Evolution of Hydrodynamic Instability on Planar Jelly Interface Driven by Explosion

ABSTRACT

A high precision numerical algorithm MVPPM (multi-viscous-fluid piecewise parabolic method) is proposed and applied to study the multi-viscous-fluid dynamics problems. Three planar jelly experiments with periodic cosine perturbation on the initial interface have been conducted and numerically simulated by MVPPM. Good agreement between experimental and numerical results has been achieved, including the shape of jelly interface, the displacements of front face of jelly layer, bubble top and spike head. The effects of initial conditions (including amplitude and wave length of perturbation, thickness of jelly layer, etc.) on the evolution of the jelly interface have been numerically analyzed. It is found that the key affecting factors are the perturbation amplitude and thickness of jelly layer. The hydrodynamic instability on double planar jelly layers driven by explosion has been investigated numerically to examine their laws of evolution, and an interesting phenomenon is observed.

A high precision numerical algorithm MVPPM (multi-viscous-fluid piecewise parabolic method) is proposed and applied to study the multi-viscous-fluid dynamics problems. Three planar jelly experiments with periodic cosine perturbation on the initial interface have been conducted and numerically simulated by MVPPM. Good agreement between experimental and numerical results has been achieved, including the shape of jelly interface, the displacements of front face of jelly layer, bubble top and spike head. The effects of initial conditions (including amplitude and wave length of perturbation, thickness of jelly layer, etc.) on the evolution of the jelly interface have been numerically analyzed. It is found that the key affecting factors are the perturbation amplitude and thickness of jelly layer. The hydrodynamic instability on double planar jelly layers driven by explosion has been investigated numerically to examine their laws of evolution, and an interesting phenomenon is observed.

Cite this paper

nullT. Wang, J. Bai, W. Huang, Y. Jiang, L. Zou, P. Li and D. Tan, "Evolution of Hydrodynamic Instability on Planar Jelly Interface Driven by Explosion,"*World Journal of Mechanics*, Vol. 2 No. 3, 2012, pp. 152-161. doi: 10.4236/wjm.2012.23018.

nullT. Wang, J. Bai, W. Huang, Y. Jiang, L. Zou, P. Li and D. Tan, "Evolution of Hydrodynamic Instability on Planar Jelly Interface Driven by Explosion,"

References

[1] R. D. Richtmyer, “Taylor Instability in Shock Acceleration of Compressible Fluids,” Communications on Pure and Applied Mathematics, Vol. 13, No. 2, 1960, pp. 297- 319. doi:10.1002/cpa.3160130207

[2] E. E. Meshkov, “Instability of the Interface of Two Gases Accelerated by a Shock Wave,” Soviet Fluid Dynamics, Vol. 4, No. 5, 1969, pp. 101-104. doi:10.1007/BF01015969

[3] G. I. Taylor, “The Instability of Liquid Surfaces When Accelerated in a Direction Perpendicular to Their Planes. I,” Proceedings of the Royal Society A, Vol. 201, No. 1065, 1950, pp. 192-196. doi:10.1098/rspa.1950.0052

[4] D. H. Sharp, “An Overview of Rayleigh-Taylor Instability,” Physica D, Vol. 12, No. 1-3, 1984, pp. 3-18. doi:10.1016/0167-2789(84)90510-4

[5] S. Chandrasekhar, “Hydrodynamic and Hydromagnetic Stability,” Oxford Univer-sity Press, London, 1961.

[6] G. Dimonte and M. B. Schneider, “Turbulent Rayleigh- Taylor Instability Experiments with Variable Acceleration,” Physical Review E, Vol. 54, No. 4. 1996, pp. 3740-3743. doi:10.1103/PhysRevE.54.3740

[7] Y. A. Kucherenko1, A. P. Pylaev, V. D. Murzakov, et al., “Expe-rimental Study into the Asymptotic Stage of the Separtion of the Turbulized Mixtures in Gravitationally Stable Mode,” In: R. Young, J. Glimm and B. Boston, Eds., Proceeings of the 5th International Workshop on Compressible Turbulent Mixing, Stony Brook University Press, New York, 1996.

