Shock Waves

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The interaction of a cylindrical shock wave segment with a converging–diverging duct

  • B. B. NdebeleEmail author
  • B. W. Skews
Original Article


In this study, we present the results of an investigation of the propagation of cylindrical shock wave segments in converging–diverging channels. Three planar-symmetric channels were used: two formed from a pair of walls with circular wall profiles (radii 150 mm and 225 mm) and a third following a third-order-polynomial profile. In contrast to the circular walls, which are convex, the polynomial wall has both convex and concave curved sections. A plane shock was generated using a conventional shock tube after which a 165-mm-radius cylindrical shock segment was formed by allowing the plane shock to pass through a circular arc-shaped annular space. This shock was then allowed to propagate in the converging–diverging channel formed from the two walls described earlier. The resulting shock evolution was captured using a z-type schlieren technique. Whitham’s geometric shock dynamics (GSD) and computational fluid dynamics (CFD) were used to create numerical models of the shock’s propagation. Comparisons between the three methods (experiment, CFD, and GSD) were made. In general, qualitative agreements between the three methods were observed (with slight discrepancies). For example, a shock with an initial Mach number of 1.37 interacting with a 150-mm-radius wall exhibited high curvature at the shock’s central position towards the channel’s throat (as observed in experiment), an observation which was not replicated by either CFD or GSD. Quantitatively, there were significant differences (before accounting for experimental errors). On comparing centreline shock Mach numbers between the three methods, CFD results were closer to experimental results, while GSD results were consistently higher but within the experimental data error bounds. However, the general trend was the same in all three, i.e., the shock strengthens and weakens in the converging and diverging sections, respectively.


Cylindrical shock wave Converging–diverging duct Diffraction Reflection Geometric shock dynamics 

List of symbols

\((\alpha , \beta )\)

Coordinates on a curvilinear coordinate system


Coordinates on a Cartesian plane

\(\epsilon _{\mathrm {M}}, \epsilon _{\mathrm {P}}\)

Error in shock Mach number and position, respectively

\(\eta (M)\)

Modification factor of Whitham’s theory

\(\gamma \)

Ratio of specific heat capacities

\(\theta \)

Shock orientation

\(\mathbf {n}_{i}\)

Normal on the shock front


Channel cross-sectional area


Speed of sound ahead of the shock

\(f_1, f_2\)

Property value (e.g., density) for a fine and course mesh (relatively), respectively


Shock Mach number

\(M_{\mathrm {0}}\)

Initial shock Mach number


Position of probe point


Shock radius, or ratio between course and fine mesh sizes



u,  v

x and y component of velocity


Grid Convergence Index



Funding was provided by the South African National Research Foundation.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of the WitwatersrandJohannesburgSouth Africa

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