is a two-dimensional method for the computation of large-scale flow in pipelines with L/D = O(1 - 10,000), which is based on weakly nonlinear inviscid multiphase flow. This capability (referred to as "Long-Pipe") is useful for understanding and predicting the initial development and nonlinear evolution of interfacial disturbances over long distances. The other numerical method carries out direct numerical simulations (DNS) of the Navier-Stokes equations. This capability (called "Short-Pipe") is applicable to short pipes with length-diameter ratio L/D = O(1-10), and is used for understanding detailed turbulent flow dynamics of roll waves and fully developed slugs and for developing effective turbulent/viscous models for larger-scale applications. The combined framework will be useful for developing effective physics-based modeling and strategies for improving full-scale viscous applications such as RANS and phenomenological models for multiphase roll wave and slug flow prediction in long pipes. Long-Pipe simulations are performed to understand the nonlinear effects upon the instabilities of two-phase stratified flows and to study the mechanisms for the development of large-amplitude long interfacial waves by resonant nonlinear wave-wave interactions. Comparisons have been made between Long-Pipe predictions, existing theoretical analysis, and laboratory observations on the initial growth and subsequent nonlinear evolution of wavy flows with good agreement being observed in Campbell (2009). Short-Pipe simulations are conducted to understand the detailed flow dynamics in the later stage of development of breaking waves and slug formation. Of critical interest is the development and validation of physics-based turbulence and interface closure models appropriate for use in the simulations of violent interfacial flow in pipelines.
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: VID00229
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: 944-Hendrickson-ICMF2010.pdf

Full Text



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


Understanding and Prediction of Hydrodynamics of Multiphase Flow Using CFD


K. Hendrickson*, B. K. Campbell*, Y. Liu*, and R. Robertst

Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
t Chevron Corporation, ETC Advanced Production Systems, 1400 Smith Street, Houston, TX 77002
khendrk@mit.edu, bcampb@mit.edu, yuming@mit.edu, and roberrml@chevron.com
Keywords: direct numerical simulation, nonlinear interfacial instability, slug flow initiation and evolution




Abstract

This work describes the development of two computational capabilities designed to investigate the hydrodynamics
of multiphase (gas/liquid) flow in support of the design of deepwater pipelines and the development of advanced
flow-monitoring devices. Of particular interest is understanding the transition and nonlinear evolution of stratified
flows into liquid slugs in horizontal or incline pipes.

Two complementary computational capabilities are developed. The first scheme is a two-dimensional method for the
computation of large-scale flow in pipelines with L/D = 0(1 10,000), which is based on weakly nonlinear inviscid
multiphase flow. This capability (referred to as "Long-Pipe") is useful for understanding and predicting the initial
development and nonlinear evolution of interfacial disturbances over long distances. The other numerical method
carries out direct numerical simulations (DNS) of the Navier-Stokes equations. This capability (called "Short-Pipe")
is applicable to short pipes with length-diameter ratio L/D = O(1-10), and is used for understanding detailed turbulent
flow dynamics of roll waves and fully developed slugs and for developing effective turbulent/viscous models for
larger-scale applications. The combined framework will be useful for developing effective physics-based modeling
and strategies for improving full-scale viscous applications such as RANS and phenomenological models for
multiphase roll wave and slug flow prediction in long pipes.

Long-Pipe simulations are performed to understand the nonlinear effects upon the instabilities of two-phase stratified
flows and to study the mechanisms for the development of large-amplitude long interfacial waves by resonant
nonlinear wave-wave interactions. Comparisons have been made between Long-Pipe predictions, existing theoretical
analysis, and laboratory observations on the initial growth and subsequent nonlinear evolution of wavy flows with
good agreement being observed in Campbell (2009). Short-Pipe simulations are conducted to understand the detailed
flow dynamics in the later stage of development of breaking waves and slug formation. Of critical interest is the
development and validation of physics-based turbulence and interface closure models appropriate for use in the
simulations of violent interfacial flow in pipelines.


