Skip to main content
SpringerLink
Log in

Shock Waves

  • Chapter
  • First Online:
Book cover Waves and Compressible Flow

Part of the book series: Texts in Applied Mathematics ((TAM,volume 47 ))

  • 2246 Accesses

Abstract

The time has come to face up to the task of making a mathematical model that can deal with flows containing shock waves or shocks, across which the various dependent physical variables themselves have discontinuities. Such discontinuities are often called jump discontinuities in contrast to situations in which only the derivatives of the physical variables have discontinuities.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 19.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 29.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 39.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    If the interior of C is not a convex region, we may have to divide it up into convex regions in order to do the integration illustrated in Figure 6.3, but the final result (6.4) will still hold.

  2. 2.

    The basic idea of a weak solution to (6.1) is to replace (6.4) by \(\iint _{S}(f\partial \phi /\partial x +\rho \partial \phi /\partial t)\,dxdt = 0\), where S is an arbitrary fixed region in the (x, t) plane and ϕ is any suitably smooth test function (see [43]).

  3. 3.

    An alternative way of describing the flow through an oblique shock graphically is to use the shock polar as in Exercise 6.8.

  4. 4.

    In view of the dimensionality arguments of Section 5.4.2 and Exercise 5.19, this phenomenon should not occur for a corner in an infinite wall. However, because the flow behind the shock at A is subsonic, the equations are locally elliptic, and the solution will depend on conditions downstream that will, in practice, involve a length scale which determines the stand-off distance and the shock curvature. We will see another example of such a configuration at the end of Section 6.2.2.

  5. 5.

    Practical observations indicate that the transition from an undular bore to a turbulent bore occurs as the ratio of the increase in depth to the original depth increases through a value of around 0.3.

  6. 6.

    Note that we have implicitly assumed that the flow is compressible. Clearly an incompressible flow can flow smoothly through any slowly varying nozzle.

  7. 7.

    Depending on the aerofoil shape and angle of incidence, the leading edge could emit either two weak shock waves or a weak shock and a weak expansion wave.

  8. 8.

    This Newtonian limit is not so unrealistic because molecular effects in hypersonic flows cause γ to decrease below 1.4 for air.

  9. 9.

    Figures 6.28, 6.29 and 6.30 are taken from [47], see also [38].

    Fig. 6.28
    figure 28

    Plot of n i from (6.78).

    Full size image
    Fig. 6.29
    figure 29

    Numerical solution for n i and \(\bar{u}\) in a shock tube when n r is O(1).

    Full size image
    Fig. 6.30
    figure 30

    Level curves for u in the shock tube when n r  < < 1

    Full size image

References

  1. Anderson, J. D. (1989). Hypersonic and high temperature gas dynamics. New York: McGraw-Hill.

    Google Scholar 

  2. Chapman, C. J. (2000). High speed flow. Cambridge: Cambridge University Press.

    MATH  Google Scholar 

  3. Courant, R., & Friedrichs, K. O. (1948). Supersonic flow and shock waves. New York: Interscience.

    MATH  Google Scholar 

  4. Davison, L. (2010). Fundamentals of shock wave propagation in solids. New York: Springer.

    Google Scholar 

  5. Edwards, C. M., Howison, S. D., Ockendon, H., & Ockendon, J. R. (2008). Non-classical shallow water flows. IMA Journal of Applied Mathematics, 73, 137–157.

    Article  MathSciNet  MATH  Google Scholar 

  6. Garabedian, P. R. (1964). Partial differential equations. New York: John Wiley and Sons.

    MATH  Google Scholar 

  7. Germain, P., & Lee, E. H. (1973). On shockwaves in elasto-plastic solids. Journal of the Mechanics and Physics of Solids, 21, 359–382.

    Article  MATH  Google Scholar 

  8. Guderley, K. G. (1962). The theory of transonic flow. Elmsford, NY: Pergamon. Translated by J. R. Moszynski.

    MATH  Google Scholar 

  9. Hayes, W., & Probstein, R. (1959). Hypersonic flow. New York: Academic Press.

    MATH  Google Scholar 

  10. Howell, P. D., Ockendon, H., Ockendon, J. R. (2012). Mathematical modelling of elastoplasticity at high stress. Proceedings of the Royal Society A, 468, 3842–3863.

    Google Scholar 

  11. Keyfitz, B.-L. (1999). Conservation laws, delta shocks and singular shocks. In Nonlinear theory of generalised functions. Research notes in mathematics (pp. 99–111). Boca Raton: Chapman and Hall.

    Google Scholar 

  12. Lax, P. D. (1953). Nonlinear hyperbolic equations. Communications on Pure and Applied Mathematics, 6, 231–258.

    Article  MathSciNet  MATH  Google Scholar 

  13. Leveque, R. J. (2004). The dynamics of pressureless dust clouds and delta waves. Journal of Hyperbolic Differential Equations, 1, 315–327.

    Article  MathSciNet  MATH  Google Scholar 

  14. Liepmann, H. W., & Roshko, A. (1957). Elements of gas dynamics. New York: John Wiley and Sons.

    Google Scholar 

  15. Mora, P. (2003). Plasma expansion into a vacuum. Physical Review Letters, 90, 185002.

    Article  Google Scholar 

  16. Newton, I. (1871). Principia. New York: Daniel Adee. Translated by A. Motte (1846).

    Google Scholar 

  17. Ockendon, J., Howison, S., Lacey, A., & Movchan, A. (1999). Applied partial differential equations. Oxford: Oxford University Press.

    MATH  Google Scholar 

  18. Perego, M., Howell, P. D., Gunzburger, M. D., Ockendon, J. R., & Allen, J. E. (2013). The expansion of a collisionless plasma into a plasma of lower density. Physics of Plasmas, 20, 052101.

    Article  Google Scholar 

  19. Van Dyke, M. (1975). Perturbation methods in fluid dynamics. Stanford, CA: Parabolic.

    Google Scholar 

  20. Van Dyke, M. (1982). An album of fluid motion. Stanford, CA: Parabolic.

    Google Scholar 

  21. Whitham, G. B. (1974). Linear and nonlinear waves. New York: Wiley.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Oxford, UK

    Hilary Ockendon & John R. Ockendon

Authors
  1. Hilary Ockendon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John R. Ockendon
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Science+Business Media New York

About this chapter

Cite this chapter

Ockendon, H., Ockendon, J.R. (2015). Shock Waves. In: Waves and Compressible Flow. Texts in Applied Mathematics, vol 47 . Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3381-5_6

Download citation

Publish with us

Policies and ethics

Search

Navigation