analytical and level-set method based numerical study on oil–water smooth/wavy stratified-flow in...

International Journal of Multiphase Flow 38 (2012) 99–117

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International Journal of Multiphase Flow

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Analytical and level-set method based numerical study on oil–watersmooth/wavy stratified-flow in an inclined plane-channel

Vinesh H. Gada, Atul Sharma ⇑Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai 400 076, India

a r t i c l e i n f o

Article history:Received 2 May 2011Received in revised form 26 August 2011Accepted 30 August 2011Available online 12 September 2011

Keywords:Oil–water flowSmooth-stratified flowWavy-stratified flowFlow transitionLevel set methodDevelopment length

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⇑ Corresponding author. Tel.: +91 22 25767505; faxE-mail address: [email protected] (A. Sharma)

a b s t r a c t

Level-set method based transient 2D simulation of developing oil–water smooth-stratified (SS) andwavy-stratified (WS) flow in a horizontal and inclined plane-channel is done; for various inlet-velocity(uin), inlet-interface-height (Hin), inclination-angle (h), reduced surface-tension (r 6 row) and reducedgravity (g 6 ge). At the certain critical value of the governing parameters, SS to WS flow transition is cap-tured, analyzed and reasoned. Increase in uin, Hin and h and a decrease in row and ge are found to desta-bilize the SS flow. For fully-developed SS flow, a detailed analytical solution of interface-height,maximum axial-velocity, its transverse location, pressure-gradient and wall-shear stress is proposedfor an inclined channel; an excellent agreement between the present analytical and numerical resultsis also reported. The SS flow analytical-results are also compared with the time-averaged WS flow numer-ical-results (near the channel-outlet), to understand the reasons for flow-transition. Flow development isstudied with the help of axial variation of interface-height and maximum (over the cross-section of thechannel) axial velocity; time-averaged for WS flow. Furthermore, the variation of development-length forSS flow and WS flow is studied. Finally, stretching of interface-area due to onset of WS flow is presentedwith the help of interfacial-area-concentration. Its calculation directly from the Dirac-delta function ofthe level set method is a novel-contribution.

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1. Introduction

Study of two-phase flow in a conduit or channel is widelyencountered in various process industries. During the flow, theinterface attains different configuration and shape based on theprevalent operating conditions such as inlet flow-rates, inletvoid-fraction, and inclination. Hence, such flow is classified intovarious flow patterns/regime (Brennen, 2005) depending on theinterface configuration. Moreover, based on the prevalent flow-regime, accurate correlations of pressure-drop, interfacial-areaand liquid-holdup are required for efficient design of pipeline.Thus, for a given set of operating conditions, prediction of the typeof flow-regime and subsequent calculation of the engineeringparameters is of paramount importance.

For the present work, stratified flow regime is considered;wherein a continuous interface exists between two fluids flowingco-currently at low velocities, with lighter-fluid on top of hea-vier-fluid. Moreover, depending upon the governing parameters,two sub-regimes are possible within the stratified flow regime:smooth-stratified (SS) and wavy-stratified (WS) flow. Severalexperimentalists (Charles and Lilleleht, 1965a,b; Angeli andHewitt, 2000; Wegmann and von Rohr, 2006; Lum et al., 2006;

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Rodriguez and Oliemans, 2006; Mandal et al., 2007; Grassi et al.,2008) have proposed flow-regime maps and correlations of engi-neering-parameters, for two-phase flow in a pipe/channel. More-over, researchers have also attempted semi-analytical (Landman,1991; Sadatomi et al., 1993; Ullmann et al., 2004; Ullmann andBrauner, 2006) and analytical (Tang and Himmelblau, 1963; Akaiet al., 1981; Shoham and Taitel, 1984; Issa, 1988; Newton andBehnia, 2000; Datta, 2010) models applicable for fully-developedtwo-phase flow. Apart from fully developed results, few research-ers (Taitel and Dukler, 1976; Brauner, 2001, 2002) have proposedmodels to predict the flow regime transition.

Several two-dimensional numerical studies on SS two-phaseflow, for developing as well as fully developed region, inplane-channel are found (Khomami, 1990; Zubkov et al., 2005;Yap et al., 2005; Datta, 2010). Recent work by Datta (2010) consistsof detailed non-dimensional analytical and numerical studies. Theydeveloped analytical solution for density-matched fully-developedSS flow in a horizontal plane-channel and square-duct, wherenon-dimensional equations for fully-developed interface-height,velocity-profile, pressure-gradient and wall-shear-stress werederived from simplified Navier–Stokes equations. Moreover, forthe first time, a detailed level set method (LSM) based numericalinvestigation was also performed to study the developing region ofsmooth-stratified flow, at various viscosity-ratio and Reynolds-number. For fully-developed flow, they showed an excellent

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100 V.H. Gada, A. Sharma / International Journal of Multiphase Flow 38 (2012) 99–117

agreement between their analytical and numerical results. For thedeveloping flow in horizontal plane-channel, they proposed a cor-relation of hydrodynamic development length as a function of vis-cosity ratio and Reynolds number.

Very few numerical works exist for WS flow (Cao et al., 2004;Heidarinejad and Sani, 2010) and other two-phase flow regimes(De Schepper et al., 2008; Parvareh et al., 2010). For a two-dimensional plane periodic channel, Cao et al. (2004) studied theKelvin–Helmholtz (shear) instability at the interface of a two-fluiddensity-matched viscosity-stratified flow at small to mediumReynolds numbers; where the governing equations for flow weresolved using finite-difference method and the interface evolutionwas modeled using front-tracking method. Another notable workis that of De Schepper et al. (2008), where they demonstrated thecapability of 3D volume-of-fluid based simulation, using a com-mercial software, to capture all the horizontal flow-regimes re-ported in the Baker-chart during vapor–liquid/gas–liquid flow ina pipe.

From the literature review on numerical simulation of stratifiedflow, it is concluded that most of the work related to developingand fully-developed region is restricted to SS flow regime.Although some computational studies are done for periodicallyfully-developed WS flow, no work is found for developing WS flowexcept that of De Schepper et al. (2008) and Parvareh et al. (2010)for pipe. Furthermore, no numerical study dealing with the effectof various governing parameters on the SS to WS flow transitionis found. A detailed analytical solution is available for fully-developed stratified flow in a horizontal plane-channel (Datta,2010) but not in an inclined channel.

SS/steady to WS/periodic flow transition leads to increase in thetwo-phase pressure drop; however, the heat and mass transfer fromthe interface and channel-wall also increases (Frisk and Davis,1972). Thus, the SS and WS flow are preferred according to an engi-neering application. For oil–water stratified flow, the flow-transi-tion (SS to WS and vice versa) can be controlled by changingvarious governing parameters, i.e. inlet-velocity, inlet water-frac-tion, inclination of the channel, surface-tension and accelerationdue to gravity. This is the motivation of the present work. Anothermotivation of the present work is to propose a detailed analyticalsolution for fully-developed SS flow in an inclined channel.

The primary objective of the present work is to investigate theeffect of inlet-velocity and interface-height/water-fraction at theinlet, on the SS to WS flow-transition; for developing oil–waterflow in a horizontal plane-channel. For inclined channel, the effectof reduced coefficient of surface-tension and gravitational-acceleration on the flow-transition is studied. Furthermore, forfully-developed flow, the objective is to compare the numericallyobtained steady and time-averaged results for SS and WS flow,respectively; with the proposed analytical solution for SS flow.For developing flow, the objective is to study the axial-variationof interface-height and maximum value (over the cross-section ofthe channel) of axial and transverse velocity. Furthermore, theobjective is to study the variation of development-length withchange in the governing parameters; for SS as well as WS flow.Finally, the objective is to propose a calculation procedure forinterfacial-area-concentration using level-set (LS) method andquantify its variation during the transition of SS to WS flow. Anovel level-set method based in-house code, developed in ourprevious work (Gada and Sharma, 2011), is used for the numericalsimulation.

2. Physical model for smooth and wavy stratified flow

Flow in a plane-channel, in-between two parallel plates of infi-nite breadth, essentially simplifies 3D flow to 2D flow. Physically,

such flow situation is realized in a rectangular-duct with highbreadth-to-height ratio. As already mentioned, stratified flow re-gime can be smooth or wavy. Hence, to understand the character-istics of both the stratified flow regimes, standard terminology andthe flow features are discussed with the help of Fig. 1a and b, fordeveloping SS and WS flow in an inclined plane-channel,respectively.

The channel walls, separated by a distance H, are represented bythick solid lines in Fig. 1. Furthermore, it is inclined at an upwardangle h with horizontal direction; the gravity acts in verticallydownward direction. Note that the x- and y-coordinates are alongand perpendicular to the channel wall, respectively. Both the fluidsenter and leave the channel from left and right boundary, respec-tively; with heavier-fluid (water/fluid 1) below the lighter-fluid(oil/fluid 2). The amount of oil and water entering the channel isconstant at all times, which leads to a constant value of interfaceheight at the inlet i.e. hin. Its non-dimensional value, Hin = hin/H,represents inlet water-fraction as well as non-dimensional volu-metric-flow-rate of water, Q �1 ¼ Q1=ðQ 1 þ Q 2Þ; as both the fluidsenter with the same inlet velocity, uin (Fig. 1). Uniform inlet veloc-ity for the study of flow development of single-fluid flow is com-monly used. In the present work, the uniform inlet velocity isused for the two-fluid flow. However, the inlet velocity is takenas equal in the two fluids; to avoid shear at the interface. This isbecause the inlet shear can effect the flow-development andflow-transition. The scope of the present work is limited to the per-turbation-free and zero-shear inlet velocity profile. Any othervelocity-profile at the inlet will introduce shear and will effectthe results, which can be studied in future work.

