A Review of the Dynamics of Cavitating Pumps
Christopher Earls Brennen
∗
April 13, 2012
Abstract
This paper presents a review of some of the recent developments in our understanding of the dynamics and instabilities
caused by cavitation in pumps. Focus is placed on presently available data for the transfer functions for cavitating pumps
and inducers, particularly on the compliance and mass flow gain factor which are so critical for pump and system stab
ility.
The resonant frequency for cavitating pumps is introduced and contexted. Finally emphasis is placed on the paucity of
our understanding of pump dynamics when the device or system is subjected to global osc
illation.
Nomenclature
a
= global acceleration of the pumping system
C
= cavitation compliance
C
∗
,M
∗
= quadratic compliance and mass flow gain factor coefficients
f
= frequency in Hz
h
= inducer blade tip spacing, 2
πR
t
/N
j
=(
−
1)
1
2
K
,M
= bubbly flow model parameters
L
= pump inertance
L
s
= length of the cavitation zone
m
= complex fluctuating mass flow rate
M
= mass flow gain factor
N
= number of main inducer blades
P
= complex fluctating total pressure
R
= pump resistance
R
t
= radius of the inducer tip
TP
ij
= pump transfer function elements
U
s
= velocity in the cavitation zone
U
t
= velocity of the inducer tip
φ
1
= inlet flow coefficient
σ
= cavitation number
ρ
= liquid density
τ
C
,τ
M
= phase lags of the compliance and mass flow gain factor
ω
= dimensionless frequency,
ωh/U
t
ω
= dimensional radian frequency
Ω
= frequency of rotation of the pump
Ω
P
= natural cavitation frequency of the inducer
Ω
P
= dimensionless natural cavitation frequency of the inducer, Ω
P
h/U
t
Subscripts:
1
= inlet to a component
2
= discharge from a component
B
= pertaining to the tip clearance flow
∗
Hayman Professor of Mechanical Engineering Emeritus, Ca
lifornia Institute of Technology, Pasadena, CA 91125, USA.
1
Figure 1: Scale model of the low pressure liquid oxygen pump impeller for the Space Shuttle Main Engine (SSME) in moderate
cavitating conditions in water.
1 Introduction
Since the first experimental measurements many years ago of the complete dynamic transfer function for a cavitating pump
(Ng and Brennen 1976, Brennen
et al.
1982) there has been a general recognition of the importance of various components
of these transfer functions (particularly the cavitation compliance and mass flow gain factor) in determining the dynamic
characteristics and instabilities of systems incorporating such pumps (see for example Rubin 1966 & 1970, Oppenheim and
Rubin 1993, Tsujimoto
et al.
2001, Dotson
et al.
2005). The present paper attempts to summarize some of the r
ecent
understandings and to evaluate the current state of knowledge of transfer functions for cavitating pumps.
2 Pump Transfer Function Data
The linear dynamic transfer matrix for a pump is denoted here by
TP
ij
and is defined by
{
P
2
m
2
}
=
[
TP
11
TP
21
TP
12
TP
22
]{
P
1
m
1
}
(1)
where
P
and
m
are the complex, linearized fluctuating total pressure and mass flow rate and subscripts 1 and 2 refer to the
pump inlet and discharge respectively. In general
TP
ij
will be a function of the frequency,
ω
, of the perturbations and the
mean flow conditions in the pump including the design, the cavitation number,
σ
, and the flow coefficient. In this review
we will focus primarily on the second of these equations and on
TP
21
and
TP
22
since cavitation has a major effect on these
characteristics and they therefore have a critical influence on the potential instab
ilities in the fluid system in which the pump
is installed. But it is valuable in passing to note that
TP
12
=
−
R
−
jωL
where
R
is the pump resistance and
L
is the pump
inertance (valuable measurements of these dynamic characteristics for a non-cavitating pump were first made by Ohashi,
1968, and by Anderson
et al.
, 1971). In the absence of cavitation and compressibility effects
TP
11
= 1 but its departure from
unity due to cavitation is also important in pump dynamics. We should also note that Rubin (2004) argues convincingly
that, from the point of utilizing these transfer functions, it is preferable to transform them and to use
m
2
rather than
m
1
in
the upper equation; here, however, we focus on the lower equation.
The transfer function and other pump dynamic characteristics presented in this paper are non-dimensionalized in the
manner of Brennen
et al.
