Numerical Investigation of Slug Flow in a Horizontal Pipe

NUMERICAL INVESTIGATION OF SLUG FLOW IN A HORIZONTAL PIPE USING AMULTI-SCALE TWO-PHASE APPROACH TO INCORPORATE GAS ENTRAINMENT

EFFECTS

Stefan Wenzel∗, Marek Czapp†, Thomas SattelmayerLehrstuhl für Thermodynamik

Technische Universität München85748 Garching, Germany

Email: [email protected]

ABSTRACTNumerical as well as experimental investigations of the

highly intermittent slug flow regime of a gas-liquid mixture inhorizontal pipes are of particular interest for nuclear reactorsafety in post loss-of-coolant accident (LOCA) situations. Thestrong variation of governing interfacial length scales, as theyare characterizing the slug flow regime, pushes common numer-ical multi-phase approaches to their limits, since they are de-signed either for interface capturing or for modeling the sub-grid behavior of the dispersed mixture. In this work an en-hanced hybrid two-phase flow solver is employed to investigatethe global and local characteristics of adiabatic, horizontal slugflows in a water-air system. A dynamic switching algorithm foran interface capturing procedure is introduced to examine seg-regated and dispersed parts in the same flow domain. The inter-facial area transport equation (IATE) is used to detect dispersedflow regions as well as to determine variable bubble sizes andtheir distribution within the slug body. Experimental results ofvideometry measurements on a horizontal, 10 m long pipe withan inner diameter of 54 mm at atmospheric pressure and roomtemperature are compared with numerical results of the same ge-ometry in terms of global characteristics such as slug frequencyand onset position. Local properties, such as the interfacial areadensity in the slug body, are also examined. This study demon-strates the capability of a coupled multiscale approach based on

∗Address all correspondence to this author.†Dantec Dynamics GmbH, Kaessbohrer Str. 18, 89077 Ulm, Germany

the Euler-Euler two-fluid model (TFM) for the simulation of slugflow in horizontal pipes with a high amount of entrainment.

NOMENCLATURE

Latin LettersA [m2] surface areaa [m-1] surface area densityC [-] coefficientD [m] pipe diameterd [m] bubble diameterF [N] force vectorFr [-] Froude numberg [m/s2] gravity vectorL [m] pipe lengthn [-] surface normal vectorp [Pa] pressureR [m-1s-1] interfacial area sourceRe [-] Reynolds numbert [s] timeU [m/s] velocity vectorV [m3] volume

Greek Lettersα [m-3] volume fractionδ [-] detection function

Proceedings of the 2016 24th International Conference on Nuclear Engineering ICONE24

June 26-30, 2016, Charlotte, North Carolina

ICONE24-60259

1 Copyright © 2016 by ASME

Γ [-] signal strengthκ [m-1] curvatureλ [-] friction coefficientµ [kg/m s] dynamic viscosityΠ [-] viscous shear contributionϕ [rad] angle between surface normal vectorsρ [kg/m3] densityτττ [kg/m s2] stress tensor

Sub- and Superscriptsc compressionD dragd dispersedG gasg growthi interfacej control variablek conditioned fluidL liquidM macroscopicm microscopicp particler regionals surface tensions scalingse segregatedsm sauter meanT turbulentµ viscous

INTRODUCTIONSlug flow of gas-liquid mixture in horizontal configurations

is well known to exist over a wide range of flow rate combina-tions as demonstrated by classical flow-pattern maps [1, 2]. Thishighly intermittent flow regime is characterized by alternatingliquid slugs, occupying the entire cross section of the flow do-main followed by elongated gas structures, propagating on a liq-uid film of varying height.

The intermittency of this flow pattern can create strong pres-sure pulsations and impulse-like forces during the formation andpropagation of the slugs, especially when passing hydraulic com-ponents or when the flow changes its direction. The liquid slugbody may be enriched with dispersed bubbles as a result of gasentrainment, depending on the gas-liquid superficial velocity ra-tio. Superficial velocity denotes the hypothetical velocitiy calcu-lated as if the regarded fluid filled the entire flow domain withoutthe presence of its partner. The amount of entrainment and itsdistribution within the slug body considerably effects its stabil-ity. Large agglomerations of gaseous structures are eventuallyable to bridge the liquid slug and lead to its disintegration [3].

