cabook.DVI - chap15.pdf

SYSTEM INSTABILITIES

15.1 INTRODUCTION

One of the characteristics of multiphase flows with which the engineer has to

contend is that they often manifest instabilities that have no equivalent in

single phase flow (see, for example, Boure

et al.

1973, Ishii 1982, Gouesbet

and Berlemont 1993). Often the result is the occurence of large pressure,

flow rate or volume fraction oscillations that, at best, disrupt the expected

behavior of the multiphase flow system (and thus decrease the reliability

and life of the components, Makay and Szamody 1978) and, at worst, can

lead to serious flow stoppage or structural failure (see, for example, NASA

1970, Wade 1974). Moreover, in many systems (such as pump and turbine

installations) the trend toward higher rotational speeds and higher power

densities increases the severity of the problem because higher flow velocities

increase the potential for fluid/structure interaction problems. This chapter

will focus on internal flow systems and the multiphase flow instabilities that

occur in them.

15.2 SYSTEM STRUCTURE

In the discussion and analysis of system stability, we shall consider that the

system has been divided into its components, each identified by its index,

as shown in figure 15.1 where each component is represented by a box. The

connecting lines do not depict lengths of pipe which are themselves com-

ponents. Rather the lines simply show how the components are connected.

More specifically they represent specific locations at which the system has

been divided up; these points are called the nodes of the system and are

denoted by the index,

Typical and common components are pipeline sections, valves, pumps,

344

Figure 15.1.

Flow systems broken into components.

Figure 15.2.

Typical component characteristics, Δ

( ̇

turbines, accumulators, surge tanks, boilers, and condensers. They can be

connected in series and/or in parallel. Systems can be either open loop or

closed loop as shown in figure 15.1. The mass flow rate through a component

will be denoted by ̇

and the change in the total head of the flow across

the component will be denoted by Δ

defined as the total pressure at inlet

minus that at discharge. (When the pressure ratios are large enough so that

the compressibility of one or both of the phases must be accounted for, the

analysis can readily be generalized by using total enthalpy rather than total

pressure.) Then, each of the components considered in isolation will have a

performance characteristic in the form of the function Δ

( ̇

)asdepicted

graphically in figure 15.2. We shall see that the shapes of these character-

istics are important in identifying and analysing system instabilities. Some

345

Figure 15.3.

Typical system characteristic, Δ

( ̇

), and operating point.

of the shapes are readily anticipated. For example, a typical single phase

flow pipe section (at higher Reynolds numbers) will have a characteristic

that is approximately quadratic with Δ

∝

. Other components such as

pumps, compressors or fans may have quite non-monotonic characteristics.

The slope of the characteristic,

∗

,where

∗

ρg

(15.1)

is known as the component resistance. However, unlike many electrical com-

ponents, the resistance of most hydraulic components is almost never con-

stant but varies with the flow, ̇

Components can readily be combined to obtain the characteristic of

groups of neighboring components or the complete system. A parallel com-

bination of two components simply requires one to add the flow rates at

thesameΔ

, while a series combination simply requires that one add the

values of the two components at the same flow rate. In this way one

can synthesize the total pressure drop, Δ

( ̇

), for the whole system as a

function of the flow rate, ̇

. Such a system characteristic is depicted in fig-

ure 15.3. For a closed system, the equilibrium operating point is then given

by the intersection of the characteristic with the horizontal axis since one

must have Δ

= 0. An open system driven by a total pressure difference of

(inlet total pressure minus discharge) would have an operating point

where the characteristic intersects the horizontal line at Δ

=Δ

.Since

these are trivially different we can confine the discussion to the closed loop

case without any loss of generality.

In many discussions, this system equilibrium is depicted in a slightly dif-

346

Figure 15.4.