Nomenclature Re = pjUID Reynolds number
t Time (s)
Roman symbols U Fluid current speed (ms 1)
Fr p- Froude number We p U D Weber number
g Gravitational acceleration (ms 2) x Horizontal spatial coordinate (m)
h Equilibrium fluid depth (m) y Vertical spatial coordinate (m)
D Channel depth (m)

1 _Greek symbols
k Wavenumber (m 1) f st
c a Void fraction
N = L Viscosity ratio
N iscs it Pressure forcing coefficient
P Pressure (Nm 2) 6 Dirac delta function
R = Density ratio
P1











ir Free surface elevation (m)
o/0 Initial free surface elevation (m)
7 Surface tension (Nm 1)
SInterface curvature ()
p Viscosity (N-s-m 2)
w Frequency (s 1)
T Viscous stress tensor
p Potential function (ms- 2)
p Level set function
p Density (kg-m-3)

Subscripts
u Upper fluid
1 Lower fluid



Introduction

The transition from stratified flow to more complex flow
regimes represents a significant theoretical and engineer-
ing challenge. The offshore oil industry's reliance on
long pipelines has raised interest in understanding the
complex flow physics which can arise in the pipeline
during the transport of multiphase mixtures from off-
shore to land based processing facilities. Variations in
parameters such as flow rates, pressure drop, pipe incli-
nation (due to subsea terrain) can lead to the formation
of liquid slugs within the pipe. The presence of these
slugs create igmiitk.iiI design and flow assurance chal-
lenges. Being able to consistently and accurately predict
the conditions which cause these transitions is necessary
for robust design of engineering systems and remains an
active area of research.
The theoretical prediction of the transition to slug flow
has traditionally been addressed through linear stability
analysis of a stratified state. Numerous stability mod-
els have been generated such as Taitel & Dukler (1976),
Lin & H.iII.ili\ (1986), Barnea & Taitel (1993), and Fu-
nada & Joseph (2001). These models include a wide
range of physical features such as experimental correla-
tions, surface tension, normal viscous stresses, and in-
terfacial friction factors. A survey by Mata et al. (1993)
found that comparisons between the different stability
models resulted in a wide range of stability predictions
with generally poor comparisons to experimental mea-
surements. While linear instability techniques may give
insight into initial instabilities, these results are not valid
once the waves reach finite amplitude states. Being able
to predict the transition from finite amplitude waves to
slug flow remains a difficult challenge which cannot be
solved without robust nonlinear analysis.
Later stages of the transition process are complicated
by the effects of turbulence. Once the waves gen-


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


rated by the initial instability reach large amplitude
states, strong viscous and turbulent mechanisms gener-
ate significantly more complex flow physics such as bub-
ble/spray formation and breaking waves. As the waves
eventually bridge the pipe diameter, the slug body elon-
gates and becomes turbulent and bubbly. The presence
of such effects makes it unlikely that accurate predic-
tions of slug formation can be made without a detailed
understanding of the turbulent flow physics.
Recently, a study by Valluri et al. (2008) carried
out direct numerical simulations (DNS) of the Navier-
Stokes equations to examine the initial evolution of the
interface into slug-type structures in a pressure driven
channel flow. Good agreement was found with the linear
Orr-Sommerfeld instability predictions and it was found
that large amplitude structures evolve through a non-
linear interfacial mechanism. While these simulations
were stopped prior to the wave touching the channel lid,
these results show that DNS will be able to address is-
sues which could not be easily examined theoretically or
experimentally.
In addition to slug formation conditions, predicting
the length of the slug body, the pressure drop along the
pipeline, and frequency of slugging is important for en-
gineering design. Numerical methods, such as OLGA,
have been developed that can carry out the large scale
simulations needed for modeling flow through pipelines.
These methods are based on phenomenological models,
area averaged techniques or rely on RANS codes. Due
to the length scales involved in the problem, these ap-
proaches are necessary in order to be computationally
tractable; however, as a consequence, they are not ca-
pable of resolving the important small scaled turbulent
flow physics. More accurate predictions could be made
from these industrial codes if these phenomenological
models could be supplemented with detailed physics-
based, multiphase closure models obtained from DNS,
which capture all relevant length and time scales of the
problem or large eddy simulations (LES), which uses
physics-based closure models to approximate the sub-
grid scale physics.
This paper describes two numerical methods which
have been developed for the purpose of determining
some of the underlying physical mechanisms which
cause the transition from a stratified flow to large am-
plitude wavy states and the eventual formation of liquid
slugs. One method captures the nonlinear evolution of
instabilities grown from a flat interface using a potential
flow based high-order pseudo-spectral method. The sec-
ond method carries out direct numerical simulations of
the two-fluid Navier-Stokes equations for the purpose of
growing a large amplitude wave into a slug. This code
can illuminate the dominant physical processes involved
in the later stages of slug development, examine the con-











