Anand B. DESAMALA， Vinayak VIJAYAN， Anjali DASARI， Ashok Kumar DASMAHAPATRA，Tapas K. MANDAL

Department of Chemical Engineering， Indian Institute of Technology Guwahati， Guwahati 781039， India，E-mail: desamalaanand@gmail.com

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Prediction of oil-water flow patterns， radial distribution of volume fraction，pressure and velocity during separated flows in horizontal pipe*

Anand B. DESAMALA， Vinayak VIJAYAN， Anjali DASARI， Ashok Kumar DASMAHAPATRA，Tapas K. MANDAL

Department of Chemical Engineering， Indian Institute of Technology Guwahati， Guwahati 781039， India，E-mail: desamalaanand@gmail.com

Flow of two immiscible fluids gives rise to variety of flow patterns， which influence transportation process. In this work，we present detailed analysis on the prediction of flow pattern maps and radial distribution of volume fraction， pressure and velocity of a pair of immiscible liquids through a horizontal pipeline by computational fluid dynamics （CFD） simulation using ANSYS FLUENT 6.3. Moderately viscous oil and water have been taken as the fluid pair for study. Volume of fluid （VOF） method has been employed to predict various flow patterns by assuming unsteady flow， immiscible liquid pair， constant liquid properties， and co-axial flow. From the grid independent study， we have selected 47 037 number of quadrilateral mesh elements for the entire geometry. Simulation successfully predicts almost all the flow patterns （viz.， plug， slug， stratified wavy， stratified mixed and annular）， except dispersion of oil in water and dispersion of water in oil. The simulated results are validated with experimental results of oil volume fraction and flow pattern map. Radial distribution of volume fraction， pressure and velocity profiles describe the nature of the stratified wavy， stratified mixed and annular flow pattern. These profiles help to developing the phenomenological correlations of interfacial characteristics in two-phase flow.

computational fluid dynamics （CFD） simulation， volume of fluid （VOF） method， flow pattern transition， prediction of flow transition boundary， viscous oil-water flow

Introduction

The transportation of crude oil through pipelines results in enormous pumping cost owing to its high viscosity， which causes huge pressure drop during flow. Addition of another fluid （with low viscosity than oil）through the same pipe line， reduces pressure drop significantly and in turn pumping cost. In such twophase flow， the phases （viz.， oil and water） arrange themselves in various patterns known as flow patterns or flow regimes. Flow patterns are usually of three categories: separated （viz.， stratified smooth， stratified wavy， stratified mixed and core-annular flow）， plug/ slug and dispersed （oil in water and water in oil）. These flow patterns greatly influence pressure drop and holdup characteristics of pipeline transportation， heat and mass transfer characteristics of upstream operation in refinery （desalter and distillation units）， and kinetics between two immiscible phases. Each of the flow patterns has their own importance in different fields. Separated patterns especially， annular flow is preferable in pipeline transportation as the pressure drop for such flow is least as compared to other flow patterns，whereas， dispersed flows are preferable from heat and mass transfer point of view. Knowledge on transition boundaries of different flow patterns is helpful in guiding accurate design of various unit operations and processes.

Several studies have been conducted to predict transition boundaries by experiments［1-6］， analytical models［7-9］and computational fluid dynamics （CFD）simulations［10-13］for various systems. Experimentally，identification of flow pattern maps is usually done by visualization， imaging［4］and probe techniques. Probe techniques are of either intrusive［1，3］or non-intrustive［2］type. Analytical models are limited to predict selectiveflow pattern maps， rather than the complete map. For example， Brauner［7］have successfully predicted the transition boundary of stratified to dispersed flow，Brauner and Maron［8］have predicted transition boundary of wavy stratified to stratified mixed flow， and Soleimani and Hanratty［9］have predicted transition boundary for stratified to intermittent （viz.， plug and slug） flow. CFD simulations are also being done to intricately understand the hydrodynamics in multiphase flow［14，15］. The success of CFD simulation greatly depends on the models used in the simulation. Prediciton of core-annular flow in a vertically downward circular pipeline［11］based on VOF method， differs from the prediction of Ko et al.［10］， who have used different turbulence models for simulation in predicting core-annular flow. Further， Al-Yaari and Abu-Sharkh［16］have predicted the stratified flow pattern for oil-water flow through a horizontal pipeline employing the volume of fluid （VOF） method along with re-noramalization group （RNG）k-εturbulence model. In addition to these， there are few studies that determine the velocity， void fraction and pressure drop profiles of two-phase flow in pipes［17，18］. Ghorai et al.［17］have performed extensive numerical simulations to predict the flow field characteristics like gas velocity， volume fraction of liquid. They have also developed a correlation between interfacial friction factor and wall friction factor for stratified wavy flow. Sidi-Ali et al.［18］have performed 3-D simulations to investigate interfacial friction factor of horizontal stratified two-phase flow using Fluent. They have noticed good agreement with experimental results. Past literature reveals that development of a generalised flow pattern map for liquidliquid flow is quite complicated. A single analytical model also cannot predict all the flow pattern maps. VOF method being used largely for the prediction of stratified or core-annular flow. In this work， we present detailed CFD simulations to investigate all probable flow patterns （except dispersion） for moderately viscous oil-water flow through a horizontal pipeline. VOF successfully accounts for the interfacial interaction between the phases. We have also validated our simulation results with the experimental data obtained from experiments on equivalent systems with a wide range of superficial velocities of both oil and water， covering all the flow patterns. Subsequently， efforts have been made to understand the profile of volume fraction，pressure drop and velocity of a separated flow （stratified wavy， stratified mixed and annular flow） using VOF method.