Alternate presentation of figure 15.3.

ferent but completely equivalent way by dividing the system into two series

elements, one of which is the

pumping

component,

pump

, and the other

is the

pipeline

component,

line

. Then the operating point is given by

the intersection of the

pipeline

characteristic, Δ

line

,andthe

pump

charac-

teristic,

−

pump

, as shown graphically in figure 15.4. Note that since the

total pressure increases across a pump, the values of

−

pump

are normally

positive. In most single phase systems, this depiction has the advantage

that one can usually construct a series of quadratic

pipeline

characteristics

depending on the valve settings. These

pipeline

characteristics are usually

simple quadratics. On the other hand the pump or compressor characteristic

can be quite complex.

15.3 QUASISTATIC STABILITY

Using the definitions of the last section, a quasistatic analysis of the stabil-

ity of the equilibrium operating point is usually co

nducted in the following

way. We consider perturbing the system to a new mass flow rate

greater

than that at the operating point as shown in figure 15.4. Then, somewhat

heuristically, one argues from figure 15.4 that the total pressure rise across

the

pumping

component is now less than the total pressure drop across the

pipeline

and therefore the flow rate will decline back to its value at the oper-

ating point. Consequently, the particular relationship of the characteristics

in figure 15.4 implies a stable operating point. If, however, the slopes of the

two components are reversed (for example, Pump B of figure 15.5(a) or the

operating point

of figure 15.5(b)) then the operating point is unstable

since the increase in the flow has resulted in a

pump

total pressure that now

exceeds the total pressure drop in the

pipeline

. These arguments lead to the

347

Figure 15.5.

Quasistatically stable and unstable flow systems.

conclusion that the operating point is stable when the slope of the system

characteristic at the operating point (figure 15.3) is positive or

∗

(15.2)

The same criterion can be derived in a somewhat more rigorous way by

using an energy argument. Note that the net flux of flow energy out of each

component is ̇

. In a straight pipe this energy is converted to heat

through the action of viscosity. In a pump ̇

(

−

) is the work done on

the flow by the pump impeller. Thus the net energy flux out of the whole

system is ̇

and, at the operating point, this is zero (for simplicity we

discuss a closed loop system) since Δ

= 0. Now, suppose, that the flow

rate is perturbed by an amount

. Then, the new net energy flux out of

the system is Δ

where

=( ̇

)

≈

(15.3)

Then we argue that if

is positive and the perturbed system therefore

dissipates more energy, then it must be stable. Under those circumstances

one would have to add to the system a device that injected more energy

into the system so as to sustain operation at the perturbed state. Hence the

criterion 15.2 for quasistatic stability is reproduced.

348

15.4 QUASISTATIC INSTABILITY EXAMPLES

15.4.1 Turbomachine surge

Perhaps the most widely studied instabilities of this kind are the surge insta-

bilities that occur in pumps, fans and compressors when the turbomachine

has a characteristic of the type shown in figure 15.5(b). When the machine is

operated at points such as

the operation is stable. However, when the tur-

bomachine is throttled (the resistance of the rest of the system is increased),

the operating point will move to smaller flow rates and, eventually, reach the

point

at which the system is neutrally stable. Further decrease in the flow

rate will result in operating conditions such as the point

that are qua-

sistatically unstable. In compressors and pumps, unstable operation results

in large, limit-cycle oscillations that not only lead to noise, vibration and

lack of controllability but may also threaten the structural integrity of the

machine. The phenomenon is known as compressor, fan or pump surge and

for further details the reader is referred to Emmons

et al.

(1955), Greitzer

(1976, 1981) and Brennen (1994).

15.4.2 Ledinegg instability

Two-phase flows can exhibit a range of similar instabilities. Usually, however,

the instability is the result of a non-monotonic

pipeline

characteristic rather

than a complex

pump

characteristic. Perhaps the best known example is the

Figure 15.6.

Sketch illustrating the Ledinegg instability.

