Numerical Analysis of Stratified and Slug Flows

Department of Physics, College of Science and Arts at ArRass, Qassim University, Buraydah, Saudi Arabia Department of Physics, University of Tunis, Tunis, Tunisia Department of Physics, University of Tunis El-Manar, Tunis, Tunisia Department of Mathematics, College of Science and Arts at ArRass, Qassim University, Buraydah, Saudi Arabia Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Oran 31000, Algeria Department of Mathematics, College of Science, Qassim University, Buraydah, Saudi Arabia Department of Computer Science, College of Science and Arts at ArRass, Qassim University, Buraydah, Saudi Arabia Department Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, China


Introduction
e stratified and the slug flows are particular two-phase phenomena existing in many applications such as petroleum transportation and chemical microreactors (see Desir et al. [1] and Santos et al. [2]). Using the slug flow, the transfer distance is reduced and the mixing process is enhanced along a microchannel. In the production pipelines, the pressure drop has a great effect on the operational costs and depends on the phase flow rates, the pipe diameter, and fluid properties (surface tension, density, and viscosity). e two-phase flow can be divided into two kinds: the gas-liquid flow and the liquid-liquid flow. e gas-liquid flow regimes are the most studied regimes numerically and experimentally [3]. Since the pioneer experimental study of a liquid-liquid flow regime conducted by Charles et al. in [4], many other investigations have been conducted. Following the experimental investigation of Angeli and Hewitt in [5], different flow characteristics of two-phase immiscible fluids are measured including phase distribution and phase holdup of oil and water in horizontal pipe. Based on the experimental study conducted by Elseth in [6], the stratified flow occurs under low mixture velocity, and when the mixture velocity increases, the dispersed regime is observed. e contribution of the wall film associated with the slug flow on the mass transfer was demonstrated experimentally by Arsenjuk et al. in [7]. e available analytical solution is limited to a particular flow pattern. In the literature, different formulations for the determination of the pressure drop are available for core-annular flow with a laminar core and turbulent annulus [8].
e recent advances in computational fluid dynamics (CFD) have led to various numerical studies of the complex unsteady two-phase flow, overcoming the lack of laboratory information in some operating conditions. Al-Jadidi in [9] investigated a three-dimensional (3D) two-phase model with a large eddy simulation (LES) to better reproduce the turbulent structure to simulate the heavy oil and water flow regimes. e primary annular flow is identified and distinguished into three configurations: sudden contraction or sudden expansion and horizontal channel orifice. Carlos in [10] developed a numerical model of oil and water flow in horizontal pipes in order to study the transition between semidispersed and fully dispersed flows.
e T-junction geometry in 2D and 3D models is the most used method to investigate the CAF and stratified and slug flows [11,12]. Senapati and Dewangan in [13] used a 2D T-junction model to study the stratified flow with three different approaches to capture the interface between the oil and water phase. e coupled level set and volume of fluid (CLSVOF) can better reproduce the flow characteristics. For all these simulations, the k − ω turbulence model was adopted. 3D Reynolds Average Navier-Stokes (RANS) equations combined with the VOF method were used by Al-Yaari and Abu-Sharkh in [14] to simulate the oil and water stratified flow. e oil enters perpendicular to the water inlet along a T-junction configuration. After a comparative study between the different k − ε turbulence variants, the re-normalization group (RNG) k − ε is chosen. e pressure drop associated to the slug flow was investigated experimentally by Kashid and Agar in [15] using a Y-junction as inlet configuration of the oil and water fluids. Desamala et al. in [16] conducted an experimental study to identify the transition of various flow regimes: plug to slug flow, slug to stratified wavy flow, and stratified wavy to stratified mixed flow. Based on 2D Tjunction inlet configuration, the different transition flow patterns were validated. e RANS coupled with the VOF model and the k − ε turbulence model were used. e majority of numerical studies focused on the effects of the mixture velocity, the water-cut, the fluid properties, and the wetting properties on the flow regime. e two and three-dimensional flow nature is rarely analyzed. In this study, the numerical results of 2D and 3D models are compared for both turbulent stratified and laminar slug flows. is comparative study permits to identify the effect of the lateral dimension on the prediction of the interface interaction for oil-water stratified flow and kerosene-water slug flows.

