The pitot-static tube for high speed flow

In chapter 5 I used the pitôt tube as an example of the use of the energy equation. The treatment was restricted to the case of measuring the velocity of a flowing fluid when density changes are small enough to ignore or zero. If a pitôt tube is to be used for high speed flows including supersonic flow the effects of density changes must be taken into account. This gives an opportunity to look at the physics for a single device that could be used over the whole range of flow and there are few such devices.

 

Both the pitôt-static tube and pitôt tubes with separate static pressure connections to holes in the outer surface of the aeroplane are used on aeroplanes although the extraordinary advances in avionics means that it is more likely that a GPS will be used these days for non-aerobatic aeroplanes. Our physics will only let us estimate the rise in pressure with speed for the facing tube and the actual pressure rise will depend on the profile of the facing tube. That profile will be found by experiment as will the method used to detect the free air pressure. So we need to find the effect of the variation in density  with speed  on the pressure at the facing tube.

 

It makes sense to start with the pitôt-static tube when used for subsonic speeds.

 

The pitôt tube in subsonic flow

Figure 14-25 shows a facing tube that is actually moving at velocity  through still air but has been shown stationary with the air moving at  for the purposes of analysis. The still air at point 1 has properties . The point 1 is on a flow line that is aligned with the centre line of the pitôt tube and the air flowing through 1 will come to rest at 2 in the plane of the inlet to the tube. Then its properties will be  where suffix s indicates the stagnation condition.

 

If we modelled the flow as adiabatic and reversible it is possible to derive an expression for the rise in pressure  and this can be used with a coefficient to produce a calibration expression. It is based on the expression for the rise in pressure produced when an incompressible fluid moving at velocity  is brought to rest. That expression is that the rise in pressure .

 

Then we can write that , where  is a factor to be found from analysis and, that the actual rise  where  is a coefficient to be found by experiment.

 

In order to find an expression for  all we need is an expression for  and it turns out to be better to work in terms of Mach number so that a graph of  can be plotted.

 

We have  and from this . We can change to Mach number notation .

We know that  so

                                                

 

This can be manipulated to give

Using  gives . We also have .

Then  and this gives the result

 

We actually want  and then the factor  is given by :- . However  and then:-

                                  

 

This can be plotted using Mathcad as in graph 14-2. It becomes clear that the effect of the variation of density with speed is small, less than 1% up to Mach 0.2, which is 80 m/s for a temperature of 20°C, and only 4% up to Mach 0.4.

 

It seems to me that the value of the coefficient C is dependent on the design of the nose of the pitôt tube and on the arrangements of the static tapping. Inspection of expensive high-performance gliders that can fly at 70 m/s leads me to think that these arrangements are not critical.

 

It is interesting that this factor is a function of  which means, of course, that it is really a function of . The use of the non-dimension group has again reduced the burden of data storage and no doubt values for the experimentally determined coefficient would be stored in the same way.

 

The pitôt tube in supersonic flow.

If the pitôt tube is to be used on a supersonic aeroplane it will have to be fitted so that the flow at its tip is not affected by the flow over the aeroplane. Early supersonic fighters had the pitôt tube on a long stalk either at the nose or attached to a wing but they are not so fitted on modern fighter aeroplanes. The pitôt tube in supersonic flow must be designed for a shock wave to form in front of its tip and when Rayleigh analysed this he chose to treat the shock wave as a plane shock wave in which the speed relative to the nose dropped as it passed through the shock wave and then treated the air in the small space between the wave and the nose as coming to rest with a further rise in pressure. As the hole in the pitôt tube cannot be too small if it is not to be clogged with insect debris and as the shock wave must be far enough in front of the nose to permit the compression the wall of the tube must be relatively thick. I do not know its profile but it has been shown as a “flat” conical shape like that shown in figure 14-26.

 

There is still the need to detect the value of  and we have seen that the pressure falls quite quickly with distance from the nose of the pitôt tube. If the long probe used originally is to incorporate the static tapping to make it a pitôt-static tube it would have to be much further away from the nose than would be the case for subsonic flow. Pitôt-static tubes were very long.

 

In figure 14-26 I have shown a flow line starting at 1, where conditions are  where these are the steady flow conditions, and reaching the shock wave at 2 where conditions change to . Then the flow crosses the gap between the shock wave and the tube and comes to rest at the stagnation conditions . We need an expression for  as before but now we have to find the expression from two steps.

 

Step 1 to 2. Across the shock wave continuity gives :-  and momentum gives:-  and as we have  we can write:-

                                             

 

Changing to Mach numbers:-  and

                                             

Dividing by  gives :-                and

                                                 

                                                       

This takes us through the shock wave, now we have to bring the air to rest. Then:-

                                        

We need to eliminate  in favour of  and we can use .

This can be substituted in the expression for  to give a frightening looking expression that will simplify to give :-       . This expression is attributed to Lord Rayleigh (1842-1919).

 

If you choose to continue with the rational expression where  we get for supersonic flow 

 

The graph of  can be plotted using Mathcad. It is given in graph 14-3. I used the basic expressions and not Rayleigh’s expression and brought forward the graph for subsonic flow. The expressions can easily be identified in the Mathcad print out. The two traces clearly join at a value of k=1.276.

	Graph 14-3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The physics leads to a continuous relationship between  but I think that one might expect some hiccups in the transition from subsonic to supersonic flow.