349

Ledinegg instability (Ledinegg 1983) which is depicted in figure 15.6. This

occurs in boiler tubes through which the flow is forced either by an imposed

pressure difference or by a normally stable pump as sketched in figure 15.6. If

the heat supplied to the boiler tube is roughly independent of the flow rate,

then, at high flow rates, the flow will remain mostly liquid since, as discussed

in section 8.3.2,

/ds

is inversely proportional to the flow rate (see equation

8.24). Therefore

remains small. On the other hand, at low flow rates, the

flow may become mostly vapor since

/ds

is large. In order to construct

the Δ

( ̇

) characteristic for such a flow it is instructive to begin with

the two hypothetical characteristics for all-vapor flow and for all-liquid flow.

The rough form of these are shown in figure 15.6; since the frictional losses

at high Reynolds numbers are proportional to

ρu

= ̇

/ρ

, the all-vapor

characteristic lies above the all-liquid line because of the different density.

However, as the flow rate, ̇

, increases, the actual characteristic must make

a transition from the all-vapor line to the all-liquid line, and may therefore

have the non-monotonic form sketched in figure 15.6. This may lead to

unstable operating points such the point

. This is the Ledinegg instability

and is familiar to most as the phenomenon that occurs in a coffee percolator.

15.4.3 Geyser instability

The geyser instability that is so familiar to visitors to Yellowstone National

Park and other areas of geothermal activity, has some similarities to the

Ledinegg instability, but also has important differences. It has been studied

in some detail in smaller scale laboratory experiments (see, for example,

Nakanishi

et al.

1978) where the parametric variations are more readily

explored.

The geyser instability requires the basic components sketched in figure

15.7, namely a buried reservoir that is close to a large heat source, a vertical

conduit and a near-surface supply of water that can drain into the conduit

and reservoir. The geyser limit cycle proceeds as follows. During the early

dormant phase of the cycle, the reservoir and conduit are filled with water

that is being heated by the geothermal source. Once the water begins to boil

the vapor bubbles rise up through the conduit. The hydrostatic pressure in

the conduit and reservoir then drop rapidly due to the reduced mixture

density in the conduit. This pressure reduction leads to explosive bo

iling

and the eruption so widely publicized by

Old Faithful

. The eruption ends

when almost all the water in the conduit and reservoir has been ejected.

350

Figure 15.7.

Left: The basic components for a geyser instability. Right:

Laboratory measurements of geysering period as a function of heat supply

(200

, 330

, 400

) from experiments (open symbols) and

numerical simulations (solid symbols). Adapted from Tae-il

et al.

(1993).

The reduced flow then allows sub-cooled water to drain into and refill the

reservoir and conduit. Due to the resistance to heat transfer in the rock

surrounding the reservoir, there is a significant time delay before the next

load of water is heated to boiling temperatures. The long cycle times are

mostly the result of low thermal conductivity of the rock (or other solid

material) surrounding the reservoir and the consequent low rate of transfer

of heat available to heat the sub-cooled water to its boiling temperature.

The dependence of the geysering period on the strength of the heat source

and on the temperature of the sub-cooled water in the water supply is exem-

plified in figure 15.7 which presents results from the small scale laboratory

experiments of Tae-il

et al.

(1993). That figure includes both the experimen-

tal data and the results of a numerical simulation. Note that, as expected,

the geysering period decreases with increase in the strength of the heat

source and with the increase in the temperature of the water supply.

15.5 CONCENTRATION WAVES

There is one phenomenon that is sometimes listed in discussions of multi-

phase flow instabilities even though it is not, strictly speaking, an instability.

We refer to the phenomenon of concentration wave oscillations and it is valu-

351

Figure 15.8.

Sketch illustrating a concentration wave (density wave) oscillation.

able to include mention of the phenomenon here before proceeding to more

complex matters.

Often in multiphase flow processes, one encounters a circumstance in

which one part of the circuit contains a mixture with a concentration that is

somewhat different from that in the rest of the system. Such an inhomogene-

ity may be created during start-up or during an excursion from the normal

operating point. It is depicted in figure 15.8, in which the closed loop has

been arbitrarily divided into a

pipeline

component and a

pump

component.

