Spiky Development at the Interface in Rayleigh-Taylor Instability: Layzer Approximation with Second Harmonic

Affiliation(s)

Department of Instrumentation Science, Centre for Plasma Studies, Jadavpur University, Kolkata, India.

Department of Instrumentation Science, Centre for Plasma Studies, Jadavpur University, Kolkata, India.

ABSTRACT

Layzer’s approximation method for investigation of two fluid interface structures associated with Rayleigh Taylor instability for arbitrary Atwood number is extended with the inclusion of second harmonic mode leaving out the zeroth harmonic one. The modification makes the fluid velocities vanish at infinity and leads to avoidance of the need to make the unphysical assumption of the existence of a time dependent source at infinity. The present analysis shows that for an initial interface perturbation with curvature exceeding , where *A *is the Atwood number there occurs an almost free fall of the spike with continuously increasing sharpening as it falls. The curvature at the tip of the spike also increases with Atwood number. Certain initial condition may also result in occurrence of finite time singularity as found in case of conformal mapping technique used earlier. However bubble growth rate is not appreciably affected.

Cite this paper

R. Banerjee, L. Mandal, M. Khan and M. Gupta, "Spiky Development at the Interface in Rayleigh-Taylor Instability: Layzer Approximation with Second Harmonic,"*Journal of Modern Physics*, Vol. 4 No. 2, 2013, pp. 174-179. doi: 10.4236/jmp.2013.42024.

R. Banerjee, L. Mandal, M. Khan and M. Gupta, "Spiky Development at the Interface in Rayleigh-Taylor Instability: Layzer Approximation with Second Harmonic,"

References

[1] J. D. Lindl, P. Amendt, R. L. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffmann, O. L. Landen and L. J. Suter, “The Physics Basis for Ignition Using Indirect-Drive Targets on the National Ignition Facility,” Physics of Plasmas, Vol. 11, No. 2, 2004, pp. 339-491. doi:10.1063/1.1578638

[2] D. Batani, W. Nazakov, T. Hall, T. Lower, M. Koenig, B. Faral, A. B. Mounaix, N. Grandjouan, “Foam-Induced Smoothing Studied through Laser-Driven Shock Waves,” Physical Review E, Vol. 62, No. 6, 2000, pp. 3573-3582. doi:10.1103/PhysRevE.62.8573

[3] R. Dezulian, F. Canova, S. Barbanotti, et al., “Hugoniot Data of Plastic Foams Obtained from Laser-Driven Shocks,” Physical Review E, Vol. 73, No. 4, 2006, pp. 047401-1-047401-4. doi:10.1103/PhysRevE.73.047401

[4] S. Atzeni and J. Meyer-Ter-Vehn, “The Physics of Inertial Fusion: Beam Plasma Interaction, Hydrodynamics, Hot Dense Mater,” Oxford University, London, 2004.

[5] D. Layzer, “On the Instability of Superposed Fluids in a Gravitational Field,” Astrophysical Journal, Vol. 122, No. 1, 1955, pp. 1-12. doi:10.1086/146048

[6] J. Hecht, U. Alon and D. Shvarts, “Potential Flow Models of Rayleigh-Taylor and Richtmyer-Meshkov Bubble Fronts,” Physical Fluids, Vol. 6, No. 12, 1994, pp. 4019-4030. doi:10.1063/1.868391

[7] Q. Zhang, “Analytical Solutions of Layzer-Type Approach to Unstable Interfacial Fluid Mixing,” Physical Review Letters, Vol. 81, No. 16, 1998, pp. 3391-3394. doi:10.1103/PhysRevLett.81.3391

[8] V. N. Goncharov, “Analytical Model of Nonlinear, Single-Mode, Classical Rayleigh-Taylor Instability at Arbitrary Atwood Numbers,” Physical Review Letters, Vol. 88, No. 13, 2002, pp. 134502-1-134502-4. doi:10.1103/PhysRevLett.88.134502

