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A review of the dynamics of cavitating pumps

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A review of the dynamics of cavitating pumps

C E Brennen

Hayman Professor of Mechanical Engineer

ing, Emeritus, California Institute of

Technology, Psadena, California 91125, USA

brennen@caltech.edu

Abstract

. This paper presents a review of som

e 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 f

unctions for cavitating pumps and inducers,

particularly 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 contexted.

Finally emphasis is placed on the paucity of ou

r understanding of pump dynamics when the

device or system is subjected to global oscillation.

1. Introduction

Since the first experimental measurements many y

ears 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) i

n determining the dynamic characteristics and

instabilities of systems incorporating such pum

ps (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 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 by TP

and is defined by

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

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

these equations and on TP

and TP

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 TP

=-R-j



L 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

= 1 but its departure from unity due to cavitation is also

important in pump dynamics.

26th IAHR Symposium on Hydraulic Machinery and Systems

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IOP Conf. Series: Earth and Environmental Science

(2012) 012001

doi:10.1088/1755-1315/15/1/012001

Published under licence by IOP Publishing Ltd

The transfer function and other pump dynamic ch

aracteristics presented in this paper are non-

dimensionalized in the manner of Brennen

et al

. (1982). Specifically the frequency,



, is non-

dimensionalized as



’=



h/U

where h is the peripheral blade tip spacing at the inlet to the pump or

inducer (h=2



/N where R

is the inlet tip radius and N is the number of main blades) and U

is the

inlet tip speed (U



where



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

and TP

,atlow

frequency in a power series in j



The compliance, C, and mass flow gain factor, M, are non-dimensionalized by

Note that the above non-dimensionalization scheme differs from that used in Brennen (1994) but is

preferred since each blade produces cav

itation that contributes to C and M.

Those first experimental measurements of the com

plete 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

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

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).

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.

Measured transfer functions for that 10.2

diameter SSME inducer operating in water at 6000

rpm

a flow coefficient of



=0.07 and various cavitation numbers,



, 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 w

ith it and therefore polynomial fits in the Laplace

variable j



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

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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 uncertainty 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



. 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.

Figure 2.

Left: Typical transfer functions for a cavitating inducer obtained by Brennen

et al

. (1982)

for the 10.2

diameter SSME inducer operating in water at 6000

rpm

and a flow coefficient of



=0.07. 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

. (1982).

Before further discussion of this data collection w

e 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 freque

ncies 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 recent times has been "cavitation

surge". It typically occurs at low cavitation num

bers 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

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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 inducer 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



)

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, an

d the cavitation compliance, C, clearly exhibits a

natural frequency,



,givenby

Using the data for the SSME LOX inducer from Brennen (1994) we can approximate L and C by

so that, substituting into Equation 5.,

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 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 thi

s frequency. The study of Hori and Brennen (2011)

discussed later in this paper shows, however, that m

ajor instabilities or resonances can occur when this

natural frequency for a cavitating pump coincides w

ith other system frequencies such as an organ pipe

mode in a suction or discharge tube.

Figure 3.

Dimensionless cavitation compliance (left) and mass flow gain factor (right) plotted against

tip cavitation number for: [a] Brennen

et al

. (1982) SSME 10.2

model inducer in water (solid blue

squares) [b] Brennen

et al

. (1982) SSME 7.6

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).

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Figure 4.

Left: Non-dimensional cavitation surge freque

ncy 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



)

1/2

. Adapted from Braisted and Brenne

n (1980). Right: A dynamic model of

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

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 expansion of Equati

ons 1. and 2. 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 fo

llow Rubin in writing the expansions of Equations 1.

and 2. up to and including the quadratic order as

where



and





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 appa

rent importance of these terms, we have extracted

values of



and





from the data of Figure 2. (right) and plotted them against cavitation number in

Figure 5.

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. Fi

gure 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 curre

nt discussion). It is particularly interesting to

observe that the reduced frequency plotted horizontally is defined as f L

where f is the perturbation

frequency (in

)andL

and U

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

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reduced frequency increases beyond a value of a

bout 0.1. Note, also, that the frequency, f L

unity 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.

Figure 5.

Non-dimensional time lags for the compliance,



, and the mass flow gain factor,





,as

functions of the cavitation number for the SSME

10.2cm model inducer in water. Taken from the data

of Brennen

et al

. (1982).

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.

Figure 6.

Real and imaginary parts of the dimensionless compliance (per bubble) of a stream of

cavitating bubbles as functions of a reduced fre

quency based on the length of the cavitation zone, L

and its typical velocity, U

. Results shown for seve

ral cavitation numbers,



, and bubble nuclei size, r

From Brennen (1973).

