Brennen Ch.E. Fundamentals of Multiphase Flow

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tems of a liquid-propelled rocket engine), it is very important to be able to

evaluate the stability of that system and knowledge of the transfer function

for the cavitating pumps is critical to that analysis (Rubin 1966).

For simplicity in this analytical model, the pump inlet and discharge ﬂows

are assumed to be purely incompressible liquid (density ρ

). Then the inlet

and discharge ﬂows can be characterized by two ﬂow variables; convenient

choices are the total pressure, p

, and the mass ﬂow rate, ˙m

,wherethe

inlet and discharge quantities are given by i =1 andi = 2 respectively as

described in section 15.7.2. Consider now the form of the transfer function

(equation 15.10) connecting these ﬂuctuating quantities. As described in

section 15.7.2 the transfer function will be a function not only of frequency

but also of the pump geometry and the parameters deﬁning the mean ﬂow

(see section 8.4.1). The instantaneous ﬂow rates at inlet and discharge will

be diﬀerent because of the rate of change of the total volume, V , of cavitation

within the pump. In the absence of cavitation, the pump transfer function

is greatly simpliﬁed since (a) if the liquid and structural compressibilities

are are neglected then ˙m

=˙m

and it follows that T

=0,T

=1and(b)

since the total pressure diﬀerence across the pump must be independent of

the pressure level it follows that T

= 1. Thus the non-cavitating transfer

function has only one non-trivial component, namely T

where −T

known as the pump impedance. As long as the real part of −T

, the pump

resistance, is positive, the pump is stable at all frequencies. Instabilities only

occur at oﬀ-design operating points where the resistance becomes negative

(when the slope of the total pressure rise against ﬂow rate characteristic

becomes positive). Measurements of T

(which is a function of frequency)

can be found in Anderson et al. (1971) and Ng and Brennen (1976).

A cavitating pump is much more complex because all four elements of [T ]

are then non-trivial. The ﬁrst complete measurements of [T ] were obtained

by Ng and Brennen (1976) (see also Brennen et al. 1982). These revealed

that cavitation could cause the pump dynamic characteristics to become

capable of initiating instability in the system in which it operates. This

helped explain the cavitation surge instability described in section 15.6.2.

Recall that cavitation surge occurs when the pump resistance (the real part

of −T

) is positive; thus it results from changes in the other elements of [T ]

that come about as a result of cavitation.

A quasistatic approach to the construction of the transfer function of a

cavitating pump was ﬁrst laid out by Brennen and Acosta (1973, 1976) and

proceeds as follows. The steady state total pressure rise across the pump,

∆p

, ˙m) and the steady state volume of cavitation in the pump, V (p

, ˙m)

379

Figure 16.11. Typical measured transfer functions for a cavitating pump

operating at ﬁve diﬀerent cavitation numbers, σ =(A)0.37, (C) 0.10, (D)

0.069, (G) 0.052 and (H) 0.044. Real and imaginary parts which are denoted

by the solid and dashed lines respectively, are plotted against the non-

dimensional frequency, ω/Ω; r

is the impeller tip radius. Adapted from

Brennen et al. (1982).

will both be functions of the mean mass ﬂow rate ˙m. They will also be

functions of the inlet pressure (or, more accurately, the inlet pressure minus

the vapor pressure) because this will change the cavitation number and the

total pressure rise may depend on the cavitation number as discussed in

section 8.4.1. Note that V is not just a function of cavitation number but

also depends on ˙m because changing ˙m changes the angle of incidence on

the blades and therefore the volume of cavitation bubbles produced. Given

these two functions we could then construct the quasistatic or low frequency

form of the transfer function as

[T ]=







d(∆p

)

˙m

d∆p

d ˙m

iωρ

˙m

1+iωρ

d ˙m







(16.22)

380

The constant K

∗

= −ρ

(dV/dp

)

˙m

is known as the cavitation compliance

while the constant M

∗

= −ρ

(dV/d ˙m)

is called the cavitation mass ﬂow

gain factor. Later, we comment further on these important elements of the

transfer function.

Typical measured transfer functions (in non-dimensional form) for a cavi-

tating pump are shown in ﬁgure 16.11 for operation at four diﬀerent cavita-

tion numbers. Note that case (A) involved virtually no cavitation and that

the volume of cavitation increases as σ decreases. In the ﬁgure, the real and

imaginary parts of each of the elements are shown by the solid and dashed

lines respectively, and are plotted against a non-dimensional frequency. Note

that both the compliance, K

∗

, and the mass ﬂow gain factor, M

∗

,increase

monotonically as the cavitation number decreases.

In order to model the dynamics of the cavitation and generate some un-

derstanding of data such as that of ﬁgure 16.11, we have generated a simple

bubbly ﬂow model (Brennen 1978) of the cavitating ﬂow in the blade pas-

sages of the pump. The essence of this model is depicted schematically in

ﬁgure 16.12, that shows the blade passages as they appear in a developed,

cylindrical surface within an axial-ﬂow impeller. The cavitation is modeled

as a bubbly mixture that extends over a fraction, , of the length of each

blade passage before collapsing at a point where the pressure has risen to

a value that causes collapse. This quantity, , will in practice vary inversely

with the cavitation number, σ, (experimental observations of the pump of

ﬁgure 16.11 indicate  ≈ 0.02/σ) and therefore  is used in the model as a

surrogate for σ. The bubbly ﬂow model then seeks to understand how this

ﬂow will respond to small, linear ﬂuctuations in the pressures and mass ﬂow

rates at the pump inlet and discharge. Pressure perturbations at inlet will

Figure 16.12. Schematic of the bubbly ﬂow model for the dynamics of

cavitating pumps (adapted from Brennen 1978).