As indicated, a portion of the flow has a mass quality that is larger by Δ

than the mass quality in the rest of the system. Such a perturbation could

be termed a concentration wave though it is also called a density wave or a

continuity wave; more generally, it is known as a kinematic wave (see chap-

ter 16). Clearly, the perturbation will move round the circuit at a speed that

is close to the mean mixture velocity though small departures can occur in

vertical sections in which there is significant relative motion between the

phases. The mixing processes that would tend to homogenize the fluid in

the circuit are often quite slow so that the perturbation may persist for an

extended period.

It is also clear that the pressures and flow rates may vary depending

on the location of the perturbation within the system. These fluctuations

in the flow variables are termed concentration wave oscillations and they

arise from the inhomogeneity of the fluid rather than from any instability

in the flow. The characteristic frequency of the oscillations is simply related

352

to the time taken for the flow to complete one circuit of the loop (or some

multiple if the number of perturbed fluid pockets is greater than unity). This

frequency is usually small and its calculation often allows identification of

the phenomenon.

One way in which concentration oscillations can be incorporated in the

graphical presentation we have used in this chapter is to identify the compo-

nent characteristics for both the mass quality,

, and the perturbed quality,

+Δ

, and to plot them using the volume flow rate rather than the mass

flow rate as the abscissa. We do this because, if we neglect the compressibil-

ity of the individual phases, then the volume flow rate is constant around the

circuit at any moment in time, whereas the mass flow rate differs according

to the mass quality. Such a presentation is shown in figure 15.8. Then, if the

perturbed body of fluid were wholly in the pipeline section, the operating

point would be close to the point

. On the other hand, if the perturbed

body of fluid were wholly in the pump, the operating point would be close

to the point

. Thus we can see that the operating point will vary along a

trajectory such as that shown by the dotted line and that this will result in

oscillations in the pressure and flow rate.

In closing, we should note that concentration waves also play an important

role in other more complex unsteady flow phenomena and instabilities.

15.6 DYNAMIC MULTIPHASE FLOW INSTABILITIES

15.6.1 Dynamic instabilities

The descriptions of the preceding sections were predicated on the frequency

of the oscillations being sufficiently small for all the components to track up

and down their steady state characteristics. Thus the analysis is only appli-

cable to those instabilities whose frequencies are low enough to lie within

some quasistatic range. At higher frequency, the effective resistance could

become a complex function of frequency and could depart significantly from

the quasistatic resistance. It follows that there may be operating points at

which the total

dynamic

resistance over some range of frequencies is nega-

tive. Then the system would be dynamically unstable even though it may be

quasistatically stable. Such a description of dynamic instability is instructive

but overly simplistic and a more systematic approach to this issue will be de-

tailed in section 15.7. It is nevertheless appropriate at this point to describe

two examples of dynamic instabilities so that reference to these examples

can be made during the description of the transfer function methodology.

353

15.6.2 Cavitation surge in cavitating pumps

In many installations involving a pump that cavitates, violent oscillations

in the pressure and flow rate in the entire system can occur when the cavi-

tation number is decreased to a value at which the volume of vapor bubbles

within the pump becomes sufficient to cause major disruption of the flow and

therefore a decrease in the total pressure rise across the pump (see section

8.4.1). While most of the detailed investigations have focused on axial pumps

and inducers (Sack and Nottage 1965, Miller and Gross 1967, Kamijo

et al.

1977, Braisted and Brennen 1980) the phenomenon has also been observed

in centrifugal pumps (Yamamoto 1991). In the past this surge phenomenon

was called

auto-oscillation

though the modern term

cavitation surge

is more

appropriate. The phenomenon is described in detail in Brennen (1994). It

can lead to very large flow rate and pressure fluctuations. For example in

boiler feed systems, discharge pressure oscillations with amplitudes as high

as 14

bar

have been reported informally. It is a genuinely dynamic instability

in the sense described in section 15.6.1, for it occurs when the slope of the

pump total pressure rise/flow rate characteristic is still strongly negative

and the system is therefore quasistatically stable.