[8] J. W. Jacobs and J. M. Sheeley, “Experimental Study of Incompress-ible Richtmyer-Meshkov Instability,” Physics of Fluids, Vol. 8, No. 2, 1996, pp. 405-415. doi:10.1063/1.868794

[9] E. E. Meshkov, N. V. Nevmerz-hitsky, V. A. Pavlovskii, et al., “Jelly Technique Applications in Evolution Study of Hydrodynamic Instabilities on Unstable Plane and Cylindrical Surfaces,” In: R. Young, J. Glimm and B. Boston, Eds., Prceedings of the 5th International Workshop on Compressible Turbulent Mixing, Stony Brook University Press, New York, 1996.

[10] L. Houas, G. Jourdan, L. Schwaederlé, et al., “A New Large Cross-Section Shock Tube for Studies of Turbulent Mixing Induced by Interfacial Hydrodynamic Instability,” Shock Waves, Vol. 13, No. 5, 2003, pp. 431-434. doi:10.1007/s00193-002-0173-y

[11] S. H. R. Hosseini and K. Takayama, “Experimental Study of Richtmyer-Meshkov Insta-bility Induced by Cylindrical Shock Waves,” Physics of Fluids, Vol. 17, No. 8, 2005, p. 084101. doi:10.1063/1.1964916

[12] P. M. Rightley, P. Vorobieff, R. Martin, et al., “Experi- mental Observations of the Mixing Transition in a Shock- Accelerated Gas Curtain,” Physics of Fluids, Vol. 11, No. 1, 1999, pp. 186-200. doi:10.1063/1.869911

[13] B. J. Balakumar, G. C. Orlicz, C. D. Tomkins, et al., “Si- multaneous Particle-Image Velocimetry-Planar Laser-In- duced Fluorescence Measurements of Richtmyer-Meshkov instability Growth in a Gas Curtain with and without Re- shock,” Physics of Fluids, Vol. 20, No. 12, 2008, p. 124103. doi:10.1063/1.3041705

[14] J. W. Jacobs, “The Dynamics of Shock Accelerated Light and Heavy Gas Cylinders,” Physics of Fluids, Vol. 5, No. 9, 1993, pp. 2239-2247. doi:10.1063/1.858562

[15] C. D. Tomkins, K. P. Prestridge, P. M. Rightley, et al., “A Quantitative Study of the Interaction of Two Richtmyer- Meshkov Unstable Gas Cylinders,” Physics of Fluids, Vol. 15, No. 4, 2003, pp. 986-1004. doi:10.1063/1.1555802

[16] S. Kumar, P. Vorobieff, G. Orlicz, et al., “Complex Flow Morphologies in shock-Accelerated Ga-seous Flows,” Phy- sica D, Vol. 235, No. 1-2, 2007, pp. 21-28. doi:10.1016/j.physd.2007.04.023

[17] S. Gupta, S. Zhang and N. J. Zabusky, “Shock Interaction with a Heavy Gas Cylinder: Emergence of Vortex Bilay- ers and Vortex- Accelerated Ba-roclinic Circulation Gen- eration,” Laser and Particle Beams, Vol. 24, No. 3, 2003, pp. 443-448.

[18] S. Zhang, N. J. Za-busky, G. Z. Peng, et al., “Shock gases Cylinder Interactions: Dynamically Validated Initial Con- ditions Provide Excellent Agreement between Experiments and Numerical Simulations to Late-Intermediate Time,” Physics of Fluids, Vol. 16, No. 5, 2004, pp. 1203-1216. doi:10.1063/1.1651483

[19] S. Andronov and E. E. Meshkov, “Computational and Ex- perimental Studies of Hydro-Dynamic Instabilities and Turbulent Mixing,” Los Alamos National La-boratory, Los Alamos, 1995.

