untitled - fe_135_6

Christopher Earls Brennen

Hayman Professor of Mechanical Engineering

Emeritus,

California Institute of Technology,

Pasadena, CA 91125

A Review of the Dynamics

of Cavitating Pumps

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 pres-

ently available data for the transfer functions for cavitating pumps and inducers, particu-

larly on the compliance and mass flow gain factor, which are so critical for pump and

system stability. The resonant frequency for cavitating pumps is introduced and con-

texted. Finally, emphasis is placed on the paucity of our understanding of pump dynamics

when the device or system is subjected to global oscillation.

[DOI: 10.1115/1.4023663]

1 Introduction

Since the first experimental measurements many years ago of

the complete dynamic transfer function for a cavitating pump

[

], 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 incorporat-

ing such pumps (see, for example, Refs. [

–

]). The present paper

attempts to summarize some of the recent 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

and is defined by

(1)

where

and

are the complex, linearized fluctuating total pres-

sure and mass flow rate and subscripts 1 and 2 refer to the pump

inlet and discharge, respectively (the reader is referred to Ref. [

]

for extensive discussion of hydraulic transfer functions and their

properties). In general,

will be a function of the frequency,

of the perturbations and the mean flow conditions in the pump,

including the design, the cavitation number,

, and the flow coeffi-

cient. In this review, we will focus primarily on the second of

these equations and on

and

, since cavitation has a major

effect on these characteristics and they therefore have a critical

influence on the potential instabilities in the fluid system in which

the pump is installed [

]. But it is valuable in passing to note that

, where

is the pump resistance and

is the

pump inertance (valuable measurements of these dynamic charac-

teristics for a noncavitating pump were first made by Ohashi [

]

and by Anderson et al. [

]). In the absence of cavitation and

compressibility effects,

1, but its departure from unity due

to the dependence of the pressure rise on cavitation number at

fixed flow rate may also be important, particularly at very low

cavitation numbers.

The transfer function and other pump dynamic characteristics

presented in this paper are nondimensionalized in the manner of

Brennen et al. [

]. Specifically, the frequency,

, is nondimen-

sionalized as

, where

is the peripheral blade tip

spacing at the inlet to the pump or inducer (

, where

is the inlet tip radius and

is the number of main blades) and

is the inlet tip speed (

X

, where

X

is the rotational speed in

rad/s). Then, the compliance,

, and mass flow gain factor,

, are

defined by expanding the transfer function elements,

and

, at low frequency in power series in

þð

...

(2)

þð

...

(3)

The compliance,

, and mass flow gain factor,

, are nondimen-

sionalized by

X

and

X

(4)

Note that the above nondimensionalization scheme differs from

that used in Brennen [

] but is preferred, since each blade pro-

duces cavitation that contributes to

and

Those first experimental measurements of the complete

dynamic transfer function for a cavitating pump [

] were carried

out in water with a series of model inducers, 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 Fig.

, which illustrates the tip clearance backflow and

cavitation that is typical of many inducers [

Measured transfer functions for that 10.2-cm-diameter SSME

inducer operating in water at 6000 rpm, a flow coefficient of

07, and various cavitation numbers,

, are reproduced in

Fig.

(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 uncertainty associ-

ated with it (primarily because of the difficulty of accurately

measuring the unsteady flow rates [

]) and therefore polynomial

fits in the Laplace variable

were produced in order to extract

quantities like

, and

(the polynomial fits to Fig.

(left)

are shown in Fig.

(right)).

An up-to-date collection of the available data on the compli-

ance and the mass flow gain factor is presented in Fig.

, where

those quantities are plotted against the cavitation number. The

data on the 10.2-cm SSME inducer in water was extracted from

Fig.

; the data for the 7.6-cm SSME inducer is more scattered

because of greater uncertainty in the measurements of the transfer

functions for that smaller model [

]. The old J2 oxidizer data was

derived by Brennen and Acosta [

] using test data and a heuristic

dynamic model of the test facility. The LE-7 test data in liquid

nitrogen was obtained by Shimura [

]. The LE-7A data is the

only LOX data and was also extracted from test data by Hori and

Brennen [

]. For a full description of the derivation of the J2,

LE-7, and LE-7A data extraction, the reader is referred to

Contributed by the Fluids Engineering Division of ASME for publication in the

OURNAL OF

LUIDS

NGINEERING

. Manuscript received March 26, 2012; final

manuscript received November 30, 2012; published online April 8, 2013. Assoc.

Editor: Olivier Coutier-Delgosha.

