Why is thrust said to be constant over speed for a jet engine?

Question

I'm just interested in some basic facts regarding flight. On reading the rather "low level" book "Understanding Flight, David F. Anderson, Scott Eberhardt), I recently came up to the following diagram:

Why is thrust approximately constant with speed for a jet engine? Everything gets mixed up in my head as follows:

Thrust is approximately $F_t=\dot m \cdot \left( v_{gas}-v_a\right) $, where the speeds are outlet speed on the nozzle and true airspeed, respectively.

https://en.wikipedia.org/wiki/Turbojet#Net_thrust

So there are really a lot of parameters. What of them are assumed to kept constant in the right diagram and what are changing? Without that information, the diagram seems meaningless for me. For my naïve opinion, a change of thrust is controlled by changing the thrust lever and thereby changing the fuel content within the combustion chamber. Lets say the lever is at its max or a fixed position (whatever this means). Then while increasing speed the factor $v_{gas}-v_a$ gets smaller, assuming the outlet speed is constant.

So to have still the same thrust there are three possibilities':

• outlet speed increases
• mass stream increases
• both of them increase

I can imagine that mass stream increases "somehow" with airspeed (keeping fuel injection constant), so there might be some cancellation of the speed factor. .But is this the whole story? Why is it canceling completely?

On writing this question another question arises:

Considering an ideal engine without losses, total power is

$P_{tot} = \frac{1}{2} \dot m \cdot (v_{gas}^2-v_a^2) = \frac{1}{2} \dot m \cdot (v_{gas}-v_a)(v_{gas}+v_a) = \frac{F_t}{2}\cdot (v_{gas}+v_a)$

Lets say, the power lever is on a fixed position. Is this position an indicator for the total power of the engine? In other words: is total power to the engine constant, when the lever is on a fixed position? I ask, because when assuming constant thrust this formula would mean, that total power (as well as propulsive power) must also increase, because at higher TAS of the plane, gas outlet speed would (up to my poor understanding) rather increase instead of decrease. So what happens, when the thrust lever is on a fixed position? Could it be, that there is a governor behind, taking control of thrust? In this case the right picture has no meaning at all, because while increasing TAS the governor automatically injects more fuel.

There are so many questions and in fact no understanding of anything...

Answer 1

There are several effects which in combination make constant thrust a good approximation at subsonic speed.

Thrust is created by accelerating a working mass in opposite direction. Net thrust is the difference between the impulse of the air flowing towards the engine and the combined impulse of burnt fuel and the air exiting the engine (and propeller, if one is fitted), derived after the time. Since that impulse is the product of mass and speed, you can either accelerate a large mass by a small speed difference, like a propeller does, or a small mass by a large speed difference, like a turbojet does.

When flying faster, the entry impulse of a propeller quickly grows large relative to the exit impulse, so thrust goes down with the inverse of speed. On the other hand, the high exit speed of a turbojet results only in a small increase of the entry impulse relative to the exit impulse while speed increases.

But if that were all, even the thrust of a turbojet engine would drop when speed increases. But there is a second effect which helps to let thrust grow with speed. With the square of speed, to be precise. That is the ram effect which helps to precompress the air entering the engine. At subsonic speed, this just about compensates for the loss of thrust: At low speed, the growing entry impulse lets thrust drop a bit but at higher subsonic speed the ram effect becomes larger and raises thrust again, such that a constant thrust becomes a good approximation. However, at supersonic speed the ram effect becomes dominant and thrust grows with speed squared – until the absolute internal pressure becomes too high so the engine must be throttled (or the aircraft needs to fly higher) or the shock losses in the intake become too large and thrust drops again.