Cohen I.M., Kundu P.K. Fluid Mechanics

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Chapter 16

Compressible Flow

1. Introduction......................... 713

Criterion for Neglect of Compressibility

Effects............................ 714

Classiﬁcation of Compressible

Flows............................. 715

Useful Thermodynamic Relations .... 716

2. Speed of Sound ...................... 717

3. Basic Equations for

One-Dimensional Flow .............. 721

Continuity Equation................. 721

Energy Equation .................... 721

Bernoulli and Euler

Equations ........................ 723

Momentum Principle for a Control

Volume ........................... 723

4. Stagnation and Sonic Properties ..... 724

Table 16.1: Isentropic Flow of

a Perfect Gas (γ = 1.4)............ 727

5. Area–Velocity Relations in

One-Dimensional Isentropic Flow .... 729

Example 16.1 ....................... 732

6. Normal Shock Wave ................. 733

Normal Shock Propagating in a Still

Medium .......................... 738

Shock Structure .................... 738

7. Operation of Nozzles at Different

Back Pressures ..................... 741

Convergent Nozzle .................. 741

Convergent–Divergent Nozzle ....... 741

Example 16.2 ...................... 742

Table 16.2: One-Dimensional Normal-

shock Relations (γ = 1.4)........ 745

8. Effects of Friction and Heating in

Constant-Area Ducts ............... 747

Effect of Friction ................... 748

Effect of Heat Transfer ............. 749

Choking by Friction or Heat

Addition ............ ............ 750

9. Mach Cone ......................... 750

10. Oblique Shock Wave................ 752

Generation of Oblique Shock

Waves ........................... 754

The Weak Shock Limit ............. 756

11. Expansion and Compression in

Supersonic Flow.................... 756

12. Thin Airfoil Theory in Supersonic

Flow ............................... 758

Exercises ........................... 761

Literature Cited .................... 763

Supplemental Reading .............. 763

1. Introduction

To this point we have neglected the effects of density variations due to pressure

changes. In this chapter we shall examine some elementary aspects of ﬂows in which

the compressibility effects are important. The subject of compressible ﬂows is also

713

DOI: 10.1016/B978-0-12-381399-2.50016-2

714 Compressible Flow

called gas dynamics, which has wide applications in high-speed ﬂows around objects

of engineering interest. These include external ﬂows such as those around airplanes,

and internal ﬂows in ducts and passages such as nozzles and diffusers used in jet

engines and rocket motors. Compressibility effects are also important in astrophysics.

Two popular books dealing with compressibility effects in engineering applications

are those by Liepmann and Roshko (1957) and Shapiro (1953), which discuss in

further detail most of the material presented here.

Our study in this chapter will be rather superﬁcial and elementary because this

book is essentially about incompressible ﬂows. However, this small chapter on com-

pressible ﬂows is added because a complete ignorance about compressibility effects

is rather unsatisfying. Several startling and fascinating phenomena arise in compress-

ible ﬂows (especially in the supersonic range) that go against our intuition developed

from a knowledge of incompressible ﬂows. Discontinuities (shock waves) appear

within the ﬂow, and a rather strange circumstance arises in which an increase of ﬂow

area accelerates a (supersonic) stream. Friction can also make the ﬂow go faster and

adding heat can lower the temperature in subsonic duct ﬂows. We will see this later in

this chapter. Some understanding of these phenomena, which have no counterpart in

low-speed ﬂows, is desirable even if the reader may not make much immediate use of

this knowledge. Except for our treatment of friction in constant area ducts, we shall

limit our study to that of frictionless ﬂows outside boundary layers. Our study will,

however, have a great deal of practical value because the boundary layers are espe-

cially thin in high-speed ﬂows. Gravitational effects, which are minor in high-speed

ﬂows, will be neglected.

Criterion for Neglect of Compressibility Effects

Compressibility effects are determined by the magnitude of the Mach number

deﬁned as

M ≡

where u is the speed of ﬂow, and c is the speed of sound given by



∂p

∂ρ



where the subscript “s” signiﬁes that the partial derivative is taken at constant entropy.

To see how large the Mach number has to be for the compressibility effects to be

appreciable in a steady ﬂow, consider the one-dimensional steady ﬂow version of the

continuity equation ∇ · (ρu) = 0, that is,

∂ρ

∂x

+ ρ

∂u

∂x

= 0.

