F18 – Lecture Notes

1. Prandtl-Meyer Waves

2. Shock-Expansion Theory (Supersonic Airfoils)

Reading: Anderson 9.6, 9.7

Prandtl-Meyer Waves

Wave ﬂow relations

An expansion fan, sometimes also called a Prandtl-Meyer expansion wave, can be considered

as a continuous sequence of inﬁnitesimal Mach expansion waves. To analyze this continuous

change, we will now consider the ﬂow angle θ to be a ﬂowﬁeld variable, like M or V .

Across each Mach wave of the fan, the ﬂow direction changes by dθ, while the speed changes

by dV . Oblique-shock analysis dictates that only the normal velocity component u can

change across any wave, so that dV must be entirely due to the normal-velocity change du.

�

1

1

2

2

1 2

M M

V

V

dV

du

u

V

u

µ

V

� �

V

dV

du

µ

Mach wave

d

�

dV

µ

tan

From the u-V and du-dV velocity triangles, it is evident that dθ and dV are related by

dV 1

dθ =

tan µ V

assuming dθ is a small angle. With sin µ = 1/M , we have

1 cos µ

1

− sin

2

µ

1

− 1/M

2

√

M

2

− 1= = = =

tan µ sin µ sin µ 1/M

and so the ﬂow relation above becomes

dV

dθ =

√

M

2

− 1 (1)

V

This is a diﬀerential equation which relates a change dθ in the ﬂow angle to a change dV in

the ﬂow speed throughout the expansion fan.

Prandtl-Meyer Function

The diﬀerential equation (1) can be integrated if we ﬁrst express V in terms of M .

V = M a = M a

o

1 +

γ −1

M

2

−1/2

2

1

ln V = ln M + ln a

o

−

1

2

ln

1 +

γ −1

2

M

2

dV

V

=

dM

M

−

1

2

1 +

γ −1

2

M

2

−1

γ −1

2

2M dM

dV

1

dM

=

V

1 +

γ−1

2

M

2

M

Equation (1) then becomes

dM

dθ =

√

M

2

− 1

(2)

1 +

γ−1

M

2

M

2

which can now be integrated from any point 1 to any point 2 in the Prandtl-Meyer wave.

θ

2

M

2

√

M

2

− 1

dM

dθ =

θ

1

M

1

1 +

γ−1

M

2

M

2

θ

2

− θ

1

= ν(M

2

) − ν(M

1

) (3)

γ +1

where ν(M ) ≡

γ −1

arctan

γ −1

(M

2

− 1) − arctan

√

M

2

− 1 (4)

γ +1

Here, ν(M ) is called the Prandtl-Meyer function, and is shown plotted for γ = 1.4 .

v(M) deg.

50

45

40

35

30

25

20

15

10

5

0

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

M

Equation (3) can be applied to any two points within an expansion fan, but the most common

use is to relate the two ﬂow conditions before and after the fan. Reverting back to our

previous notation where θ is the total turning of the corner, the relation between θ and the

upstream and downstream Mach number is

θ = ν(M

2

) − ν(M

1

) (5)

2

ν(

)

θ

�

�

M

θ

1

M

M

2

M

M

1

M

2

This can be considered an implicit deﬁnition of M

2

(M

1

, θ), which can be evaluated graphi-

cally using the ν(M) function plot, as shown in the ﬁgure.

Shock-Expansion Theory

The combination of oblique-shock relations and Prandtl-Meyer wave relations constitutes

Shock-Expansion Theory , which can be used to determine the ﬂow properties and forces

about simple 2-D shapes in supersonic ﬂow.

Flat-plate supersonic airfoil

A ﬂat plate is the simplest supersonic airfoil. When set at an angle of attack α, the leading

edge point eﬀectively is a convex corner to the upper surface ﬂow, with turning angle θ = α.

The upper ﬂow then passes through the resulting Prandtl-Meyer expansion, which increases

its Mach number from M

1

= M

�

, to a larger value M

2

= M

u

. The latter is implicitly

determined via the Prandtl-Meyer relation (5).

α = ν(M

u

) − ν(M

�

) → M

u

(M , α)

The corresponding upper-surface pressure is then given by the isentropic relation.

1 +

γ−1

M

2

γ/(γ−1)

2

�

p

u

= p

1 +

γ−1

M

2

u

2

Conversely, the leading edge is a concave corner to the bottom surface ﬂow, which then sees

a pressure rise through the resulting oblique shock. The lower-surface Mach number M

and

pressure p

are obtained from oblique-shock relations, with M

1

= M

�

, θ = α as the inputs.

M

p

p

u

p

l

α

L’

D’

R’

c

The pressure diﬀerence produces a resultant force/span R

�

acting normal to the plate, which

can be resolved into lift and drag components.

R

�

= (p

− p

u

) c

L

�

= (p

− p

u

) c cos α

D

�

= (p

− p

u

) c sin α

3

It’s worthwhile to note that this supersonic airfoil has a nonzero drag even though the ﬂow is

being assumed inviscid. Hence, d’Alembert’s Paradox deﬁnitely does not apply to supersonic

2-D ﬂow. The drag D

�

computed here is associated with the viscous dissipation occurring

in the oblique shock waves, and hence is called wave drag . This wave drag is an additional

drag component in supersonic ﬂow, and must be added to the usual viscous friction drag,

and also the induced drag in 3-D cases.

4