Nonlinear waves in counter-current gas–liquid film flow

Abstract

We investigate the dynamics of a thin laminar liquid film flowing under gravity down the lower wall of an inclined channel when turbulent gas flows above the film. The solution of the full system of equations describing the gas-liquid flow faces serious technical difficulties. However, a number of assumptions allow isolating the gas problem and solving it independently by treating the interface as a solid wall. This permits finding the perturbations to pressure and tangential stresses at the interface imposed by the turbulent gas in closed form. We then analyse the liquid film flow under the influence of these perturbations and derive a hierarchy of model equations describing the dynamics of the interface, i.e. boundary-layer equations, a long-wave model and a weakly nonlinear model, which turns out to be the KuramotoSivashinsky equation with an additional term due to the presence of the turbulent gas. This additional term is dispersive and destabilising (for the counter-current case; stabilizing in the co-current case). We also combine the long-wave approximation with a weighted-residual technique to obtain an integral-boundary-layer approximation that is valid for moderately large values of the Reynolds number. This model is then used for a systematic investigation of the flooding phenomenon observed in various experiments: as the gas flow rate is increased, the initially downward-falling film starts to travel upwards while just before the wave reversal the amplitude of the waves grows rapidly. We confirm the existence of large-amplitude stationary waves by computing periodic travelling waves for the integral-boundary-layer approximation and we corroborate our travelling-wave results by time-dependent computations.

Absolute and convective instabilities in counter-current gas–liquid film flows

“…As the gas flow rate is further increased, the system again becomes convectively unstable with the localised disturbances propagating upwards. We find that the upper limit of the absolute instability region is close to the 'flooding' point associated with the appearance of large-amplitude standing waves, as obtained in Tseluiko & Kalliadasis (2011), and our analysis can therefore be used to predict the onset of flooding. We also find that an increase in the angle of inclination of the channel requires an increased gas flow rate for the onset of absolute instability.…”

“…AlthoughÓ Náraigh et al (2011Náraigh et al ( , 2013 found that the waviness induced Reynolds stresses do not play a significant role in the stability of the interface for the parameter values they considered, it should be noted that the simple quasi-laminar assumption does not provide good agreement with experiments -e.g. by Thorsness et al (1978) -as shown in Tseluiko & Kalliadasis (2011). Also, the conclusion ofÓ Náraigh et al (2011Náraigh et al ( , 2013 on the impact of waviness induced Reynolds stresses on the interfacial instability seems to be in contradiction with the results presented in Tseluiko & Kalliadasis (2011).…”

“…Also, the conclusion ofÓ Náraigh et al (2011Náraigh et al ( , 2013 on the impact of waviness induced Reynolds stresses on the interfacial instability seems to be in contradiction with the results presented in Tseluiko & Kalliadasis (2011). Therefore, an improved version of the quasi-laminar assumption needs to be used, as in Tseluiko & Kalliadasis (2011). From the modelling point of view, the fact thatÓ Náraigh et al (2011Náraigh et al ( , 2013 assume that the turbulence enters only through the base velocity results in a linearised problem that completely ignores the turbulent Reynolds stresses, i.e.…”

“…by Thorsness et al (1978) -as shown in Tseluiko & Kalliadasis (2011). Also, the conclusion ofÓ Náraigh et al (2011Náraigh et al ( , 2013 on the impact of waviness induced Reynolds stresses on the interfacial instability seems to be in contradiction with the results presented in Tseluiko & Kalliadasis (2011). Therefore, an improved version of the quasi-laminar assumption needs to be used, as in Tseluiko & Kalliadasis (2011).…”

“…As mentioned earlier, in a recent study, Tseluiko & Kalliadasis (2011) developed a hierarchy of non-local low-dimensional models, namely a long-wave model and a WIBL model to analyse flooding in counter-current gas-liquid flows, focussing mainly on singlehump solitary waves and periodic travelling waves. Their WIBL model is based on the methodology of Ruyer-Quil & Manneville (1998 for falling liquid films, obtained by combining a long-wave expansion with a polynomial representation of the velocity field, with the integral-boundary-layer approximation and the method of weighted residuals.…”