Oblique Shock Waves

Generally, a shock wave is not normal to the flow. In Figure 5.11, for example, the wave becomes oblique to the flow as one moves away from the nose. Let us therefore examine this more general case in the same manner as we did the normal shock wave.

Figure 5.12 pictures a flow passing through an oblique shock wave that is

In view of Equation 5.41, the energy equation becomes

V? V?

CpT, + ^=CpT2 + -^

Equations 5.39, 5.40, and 5.42 are identical to Equations 5.25, 5.26, and 5.28 if Vin is replaced by Vi and V2„ by V2. Thus, all of the relationships previously derived for a normal shock wave apply to an oblique shock wave if the Mach numbers normal to the wave are used. These relationships, together with the fact that the tangential velocity remains unchanged through the wave, allow us to determine the flow conditions downstream of the wave as well as the angles в and S.

2.0.

1.92

M2„ = 0.69

Now we must be careful, because the tangential velocity is constant across the wave, not the tangential Mach number. To obtain M2f, we write

or, in this case,

Notice for this example that

M2 = VMi + Ml, = 1.31

Thus the flow is still supersonic after it has passed through the wave, unlike the flow through a normal shock wave.

As an exercise, repeat the foregoing example, but with а 6 of 76.5°.

Surprisingly, the same turning angle is obtained for the same upstream Mach number but different wave angle, 0. For this steeper wave, which is more like a normal shock, the flow becomes subsonic behind the wave, M2 being equal to 0.69.

The deflection angle, 3, as a function of 6, for a constant Mach will appear as shown qualitatively in Figure 5.13. For a given Mi, a maximum deflection angle exists with a corresponding shock wave angle. For deflections less than the maximum, two different в values can accomplish the same deflection. The oblique shock waves corresponding to the higher 6 values are referred to as strong waves, while the shock waves having the lower в values are known as weak waves. There appears to be no analytical reason for rejecting either possible family of waves but, experimentally, one finds only the weak oblique shock waves. Thus, the flow tends to remain supersonic through the wave unless it has no other choice. If, for a given Mb the boundary of the airfoil requires a turning greater than Smax, the wave will become detached, as illustrated in Figure 5.11. The flow then becomes subsonic just behind the normal part of the wave and navigates around the blunt nose under the influence of pressure gradients propagated ahead of the

ЯІ{|игв 5.13 Relationship between the shock wave angle, в, and the deflection angle, s.

airfoil in the subsonic flow region. It then accelerates downstream, again attaining supersonic speeds.

An explicit relationship for 8 as a function of Mi and 6 can be obtained by applying the equations for the normal shock wave to the Mach number normal components of the oblique shock wave, Mt sin в and M2 sin (6 ~ 8). After a considerable amount of algebraic reduction, one obtains the result

Mi2 sin 20-2 cot в 6 2 + Mi2(y + cos 26)

_ _ Mi2 sin 26-2 cot 0 ~ iO + Mi2(7 + 5 cos 26) (for у = 7/5)

This equation is presented graphically in Figure 5.14 (taken from Ref. 5.7) for a range of Mach numbers and shock wave angles from 0 to 90°. 0 values

lying below the broken line correspond to weak oblique shock waves. This dividing line is close to but slightly below the solid line through the maximum deflection angles.