For the SS flow, it is seen from Fig. 1a that the two-fluid inter-face and the extent of boundary-layer (BL) are represented by solidand dash-dot line, respectively. It is seen that, there is boundary-layer flow between channel-wall and the dash-dot lines; and coreflow within the two dash-dot lines. Owing to the density and vis-cosity difference of the two-phases, the BL growth from the chan-nel walls is asymmetric. Moreover, the boundary layers coincide atthe interface, to ensure the continuity of shear-stress across theinterface. The two-fluid region from the inlet to the axial locationat which BLs meet is known as inlet-region. Once the BLs meet atthe interface, the axial-velocity in the core increases till some axialdistance downstream, called as filled-region. The filled region wasproposed by Mohanty and Asthana (1978). They mentioned thatthe boundary layers meet at the end of the inlet region but thevelocity profiles are not yet similar. The filled-region is definedas the extent of the developing-region between the location wherethe boundary-layers meet and the velocity profile becomes invari-ant in the axial direction. The length of the filled-region signifiesthe length required to adjust the completely viscous velocity pro-file, after the meeting of the boundary-layers from the top and bot-tom wall of the channel. Further downstream, the region is calledfully-developed region where the interface height is also invariantin the axial-direction. The summation of axial length of inlet andfilled region is known as development-length for fully-developedSS flow, lfd,SS, shown in Fig. 1a. The fully-developed axial velocityprofile is also shown in the figure, with the maximum value repre-sented as umax.

For the WS flow, shown in Fig. 1b, the domain setup is similar tothat used to describe SS flow. Here, because of different modes ofinstability (discussed in Section 8.4), steady to unsteady flow tran-sition takes place; owing to which a constant interface-height isnot reached in WS flow and the flow is never acceleration free. Fur-thermore, it is found in our numerical study (discussed in Section10.1) that, the axial variation of time-averaged interface has adecreasing and increasing trend; with a minimum at certain axiallocation. Up-to this axial location, the instantaneous interface isfound to be nearly-coinciding with time-averaged interface; and

(a)

(b)Fig. 1. Physical model for developing oil–water stratified flow in an inclined plane channel: (a) SS and (b) WS flow. Channel wall, interface and boundary layers arerepresented by thick-solid, solid and dash-dot lines, respectively. For WS flow, time-averaged interface is represented by dashed line.

V.H. Gada, A. Sharma / International Journal of Multiphase Flow 38 (2012) 99–117 101

it becomes wavy further downstream shown in Fig. 1b. Thus, theregion of monotonic decrease of the interface-height can be con-sidered as SS flow and thereafter it is WS flow, shown in Fig. 1b.The axial-length of the SS flow region, shown in the figure, is de-fined here as the development-length for WS flow, lfd,WS. This isthe length of SS flow required for the incipience of WS flow.

3. Mathematical formulation

Although the present study is for a particular fluid combination(oil–water), the present mathematical formulation and in-housecode are non-dimensional. The heavier/bottom fluid (water, de-noted by subscript 1) is taken as the reference fluid. Moreover,the channel height, H and inlet velocity, uin are taken as character-istic length and velocity scale, respectively. Lighter-fluid (oil) is de-noted by subscript 2.

3.1. Level set method

Level set method (LSM) (Sussman et al., 1994) is an Euleriancomputational technique for capturing moving boundaries orinterfaces. In this method, three functions are defined in theflow domain: level-set-function, /, for interface representation;Heaviside-function, H(/), to calculate fluid properties; and Dirac-delta-function, d(/), to model the effect of surface tension forcein the momentum equation. The LS function (/) is defined as asigned normal distance function measured from the interface; itis equal to zero at the interface, positive in fluid 1 (water) and neg-ative in fluid 2 (oil). The LS field, thus defined, is smooth and theexact instantaneous interface position can be captured by locatingthe zero level set.

In our earlier work (Gada and Sharma, 2009), the Heaviside-function at the centroid (face-centers) was physically interpretedas volume-fraction (cell surface-area fraction) occupied by the ref-erence fluid for a control-volume (CV). The Dirac-delta-functionwas shown to be the ratio of interface-area in the CV to the volumeof the CV. To avoid numerical instability in LSM, instead of sharpchange of H(/) across and d(/) at the interface, the interface is dif-fused having a finite thickness (2� = 3DX); across which the fluidproperties and surface tension force are smoothened.

3.2. Conservation Equations

The Navier–Stokes equations are coupled with LS method byinvoking the single field formulation (Juric and Tryggvason,1998; Gada and Sharma, 2009, 2011). The fluids are assumed tobe incompressible and the individual fluid properties are invariantin space and time. The interface is taken to be thin and mass-less.The surface-tension force, whose coefficient is assumed to be con-stant, is modeled in the momentum equation as a volumetricsource term (Brackbill et al., 1992; Chang et al., 1996) and isnon-zero only within the diffused interface.

In our earlier work (Gada and Sharma, 2009), the continuity andthe LS advection equation were derived from volume-conservationand mass-conservation laws, respectively. The non-dimensionalform of the governing equations are given as.

Volume conservation (continuity) equation:

r � U!¼ 0 ð1Þ

Mass conservation (level-set advection) equation:

@/@sþ U!�r/ ¼ 0 ð2Þ


Momentum conservation equation:

@ q�m U!� �

@sþr � q�m U

!U!� �

¼ �rP þ 1Rer � 2l�mD

� �þ q�m G

!

Fr2 þ1

Wejnd�ð/Þ ð3Þ

where rate of deformation tensor and gravity vector are given asD ¼ 0:5ðrU

!þ ðrU!ÞTÞ and G

!¼ ð� sin hi;� cos hjÞ, respectively. Fur-thermore, the interface unit normal vector and curvature are ex-pressed as n ¼ r/=jr/j and j ¼ �r � n, respectively. The meannon-dimensional density and viscosity are calculated as

q�m ¼ H�ð/Þ þ vð1� H�ð/ÞÞl�m ¼ H�ð/Þ þ gð1� H�ð/ÞÞ

ð4Þ

where H�(/) and d�(/) are smoothened Heaviside and Dirac-deltafunction, respectively (Gada and Sharma, 2009).

For the governing equations, the non-dimensional variables areexpressed as velocity: U

!¼ ~u=uin, length: X = x/H, time: s = tuin/H,pressure: P ¼ p=q1u2

in, density-ratio: v = q2/q1 and viscosity-ratio:g = l2/l1. The non-dimensional governing parameters are theReynolds number, Froude number and Weber number, given as

Re ¼ q1uinHl1

; Fr ¼ uinffiffiffiffiffiffigH

p ; We ¼ q1u2inH

rð5Þ

3.3. Subsidiary equation

Reinitialization equation is a subsidiary equation. The purposeof this equation is to regain the level set function as a normal dis-tance function, within a region of constant thickness 2� of diffused-interface, without altering the / = 0 level (interface) obtained afterthe advection of interface at each time-step. In the present work,an improved constraint based reinitialization procedure of Suhand Son (2008) is used.

4. Numerical methodology

In the present work, a novel dual-grid level set method(DGLSM), proposed in our earlier work (Gada and Sharma, 2011),is used. In DGLSM, coupling of finer grid for interface with a rela-tively coarser grid for flow is used. This is based on our hypothesesthat the grid required to capture the interface is finer as comparedto that required for flow. Qualitative and quantitative results of theDGLSM were compared against published results, for three strin-gent test problems. DGLSM was shown to be nearly as accurateas the completely refined traditional LS method at a substantiallyless computational expense.

The computational domain used for the present numericalstudy, is shown in Fig. 2. The non-dimensional length of thedomain, L = l/H is obtained after a domain-length independencestudy (discussed in Section 7). Here, the height of the channel(H) is considered as the length scale, hence the non-dimensionalchannel-height is unity. Note that the X-coordinate in the figure

Fig. 2. Computational domain, boundary conditions and initial interface shap

corresponds to the direction of the inclined channel, i.e., inclinedat an angle h from the horizontal direction. Fig. 2 shows that, waterand oil enter the channel, at uniform non-dimensional velocityUin = 1. At the exit of the domain, convective outflow boundarycondition is used. No-slip boundary condition is used at the chan-nel walls. The velocity and pressure field are initialized as zero. Theinterface is initialized to be parallel to channel walls at a distanceHin from bottom wall; with level set function given as / = Hin � Y. Itis ensured here that Hin does not vary with time at the inlet.

The validation of the present code with the analytical, experi-mental, and numerical results is given in our previous work (Gadaand Sharma, 2011); for three standard test cases of two phase flow(dam-break simulation, Rayleigh–Taylor instability and filmboiling). Here, the validation is done for SS and WS flow. For SSflow, Table 1 shows an excellent agreement, of present analytical(discussed in Appendix A) and numerical results against thenumerical result of Khomami (1990); for fully-developed maxi-mum U-velocity, its Y-coordinate and interface-height. For period-ically fully-developed WS flow in a channel, Fig. 3a–f shows goodagreement between the present results and that obtained by Caoet al. (2004); for the instantaneous interface shape at a timeinstant, s = 0.7, 2.1 and 2.8. The simulation is performed on the do-main of size 1 � 1 at g = 10, v = 1, Re = 5 and We = 18.8. The initialheight of the interface is set as 0.375 and constant pressure drop isspecified across the channel, so that the instability develops over aperiod of time.

5. Parametric details

The governing parameters considered in the present work, for aplane channel, are those used by Wegmann and von Rohr (2006) intheir experimental study for a oil–water flow in a pipe. They havepredicted various two-phase flow regime by varying inlet-velocityof oil/water, uin, and inlet water-fraction. Note that the hydraulicdiameter of the plane-channel here is kept equal to that of the pipe,considered by Wegmann and von Rohr (2006). Hence, thedimensional channel-height (characteristic-length) is taken asH = 2.8 mm. Furthermore, the properties of water (fluid 1) and par-affin oil (fluid 2) are taken as q1 = 997.85 kg/m3, l1 = 0.89 mPa s,q2 = 819.25 kg/m3, l2 = 4.75 mPa s; and the coefficient of surfacetension is taken as row = 62.2 mN/m.

In the present study, parametric investigation is done to studythe effect of inlet-velocity, inlet-interface-height/water-fraction,surface-tension and gravitational-acceleration on the SS to WSflow transition and engineering parameters.

For a horizontal channel (h = 0�), the governing parameters areas follows:

Inlet-velocity: uin = 0.1, 0.3, 0.4 and 0.5 m/sInlet-interface-height: Hin = 0.1–0.7 in steps of 0.1; and 0.7 to0.9 in steps of 0.025

For inclined channel, the governing parameters are as follows:

Inlet-velocity: uin = 0.5 m/s

e, for simulation of developing oil–water SS/WS flow in a plane-channel.

Table 1Comparison of present fully-developed SS flow results against the numerical results of Khomami(1990) at v = 1, g = 10 and Re = 10.

