Mechanics of Fluids

506 Compressible flow of gases
Equations 11.34–11.36 differ from those for a normal shock only in using the
normal component of flow velocity in place of the full velocity. Therefore
the subsequent equations developed for the normal shock are applicable to
the oblique shock if u sin β is substituted for u, and M sin β for M.
The angles β1 and β2 of Fig. 11.9 are related by
tan β2
=
tan β1
u2n
=
u1n
ρ1
=
ρ2
p1T2
p2T1
With the aid of eqns 11.24, 11.26 and 11.29 this expression becomes
tan β2
tan β1
= 2 + (γ − 1)M2 1 sin β1
(γ + 1)M2 1 sin2 β1
and elimination of β2 in favour of the angle of deflection (β1 – β2) then gives
tan (β1 − β2) = 2 cot β1(M 2 1 sin2 β1 − 1)
M 2 1 (γ + cos 2β1) + 2
(11.37)
Equation 11.37 shows that β1 − β2 = 0 when β1 = 90 ◦ (a normal shock
wave) or when M1 sin β1 = 1. This second condition is the limiting case when
the normal velocity component = a1 and the pressure rise is infinitesimal; β1
is then the Mach angle (Section 11.4.1).
Equation 11.37 is plotted in Fig. 11.10 from which it is seen that, for a
given value of M1, the deflection has a maximum value, and that a particular
deflection below this maximum is given by two values of β1.
As an example we may consider a wedge-shaped solid body of semi-vertex
angle β1 − β2 placed symmetrically in a uniform supersonic flow of Mach
number M1 (Fig. 11.11). On reaching the wedge the flow is deflected through
the angle β1 − β2 and if this is less than the maximum for the given value
of M1 an oblique shock wave is formed at the nose of the wedge as shown
in Fig. 11.11a. Now Fig. 11.10 shows that two values of β1 will satisfy
the oblique shock equations, and the question arises: which one occurs in
practice?
For the smaller value of β1 the normal component M1 sin β1 would be
smaller and so (as shown by Figs 11.7 and 11.8) the corresponding pressure
rise and entropy increase would also be smaller. The wave that has this
smaller value of β1 and is known as a weak, or ordinary, oblique shock
wave, is the one that usually appears. The strong, or extraordinary, wave,
corresponding to the larger value of β1, occurs only if the boundary conditions
of the flow are such as to require the greater pressure rise across the
strong wave.
Figure 11.10 also shows that for a given value of M1 there is a maximum
value of β1 − β2 for which an oblique shock wave is possible. This maximum
increases from zero when M1 = 1 to arccosec γ (= 45.6 ◦ for air) when
M1 =∞. (This is a mathematical rather than physical result since for M1 =∞
there would be infinite rises for pressure and temperature which would invalidate
the assumption of a perfect gas.) Conversely it may be said that for a
specified deflection angle (less than arccosec γ ) an oblique shock wave is possible
only if M1 exceeds a certain minimum value. If the semi-vertex angle of