[9] S.-I. Sohn and Q. Zhang, “Late Time Behavior of Bubbles at Unstable Interfaces in Two Dimensions,” Physical Fluids, Vol. 13, No. 11, 2001, pp. 3493-3495. doi:10.1063/1.1410119

[10] S.-I. Sohn, “Analytic Solutions of Unstable Interfaces for All Density Ratios in Axisymmetric Flows,” Journal of Computational and Applied Mathematics, Vol. 177, No. 2, 2005, pp. 367-374. doi:10.1016/j.cam.2004.09.026

[11] S.-I. Sohn, “Simple Potential-Flow Model of Rayleigh-Taylor and Richtmyer-Meshkov Instabilities for All Density Ratios,” Physical Review E, Vol. 67, No. 2, 2003, pp. 026301-1-026301-5. doi:10.1103/PhysRevE.67.026301

[12] K. O. Mikaelian, “Limitations and Failures of the Layzer Model for Hydrodynamic Instabilities,” Physical Review E, Vol. 78, No. 1, 2008, pp. 015303-1-015303-4. doi:10.1103/PhysRevE.78.015303

[13] P. Ramaprabhu, G. Dimonte, Y.-N. Young, A. C. Calder and B. Fryxell, “Limits of the Potential Flow Approach to the Single-Mode Rayleigh-Taylor Problem,” Physical Review E, Vol. 74, No. 6, 2006, pp. 066308-1-066308-10. doi:10.1103/PhysRevE.74.066308

[14] P. Clavin and F. Williams, “Asymptotic Spike Evolution in Rayleigh? Taylor Instability,” Journal of Fluid Mechanics, Vol. 525, No. 5, 2005, pp. 105-113. doi:10.1017/S0022112004002630

[15] L. Duchemin, C. Josserand and P. Clavin, “Asymptotic Behavior of the Rayleigh-Taylor Instability,” Physical Review Letter, Vol. 94, No. 22, 2005, pp. 224501-1-224501-4. doi:10.1103/PhysRevLett.94.224501

[16] S. I. Abarzhi, K. Nishihara and J. Glimm, “Rayleigh? Taylor and Richtmyer? Meshkov Instabilities for Fluids with a Finite Density Ratio,” Physical Review A, Vol. 317, No. 5-6, 2003, pp. 470-475. doi:10.1016/j.physleta.2003.09.013

[17] S.-I. Sohn, “Bubble Interaction Model for Hydrodynamic Unstable Mixing,” Physical Review E, Vol. 75, No. 6, 2007, pp. 066312-1-066312-12. doi:10.1103/PhysRevE.75.066312

[18] S.-I. Sohn, “Effects of Surface Tension and Viscosity on the Growth Rates of Rayleigh-Taylor and Richt-Myer-Meshkov Instabilities,” Physical Review E, Vol. 80, No. 5, 2009, pp. 055302-1-55302-4. doi:10.1103/PhysRevE.80.055302

[19] M. R. Gupta, S. Roy, M. Khan, H. C. Pant, S. Sarkar and M. K. Srivastava, “Effect of Compressibility on the Rayleigh? Taylor and Richtmyer? Meshkov Instability Induced Nonlinear Structure at Two Fluid Interface,” Physics of Plasmas, Vol. 16, No. 3, 2009, pp. 032303-1-032303-12. doi:10.1063/1.3074789

[20] R. Betti and J. Sanz, “Bubble Acceleration in the Ablative Rayleigh-Taylor Instability,” Physical Review Letter, Vol. 97, No. 20, 2006, pp. 205002-1-205002-4. doi:10.1103/PhysRevLett.97.205002

[21] M. R. Gupta, L. Mandal, S. Roy and M. Khan, “Effect of Magnetic Field on Temporal Development of Rayleigh? Taylor Instability Induced Interfacial Nonlinear Structure,” Physics of Plasmas, Vol. 17, No. 1, 2010, pp. 012306-1-012306-12. doi:10.1063/1.3293120

