Rayleigh-Taylor stability for a normal shock wave-density discontinuity interaction. (English) Zbl 0592.76091
Summary: The solution for the perturbation growth of a shock wave striking a density discontinuity at a material interface is developed. The Laplace transformation of the perturbation results in an equation which has a simple solution for weak shock waves. The solution for strong shock waves may be given by a power series. It is assumed that the equation of state is that of an ideal gas. The four independent parameters of the solution are the ratio of specific heat for each material, the density ratio at the interface, and the incoming shock strength.
Properties of the solution which are investigated include the asymptotic behavior at large times of the perturbation velocity at the interface, the vorticity near the interface, and the rate of decay of the solution at large distances from the interface. The last is much weaker than the exponential decay in an incompressible fluid. The asymptotic solution near the interface, in addition to a constant term, consists of a number of slowly decaying discrete frequencies. The number is roughly proportional to the logarithm of the density ratio at the surface for strong shocks, and decreases with shock strength. For weak shocks the solution is compared with results for an incompressible fluid. Only interface perturbation velocities which tend to zero at large times lead to a limited deformation of the interface. It is found that these are possible only for density ratios less than about 1.5.
Properties of the solution which are investigated include the asymptotic behavior at large times of the perturbation velocity at the interface, the vorticity near the interface, and the rate of decay of the solution at large distances from the interface. The last is much weaker than the exponential decay in an incompressible fluid. The asymptotic solution near the interface, in addition to a constant term, consists of a number of slowly decaying discrete frequencies. The number is roughly proportional to the logarithm of the density ratio at the surface for strong shocks, and decreases with shock strength. For weak shocks the solution is compared with results for an incompressible fluid. Only interface perturbation velocities which tend to zero at large times lead to a limited deformation of the interface. It is found that these are possible only for density ratios less than about 1.5.
MSC:
76L05 | Shock waves and blast waves in fluid mechanics |
76E99 | Hydrodynamic stability |
76M99 | Basic methods in fluid mechanics |
76N15 | Gas dynamics (general theory) |
Keywords:
perturbation growth of a shock wave; density discontinuity; material interface; Laplace transformation; solution for weak shock waves; solution for strong shock waves; power series; ideal gas; shock strength; asymptotic behavior at large times; vorticity near the interface; rate of decay of the solution; asymptotic solution near the interface; slowly decaying discrete frequenciesReferences:
[1] | S. Chandrasekhar,Hydrodynamic and Hydromagnetic Stability(Clarendon, Oxford, 1961), Chap. X. |
[2] | Richtmyer, Commun. Pure Appl. Math. XIII pp 297– (1960) |
[3] | Meyer, Phys. Fluids 15 pp 753– (1972) |
[4] | Ye. Meshkov, NASA Technical Translation NASA TTF-13, 074 (1970). |
[5] | Kirpatrick, Nucl. Fusion 15 pp 333– (1975) · doi:10.1088/0029-5515/15/2/019 |
[6] | G. S. Fraley, W. P. Gula, D. B. Henderson, R. L. McCrory, R. C. Malone, R. J. Mason, and R. L. Morse, inPlasma Physics and Controlled Nuclear Fusion Research(IAEA, Vienna, 1975), Vol. II, p. 543. |
[7] | Fraley, Phys. Fluids 19 pp 1495– (1976) |
[8] | L. Spitzer,Physics of Fully Ionized Gases(Interscience, New York, 1962). · Zbl 0074.45001 |
[9] | R. Courant and K. O. Friedrichs,Supersonic Flow and Shock Waves(Interscience, New York, 1968), p. 8. · Zbl 0041.11302 |
[10] | Freeman, Proc. R. Soc. London Ser. A 228 pp 341– (1954) |
[11] | Briscoe, J. Fluid Mech. 31 pp 529– (1968) |
[12] | A. E. Roberts (private communication). |
[13] | Freeman, J. Fluid Mech. 2 pp 397– (1957) |
[14] | Lapworth, J. Fluid Mech. 6 pp 469– (1959) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.