Oblique Shock

• 
  The discontinuities in supersonic flows do not always exist as normal to the flow direction. There are oblique shocks which are inclined with respect to the flow direction. Refer to the shock structure on an obstacle, as depicted qualitatively in Fig.41.6.
• The segment of the shock immediately in front of the body behaves like a normal shock.
  
• Oblique shock can be observed in following cases-

  1. Oblique shock formed as a consequence of the bending of the shock in the free-stream direction (shown in Fig.41.6)
  2. In a supersonic flow through a duct, viscous effects cause the shock to be oblique near the walls, the shock being normal only in the core region.
  3. The shock is also oblique when a supersonic flow is made to change direction near a sharp corner

Fig 41.6 Normal and oblique Shock in front of an Obstacle

• The relationships derived earlier for the normal shock are valid for the velocity components normal to the oblique shock. The oblique shock continues to bend in the downstream direction until the Mach number of the velocity component normal to the wave is unity. At that instant, the oblique shock degenerates into a so called Mach wave across which changes in flow properties are infinitesimal. 
  
• Let us now consider a two-dimensional oblique shock as shown in Fig.41.7 below

Fig 41.7 Two dimensional Oblique Shock

For analyzing flow through such a shock, it may be considered as a normal shock on which a velocity  (parallel to the shock) is superimposed. The change across shock front is determined in the same way as for the normal shock. The equations for mass, momentum and energy conservation , respectively, are

(41.16)

(41.17)

(41.18)

These equations are analogous to corresponding equations for normal shock. In addition to these, we have

and

Modifying normal shock relations by writing  and  in place of  and  , we obtain 

(41.19)

(41.20)

(41.21)

Note that although <1,  might be greater than 1. So the flow behind an oblique shock may be supersonic although the normal component of velocity is subsonic. 
In order to obtain the angle of deflection of flow passing through an oblique shock, we use the relation

Having substituted  from Eq. (41.20), we get the relation (see steps here)

 

(41.22)

Sometimes, a design is done in such a way that an oblique shock is allowed instead of a normal shock. The losses for the case of oblique shock are much less than those of normal shock.This is the reason for making the nose angle of the fuselage of a supersonic aircraft small.