(1982). Specifically the frequency,
ω
, is non-dimensionalized as
ω
=
ωh/U
t
where
h
is the
peripheral blade tip spacing at the inlet to the pump or inducer (
h
=2
πR
t
/N
where
R
t
is the inlet tip radius and
N
is
the number of main blades) and
U
t
is the inlet tip speed (
U
t
=Ω
R
t
where Ω is the rotational speed in
rad/s
). Then the
compliance,
C
, and mass flow gain factor,
M
, are defined by expanding the transfer function elements,
TP
21
and
TP
22
,at
low frequency in power series in
jω
:
TP
21
=
−
jωC
+(
jω
)
2
C
∗
+
...
(2)
TP
22
=1
−
jωM
+(
jω
)
2
M
∗
+
...
(3)
2
Figure 2: Left: Typical transfer functions for a cavitating inducer obtained by Brennen
et al.
(1982) for the 10
.
2
cm
diameter
SSME inducer operating in water at 6000
rpm
and a flow coefficient of
φ
1
=0
.
07. Data is shown for four different cavitation
numbers,
σ
=(A)0
.
37, (C) 0
.
10, (D) 0
.
069, (G) 0
.
052 and (H) 0
.
044. Real and imaginary parts are denoted by the solid and
dashed lines respectively. The quasistatic pump resistance is indicated by the arrow. Right: Polynomial curves fitted to the
data on the left. Adapted from Brennen
et al.
(1982).
The compliance,
C
, and mass flow gain factor,
M
, are non-dimensionalized by
CN
Ω
2
4
π
2
R
t
and
MN
Ω
2
π
(4)
Note that the above non-dimensionalization scheme differs from that used in Brennen (1994) but is preferred since each blade
produces cavitation that contributes to
C
and
M
.
Those first experimental measurements of the complete dynamic transfer function for a cavitating pump (Ng and Brennen
1976, Brennen
et al.
1982) were carried out in water with a series of model i
nducers including a scale model of the low pressure
LOX inducer in the Space Shuttle Main Engine (SSME). A typical photograph of the 10
.
2
cm
diameter version of that inducer
under moderate cavitating conditions is included as figure 1 which
illustrates the tip clearance backflow and cavitation that
is typical of many inducers (Brennen 1994).
Measured transfer functions for that 10
.
2
cm
diameter SSME inducer operating in water at 6000
rpm
,aflowcoefficient
of
φ
1
=0
.
07 and various cavitation numbers,
σ
, are reproduced in figure 2 (left) where the four transfer functions elements
are each plotted against a dimensionless frequency, the real parts as the solid lines and the imaginary parts as dashed lines.
We should note that this data necessarily has substantial uncertainity associated with it and therefore polynomial fits in
the Laplace variable
jω
were produced in order to extract quantities like
R
,
L
,
C
and
M
(the polynomial fits to figure 2
(left) are shown on figure 2 (right)). An up-to-date collection of the available data on the compliance and the mass flow
gain factor is presented in figure 3 where those quantities are plotted against the cavitation number. The data on the SSME
inducers in water is extracted from figure 2 while the J2 oxidizer data was derived by Brennen and Acosta (1976) using test
data and a heuristic dynamic model of the test facility. The LE-7 test data in liquid nitrogen was obtained by Shimura
(1995). The LE-7A data is the only LOX data and was also extracted from test data by Hori and Brennen (2011). All of this
data is subject to significant uncertainity though the original SSME data is probably the most reliable since it is based on
measurements of the complete dynamic transfer function. Nevertheless, with one exception, both the compliance and mass
flow gain factor data exhibit significant consistency in which both
C
and
M
are inversely proportional to
σ
. The exception
is the LE-7A LOX data for the mass flow gain factor; whether this discrepancy is within the uncertainity band or an actual
3
Figure 3: Dimensionless cavitation compliance (left) and mass flow gain factor (right) plotted against tip cavitation number
for: [a] Brennen
et al.
(1982) SSME 10.2cm model inducer in wate
r (solid blue squares) [b] Brennen
et al.
(1982) SSME
7.6cm model inducer in water (open blue squares) [c] Brennen & Acosta (1976) J2-Oxidizer (solid green circles) analysis [d]
Hori & Brennen (2011) LE-7A LOX data (solid red triangles) [e] Shimura (1995) LE-7 LN2 data (open red triangles).
LOX thermal effect remains to be seen.