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Refs. [

–

] and references therein. All of this data is subject to

significant uncertainty (due to a combination of uncertainty in the

hypothesized dynamic model or in the measurement of the

unsteady flow rates), though the original SSME data is probably

the most reliable, since it is based on measurements of the

complete dynamic transfer function and not on any hypothesized

dynamic model. Nevertheless, with one exception, both the com-

pliance and mass flow gain factor data exhibit significant consis-

tency, in which both

and

are inversely proportional to

. The

exception is the LE-7A LOX data for the mass flow gain factor;

whether this discrepancy is within the uncertainty band or an

actual 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 received 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 oscil-

lations in the system of which the pump is a part [

–

]. In

the early days, this was known as “auto-oscillation,” but the pre-

ferred name in recent times has been “cavitation surge.” It typi-

cally 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, Refs. [

–

]). Fig-

ure

reproduces data on the frequencies of oscillation observed

for the model SSME inducer and for a helical inducer by Braisted

and Brennen [

]; they also plotted a rough empirical fit to that

data, which approximated the dimensionless surge frequency by

. 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 Fig.

designed to

simulate the parallel streams of main flow and tip clearance flow)

that incorporates both the pump inertance,

, and the cavitation

compliance,

, clearly exhibits a natural frequency,

X

, given by

X

(5)

Using the data for the SSME LOX inducer from Brennen [

], we

can approximate

and

X

(6)

Fig. 2 Left: Typical transfer functions for a cavitating inducer obtained by Brennen et al. [

] for the 10.2-cm-diameter SSME

inducer operating in water at 6000rpm and a flow coefficient of

07. Data is shown for four different cavitation numbers,

(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. [

Fig. 1 Scale model of the low pressure liquid oxygen pump

impeller for the space shuttle main engine (SSME) in moderate

cavitating conditions in water

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so that, substituting into Eq.

(5)

X

X

X

X

(7)

This is precisely the same as the result proposed empirically by

Braisted and Brennen [

] and shown on the left in Fig.

.Wewill

refer to this as the natural frequency of a cavitating pump. Indeed

the data of Fig.

(left) displays further detail 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 fre-

quency. The study of Hori and Brennen [

] 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 [

–

] have pointed out that the compli-

ance and mass flow gain factor may become complex as the fre-

quency 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

and

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 pres-

sure 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

(8)

(9)

where

and

are the nondimensional 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 uncertainty than 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

and

from the data of Fig.

(right) and

plotted them against cavitation number in Fig.

. Note that the

uncertainties in this data probably exceed 50%. Nevertheless, we

might suggest that the phase lags appear to be roughly independ-

ent of the cavitation number and to be somewhat greater for the

compliance than for the mass flow gain factor.

Fig. 3 Dimensionless cavitation compliance (left) and mass flow gain factor (right) plotted against tip cavitation number for:

(

) Brennen et al. [

] SSME 10.2-cm model inducer in water (solid squares); (

) Brennen et al. [

] SSME 7.6-cm model inducer in

water (open squares); (

) Brennen and Acosta [

] J2-Oxidizer (circles) analysis; (

) Hori and Brennen [

] LE-7A LOX data

(solid triangles); (

) Shimura [

] LE-7 LN2 data (open triangles)

Fig. 4 Left: Nondimensional cavitation surge frequency as a function of cavitation number for the SSME model inducers at var-

ious speeds and flow coefficients, as shown. The theoretical prediction is the dashed line,

(adapted from [

]). Right: A

dynamic model of the main flow and the parallel tip clearance backflow in a cavitating inducer.

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It may be valuable to make some tentative suggestions regard-

ing these phase lags. It seems physically reasonable to envisage

that a stream of cavitating bubbles (for example, that carried for-

ward 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 [

] investigated the compliance of a simple stream of

cavitating bubbles at various frequencies, cavitation numbers, and

cavitation nuclei sizes. Figure

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

Fig.

are not relevant to the current discussion). It is particularly

interesting to observe that the reduced frequency plotted horizon-

tally is defined as

, where

is the perturbation frequency (in

Hz) and

and

are, respectively, the length and velocity of the

simple stream of cavitating bubbles studied. Note from Fig.

that

the phase lag becomes important when the reduced frequency

increases beyond a value of about 0.1. Note also that the fre-

quency,

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. [

] and

Fig.

. For a 10.2-cm-diameter inducer at a speed of 6000 rpm, a

flow coefficient of

07 (so that

200 cm

s), and an

estimate length

of about 10 cm, the actual frequency that corre-

sponds to

1is

20 Hz. This corresponds well to the

frequency in Fig.

