Chapter 14 Flow round bodies moving at supersonic speed
Introduction
As I was writing chapter 13 I thought that I would include the pitôt-static tube when used to measure speeds of supersonic flow. When I tried to describe it I found that I needed to take a much closer look at what happens around bodies like bullets and aeroplanes. It proved to be rewarding.
I think that the only thing that is generally well known about objects moving at supersonic speed is that shock waves are produced and that somehow these make sonic booms. Such shock waves will be propagated from the body and the shape of these waves will depend to some extent on the shape of the object. The size and intensity of the waves will be greater for larger objects and one would not expect a bullet to produce the same wave pattern as a supersonic aeroplane. We require an explanation of these waves but not to the depth required by someone working in this field.
The investigation of high speed flow round solid bodies would be a lot easier if we had some means of visualising the flow like those used for low speed flows. Occasionally aeroplanes produce interesting clouds of water vapour and a good photograph of this will be viewed over and over again rather like the Tacoma Narrows Bridge but the fact is that these odd photographs do not tell us very much at all. We have to bring supersonic flow indoors into wind tunnels and find ways of “seeing” the flow and of investigating systematically.
There are snags. Small supersonic tunnels often do not run continuously and have obvious limitations. On the other hand a supersonic tunnel having a cross section of, say, 5 square metres running at Mach 2 will require powers measured in megawatts and can only be supplied with power at night. It is an expensive facility. There will be extensive instrumentation to measure speeds, pressures, temperatures etc. and means to observe and photograph the density patterns produced round models in the tunnel. None of the laboratory work is easy to carry out and, if experimental data is to be interpreted correctly, the levels of uncertainty in test data makes it very important to have good mental models of the physics and of the observable behaviour of air moving at very high speed.
It seems that there is no new physics for this flow and we shall need to use the physics that has been established for nozzles and shock waves. I want to try to create a mental model of supersonic flow and that will mean combining this physics with an understanding of pictures taken using schlieren equipment.
So I must start with the schlieren apparatus.
The schlieren apparatus
Anyone wanting to study supersonic flow will want is to be able to “see” how air behaves in some relevant application. The schlieren apparatus has been developed for this purpose and utilises the fact that light passing through air will be refracted at any place where there is a density gradient and if there is uniform density there will be no refraction. This effect can be seen in a bath that has been filled with water when light shines on the bath. I used to have access to a schlieren apparatus and a supersonic tunnel and I always used a candle flame as a way of producing regions of fluctuating density in order to get the equipment set up properly. The candle flame does not produce large density differences but the density gradients show up very well. Supersonic flow is characterised by its shock waves and compression waves in which there are very large density gradients and by its expansion waves where the density changes but much more gradually.
It
is 30 years since I used this apparatus and no doubt there have been many
refinements in those years. I will give the principle of operation of the one
that I used because there are fundamental consequences of the way in which it
operates.
The schlieren equipment is not portable like say a camera. It is obviously capable of “seeing” the shock waves made by a bullet that passes through its main beam but its main application is its use with a supersonic tunnel. The tunnel must have optically flat windows of optical quality glass and somehow the models have to be mounted between the two windows. That, in itself, is a technical problem and, in the tunnel that I used the windows had matching slots cut through them and the models were made with lugs to go into these slots. Mounting models was not a job for the ham-fisted.
I
will describe the schlieren equipment by starting at the lamp and ending at the
video camera. The lamp should be a point source of light but we make do with a quartz
halogen bulb or the like. The light from the bulb is focussed by a lens at a
point where there are two knife-edges that come together to form a slit through
which some of the light passes. In figure 14-2 I have drawn the arrangement of
the fitting used to give this adjustable slit. There is a ring mounted on a
base that will go on a adjustable stand of some sort. A plate with a bevelled
edge to give a very sharp, well-defined edge is fixed in this ring with its
edge vertical. The fitting has a similar moving plate mounted in the ring so
that can be adjusted with a screw and a thumb-nut to give a narrow slit to let
light through. Usually the light source and the slit are made into one unit so
that it can be moved without upsetting the adjustment of the two components.
Now the light source and the slit must be moved so that the slit is at the focus of a concave mirror that is silvered on its concave surface and adjustable for direction and azimuth. This produces a beam of light with parallel rays that can be directed to pass through the schlieren windows as shown in figure 14-1
As
this beam passes through the air that is flowing at supersonic speed over the
model, density gradients divert rays of light that pass through them. Now the
beam, with its deflected rays, falls on the second mirror that focuses the beam
on to a second fitting shown in fig 14-3 with one moveable plate. See the
arrangement in figure 14-1. This plate is carefully adjusted so that it
intercepts only those rays that have been diverted towards it and prevents them
from going on to the video camera for display on the VDU. This produces black
lines on the screen. However some of the light that has been diverted in the
opposite direction goes on to appear on the screen and create regions of
brighter light than the background level. Furthermore the fact that the slit is
vertical makes the detection of areas of changing densities biased in one
direction. I have copied a schlieren picture from Prandtl as figure 14-4 and it
is evident that the apparatus has correctly found the waves that cross and are
compression waves and expansion waves but has failed to recognise the upper and
lower edges of the jet to be of the same character.
