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Nonlinear asymptotic theory of hypersonic flow past a circular cone

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Summary

The hypersonic small-disturbance theory is reexamined in this study. A systematic and rigorous approach is proposed to obtain the nonlinear asymptotic equation from the Taylor-Maccoll equation for hypersonic flow past a circular cone. Using this approach, consideration is made of a general asymptotic expansion of the unified supersonic-hypersonic similarity parameter together with the stretched coordinate. Moreover, the successive approximate solutions of the nonlinear hypersonic smalldisturbance equation are solved by iteration. Both of these approximations provide a closed-form solution, which is suitable for the analysis of various related flow problems. Besides the velocity components, the shock location and other thermodynamic properties are presented. Comparisons are also made of the zeroth-order with first-order approximations for shock location and pressure coefficient on the cone surface, respectively. The latter (including the nonlinear effects) demonstrates better correlation with exact solution than the zeroth-order approximation. This approach offers further insight into the fundamental features of hypersonic small-disturbance theory.

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Abbreviations

a :

speed of sound

H :

unified supersonic-hypersonic similarity parameter,\(\sqrt {(M_\infty ^2 - 1)} \delta\)

K δ :

hypersonic similarity parameter, M∞δ

M∞ :

freestream Mach number

P :

pressure

T :

temperature

S :

entropy

u, v :

radial, polar velocities

V ∞ :

freestream velocity

β:

shock angle

δ:

cone angle

ϱ:

density

ε:

density ratio, ϱ∞/ϱ(β)

γ:

ratio of specific heats

θ:

polar angle

\(\bar \theta\) :

stretched polar angle, θ/δ

α(δ), σ(δ), λ(δ):

gage functions

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Author notes

  1. Y. T. Chou &  S. C. Lin

    Present address: Department of Mechanical Engineering, National Taiwan University of Science and Technology, 43 Keelung Road, Section 4, Taipei, Taiwan, ROC

  2. C. K. Feng

    Present address: Department of Aerospace Engineering, Tamkang University, Tamsui, Taipei, Taiwan, ROC

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    1. Y. T. Chou
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    2. S. C. Lin
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    3. C. K. Feng
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    Chou, Y.T., Lin, S.C. & Feng, C.K. Nonlinear asymptotic theory of hypersonic flow past a circular cone. Acta Mechanica 130, 1–15 (1998). https://doi.org/10.1007/BF01187039

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