This mechanism is denoted as blow-through and significantly ef-fects the mechanical load on the confining structure. Since slugflow pattern is expected to occur during LOCA events in lightwater reactors, an appropriate description and prediction of thisphenomenon is of high interest for nuclear reactor safety analy-sis. Furthermore, its frequent appearance under horizontal andnearly horizontal conditions of multi-phase systems, highlightsits significance for many industrial applications, e.g. in oil in-dustry and chemical plants. Hence, in the last decades, numer-ous investigations have been made towards the development ofnumerical and mechanistic models for slug flow.

Traditional models, such as the unit-cell approach [3,4] pro-vide only global, averaged information of the flow pattern (e.g.characteristic slug length and velocity, overall pressure drop).Fully developed and quasi steady-state conditions are assumed.Issa and Kempf [5] demonstrated the capability of the one dimen-sional two-fluid model (TFM) to capture the mechanisms leadingto slug generation under mathematically well-posed conditions.However, they neglected the presents of entrained gas withinthe slug body. Secondary flow effects that influences the originof liquid slugs in circular pipes were successfully accomplishedby Frank [6] and Czapp et al. [7] utilizing the Volume of Fluid(VOF) method for three dimensional simulations. Using this ap-proach, the consideration of entrainment is limited to an accept-able numerical grid resolution. Since classical VOF methods arenot able to model the sub-grid behavior of dispersed particles, aresolution of the smallest length-scales would require a unaccept-able cell size, several times smaller than the smallest immersedparticle of interest. It turns out that slug flow with a consider-able amount of entrainment is indeed a multi-scale multi-phasephenomenon which pushes common numerical multi-phase ap-proaches to their limits.

In order to arrive at a more general multi-phase flow model,several groups suggested the coupling of interface resolving andinterface averaging techniques. Cerne et al. [8, 9] explicitly cou-pled TFM and VOF methods for the first time, while amongothers Štrubelj et al. [10, 11] partially utilized the VOF methodwithin the TFM framework. Marschall and Hinrichsen [12] de-rived a general multi-scale two-phase flow model frameworkwhich is able to simulate dispersed as well as segregated flows.However, the flow type has to be specified as an input parame-ter. Hänsch et al. [13] presented a generalized two-phase flow(GENTOP) model based on the inhomogeneous multiple sizegroup (MUSIG) model [14], where several transport equationsare solved for groups of various sized, spherical bubbles. Thedynamic evolution of the particle size field is considered us-ing a population balance model. Additionally, a second con-tinues phase for non-dispersed parts of the gaseous portion isincluded to represent segregated parts within the flow domain.Wardle and Weller [15] developed a hybrid multi-phase CFDsolver that became part of the official release of the open sourceCFD toolkit OpenFOAM. It is based on an Eulerian multi-fluid


solution framework with an optional, generalized interface cap-turing methodology that is selected by the user when segregatedflow regime is expected. The same group suggested an extensionof this solver for a dynamic recognition of dispersed and segre-gated flow parts during execution [16]. It remains unclear how toarrive at a reliable and physically valid switching criteria.

In the present contribution a coupled multi-scale approachbased on the Euler-Euler two-fluid model is presented. Advan-tages and limitations in the simulation of slug flow with consid-erable amount of entrainment are evaluated.

DESCRIPTION OF THE COMPUTATIONAL METHODSThe implementation presented in this contribution has been

carried out extending the multiphaseEulerFoam solver, in-cluded in the OpenFOAM CFD toolbox version 2.3. A de-tailed description of the basic solver is given by Wardle andWeller [15].

Scale Separation MethodThe solver is capable to simulate a number of k incompress-

ible fluids in an Eulerian solution framework. One mass and mo-mentum equation is solved for every involved fluid k given by

∂αk

∂ t+∇• (Ukαk) = 0, (1)

∂ (ρkαkUk)

∂ t+∇• (ρkαkUkUk) =

−αk∇p+∇•(αk(τττ

µ

k + τττTk ))+ρkαkg+∑

jF j

k,(2)

where the fluid density, its phase fraction and velocity vector aredenoted as ρk, αk and Uk respectively and g denotes the gravityvector. The viscous and turbulent stress tensors are given by τττ

µ

kand τττT

k . The interfacial transfer terms F jk on the r.h.s. of Eqn. (2)

are dynamically de- and reactivated with respect to the local mor-phology of the multi-phase mixture using a regional detectionfunction to account for the scale variation and resolvability ofthe interfacial structures. A interface capturing methodology isapplied on segregated parts of the flow domain following the con-ceptual approach of the VOF interface compression method ac-cording to Rusche and Weller [17]. For this purpose the volumefraction transport equation for fluid k