381

cause pressure waves to travel through the bubbly mixture and this part of

the process is modeled using a mixture compressibility parameter, K

∗∗

,that

essentially ﬁxes the wave speed. In addition, ﬂuctuations in the inlet ﬂow

rate produce ﬂuctuations in the angle of incidence that cause ﬂuctuations in

the rate of production of cavitation at inlet. These disturbances would then

propagate down the blade passage as kinematic or concentration waves that

travel at the mean mixture velocity. This process is modeled by a factor of

proportionality, M

∗∗

, that relates the ﬂuctuation in the angle of incidence

to the ﬂuctuations in the void fraction. Neither of the parameters, K

∗∗

, can be readily estimated analytically; they are, however, the two key

features in the bubbly ﬂow model. Moreover they respectively determine

the cavitation compliance and the mass ﬂow gain factor; see Brennen (1994)

for the speciﬁc relationships between K

∗∗

and K

∗

and between M

∗∗

and

∗

. Comparison of the model predictions with the experimental measure-

ments indicate that K

∗∗

=1.3andM

∗∗

=0.8 are appropriate values and,

with these, the complete theoretical transfer functions for various cavitation

numbers are as depicted in ﬁgure 16.13. This should be compared with the

experimentally obtained transfer functions of ﬁgure 16.11. Note that, with

only a small number of discrepancies, the general features of the experi-

mental transfer functions, and their variation with cavitation number, are

reproduced by the model.

Following its veriﬁcation, we must then ask how this knowledge of the

pump transfer function might be used to understand cavitation-induced in-

stabilities. In a given system, a stability analysis requires a complete model

(transfer functions) of all the system elements; then a dynamic model must

be constructed for the entire system. By interrogating the model, it is then

possible to identify the key physical processes that promote instability. In

the present case, such an interrogation leads to the conclusion that it is the

formation and propagation of the kinematic waves that are responsible for

those features of the transfer function (in particular the mass ﬂow gain fac-

tor) that lead to cavitation-induced instability. In comparison, the acoustic

waves and the cavitation compliance have relatively benign consequences.

Hence a more complete understanding of the mass ﬂow gain factor and the

kinematic wave production processes that contribute to it will be needed to

enhance our ability to predict these instabilities.

382

Figure 16.13. Theoretical transfer functions calculated from the bubbly

ﬂow model for comparison with the experimental results of ﬁgure 16.11.

The calculations use K

∗∗

=1.3andM

∗∗

=0.8 (adapted from Brennen et

al. 1982).

16.5 TWO-DIMENSIONAL KINEMATIC

WAV E S

Noting that all of the above analyses are for simple one-dimensional ﬂow, we

should add a footnote on the nature of kinematic waves and shocks in a more

general three-dimensional ﬂow. Though the most general analysis is quite

complex, a relative simple extension of the results of the preceding sections

is obtained when attention is restricted to those ﬂows in which the direction

of the relative velocity vector, u

ABi

, is everywhere the same. Such would be

the case, for example, for a relative velocity caused by buoyancy alone. Let

that be the 1 direction (so that u

AB2

= 0) and consider, therefore, a planar

ﬂow in the 12 plane in which, as depicted in ﬁgure 16.14, the kinematic

wave or shock is inclined at an angle, θ, to the 2 direction and is moving at

avelocity,q

, normal to itself. It is readily shown that the volume ﬂux of

383

Figure 16.14. A two-dimensional kinematic wave or shock.

any component, N , normal to and relative to the shock is

sin θ + u

cos θ − q

) (16.23)

and the total volume ﬂux, j

,relativetotheshockis

= j

sin θ + j

cos θ − q

(16.24)

Now consider a multiphase ﬂow consisting of two components A and B,with

velocity vectors, u

, u

, with volume ﬂux vectors, j

, j

,withvolume

fractions, α

= α, α

=1− α, and with a drift ﬂux vector, u

ABi

where, in

the present case, u

AB2

= 0. If the indices a and b denote conditions on the

two sides of the shock, and if the individual volume ﬂuxes into and out of

the shock are equated, we obtain, two relations. The ﬁrst relation simply

states that the total volume ﬂux, j

, must be the same on the two sides of

the shock. The second relation yields:

= j

+cosθ



AB1a

− j

AB1b

− α



(16.25)

which, in the case of a inﬁnitesimal wave, becomes

= j

+cosθ

AB1

dα



(16.26)

These are essentially the same as the one-dimensional results, equations

16.13 and 16.7, except for the cos θ. Consequently, within the restricted

class of ﬂows considered here, the propagation and evolution of a kinematic

wave or shock in two- and three- dimensions can be predicted if the drift

ﬂux function, j

AB1

(α), is known and its direction is uniform.

384

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