An improvement to Prandtl's 1933 model for minimizing induced drag

Abstract

We consider Prandtl's 1933 model for calculating circulation distribution function $Γ$ 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 $Γ\geq 0$ 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.

Figures (2)

• Figure 1: A geometric interpretation of the Prandtl's 1933 solution to the minimimal induced drag problem. Note that the region that Prandtl prandtl_33 considered corresponds to the part of blue region above the line $a_2=-a_0/3$. The part below the line (which corresponds to the case when $\Gamma$ can change sign) was obtained by a numerical simulation of the constraint \ref{['A2']}.
• Figure 2: A geometric interpretation of the optimal solution of the problem \ref{['new_inf']}, and comparison to Prandtl's 1933 solution to the minimimal induced drag problem. Here the region above the line $a_2=-a_0/3$ represent the nonnegativity assumption $M\geq 0$ (see \ref{['noncol']}), the blue region represents the total weight constraint \ref{['A2']}, and the red region represents the "noncollapse" condition (i.e., the second part of \ref{['noncol']}). The blue dots at the top-left corner indicate that the nonphysical vertical blue regions continue to appear and concentrate aa $a_0\to 0^+$.

Theorems & Definitions (6)

• Theorem 1: Minimizer of the Prandtl problem
• proof : Proof of Theorem \ref{['T01']}.
• Theorem 2: Minimizer of the minimal induced drag problem \ref{['new_inf']}
• proof
• Lemma 3: Convex sets in Banach spaces and weak limits
• proof