Ultraefficient reduced model for countercurrent two-layer flows

We investigate the dynamics of two superposed layers with density contrast flowing countercurrent inside a channel, when the lower layer is much thinner than the wavelength of interfacial waves. We apply a low-dimensional film model to the bottom (heavier) layer and introduce a fast and efficient method to predict the onset of flow reversal in this phase. We study three vertical scenarios with different applied pressure gradients and compare the temporal growth rates of linear and weakly nonlinear waves to the Orr-Sommerfeld problem and to the weakly nonlinear theory, respectively. At the loading point, i.e., when a large wave hump stands at the interface, our spatiotemporal analysis shows that the system is absolutely unstable. We then present profiles of nonlinear saturated waves, pressure field, and streamline distribution in agreement with direct numerical simulation. The reduced model presented here allows us to explore the effect of the upper-layer speed on the wave pattern, showing that the wave profile is very sensitive when the mean film thickness, rather than the liquid flow rate, is maintained constant in the simulation. In addition, we show the strong effect of surface tension on both the maximum wave hump and the crest steepness before the loading point. Finally, we reveal how the nonlinear wave speed affects the vortex distribution within the lower layer by analyzing the stream function under different scenarios.