Vortex development on slender missiles at supersonic speeds

Extension and Enhancement of the Allen–Eggers Analytical Ballistic Entry Trajectory Solution

“…The results differ slightly from those presented in Ref. [2] due to use of current conventions for the definition of flight-path angle and ballistic coefficient.…”

“…W HEN planetary entry was first studied in the middle of the twentieth century, analytical approximations were developed to enable vehicle and mission designers to evaluate vehicle trajectories and performance with minimal or no computer usage. Seminal examples of analytical approximations include, in chronological order, Sänger and Bredt's expressions for lifting entry [1], the Allen-Eggers approximation for ballistic entry [2], Chapman's Z-function [3], Loh's "second-order" approximation [4], and those developed by Vinh et al [5]. Analytical and hybrid analytical-numerical approximate solutions utilize assumptions to make the nonlinear equations of motion soluble.…”

“…Harry "Harvey" Julian Allen and Alfred J. Eggers, working at the Ames Aeronautical Laboratory, first documented their approximate solution for ballistic (nonlifting) entry in a classified National Advisory Committee for Aeronuatics (NACA) research memorandum in 1953 [7]. The memorandum was declassified in 1957 [7], and subsequently republished as a NACA report in 1958 [2]. Allen and Eggers developed their approximate analytical solution "to determine means available to the designer for minimizing aerodynamic heating" for missile applications [2].…”

“…The memorandum was declassified in 1957 [7], and subsequently republished as a NACA report in 1958 [2]. Allen and Eggers developed their approximate analytical solution "to determine means available to the designer for minimizing aerodynamic heating" for missile applications [2]. The Allen-Eggers solution is based on the insight that, for ballistic entry at a sufficiently steep initial flight-path angle, the gravitational force may be neglected because its magnitude is small relative to the magnitude of the drag force [2].…”

“…Allen and Eggers developed their approximate analytical solution "to determine means available to the designer for minimizing aerodynamic heating" for missile applications [2]. The Allen-Eggers solution is based on the insight that, for ballistic entry at a sufficiently steep initial flight-path angle, the gravitational force may be neglected because its magnitude is small relative to the magnitude of the drag force [2]. Neglecting gravity yields a closed-form analytical relationship between velocity and altitude from which Allen and Eggers were able to derive closed-form analytical expressions for acceleration and heating as functions of velocity, including state values at peak conditions of interest.…”