Induced Drag Coefficient

Induced drag is the drag a wing cannot avoid paying for the lift it makes, and it is the one term in aircraft performance that behaves backwards. Every other kind of drag rises with speed. This one FALLS with speed, and it falls fast — as the fourth power of it, once you follow the chain through.

Here is that chain, because it explains almost everything an aeroplane does slowly. In level flight the lift must equal the weight, so $C_L$ must rise as the speed falls: halve the speed and $C_L$ must quadruple. The induced drag coefficient goes as $C_L^2$, so it goes up sixteenfold. The drag FORCE is that coefficient times $q$, and $q$ has fallen to a quarter, so the induced drag force ends up four times larger. Meanwhile the parasite drag force has fallen to a quarter. The two curves cross, and the total has a minimum, and everything about how an aeroplane is flown at low speed comes out of that shape.

The formula is Prandtl's, from the lifting-line theory he developed at Göttingen and published in the NACA report series in the early 1920s, and it is one of the genuinely beautiful results in engineering: $C_{Di} = C_L^2/(\pi \cdot AR \cdot e)$. The $e$ is the span efficiency factor, and it is 1 for an elliptical lift distribution — Prandtl's own result, that the elliptical distribution minimises induced drag for a given span and lift, is why the Spitfire had the wing it had. Real wings, with taper instead of a true ellipse, twist, a fuselage interrupting the middle and flaps disturbing the inboard sections, come in at 0.7 to 0.85.

The classic mistake here is simply forgetting the term. A drag estimate built from a cruise $C_D$ and applied to an approach will understate the drag badly, because in cruise induced drag might be 15% of the total and on final approach it can be more than half. This is the physics behind the REGION OF REVERSED COMMAND, where flying slower requires more power, and it is where a dragged-in low approach ends up needing full throttle just to hold altitude — with nothing left for the go-around. The equation predicts it exactly, and the aeroplane obeys.

One caution on the number $e$. Span efficiency, which is what this equation wants, can never exceed 1. If you fit a constant to TOTAL measured drag instead of the induced part alone, you get an OSWALD efficiency factor, which absorbs the lift-dependent share of the profile drag as well and is always smaller. The two are widely written with the same letter and are not the same quantity — a figure above 1 is nearly always a sign of one being used in the other's place.