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AIAA-00-0858Microgravity Geyser and Flow FieldPredictionR.J Thornton and J.I. HochsteinThe University of Memphis
Memphis, TN
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AIAA-00-0858
American Institute of Aeronautics and Astronautics1
MICROGRAVITY GEYSER AND FLOW FIELD PREDICTIONR.J. Thornton*, and J.I. Hochstein†,
The University of Memphis
Memphis, Tennessee
ABSTRACT
Modeling and prediction of flow fields and geyser formation in microgravity cryogenic propellant tanks
was investigated. A computational simulation was used
to reproduce the test matrix of experimental results
performed by other investigators, as well as to model
the flows in a larger tank. An underprediction of geyser height by the CFD code led to a sensitivity study to
determine if variations in surface tension coefficient,
contact angle, or jet pipe turbulence significantly effect
the simulations, it was determined that computationalgeyser height is not sensitive to slight variations in any
of these items. An existing empirical correlation based
on dimensionless parameters was re-examined in aneffort to improve the accuracy of geyser prediction.
This resulted in the proposal for a re-formulation of two
dimensionless parameters used in the correlation; the
non-dimensional geyser height and the Bond number. It
was concluded that the new non-dimensional geyser height shows little promise. Although further data will
be required to make a definite judgement, the
reformulation of the Bond number provided correlations that are more accurate and appear to be
more general than the previously established correlation.
NOMENCLATURE
a = Acceleration
Bo = Bond Number
D = Diameter F = Flow Characterization Parameter G = Non-Dimensional Geyser Height R = Radius
We = Weber Number
ρ = Density
σ = Surface Tension Coefficient
Subscripts
AYD = Aydelott
j = Jet at the Liquid-Vapor Interface
* Research Assistant, Mechanical Engineering, Member AIAA† Professor/Chair, Mechanical Engineering, Member AIAA
Copyright © 2000 The American Institute of Aeronautics and Astronautics Inc.
All rights reserved.
o = Jet Pipe Outlet RF = Re-Fitt = Tank
TH = Thornton-Hochstein
tj = Tank/Jet
INTRODUCTION
Often spacecraft missions require the vehicle to store
rocket propellants for future use. In the case of
cryogenic liquid propellants, tank self-pressurizationcan be a significant problem. Despite insulation,
incident solar radiation heats the cryogenic fluid
causing liquid to vaporize and raise the pressure inside
the tank. If this self-pressurization were allowed to
continue unchecked, the tank would rupture. A stronger tank could help, but this would require an undesirable
increase in vehicle mass. Another solution to the tank
pressurization problem is to vent the tank. Tank ventingis undesirable because it wastes valuable propellant. In
addition, due to the lack of gravity to positively orient
the propellant in a predictable manner, it would beimpossible to locate a vent where it could be certain
that only vapor would be vented. In fact, even in partially filled tanks, it is possible that the entire tank
surface could be wetted due to the influence of surface
tension in a microgravity environment.
An alternative solution to the tank self-pressurization
problem is a Thermodynamic Vent System, or TVS1, 2.The TVS would extract a small portion of the bulk
liquid from the tank and pass it through a Joule-
Thomson valve resulting in a reduction in temperature
as well as pressure. Once cooled, the fluid would berouted through a heat exchanger, which is used to cool
a separate flow of propellant extracted from the bulk
liquid. If the fluid leaving the Joule-Thomson valve is a
two-phase mixture, it will continue changing phase inthe heat exchanger until it is completely vaporized and will eventually be sacrificially vented overboard. The
other stream of cooled liquid could then be pumped
from the heat exchanger back into the tank and injected through an axial jet pipe located at the fore end of the
tank. The jet flow provides several benefits including
mixing of the bulk liquid, which helps to reduce
temperature gradients and in turn helps prevent
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American Institute of Aeronautics and Astronautics2
evaporation of the propellant. Introduction of the cooled
liquid also reduces the temperature of the bulk fluid. Inaddition, if the jet has a moderate amount of momentum
it can cause the formation of a geyser at the
liquid/vapor interface. The increased surface area of the
free surface due to the formation of a geyser would help
promote condensation, thus reducing the pressure evenfurther. At higher levels of jet momentum, the geyser
will strike the opposite end of the tank and either form a
separate pool or “roll” down the tank walls re-mixingwith the bulk fluid. As the fluid comes around the tank
walls, there will also be a cooling effect on the wall.
However, the addition of excessive kinetic energy tothe bulk fluid would eventually result in undesirable
heat generation through viscous dissipation.