Before further discussion of this data collection we digress briefly to introduce a property in the dynamics of cavitating
pumps that has not r
eceived sufficient attention in the past, namely the fundamental resonant frequency of a cavitating
pump.
3 Resonant Frequency of a Cavitating Pump
It has been known for a long time that a cavitating inducer or pump may exhibit a violent surge oscillation at subsynchronous
frequencies that results in very large pressure and flow rate oscillations in the system of which the pump is a part (Sack and
Nottage 1965, Rosemann 1965, Natanzon
et al.
1974, Miller and Gross 1967, Braisted & Brennen 1980, Brennen 1994, Zoladz
2000). In the early days, this was known as ”auto-oscillation” but the preferred name in r
ecent times has been ”cavitation
surge”. It typically occurs at low cavitation numbers just above those at which cavitation head loss becomes severe. Often
it is preceded by a rotating cavitation pattern (see, for example, Kamijo
et al.
1994, Tsujimoto
et al.
1993, Hashimoto
et al.
1997, Zoldaz 2000). Figure 4 reproduces data on the frequencies of oscillation observed for the model SSME i
nducer and for
a helical inducer by Braisted and Brennen (1980); they also plotted a rough empirical fit to that data which approximated
the dimensionless surge frequency by (5
σ
)
1
2
. More recently we recognize that this ”natural frequency of a cavitating pump”
has a more fundamental origin as follows:
Almost any reasonable, proposed dynamic model for a cavitating inducer or pump (such as that on the right of figure 4
designed to simulate the parallel streams of main flow and tip clearance flow) which incorporates both the pump inertance,
L
, and the cavitation compliance,
C
, clearly exhibits a natural frequency, Ω
P
,givenby
Ω
P
=
1
(
LC
)
1
2
(5)
Using the data for the SSME LOX inducer from Brennen (1994) we can approximate
L
and
C
by
L
≈
10
R
t
and
C
≈
0
.
05
R
t
σ
Ω
2
(6)
so that, substituting into equation 5,
Ω
P
Ω
≈
(2
σ
)
1
2
or Ω
P
≈
Ω
P
h
U
t
≈
(5
σ
)
1
2
(7)
This is precisely the same as the result proposed empirically by Braisted and Brennen (1980) and shown on the left in figure
4. We will refer to this as the natural frequency of a cavitating pump. Indeed the data of figure 4 (left) displays further detail
4
Figure 4: Left: Non-dimensional cavitation surge frequency as a function of cavitation number for the SSME model inducers
at various speeds and flow coefficients as shown. The theoretical prediction is the dashed blue line, (5
σ
)
1
2
. Adapted from
from Braisted and Brennen (1980). Right: A dynamic model of the main flow and the parallel tip clearance backflow in a
cavitating inducer.
of this cavitating pump property. There is a manifest trend for the frequency to decrease somewhat with flow coefficient and
this seems certain to be the result of an increasing volume of cavitation and increasing compliance as the blades are loaded
up at lower flow coefficients.
It is important to emphasize that this does not necessarily mean that the major system instability oscillations occur at
this frequency. The study of Hori and Brennen (2011) discussed later in this paper shows, however, that major instabilities
or resonances can occur when this natural frequency for a cavitating pump coincides with other system frequencies such as
an organ pipe mode in a suction or discharge tube.
4 Phase lags in the Cavitation Dynamics
Several researchers (Brennen 1973, Otsuka
et al.
1996, Rubin 2004) have pointed out that the compliance and mass flow gain
factor may become complex as the frequency increases and that this can have important consequences for launch vehicles.
This is clearly equivalent to significant values of the quadratic terms in the expansions 8 and 9 but Rubin puts the values of
C
∗
and
M
∗
in terms of a compliance phase lag and a mass flow gain factor phase lag. One can visualize these phase lags
as delays in the cavitation volume response to the pressure and incidence angle perturbations respectively. In this paper we
will follow Rubin in writing the expansions 8 and 9 up to and including the quadratic order as
TP
21
=
−
jωC
{
1
−
jω
τ
C
}
(8)
TP
22
=1
−
jωM
{
1
−
jω
τ
M
}
(9)
where
τ
C
and
τ
M
are the non-dimensional compliance phase lag and mass flow gain factor phase lag respectively. Data on
these quadratic terms in the frequency expansions is, of course, subject to even greater uncertainity that the linear terms
that lead to the compliance and mass flow gain factor. Nevertheless, in the light of the increasingly apparent importance of
these terms, we have extracted values of
τ
C
and
τ
M
from the data of figure 2 (right) and plotted them against cavitation
number in figure 5. Note that the uncertainties in this data probably exceed 50%. Nevertheless we might suggest that the
phase lags appear to be roughly independent of the cavitation number and to be somewhat greater for the compliance than
for the mass flow gain factor.
