Kinematics of a shock wave of arbitrary strength in a non-ideal gas
Authors:
Manoj Pandey and V. D. Sharma
Journal:
Quart. Appl. Math. 67 (2009), 401-418
MSC (2000):
Primary 35L50, 35L67, 35L65, 76L05
DOI:
https://doi.org/10.1090/S0033-569X-09-01111-5
Published electronically:
May 5, 2009
MathSciNet review:
2547633
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Singular surface theory is used to study the evolutionary behaviour of an unsteady three-dimensional motion of a shock wave of arbitrary strength propagating through a non-ideal gas. The dynamical coupling between the shock front and the induced discontinuities behind it is investigated by considering an infinite system of transport equations governing the strength of a shock wave and the induced discontinuities behind it. This infinite system, when subjected to a truncation approximation, efficiently describes the shock motion. Disturbances propagating on the shock and the onset of shock-shocks are briefly discussed. For a two-dimensional shock motion, our transport equations bear a structural resemblance to those of geometrical shock dynamics. Attention is drawn to the connection between the transport equation obtained by using the truncation rule and the one obtained by using the characteristic rule. The effects of van der Waals’ excluded volume and wavefront geometry on the evolutionary behaviour of shocks are discussed.
References
- R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Springer-Verlag, New York-Heidelberg, 1976. Reprinting of the 1948 original; Applied Mathematical Sciences, Vol. 21. MR 0421279
- Lighthill, M. J., Higher approximations in aerodynamic theory, Princeton University Press, Princeton, N.J., (1960).
- R. E. Meyer, Theory of characteristics of inviscid gas dynamics, Handbuch der Physik, Vol. 9, Part 3, Springer-Verlag, Berlin, 1960, pp. 225–282. MR 0119653
- Ardavan - Rhad, H., The decay of a plane shock wave, J. Fluid Mech. 43 (1970), 737-751.
- L. Sirovich and T. H. Chong, Approximate solution in gasdynamics, Phys. Fluids 23 (1980), no. 7, 1291–1295. MR 578116, DOI https://doi.org/10.1063/1.863138
- Sharma, V. D., Ram, R. and Sachdev, P., Uniformly valid analytical solution to the problem of decaying shock wave, J. Fluid Mech. 185 (1987) 153-170.
- Chong, T. H. and Sirovich, L., Numerical integration of the gasdynamic equation, Phys. Fluids 23 (1980), 1296-1300.
- Lewis, T. S. and Sirovich, L., Approximate and exact numerical supersonic flow over an airofoil, J. Fluid Mech. 112 (1981), 265-282.
- G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0483954
- Collins, R. and Chen, H. T., Propagation of shock wave of arbitrary strength in two half planes containing a free surface, J. Comp. Phys. 5 (1970), 415-422.
- Collins, R. and Chen, H. T., Motion of a shock wave through a non-uniform fluid, In Proc. 2nd Int. Conf. on Numerical Methods in Fluid Dynamics (ed. M. Holt). Lecture Notes in Physics. Vol. 8, pp. 264-269. Springer.
- C. J. Catherasoo and B. Sturtevant, Shock dynamics in nonuniform media, J. Fluid Mech. 127 (1983), 539–561. MR 698526, DOI https://doi.org/10.1017/S0022112083002876
- Henshaw, W. D., Smyth, N. F. and Schwendeman, D. W., Numerical shock propagation using geometrical shock dynamics, J. Fluid Mech. 171 (1986), 519-545.
- D. W. Schwendeman, A numerical scheme for shock propagation in three dimensions, Proc. Roy. Soc. London Ser. A 416 (1988), no. 1850, 179–198. MR 936120
- J. E. Cates and B. Sturtevant, Shock wave focusing using geometrical shock dynamics, Phys. Fluids 9 (1997), no. 10, 3058–3068. MR 1472445, DOI https://doi.org/10.1063/1.869414
- D. W. Schwendeman, A new numerical method for shock wave propagation based on geometrical shock dynamics, Proc. Roy. Soc. London Ser. A 441 (1993), no. 1912, 331–341. MR 1219259, DOI https://doi.org/10.1098/rspa.1993.0064
- Nunziato, J. W. and Walsh, E. K., Propagation and growth of shock waves in inhomogeneous fluids, Phys. Fluids 15 (1972), 1397-1402.
- Chen, P. J., Selected topics in wave propagation, Noordhoff, Leyden (1976).
- Wright, T. W., An intrinsic description of unsteady shock waves, Q. J. Mech. Appl. Math. 29 (1976), 311-324.