Let us consider the corresponding reduced frequenc

y for the backflow cavitation in the experiments

of Brennen

et al

. (1982) and Figure 2. For a 10.2

diameter inducer at a speed of 6000

rpm

,aflow

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coefficient of



=0.07 (so that U

is approximately 200

cm/s

) and an estimate length L

of about 10

the frequency that corresponds to a value of unity for f L

is f=20

. 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 explan

ation 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

. (1996) show that a blade cavitation

model can also yield complex compliances and mas

s 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 cavitatio

n in the complex geometry of an inducer (and, in

particular, for the backflow cavitation) will be need

ed before this approach will provide 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 mode

l of Brennen (1978) (see Figure 7. (left)) incorporated several of

the basic phenomena that we now know are inhere

nt in the dynamic response of an inducer or pump.

In particular, the compliance of the bubbly stream w

ithin the flow (though the compressibility of that

bubbly flow had to be represented by a empirical cons

tant, 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 respectiv

ely 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.

Figure 7.

Left: Schematic of the bubbly flow model for the dynamics of cavitating pumps. Right:

Transfer functions for the SSME inducer at



=0.07 calculated from the bubbly flow model. Adapted

from Brennen (1978).

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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 analy

sis presented earlier by Brennen and Acosta (1976).

That analysis has the advantage that it does not cont

ain 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 cas

cades. 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.

Figure 8.

Dimensionless cavitation compliance (left) and mass flow gain factor (right) plotted against

tip cavitation number for: [a] Brennen

et al

. (1982) SSME 10.2

model inducer in water (solid blue

squares) [b] Brennen

et al

. (1982) SSME 7.6

model inducer in water (open blue squares) [c]

Brennen (1978) bubbly flow model 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).

6. Resonances in globally oscillating systems

The research literature clearly exhibits a strong bi

as 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) recently constructed a time-domain model for prototypical pumping systems in order

to examine the response of those systems to glob

ally 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 r

esonance when subjected to global oscillation.

Hori and Brennen (2011) 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 Figure 9., 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 facility. The fourth

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configuration is the flight hardware. All four

configurations include the same LE-7A turbopump

whose cavitation compliance and mass flow gain f

actor 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 t

he cavitation surge (and other dynamic responses)

observed in the ground-based tests are similarly triggered by random pressure noise.

Figure 9.

The four hydraulic system configurati

ons whose dynamic responses are compared.

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, 13. In the case o

f 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 freque

ncy 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 non-dime

nsional frequencies of

0.13 and 0.31; these correspond to the second and the fourth organ pipe modes of the suction line.

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Figure 10.

Model calculations (upper graphs) and test f

acility 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.

Sample results for the second configuration, a di

fferent cold-test facility with an accumulator, are

presented in Figure 11. The most obvious change fro

m the first configuration is the appearance of a

natural resonant oscillation of the flow between th

e 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 decrea

ses and the cavitation compliance increases, the

frequency of this natural cavitation surge decreases. For



>0.040, the inducer pressure fluctuations

involved are very small. But when



is reduced to 0.037, a double resonance occurs involving the

natural frequency of the cavitating pump, the fre

quency 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, subs

tantial increase in the magnitude of the pressure

oscillations. With further decrease in



to 0.035, the fluctuation magnitude decreases again as the

double resonance has passed. The corresponding e

xperimental 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 turb

opump). 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.

26th IAHR Symposium on Hydraulic Machinery and Systems

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Figure 11.

Model calculations (upper graphs) and test f

acility 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

Figure 12.

Model calculations (upper graphs) and test f

acility measurements (lower graphs) of the

pump inlet pressure (left) and the inducer discharge pressure (right) from the hot-firing engine test, the

third configuration.

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Figure 13.

Model calculations for the flight configurat

ion subject to global accele

ration. Upper graphs:

in the absence of pump cavitation. Lower graphs: when the pump cavitation number is



=0.02.

Pressure amplitudes (left) and flow rate amplitude

s (right) over a wide range of different oscillation

frequencies and an oscillating acceleration magnitude of 0.1

m/s

. Solid, dashed and dotted lines

respectively present the pump discharge, inducer inlet and tank outlet quantities.