Problem Description
In order to validate and examine the stratified and slug flows in 2D and 3D, we consider the following geometric configurations previously studied experimentally. Figure 1 illustrates the T-junction adopted to study the water and oil stratified flow regime using a 2D model. e flow direction is along the z-axis. e diameter of the pipe is the same as in the experimental study conducted by Elseth in [6] and is equal to D � 0.05575 m. e length of the main pipe is L 1 � 5 m, and the branch pipe dimensions are as follows: e stratified flow is simulated for the mixture velocity U m � 0.68 m/s and the water-cut C w � 0.5, which corresponds to the following inlet velocities of water and oil imposed, respectively, at boundaries (1) and (2): In the present 3D model, we suppose that the stratified flow is axisymmetric. Hence, the computational domain is composed of only the half cylinder as shown in Figure 2.
e geometric configuration of T-junction to simulate the 2D slug flow is illustrated in Figure 3. e same configuration was investigated experimentally by Cherlo et al. in [17]. e main and the branch microchannels meet at a right angle. e diameter of the microchannel is D � 590 μm. e water and kerosene inlets have the following length: B 4 � 5 D. e length of the main channel L � 17 D. For the 3D slug flow model, the dimension of the rectangular pipe in the x-direction is equal to h � 500 μm. e inlet for water and kerosene at boundaries (1) and (2) is U w,in � U k,in � 10 ml/h .

Governing Transport Equations.
For turbulent flow, the transport equations for the two immiscible phases are the Reynolds Average Navier-Stokes (RANS) equations written in terms of the mixture properties. ese equations concern the average mass conservation equation and the average momentum conservation equations and can be written in the following form [11]: where ρ is the density of the mixture and µ is the dynamic molecular viscosity of the mixture defined by 2 Mathematical Problems in Engineering where ρ 1 and ρ 2 are, respectively, the densities of phases 1 and 2; μ 1 and μ 2 are, respectively, the dynamic viscosities of phases 1 and 2; α 1 and α 2 are the volume fractions of each phase such that α 1 + α 2 � 1; U i are the mean velocity components; P is the mean pressure; μ t is the dynamic turbulent viscosity (μ t � 0 for the laminar slug flow); g i is the gravity acceleration field; and F i are the components of external force per unit volume associated to the interfacial tension.
Depending on the flow configuration, the better closure turbulent model can be chosen. In the present study, the k − ω model with shear stress transport (SST) is adopted as indicated by Shi et al. in [11]. Following this model, the turbulent viscosity μ t is defined as follows [18]: Two added transport equations are needed to compute μ t : for the turbulent kinetic energy, k, and for the specific dissipation rate, ω: where P k is the volumetric production rate of k: where f μ , f 1 , and f 2 are low Reynolds damping functions. e turbulent dissipation rate is given by e following constants are commonly used: At high Reynolds numbers, the damping functions are set to unity and given by where R e,t is the turbulent Reynolds number defined as At high Reynolds numbers, the damping functions f μ , f 1 , and f 2 tend to unity. e surface tension at the interface between the two phases (the oil-water for stratified flow and the kerosenewater for the slug flow) can be transformed to a continuum surface force (CSF) as indicated by Brackbill et al. [19]. Following this model, the surface tension force is proportional to the curvature κ: where κ is the interface curvature computed by the divergence of the unit normal as e surface normal is given by the gradient of the volume fraction of the first phase: For the stratified flow, investigated experimentally by Elseth in [6], the surface tension between the oil and water phases is σ � 0.043N/m. For the slug flow, investigated experimentally by Cherlo et al. in [17], the surface tension between the kerosene and water phases is σ � 0.045N/m and the contact angle θ �

Initial and Boundary Conditions.
For the stratified flow, the horizontal pipe is filled with water at the initial time of simulation (t � 0 s). In the case of the slug flow, the rectangular cross section channel is filled with the kerosene phase. For all simulated cases, at the pipe wall, a no-slip boundary condition is considered. At the outlet, denoted as (3) as indicated in Figures 1-3, zero pressure is specified. For the turbulent stratified flow, the turbulent kinetic energy and the specific dissipation rate at the inlet are calculated by the following equations: where H is a characteristic inlet dimension as the hydraulic radius. order to satisfy that y + < 5. For the laminar slug flow, a local grid refinement is needed to well predict the velocity gradient near the wall. e grid numbers are 5100 and 71,400, respectively, for 2D and 3D T-junction geometries. e velocity-pressure coupling is solved by the PISO algorithm (pressure implicit with splitting of operator). For all solved variables, the residual is fixed to 10 − 5 . e geometric reconstruction (piecewise-linear) method is chosen to accurately capture the interface variation between the two immiscible phases. e same method was adopted by Shi et al. in [20]. A variable time step is chosen with initial time step Δt � 10 − 5 s. For all simulated cases, the fully developed flow (FDF) is attained if the maximum axial velocity is unchanged with time.