Fig.1 Schematic representation of experimental setup （WT-water tank， OT-oil tank， CP-centrifugal pump， GP-gear pump， C1 to C3-control valves， G1 to G4-gate valves， RM1， RM2-rotameter， PT1 to PT2-pressure taps， B1 to B2-ball valves，MP-manometer panel， EN-entry section， EX-exit section， TS-test section， S-separator）

Fig.2 Arrangement of conductance probe and phase configuration

Fig.3（a） Schematic diagram of the model geometry

Fig.3（b） Mesh geometry of the entry section and pipeline

1. Experimentation

The schematic representation of the experimental setup is shown in Fig.1， consisting of an entry section（EN）， a test section （TS） and an exit section （EX） in the direction of the flow. Exit section is connected with a separator （S）， where oil separates from water due to density difference between oil and water. A centrifugal pump （CP） and a Gear pump （GP） are used for water and oil respectively. The detailed dimensions of the test section and specification of test fluids used in the present work are the same as Dasari et al.［19］.

Experiments are performed to detect all the flow patterns as discussed by Dasari et al.［19］. Here， a separate experiment is carried out to investigate annular flow using conductance probe. Two conductance probes are used in the present experimental investigation of core-annular flow. Two electrodes of one probe are placed diametrically opposite of the pipe cross section，to detect the presence of water in the annular region. Another probe is mounted on the top wall of the pipe along a length， to detect the presence of a continuous water layer on the inner pipe wall. The schematic representation of this probe and cross sectional view of flow configuration is shown in Figs.2（a） and 2（b） respectively. The electrical circuit consists of an LED， a resistor and a 9 V battery in series. The circuit will be completed through 1-2 in circuit-I and 3-4 in circuit-II，if water present in between. These circuits are tested by running water and oil separately. LED in this circuit glows when only water is flowing in the pipe because water is a good conductor of electricity. When only oil is flowing， LED does not glow since oil is a very poor conductor of electricity. Glowing of LED's in both circuits confirms the existence of water layer in annulus position， and hence confirms the existence of coreannular flow.

Experiments are carried out by gradual increasing the oil velocity at constant water velocity. After attaining the steady state， image of flow pattern is captured using a digital camera （DSC-HX100V， Sony） at the test section and glowing of LED is noted down. To check the reproducibility， experiments are repeated thrice and 99% accuracy is observed. After completion of a set of experiment， water velocity is changed to next higher value and repeated the same procedure as mentioned above.

2. Model development

Figure 3（a） shows detailed dimensions of the conduit geometry having internal diameter 0.025 m and length 7.16 m considered in the present computational work to mimic the actual experimental test loop. Oil and water （viscosity ratio: 107 and density ratio: 0.89）were introduced into the pipe through a T-junction at the entry section where water and oil enter into the pipe from the horizontal and vertical directions， respectively. The CFD software package of ANSYS FLUENTTMhas been used for simulation. We have modeled 2-D pipeline in GAMBIT and exported to FLUENT. Figure 3（b） represents the meshed model of pipeline in x-yplane. In the present work， the mesh consists of 44 330 and 2 707 quadrilateral mesh elements for pipeline and entry section respectively，which provides grid independence solution as discussed in Section 3.2.