tact problem as the wave touches the pipe wall, and pro-
vide physical models for the turbulent behavior inside of
the slug body.


Numerical Methods

In order to accurately simulate the development of large
amplitude disturbances and the transition to slug flow, it
is necessary to capture the appropriate physics inherent
to the problem. In order to do this, while still remain-
ing computationally tractable, two separate numerical
schemes are described. The first method, referred to as
"Long-Pipe", examines the evolution of the flow using a
potential flow formulation. The efficiency and accuracy
of this method makes it computationally inexpensive to
examine the initial growth and nonlinear evolution of a
flat interface into large amplitude waves. Once the am-
plitude of the disturbances becomes large, viscous (tur-
bulent) effects become dominant and are examined by a
secondary method referred to as "Short-Pipe". Through
direct numerical simulations, more complicated phe-
nomena such as spray formation, wave breaking, and the
early stages of slug flow development can be examined.
While the solution generated by Long-Pipe does not
satisfy the same conditions as a viscous solution (conti-
nuity of velocity and stress at the interface and no slip
at the walls), it is successful in identifying key physics
which should be incorporated into the Short-Pipe initial
condition. Following this procedure allows for the ex-
amination of realistic large amplitude disturbances with-
out the 'igiiii.iiill computational expense of using DNS
to simulate the growth of nonlinear waves from a flat in-
terface. In the following sections, the formulation and
capabilities of both of these numerical schemes shall be
discussed.
Together these codes can aide in understanding the
physical processes governing multiphase flow regime
transitions and develop physics-based models used for
industrial applications.


Long-Pipe

The formulation of Long-Pipe assumes that the flow is
composed of two incompressible immiscible fluids in a
horizontal channel. The coordinate system is located at
the equilibrium position between the two fluids with y-
directed upwards. The vertical displacement of the in-
terface away from the equilibrium position is described
by the function y = n(x, t). The upper and lower fluids
have densities p, and p, and flow with constant uniform
currents Uu and U1 respectively. The initial depths of
each of the fluid layers are hu and hi. Both fluids are
assumed to be irrotational and can be described by po-


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


tential functions cp and p which are governed by


V2 0 < y < h


V2p 0


h < y < TK.


No flux conditions are enforced at the channel walls re-
quiring


y h.


0 y = -h.


Kinematic constraints are enforced between the two flu-
ids requiring that the interface remain material for all
time


y = (5)

y = (6)


at ( 5x) ax


The dynamic boundary condition represents a balance
between the fluid pressures, surface tension, and can in-
corporate a slope coherent pressure forcing term which
models the effects of shear stress at the interface

