Core–annular flow in a periodically constricted circular tube. Part 2. Nonlinear dynamics

Abstract

Nonlinear dynamics of the concentric, two-phase flow of two immiscible fluids in a circular tube of variable cross-section is studied for parameter values where the steady core–annular flow (CAF) is linearly unstable. The simulations are based on a pseudo-spectral numerical method. They are carried out assuming axial symmetry, that the total flow rate remains constant and that all dependent variables are periodic in the axial direction, which includes the minimum necessary number of repeated units so that the obtained solution is independent of this number. The time integration originates with the numerically computed steady CAF or the steady CAF seeded with either the most unstable mode or random small disturbances. Only a limited number of the most interesting cases are presented. For the most part, the values of the majority of the dimensionless parameters are such that oil flows in the centre of the tube driven by an applied pressure gradient against gravity, whereas water is flowing in the annulus. It is shown that, whereas the steady (unstable) solution may indicate that the heavier water flows countercurrently with respect to the oil, the time periodic (observable) solution may indicate the same, albeit at a much smaller core flow rate or that concurrent flow occurs. This is due to the water being trapped between the large-amplitude interfacial waves that are generated and being convected by the oil. It is also shown that increasing the inverse Weber number increases the wave amplitude to the point that the flow of the core fluid may become discontinuous with a mechanism that depends on the viscosity ratio between the two fluids. Increasing the amplitude of the sinusoidal variation of the tube leads to a combination of travelling and standing waves, which interact to produce a time periodic solution with a long period associated with the time it takes the travelling wave to travel through the computational domain and a second much shorter period that is related to their interaction time. Qualitative agreement has been obtained upon comparing our numerical simulations with limited experimental reports, even though the experimental conditions were not identical to those in our model.