The following continuity and momentum equations are used to describe the motion of oil and waterin horizontal pipeline，

whereρ，U，t，S are density， velocity， time， mass source respectively andF is considered to be surface force. In the present case “S ” is zero and “g” is gravitational acceleration， -ve iny-direction （Figs.3（a） and 3（b））. We have selected VOF approach for twophase modeling in FLUENT in which two fluids share a well-defined interface［20］. VOF solves a single set of momentum equations and then tracked the interface by solving the following volume fraction equation satisfying the continuity equation with the restriction，

where αq，Uqand Sαqare volume fraction， velocity and source term of volume fraction ofqthfluid in（Sαqis zero in present case） a cell respectively. At the interface， volume fraction is0＜αq＜1.

Among the available turbulent models in FLUENT［20］the k-εmodel has been selected to simulate the flow patterns. In this model， the turbulent kinetic energy and viscous dissipation rates have been calculated using the following equations to obtain the turbulent viscosity in the flow field.

wherek，ε，μtandU are the turbulent kinetic energy， dissipation rate， eddy viscosity and velocity respectively.Eijis defined as

The constants are taken as:Cμ=0.09，σk=1.0， σε=1.3，C1ε=1.44，C2ε=1.92［20］. The inlets are modeled with the velocity inlet， and the outlet is modeled employing the pressure outlet boundary conditions. The diffusion flux variables at the exit are taken as zero. No-slip and impermeable（U =0）conditions are assumed for the walls of the pipe. In the present model， a constant surface tension coefficient is incorporated in VOF method using continuum surface force（CSF） model［21］. A contact angle of 8.5o［22］between the phases and the wall is also specified to count wall adhesion.

The convergence is usually decided based on the residuals of continuity and observable variables， such as velocity components， and convergence is achieved when the residuals values are ～10-3. In case of turbulent energy and dissipation rate， residuals are typically～10-6［20］. In the present study， the numerical computation is considered to be converged when the residuals of continuity， velocity components （XandY component） are ～10-3or decrease continuously even after 40-50 iterations and residuals of turbulent energy and dissipation rate are below 10-6. Oil and water are introduced at their respective inlets and the unsteady state simulation is started. The superficial velocities of both the phases corresponding to a given experimental condition are set as inlet conditions. After getting a desired convergence， the flow of both phases are tracked to get the flow pattern.

3. Results and discussions

3.1 Experimental results

During the experiments， the oil superficial velocity （USO）has been varied from 0.015 m/s-1.25 m/s and the water superficial velocity（USW）has been varied from 0.1 m/s-1.1 m/s to obtain transition boundaries of all the flow patterns. Seven different flow patterns have been observed: plug flow （P）， slug flow（S）， stratified wavy flow （SW）， stratified mixed flow（SM）， annular flow （A）， dispersion of oil in water（DO/W） and dispersion of water in oil （DW/O）. The details about these flow patterns， except the annular flow，are not discussed here to avoid repetition as they have been reported by Dasari et al.［19］. In the present work，annular flow is observed in a small velocity range of oil，USO=0.27 m/s-0.60 m/sand water，USW= 0.40 m/s-0.63m/s. Experimental image of annular flow observed at USO=0.40 m/s，USW=0.45 m/sis shown in Fig.4. Flow pattern map reported by Dasari et al.［19］is modified by introducing annular flow and shown in Fig.5. The figure shows that the annular flow is sandwiched in stratified mixed flow region.

Fig.4 Experimental image annular flow at USO=0.40 m/s，USW=0.45 m/s

Fig.5 Experimental flow pattern map

3.2 Grid independence study

To find the optimum mesh size for the simulation，we carried out grid independence study by taking three different grid sizes: 14 732， 47 037 and 63 191. The test simulations have been carried out on stratified wavy flow with the following superficial velocities: water，USW=0.23m/sand oil，USO=0.10 m/s. The result of grid independence study is presented in Fig.A1 （Annexure）. In these figures， black and gray color represents the presence of water and oil respectively. Figure A1 depicts that the system with 14 732 cells was not able to predict the stratified flow pattern and hence， additional cells are needed. System with 47 037 and 63 191 cells closely predict the oil volume fraction with a clear interface. Therefore， based on the contour results of oil volume fraction， 47 037 cells are chosen as the optimum number of cells to be used in the present simulation. Using this optimum mesh size（0.00211 m）， step size for time in VOF method is fixed at 0.001 s for each simulation， satisfying the criteria:（time steps×velocity） ＜ mesh size.

