Introduction to Compressible Fluid Flow, Second Edition, 2nd Edition

39

3

Some Fundamental Aspects of Compressible Flow

Introduction

It was indicated in the previous chapters that compressibility effects become important in

a gas ow when the velocity in the ow is high. An attempt will be made in this chapter to

show that it is not the value of the gas velocity itself but rather the ratio of the gas velocity

to the speed of sound in the gas that determines when compressibility is important. This

ratio is termed the Mach number, M, i.e.,

M

V

a

==

gasvelocity

speed of sound

(3.1)

where a is the speed of sound.

If M < 1, the ow is said to be subsonic, whereas if M > 1, the ow is said to be super-

sonic. If the Mach number is near 1 and there are regions of both subsonic and supersonic

ow, the ow is said to be transonic. If the Mach number is very much greater than 1, the

ow is said to be hypersonic. Hypersonic ow is normally associated with ows in which

M > 5.

As will be shown later in this chapter, the speed of sound in a perfect gas is given by

a

p

RT==

γ

γ

ρ

(3.2)

The speed of sound in a gas depends therefore only on the absolute temperature of the gas.

Isentropic Flow in a Streamtube

To illustrate the importance of the Mach number in determining the conditions under

which compressibility must be taken into account, isentropic ow, i.e., frictionless adia-

batic ow, through a streamtube will be rst considered. The changes in the ow variables

over a short length, dx, of the streamtube shown in Figure 3.1 are considered.

40 Introduction to Compressible Fluid Flow

The Euler equation, Equation 2.8, derived in the previous chapter by applying the con-

servation of momentum principle and ignoring the effects of friction gives

dp

p

V

p

dV

V

=−

ρ

2

(3.3)

Using the expression for the speed of sound, a, given in Equation 3.2 allows this equation

to be written as

dp

p

V

a

dV

V

=−γ

2

2

(3.4)

but M = V/a, so this equation can be written as

dp

p

M

dV

V

=−γ

2

(3.5)

This equation shows that the magnitude of the fractional pressure change, dp/p, induced

by a given fractional velocity change, dV/V, depends on the square of the Mach number.

Next consider the energy equation. Since adiabatic ow is being considered, Equation

2.17 gives

dT

T

V

cT

dV

V

p

=−

2

(3.6)

which can be rearranged to give

dT

T

R

c

M

dV

V

p

=−

γ

2

(3.7)

However, since R = c

p

– c

v

, i.e., since R/c

p

= 1 – 1/γ, it follows that

γ

γ

R

c

p

=−

1

(3.8)

p + dp

ρ + dρ

V + dV

T + dT

V

T

ρ

p

FIGURE 3.1

Portion of streamtube considered.

41Some Fundamental Aspects of Compressible Flow

Equation 3.7 can therefore be written as

dT

T

M

dV

V

=− −()γ 1

2

(3.9)

This equation shows that the magnitude of the fractional temperature change, dT/T,

induced by a given fractional velocity change, dV/V, also depends on the square of the

Mach number.

Lastly, consider the equation of state. As shown in the previous chapter, this gives

dp

p

dd

T

T

=+

ρ

ρ

(3.10)

Combining this equation with Equations 3.5 and 3.9 then gives

d

M

dV

V

M

dV

V

M

dV

V

ρ

ρ

=− +− =−γγ

222

1()

(3.11)

This equation indicates that

d

dV V

M

ρρ

/

/

=−

2

From this equation, it will be seen that for a given fractional change in velocity, i.e., for a

given dV/V, the corresponding induced fractional change in density will also depend on

the square of the Mach number. For example, at Mach 0.1, the fractional change in density

will be 1% of the fractional change in velocity; at Mach 0.33, it will be about 10% of the frac-

tional change in velocity; whereas at Mach 0.4, it will be 16% of this fractional change in

velocity. Therefore, at low Mach numbers, the density changes will be insignicant but as

the Mach number increases, the density changes, i.e., compressibility effects, will become

increasingly important. Hence, compressibility effects become important in high Mach

number ows. The Mach number at which compressibility must start to be accounted for

depends very much on the ow situation and the accuracy required in the solution. As a

rough guide, it is sometimes assumed that if M > 0.5, then there is a possibility that com-

pressibility effects should be considered.

It should also be noted that Equation 3.9 gives

dT T

dV V

M

/

/

=− −

()

γ 1

2

This indicates that if the Mach number is high enough for density changes in the ow to

be signicant, the temperature changes in the ow will also be important.

It should be clear from the above results that the Mach number is the parameter that

determines the importance of compressibility effects in a ow.