The behaviour of the flow in a nozzle as the back-pressure is increased

So far I have introduced the physics of the plane shock wave in the divergence without saying anything more than the simple statement that it can exist. We need to know more than that.

 

The physics for reversible, adiabatic, one-dimensional flow shows us that when there is no shock wave in the nozzle the pressure at which the fluid leaves the nozzle depends only on the inlet conditions and the dimensions of the nozzle. However the back-pressure is clearly a quantity that can change and it could be below or above the pressure at exit from the nozzle or be greater than the exit pressure. We have already seen from the worked example that a shock wave can form at the exit and that the pressure at which the fluid leaves the nozzle and could be 3.14 bar with the flow in the divergence still wholly supersonic and falling to 0.87 bar just before the shock wave. We need an explanation for this behaviour.

 

Graph13-8 is not really suitable for the next stage of this text so I have redrawn it as graph 13-16 with the divergence truncated so that the pressure at exit would be 1 bar absolute. This is just an arbitrary decision taken to give me increments in pressure that can be seen and can be labelled. Notice how short the divergence is for an exit Mach number of more than 2.

 

I have to start with the green line A-B-C in the diagram. It is the pressure distribution for typical subsonic flow throughout the nozzle. A represents the inlet pressure, C is the back-pressure at exit and B is the lowest pressure, that will always occur in the throat. We have seen this pressure distribution before for the Venturi meter. If the back-pressure is raised, the pressure at B will increase but, if the backpressure is lowered, the pressure at B will fall until the conditions at the throat are such that the velocity at the throat is equal to  where  is the absolute temperature in the throat and the pressure distribution will be the black line A-T-D. So D represents the lowest value that the backpressure can have and still be consistent with subsonic flow through the nozzle. It is called the upper critical back-pressure.

 

In fact it is much more likely that when the velocity at the throat goes to  the expansion through the divergence will be supersonic and the pressure distribution will follow the line T-E and, because the nozzle has been truncated to produce these conditions, the exit pressure will be 1 bar and equal to the backpressure. The pressure at E is called the lower critical backpressure.

 

It is what happens in between E and D that is now of interest because it is obvious that the back-pressure can have any value between 0 and that at D and still have a speed of  in the throat. We have already seen that there can be a plane shock wave in the divergence of a convergent-divergent nozzle. Now I can add lines to scale on graph 13-14 to give graph 13-17 and to show the pressure distribution when a shock wave is present.

A typical line is A-T-J-K-L. A-T-J is the normal expansion. J-K is the rise in pressure in a plane shock wave and K-L is a subsonic expansion to the exit where the pressure just reaches the back pressure.

 

I have drawn a second line A-T-F-G-H and shown the special case of a shock wave at exit as A-T-E-P.

 

There is a line T-K-G-P, shown dotted, that is the locus of the points representing the pressure just after the shock wave. Points K-G and P can be calculated using the model for a plane shock wave and the lines K-L and G-H using the reversible, adiabatic, one-dimensional model.