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.