[22] R. Banerjee, L. Mandal, S. Roy, M. Khan and M. R. Gupta, “Combined Effect of Viscosity and Vorticity on Single Mode Rayleigh? Taylor Instability Bubble Growth,” Physics of Plasmas, Vol. 18, No. 2, 2011, pp. 022109-1-022109-5. doi:10.1063/1.3555523

[23] T. Yoshikawa and A. Balk, “A Conformal-Mapping Model for Bubbles and Fingers of the Rayleigh-Taylor Instability,” Mathematical and Computer Modelling, Vol. 38, No. 1-2, 2003, pp. 113-121. doi:10.1016/S0895-7177(03)90009-4

[24] S. Tanveer, “Singularities in Water Waves and Rayleigh-Taylor Instability,” Proceedings of the Royal Society of London A, Vol. 441, No. 1913, 1993, pp. 501-525.

[1] J. D. Lindl, P. Amendt, R. L. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffmann, O. L. Landen and L. J. Suter, “The Physics Basis for Ignition Using Indirect-Drive Targets on the National Ignition Facility,” Physics of Plasmas, Vol. 11, No. 2, 2004, pp. 339-491. doi:10.1063/1.1578638

[2] D. Batani, W. Nazakov, T. Hall, T. Lower, M. Koenig, B. Faral, A. B. Mounaix, N. Grandjouan, “Foam-Induced Smoothing Studied through Laser-Driven Shock Waves,” Physical Review E, Vol. 62, No. 6, 2000, pp. 3573-3582. doi:10.1103/PhysRevE.62.8573

[3] R. Dezulian, F. Canova, S. Barbanotti, et al., “Hugoniot Data of Plastic Foams Obtained from Laser-Driven Shocks,” Physical Review E, Vol. 73, No. 4, 2006, pp. 047401-1-047401-4. doi:10.1103/PhysRevE.73.047401

[4] S. Atzeni and J. Meyer-Ter-Vehn, “The Physics of Inertial Fusion: Beam Plasma Interaction, Hydrodynamics, Hot Dense Mater,” Oxford University, London, 2004.

[5] D. Layzer, “On the Instability of Superposed Fluids in a Gravitational Field,” Astrophysical Journal, Vol. 122, No. 1, 1955, pp. 1-12. doi:10.1086/146048

[6] J. Hecht, U. Alon and D. Shvarts, “Potential Flow Models of Rayleigh-Taylor and Richtmyer-Meshkov Bubble Fronts,” Physical Fluids, Vol. 6, No. 12, 1994, pp. 4019-4030. doi:10.1063/1.868391

[7] Q. Zhang, “Analytical Solutions of Layzer-Type Approach to Unstable Interfacial Fluid Mixing,” Physical Review Letters, Vol. 81, No. 16, 1998, pp. 3391-3394. doi:10.1103/PhysRevLett.81.3391

[8] V. N. Goncharov, “Analytical Model of Nonlinear, Single-Mode, Classical Rayleigh-Taylor Instability at Arbitrary Atwood Numbers,” Physical Review Letters, Vol. 88, No. 13, 2002, pp. 134502-1-134502-4. doi:10.1103/PhysRevLett.88.134502

[9] S.-I. Sohn and Q. Zhang, “Late Time Behavior of Bubbles at Unstable Interfaces in Two Dimensions,” Physical Fluids, Vol. 13, No. 11, 2001, pp. 3493-3495. doi:10.1063/1.1410119

[10] S.-I. Sohn, “Analytic Solutions of Unstable Interfaces for All Density Ratios in Axisymmetric Flows,” Journal of Computational and Applied Mathematics, Vol. 177, No. 2, 2005, pp. 367-374. doi:10.1016/j.cam.2004.09.026

[11] S.-I. Sohn, “Simple Potential-Flow Model of Rayleigh-Taylor and Richtmyer-Meshkov Instabilities for All Density Ratios,” Physical Review E, Vol. 67, No. 2, 2003, pp. 026301-1-026301-5. doi:10.1103/PhysRevE.67.026301