[20] B. Goodwin an S. Weir, “Rayleigh-Taylor Instability Ex- periments in a Cylindrically Convergent Geometry,” Los Alamos National Laboratory, Los Alamos, 1996.

[21] K. O. Mikaelian, “Rayleigh-Taylor and Richtmyer-Meshkov Instabilities and Mixing in Stratified Spherical Shells,” Physical Review A, Vol. 42, No. 6, 1990, pp. 3400-3420. doi:10.1103/PhysRevA.42.3400

[22] K. O. Mikaelian, “Ray-leigh-Taylor and Richtmyer-Meshkov Instabilities and Mixing in Stratified Cylindrical Shells,” Physics of Fluids, Vol. 17, No. 9, 2005, pp. 094105. doi:10.1063/1.2046712

[23] N. C. Hearn, T. Plewa, R. P. Drake, et al., “Flash Code Simulation of Rayleigh-Taylor and Richtmyer-Meshkov Instabilitites in Laser-Driven Experiments,” Astrophysics and Space Science, Vol. 307, No. 1-3, 2007, pp. 227-231. doi:10.1007/s10509-006-9226-5

[24] H. Lee, H. Jin, Y. Yu, et al., “On Validation of Turbulent Mixing Simulations for Ray-leigh-Taylor Instability,” Phy- sics of Fluids, Vol. 20, No. 1, 2008, p. 012102. doi:10.1063/1.2832775

[25] A. Marocchino, S. Atzeni and A. Schiavi, “Numerical Stu- dy of the Ablative Richtmyer-Meshkov Instability of La- ser-Irradiated Deuterium and Deuterium-Tritium Targets,” Physics of Plasmas, Vol. 17, No. 11, 2010, p. 112703. doi:10.1063/1.3505112

[26] L. Biferale, F. Mantovani and M. Sbragaglia, “High Re- solution Numerical Study of Ray-leigh-Taylor Turbulence Using a Thermal Lattice Boltzmann Scheme,” Physics of Fluids, Vol. 22, No. 11, 2010, p. 115112. doi:10.1063/1.3517295

[27] L. F. Wang, W. H. Ye, W. S. Don, et al., “Formation of Large-Scale Structures in Ablative Kel-vin-Helmholtz In- stability,” Physics of Plasmas, Vol. 17, No. 12, 2010, p. 122308. doi:10.1063/1.3524550

[28] M. A. Ullah, W. B. Gao and D. K. Mao, “Numerical Simu- lations of Rich-tmyer-Meshkov Instabilities Using Conser- vative Front-Tracking Method,” Applied Mathematics and Mechanics, Vol. 32, No. 1, 2011, pp. 119-132. doi:10.1007/s10483-011-1399-x

[29] T. Wang, J. S. Bai, P. Li, et al., “The Numerical Study of Shock-Induced Hydrodynamic Instability and Mixing,” Chinese Physics B, Vol. 18, No. 3, 2009, pp. 1127-1135. doi:10.1088/1674-1056/18/3/048

[1] R. D. Richtmyer, “Taylor Instability in Shock Acceleration of Compressible Fluids,” Communications on Pure and Applied Mathematics, Vol. 13, No. 2, 1960, pp. 297- 319. doi:10.1002/cpa.3160130207

[2] E. E. Meshkov, “Instability of the Interface of Two Gases Accelerated by a Shock Wave,” Soviet Fluid Dynamics, Vol. 4, No. 5, 1969, pp. 101-104. doi:10.1007/BF01015969

[3] G. I. Taylor, “The Instability of Liquid Surfaces When Accelerated in a Direction Perpendicular to Their Planes. I,” Proceedings of the Royal Society A, Vol. 201, No. 1065, 1950, pp. 192-196. doi:10.1098/rspa.1950.0052

[4] D. H. Sharp, “An Overview of Rayleigh-Taylor Instability,” Physica D, Vol. 12, No. 1-3, 1984, pp. 3-18. doi:10.1016/0167-2789(84)90510-4

[5] S. Chandrasekhar, “Hydrodynamic and Hydromagnetic Stability,” Oxford Univer-sity Press, London, 1961.

