fluid_mechanics

628 Chapter 11 ■ Compressible Flow
T
T
y
p 0, y
T 0 =
constant
y
b
Normal shock
x
p 0, x
p*
Fanno line
T* =
constant
Normal shock
x
a
Rayleigh line
p a
T a
s
s
(a)
(b)
T
p 0, x p 0, y
T 0
y
Normal shock
x
(c)
s
F I G U R E 11.26 (a) The normal
shock in a Fanno flow. (b) The normal shock in a
Rayleigh flow. (c) The normal shock in a frictionless
and adiabatic flow.
But from Eq. 11.123 for Rayleigh flow we get
and
p y
1 k
p
2
a 1 kMa y
p x
1 k
p
2
a 1 kMa x
Thus, by combining Eqs. 11.137, 11.138, and 11.139 we get
p y
p x
1 kMa x 2
1 kMa y
2
(11.138)
(11.139)
(11.140)
Equation 11.140 can also be derived starting with
Ratios of thermodynamic
properties
across a normal
shock are functions
of the Mach
numbers.
and using the Fanno flow equation 1Eq. 11.1072
As might be expected, Eq. 11.140 can be obtained directly from the linear momentum equation
since rV 2 p V 2 RT kV 2 RTk k Ma 2 .
For the Fanno flow of Fig. 11.26a,
p y
p x
a p y
p* b ap* p x
b
p
p* 1 Ma e 1k 122
12
1 31k 1224Ma f 2
p x r x V x 2 p y r y V y
2
T y
T x
a T y
T* b aT* T x
b
(11.141)