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Taylor-Maccoll

The Taylor-Maccoll equations describe steady, inviscid, irrotational supersonic flow over a right-circular cone at zero angle of attack.

This is the high-fidelity conical-flow model in gasdyn and is used to map between freestream Mach number \(M_\infty\), cone half-angle \(\theta_c\), and shock angle \(\beta\).

Physical Setup

Consider a right-circular cone with half-angle \(\theta_c\) in a uniform freestream at Mach number \(M_\infty > 1\). For an attached conical shock:

  • The shock sits at angle \(\beta\) from the cone axis.
  • The post-shock flow between the shock and cone surface is isentropic.
  • The flow is conically self-similar, so state variables depend only on polar angle \(\theta\), not radius \(r\).

Solver Formulation Used in gasdyn

gasdyn solves the first-order Taylor-Maccoll ODE system for \([V_r(\theta), V_\theta(\theta)]\):

\[ \frac{dV_r}{d\theta} = V_\theta \]
\[ \frac{dV_\theta}{d\theta} = \frac{V_\theta^2 V_r - \dfrac{\gamma - 1}{2}(1 - V^2)\!\left(2 V_r + \dfrac{V_\theta}{\tan\theta}\right)} {\dfrac{\gamma - 1}{2}(1 - V^2) - V_\theta^2} \]

where \(V^2 = V_r^2 + V_\theta^2\) and velocities are non-dimensionalized by the maximum speed implied by stagnation enthalpy.

Initial conditions are taken at the shock from oblique-shock relations. Integration proceeds from \(\theta = \beta\) toward the axis, and the cone surface is detected by:

\[ V_\theta(\theta_c) = 0 \]

Derivation Details

Derivation details

Starting from the steady axisymmetric Euler equations in spherical coordinates, conical self-similarity and irrotationality give the key identity:

\[ \frac{dV_r}{d\theta} = V_\theta \]

With isentropic closure and non-dimensional energy,

\[ a^{*2} = \frac{\gamma - 1}{2}\left(1 - V_r^2 - V_\theta^2\right) \]

the continuity + momentum equations reduce to the Taylor-Maccoll equation shown above. The numerical solve is then posed as a first-order IVP beginning at the post-shock state \((\theta = \beta)\), where the components are:

\[ V_{r,0} = V_0 \cos(\beta - \delta), \qquad V_{\theta,0} = -V_0 \sin(\beta - \delta) \]

with \(\delta\) from the \(\theta\)-\(\beta\)-\(M\) relation and \(V_0 = 1 / \sqrt{1 + 2/[(\gamma-1)M_2^2]}\).

The cone angle is the event location where \(V_\theta\) crosses zero.

References

Taylor, G. I. and Maccoll, J. W. (1933). The Air Pressure on a Cone Moving at High Speeds. Proceedings of the Royal Society A, 139(838), 278-311. https://doi.org/10.1098/rspa.1933.0017

Anderson, J. D. (2003). Modern Compressible Flow: With Historical Perspective, 3rd ed. McGraw-Hill, Section 10.4.