Published numerical results (Khomami, 1990) Present results

Analytical Numerical

Umax 1.770 1.718 1.695YUmax 0.260 0.262 0.250Hfd 0.370 0.388 0.392

0 0.25 0.5 0.75 10.3

0.4

0.5

0.6

0.3

0.6

0.5

0.4

10 0.25 0.5 0.75

10.750.50 0.250.3

0.4

0.5

0.6

10.750.50 0.250.3

0.5

0.4

0.6

0 0.25 0.5 0.75 10.3

0.4

0.5

0.6

0 0.25 0.5 0.75 10.3

0.4

0.5

0.6

(a) (d)

(b) (e)

(c) (f)

Domain Length (L)

SSWS

(g)

Hin

H

0.675

0.65

0.7

0.725

0.75

0.775

0.8

0.825

0.85

Grid Size per Unit Length30 40 50 60 70 80 20 30 40 50 60 70

(h)

Hin

H

0.675

0.65

0.7

0.725

0.75

0.775

0.8

0.825

0.85

Fig. 3. Comparison of instantaneous interface for simulation of WS flow in an periodic channel: (a–c) present and (d–f) published results (Cao et al., 2004) at a non-dimensional time instant of (a and d) 0.7, (b and e) 2.1 and (c and f) 2.8. Flow-regime dependence on (g) domain-length (with 50 grids per unit length) and (h) grid-size perunit length (on a domain length of 60); obtained at uin = 0.5 m/s and 0.7 6 Hin 6 0.8.


Inlet-interface-height: Hin = 0.725Angle of Inclination: h = 2.5�, 5�, 7.5� and 10�Coefficient of surface-tension: r = row, 0.95row, 0.85row, 0.8row

and 0.75row at g = ge

Gravitational-acceleration: g = ge, 0.5ge, 0.2ge and 0.1ge atr = 0.85row

where ge is the gravitational-acceleration on earth. The reduc-tion in surface-tension can be obtained by using surfactants andthe reduction in gravitational-acceleration is found in the outer-space. Note that, the effect of varying surface-tension is also stud-ied for horizontal-channel.

Although the parametric details are in dimensional form, theresults are obtained in non-dimensional form; using a non-


dimensional formulation and code (refer Sections 3 and 4). Theabove values of the dimensional governing parameters areincorporated by corresponding value of Re, We and Fr (Eq. (5)).Furthermore, the non-dimensional properties in oil-phase aretaken as q�2 ¼ v ¼ 0:835 and l�2 ¼ g ¼ 5:326; and that in thewater-phase as q�1 ¼ 1 and l�1 ¼ 1.

6. Calculation of non-dimensional engineering parameters

For fully-developed flow, the non-dimensional parameters ofinterest are U-velocity profile, its maximum value over the cross-section of the channel Umax, corresponding Y-coordinate YUmax,interface-height Hfd, pressure gradient dP/dX, and Poiseuille num-ber at the wall PoB/PoT. For fully-developed flow, the pressure andgravitational force balance the viscous force. Thus, instead of iner-tial scale, here, the pressure and wall shear-stress are non-dimen-sionalized with viscous scale as P = pH/l1uin and Pow = 2syx,wH/l1uin, respectively (see Appendix A for details). For numerical-study of developing-flow, the pressure and wall shear-stress arenon-dimensionalized by inertial scale as P ¼ p=q1u2

in andCfw ¼ 2syx;w=q1u2

in, respectively. Hence, the numerically calculatedfully-developed P, Cf,B and Cf,T are multiplied with Re to obtain P,PoB and PoT, respectively.

Numerical differentiation is done to obtain the non-dimen-sional pressure-gradient and wall shear-stress. Hfd is obtained asthe Y-coordinate of the fully-developed horizontal interface (repre-sented by / = 0). For WS flow, the time-averaged results are ob-tained at various axial locations, close to the outlet of thechannel, and negligible change in the results is obtained. Finally,the non-dimensional quantities are presented here for an axiallocation at X = 55; close to the channel-outlet.

An engineering parameter which has a very important role inthe efficient transportation of oil through pipelines is the pressuregradient reduction factor (PGRF). It is defined as the ratio of thepressure-gradient for the more-viscous-phase flowing alone to thatfor the flow of both the phases through the channel; while the flowrate of more-viscous-phase is kept same in both the cases (Charlesand Lilleleht, 1965a). The role of PGRF is to quantify the amount ofsaving in the pumping power, by using less-viscous-phase (such aswater) during transportation of more viscous phase (such as oil) ina conduit. Thus, PGRF value greater than one signifies reduction inthe pumping cost. The PGRF for WS flow is calculated from thetime-averaged pressure drop.

The development length for SS flow (Lfd,SS = lfd,SS/H) is obtainedhere as the axial length at which Umax reaches 99% of the fully-developed value; Umax near the channel-outlet which is almostequal to analytical value. The development for WS flow(Lfd,WS = lfd,WS/H) is obtained as the axial location where the time-averaged interface-height reaches its minimum value.

7. Domain-length and grid independence study

In the present work, the onset of smooth to wavy flow transi-tion and the values of engineering parameters should be

Table 2Domain-length independence-study for SS and WS flow at Hin = 0.725 and 0.75, respectivechannel, with square control volumes.

Domain length (L) SS flow

Lfd Hfd

50 42.111 0.56660 47.588 0.56470 47.670 0.563

Analytical 0.558

independent of the domain-length and grid-size. For the largest in-let velocity, uin = 0.5 m/s, a flow regime map in Fig. 3g and h showsthat the onset of flow transition at Hin = 0.75 is independent of do-main-length and grid-size except for the smallest domain size ofL = 40. Thus, the onset of WS flow is more sensitive to domain-length as compared to grid-size.

Table 2 shows the variation of engineering parameters, for SSand WS flow for various domain-length, at a constant grid-size of50 control volumes per unit length of the channel. Maximum dif-ference is seen for the development-length of the SS flow. As thedomain-length is increased from 50 to 60, it reduces from 11.66%to 0.17%; when compared to the result on a domain-length of 70.Thus, a domain-length, L = 60, is used. A grid size of 50 per unitlength is found sufficient for grid-independent results. Thus, agrid-size of 3000 � 50 (with square control volumes) forNavier–Stokes and 6000 � 100 for LS equations is used on adomain-size of 60 � 1 for all the DGLSM simulations.

8. Smooth-stratified to wavy-stratified flow-transition

In this section, the effect of different governing parameters onthe transition from SS (steady) to WS (unsteady) flow is discussed.For the SS flow, the fully-developed interface is parallel to channelwall, i.e., interface-height, Hi, does not vary in the axial direction.Whereas, for WS flow, the interface-height varies not only alongthe axial direction but also with time. Here, the instantaneousinterface plots and the temporal variation of interface-height (atan axial location X = 55) are used to monitor the unsteadiness.

8.1. Effect of inlet-velocity and inlet-interface-height for horizontalchannel

The effect on the instantaneous interface shape is shown inFig. 4, at a non-dimensional time instant of s = 250. The temporalvariation of the interface is shown in Animation 1 (see ElectronicAnnex 1 in the online version of this article). With increasing inletinterface-height, the figure and the animation shows onset of WSflow at a critical inlet-interface-height, Hin,c, of 0.875, 0.775 and0.75 for uin = 0.3, 0.4 and 0.5 m/s, respectively. With increasing in-let-velocity at Hin = 0.85, Fig. 4a and g shows the onset of WS flowat uin,c = 0.4 m/s. For the SS flow, it can be seen that the interface-height decreases in the axial direction and reaches a constant fully-developed value. Whereas, for the WS flow, instead of reaching aconstant value, the interface becomes wavy after a certain axial-location. However, the axial location at which the interface be-comes wavy decreases with increasing Hin (Fig. 4d–h atuin = 0.4 m/s) and increasing uin (Fig. 4c, h and m at Hin = 0.9); indi-cating that the unsteadiness moves upstream. The figures alsoshow that the waviness of the interface increases with increasingHin and uin. The increased waviness may result in the interfacetouching the top wall, initiating slug flow. However, note that,the interface does not touch the top wall in the present case,although it reaches very close to it.

ly, and uin = 0.5 m/s. Grid-size is taken as 50 per unit non-dimensional length of the

WS flow

�dP/dX Umax Hfd Umax

31.362 1.707 0.588 1.69329.082 1.730 0.609 1.72628.947 1.738 0.609 1.729

29.499 1.743

Fig. 4. Instantaneous interface for various inlet-velocity t (m/s) {(a–c) 0.3 (d–h) 0.4 (i–m) 0.5} and inlet-interface-height in a horizontal channel at non-dimensional times = 250.


Fig. 5a–d shows the temporal variation of interface-height Hi, atan axial location X = 55, for various Hin and uin. For all Hin, the fig-ures show that the interface-height remains close to its inlet valuefor quite some time. Thereafter, for Hin = 0.1, the interface-heightincreases slightly to its fully-developed value (Hfd). Whereas, it re-mains almost constant for Hin = 0.3. For Hin P 0.5, Fig. 5b–d showsthat the interface-height oscillates, followed by a sharp decrease inits value within a short span of time. Thereafter, it reaches to a con-stant fully-developed interface-height (Hfd) for SS flow and oscil-lates about a mean value for WS flow. The sharp decrease in theinterface-height from its inlet value is due to the formation of awavy-hump which passes through X = 55 (Animation 1, see Elec-tronic Annex 1 in the online version of this article). After the sharpdecrease, for Hin = 0.7–0.85 at uin = 0.3 m/s, Fig. 5b shows that thelow amplitude oscillations are observed about the newly attainedinterface-height. It can be seen that the oscillations eventuallydie down with time; leading to a steady state value. However, forHin = 0.875 and 0.9, the oscillations are sustained and thus, the flowis classified as WS flow.

The flow regimes obtained for horizontal channel with varyinginlet interface-height (water-fraction) and inlet-velocity are over-lapped on flow regime map of Wegmann and von Rohr (2006)for horizontal pipe in Fig. 5e. Note that, the overlapped regime

map is only intended for qualitative comparison; as the flow ratesentering the channel are different from those in case of pipe. Inaddition to that, the geometrical difference between a channeland a pipe will cause different flow-phenomena and thus, the tran-sition boundaries. For the horizontal channel, Fig. 5e shows thatthe flow transition takes place with increasing Hin except foruin = 0.1 m/s. Here also, it can be seen that the critical value ofHin, for the onset of WS flow, decreases with increasing uin.

