Statistical Characterization of Transient Energy Growth in Rayleigh-Taylor Instability under Multimodal Perturbations

A novel statistical framework developed by Frame and Towne (2024) is employed and extended to examine how initial conditions influence the linear evolution of the Rayleigh-Taylor instability. The analysis focuses on the evolution of the mean energy for two representative classes of initial perturbations. The first class, based on a Gaussian spatial correlation, reveals a time delay in the onset of instability growth and a non-monotonic dependence of this delay on the perturbation correlation length. This behavior suggests a low-pass-filter-like selection of modes governing the evolution of the mean energy. The second class consists of perturbations formed as linear combinations of Fourier modes with random phase shifts and amplitudes drawn from a prescribed spectrum. For this case, an analytical expression for the mean energy growth is derived, linking the spectral content of the initial perturbations to the temporal evolution of the energy through the initial spectral amplitudes. This formulation enables direct comparison with ensembles of three-dimensional direct numerical simulations, demonstrating both the framework's predictive capability and the limitations imposed by the problem's non-autonomous nature. Finally, the effects of viscosity and density stratification are analyzed, showing how these factors modify the time-delay mechanism and influence the early-stage evolution of the instability.