Abstract
The numerical solution of several examples of plane shock waves using artificial viscosity and their comparison with theoretical predictions is the dominant feature of this chapter. The Lagrangian form of the equations in plane geometry is derived and after a short introduction to finite difference representations of differential equations, the discrete form of the equations is presented. Numerical solutions involving plane shocks arising from various forms of piston motion are presented, discussed and compared with the predictions of the Rankine-Hugoniot equations of Chap. 3. Reflected shocks are also considered and the numerical results are compared with the theoretical predictions. Piston withdrawal from a tube that generates an expansion wave is also discussed and the numerical results are compared with the predictions based on the method of characteristics presented in Chap. 2. Further discussion of this piston withdrawal problem is presented in Appendix A and the numerical results are compared with the method of characteristics in the case where the expansion fan reflects at an end wall. Numerical results arising from an analysis of the shock tube are also presented and further numerical results for a closed shock tube can be found in Appendix B. New sections dealing with the effects of amplitude on wave propagation, short duration piston motion and shock decay are included, together with some numerical results of shock wave interactions for overtaking and colliding shock waves.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Partial derivatives are used here to indicate the changes in position and time of specific particles; nonetheless, it should be understood that these partial derivatives imply that we are in fact following the path taken by specific particles of fluid according to the Lagrangian description.
References
J. Von Neumann, R.D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys. 21, 232 (1950)
C.F. Sprague III, The Numerical Treatment of Simple Hydrodynamic Shocks Using the Von Neumann-Richtmyer Method, LA-1912 Report (Los Alamos Scientific Laboratory of the University of California, Los Alamos, 1955)
Ya.B. Zel’dovich, Yu.P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, (Dover Publications, Inc., Mineola, 2002), Chapter 1
J.D. Anderson Jr., Computational Fluid Dynamics: The Basics with Applications (McGraw-Hill, New York, 1995)
T.J. Chung, Computational Fluid Dynamics, 2nd edn. (Cambridge University Press, New York, 2010), Chapter 3
J.D. Ramshaw, Elements of Computational Fluid Dynamics (Imperial College Press, London, 2011)
K.W. Morton, D.F. Mayers, Numerical Solution of Partial Differential Equations (Cambridge University Press, New York, 1994)
D. Mihalas, B. Weibel-Mihalas, Foundations of Radiation Hydrodynamics (Dover Publications Inc., New York, 1999), p. 273
J.D. Anderson, Modern Compressible Flow with Historical Perspective, 3rd edn. (McGraw-Hill, 2003), Chapter 7
H.W. Liepmann, A. Roshko, Elements of Gas Dynamics, (Dover Pub. Inc., Mineola, 1956), Chapter 3
N. Curle, H.J. Davies, Modern Fluid Dynamics, vol. 2, (Van Nostrand Reinhold Company, London, 1971), Section 3.4
G.B. Whitham, Linear and Nonlinear Waves, (John Wiley & Sons, Inc., New York, 1999), Chapter 6
S. Chandrasekhar, On the Decay of Plane Shock Waves, Report No. 423 (Ballistics Research Laboratory, Aberdeen Proving Ground, Maryland, 1943)
K.O. Friedrichs, Formation and Decay of Shock Waves, Report: NYU IMM-158 Institute for Mathematics and Mechanics (New York University, New York, 1947)
A. Kantrowitz, One-Dimensional Treatment of Non-steady Gas Dynamics, Section C, in Fundamentals of Gas Dynamics, ed. by H. W. Emmons, vol. 2, (Princeton University Press, Princeton, 1958)
L.D. Landau, E.M. Lifshitz, Fluid Mechanics, (Pergamon Press, London, 1966), Chapter 10
R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves: A Manual on the Mathematical Theory of Non-Linear Wave Motion (New York, Courant Institute of Mathematical Sciences, New York University, 1944), p. 115
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Prunty, S. (2021). Numerical Treatment of Plane Shocks. In: Introduction to Simple Shock Waves in Air. Shock Wave and High Pressure Phenomena. Springer, Cham. https://doi.org/10.1007/978-3-030-63606-7_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-63606-7_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-63605-0
Online ISBN: 978-3-030-63606-7
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)