Prediction of Atomization in Two-Phase Turbulent Flow

The interfacial behavior of a two phase stratified flow can be classified as smooth, wavy, or rolling.  An initial linear instability of the smooth interface leads to periodic waves and a second instability of the wavy interfaces leads to formation of roll waves.  It is the existence of roll waves that provides the mechanism for atomization, and so prediction of this second transition is the goal of this study.  Ultimately, the work presented here is to aid in the design of future pipelines to reduce failures, reduce maintenance expenses, and ensure platform safety.

Previous experimental work (Bruno and McCready, 1988) had shown that roll waves grow out of a linearly unstable long wave.  However, quantitatively correct predictions had not been possible because previous stability analyses had not correctly described the wavy interface basestate nor accounted for the turbulent gas flow. 

In this work a k-epsilon turbulence model is used and adapted for gas-liquid flows.  Models of this type are commonly used for engineering calculations and are general enough to apply to all conditions found in oil-gas pipelines.  Since short waves always exist before long waves, finding the neutral stability of long waves requires the basestate velocity to account for the effects of a wavy interface.  This effect, similar to single phase flow with one roughened wall, is a shift of the maximum velocity away from the roughened surface.  This effect is well documented in literature (Akai 1981, Krogstad 1992, Lorencez 1997). For gas-liquid flow an alteration was needed to account for the fact that the relaminarization at the interface is not the same as at the wall.  By adjusting the effective y⁺ at the interface, the base state calculations match the asymmetric gas-liquid profiles found in literature.

The effect of turbulence is not confined to the basestate.  The Orr-Sommerfeld equation is augmented with terms that arise from eddy viscosity arguments.  To solve the linear stability problem, as is common with a highly non-linear coupled system of differential equations, an iterative method was implemented, which requires an initial guess of the three dependent variables: velocity (u), turbulent kinetic energy (k), and turbulent dissipation (e).  Profiles produced by a smooth interface proved to be insufficient as initial guesses.  Furthermore, in any given iteration, a value of k and e had to be given as a boundary condition.  To remedy the problem, the values of k and e at the interface were set in proportion to the values of k and e at the effective y⁺ near the wall.  Keeping these values merely proportional to the local turbulence maximums decoupled the interface and the wall regions in the sense that the profile near the interface was allowed to have a different magnitude of turbulence than the wall, which is what is observed in smooth interface calculations and expected because of the difference in velocities. 

After ensuring numerical accuracy for gas Reynolds numbers as large as 10⁶  and applying multiple numerical and solution methods as a means of verification, the effective y⁺ at the interface can be adjusted as the single empirical variable in order to match the atomization data. The calculations are compared in the figure below with data taken from Ivan Mantilla (Tulsa, 2007).   It is expected that this single parameter, which is the effective surface roughness, can be predicted from the linear growth rates of the smooth interface.

This talk will describe the implementation of the numerical scheme, the verification, by comparison with experiments, of the correctness of solution to the basestate problem and predictions of atomization onset for a range of conditions again in comparison with experimental results. 

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Funding provided by the Chevron Energy Technology Company