Basics of Fluid Mechanics, 2014a

488 CHAPTER 12. COMPRESSIBLE FLOW 2–DIMENSIONAL
Unlike in the normal shock, here there are three possible pairs 1 of solutions to
these equations. The first is referred to as the weak shock; the second is the strong
shock; and the third is an impossible solution (thermodynamically) 2 . Experiments and
experience have shown that the common solution is the weak shock, in which the shock
turns to a lesser extent 3 .
tan θ
tan(θ − δ) = U 1n
(12.7)
U 2n
The above velocity–geometry equations can also be expressed in term of Mach number,
as
and in the downstream side reads
sin θ = M 1n
M 1
(12.8)
sin(θ − δ) = M 2n
(12.9)
M 2
Equation (12.8) alternatively also can be expressed as
cos θ = M 1t
(12.10)
M 1
And equation (12.9) alternatively also can be expressed as
cos (θ − δ) = M 2t
M 2
(12.11)
The total energy across a stationary oblique shock wave is constant, and it follows
that the total speed of sound is constant across the (oblique) shock. It should be noted
that although, U 1t = U 2t the Mach number is M 1t ≠ M 2t because the temperatures
on both sides of the shock are different, T 1 ≠ T 2 .
As opposed to the normal shock, here angles (the second dimension) have to
be determined. The solution from this set of four equations, (12.8) through (12.11),
is a function of four unknowns of M 1 , M 2 , θ, and δ. Rearranging this set utilizing
geometrical identities such as sin α = 2 sin α cos α results in
Angle Relationship
[
M 2 1 sin 2 ]
θ − 1
tan δ = 2 cot θ
M 2 1 (k + cos 2θ)+2
(12.12)
1 This issue is due to R. Menikoff, who raised the solution completeness issue.
2 The solution requires solving the entropy conservation equation. The author is not aware of
“simple” proof and a call to find a simple proof is needed.
3 Actually this term is used from historical reasons. The lesser extent angle is the unstable angle
and the weak angle is the middle solution. But because the literature referred to only two roots, the
term lesser extent is used.