As previously stated, cavitation surge occurs when the region of cavitation

head loss is approached as the cavitation number is decreased. Figure 15.9

Figure 15.9.

Cavitation performance of a SSME low pressure LOX pump

model showing the approximate boundaries of the cavitation surge region

for a pump speed of 6000

rpm

(from Braisted and Brennen 1980). The flow

coefficient,

, is based on the impeller inlet area.

354

Figure 15.10.

Data from Braisted and Brennen (1980) on the ratio of

the frequency of cavitation surge,

, to the frequency of shaft rotation, Ω,

for several axial flow pumps: for SSME low pressure LOX pump models:

diameter:

(9000

rpm

) and + (12000

rpm

), 10

diameter:

(4000

rpm

)and

(6000

rpm

); for 9

◦

helical inducers: 7

diameter:

∗

(9000

rpm

): 10

diameter:

(with suction line flow straightener)

and

(without suction line flow straightener). The flow coefficients,

are based on the impeller inlet area.

provides an example of the limits of cavitation surge taken from the work

of Braisted and Brennen (1980). However, since the onset is sensitive to the

detailed dynamic characteristics of the system, it would not even be wise

to quote any approximate guideline for onset. Our current understanding

is that the methodologies of section 15.7 are essential for any prediction of

cavitation surge.

Unlike compressor surge, the frequency of cavitation surge,

, scales with

the shaft speed of the pump, Ω (Braisted and Brennen 1980). The ratio,

Ω, varies with the cavitation number,

(see equation 8.31), the flow

coefficient,

(see equation 8.30), and the type of pump as illustrated in

figure 15.10. The most systematic variation is with the cavitation number

and it appears that the empirical expression

Ω=(2

)

(15.4)

provides a crude estimate of the cavitation surge frequency. Yamamoto

(1991) demonstrated that the frequency also depends on the length of the

suction pipe thus reinforcing the understanding of cavitation surge as a sys-

tem instability.

355

15.6.3 Chugging and condensation oscillations

As a second example of a dynamic instability involving a two-phase flow we

describe the oscillations that occur when steam is forced down a vent into a

pool of water. The situation is sketched in figure 15.11. These instabilities,

forms of which are known as

chugging

and

condensation oscillations

,have

been most extensively studied in the context of the design of pressure sup-

pression systems for nuclear reactors (see, for example, Wade 1974, Koch

and Karwat 1976, Class and Kadlec 1976, Andeen and Marks 1978). The

intent of the device is to condense steam that has escaped as a result of

the rupture of a primary coolant loop and, thereby, to prevent the build-

up of pressure in the containment that would have occurred as a result of

uncondensed steam.

The basic components of the system are as shown in figure 15.11 and

consist of a vent or pipeline of length,

, the end of which is submerged

to a depth,

, in the pool of water. The basic instability is illustrated in

figure 15.12. At relatively low steam flow rates the rate of condensation

at the steam/water interface is sufficiently high that the interface remains

within the vent. However, at higher flow rates the pressure in the steam

increases and the interface is forced down and out of the end of the vent.

When this happens both the interface area and the turbulent mixing in

the vicinity of the interface increase dramatically. This greatly increases the

Figure 15.11.

Components of a pressure suppression system.

356

Figure 15.12.

Sketches illustrating the stages of a condensation oscillation.

condensation rate which, in turn, causes a marked reduction in the steam

pressure. Thus the interface collapses back into the vent, often with the same

kind of violence that results from cavitation bubble collapse. Then the cycle

of growth and collapse, of oscillation of the interface from a location inside

the vent to one outside the end of the vent, is repeated. The phenomenon is

termed condensation instability and, depending on the dominant frequency,

the violent oscillations are known as

chugging

condensation oscillations

(Andeen and Marks 1978).

