Page 250 - 48Fundamentals of Compressible Fluid Mechanics

212 CHAPTER 13. OBLIQUE-SHOCK

The density and normal velocities ratio can be found from the following equation

!
(13.14)
!

"$

The temperature ratio is expressed as E
"

(13.15)

Prandtl’s relation for oblique shock is

E E E

(13.16)
" "
"
The Rankine–Hugoniot relations are the same as the relationship for the normal

E

E
shock
$
B

(13.17)
!
!
!
!
E E
E E
13.4 Solution of Mach Angle
The oblique shock orientated in coordinate perpendicular and parallel shock plan is
like a normal shock. Thus, the properties relationship can be founded by using the
normal components or utilizing the normal shock table developed earlier. One has
to be careful to use the normal components of the Mach numbers. The stagnation
temperature contains the total velocity.
Again, as it may be recalled, the normal shock is one dimensional problem,
thus, only one parameter was required (to solve the problem). The oblique shock
is a two dimensional problem and two properties must be provided so a solution
can be found. Probably, the most useful properties, are upstream Mach number,
and the deflection angle which create somewhat complicated mathematical
procedure and it will be discussed momentarily. Other properties combinations
provide a relatively simple mathematical treatment and the solutions of selected
pairs and selected relationships are presented.

13.4.1
Again, this set of parameters is, perhaps, the most common and natural to exam-
Upstream Mach number,
, and deflection angle,
ine. Thompson (1950) has shown that the relationship of shock angle is obtained
from the following cubic equation:
(13.18)

. , , ,
.