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PHYSICS OF FLUIDS 27, 092101 (2015)
Using statistical learning to close two-fluid multiphase flow
equations for a simple bubbly system
Ming Ma, Jiacai Lu, and Gretar Tryggvasona)
University of Notre Dame, Notre Dame, Indiana 46556, USA
(Received 5 February 2015; accepted 21 August 2015; published online 15 September 2015)
Direct numerical simulations of bubbly multiphase flows are used to find closure
terms for a simple model of the average flow, using Neural Networks (NNs). The flow
considered consists of several nearly spherical bubbles rising in a periodic domain
where the initial vertical velocity and the average bubble density are homogeneous in
two directions but non-uniform in one of the horizontal directions. After an initial
transient motion the average void fraction and vertical velocity become approxi-
mately uniform. The NN is trained on a dataset from one simulation and then used to
simulate the evolution of other initial conditions. Overall, the resulting model predicts
the evolution of the various initial conditions reasonably well. C 2015 AIP Publishing
LLC. [http://dx.doi.org/10.1063/1.4930004]
I. INTRODUCTION
Direct numerical simulations (DNS) of multiphase flows, where every continuum length and
time scale are fully resolved, are starting to provide new insight into various aspects of the dy-
namics of such flows.1–3
For most flows of industrial interest, however, DNS require too fine grid
resolution to be practical on current computers. Furthermore, since the small scale behavior often
exhibits significant universality, it is likely that in many cases the small scales can be modeled rather
than fully resolved. For single phase flows, DNS have resulted in significant progress in modeling
unresolved scales, including how they depend on the large scale resolved flow and how the small
scales in turn influence the larger scales. Developments of new models have, in particular, lead to
new questions that have motivated new simulations.
Modeling of multiphase flows generally builds on the two-fluid model,4,5
where separate equa-
tions are solved for the average motion of the different fluids. The current status of such models
has been reviewed by a number of authors.6,7
Although DNS of multiphase flows have progressed
rapidly during the last two decades, attempts to use DNS results to help modeling of multiphase
flows are still rare. The most intense effort appears to have been by a French group, who has
published several papers on LES’s (large eddy simulations) modeling of multiphase flows, including
the derivation of the filtered LES equations, identification of the unresolved terms, and comparisons
of model results with DNS.8–11
In the present paper we explore the use of Neural Networks (NNs) to obtain closure relations
for the two-fluid equations for the average motion for a simple unidirectional bubbly flows in a peri-
odic domain, by mining DNS results. We follow the evolution of one initial bubble distribution and
velocity profile using DNS and then fit the data to produce the closure relations for the equations for
the average flow. The model is then used to predict the evolution of other initial conditions.
Neural networks have been used extensively to correlate experimental results for multiphase
flows. Such applications include the identification of flow regimes in gas-liquid flows12–14
and
gas-liquid-fiber slurries,15
as well as to estimate the proportions of oil, water, and gas in oil transfer
pipelines from noninvasive but indirect measurements.15–18
Indeed, one such data set19
has been
used as an example in a popular introduction to machine learning.20
Use of neural networks for
correlation of experimental data for single phase flows includes predictions of aerodynamic coeffi-
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092101-2 Ma, Lu, and Tryggvason Phys. Fluids 27, 092101 (2015)
cients,21
processing of experimental results to reproduce the dynamics in models of the 2D flow,22
and correlation of experimental results for the mean velocity and turbulence intensities for channel
flows.23
Several authors have examined the use of neural networks for processing of data from single
phase DNS. Lee et al.24
used NN to develop an adaptive controller for turbulence control for drag
reduction; Milano and Koumoutsakos25
used NN for the reconstruction of the near wall flow in a
turbulent channel flow and showed that the neural network results compared favorably with recon-
struction by linear proper orthogonal decomposition; Sarghini et al.26
developed a subgrid scale
(SGS) model for LES, where the turbulent viscosity coefficient is found using the derivatives of
the instantaneous velocities of the turbulent flux and the Reynolds tensor components using NN;
Moreau et al.27
used NN for optimal estimation of subgrid models for large-eddy simulation of
turbulence; and Esau and Rieber28
used NN to parameterize the flow-to-surface interactions due to
unresolved surface features in coarse LES.
