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Shock Waves

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

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

Abstract

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.

Keywords

Cylindrical shock wave Diffraction Reflection Geometric shock dynamics 

List of symbols

A

Cross-sectional area between two rays

\(a_1,a_2\)

Speed of sound ahead of and behind the shock front

c, \(c_{i}\)

Speed of disturbances propagating on the shock front

f

Parameter for grid convergence study

M

Shock Mach number

\(M_0\)

Initial shock Mach number

N

Number of grid elements

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

General normal vector and normal vector at point i, respectively

p

Order of numerical method

r

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

\(r_{i}\)

Shock radius

\(t_{\mathrm{c}}\)

Time taken for a cylindrical shock to collapse to its centre

\(u_1, u_2\)

Flow velocity ahead of and behind the shock front

\(W_{\mathrm{s}}\)

Shock speed

(xy)

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

Notes

Acknowledgements

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

References

  1. 1.
    Skews, B.W.: The shape of a diffracting shock wave. J. Fluid Mech. 29(2), 297–304 (1967).  https://doi.org/10.1017/S0022112067000825 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).  https://doi.org/10.1017/S002211205700004X MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Milton, B.E.: Mach reflection using ray-shock theory. AIAA J 13, 1531–1533 (1975).  https://doi.org/10.2514/3.60566 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).  https://doi.org/10.1017/S0022112081002176 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).  https://doi.org/10.1098/rspa.1949.0113 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).  https://doi.org/10.1017/S0022112057000130 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chester, W.: CXLV. The quasi-cylindrical shock tube. Lond. Edinb. Dublin Philos. Mag. J. Sci. 45(371), 1293–1301 (1954).  https://doi.org/10.1080/14786441208561138 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).  https://doi.org/10.1115/1.2910291 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).  https://doi.org/10.1017/S0022112086001568 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).  https://doi.org/10.1007/s00193-014-0541-4 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).  https://doi.org/10.1063/1.4999349 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).  https://doi.org/10.1007/s00193-018-0812-6 CrossRefGoogle Scholar

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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