In a recent 3D study of SS flow in a duct by Datta et al. (2011), itwas found that the effect of governing parameters on the engineer-ing parameters is more pronounced for 3D flow in a square-duct ascompared to 2D flow in plane-channel; due to the additionalconfinement for the 3D flow. Thus, it is conceived here that thetransition boundaries will move towards the lower values ofgoverning-parameters for 3D flow situation. This is also seen inthe flow regime map in Fig. 5e, where the SS to WS flow transitionboundary for pipe is lower as compared to the channel.

Although the scope of the present investigation is limited to SSand WS flow, it is quite exhaustive in terms of number of simula-tion (60) done, results presented and the analysis. Extending thepresent work to other flow regimes will require longer domain-sizeand larger grid-size, with increasing Reynolds number. This willalso require more computational time and can be studied in future.

Hin= 0.10

0.30

0.50

0.70

0.775

0.90

0.75

Hin= 0.10

0.30

0.75

0.50

0.70

0.900.850.80

τ0 50 100 150 200

uin = 0.4 m/s(c)

Hi

0

0.2

0.4

0.6

0.8

1

Hi

u in(m

/s)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5Experimental - PipeSS Flow - ChannelWS Flow - Channel

DispersedWater Cont.

Stratified (SS + WS)

Intermittent (Slug + Plug)

Dispersed

DispersedOilCont.

(e) Hi

0

0.2

0.4

0.6

0.8

1

(a) uin = 0.1 m/s

in

(b) uin = 0.3 m/s

Hin= 0.10

0.30

0.725

0.50

0.70

0.75

0.90

Hin = 0.10

0.30

0.75

0.50

0.70

0.875

0.80

0.90

τ0 50 100 150 200u = 0.5 m/s(d)

Fig. 5. (a–d) Temporal variation of interface-height, at X = 55, for various inlet-interface-height and inlet-velocity. (e) Flow regime map of the present and experimental(Wegmann and von Rohr, 2006) results, for a horizontal channel and a pipe, respectively.

σ/σ ow

θ0 2.5 5 7.5 10

θ0 2.5 5 7.5 100.7

0.8

0.9

1

1.1 SS FlowWS Flow

(a)

g/g e

0

0.2

0.4

0.6

0.8

1

1.2

(b)Fig. 6. Flow regime map demarcating SS and WS flow for horizontal/inclined channel under the influence of (a) reduced coefficient of surface-tension (for g = ge) (b) reducedgravitational-acceleration (for r = 0.85row); at uin = 0.5 m/s, Hin = 0.725.



8.2. Effect of reduced surface-tension for horizontal and inclinedchannel

For a horizontal channel (h = 0�), the temporal variation of inter-face-height at X = 55 (not shown here) showed that the onset ofWS flow occurs by reducing the coefficient of surface tension to80% of its standard value (row). Moreover, it was found that thiscritical surface-tension value is sufficient enough for the onset ofWS flow for a channel inclined at 2.5� and 5�; whereas, it increasesto 85% of row, for the channel inclined at 7.5� and 10�. The flow re-gime map demarcating the SS and WS flow for the combination ofr and h is shown in Fig. 6a. It shows that the critical value of coef-ficient of surface-tension, for the onset of WS flow, increases withincreasing inclination. The figure also shows the onset of WS flowwith increasing h at r = 0.85row.

8.3. Effect of reduced gravity for inclined channel

Note that this study is done at r = 0.85row which correspondsto SS flow at h = 0�, 2.5� and 5� and WS flow at h = 7.5� and 10�. Itwas found that, the SS flow in an inclined channel can be con-verted to WS flow by reducing the gravitational-acceleration.However, the onset of WS flow was found to occur at a larger va-lue of gravitational acceleration g = 0.2ge at h = 5� as compared tog = 0.1ge required at h = 2.5�. The flow regime map demarcatingthe SS and WS flow for the combination of g and h is shown inFig. 6b. It can be seen that the reduction in g required for the on-set of WS flow decreases with increased inclination of the chan-nel. The figure also shows onset of WS flow with increasing h,at constant g.

8.4. Reasons for the flow-transition

Flow regime map in Figs. 5 and 6 show that the onset of WSflow occurs with an increase in Hin, uin and h and with a decreasein r and g. In this section, probable reasons for SS to WS flow-tran-sition will be discussed.

Using linear stability analysis of two superposed fluids in planePoiseuille flow, the mechanism of generation of the interfacialwaves during WS flow is attributed to several modes (Boomkampand Miesen, 1996). Among the several modes mentioned, twomodes are dominant for stratified-flow in horizontal channel: First,long-wave interfacial-mode (Yih, 1967; Hooper, 1985; Renardy,1985; Joseph et al., 1984) and Second, short-wave shear-mode(Yiantsios and Higgins, 1988a,b) of Tollmien–Schlichting type.The first mode occurs mainly due to jump in viscosity across theinterface; whereas, the second mode occurs if the Re is sufficientlylarge. Both the modes of instability are encountered in the presentstudy, done at moderate Reynolds number.

Due to long-wave interfacial-mode, long-wavelength (greaterthan the channel height) of the interface is seen in Fig. 4b, e andj at the critical Hin for onset of WS flow. In the published literature(Boomkamp and Miesen, 1996; Hooper, 1985; Renardy, 1985;Joseph et al., 1984), this is attributed to thin layer effect, i.e., a thinlayer of more viscous fluid near to a solid boundary is linearlyunstable to an long-wavelength interfacial-mode. Yiantsios andHiggins (1988b) showed the possibility of multiple wavelengthsdue to existence of more than one interfacial-mode during WSflow. The multiple-wavelengths are seen here in Fig. 4e–h, withincreasing Hin at uin = 0.4 m/s.

Due to short-wave shear-mode, a reduction of interfacial-wave-length is seen in Fig. 4c, h and m with increasing uin at a constantHin = 0.9. In the present work, increase in Re occurs with an in-crease in the uin. Then, the stabilizing viscous, surface-tensionand gravitational effects are suppressed with an onset of wavinessat the interface. Thus, increase in the Hin and uin, both destabilize

the flow and smaller Hin is required for the onset of WS flow at alarger uin (Figs. 4 and 5).

Fully-developed interface-height is affected by viscosity ratioalone in case of horizontal-channel; whereas, for inclined channel,it is also affected by density-ratio. Thus, increase in interface-height occurs with increasing Hin for horizontal-channel and alsoby increasing upward inclination for inclined-channel (also seenin Fig. 9a–c). This is due to the fact that the bottom-fluid/water,being heavier, experiences more gravitational resistance as com-pared to lighter top-fluid/oil; which leads to retardation of flowin water. Due to the reduced velocity in water, the interface heightmust rise to ensure mass-conservation; reducing the thickness ofthe oil-layer near the wall. Moreover, with upward-inclination,the stabilizing gravitational effect in the transverse direction re-duces (Boomkamp and Miesen, 1996). The combined effect of theseoccurrences leads to the onset of WS flow.

The role of surface tension is to resist the stretching of interface;by the action of cohesion in the fluid particles. The present flowconfiguration is stable under static condition, as the heavier fluidis below the lighter fluid. However, under dynamic condition, theinteraction of inertia, viscous, gravitational and surface-tensionforce at the interface causes it to become unstable. For this situa-tion, the surface tension force tries to maintain smooth interface.Reduction of surface-tension reduces the cohesion of fluid parti-cles. Then, the interface becomes prone to stretching, with an in-crease in surface area under the effect of interfacial-shear-stress,resulting in the SS to WS flow transition.

When the channel is inclined and the gravitational-accelerationis reduced, the self-weight of the fluids, which stabilizes the flow,reduces; resulting in the flow-transition. Yiantsios and Higgins(1988b) showed that reduction of gravitational-acceleration desta-bilizes the flow.

Thus, increase in h and decrease in r and g, both destabilize theflow and smaller reduction in r and g is required for the onset ofWS flow at larger h, shown in Fig. 6. These results are consistentwith the results found in the published literature, discussed above.

9. Analytical and numerical results for fully-developedstratified flow

In this section, for the fully-developed SS flow, the effect of in-let-velocity and inlet-interface-height, upward inclination, surface-tension and gravitational-acceleration on the present analytical(given in Appendix-A) and numerical results are compared forvarious governing parameters; to develop confidence in both theresults. Numerically, SS to WS flow transition occurs at a criticalvalue of governing parameters. Here, the numerically obtainedtime-averaged WS flow results are compared with the analyticallyobtained SS flow results. The time-averaged WS flow results are gi-ven here for two reasons: first, to quantify a change in the resultsdue to the flow-transition (SS to WS); and second, to reason theflow-transition from the difference in the WS numerical and SSanalytical results.


With increasing inlet-interface-height at uin = 0.1, 0.3, 0.4 and0.5 m/s, Fig. 7 shows an excellent agreement between the presentfully-developed SS flow analytical and numerical results; forvariation of interface-height, the maximum U-velocity and itsY-coordinate. Fig. 8 shows similar agreement in the results, fornon-dimensional pressure-gradient, Poiseuille-number on the topand bottom wall of the channel and pressure gradient reductionfactor (PGRF). It can be seen in both the figures that the SS flow


results vary with Hin but not with uin; whereas, the time-averagedWS flow results are dependent on both the input-parameters.

Fig. 7a–d shows that the fully-developed interface-heightHfd > Hin (Hfd < Hin) for Hin < 0.302 (Hin > 0.302). This indicates that

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1Numerical SS: Hfd

Numerical WS: Hfd

Analytical SS: Hfd

Hfd = Hin

HinHin

HinH

HinHin

HinHin

0 0.25 0.5 0.75 1

Numerical SS: YUmax

Numerical WS: YUmax

Analytical SS: YUmax

(a)

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

(c)

Hfd

,YU

max

Hfd

,YU

max

Hfd

,YU

max

Hfd

,YU

max

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

(b)

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

(d)Fig. 7. Variation of fully-developed SS and WS flow results for a horizontal channel ((a–(b, f, j) 0.3 (c,g,k) 0.4 (d,h, l) 0.5}.

interface-height increases (decreases) along the axial-direction inthe developing flow region; with the maximum fully-developedU-velocity obtained in oil (water) for Hin < 0.302 (Hin > 0.302). Thisis shown in Fig. 7a–d with YUmax > Hfd for Hin < 0.302 and YUmax < Hfd

HinHin

HinHin

HinHin

HinHin

0 0.25 0.5 0.75 11.25

1.5

1.75

2Numerical SSNumerical WSAnalytical SS

(e)

0 0.25 0.5 0.75 11.25

1.5

1.75

2

(h)

0 0.25 0.5 0.75 11.25

1.5

1.75

2

(f)

Um

axU

max

Um

axU

max

0 0.25 0.5 0.75 11.25

1.5

1.75

2

(g)

d) Hfd, YUmax and (e–h) Umax) with increasing Hin, at a constant uin (m/s) {(a,e,i) 0.1


for Hin > 0.302. Moreover, it is interesting to see in the figures thatYUmax = Hfd = Hin at Hin = 0.302, indicating that the interface heightis invariant in the developing flow region and the maximumU-velocity is obtained at the interface.