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 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. [

]

show that a blade cavitation model can also yield complex com-

pliances and mass flow gain factors that correspond to time lags

qualitatively similar to those presented in Fig.

5 Comments on Some Analytical Models

We comment in the conclusions on the difficulties with any

detailed computational fluid dynamics (CFD) approach that aims

Fig. 5 Nondimensional time lags for the compliance,

, and

the mass flow gain factor,

, as functions of the cavitation

number for the SSME 10.2-cm model inducer in water. Taken

from the data of Brennen et al. [

Fig. 6 Real and imaginary parts of the dimensionless compliance (per bubble) of a stream of cavitating bubbles as func-

tions of a reduced frequency based on the length of the cavitation zone,

, and its typical velocity,

. Results shown for

several cavitation numbers,

, and bubble nuclei size,

(from Ref. [

]).

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to predict the dynamic transfer function for a cavitating inducer. It

seems clear that much progress in developing reduced order mod-

els for cavitation in the complex geometry of an inducer (and, in

particular, for the backflow cavitation) will be needed before this

approach will provide practical and useful guidance. However, in

the short-term crude, one-dimensional models and lumped param-

eter models (see, for example, Cervone et al. [

]) guided by the

existing data base can give useful benchmarks. The bubbly flow

model of Brennen [

] (see Fig.

(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 com-

pliance of the bubbly stream within the flow (though the compres-

sibility of that bubbly flow had to be represented by an empirical

constant,

) and the magnitude of the void fraction fluctuations

produced by the fluctuating angle of attack (represented by a sec-

ond empirical factor of proportionality,

). 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 Fig.

(right), and

the similarity with the transfer functions in Fig.

(right) is encour-

aging, even though the two constants

and

were empirically

chosen.

The measured compliances and mass flow gain factors for the

SSME inducers and for the J2 oxidizer inducer are reproduced in

Fig.

in order to compare that data with several predictions from

the bubbly flow model (dashed lines for several choices of

and

). The predictions appear to provide a useful benchmark for

future data evaluation and comparison.

Figure

also includes predictions from the blade cavitation

analysis presented earlier by Brennen and Acosta [

]. That anal-

ysis has the advantage that it does not contain any empirical pa-

rameter as such. However, it assumes that all the cavitation is

contained within a single cavity attached to each blade. Moreover,

the comparisons in Fig.

suggest that such a model does not yield

Fig. 7 Left: Schematic of the bubbly flow model for the dynamics of cavitating pumps. Right: Transfer functions for the SSME

inducer at

07 calculated from the bubbly flow model (adapted from Ref. [

]).

Fig. 8 Dimensionless cavitation compliance (left) and mass flow gain factor (right) plotted against tip cavitation number for:

(

) Brennen et al. [

] SSME 10.2-cm model inducer in water (solid squares); (

) Brennen et al. [

] SSME 7.6-cm model inducer in

water (open squares); (

) Brennen [

] bubbly flow model results (short dash lines); (

) Brennen and Acosta [

] SSME low

pressure oxidizer turbopump blade cavitation prediction (dot dash line); (

) Brennen and Acosta [

] J2-Oxidizer data (circles);

(

) Brennen and Acosta [

] J2-Oxidizer blade cavitation prediction (long dash line); (

) Yonezawa et al. [

] quasistatic CFD

cascade data (diamonds)

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Fig. 9 The four hydraulic system configurations whose dynamic responses are compared (reproduced from

Ref. [

])

Fig. 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 (reproduced

from Ref. [

])

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very useful results, which is not surprising when photographs,

such as Fig.

, indicate that the cavitation is primarily bubbly cav-

itation and not blade cavitation [

Also included in Fig.

are some quasistatic compliances and

mass flow gain factors very recently derived by Yonezawa et al.

[

] 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 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 differ-

ence between such 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. Fol-

lowing the approach originally developed by Rubin [

], Hori and

Brennen [

] recently constructed a time-domain model for proto-

typical pumping systems in order to examine the response of those

systems to globally imposed acceleration,

(

). 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 sys-

tem exhibits serious resonance when subjected to global oscillation.

Hori and Brennen [

] constructed dynamic models for four

different configurations used during the testing and deployment of

the LOX turbopump for the Japanese LE-7A rocket engine. As

sketched in Fig.

