Basics of Fluid Mechanics, 2014a

12.2. OBLIQUE SHOCK 489
The relationship between the properties can be determined by substituting M 1 sin θ
for of M 1 into the normal shock relationship, which results in
Pressure Ratio
P 2
= 2 kM 1 2 sin 2 θ − (k − 1)
P 1 k +1
(12.13)
The density and normal velocity ratio can be determined by the following equation
ρ 2
ρ 1
= U 1n
U 2n
=
Density Ratio
(k +1)M 1 2 sin 2 θ
(k − 1)M 1 2 sin 2 θ +2
(12.14)
The temperature ratio is expressed as
Temperature Ratio
T 2
= 2 kM 1 2 sin 2 θ − (k − 1) [ (k − 1)M 2 1 +2 ]
T 1 (k +1) 2 M 1
(12.15)
Prandtl’s relation for oblique shock is
U n1 U n2 = c 2 − k − 1
k +1 U t 2 (12.16)
The Rankine–Hugoniot relations are the same as the relationship for the normal shock
12.2.1 Solution of Mach Angle
P 2 − P 1
ρ 2 − ρ 1
= k P 2 − P 1
ρ 2 − ρ 1
(12.17)
Oblique shock, if orientated to a coordinate perpendicular and parallel shock plane is
like a normal shock. Thus, the relationship between the properties can be determined by
using the normal components or by 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, the normal shock is a one–dimensional problem, thus, only one parameter
is required (to solve the problem). 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, M 1 and the deflection angle, which create a
somewhat complicated mathematical procedure, and this will be discussed later. Other
combinations of properties provide a relatively simple mathematical treatment, and the
solutions of selected pairs and selected relationships will be presented.