Linear and nonlinear stability of two fluid columns of different densities and viscosities subject to gravity

We study the stability of a vertical interface separating two miscible fluid columns of different densities and viscosities subject to gravity. This flow possesses a time-dependent laminar one-dimensional base state with the interface thickness growing as the square root of time (by diffusion). First, numerical integration of the linearized initial value problem is carried out as a function of vertical and spanwise wavenumber, initial time, density and viscosity ratios. Adjoint-based optimization is performed to determine the linearly optimal perturbations (LOPs) that lead to maximum growth of total energy disturbances in finite time. Results indicate that the perturbation energy growth rate at small wavenumbers (less affected by viscosity initially) is dominated by two-dimensional modes (no spanwise variation). Substantial transient growth is observed at higher wave modes initially, followed by asymptotic decay of perturbations at large time. Finally, nonlinear direct simulation of the single-mode LOPs are studied in terms of perturbation energy evolutions, energy cascade, and other statistics relevant to the flow.