3.3 Simulation results

Simulations have been performed by taking different data points along the transition boundaries observed from the experimental flow pattern map. Using VOF we successfully predicts plug， slug， stratified wavy， stratified mixed， annular flow patterns， and transitions boundary of （1） plug flow to slug flow， （2） slug flow to stratified wavy flow， （3） stratified wavy flow to stratified mixed flow， and （4） a region of annular flow instead of its transition boundaries. The other transitions （viz.， SM to DW/Oand SM to DO/W） observed at higher phase velocities are difficult to predict by using VOF. The reason can be attributed to the method of interface reconstruction scheme of VOF model［11］， which fails to capture the waviness of oilwater interface properly. Therefore， the transition boundaries observed at higher phase velocities are not simulated in the present study. Below， we discuss prediction of different transition boundaries as mentioned above.

3.3.1 Plug to slug transition boundary （P-S）

To show the transition between plug and slug flow patterns， simulation is run at USW=0.23m/s，USO=0.04 m/sfor plug flow and USW=0.23m/s，USO=0.05 m/sfor slug flow. The simulated result at these boundary conditions is validated with the experimental images captured at the same flow conditions as shown in Fig.6. The volume fraction contours of oil and water are plotted in all the cases to mimic the interfacial morphology of flow patterns. In these figures，black and gray color represents the presence of water and oil respectively. Figure 6（a） shows the contours of oil fraction for plug flow at USW=0.23m/s，USO= 0.04 m/sand Fig.6（b） shows the experimental image taken at the same velocity. Similarly， the simulated result of slug flow atUSW=0.23m/s，USO=0.05 m/s is shown in Fig.6（c） and the experimental image taken at the same velocity is shown in Fig.6（d）. Figures 6（a）and 6（c） portray the prediction of transition from plugto slug flow using CFD simulation. A good agreement between the simulated result and experimental result is observed. Present simulation results based on the VOF method， is also in accordance with the experimental results reported by Chakrabarti et al.［2］.

Fig.6 Transition of plug to slug flow （P-S）

Fig.7 Transition of slug to stratified wavy flow （S-SW）

Fig.8 Transition of stratified wavy to stratified mixed flow （SW-SM）

3.3.2 Slug to stratified wavy flow transition boundary（S-SW）

It is well known that at the slug flow， if oil velocity is increased at a constant water velocity， the individual slugs are coalesced leading to a stratified flowpattern. The waviness at the interface can be accounted for by taking into consideration of the interfacial tension and interfacial shear. To show the transition，simulations have been carried out at USW=0.23m/s，USO=0.078 m/sfor slug flow and USW=0.23m/s，USO=0.10 m/sfor stratified wavy flow. The transition boundary of S-SW from experiment and simulation is shown in Fig.7. The transition from slug to wavy stratified flow pattern at USW=0.23m/sis clearly visible and good agreement with experimental result is observed.

3.3.3 Stratified wavy to Stratified mixed flow transition boundary （SW-SM）

The transition of SW-SM is obtained from the simulation and experiment at USW=0.2 m/s，USO= 0.17 m/s for stratified wavy flow and USW=0.2 m/s，USO=0.21m/sfor stratified mixed flow （Fig.8）. The simulation result （Fig.8（a）） shows stratified flow with clear interface between the two phases with some wavy nature. As the oil flow rate increases， the wave amplitude increases with bubble entrainment at critical wave amplitude as shown in Fig.8（c）. Experimental photographs in Figs.8（b） and 8（d） clearly demonstrate this phenomenon of bubble entrainment at the interface of oil and water. A good agreement is achieved between experimental and simulated results. Unlike previous literature work［16］， we have successfully predicted both stratified and stratified mixed flow with good accuracy.

Fig.9 Annular flow

3.3.4 Region of annular flow （A）

During the annular flow， oil flows as core in the center and a thin layer of water flows as annulus in a pipe. Five different transitions are observed around the annular flow （Fig.5）: slug， stratified wavy， stratified mixed and dispersion of both oil and water. In the present work， prediction of these transition boundaries is not made due to the limitation of the VOF method at higher phase fractions （see Section 3.3）. However， the region of annular flow is simulated successfully. Simulation and experimental result of annular flow at flow condition USW=0.45 m/s，USO=0.40 m/sis shown in Fig.9（a） and Fig.9（b） respectively， showing a good agreement. Simulations have been done at various combinations of velocities to get the entire region of annular flow. Similar prediction of core-annular flow has been reported for a horizontal flow through a pipeline with a sudden change in cross-sectional area and a return bend［12］.