H

PoT,

PoB

0.25 00

20

40

60

NumNumAnaly

(e)Hin

Hin

Hin

Hin

-dΠ

/dX

0.25 0.5 0.75 10

20

40

60

Numerical SSNumerical WSAnalytical SS

(a)

-dΠ

/dX

0.25 0.5 0.75 10

20

40

60

(b)

-dΠ

/dX

0.25 0.5 0.75 10

20

40

60

(c)

-dΠ

/dX

0.25 0.5 0.75 10

20

40

60

(d) H

PoT,

PoB

0.25 00

20

40

60

(h)

H

PoT,

PoB

0.25 00

20

40

60

(g)

H

PoT,

PoB

0.25 00

20

40

60

(f)

Fig. 8. Variation of fully-developed SS and WS flow results for a horizontal channel {(a–{(a,e, i) 0.1 (b, f, j) 0.3 (c,g,k) 0.4 (d,h, l) 0.5}.

Furthermore, with increasing Hin, Fig. 7e–h shows that Umax

decreases; from its value 1.5 at Hin = 0 to its minimum value ofUmax = 1.381 at Hin = 0.029. With further increase in Hin, the Umax

increases and reaches the value of 1.5 again at Hin = 0.302 (at which

Numerical SS: Po T

Numerical WS: Po T

Analytical SS: Po T

in

.5 0.75 1

erical SS: PoB

erical WS: PoB

tical SS: PoB

Hin

PGR

F

0.25 0.5 0.75 10

1


(i)

in

.5 0.75 1

in

.5 0.75 1

in

.5 0.75 1

Hin

PGR

F

0.25 0.5 0.75 10

1

2

(k)

Hin

PGR

F

0.25 0.5 0.75 10

1

2

(j)

Hin

PGR

F

0.25 0.5 0.75 10

1

2

(l)

d) �dP/dX, (e–h) PoT, PoB and (i–l) PGRF} with increasing Hin, at a constant uin (m/s)


YUmax = Hfd = Hin), attaining a maximum value of 1.789 atHin = 0.862; and then, decreases to 1.5 at Hin = 1.

Fig. 8 shows that the non-dimensional pressure gradient andtop/bottom wall Poiseuille number decreases with increasing Hin.At Hin = 0 (Hin = 1) i.e. single phase oil (water) flow, the maximum(minimum) pressure gradient andPoiseuille number values areseen in the figures. With increasing Hin, Fig. 8e–h shows asymptoticdecrease in PoB; due to negligible change in non-dimensionalvelocity gradient (oU/@Y) at the bottom wall at larger Hin. The con-stant value is much smaller than the single phase oil flow, i.e.,Hin = 0. This is due to the fact that water (which is less viscous thanoil) comes in contact with the bottom wall and thus, leads toreduction of the bottom wall shear stress and the Poiseuille num-ber. Thus, PoB < PoT is observed for all values of 0 < Hin < 1 andPoB = PoT is observed for Hin = 0 and 1. At larger values of Hin, thevelocity gradient at the top-wall decreases sharply with increasingHin; leading to sharp decrease in PoT, shown in Fig. 8e–h.

The reduction in pressure gradient and thus, the pumpingpower is quantified by the PGRF (defined in Section 6), shown in

Numerical SS: YUmax

Numerical WS: YUmax


(a) θ0 15 30 45

θ0 15 30 45

θ0 15 30 45

0 15

0 15

0 15

Numerical SS: Hfd

Numerical WS: Hfd

Analytical SS: Hfd

(b)

(f)

Um

ax

1.5

1.75

2

Um

ax

1.5

1.75

2

Um

ax

1.5

1.75

2

(d)

(c)

(e)

0

0.5

1

Hfd

,YU

max

0

0.5

1

Hfd

,YU

max

0

0.5

1

Hfd

,YU

max

Fig. 9. Variation of fully-developed SS and WS flow results {(a–c) Hfd, YUmax (d–f) Umax (analytical results are obtained at r = 0; whereas, numerical results are for various r/row

Fig. 8i–l with increasing Hin. It is observed that the PGRF is 1 atHin = 0 i.e. single phase oil. With increasing Hin, the figure showsthat the PGRF increases reaching to its maximum value of 1.66 atHin = 0.098. Further increase in Hin results in decrease of PGRF. AtHin = 0.493, PGRF reaches 1 indicating that there will be no savingin pumping cost; as compared to single-phase oil flow. Finally,for Hin > 0.493, the PGRF is always less than 1, indicating morepumping power as compared to single more-viscous phase flow.For plane-channel, saving in pumping cost can be obtained withwater occupying less than 49.3% (almost half) of the channel-height; with maximum saving for the water-fraction of 9.8%. Thus,wetting the bottom-wall by water (less viscous fluid) can be usedas an efficient technique for transportation of oil (more viscousfluid).

To quantify the change in the fully-developed results due to theflow-transition (SS to WS), Fig. 7a–d shows that the time-averagedinterface-height for the WS numerical-solution is higher as com-pared to steady SS analytical-solution; and the difference increaseswith increase in inlet-velocity (inertia) at a constant Hin. For the

θ30 45

θ30 45

θ30 45

θ0 15 30 45

θ0 15 30 45

θ0 15 30 45


0

50

100

150

(h)

-dΠ

/dX

-dΠ

/dX

-dΠ

/dX

0

50

100


(g)

0

50

100

150

(i)g–i) �dP/dX} with increasing h, at a constant uin = 0.5 m/s and Hin = 0.725. Present= (a,d,g) 1 (b,e, h) 0.80 (c, f, i) 0.75.


WS flow, the increase in Hfd leads to higher flow area in water-phase. Thus, for WS as compared to SS flow, the Umax value (whichis realized in the water phase) decreases (Fig. 7e–h) while YUmax va-lue increases (Fig. 7a–d).

Fig. 8a–d shows that the non-dimensional pressure-gradient forWS, as compared to SS flow, is much larger; and increases sharplywith increasing Hin. The figure also shows that the pressure-gradi-ent for SS flow remains constant whereas that for the WS flow in-creases with increasing uin, at a constant Hin. For uin = 0.5 m/s atlarger values of Hin, Fig. 8d shows that the magnitude of two-phasepressure-gradient for WS flow is nearly equal to the pressure-gra-dient corresponding to single phase oil flow, i.e., Hin = 0. The reasonfor increased pressure-gradient is a substantial increase in top-wallPoiseuille number, for WS as compared to SS flow, shown inFig. 8f–h. The figure also shows that the bottom wall shear stressis least affected due to WS flow. This is because the wavy interfaceis closer to top-wall. Thus, it has more effect on the top as com-pared to the bottom wall Poiseuille number; the PGRF for WS flow

Numerical SS: YUmax

Numerical WS: YUmax


Numerical SS: Hfd

Numerical WS: Hfd

Analytical SS: Hfd

(a) θ0 15 30 45

0

0.5

1

Hfd

,YU

max

(b) θ0 15 30 45

0

0.5

1

Hfd

,YU

max

(c) θ0 15 30 45

0

0.5

1

Hfd

,YU

max

0 15

Um

ax

1.5

1.75

2

(d)

0 15

Um

ax

1.5

1.75

2

(f)

0 15

Um

ax

1.5

1.75

2

(e)

Fig. 10. Variation of fully-developed SS and WS flow results {(a–c) Hfd, YUmax (d–f) Umax (g–numerical study) and various values of gravitational acceleration g/ge = (a,d,g) 1 (b,e,h)

becomes smaller than that for SS flow, indicating further increasein pumping power.

Here, an attempt is made to reason the flow transition from theanalytical fully-developed SS flow results. In this regard, the varia-tion of SS results with increasing Hin are analyzed. It is interestingto note that the flow transition occurs close to Hin value corre-sponding to the maximum of Umax variation in Fig. 7e–h. Further-more, Fig. 8f–h shows that the transition occurs at a Hin valuecorresponding to sharp decrease in the PoT. Thus, with increasingHin, the sharp increase in Umax and decrease in PoT, affect the inter-facial-shear-stress, which probably leads to the SS to WS flow-transition.

9.2. Effect of reduced surface-tension for horizontal and inclinedchannel

The effect of surface-tension on the results for fully-developedSS and time-averaged WS flow (at X = 55) is studied at



θ30 45

θ30 45

θ30 45

θ0 15 30 45

-dΠ

/dX

0

50

100

150

(g)

θ0 15 30 45

-dΠ

/dX

0

50

100

150

(h)

θ0 15 30 45

-dΠ

/dX

0

50

100

150

(i)i)�dP/dX} with increasing h, at a constant uin = 0.5 m/s, Hin = 0.725, r = 0.85row (for0.20 (c, f, i) 0.10.


uin = 0.5 m/s and Hin = 0.725. For SS flow, Fig. 9 shows an excellentagreement between the variation of present analytical and numer-ical results (interface height, the maximum U-velocity, its Y-coor-dinate and pressure gradient) with increasing inclination atvarious values of coefficient of surface tension. It is seen in the fig-ure that the SS numerical results vary only with h and not with r/row; although the surface-tension was not neglected during simu-lations. This observation validates the assumption regarding sur-face tension (considered as zero) in the analytical solution. Thus,for SS flow, the role of surface tension on the results are negligibleas the interface is horizontal in the fully-developed region; how-ever, it affects the SS to WS flow transition. The time-averagedWS flow results, in Fig. 9, vary with h as well as r/row.

It is observed in Fig. 9 that all the quantities (for SS as well asWS flow) exhibit nearly-linear variation with h: Hfd, YUmax, pres-sure-gradient increases whereas Umax decreases. However, themagnitude of variation for all quantities except pressure-gradientis less (due to low density ratio i.e. v? 1).