So the schlieren equipment produces lighter and darker area on the screen and often for the same type of wave. This means that interpreting schlieren pictures can only be done properly in the context of the application and that the interpretation improves with experience.
Then
there is the problem of definition. Because the schlieren equipment is most
frequently used with wind tunnels and the flow is normally two-dimensional I do
not think that anyone would expect the various waves to be exactly
two-dimensional so the rays passing through the flow will become blurred. When
one thinks of the fact the shock waves are very thin it becomes evident that we
shall never get very accurate photographs.
We have all seen schlieren pictures of bullets in flight. Ignoring the problems of high-speed photography we must now be looking at three-dimensional flow. This means that the shock waves are cone shaped and we are looking through those cones. If the rays of the schlieren equipment are horizontal we shall be looking at the upper and lower tangents to the cones and inevitably there will be shading inside the cones. See figure 14-5.
The definition of the photos is further dependent on the width of the slit and the narrower the slit the better the definition. This means that when the best definition is obtained the light level is very low. This can be overcome with digital cameras etc. but ultimately the accuracy of the knife-edges will set yet more limits.
Experimentalists have been ingenious and skilful at showing flow patterns as distinct from wave patterns in the low velocity flow of air and water round and through various objects. They have used smoke, dye, solid particles and any method that they can think of. I do not think that any comparable methods have been produced for the supersonic flow of gases. The schlieren equipment shows us the density gradients but not the flow lines and we must count ourselves fortunate to have it to help us.
In the wider context of science that means that we are heavily dependent on a difficult technique for all our visual material and we must expect to have to make a greater input from mental modelling and mathematical modelling than we needed to for, say, open channels. In doing so we have to take notice of the fact that we may be being misled by the effects of having the air moving round the body and not the body moving through the still air. [1]
The flow round a bullet
I found that I needed to look at a real schlieren picture and not just a diagram with someone else’s interpretation of it. I needed an example of a body moving supersonically through still air. The obvious first object is a bullet because there are plenty of pictures available.
Figure
14-6 that comes from Prandtl’s book “Fluid Dynamics” published in 1952 and it
is still my main reference book. It is clear that there is a bow wave just like
that for a boat and it looks like a shock wave. We shall see that it is indeed
a shock wave in the immediate vicinity of the bullet but it decays to become an
acoustic wave. This bow wave and the rest of the whole pattern moves through the
otherwise still air at the same speed as the bullet.
It seems to me that we need all
the help that we can get to interpret this picture. Let me draw your attention
to the photograph in figure 14-7. It is a picture of a model boat[2].
It weighed about 10 kg or 22 pounds and the electrical input to the motor that
drove the airscrew was about 70 watts. So this heavy boat was driven by less
than 50 watts. The wave pattern is obvious and, if another photograph had been
taken 10 seconds later it would have looked to be exactly the same. The wave
pattern moves at the same speed as the boat. The wave pattern is being
generated continuously. A look at the flow pattern in the top right hand corner
shows that the waves are dying away but they are a long way from the point at
which they were generated and they will go much further yet. It is clear that
there is a bow wave generated at the bow and a much smaller wave generated at
the stern.[3]
If you suddenly stop the boat the stern wave hits the stern and the bow wave
carries on spreading with a wave in the form of a circular arc spreading from
the stem and joining the two 
now-detached
waves. If you stand on the bank and just look at the bow wave as it approaches
it is very obviously travelling sideways at right angles to its length. If it
strikes a vertical wall it will be reflected with almost no change in magnitude
only in direction. There can be no doubt that the flow round a boat bears a
marked similarity to the flow pattern that is so evident in figure 14-6 but it
is never wise to read too much into such comparisons because the wave pattern
created by the bullet is three-dimensional as I have shown in figure 14-8. That
makes a big difference because the energy in the wave is spread out more
quickly with distance.
Figure 14-8 also tells us that this ever-growing wave is what happens in the air when a bullet moves through it at high speed and by that I mean in excess of where is the absolute temperature of the still air. We have to account for this behaviour. I have spent weeks on this single chapter and I have come to the conclusion that, if I am to offer an explanation of this pattern of flow, I must start with the generation of the wave.