The incompressibility assumption requires that

∂ρ

∂x

 ρ

∂u

∂x

1. Introduction 715

or that

δρ



δu

. (16.1)

Pressure changes can be estimated from the deﬁnition of c, giving

δp  c

δρ . (16.2)

The Euler equation requires

uδu 

δp

. (16.3)

By combining equations (16.2) and (16.3), we obtain

δρ



δu

From comparison with equation (16.1) we see that the density changes are negligible if

= M

 1.

The constant density assumption is therefore valid if M<0.3, but not at higher Mach

numbers.

Although the signiﬁcance of the ratio u/c was known for a long time, the Swiss

aerodynamist Jacob Ackeret introduced the term “Mach number,” just as the term

Reynolds number was introduced by Sommerfeld many years after Reynolds’ exper-

iments. The name of the Austrian physicist Ernst Mach (1836–1916) was chosen

because of his pioneering studies on supersonic motion and his invention of the

so-called Schlieren method for optical studies of ﬂows involving density changes;

see von Karman (1954, p. 106). (Mach distinguished himself equally well in philos-

ophy. Einstein acknowledged that his own thoughts on relativity were inﬂuenced by

“Mach’s principle,” which states that properties of space had no independent exis-

tence but are determined by the mass distribution within it. Strangely, Mach never

accepted either the theory of relativity or the atomic structure of matter.)

Classiﬁcation of Compressible Flows

Compressible ﬂows can be classiﬁed in various ways, one of which is based on the

Mach number M. A common way of classifying ﬂows is as follows:

(i) Incompressible ﬂow: M<0.3 everywhere in the ﬂow. Density variations due

to pressure changes can be neglected. The gas medium is compressible but the

density may be regarded as constant.

(ii) Subsonic ﬂow: M exceeds 0.3 somewhere in the ﬂow, but does not exceed 1

anywhere. Shock waves do not appear in the ﬂow.

716 Compressible Flow

(iii) Transonic ﬂow: The Mach number in the ﬂow lies in the range 0.8–1.2. Shock

waves appear and lead to a rapid increase of the drag. Analysis of transonic

ﬂows is difﬁcult because the governing equations are inherently nonlinear,

and also because a separation of the inviscid and viscous aspects of the ﬂow

is often impossible. (The word “transonic” was invented by von Karman and

Hugh Dryden, although the latter argued in favor of having two s’s in the word.

von Karman (1954, p. 116) stated that “I ﬁrst introduced the term in a report to

the U.S. Air Force. I am not sure whether the general who read the word knew

what it meant, but his answer contained the word, so it seemed to be ofﬁcially

accepted.”)

(iv) Supersonic ﬂow: M lies in the range 1–3. Shock waves are generally present.

In many ways analysis of a ﬂow that is supersonic everywhere is easier than an

analysis of a subsonic or incompressible ﬂow as we shall see. This is because

information propagates along certain directions, called characteristics, and a

determination of these directions greatly facilitates the computation of the ﬂow

ﬁeld.

(v) Hypersonic ﬂow: M>3. The very high ﬂow speeds cause severe heating in

boundary layers, resulting in dissociation of molecules and other chemical

effects.

Useful Thermodynamic Relations

As density changes are accompanied by temperature changes, thermodynamic prin-

ciples will be constantly used here. Most of the necessary concepts and relations have

been summarized in Sections 8 and 9 of Chapter 1, which may be reviewed before

proceeding further. Some of the most frequently used relations, valid for a perfect gas

with constant speciﬁc heats, are listed here for quick reference:

Equation of state p = ρRT,

Internal energy e = C

Enthalpy h = C

Speciﬁc heats C

γR

γ − 1

− C

= R,

Speed of sound c =



γRT,

Entropy change S

− S

= C

− R ln

, (16.4)

= C

− R ln

. (16.5)

2. Speed of Sound 717

An isentropic process of a perfect gas between states 1 and 2 obeys the following

relations:









γ −1





(γ −1)/γ

Some important properties of air at ordinary temperatures and pressures are

R = 287 m

/(s

K),

= 1005 m

/(s

K),

= 718 m

/(s

K),

γ = 1.4.

These values will be useful for solution of the exercises.

2. Speed of Sound

We know that a pressure pulse in an incompressible ﬂow behaves in the same way

as that in a rigid body, where a displaced particle simultaneously displaces all the

particles in the medium. The effects of pressure or other changes are therefore instantly

felt throughout the medium. A compressible ﬂuid, in contrast, behaves similarly to

an elastic solid, in which a displaced particle compresses and increases the density of

adjacent particles that move and increase the density of the neighboring particles, and

so on. In this way a disturbance in the form of an elastic wave,orapressure wave,

travels through the medium. The speed of propagation is faster when the medium is

more rigid. If the amplitude of the elastic wave is inﬁnitesimal, it is called an acoustic

wave,orasound wave.