9.3. Effect of reduced gravity for inclined channel

The effect of reduced gravity on the fully-developed analyticalresults for SS and numerical results for SS as well as time-averagedWS flow (at X = 55) is studied at uin = 0.5 m/s and Hin = 0.725. Forthe numerical study, coefficient of surface-tension, r = 0.85row isconsidered; which corresponds to the case where both SS andWS flow are obtained with increasing h, shown in Fig. 6a. Sur-face-tension is neglected in the analytical study. For SS flow,Fig. 10 shows an excellent agreement between the variation ofpresent analytical and numerical results with increasing inclina-tion at various values of g. The figure also shows that all the engi-neering-parameters for SS flow exhibit nearly-linear variation: Hfd,YUmax, pressure-gradient increases (decreases) whereas Umax de-creases (increases), with increasing h (decreasing g). Decreasing gresults in decrease in the slope of the linear-variation with h (seenin Fig. 10); indicating the reduction in the effect of inclination onthe fully-developed SS flow results, under the influence of reducedgravity.

For WS flow in a channel inclined at an angle h, the time-aver-aged results in Fig. 10 also show the linear variation, with

Numerical, Developing SSNumerical, Developing WSAnalytical, Fully-Developed SS

uin = 0.1 m/s = 0.4 m/s = 0.5 m/s

Hi

X0 20 40 60(a)

U max

0.5

1

1.5

2

X0 20 40 60(c)

0.4

0.6

0.8

1

Fig. 11. Axial variation of (a and b) interface-height, (c and d) maximum axial velocity foruin. The figure also shows analytical results of fully-developed SS flow as a horizontal li

increasing h and increasing g; similar to SS flow with a differentslope of variation.

10. Numerical results for developing stratified flow

In this section, the axial variation of two-phase flow propertiesand the development length for SS and WS flow is studied.


Fig. 11 shows the axial variation of interface-height, maximumvalue of U velocity over the cross-section of the channel, for stea-dy-state SS and time-averaged WS flow in a horizontal channel.The asymptotic value of numerical results for developing SS floware compared with fully-developed SS flow analytical results.

For the SS flow cases, Fig. 11a and b shows an asymptotic de-crease in the interface-height and Fig. 11c and d shows an asymp-totic increase in the maximum U-velocity along the axial direction;and finally, they reach to a fully developed value.

Thus, for oil–water SS flow at Hin = 0.85 and 0.9 and uin =0.1 m/s, it can be seen that the interface-height reduces to the SSfully-developed value of 0.6601 and 0.7129, respectively; with al-most same reduction in the interface-height from inlet to fully-developed value, i.e., Hin � Hfd � 0.19 for both the values of Hin.Moreover, the maximum non-dimensional U-velocity, which oc-curs in water, increases from 1 to nearly 1.78 for both the valuesof Hin (Fig. 11c and d).

Similar to SS flow, for WS flow, decrease in Hi and increase inUmax is seen up-to certain axial length. However, further down-stream, Fig. 11 shows increase in Hi, and decrease in Umax. Thus,the trend of variation changes across an axial-location which cor-responds to minimum of Hi and maximum of Umax; occurring at al-most same axial-length.

As the perturbation-free fluids with no-shear at interface enterthe domain at the inlet, behavior of interface from the inlet till acertain location is similar to that found for SS flow i.e. monotonicdecrease (increase) in Hi (Umax). At a certain axial location, theinstability initiates which is non-linear with random-wavelengthsand pointed crests at interface. The existence of pointed crests is

U max

0.5

1

1.5

2

X0 20 40 60(d)

X

Hi

0 20 40 60(b)0.4

0.6

0.8

1

developing SS and WS (time-averaged) flow, at various Hin {(a,c) 0.85 (b,d) 0.9} andne.

uin= 0.1 m/s0.3 m/s0.4 m/s0.5 m/s

L fd L fd

θ0 5 10

σ = 1.00σow0.95σow0.85σow0.80σow0.75σow

Hin0 0.2 0.4 0.6 0.8 10

25

50Numerical SSNumerical WS

46

46.5

47

47.5

48

(a)

θ

g = 1.00ge0.75ge0.50ge0.20ge0.10ge

0 5 10

L fd

46.5

47

47.5

48

(c)

(b)

Fig. 12. Variation of development-length for SS and WS flow with increasing (a) Hin and (b and c) h, at various (a) uin, (b) r and (c) g.


reported in the experimental work of Charles and Lilleleht (1965b)and in the numerical simulation of non-linear instability by Caoet al. (2004). Due to the non-linear growth of the interface, thereis periodic constriction and expansion of the oil-layer, which leadsto periodic increase and decrease of the axial-velocity in oil-phase.Moreover, owing to the non-symmetric shape and pointed crests ofthe interface, the time-averaged maximum-axial-velocity in theoil-phase for WS case is greater as compared to its value for SScase. Thus, the time-averaged averaged interface-height increasesas compared to its value at the axial location for the onset of wav-iness (which is near to the value predicted by analytical SS solu-tion) but it remains less than the inlet interface-height, shown inFig. 11. Moreover, due to the increase of time-averaged interfaceheight as compared to the SS value, the time-averaged Umax, whichis found in water, reduces as compared to the SS value, shown inFig. 11.

Thus, the smooth and wavy regions of the flow can be demar-cated by locating the axial-distance at which the interface-heightis minimum; defined here as the development length for WS flow,which decreases with increasing inlet-velocity (Fig. 11a and b).

10.2. Development-length

The procedure for determination of the development-length forSS and WS flow is discussed in Section 6. For horizontal-channel,Fig. 12 shows that the development-length increases for SS flowand decreases for WS flow, with increasing Hin and uin. Increasein the inertia of the flow reduces the growth rate of boundary-layerin the axial-direction, resulting in the increase in the development-length for the SS flow. Whereas, for WS flow, it causes the onset ofwaviness at smaller axial-length, resulting in the decrease in thedevelopment-length. For both SS and WS flow in an inclined-chan-nel, Fig. 12b and c shows that the development length decreases

slightly, with increasing h and decreasing r and g. Thus, the axiallocation of the onset of waviness in the developing interface ofthe WS flow moves upstream with increasing Hin, uin, h anddecreasing r and g.

11. Numerical results for interfacial-area-concentration

Interfacial-area during two-phase flow situation indicates theavailable area for interfacial mass, momentum and energy transferbetween the two phases. Moreover, its accurate knowledge for aparticular flow situation is crucial for two-fluid model formulation(Hibiki and Ishii, 2000). There have been various experimentalstudies (Kocamustafaogullari et al., 1994; Hibiki et al., 2006) tomeasure the interfacial-area for different two-phase flow regimes.In the present work, interfacial-area is calculated using the levelset method for the first time. The calculation is based on our phys-ical interpretation of Dirac delta function, d�(/i,j,), as the ratio ofinterface area inside the control-volume (CV, with running indicesi, j) and the volume of the CV (Gada and Sharma, 2009) called asinterfacial-area-concentration. Thus, on a flow domain of volumeX, it is calculated as

ai ¼R

X d�ð/ÞdVRX dV

�P

i;jd�ð/i;j;ÞDVi;jPi;jDVi;j

ð6Þ

For the present 2D study, X = La � 1, where La = 10; corresponds tothe flow region between X = 50–60. For all the cases (SS and WSflow), this region of the domain ensures that results obtained arefully-developed as Lfd 6 50 (refer Fig. 12). The numerator anddenominator in Eq. (6) correspond to the area of the interface andtotal volume of the domain considered in the calculation of ai,respectively. Both are equal to 10 for fully-developed SS flow,resulting in ai = 1; not plotted here. Whereas, for WS flow, the

Hin

a i a i

0.90.8

uin= 0.3 m/s0.4 m/s0.5 m/s

θ0 5 101

1.01 σ = 0.85σow0.80σow0.75σow

1

1.01

1.02

1.03

(a) (b)a i

θ105

g = 1.00ge0.75ge0.50ge0.20ge0.10ge

1

1.01

(c)Fig. 13. Variation of interfacial-area-concentration for WS flow region (50 6 X 6 60) with increasing (a) Hin and (b and c) h, at various (a) uin, (b) r and (c) g.


interface is always stretched and ai is shown in Fig. 13 for all cases.Fig. 13 shows that the interfacial-area-concentration increases(indicating that the waviness of the interface increases) withincreasing Hin, uin and h; and decreasing surface-tension and gravi-tational-acceleration. Quantitatively, Fig. 13a shows that, increasein inertia or water-fraction leads to maximum stretching of inter-face; with ai = 1.025 (2.5% stretching) at highest values of uin andHin considered here.

12. Conclusions

SS to WS flow transition is captured, by a novel dual-grid level-set method based 2D numerical-simulation for developing flow ina horizontal and inclined channel, at certain critical value of inlet-velocity (uin), inlet-interface-height (Hin), inclination-angle (h), re-duced surface-tension (r 6 row) and reduced gravity (g 6 ge). Forthe onset of WS flow under the combined effect, it is found thatlower Hin and uin is required at higher uin and Hin, respectively;and a smaller reduction in row and ge is required at higher h. Thisis because an increase in uin, Hin and h, and a decrease in row andge are found to destabilize the flow. Increase in the uin and Hin in-creases the inertia of the flow which creates instability in the flow.Increase in h reduces the inertia of the two fluids by a differentamount because of density difference, leading to the increase inthe interface height; aiding the instability in flow. Reduction ofcoefficient of surface-tension reduces the cohesion of the fluid par-ticles and increases the stretching of the interface, leading to wavyinterface. Reduction in gravitational acceleration reduces the stabi-lizing-effect caused by the self-weight of the two-fluids, resultingin the onset of WS flow.

For fully-developed SS two-fluid flow in an inclined channel, adetailed analytical solution is proposed here and an excellentagreement is reported with the numerical results; for variation ofinterface-height, maximum U-velocity, its Y-coordinate, non-dimensional pressure-gradient, wall-shear stress and pressure-gradient-reduction-factor, at various values of the governingparameters. For fully-developed SS flow in a plane-channel, savingin pumping cost is obtained with water occupying less than 49.3%(almost half) of the channel-height; with a maximum saving at awater-fraction of 9.8%. The pumping cost is more for WS as com-pared to SS flow.