∂αk

∂ t+∇• (Ukαk)+δr ·∇• (Uc αk(1−αk)) = 0 (3)

is extended by a third term on the l.h.s. of Eqn. (3) which onlyacts in the interface transition region and parallel to ∇αk, such

that

Uc =Cc|Uk|∇αk

|∇αk|, (4)

where Cc is the compression coefficient, proposed to be0≤Cc ≤1 and a value of 0.8 was applied in the present study.The regional detection function δr, that will be defined later on,enables the interface compression to be only applied in flow situ-ation where the local computational grid density allows interfacecapturing. To detect these regions properly, δr has to hold infor-mation about the macroscopic shape of the interfacial structureas well as information about the locally unresolved interfacialmorphology. The former can be obtained by calculating the cur-vature of the interface based on the divergence of the ∇αk field

κ =−∇• ∇αk

|∇αk|. (5)

This approach is widely used for Eulerian based multi-phasesolvers, to calculate the volumetric surface tension force Fs

k ac-cording to Brackbill et al. [18]. However, using this methodalone can lead to irregular artifacts in the resulting volumetriccurvature field, especially in regions with low mixture quan-tity [19] (αk . 0.01). Consequently, in the current work theangles ϕ between the gradient based surface normal vectorsnk=∇αk/|∇αk| are averaged within an 3x3x3 matrix of the sur-rounding cells and divided by the average distance to their neigh-boring cell centers ∆x, such that the overall curvature becomesκM=∆ϕ/∆x. To minimize the additional effort of this geometri-cal operation, the calculation is limited to near-interface regionswith at least 1 %� of mixture quantity.

Since information about unresolved curvature of the inter-facial structures is a priori unknown, a closure model is neededto include these portion in the superior method. Marschall [20]suggests the concept of local instantaneous isotropic interfacemorphology (IIM), stating that the interfacial area density ai isassumed to be dominated by contributions of unresolved struc-tures and is not allowed to vary within an infinitesimal averag-ing volume with constant αk, resulting in ai ≡ |∇αk|. SubjectingEqn. (5) to a further analysis

κ =− 1|∇αk|

[∇

2αk−

∇αk

|∇αk|•∇|∇αk|︸︷︷︸

=nk•∇ai

], (6)

a scale splitting can be achieved through

κ = −∇2αk

|∇αk|︸︷︷︸macroscopic

+d(ai)

dαk︸︷︷︸microscopic

, (7)


where the first term on the r.h.s. of Eqn. (7) is accounting for thecontribution of the macroscopic curvature κM of the interfacetransition region and second one is related to unresolved struc-tures κm within this region. Transferring this circumstance to thebasic concept of the two-fluid model, describing non-resolveddisperse particles inside a control volume, the latter term can bedynamically modeled using a dynamic correlation for these un-resolved structures. As mentioned in the introduction, variousmethods are reported to obtain a time and space variant descrip-tion of dispersed particle size. Bearing Eqn. (7) in mind, it seemsreasonable to directly involve a transport model for this unre-solved contribution of the interfacial curvature.

Interfacial Area Transport EquationThe unresolved interfacial structures in horizontal slug flow

regime appear either to be arbitrary micro-scale disturbanceswith various amplitudes at the interface of segregated flow re-gions or as dispersed bubbles entrained in the liquid slug body.In the presented method, the latter case is treated as a bubblyflow region within a general segregated flow phenomenon. Thisallows the utilization of the widely used and extensively vali-dated approaches for simulating dispersed bubbly flow in thismulti-scale Euler-Euler multi-phase framework.

Using this approach, the non-resolved sub-scale curva-ture κm in the bubbly regions can be interpreted as the sum of theinverse radii of a number of n equal, spherical particles insidethe averaging volume. Accordingly, it is related to the interfa-cial area density ai via the fluid volume fraction and the Sautermean diameter dsm of these dispersed bubbles. Representing theamount of interfacial area ai=(nAp)/Vc and fluid volume fractionα=(nVp)/Vc within the control volume Vc by n equal, sphericalparticles of size dsm and volume Vp with particle surface area Apit leads, according to Ishii and Hibiki [21], to the expression

κm =ai

αk=

6dsm

. (8)

