Statistical Characterization of Energy Growth in Rayleigh–Taylor Instability

This study presents a statistical characterization of transient energy growth in RTI, focusing on broadband and multimodal perturbations at early times. Using the incompressible variable-density formulation for two miscible components, linearized about a diffusive interface base state, we derive a velocity-density formulation and develop an accurate spectral linear stability solver, verified to characterize the statistical transient evolution of RTI. To elucidate how initial spectra shape early-time dynamics, we employ two types of correlation statistics. First, a semi-isotropic Gaussian spectrum defined by an in-plane correlation length isolates the fundamental mechanisms governing mean energy decay and subsequent growth. Second, ensembles of Fourier modes with random phases and amplitudes corroborate our predictions against direct numerical simulations (DNS) in the Boussinesq limit.

The mean energy initially decays due to stable modes, reaches a distinct minimum, then transitions to exponential growth; the depth of that minimum varies nonmonotonically with the filtered spectral fraction. Asymptotic expansions clarify how the in-plane length influences this behavior. For the Fourier ensemble, we derive an expression that directly links mean energy growth to spectral amplitudes, enabling a quantitative comparison with DNS results. We also report parametric studies of Atwood number, viscosity, and interface thickness. By leveraging the known statistics of initial perturbations, this framework provides a precise, robust characterization of RTI’s transient dynamics, revealing a nontrivial dependence of the energy trajectory on the spectral content of the incoming perturbations.