## 1 Introduction  
An annular flow pattern is observed when a gas-liquid mixture with a high fraction of gas phase flows in a duct. This flow is characterized by existence of a continuous gas core and a continuous liquid film on the duct walls. The dispersed phase consists of liquid droplets torn from the film surface and entrained into the gas core, and the gas bubbles are entrapped by the liquid film. Annular flow occurs in many industrial and power applications such as heat exchangers, steam generators, absorbers, oil-and-gas pipelines, as well as nuclear reactors. The efficiency and safety of the operation of these facilities demand a prediction of the overall flow characteristics, including the pressure drop, film thickness, entrained fraction, heat transfer and the critical heat flux. At present, the predictions are based on empirical correlations, which simplify the flow physics significantly. The interaction of the gas stream and liquid film leads to the creation of a complex system of waves on the film surface. The dominant structures in this system are so-called disturbance waves, which are large lumps of liquid traveling with a high speed over a thin residual layer of liquid (\"base film\"). The disturbance waves generate small-scale ripple waves which propagate over both the base film and disturbance waves. Both disturbance waves and ripples affect the interfacial shear, the mixing of liquid, heat transfer and turbulence in both phases; they are necessary for the entrainment of liquid and gas entrainment. Physical modeling of integral flow parameters requires the modeling of disturbance waves and ripples as a necessary first step. The properties of disturbance waves and ripples were measured in many experimental works. It is well known that the average velocity and frequency of the disturbance waves grow linearly with the superficial gas velocity, whilst the height of the waves decreases with gas speed, along with film thickness in general [1; 2; 3; 4]. An increase in liquid flow rate leads to an increase in all the aforementioned characteristics. The properties of the waves are affected by many other parameters, such as duct shape [5] and hydraulic diameter [6; 7; 8], flow orientation [9], liquid viscosity [10; 11] and surface tension [12; 13; 14], gas density [15; 16; 17], etc. However, apart from empirical approaches with their obvious limitations [18; 19; 20; 21], the prediction of wave properties by numerical or semi-analytical models is far from successful yet. One of the problems with modeling is that the wave structure of the liquid film is also affected by the distance from the point of primary contact between the phases. Even if the entrance of liquid is smooth--i.e., when the liquid is introduced as a film from the very beginning--the film undergoes a cascade of qualitative and quantitative changes. It is hardly possible to produce a working model which ignores the stage of flow development. At least, the numerical model with a long computational domain starting from the inlet [22] reproduces the experimental characteristics of disturbance waves much better than those employing periodic boundary conditions in the flow direction (e.g., [23; 24]). In general, a disturbance wave should be modeled starting from an infinitesimal perturbation near the inlet, passing through all the stages of evolution observed in the experiments. Those stages are described below. Very close to the inlet, the film surface is smooth; high-frequency initial waves appear here and grow in amplitude as they propagate downstream. Further downstream coalescence of the initial waves leads to the formation of large-scale disturbance waves which produce ripples on their rear slopes [25]. The disturbance waves coalesce with each other, which reduces their frequency with downstream distance [26; 27], whilst their velocity grows downstream [28]. It takes a distance of order of 100 pipe diameters to reach relatively stabilized waves properties [28].