Yamaguchi H. Engineering Fluid Mechanics (Fluid Mechanics and Its Applications)

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222 4 Perfect Flow

Bibliography

Some fundamental aspects of turbomachinery with working examples are

given in

H.S. Bean (edited), Fluid Meters: their theory and application (6th

Edition), American Society of Mechanical Engineers, New York, 1971.

A.H. Shapiro, The Dynamics and Thermodynamics of Compressible

Fluid Flow, Ronald Press, New York, 1953.

M. Massound, Engineering Thermofluids, Springer, 2005.

M.C. Potter, D.C. Wiggent, M. Hondzo, Mechanics of Fluids (2nd Edi-

tion), Prentice-Hall Inc., 1997.

S.L. Dixon, Fluid Mechanics and Thermodynamics of Turbomachinery

(2nd Edition), Pergamon Press, 1975.

T. Nishiyama, Fluid Mechanics, Vol. I, II (13th Edition), Nikan Ko-

gyo-Shinbunsha, Tokyo, 1989 (in Japanese).

J. Katz and A. Plotkin, Low-Speed Aerodynamics (2nd Edition), Cam-

bridge University Press, 2001.

Great deal of detailed data are available and to be useful for designers of

turbomachines. There are useful correlations and definitions for pump de-

sign, which are found in

C. Pfleidener, Die Kreiselpumpen für Flüssigkeiten and Gase, Springer,

1961.

J.A. Schetz and A.E. Fuhs (edited), Handbook of Fluid Dynamics and

Fluid Machinery, Vol. I, II and III, John Wiley & Sons Inc., 1996.

Phenomenological treatment for cavitation is given in greater extend in

13.

C.T. Crowe (edited), Multiphase Flow (Handbook), Taylor & Francis,

CRC Press, 2006.

10.

H. Blasius, Funktionentheoretische Methoden in der Hydrodynamik,

Math. Zeitschrift, 58, 90–110, 1910.

11.

D.N. Roy, Applied Fluid Mechanics, Ellis Horwood Ltd, 1988.

12.

W. Thomson, (L. Kelvin), On Vortex Motion. Trans. Roy. Soc. Edinburgh.,

25, 217–260, 1869.

Rayleigh-Plesset equation for bubble growth is referred from

14.

M.S. Plesset, The dynamics of cavitation bubbles, J. Appl. Mech., 1949.

15.

M.S. Plesset and A. Prosperetti, Bubble dynamics and cavitation, Ann.

Rev. Fluid Mech., 9, 1977.

Estimation of efficiencies in turbomachines are introduced in

16.

C.P. Kittredge, Estimating the efficiency of prototype pumps from

model tests, Trans, ASME, J. Engg. Power, 90(2), 1968.

17.

A.J. Stepanoff, Centrifugal and Axial Flow Pumps, John Wiley & Sons

Inc., 1957.

Bibliography 223

5. Compressible Flow

There are many physical processes, when dealing with both gases and liq-

uids, where the density variations are of major importance in determining

the character of the flow. In both cases, with regard to high-speed flow, the

effects on compressibility cannot be ignored; as a result, a new class of

flow effects appears. Such high-speed flows are termed compressible flows.

In practice, indeed appreciable density variations are mostly seen in high-

speed gas flows. In this chapter we will consider the one dimensional

flows of ideal gases, which are important when tackling most physics

problems, since they are often involved in compressible flows and are

found to be a good approximation of many actual flows.

In applying the one dimensional assumption, we shall restrict the

physical situation exclusively to internal flows, where the quantities (such

as density, pressure, temperature, velocity and etc.) are uniform over any

cross section, and vary only along the channel. Looking at compressible

flows, we will consider three conservation laws for mass, momentum, and

energy, which are supplemented by thermodynamics state equations to-

gether with the definition of sonic speed (related to Mach number). In this

chapter, we shall begin to look at the speed of sound which will lead to a

better understanding of the role it plays and its effects on more general

compressible flows.

5.1 Speed of Sound and Mach Number

If an infinitesimal disturbance occurs in a fluid, the disturbance will

propagate through the fluid at a well-defined velocity called the sonic ve-

locity or speed of sound (sometimes called the acoustic velocity). The

speed of sound depends upon the properties of the fluid, and may be de-

termined by considering the equation of continuity from Eq. (2.1.5) and

225

the Euler equation from Eq. (4.2). In the case of one dimensional assump-

tion, the equations are written by ignoring the effect of the gravity (body

force term)

5 Compressible Flow



(5.1.1)





(5.1.2)

Here,

is the velocity component in the direction of

(one dimensional).

