Basics of Fluid Mechanics, 2014a

12.2. OBLIQUE SHOCK 501
or explicitly
M 2 2 = (k +1)2 M 1 4 sin 2 θ − 4(M 1 2 sin 2 θ − 1)(kM 1 2 sin 2 θ +1)
(
2 kM1 2 sin 2 θ − (k − 1) )( (k − 1) M 1 2 sin 2 θ +2 ) (12.57)
The ratio of the total pressure can be expressed as
Stagnation Pressure Ratio
[
P 02 (k 2 +1)M1 sin 2 ]
θ
k [ ] 1
k−1 k−1
k +1
=
P 01 (k − 1)M 2 1 sin 2 θ +2 2kM 2 1 sin 2 θ − (k − 1)
(12.58)
Even though the solution for these variables, M 1 and θ, is unique, the possible range
deflection angle, δ, is limited. Examining equation (12.51) shows that the shock angle,
θ , has to be in the range of sin −1 (1/M 1 ) ≥ θ ≥ (π/2) (see Figure
√
12.8). The range
of given θ, upstream Mach number M 1 , is limited between ∞ and 1/ sin 2 θ.
Defection angle
strong
solution
subsonic
weak
solution
θ min = sin −1 1 M 1
possible solution
1.0