Fig.10 Comparison of experimental and predicted oil volume fraction

Fig.11 Validation of simulation results with experimental results （Dotted line 1-P/S transition boundary， Dashed line 2-S/SW transition boundary， Dash dotted line 3-SW/ SM transition boundary， Black color solid line 4-Annular flow transition， Circled data points are simulation data points along the transition boundaries of the current work）

3.4 Validation

In order to validate the simulation， the results are compared with the experimental data of the present work. For this， oil volume fraction of separated flow has been simulated and compared with the experimental result as shown in Fig.10. Area average volume fraction is calculated from simulation and considered as equivalent to the experimental volume fraction. The figure shows a good agreement with the experiments. The figure shows that the model predicts volume fraction of oil with an average absolute error of 11.7%. We have also validated the simulated results with experimental flow pattern map. Following this， the simulated points （of different flow patterns） have been superimposed on the horizontal flow pattern map（Fig.11）. Marked （by circle） data points represent simulated data while solid and broken lines are experimental transition boundaries. Line 1， 2， 3 and 4 represent the transition boundary of plug to slug， slug to stratified wavy， stratified wavy to stratified mixed and boundary around annular flow respectively. The circled data points around the respective boundary are simulated results of respective flow regimes. Figure 11 depicts a good matching between the simulation and the experimental flow patterns.

After validation with experiments， the model is used to generate useful information on flow characteristics （viz.， volume fraction， pressure and velocity profile） of a separated flow （viz.， stratified wavy， stratified mixed and annular flow） which is encountered commonly in chemical and petroleum industries. It may help in understanding the mechanism of flow phenomena and to determine the interfacial parameters such as interfacial velocity， and shear. Using the interfacial velocities prediction accuracy of two-phase pressure drop may be improved. The detailed analysis of modelling and importance of flow characteristics are discussed in the next section.

Fig.12 Schematic representation of radial positions of a circular pipe under consideration

4. Radial distribution of volume fraction， pressure and velocity of a separated flow

In the following section， the volume fraction profile， local pressure across a cross section and velocity profile along the radius of a seperated flow at different phase velocities have been reported. The schematic representation of radial positions of a circular pipe under consideration is shown in Fig.12. All the contours of seperated flow have been shown by considering a line across the cross section at L/ D=120（Fig.12）.This ratio（L/ D）is chosen to ensure the fully developed flow as reported by Dasari et al.［19］.

Fig.13 Radial distribution of oil volume fraction

4.1 Volume fraction profile

The volume fraction profile helps to characterize the two-phase flows and is used to estimate numerous other important parameters， such as two-phase density and two-phase viscosity， relative average velocity of the two phases， and predicting flow pattern transitions and pressure drop. The radial behavior of oil volume fraction for stratified wavy （Fig.13（a））， strafied mixed（Fig.13（b）） and annular （Fig.13（c）） flow is shown in Fig.13. Figure 13（a） shows the variation of oil fractionwith increasing oil velocity （USO=0.2 m/s and 0.3 m/s）at a constant water velocity（USW=0.23m/s）. During stratified wavy flow， oil phase thickness increases as oil velocity increases.

Fig.14 Radial distribution of total pressure

Similarly， oil volume fraction profile of stratified mixed flow pattern is shown in Fig.13（b） at constant water velocity （USW=0.33m/s）and different oil velocities （USO=0.2 m/s and 0.3 m/s）. The figures depict the distinct three layers: （1） a continuous water layer at bottom， （2） a continuous oil layer at top and （3）a mixed layer of oil droplets in continuous water inbetween continuous layer of oil and water. The profile is bimodal due to the presence of a mixed layer （3rd layer）. The small hub of the plot indicates the oil droplets in continuous water. During annular flow， the oil phase flows along the center as core and water phase flows through annulus between pipe wall and oil core（Fig.4）. This is confirmed from the radial distribution of oil volume fraction which is shown in Fig.13（c）. These results show good acceptability with the experimental results with an average error of 11.7% （Fig.10）.