Having to some extent validated the model calculations, Hori and Brennen (2011) then turned to

the flight configuration. First the response of the flight configuration

without

imposed acceleration was

investigated and only very small pressure oscillat

ions (less than 0.01% of inducer 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 s

table 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

at various non-dimensional frequencies ranging from 0 to 0.5. The

magnitude 0.1

m/s

would be characterisitic of the backgroun

d 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 inducer tip dynamic pr

essure 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 acceleration magnitude of 0.1

m/s

for a range

of oscillation frequencies), when the pump is cav

itating. The lower graphs of Figure 13. present the

results for the lowest cavitation number examined namely



=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 a

re 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 inducer at all

frequencies and the largest pressure amplitudes occur at the inducer discharge. Thus the flow rate

oscillation between the accumulator and the indu

cer dominates the overall response and excites the

26th IAHR Symposium on Hydraulic Machinery and Systems

IOP Publishing

IOP Conf. Series: Earth and Environmental Science

(2012) 012001

doi:10.1088/1755-1315/15/1/012001

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 cavitatio

n 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 duplicatin

g 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.

8. Conclusion

In concluding this review we should remark that despite significant progress in understanding the

dynamics of cavitation in pumps and inducers, there i

s 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 t

he most glaring gaps in our knowledge.

In terms of accomplishments we do have a reas

onable 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), cav

itation 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 studied and are well known, for example, in

the context of cryogenic pumps (see, for example, Br

ennen 1994); thermal effects in liquid oxygen are

important and they are pervasive in liquid hydroge

n pumps. But, apart from some preliminary tests

(Brennen

et al.

1982, Yoshida

et al.

2009, 2011) and some very limited t

heoretical considerations

(Brennen 1973), little is really known about the therm

al effects on the dynamic characteristics of

cavitating pumps. Testing in fluids other than wate

r is very limited though the recent work of Yoshida

et al.

(2011) in liquid nitrogen suggests little thermal e

ffect 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 capab

ility. (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 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

factor this data has very large uncertainties ass

ociated with it and we have little knowledge about how

the values scale with speed or size. These effects a

nd their uncertainty strongly suggest that a more

extensive transfer function data base is needed th

at would not only examine the thermal effects but

also extend the data to higher speeds. Such experimental investigations should also investigate the

non-linear effects that obviously limit the amplit

ude of the cavitation instabilities and the resonant

responses.

Another major gap in our current understanding has been evident for some time through the work

of Rubin and others on the response of pump syste

ms 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

26th IAHR Symposium on Hydraulic Machinery and Systems

IOP Publishing

IOP Conf. Series: Earth and Environmental Science

(2012) 012001

doi:10.1088/1755-1315/15/1/012001

with a pump loop experiment mounted on a shaker tab

le that can impose substantial global oscillations

up to frequencies of the order of 50

or more. Given the availability of huge shaker tables for

earthquake engineering research and the known dest

ructive consequences of instabilities such as the

Pogo instability of liquid-propelled rockets, it is surprising that such experiments have not been

carried out in the past.

Finally, I can anticipate that some will promote t

he use of computational models for cavitating

flows in order to try to bridge these gaps. Though

there have been some valuable efforts to develop

CFD methods for cascades (see, for example, Iga

et al.

2004), the problem with this suggestion is that

accurate numerical treatments for cavitating pumps that will adequately represent both the non-

equilibrium 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 backfl

ow are many years away. It seems clear that much

progress will be needed in the development of r

educed-order models for cavitation before the

computational approach can produce useful, practical results.

Acknowledgments

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 collaborator Allan Acosta as well as to numerous students at

Caltech, particularly S.L.Huang and David Braiste

d. The numerous constructive discussions with long

time associates Loren Gross, Henry Stinson, Sheldon Rubin, Jim Fenwick, Tom Zoladz, Kenjiro

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

C*,M*

K’, M’

Global acceleration of the pumping system

Cavitation compliance

Quadratic compliance and mass flow gain

factor coefficients

Frequency [Hz]

Inducer blade tip spacing, 2



(-1)

1/2

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 fluctating 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

Dimensional radian frequency

Dimensionless frequency,



h/U

Frequency of rotation of the pump

Natural cavitation frequency of the inducer

Dimensionless natural cavitation frequency of t

he inducer,



h/U

References

[1] Ng S L and Brennen C E 1978

ASME J. Fluids Eng

100

, 166-176.

[2] Brennen C E, Meissne

C, Lo E Y and Hoffman G S 1982

ASME J. Fluids Eng

104

, 428-433.

[3] Rubin S 1966

AIAA J. Spacecraft and Rockets

, No. 8, 1188-1195.

[4] Rubin S 1970 Prevention of coupled str

ucture-propulsion instability (POGO)

NASA SP

-8055.

[5] Oppenheim B W and Rubin S 1993

AIAA J. Spacecraft and Rockets

, No. 3, 360-373.