- Maslov, V. P., Propagation of shock waves in an isentropic non viscous gas, J. Soviet Math. 13 (1980), 119-163.
- M. A. Grinfel′d, Ray method of calculating the wave front intensity in nonlinearly elastic material; Russian transl., J. Appl. Math. Mech. 42 (1978), no. 5, 883–898. MR 620884
- A. M. Anile and G. Russo, Generalized wavefront expansion. I. Higher order corrections for the propagation of weak shock waves, Wave Motion 8 (1986), no. 3, 243–258. MR 841962, DOI https://doi.org/10.1016/S0165-2125%2886%2980047-6
- Y. B. Fu and N. H. Scott, One-dimensional shock waves in simple materials with memory, Proc. Roy. Soc. London Ser. A 428 (1990), no. 1875, 547–571. MR 1050540
- V. D. Sharma and Ch. Radha, On one-dimensional planar and nonplanar shock waves in a relaxing gas, Phys. Fluids 6 (1994), no. 6, 2177–2190. MR 1275083, DOI https://doi.org/10.1063/1.868220
- Renuka Ravindran, Strong shock approximation in the new theory of shock dynamics, Quart. J. Mech. Appl. Math. 50 (1997), no. 2, 251–259. MR 1451070, DOI https://doi.org/10.1093/qjmam/50.2.251
- Ch. Radha, V. D. Sharma, and A. Jeffrey, An approximate analytical method for describing the kinematics of a bore over a sloping beach, Appl. Anal. 81 (2002), no. 4, 867–892. MR 1929550, DOI https://doi.org/10.1080/0003681021000004474
- Jürgen Batt and Renuka Ravindran, Calculation of shock using solutions of systems of ordinary differential equations, Quart. Appl. Math. 63 (2005), no. 4, 721–746. MR 2187929, DOI https://doi.org/10.1090/S0033-569X-05-00976-6
- C. Truesdell and R. Toupin, The classical field theories, Handbuch der Physik, Bd. III/1, Springer, Berlin, 1960, pp. 226–793; appendix, pp. 794–858. With an appendix on tensor fields by J. L. Ericksen. MR 0118005
- V. V. Menon, V. D. Sharma, and A. Jeffrey, On the general behavior of acceleration waves, Applicable Anal. 16 (1983), no. 2, 101–120. MR 709814, DOI https://doi.org/10.1080/00036818308839462
- Sharma, V. D. and Menon, V. V., Further comments on the behavior of acceleration waves of arbitrary shape, J. Math. Phys. 22 (1981), 683-684.
- Best, J. P., A generalization of the theory of geometrical shock dynamics, Shock Waves, 1 (1991), 251-273.
References
- Courant, R. and Friedrichs, K. O., Supersonic flow and shock waves, Springer, New York (1999). MR 0421279 (54:9284)
- Lighthill, M. J., Higher approximations in aerodynamic theory, Princeton University Press, Princeton, N.J., (1960).
- Meyer, R. E., In Handbuch der Physik, Vol. IX, edited by S. Flugge (Springer-Verlag, Berlin) (1960). MR 0119653 (22:10414)
- Ardavan - Rhad, H., The decay of a plane shock wave, J. Fluid Mech. 43 (1970), 737-751.
- Sirovich, L. and Chong, T. H., Approximate solution in gasdynamics, Phys. Fluids 23 (1980), 1291-1295. MR 578116 (81d:76071)
- Sharma, V. D., Ram, R. and Sachdev, P., Uniformly valid analytical solution to the problem of decaying shock wave, J. Fluid Mech. 185 (1987) 153-170.
- Chong, T. H. and Sirovich, L., Numerical integration of the gasdynamic equation, Phys. Fluids 23 (1980), 1296-1300.
- Lewis, T. S. and Sirovich, L., Approximate and exact numerical supersonic flow over an airofoil, J. Fluid Mech. 112 (1981), 265-282.
- Whitham, G. B., Linear and nonlinear waves, Wiley, New York (1974). MR 0483954 (58:3905)
- Collins, R. and Chen, H. T., Propagation of shock wave of arbitrary strength in two half planes containing a free surface, J. Comp. Phys. 5 (1970), 415-422.
- Collins, R. and Chen, H. T., Motion of a shock wave through a non-uniform fluid, In Proc. 2nd Int. Conf. on Numerical Methods in Fluid Dynamics (ed. M. Holt). Lecture Notes in Physics. Vol. 8, pp. 264-269. Springer.
