Tryggvason

DNS of Multiphase Flows
Grétar Tryggvason,
Ming Ma & Jiacai Lu
University of Notre Dame
DNS Assisted Modeling of
Bubbly Flows in Vertical
Channels
2015 NETL Workshop on Multiphase Flow Science
Lakeview Resort, Morgantown, WV, August 12, 2015
Work supported by NSF & DOE (CASL & PSAAP II)

Bubbles in Vertical Channels

DNS of large systems of disperse multiphase
systems (hundreds of bubbles, drops and particles
in turbulent flows) are rapidly becoming relatively
routine and have been used to explore the
elementary aspects of several systems.
While further studies of relatively simple systems,
as well as the development of more accurate,
robust, and efficient methods is important, current
progress provides new opportunities. Those
include:
• Using the current capabilities to greatly advance
modeling of multifluid and multiphase systems,
and
• Develop methods for much more complex flows
and explore new problems.
g
W
L
Direct Numerical
Simulations:
Fully resolved and
verified simulation of a
validated system of
equations that include
non-trivial length and
time scales

The method has been used to
simulate many problems and
extensively tested and
validated
Tracked front to advect the
fluid interface and find surface
tension
Fixed grid used for the solution
of the Navier-Stokes equations
Front Tracking
Numerical Method
r
¶u
¶t
+ rÑ× uu = -Ñp+ f + Ñ× m Ñu + ÑT
u( )+ sFò knd x - xf( )da
Singular interface term
Ñ×u = 0
Dr
Dt
= 0;
Dm
Dt
= 0

Upflow
Downflow
Spherical Bubbles in Vertical Channels
For nearly spherical bubbles in laminar
and weakly turbulent flows the flow
consists of a homogeneous core
where the mixture is in hydrostatic
equilibrium and a wall-layer.
For upflow the wall-layer is bubble rich
and the total flow rate depends
strongly on the deformability of the
bubbles.
For downflow the wall-layer has no
bubbles and the velocity profile is
easily found for both laminar and
turbulent flow. For downflow the exact
size of the bubbles plays only a minor
role, as long as they remain nearly
spherical.
For upflow deformable bubbles stay
away from walls, completely changing
the flow structure

YUmean
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
Fluid velocity
Average bubbles’ velocity
Upflow
Y
Umean
0 0.5 1 1.5 2
-1
-0.8
-0.6
-0.4
-0.2
0
Fluid velocity
Analytical fluid velocity
averaged bubbles’ velocity
Downflow
The average average void fraction (left) and the vertical liquid velocity (right)
profile across the channel for the upflow (top) and the downflow (bottom).
The red line shows the hydrostatic model for the void fraction.
Bubbles in Laminar Channel Flows
Y
Voidfraction
0 0.5 1 1.5 2
0
0.1
0.2
0.3
simulation
model
Downflow
Y
Voidfraction
0 0.5 1 1.5 2
0
0.1
0.2
0.3
0.4
0.5
0.6
simulation
model
Upflow
2tw = -bW , (4)
annel. This relationship, together with a constant flow rate, can be used to check whether
state. The local density is a function of the void fraction, r = erg + 1-e( )rl , and the
and the average channel void fraction, eav, are related by rav = eavrg + 1-eav( )rl .
d
dx
t (x)- b - gDr eav -e(x)( ) = 0 , (5)
ensity difference. Putting the shear equal to zero, as it is in the center of the channel,
ec = eav +
b
gDr
. (6)
e void fraction in the center will therefore be less than eav, but for downflow, where
center will be larger than eav . If the pressure gradient is sufficiently large for upflow,
e walls and the void fraction in the center becomes zero. The discussion below is limited
b gDr > 0 . For downflow, the only limitation is that the void fraction in the center
hough in practice the assumption of small spherical bubbles is likely to break down at
that increase the void fraction in the center come from the walls, resulting in a bubble
s can be found by mass conservation:
eavW = ec W - 2d( ). (7)
n in the core region gives the wall-layer thickness:
bW
© 2014 The Japan Society of Mechanical
continues until the weight of the mixture is balanced
average shear is reduced to zero and the migration of
ces the weight of the fluid mixture there, or increases
ure gradient that drives the motion downward. The
ation stops. Thus, for both upflow and downflow the
h that it matches the imposed pressure gradient, and
6). At steady state a force balance on an elementary
= 0 . (3)
ressure gradient and the last term is the weight of the
ixture, or the sign of b = dp / dy+ ravg, determines
he weight of the mixture and the flow is upward, but
the wall shear must balance the net pressure gradient

Flow Flow
Spherical Deformable
Turbulent Upflow: Effect of Deformability
M=1.54 ✕ 10-10
Eo=0.45
M=1.54 ✕ 10-7
Eo=4.5
The path of the bubbles (the vertical
coordinate versus time) and iso-
contours of the instantaneous
vertical velocity in a plane through
the middle of the channel for the
upflow of nearly spherical (left) and
much more deformable (right)
bubbles at one time when the flow is
approximately at steady state.
J. Lu, S. Biswas, and G. Tryggvason, “A DNS study
of laminar bubbly flows in a vertical channel,” Int’l J.
Multiphase Flow 32, 2006, 643-660.
L. Lu and G. Tryggvason. “Effect of Bubble
Deformability in Turbulent Bubbly Upflow in a
Vertical Channel.” Physics of Fluids. 20 040701
(2008).
J. Lu and G. Tryggvason. Dynamics of nearly
spherical bubbles in a turbulent channel upflow.
Journal of Fluid Mechanics 732 (2013), 166-189.