It may be valuable to make some tentative suggestions regarding these phase lags. It seems physically reasonable to
envisage that a stream of cavitating bubbles (for example that carried forward by the backflow) would not respond immediately
to the inlet pressure and flow rate fluctuations but would exhibit a phase lag delay that would increase with the frequency of the
perturbations. Brennen (1973) investigated the compliance of a simple stream of cavitating bubbles at various frequencies,
cavitation numbers and cavitation nuclei sizes. Figure 6 reproduces several figures from that paper which show that the
compliance becomes increasingly complex as the frequency of the perturbations increases, and that the negative imaginary
parts of the compliance which develop as the frequency increases represent just the kind of phase lag that we are addressing
here (the magnitudes of the compliance in figure 6 are not relevant to the current discussion). It is particularly interesting
5
Figure 5: Non-dimensional time lags for the compliance,
τ
C
, and the mass flow gain factor,
τ
M
, as functions of the cavitation
number for the SSME 10.2cm model inducer in water. Taken from the data of Brennen
et al.
(1982).
Figure 6: Real and imaginary parts of the dimensionless compliance (per bubble) of a stream of cavitating bubbles as functions
of a reduced frequency based on the length of the cavitation zone,
L
s
, and its typical velocity,
U
s
. Results shown for several
cavitation numbers,
σ
, and bubble nuclei size,
r
N
. From Brennen (1973).
to observe that the reduced frequency plotted horizontally is defined as
fL
s
/U
s
where
f
is the perturbation frequency (in
Hz
)and
L
s
and
U
s
are respectively the length and velocity of the simple stream of cavitating bubbles studied. Note from
figure 6 that the phase lag becomes important when the reduced frequency increases beyond a value of about 0
.
1. Note, also,
that the frequency,
fL
s
/U
s
≈
1, is a
kinematic
frequency associated with the entry and exit of bubbles from the cavitating
zone rather than a
dynamic
frequency associated with the oscillation of the cavitation volume.
Let us consider the corresponding reduced frequency for the backflow cavitation in the experiments of Brennen
et al.
(1982)
and figure 2. For a 10
.
2
cm
diameter inducer at a speed of 6000
rpm
,aflowcoefficientof
φ
1
=0
.
07 (so that
U
s
≈
200
cm/s
)and
an estimate length
L
s
of about 10
cm
the actual frequency that corresponds to
fL
s
/U
s
=1is
f
=20
Hz
. This corresponds
well to the frequency in figure 2 at which the imaginary parts of the compliance are observed to become well developed.
However, this proposed physical explanation of the compliance phase lag also has some worrying implications. It suggests that
the scaling of the phase lags may be a cause for concern for, at much higher rotational speeds, the phase lag would be much
6
Figure 7: Left: Schematic of the bubbly flow model for the dynamics of cavitating pumps. Right: Transfer functions for the
SSME inducer at
φ
1
=0
.
07 calculated from the bubbly flow model. Adapted from Brennen (1978).
smaller and, consequently, any stability benefit that might accrue from it would be much smaller. However, in the absence of
any hard evidence for the scaling of these quadratic effects, all we can conclude at present is that more measurements over a
broader range of rotational speeds is needed in order to establish appropriate scaling for the phase lags.
We should note before leaving this topic that Otsuka
et al.
(1996) show that a blade cavitation model can also yield
complex compliances and mass flow gain factors that correspond to time lags qualitatively similar to those presented in figure
5.
5 Comments on some analytical models
We comment in the conclusions on the difficulties with any detailed CFD approach that aims to predict the dynamic transfer
function for a cavitating inducer. It seems clear that much progress in developing reduced order models for cavitation in
the complex geometry of an inducer (and, in particular, for the backflow cavitation) will be needed before this approach will
provided practical and useful guidance. However, in the short term crude, one-dimensional models and lumped parameter
models (see, for example, Cervone
et al.