The use of neural networks to develop reduced order models for multiphase materials from
simulation data has a long history for solid mechanics applications29
and recent examples include
models for damage in concrete30,31
and bones.32,33
For multiphase flows the only application that we
are aware of where neural networks are used to extract closure relationships from DNS is the thesis
by Lu,34
where drag laws are found for particles accelerated by a shock.
II. PROBLEM SETUP
The computational domain is a rectangular box and gravity acts in the negative y direction. The
initial average velocity and void fraction depend only on the x coordinate, and not on y or z. As
the flow evolves, the various averaged quantities are therefore functions of x only. Assuming that
the gas phase has zero density and viscosity, that fluctuations in the viscous term can be neglected,
and that the bubbles remain spherical, the equations for the gas volume fraction, αg, and the average
vertical velocity of the liquid, v l, are easily derived by averaging over the y and z coordinates, as
shown in the Appendix. The results are
∂αg
∂t
+
∂
∂x
Fg = 0, (1)
for the gas volume fraction, and
∂
∂t
αl v l+
∂
∂x
αl u l v l
= −
1
ρl
dp0
dy
− gyαl + νl
∂
∂x
αl
∂ v l
∂x
−
∂
∂x
αl u′
v′
l (2)
for the average vertical velocity of the liquid. Here, Fg = αg u g, where u g is the average
horizontal velocity of the gas, is the horizontal gas flux, αl is the liquid volume fraction, and
u′
v′
l are the “streaming” stresses. Since αg + αl = 1, the first equation also gives αl. In addition,
αg u g + αl u l = 0, so the nonlinear term in the second equation can be written as αl u l v l
= −αg u g v l = −Fg v l. Since there are no walls, the pressure gradient is set to ρavggy to match
the weight of the mixture. The model equations are solved by a simple first order finite volume
method with sufficiently fine grid and small time steps to ensure that the solution has converged.
Since we made a few assumptions in deriving the average equations that are not satisfied exactly
by the DNS (such as ignoring the density and viscosity of the gas and assuming exactly spherical
bubbles), we have computed each term in the model equations from the DNS data to check how
well it is satisfied. The residual (computed as the difference between the left and right hand side) is
always less than 8% of the magnitude of left or right hand side.
To develop models for the closure terms, the gas flux Fg (or the average horizontal gas velocity
u g) and the streaming stresses u′
v′
l, we use the results of DNS of one system to “teach” the
model how they depend on the average variables. Then we evaluate how good the model is by
computing how other initial bubble distributions and velocities evolve and comparing the results
with DNS. To keep the development as simple as possible, in the present work we make the rather
significant assumption that the unknown closure terms depend only on the instantaneous average
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variables (and not on any average descriptions of the unresolved flow, such as the turbulent kinetic
energy, or on any history effect). The closure terms obviously cannot depend on the velocity, but
only its gradient. They can, however, depend on the void fraction, and in addition we include the
gradient of the void fraction. In principle the closure terms could depend on higher derivatives of the
velocity and the void fraction, but we have assumed that they do not. We have tested our selection of
variables on which the closure terms depend and found that the three that we have selected improve
the correlation coefficient and reduce the error between the DNS and model results compared to
selecting fewer variables. Our tests also showed that adding other variables, such as the second
derivative of the velocity, did not improve the results. Specifically, we assume that
Fg = f (x), u′
v′
l = g(x), with x = αg,
∂αg
∂x
,
∂ v l
∂x
, (3)
where f and g are the functions that are determined using data from the DNS. These functional
relationships can be found in many ways35
but here we use NN. The multilayer NN is a powerful
nonlinear regression technique, typically with a three-layer architecture called input layer, hidden
layer, and output layer. Each unit in the hidden layer is transformed by a nonlinear function of a
linear combination of all the variables in the input layer. Once the hidden units are defined, another
linear combination connects all the hidden units to the output units. Take the NN modeling of f (x)
for the Fg as an example: suppose there are m input units and n hidden units, then
hi(x) = s(a0i +
m
j=1
xjaji), (4)
f (x) = b0 +
n
i=1
bihi(x), (5)
where we use the hyperbolic tangent sigmoid function s(u) = 2/(1 + e−2u
) − 1, as the transfer func-
tion. The adjustable coefficients a0i,aji,b0,bi are found during the training of the NN.