A morphology has been defined based on four distinct
flow patterns that occur when an axial jet is injected into the bulk liquid region of a propellant tank in a
microgravity environment
1
. These flow patterns aredefined as follows (Figs. 1 and 2):
I. Dissipation of the jet in the bulk liquid region.
II. Geyser Formation.
III. Collection of jet liquid in the aft end (opposite of the jet inlet) of the tank.
IV. Liquid circulation over the aft end of the tank and
down the tank walls re-mixing with the bulk liquid.
Fig.1. Flow Patterns I and II.
The ability to predict the flow pattern for any given
combination of tank geometry, tank fill level,
gravitational acceleration, and jet momentum is crucial
for optimization and implementation of a TVS. A studyof geysers and their formation is therefore a necessary
task in the development process of the TVS concept.
Fig. 2. Flow Patterns III and IV.
COMPUTATIONAL MODEL, ECLIPSE
The complexity and expense of microgravityexperimentation severely limits the amount of data that
can be, and has been, collected on microgravity geyser formation. Therefore, an attractive alternative is the use
of CFD, computational fluid dynamics, to model the
geyser flows. The use of a CFD code can significantlyreduce the cost of investigating the microgravity geyser
phenomena, as well as allow for easy manipulation of
parameters such as tank size, fluid properties, jetmomentum levels, and gravitational acceleration levels.
The ECLIPSE code was used for the computational
studies in this work. ECLIPSE was chosen for its
previously demonstrated ability to model themicrogravity geyser flows of interest here3, 4. ECLIPSE
has evolved from the RIPPLE5 code developed at Los
Alamos National Laboratory. RIPPLE is a descendent
of the NASA VOF2D code, and was developed tomodel “transient, two-dimensional, laminar,
incompressible fluid flows with free surfaces of general
topology.”5
The flow field is discretized into finitevolumes to form a regular non-uniform mesh. RIPPLE
models free surfaces with volume of fluid (VOF) data
on the mesh, and a continuum surface force (CSF)
model is used to model surface tension. Staggered grid
differential equation approximations result in a systemof algebraic equations that are solved by a two step
projection method employing an incomplete Choleskyconjugate gradient (ICCG) solution technique for the
pressure Poisson equation (PPE)5. One of the features
added to RIPPLE to produce ECLIPSE is the
implementation of the two-equation Jones-Launder-
Pope k −ε turbulence model.
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American Institute of Aeronautics and Astronautics3
Computational Model Fidelity and Sensitivity
It has been demonstrated that ECLIPSE and its predecessors do a credible job of predicting geyser flow
pattern as defined by the four-pattern morphology
previously presented 1. Given the dearth of low-gravity
geyser data, test cases in these previous studies
included configurations with gravity levels as high as 1-g. The focus of the present research was on low-gravity
environments with completely turbulent jets, which
reduced the pool of available experimental data to only14 test cases in which a geyser was formed, (Flow
Pattern II). Fig. 3 shows the measured geyser height,
displayed in dimensionless form, as a function of theWeber number and Bond number as defined by
Aydelott2.
Boa R
j
= =
ρ
σ
2
acceleration force
surface tension force(1)
WeV R
D
o o
j
= =
ρ
σ
2 2inertia force
surface tension force(2)
Gh
R
g
t
= =
geyser height
tank radius(3)
Also displayed in Fig. 3 are the geyser heights predicted
by ECLIPSE for these cases. Although the correct flow pattern is indeed predicted, the computational
predictions consistently underpredict geyser height.
This underprediction averages 44% with a standard
deviation of only 9.4% about this mean. Although this
level of agreement should be useful for predictingtrends, and has been useful for predicting flow patterns,
it prompted an investigation into the sensitivity of the
computational simulation to user specified parameters;surface tension coefficient, contact angle, and inlet
turbulence quantities.
Aydelott’s experiments were conducted in plexiglas
tanks that were cleaned with the utmost care prior to
filling with ethanol; even a small amount of
contaminant can significantly alter the surface tension
coefficient, σ . The literature does not indicate that any
of the physical properties of the fluid were measured
and it appears that all calculations related to theexperiments have been performed using standard
published values for these properties. To study the
sensitivity of the simulation to the value of σ ,simulations of a case with a measured geyser height of
6.5 cm and a computationally predicted height of 3.66
cm were rerun with σ specified to be 90% and 80% of
Fig. 3. Aydelott’s experimental non-dimensional geyser heights1 and computational simulations of these
experiments as a function of Weber and Bond number.
Experimental Non-Dimensional Geyser Height
Computational Non-Dimensional Geyser Height
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American Institute of Aeronautics and Astronautics4
the published value for ethanol. Reducing sigma by
20% resulted in an 8% (0.31 cm) increase incomputationally predicted geyser height. Although this
is a small improvement, it does not fully account for the
difference between predicted and measured heights.
