When a gas and a liquid flow together in parallel streams, the interface between them is flat at low gas velocities. At higher gas velocities, it becomes unstable to small perturbations, and waves appear on it. Wavy flow is very common in nature as well as in industrial applications, typical examples are waves on the sea and wavy flow down an inclined plane in an absorption column.

Depending, principally, on the geometry and the flow rates of the fluids, the interfacial waves can take a variety of shapes and sizes. Typical wave patterns occurring in horizontal channel flows are shown in Figure 1 [Hanratty (1983)] as a function of the air velocity and the liquid Reynolds Number, Re_{L}, which is based on the average flow velocity and the hydraulic diameter. It can be seen that for low water flow rates, i.e., small Re_{L}, only the surface tension-induced capillary waves prevail. At higher Re_{L}, two-dimensional, three-dimensional and roll waves may occur depending on the air velocity. The wave patterns occurring in vertical upward flow are shown in Figure 2 [Hewitt and Hall-Taylor (1970)] as a function of the superficial velocities of the two phases. Distinguishing features of this flow are the disturbance waves which are intermittent waves having an amplitude of up to five times the mean thickness of the film. These occur only when the liquid film Reynolds number is greater than a critical value of about 500.

The onset of wavy flow can be calculated using stability analyses which investigate the stability of the system to a perturbation. The simplest of these is the *Kelvin-Helmholtz instability* for incompressible, inviscid flows. More accurate analyses take the form of a linear stability analysis of the *Orr-Sommerfeld* type which includes the effect of viscosity but is valid only for small perturbations. Real waves are the result of nonlinear effects some of which can be taken into account successfully [Hanratty (1983)].

Interfacial waves have a major effect on the transfer processes between the two phases [Hanratty (1991)]. The presence of surface waves can increase the *interfacial friction factor* by more than an order of magnitude; this increase is normally given in the form of empirical correlations. The waves are also a source of *droplet entrainment*, and can also affect the turbulence characteristics in the gas phase [Cohen and Hanratty (1968)]. They also *enhance the heat and the mass transfer* rates across the interface. Typically, the heat transfer coefficient for wavy films can be about 20 to 50% higher than that for a smooth film. The effect of waves on mass transfer has also been established experimentally [Brauner and Maron (1982)], although there are few correlations which take account of the wave effect explicitly.

#### REFERENCES

Cohen, L. S. and Hanratty, T. J. (1968) Effect of waves at a gas-liquid interlace on a turbulent air flow, *J. Fluid Mech.*, 31, 467-479.

Brauner, N. and Maron, D. M. (1982) Characteristics of inclined thin films, waviness and the associated mass transfer, *Int. J. Heat Mass Transfer*, 25(1), 99-110. DOI: 10.1016/0017-9310(82)90238-1

Hanratty, T. J. (1983) Interfacial instabilities caused by air flow over a thin liquid layer. *Waves on Fluid and Fluid Interfaces*, 221-259, R. E. Meyer, Ed., Academic Press.

Hanratty, T. J. (1991) Separated flow modelling and interfacial transport phenomena, *Applied Scientific Research*, 48, 353-390.

Hewitt, G. F. and Hall-Taylor, N. S. (1970) *Annular Two-phase Flow*, Pergamon Press.