Closed-Form Expressions for Five-Digit Reflex Camber-Line Design Parameters

TL;DR

This paper derives closed-form analytical expressions for the design integrals of the NACA five-digit reflex camber-line, enabling precise and efficient calculations for aircraft design without reliance on numerical quadrature or tabulated data.

Contribution

It provides the first closed-form solutions for all lift and zero-moment integrals of the NACA five-digit reflex camber-line, improving accuracy and computational efficiency.

Findings

Analytical expressions match numerical quadrature to machine precision.

Closed-form solutions satisfy the zero-moment condition with minimal residuals.

Comparison shows significant improvement over historical tabulations.

Abstract

Despite nearly a century of use in tailless and flying-wing aircraft, the NACA five-digit reflex camber-line family lacks published closed-form expressions for the governing design integrals; practitioners have instead relied on numerical quadrature and tabulated constants available only for a limited set of standard configurations. This paper addresses that gap by deriving closed-form analytical expressions for all lift and zero-moment integrals in terms of elementary functions of the breakpoint and maximum-camber-location parameters. A trigonometric substitution eliminates the endpoint singularities introduced by the Glauert transformation, yielding integrals expressible as inverse trigonometric functions combined with explicit polynomials. The breakpoint parameter is determined by solving a single transcendental equation, and the remaining camber-line constants follow from direct evaluation without numerical integration. Independent verification by numerical quadrature confirms agreement with the analytical expressions to machine precision. Comparison with historical tabulations shows that the closed-form values satisfy the zero-moment design condition to machine precision, whereas substituting the tabulated breakpoints back into the same condition yields residuals nine to thirteen orders of magnitude larger, consistent with the rounding and computational precision available when the original tabulations were produced.