Further, although the lower value improves the
predictions, and the lowered value might be appropriateif there was a slight contamination of the ethanol, there
is no evidence in the literature that there was
contamination and there is no rationale for selecting aspecific value for sigma other than the standard
published value. In fact, because the value of σ is sosensitive to even slight contamination of the fluid, it is
somewhat comforting to know that the predicted geyser
height is not particularly sensitive to the value of this parameter.
The second parameter to be examined in the sensitivity
study is the contact angle between the liquid and the
tank wall, α . Actual propellant/aluminum/vapor combinations and the experiment ethanol/plexiglas/air
combinations exhibit very small contact angles. Toavoid divide-by-zero difficulties in the simulations, the
contact angle in the standard simulations has been
specified to be 2 degrees. A sequence of simulations
was run with increasing values of α for the same test
case that was used for the σ -sensitivity study.
Increasing α all the way to 30 degrees resulted in anincrease in predicted geyser height of only 0.3%. It is
therefore concluded that geyser height is relatively
insensitive to contact angle.
The final group of parameters examined as part of thesensitivity study are the turbulence kinetic energy, κ ,
and the turbulence energy dissipation rate, ε , at the exitof the jet-pipe. Standard simulations are run using
values computed using published correlations for fully
developed flow in a pipe. To examine the sensitivity of
geyser height to this fairly crude assumption, three
additional simulations were performed. The first used values for these parameters were simply half of the
values computed from the correlations. The second and
third used values of κ were selected to produce aturbulent viscosity at the pipe outlet equal to 10% and
1% (respectively) of the fully developed value. The
increase in geyser height was less than 8% for any of these cases. Although this is an improvement in the
predictive accuracy of the computational simulation,
there is no clear rationale for using these reduced
values. Again, as concluded for the σ -sensitivity study,
it is comforting to show that geyser height is not
particularly sensitive to the specified value of κ or ε and
all standard simulations for this study were performed
using values computed using the standard correlationsfor full developed pipe flow.
What conclusions can be reached about the fidelity of
the computational model? In the past, thecomputational model has been asked to simply predict
which of the four flow patterns will be produced by a
given jet/tank/filling combination and it still does so
with good reliability. For the present research, the
more challenging task of accurately predicting geyser height for flow pattern II configurations was selected as
the fidelity criterion. The computational simulation
consistently underpredicts geyser height with a standard deviation about the mean error of less than 10%.
Although the source of this error has not been
identified, it seems to be consistent. A sensitivity study
to examine the influence of σ , α , κ inlet , and ε inlet on
simulated geyser height was conducted because it wasfelt that these parameters were the most likely to have
different values between the simulation and the
experiment. Although the simulations predictive
accuracy could be improved by changing σ , κ inlet , and
ε inlet in a reasonable manner, a sound justification for making these changes could not be established. It is
reassuring to note that although the predictions
improved, the predicted geyser height is not particularlysensitive to these parameters that may not be known á
priori with great precision in a spacecraft design
environment. Based on all of this information, it was
concluded that ECLIPSE is still a very useful tool for predicting jet-induced flow patterns in a propellant
tank, that it consistently underpredicts geyser height,
and that it should be useful for exploring the
relationships between the various parameters that
influence geyser production in a low-gravity
environment.
DIMENSIONLESS MODELING
Small scale testing was conducted during the late
1970’s and 1980’s to investigate the geyser phenomena.
Through drop tower testing performed in the NASA
LeRC Zero Gravity Facility, Aydelott2 determined that
the inertia, acceleration, and surface tension forces arethe primary factors affecting geyser formation in a
microgravity environment. From these forces, he
deduced that the main parameters in the dimensional
analysis should be the jet velocity, liquid density and surface tension, acceleration environment, geyser
height, and selected characteristic lengths. Using theBuckingham Theorem, Aydelott established three
dimensionless groups pertinent to predicting geysers
and geyser height. These dimensionless groups are the
jet-Bond number Bo, the Weber number We, and a non-
dimensional geyser height G. Aydelott formed hisdimensionless groups as presented in Equations (1), (2),
and (3) above.
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Aydelott concluded from his data that a linear relation
existed between the non-dimensional geyser height and the Weber number. He also assumed that the non-
dimensional geyser height should be inversely
proportional to the Bond number. Through a least-
squares deviation curve fit of turbulent experimental
data the following geyser height prediction correlationwas developed by Aydelott:
F We
Bo=
− +
+
0 5 16
1 0 6
. .
