fluid_mechanics

608 Chapter 11 ■ Compressible Flow
r
T
–0.5 0
x, m
+0.5
–0.5 0
x, m
+0.5
(a)
(c)
p I
Ma
1.0
p
p II
–0.5
0
x, m
+0.5
–0.5 0
x, m
+0.5
(b)
F I G U R E 11.12 (a) The variation of duct radius with axial distance. (b) The variation of
Mach number with axial distance. (c) The variation of temperature with axial distance. (d) The variation
of pressure with axial distance.
(d)
Shock waves
Photographs courtesy
of NASA.
V11.6 Supersonic
nozzle flow
flow solutions are represented in Fig. 11.12. When the pressure at x 0.5 1exit2 is greater
than or equal to p I indicated in Fig. 11.12d, an isentropic flow is possible. When the pressure
at x 0.5 is equal to or less than p II , isentropic flows in the duct are possible. However, when
the exit pressure is less than p I and greater than p III as indicated in Fig. 11.13, isentropic flows
are no longer possible in the duct. Determination of the value of p III is discussed in Example
11.19.
Some possible nonisentropic choked flows through our converging–diverging duct are
represented in Fig. 11.13. Each abrupt pressure rise shown within and at the exit of the
flow passage occurs across a very thin discontinuity in the flow called a normal shock wave.
Except for flow across the normal shock wave, the flow is isentropic. The nonisentropic flow
equations that describe the changes in fluid properties that take place across a normal shock
wave are developed in Section 11.5.3. The less abrupt pressure rise or drop that occurs after
the flow leaves the duct is nonisentropic and attributable to three-dimensional oblique shock
waves or expansion waves 1see margin photograph2. If the pressure rises downstream of the
duct exit, the flow is considered overexpanded. If the pressure drops downstream of the duct
exit, the flow is called underexpanded. Further details about over- and underexpanded flows
and oblique shock waves are beyond the scope of this text. Interested readers are referred to
p
p I
p III
F I G U R E 11.13 Shock formation in
converging – diverging duct flows.
x
p II