[6] G. Dimonte and M. B. Schneider, “Turbulent Rayleigh- Taylor Instability Experiments with Variable Acceleration,” Physical Review E, Vol. 54, No. 4. 1996, pp. 3740-3743. doi:10.1103/PhysRevE.54.3740

[7] Y. A. Kucherenko1, A. P. Pylaev, V. D. Murzakov, et al., “Expe-rimental Study into the Asymptotic Stage of the Separtion of the Turbulized Mixtures in Gravitationally Stable Mode,” In: R. Young, J. Glimm and B. Boston, Eds., Proceeings of the 5th International Workshop on Compressible Turbulent Mixing, Stony Brook University Press, New York, 1996.

[8] J. W. Jacobs and J. M. Sheeley, “Experimental Study of Incompress-ible Richtmyer-Meshkov Instability,” Physics of Fluids, Vol. 8, No. 2, 1996, pp. 405-415. doi:10.1063/1.868794

[9] E. E. Meshkov, N. V. Nevmerz-hitsky, V. A. Pavlovskii, et al., “Jelly Technique Applications in Evolution Study of Hydrodynamic Instabilities on Unstable Plane and Cylindrical Surfaces,” In: R. Young, J. Glimm and B. Boston, Eds., Prceedings of the 5th International Workshop on Compressible Turbulent Mixing, Stony Brook University Press, New York, 1996.

[10] L. Houas, G. Jourdan, L. Schwaederlé, et al., “A New Large Cross-Section Shock Tube for Studies of Turbulent Mixing Induced by Interfacial Hydrodynamic Instability,” Shock Waves, Vol. 13, No. 5, 2003, pp. 431-434. doi:10.1007/s00193-002-0173-y

[11] S. H. R. Hosseini and K. Takayama, “Experimental Study of Richtmyer-Meshkov Insta-bility Induced by Cylindrical Shock Waves,” Physics of Fluids, Vol. 17, No. 8, 2005, p. 084101. doi:10.1063/1.1964916

[12] P. M. Rightley, P. Vorobieff, R. Martin, et al., “Experi- mental Observations of the Mixing Transition in a Shock- Accelerated Gas Curtain,” Physics of Fluids, Vol. 11, No. 1, 1999, pp. 186-200. doi:10.1063/1.869911

[13] B. J. Balakumar, G. C. Orlicz, C. D. Tomkins, et al., “Si- multaneous Particle-Image Velocimetry-Planar Laser-In- duced Fluorescence Measurements of Richtmyer-Meshkov instability Growth in a Gas Curtain with and without Re- shock,” Physics of Fluids, Vol. 20, No. 12, 2008, p. 124103. doi:10.1063/1.3041705

[14] J. W. Jacobs, “The Dynamics of Shock Accelerated Light and Heavy Gas Cylinders,” Physics of Fluids, Vol. 5, No. 9, 1993, pp. 2239-2247. doi:10.1063/1.858562

[15] C. D. Tomkins, K. P. Prestridge, P. M. Rightley, et al., “A Quantitative Study of the Interaction of Two Richtmyer- Meshkov Unstable Gas Cylinders,” Physics of Fluids, Vol. 15, No. 4, 2003, pp. 986-1004. doi:10.1063/1.1555802

[16] S. Kumar, P. Vorobieff, G. Orlicz, et al., “Complex Flow Morphologies in shock-Accelerated Ga-seous Flows,” Phy- sica D, Vol. 235, No. 1-2, 2007, pp. 21-28. doi:10.1016/j.physd.2007.04.023

[17] S. Gupta, S. Zhang and N. J. Zabusky, “Shock Interaction with a Heavy Gas Cylinder: Emergence of Vortex Bilay- ers and Vortex- Accelerated Ba-roclinic Circulation Gen- eration,” Laser and Particle Beams, Vol. 24, No. 3, 2003, pp. 443-448.