Kocamustafaogullari and Ishii [22] deduced the interfacial areatransport equation by considering the particle density trans-port equation analogously to Boltzmann’s transport equation forspherical bubbly flow without phase change to

∂ai

∂ t+∇• (aiUi) =

23

(ai

αG

){∂αG

∂ t+∇• (αGUG)

}+12π

(αG

ai

)2

∑j

Rj.(9)

The total rate of change in the interfacial area concentration onthe l.h.s. of Eqn. (9) is determined by the rate of change due to

changes in particle volume in consequence of pressure variationand particle interaction respectively on the r.h.s. of the equation.This approach allows to account for dynamic changes in ai dueto the coalescence and disintegration of particles in the local bub-bly flow regions of the slug body. Since the interfacial transferterms for mass and momentum are proportional to the interfacialarea density, an integration of this method in the computationalsolution procedure has great potential to enhance the predictionquality of the resulting solver. The particle interaction terms Rj inEqn. (9) are modeled according to Wu et al. [23]. A major draw-back using this model in the current application is that it wasdeveloped for vertical bubbly flow. It remains questionable if theempirical correlations that are part of the approach are justifiedfor horizontal conditions. However, for the lack of appropriateinvestigations of particle coalescence and disintegration in hori-zontal flow and since the interactions are assumed to be rather in-fluenced by local conditions than by the global appearance of thedomain, the selection seems warranted. The model consists ofthree mechanistic processes, which are coalescence due to ran-dom collisions, coalescence due to wake entrainment and break-up due to turbulent impact, where the latter is assumed to playa major role especially in the highly turbulent slug front regionwhile wake entrainment seems of minor importance under hori-zontal conditions.

Dynamic Regional Detection MethodFollowing the preceding argumentation of scale separation

due to splitting of the curvature contributions, a regional detec-tion function δr can be deduced. By contemplating the mor-phologies of a complete slug unit, including the slug body andthe trailing elongated gas structure, three general region typesare distinguishable, that differ in the proportion of the curvaturecontributions κM and κm:

Segregated region, in which macroscopic as well as mi-croscopic contributions of the interfacial curvature are small(e.g. the liquid film under the elongated gas structure).Disturbed stratified region, where the contribution of mi-croscopic curvature dominates a generally stratified flowmorphology (e.g. near the slug front).Dispersed region, where parts of one phase are fully dis-integrated into small particles in the other phase and bothcontributions are high (e.g. bubble entrainment in the slugbody).

The resolvability of the local interface morphology can be as-sessed by comparing κM and κm to the local cell size. If thecurvature of a macroscopic structure in segregated flow regionsexceeds a critical value κcrit that correlates with the inverse of thelocal grid spacing, its shape can not be represented by a interfacecapturing method anymore. Hence, the structure will disintegrateand will be treated as disperse region by the algorithm. The fur-


ther evolution of this region is determined by the interfacial areatransport equation that involves the modeling of break up andcoalescence of the dispersed particles. It eventually leads to adecrease of the transported microscopic curvature field κm dueto coalescence, until a resolution of a structure with the inter-face capturing method is valid again. The characteristic lengthscale of the interfacial structure should be several times largerthen the actual grid spacing to reliably represent it by the inter-face capturing method. Cerne et al. [9] suggested that the edgelength of a hexahedral, Cartesian cell should be at least five timessmaller then the diameter of a spherical particle to represent itsshape properly. This argumentation determines the scaling coef-ficient Cs in the calculation of κcrit = 1/(Cs∆x) and is adapted inthe presented work to be 4. Accordingly, in proximity to a slugbody both parameters have to fulfill the critical curvature crite-ria κM,κm < κcrit to detect the actual, local region as resolvable.For reasons of stability, a smoothed transition function was cho-sen such that the regional detection function in compliance to thework of Hänsch et al. [13], becomes

δr =(0.5tanh(Cg∆x [κcrit −κM])+0.5

)·(0.5tanh(Cg∆x [κcrit −κm])+0.5

),

(10)

where Cg denotes the growth coefficient that determines thesmoothness of the transition and a value of 50 was applied in thepresent study. The first term of Eqn. (10) represents the macro-scopic criterion that leads to the disintegration of the interfacialstructures if κM > κcrit , while the second term enables the inter-face capturing again for regions where κm < κcrit .

