Journal of Modern Physics, 2013, 4, 174-179 http://dx.doi.org/10.4236/jmp.2013.42024 Published Online February 2013 (http://www.scirp.org/journal/jmp) Spiky Development at the Interface in Rayleigh-Taylor Instability: Layzer Approximation with Second Harmonic Rahul Banerjee 1,2* , Labakanta Mandal 1 , Manoranjan Khan 1 , Mithil Ranjan Gupta 1 1 Department of Instrumentation Science, Centre for Plasma Studies, Jadavpur University, Kolkata, India 2 St. Paul’s Cathedral Mission College, Raja Rammohan Roy Sarani, Kolkata, India Email: * rbanerjee.math@gmail.com Received June 19, 2012; revised October 20, 2012; accepted November 13, 2012 ABSTRACT Layzer’s approximation method for investigation of two fluid interface structures associated with Rayleigh Taylor in- stability 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   1 2 A , 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 in- creases 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. Keywords: Rayleigh Taylor Instability; Bubble; Spike; ICF; Supernova 1. Introduction Hydrodynamic instabilities such as Rayleigh Taylor in- stability (RTI) which sets in when a lighter fluid supports a heavier fluid against gravity or Richtmyer Meshkov instability (RMI) which is initiated when a shock passes an interface between two fluids with different acoustic impedances are of increasing importance in a wide range of physical phenomena starting from inertial confinement fusion (ICF) to astrophysical ones like supernova explo- sions. In ICF, the capsule shell undergoes the RTI both in the acceleration and deceleration phases. RTI can retard the formation of the hot spot by the cold RTI spike of capsule shell resulting in the destruction of the ignition hot spot or autoignition [1-4]. The hydrodynamic insta- bilities lead to development of heavy fluid “spikes” pene- trating into the lighter fluid and “bubbles” of lighter fluid rising through the heavier fluid. Different approaches have been used for the study of such problems. Among these Layzer’s [5] approach applied to single mode po- tential flow model [6-11] is a useful one giving approxi- mate estimate of both Rayleigh Taylor and Richtmyer Meshkov instability evolution. The bubbles were shown by Zhang [7] to rise at a rate tending asymptotically to a terminally constant velocity while spikes were shown to descend with a constant acceleration. However, whether for bubbles or for the spikes, Zhang’s analysis was ap- plicable only for Atwood number A = 1, i.e., only for fluid-vacuum interface. An extension to arbitrary value of Atwood number A was done by Goncharov [8]. With- in limitations of Layzer’s model as pointed out by Mi- kaelian [12] bubbles were shown to rise with a velocity tending to an asymptotic value dependent on A and hav- ing a fairly close agreement with the simulation results of Ramaprabhu et al. [13]. But the spikes were found to descend with a terminal constant velocity in contrast to a constant acceleration as obtained by Zhang [7] for A = 1. Asymptotic spike evolution in Rayleigh Taylor insta- bility behaving almost as a free fall was obtained by Clavin and Williams [14] and also by Duchemin et al. [15] by conformal mapping method. Associated with the free fall of the spike, the surface curvature of the spike was also found to increase with time (i.e. the spike shar- pens as it falls). The present paper described the dynamics of bubble and spike tips arising at the two fluid interfacial structure due to RTI with extended Layzer’s model replacing the zeroth harmonic term [8] by second harmonic term to satisfy the condition that the fluid velocity vanishes at * Corresponding author. Copyright © 2013 SciRes. JMP