.(4)
where F is the flow characterization parameter and is
expected to equal the dimensionless geyser height, G.
To formulate a reliable correlation applicable to
propellant tank microgravity geyser prediction ingeneral, it is necessary to not only recognize the proper
dimensionless groups, but also identify the proper length scales in these parameters. As seen in Equations
(1), (2), and (3) Aydelott uses three different length
scales, the tank radius Rt , jet pipe radius Ro, and the jetradius at its point of impact with the liquid-vapor
interface R j. The shape and curvature of the liquid-
vapor interface once a geyser begins to form becomes
quite complex (Fig. 4). Therefore, the determination of an appropriate length scale must take into account
whether the formation of a geyser is dependent on
“local” effects (i.e. in the vicinity of the actual geyser),“global” effects (i.e. tank scale effects), or a
combination of the two.
Fig. 4. Typical free surface shape of a microgravity
geyser.
Non-Dimensional Geyser Height, G
Aydelott’s experimental correlation was formulated with Weber and Bond numbers dependent only on the
jet properties making them “local” parameters. If the
“local” effect only model is accepted, and Aydelott’s
original formulations of Bo and We are assumed to be
appropriate, it seems reasonable that the non-dimensional geyser height should also be scaled with
the jet radius instead of the tank radius. It should be
noted that all of Aydelott’s experiments were conducted in configurations with the same tank radius to jet radius
ratio. Therefore, it is possible that Aydelott’s
correlation may not perform well when applied to tankswith a tank to jet radius ratio that is different from the
experiment configuration; while a reformulation of the
non-dimensional geyser height to include the jet radius
may provide improved accuracy.
The proposed new non-dimensional geyser height uses
the jet diameter at the liquid-vapor interface determined by approximate expressions2.
Gh
DTH
g
j
= (5)
Several analyses have been made to determine the
validity of this new non-dimensional geyser height,
GTH , utilizing Aydelott’s experimental data as well as
computational data.
Bond Number, Bo
The Bond number is the ratio of acceleration force tosurface tension force, and can have both “local” and
“global” effects on geyser formation. In a no geyser
situation, the tank Bond number will dictate the shapeof the free surface, and at least partly so in a geyser
situation. Three different formations of the Bond
number were considered for the geyser problem. The
jet-Bond number Bo or Bo j (Equation (1) used byAydelott), the tank Bond number Bot (using the tank
diameter as the characteristic length scale, Equation
(7)), and a combination tank/jet-Bond number Botj (using the tank and jet diameters, Equation (8)).
Bo a Dt
t =
ρ
σ
2
(7)
Boa
tj
D Dt j
=
ρ
σ
c h(8)
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Although in certain circumstances, the “local” or
“global” surface tension effects may dominate the physics of geyser formation, most cases are likely to
involve both. Incorporating the tank or tank/jet-Bond
number may be a feasible way to account for both
“local” and “global” surface tension effects. An
investigation was conducted to determine if either Bot or Botj could be used to produce a correlation with
better predictive ability than the existing correlation
based on Bo j.
Weber Number, We
The Weber number is the ratio of the inertia force to
surface tension force. In the geyser problem, the Weber
number primarily describes the interaction between the
momentum of the jet impacting on the free surface and the restoring force due to surface tension. Therefore, the
jet-Weber number (Equation (2)) used by Aydelott still
appears to be the appropriate formulation, and has
therefore been employed throughout this study.
NON-DIMENSIONAL GEYSER HEIGHT STUDY
A combination of experimental and computational data
was used to evaluate the predictive utility of the proposed dimensionless geyser height. The
experimentally measured and computationally predicted
geyser data presented in Fig. 3 was supplemented with
the results of several additional simulations. Twelve
simulations were run using the same tank as the
experiment but with parameter values specified to provide a better spread of data over the We- Bo space of
Fig. 3 than was available from the experiment. Another set of cases were defined to almost duplicate the
experiment conditions except for a tank with a diameter
equal to 150% of the diameter of the experiment tank.
It should be remembered that the computationalsimulations consistently underpredict geyser height.
Therefore, in the following evaluations, computational
and experimental data are segregated as appropriate.
All of the correlations developed in this study were produced using a least-squares regression method. The
root mean square (RMS) error between a correlation’s
predicted geyser height and that measured in theexperiment, or predicted directly by the simulations,
was selected as the measure of quality for this study.