- Catherasoo, C. J. and Sturtevant, B., Shock dynamics in nonuniform media, J. Fluid Mech. 127, (1983) 539-561. MR 698526 (84e:76044)
- Henshaw, W. D., Smyth, N. F. and Schwendeman, D. W., Numerical shock propagation using geometrical shock dynamics, J. Fluid Mech. 171 (1986), 519-545.
- Schwendeman, D. W., A numerical scheme for shock propagation in three dimensions, Proc. R. Soc. London Ser. A 416 (1988), 179-198. MR 936120 (89c:76077)
- Cates, J. E. and Sturtevant. B., Shock wave focusing using geometrical shock dynamics, Phys. Fluids 9 (1997), 3058-3068. MR 1472445
- Schwendeman, D. W., A new numerical method for shock wave propagation on geometrical shock dynamics, Proc. R. Soc. London, A 441 (1993), 331-341. MR 1219259 (94i:76046)
- Nunziato, J. W. and Walsh, E. K., Propagation and growth of shock waves in inhomogeneous fluids, Phys. Fluids 15 (1972), 1397-1402.
- Chen, P. J., Selected topics in wave propagation, Noordhoff, Leyden (1976).
- Wright, T. W., An intrinsic description of unsteady shock waves, Q. J. Mech. Appl. Math. 29 (1976), 311-324.
- Maslov, V. P., Propagation of shock waves in an isentropic non viscous gas, J. Soviet Math. 13 (1980), 119-163.
- Grinfeld, M. A., Ray method of calculating the wavefront intensity in nonlinearly elastic material, PMM 42 (1978), 958-977. MR 620884 (82j:73024)
- Anile, A. M. and Russo, G., Generalized wavefront expansion I : Higher order corrections for the propagation of weak shock waves, Wave Motion 8 (1986), 243-258. MR 841962 (87h:76081)
- Fu, Y. B. and Scott, N. H., One dimensional shock waves in simple materials with memory, Proc. R. Soc. Lond. A 428 (1990), 547-571. MR 1050540 (91k:73048)
- Sharma, V. D. and Radha, Ch., On one-dimensional planar and non-planar shock in a relaxing gas, Phys. Fluids 6 (1994), 2177-2190. MR 1275083 (95b:76053)
- Ravindran, Renuka, Strong shock approximation in the new theory of shock dynamics, Quart. J. Mech. Appl. Math. 50 (1997), 251-259. MR 1451070 (98d:76101)
- Radha, Ch., Sharma, V. D., and Jeffrey, A., An approximate analytical method for describing the kinematics of a bore over a sloping beach, Appl. Anal. 81 (2002), 867-892. MR 1929550 (2003h:76054)
- Batt, Jürgen and Ravindran, Renuka, Calculation of shock using solutions of systems of ordinary differential equations, Quart. Appl. Math. 63 (2005), 721-746. MR 2187929 (2007f:35245)
- Truesdell, C. and Toupin, R. A., The classical field theories, In Handbuch der Physik, Vol. III, Berlin, Springer-Verlag (1960). MR 0118005 (22:8778)
- Menon, V. V, Sharma, V. D., and Jeffrey, A., On the general behavior of acceleration waves, Applicable Analysis 16 (1983), 101-120. MR 709814 (84h:73014)
- Sharma, V. D. and Menon, V. V., Further comments on the behavior of acceleration waves of arbitrary shape, J. Math. Phys. 22 (1981), 683-684.
- Best, J. P., A generalization of the theory of geometrical shock dynamics, Shock Waves, 1 (1991), 251-273.
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2000):
35L50,
35L67,
35L65,
76L05
Retrieve articles in all journals
with MSC (2000):
35L50,
35L67,
35L65,
76L05
Additional Information
Manoj Pandey
Affiliation:
Department of Mathematics, I.I.T. Bombay, Powai Mumbai, India 400076
Email:
mkp@math.iitb.ac.in
V. D. Sharma
Affiliation:
Department of Mathematics, I.I.T. Bombay, Powai Mumbai, India 400076
Email:
vsharma@maths.iitb.ac.in
Keywords:
Singular surface theory,
non-ideal gas,
shock wave,
geometrical shock dynamics
Received by editor(s):
March 21, 2007
Published electronically:
May 5, 2009
Additional Notes:
Research support from ISRO-IIT Bombay, Space Technology Cell (Ref. No. 05-IS001) is gratefully acknowledged.
Article copyright:
© Copyright 2009
Brown University
The copyright for this article reverts to public domain 28 years after publication.