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[1]. 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.[2] 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 :- [3]. 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.