The modeling of the interfacial forces obviously have to dif-fer in the described regions, including surface tension and an ap-propriate drag force formulation in segregated parts as well asdrag and non-drag forces in dispersed regions. To account forthis circumstance the interfacial transfer terms F j

k in Eqn. (2) arescaled by δr and 1-δr for segregated and dispersed regions, re-spectively.

Methodologically compatible techniques for dynamicswitching between the approaches are reported by [8,13,16]. Thepresented contribution includes the direct dynamic modeling ofthe unresolved interfacial curvature and reduces the detection cri-terion to one critical parameter to simplify the approach.

EXPERIMENTAL FACILITYExperiments were carried out on a horizontal two-phase flow

facility for an isothermal air-water mixture. The test section, asthe main part of this facility, consists of a L=10 m long, trans-parent polycarbonate pipe. The pipe has an inner diameter D of54 mm and is attached to a phase separator at its end. It con-sist of multiple pipe segments, which are connected by flanges

M

temperaturesensor

heatexchanger

rootsblower

phaseseparator

test section

camera

LED light source

measuringorifice

mixingdevice

filter three-way valve

inductiveflow meter

water pump

watertank

FIGURE 1: SCHEMATIC OF THE EXPERIMENTAL FACIL-ITY IN VIDEOMETRY CONFIGURATION.

to allow the mounting of measurement techniques. A mixing-device at the inlet directs both fluids into the test section. Theinlet cross sections of the fluids are equally sized and divided bya 1 mm thin separation plate, whereas the axial position x = 0 islocated at its end. Water is supplied by a centrifugal pump withina closed loop, consisting of a tank and a bypass, whereas a roots-blower, a heat exchanger, and a bypass provide environmentalair. Experiments were run at constant fluid superficial veloci-ties. Temperature, pressure and flow rates of gas and liquid aremeasured with PT100 sensors, static pressure gauges and flowmeters, respectively. A high speed camera setup is attached toallow highly time resolved videometic measurements. Figure 1shows the experimental facility in videometry configuration.

Videometry MeasurementsTo obtain qualitative information about the vertical distribu-

tion of the interfacial area density and its variation in time withinthe slug body, high-speed image recording was applied to thetest section as illustrated in Fig. 1 at 75 % of its total length.The flow was investigated at quasi steady state condition over amaximum time range of one minute with an exposure frequencyof up to 1 kHz adapted to the mixture velocity. The observedpart of the test section was enclosed in a water filled, rectangularbox in order to avoid light refraction and was illuminated by ahomogenized LED panel. An image processing algorithm wasdeveloped, that allows the detection of the gas-liquid interfaceand hence the quantitative analysis of the global characteristicsof the actual slug pattern, such as slug length, front and end ve-locity as well as slug frequency. Proceeding from this data, itis possible to track the slugs in a Lagrangian sense, where therelative position η = 0 denotes the front of an developed slugand η = 1 denotes its end. Information about local, time varyingproperties at relative positions within the propagating slug bodycan thereof be obtained.

Image processing allows accurate measurement of the size


100 200 300 400 500 600 700 800 900 1000

20

40

60

80

100

120

140

160

180

200

1.0

0.5

0.0

y/D

[-]

η3 = 0.5 η3 = 0.3 η3 = 0.1

0.67 0.68 0.69 0.70Axial Position, x/L [-]

Γ(η) -flow direction

?

g

FIGURE 2: VERTICAL, SPACE-AVERAGED SIGNALSTRENGTH DISTRIBUTION AT THREE RELATIVE POSI-TIONS IN THE SLUG BODY.

and number of entrained bubbles dispersed in the slug body, ifparticle overlapping is low. This circumstance is used in thecurrent study to determine a correlation for the interfacial areadensity and its vertical distribution in the slug body from thehigh-speed videometry data. The raw images are filtered andfurther processed in order to detect non-overlapping bubbles foreach time frame that involves a part of a slug body. The av-eraged equivalent diameters d of the detected bubbles are thendetermined for N discrete, equally distributed, vertical segments,where the height of the segments are adapted to the maximumdetected bubble size. This procedure finally provides an aver-aged vertical bubble size distribution for each measured jL/ jGcombinations. In order to arrive at an estimation for the inter-facial area density from the line-integrated videometric data, theproperty of a gas-liquid interface to refract light is utilized. Thesignal strength at the camera sensor decreases, when a light rayimpinges a curved interface on its way from the illuminationsource. The number of impingements and the angle of the im-pinged surfaces determine the amount of light that finally arrivesat the sensor. Both criteria are related to the number and size ofdispersed bubbles respectively, located between the light sourceand the camera sensor. The calculation of the interfacial areadensity from this approach is based on the assumption that themaximum reduction of the signal strength observed, correspondsto the close-packing of spheres of bubbles from size dj at verticalposition j within an observed discrete pipe segment volume V j.In a first approximation a linear correlation between the particlenumber density and signal strength Γ is assumed, such that

n j(Γj) =2Vj√

2 d3j

(1−

Γj

Γmax

)(11)

and the interfacial area density in the discrete averaging volumeVj can be approximated to

ai, j =n j π d

2j

4Vj. (12)

