(116c) 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.