Page 144 - 48Fundamentals of Compressible Fluid Mechanics
P. 144
106 CHAPTER 6. NORMAL SHOCK IN VARIABLE DUCT AREAS
flow is different from what was discussed before. In this case, no continuous pres-
continuous pressure
B
a discontinuous point (a shock)
will occur. As conclusion, once the flow becomes supersonic, only exact geometry
sure possibility can exists. Only in one point where
can achieve continuous pressure flow.
exist. If the back pressure,
B is smaller than
In the literature, some refers to a nozzle with area ratio such point b is
above the back pressure and it is referred to an under–expanded nozzle. In the
under–expanded case, the nozzle doesn’t provide the maximum thrust possible.
On the other hand, when the nozzle exit area is too large a shock will occur and
other phenomenon such plume will separate from the wall inside the nozzle. This
nozzle is called an over-expanded nozzle. In comparison of nozzle performance
for rocket and aviation, is that over-expanded nozzle is worse than in the under-
expanded nozzle because the nozzle’s large exit area results in extra drag.
The location of the shock is determined by geometry to achieve the right
B , is lower than the critical value
(the only value that achieve continuous pressure) a shock occurs outside of the
than the exact
back pressure. Obviously if the back pressure,
location determined in a such location that after the shock the subsonic branch will
matches the back pressure.
nozzle if needed. If the back pressure is within the range of
to
First example is pressed
for academic reasons. It has
troat
to be recognized that the shock 8)9 :<; =?>@A BDC$E.F
wave isn’t easily visible (see
for Mach’s photography tech-
exit
niques). Therefore, this ex- point "e"
ample provides an demonstra-
tion of the calculations for re- x y
quired location even it isn’t real- "!$#&% ')(&*,+.- /1023 4"5$6&7
istic. Nevertheless, this exam-
ple provide the fundamentals to Fig. 6.2: A nozzle with normal shock
explain the usage of the tools
(equations and tables) that were developed so far.
Example 6.1:
A large tank with compressed air is attached into a converging–diverging nozzle at
] and the
]. The shock occurs in a location where the cross section area
. Nozzle throat area is 3[
]. Calculate the back pressure and the temperature of the flow (It should
exit area is 9[
be noted that the temperature of the surrounding is irrelevant in this case.) Also
is 6[
determine the critical points for the back pressure (point “a” and point “b”.
and temperature of
pressure
SOLUTION
Since the key word “large tank” was used that means that the stagnation tempera-
ture and pressure are known and equal to the conditions in the tank.