Figure 14-9 shows a bullet travelling along the barrel of a gun after it has been fired. The bullet acts as a piston and is accelerated by the expanding gases behind it. It must cause the air in the open end of the barrel to accelerate and, if the bullet moves at a speed in excess of where is the absolute temperature of the still air, a shock wave will form in the barrel and pressure will rise behind the wave. That shock wave travels faster than and cannot come to rest in the barrel. Instead the shock wave bursts out of the end of the barrel with a pressure of 1 bar in front of it and high pressure behind it. When it leaves the barrel the air will spread under the influence of this high pressure to form a shock wave and the high pressure will immediately collapse. In this process, air from the barrel will be diverted in all directions and air behind the shock wave will mix with it. The mass of air involved is small compared with that produced by the explosive charge so it will never be the dominant feature of the flow. I have shown the wave in two positions A and B and it is evident from the rapid increase in space for the flow that the wave will quickly die away. The shock wave will continue forwards at decreasing speed and spread like the shock wave from an explosion to create a hemispherical acoustic wave in front of it and possibly a hemispherical expansion wave behind it to complete a sphere. The wave will die away quickly as it expands but, at the same time, the bullet will emerge from the barrel to create its own cone shaped shock wave. It is this wave that we can see in figure 14-6. However there is still the gas that resulted from the burning of the explosive charge following the bullet along the barrel. It will be released suddenly when the tail of the bullet clears the end of the barrel. This is a much more powerful burst than that produced by the air ahead of the bullet and it will create another much stronger hemispherical shock moving forwards and possibly a hemispherical expansion wave moving backwards. The main spherical shock wave will slow to the speed of and the fast moving bullet will set up its well-known pattern ahead of it.
You
can see all of this in figure 14-10 although it is the spherical wave from the
explosion that we can see. This picture comes from the internet but there is no
information that might let us interpret it properly. The fact that the two
halves of the spherical wave are different in that one is black and one white
may be an artefact of the equipment. However it may be that the black one is an
expansion wave created by the cloud of gas moving quickly from right to left
and the white one the shock wave. If this is the case the expansion wave will
die away but the shock wave will decay into an acoustic wave that appears to
travel radially from a point near to the end of the barrel.
I think that the most important thing about this photograph is that the wave is spherical and I will come back to this later.
Figure 14-11 is also of a gun being fired but this gun appeared to release some of the gas from the charge at the stock end of the barrel. The result is two spherical waves one almost entirely inside the other. The small one is just moving through the large one. They do not appear to combine.
The above is just a collection of assertions on my part and they must be justified. I think that this is possible using the physics of adiabatic, one-dimensional flow and of the plane shock wave. I want to use the schlieren picture of a shock wave in figure 14-6 as my example.
Returning
to figure 14-6, that comes from Prandtl’s book “Fluid Dynamics” published in
1952 and still my main reference book, it is clear that there is a bow wave
just like that for a boat and it looks like a shock wave. We shall see that it
is only a shock wave in the immediate vicinity of the bullet and that it decays
to become an acoustic wave. This bow wave and the rest of the whole pattern
moves through the otherwise still air at the same speed as the bullet just like
the surface waves in figure 14-7. The picture in 14-6 gives us no idea how
thick the shock wave may be but it must be remembered that the rays of light,
close to, but inside the shock wave, are passing twice through regions of
intense density gradient and will be deflected as well. There are other
elements to figure 14-6 but I will look at those later in this text.
The dominant feature is the bow shock wave and its acoustic wave and I will start with this. To this end I have drawn this bow wave in several positions representing equal increments of time from right to left. This gives the figure 14-12.
Clearly the point of generation of the shock wave must move forward with the bullet so that the whole wave is moving through the formerly still air. It is inconceivable that the passage of the wave will leave all the air moving at high speed and the wave must move through the air creating a disturbance as it passes but leaving no significant net effect. If the power involved in creating the surface waves in figure 14-7 is so small it is unlikely that there is much energy in the wave from the bullet.
There is an observation to be made about the shape of this shock wave. In order to draw this diagram I enlarged the schlieren photograph and cut a draughting template from 0.8 mm ply. It was clear that the shape is not the cross-section of a cone, although, outside the cylindrical region that I have separated with dotted lines, the shape does appear to be conical. This suggests that the shock wave is generated at the very front of the bullet and that it’s shape is the result of the shock wave increasing in diameter at a rate that is not the same everywhere but that ultimately becomes steady as it moves outwards whilst moving forwards at a steady rate when it has decayed to an acoustic wave. The transient phase takes place within the central cylinder of space indicated by the dotted lines and appears to be complete in a radial distance of about 2.5 times the diameter of the bullet. The creation of the shock wave and its decay to an acoustic wave is all over quite quickly. I think that it is important to recognise that in reality the air outside my dotted lines is undisturbed except for the passage of the acoustic wave and that, whatever real disturbance the bullet makes, is inside the dotted lines. It looks as though very little, if any, air actually moves at a speed that is equal to or greater than where is the local absolute temperature of the air.
I am now in a position to consider how one might try to use the physics we have to describe what goes on in a shock wave of this sort.
The only real information that we have is that a plane shock wave is possible in a convergent-divergent nozzle and that, by definition, in that shock wave, the speed drops from supersonic to subsonic, the pressure and temperature both rise and, not least, energy is lost. The wave created by this bullet looks like a shock wave of the same character as a shock wave in a nozzle. If it is, we can use the physics of the plane shock wave to deal with this new shock wave. I think that the first step is to note that the shock wave in a nozzle appears to be plane and to be moving upstream through the fluid in the nozzle at the speed of the flow and that equality makes it stationary. If we represented the wave by a straight line, the wave must be moving in a direction at right angles to that line. Now, for the bullet, we have a wave that, once beyond the cylinder represented by my dotted lines in figure 14-12 moves everywhere normally to the surface of a cone. It follows that everywhere, including the curved part, either a shock wave or an acoustic wave moves at right angles to the surface by which it is represented. This leads me to re-draft figure 14-12 to become figure 14-13.
