A numerical study of the influence of initial perturbations on the turbulent Rayleigh–Taylor instability

Abstract

The effect of initial conditions on the growth rate of turbulent Rayleigh–Taylor (RT) mixing has been studied using carefully formulated numerical simulations. A monotone integrated large-eddy simulation (MILES) using a finite-volume technique was employed to solve the three-dimensional incompressible Euler equations with numerical dissipation. The initial conditions were chosen to test the dependence of the RT growth coefficient ($\alpha_{b})$ and the self-similar parameter ($\beta_{b}\,{=}\,\lambda_{b}/h_{b})$ on (i) the amplitude, (ii) the spectral shape, (iii) the longest wavelength imposed, and (iv) mode-coupling effects. With long wavelengths present in the initial conditions, $\alpha _{b}$ was found to increase logarithmically with the initial amplitudes, while $\beta_{b}$ is less sensitive to amplitude variations. The simulations are in reasonable agreement with the predictions for $\alpha_{b}$ from a recently proposed model, but not for $\beta_{b}$. In the opposite limit where mode-coupling dominates, no such dependence on initial amplitudes is observed, and $\alpha_{b}$ takes a universal lower-bound value of ${\sim}\,0.03\,{\pm}\,0.003$. This may explain the low values of $\alpha _{b}$ reported by most numerical simulations that are initialized with annular spectra of short-wavelength modes and hence evolve purely through mode-coupling. Small-scale effects such as molecular mixing and kinetic energy dissipation showed a weak dependence on the structure of initial conditions. Initial density spectra with amplitudes distributed as $k^{0}$, $k^{-1}$ and $k^{-2}$ were used to investigate the role of the spectral slopes on the development of turbulent RT mixing. Furthermore, in a separate study, the longest wavelength imposed in the initial wavepacket was also varied to determine its effect on $\alpha_{b}$. It was found that the slopes of the initial spectra, and the longest wavelength imposed had little effect on the RT growth parameters.