×

Linear Rayleigh-Taylor instability analysis of double-shell Kidder’s self-similar implosion solution. (English) Zbl 1378.76029

Summary: This paper generalizes the single-shell Kidder’s self-similar solution to the double-shell one with a discontinuity in density across the interface. An isentropic implosion model is constructed to study the Rayleigh-Taylor instability for the implosion compression. A Godunov-type method in the Lagrangian coordinates is used to compute the one-dimensional Euler equation with the initial and boundary conditions for the double-shell Kidder’s self-similar solution in spherical geometry. Numerical results are obtained to validate the double-shell implosion model. By programming and using the linear perturbation codes, a linear stability analysis on the Rayleigh-Taylor instability for the double-shell isentropic implosion model is performed. It is found that, when the initial perturbation is concentrated much closer to the interface of the two shells, or when the spherical wave number becomes much smaller, the modal radius of the interface grows much faster, i.e., more unstable. In addition, from the spatial point of view for the compressibility effect on the perturbation evolution, the compressibility of the outer shell has a destabilization effect on the Rayleigh-Taylor instability, while the compressibility of the inner shell has a stabilization effect.

MSC:

76E17 Interfacial stability and instability in hydrodynamic stability
35Q30 Navier-Stokes equations
76M20 Finite difference methods applied to problems in fluid mechanics
76M55 Dimensional analysis and similarity applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Rayleigh, L. Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. 14(1), 170–177 (1883) · JFM 15.0848.02 · doi:10.1112/plms/s1-14.1.170
[2] Taylor, G. I. The instability of liquid surface when accelerated in a direction perpendicular to their planes. Proc. R. Soc. Lond. A 201(1065), 192–196 (1950) · Zbl 0038.12201 · doi:10.1098/rspa.1950.0052
[3] Richtmyer, R. D. Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Math. 13(2), 297–319 (1960) · doi:10.1002/cpa.3160130207
[4] Meshkov, E. E. Instability of the interface of two gases accelerated by a shock wave. Fluid Dyn. 4(5), 151–157 (1969)
[5] Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability, Oxford University Press, London (1961)
[6] Kidder, R. E. Laser-driven compression of hollow shells: power requirements and stability limitations. Nucl. Fusion 16(1), 3–14 (1976) · doi:10.1088/0029-5515/16/1/001
[7] Breil, J., Hallo, L., Maire, P. H., and Olazabal-Loumé, M. Hydrodynamic instabilities in axisymmetric geometry self-similar models and numerical simulations. Laser Part. Beams 23, 155–160 (2005) · doi:10.1017/S0263034605050251
[8] Jaouen, S. A purely lagrangian method for computing linearly-perturbed flows in spherical geometry. J. Comp. Phys. 225(1), 464–490 (2007) · Zbl 1343.76029 · doi:10.1016/j.jcp.2006.12.008
[9] Shui, H. S. One-Dimensional Hydrodynamic Finite Difference Method (in Chinese), National Defense Industry Press, Beijing (1998)
[10] Sharp, D. H. An overview of Rayleigh-Taylor instability. Physica D 12, 3–18 (1984) · Zbl 0577.76047 · doi:10.1016/0167-2789(84)90510-4
[11] Després, B. Lagrangian systems of conservation laws. invariance properties of Lagrangian systems of conservation laws, approximate Riemann solvers and the entropy condition. Numer. Math. 89(1), 99–134 (2001) · Zbl 0990.65098 · doi:10.1007/PL00005465
[12] Livescu, D. Compressible effects on the Rayleigh-Taylor instability growth between immiscible fluids. Phys. Fluids 16(1), 118–127 (2004) · Zbl 1186.76332 · doi:10.1063/1.1630800
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.