It seemed to me that the first thing to do was to see how this pattern was actually brought into existence. This meant including the end of the barrel from which the bullet was fired. I have shown that things will not be as simple as I have shown them here but there can be no doubt that the shock wave and the acoustic wave into which it decays, will be produced by the bullet pushing air out of its path. I already had my pattern of wave profiles from figure 14-12 and, if the wave is to flow everywhere at right angles to itself, the path through space of any point on the wave must be curved. I interpolated four extra profiles in black after my point on the diagram and drew a path that crossed each of these at right angles. It gives the shape of the line in red for moving to its instantaneous position and then onwards in a straight line. I then drew the rest of the net in red. The profiles are drawn at intervals of ¼ of the length of the bullet so this whole pattern as I have drawn it is created in the very short time that it takes for the bullet to travel 3.25 times its own length. The pattern will go on growing as the bullet continues its flight.
Figure 14-13 is for the bullet moving through air that is stationary and the bullet must give the air that is in the immediate vicinity of its nose a velocity, that is, a speed and a direction. If the direction is always at right angles to the wave itself then I can locate vectors in position and direction but not in magnitude. If I now draw another wave profile and let the vectors terminate at this new profile this will give the magnitudes of the speeds to some scale. The two give velocities of the wave that are drawn in green. The speed of the acoustic wave is where is the absolute temperature of the air. The greatest speed of the wave is at the nose and is the same as that of the bullet. This means that a right-angled triangle can be drawn between and the speed of the bullet as shown in figure 14-13 and the sine of the Mach angle equals the ratio :- [4]. We could draw a similar diagram for any point on the shock wave up to the transition to an acoustic wave.
All this is about the behaviour of the wave and not about the behaviour of the air. The two seem to be almost independent but this cannot be the case so I must now look again to try to decide what actually happens to the air and this means that figure 14-13 must be adapted to represent a bullet that is stationary with the air moving over it because that is the way that the physics of shock wave has been deduced and it is not easy to see an alternative.
If we have to start thinking of the bullet and the wave being stationary and the air flowing past it, our starting point must be that what appears to be one wave is really two consecutive waves, the shock wave and the acoustic wave. The speed at which the acoustic wave travels at right angles to itself is always equal to which, in effect, defines the wave. The speed at which the rest of the wave travels at right angles to the tangent to itself, is everywhere greater than and therefore supersonic and the wave is a shock wave and, as such, can be treated using the physics that we already have for a shock wave. This shock wave is, as we have seen, in the immediate vicinity of the bullet. It will affect the air as it travels through it and our physics will tell us how.
First
we must have a way to transfer the physics that was deduced for a plane shock
wave in a nozzle to a curved shock wave in a flow of air. I think that we must
accept the idea that the wave moves at right angles to the tangent to itself.
In figure 14-14 the tangent to the wave is the reference line as shown. The
flow of air relative to the, now fixed, wave moves with the velocity of the
bullet and is shown as vector .
That vector has a component normal to the tangent, that is, vector .
This must be the speed at which the shock wave is advancing through the moving
air so that the shock wave appears to be fixed in space. We can now use the
relationships that were derived for a plane shock wave in a nozzle to deduce
the speed, density, temperature and pressure after the shock wave and in this
vector diagram it gives a value for that is inevitably smaller than and subsonic. Now we are trying to find out
what happens to the air as it flows through the shock wave and we need another
piece of information. Vector has another component that is tangential to the wave and it is
common to suppose that any flow of energy along the wave during the
establishment of the flow pattern has ceased and that this component will not change
as the air flows through the shock wave. I think that the fact that the waves
are spherical in figures 14-10 and 14-11 can only come about if there is some
shuffling of energy along the wave to make the properties all over the surface
of the wave uniform. So component reappears as and can be combined with to give .
It follows that in passing through the shock wave the flow is diverted and of course
there will be changes in the properties of the air.
I have seen several pictures of the bow shock waves round bullets. They are not all the same shape and the important determinant is the profile of the nose of the bullet. Some bullets have flat noses although I do not know why. Then the shock wave has a blunt “nose” as well and is clearly well in front of the nose of the bullet. The shape of the wave is the consequence of the need to make space for air at subsonic speeds (relative to the local absolute temperature) to flow inside the shock wave and round the bullet. There must be some flow pattern in the vicinity of the nose that can be represented by a set of flow lines just as we do for any other flow. Those flow lines will start at the shock wave.
If one diagram can be drawn to show the flow through the shock wave so can others for several angles for tangents around the “nose” of the shock wave.
I set up a calculating program Appendix 14-1 in Mathcad. In this x is the Mach number and is the angle of the wave. It simply calculates the velocities etc., needed to draw the diagrams in figure 14-15.