R [i + 1|P, 2| + U. 0,4+ 1 1


at 2 0x FF,2 9
2 Nj~ 2 09 2 1 r/x
Re [ Y2 y2 We (1i +, ,)3/2
Y = (7)

where B is a dimensionless pressure forcing coefficient,
Re is the Reynolds number, and We is the Weber num-
ber.
To solve this system of equations, an efficient high
order spectral method developed by Campbell (2009)
was utilized. This method efficiently tracks the evolution
of a large number of wave components in a broadband
spectrum and accounts for their nonlinear interactions up
to an arbitrary high order of nonlinearity using a pseudo-
spectral approach. Validation tests have shown that the
method converges exponentially with interaction order
and the number of wave modes (or grid refinement).
Due to the efficiency of this method, Long-Pipe is ca-
pable of the simulation of flow physics over large scale
(L/D ~ 1000). It can be utilized to generate further de-
veloped nonlinear initial conditions for direct numerical
simulations which would significantly reduce the com-
putational expense of Short-Pipe by not requiring that it
simulate disturbances from a flat interface.







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


Short-Pipe

Short-Pipe carries out direct numerical simulations of
the incompressible, two-fluid Naiver-Stokes equations
for a domain with variable fluid properties. To account
for these effects, this Cartesian Grid method captures
the interface using the level set method, which advects
a signed distance function p (Y) for which p (7) 0
identifies the interface between the two fluids p (7) > 0
in the lower fluid and p (7) < 0 in the upper fluid. This
level set function is governed by


+ u V =- 0


where i is the fluid velocity. Knowing implicitly the lo-
cation of = 0 then allows for all of the fluid properties
within the domain to be identified by:

p (0) = p./pi + (1 p/lp)H (0) (9)
S( /) + //i + (1 1p/1)H () (10)

where H (y) is a Heaviside function. Applying this
technique then allows for the Naiver-Stokes equation to
be cast in the form:


D [P()]
Dt


1V V
-VP + V (2p (0) 7)
Re


p() 1
p()k+ 1K6 () V (11)


where is the material derivative and 6(0) is a Dirac
delta function.
As in Valluri et al. (2008), the level set implementa-
tion is smoothed over a few grid points c and the Heavi-
side function H (&) becomes

(1 7 e lower fluid
H(4; e) f(; e) f E interface region .
0 7 e upper fluid
(12)
and f(0; c) is a smoothing function evaluating to 1, 1/2
and 0 for p ec, 0, -e, respectively.

f( ; ) = (+ + sin( )) for < .
(13)
For this work, these equations are solved in two di-
mensions using a second-order finite volume method on
a structured grid over a rectangular domain. A third or-
der QUICK scheme is used to calculate the advection
terms in the Navier-Stokes equations. The pressure is
calculated using a Chorin projection method where the
continuity equation is used to project the velocity onto a
divergence-free field. A V-cycle multigrid solver is used
to solve the resulting Poisson equation.


The level set equation is advected using a conservative
form and central differences. While typical implemen-
tations use high-order WENO schemes for this equation
(Osher & Fedkiew 2003), experience has shown that this
type of implementation is not necessary if the interface
does not form sharp corners and the additional numer-
ical dissipation and volume loss of the WENO scheme
is avoided. Through the course of advection, the level
set function loses its signed distance property and re-
quires periodic reinitialization to a signed distance func-
tion. The reinitialization stage is performed through a
psuedo-time integration of a partial differential equation
as in Sussman & Fatemi (1999). Finally, time integra-
tion of the Navier Stokes and level set equations is im-
plemented using an explicit second order total variation
dimension (TVD) Runge-Kutta scheme as proposed in
Trygvasson et al. (2007).
Periodic boundary conditions were imposed along the
length of the domain and both no flux and no slip con-
ditions were imposed along the channel walls. A fixed
pressure gradient, with amplitude P(t), along the length
of the channel is applied to enforce desired upper and
lower velocities. P(t) may be constant or a tanh func-
tion. For all of the cases presented here, the flow is initi-
ated from a small-amplitude two phase Airy wave solu-
tion (Hendrickson 2004). If P(t) is constant, a mapped
linear Poisuielle flow solution is added to the Airy wave
solution.
Due to the fact that this code carries out direct numeri-
cal simulations of the Navier-Stokes equations and is re-
quired to resolve all small scale physics, this code is lim-
ited to relatively small domain lengths (L/D ~ 1 10).