[18] S. Zhang, N. J. Za-busky, G. Z. Peng, et al., “Shock gases Cylinder Interactions: Dynamically Validated Initial Con- ditions Provide Excellent Agreement between Experiments and Numerical Simulations to Late-Intermediate Time,” Physics of Fluids, Vol. 16, No. 5, 2004, pp. 1203-1216. doi:10.1063/1.1651483

[19] S. Andronov and E. E. Meshkov, “Computational and Ex- perimental Studies of Hydro-Dynamic Instabilities and Turbulent Mixing,” Los Alamos National La-boratory, Los Alamos, 1995.

[20] B. Goodwin an S. Weir, “Rayleigh-Taylor Instability Ex- periments in a Cylindrically Convergent Geometry,” Los Alamos National Laboratory, Los Alamos, 1996.

[21] K. O. Mikaelian, “Rayleigh-Taylor and Richtmyer-Meshkov Instabilities and Mixing in Stratified Spherical Shells,” Physical Review A, Vol. 42, No. 6, 1990, pp. 3400-3420. doi:10.1103/PhysRevA.42.3400

[22] K. O. Mikaelian, “Ray-leigh-Taylor and Richtmyer-Meshkov Instabilities and Mixing in Stratified Cylindrical Shells,” Physics of Fluids, Vol. 17, No. 9, 2005, pp. 094105. doi:10.1063/1.2046712

[23] N. C. Hearn, T. Plewa, R. P. Drake, et al., “Flash Code Simulation of Rayleigh-Taylor and Richtmyer-Meshkov Instabilitites in Laser-Driven Experiments,” Astrophysics and Space Science, Vol. 307, No. 1-3, 2007, pp. 227-231. doi:10.1007/s10509-006-9226-5

[24] H. Lee, H. Jin, Y. Yu, et al., “On Validation of Turbulent Mixing Simulations for Ray-leigh-Taylor Instability,” Phy- sics of Fluids, Vol. 20, No. 1, 2008, p. 012102. doi:10.1063/1.2832775

[25] A. Marocchino, S. Atzeni and A. Schiavi, “Numerical Stu- dy of the Ablative Richtmyer-Meshkov Instability of La- ser-Irradiated Deuterium and Deuterium-Tritium Targets,” Physics of Plasmas, Vol. 17, No. 11, 2010, p. 112703. doi:10.1063/1.3505112

[26] L. Biferale, F. Mantovani and M. Sbragaglia, “High Re- solution Numerical Study of Ray-leigh-Taylor Turbulence Using a Thermal Lattice Boltzmann Scheme,” Physics of Fluids, Vol. 22, No. 11, 2010, p. 115112. doi:10.1063/1.3517295

[27] L. F. Wang, W. H. Ye, W. S. Don, et al., “Formation of Large-Scale Structures in Ablative Kel-vin-Helmholtz In- stability,” Physics of Plasmas, Vol. 17, No. 12, 2010, p. 122308. doi:10.1063/1.3524550

[28] M. A. Ullah, W. B. Gao and D. K. Mao, “Numerical Simu- lations of Rich-tmyer-Meshkov Instabilities Using Conser- vative Front-Tracking Method,” Applied Mathematics and Mechanics, Vol. 32, No. 1, 2011, pp. 119-132. doi:10.1007/s10483-011-1399-x

[29] T. Wang, J. S. Bai, P. Li, et al., “The Numerical Study of Shock-Induced Hydrodynamic Instability and Mixing,” Chinese Physics B, Vol. 18, No. 3, 2009, pp. 1127-1135. doi:10.1088/1674-1056/18/3/048