I drew diagrams for 90° to 30° by 10° steps. They are in figure 14-15.
The diagrams in figure 14-15 are for a notionally stationary bullet with air flowing past it at Mach 2. This speed is an entirely arbitrary choice and the diagrams can easily be drawn for some other approach velocity. In the diagrams the black arrows are the flow lines for the air before and after the shock wave. The red arrows are the velocities before and after the shock wave. The colour-coding matches figure 14-14. When =90° the shock wave is normal to the approach flow and the red and black arrows would be coincident.
Now let me interpret the physics that these diagrams represent. The important diagram is the eighth one and I must explain how it was obtained. The first 7 diagrams are all just re-drafts of figure 14-14. When I looked at them I found that I really wanted to know the absolute velocity, that is, relative to Earth, of the air after the shock wave. This can be found simply by subtracting the absolute velocity of the bullet vectorially from the exit velocity relative to the bullet. With this in mind I added the vectors representing the velocity of the bullet in blue to each diagram. All of them formed triangles with the normal component to the shock wave of the absolute velocity of the air at approach and the remaining side is the absolute velocity of the air and it is normal to the wave. I transferred these velocities at the appropriate angle to the seventh diagram and added an arc having a radius equal to the value of where is the temperature of the undisturbed air. This shows that, according to the physics that was used, the only air that actually moves at a speed in excess of is between the shock wave and the bullet 90° and just under 70°. All the rest moves subsonically. The air in this region between 90° and just under 70° is moving slower than where is the local absolute temperature so nowhere does the air move at supersonic speed except I suppose in the boundary layer. The highest speed attained by the air is about 430 m/s.
When the angle of the wave is 30° the air is only affected by the passage of the shock wave and that is transient, otherwise it is stationary. The wave is just an ordinary acoustic wave.
The application of the physics also shows that, for the angle of 90°, the velocity drops from 686 m/s to 257 m/s and the pressure rises to 4.5 bar as the air flows through the shock wave. This causes me a problem. Others talk of these bow shock waves as being attached or detached. By implication this must mean that they can conceive a process by which the shock wave actually occurs at the nose or that the nose pokes through the shock wave. The schlieren equipment in not sensitive enough to show a very thin gap between the shock wave and the nose when the nose is pointed but it can if the nose is blunt. I am going to proceed as if the shock wave is always detached however small the gap may be. So between the shock wave and the nose of the bullet there is high pressure and it will get still higher when the air comes to rest relative to the nose of the bullet.
So let me try to extract more information from this result. It is the diagram for 30° that is my first interest. It is saying that there can be a shock wave in which there is no change in velocity, no change in pressure and no change in direction as the flow passes through the wave. It seems to me that, strictly, this can only occur at infinity and then that angle of the wave is 30° but in practice it occurs only a few bullet diameters out. (The angle 30° has no special significance. It would have been different for some other value of Mach number.) I think that this means that the shock wave created by this bullet would fit into a right cone with apex angle 60° and be asymptotic to the cone at infinity. This gives a better interpretation of figure 14-7.
In figure 14-16 I have drawn the shock wave and the profile of the bullet for an approach flow at Mach 2. I do not know the precise position of the shock wave relative to the nose of the bullet but I take it that there must be some gap however small it may be. The diagrams in figure 14-15 show that the flow along the centre line is not diverted and, after the shock wave, is moving at a pressure of the order of 4.5 bar absolute and at a lower speed than the bullet. As the shock wave curves through 50° the pressure of the air flowing through the shock wave drops to 1.7 bar. I have drawn lines that are normal to the oblique shock wave in the region of the nose. At each intersection between these lines and the shock wave the air passing through the shock wave will be diverted upwards in my diagram. It will enter a region in which air is already flowing between the shock wave and the bullet and the shape of the shock wave will change to accommodate this flow. When it has settled down it will have a steady flow pattern something like the one that I have drawn in red.
That said it seems to me that the outcome of applying the physics of the reversible, adiabatic one-dimensional flow and the physics of the non-reversible flow through a shock wave to this bullet is quite convincing.
It seems to me that this shock wave is nothing whatever to do with sound which, after all, only exists because we have ears that are exceptionally good instruments to measure tiny, tiny changes in pressure. The shock wave has its origin in the very small region at the nose of the bullet when the air is suddenly made to divide, turn through 90°, and flow sideways in all directions. In that division an outward radial force acts continually on the air flowing towards and then away from the nose just as I drew in figure 4-4 in chapter 4. That force will give the air momentum in the radial direction and there is nothing anywhere in the whole system to generate a force to reduce this momentum. It is just spread through an ever-increasing mass of air to attenuate the acoustic wave. The pressure wave so produced is very directional and it has nothing in common with a steam whistle on a locomotive that emits pressure waves in all directions. I am not at all impressed by animated Doppler effects; they seem to me to mistake a simple geometrical construction for a piece of physics. The creation of a shock wave is certainly an event and might even be regarded as a singularity. Other than that the flow is what one might expect.
