A new short-wave instability mode in gas-sheared falling liquid films

We study the linear stability of a laminar falling liquid film flowing in an inclined channel under the effect of a turbulent counter-current gas flow. In such systems, surface waves resulting from interfacial instability may strongly enhance interphase heat/mass transfer, but can also trigger flooding events, such as obstruction of the channel cross-section, droplet entrainment, or wave reversal. In a recent article (Ishimura et al., J. Fluid Mech., vol. 971, A37, 2023), we discovered, via linear stability calculations, a new short-wave instability mode, which is triggered by the turbulent gas flow. These calculations were based on the Navier-Stokes equations in the liquid and the Reynolds averaged Navier-Stokes (RANS) equations in the gas, using a temporal stability formulation. The short-wave instability mode gives rise to upward-travelling ripples, which have been observed in experiments, and our linear predictions were shown to be in good agreement with the measured wave speed and wavelength. Further, upon increasing the counter-current gas flow rate, the short-wave mode merges with the classical long-wave Kapitza mode, usually observed in falling liquid films. In the current contribution, we extend our previous work in two ways. Firstly, we determine how the onset of the short-wave instability is affected by the relevant control parameters, i.e. the liquid and gas Reynolds numbers, the channel height, the inclination angle, and the Kapitza number. In particular, we find that the short-wave mode disappears as the inclination angle is increased. Further, we perform stability calculations based on a spatio-temporal formulation, i.e. by differentiating the Orr-Sommerfeld eigenvalue problem with respect to the complex wavenumber, allowing to solve for the group velocity. Further, we show that the merged and long-wave modes can exhibit downward-convective, upward-convective, and absolute instability behavior, whereas the short wave mode is always upward convective.