= Lift-constrained maximum lift-to-drag ratio
A \b[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 $k$, define
$$
f(x)=\frac{x}{c+kx^2},\qquad x=C_L,\qquad 0<\ell\le x\le u.
$$
Its derivative is
$$
f'(x)=\frac{c-kx^2}{(c+kx^2)^2}.
$$
The ratio increases up to $x_*=\sqrt{c/k}$ and decreases afterward. The exact constrained solution is therefore
$$
\boxed{x_{\rm opt}=\min\{u,\max\{\ell,\sqrt{c/k}\}\},\qquad
f_{\rm constrained}=\frac{x_{\rm opt}}{c+kx_{\rm opt}^2}.}
$$
This 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 $x_*\simeq0.6325$ 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> $S$,
$$
C_L(V)=\frac{2W}{\rho SV^2},\qquad
V\in[V_{\min},V_{\max}]
\ \Longrightarrow
C_L\in\left[\frac{2W}{\rho SV_{\max}^2},\frac{2W}{\rho SV_{\min}^2}\right].
$$
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.<Induced Drag Coefficient by NASA Glenn Research Center>[Drag Coefficient and total-drag equations]{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.
Back to article page