Software packages to train neural networks are widely available and we have used the neural
network toolbox in Matlab. Once the training is done, Matlab returns functions that relate the input
variables to the output and here the fitting results in two functions, one for the gas flux and another
for the streaming stresses. These functions are then used in a Matlab program to solve Equations (1)
and (2).
The DNS are done using a finite-volume/front-tracking code that has been used in several
previous studies.1,3,36–38
The Navier-Stokes equations are solved by a second-order accurate projec-
tion method, using centered-differences on a fixed, staggered grid. In order to keep the boundary
between the different fluids sharp, to advect the density and the viscosity fields, and to accu-
rately compute the surface tension, the fluid interface is tracked by connected marker points (the
“front”). The front points, which are connected to form an unstructured surface grid, are advected
by the fluid velocity, interpolated from the fixed grid. As the front deforms, surface markers are
dynamically added and deleted. The surface tension is represented by a distribution of singularities
(delta-functions) located at the front. The gradients of the density and viscosity become delta func-
tions when the change is abrupt across the boundary. To transfer the front singularities to the fixed
grid, the delta functions are approximated by smoother functions, with a compact support, on the
fixed grid. At each time step, after the front has been advected, the density and the viscosity fields
are reconstructed by integration of the smooth grid-delta function. The surface tension is then added
to the nodal values of the discrete Navier-Stokes equations. Finally, an elliptic pressure equation
is solved by a multigrid method to impose a divergence-free velocity field. The method has been
applied to several multifluid problems and tested and validated in a number of ways.39
We note that
the DNS and the average model are complementary but not competing approaches, where the DNS
provide input for the average model. For the simple problem examined here, the DNS calculations
generally take several days on a parallel cluster while the average model takes only a few minutes
on a laptop computer. For more complex problems, where DNS are out of reach, we would expect
to use closure relationship developed for simples systems that nevertheless spanned the situations
encountered.
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Here, we examine just one set of material parameters and take the liquid density to be
ρl = 0.25, the liquid viscosity to be µl = 0.008, and surface tension to be σ = 0.4. The bubble
density and viscosity are one-twentieth of the values for the liquid, or ρg/ρl = µg/µl = 0.05. The
bubbles are all of the same size, with a diameter d = 0.18, but their number is varied to give
different volume fractions. The gravity acceleration is gy = 1. These parameters result in a Morton
number of M = gy µ4
l/ρlσ3
= 2.56 × 10−8
and an Eötvös number of Eo = ρlgyd2
/σ = 0.2025. The
domain size is 2.0 × 1.5 × 1.0 computational units in the x, y, and z direction, and the DNS have
been done using a 256 × 192 × 128 uniformly spaced grid. Simulations with both finer and coarser
grids show that the results are essentially fully converged.
III. RESULTS
A. Training of the model
The data used to “train” the NN ideally need to contain those situations that the systems that
we want to model are likely to encounter. For nearly spherical bubbles, as we study here, we expect
FIG. 1. The bubble distribution for the training case at six times (0.0, 1.25, 2.5, 5.0, 10.0, 20.0). The color shows the
magnitude of the vertical velocity, mapping the interval [−0.9,0.9] to the color interval blue to red. The first frame shows the
initial condition. The bubbles initially concentrated at center, where the velocity is lowest. The large bubble concentration
then increases the velocity there, causing the bubbles to spread again, eventually making the velocity nearly uniform, as seen
in the last frame.
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the dynamics of the system to be dominated by the lateral motion of the bubbles from regions of
high velocity to regions of low velocity, and the buoyant increase of the vertical velocity of the
liquid where the void fraction is high. We have experimented with several different combinations of
training and testing cases, but for the results discussed here, the training case consists of a uniform
distribution of 98 bubbles, resulting in an average void fraction of 9.98% and an initial velocity
profile given by v(x) = 1.0 × cos(πx). For the parameters used here the slip velocity of the bubbles
is about 5 in computational units, so the variations in the liquid velocity are slightly less than half
the bubble slip velocity.