The size of the averaging volume V j is determined by its verticalposition, due to the cylindrical geometry of the pipe. Figure 2shows a raw image of a slug front with overlapped representa-tions of the vertical interfacial area density distribution at threerelative positions η in the slug body. It has to be emphasized, thatthe described method still only provides a coarse approximationto a quantitative representation of ai.

SIMULATION SETUPNumerical simulations were carried out on a three dimen-

sional computational domain, that corresponds to the geometryof experimental test section. To reduce the computational ef-fort, a symmetry plane bounds the domain in depth and separatesthe pipe cross section vertically. The O-shaped, block structuredmesh consists of six equal distributed blocks in a Cartesian co-ordinate system, where the x-coordinate corresponds to the po-sition in mean flow direction and the y-coordinate correspondsto the position along the pipe diameter parallel to g. To evalu-ate mesh dependency, one flow rate configuration was simulatedon four grids with increasing mesh resolution. Figure 3 showsthe mean onset position of the slugs predicted by the differentaveraged cell length 3

√Vc to pipe diameter D ratios. The dashed

line and the error bars correspond to the standard deviation in ex-periment and simulation, respectively. It can be recognized thatthe predicted slug onset position converges to the experimentalvalue with increasing mesh resolution and only small deviationsare observed for 3

√Vc/D<0.08. Hence, the later presented nu-

merical simulations were all carried out for a mesh density of3√

Vc/D=0.063 which corresponds to a mesh of 470400 cells.Since turbulence-enforced, small interface disturbances are

acknowledged from experimental observations to decisively in-fluence the origin of slugs [24], a large eddy simulation (LES)treatment of turbulence with Smagorinsky sub-grid modeling

1

2

3

4

5

6

0.04 0.06 0.08 0.1 0.12 0.14

Ons

etPo

sitio

n,x/

L[-

]

Grid Resolution, 3√

Vc/D [-]

simulationexperiment

FIGURE 3: SLUG ONSET POSITION FOR DIFFERENTMESH DENSITY FOR jL/ jG = 0.5.


0.0

0.5

1.0

0 0.2 0.4 0.6 0.8 1

y/D

[-]

0.0

1.0

2.0

0 0.2 0.4 0.6 0.8 1

Fr[-

]

Axial Position, x/L [-]

interface position

critical condition

FIGURE 4: INTERFACE POSITION (TOP) AND FROUDENUMBER PROGRESSION (BOTTOM) FOR AN INITIALSTATE WITH HYDRAULIC JUMP CONDITION.

was used to resolve large turbulent structures near the interface.Although the choice of the LES sub-gird model as well as meshquality and density is known to be of great importance for theturbulence representation, detailed investigations of the influenceof these parameters will be addressed in the future, since the pri-mary purpose for the choice of LES turbulence modeling in thiswork was the representation of arbitrary interface disturbances.

It turned out that an initialization procedure has to be ap-plied to the calculations that is equal to the one conduced in theexperiments to arrive at realistic initial conditions. This proce-dure involves a filling of the pipe and careful ramping up ofthe inlet velocity until steady state conditions are achieved. Athree dimensional turbulent velocity profile was imposed at theinlet. The profile varies in time during the initialization process,whereas its maximum value follows

Uk(t) = U0,k +U0,k

|U0,k|

(jk−|U0,k|

2

)(tanh(t− t0,k)+1

). (13)

U0,k, jk, t0,k are denoted as the initial velocity vector, the finalsuperficial velocity and the point in time at maximum velocitygradient during the ramp up, respectively. By following this pro-cedure a static hydraulic jump could be achieved that results ina sudden increase in the liquid height and thus accelerates thegaseous phase, which supports the growth of surface instabili-ties. The position and height of this jump depends on both theliquid and gas flow rate and only occurs when the liquid flow ful-fills the critical condition of Froude number Fr>1 at the inlet andreaches Fr<1, due deceleration, within the length of the domain.An initial state with hydraulic jump before the onset of sluggingis shown in Fig. 4 at the symmetry plan of the three dimensionaldomain. To obtain statistical relevant data, the total physical timespace of the transient simulation was 40 s with an initializationtime of 10 s to arrive at quasi steady state conditions.

