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

TL;DR

This work derives closed-form expressions for all moment and lift integrals governing NACA 5-digit reflex camber lines, eliminating the need for numerical quadrature. By mapping to the $x$-domain and applying a trig substitution, the integrals reduce to combinations of $\arcsin(\sqrt{r})$, $\arccos(\sqrt{r})$, and explicit polynomials in $r$ and $x_{mc}$; the breakpoint $r$ is found from a single transcendental equation and $k_1$ is computed directly, with $k_2/k_1$ from a closed-form ratio. The analytical results are validated against NASA tabulations and independent quadrature, showing machine-precision agreement for the moment integrals and high fidelity for $k_1$ and $k_2/k_1$ when accounting for tabulation rounding and sensitivity. The closed-form framework improves reproducibility and efficiency, enables gradient-based optimization, and extends readily to additional reflex camber-line families and extended parameter tables.

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.

Figures (2)

• Figure 1: Validation of breakpoint parameter $r$: (left) analytical and reference Ladson_Brooks_1975 values versus $x_{mc}$; (upper right) discrepancy $r_{\mathrm{anal}}-r_{\mathrm{ref}}$; (lower right) moment-condition residual versus $r$.
• Figure 2: Validation of scaling constant $k_1$: (left) analytical and reference Ladson_Brooks_1975 values versus $x_{mc}$; (upper right) discrepancy $k_{1,\mathrm{anal}} - k_{1,\mathrm{ref}}$; (lower right) recovered $C_{l,i}$.