2009) guided by the existing data base can give useful benchmarks. The bubbly flow
model of Brennen (1978) (see figure 7 (left)) incorporated several of the basic phenomena that we now know are inherent in
the dynamic response of an inducer or pump. In particular, the compliance of the bubbly stream within the flow (though
the compressibility of that
bubbly flow had to be represented by a empirical constant,
K
) and the magnitude of the void
fraction fluctuations produced by the fluctuating angle of attack (represented by a second empirical factor of proportionality,
M
). These two features respectively lead to dynamic waves and to kinematic waves in the bubbly blade passage flow. A
typical transfer function derived from the bubbly flow model is reproduced in figure 7 (right) and the similarity with the
transfer functions in figure 2 (right) is encouraging even though the two constants
K
and
M
were empirically chosen.
The measured compliances and mass flow gain factors for the SSME inducers and for the J2 oxidizer inducer are reproduced
in figure 8 in order to compare that data with several predictions from the bubbly flow model (dashed blue lines for several
choices of
K
and
M
). The predictions appear to provide a useful benchmark for future data evaluation and comparison.
Figure 8 also includes predictions from the blade cavitation analysis presented earlier by Brennen and Acosta (1976).
That analysis has the advantage that it does not contain any empirical parameter, as such. However, it assumes that all
the cavitation is contained within a single cavity attached to each blade. Moreover the comparisons in figure 8 suggest that
such a model does not yield very useful results which is not surprising when photographs such as figure 1 indicate that the
cavitation is primarily bubbly cavitation and not blade cavitation (Brennen 1994).
Also included in figure 8 are some quasistatic compliances and mass flow gain factors very recently derived by Yonezawa
et al
(2012) from steady CFD calculations of the cavitating flow in linear cascades. They have also performed calculations at
a series of flow coefficients that show a general trend of increasing compliance and mass flow gain factor as the flow coefficient
7
Figure 8: Dimensionless cavitation compliance (left) and mass flow gain factor (right) plotted against tip cavitation number
for: [a] Brennen
et al.
(1982) SSME 10.2cm model inducer in wate
r (solid blue squares) [b] Brennen
et al.
(1982) SSME 7.6cm
model inducer in water (open blue square
s) [c] Brennen (
1978) bubbly flow mo
del results (dashed blue lines) [d] Brennen &
Acosta (1976) SSME LPOTP blade cavitation prediction (dot-dash blue line) [e] Brennen & Acosta (1976) J2-Oxidizer data
(solid green circles) [f] Brennen & Acosta (1976) J2-Oxidizer blade cavitation prediction (dot-dash green line) [g] Yonezawa
et al.
(2012) quasistatic CFD cascade data (solid red diamonds).
is decreased.
6 Resonances in Globally Oscillating systems
The research literature clearly exhibits a strong bias toward investigations of flow instabilities in systems which are essentially
at rest, usually in a research laboratory test stand. While this bias is understandable, it can be misleading for it tends to
mask the difference between such a flow instability and the resonant response in a flow system subject to global fluctuation.
This is particularly an issue with launch vehicle propulsion systems for they can exhibit some serious resonances with the
oscillating vehicle structure. Following the approach originally developed by Rubin (1966), Hori and Brennen (2011) r
ecently
constructed a time-domain model for prototypical pumping systems in order to examine the response of those systems to
globally imposed acceleration,
a
(
t
). We review those results here for they present a case in which the static ground based
systems appear free of serious instability but the same system exhibits serious resonance when subjected to global oscillation.
Hori and Brennen (2011) constructed dynamic models for four different configurations used during the testing and de-
ployment of the LOX turbopump for the Japanese LE-7A rocket engine. As sketched in figure 9, these configurations include
three ground-based fac
ilities, two cold-test facilities (one with a suction line accumulator and the other without), and a
hot-fire engine test facility. The fourth configuration is the flight hardware. All four configurations include the same LE-7A
turbopump whose cavitation compliance and mass flow gain factor were extracted from the ground tests and were included
in figure 3.
The dynamic model for these LE-7A turbopump systems incorporated the time domain equivalent of the pump transfer
function including pump cavitation compliance and mass flow gain factor terms as well as the known steady pump performance
characteristic. It also included lumped parameter models for the storage tank (fuel or oxidizer), the accumulator, and the
valves, as well as compressible, frictional flow equations for the flows in the feedlines. The assumed boundary conditions
at inlet to and discharge from these hydraulic systems were an assumed storage tank pressure and the back pressure in the
combustion chamber or catchment tank. Additional, pseudo-pressure terms (Batchelor, 1967) were included in the flight
configuration to account for the globally-imposed acceleration,
a
. These model equations were solved numerically in the
time domain using the traditional methods of fluid transients (Wylie
et al.