, these configurations include three ground-

based facilities, two cold-test facilities (one with a suction line

accumulator and the other without), and a hot-fire engine test fa-

cility. The fourth configuration is the flight hardware. All four

configurations include the same LE-7A turbopump, whose cavita-

tion compliance and mass flow gain factor were extracted from

the ground tests and were included in Fig.

The dynamic model for these LE-7A turbopump systems incor-

porated 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 characteris-

tic. It also included lumped parameter models for the storage tank

(fuel or oxidizer), the accumulator, and the valves, as well as com-

pressible, 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 pseudopressure terms [

] were included in the

flight configuration to account for the globally imposed accelera-

tion,

. These model equations were solved numerically in the

time domain using the traditional methods of fluid transients

[

], 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 trig-

gered by random pressure noise.

7 Comparing the System Response and Stability

The results that Hori and Brennen [

] obtained for the four

LE-7A test systems are presented in Figs.

–

. In the case of

Fig. 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 (reproduced

from Ref. [

])

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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 Fig.

and show excellent agreement. For a cavita-

tion 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 nondimen-

sional 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

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 nondimensional 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 Fig.

. The

most obvious change from the first configuration is the appearance

of a natural resonant oscillation of the flow between the accumula-

tor 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

040, the in-

ducer pressure fluctuations involved are very small. But when

reduced to 0.037, a double resonance occurs, involving the natural

frequency of the cavitating pump, the frequency of oscillation 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

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 calcu-

lations; 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 and 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 Fig.

. 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 in-

ducer 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 [

] then turned to the flight configuration. First, the

response of the flight configuration without imposed acceleration

was investigated and only very small pressure oscillations (less

than 0.01% of inducer tip dynamic pressure) and flow rate oscilla-

tions (less than 0.01% of mean flow) were calculated. Thus, like

Fig. 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 (reproduced from Ref. [

])

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the first three configurations, the flight configuration is very stable

in a nonaccelerating frame. Then, the model was used to examine

the response of the flight configuration in a sinusoidally accelerat-

ing frame with an acceleration amplitude of 0

at various

nondimensional frequencies ranging from 0 to 0.5. The magnitude

would be characteristic of the background excitation

experienced in the rocket environment. Typical model results

under noncavitating conditions are shown in the upper graphs of

Fig.

and are similar in magnitude to the results for the ground-

based calculations; the conclusion is that, in the absence of cavita-

tion, the system response is quite muted with pressure oscillation

magnitudes less than 0.05% of inducer tip dynamic pressure and

flow rate oscillation magnitudes less than 0.02% of mean flow.

Finally, Hori and Brennen [

] present their key result, namely,

the response of the flight configuration to the same range of global

oscillation (an acceleration magnitude of 0

for a range of

oscillation frequencies) when the pump is cavitating. The lower

graphs of Fig.

present the results for the lowest cavitation num-

ber examined, namely,

02. It is clear that the result is a vio-

lent 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 inducer 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 in-

ducer at all frequencies, and the largest pressure amplitudes occur

at the inducer discharge. Thus, the flow rate oscillation between

the accumulator and the inducer 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” coin-

cides 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 fre-

quencies, the response dies off rather rapidly.

Thus, the model calculations demonstrate how a violent reso-

nant 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 con-

dition—makes accurate model calculations an almost essential

design tool.

8 Concluding Remarks

In concluding this review, we should remark that, despite sig-

nificant progress in understanding the dynamics of cavitation in

pumps and inducers, there is much that remains to be accom-

plished before an adequate pump system design procedure is

Fig. 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

02. Pressure ampli-

tudes (left) and flow rate amplitudes (right) over a wide range of different oscillation frequencies and an

oscillating acceleration magnitude of 0

. Solid, dashed, and dotted lines, respectively, present the

pump discharge, inducer inlet, and tank outlet quantities (reproduced from Ref. [

]).

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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), and cavita-

tion number. The effect of flow coefficient is less well-

established, due to the limited range of flow coefficients for which

transfer functions have been measured. However, most of that

data is in water at roughly normal temperatures. Therefore, the

first major deficiency is the lack of experimental data for the ther-

mal effects on the dynamics. Thermal effects on cavitation and on

the steady state performance of pumps have been extensively

studied and are well known; for example, in the context of cryo-

genic pumps (see, for example, Brennen [

]), thermal effects in

liquid oxygen are important and they are pervasive in liquid

hydrogen pumps. But apart from some preliminary tests [

]

and some very limited theoretical considerations [

], little is

really known about the thermal effects on the dynamic character-

istics of cavitating pumps. Testing in fluids other than water is

very limited, though the recent work of Yoshida et al. [

] in liq-

uid 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 nonaqueous 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 con-

structively 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 stability 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 fac-

tor, 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 sug-

gest that a more extensive transfer function data base is needed

that would not only examine the thermal effects but also

extend the data to higher speeds. Such experimental investiga-

tions should also investigate the nonlinear effects that obvi-

ously limit the amplitude of the cavitation instabilities and the

resonant responses.

