An Improvement to Prandtl’s 1933 Model for Minimizing Induced Drag

Abstract

We consider Prandtl’s 1933 model for calculating circulation distribution function $\Gamma$ of a finite wing which minimizes induced drag, under the constraints of prescribed total lift and moment of inertia. We prove existence of a global minimizer of the problem without the restriction of nonnegativity $\Gamma \ge 0$ in an appropriate function space. We also consider an improved model, where the prescribed moment of inertia takes into account the bending moment due to the weight of the wing itself, which leads to a more efficient solution than Prandtl’s 1933 result.

Appendices

Chebyshev Polynomials

We recall [3, Section 22] Chebyshev polynomials $T_n$, $U_n$ of the first kind and second kind, respectively, defined by

$$\begin{aligned} T_0(x) = U_0 (x) :=1, \quad T_1(x) :=x, \quad U_1 (x) :=2x, \end{aligned}$$

(A.27)

and by

$$\begin{aligned} \begin{aligned} T_{n+1} (x)&:=2x T_n (x) -T_{n-1} (x),\\ U_{n+1} (x)&:=2x U_n (x) - U_{n-1} (x) \end{aligned} \end{aligned}$$

(A.28)

for $n\ge 1$. The Chebyshev polynomials are related to each other in a number of remarkable ways. Here we only list those properties used in the note. First, we have

$$\begin{aligned} U_n' = \frac{xU_n - (n+1) T_{n+1}}{1-x^2} \end{aligned}$$

(A.29)

for all n. Second,

$$\begin{aligned} \int _{-1}^1 \frac{T_n (y) }{\sqrt{1-y^2} (x-y) } \textrm{d}y = - \pi U_{n-1} (x) \end{aligned}$$

(A.30)

for all. Finally, the $U_n$’s form an orthonormal basis of H (recall (13)), with

$$\begin{aligned} \int _{-1}^1 U_n (x) U_m(x) \sqrt{1-x^2} \textrm{d}x = {\left\{ \begin{array}{ll} 0\qquad &{}m\ne n,\\ \pi /2 &{} m=n. \end{array}\right. } \end{aligned}$$

(A.31)

Convex Sets in Banach Spaces

Here we prove the following fact relating convex sets in Banach spaces and weak convergence.

Lemma B.1

(Convex sets in Banach spaces and weak limits) Let X, Y be real Banach spaces. Suppose that $x_k \rightharpoonup x$ in X as $k\rightarrow \infty$, and that $A_k,A \in B(X,Y)$ are such that $\Vert A_k - A \Vert _{B(X,Y)} \rightarrow 0$ as $k\rightarrow \infty$. Let $K\subset Y$ be a closed convex set. Then,

$$\begin{aligned} \text { if }A_kx_k \in K\text { for all }k,\qquad \text { then }\qquad Ax \in K. \end{aligned}$$

Proof

Let $l\in Y^*$ denote any supporting hyperplane of K in Y, i.e.

$$\begin{aligned} K\subset \{ y\in Y :l(y)\le l_0 \} \end{aligned}$$

for some $l_0\in R$. Since $\{ x_k \}$ converges weakly, it is bounded, and so there exists $M>0$ such that $\Vert x_k \Vert \le M$ for all k. Given $\varepsilon >0$ we let k be sufficiently large so that

$$\begin{aligned} |l(A(x-x_k)|\le \frac{\varepsilon }{2} \qquad \text { and }\qquad \Vert A-A_k \Vert \le \frac{\varepsilon }{2\Vert l \Vert _{Y^*}M}. \end{aligned}$$

(B.32)

Note that such choice is possible by assumptions, since $l\circ A \in X^*$. We obtain

$$\begin{aligned} \begin{aligned} l(Ax)&= l(A(x-x_k)) + l((A-A_k)x_k ) + l(A_k x_k ) \\&\lesssim |l(A(x-x_k))| + \Vert l \Vert _{Y^*} \Vert A- A_k \Vert M + l_0 \\&\le l_0 + \varepsilon , \end{aligned} \end{aligned}$$

where we used the assumption $A_kx_k\in K$ in the second line and (B.32) in the last. Taking $\varepsilon \rightarrow 0$ we thus have that $l(Ax) \le l_0$, and the claim follows, since every closed convex set in Y equals to the intersection of all its supporting hyperplanes (see [16, Corollary 21.8], for example). $\square$