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

135

6

Oblique Shock Waves

Introduction

Attention is now turned from normal shock waves, which are straight and in which the

ow before and after the wave is normal to the shock, to oblique shock waves. Such shock

waves are, by denition, also straight but they are at an angle to the upstream ow and, in

general, they produce a change in ow direction as indicated in Figure 6.1.

The oblique shock relations can be deduced from the normal shock relations by noting

that the oblique shock can produce no momentum change parallel to the plane in which

it lies. To show this, consider the control volume shown in Figure 6.2. In this gure, the

components of the velocity parallel to the wave are L

1

and L

2

while the components normal

to the wave are N

1

and N

2

as shown.

Because there are no changes in the ow variables in the direction parallel to the wave

there is no net force on the control volume parallel to the wave and there is, consequently,

no momentum change parallel to the wave. Because there is no momentum change paral-

lel to the shock, L

1

must equal L

2

. Hence, if the coordinate system moving parallel to the

wave front at a velocity L = L

1

= L

2

is considered, the ow in this coordinate system through

the wave is as shown in Figure 6.3.

In this coordinate system, the oblique shock has been reduced to a normal shock and

the normal shock relations must therefore apply to the velocity components N

1

and N

2

.

Further, since the scalar ow properties p, ρ, and T are unaffected by the coordinate system

used, the Rankine–Hugoniot relations must apply without any modication to oblique

shocks. Thus, all the properties of oblique shocks can be obtained by the modication and

manipulation of the normal shock relations provided that the angle of the shock relative

to the upstream ow is known. However, it is more instructive and, in some respects, sim-

pler to deduce these oblique shock relations from the fundamental conservation of mass,

momentum, and energy laws, using the normal shock relations when a parity is formally

established.

Oblique Shock Wave Relations

Consider again ow through a control volume that spans the shock wave and which, with-

out any loss of generality, can be assumed to have unit area parallel to the oblique shock

wave. This control volume is shown in Figure 6.4. As shown in this gure, β is dened as

the shock wave angle and δ is the change in ow direction induced by the shock wave.

136 Introduction to Compressible Fluid Flow

The conservation of mass, momentum, and energy principles are now applied to the con-

trol volume shown in Figure 6.4. Since there is no change in velocity parallel to the wave,

the L velocity components can carry no net mass into the control volume. Conservation of

mass therefore gives

ρ

1

N

1

= ρ

2

N

2

(6.1)

The conservation of momentum equation is applied in the direction normal to the shock,

giving

pp

NN

12 22

2

11

2

−= −

ρρ

(6.2)

Flow upstream

of sh

ock wave

Flow downstream

of shock wave

Oblique

shock wave

Change in flow

direction through

oblique shock wave

FIGURE 6.1

An oblique shock wave.

L

L

N

N

V

V

1

1

1

2

2

2

Oblique

shock wa

ve

Control volume

considered

FIGURE 6.2

Control volume considered.

137Oblique Shock Waves

Because there are no gradients of temperature upstream and downstream of the shock

wave, the ow through the control volume must, as in the case of the normal shock, be

adiabatic. The energy equation therefore gives

2

1

2

1

1

1

1

2

2

2

2

2

γ

γρ

γ

γρ−

+=

−

+

p

V

p

V (6.3)

N

N

1

2

FIGURE 6.3

Flow normal to an oblique shock wave.

Oblique

shock wave

Control volume

with unit frontal area

V

V

L

L

N

N

β

β

1

1

1

2

2

2

δ

FIGURE 6.4

Control volume used in the analysis of ow through an oblique shock wave.