Long-Pipe Numerical Results

Long-Pipe was validated through several rigorous tests.
The first validation trial demonstrated convergence of
the method against the analytic solution of the linear
Kelvin-Helmholtz problem for both stable and unsta-
ble waves over a large range of wavenumbers. It was
found that the numerical solutions converged exponen-
tially to the analytic solutions with the refinement of the
time step or the grid spacing. This process validated the
numerical scheme for first order expansions.
As a secondary validation of Long-Pipe, a comparison
was made to a second order nonlinear solution of Eqns.
1-7 obtained by Campbell (2009). It was found that for
second order nonlinearity, an interface composed of two
specially selected linear modes with wavenumbers and
frequencies (ki, wc) and (k2, w2) could satisfy a triad
resonance condition of the form


k2 k = k3
W2 W1 = 3.







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


10-2
10.

i10-
10'

S10'


10' Fo k


200 300 400 500
Nondimensional Time


Figure 1: Comparisons of wave amplitude variations in
a resonant triad between the Long-Pipe sim-
ulation (circle) and theoretical multiple-scale
solution (solid line).


with (k3, W3) satisfying the linear dispersion relation-
ship. When these conditions are satisfied, a set of non-
linear ordinary differential equations of the form

da B23a2a (16)
dt


B13saa3

B12a a2


(where the represents the complex conjugate) are ob-
tained by use of the method of multiple scales which de-
scribe the resonant interactions. Comparisons between
the solution of these nonlinear interaction equations and
the numerical method validate the accuracy of Long-
Pipe for calculating nonlinear solutions. A case with
D 2.54 cm, U, 3.5m, U, (p,.I iI" a 0.8,
R = 1.23 x 10-3, g 9.81 -, ki 27m- and
ki2 m1 is shown in Figure 1. In this trial, surface
tension, viscosity, and the slope coherent pressure forc-
ing were not included. For this trial, a (0) a 2(0) -
10 5m and a3(0) Om. Good agreement is observed
between the theoretical and numerical solutions.
Traditionally, linear stability analysis is carried out
to determine if it is possible for long waves to be-
come unstable; however, it is commonly observed ex-
perimentally that flows predicted to be stable to Kelvin-
Helmholtz experience wave growth. Through Long-Pipe
simulations, it was found that long waves could form
due to the transfer of energy from unstable short waves
to stable long waves through nonlinear resonant wave-
wave interactions. Consider the case where D 0.5 m,
U, = 10m, U 1m, a 0.3, R 1.23 x 10-3,
g = 9.81m, ki = 327m- k2 = 347m-1, al(0)


Figure 2: Time variation of the modal amplitudes of
waves while undergoing resonant interactions
with k2 and 2i _. being unstable to Kelvin-
Helmholtz instability.


a2(0) 10 m, a3(0) 0 and a 7.34 x 10 2N
For these flow conditions, all wavenumbers less than
k 347m 1 are stable. From linear theory, long waves
(A ~ O(D)) are predicted to be stable.
If nonlinear resonant interactions are considered, it is
found that k2 satisfies a second harmonic resonance (the
phase velocity of k2 and its second harmonic are equal).
This is unique because 2i _. is linearly unstable and re-
sults in the transfer of energy between the wave modes
k2 and 2i _.. Simultaneously, k2 and kl form a triad res-
onance with the difference component k = k2 k1.
This leads to the resonant exchange of energy amongst
the modes involved in the triad. These two resonant
mechanisms occur simultaneously and rapid growth of
the longest wave component k3 is found to occur as a
result of this cascaded of energy from the unstable short
waves to the stable long waves. When the interactions
begin, k2 generates its second harmonic 2li due to the
inherent nonlinear interactions of the interface. Since
2i _. falls in the unstable wavenumber range, it begins to
grow exponentially with time. Due to the second har-
monic resonance, energy from the linear instability of
the -l _2 mode is transferred back to k2. However, since
k2 is simultaneously involved in a triad resonance with
ki and k3, the energy gained by k2 from the second
harmonic resonance is distributed among the resonant
triad. This cascade of energy through the coupled res-
onant mechanisms results in rapid growth of the long
mode k3 as can be seen in Figure 2. It can be seen that
all wave modes can grow simultaneously and can reach
amplitudes which are several orders of magnitude larger
than the initial values.
It should be noted that these resonance conditions are







