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The reflection of cylindrical shock wave segments on cylindrical concave wall segments

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Abstract

The reflection of a two-dimensional cylindrical shock wave segment on a concave-cylindrical wall segment was investigated from an experimental and numerical perspective. Qualitatively, the experimental results show that a cylindrical shock behaves similarly to a plane shock when propagating up a concave-cylindrical wall segment. Thus, whatever differences exist must be quantitative. The length of the cylindrical shock’s Mach stem was plotted against the subtending angle. From the plots, two limits are highlighted, one where the shock’s radius is much larger than the wall’s radius and another where the wall has the larger radius. The former being akin to a plane shock interacting with a cylindrical wall segment. An increase in initial shock Mach number was observed to affect the type of Mach reflection that is formed (whether it is an inverse or stationary Mach reflection) as well as the transition point to a transitioned regular reflection. An expression which relates the shock’s Mach stem to the subtending angle was derived. Comparisons between the expression’s predictions, experimental and CFD data were completed, and they showed good correlation. Further, the expression was shown to reduce to that derived by Itoh et al. when the shock’s radius was much larger than the wall’s radius.

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Abbreviations

\(M_0\) :

Initial shock Mach number

\(M_\mathrm{s}\) :

Plane shock Mach number

\(M_\mathrm{w}\) :

Wall shock Mach number

\(M_1\) :

Mach stem Mach number

\(u_1\) :

Fluid velocity behind the shock

\(a_1\) :

Sound speed behind the shock

\(a_0\) :

Sound speed ahead of the shock

\(R_\mathrm{w}\) :

Wall radius

\(R_\mathrm{s}\) :

Initial shock radius

r :

Shock radius

\(\lambda \) :

Mach stem length

\(\mathrm{d}s\) :

Arc length element

\(\phi \) :

Angle subtended by the Mach stem

\(\theta _\mathrm{F}\) :

Angle when Mach reflection first forms

\(\theta \) :

Shock orientation

\(\theta _\mathrm{w}\) :

Shock orientation at the wall

\(\eta (M)\) :

Modification to Whitham’s theory

\(\phi ^*\) :

Angle when the shock transitions from MR to RR

u(r):

Gas speed behind the shock

a(r):

Speed of sound behind the shock

\(u_\mathrm{s}(r)\) :

Shock speed

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Acknowledgements

This research was supported by the South African National Research Foundation.

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Authors and Affiliations

  1. University of the Witwatersrand, 1 Jan Smuts Av. Braamfontein, Johannesburg, 2000, South Africa

    B. B. Ndebele & B. W. Skews

Authors
  1. B. B. Ndebele
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  2. B. W. Skews
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Correspondence to B. B. Ndebele.

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Communicated by O. Igra and A. Higgins.

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Ndebele, B.B., Skews, B.W. The reflection of cylindrical shock wave segments on cylindrical concave wall segments. Shock Waves 28, 1185–1197 (2018). https://doi.org/10.1007/s00193-018-0812-6

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  • DOI: https://doi.org/10.1007/s00193-018-0812-6

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