The frequency of the phenomenon tends to lock in on one of the natural

modes of oscillation of the system in the absence of condensation. There are

two obvious natural modes. The first, is the manometer mode of the liquid

inside the end of the vent. In the absence of any steam flow, this manometer

mode will have a typical small amplitude frequency,

g/h

)

,where

the acceleration due to gravity. This is usually a low frequency of the order

of 1

or less and, when the condensation instability locks into this low

frequency, the phenomenon is known as

chugging

. The pressure oscillations

resulting from chugging can be quite violent and can cause structural loads

357

Figure 15.13.

The real part of the input impedance (the input resistance)

of the suppression pool as a function of the perturbation frequency for

several steam flow rates. Adapted from Brennen (1979).

that are of concern to the safety engineer. Another natural mode is the first

acoustic mode in the vent whose frequency,

, is approximately given by

πc/

where

is the sound speed in the steam. There are also observations

of lock-in to this higher frequency. The oscillations that result from this are

known as

condensation oscillations

and tend to be of smaller amplitude than

the chugging oscillations.

Figure 15.13 illustrates the results of a linear stability analysis of the sup-

pression pool system (Brennen 1979) that was carried out using the transfer

function methodology described in section 15.7. Transfer functions were con-

structed for the vent or downcomer, for the phase change process and for

the manometer motions of the pool. Combining these, one can calculate the

input impedance of the system viewed from the steam supply end of the

vent. A positive input resistance implies that the system is absorbing fluc-

tuation energy and is therefore stable; a negative input resistance implies

an unstable system. In figure 15.13, the input resistance is plotted

against

the perturbation frequency for several steam flow rates. Note that, at low

steam flow rates, the system is stable for all frequencies. However, as the

steam flow rate is increased, the system first becomes unstable over a narrow

range of frequencies close to the manometer frequency,

. Thus chugging is

predicted to occur at some critical steam flow rate. At still higher flow rates,

the system also becomes unstable over a narrow range of frequencies close

to the first vent acoustic frequency,

; thus the possibility of condensation

358

oscillations is also predicted. Note that the quasistatic i

nput resistance at

small frequencies remains positive throughout and therefore the system is

quasistatically stable for all steam flow rates. Thus, chugging and conden-

sation oscillations are true, dynamic instabilities.

It is, however, important to observe that a linear stability analysis can-

not model the highly non-linear processes that occur during a

chug

and,

therefore, cannot provide information on the subject of most concern to the

practical engineer, namely the magnitudes of the pressure excursions and

the structural loads that result from these condensation instabilities. While

models have been developed in an attempt to make these predictions (see, for

example, Sargis

et al.

1979) they are usually very specific to the particular

problem under investigation. Often, they must also resort to empirical infor-

mation on unknown factors such as the transient mixing and condensation

rates.

Finally, we note that instabilities that are similar to chugging have been

observed in other contexts. For example, when steam was injected into the

wake of a streamlined underwater body in order to explore underwater jet

propulsion, the flow became very unstable (Kiceniuk 1952).

15.7 TRANSFER FUNCTIONS

15.7.1 Unsteady internal flow methods

While the details are beyond the scope of this book, it is nevertheless of

value to conclude the present chapter with a brief survey of the transfer

function methods referred to in section 15.6. There are two basic approaches

to unsteady internal flows, namely solution in the time domain or in the fre-

quency domain. The traditional time domain or

water-hammer

methods for

hydraulic systems can and should be used in many circumstances; these

are treated in depth elsewhere (for example, Streeter and Wylie 1967, 1974,

Amies

et al.

1977). They have the great advantage that they can incorporate

the nonlinear convective inertial terms in the equations of fluid flow. They

are best suited to evaluating the transient response of flows in long pipes

in which the equations of the flow and the structure are well established.

