Change of Total Pressure across a Shock

There is no heat added to or taken away from the flow as it traverses a shock wave; that is, the flow process across the shock wave is adiabatic. Therefore, the total temperature remains the same ahead of and behind the wave:

T02 = T01.

Now, it is important to note that Equation (9.152), valid for a perfect gas, is a special case of the more general result that the total enthalpy is constant across a normal shock, as given by Equation (9.135). For a stationary normal shock, the total enthalpy is always constant across the wave which, for calorically or thermally perfect gases, translates into a constant total temperature across the shock. However, for a chemically reacting gas, the total temperature is not constant across the shock. Also, if the shock wave is not stationary (that is, for a moving shock), neither the total enthalpy nor the total temperature are constant across the shock wave.

For an adiabatic process of a perfect gas, we have:

p01

S02 — S01 = R ln — .

p02

In the above equation, all the quantities are expressed as stagnation quantities. It is seen from the equation that the entropy varies only when there are losses in pressure. It is independent of velocity and hence there is nothing like stagnation entropy. Therefore, the entropy difference between states 1 and 2 is expressed, without any reference to the velocity level, as:

s2 – s = R ln —. (9.153)

P02

The exact expression for the ratio of total pressure may be obtained from Equations (9.153) and (9.151) as:

Equation (9.154) is an important and useful equation, since it connects the stagnation pressures on either side of a normal shock to flow Mach number ahead of the shock. Also, we can see the usefulness of Equation (9.154) from the application aspect. When a pitot probe is placed in a supersonic flow facing the flow, there would be a detached shock standing ahead of probe nose and, therefore, the probe measures the total pressure behind that detached shock. However, the portion of the shock ahead of a pitot probe mouth can be approximated as a normal shock. Thus, what a pitot probe facing a supersonic flow measures is the total pressure P02 behind a normal shock. Knowing the stagnation pressure ahead of the shock, which is the pressure in the reservoir, for isentropic flow up to the shock, we can determine the flow Mach number ahead of the shock with Equation (9.154).