What I have not drawn is any representation of the pressure distribution. The physics suggests a very large pressure gradient in the region just round the nose but it is also a region in which there is a sharp change in direction in every flow line and therefore quite complex.
There will be a boundary layer in the flow in which there will be large velocity gradients and these will create a skin drag that must be added to a net force produced by the over pressure on the nose. A really pointed nose will reduce this latter force and the intensity of the shock wave.
Whilst I have been writing this chapter I have looked at several schlieren pictures of bullets. Without exception they show too little of the whole pattern to be really useful. Perhaps this is caused by the constraints of schlieren equipment but in my experience most pictures of real flow patterns are cropped much too close to the object involved. It is as if the person cropping the pictures believes that there is no information in the outer parts of the pattern. This is never the case and for shock patterns made by bullets we cannot deduce the speed of the bullet unless the outer region is shown. Often we do not have enough even to know the shape of the shock wave in the vicinity of the bullet, only the shape at the nose. I think that the shape of the wave varies greatly with the Mach number but especially for values in the range 1 to say 1.25. Then shape also varies with the shape of the nose being quite different for pointed bullets and flat nosed bullets. Any real projectile will have a nose that is blunt when compared with the size of molecules of gas. A sewing needle fired point first at supersonic speed would produce a shock wave but only a very small one. By comparison some bullets have flattened noses and these produce a much larger shock wave than a pointed bullet and one where the schlieren equipment will show a gap between the shock wave and the nose of the bullet. One wonders what size shock wave the space shuttle produces on its return to Earth from space.
It seems to me that this application of physics to the schlieren picture in figure 14-6 gives a better insight to the phenomenon of the shock wave than the picture by itself. It is a foundation for looking at other objects moving at supersonic speeds through still air and the physics will apply to two dimensional patterns as well. I do not think that I can take it any further. However there are other features to the photograph of the bullet that need interpretation.
We are left with the other features on the schlieren photograph of the bullet. I have repeated it in figure 14-17 with the photo skewed to roughly match the photo of a model boat in figure 14-18. From the physics the bow wave appears to be a shock wave that, in a very short distance decays to an acoustic wave. The schlieren equipment does not distinguish between them because thay are both density gradients. Nowhere, except in the boundary layer, is any air moving at a speed relative to Earth greater than where is the local absolute temperature. There will be no more shock waves. However there are other waves in evidence.
I am wary of making comparisons between two different systems. The bullet is submerged in air, the boat is afloat on a surface of separation. However the comparison gives an opportunity to see the same mechanism at work in two different applications. The boat is flat-bottomed with vertical sides and the plan view is made up of two arcs of circles. It is driven by an airscrew. Clearly the boat is leaving an eddying wake behind. It comes from the thick eddying boundary layer that forms around the submerged part of the hull. It is not caused by an under water propeller.
The
bow wave is evident and the surface is fairly flat aft of amidships. Over the
rear part of the hull the water on the two sides of the hull is moving inwards
and will inevitably collide. When this happens the two lots of lateral momentum
are destroyed in the creation of a force that can only divert the flow upwards
as in figure 14-19 to form a wave in which the energy in the two converging
flows reappear as potential energy. That wave spreads out as can be seen in
figure 14-18. It is too much like figure 14-17 to be coincidence.
The bullet is a solid of revolution with a square base. The flow over it may be everywhere subsonic but it is still at high speed. There can be no question of the flow curling round the circular rear corner to converge to the centre. It breaks away and the pressure on the base of the bullet drops to a very small value and so creates a pressure gradient that acts radially inwards. It produces a wave of expansion that widens as it progresses towards the centre. When the wave reaches the centre it is quite wide axially and it produces a rise in pressure over a significant length. This creates a compression wave that becomes flat fronted and behaves like an acoustic wave. In the cone of that wave there is an eddying wake.
The
acoustic wave from the nose produces a sonic boom that actually sounds like a
whip-crack and the compression wave produces a second whip-crack.
There are two other faint expansion waves but these are generated by a change in the radius of curvature of the bullet and die away quite quickly. The two oblique acoustic waves may persist over long distances.
I think that I have extracted as much as I can from Prandtl’s schlieren picture.
The expansion wave.
Shock waves are the consequence of compression and there are obvious points at which this can occur. Expansion or rarefaction waves are the result of expansion and they crop up wherever the flow is subject to a transverse pressure gradient. They can occur when jets emerge from nozzles, where there are discontinuities in a surface and where there are changes in the radius of curvature of a surface. For an engineer they do not present the same problem as a shock wave because a shock wave must taken into account at the design stage and the expansion wave will be dealt with in the development stage.
I
think that one must be careful with is expansion wave when trying to find some
simple system on which an analysis might be based. Prandtl and Meyer imagined a
frictionless flow at supersonic speed along a plane solid boundary. There is no
real system that corresponds with this, it is highly idealised. So the physics
that is used for the expansion wave is for a very special notional case. It is
the only thing we have to give the basic behaviour of an expansion wave and we
must adapt it very carefully as required by circumstance.
