Development of Maximum Conical Shock Angle Limit for Osculating Cone Waveriders

TL;DR

This paper derives mathematical limits for the maximum conical shock angle in osculating cone waveriders, enhancing design constraints for hypersonic vehicle optimization.

Contribution

It introduces new geometrical conditions and closed-form expressions to determine the maximum cone shock angle for successful waverider generation.

Findings

Maximum cone shock angle is lower than the attached shock angle limit.

Closed-form solutions are provided for power law curves.

Numerical procedures are developed for Bezier curves.

Abstract

Hypersonic waveriders are special shapes with leading edges coincident with the body's shock wave, yielding high lift-to-drag ratios. The waverider geometry results from streamline tracing using the solutions of a basic flow field such as the wedge or the cone for specified shock and base curves. The base and shock curves can be independently prescribed in the osculating cone method enabling a larger design space. Generally, low values of the conical shock angle (9-15 degrees) are used. The lack of any method to limit the maximum cone angle for osculating cone waverider motivates this study. Mathematical expressions are derived for geometrical conditions that result in successful osculating cone waverider generation. A power law curve and a Bezier curve are analyzed. Closed-form expressions for the maximum cone shock angle are obtained for the power law curve. A numerical procedure to solve the same for the Bezier curve is developed. The results, for a typical Mach number of 6.0, evidently show that the maximum cone shock angle for successful waverider generation is significantly lower than the maximum angle for attached shock solutions. The limiting conditions developed will be essential in constraining the waverider sample space for automated multiobjective optimization routines.