For developing flow, it is shown that the axial-variation of time-averaged interface-height and maximum in the V-velocity profiledecreases, reaches a minimum; and then, increases to an asymp-totic value near the outlet region. Opposite trend is found in theaxial-variation of maximum in the U-velocity profile. Thus, thetime-averaged WS flow results asymptotes to a constant value nearthe channel-outlet. The asymptotic value of interface-height in-creases, maximum U-velocity (Umax) decreases, non-dimensionalpressure-gradient and top-wall shear-stress (PoT) increases; whencompared with the steady-state SS flow analytical results. The in-crease/decrease is found to be more with an increase in uin, Hin

and h, and a decrease in row and ge. With increasing Hin, the varia-tion of analytical results shows that the onset of WS flow occursnear the sharp increase and decrease of Umax and PoT variation,respectively; indicating a sharp variation in interfacial shear-stressprobably leading to the flow-transition.

Development-length of the SS flow increases with increasing Hin

and uin, and decreases slightly with increasing h. For WS flow,development-length is defined here as the channel-length whichcorresponds to the minimum in the axial-variation of time-averaged


interface-height. This axial location also corresponds to onset ofwaviness of the interface; lies in between the SS flow in the up-stream and WS flow in the downstream region of the channel. Thedevelopment-length for the WS flow is found to decrease with an in-crease in Hin, uin, h and a decrease in row and ge; instability move up-stream inside the channel. The calculation of interfacial-area-concentration, an important parameter for heat and mass transfer,is proposed here using level set method for the first time. For theWS flow, it is found that the interfacial-area-concentration increaseswith an increase in Hin, uin, h and a decrease in row and ge.

Although developing WS flow and the flow-transition study isdone here for the first time, a more detailed analysis needs to bedone on a more commonly used configuration such as pipe; whichis the objective of the future work.

Acknowledgment

The present work is part of a research project funded by boardof research in nuclear sciences (India) under Project No. 2007/36/14-BRNS/718.

Appendix A. Analytical solution for fully-developed smooth-stratified flow in an inclined channel

Assuming that the pressure gradient in x-direction is same forthe two fluids and does not vary in the transverse (y) direction,the simplified momentum equations are given as

For fluid 1 :@syx;1

@y¼ @p@xþ q1g sin h and

@p@yþ q1g cos h ¼ 0

For fluid 2 :@syx;2

@y¼ @p@xþ q2g sin h and

@p@yþ q2g cos h ¼ 0

ð7Þ

The no-slip at the channel-walls and continuity of velocity aswell as shear stress at the interface are the boundary-conditions:y = 0) u1 = 0, y = H) u2 = 0 and y = hfd) u1 = u2 and syx,1 = syx,2.Substituting syx = l@u/@y in Eq. (7), it is observed that x-momen-tum equation is reduced to 1D second-order ordinary differentialequation (for u-velocity as a function of y-coordinate) with a con-stant coefficient (@p/@x + qgsinh); for each fluid. Integration in y-direction results in four constants (two for each fluid); obtainedby the four boundary-conditions. Thus, the expressions for shearstress and velocity profile are obtained.

In context of the present work, under fully-developed condi-tions of SS flow in an inclined channel, the pressure and gravita-tional force balance the viscous force (refer Eq. (7)). Thus, in thepresent work, the pressure and gravitational force are non-dimen-sionalized with viscous scale as

Non-Dimensional pressure : P � pl1uin=H

¼ q1uinHl1

� pq1u2

in

¼ ReP

Archimedes Number : Ar � q1gHl1uin=H

¼ q1uinHl1

� q1gHq1u2

in

¼ Re

Fr2

where P ¼ p=q1u2in and Fr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiuin=gH

p(Froude number) are the

pressure and gravitational force non-dimensionalized by inertia

Q �2 ¼�@P@X �H4

fdðg�1Þþ2H3fdð3g�2Þ�3H2

fdð3g�2Þþ4Hfdðg�1Þþ1� �

þAr sinh H4fdðvþ

n12g½1þðg�

force. Ar is a non-dimensional number called as Archimedes num-ber, representing the ratio of gravitational and viscous force. Tra-ditionally, Ar is used in problems involving buoyancy,fluidization or motion due to density difference. Here, it is usedfor fully-developed flow in an inclined channel probably for thefirst time. Thus, Re and Fr are eliminated in the analytical resultsby using P and Ar.

In the analytical study here, the known quantities are flow-rate-ratio at inlet (calculated by knowing inlet-interface-height and in-let-velocity), inclination and property ratios; and the unknownquantities are fully-developed velocity-profile, pressure-gradient,interface-height and wall shear-stress. According to a procedureused here, firstly, the expression of velocity profile is obtained interms of property ratios, inclination, pressure-drop and fully-developed interface height; by integrating the x-momentum equa-tion and using the boundary conditions. Thereafter, the expressionof volumetric flow rates of individual fluids is obtained, in terms ofproperty-ratios, inclination, pressure-gradient and fully-developedinterface-height; by integrating the velocity field in respective lim-its. Finally, using the assumption that the two-fluid pressure-gradi-ent is same in both the fluids; the pressure-gradient is eliminatedfrom the expression of volumetric flow rates to derive the expres-sion of fully-developed interface height in terms of property-ratios,inclination and individual flow-rates of the fluids (which areknown quantities). Once the fully-developed interface-height isknown, all the other quantities can be calculated sequentially.The final non-dimensional expressions obtained in the present der-ivation are given here.

The non-dimensional form of velocity profiles in the two-fluidsare obtained as

U1 �u1

uin¼�1

2dPdX

1þH2fdðg�1Þ

1þHfd g�1ð ÞY�Y2

" #

þAr sinh2

H2fd½g�vþ2gðv�1Þ�þvþ2Hfdðv�1Þ½Hfdðg�1Þþ1�

1þHfdðg�1Þ YþY

( )

ð8Þ

U2 �u2

uin¼ � 1

2gdPdX

1þ H2fdðg� 1Þ

1þ Hfdðg� 1Þ ðY � 1Þ � ðY2 � 1Þ" #

þ Ar sin h2g

H2fd½g� vþ 2gðv� 1Þ� þ v

1þ Hfdðg� 1Þ ðY � 1Þ þ vðY2 � 1Þ( )

ð9Þ

where Y = y/H, Hfd = hfd/H, P = pH/l1uin, Ar = q1gH2/l1uin.Integrating U1 from 0 to Hfd,the non-dimensional volumetric

flow rate of fluid 1 Q �1 ¼ Q1=uinH� �

is obtained as

Q �1 ¼�H2fd

�@P@X 3�2HfdþH2

fdðg�1Þh i

þAr sinh H2fdð3vþg�4ÞþHfdð4�6vÞþ3v

n o12½1þðg�1ÞHfd�

ð10Þ

Similarly, integrating U2 from Hfd to 1 the non-dimensional volu-metric flow rate of fluid 2 Q �2 ¼ Q2=uinH

� �is obtained as

3g�4gvÞþH3fdð12gv�6g�4vÞþH2

fdð3gþ6v�12gvÞþ4Hfdvðg�1Þþvo

1ÞHfd�ð11Þ


The pressure-gradient, which is same in both the phases is obtainedfrom equation of Q �1 and Q �2 as

@P@X¼�

12Q �1½1þHfdðg�1ÞþAr sinhH2fd H2

fdð3vþg�4Þþ2Hfdð2�3vÞþ3vn o

H2fd 3�2HfdþH2

fdðg�1Þh i

ð12Þ

@P@X¼ �

12gQ �2½1þ Hfdðg� 1Þ� þ Ar sin hfH4fdðvþ 3g� 4gvÞ þ H3

fdð12gv� 6g� 4vÞ þ H2fdð3gþ 6v� 12gvÞ þ 4Hfdvðg� 1Þ þ vg

�H4fdðg� 1Þ þ 2H3

fdð3g� 2Þ � 3H2fdð3g� 2Þ þ 4Hfdðg� 1Þ þ 1

ð13Þ

Equating Eqs. (12) and (13), the fully developed interface height isobtained as 8th order polynomial; solved using bisection method.

H8fdAr sin h½4ð1� vÞðg� 1Þ2� þH7

fdAr sin h½ð1� vÞðg� 1Þð20� 12gÞAr�þH6

fdAr sin h½ð1� vÞð40� 48gþ 12g2Þ� þH5fd Ar sin hð1� vÞð�40½

þ32g� 4g2Þ þ 12 Q �1 þ gQ �2� �

ðg� 1Þ2iþH4

fd Ar sin hð1� vÞð20� 8gÞ½

þ12 Q �1ð11g� 6g2 � 5Þ þ gQ �2ð1� gÞ� ��

þH3fd �4Ar sin hð1� vÞ½

þ12 Q �1ð9g2 � 21gþ 10Þ þ gQ �2ð3g� 5Þ� ��

þH2fd 12 Q �1ð17g� 4g2 � 10Þ

��þ3gQ �2

��þHfd 12Q �1ð5� 5gÞ

� �� 12Q �1 ¼ 0 ð14Þ

Here, the shear stress is non-dimensionalized using viscousscale instead of inertia, resulting in Poiseuille number (Churchill,1988):

Po1 �2syx;1

l1uin=H¼ Cf ;1Re ¼ 2

@U1

@Y

Po2 �2syx;2

l1uin=H¼ Cf ;2Re ¼ 2g

@U2

@Y

The non-dimensional form of shear stress, represented by Poiseuillenumber, Po1(Y) and Po2(Y), are obtained by taking the derivative ofvelocity profile in Eqs. (8) and (9), respectively. By substituting Y = 0and Y = 1 in the expressions for Po1(Y) and Po2(Y), respectively; thenon-dimensional wall shear stress is obtained for the bottom andtop wall of the channel as

PoB ¼�2dPdX

1þH2fdðg�1Þ

2 1þHfdðg�1Þ� �

" #þ2Ar

� sinhH2

fdðg�vþ2gðv�1ÞÞþvþ2Hfdðv�1Þ½Hfdðg�1Þþ1�2½1þHfdðg�1Þ�

( )ð15Þ

PoT ¼�2dPdX

1þH2fdðg�1Þ

2½1þHfdðg�1Þ��1

" #þ2Ar sinh

H2fd½g�vþ2gðv�1Þ�þv

2½1þHfdðg�1Þ� þv( )

ð16Þ

If Hfd < Hin, the Y-coordinate of maximum velocity in the U-velocityprofile i.e. YUmax is obtained by setting dU1/dY = 0 in Eq. (8) as

YUmax;1 ¼�� @P

@X 1þH2fdðg�1Þ

h iþAr sinh H2

fd ½g�vþ2gðv�1Þ�þvþ2Hfdðv�1Þ½1þHfdðg�1Þ�n o

2½1þHfdðg�1Þ� @P@X þAr sinh� �

8<:

9=; ð17Þ

If Hfd P Hin, YUmax is obtained by setting dU2/dY = 0 in Eq. (9) as

YUmax;2 ¼��@P

@X 1þH2fdðg�1Þ

h iþAr sinh H2

fd½g�vþ2gðv�1Þ�þvn o

2½1þHfdðg�1Þ� @P@X þvAr sinh� �

8<:

9=; ð18Þ

Maximum U velocity Umax is calculated using either YUmax,1 (YUmax,2)in Eq. (8) (Eq. (9)) if Hfd < Hin (Hfd P Hin).