[12] K. O. Mikaelian, “Limitations and Failures of the Layzer Model for Hydrodynamic Instabilities,” Physical Review E, Vol. 78, No. 1, 2008, pp. 015303-1-015303-4. doi:10.1103/PhysRevE.78.015303

[13] P. Ramaprabhu, G. Dimonte, Y.-N. Young, A. C. Calder and B. Fryxell, “Limits of the Potential Flow Approach to the Single-Mode Rayleigh-Taylor Problem,” Physical Review E, Vol. 74, No. 6, 2006, pp. 066308-1-066308-10. doi:10.1103/PhysRevE.74.066308

[14] P. Clavin and F. Williams, “Asymptotic Spike Evolution in Rayleigh? Taylor Instability,” Journal of Fluid Mechanics, Vol. 525, No. 5, 2005, pp. 105-113. doi:10.1017/S0022112004002630

[15] L. Duchemin, C. Josserand and P. Clavin, “Asymptotic Behavior of the Rayleigh-Taylor Instability,” Physical Review Letter, Vol. 94, No. 22, 2005, pp. 224501-1-224501-4. doi:10.1103/PhysRevLett.94.224501

[16] S. I. Abarzhi, K. Nishihara and J. Glimm, “Rayleigh? Taylor and Richtmyer? Meshkov Instabilities for Fluids with a Finite Density Ratio,” Physical Review A, Vol. 317, No. 5-6, 2003, pp. 470-475. doi:10.1016/j.physleta.2003.09.013

[17] S.-I. Sohn, “Bubble Interaction Model for Hydrodynamic Unstable Mixing,” Physical Review E, Vol. 75, No. 6, 2007, pp. 066312-1-066312-12. doi:10.1103/PhysRevE.75.066312

[18] S.-I. Sohn, “Effects of Surface Tension and Viscosity on the Growth Rates of Rayleigh-Taylor and Richt-Myer-Meshkov Instabilities,” Physical Review E, Vol. 80, No. 5, 2009, pp. 055302-1-55302-4. doi:10.1103/PhysRevE.80.055302

[19] M. R. Gupta, S. Roy, M. Khan, H. C. Pant, S. Sarkar and M. K. Srivastava, “Effect of Compressibility on the Rayleigh? Taylor and Richtmyer? Meshkov Instability Induced Nonlinear Structure at Two Fluid Interface,” Physics of Plasmas, Vol. 16, No. 3, 2009, pp. 032303-1-032303-12. doi:10.1063/1.3074789

[20] R. Betti and J. Sanz, “Bubble Acceleration in the Ablative Rayleigh-Taylor Instability,” Physical Review Letter, Vol. 97, No. 20, 2006, pp. 205002-1-205002-4. doi:10.1103/PhysRevLett.97.205002

[21] M. R. Gupta, L. Mandal, S. Roy and M. Khan, “Effect of Magnetic Field on Temporal Development of Rayleigh? Taylor Instability Induced Interfacial Nonlinear Structure,” Physics of Plasmas, Vol. 17, No. 1, 2010, pp. 012306-1-012306-12. doi:10.1063/1.3293120

[22] R. Banerjee, L. Mandal, S. Roy, M. Khan and M. R. Gupta, “Combined Effect of Viscosity and Vorticity on Single Mode Rayleigh? Taylor Instability Bubble Growth,” Physics of Plasmas, Vol. 18, No. 2, 2011, pp. 022109-1-022109-5. doi:10.1063/1.3555523

[23] T. Yoshikawa and A. Balk, “A Conformal-Mapping Model for Bubbles and Fingers of the Rayleigh-Taylor Instability,” Mathematical and Computer Modelling, Vol. 38, No. 1-2, 2003, pp. 113-121. doi:10.1016/S0895-7177(03)90009-4

[24] S. Tanveer, “Singularities in Water Waves and Rayleigh-Taylor Instability,” Proceedings of the Royal Society of London A, Vol. 441, No. 1913, 1993, pp. 501-525.