Figure 1 shows the bubbles and the average vertical velocity at several times, starting with the
initial conditions. The bubbles initially migrate to the region where the velocity is lowest (middle
of the domain), then the increased void fraction increases the buoyancy there and thus the velocity.
Once the average vertical velocity in the middle is higher than elsewhere in the domain, the bubbles
spread out again, eventually resulting in a nearly uniform velocity and a nearly homogeneous distri-
bution of the bubbles. Thus, this DNS data set contains both situations where the bubbles gather
together to increase the void fraction locally and situations where the bubbles move to decrease the
local void fraction.
To gather data for the “training” of the NN, the DNS results are averaged at time intervals
equal to δt = 0.02, over 256 vertical planes, perpendicular to the x axis. While the simulations are
run out to t = 25, we only use data for the time interval 0–5. Thus, there are 256 × 251 = 64 256
samples, each containing the gas flux, the streaming stresses, the void fraction, the gradient of the
void fraction, and the gradient of the average liquid velocity. The data set is divided into three parts
by random sampling, one for the actual fitting or training (70% or 44 980 samples), another for
FIG. 2. The distribution of the error for the gas flux (top) and streaming stresses (bottom).
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validation (15% or 9638 samples), and one for testing (15% or 9638 samples). The fitting starts by
iteratively adjusting the parameters of the NN to fit the first part of the data. Usually the fitting of the
data set improves with the iteration count, but more iterations also result in overfitting and degrade
the ability of the model to predict new data. The validation data are used to detect when this starts
to happen and to stop the iterations. Finally, the model is checked by comparing its predictions with
the testing part of the data set.
We have experimented extensively with the various parameters and network structure govern-
ing the fitting. We have, for example, tried splitting the data set differently and generally find that
the fitting is insensitive to how we divide it up. The Matlab toolbox uses a three-layer architec-
ture neural network with feed-forward back-propagation and the Levenberg-Marquardt method to
optimize the weight coefficients, which connects the three inputs, ten neurons, and one output for
each closure relation. We have experimented with different solution algorithms, different number of
neurons, as well as with more hidden layers, and find that a single hidden layer with ten neurons and
the solution algorithm used here give a good balance between accuracy and robustness. The starting
values for the weights are chosen to be random numbers near zero.
FIG. 3. The reduction in the error versus number of iterations (epochs) for the gas flux (top) and the streaming stresses
(bottom).
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To examine the quality of the fit, Figure 2 shows the distribution of the error for both the gas flux
(top) and the streaming stresses (bottom). The abscissa is the error and the ordinate is the number
of points with a given error. The training data are blue, the validation data are green, and the test
data are red. The vertical black line in the center is zero error and it is clear that most of the points
have prediction errors close to zero. This is, of course, particularly true for the training data, but
even for the test data, the number of points with relatively large errors is small. We have compared
the distribution of the error with a Gaussian and find that while the error for the bubble flux is very
close to Gaussian, the error for the streaming stresses is slightly skewed. In Figure 3, the reduction
in mean square error versus number of iterations (epochs, in neural network terminology) is shown,
demonstrating that the error is reduced quickly for both the gas flux and the streaming stresses.
If the original data were perfect and the model fitted it exactly, then the model would produce
the original data. This is, of course, not the case and to examine how well the model and the data
align; in Figure 4, we plot a scatter plot of the predicted gas fluxes (top) and the streaming stresses
(bottom) versus the fluxes and stresses for the part of the data set used for the training in the left
row, and in the right row for the part of the data reserved for testing the model (that is, the points
not used to construct the neural network, nor determine when to stop the iterations). The diagonal
solid line is the perfect agreement line where the predicted fluxes and stresses are exactly equal to
the DNS fluxes and stresses, and the dashed line, which has a slightly different slope, is the best
fitting line. The correlation coefficient is shown above each frame. For the gas fluxes, the solid
and the dash lines are nearly parallel and the correlation coefficient is high. The correlation for
the streaming stresses is not quite as good, but nevertheless it is still reasonably high. While the
correlation coefficient for the test data is slightly lower than for the training data, they are close in
both cases.