However, they encounter great difficulties when either the geometry is com-

plex (for example inside a pump), or the fluid is complex (for example in

a multiphase flow). Under these circumstances, frequency domain methods

have distinct advantages, both analytically and experimentally. Specifically,

unsteady flow experiments are most readily conducted by subjecting the

component or device to fluctuations in the flow over a range of frequen-

359

cies and measuring the fluctuating quantities at inlet and discharge. The

main disadvantage of the frequency domain methods is that the nonlin-

ear convective inertial terms cannot readily be included and, consequently,

these methods are only accurate for small perturbations from the mean flow.

While this permits evaluation of stability limits, it does not readily allow

the evaluation of the amplitude of large unstable motions. However, there

does exist a core of fundamental knowledge pertaining to frequency domain

methods (see for example, Pipes 1940, Paynter 1961, Brown 1967) that is

summarized in Brennen (1994). A good example of the application of these

methods is contained in Amies and Greene (1977).

15.7.2 Transfer functions

As in the quasistatic analyses described at the beginning of this chapter,

the first step in the frequency domain approach is to identify all the flow

variables that are needed to completely define the state of the flow at each

of the nodes of the system. Typical flow variables are the pressure,

,(or

total pressure,

) the velocities of the phases or components, the volume

fractions, and so on. To simplify matters we count only those variables that

are not related by simple algebraic expressions. Thus we do not count both

the pressure and the density of a phase that behaves barotropically, nor

do we count the mixture density,

, and the void fraction,

,inamixture

of two incompressible fluids. The minimum number of variables needed to

completely define the flow at all of the nodes is called

the order of the system

and will be denoted by

. Then the state of the flow at any node,

,is

denoted by the vector of state variables,

{

}

→

. For example,

in a homogeneous flow we could choose

,tobethe

pressure, velocity and void fraction at the node

The next step in a frequency domain analysis is to express all the flow

variables,

{

}

→

, as the sum of a mean component (denoted by

an overbar) and a fluctuating component (denoted by a tilde) at a frequency,

. The complex fluctuating component incorporates both the amplitude and

phase of the fluctuation:

{

(

s, t

)

}

{

(

)

}

{

(

s, ω

)

}

iωt

(15.5)

for

→

where

is (

−

and

denotes the real part. For example

(

s, t

)= ̄

(

s, ω

)

iωt

(15.6)

(

s, t

(

s, ω

)

iωt

(15.7)

360

(

s, t

)= ̄

(

s, ω

)

iωt

(15.8)

Since the perturbations are assumed linear (

)

they can be readily superimposed, so a summation over many frequencies

is implied in the above expressions. In general, the perturbation quantities,

{

}

, will be functions of the mean flow characteristics as well as position,

, and frequency,

The utilization of transfer functions in the context of fluid systems owes

much to the pioneering work of Pipes (1940). The concept is the following.

If the quantities at inlet and discharge are denoted by subscripts

=1and

= 2, respectively, then the transfer matrix, [

], is defined as

{

}

]

{

}

(15.9)

It is a square matrix of order

. For example, for an order

= 2 system

in which the independent fluctuating variables are chosen to be the total

pressure, ̃

, and the mass flow rate,

, then a convenient transfer matrix

(15.10)

In general, the transfer matrix will be a function of the frequency,

,ofthe

perturbations and the mean flow conditions in the device. Given the transfer

functions for each component one can then synthesize transfer functions for

the entire system using a set of simple procedures described in detail in

Brennen (1994). This allows one to pro

ceed to a determination of whether

or not a system is stable or unstable given the boundary conditions acting

upon it.

The transfer functions for many simple components are readily identified

(see Brennen 1994) and are frequently composed of impedances due to fluid

friction and inertia (that primarily contribute to the real and imaginary

parts of

respectively) and compliances due to fluid and structural com-

pressibility (that primarily contribute to the imaginary part of

). More

complex components or flows have more complex transfer functions that can

often be determined only by experimental measurement. For example, the

dynamic response of pumps can be critical to the stability of many internal

flow systems (Ohashi 1968, Greitzer 1981) and consequently the transfer

functions for pumps have been extensively explored (Fanelli 1972, Anderson

et al.