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


1 3 7
x (M)


Figure 3: Large amplitude long wave generated by
spectra of nonlinear resonant interactions with
maximum disturbance amplitude ~ 0.25h,.


not a rare occurrence. For a given spectrum, it is possi-
ble for there to be many wave modes which satisfy the
triad resonance condition. When these additional wave
modes are considered, ignikii.lli amounts of energy are
transferred to the long wave components. Consider the
case where the initial interface is composed of all modes
between k = 327rm landk2 347m 1 and all of the
higher harmonics as well as all sum and difference inter-
actions are permitted. For this band of wavenumbers,
nearly all components form a triad resonance (or near
resonance) with other components in the spectrum and
participate in the transfer of energy from short to long
waves through a similar mechanism as was previously
shown. Simulations produced by Long-Pipe show that
the resulting nonlinear evolution of the interface gen-
erates large amplitude long waves as shown in Figure
3. This long wave has an amplitude O(0.25h,) and a
wavelength O(2D).
In the last simulation, the numerical method had is-
sues due to the growth of very short waves. Due to
the linear instability, the short waves grew quickly and
reached large steepness values. Since the interface was
represented through a Fourier series, it was possible for
the short waves to reach the breaking point and stop
the simulation. To overcome this, a wave breaking fil-
ter based on the wave steepness was applied. This fil-
ter acted on the short waves by monitoring their modal
steepness (ak). Once a short wave component reached
a particular threshold value, the amplitude of that com-
ponent was reset to a small value ~ O(0.la). Several
tests were conducted to determine the effect that this
threshold value had on the global solution and the dif-
ferences were found to be iniiigilii,.i Removing the
short waves was not acceptable because the short wave
components generated the unstable energy which was
transferred to the long wave components. Removing the
short waves completely cripples the efficiency of the res-


a 20 30 4 0 W M EU W

Figure 4: Standard decay test of two-phase Airy wave
using Short-Pipe. (green) r(xp, t) (blue) the-
oretical decay envelope.


onant mechanism. It is desirable to extend these simu-
lations to the case of a broadband spectrum because as
the number of unstable modes increases, so will the am-
plitude and growth rate; however, a more robust filtering
technique is needed for this.


Short-Pipe Numerical Results

Of particular interest for Short-Pipe simulations is en-
suring that the dissipation associated with the numerical
scheme is not excessive compared to the viscous dissi-
pation. A standard test for single phase viscous flow
solvers is to simulate an Airy wave and compare the
amplitude decay with theoretical values. In a similar
test simulating a two-phase Airy wave, Short-Pipe was
shown to predict the appropriate viscous dissipation over
a simulation of 36 wave periods. Figure 4 is a wave
probe record with Re=1000, We=oo, Fr=l, pu/pl=O.Ol,
pt,/bi=0.02 and wave amplitude of a 0.1/(27). The
theoretical decay rate is also shown (Hendrickson 2004).
The simulation was performed on a 1282 grid with dt ~
0.006 on 1 node of a dual-processor, dual-core Athlon-
based cluster, utilizing all four cores through a parallel
MPI implementation, in ~ 15 minutes for 15,000 time
steps.
As the purpose of Short-Pipe is to understand the de-
tailed flow dynamics in the later stages to develop and
validate physics-based turbulence models, the simula-
tions of multiphase pipe flows over a range of operating
parameters is critical. The DNS datasets generated as a
part of this effort will be used for a priori analysis to de-
velop and validate physics-based turbulence models for
use in larger scale simulations such as Large Eddy Sim-
ulation (LES) and RANS. To date, Short-Pipe has been
utilized to generate over a dozen datasets within two
classes of runs. The first class of runs, called Case A, are