1993, Brennen 1994) including the method of
characteristics for the feedlines. Low-level white noise pressure perturbations were injected at the pump inlet in order to
provide a trigger for potential cavitation surge, should that be inclined to occur. This technique is based on the assumption
that the cavitation surge (and other dynamic responses) observed in the ground-based tests are similarly triggered by random
pressure noise.
8
7 Comparing the System Response and Stability
The results that Hori and Brennen (2011) obtained for the four LE-7A test sytems are obtained are presented in figures 10,
11, 12, and 13. In the case of the first three ground-based configurations comparison is made with pressure spectra obtained
during the system testing.
The calculated and measured spectra for the first configuration are shown in figure 10 and show excellent agreement.
For a cavitation number greater than 0.04, the pressure fluctuations are very small indeed. However, when the cavitation
number is decreased into the range 0
.
033 to 0
.
020, pressure fluctuations at a non-dimensional frequency of 0
.
22 become
dominant; as described earlier this is the natural frequency of the cavitating pump and the increase occurs when there is a
resonance between that natural frequency (which decreases as
σ
decreases) and the third organ pipe mode of oscillation of
the suction line. However, even these resonant pressure oscillations are inconsequential; for example the amplitude at the
inducer discharge is less than 0.4% of inducer tip dynamic pressure. Note that the spectra also include very small pressure
fluctuations at non-dimensional frequencies of 0.13 and 0.31; these correspond to the second and the fourth organ pipe modes
of the suction line.
Sample results for the second configuration, a different cold-test facility with an accumulator, are presented in figure 11.
The most obvious change from the first configuration is the appearance of a natural resonant oscillation of the flow between the
accumulator and the cavitation in the pump. This occurs because of the short length (and therefore small inertance) of fluid
between the accumulator and the cavitation. As the cavitation number decreases and the cavitation compliance increases,
the frequency of this natural cavitation surge decreases. For
σ>
0
.
040, the inducer pressure fluctuations involved are very
small. But when
σ
is reduced to 0.037, a double resonance occurs involving the natural frequency of the cavitating pump, the
frequency of oscillating of the fluid between the accumulator and the pump
and
the third organ pipe mode of the feedline
between the tank and the pump. This double resonance results in a sudden, substantial increase in the magnitude of the
pressure oscillations. With further decrease in
σ
to 0.035, the fluctuation magnitude decreases again as the double resonance
has passed. The corresponding experimental spectra exhibit good qualitative agreement with the model calculations; the
higher harmonics observed in the test and which do not appear in the model calculations are probably caused by non-linear
effects. However, despite this double resonance, both the tests and the calculations exhibit very small pressure oscillation
amplitudes, less than 1% of inducer tip dynamic pressure and, as in the first configuration, this magnitude is inconsequential.
Spectra for the third configuration, the hot-firing engine test are shown in figure 12. As in the second configuration, the
response is dominated by a strong resonance of the fluid between the pump and the accumulator. The frequency of this
resonance decreases from 0.5 to 0.2 as the cavitation number is decreased from 0.05 to 0.02 (the frequencies are higher than
in the second configuration because the accumulator is much closer to the turbopump). Again, the model results appear to
simulate the test data very well, matching both the frequency and the amplitude. However, the pressure amplitudes are still
very small, less than 0.01% of the inducer tip dynamic pressure. Even when the peak frequency matches one of the suction
line organ pipe frequencies, no large pressure oscillation magnitudes occur because the suction line in the hot-firing engine
test facility is very long and the suction line resistance is large.
Having to some extent validated the model calculations, Hori and Brennen (2011) then turned to the flight configuration.
Figure 9: The four hydraulic system configurations whose dynamic responses are compared.