4.2 Pressure profile

Pressure drop in two-phase flow is a major design parameter， governing the pumping cost to transport two-phase fluids. Variation of localpressure （mixture）along the radial direction for separated flow is shown in Fig.14. Figures 14（a） to 14（c） depicts numerically predicted pressure variation along axial position at L/ D=120. For stratified wavy flow， the pressure will be maximum at the oil-water interface due to slip between the phases. Figure 14（a） depicts the same showing maximum pressure at interface. Simillar observation is also recorded in case of stratified mixed flow（Fig.14（b））. During annular flow， total pressure decreases with increase in oil velocity at constant water velocity as shown in Fig.14（c）. The figure （Fig.14（c））depicts maximum pressure at top interface of annular flow due to higher shear at that interface.

4.3 Velocity profile

The velocity profile depends on the type of flow（flow through confined conduit， open channel flow and flow through parallel plates， etc.）. It may give an accurate value of kinetic turbulent energy， turbulent dissipation rate， momentum correction factor， etc. Ghorai et al.［17］and Sidi-Ali et al.［18］predicted the velocity profile for gas-liquid two-phase flow. Till now there is no such literature that describes the velocity profile of liquid-liquid flow using CFD. In the present work， efforts have been made to understand the radial distribution of velocity for stratified wavy （Fig.15（a）），stratified mixed （Fig.15（b）） and annular flow（Fig.15（c））. The velocity profiles of stratified wavy flow is drawn for USO=0.2 m/sand USO= 0.3m/s at constant water velocity （USW=0.23m/s）in Fig.15（a）. Similarly Fig.15（b） shows the velocity profile of stratified mixed flow at USW=0.33m/s，USO=0.2 m/sand USW=0.33m/s，USO=0.3m/s. In stratified wavy and stratified mixed flow， oil and water phase are in laminar and turbulent flow regime resepectively. Hence， the velocity is high for water phase and low for oil phase （see Figs.15（a）， 15（b））. In annular flow， the velocity profiles （atUSW=0.4 m/s，USO=0.4 m/sand USW=0.4 m/s，USO=0.5 m/s）look like an inverted “U” showing maximum velocity at the centre （which is occupied by oil phase） as shown in Fig.15（c）. The same trend has been reported by Kaushik et al.［13］.

Fig.15 Radial distribution of velocity profiles

5. Conclusion

Computational fluid dynamics study has been conducted to predict the transition boundaries of viscous oil-water （viscosity ratio: 107 and density ratio: 0.89） two-phase flow through a horizontal pipeline having internal diameter 0.25 m and length 7.16 m. Based on grid independence study， 47 037 numbers of cells are selected as optimum number of cells for the present simulation. We have successfully simulated plug flow， slug flow along with other separated flow patterns （viz.， stratified wavy， stratified mixed and annular flow）， and their transition boundaries. The VOF technique successfully predicts transition boundaries of plug to slug， slug to stratified wavy and stratified wavy to stratified mixed flow. Experimentally， we have observed seven different flow patterns （plug，slug， stratified wavy， stratified mixed， annular， oil dispersed in water and water dispersed in oil） in the velocity range of oil，USO=0.015 m/s-1.250 m/sand water，USW=0.1m/s-1.1m/srespectively. It is to be mentioned that we have observed annular flow in the following velocity range:USO=0.27 m/s-0.60 m/sand USW=0.40 m/s-0.63m/s. Simulation results show a good agreement with experimental results in predicting of volume fraction （with an absolute error of 11.7%）， flow patterns and transition boundaries. We have simulated annular flow and its region （not all the boundaries related to the annular flow） with an appreciable accuracy. The VOF method also predicts the volume fraction， pressure and velocity profile of the separated flow. These profiles clearly depict the nature of the flow patterns like stratified wavy， stratified mixed and annular flow. The profiles also helpful in determining the interfacial characteristics like interfacial thickness， thickness of the fluid phase and size of the droplet present at the interface. The present findings reveal the ability of VOF to predict the hydrodynamics of viscous oil-water two-phase flow.

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10.1016/S1001-6058（16）60670-4

Appendix

Fig.A1 Grid independence study of stratified wavy flow at USW=0.23m/sand USO=0.10 m/s

November 6， 2014， Revised July 27， 2015）

* Biography: Anand B. DESAMALA （1985-）， Male， Ph. D.

Tapas K. MANDAL，

E-mail: tapasche@iitg.ernet.in

2016，28（4）:658-668