By aircraft design, excluding engine power, how do you make planes that have a tight turn radius while maintaining speed?

Question

What are all factors in design, excluding engine power and acceleration, thrust vectoring and flight computers, that makes a plane that have a real tight turning radius while maintaining speed? Like larger elevator, light weight etc. Where should the centre of gravity be placed? What are all aspects I need to know?

Answer 1

In a turn you have two opposing effects: In order to tighten the turn, more of the lift needs to be tilted sideways for the needed centripetal force. At the same time, you want to fly slowly so the turn rate and with it the centripetal force stays manageable.

Starting with the equation for the turn radius: $$R = \sqrt{\frac{m^2}{(\frac{\rho}{2}\cdot c_{L}{\cdot}S)^2 - (\frac{m\cdot g}{v^2})^2}} = \frac{v}{\omega} = \frac{v^2}{g\cdot\sqrt{n_z^2-1}}$$ you can see that radius $R$ is directly proportional to speed squared for the same load factor. Hence the desire to fly slowly. If you now plot turn radius in a Cartesian coordinate system with speed on the X axis and turn rate on the Y axis, lines of equal radius fan out from the origin like this:

Now there are two boundaries which determine where in this Radius-turn rate relationship an airplane can fly:

1. Minimum speed at a given load factor (bold green line below), and
2. Maximum sustainable load factor, which is limited by the structure and engine thrust (bold red line below).

Since compensation of the weight still needs some vertical component of lift (which explains the -1 in the denominator above), the green line creeps up to smaller radii as load factor increases (and the weight becomes less significant compared to the centripetal force). As soon as the airplane flies faster than minimum speed (the area to the right of the green curve), radius suffers.

This means the tightest turns are where the bold red and green lines meet. This minimum radius point can be shifted to smaller radii by:

• Lower wing loading. This will allow the airplane to fly more slowly.
• Higher maximum lift coefficient, for the same reason.
• Higher structural limits, so a higher load factor can be flown.

Of course, it must be possible to power the airplane at the minimum radius point and to trim the resulting pitch speed. Since engine power can be excluded here, the factors which help with a high pitch rate are:

• Low natural stability. The center of gravity should only be slightly ahead of the neutral point.
• Low stick forces over load factor, if you have manually operated controls.
• Short tail lever arm, which keeps pitch damping low.

If you have trouble to achieve enough thrust at the minimum radius point, try to increase wingspan in order to reduce induced drag. At this point induced drag has by far the biggest share of overall drag.

Answer 2

This generally means a short wingspan with an extra-strong wing spar structure (to take the lift and G loads), a wing with either high-lift devices that deploy in the turn or a relatively thick wing profile, no fuel tanks or excess mass in the wing itself (to minimize the axial moment of inertia of the airframe), big ailerons (to generate a large rolling moment), boosted controls (so a human is capable of working those big control surfaces) and a big engine (to maintain altitude & airspeed during the turn).

You'll find most of these design features in an aerobatic plane or a fighter.

Answer 3

You specifically talk about sustained turn -- a turn without loosing speed (and altitude). This is in contrast to an instantaneous turn -- a turn where you are allowed to lose speed and/or altitude.

Peter Kämpf's answer addresses all the aspects of an instantaneous turn. This is primarily a kinematic problem - i.e. the geometry of motion - not a lot to do with the airplane itself. This means that the curves and behavior are the same for all airplanes for things like the relationship between turn radius, turn rate, and speed -- or turn rate, load factor (g's) and speed.

For an instantaneous turn, the two constraints are stall (CLmax and wing loading) and structure (max g's -- nmax).

However, for an instantaneous turn, you need to consider the specific excess power. Largely determined by how powerful the engines are -- but also the weight of the aircraft and how clean you are (how little drag).

$P_s=\frac{V(T-D)}{W}$

$P_s$ Specific excess power

$V$ Airspeed

$T$ Thrust

$D$ Drag

$W$ Weight

$P_s$ is the energy maneuverability of the aircraft. To maintain a sustained turn, you need to look at $P_s=0$. Next, you need to dig into the drag...

$D=C_D\,q\,S_\mathrm{ref}$

$C_D$ Drag coefficient

$q$ Dynamic pressure $q=0.5\,\rho\,V^2$

$\rho$ Density of air

$S_\mathrm{ref}$ Wing reference area

We will assume a very simple form of a drag polar where the drag coefficient is made of a parasite and induced term...

$C_D=C_{D,0}+\frac{{C_L}^2}{\pi\,e\,AR}$

$C_{D,0}$ Parasite drag coefficient

$C_L$ Lift coefficient

$e$ Aerodynamic efficiency factor

$AR$ Aspect ratio

Finally, to look closer at the induced drag, we have to look at the lift coefficient.

$C_L=\frac{n\,W\,\cos(\theta)}{q\,S_\mathrm{ref}}$

This is a little different than a normal lift coefficient equation.

$n$ Load factor, how many g's are you pulling in the turn

$\theta$ Flightpath angle, if you're not climbing or diving, it is zero. We will use this case.

OK, lets put it all together...

To have excellent sustained turn capability, you need to be able to maintain $P_s=0$ at the speed, altitude, and turn (load factor) that you desire.

You can achieve this with lots of thrust, minimal drag, and light weight.

$P_s=\frac{V(T-D)}{W}$

To minimize the drag, you need a clean airplane. You first want the parasite drag to be a minimum.

$C_{D,0}$

But importantly, you also want the induced drag to be low. This will tend to increase the aspect ratio (vs. an airplane not optimzied for sustained turn) and will have some other effects.

In fact, although we normally think about maximizing the lift-to-drag ratio of an airplane in cruise ($n=1$), for sustained turn, you want to improve the lift-to-drag ratio during the turn.

I believe the experimental Grumman X-29 was designed to hit best $L/D$ during a combat maneuver -- i.e. combat altitude, transonic speeds, and $n=3$ g's!