fluid_mechanics

11.5 Nonisentropic Flow of an Ideal Gas 631
E XAMPLE 11.17
Stagnation Pressure Drop across a Normal Shock
GIVEN Designers involved with fluid mechanics work hard at
minimizing loss of available energy in their designs. Adiabatic,
frictionless flows involve no loss in available energy. Entropy
remains constant for these idealized flows. Adiabatic flows with
friction involve available energy loss and entropy increase. Generally,
larger entropy increases imply larger losses.
FIND For normal shocks, show that the stagnation pressure
drop 1and thus loss2 is larger for higher Mach numbers.
SOLUTION
We assume that air 1k 1.42 behaves as a typical gas and use Fig.
D.4 to respond to the above-stated requirements. Since
1 p 0,y
p 0,x
p 0,x p 0,y
p 0,x
we can construct the following table with values of
from Fig. D.4.
p 0,yp 0,x
COMMENT When the Mach number of the flow entering the
shock is low, say Ma x 1.2, the flow across the shock is nearly
isentropic and the loss in stagnation pressure is small. However,
as shown in Fig. E11.17, at larger Mach numbers, the entropy
change across the normal shock rises dramatically and the stagna-
1
tion pressure drop across the shock is appreciable. If a shock occurs
at Ma x 2.5, only about 50% of the upstream stagnation
pressure is recovered.
In devices where supersonic flows occur, for example, highperformance
aircraft engine inlet ducts and high-speed wind tunnels,
designers attempt to prevent shock formation, or if shocks
must occur, they design the flow path so that shocks are positioned
where they are weak 1small Mach number2.
Of interest also is the static pressure rise that occurs across a
normal shock. These static pressure ratios, p yp x , obtained from
Fig. D.4 are shown in the table for a few Mach numbers. For a developing
boundary layer, any pressure rise in the flow direction is
considered as an adverse pressure gradient that can possibly cause
flow separation 1see Section 9.2.62. Thus, shock–boundary layer
interactions are of great concern to designers of high-speed flow
devices.
0.8
p 0,x – p 0,y _________
p 0,x
0.6
0.4
0.2
0
0 1 2 3 4 5 6
F I G U R E E11.17
Ma x
p 0,x
p 0,x p 0,y
Ma x p 0,yp 0,x
Ma x p yp x
1.0 1.0 0 1.0 1.0
1.2 0.99 0.01 1.2 1.5
1.5 0.93 0.07 1.5 2.5
2.0 0.72 0.28 2.0 4.5
2.5 0.50 0.50 3.0 10
3.0 0.33 0.67 4.0 18
3.5 0.21 0.79 5.0 29
4.0 0.14 0.86
5.0 0.06 0.94
E XAMPLE 11.18
Supersonic Flow Pitot Tube
GIVEN A total pressure probe is inserted into a supersonic air
flow. A shock wave forms just upstream of the impact hole and
head as illustrated in Fig. E11.18. The probe measures a total
pressure of 60 psia. The stagnation temperature at the probe head
is 1000 °R. The static pressure upstream of the shock is measured
with a wall tap to be 12 psia.
Wall static pressure tap
Supersonic
flow
Stagnation
pathline
x
y
Shock
wave
FIND
Determine the Mach number and velocity of the flow.
Total
pressure probe
F I G U R E E11.18