1971, Brennen and Acosta 1976). Under stable operating conditions

(see sections 15.3, 16.4.2) and in the absence of phase change, most pumps

can be modeled with resistance, compliance and inertance elements and they

361

are therefore dynamically passive. However, the situation can be quite dif-

ferent when phase change occurs. For example, cavitating pumps are now

known to have transfer functions that can cause instabilities in the hydraulic

system of which they are a part. Note that under cavitating conditions, the

instantaneous flow rates at inlet and discharge will be different because of

the rate of change of the total volume,

, of cavitation within the pump and

this leads to complex transfer functions that are described in more detail in

section 16.4.2. These characteristics of cavitating pumps give rise to a vari-

ety of important instabilities such as cavitation surge (see section 15.6.2) or

the Pogo instabilities of liquid-propelled rockets (Brennen 1994).

Much less is known about the transfer functions of other devices involv-

ing phase change, for example boiler tubes or vertical evaporators. As an

example of the transfer function method, in the next section we consider a

simple homogeneous multiphase flow.

15.7.3 Uniform homogeneous flow

As an example of a multiphase flow that exhibits the solution structure

described in section 15.7.2, we shall explore the form of the solution for

the inviscid, frictionless flow of a two component, gas and liquid mixture

in a straight, uniform pipe. The relative motion between the two compo-

nents is neglected so there is only one velocity,

(

s, t

). Surface tension is

also neglected so there is only one pressure,

(

s, t

). Moreover, the liquid

is assumed incompressible (

constant) and the gas is assumed to behave

barotropically with

∝

. Then the three equations governing the flow are

the continuity equations for the liquid and for the gas and the momentum

equation for the mixture which are, respectively

∂

∂t

−

∂

∂s

[(1

−

)

] = 0

(15.11)

∂

∂t

(

∂

∂s

(

αu

) = 0

(15.12)

∂u

∂t

∂u

∂s

−

∂p

∂s

(15.13)

where

is the usual mixture density. Note that this is a system of order

= 3 and the most convenient flow variables are

and

.Theserelations

362

yield the following equations for the perturbations:

−

iω

∂

∂s

[(1

−

) ̃

−

] = 0

(15.14)

iω

+ ̄

∂

∂s

+ ̄

∂

∂s

+ ̄

∂

∂s

= 0

(15.15)

−

∂

∂s

= ̄

iω

+ ̄

∂

∂s

(15.16)

where ̃

= ̃

. Assuming the solution has the simple form

⎧

⎨

⎩

⎫

⎬

⎭

⎧

⎨

⎩

iκ

⎫

⎬

⎭

(15.17)

it follows from equations 15.14, 15.15 and 15.16 that

−

)

(15.18)

(

)

(

)

+ ̄

ακ

= 0

(15.19)

(

)

= 0

(15.20)

Eliminating

and

leads to the dispersion relation

(

)

−

(

)

= 0

(15.21)

The solutions to this dispersion relation yield the following wavenumbers

and velocities,

−

ω/κ

, for the perturbations:

−

ω/

which has a wave velocity,

= ̄

. This is a purely kinematic wave, a

concentration wave that from equations 15.18 and 15.20 has

=0and

so that there are no pressure or velocity fluctuations associated with this type of

wave. In other, more complex flows, kinematic waves may have some small pres-

sure and velocity perturbations associated with them and their velocity may not

exactly correspond with the mixture velocity but they are still called kinematic

waves if the major feature is the concentration perturbation.

,κ

−

ω/

( ̄

)where

is the sonic speed in the mixture, namely

(

)

1

2

. Consequently, these two modes have wave speeds

= ̄

and

are the two acoustic waves traveling downstream and upstream respectively.

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