1 A pitôt-static tube is used as the detecting element of an air speed indicator on an aeroplane that flies at Mach 1.5 at a height of 10,000 m where the pressure is 0.26 bar. A shock wave forms in front of the pitôt-static tube and if this wave is treated as a plane shock wave an estimate can be made of the stagnation pressure after the shock wave and therefore the pressure exerted on the facing tube. The difference between this pressure and the free stream pressure can be measured using a sensitive pressure gauge and displayed as speed. Calculate this pressure difference.
2 An explosion creates a spherical shock wave that expands radially into still air at 1 bar and a temperature of 15°C. An estimate of the speed of the wave can be made by treating the wave as a plane shock wave. Calculate the speed of this wave when the pressure just inside the wave is 14 bar.
3 (a) Show that for a compressible fluid the stagnation pressure corresponding to a pressure p and a Mach number is given by :-
(b) A tube that is open at one end and closed at the other is set up in a stream of air moving at supersonic speed. The tube is in line with the direction of flow. A detached shock wave forms in front of the tube. The Mach number immediately upstream of the wave is 2.8 and the pressure and temperature inside the tube is found to be 2.5 bar and 15°C.
Treat the shock wave as a plane shock wave and calculate the velocity of the air approaching the shock wave.
Solutions
(1)
We can use Mathcad to find property ratios for a stationary shock wave. The shock wave is advancing at the speed corresponding to Mach 1.5 and the approach flow to it is at Mach 1.5. Then the ratio of . It follows that .
This air is still moving relative to the pitôt static tube and some will be brought to rest by the tube and produce the stagnation pressure. If the compression to bring the air to rest is regarded as reversible and adiabatic we can use the ratios for a nozzle to find the stagnation pressure . First we need the value of the Mach number after the shock wave. This is 0.701 from Mathcad.
Then
and . Then
if the surrounding air pressure can be detected at the static tapping the
pressure difference between the facing tube and the static pressure is .
(2)
In order to use our physics we must regard the shock wave as stationary and the air flowing through it. Then it flows from right to left in the diagram. .Using Mathcad and by trial .
Then
as .
(3)(a) The steady flow energy equation gives:-
At the stagnation conditions where is the pressure at velocity and is the stagnation temperature. Dividing by gives :- .
We
have and then .
Using :-
(b) The temperature in the tube is higher than that of the
approaching air by the rise in the shock wave plus the rise in the compression
to bring the air to rest at the open end of the tube.
From Mathcad ;
;
.
Now we have to work back from the 25°C
in the tube. For a reversible adiabatic compression between the shock wave and
the end of the tube to give the stagnation temperature 25°C we
can use .
This gives:- .
We have so
