The plane shock wave

It is a matter of observation that such a shock wave can occur in the divergent cone of a nozzle and, in a very short distance, supersonic flow is reduced to subsonic flow. Figure 13-12 shows the main features of the plane shock wave.

 

We have seen the parallel to this in the Venturi flume where, in a hydraulic jump, the flow changes from rapid flow to tranquil flow. The jump would have started as rolling wave (like a bore) that would have formed downstream and moved upstream against the flow until its speed is equal to that of the flow and that must occur in the divergence. The speed of the bore is greater than the critical speed for the flow.

 

In the nozzle we have a wave that has come to rest in a supersonic flow and its speed relative to the flow must also be supersonic.

 

I want to look at the physics of a plane shock wave in the divergence of a convergent-divergent nozzle.

 

In figure 13-13 I have drawn a plane shock wave at some arbitrary position in the divergence that I have been using throughout. The shock wave will be about 0.00005 mm thick. That seems to be very thin to us but in every square centimetre of the shock wave there will be about  molecules. The 1.5 may not be accurate but the telling figure is the power 15 To all those molecules a shock wave is nothing special.

 

In the shock wave the properties of the gas change from  to  and the velocity changes from  that is supersonic to  that is subsonic. The pressure, the density and the temperature all increase but the velocity falls. The kinetic energy of the mass centre of the gas is depleted as it flows through the shock wave and some proportion of the energy is absorbed into the molecular structure of the gas. In the end we shall find that, for this model of a shock wave, . This result suggests an underlying order and is certainly very elegant.

 

The physics of the normal shock wave is so well established that the property relationships that come from it have been tabulated and published. This means that we need to follow the set procedure that leads to these tables to get an understanding of what has been done. So let us look at the physics of a shock wave in a nozzle by treating the flow up to the shock wave as reversible, adiabatic, one-dimensional flow, the flow through the shock wave as non-reversible, and the flow through the rest of the divergence as reversible, adiabatic, one-dimensional flow again. As the wave is very thin we regard the area of cross-section as the same on the two sides of the wave.

 

Now, as so often in these analyses, we start with momentum and equate the net force on the gas in the wave to the change in momentum/second that takes place. This gives :-

                                             

Now we must follow the set sequence to get the required outcome. First  :-

                                                                                                   (1)

Then we can use the energy equation in the form for one-dimensional flow and write :-.

                     where  is the stagnation temperature at inlet to the nozzle.

However  and  and therefore  or . It then follows that :-                                                                         (3¢)

At this point we have a choice between proceeding in ordinary properties, changing to Mach number or to do both. I will do both in turn and start with properties.

So proceeding  and hence . Then :-

                                                                                    (2¢)    

                                                                                           (2)

Now we have to give some thought to the way in which we would like the result to turn out. As always we want the outcome to involve only physical constants and measurable quantities. We cannot measure velocity and certainly not just before and just after a shock wave nor can we measure temperature, pressure and density with any confidence. This means that we really want our answer to give ways to calculate these quantities from the stagnation conditions at entry to the nozzle. So let us first get rid of the velocities in favour of  and to find  using reversible adiabatic, one-dimensional flow to the shock wave.

 

Equation (1) gives :-  and multiplying by  gives:-

                             

                  

                          

                                                              from (2)

Then :-

                    

The next step is to multiply by  to give :-

      

                       

and then to divide by  when :-

                                  and

                                               

 

This is one form of the Rankine-Hugoniot expression that relates the pressures and densities before and after the shock wave. If the position of the shock wave is known the properties of the flow upstream of the shock wave can be found and the Rankine-Hugoniot expression will take us through the shock wave to give new start conditions.

 

It can be rearranged quite simply to give  in terms of .

                        

                            

                                                 

 

 

For air these become  and .

Other forms of these equations can be derived but none of them eliminate the densities so we must use these equations and trial methods if necessary. In order to link the two sides of the shock wave we need another relationship. The one that we can have is between the velocities on either side of the shock wave.

 

I have already said that there is a rather elegant relationship between the velocities before and after the shock wave. Now I have to produce it.

 

Go back to equation (2¢) and it can be extended to include the throat.

 

                                                       

                                                                  

                                                                  

                                                     (4)

We have from (1) :-                        

Therefore                               

or :-                                                                                              (5)

but we have from (4) :-                            and :-

                                                               .

From these two equations :-

                                      

                                       

                                                       

 and finally                or                           

This also means that :-