We shall now ﬁnd an expression for the speed of propagation of sound.

Figure 16.1a shows an inﬁnitesimal pressure pulse propagating to the left with speed c

into a still ﬂuid. The ﬂuid properties ahead of the wave are p, T , and ρ, while the ﬂow

speed is u = 0. The properties behind the wave are p + dp, T + dT, and ρ + dρ,

whereas the ﬂow speed is du directed to the left. We shall see that a “compression

wave” (for which the ﬂuid pressure rises after the passage of the wave) must move

the ﬂuid in the direction of propagation, as shown in Figure 16.1a. In contrast, an

“expansion wave” moves the ﬂuid “backwards.”

To make the analysis steady, we superimpose a velocity c, directed to the right,

on the entire system (Figure 16.1b). The wave is now stationary, and the ﬂuid enters

the wave with velocity c and leaves with a velocity c − du. Consider an area A on

the wavefront. A mass balance gives

Aρc = A(ρ + dρ)(c − du).

718 Compressible Flow

Figure 16.1 Propagation of a sound wave: (a) wave propagating into still ﬂuid; and (b) stationary wave.

Because the amplitude is assumed small, we can neglect the second-order terms,

obtaining

du = c(dρ/ρ). (16.6)

This shows that du > 0ifdρ is positive, thus passage of a compression wave leaves

behind a ﬂuid moving in the direction of the wave, as shown in Figure 16.1a.

Now apply the momentum equation, which states that the net force in the

x-direction on the control volume equals the rate of outﬂow of x-momentum minus

the rate of inﬂow of x-momentum. This gives

pA − (p + dp)A = (Aρc)(c − du) − (Aρc)c,

where viscous stresses have been neglected. Here, Aρ c is the mass ﬂow rate. The ﬁrst

term on the right-hand side represents the rate of outﬂow of x-momentum, and the

second term represents the rate of inﬂow of x-momentum. Simplifying the momentum

equation, we obtain

dp = ρc du. (16.7)

Eliminating du between equations (16.6) and (16.7), we obtain

dρ

. (16.8)

2. Speed of Sound 719

If the amplitude of the wave is inﬁnitesimal, then each ﬂuid particle undergoes a

nearly isentropic process as the wave passes by. The basic reason for this is that

the irreversible entropy production is proportional to the squares of the velocity and

temperature gradients (see Chapter 4, Section 15) and is therefore negligible for

weak waves. The particles do undergo small temperature changes, but the changes

are due to adiabatic expansion or compression and are not due to heat transfer from

the neighboring particles. The entropy of a ﬂuid particle then remains constant as a

weak wave passes by. This will also be demonstrated in Section 6, where it will be

shown that the entropy change across the wave is dS ∝ (dp)

, implying that dS goes

to zero much faster than the rate at which the amplitude dp tends to zero.

It follows that the derivative dp/dρ in equation (16.8) should be replaced by the

partial derivative at constant entropy, giving



∂p

∂ρ



(16.9)

For a perfect gas, the use of p/ρ

= const. and p = ρRT reduces the speed of sound

(16.9) to

c =



γp



. (16.10)

For air at 15

◦

C, this gives c = 340 m/s. We note that the nonlinear terms that we

have neglected do change the shape of a propagating wave depending on whether it

is a compression or expansion, as follows. Because γ>1, the isentropic relations

show that if dp > 0 (compression), then dT > 0, and from equation (16.10) the

sound speed c is increased. Therefore, the sound speed behind the front is greater

than that at the front and the back of the wave catches up with the front of the wave.

Thus the wave steepens as it travels. The opposite is true for an expansion wave, for

which dp < 0 and dT < 0soc decreases. The back of the wave falls farther behind

the front so an expansion wave ﬂattens as it travels.

Finite amplitude waves, across which there is a discontinuous change of pressure,

will be considered in Section 6. These are called shock waves. It will be shown that

the ﬁnite waves are not isentropic and that they propagate through a still ﬂuid faster

than the sonic speed.

The ﬁrst approximate expression for c was found by Newton, who assumed that

dp was proportional to dρ, as would be true if the process undergone by a ﬂuid

particle was isothermal. In this manner Newton arrived at the expression c =

√

RT .