FIG. 4. The value for the gas fluxes (top) and the streaming stresses (bottom) predicted by the model versus the DNS data.
The left column shows the training data and the right column shows the test data.
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After using the NN to ﬁnd f and g, we run the model to predict the evolution of the original
data set and in Figure 5 we show the average velocity and void fraction as found by averaging the
DNS data and as predicted by the average model, at six times. Since the ﬂow evolves fast in the
beginning and more slowly later, the frames are at non-equal time intervals. The average velocity is
shown in the left column and the void fraction in the right one, with the initial conditions in the top
FIG. 5. The average vertical velocity (left) and void fraction (right) as found by averaging the DNS training data (solid line)
and as predicted by the average model (dashed line), at the six times shown in Figure 1.
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FIG. 6. The root mean square difference between the local vertical velocity and the average velocity (top), and the root mean
square difference between the local void fraction and the average void fraction (bottom), versus time, as found by averaging
the DNS training data and as predicted by the average model.
row. The initially uniform void fraction quickly increases in the middle of the domain, where the ve-
locity is lowest, as the bubbles move down the velocity gradient. The buoyancy there increases the
local velocity, eventually producing a velocity slightly above the average, as seen in the fifth frame.
This reverses the gas flux, first reducing the void fraction in the middle and eventually producing a
more uniform, but unsteady, void fraction distribution. Although the DNS data for the void fractions
show slightly more fluctuations than the model predicts at late stage, overall the model reproduces
the averaged DNS results.
To quantify the difference between the model predictions and the averaged DNS data, in
Figure 6 we show an integral comparison between the DNS results and the model results by plotting
the root mean square difference between the local void fraction and the liquid velocity and the
average value (or the variance), integrated across the domain, versus time. More specifically, those
TABLE I. The various initial conditions used to train and test the model for
the closure terms. The last two columns are defined by Equations (8) and
(9).
Case
Initial
velocity
Initial void fraction
(%) MSD( v rms) MSD(αrms)
Train cos πx Uniform αg = 9.98 0.027 39 0.007 075
Test 1 cos πx Uniform αg = 4.99 0.033 47 0.009 842
Test 2 2 cos πx Uniform αg = 9.98 0.041 35 0.017 491
Test 3 2 cos πx Uniform αg = 4.99 0.075 50 0.007 843
Test 4 cos 2πx Uniform αg = 9.98 0.036 65 0.010 543
Test 5 cos 2πx Uniform αg = 4.99 0.034 51 0.007 719
Test 6 cos πx Centered αg = 4.99 0.024 64 0.016 551
Test 7 sin πx Centered αg = 4.99 0.044 32 0.010 229
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are deﬁned by
v rms =
1
l
( v (x) − v avg)2dx, (6)
FIG. 7. The average vertical velocity (left) and void fraction (right) as found by averaging the DNS training data (solid line)
and as predicted by the average model (dashed line), at six times (0.0, 1.25, 2.5, 5.0, 10.0, 20.0), for test 7 in Table I.
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αrms =
1
l
(α(x) − αavg)2dx, (7)
where l is the period in x direction. For our specific setup, the average velocity ( v avg) is, of course,
zero. The agreement is reasonably good, as we expect from Figure 5. Notice that the root mean
square difference for both the void fraction and the vertical average liquid velocity does not become
zero, for the time examined, and that the model predicts the level of the non-zero root mean square
difference, as well as the time scale of the initial change.
B. Applications to different initial conditions
We do, of course, expect the model to reproduce the original data reasonably well, since the
model was derived using precisely these data. To see if the relationship has a more general validity,
we have evolved several different initial configurations using the model, with f and g found using
the training set described above, and compared the results with DNS of the same initial conditions.
