The Extension of Analytic Hypersonic Force Coefficients for Conceptual Design Using the Divergence Theorem

Abstract

This study investigates the type and performance of analytic Newtonian aerodynamic solutions made possible using the Divergence Theorem. A reformulation of the Newtonian surface pressure calculation enables a mathematically equivalent divergence calculation to be performed as a substitute. This manipulation enables analytic force coefficients to be derived for shapes of increasing complexity while also reducing computational cost when compared to existing analytic solutions. The divergence solutions are obtained by converting the physical flow field into a mathematical flow field that is constant in direction but with magnitude that is equivalent to the Newtonian pressure coefficient. This unique property allows various mathematical techniques that are not available with the traditional Newtonian calculation to be performed that further reduce computational cost. The results of this investigation enable the construction of analytic relations for new hypersonic configurations of interest, and this approach serves as the foundation to construct efficient hybrid exact-approximate solutions for more complex configurations. Comparisons of the current analytic database to a state-of-the-art hypersonic design tool illustrate the computational advantages of the analytic relations to support hypersonic conceptual design.

The Construction of Analytic Hypersonic Pitch Moment Coefficients Using a Curl Transformation

“…This property greatly reduces the complexities of the analytic integration process in manner similar to the benefits observed in prior work that investigated the construction of analytic force coefficients. 32 The pitch moment coefficients for the configurations shown in Fig. 11 and the same range of c.g.…”

“…32 In this approach, the traditional Newtonian surface integration is converted to an equivalent calculation of a mathematical flux that passes through the unshadowed surface of the vehicle. As an example, Eq.…”

“…32 In this prior work, analytic force coefficients were constructed using an appropriate integration order across specified volumetric regions. Additionally, mathematical techniques were utilized to improve the analytic integration process.…”

Hybrid Exact-Approximate Analytic Hypersonic Aerodynamic Relations for General Vehicle Shapes

“…13 Additional relations for quadratic-and parabolic-based shapes were also constructed through the use of boundaryvolume techniques and other transformations. 14,15,16 Since many hypersonic vehicles of interest can be constructed through the superposition of these basic shapes as shown in Fig. 4, 17,18,19,20,21,22 the hypersonic aerodynamics of these systems can also be expressed analytically through the appropriate combination of the analytic aerodynamic relations.…”

“…For many complex shapes, no exact analytic force or moment solution can be constructed, and this is a consequence of the fundamental challenges that exist when performing complex integrations symbolically. 14,15,16 To expand the current analytic aerodynamic database to also include general vehicle configurations, hybrid exactapproximate analytic solutions are constructed for Bezier curves of revolution and Bezier surfaces. Note that these relations are generically parametrized by an appropriate set of control nodes.…”