Elliptical planform (idea)

Proof that an elliptic planform makes the most aerodynamically efficient wing (assuming low speeds)

We begin by stating, without proof, that the equation for induced drag C_Di over a wing with circulation Γ(y) in a flow with density ρ_∞ and a downwash velocity of w(y). y represents position on the wing measured from the root, varying from -b/2 to +b/2, where b is wingspan.

D_i = -∫_-b/2^+b/2ρ_∞w(y)Γ(y)dy

Next, we take the elliptical distribution of circulation around a wing, which depends on y:

Γ(y) = Γ₀√(1 - (y/(b/2))²)

Finally, the downwash velocity for this planform simplifies to:

w(y) = -Γ₀/(2b)

We may now substitute into the integral expression:

D_i = -∫_-b/2^+b/2ρ_∞(-Γ₀/(2b))(Γ₀√(1 - (y/(b/2))²))

We introduce the coordinate transfomation y = -s cos φ, and simplify the expression to:

D_i = (ρ_∞Γ₀²/(2b))∫₀^π(b/2) sin φ√(1 - cos² φ) dφ
D_i = (π/8)ρ_∞Γ₀²

We now apply the formula for the drag coefficient:

C_Di = D_i/(0.5*ρ_∞V_∞²S)
C_Di = πΓ₀²/(4V_∞²S)

This is all well and good, but the circulation at the wing root is not a practical design variable. We then state, without proof, that based on the expression for the lift coefficient of the same wing:

Γ₀ = 2C_LV_∞S/(πb)

Thus, we can substitute into our expression for the drag coefficient:

C_Di = {π/(4V_∞²S)}{2C_LV_∞S/(πb)}²
C_Di = C_L² / (π*AR)

We have now derived the coefficient of induced drag for a wing with an elliptical circulation, and thus, elliptical lift distribution and planform (these follow directly from the Kutta-Joukowski lift theorem and the proportionality of chord to circulation).

Now, *poof* I present the formula for the coefficient of induced drag in general:

C_Di = C_L² / (π*AR*e)

where the airplane efficiency factor e ≤ 1. Note then that for the elliptical planform, e = 1. Therefore, a wing with an elliptical planform is the most efficient (provided aspect ratio and lift coefficient are fixed), because it has the least possible value for its coefficient of induced drag.