I 1 1 I I I I I


0l01

o 005








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


100
80
60
40
20
20
-40
60
80
100


(a)


200
160
120
80
40
-40
-80
-120
-160
200


Figure 5: Case A simulation of a two-phase air-water
Airy wave with a pressure gradient by Short-
Pipe. U,,=0.62 m/s, U,1=0.015 m/s, and
a=0.5. Re=2000, Fr=l, We=oo. In (a), black
line is interface between two fluids and chan-
nel wall.



for air and water at a Reynolds number of 2000, which
equates to a channel height of only 1.25 cm. While
Short-Pipe is capable of including the effects of surface
tension, none are currently included in these classes.
For Case A, Us,/Us8 ranged from 20-100 resulting in
mainly entraining events. Figure 5 is a simulation of
a two-phase Airy wave with wavelength equal to the
pipe diameter that has the pipe pressure increased from
zero over a period of time resulting in Us,=0.62 m/s and
U,1=0.015 m/s. Figure 5a shows the transverse vortic-
ity contours while Figure 5b shows the velocity profile,
where select vertical slices are shown for clarity. Flow is
from left to right. Separation off of the crests impacts the
back face of the downstream wave. There is evidence of
an entrainment event as the surface begins to pinch off
at the crest of the wave. The resolution for this case
is 512x256 and is not sufficient to resolve the interaction
of the crest separation vortex impacting the interface and
thus the simulation stops.
The second class of runs, called Case B, are for
high-viscosity oil (0.181 PA-s) in a 5 cm channel with
Reynolds number 0(200) (Gokcal et al. 2006). These
cases have Us,/U,1 of 0.1-10 and result in mainly elon-
gated bubble events, where the interface attaches to the
channel wall and produces little to no entrainment. Fig-
ure 6 is a simulation of a small amplitude two-phase Airy


S11100
60
40
S20
-20
6-40
100



(b)


Figure 6: Case B simulation of a two-phase Airy wave
of air with high-viscosity oil with a pres-
sure gradient by Short-Pipe. U,,=0.09 m/s,
Us,=0.8 m/s, and a=0.9. Re=200, Fr=l,
We=oo. In (a), black line is interface between
two fluids and channel wall.



wave (ka 0.01) with a mapped linearized Poisuielle
flow added to the velocity profile. The wave is forced
via slope-coherent pressure forcing until it touches the
top of the channel, forming an air cavity. Shown in Fig-
ure 6a is the transverse vorticity contours and in Figure
6b is the velocity field. Again, select vertical slices are
shown for clarity. The boundary layers for the walls and
interface are clearly visible as is the formation of a jet
inside the air cavity.


Conclusions

Two complementary computational capabilities, Long-
Pipe and Short-Pipe, form a framework that is used
to understand the transition and nonlinear evolution of
multiphase pipe flows in support of the design of deep-
water pipelines. Long-Pipe is used to understand the
initial growth and nonlinear evolution of a flat interface
into large amplitude waves and Short-Pipe is used to un-
derstand complex interfacial phenomena such as wave
breaking, slug flow and entrainment. These two capa-
bilities will be used in tandem to understand the physi-
cal processes governing the transition of multiphase flow
regimes and develop physics-based models for industrial
applications.
Preliminary studies using Long-Pipe have demon-


