Bamboo Waves Research Articles

Inception and termination of the core‐annular flow pattern for oil‐water downflow through a vertical pipe

AbstractDifferent flow patterns for lube oil–water and for kerosene‐water downflow through a vertical glass tube have been analyzed with the help of flow visualization. Core‐annular flow is the dominant flow regime, with oil forming the core, and water is forming the wall film. When the velocities are increased, transition to slug flow and transition to dispersed flow are found. The waves found during the transition to slug flow depend on oil viscosity: axisymmetric bamboo waves are seen in kerosene‐water downflow and the waves are asymmetric in case of lube oil–water flow where they have a cork‐screw shape. Based on the experimental observations, simple mathematical models have been proposed for predicting the flow pattern transition curves. © 2011 American Institute of Chemical Engineers AIChE J, 2012

Level Set Based Simulations of Two-Phase Oil–Water Flows in Pipes

We simulate the axisymmetric pipeline transportation of oil and water numerically under the assumption that the densities of the two fluids are different and that the viscosity of the oil core is very large. We develop the appropriate equations for core-annular flows using the level set methodology. Our method consists of a finite difference scheme for solving the model equations, and a level set approach for capturing the interface between two liquids (oil and water). A variable density projection method combined with a TVD Runge---Kutta scheme is used to advance the computed solution in time. The simulations succeed in predicting the spatially periodic waves called bamboo waves, which have been observed in the experiments of [Bai et al. (1992) J. Fluid Mech. 240, 97---142.] on up-flow in vertical core flow. In contrast to the stable case, our simulations succeed in cases where the oil breaks up in the water, and then merging occurs. Comparisons are made with other numerical methods and with both theoretical and experimental results.

Stationary core-annular flow through a horizontal pipe

A theoretical investigation has been made of core-annular flow: the flow of a high-viscosity liquid core surrounded by a low-viscosity annular liquid layer through a horizontal pipe. Special attention is paid to the question how the buoyancy force on the core, caused by a possible density difference between the core and the annular layer, is counterbalanced. From earlier studies it is known that at the core surface ripples are present that have the shape of "bamboo" waves or "snake" waves. They generate pressure variations and secondary flows in the annular layer that can cause a net hydrodynamic force on the core. Using hydrodynamic-lubrication theory (assuming the core to be rigid) it has been shown that for snake waves the lubrication force can counterbalance the buoyancy force. For bamboo waves that is not the case.

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.

Merging of drops to form bamboo waves

Topological transitions in core-annular flow of oil and water are described. Drops of one liquid are suspended in the second liquid and transported through a pipeline under a pressure gradient. We examine the merging of an array of drops in this two-fluid flow. Larger drops are found to merge and the flow undergoes a topological transition to core-annular flow. In core-annular flow of two liquids, one liquid lies at the core and the other surrounds it. For the case of large oil bubbles in water, a steady core-annular flow with bamboo waves on the interface is achieved. For the case of large water bubbles in oil, the ensuing core-annular flow has continually breaking waves at the interface.

Steady flow and interfacial shapes of a highly viscous dispersed phase

A perturbation theory for the steady flow of immiscible liquids is developed when the dispersed phase is much more viscous than the continuous phase, as is the case in emulsions of highly viscous bitumen in water and in water lubricated pipelines of heavy crude. The perturbation is nonsingular, but nonstandard; the partitioning of the boundary conditions at different orders is not conventional. At zero-th order the dispersed phase moves as a rigid solid with an as yet unknown, to-be-determined, pressure. The flow of the continuous phase at zero-th order is determined by a Dirichlet problem with prescribed velocities on a to-be-iterated interfacial boundary. The first order problem in the dispersed phase is determined from the solution of a Stokes flow problem driven by the previously determined shear strain on the as yet undetermined interfacial boundary. This Stokes problem determines the unknown, to-be-determined, lowest order pressure distribution. At this point we have enough information to test the balance of normal stresses at lowest order; by iterating the interface shapes we may now complete the description of the lowest order problems. The perturbation sequence in powers of the viscosity ratio has a similar structure at every order and all the problems may be solved sequentially with the caveat that interface shape must be determined iteratively in each perturbation loop. A perturbation solution for the wavy interfacial shapes on core-annular flows of very viscous oil is presented and the results are compared with experiments and a simpler approximation in which the core moves as rigid, but deformable body with no secondary motions. The perturbation theory gives rise to an accurate description of the bamboo waves observed in experiments when the holdup ratio measured in the experiments is assumed in the theoretical calculation. The perturbation solution and the rigid body approximation are in a relatively good agreement with errors of the order 10% in the flow curves and wave shapes; the error is associated with the neglect of the secondary motion in the rigid-deformable core approximation.

