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

Chapter 17 | 859

Analysis Because of assumption 2, we approximate the oblique shock

deflection angle as equal to the wedge half-angle, i.e., ud
10°. With

Ma 1 
2.0 and u
10°, we solve Eq. 17–46 for the two possible values of

oblique shock angle b: Bweak
39.3°and Bstrong
83.7°. From these values,

we use the first part of Eq. 17–44 to calculate the upstream normal Mach

number Ma1,n,

Weak shock:

Strong shock:

We substitute these values of Ma1,ninto the second equation of Fig. 17–40

to calculate the downstream normal Mach number Ma2,n. For the weak

shock, Ma2,n
0.8032, and for the strong shock, Ma2,n
0.5794. We also

calculate the downstream pressure for each case, using the third equation of

Fig. 17–40, which gives

Weak shock:

Strong shock:

Finally, we use the second part of Eq. 17–44 to calculate the downstream

Mach number,

Weak shock:

Strong shock:

The changes in Mach number and pressure across the strong shock are

much greater than the changes across the weak shock,as expected.

Discussion Since Eq. 17–46 is implicit in b, we solve it by an iterative

approach or with an equation solver such as EES. For both the weak and

strong oblique shock cases, Ma1,nis supersonic and Ma2,nis subsonic. How-

ever, Ma 2 is supersonicacross the weak oblique shock, but subsonicacross

the strong oblique shock. We could also use the normal shock tables in

place of the equations, but with loss of precision.

Ma 2 

Ma2,n

sin 1 bu 2

0.5794

sin 1 83.7°10° 2

0.604

Ma 2 

Ma2,n

sin 1 bu 2

0.8032

sin 1 39.3°10° 2

1.64

P 2

P 1

2 kMa^2 1,nk 1

k 1

SP 2 
 1 75.0 kPa 2

21 1.4 21 1.988 22 1.4 1

1.4 1

333 kPa

P 2

P 1

2 kMa^2 1,nk 1

k 1

SP 2 
 1 75.0 kPa 2

21 1.4 21 1.267 22 1.4 1

1.4 1

128 kPa

Ma1,n
Ma 1 sin b S Ma1,n
2.0 sin 83.7°
1.988

Ma1,n
Ma 1 sin b S Ma1,n
2.0 sin 39.3°
1.267

EXAMPLE 17–12 Prandtl–Meyer Expansion Wave Calculations

Supersonic air at Ma 1 
2.0 and 230 kPa flows parallel to a flat wall that

suddenly expands by d
10° (Fig. 17–49). Ignoring any effects caused by

the boundary layer along the wall, calculate downstream Mach number Ma 2

and pressure P 2.

Ma 1 
 2.0

Ma 2

d 
 10 °

u

FIGURE 17–49

An expansion fan caused by the sudden

expansion of a wall with d
 10  .

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