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In the book of Corke, Chapter 7 about engine selection, he speaks about the Mach factor which is a factor larger than 1 in supersonic to take into account the ram effect.

In his case study he takes 2,39 for a cruise speed of Mach 2.1.

He mentions that there exists algorithms and books that allow to compute this Mach factor.

I am looking for a figure showing this Mach factor in function of the Mach number. I have been looking like crazy.

If anyone has sources or this type of chart, it would be great.

Thanks

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  • $\begingroup$ Ram pressure raises the pressure level in the engine, increasing mass flow and thrust. See this answer for more, and this one discusses the pressure increase of Concorde, which flew close to Mach 2.1. However, that 2.39 factor I cannot explain. $\endgroup$
    – Peter Kämpf
    May 22, 2018 at 23:15
  • $\begingroup$ @PainAndGain. Hi, just curious, if you were looking like crazy for an explanation, did our answer help? Cheers. $\endgroup$
    – Penguin
    May 29, 2018 at 11:43
  • $\begingroup$ Hi @Penguin, thank you for your answer. Actually your answer, although very complete, confused me a little. How would you define the Mach Factor mentioned in Corke ? Also, I am looking for figures of existing data or algorithms. I understand the physics behind, it's existing data that is really hard to find. I also couldn't access figure 10.35. Thank you ! $\endgroup$
    – PainAndGain
    May 30, 2018 at 19:03
  • $\begingroup$ @PainAndGain. Hi ... to be clear, I can't see what Corke says in his book, all I know of its contents is what you have said above. I had thought his "Mach factor" was the ram pressure rise due to forward motion of the vehicle. But now reading the statement in you question ("factor of increased thrust due to the ram effect"), I now think Corke is not talking about the ram effect, but the effect of the ram effect on thrust... So, my answer below, which is only about the existence of ram effect, not how it changes thrust, is not really hitting the relevant aspect... Cheers. $\endgroup$
    – Penguin
    Jun 15, 2018 at 11:19

2 Answers 2

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It sounds like what you are describing are the isentropic compressible flow equations. See equation 6 & 7.

At Mach 2.1, these give a stagnation pressure rise of 9.14, and a stagnation temperature rise of 1.88, since air has a value of 1.4 for gamma.

But, the air does not come to complete rest (stagnation) in the intake, hence these values are not obtained.

What you need to know, is the actual Mach number at the engine inlet, which is I assume where you are suggesting is the location of a total pressure 2.39 times higher than the freestream total pressure. Typically, this is around Mach 0.4. But, since the freestream flow is supersonic, that means it will have had to go through some shock waves, to get to a subsonic speed. This means there will be pressure losses, associated with those shock waves. Normal shock waves (a shock wave perpendicular to the flow) result in higher total pressure losses than oblique shock waves. Hence, supersonic aircraft have pointed cones in their intakes to create oblique shock waves, to minimise the pressure losses. To calculate the total pressure losses from these, you need to know the geometry of these cones to know if they cause 1, 2, or maybe even 3 oblique shock waves, as the pressure loss of each shock depends on the strength of each shock wave.

Jack Mattingly, in his text book "Aircraft Engine Design" gives a good description of how to calculate the losses associated with oblique intake shock waves. It's fairly complex. See here. The figure you want is 10.35, but that wasn't shown in the free preview when I tried it, but maybe the pages they hide are random, and you will have better luck.

So, in summary, to verify the 2.39 factor, you need to know the Mach number at the engine fan inlet, and be able to calculate the total pressure losses of the shock waves, which requires info of the intake geometry.

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The answer is that the ram factor represents the increase in mass flow to the engine. The increase in mass flow due to the compressible Mach number is

m_dot ~ P/sqrt(T) where P and T are the total pressures and temperatures. These can be found from isentropic relations.

(P_t/P_s)= (1 + ((gamma-1)/2)M^2)^(gamma/(gamma-1))

and

(T_t/T_s)= (1 + ((gamma-1)/2)M^2)

Therefore m_dot ~ (1 + ((gamma-1)/2)M^2)^(gamma/(gamma-1))/(1 + ((gamma-1)/2)M^2)^0.5

At Mach 2.0, m_dot ~ 5.8 which is the ram effect multiplied by the thrust at altitude.

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