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Hi All,

I have a theoretical question on the Specific Range (=range per unit weight of fuel) vs. Mach number....

At constant altitude and constant weight, the curve is like a dome where the max SR is at Maximum Range Mach number (MMR)..... then the higher the speed the lower the SR and below this MMR, the lower the speed the lower the SR too....

How get we get theoretically to the dome curve, can anybody give me the steps from the basic aerodynamics equations?

All the best,

Paco

I have a theoretical question on the Specific Range (=range per unit weight of fuel) vs. Mach number....

At constant altitude and constant weight, the curve is like a dome where the max SR is at Maximum Range Mach number (MMR)..... then the higher the speed the lower the SR and below this MMR, the lower the speed the lower the SR too....

How get we get theoretically to the dome curve, can anybody give me the steps from the basic aerodynamics equations?

All the best,

Paco

Here's my theory:

At MMR for a specific weight, the airplane flys at an aerodynamicly-seen optimal angle with as less drag as possible.

If you increase speed, the drag increases too, so your speed benefit is NOT proportional to fuelburn. In fact, lets say a 5% bump in speed does not go along with 5% more fuelburn but more like something around 10% more fuel burn. You're basically travelling in less time but hence your fuel burn is higher, your range decreases.

If you decrease speed, the AoA changes too. That means, you actuallly "put more surface in the wind" causing more drag. However, your decrease of speed (lets say 5%) is again not proportional with your decrease of fuelburn which is only 2% for example due to the higher drag caused by a not optimal angle of the airplane.

But again, that's only my theory, I have not studied aerodynamics.

At MMR for a specific weight, the airplane flys at an aerodynamicly-seen optimal angle with as less drag as possible.

If you increase speed, the drag increases too, so your speed benefit is NOT proportional to fuelburn. In fact, lets say a 5% bump in speed does not go along with 5% more fuelburn but more like something around 10% more fuel burn. You're basically travelling in less time but hence your fuel burn is higher, your range decreases.

If you decrease speed, the AoA changes too. That means, you actuallly "put more surface in the wind" causing more drag. However, your decrease of speed (lets say 5%) is again not proportional with your decrease of fuelburn which is only 2% for example due to the higher drag caused by a not optimal angle of the airplane.

But again, that's only my theory, I have not studied aerodynamics.

quit a.net 07/2016

Quoting Klemmi85 (Reply 1):
But again, that's only my theory, I have not studied aerodynamics. |

That's a good description. What you're describing is the combination of the drag polar and the range equation.

Quoting Paco1980 (Thread starter):How get we get theoretically to the dome curve, can anybody give me the steps from the basic aerodynamics equations? |

Start with the Brueget range equation. Assume SFC, TOW, and LW don't change. Then, for various speeds (Mach numbers), get L/D from your drag polar, and then plug that into the range equation. The "dome" in the drag polar causes the dome in the range, skewed a little bit to the left (more range) by speed.

Tom.

- phollingsworth
**Posts:**759**Joined:**

Quoting Paco1980 (Thread starter):Hi All,
I have a theoretical question on the Specific Range (=range per unit weight of fuel) vs. Mach number.... At constant altitude and constant weight, the curve is like a dome where the max SR is at Maximum Range Mach number (MMR)..... then the higher the speed the lower the SR and below this MMR, the lower the speed the lower the SR too.... How get we get theoretically to the dome curve, can anybody give me the steps from the basic aerodynamics equations? |

The most simple answer is that it is a function of the effects of transonic flow over the aircraft wing and body, i.e. drag divergence.

In incompressible flow (and for the most part isentropic compressible flow), i.e. M is effectively 0, there is no maximum range Mach number. If we take the idea jet engine performance, the TSFC does not change appreciably with speed and the maximum L/D remains constant. Then the faster you go the better specific range you have. For a given aircraft configuration and weight there is a specific dynamic pressure (and as such and Indicated Airspeed) that maximizes specific range. Since the true airspeed is a function of indicated airspeed and altitude you would fly at a given IAS and increase your altitude until your engine thrust drops to the point that you have reached your ceiling. As you burn fuel your weight decreases and the best range IAS goes down. In this case you just climb; hence the "cruise-climb" flight profile. If we introduce compressibility, and hence the concept of Mach number, then we can start relating things with respect to Mach number. Keeping the idea of the ideal jet and the isentropic flow then the effect of compressibility on L/D is fairly small, as the effect on drag is compensated for by a proportional effect on lift. However, the total drag will increase faster that the total thrust. This will lower you cruising altitude and push down your specific range.

The problem, in the real world, is that if you get supersonic flow over the wing it very quickly becomes anisentropic (entropic). This means that you have shock waves, Mach buffeting, and other effects. This gives you a Mach number where the drag diverges. This means to fly at a faster Mach number than the Mdd your drag increases disproportionately to your lift, killing your L/D. This causes the dome. Since M=V/a, as you climb at a fixed EAS (Equivalent now that compressibility is being accounted for) the true airspeed increases, i.e. V, and your M increase. Also as you climb the speed of sound decreases until you reach the tropopause at which point it remains flat. This means that there is a point where for a given weight you don't want to fly higher as the Mach number would lead to increased drag. The other effect on this is that as you get lighter you actually want to fly at a slower IAS.

I hope this helps. If necessary I can give you some references.

Quoting Tdscanuck (Reply 2):What you're describing is the combination of the drag polar and the range equation. |

Fine, now my theory has a name

Quoting Phollingsworth (Reply 3): |

Good explanation! This one sounds way more professional than mine, may I ask what job you do?

quit a.net 07/2016

- phollingsworth
**Posts:**759**Joined:**

Quoting Tdscanuck (Reply 2):Start with the Brueget range equation |

While a great equation, especially for quick calculations, it is one of the many examples where it is named after someone other than the person who first formulated it.

Quoting Klemmi85 (Reply 4):Good explanation! This one sounds way more professional than mine, may I ask what job you do? |

University Lecturer. Though I have done other jobs in aerospace in the past.

Thanks to all of you for your answers and sorry for not having posted anything earlier.

Phollingsworth, would you know any website or any PDF file that would treat the subject into more details (including equations and graphics...) ?

Cheers,

Paco

Phollingsworth, would you know any website or any PDF file that would treat the subject into more details (including equations and graphics...) ?

Cheers,

Paco

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