The stability of core-annular flow (CAF) in pipes is analysed using the linear theory of stability. Attention is confined to the potentially stable case of lubricated pipelining with the less viscous liquid, say water, in the annulus. The effects of surface tension and density are included, but gravity is excluded. We find upper and lower branches of the neutral curve in a Reynolds number (ℝ) vs. wavenumber (α) plane. A window of parameters is identified in which CAF is stable to small disturbances. When ℝ is below the lower critical value, CAF is destabilized by surface tension and long waves break up into slugs and bubbles. The sizes of slugs and bubbles of oil in water observed by Charles, Govier & Hodgson (1961) are given by the wavelength of the fastest growing long wave. This long-wave instability is a capillary instability, modified by shear, which reduces to Rayleigh's instability in the appropriate limit. At higher ℝ, the capillary instability is stabilized by shear. At yet higher ℝ, above the upper critical value, the flow is unstable to generally shorter waves which leads to emulsification, water droplets in oil. The theory agrees with experiments. The analysis seems to be applicable to the design of lubricated pipelines; for example, there is an optimum viscosity ratio for stability, greater stability can be obtained by using heavy liquid as a lubricant when the flow is unstable to capillary modes on the lower branch and by using light liquids when the flow is unstable to emulsifying disturbances on the upper branch.

Lubricated pipelining: Stability of core-annular flow

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.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112092000041

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

Using a mechanistic approach, equations are developed to describe the stratified cocurrent flow of gas-liquid mixtures in horizontal pipes. An iterative method is suggested for their solution.

Experimental data were collected using air-oil mixtures in a one inch 1. D. by 101 ft. long acrylic pipe. Pressure drop and average in situ liquid volume fraction are presented as a function of the superficial gas velocity and are compared with predicted results from this study and other models reported in the literature.

Visual observation of flow patterns and their transitions are indicated on the tentative flow pattern map of Govier and Aziz(1).

An analysis of horizontal stratified two phase flow in pipes

A two-fluid model for stratified flows with curved interfaces

Determination of the interface curvature in stratified two-phase systems by energy considerations

Complete numerical solutions were determined in dimensionless form, of the velocity profiles for cocurrent laminar stratified flow of two immiscible Newtonian liquids in a circular pipe, for viscosity ratios of 1, 10, 100 and 1000, at eight positions of the horizontal liquid-liquid interface, which was incorporated into the relaxation procedure by a twodimensional finite difference method described by D. N. de G. Allen. The velocity profiles were used to compute the theoretical hold-up ratios, pressure gradient reduction (or volumetric flow rate enhancement) factors and power reduction factors for the various viscosity ratios and interface positions; the maximum reduction factors for each viscosity ratio; and the input volume fractions of the two liquids corresponding to each of the above.



The theoretical results for laminar flow were compared with experimental data in the literature, in particular those of Russell, Hodgson and Govier for the horizontal flow of a mineral oil and water (viscosity ratio = 20.1) in a circular pipe. Good agreement was obtained for both hold-up and pressure drop data when both water and oil were in laminar flow. As the water entered the transitional and turbulent regions, however, a deviation between the experimental and the computed results developed, the deviation increasing in the anticipated direction as the Reynolds number of the water increased.



On a determine, sous forme non dimensionnelle, des solutions numeriques completes de profils de vitesse pour l'ecoulement laminaire, stratifie et concourant de deux liquides newtoniens immiscibles, dans un tuyau circulaire, en utilisant les rapports de viscosite de 1, 10, 100 et 1,000, pour huit positions differentes de l'interface horizontale liquide-liquide; on a tenu compte de la position de cette derniere en appliquant le processus de relaxation a la methode de difference finie a deux variables decrite par D. N. de G. Allen. A partir des profils de vitesse les auteurs ont calcule, pour les differents rapports de viscosite et les diverses positions de l'interface, le rapport theorique de retention, les facteurs de reduction des pertes de charge (ou d'accroissement du taux d'ecoulement volumetrique) et les facteurs de reduction de puissance; ils ont calcule de měme le facteur maximum de reduction pour chaque rapport de viscosite et egalement les fractions volumetriques respectives des deux liquides a l'entree dans chacun des cas cihaut mentionnes.



Les resultats theoriques pour l'ecoulement laminaire se sont montres en parfait accord avec les donnees experimentales rapportees dans la litterature, tant pour la retention que pour les pertes de charge, quand l'ecoulement laminaire prevalait pour l'eau et l'huile. Cependant, a mesure que l'on passait par la periode de transition et par la zone de turbulence pour l'eau, on a constate une deviation entre les valeurs experimentales et les resultats obtenus par calcul; cette deviation croissait dans la direction anticipee a mesure que le nombre de Reynold pour l'eau augmentait.

/pdf/numerical-analysis-of-stratified-laminar-flow-of-two-3405lvqjde.pdf

Alan R. Gemmell

Papers

Numerical analysis of stratified laminar flow of two immiscible Newtonian liquids in a circular pipe