Page 304 - Aerodynamics for Engineering Students

286 Aerodynamics for Engineering Students

I

PIP0 0.6 ,---D
0.4
0.2

Fig. 6.4

Equally it is necessary for the convergent-divergent tube of Fig. 6.1 to be choked
before the divergent portion will maintain supersonic conditions. If this condition is
not realized, the flow will accelerate to a maximum value in the throat that is less
than the local sonic speed, and then decelerate again in the divergent portion,
accompanied by a pressure recovery. This condition can be schematically shown by
the curves A in Fig. 6.4 that are plots of p/po against tube length for increasing mass
flow magnitudes. Curves B and C result when the tube is carrying its maximum flow.
Branch B indicates the pressure recovery resulting from the flow that has just reached
sonic conditions in the throat and then has been retarded to subsonic flow again in
the divergent portion. Branch B is the limiting curve for subsonic flow in the duct and
for mass flows less than the maximum or choked value. The curve C represents the
case when the choked flow is accelerated to supersonic velocities downstream of the
throat.
Considerations dealt with so far would suggest from the sketch that pressure ratios
of a value between those of curves B and C are unattainable at a given station
downstream of the throat. This is in fact the case if isentropic flow conditions are
to be maintained. To arrive at some intermediate value D between B and C implies
that a recompression from some point on the supersonic branch C is required. This is
not compatible with isentropic flow and the equations dealt with above no longer
apply. The mechanism required is called shock recompression.

Example 6.4 A wind-tunnel has a smallest section measuring 1.25m x 1 m, and a largest
section of 4m square. The smallest is vented, so that it is at atmospheric pressure. A pressure
tapping at the largest section is connected to an inclined tube manometer, sloped at 30" to the
horizontal. The manometer reservoir is vented to the atmosphere, and the manometer liquid
has a relative density of 0.85. What will be the manometer reading when the speed at the
smallest section is (i) 80ms-' and (ii) 240ms-'? In the latter case, assume that the static
temperature in the smallest section is 0 "C, (273 K).
Denote conditions at the smallest section by suffix 2, and the largest section by sufix 1. Since
both the smallest section and the reservoir are vented to the same pressure, the reservoir may
be regarded as being connected directly to the smallest section.
Area of smallest section A2 = 1.25mZ
Area of largest section A1 = 16 mz