The cases are listed in Table I, and in Figure 7, we show the average velocity and void fraction at
six times for case 7, as predicted by the model and found from the DNS results. The velocity field is
the same as in the training case—shifted by half a period by using sin(πx) instead of cos(πx)—but
the bubbles are concentrated in the middle of the domain where the average void fraction is about
20%. The average void fraction over the whole domain is 4.99%. The evolution is different from
the training case in that the bubble distribution is initially moved to the right, to where the velocity
is lowest. Once the bubbles are there, the increased buoyancy increases the velocity and eventually
both the velocity and the void fraction become relatively uniform. The root mean square difference
between the local void fraction and the liquid velocity and the average value, integrated across the
domain, is shown versus time in Figure 8, and while the agreement is not quite as good as for the
training data (Figure 6), it is clear that the model still predicts the DNS data reasonably well.
We have examined all the initial conditions listed in Table I in the same way and find similar re-
sults. In Figure 9, we plot the difference between the root mean square differences, or the variances,
FIG. 8. The root mean square difference between the local vertical velocity and the average velocity (top), and the root mean
square difference between the local void fraction and the average void fraction (bottom), versus time, as found by averaging
the DNS training data and as predicted by the average model, for test 7 in Table I.
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FIG. 9. The absolute value of the difference between the variance predicted by the model and found from the DNS data
versus time for all the initial conditions in Table I. The results for the average velocity are shown in the top frame and for
the void fraction in the bottom frame. The differences have been normalized by the maximum value of v rms and αrms in the
DNS results.
as predicted by the model and found from the DNS data, versus time, for all the tests in Table I (that
is, the difference between the dashed and the solid curves in Figures 6 and 8). Again, the agreement
is worse than for the training data (shown by the blue solid line), but in all cases the differences
between the model predictions and the DNS data are of comparable order of magnitude. The results
are condensed even further in the two rightmost columns in Table I where the mean square different
of the variances, defined by
MSD( v rms) =
1
T
( v rms)DNS − ( v rms)model
2
dt, (8)
MSD(αrms) =
1
T
(αrms)DNS − (αrms)model
2
dt, (9)
is listed, for T = 25.
We have examined a number of other quantities to evaluate the performance of the model, such
as the various correlation coefficients. In many cases we find that those are not particularly useful,
since often most of the data points are clustered relatively randomly around the average value and
the main contribution to the fit comes from relatively few points far from the average. For the
particular setup used here, the largest values of the fluxes and the streaming stresses occur relatively
early in the simulations so the early times contribute most to the fitting, and this is the reason that for
the results presented here, we have used only the data for the first 5 time units for the fitting. In other
simulations we have used the whole set and find similar results.
We have also repeated the study presented here by swapping the training case for one of the
test cases from Table I and find that the results are comparable. When we use data from two cases
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with different setups (such as the current training set and test case 6) for the training we find that
while the training error does not decrease significantly, the robustness of the model increases in
the sense that the errors for all the test cases are comparable. This is probably not surprising. In
general we find that the closure for the gas flux, Fg, and how it is modeled, is critical for the correct
prediction of the evolution of the flow, including both its structure at any given time and the rate
at which it changes. We have, for example, tested the sensitivity by reducing the fluxes by half,
as well as doubling them, and find that doing so results in much worse agreement with the DNS
results, particularly for the void fraction. The streaming stresses, on the other hand, play a much
smaller role, at least for the parameter values examined here, and we find, for example, that taking
them to be zero results in relatively minor increase in the difference between the DNS and the
model results. The closure relationships found here are, of course, only valid for the particular set
of governing parameters selected here. If we make the bubbles larger or smaller, the Eötvös (Eo)
number changes, and if it becomes sufficiently high, the bubbles are no longer nearly spherical
and we would expect the closure relationships to change. However, if the bubbles remain nearly
spherical, the dynamics does not depend on Eo and we expect reasonably good agreement. We
could, of course, expand the dataset to include the governing parameters, in which case we expect to
be able to use the model for a larger range of situations.
IV. CONCLUSION
Here, we have examined the use of neural networks to fit DNS data to develop closure relations
for the average two-fluid equations for a very simple bubble system. The neural network is trained
on one DNS data set and then applied to other flows where the initial velocity and void fraction are
different, but for which we also have DNS results to compare with. Overall, the average equations
with the neural network closure reproduce the main aspects of the DNS results, including the gener-
ally rapid change initially as well as the level of unsteadiness remaining after the initial adjustment.