Infinitesimal disturbances are thought to occur in a flow at rest (or the un-

disturbed fluid is moving relative to the propagations of disturbances as

uuu



), assuming 0

u so that

(5.1.3)

and

ppp



(5.1.4)



(5.1.5)

Thus,

u ,

p and

are the properties of undisturbed fluid and

, p

and

are the infinitesimal disturbances of velocity, pressure and

density respectively. Equations (5.1.3), (5.1.4) and (5.1.5) are substituted

into Eqs. (5.1.1) and (5.1.2). Simplifying and neglecting higher order terms

with the conditions of

tww

and xp ww

being zero, we have a set of

equations for the disturbances



(

5.1.6)

and



(5.1.7)

Eliminating

from Eqs. (5.1.6) and (5.1.7), we have an equation relating

and p

as follows

t w

(5.1.8)

In order to make up a closed mathematical model, which deals with the

propagation of disturbance, we need to relate

and p

. In engineering

fluid mechanics it is reasonable to assume that the thermodynamic process

226

5.1 Speed of Sound and Mach Number

is in the state of equilibrium and the ideal gas law approximation is valid

for low and moderate density gases with a lower temperature gradient

process. In view of this consideration, we have the equation of state

RTp

(5.1.9)

RTpv

(5.1.10)

v is the specific volume,

is the ideal (specific) gas constant and

the absolute temperature. Equation (5.1.9) can be further written in differ-

ential form

dTRȡpdȡdp



(5.1.11)

Next we must introduce the disturbance in addition to Eqs. (5.1.4) and

(5.1.5), and that of temperature

TTT



(5.1.12)

These disturbances are then added to Eq. (5.1.11), simplifying and neglect-

ing high order terms with the conditions of

pddp

and

(5.1.13)

We can reduce this to the following relation

(5.1.14)

Assuming that



ddp is kept constant, we can eliminate p

by substitut-

ing Eq. (5.1.14) into Eq. (5.1.8), so that we have

t w

(5.1.15)

The hyperbolic partial differential equation, Eq. (5.1.15), is a wave equa-

tion for

. It is generally known from the wave equation that mathemati-

cally

has a solution, which is expressed as



atxgatxf 

(5.1.16)

f and g are the wave functions, for example



atx r

sin

. In Eq.

(5.1.16),

propagates with the speed of sound a , that is expressed by

227

5 Compressible Flow

(5.1.17)

It must be emphasized that the disturbance

is infinitesimal, and the

process is regard as reversible, adiabatic and hence isentropic.

Equation (5.1.17), the definition of the speed of sound, can be further

rewritten by other fluid properties. When the bulk modulus

of the fluid

is considered, from the definition of

, we can write

Kdp 

dȡ

(5.1.18)

and

dȡ

(5.1.19)

So, Eq. (5.1.17) may be written as

(5.1.20)

Furthermore, since the process is adiabatic, we have the relationship

const.

(5.1.21)

const.

(5.1.22)

Here,

k is the specific heat ratio defined as

cck . Equation (5.1.22)

may be demonstrated as a differential form like so

dȡ

(5.1.23)

Subsequently, Eq. (5.1.17) can be written

228

5.1 Speed of Sound and Mach Number

(5.1.24)

Additionally, using the equation of state of ideal gases with reference to Eq.

(5.1.9), we can also write Eq. (5.1.24) with the following formula

kRTa

(5.1.25)

As it is generally directed, the speed of sound is the rate at which a

very small disturbance travels through a fluid. In other words, from the

under laying physical phenomena, we are to understand that small (infini-

tesimally small) disturbances in pressure (or density) propagate, as sound

waves, with the speed of sound in a fluid. We should next consider how the

sound wave (the wave of pressure disturbance) may be propagated when a

sound source is at rest in a stationary fluid, or traveling in a straight line

with a velocity

in a stationary fluid.

As shown in Fig. 5.1(a), the sound wave propagates radially in all di-

rections with the spherical wave fronts for different transmission times,

forming concentric spheres. When the sound source moves with a speed of

au  , the waves are no longer concentric, as shown in Fig. 5.1(b), and the

spherical wave fronts for different transmission times nest with no inter-

section. An observer in front of the moving source may experience a Dop-

pler effect after the initial wave front reaches the observer. This situation is

called subsonic (in the case of flows, it is called the subsonic flow).

