Velocity ratio and pressure ratio across a shock wave in terms of Mach number

We already have as equation (3¢):-

                                           .

It can be applied to the conditions at the throat and to any other section but it is the condition just before the shock wave that we need and is section 1  :-

                              

When this is divided by  it gives:-

                                             which gives :-

 

                                             and :-

                                                              

Now we can use the relationship  and note that

Therefore                  

 

And finally                                         

 

This is quite an important equation because it can now be used with Rankine-Hugoniot to relate pressure ratio over a shock wave with

 

First use continuity applied between the two sides of the shock wave.

 and as  is unchanged :- . This can be used to eliminate the inconvenient density terms in favour of .

             becomes . This can actually be simplified!

the last term cancels out  and if we divide through by  we get :-

                         

                         

                         

                         

                                     

 

This is a remarkable outcome from such an unlikely looking expression.

 

We need a relationship between the temperatures before and after the shock wave. It is inevitable that a the energy exchanges in a shock wave are adiabatic and then the expression that can be applied through the shock wave is the energy equation in the form:-  where  and  are the absolute temperatures just before and just after the shock and  is the stagnation temperature. It is interesting that the stagnation temperature is the same before and after the shock wave but this follows from the fact that, even though there is depletion of kinetic energy in the shock wave, it reappears in the internal energy of the flowing gas..

 

We can write  and then :-

                              

                            

 and then

Substituting we get  which is a considerable simplification.

 

This ratio can also be expressed in terms of

                               

But, using   

                    

                                            

 

These are all the ratios that are required to calculate the properties before and after a shock wave. It is the value of  that determines all the other properties before and after the shock wave. The value of  is of course dependent on  and  and, for a nozzle, these can be determined from reversible, adiabatic flow up to the shock wave.

 

At this point we need a worked example to show how one might use all these ratios. A suitable example follows in blue.

 

The initial calculations on the design of a convergent-divergent nozzle are to be made using the reversible, adiabatic one-dimensional model. Air is to be supplied to the nozzle at 5 bar and 40°C, the throat diameter is to be 6 mm and the Mach number at exit is to be 1.8.

 

Calculate using Mathcad or tables :-

(i)                the diameter of the nozzle at exit,

(ii)              the mass flow of air,

(iii)            the pressure at exit,

(iv)             the back pressure at which a plane shock wave could occur at exit.

 

Faced with this example and all the convoluted expressions that require lots of calculation one must think of ways to reduce the labour involved and reduce the opportunity for calculating error at the same time. There are tables in which all these calculations have been done and published. They were created around 1970 and are copyright. Since that time pc’s have evolved and these can reduce the work.

 

I am sure that there are people who can find a more elegant way to use Mathcad but all that engineers want is an answer. Basically this calculating programme finds the ratios between properties for any value of the Mach number in a nozzle for any value of . Mathcad will not accept a / in the designation of a variable so I have used z instead to mean “divided by”. The upper group of ratios is for a reversible, adiabatic, one-dimensional model of an expansion. The lower group is for a plane shock wave. I have put  and let  to give the values of the ratios at the throat but when  is given some new value all the ratios are recalculated instantly..

 

 

Now we can attempt the worked example.

(i) Change to 1.8 in Mathcad and this gives the area ration between throat and exit as 1.439. As areas are proportional to diameter squared .

(ii)  The programme above gives the ratios for the throat.

   Therefore  and

 Therefore .

(iii) At exit  =1.8 and  Therefore

(iv) This question is pointing out that for the flow in the divergence to be free from shock waves the back pressure must be lower that that at which a shock wave forms at the exit. If a shock wave were to be at the exit the value of  before the shock wave would be 1.8 and the ratio of pressures through the shock wave will be 3.613. So :-

                                         The back-pressure

 

The specimen example shows that Mathcad can reduce the burden of calculation but, in examinations, it may be desirable to use tables. For real engineering the Mathcad programme or something like it will speed the design process.