Basics of Fluid Mechanics, 2014a

11.5. NORMAL SHOCK 411
The density and upstream Mach number relationship can be obtained in the same
fashion to became
Shock Density Ratio
ρ y
= U x
= (k +1)M x 2
ρ x U
2
(11.92)
y 2+(k − 1)M x
The fact that the pressure ratio is a function of the upstream Mach number, M x , provides
additional way of obtaining an additional useful relationship. And the temperature
ratio, as a function of pressure ratio, is transformed into
Shock Temperature Ratio
⎛
( )
k +1
T y Py
= ⎜ k − 1 + P ⎞
y
P x
T x P x
⎝
1+ k +1
⎟
P y
⎠
k − 1 P x
(11.93)
In the same way, the relationship between the density ratio and pressure ratio is
Shock P − ρ
( k +1
1+
ρ x k − 1
= ( )
ρ y k +1
+
k − 1
)(
Py
(
Py
P x
)
P x
)
(11.94)
which is associated with the shock wave.
11.5.1.1 The Star Conditions
The speed of sound at the critical condition can also be a good reference velocity. The
speed of sound at that velocity is
c ∗ = √ kRT ∗ (11.95)
In the same manner, an additional Mach number can be defined as
M ∗ = U c ∗ (11.96)
11.5.2 Prandtl’s Condition
It can be easily observed that the temperature from both sides of the shock wave is
discontinuous. Therefore, the speed of sound is different in these adjoining mediums.
It is therefore convenient to define the star Mach number that will be independent of
the specific Mach number (independent of the temperature).