Experiment Data - Experiment Correlations
The first correlation presented in Table 1 is the original
correlation, (Equation (4)), developed by Aydelott for
all of his experiments including cases not part of the present study. The second correlation is a refit of the
original formulation using only the data presented inFig. 3. The third correlation uses the same formulation
but replaces the tank radius with the jet impact diameter
to form the new dimensionless geyser height, GTH .As can be seen from Table 1, the refit of the original
correlation provided a slight improvement in the
correlation’s accuracy for the Fig. 3 data, while the use
of the newly proposed non-dimensional geyser height
resulted in a decrease in accuracy. Fig. 5 presents themeasured non-dimensional geyser heights (G) as a
function of these three flow characterization parameters
(F ). It also shows three straight lines representing alinear fit to each of the data sets using these three
formulations.
Table 1. Experimental data correlation results.
Fig. 5. Experimental non-dimensional geyser height asa function of the three formulations of Table 1
(lines are linear fit to each correlation’s data set).
Computational Data - Experiment Correlations
The experimental tests were all simulated with
ECLIPSE, as were similar cases with the 50% enlarged tank. The geyser heights of these simulations where
predicted using the same three correlations of Table 1
developed strictly from experiment data. Although thequality of the predictions was significantly lower than
Table 1 because the simulations consistently
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7
Flow Characterization Parameter F
N o n - D i m e n s i o n a l G e y s e r H e i g h t G
New G Correlation
Original G Refit Correlation
Original Correlation
Original G Refit Correlation
New G Correlation
Original Correlation
Corelation Expression RMS Error
Aydelott 0.203
Aydelott Refit 0.190
New
Formulation
G TH
0.388
F We
Bo AYD =
− +
+
0 5 16
1 0 6
. .
.
F
We
Bo AYD RF /
. .
.=
− +
+
0 34 145
1 0 63
F We
BoTH =
− +
+
3 43 5 69
1 198
. .
.
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American Institute of Aeronautics and Astronautics7
underpredict geyser height, the relative performance of
the correlations was not different from that of Table 1.
Computational Data - Computational Correlations
Computational Data for the Experiment Tank
Geyser heights predicted by the computationalsimulations for the original experiment conditions were
used to refit the original correlation formulation, and
the new non-dimensional geyser height formulation, to produce the results presented in Table 2. The
experiment tank has a 0.4 cm diameter jet pipe,
hemispherical heads, a 10 cm diameter, and a total
length of 20 cm. The new formulation does not
outperform the original formulation.
Table 2. Correlations from original tank computational
data only.
Computational Data for the Enlarged Tank
The conditions of the experiment cases were
reproduced in a tank with a diameter equal to 150% of
the diameter of the experiment tank. This “enlarged”tank has a 15 cm diameter and a total length of 25 cm.
The resulting data was again used to fit the two
correlations and once again the new formulation did not
outperform the original one.
Table 3. Correlations from enlarged tank computational
data only.
Correlations Based on Data from Both Tanks Applied to Both Tanks
The computational data from both tanks was used to fit
correlations for both non-dimensional geyser heights.Table 4 outlines these results. Fig. 6 is a plot of the non-
dimensional geyser height and flow characterization
parameter for both tanks.
Fig. 6. Computational non-dimensional geyser height
for both tanks as a function of the two formulations of Table 4 (lines are linear fit to each correlation’s data
set).
Table 4. Correlations from computational data of both
tanks.
Original Tank Correlations Applied to Enlarged Tank
In an actual design situation, the correlation from a test
tank will be applied to a different sized tank. To seehow the two versions of the correlations perform when
applied to tanks of differing size, the original tank
computational correlations (Table 2) were used to
predict the geyser height in the enlarged tank. Table 5outlines these results.
Table 5. Results of the original tank correlationsapplied to the enlarged tank data.
0
1
2
3
4
5
0 1 2 3 4 5
Flow Characterization Parameter F
N o n - D i m e n s i o n a l G e y s e r H e i g h t G
New G Correlation
Original Correlation
Original Correlation
New G Correlation
Correlation Expression RMS Error
Aydelott 0.155
New
Formulation
G TH
0.172
F We
Bo AYD =
− +
+
0 014 0 73
1 0 71
. .
.
F We
BoTH =
− +
+
0 48 176
1 116
. .
.
Correlation Expression RMS Error
Aydelott 0.121
New
Formulation
G TH
0.158
F We
Bo AYD =
− +
+
0 028 0 66
1 114
. .
.
F We
BoTH =
− +
+
0 29 182
1 119
. .
.
Correlation Expression RMS Error
Aydelott 0.160
New
Formulation
G TH
0.179F We
BoTH =
− +
+
0 43 185
1 122
. .
.
F We
BoTH =
+
0 66
1 083
.
.