He attributed the discrepancy of this formula with experimental measurements as

due to “unclean air.” The science of thermodynamics was virtually nonexistent at the

time, so that the idea of an isentropic process was unknown to Newton. The correct

expression for the sound speed was ﬁrst given by Laplace.

720 Compressible Flow

To show explicitly that small disturbances in a compressible ﬂuid obey a wave

equation, we consider a slightly perturbed uniform ﬂow in the x-direction so that

u = U

∞

+ u



), p = p

∞

(1 + p



), ρ = ρ

∞

(1 + ρ



), and so on

where the perturbations ()



are all << 1. We substitute this assumed ﬂow into the

equations for conservation of mass, momentum, and energy. We shall neglect the

effects of viscous stresses and heat conduction here but we will include them at

the end of Section 6, where they are determinative of shock structure. We may write

the conservation laws in the form

Dρ/Dt + ρ∇·u = 0

ρDu/Dt +∇p = 0

ρDh/Dt −Dp/Dt = 0

where body forces have also been neglected and D/Dt denotes the derivative follow-

ing the ﬂuid particle, D/Dt = ∂/∂t +u ·∇. Substituting the assumed ﬂow into mass

conservation ﬁrst,

∞

∂ρ



/∂t +ρ

∞

∂ρ



/∂x +ρ

∞



·∇ρ



+ρ

∞

∇·u



+ρ

∞



∇·u



= 0.

We neglect the squares and products of the perturbations, leaving

∂ρ



/∂t + U

∞

∂ρ



/∂x + U

∞

∇·u



= 0.

Similarly, the momentum equation yields

∂u



/∂t + U

∞

∂u



/∂x +[p

∞

/(ρ

∞

)]∇p



= 0.

We may eliminate u



by taking the divergence of the momentum equation and sub-

stituting into mass conservation, giving,

(∂/∂t +U

∞

∂/∂x)∇·u



=−(1/U

∞

)(∂/∂t +U

∞

∂/∂x)



=−[p

∞

/(ρ

∞

)]∇



The energy equation is put in terms of p



,ρ



for a perfect gas with constant speciﬁc

heats, h = (C

/R)(p/ρ) and C

/R = γ /(γ −1). This results in D/Dt(p/ρ

) = 0

but p/ρ

= (p

∞

/ρ

∞

)(1 +p



−γρ



), with squares and products of the perturbations

neglected. Then (∂/∂t + U

∞

∂/∂x)p



− γ(∂/∂t + U

∞

∂/∂x)ρ



= 0. Using this to

eliminate ρ



, (∂/∂t +U

∞

∂/∂x)



= (γp

∞

/ρ

∞

)∇



= c

∇



. This is a classical

linear wave equation for p



. We can translate this back to a frame at rest by a Galilean

transformation, (x,y,z,t) → (x



) with t



= t + x/U

∞



= x,y



y,z



= z. Thus ∂/∂t



= ∂/∂t + U

∞

∂/∂x and we are left with

∂

p/∂t

= c

∇

(primes suppressed), as seen in Section 7.2, p.200. The solution in one dimension is

given there and it is seen that c is the wave speed.

3. Basic Equations for One-Dimensional Flow 721

3. Basic Equations for One-Dimensional Flow

In this section we begin our study of certain compressible ﬂows that can be analyzed

by a one-dimensional approximation. Such a simpliﬁcation is valid in ﬂow through a

duct whose centerline does not have a large curvature and whose cross section does

not vary abruptly. The overall behavior in such ﬂows can be studied by ignoring the

variation of velocity and other properties across the duct and replacing the property

distributions by their average values over the cross section (Figure 16.2). The area of

the duct is taken as A(x), and the ﬂow properties are taken as p(x), ρ(x), u(x), and

so on. Unsteadiness can be introduced by including t as an additional independent

variable. The forms of the basic equations in a one-dimensional compressible ﬂow

are discussed in what follows.

Continuity Equation

For steady ﬂows, conservation of mass requires that

ρuA = independent of x.

Differentiating, we obtain

dρ

= 0. (16.11)

Energy Equation

Consider a control volume within the duct, shown by the dashed line in Figure 16.2.

The ﬁrst law of thermodynamics for a control volume ﬁxed in space is





e +



dV +





e +



ρu



−



q · dA,

(16.12)

where u

/2 is the kinetic energy per unit mass. The ﬁrst term on the left-hand side

represents the rate of change of “stored energy” (the sum of internal and kinetic

Figure 16.2 A one-dimensional ﬂow.