While it can be argued that the present problem is sufficiently simple so that success is likely, we
feel that the small size of the training data and the relatively high level of fluctuations observed
in the DNS results resulted in better agreement than we expected. The approach taken here should
allow us to quantify the uncertainty of the model and how it depends of the size of the training
data.21,40
However, this quantification, as well as propagating this41
and other uncertainties,42
such
as in the initial conditions, to the final predictions, remains to be explored.
Neural networks are, of course, not the only statistical learning technique35,42
that we could
have used and we have also experimented with linear regressions where we postulate a functional
form for the relationship with coefficients that are determined from the data. Our current belief is
that the neural networks are more general and require less user input. We also note that there are
several versions of neural networks and related approaches20
and we have made no effort to find
the best one. Perhaps the biggest drawback of the neural network is that there is little physics in
the fitting. This is, presumably, both a strength and a weakness. We do not need to know much
about the physics to obtain the closures, but conversely a knowledge of the physics is of little help,
at least for the approach taken here. However, instead of looking for the complete closure term as
we have done here, we could presumably postulate a relationship and use neural networks to fit
unknown coefficients. Furthermore, although here we have imported the results of the NN directly
into the model for the average flow, the results could also be used to examine directly how the
various closure terms depend on the resolved variables. Such fits could then be used to assist in the
generation of more “mechanistic” models.
We emphasize that the system considered here is a very simple one. First of all it is homoge-
neous in two directions so the averaged equations have only one spatial coordinate. Second, it is
periodic in all directions and there are no walls, and third, no average descriptions of unresolved
quantities appear to be necessary (which might be governed by their own evolution equations with
unknown closure terms). Furthermore, the gas flux appears to be a function of the local state only
and no evolution equation for the bubble velocity needs to be solved. Most multiphase flows are
considerably more complex than the system considered here, but we believe that there is nothing
that intrinsically limits using statistical learning for the closure terms to simple systems only.
to IP: 66.254.236.59 On: Wed, 20 Jan 2016 22:45:13

ACKNOWLEDGMENTS
This work is supported by the National Science Foundation Grant No. CBET-1335913 and
the Consortium for Advanced Simulations of Light Water Reactors (CASL). This research used
resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory,
which is supported by the Office of Science of the U.S. Department of Energy under Contract No.
DE-AC05-00OR22725.
APPENDIX: PLANAR AVERAGE OF THE VERTICAL MOMENTUM EQUATIONS
Derivation of Equation (1) can be found in most standard texts on multiphase flow model-
ing.6
We thus only outline the derivation of Equation (2), which is a simplified version of the
general multifluid momentum equation. To keep the derivation as simple as possible we assume
two-dimensional flow in the x − y plane, where the averaging is over the y-direction. Averaging the
full equations over the y − z plane produces the same results (see supplementary material43
). The
y-momentum equation, written for the whole flow field,
∂ρv
∂t
+
∂ρuv
∂x
+
∂ρv2
∂ y
= −
dpo
dy
−
∂p′
∂ y
− ρgy +
∂
∂x
µ
∂v
∂x
+
∂u
∂ y
+
∂
∂ y
2µ
∂v
∂ y
, (A1)
where we have written the pressure gradient as a sum of the imposed gradient, dp0/dy and a
periodic perturbation pressure, dp′
/dy. The density and viscosity of the gas can be taken as zero, so
that we can write ρ = ρl H and µ = µl H, where
H =



1 in liquid,
0 in gas.