Correlation Expression RMS Error
Aydelott 0.200
New
Formulation
G TH
0.219
F
We
Bo AYD =
− +
+
0 014 0 73
1 0 71
. .
.
F We
BoTH =
− +
+
0 48 176
1 116
. .
.
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American Institute of Aeronautics and Astronautics8
Enlarged Tank Correlation Applied to Original Tank
The enlarged tank computational correlations (Table 3)were used to predict geyser height in the original tank.
This is the only case where the newly proposed non-
dimensional geyser height matched the accuracy of the
original non-dimensional geyser height correlation.
Table 6. Results of the enlarged tank correlations
applied to the original tank data.
G Study Summary
As can be seen from Tables 1-6, the desired
improvement using the jet diameter normalized non-
dimensional geyser height was not realized. In all but
one case (the enlarged tank correlations applied to the
original tank), the original correlation was moreaccurate in predicting geyser height. It is possible that
as more data for a wider variety of configurations
becomes available it may appropriate to reinvestigate
the GTH formulation for the non-dimensional geyser height parameter. However, given the currently
available data, the present study leads to the conclusion
that the original formulation is superior and should beused.
BOND NUMBER RE-FORMULATION
To determine if the tank-Bond number, Bot , or the
tank/jet-Bond number, Botj , can provide improved
geyser prediction, correlations were refitted in a manor similar to the non-dimensional geyser height study.
These refitted correlations were based-on the original
experimental data as well as the computational predictions for both the original and the enlarged tank.
Experimental Data - Experiment Correlations
Again, the experiment data was used as a starting point
for the correlation study. Table 7 presents a summary of the performance of the Bot correlation, the Botj
correlation, and the original Aydelott correlation ( Bo j),in predicting experimentally measured geyser height.
The Bot correlation shows a slight improvement over
the original correlation whereas the Botj correlationshows a whopping 35% improvement in predictive
accuracy. Ideally, a correlation would result in a direct
prediction of G, (i.e. F = G). This is precisely the result
indicated in Fig. 7 where it can be seen that the linear
fit to both the Bot and the Botj correlation data sets haveslopes near unity and would pass nearly through the
origin.
Table 7. Experimental data correlations for various
Bond number formulations.
Fig. 7. Experimental non-dimensional geyser height as
a function of the three Bond number correlationversions of Table 7 (lines are linear fit to each
correlation’s data set).
Computational Data - Computational Correlations
Again, to study the effects of different tank configurations the computational data from the original
and enlarged tanks were used in various combinations
to study the performance of the proposed correlations.
Original Tank DataTable 8 presents the predictive performance of the three
formulations using coefficients determined fromsimulated geyser heights for the original tank when the
correlations are applied to precisely that data set. Again
the Bot correlation slightly outperformed the originalcorrelation and the Botj correlation was the best
performer resulting in a 19% improvement over the Bo j
correlation.
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3 3.5
Flow Characterization Parameter F
N o n - D i m e n s i o n a l G e y s e r H e i g h t G
Jet-Bo Correlation (Aydelott)
Tank-Bo Correlation
Tank/Jet-Bo Correlatioin
Jet-Bo Correlation
Tank/Jet-Bo Correlation
Tank-Bo Correlation
Correlation Expression RMS Error
Aydelott 0.223
New
Formulation
G TH
0.221
F We
Bo AYD =
− +
+
0 028 0 66
1 114
. .
.
F We
BoTH =
− +
+
0 29 182
1 119
. .
.
Corelation Expression RMS Error
Tank-Bond
Number 0.234
Tank/Jet-
Bond
Number
0.161
Jet Bond
Number 0.250
F We
Bo Bo
t t
=− +
+
0 57 160
1 0 016
. .
.
F We
Bo Bo
tjtj
=− +
+
0 60 186
1 0 079
. .
.
F We
Bo Bo
j j
=− +
+
05 16
1 0 6
. .
.
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American Institute of Aeronautics and Astronautics9
Table 8. Original tank computational data correlations
for various Bond number formulations.
Enlarged Tank Data
Table 9 presents the predictive performance of the three
formulations using coefficients determined from
simulated geyser heights for the enlarged tank when thecorrelations are applied to precisely that data set. In this
instance, both the Bot and the Botj correlationssignificantly outperformed the Bo j correlation by 49%
and 29% respectively.
Table 9. Enlarged tank computational data correlations
for various Bond number formulations.
Correlations Based on Data from Both Tanks Applied to Both Tanks
The computational data from both tanks was used to fit
all three correlations. Table 10 and Fig. 8 present the performance of these correlations. All three correlations
seem to provide the desired “direct” prediction of
geyser height, (G = F ). However, an error reduction of
approximately 30% can be noted for either the Bot or
the Botj correlations compared to the Bo j correlation.