By substituting Ar = 0 or h = 0 or v = 1, the analytical solution forthe inclined channel (Eqs. (8)–(18)) reduces to two-fluid SS flowsolution for a horizontal-channel; derived by Datta (2010). Fur-

thermore, by substituting v = 1, g = 1, Ar = 0 and Q �1 ¼ Q �2 ¼ 0:5,the analytical solution degenerates to the solution for single-fluidflow: Hfd = YUmax = 0.5, U1 = U2 = 6Y (1 � Y),Umax = 1.5, dP/dX = �12and PoB = �PoT = 12.

Appendix B. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.ijmultiphaseflow.2011.08.015.

References

Akai, M., Inoue, A., Aoki, S., 1981. The prediction of stratified two-phase flow with atwo-equation model of turbulence. Int. J. Multiphase Flow 7, 21–39.

Angeli, P., Hewitt, G.F., 2000. Flow structure in horizontal oil–water flow. Int. J.Multiphase Flow 26, 1117–1140.

Boomkamp, P.A.M., Miesen, R.H.M., 1996. Classification of instabilities in paralleltwo-phase flow. Int. J. Multiphase Flow 22, 67–88.

Brackbill, J.U., Kothe, D.B., Zemach, C., 1992. A continuum method for modelingsurface tension. J. Comput. Phys. 100, 335–354.

Brauner, N., 2001. The prediction of dispersed flows boundaries in liquid–liquid andgas–liquid systems. Int. J. Multiphase Flow 27, 885–910.

Brauner, N., 2002. Liquid–liquid two-phase flow systems. In: Bertola, V. (Ed.),Modeling and Control of Two-phase Flow Phenomena. CISM Center, Udine,Italy.

Brennen, C.E., 2005. Fundamentals of Multiphase Flow. Cambridge University Press,Cambridge.

Cao, Q., Sarkar, K., Prasad, A.K., 2004. Direct numerical simulations of two-layerviscosity-stratified flow. Int. J. Multiphase Flow 30, 1485–1508.

Chang, Y.C., Hou, T.Y., Merriman, B., Osher, S., 1996. A level set formulation ofEulerian interface capturing methods for incompressible fluid flows. J. Comput.Phys. 124, 449–464.

Charles, M.E., Lilleleht, L.U., 1965a. Co-current stratified laminar flow of twoimmiscible liquids in a rectangular conduit. Can. J. Chem. Eng. 43, 110–116.

Charles, M.E., Lilleleht, L.U., 1965b. An experimental investigation of stability andinterfacial waves in co-current flow of two liquids. J. Fluid Mech. 22, 217–224.

Churchill, S.W., 1988. Viscous Flows: The Practical Use of Theory, first ed.Butterworth, New York.

Datta, D., 2010. Analytical and level set method based numerical study for two-phase stratified flow in a plane channel dissipating uniform heat flux and asquare duct, Ph.D. Thesis, Indian Institute of Technology Bombay, Mumbai,India.

Datta, D., Gada, V.H., Sharma, A., 2011. Analytical and level-set method-basednumerical study for two-phase stratified flow in a plane channel and a squareduct. Numer. Heat Transfer A 60, 347–380.

De Schepper, S.C.K., Heynderickx, G.J., Marin, G.B., 2008. CFD modeling of all gas-liquid and vapor-liquid flow regimes predicted by the Baker chart. Chem. Eng. J.138, 349–357.

Frisk, D.P., Davis, E.J., 1972. The enhancement of heat transfer by waves in stratifiedgas–liquid flow. Int. J. Heat Mass Transfer 15, 1537–1552.

Gada, V.H., Sharma, A., 2009. On derivation and physical-interpretation of level setmethod based equations for two-phase flow simulations. Numer. Heat TransferB 56, 307–322.

Gada, V.H., Sharma, A., 2011. On a novel dual grid method for two-phase flowsimulation. Numer. Heat Transfer B 59, 26–57.

Grassi, B., Strazza, D., Poesio, P., 2008. Experimental validation of theoretical modelsin two-phase high-viscosity ratio liquid-liquid flows in horizontal and slightlyinclined pipes. Int. J. Multiphase Flow 34, 950–965.

Heidarinejad, G., Sani, A.E., 2010. Two-dimensional modeling for stability analysis oftwo-phase stratified flow. Int. J. Numer. Meth. Fluids 62, 733–746.

Hibiki, T., Ishii, M., 2000. One-group interfacial area transport of bubbly flows invertical round tubes. Int. J. Heat Mass Transfer 43, 2711–2726.

Hibiki, T., Ho Lee, T., Young Lee, J., Ishii, M., 2006. Interfacial area concentration inboiling bubbly flow systems. Chem. Eng. Sci. 61, 7979–7990.

Hooper, A.P., 1985. Long wave instability at the interface between two viscousfluids: thin layer effects. Phys. Fluids 28, 1613.

Issa, R.I., 1988. Prediction of turbulent, stratified, two-phase flow in inclined pipesand channels. Int. J. Multiphase Flow 14, 141–154.

Joseph, D.D., Renardy, M., Renardy, Y., 1984. Instability of the flow of twoimmiscible liquids with different viscosities in a pipe. J. Fluid Mech. 141,309–317.



Juric, D., Tryggvason, G., 1998. Computations of boiling flows. Int. J. Multiphase Flow24, 387–410.

Khomami, B., 1990. Interfacial stability and deformation of two stratified power lawfluids in plane Poiseuille flow. J. Non-Newton. Fluid Mech. 37, 19–36.

Kocamustafaogullari, G., Huang, W.D., Razi, J., 1994. Measurement and modeling ofaverage void fraction bubble size and interfacial area. Nucl. Eng. Des. 148, 437–453.

Landman, M.J., 1991. Non-unique holdup and pressure drop in two-phase stratifiedinclined pipe flow. Int. J. Multiphase Flow 17, 377–394.

Lum, J.Y.-L., Al-Wahaibi, T., Angeli, P., 2006. Upward and downward inclined oil–water flows. Int. J. Multiphase Flow 32, 413–435.

Mandal, T.K., Chakrabarti, D.P., Gas, G., 2007. Oil–water flow through differentdiameter pipes – similarities and differences. Trans. IChemE A 85, 1123–1128.

Mohanty, A.K., Asthana, L., 1978. Laminar flow in the entrance region of a smoothpipe. J. Fluid Mech. 90, 433–447.

Newton, C.H., Behnia, M., 2000. Numerical calculation of turbulent stratified gas–liquid pipe flows. Int. J. Multiphase Flow 26, 327–337.

Parvareh, A., Rahimi, M., Alizadehdakhel, A., Alsairafi, A.A., 2010. CFD and ERTinvestigations on two-phase flow regimes in vertical and horizontal tubes. Int.Commun. Heat Mass Transfer 37, 304–311.

Renardy, Y., 1985. Instability at the interface between two shearing fluids in achannel. Phys. Fluids 28, 3441.

Rodriguez, O.M.H., Oliemans, R.V.A., 2006. Experimental study on oil–water flow inhorizontal and slightly inclined pipes. Int. J. Multiphase Flow 32, 323–343.

Sadatomi, M., Kawaji, M., Lorencez, C.M., Chang, T., 1993. Prediction of liquid leveldistribution in horizontal gas–liquid stratified flows with interfacial levelgradient. Int. J. Multiphase Flow 19, 987–997.

Shoham, O., Taitel, Y., 1984. Stratified turbulent–turbulent gas–liquid flow inhorizontal and inclined pipes. AIChE J. 30, 377–385.

Suh, Y., Son, G., 2008. A level-set method for simulation of a thermal inkjet process.Numer. Heat Transfer B 54, 138–156.

Sussman, M., Samereka, P., Osher, S., 1994. A level set approach for computingsolutions to incompressible two-phase flows. J. Comput. Phys. 114, 146–159.

Taitel, Y., Dukler, A.E., 1976. A model for predicting flow regime transitions inhorizontal and near horizontal gas–liquid flow. AIChE J. 22, 47–55.

Tang, Y.P., Himmelblau, D.M., 1963. Velocity distribution for isothermal two-phaseco-current laminar flow in a horizontal rectangular duct. Chem. Eng. Sci. 18,143–148.

Ullmann, A., Brauner, N., 2006. Closure relations for two-fluid models for two-phasestratified smooth and stratified wavy flows. Int. J. Multiphase Flow 32, 82–105.

Ullmann, A., Goldstein, A., Zamir, M., Brauner, N., 2004. Closure relations for theshear stresses in two-fluid models for laminar stratified flow. Int. J. MultiphaseFlow 30, 877–900.

Wegmann, A., von Rohr, P.R., 2006. Two phase liquid–liquid flows in pipes of smalldiameters. Int. J. Multiphase Flow 32, 1017–1028.

Yap, Y.F., Chai, J.C., Toh, K.C., Wong, T.N., Lan, Y.C., 2005. Numerical modeling ofunidirectional stratified flow with and without phase change. Int. J. Heat MassTransfer 48, 477–486.

Yiantsios, S.G., Higgins, B.G., 1988a. Numerical solution of eigenvalue problemsusing the compound matrix method. J. Comput. Phys. 74, 25–40.

Yiantsios, S.G., Higgins, B.G., 1988b. Linear stability of plane Poiseuille flow of twosuperposed fluids. Phys. Fluids 31, 3225.

Yih, C.-S., 1967. Instability due to viscosity stratification. J. Fluid Mech. 27, 337–352.Zubkov, P.T., Serebryakov, V.V., Son, E.E., Tarasova, E.N., 2005. Steady-state flow of

two viscous immiscible incompressible fluids in a plane channel. High Temp.43, 769–774.

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