(A2)
The planar phasic average of the liquid velocity and volume fraction is defined as
v l = 1
αl L Hvdy, αl = 1
L Hdy, where L is the period in the y direction. The average of the left
hand side is
1
L
∂ρl Hv
∂t
+
∂ρl Huv
∂x
+
∂ρl Hv2
∂ y
dy
= ρl
∂
∂t
1
L
Hvdy + ρl
∂
∂x
1
L
Huvdy + ρl
∂
∂ y
1
L
Hv2
dy
= ρl
∂
∂t
αl v l + ρl
∂
∂x
αl uv l + 0. (A3)
The pressure and the gravity terms are
1
L
dpo
dy
+
∂p′
∂ y
+ ρl Hgy dy =
dpo
dy
+ 0 + ρlgyαl. (A4)
The average of the viscous term is
1
L
∂
∂x
µl H
∂v
∂x
+
∂u
∂ y
+
∂
∂ y
2µl H
∂v
∂ y
dy
=
∂
∂x
µl
L
H
∂v
∂x
+
∂u
∂ y
dy +
1
L
2µl H
∂v
∂ y
L
0
= µl
∂
∂x
αl
∂v
∂x
+
∂u
∂ y
=
∂
∂x
µl
L
∂Hv
∂x
+
∂Hu
∂ y
− v
∂H
∂x
− u
∂H
∂ y
dy
= µl
∂2
∂x2
(αl v l) + 0 −
∂
∂x
µl
L
v
∂H
∂x
+ u
∂H
∂ y
dy . (A5)
If we decompose the velocity field into a mean and fluctuation part, ul = u l + u′
l, and
vl = v l + v′
l , and use that
∂α
∂x
=
1
L
∂H
∂x
dy, (A6)
to IP: 66.254.236.59 On: Wed, 20 Jan 2016 22:45:13

the viscous term is
µl
∂2
∂x2
(αl v l) −
∂
∂x
µl
L
( v l + v′
l )
∂H
∂x
+ ( u l + u′
l)
∂H
∂ y
dy
= µl
∂2
∂x2
(αl v l) −
µl
L
∂
∂x
v l
∂H
∂x
dy + u l
∂H
∂ y
dy + v′
l
∂H
∂x
+ u′
l
∂H
∂ y
dy
= µl
∂2
∂x2
(αl v l) − µl
∂
∂x
v l
∂α
∂x
+ 0 +
1
L
(v′
l nx + u′
lny)δdy
= µl
∂
∂x
αl
∂ v l
∂x
+ µl
∂
∂x
v l
∂α
∂x
− µl
∂
∂x
v l
∂α
∂x
+
1
L
(v′
l nx + u′
lny)δdy
= µl
∂
∂x
αl
∂ v l
∂x
−
µl
L
∂
∂x int
(v′
l nx + u′
lny). (A7)
Here, we use that ∂H
∂x = δnx and ∂H
∂y = δny, where δ is a delta function and nx, ny are the normal
components of the normal vector. The summation in the last line of Equation (A7) is over the
interfaces (bubble surfaces that cross the x = constant plane), using Equations (A3), (A4), (A7), and
the averaged equation for the vertical momentum is
ρl
∂
∂t
αl v l +ρl
∂
∂x
αl uv l
= −
dpo
dy
− ρlgyαl + µl
∂
∂x
αl
∂ v l
∂x
−
µl
L
∂
∂x int
(v′
l nx + u′
lny). (A8)
Dividing by the liquid density, and introducing the kinematic viscosity νl = µl/ρl, gives
∂
∂t
αl v l +
∂
∂x
αl uv l = −
1
ρl
dpo
dy
− gyαl + νl
∂
∂x
αl
∂ v l
∂x
−
νl
L
∂
∂x int
(v′
l nx + u′
lny). (A9)
If we write uv l = u l v l + u′
v′
l and ignore the fluctuating interface viscous terms, we have
∂
∂t
αl v l +
∂
∂x
αl u l v l = −
1
ρl
dpo
dy
− gyαl + νl
∂
∂x
αl
∂ v l
∂x
−
∂
∂x
αl u′
v′
l. (A10)
Here, we have ignored
νl
L
∂
∂x int
(v′
l nx + u′
lny). (A11)
It is likely to be of most importance near the walls, and for the present setup, we have no walls. We
have also left out the surface tension term in the derivation above. For spherical bubbles it is easily
shown that the average of this term is zero. The average of the y-component is
1
L
σκnyδdy =
1
L
σκo
int
ny =
1
L
σκo
bub
(ntop
y + nbot
y ), (A12)
which is zero, since for a spherical bubble ntop
y = −nbot
y .
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