Fundamentals of Flight

In one line: An airplane in steady flight balances four forces: lift against weight and thrust against drag. A few equations then predict how fast it can fly, how high, and how quickly it can climb.

Core idea

Early aircraft designers worried mostly about getting off the ground. Once the Wright brothers succeeded in 1903, the questions became about performance: top speed, climb rate, ceiling, range and endurance. John Anderson’s standard introductory text [1] builds these answers from a force balance and a description of drag.

In steady, level flight:

• lift equals weight
• thrust equals drag

How it works

Lift and drag both scale with dynamic pressure, $\frac{1}{2}\rho V^2$, and wing area $S$:

$L=\frac{1}{2}\rho V^{2}S{C}_{L}D=\frac{1}{2}\rho V^{2}S{C}_D$

The lift coefficient $C_L$ rises with the wing’s angle of attack until the flow separates and the wing stalls.

The drag polar. Total drag has two parts:

• parasite drag: skin friction and form drag, which grows with speed
• induced drag: the cost of making lift, which shrinks with speed

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

Here $C_{D,0}$ is the zero-lift drag, $e$ the Oswald efficiency factor, and $AR$ the wing’s aspect ratio, its span squared divided by its area. Long, slender wings, as on gliders, cut induced drag.

Power required versus power available.

• The power required to fly is drag times speed. Plotted against speed, it forms a U-shaped curve.
• A propeller aircraft’s power available is the engine’s power times the propeller’s efficiency. It falls with altitude as the air thins.

Where the two curves cross at high speed is the maximum speed. The gap between them is excess power, which sets the rate of climb: excess power divided by weight. The absolute ceiling is the altitude where the excess power falls to zero.

Example: a light aircraft weighing 2,000 lb, with $C_{D,0}=0.03$, $e=0.8$ and $AR=7$, has a maximum lift-to-drag ratio of about 12. At its most efficient speed, roughly 70 knots at sea level, it needs only about 165 lb of thrust to hold up a ton.

In practice

The course’s first project asks a team to predict a popular training aircraft’s speed, ceiling and time to climb from published data and these relations. Then they compare the results with the manufacturer’s numbers. The exercise shows how much performance follows from a few parameters, and how sensitive the predictions are to assumptions such as propeller efficiency. The same reasoning drives real trade-offs between wing size, weight and engine power (see Trade-off Analysis).

Questions to ask the experts

• What drag polar is assumed, and how was it measured or estimated?
• How does performance degrade with altitude, temperature and added weight?
• Where is the stall margin at the slowest operating speeds?

Connections

• The discipline: Mechanical and Aerospace Engineering. Detailed aerodynamics: Computational Fluid Dynamics.
• Performance as requirements: Product Specifications. Balancing them: Trade-off Analysis.
• Capabilities exercised: Inquiry, Connecting Across Disciplines, Decision Making.

Sources

• Anderson - Introduction to Flight [1]

[1]

J. D. Anderson Jr., Introduction to Flight. New York, NY, USA: McGraw-Hill.