Table 10. Both tank computational data correlations for
various Bond number formulations.
Fig. 8. Computational non-dimensional geyser height
for both tanks as a function of the three Bond number
correlation versions of Table 10 (lines are linear fit to
each correlation’s data set).
Original Tank Correlations Applied to Enlarged Tank
Again, the correlations developed from the original tank
computational data were applied to the enlarged tank.
This was done to give some indication of theapplicability of the correlations developed from a test
tank to a different size tank. Table 11 shows the results,
this time with the Bot correlation yielding an RMS error
reduction of 38% over the Bo j correlation.
0
0.5
1
1.5
2
0 0.5 1 1.5 2
Flow Characterization Parameter F
N o n - D i m e n s i o n a l G e y s e r
H e i g h t G
Jet-Bo Correlation
Tank-Bo Correlation
Tank/Jet-Bo Correlation
Jet-Bo Correlation
Tank-Bo Correlation
Tank/Jet-Bo Correlation
Corelation Expression RMS Error
Tank-Bond
Number 0.131
Tank/Jet-
Bond
Number
0.125
Jet Bond
Number 0.155
F We
Bo Bo
t t
=− +
+
0 25 0 95
1 0 023
. .
.
F We
Bo Bo
tjtj
=− +
+
0 20 0 96
1 0 081
. .
.
F We
Bo Bo
j j
=− +
+
0 014 0 73
1 0 71
. .
.
Corelation Expression RMS Error
Tank-Bond
Number 0.062
Tank/Jet-
Bond
Number
0.086
Jet Bond
Number 0.121
F We
Bo Bo
t t
=− +
+
014 0 76
1 0 013
. .
.
F We
Bo Bo
tjtj
=− +
+
0 082 0 72
1 0 062
. .
.
F We Bo
Bo
j j
=− +
+
0 028 0 661 114. .
.
Corelation Expression RMS Error
Tank-Bond
Number 0.108
Tank/Jet-
Bond
Number
0.116
Jet Bond
Number 0.160
F We
Bo Bo
t t
=− +
+
014 0 78
1 0 015
. .
.
F We
Bo Bo
tjtj
=− +
+
0 084 0 77
1 0 062
. .
.
F We
Bo Bo
j j
=
+
0 66
1 083
.
.
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American Institute of Aeronautics and Astronautics10
Table 11. Original tank computational data correlations
applied to the enlarged tank.
Enlarged Tank Correlations Applied to Original Tank
Likewise, the enlarged tank correlations were applied to
the original tank with similar results, (Table 12).
Table 12. Enlarged tank computational data correlations
applied to the original tank.
Bond Number Study Summary
As can be seen from Tables 7-12, the Bot and Botj correlations consistently outperform the Bo j correlations
in prediction of geyser height. This holds true for both
the experimental and the computational data sets.
Figure 7 shows the improved predictive capability withthe best performance produced by the Botj correlation.
F We
Botj
=
− +
+
0 6 186
1 0 079
. .
.(9)
Based on the data and analyses presented, it isrecommended that this new correlation be used for the
design of future spacecraft propellant management
systems.
SUMMARY AND CONCLUSIONS
Design of future spacecraft propellant management
systems employing a Thermodynamic Vent System
(TVS) to suppress tank self-pressurization will require
prediction of geyser formation in the propellant pool.
Although an experiment-based correlation exists for predicting TVS jet-induced flow patterns in a propellanttank, the data on which this correlation has been
developed is very limited due to the high cost of
experimentation in a reduced-gravity environment.
Although very challenging, computational simulation provides a very attractive alternative to experimentation
for gaining insight into the geyser formation process
and for developing improved correlations for design.
The existing correlation assumes that the geyser heightcan be predicted if the jet-Bond number and the jet-
Weber number are known. To begin the present study,
the limited data set of drop-tower experiments wassimulated using the ECLIPSE code. Comparison
between experiment and simulation show a consistent
underprediction of geyser height within a relatively
narrow error band. A sensitivity study was undertakento determine if the underprediction was due to an
unexpectedly strong dependence of simulated geyser
height on user-specified parameters that may not be
precisely known for the experiments and also may not
be well know for real applications. This study showed that geyser height is nearly independent of contact
angle. Although it depends on the value of surfacetension coefficient and the jet inlet turbulence, geyser
height is not unexpectedly sensitive to these parameters.
