 Lift-constrained maximum lift-to-drag ratio

A lift-constrained maximum lift-to-drag ratio is the best ratio available within an allowed range of lift coefficients, rather than over every coefficient admitted algebraically by a drag polar. It is a constrained optimization problem whose feasible set must be specified before claiming an attainable optimum.

For the parabolic drag polar with positive $c=C_{D0}$ and $k$, defineIts derivative isThe ratio increases up to $x_{\ast }=\sqrt{c/k}$ and decreases afterward. The exact constrained solution is thereforeThis proof handles an interior optimum and either active endpoint; simply inserting an upper limit into the unconstrained formula does not.

An original example takes $c=0.020$ and $k=0.050$, giving $x_{\ast }\simeq 0.6325$ and unconstrained ratio 15.8114. If the allowed interval is $[0.20,0.45]$, the maximum occurs at 0.45 and is about 14.9378. If the allowed interval is instead $[0.90,1.20]$, it occurs at the lower endpoint 0.90 and is about 14.8760. A lower bound can be active as well as an upper bound.

In steady level flight, a speed interval can induce a lift-coefficient interval. With fixed gravitational weight $W$, mass density $\rho$ and wing area $S$,The endpoints reverse because the function decreases with speed. Additional limits, including the usable range before aerodynamic stall, require intersecting feasible intervals.

The simple finite-wing contribution motivates a quadratic polar, but changing configuration, Mach number or Reynolds number can change its fitted parameters.^[ref] Restricting the mathematical interval does not validate the polar across that interval. Wind and ground-distance objectives also define a different optimization from the aerodynamic ratio considered here.