9
0
0.2
0.4
0.6
0.8
1.0
0.02
0.03
0.04
0
0.5
1.0
Cavitation Numbe
r
Frequency
Pressure Amplitude
[% Inducer Tip Dynamic Pressure]
0
0.2
0.4
0.6
0.8
1.0
0.02
0.03
0.04
0
0.5
1.0
Cavitation Numbe
r
Frequency
Pressure Amplitude
[% Inducer Tip Dynamic Pressure]
0
0.2
0.4
0.6
0.8
1.0
0.04
0.03
0.022
0
0.5
1.0
Cavitation Numbe
r
Frequency
Pressure Amplitude
[% Inducer Tip Dynamic Pressure]
0
0.2
0.4
0.6
0.8
1.0
0.04
0.03
0.022
0
0.5
1.0
Cavitation Numbe
r
Frequency
Pressure Amplitude
[% Inducer Tip Dynamic Pressure]
Figure 10: Model calculations (upper graphs) and test facility measurements (lower graphs) of the pump inlet pressure (left)
and the inducer discharge pressure (right) from the cold test facility without an accumulator, the first configuration.
0
0.2
0.4
0.6
0.8
1.0
0.02
0.03
0.04
0
0.5
1.0
Cavitation Number
Frequency
Pressure Amplitude
[% Inducer Tip Dynamic Pressure]
0
0.2
0.4
0.6
0.8
1.0
0.02
0.03
0.04
0
0.5
1.0
Cavitation Number
Frequency
Pressure Amplitude
[% Inducer Tip Dynamic Pressure]
0
0.2
0.4
0.6
0.8
1.0
0.04
0.03
0.022
0
0.5
1.0
Cavitation Number
Frequency
Pressure Amplitude
[% Inducer Tip Dynamic Pressure]
0
0.2
0.4
0.6
0.8
1.0
0.04
0.03
0.022
0
0.5
1.0
Cavitation Number
Frequency
Pressure Amplitude
[% Inducer Tip Dynamic Pressure]
Figure 11: Model calculations (upper graphs) and test facility measurements (lower graphs) of the pump inlet pressure (left)
and the inducer discharge pressure (right) from the cold test facility with an accumulator, the second configuration.
10
0
0.2
0.4
0.6
0.8
1.0
0.02
0.03
0.04
0
0.05
0.10
Cavitation Numbe
r
Frequency
Pressure Amplitude
[% Inducer Tip Dynamic Pressure]
0
0.2
0.4
0.6
0.8
1.0
0.02
0.03
0.04
0
0.5
1.0
Cavitation Numbe
r
Frequency
Pressure Amplitude
[% Inducer Tip Dynamic Pressure]
0
0.2
0.4
0.6
0.8
1.0
0.05
0.04
0.03
0.02
0
0.05
0.10
Cavitation Numbe
r
Frequency
Pressure Amplitude
[% Inducer Tip Dynamic Pressure]
0
0.2
0.4
0.6
0.8
1.0
0.05
0.04
0.03
0.02
0
0.5
1.0
Cavitation Numbe
r
Frequency
Pressure Amplitude
[% Inducer Tip Dynamic Pressure]
Figure 12: Model calculations (upper graphs) and test facility measurements (lower graphs) of the pump inlet pressure (left)
and the inducer discharge pressure (right) from the hot-firing engine test, the third configuration.
First the response of the flight configuration
without
imposed acceleration was investigated and only very small pressure
oscillations (less than 0.01% of i
nducer tip dynamic pressure) and flowrate oscillations (less than 0.01% of mean flow) were
calculated. Thus, like the first three configurations, the flight configuration is very stable in a non-accelerating frame.
Then the model was used to examine the response of the flight configuration in a sinusoidally accelerating frame with an
acceleration amplitude of 0
.
1
m/s
2
at various non-dimensional frequencies ranging from 0 to 0.5. The magnitude 0
.
1
m/s
2
would be characterisitic of the background excitation experienced in the rocket environment. Typical model results under
non-cavitating conditions are shown in the upper graphs of figure 13 and are similar in magnitude to the results for the
ground-based calculations; the conclusion is that, in the absence of cavitation, the system response is quite muted with
pressure oscillation magnitudes less than 0.05% of i
nducer tip dynamic pressure and flow rate oscillation magnitudes less
than 0.02% of mean flow.
Finally Hori and Brennen (2011) present their key result, namely the response of the flight configuration to the same
range of global oscillation (an a
cceleration magnitude of 0
.
1
m/s
2
for a range of oscillation frequencies), when the pump is
cavitating. The lower graphs of figure 13 present the results for the lowest cavitation number examined namely
σ
=0
.
02.