512 x 384 x 256 grids, 192 cores)
Re+ = 250; void faction = 3%
Turbulent Upflow: Different bubble sizes
J. Lu & G. Tryggvason:
JFM 732 (2013), 166-189

Closure Relations
by Statistical
Learning using
DNS Data
9
With
Ming Ma & Jiacai Lu

Averaged vertical momentum of the liquid:
Horizontal flux of bubbles
Void fraction and phase averaged velocity
A simple description of the average flow is derived
by integrating the vertical momentum equation and
taking the density and viscosity of the gas is zero
Finding Closure Terms by Data Mining
Obviously:
g
W
L

By averaging the DNS results over planes parallel to the walls, we construct
the Table above with quantities that are known and unknown in the averaged
equations. Using Neural Networks, we fit the data, resulting in:
Fg < ¢u ¢v > fs ag
¶ag
¶x
¶ < v >l
¶x
dw kt et a aij
“Closure” variables
needed for models
of the average flow
Resolved average
variables
Quantities
summarizing the state
of the unresolved flow
Fb
= f1
x( ); < u'v' >= f2
x( ); Fs
= f3
x( ); x = a,
¶a
¶x
,
¶< v >
¶x
,dw
æ
è
ç
ö
ø
÷
Not
include
yet
These relationships are used when solving the average
equations for the void fraction and the vertical liquid velocity
Data obtained by averaging the DNS results
Closure Terms by Statistical Learning

The bubbles
are rapidly
pushed to the
walls by the lift force. The
flow then slowly slows down
and finally some of the bubbles are
pushed back into the middle to establish
an hydrostatic equilibrium in the bulk
To generate a data base that can be mined for closure
laws, we have done DNS of the transient evolution
of an initially parabolic laminar flow with a
uniform distribution of bubbles that
remain nearly spherical. The
domain is bounded by two
vertical walls and periodic
in the streamwise
and spanwise
direction

Averaged
DNS results
and Model
predictions
using the
ANN closure
terms at
several
different
times

Flows without walls:The predicted closure terms versus the DNS data for the training part of the data (top)
and the part retained for testing (bottom) for both the gas flux (left) and the streaming stresses (right). The
solid line is perfect agreement and the dashed line is best fit.
The convergence
of the fit for both
the gas flux and
the streaming
stresses
The
distributio
n of the
error in
the data

The closure
relations
derived from
the upflow
cases
applied to
downflow

More Complex
Gas-Liquid
Flows
16
With
Ming Ma & Jiacai Lu

500 bubbles of different sizes in channel flow with
Re+=500 computed on a 1024 × 768 × 512 grid
using 2048 processors on the Titan
Turbulent Bubbly Channel Flow

Turbulent Bubbly Channel Flow

Flow Regime Transition

                                Time = 10.0

                                Time = 20.0

                                                                                            Time = 30.0

                                                                                             Time = 40.0

                                                                                                                                   Time = 70.0


Latest results:   Time = 100.0                               Time=100.0                             Time=80.0                                            Time=70.0                                       Time=62.5
Time = 10.0

Time = 20.0

Time = 20.0


Time = 40.0



                                                                                                                                   Time = 70.0


Latest results:   Time = 100.0                               Time=100.0                             Time=80.0                                            Time=70.0

Latest results:   Time = 100.0                       Time=100.0                             Time=97.0                                          Time=80.0


Latest results:   Time = 100.0                       Time=100.0

A =
1
Vol
nnda
S
ò
Low We High We
High We
Low We

DNS of Multiphase FlowsThis channel is the same as the one used in the simulation of microbubble-driven drag reduction. The initial number of
bubble is 60, with an initial diameter of 0.4.
T=0.0 T=10.0 T=20.0
T=30.0 T=40.0 T=50.0
This channel is the same as the one used in the simulation of microbubble-driven drag reduction. The initial number of
bubble is 60, with an initial diameter of 0.4.
T=0.0 T=10.0 T=20.0
T=30.0 T=40.0 T=50.0
T=60.0 T=70.0 T=8.0
T=60.0 T=70.0 T=8.0
T=90.0 T=99.0
Flow Regime Transition
Does the evolution
depend on the details
of the coalescence?

The effect of applying a top hat filter with a size
slightly larger than the diameter of the smallest
bubbles to both the velocity and the interface. Large
bubbles and vortical structures are smoothed and
small bubbles become point particles
“LES-like” filtering
tribution has
sipation rate
city squared
iddle of the
shows most
state results
we expect its
ticeable that
not changed
at care must
for turbulent

Modeling Challenges and Opportunities:
The enormous of amount of data generated by DNS—and increasingly by
experiments—will allow reduced order models that involve large number of
variables and complex relationships between the resolve and unresolved
variables and are applicable to complex flows
Determining complex nonlinear relationships from massive data involving a
range of physical scales using modern statistical learning is becoming easier
Modeling challenges will therefore shift to the development of more
sophisticated and comprehensive models, the identification of the
appropriate variables, and the incorporation and propagation of physical and
model uncertainties
The inclusion of limiting cases, such as where the relationships are known,
or the scaling is understood, in fitting is currently difficult but is likely to
become increasingly important

Tryggvason

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