Prandtl and Meyer considered the case of a jet emerging from a nozzle at a pressure that exceeded the pressure surrounding the jet. I have drawn this case in figure 14-21. The flow up to the exit is steady at and at pressure and it emerges into surroundings at pressure that is lower by a small increment. Clearly the air flowing in contact with the solid wall of the nozzle will suddenly be subjected to this small pressure difference to make it accelerate sideways. It will start to move sideways immediately and in doing so expose the next layer of the flow to the sideways force and the effect of passing the edge will be propagated away from the edge at . The effect will be the appearance of a line at an angle where and is the Mach number of the flow. This line is not a shock wave. It marks the small decrease in density that could be detected by schlieren equipment. At this line the pressure drops and the velocity increases.
Prandtl and Meyer found that the drop in pressure at this line could not be finite but only infinitesimally small and that it is necessary to think of the net result of the emergence of the jet into the open surroundings as taking place in an infinite number of small steps. In figure 14-21 I have drawn the result of many infinitesimal steps. The point to be made is that in each step the direction of every flow line changes by a small angle and the velocity increases. This means that each small step turns the line across the flow by a small angle and ultimately by the finite angle shown. When the pressure falls to the turning will cease and the flow will continue steadily at the new angle which will equal .
This can often be seen on Schlieren photographs.
The wing in supersonic flight
We are accustomed to the idea of fighter aeroplanes being capable of supersonic flight. The only civil supersonic aeroplane was the Anglo-French Concorde. It lasted 30 years and was not replaced. There is an obvious advantage in flying at high speed and Concorde cruised at about 2,000 km/hr or 1300 mph. Nevertheless modern airliners fly much more slowly at about 1,000 km/hr or 650 mph or about Mach 0×85. There are good reasons for this.
There is the social problem of the sonic boom but that may be more in the mind than the fact.[5] Even so Concorde was not allowed to fly supersonically over land. There are real engineering and logistic problems. The main engineering problem is that the supersonic aeroplane must be able to fly at subsonic speeds in order to take off and land. Unfortunately the lift force generated by the wing is in a very different place for subsonic and supersonic flight. Some military aeroplanes, the Tomcat for example, have swing-wings and others have dodges to overcome this problem but Concorde moved its fuel about to move its centre of gravity to re-balance the aeroplane. When Concorde flew at low speed the angle of attack of the wings was very high and the engines were used to overcome the very large drag. Still the aeroplane took off and landed at high speed. Concorde also needed preferential landing slots to avoid excessive wastage of fuel in a landing “stack” when compared with a subsonic aeroplane.
It is evident from the foregoing that the shock waves and the attendant sonic booms that would be created by a large supersonic aeroplane would be a serious impediment to its acceptability. The shock waves present a serious problem to designers. This means that any design must aim to minimise the number and intensity of these waves. There are two unavoidable but very loud waves, one created by the nose, however sharp that may be, and the other at the tail. There will be other waves emanating from the engine intakes etc.. They all become more intense as the size of the aeroplane increases. Concorde carried about 100 passengers but a new supersonic civil aeroplane would have to carry many more in order to compete with modern subsonic aeroplanes carrying 500 passengers. It may well be that the supersonic civil aeroplane would be too wide, too heavy and too legally restricted to be viable.
Military aeroplanes are small and have their own landing facilities. They have their own design constraints not least that they are always built in an atmosphere of haste generated by political competition between countries.
Un-swept wings
The only application for un-swept wings in supersonic flight is in military hardware such as missiles that have no take-off and landing requirements. The wings and other flight surfaces are normally very small and used only once.
The basic sections that are used are the double wedge shown in figure 14-14 and the double arc.
I have drawn the wave pattern for a double wedge in figure 14-22 for about Mach 2. There is a shock wave off the leading edge and a compression wave from the trailing edge and expansion waves off the apexes. The bow shock wave and the trailing edge compression waves are generated in different ways as we have seen with the bullet. They are now not conical like that for a bullet but shaped like a folded piece of paper.
At the apex the flow breaks away but the inevitable pressure gradient that forms just after the apex bends the flow progressively in the expansion wave. Then the compression wave at the trailing edge forms in the collision between two flows along the after part of the wing.
I have drawn this double wedge wing at zero angle of attack but, if the wing is to lift it will have to be set at some small angle to the flight path and that will make the flow pattern asymmetrical.
It is not at all a suitable shape for subsonic lift, nor are the acoustic waves desirable for either civil or military use. Thin wings, that might be seen as derivations from subsonic wing sections, have been designed for “ordinary” supersonic flight.
Swept wings
Swept wings originated in designs for subsonic aeroplanes and were first used practically on the Boeing B 52 bomber. The sweep back was about 35°. It is now clear that, for optimum performance of a wing, there is a direct link between the speed of the aeroplane and the sweep back. The sweep back increases to about 60° at Mach 2 and more as the speed increases again. But there seem to be some limit on the further increase in sweep back that may be the result of the need to mount munitions externally on the wings. It is as if Mach 2 is a high enough speed for combat aeroplanes.
