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.
The nonlinear dynamics of the concentric, two-phase flow of two immiscible fluids in a circular tube is studied when the viscosity ratio of the fluid in the annulus to that in the core of the tube, μ, is larger than or equal to unity. For these values of the viscosity ratio the perfect core-annular flow (CAF) is linearly unstable and it is necessary to keep the ratio of the thickness of the annulus to the radius of the tube small so that the solutions remain uniformly bounded. The simulations are based on a pseudospectral numerical method while special care has been taken in order to minimize as far as possible the effect of the boundary conditions imposed in the axial direction allowing for multiple waves of different lengths to develop and interact. The time integration originates with the analytical solution for the pressure driven, perfect CAF or the perfect CAF seeded with either the most unstable mode or random disturbances. Quite regular wave patterns are predicted in the first two cases, whereas multiple unstable modes grow and remain even after saturation of the instability in the last case. The resulting waves generally travel in the same direction and faster than the undisturbed interface, except for the case with μ=1 for which they are stationary with respect to it. Depending on parameter values, waves move with the same velocity or interact with each other exchanging their amplitudes or merge and split giving rise to either chaotic or organized solutions. For fluids of equal viscosities and densities (μ=ρ=1) and for a Reynolds number, Re(≡Λρ̂1R̂2Ŵ0/μ̂1)=0.0275 and an inverse Weber number, W(≡T̂/(ρ̂1Ŵ02R̂2))=145.4, both based on the properties of the inner fluid, the tube radius, R̂2, and the average flow velocity, Ŵ0, small amplitude waves are predicted. The increase of μ by almost two orders of magnitude does not affect their amplitudes, but increases their temporal period linearly. Varying W by more than three orders of magnitude increases their amplitudes proportionately, while their period increases with the logarithm of W. Similar to that is the effect of increasing Re. The present analysis confirms and extends results based on long wave expansions, which lead to the Kuramoto–Sivashinsky equation and modifications of it.
The concentric, two-phase flow of two immiscible fluids in a tube of sinusoidally varying cross-section is studied. This geometry is often used as a model to study the onset of different flow regimes in packed beds. Neglecting gravitational effects, the model equations depend on five dimensionless parameters: the Reynolds and Weber numbers, and the ratios of density, viscosity and volume of the two fluids. Two more dimensionless numbers describe the shape of the solid wall: the constriction ratio and the ratio of its maximum radius to its period. In addition to the effect of the Weber number, which depends on both the fluid and the flow, the effect of the Ohnesorge number J has been examined as it characterizes the fluid alone. The governing equations are approximated using the pseudo-spectral methodology while the Arnoldi algorithm has been implemented for computing the most critical eigenvalues that correspond to axisymmetric disturbances. Stationary solutions are obtained for a wide parameter range, which may exhibit flow recirculation at the expanding portion of the tube. Extensive calculations are made for the dependence of the neutral stability boundaries on the various parameters. In most cases where the steady solution becomes unstable it does so through a Hopf bifurcation. Exceptions to this are cases where the viscosity ratio is O(10−3) and, then, the most unstable eigenvalue remains real. Generally, steady core–annular flow in this geometry is more susceptible to instability than in a straight tube and, in similar ranges of the parameters, it may be generated by different mechanisms. Decreasing the thickness of the annular fluid, inverse Weber number or the Ohnesorge number or the density of the core fluid stabilizes the flow. For stability reasons, the viscosity ratio must remain strictly below unity and it has an optimum value that maximizes the range of allowed Reynolds numbers.
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