To more fully populate the Bo-We space on whichdimensionless geyser height depends, twelve additional
computation-only data points were acquired, (using
ECLIPSE), to evaluate the predictive capability of
proposed correlations. All of the available experiment
data is for configurations with the same ratio of tank diameter to jet diameter. Therefore, another set of
computation-only data points was acquired, (using
ECLIPSE), for a larger tank/jet ratio to better evaluatethe range of applicability of the proposed correlations.
A re-evaluation of dimensionless parameters and
correlations previously used to predict geysers led to aninvestigation into reformulation of the non-dimensional
geyser height and the Bond number. Experimental and
computational data were employed to reformulate
geyser height prediction correlations. The quality of acorrelation’s prediction was judged by the RMS error
between predicted geyser height and either the
measured or computationally simulated geyser height.
Corelation Expression RMS Error
Tank-Bond
Number 0.123
Tank/Jet-
Bond
Number
0.148
Jet Bond
Number 0.200
F We
Bo Bo
t t
=− +
+
0 25 0 95
1 0 023
. .
.
F We
Bo Bo
tjtj
=− +
+
0 20 0 96
1 0 081
. .
.
F We
Bo Bo
j j
=− +
+
0 014 0 73
1 0 71
. .
.
Corelation Expression RMS Error
Tank-Bond
Number 0.145
Tank/Jet-
Bond
Number
0.152
Jet Bond
Number 0.223
F We
Bo Bo
t t
=− +
+
014 0 76
1 0 013
. .
.
F We
Bo Bo
tjtj
=− +
+
0 082 0 72
1 0 062
. .
.
F We
Bo Bo
j j
=− +
+
0 028 0 66
1 114
. .
.
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American Institute of Aeronautics and Astronautics11
Although a new formulation of the non-dimensional
geyser height was proposed after a re-examination of the physics of geyser formation, correlations base on
this new parameter showed consistently worse
performance in predicting geyser height than the
existing correlation. In only one instance did the results
of the new formulation match the original correlation’s predictive capability. It is therefore concluded that the
existing non-dimensional geyser height results in
superior predictive capability and should be used for future work.
In contrast with the failure of the dimensionless geyser height reformulation, correlations based on a new
formulation of the Bond number show great promise.
Correlations were developed based on three different
formulations of the Bond number. The existingcorrelation used the jet-Bond number, Bo j , in which thecharacteristic length is the diameter of the jet at impact
with the free surface. The first alternate formulationuses the tank diameter as the characteristic length to
produce the tank-Bond number, Bot . The characteristiclength of the second alternate formulation uses a
combination of the two, ( D Dt j
), to produce a
tank/jet-Bond number, Botj . Correlations based on these
alternate formulations consistently outperform the
correlations based on the “original” parameter, Bo j .Perhaps most striking is the 35% reduction in predictive
error when the Botj correlation is compared to the
original correlation for the experiment data on whichthe original correlation was based. It is hoped that the
correlations based on small-scale experiments will be
useful for predicting the performance of a TVS in anactual spacecraft tank. To test the sensitivity of thecorrelation’s predictive accuracy to changes in tank
size, correlations with coefficients computed for the
experiment tank were used to predict geyser formation
in a larger diameter tank, and vice-versa. In both cases,
the new Bond number formulations significantlyoutperform the existing formulation.
Future work should focus on increasing the quantity of reliable data on geyser formation in a reduced gravity
environment. Experimental data will be expensive and
time consuming to obtain. If the quality of the
computational simulation can be further improved, itmay provide a less expensive path to acquiring the
necessary information. The new correlations developed
during the present study show clearly improved
performance for the available data. These correlations,and the ECLIPSE code, are currently the best available
tools for predicting geyser formation in a reduced
gravity environment. As such, they should be used for
the design of future experiments and, where required,
for future spacecraft.
REFERENCES
1. Aydelott, J.C. “Axial Jet Mixing of Ethanol InCylindrical Containers During Weightlessness.”
NASA TP-1487, 1979.
2. Aydelott, J.C. “Modeling of Space Vehicle
Propellant Mixing.” NASA TP-2107, 1983.
3. Wendl, J.C., Hochstein, J.I., and Sasmal, G.P.
“Modeling of Jet-Induced Geyser Formation in a
Reduced Gravity Environment.” AIAA-91-0801,1991.
4. Wendl, M.C. "Jet Induced Mixing of Propellant in
Partially Filled Tanks in a Reduced Gravity
Environment." Magisterial Thesis. WashingtonUniversity, St. Louis, Missouri, 1990.
5. Kothe, D.B., Mjolsness, R.C., and Torrey, M.D.
“RIPPLE: A Computer Program for Incompressible Flows with Free Surfaces.” LA-
12007-MS, 1991