Shock Waves

pp 1–15 | Cite as

The locus of the inflection point of a diffracting cylindrical shock segment

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


A discussion on the diffraction of convex cylindrical shock wave segments is presented. Cylindrical shock waves are considered diffracting around several convex sharp corners with different wall angles (\(27.5^{\circ }, 45^{\circ }\)). It is shown that the behaviour of cylindrical shocks is qualitatively similar to that of plane shocks but quantitatively different. These differences can be attributed to the cylindrical shock’s non-constant Mach number and its varying orientation along its profile. The analysis follows the concept of a first disturbance on the shock front and its locus in the laboratory’s frame of reference. Using the concept of disturbances propagating on the shock front introduced by Whitham, a method for determining the locus of the first disturbance is presented. A comparison is made between the loci calculated using this method and those obtained from CFD simulations. Finally, an analytical method for determining the locus of the inflection point is also introduced. The results from the methods presented show good correspondence with calculations from CFD.


Cylindrical shock wave Diffraction Reflection Geometric shock dynamics 

List of symbols


Cross-sectional area between two rays


Speed of sound ahead of and behind the shock front

c, \(c_{i}\)

Speed of disturbances propagating on the shock front


Parameter for grid convergence study


Shock Mach number


Initial shock Mach number


Number of grid elements

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

General normal vector and normal vector at point i, respectively


Order of numerical method


Ratio between grid elements of a fine and coarse mesh. \(r>1\)


Shock radius


Time taken for a cylindrical shock to collapse to its centre

\(u_1, u_2\)

Flow velocity ahead of and behind the shock front


Shock speed


Rectangular coordinate system

\((\alpha ,\beta )\)

Curvilinear coordinate system

\(\gamma \)

Specific heat capacity ratio

\(\epsilon \)

Numerical error

\(\theta _{\mathrm{span}}\)

Arc angle of a cylindrical shock

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

Wall angle with respect to the x-axis

\(\nu \)

Angle of inflection of a diffracting shock



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


  1. 1.
    Skews, B.W.: The shape of a diffracting shock wave. J. Fluid Mech. 29(2), 297–304 (1967). CrossRefGoogle Scholar
  2. 2.
    Whitham, G.B.: A new approach to problems of shock dynamics Part I Two-dimensional problems. J. Fluid Mech. 2(2), 145–171 (1957). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Milton, B.E.: Mach reflection using ray-shock theory. AIAA J 13, 1531–1533 (1975). CrossRefzbMATHGoogle Scholar
  4. 4.
    Itoh, S., Okazaki, N., Itaya, M.: On the transition between regular and Mach reflection in truly non-stationary flows. J. Fluid Mech. 108, 383–400 (1981). CrossRefGoogle Scholar
  5. 5.
    Guderley, G.: Starke kugelige und zylindrische Verdichtungsstösse in de Nähe des Kugelmittelpunktes bzw de Zylinderachse. Luftfahrtforschung 19, 128–129 (1942)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Lighthill, M.J.: Diffraction of blast. I. Proc. R. Soc. A Math. Phys. Eng. Sci. 198, 454–470 (1949). MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chisnell, R.F.: The motion of a shock wave in a channel, with applications to cylindrical and spherical shock waves. J. Fluid Mech. 2(3), 286–298 (1957). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chester, W.: CXLV. The quasi-cylindrical shock tube. Lond. Edinb. Dublin Philos. Mag. J. Sci. 45(371), 1293–1301 (1954). CrossRefzbMATHGoogle Scholar
  9. 9.
    Roache, P.J.: Perspective: A new method for uniform reporting of grid independence studies. J. Fluids Eng. 116(3), 405–413 (1994). CrossRefGoogle Scholar
  10. 10.
    Henshaw, W.D., Smyth, N.F., Schwendeman, D.W.: Numerical shock propagation using geometrical shock dynamics. J. Fluid Mech. 171, 519–545 (1986). CrossRefzbMATHGoogle Scholar
  11. 11.
    Skews, B., Gray, B., Paton, R.: Experimental production of two-dimensional shock waves of arbitrary profile. Shock Waves 25, 1–10 (2015). CrossRefGoogle Scholar
  12. 12.
    Wang, H., Zhai, Z., Luo, X., Yang, J., Lu, X.: A specially curved wedge for eliminating wedge angle effect in unsteady shock reflection. Phys. Fluids 29(8), 086103 (2017). CrossRefGoogle Scholar
  13. 13.
    Ndebele, B.B., Skews, B.W.: The reflection of cylindrical shock wave segments on cylindrical concave wall segments. Shock Waves 28, 1185–1197 (2018). CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.University of the WitwatersrandJohannesburgSouth Africa

Personalised recommendations