200
160
120
80
40
40
-80
120
160
200


mD











strated it's abilities to develop large amplitude long
waves from a "nearly" flat interface using the funda-
mental physics of the system. A nonlinear resonant in-
teraction theory was found which transfers the energy
generated by linearly unstable waves across the wave
spectrum. This cascade of energy focuses energy among
the long wave modes and generates large amplitude long
waves which could possibly lead to the development of
large amplitude roll waves or slugs. The near-term goal
of Long-Pipe is to improve flow transitions criteria by
including nonlinear resonant wave interaction effects.
A state-of-the-art multiphase CFD solver, Short-Pipe,
has been developed and validated for use in DNS of
late stage flow dynamics in pipes. Its ability to sim-
ulate a range of operating parameters (fluid constitu-
tive properties, Reynolds numbers, U,,/Us, and chan-
nel height) has been established. Future applications of
Short-Pipe will be to further study simulations within the
two classes of waves presented at improved resolutions:
(i) larger Reynolds number air-water flows with large
U,g and (ii) lower Reynolds number air-high-viscosity
oil flows where slugs and elongated bubbles are known
to form. A third class of simulations are currently be-
ing investigated for SF6-water flows at Reynolds num-
ber 0(2500), as experimental results report that simi-
lar operating conditions form roll waves with igmniik.ii.i
gas entrainment in the water phase. These three classes
of simulations will be used in a priori development and
validation of physics-based turbulence closure modeling
for large-scale applications.

Acknowledgements

This research was sponsored by Chevron Corporation.

References

Barnea, D. and Taitel, Y. 1993. Kelvin-Helmholtz sta-
bility criteria for stratified flow: viscous versis non-
viscous(inviscid) approaches. Int. J. Multiphase Flow
19, 639-649.

Campbell, B. 2009. Nonlinear Effects on Interfacial
Wave Growth into Slug Flow. M.S. Thesis. Mas-
sachusetts Institute of Technology.

Funada, T. and Joseph, D.D. 2001. Viscous potential
flow analysis of Kelvin-Helmholtz instability in a chan-
nel. J. Fluid Mech. 445, 263-283.

Gokcal, B., Wang, Q., Zhang, H.-Q., and Sarica, C.
2006. Effects of High Oil Viscosity on Oil/Gas Flow Be-
havior in Horizontal Pipes. Presented at the 2006 SPE
Annual Technical Conference and Exhibition, San An-
tonio, TX, USA.


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


Hendrickson, K. 2005. Navier-Stokes Simulation of
Steep Breaking Water Waves with a Coupled Air-Water
Interface. Ph. D. Thesis. Massachusetts Institute of
Technology.

Lin, P.Y. and H.iiii.ii\ T.J. 1986. Prediction of the initi-
ation of slugs with linear stability criterion. Int. J. Mul-
tiphase Flow 12, 79-98.

Mata, C., Pereyra, E., Trallero, J.L., and Joseph, D. D.
2002. Stability of stratified gas-liquid flows. Int. J. Mul-
tiphase Flow 28, 1249-1268.

Osher, S. and Fedkiew, R. 2003. Level Set Methods and
Dynamic Implicit Surfaces. Applied Maethematical Sci-
ences vol. 153. Springer.

Sussman, M. and Fatemi, E. 1999. An efficient,
interface-preserving level set redistancing algorithm and
its application to interfacial incompressible fluid flow.
SIAM J. Sci. Comput. 20(4), 1165-1191.

Taitel,Y. and Dukler, A.E. 1976. A model for predicting
flow regime transitions in horizontal and near horizontal
gas-liquid flow. AIChE J 22, 47-55.

Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-
Rawahi, N., Tauber, W., Han, J., Nas, S., and Jan, Y. -J.
2001. A Front-Tracking Method for the Computations
of Multiphase Flow. J. Comp. Phys. 169:2, 708 759.

Valluri, P., Spelt, P.D.M., Lawrence, C.J., and Hewitt,
G.F. 2008. Numerical simulation of the onset of slug ini-
tiation in laminar horizontal channel flow. Int. J. Multi-
phase Flow 34, 206-225.




University of Florida Home Page
© 2004 - 2010 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated October 10, 2010 - Version 2.9.7 - mvs