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The following article describes how planes turn, how to compare two planes and determine which has the maneuverability advantage in different situations and a little about how to turn efficiently and tightly.

Lift mechanicsEdit

Lift equationEdit

The lift equation describes the upwards force generated by the wings as they experience airflow.

$L=\frac{1}{2}C_{L}S \rho v^{2}$

Where:

$L$= lift force

$C_{L}$= The lift coefficient, describes how much lift the cross section of the wing generates. It's dependent on the shape of your airfoil and the angle of attack.

$\rho$= Density of air changes with altitude but in most cases we'll assume the two planes being compared are co-altitude

$S$ = Wing surface area

$v$ = Airspeed

Centripetal motionEdit

All turning motion is the result of forward velocity and a perpendicular force.

The way a plane turns is no different, by rolling to point the lift into the direction it wants to turn (and pitching up to change the angle of attack to increase lift) planes use the lift from their wings to make a turn.

It's also the reason why trim is so important, left untrimmed a plane can generate excess lift (greater than the force of its weight) and 'turn' upwards or 'turn' downwards if there's a lack of lift and gravity is the dominating centripetal force. Trim allows the pilot to change the default elevator deflection angle so the lift and weight of the plane equalize and no vertical 'turning' occurs.

Wingloading is a basic measure of a plane’s maneuverability. It directly affects the stall speed (and therefore takeoff and landing speed) as well as the turn radius.

Since planes turn with lift which is a function of the wing’s surface area, the size of the wings are a good measure of a plane’s ‘liftiness’.Of course force isn’t everything, it’s relative to how heavy the plane is. That’s why wingloading, a ratio between how heavy your plane is and the size of your wings is such a good measure for a plane’s turning performance. The higher this number the heavier the plane is relative to its wings, the lower this number the more easily this plane can turn.

For wingloading comparisons you assume that the airfoil (if you're American, or 'aerofoil' if you're British) of the wing are similar enough, so there are of course uncertainties and wingloading is not the only factor involved in turn performance. However it is safe to assume that the wings of various planes are similar enough that a large wingloading difference will almost always equate to a large turning performance disparity. However it doesn't take into account high lift devices such as flaps (different flaps on various aircraft can affect turning in positive or negative ways) or slats (as in the case of Messerschmidts and Lavochkins). These devices could enable a heavier wingloaded plane to turn better than a low wingloaded plane in certain situations.

The wingloading of a plane is simple to calculate. You simply take the mass (kg) of a plane and divide by it’s wing surface area (m2). Here are some example values (using full fuel and MG/cannon load):

Bf109E: 140 kg/m2

Bf109G: 180 kg/m2

Fw190D: 200 kg/m2

Hurricane II: 120 kg/m2

Spitfire Vc: 115 kg/m2

Spitfire XVI: 150 kg/m2

The force in the centripetal motion equation can be replaced with the equation for lift:

$F=\frac{mv^{2}}{r}=L=\frac{1}{2}C_{L}S \rho v^{2}$

So: $r \propto \frac{v^{2}}{C_{L} v^{2}}\frac{m}{S} = \frac{1}{C_{L}}\frac{m}{S}$

Note that $\frac{m}{S}$ is the wingloading of the plane and that the velocity of the plane has cancelled itself out. The turn radius then is a function of the wingloading of the plane and the $C_{L}$ of the wings (which is a function of the airfoil, pitch of the plane and flap settings).

Velocity hasn't completely been eliminated however, as it influences what level of $C_{L}$ can be held. The higher the airspeed the harder it becomes for the pilot to pitch the plane.

All planes have a certain speed that they're best at turning in. Basically a plane needs to go slow enough to maximize the $C_{L}$ to fly a tight circle but needs to go fast enough to complete the turn in good time.

High speed performanceEdit

At the end of the day all the physics boils down to the centripetal acceleration equation. Acceleration in a turn is often called Gs and there exists a limit on how hard a plane can turn because even if a plane is tough enough to handle the stress of the turn, the pilot falling unconscious tends to hurt maneuverability somewhat.

