Secondary instability of the spike-bubble structures induced by nonlinear Rayleigh-Taylor instability with a diffuse interface.

Laminar-turbulent transition in Rayleigh-Taylor (RT) flows usually starts with infinitesimal perturbations, which evolve into the spike-bubble structures in the nonlinear saturation phase. It is well accepted that the emergence and rapid amplification of the small-scale perturbations are attributed to the Kelvin-Helmholtz-type secondary instability due to the high velocity shears induced by the stretch of the spike-bubble structures, however, there has been no quantitative description on such a secondary instability in literature. Moreover, the instability mechanism may not be that simple, because the acceleration or the "rising bubble" effect could also play a role. Therefore, based on the two-dimensional diffuse-interface RT nonlinear flows, the present paper employs the Arnoldi iteration and generalized Rayleigh quotient iteration methods to provide a quantitative study on the secondary instability. Both sinuous and varicose instability modes with high growth rates are observed, all of which are confirmed to be attributed to both the Rayleigh-Taylor and Kelvin-Helmholtz regimes. The former regime dominates the early-time instability due to the "rising bubble" effect, whereas the latter regime becomes more significant as time advances. Being similar to the primary RT instability [Yu et al., Phys. Rev. E 97, 013102 (2018)2470-004510.1103/PhysRevE.97.013102, Dong et al., Phys. Rev. E 99, 013109 (2019)2470-004510.1103/PhysRevE.99.013109, Fan and Dong, Phys. Rev. E 101, 063103 (2020)2470-004510.1103/PhysRevE.101.063103], the diffuse interface also leads to a multiplicity of the secondary instability modes and higher-order modes are found to exhibit more local extremes than the lower-order ones. Direct numerical simulations are carried out, which confirm the linear growth of the secondary instability modes with infinitesimal amplitudes and show their evolution to the turbulent-mixing state.