fw 190a5 flight model

[Herra Tohtori]

I will create a free-body diagram of the (relevant) forces affecting a flying aircraft in a turn when I have time for it.

I'll just note a few key factors here.

1. Maximum power of the engine is irrelevant at slow speeds.

If you were familiar with definitions of work and power you would understand this; I can show you why this is so but I don't know if you would understand the mathematics (it's reasonably simple but it does involve some grasp of differential calculus). For now, suffice to say that when an aircraft travels slower, the engines do less work per unit of time, which means by definition that their power output is reduced. Aircraft engines reach their peak power output only at maximum speed of the aircraft (same actually applies to automobiles!).

2. There is a component of thrust that is directed toward the centre of the turning circle.

This can easily be defined as

Fc = F * sin α

where F is the thrust of the propeller disk, and α is the angle of attack. Let's assume that α cannot be larger than critical angle of attack; α ≈ 15°

At critical angle of attack (maximum turn performance at any speed), the thrust toward the centre of the circle would be

Fc = F * sin 15° = 0.25 F

Hence, we can say that at most, only about quarter of the total thrust of the engine is directed inward and thus assisting in the turning radius. This, however, applies to all aircraft, not just FW-190 so it doesn't really help your point... especially as we get to point three.

3. Since we now know the assisting centripetal component of the thrust force, we can determine the assisting centripetal acceleration:

a = Fc / m = 0.25 F / m

since F/m is the thrust to mass ratio of any aircraft, we can DIRECTLY say that the thrust to mass (more commonly incorrectly expressed as thrust to weight ratio) does affect the turning performance.

Moreover, this simple exercise of physics shows us that aircraft turn harder when their engine produces more thrust.


Confusingly (or rather, not) we know that Spitfires have better acceleration and climb rate than FW-190, which means Spitfires have better thrust to mass ratio.

Which means that the expectation of the theory is that Spitfire engine can assist in turns more effectively than that of FW-190... which doesn't really help your case.


4. Quantitative analysis

How, then, does this centripetal acceleration produced by the engine thrust compare to the centripetal acceleration produced by the lift of the wings?

Well, again, simple exercise. If we assume that at certain speed v, the aircraft would be able to do a 3g turn, that means the wings produce enough force to produce 3 g's worth of acceleration (they can easily produce much, much more force up to the limit of their plasticity, in which they deform permanently, but since the discussion is about low speed performance let's keep it at that flight regime).

By contrast if we look at the maximum acceleration that the engine thrust can produce, we can immediately see that the thrust is about an order of magnitude smaller force than the lift of the wings. It's difficult to actually determine the thrust of these aircraft; however we can get some results by looking at how well they climb vertically. None of the WW2 aircraft can maintain their velocity (or increase it) in vertical climb; this means that the propellers produce less force than the aircraft's weight - their thrust/weight ratio is smaller than one.

At thrust/weight ratio of one, the engine could give the aircraft exactly 1g of acceleration. Since these aircraft get nowhere near that, let's be generous and assume the acceleration at standing start could be.. let's say 0.5 g's (it is probably less than this, but oh well...).

Now we can determine the centripetal acceleration by thrust:

ac = 0.25 a = 0.25 * 0.5 g = 0.125 g


What does this mean? Well, if a gliding aircraft at speed v can pull a 3g turn, with full power it could pull about 3.125 g turn (increasing it's turn rate and decreasing turn radius).

This applies to all powered aircraft, and the defining factor is the aircraft's thrust to mass ratio - or, unloaded acceleration by engine thrust alone.

Multiplying this by the sine of angle of attack you can directly get the assisting centripetal acceleration.

a(engine) = 0.125 g

a(lift) = 3 g

we can see that the assisting engine thrust is, at best, about 4% of the lift.

At high g-load the ratio further decreases because you can't pull critical angle of attack at high speeds - which means that most of the thrust is directed forward.


Now, if you're looking at two different planes with different thrust/mass ratios - yes, the plane with better thrust/mass ratio will provide more assisting centripetal acceleration.

However now you need to consider that the thrust/mass ratio of these aircraft had relatively small variations. What you will find is that the overwhelmingly deciding factor in turn rate is the lift/mass ratio rather than engine thrust. You might find small differences in the assisting thrust - let's say that one aircraft's engine might assist at 4% of lift, while another aircraft's engine might assist turning at 5% of the lift... but this would already mean a quite hefty 25% thrust/mass ratio difference!


Here we have shown that the engine thrust is primarily responsible for maintaining the cornering velocity (overcoming drag), and wings are primarily responsible for actually turning the aircraft.

I don't expect Gaston to really comprehend any of this, this is more for the benefit of others.

I'll make that free body diagram as soon as I can... now I must get going to school.

Toodles!