All planes can then only turn as hard as their pilot’s G-resistance allows and at high speeds even moderate turns will result in  excessive Gs. Because of the pilot's limitations at high speeds a lighter plane has lost its maneuverability edge and it’s stuck facing an opponent with heavier armament, better protection, better top speed due to horsepower (heavy planes are heavy for a reason).

It’s fundamentally incorrect to say a heavy plane turns better/tighter than a lighter one at higher speeds. Rather the lighter wingloaded plane is unable to turn tighter than the heavy plane and brings less tools to the fight. However the lighter wingloaded plane always has the advantage of being more energy efficient.

Energy efficiencyEdit

When taking the same turn, a lighter wingloaded plane also has the advantage of being able to make that turn with a shallower angle of attack than a heavier plane. Shallower pitching means less drag and better speed retention during the turn.

Not only do high angles of attack come with larger amounts of (2d) drag from the airfoil itself, the (so-called 3d) induced drag at the wingtips further contributes to energy loss during a turn.

Induced dragEdit

When wings generate lift, at the wingtips little vortices are created, they're caused by energy being wasted at the wingtips as high and low pressure air meet (wings generate lift by speeding up air over the wing so the pressure difference creates an upward force). As it is caused by lift, this drag is called lift-induced drag or induced drag for short.

During normal flight induced drag is simply one of the forms of parasitic drag, however during turning flight induced drag plays a crucial part in slowing a plane down:

$C_{Di} = \frac{C_{L}^{2}}{\pi e AR}$

Because to turn with any sort of speed a great amount of lift is needed, a high lift coefficient (resulting from pitching to increase your angle of attack) is required. This high lift coefficient increases induced drag by the 2nd power and becomes a significant contribution to drag during a turn.

Oswalds factorEdit

The Oswalds factor 'e' is a factor to quantify the effect of the shape of the wing on induced drag. Elliptical wings as found on a Spitfire reduce the chord of the wing as you near the wingtip.

The result is that there is almost no lift generated at the wingtip, no pressure difference, so wing tip vortices are almost completely eliminated. The drawback however is that these wings are much harder to produce as the constant curving along the wing complicates manufacturing. This is why the Bf 109 (mid-late models) has straight tapered wings with a rounded wing tip as a compromise.

Aspect RatioEdit

The Aspect Ratio is a measure of how wide the wingspan is compared to the size (surface area) of the wings:

$AR=\frac{b^{2}}{S}$

Another way to reduce induced drag is to use high aspect ratio wings, which is a measure of how long a wing is relative to its area. This is why gliders have extremely long wings in an effort to be as efficient as possible.

The principle is similar to changing the shape of the wing, instead of trying to reduce the lift to zero at the wingtips, increasing the length of the wing means that a smaller portion of the wing's total lift is exposed to the wingtip to be wasted as induced drag.

One example of a plane with a very high aspect ratio is the Ta-152, which also has a high taper to also try to achieve a better Oswalds efficiency factor. An example of a plane with a low aspect ratio is the A6M 'Zero' featuring very wide (chord) and short (span) wings, the Zero's wing is not very efficient. In the Zero's case a short wingspan was required as it was a carrier plane and could not afford long wings.

Powering through a turnEdit

As covered above, turns - especially high G turns - lead to energy loss through drag. However the thrust from the engine can oppose this loss of energy.

Planes eventually reach a steady turn rate, like any other time a plane is going at constant speed this is because drag (and gravity if not in level flight) and thrust have reached equilibrium. In the case of a turn, high drag caused by the turn is balanced against the power of the engine.

Some lighter planes (which generally enjoy light wingloads) also benefit from having good power to weight ratios, not necessarily using impressive amounts of horsepower, but respectable relative to their weight. These planes are able to turn well without losing too much velocity because of their relatively powerful engines. These planes are ideally suited for the 'Turn n Burn' style as they 'burn' less energy than an opponent with a weaker power to weight ratio and if at any point the fight stops being a turning fight it will also be able to recover energy by accelerating quickly.