The
use of sweep back eliminates the formation of a shock wave at the leading edge
of the wing and then the wing of a supersonic aeroplane becomes just another
wing in the range of wings for speeds from say Mach 0.5 to Mach 2.
We must see how this comes about. In figure 14-23 I have drawn a part of the leading edge of a wing having an angle of sweep back of and moving at a speed of . This speed can have components along the wing, in green, and at right angles to the leading edge in red. We know that a shock wave can form at the leading edge if the, red, normal component of the velocity is greater than . It follows that if the angle of sweep back is chosen so that the normal component to the leading edge of the speed of the aeroplane is less than there will be no shock wave formed at the leading edge.
This means that the graph 14-1 of versus M based on can be drawn for the condition when the normal component = and this graph will give the minimum value of for any value of M if an oblique shock wave is to be avoided. It more or less matches the angles used on existing aeroplanes.
This means that wing sections can have rounded leading edges and be developed for a compromise between supersonic and subsonic performance.
The trans-sonic region of flow
It is hard now to comprehend the excitement that accompanied the attempts to fly at speeds greater than Mach 1. It all happened soon after World War 2 when everyone was looking for pointers towards a new life free from the misery of war. It generated intense interest.
We have seen that the physics for supersonic flow had been established long before it was needed for flight but that same physics did not cover the region where shock waves just begin to form. I suppose that the attainment of supersonic flight might have been easier had the engines been more powerful but it was the fact of engine development that took us gradually and not suddenly into the supersonic era. Of course rockets were available but the popular interest was in aeroplanes with jet engines that they might be able to see at air shows. Supersonic flight was achieved relatively quickly but at the expense of a few deaths.
The
problem was that if you try to fly with subsonic technology just into the
transition from subsonic to supersonic flow you are inevitably faced with
unpredictable behaviour. We can only use our imagination when we try to think
about this transition phase. I have included a picture 14-24 of water flowing
at its critical depth on the sloping apron of a weir. I think that no-one would
believe this unless it is actually seen and recorded. How can the water create
these little pillars? The fact that it is creating these little pillars all
over the apron means that there is some predictability in its behaviour. It
seems to me to be reasonable to expect some unimaginable behaviour in the air
flowing past the many parts of an aeroplane attempting to fly in this
transition region and also to expect some repeatability.
The real problem turned out to be with the elevators and ailerons. These were made like those for a subsonic aeroplane with inset control surfaces and operation of these controls led to rapid and unpredictable response by the aeroplane. Redesign of these components to use all moving stabilisers made supersonic fight possible but the real answer was a big increase in thrust from the engines and to go through the trans-sonic region quickly.
Study of trans-sonic flow is difficult and outside my competence so I will stop.
The pitôt-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.
End piece
I come to the end of this chapter thinking that it is not really satisfactory. I have committed myself to writing a textbook and I cannot omit this chapter on supersonic flow. I had expected the physics to just point me in the right direction but it does not. So let me spell out what I am concerned about.
The first is that I do not know whether the behaviour of a flow of air at supersonic speed round a stationary model in a supersonic tunnel is the same as the behaviour of air when a body moves at the same supersonic speed through still air. The flow in a supersonic wind tunnel takes place in a convergent-divergent nozzle and the working section is a part of the divergence. If a model is installed in the working section there will inevitably be a shock wave formed at the forward end whatever shape it may be. Suppose that the nose is fairly blunt. The shock wave will span the whole section produce the same result as a plane shock wave in an unobstructed nozzle and the whole of the model will be in subsonic flow and no other shock wave can form subsequently. This does not preclude the formation of compression and expansion waves in high-speed, subsonic flow. Now suppose that the nose is pointed. The shock wave that would form would be like the wave round the bullet and may well decay into an acoustic wave before it reaches the tunnel walls. In the immediate vicinity of the model the flow through the shock wave at right angles to the wave will change to be subsonic but this will not happen in the acoustic wave. I think that this leaves the flow in an interesting condition that may or may not lead to the shock wave extending right across the working section and reducing all the flow to subsonic flow. This would not happen in the case of the same object moving through still air.
There is always the possibility that I have misinterpreted the Schlieren pictures and that there is no acoustic wave but a shock wave extending through the still air but the mechanics of that idea seems to be quite untenable as it would involve so much energy.
I have a second problem. Imagine a supersonic aeroplane flying at its design speed. A shock wave will form at its nose whatever that may be. Suppose that it is a pitôt tube and the shock wave will be small compared to the shock wave that would form on the nose of the fuselage in the absence of the pitôt tube. Do we get two shock waves if there is a pitôt tube? Certainly, just behind the nose of the pitôt tube, the flow will be less than where is the local temperature but it will soon decay to an acoustic wave. Then a second shock wave could form on the nose of the fuselage and this will be of a much larger size. What I cannot decide is whether this second wave affects the engine intake ducts. Do they working subsonic or supersonic flow?
To a reader this end piece may seem to be unusual but I cannot find any way to resolve these problems but I can alter this text easily in the event of authoritative information.
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
