fluid_mechanics

624 Chapter 11 ■ Compressible Flow
T
Cooling
Heating
b
a
T
b
a
T
Cooling
Normal shock
b
Heating
a
F I G U R E 11.23 (a) Subsonic Rayleigh flow. (b) Supersonic Rayleigh flow. (c) Normal
shock in a Rayleigh flow.
To quantify Rayleigh flow behavior we need to develop appropriate forms of the governing
equations. We elect to use the state of the Rayleigh flow fluid at point a of Fig. 11.22 as the reference
state. As shown earlier, the Mach number at point a is 1. Even though the Rayleigh flow being
considered may not choke and state a is not achieved by the flow, this reference state is useful.
If we apply the linear momentum equation 1Eq. 11.1102 to Rayleigh flow between any upstream
section and the section, actual or imagined, where state a is attained, we get
p rV 2 2
p a r a V a
or
p
rV 2
1 r a 2
V (11.122)
p a p a p a
a
By substituting the ideal gas equation of state 1Eq. 11.12 into Eq. 11.122 and making use of the ideal
gas speed-of-sound equation 1Eq. 11.362 and the definition of Mach number 1Eq. 11.462, we obtain
p
1 k
(11.123)
p a 1 kMa 2
This relationship is graphed in the margin for air.
From the ideal gas equation of state 1Eq. 11.12 we conclude that
__
p
T
p pa r a
1.0
(11.124)
T a
p a r
2.0
0.0
0.1
2.0
T__
T a 1.0
1.0
Ma
10
(a)
Conservation of mass 1Eq. 11.402 with constant A gives
(11.125)
which when combined with Eqs. 11.36 1ideal gas speed of sound2 and 11.46 1Mach number definition2
gives
Combining Eqs. 11.124 and 11.126 leads to
which when combined with Eq. 11.123 gives
s
Heating
Cooling
r a
(b)
r a
r V V a
r Ma T
B
T a
T
a p 2
Mab
T a
p a
s
Heating
(c)
s
(11.126)
(11.127)
0.0
0.1
1.0
Ma
10
T 11 k2Ma
2
c
T a 1 kMa d 2
(11.128)