Microsoft Word - Cengel and Boles TOC _2-03-05_.doc

where we must be careful to measure the angle relative to the newdirection

of flow downstream of the expansion, namely, parallel to the upper wall of

the wedge in Fig. 17–45 if we neglect the influence of the boundary layer

along the wall. But how do we determine Ma 2? It turns out that the turning

angle uacross the expansion fan can be calculated by integration, making

use of the isentropic flow relationships. For an ideal gas, the result is

(Anderson, 2003),

Turning angle across an expansion fan: (17–48)

where n(Ma) is an angle called the Prandtl–Meyer function(not to be con-

fused with the kinematic viscosity),

(17–49)

Note that n(Ma) is an angle, and can be calculated in either degrees or radi-

ans. Physically,n(Ma) is the angle through which the flow must expand,

starting with n
0 at Ma 
1, in order to reach a supersonic Mach number,

Ma 1.

To find Ma 2 for known values of Ma 1 ,k, and u, we calculate n(Ma 1 ) from

Eq. 17–49,n(Ma 2 ) from Eq. 17–48, and then Ma 2 from Eq. 17–49, noting

that the last step involves solving an implicit equation for Ma 2. Since there

is no heat transfer or work, and the flow is isentropic through the expansion,

T 0 and P 0 remain constant, and we use the isentropic flow relations derived

previously to calculate other flow properties downstream of the expansion,

such as T 2 ,r 2 , and P 2.

Prandtl–Meyer expansion fans also occur in axisymmetric supersonic

flows, as in the corners and trailing edges of a cone-cylinder (Fig. 17–46).

Some very complex and, to some of us, beautiful interactions involving

both shock waves and expansion waves occur in the supersonic jet pro-

duced by an “overexpanded” nozzle, as in Fig. 17–47. Analysis of such

flows is beyond the scope of the present text; interested readers are referred

to compressible flow textbooks such as Thompson (1972) and Anderson

(2003).

n 1 Ma 2 


B

k 1

k 1

tan^1 c

B

k 1

k 1

1 Ma^2  12 dtan^1 a 2 Ma^2  1 b

u
n 1 Ma 22 n 1 Ma 12

Chapter 17 | 857

FIGURE 17–46

A cone-cylinder of 12.5 half-angle in

a Mach number 1.84 flow. The

boundary layer becomes turbulent

shortly downstream of the nose,

generating Mach waves that are visible

in this shadowgraph. Expansion waves

are seen at the corners and at the

trailing edge of the cone.

Photo by A. C. Charters, Army Ballistic Research

Laboratory.

d

u

Ma 1  1

m 1

m 2

Ma 2

Expansion

waves

Oblique

shock

FIGURE 17–45

An expansion fan in the upper

portion of the flow formed by a two-

dimensional wedge at the angle of

attack in a supersonic flow. The flow

is turned by angle u, and the Mach

number increases across the expansion

fan. Mach angles upstream and

downstream of the expansion fan are

indicated. Only three expansion waves

are shown for simplicity, but in fact,

there are an infinite number of them.

(An oblique shock is present in the

bottom portion of this flow.)

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