Stratified Flow.
e main differences between stratigraphic simulations by 2D and 3D models will be analyzed in terms of axial mixture velocity and water volume fraction. e simulated 2D stratified flow characteristics are compared to the experimental results of Elseth in [6] when the fully developed flow is attained. e axial velocities at the pipe axis and at z � 72D � 4m are represented in Figure 4.
ere is a satisfactory agreement, but the faster flow in the oil region is not well reproduced. Figure 5 shows the predicted and measured water volume fractions. e sharp interface is correctly simulated. However, the existing waves at the oil and water interface cannot be predicted by the VOF model. To compare the effect of the inlet configuration, Figure 6 presents the water volume fraction in the axial plane of the pipe. e effect of the T-junction in the 2D model is only limited to the vicinity of the junction. e recirculation zone induced by the junction and by the expansion creates a water droplet having a dimension around twice the pipe diameter.
ere is no difference between these profiles far downstream.

Slug Flow.
e slug flow is validated using the time evolution water volume distribution and the static pressure profiles for the 2D and 3D models. As confirmed by Cherlo et al. in [17], the mechanism of the slug formation can be explained through the following three stages (Figures 7(a)-7(e)).
(a) e water phase enters into the top microchannel: for the 2D model, this stage continues until the water stream blocks almost the entire cross section of the main channel (Figures 7(a), 7(b), 7(f ), and 7(g)). For the 3D model, the water phase is separated from the wall at the junction (Figure 7(g)) and a thinner water interface at the main channel. (b) For the 2D model, the water phase progresses in the main channel with a continuous oil entering from the low inlet channel. e shear stress and the pressure gradients exerted by the oil phase distorted the water stream at the end of the water inlet channel. A thin water layer connecting the two water streams is formed. e oil phase flows through the space between the wall and the water phase with high velocity (Figure 7(c)). For the 3D model, the water slug breaks up before the instant t � 0.50 s and a thin kerosene layer adjacent to the wall is observed contrary to the 2D model. 2.2 mm, respectively, for 2D and 3D models. It seems that the 2D model underestimates the slug length.
In order to validate the pressure drop in the 3D model, the computed static pressure will be compared to the Young-Laplace value. e variation of the static pressure along the axis of the microchannels at the instant t � 0.80 s is represented in Figure 8.
e computed static pressure is higher in the dispersed phase (water) compared to the continuous phase (oil), and the increase of the pressure at the junction is due to the decrease of the velocity. e simulated pressure drop is around 310 Pa which approximates the Young-Laplace value calculated by the following equation: where σ is the surface tension (σ � 0.045 N/m); θ is the contact angle (θ � 1°); and R is the radius of the curvature equal to half width of the channel (R 2 � D 2 /2 � 0.295 mm). e 3D model takes into account the two radii of curvature. Hence, the pressure drop for this model is approximately twice the simulated value for the 2D model which is around 147 Pa (Figure 8). In the latest model, only one radius of curvature contributes to the pressure drop and the second radius is infinity.
To show the effect of the surface tension on the slug regime, we consider the extreme case where σ � 0N/m. Figure 9 illustrates the water volume fraction at two different instants simulated by the 2D model. Compared to the simulated case shown in Figure 7(e), the slug regime disappeared.

Conclusions and Perspectives
e present two-phase flow model is based on RANS equations coupled with the VOF model and k − ω for turbulent flow.
is model can predict the stratified flow characteristics by the 2D and 3D models. As expected, the surface tension has a crucial effect on the flow pattern. Without surface tension, the slug regime disappeared. e slug length obtained by the 2D model is slightly different from the 3D model; however, the thin film is not predicted by the 2D model.
As perspectives of this study, we can illustrate the following aspects: (1) Investigation of the slug flow for multiple ducts.
(2) Development of a two-phase model taking into account the phase transfer.

Data Availability
No data were used in this study.