Scale-Adaptive Simulation of Transient Two-PhaseFlow in Continuous-Casting Mold

ZHONGQIU LIU, ALEXANDER VAKHRUSHEV, MENGHUAI WU,ABDELLAH KHARICHA, ANDREAS LUDWIG, and BAOKUAN LI

Scale-adaptive simulation (SAS) of the transient gas–liquid two-phase flow in a laboratory-scalecontinuous-casting mold is presented. The main objective is to investigate the applicability ofthe scale-adaptive unsteady Reynolds-averaged Navier–Stokes turbulent model (URANS SAS)for predicting the transient multiscale turbulent structures in a two-phase flow. Goodquantitative agreements with the experimental data and the large eddy simulation (LES)results are obtained both for the time-averaged velocity field and for the transient turbulentcharacteristics. The introduction of the von Karman length-scale into the turbulence-scaleequation allows the SAS model to dynamically adjust to the resolved turbulent structures. TheLES-like pulsating behavior of the air gas and the large-scale liquid eddy magnitudes in theunsteady regions of flow field are captured by the SAS model. The classical � 5/3 law of powerspectrum density (PSD) of the axial velocity is kept properly for the single-phase turbulent flow.For two-phase flow, the decay of PSD is too steep at the high-frequency region; the predictedPSD obtained with SAS is damped stronger than that estimated by LES. The SAS model offersan attractive alternative to the existing LES approach or to the other hybrid RANS/LES modelsfor strongly unsteady flows.

https://doi.org/10.1007/s11663-018-1443-0� The Minerals, Metals & Materials Society and ASM International 2018

I. INTRODUCTION

DURING the continuous-casting (CC) process ofaluminum-killed steel, argon gas is usually injected intothe submerged entry nozzle (SEN) to prevent nozzleclogging due to nonmetallic inclusions (Al2O3 particles)attaching to the inner wall of the SEN. The injected gasis usually in the form of bubbles, being transported bythe steel melt into the mold. The nature of the flow ismultiphase turbulence. Understanding the gas–liquidmultiphase turbulent flow is important to control thetransport of nonmetallic inclusions, the fluctuation ofslag–steel interface (one cause of slag entrapment), andother undesired effects.[1–3]

The multiphase turbulent flow in the CC mold can becharacterized by eddies with a wide range of spatial andtemporal scales. The spatial scales of turbulent eddy arebounded by the integral scale of flow field and thediffusive action of molecular viscosity. The largest scalesare typically comparable to the characteristic length ofthe mean flow and depend on the mold geometry andcasting conditions. The smaller scales depend on thebubble dynamics and are proportional to the bubblesize. The smallest scale relates to the Kolmogorovscale[4] and is generally smaller than the bubble size.Consistently, the turbulence kinetic energy of the flowwill be dissipated into heat by the smallest scale eddies.The computational fluid dynamics (CFD) has been

widely used in designing and optimizing theCCprocess forthe past several decades. Reliable turbulence modeling isrequired to obtain the accurate flow pattern prediction.Three approaches are typically distinguished for theturbulent-flow modeling: direct numerical simulation(DNS), Reynolds-averaged Navier–Stokes (RANS), andlarge eddy simulation (LES). According to the DNSapproach, thewhole range of spatial and temporal scales ofturbulence canbe resolvedon the computational grid, fromthe smallest dissipative scale up to the integral scale,associated with the motions containing most of the kineticenergy.[5] The computational cost of DNS is extremelyhigh, and it increases with the cube of the Reynoldsnumber,[6] so that DNS is inapplicable to the high

ZHONGQIU LIU is with the School of Metallurgy, NortheasternUniversity, No. 3-11, Wenhua Road, Heping District, Shenyang 110819,China and also with the Christian Doppler Laboratory for AdvancedProcess Simulation of Solidification and Melting, MontanuniversitatLeoben, Leoben, Austria. Contact e-mail: liuzq@smm.neu.edu.cnALEXANDER VAKHRUSHEV, MENGHUAI WU, andABDELLAH KHARICHA are with the Christian DopplerLaboratory for Advanced Process Simulation of Solidification andMelting, Montanuniversitat Leoben. ANDREAS LUDWIG is with theChair of Simulation and Modeling of Metallurgical ProcessesDepartment of Metallurgy, Montanuniversitat Leoben. BAOKUANLI is with the School of Metallurgy, Northeastern University.

Manuscript submitted June 20, 2018.Article published online October 30, 2018.

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Reynolds numberflows.TheRANSmodels, e.g.,k–e,k–x,shear stress transport (SST) and Reynolds stress model(RSM), provide the mean flow results with engineeringaccuracy at moderate computational cost. However,limited by their isotropic nature and due to the eddy-vis-cosity hypothesis, even at the unsteady mode (URANS),they struggle to predict the details of flow separation oranisotropic turbulence. In LES, the turbulence flow isfiltered to obtain large- and small-scale eddies. Largeeddies are resolved directly, while small ones are modeledwith the subgrid scale (SGS) model. Currently, LES hasbeen identified as a better way to model the turbulence inthe CC process.[7–11] However, it still require rather finegrids, especially at the boundary layers. Recently, thescale-adaptive simulation (SAS) approach based on animproved length-scale equation for turbulence modelinghas been proposed by Menter and Egorov.[12–14] It is animproved URANS model allowing the formation of aturbulent spectrum by adjusting its length-scale to theresolved structures. It can offer many interesting charac-teristics like other hybrid RANS–LES combinations andgenerally requires less computational effort.

