What is the performance of a flat plate wing?

By flat plate I mean this:

Source: Physics.SE

Low performance indeed, but how much low? How would:

• The angle of attack,
• The speed range,
• The turn capability,
• The fuel consumption,
• Or other significant characteristic

compare to the ones of some reference foil, e.g. a NACA 00xx?

I cannot answer all your questions, but maybe point you to some facts to come closer to an answer.

Most important is the thickness of your parallelepiped - more than a few percent will just increase drag, without benefiting performance. It should be as thin as structurally possible. There are lots of model airplanes using flat plates for lift. The most popular are flying discs like that below:

(picture source)

Lift

The lift curve slope of a flat plate is the same as that of regular airfoils, complete with stall. They have a zero-lift angle of attack of 0° (obviously) and separated flow on the suction side. The maximum lift coefficient of 0.7 to 0.8 is reached at moderate angles of attack - details depend on the aspect ratio.

Your typical four-digit NACA airfoil will have a maximum lift coefficient of 1.2 to 1.6, depending on camber and Reynolds number (picture source).

Drag

Due to the missing nose thrust, the aerodynamic force vector will stand almost perpendicular to the plate's plane. It will be slanted slightly backwards due to friction, but the separated flow on the suction side means that friction will mostly happen on the pressure side. As a rough approximation, the drag coefficient of the flat plate is $$c_D = c_L \cdot \sin\alpha + \frac{0.43}{log\left(\frac{100}{R}\right)^{2.56}} + 0.3 \cdot\delta\cdot\cos\alpha$$ with $R$ the relative surface roughness, $\delta$ the relative thickness and $\alpha$ the angle of attack. The first part is caused by the direction of the lift vector, the second part is the friction contribution and will be dwarfed by the pressure part, and the third part is an approximation for the suction along the thick trailing edge. At $\alpha$ = 5° and a lift coefficient of 0.5, the drag coefficient will already be around 0.05.

Regular NACA airfoils will have a $c_{D0}$ of 0.004 to 0.01, depending on Reynolds number, and their $c_D$ at stall is normally between 0.02 and 0.025.

Performance

From the above it is fair to assume that drag in cruise condition will easily be 4 times higher (induced drag is not affected), and the optimum cruise condition will be at a lower speed due to the higher zero-lift drag. On top, the airplane will need a bigger engine to stay airborne, which will reduce its useful load. Expect fuel consumption to increase in line with the drag increase.

The stall speed of an aircraft with a flat plate wing will be approx. 50% higher than with a proper airfoil on the wing, and drag will increase much more at high lift coefficients than usual, which translates into very low sustained turn rates and load factors.

Only the angle of attack range will stay similar. All other parameters will be markedly affected.

• @ArtOfCode: I used to do multiplication with \cdot, but found that shift-opt-9 produces the same centered dot. Your edit did not change the appearance of the equation on my screen, but is this different in that other OS? Should I better use \cdot again for better compatibility? – Peter Kämpf Sep 30 '15 at 12:02
• The manual dot you add gets put in thanks to Unicode - just not all computers and browsers will support it. MathJax's \cdot is, IIRC, compatible in most if not all places. – ArtOfCode Sep 30 '15 at 12:05
• @PeterKämpf: shouldn‘t the MachNumber be considered as well? Looking at the wing profile of a F16 or Concord they look much more like a flat plate due to the transonic aerodynamics. – rul30 Jan 16 '18 at 5:52
• @rul30: You are right; I answered only for subsonic speed. – Peter Kämpf Jan 16 '18 at 9:27

One last note: flat plate wings work extremely well on small scale lengths, as for example in the insect world. Consider that a dragonfly can hover, perform flat turns at speed, fly inverted, and accelerate from zero to 30 MPH in a couple of seconds with two pairs of flat wings, and 250 million years +/- of selection pressure have not improved their design much.

Because a dragonfly's wings twist significantly during operation, they are not strictly speaking without camber, but anyone who has ever watched them pluck gnats out of the air cannot but be amazed at what they can accomplish, aerodynamically speaking.

• Yes the Square-Cube law does them many favours. – Koyovis Jan 16 '18 at 4:16

It depends of course: laminar or turbulent flow, and at which Reynolds number. $C_D$ for AoA zero looks like this (found here)

Flat plate aerodynamics are relevant for helicopter blades and wind turbines, and measurements have been conducted over 0 - 360 deg of AoA with results that are of interest for fixed wing as well. For instance from this document:

The $C_L$ of NACA 0012 over 180 degrees shows that the real gain in lift is within the first 10 degrees of AoA, after which the airfoil stalls, but $C_L$ starts rising again after its initial stall behaviour due to flat plate aerodynamics. The 0012 profile is quite thin, and the maximum $C_L$ is reached at 45 degrees - however if we look at 10 deg AoA, the 0012 $C_L$ is around 1 while the equivalent flat plate $C_L$ would be around 0.4 (extend the sine wave graph 15 < AoA < 170 to zero).

$C_D$ follows a similar pattern, again showing large gains until stall occurs and then continuing according to flat plate aero. at 10 deg, $C_D$ is barely above zero instead of the 0.12 value obtained when the sine curve is extended.

So the NACA profile has a much higher lift for a much lower drag than an equivalent flat plate. In order to get to a $C_L$ of 1, the AoA of a flat plate needs to be 30 deg instead of the 10 deg of the NACA. At 30 deg, $C_D$ of the wing is 0.6 instead of the 0.02 of the NACA, 30 times higher! So turn rate and other parameters are only of interest if the engines can deliver 30 times more thrust.

• Your comparison makes the flat plate look worse than it is. It reaches its optimum at a lower lift coefficient, so the comparison at c$_L$=1 is misleading. This is not to say that a flat plate is not a poor choice for subsonic aircraft - it certainly is. – Peter Kämpf Jul 17 '17 at 8:20
• Yes good point for the cruise, although it would make for much higher landing speeds if we can't get to $C_L$ of 1. – Koyovis Jul 17 '17 at 9:31
• yes, or you use a much bigger wing. – Peter Kämpf Jul 17 '17 at 14:51