# What is the relation between airspeed and altitude at fixed throttle setting in a private jet?

In the world of private jets, say a G-450, at a fixed throttle setting, does it simply go faster the higher it climbs/flies?

The basic principles needed for an answer are:

• Thrust varies linearly with air density.
• Lift at constant angle of attack (AoA) equally varies linearly with air density.
• Increasing the angle of attack from a low value will improve the lift to drag ratio (L/D), while increasing it from a higher value than the AoA at optimum L/D will decrease it.
• The aircraft will settle at the point where speed equals drag and lift equals weight. If the altitude change improves L/D, the aircraft has excess thrust to fly faster.
• Due to the atmospheric lapse rate, the speed of sound decreases with altitude.
• Above a certain Mach number ($M_{dd}$ = drag divergence Mach number), drag increases nonlinearly over dynamic pressure.

Now the answer depends on where you start:

1. Low altitude, moderate throttle setting: The aircraft will speed up, because it was flying at a low AoA in dense air. Climbing brings it closer to the polar point of optimum L/D.
2. Low altitude, high throttle setting: If the aircraft flew at close to its $M_{dd}$, climbing will put it in colder air where it needs to fly slower to maintain the same Mach number. Depending on the starting speed, the aircraft might slow down.
3. High altitude, moderate throttle setting: Now the thrust decrease with altitude will dominate the equation, because the aircraft will fly close to its optimum polar point. Once it flies above the AoA of minimum drag, climbing will slow it down.
4. High altitude, high throttle setting: If it flies above the tropopause and close to the polar point of minimum drag, the air temperature does not change with altitude, and climbing will not change flight speed. It will stay close to its $M_{dd}$. Once it flies above the polar point of minimum drag, however, climbing will again slow it down.
• Probably pedantic, but isn't drag increasing non linearly over speed, even below the divergence Mach number? Is the quadratic relation between drag & speed becoming super-quadratic at the drag divergence Mach number? Basically aren't we moving from a drag coefficient that's speed independent to a drag coefficient that increases rapidly with speed? What's the functional form of this dependence? – curious_cat Jul 27 '15 at 6:55
• @curious_cat: Yes, I should better have used dynamic pressure. Corrected. The drag increase is hard to put into a formula; I have seen approximations using the seventh power over speed, but in the end the behavior is only to capture completely by tabulation over Mach. Hint: This would be worth a question of its own. – Peter Kämpf Jul 27 '15 at 8:12
• Hint taken! :) First I will do some reading to ask a better question. – curious_cat Jul 27 '15 at 8:50

Simple answer: within normal operating altitudes, mostly yes, due to the thinner air causing less drag while the engines are producing approximately the same amount of thrust

Other simple answer: it depends, and there will be an optimal altitude above which the aircraft travels slower due to the engines losing thrust.

There are a lot of aspects that complicate it, but assuming you're basically asking 'at full throttle, does an aircraft travel faster at higher altitude?' the simple answer is yes, up to a point. Obviously that doesn't apply above the service ceiling, and the optimal altitude may be lower than this.

The optimal altitude will be a function of the engine performance and airframe.

As altitude increases, the primary effect on aircraft is the reduced density of the air it flies through. A reduction in air density has three key effects on flight:

• Power at a constant setting decreases, because the air-breathing engines we use have less air to use for combustion and to push against.
• Lift also decreases for the same reason; the aerodynamic lift surfaces (wings) are pushing less air and therefore the forces pushing the wings (and the rest of the aircraft) upward are reduced at a given angle of attack.
• Drag is decreased, because less air means less friction drag and weaker wake vortices. This is usually offset somewhat by the need to increase angle of attack, which increases induced drag.

So, as the air thins, your engines produce less power, but you need less power because your drag is reduced, but that's offset by the need to pitch up to regain lost lift. The net results are an overall decrease in aircraft performance (ability to climb/turn and max speed), but (to a point) increased fuel efficiency.

One last effect is a reduction in temperature. By the ideal gas laws, cooler air is denser than warm air at a constant pressure, however the equation is dominated by the reduction in pressure.