It is clear that the result is a violent resonant response with amplitudes about two orders of magnitude greater than in
the absence of cavitation. The pressure oscillation magnitudes are more than 2% of i
nducer tip dynamic pressure and the
flow rate oscillation magnitudes are more than 20% of mean flow. Under these cavitating conditions, the largest flow rate
magnitudes occur between the accumulator and the inducer at all frequencies and the largest pressure amplitudes occur at
the inducer discharge. Thus the flow rate oscillation between the accumulator and the i
nducer dominates the overall response
and excites the rest of the system like an oscillating piston. The suction line from the tank to the accumulator also plays
a role, albeit a secondary role. When the frequency of the “piston” coincides with an organ pipe mode of the compressible
liquid between the tank and the cavitating inducer the entire system exhibits a peak response and this happens at each of
those organ pipe modes. There is also an important global response maximum near the natural frequency of the cavitating
pump (0
.
3); at higher frequencies the response dies off rather rapidly.
Thus the model calculations demonstrate how a violent resonant response can occur in the accelerating flight environment
when pump cavitation is present and that this response can occur even when all the ground tests (and the model flight
calculations without cavitation) indicate a stable and well-behaved response. The difficulty of duplicating these adverse flight
environments in any ground test - and therefore of examining such an adverse condition - makes accurate model calculations
an almost essential design tool.
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0.01
0.1
1
0.0001
0.001
0.01
0.1
1
Acceleration Frequency
Pressure Amplitude
[% Inducer Tip Dynamic Pressure]
0.01
0.1
1
0.0001
0.001
0.01
0.1
1
Acceleration Frequency
Flowrate Amplitude [% Mean Flow]
0.01
0.1
1
0.001
0.01
0.1
1
10
Acceleration Frequency
Pressure Amplitude
[% Inducer Tip Dynamic Pressure]
0.01
0.1
1
0.01
0.1
1
10
100
Acceleration Frequency
Flowrate Amplitude [% Mean Flow]
Figure 13: Model calculations for the flight configuration subject to global acceleration. Upper graphs: in the absence of
pump cavitation. Lower graphs: when the pump cavitation number is
σ
=0
.
02. Pressure amplitudes (left) and flow rate
amplitudes (right) over a wide range of different oscillation frequencies and an oscillating a
cceleration magnitude of 0
.
1
m/s
2
.
Solid, dashed and dotted lines respectively present the pump discharge, inducer inlet and tank outlet quantities.
8 Concluding Remarks
In concluding this review we should remark that despite significant progress in understanding the dynamics of cavitation in
pumps and inducers, there is much that remains to be accomplished before an adequate pump system design procedure is
completed. It is, perhaps, most useful in these concluding remarks to identify some of the most glaring gaps in our knowledge.
In terms of accomplishments we do have a reasonable data base supporting our preliminary understanding of the scaling
of the dynamic transfer function with pump size, pump rotating speed (admittedly within a fairly narrow speed range),
cavitation number and flow coefficient. However, most of that data is in water at roughly normal temperatures. Therefore
the first deficiency is the lack of experimental data for the thermal effects on the dynamics. Thermal effects on cavitation
and on the steady state performance of pumps have been extensively study and are well known, for example, in the context
of cryogenic pumps (see, for example, Brennen 1994); thermal effects in liquid oxygen are important and they are pervasive
in liquid hydrogen pumps. But, apart from some preliminary tests (Brennen
et al.
1982, Yoshida
et al.
2009, 2011) and
some very limited theoretical considerations (Brennen 1973), little is really known about the thermal effects on the dynamic
characteristics of cavitating pumps. Testing in fluids other than water is very limited though the recent work of Yoshida
et
al.
(2011) in liquid nitrogen suggests little thermal effect on cavitation surge. The lack of data is, in large measure, due to
the absence of dynamic flow meters for non-aqueous environments. Electromagnetic flow meters have proved invaluable in
the water tests, in part, because of their unique ability to measure the cross-sectionally integrated flow rate irrespective of
axisymmetric velocity profile and, in part, because of their dynamic capability. (Electromagnetic meters for cryogenic fluids
are not out of the question and should be constructively investigated). It seems likely that thermal effects could substantially
dampen the dynamic characteristics and, if so, it would be valuable to confirm or refute this.
Another gap that has become evident in recent years and that has a significant impact on pump system stab
ility is the
effect of complex values for the compliance and mass flow gain factor. Though we have described above some very crude
data on the phase lags for compliance and mass flow gain factor this data has very large uncertainties associated with it and
we have little knowledge about how the values scale with speed or size. These effects and their uncertainty strongly suggest
that a more extensive transfer function data base is needed that would not only examine the thermal effects but also extend
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