Interfacial Waves and Shear-Stress in Vertical Upward Annular Pipe-Flow

In annular pipe-flow, the liquid flows partly as a thin wavy film along the wall, and partly as droplets entrained in the turbulent gas core. The behavior of the interface plays an important role in determining the interfacial shear-stress between the gas and the liquid, and the entrainment of droplets into the gas core. In this work, we use a conductance liquid probe to measure the thickness of the liquid film. These measurements are used to reconstruct the time-evolution of the interface and to develop a simple physically-based model for the interfacial shear-stress. The basic idea of the measurement technique is to impose an electrical potential between a pair of electrodes in contact with the liquid film and measure the resulting current, which is a function of the conductance of the liquid film, hence of its thickness. This measurement technique has been used before, e.g., Jayanti et al. (1990), however, the film thickness was measured only in a few positions. Here, we use an adaptation of the electrode-mesh sensor described in Prasser et al. (1998) with a matrix of 10 measurement locations in the axial direction and 32 in the circumferential direction, and a time resolution of 5000 Hz at each measurement location. This allows an accurate representation of the time evolution of the interface. A snapshot of the interface is shown in figure 1, where we can see the "roll waves", which play a dominant role in annular flow, e.g., Azzopardi (1997). Our results indicate that the roll waves have a random nature, with the pdf for the time-between-waves being well fitted by a gamma distribution. The random nature of the roll waves suggests that the interfacial shear-stress can be modeled by treating the interface as a surface with a random roughness. We show that the interfacial shear-stress can be determined using the classical sand-roughness theory for pipe flows. We present a simple physically-based model that allows us to extend the original Wallis correlation for the interfacial shear-stress (Wallis (1969)) for a wide range of situations, without the need to use ad-hoc "Reynolds number corrections", as often found in the literature.