[6] Tsujimoto Y, Kamijo K and Brennen C

J. Propulsion and Power

, No.3, 636-643.

26th IAHR Symposium on Hydraulic Machinery and Systems

IOP Publishing

IOP Conf. Series: Earth and Environmental Science

(2012) 012001

doi:10.1088/1755-1315/15/1/012001

[7] Dotson K W, Rubin S and Sako B H. 2005

AIAA J. Propulsion and Power

, No. 4, 619-626.

[8] Ohashi H. 1968 N

ASA TN

D-4298.

[9] Anderson D A , Blade R J and Stevens W 1971 R

esponse of a radial-bladed centrifugal pump

to sinusoidal disturbances for non-cavitating flow

NASA TN

D-6556.

[10] Brennen C E. 1994

Hydrodynamics of Pumps

. Oxford Univ. Press. and Concepts ETI.

[11] Brennen C E and Acosta A J 1976

ASME J. Fluids Eng.

, 182-191.

[12] Shimura T 1995

AIAA J. Propulsion and Power

, No. 2, 330-336.

[13] Hori S and Brennen C E. 2011

J. Spacecraft and Rockets

, No.4, 599-608.

[14] Sack L E and Nottage H B 1965

ASME J. Basic Eng.

, 917--924.

[15] Rosenmann W 1965 Experimental investigations of hydrodynamically induced shaf

forces with

a three bladed inducer

Proc. ASME Symp. on Cavitation in Fluid Machinery

, 172--195.

[16] Natanzon M S, Bl'tsev N E, Bazhanov V V and Leydervarger M R 1974

Fluid Mech., Soviet

Res

, No.1, 38--45.

[17] Mille

C D. and Gross L A 1967 A performance investigation of an eight-inch hubless pump

inducer in water and liquid nitrogen

NASA TND-3

807.

[18] Braisted D M and Brennen C E 1980

Polyphase Flow and Transport Technology

,ed.R.A.

Bajura, ASME Publ., New York, 157-166.

[19] Zoladz T 2000 Observations on rotating cav

itation and cavitation surge from the development

of the fastrac engine turbopump

36th AIAA/ASME/SAE/ASEE Joint Propulsion Conf.

AIAA-2000-3403, Huntsville, AL.

[20] Kamijo K, Shimura T and Tsujimoto Y 1994

ASME Cavitation and Gas-Liquid Flow in Fluid

Machinery and Devices

190

, 33-43.

[21] Tsujimoto Y, Kamijo K and Yoshida Y 1993

ASME J. Fluids Eng

115

, No. 1, 135-141.

[22] Hashimoto T. Yoshida M, Watanabe M, Kamijo

and Tsujimoto Y 1997

AIAA J. Propulsion

and Power

, No. 4, 488-494.

[23] Brennen C 1973

ASME J. Fluids Eng.

, 533-541.

[24] Otsuka S, Tsujimoto Y, Kamijo

and Furuya O 1996

ASME J. Fluids Eng.

118

, 400-408.

[25] Rubin S 2004 An interpretation of

ransfe

function data fo

a cavitating pump

40th

AIAA/ASME/SAE/ASEE Joint Propulsion Conference

, AIAA-2004-4025, Florida.

[26] Brennen C E 1978

J. Fluid Mech.,

, 223-240.

[27] Cervone A, Tsujimoto Y and Kawata Y 2009

ASME J. Fluids Eng

131

, No. 4.

[28] Yonezawa K, Aono J, Kang D, Horiguc

hi H, Kawata Y and Tsujimoto Y 2012

International J

of Fluid Machinery and Sysyem

(to be piblidhed).

[29] Cervone A, Tsujimoto Y and Kawata Y 2009

ASME J. Fluids Eng.

131

, No. 4.

[30] Batchelo

G K. 1967

An Introduction to Fluid Dynamics

Cambridge Univ. Press.

[31] Wylie E B, Streeter V L and Suo L. 1993

Fluid Transients in Systems

Prentice Hall.

[32] Yoshida Y, Sasao Y and Watanabe M 2009

ASME J. Fluids Eng.,

131

, No. 9.

[33] Yoshida Y, Nanri H, Kikuta K, Kazami Y, Iga Y and Ikohagi T 2011

ASME J. Fluids Eng.,

133

No. 6.

[34] Iga Y, Nohmi M, Goto A and Ikohagi T 2004

ASME J. Fluids Eng.

126

, No. 3, 419-429.

26th IAHR Symposium on Hydraulic Machinery and Systems

IOP Publishing

IOP Conf. Series: Earth and Environmental Science

(2012) 012001

doi:10.1088/1755-1315/15/1/012001