The drag force in the segregated region is modeled according

to Marschall [20]

FDG,se =−FD

L,se = λ |∇αG|2µGµL

µG +µL(UL−UG). (14)

The dimensionless friction coefficient λ accounts for tangentialinertia and shear contribution in the presence of phase slip and isindicated to be proportional to the interfacial Reynolds numberRei and the viscous shear contribution Π, which both are func-tions of the dynamic viscosities of the conditioned fluids µk. Thedrag formulation has to differ in regions which are detected asdispersed, such that

FDG,d =−FD

L,d =18

CDαGρLκm|UL−UG|(UL−UG). (15)

The drag coefficient CD is modeled using the Ishii and Zuber [25]drag model, that allows to account for variations in the shape ofthe entrained bubbles. The microscopic curvature field in thesimulation was initialized according to the results of the exper-imental bubble size measurements. Pressure-velocity couplingwas handled in the transient simulations through the PISO pro-cedure, where time steps are adapted to the maximum velocityin the domain. The backward Euler method was applied for timediscretization of the governing Equations. The Geometric Alge-braic Multi-Gird (GAMG) method with Gauss-Seidel smoothingwere used to solve for the discretized systems in terms of pres-sure and velocity.

0.1

1

1 10

Liq

uid

Supe

rfici

alV

eloc

ity,

j L[m

/s]

Gas Superficial Velocity, jG [m/s]

experimentsimulation

stratified/wavy

annular flow

slug flow

plug flow

examplecase

@

FIGURE 5: FLOW PATTERN MAP OF THE EXPERIMENTALFACILITY ACCORDING TO [2] FOR THE ANALYZED EX-PERIMENTAL AND NUMERICAL SETUPS.


0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5

Freq

uenc

ySi

mul

atio

n,f

[1/s

]

Frequency Experiment, f [1/s]

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

Ons

etPo

sitio

nSi

mul

atio

n,x/

L[-

]

Onset Position Experiment, x/L [-]

+20 %

-20 %

+20 %

-20 %

FIGURE 6: SLUG FREQUENCY (LEFT) AND SLUG ONSET POSITION (RIGHT) COMPARED TO EXPERIMENTAL DATA

The present study focuses on fully developed slug flowregime with a considerable amount of entrainment. Figure 5 de-picts the flow regime map of the experimental facility. The ex-perimental examined combinations of gas and liquid superficialvelocity as well as the corresponding simulation conditions areindicated.

RESULTS AND DISCUSSIONThe global characteristics of simulated slug flows are in

good agreement with the experimental data, as depicted in Fig. 6.The simulated and experimental slug frequency as well as on-set positions are compared to each other. The error bars in theright graph correspond to the standard deviation of the arithmeticmean values. Onset positions are slightly overestimated for lowerflow rates where the flow regime approaches wavy flow condi-tions. The quality of the onset-prediction correlates with the ac-curacies of the regional detection, since the growth of surfaceinstabilities greatly depends on the interfacial drag formulation.

Comparative calculations of the investigated test cases uti-lizing the standard OpenFOAM VOF-solver interFoam (notpresented here) were not able to reproduce the empirical ex-pected flow regime. Permanent slugging could not be achievedwith this approach using the same mesh and boundary condi-tions within the flow rate window under consideration. Theflow regime map was strongly shifted to higher liquid superfi-cial velocity. As a reason the damping of surface instabilitiesdue to the global imposed interface compression method is sus-pected. Early studies showed that a much higher grid resolutionis needed to represent slug triggering interface disturbances withthis method [7]. This circumstance emphasizes the advantage ofthe presented approach.

The introduced solver permits not just the investigation ofglobal characteristics but an examinations of local entrainmentrelated data as well. To illustrate that, Fig. 7 depicts the instan-taneous phase fraction (top), interfacial area density (mid), andregional detection function field (bottom) for jL = 0.6 m/s andjG = 1.2 m/s at t = 18.2 s. This test case is approximately lo-

0.64 0.65 0.66 0.67 0.68 0.69 0.700

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FIGURE 7: CONTOURS OF LIQUID PHASE FRAC-TION (TOP), INTERFACIAL AREA DENSITY (MID) ANDREGIONAL DETECTION FUNCTION (BOTTOM) FORjL = 0.6 m/s AND jG = 1.2 m/s.