Problems and Techniques

In this issue of Problems and Techniques, we feature articles on the formation of structure in flows involving immiscible liquids and techniques for solving nonlinear knapsack problems. Free surface and interface problems introduce the added complication of having to determine the position of unknown surfaces into a typically complex scientific computation problem. When surfaces develop intricate patterns and change topology as solutions evolve, numerical approaches must simultaneously maintain accuracy and control computational cost. In the first article, Jie Li and Yuriko Renardy describe a volume of fluid (VOF) technique for addressing instabilities in slow flows involving two immiscible, incompressible, viscous liquids subject to shearing. After describing their variant of the VOF method, Li and Renardy use it to analyze the simple shearing of two fluids in parallel flow (two-layer Couette flow), the flow of a core of fluid surrounded by a second fluid in a pipe (core-annular flow), and droplet breakup. Their results explain how the interface forms "fingers" of one fluid into the other, "bamboo waves" between the fluids, and the rupture of larger drops into trains of smaller ones. Experimental observations nicely confirm and complement the calculations. The second article, by A. Melman and G. Rabinowitz, describes a globally convergent iterative procedure for a class of nonlinear knapsack problems. As I learned, a knapsack problem involves a maximization of a concave function subject to a linear constraint. The optimization presented by Melman and Rabinowitz is motivated chemical processing where periodic servicing is necessary to remove waste by-products and to add raw materials. The operations research problem is to determine the optimal service schedule. Using quadratically convergent Newton-like iterations, the authors construct a fast solution procedure and show its global convergence for a subclass of knapsack problems. Complications arise because the function whose roots they seek is singular and has only an implicit representation. Once again, I'm pleased to thank our authors for their interesting and informative contributions.

Numerical Study of Flows of Two Immiscible Liquids at Low Reynolds Number

We review our recent results on fingering, bamboo waves, and drop breakup in low Reynolds number flows composed of two viscous liquids under shear.

Direct simulation of unsteady axisymmetric core–annular flow with high viscosity ratio

Axisymmetric pipeline transportation of oil and water is simulated numerically as an initial value problem. The simulations succeed in predicting the spatially periodic Stokes-like waves called bamboo waves, which have been documented in experiments of Bai, Chen & Joseph (1992) for up-flow. The numerical scheme is validated against linearized stability theory for perfect core–annular flow, and weakly nonlinear saturation to travelling waves. Far from onset conditions, the fully nonlinear saturation to steady bamboo waves is achieved. As the speed is increased, the bamboo waves shorten, and peaks become more pointed. A new time-dependent bamboo wave is discovered, in which the interfacial waveform is steady, but the accompanying velocity and pressure fields are time-dependent. The appearance of vortices and the locations of the extremal values of pressure are investigated for both up- and down-flows.

Direct simulation of interfacial waves in a high-viscosity-ratio and axisymmetric core–annular flow

A direct numerical simulation of spatially periodic wavy core flows is carried out under the assumption that the densities of the two fluids are identical and that the viscosity of the oil core is so large that it moves as a rigid solid which may nevertheless be deformed by pressure forces in the water. The waves which develop are asymmetric with steep slopes in the high-pressure region at the front face of the wave crest and shallower slopes at the low-pressure region at the lee side of the crest. The simulation gives excellent agreement with the experiments of Bai, Chen & Joseph (1992) on up flow in vertical core flow where axisymmetric bamboo waves are observed. We define a threshold Reynolds number and explore its utility; the pressure force of the water on the core relative to a fixed reference pressure is negative for Reynolds numbers below the threshold and is positive above. The wave length increases with the hold-up ratio when the Reynolds number is smaller than a second threshold and decreases for larger Reynolds numbers. We verify that very high pressures are generated at stagnation points on the wavefront. It is suggested that a positive pressure force is required to levitate the core off the wall when the densities are not matched and to centre the core when they are. A further conjecture is that the principal features which govern wavy core flows cannot be obtained from any theory in which inertia is neglected.

A possible explanation of ?bamboo waves? in core-annular flow of two liquids

We derive a perturbed KdV equation to model interfacial waves in viscous two-layer flows. We tentatively speculate that solutions of such an equation might be related to “bamboo waves” observed in recent experiments.

Lubricated pipelining: stability of core—annular flow. Part 5. Experiments and comparison with theory

Results are given for experiments on water-lubricated pipelining of 6.01 P cylinder oil in a vertical apparatus in up- and downflow in regimes of modest flow rates, less than 3 ft/s. Measured values of the flow rates, holdup ratios, pressure gradients and flow types are presented and compared with theoretical predictions based on ideal laminar flow and on the predictions of the linear theory of stability. New flow types, not achieved in horizontal flows, are observed: bamboo waves in upflow and corkscrew waves in downflow. Nearly perfect core–annular flows are observed in downflows and these are nearly optimally efficient with values close to the ideal. The holdup ratio in upflow and fast downflow is a constant independent of the value and the ratio of values of the flow rates of oil and water. A vanishing holdup ratio can be achieved by fluidizing a long lubricated column of oil in the downflow of water. The results of experiments are compared with computations from ideal theory for perfect core–annular flow and from the linear theory of stability. Satisfactory agreements are achieved for the celerity and diagnosis of flow type. The wave is shown to be nearly stationary, convected with the oil core in this oil and all oils of relatively high viscosity. These results are robust with respect to moderate changes in the values of viscosity and surface tension. The computed wavelengths are somewhat smaller than the average length of the bamboo waves which are observed. This is explained by stretching effects of buoyancy and lubrication forces induced by the wave. Other points of agreement and disagreement are reviewed.