Another major gap in our current understanding has been evi-

dent for some time through the work of Rubin [

] and others on

the response of pump systems in globally oscillating environments

and was particularly evident in the work of Hori and Brennen [

]

described above. There are some very real questions about the

dynamic response of cavitation and of cavitating pumps subjected

to translational or rotational acceleration. The only surefire way to

answer these questions is to conduct experiments with a pump

loop experiment mounted on a shaker table that can impose sub-

stantial global oscillations up to frequencies of the order of 50 Hz

or more. Given the availability of huge shaker tables for earth-

quake engineering research and the known destructive consequen-

ces of instabilities, such as the Pogo instability of liquid-propelled

rockets, it is surprising that such experiments have not been car-

ried out in the past.

Finally, I can anticipate that some will promote the use of com-

putational models for cavitating flows in order to try to bridge

these gaps. Though there have been some valuable efforts to de-

velop CFD methods for cascades (see, for example, Iga et al. [

]),

the problem with this suggestion is that accurate numerical treat-

ments for cavitating pumps that will adequately represent both the

nonequilibrium character of cavitation and adequately respond to

flow fluctuations are still in a very early stage of development.

Codes that can also handle the complex geometry and turbulence

of the flow in an inducer, including the tip clearance backflow, are

many years away. It seems clear that much progress will be

needed in the development of reduced-order models for cavitation

before the computational approach can produce useful, practical

results.

Acknowledgment

The author wishes to acknowledge the extensive and valuable

support provided by the NASA George Marshall Space Flight

Center, Huntsville, AL, during much of the research discussed in

this paper. I also owe a great debt to my colleague and collabora-

tor Allan Acosta as well as to numerous students at Caltech,

particularly S. L. Huang and David Braisted. The numerous con-

structive discussions with long-time associates Loren Gross,

Henry Stinson, Sheldon Rubin, Jim Fenwick, Tom Zoladz, Ken-

jiro Kamijo, Yoshi Tsujimoto, and Shusuke Hori are gratefully

acknowledged. I also appreciate the support of JAXA, the Japan

Aerospace Exploration Agency, in sponsoring the visit of Shusuke

Hori to Caltech.

Nomenclature

global acceleration of the pumping system

cavitation compliance

;

quadratic compliance and mass flow gain factor

coefficients

frequency in Hz

inducer blade tip spacing, 2

¼ð

;

bubbly flow model parameters

pump inertance

length of the cavitation zone

complex fluctuating mass flow rate

mass flow gain factor

number of main inducer blades

complex fluctuating total pressure

pump resistance

radius of the inducer tip

pump transfer function elements

velocity in the cavitation zone

velocity of the inducer tip

inlet flow coefficient

cavitation number

liquid density

;

phase lags of the compliance and mass flow gain factor

dimensionless frequency,

dimensional radian frequency

X

frequency of rotation of the pump

X

natural cavitation frequency of the inducer

X

dimensionless natural cavitation frequency of the in-

ducer,

X

Subscripts

inlet to a component

discharge from a component

pertaining to the tip clearance flow

References

[1] Ng, S. L., and Brennen, C. E., 1978, “Experiments on the Dynamic Behavior of

Cavitating Pumps,”

ASME J. Fluids Eng.

100

(2), pp. 166–176.

[2] Brennen, C. E., Meissner, C., Lo, E. Y., and Hoffman, G. S., 1982, “Scale

Effects in the Dynamic Transfer Functions for Cavitating Inducers,”

ASME J.

Fluids Eng.

104

(4), pp. 428–433.

[3] Rubin, S., 1966, “Longitudinal Instability of Liquid Rockets Due to Propulsion

Feedback (POGO),”

J. Spacecr. Rockets

(8), pp. 1188–1195.

[4] Rubin, S., 1970, “Prevention of Coupled Structure-Propulsion Instability

(POGO),” NASA Space Vehicle Design Criteria (Structures), Report No.

NASA SP-8055.

[5] Oppenheim, B. W., and Rubin, S., 1993, “Advanced Pogo Stability Analysis for

Liquid Rockets,”

J. Spacecr. Rockets

(3), pp. 360–373.

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