Fig. 5.1 Sound wave propagation

229

5 Compressible Flow

When

, i.e. the speed of the sound source is equal to the speed of

sound, the wave fronts all touch at the position of the sound source as illus-

trated in Fig. 5.1(c). In this situation, an observer may hear a sudden abrupt

sound when the source arrives at the observer, and then after that the ob-

server may experience the Doppler effect. This behavior is called sonic (in

the case of flows, it is the sonic flow).

Finally when the speed of sound source exceeds the speed of sound, a

conical surface, called the Mach corn, is formed with continuous wave

fronts, in which the intersecting spherical waves are contained. The flow is

undisturbed outside of the Mach corn and no sound reaches the observer

until the Mach corn passes. The half angle of the Mach corn is



sinsin

(5.1.26)

This is called the Mach angle, and this situation is called supersonic (in the

case of flows, it is called the supersonic flow).

The ratio of the local flow speed (or in the case of a moving sound

source the traveling speed in a stationary fluid)

u to the local speed of

sound

a is the Mach number as defined by

(5.1.27)

The Mach number is a dimensionless flow property, which characterizes

compressible flows.

5.2 Isentropic Flow

Many engineering applications associated with compressible flow can be

treated with assumptions of steady and isentropic flow of one dimensional

motion. The applications may include gases undergoing appreciable

change of density with variations in pressure and temperature through pas-

sages of varying cross-section areas. Such problems would be seen with

exhaust gases passing through gas turbines, nozzles on rocket engines and

gas flow measuring instruments, diffusers of jet engines, etc. These are in-

ternal flows where the area change is the predominant case for change of

flow conditions.

Consider the steady flow through a channel of changing area

A as

shown in Fig. 5.2. The flow is isentropic and one dimensional in

direc-

tion and has properties of the density

, the temperature

and the local

230

5.2 Isentropic Flow

speed of sound

. The continuity equation is, with reference to Eq.

(4.1.33), written as

const.

(5.2.1)

Its differential form is also written as

0 

(5.2.2)

Fig. 5.2 One dimensional isentropic compressible flow

While from the steady one dimensional Euler equation, i.e. Eq. (5.1.2) with

0 ww t

, we have



(5.2.3)

Since the flow is one dimensional, Eq. (5.2.3) can be expressed by the total

differential form

dpudu



(5.2.4)

and from Eqs. (5.1.17), (5.2.4) may be written in terms of the speed of

sound as

udu



(5.2.5)

Finally Eqs. (5.2.5) and (5.2.2) are combined to give, after introducing the

Mach number



᧩M

(5.2.6)

Equation (5.2.6) has a very important relationship for an isentropic uni-

form flow in a changing area. Equation (5.2.6) is indicating that the

231

5 Compressible Flow

Fig. 5.3 Laval tube; supersonic nozzle

In order to examine the quantitative changes of flow parameters along

the flow channel in Fig. 5.3, the isentropic flow is considered in more de-

tail by denoting that the nozzle (Laval tube) is connected with a reservoir,

where stagnation quantities, temperature

, pressure

and speed of

sound

a are defined for 0

u , and with a receiver where the (exit) flow

conditions are controlled typically by pressure.

A direct integration of Eq.

(5.2.4) along the channel for

yields an energy equation, with reference

to Eq. (4.1.27) without the scalar potential , as follows

const.



(5.2.7)

)

relation between AdA and udu changes, depending on the Mach num-

ber, i.e.

1

(subsonic), 1

(sonic) or 1!

(supersonic). One of the

important observations obtained from Eq. (5.2.6) is that for

, the

area

A takes the minimum by denoting



A . If A is decreased while keep-

ing 1d

, then there becomes 0tdu , resulting in the appearance of an ac-

celerating flow till

, where

auu . Further decreasing in area

from

A may result in no flow existing in the channel. However, if the

area is increased from

AA , du would be positive 0tdu (accelerating

again) for 1t

. If this flow process is realized, a supersonic flow

(1t

) would be achieved in the channel. This type of channel is called a

Laval tube or supersonic nozzle, which consists of converging and diverg-

ing parts with throat in between, as illustrated in Fig. 5.3.

232