0 20 40 60 80 1000

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experimentsimulation

FIGURE 8: COMPARISON OF THE SIMULATED ANDEXPERIMENTAL AVERAGED, VERTICAL INTERFACIALAREA DENSITY DISTRIBUTION FOR jL = 0.6 m/s ANDjG = 1.2 m/s AT FOUR RELATIVE POSITIONS IN THE SLUGBODY.

cated in the center of experimental matrix in Fig. 5 indicated bya jL/ jG ratio of 0.5. The slug pattern of this test case is char-acterized by immanent formation of stable slugs as well as by aconsiderable amount of entrainment and thus is chosen as rep-resentative set up. The displayed axial range in Fig. 7 is 60 cmwide, such that the slug shape is distorted in x-direction, bearingin mind that D = 5.4 cm. The red and black lines in the contourplots indicate the iso-surface of αL = 0.5.

Figure 8 shows the comparison of the averaged, simulated,vertical interfacial area density distribution with the experimentaldata of the same case, determined from the videometric measure-ments. The depicted ai distributions at four relative positions inthe slug body are averaged for all detected slugs in the simulationas well as in the experiment. To guarantee comparability, the re-sults of the simulation are line-integrated in depth. A tendency tooverestimate the interfacial area density in bulk flow of the slugbody can clearly be identified, whereas the top and bottom regionvalues tend to converge. It seems that the coalescence and dis-integration models used in the simulation overestimate the disin-

tegration of particles in the bulk of the flow, which leads to theprediction of smaller dispersed particles, that due to the momen-tum exchange between the fluids are more prevented from trav-eling upward following the buoyancy force. Nevertheless, thetendency of the dispersed fluid to agglomerate at the top of thepipe is reproduced. The end of the slug is characterized by strongfluctuations in the interfacial area density field, due to the ejec-tion of the entrained gaseous phase. If these ejections occur forhuge agglomerations of bubbles, the slug end position becomesmore diffuse and its detection loses precision. Still, the verticalposition of the jump in ai at y/D≈0.4 in position η = 0.8 is wellcaptured by the simulation. This jump corresponds to the originof the liquid film that follows the slug body in direct successionand fluctuates in height near the slug end. The other investigatedcases are showing the same principle behavior in terms of ai dis-tribution, where the described effect of its overestimation in thebulk, is less conspicuous for cases with a smaller amount of en-trainment.

CONCLUSIONThe capability of a coupled multi-scale approach based on

the Euler-Euler two-fluid model to simulate slug flow in hor-izontal pipes with a high amount of entrainment was demon-strated. Involving the interfacial area transport equation permitsthe dynamic modeling of the temporal and spatial evolution ofunresolved interfacial structures by a minimal additional com-putational effort, since only one additional transport equation issolved. The achieved information is used for scale separationand detection of resolvable and underresolved regions in termsof interfacial structures. A regional detection function is imple-mented to switch on a VOF-like interface capturing method insegregated flow parts and to apply a special treatment for the in-terface momentum transfer in these regions. The general trend inthe dispersed phase behavior within the slug body are captured,even though investigations of the local averaged interfacial areadensity distribution clearly demonstrate that more generalizedmodels are necessary to predict coalescence and disintegrationunder complex horizontal flow conditions. Good agreement wasachieved for the global characteristics and the general, morpho-logical appearance of the investigated slug flow pattern. Signif-icant improvements were obtained towards slug flow simulationcompared to the standard OpenFOAM VOF solver interFoam,since even for low liquid flow rates self-induced slug formationwith physically realistic properties was observed.

A procedure to estimate interfacial area density from video-metric measurements was presented, that uses statistical particlesize measurements and a priori known geometrical properties ofthe flow domain to obtain vertical distributions of the line inte-grated data. This method can only provide a coarse approxima-tion to a quantitative representation of the interfacial area density.


ACKNOWLEDGMENTThe presented work is funded by the German Federal Min-

istry of Economic Affairs and Energy (BMWi) on the basis of adecision by the German Bundestag (project no. 1501456) whichis gratefully acknowledged.

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Numerical Investigation of Slug Flow in a Horizontal Pipe