fluid_mechanics

11.5 Nonisentropic Flow of an Ideal Gas 627
Diverging
duct
Normal shock wave
T
y
Supersonic
flow
Section (x)
Infinitesimally thin
control volume
Section (y)
Subsonic
flow
Normal shock
Fanno line
Rayleigh line
x
s
F I G U R E 11.24
control volume.
Normal shock
F I G U R E 11.25 The
relationship between a normal shock and
Fanno and Rayleigh lines.
For the control volume containing the normal shock, no shaft work is involved and the heat
transfer is assumed negligible. Thus, the energy equation 1Eq. 5.692 can be applied to steady gas
flow through the control volume of Fig. 11.24 to obtain
ȟ V 2
2 ȟ 0 constant
or, for an ideal gas, since ȟ ȟ 0 c p 1T T 0 2 and p rRT
T 1rV22 T 2
2c p 1p 2 R 2 2 T 0 constant
(11.136)
The energy equation
for Fanno flow
and the momentum
equation for
Rayleigh flow are
valid for flow
across normal
shocks.
Equation 11.136 is identical to the energy equation for Fanno flow analyzed earlier 1Eq. 11.752.
The T ds relationship previously used for ideal gas flow 1Eq. 11.222 is valid for the
flow through the normal shock 1Fig. 11.242 because it 1Eq. 11.222 is an ideal gas property relationship.
From the analyses in the previous paragraphs, it is apparent that the steady flow of an
ideal gas across a normal shock is governed by some of the same equations used for describing
both Fanno and Rayleigh flows 1energy equation for Fanno flows and momentum equation
for Rayleigh flow2. Thus, for a given density–velocity product 1rV2, gas 1R, k2, and conditions
at the inlet of the normal shock 1T x , p x , and s x 2, the conditions downstream of the shock 1state y2
will be on both a Fanno line and a Rayleigh line that pass through the inlet state 1state x2, as is
illustrated in Fig. 11.25. To conform with common practice we designate the states upstream
and downstream of the normal shock with x and y instead of numerals 1 and 2. The Fanno and
Rayleigh lines describe more of the flow field than just in the vicinity of the normal shock when
Fanno and Rayleigh flows are actually involved 1solid lines in Figs. 11.26a and 11.26b2. Otherwise,
these lines 1dashed lines in Figs. 11.26a, 11.26b, and 11.26c2 are useful mainly as a way
to better visualize how the governing equations combine to yield a solution to the normal shock
flow problem.
The second law of thermodynamics requires that entropy must increase across a normal shock
wave. This law and sketches of the Fanno line and Rayleigh line intersections, like those of Figs.
11.25 and 11.26, persuade us to conclude that flow across a normal shock can only proceed from
supersonic to subsonic flow. Similarly, in open-channel flows 1see Chapter 102 the flow across a hydraulic
jump proceeds from supercritical to subcritical conditions.
Since the states upstream and downstream of a normal shock wave are represented by the
supersonic and subsonic intersections of actual andor imagined Fanno and Rayleigh lines, we
should be able to use equations developed earlier for Fanno and Rayleigh flows to quantify normal
shock flow. For example, for the Rayleigh line of Fig. 11.26b
p y
p x
a p y
p a
b a p a
p x
b
(11.137)