Dynamics of axisymmetric core-annular flow in a straight tube. I. The more viscous fluid in the core, bamboo waves

Nonlinear dynamics of the concentric, two-phase flow of two immiscible fluids in a circular tube is studied. The viscosity of the fluid around the axis of symmetry of the tube is larger than the viscosity of the fluid that surrounds it and gravity acts against the applied pressure gradient. A pseudo-spectral numerical method is coupled with an implicit second order time-integration scheme to solve the complete mass and momentum conservation equations as an initial value problem. The simulations originate with the analytical solution for the pressure driven, steady, core-annular flow (CAF) in a tube. In order to replicate as closely as possible the experimental conditions reported by Bai, Chen and Joseph (1992), the volume fraction of each fluid in the tube and the total flow rate of both fluids are imposed. Furthermore, the length of the tube is taken to be as long as computationally possible in order to allow for multiple waves of different lengths to develop and interact as reported in the experiments and in earlier weakly nonlinear analyses. Having performed simulations of CAF for conditions under which the reported flow charts indicate that both phases retain their integrity but the original steady flow is unstable, it was found that indeed traveling waves develop with slightly sharper crests (pointing towards the annular fluid) than troughs, the so-called “bamboo waves.” Despite the uneven interface, the flow in the core fluid closely resembles Poiseuille flow, but in the annular fluid small recirculation zones develop at the level of each crest. As the Reynolds number or the flow rate of the core fluid increase, the average wavelength and the amplitude of these waves decrease, whereas the holdup ratio of the core to the annular fluid approaches two. Their specific values for each examined case are in closer agreement with the experiments than in earlier theoretical reports. For large values of interfacial tension, waves with even different wavelength move with the same velocity, whereas for small values, they attain variable velocities and approach or repel each other but no wave merging or splitting is observed.