Simulations of the two-phase flow in the CC processhave been performed by different researchers combiningthe Euler–Euler model with the (U) RANS or LESturbulence models. The (U) RANS method has beenwidely used, especially the k–e model family. Thomaset al.[15] simulated the steady-state argon gas /molten steeltwo-phase flow based on the standard k–e model. Theinfluence of the argon gas injection on the melt flow wasinvestigated, whereas the bubble-induced turbulence(BIT) was ignored. Creech[16] quantified how the detri-mental transition from the double- to single-roll patterncould be avoided in the mold by keeping the gas ratebelow a critical value. Bai et al.[17] simulated a time-av-eraged two-phase turbulent flow in a slide-gate tundishnozzle, considering an empirical interphase drag forcebetween molten steel and argon bubbles. Ramos-Ban-deras et al.[18] studied the dynamics of a two-phasedownward flow in the SEN and its influence on thetwo-phase flow in the mold using a modified k–e model.Klostermann et al.[19] studied the gas–liquid flow in astopper rod-controlled SEN using a modified k–e modelas well, considering the Tcheu’s theory[20] for the turbu-lent distribution of the gas bubbles. Liu et al.[21] developedan Euler–Euler–Lagrangian approach to study the influ-ence of the argon gas injection on the molten steel flowand on the particle transport behavior in the mold. Themodified k–emodel with extra source term to account forthe BIT is adopted. Recently, the Euler–Euler approachcombined with RANS turbulent models has beenextended to enable the spatial variations in bubble size,and size evolution according to a population balanceframework via the MUlti-SIze Group (MUSIG)model[22–24] and Average Bubble Number Density(ABND) model.[25] Timmel et al.[26] used the SST turbu-lent model to simulate the liquid–metal two-phase flowaccording to the frequently referenced experiment at theLIMMCAST facility at Helmholtz–Zentrum

Dresden–Rossendrf (HZDR), considering the effect ofelectromagnetic brake or stirring. Sarkar et al.[27] studiedthe time-averaged meniscus flow and turbulence intensitydistribution in the mold under the influences of argoninjection and a double-ruler magnetic field. Euler–EulerLarge eddy simulation (EELES) model has been devel-oped byLiu et al.[28] to capture the transient asymmetricaltwo-phase flow in the mold. Then they used the EELESwith the dynamic SGS model to simulate the transientgas–liquid flow.[29] The EELES model can capture moredetails of the two-phase turbulent-flow characteristics,including multiscale vortex structures, vorticity distribu-tion, etc. Furthermore, a detailed review onmodeling andsimulation of CC process can be found in the recent studyfrom Thomas.[30] SAS has been successfully used tosimulate the hot buoyant jet in the cross flow in achannel,[14] turbulent combustion in a swirl burner,[14]

aerodynamic flows with massive separation,[31] and thedispersed bubbly flow in a bubble column.[32] All theresults indicated that the SASmodel leads to an improvedresult compared to (U) RANS simulations. However,relatively little study has been reported on the SASmodeling of turbulent flow in the CC mold.[33]

The current study attempts to perform a sensitivityanalysis of the SAS for modeling the transient two-phase flow in a CC mold. To the best of the author’sknowledge, the SAS is for the first time applied in thisstudy for the transient two-phase flow in the mold. Boththe experimental data measured by a laser Dopplervelocimetry reported by Iguchi and Kasai[1] and previ-ous investigations using LES modeling[29] were used tovalidate the current SAS model.

II. MATHEMATICAL MODEL

A. Euler–Euler Two-Fluid Model

In the Euler–Euler two-fluid model, conservationequations of mass and momentum for each phase mare given as follows:

@ amqmð Þ@t

þr � amqmumð Þ ¼ 0 ½1�

@ amqmumð Þ@t

þr � amqmumumð Þ¼ �r � amsmð Þ � amrPþ amqmgþ Fm; ½2�

where the lower index m denotes the phases. a, q, t,and u are the volume fraction, density, time, andvelocity, respectively. All the phases share a singlepressure field P. The terms on the right-hand side ofEq. [2], respectively, represent the stress, the pressuregradient, gravity, and interfacial forces.The stress term of m phase is described as follows:

sm ¼ �leff;m rum þ ðrumÞT � 2

3Iðr � umÞ

� �: ½3�

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The effective viscosity of the liquid phase, leff,l, iscomposed of three contributions: the molecular viscos-ity, ll; the turbulent viscosity, lT,l; and an extra termdue to bubble-induced turbulence (BIT), lBI,l. I is theunit tensor.

leff;l ¼ ll þ lT;l þ lBI;l ½4�

The effective viscosity of the gas phase is based on theeffective liquidviscosity aswasproposedby Jakobsen et al.[34]

leff;g ¼qgql

leff;l; ½5�

where the subscript l denotes the liquid phase, and gdenotes the gas phase.

In bubbly flows, the smallest scales are responsible forthe dissipation of the turbulent kinetic energy as in thesingle-phase flow. However, the bubbles can also gen-erate backscatter, i.e., energy transfer from smaller tolarger scales as reported by Dhotre et al.[35] Thecombination of both effects can yield an overallenhancement or an attenuation of the turbulenceintensity. The model proposed by Sato et al.[36] hasbeen used to take into account of the turbulence inducedby the movement of the bubbles. The defining expres-sion is

lBI;l ¼ qlCl;BIagdb ug � ul�� �� ½6�

with a model constant Cl,BI equal to 0.6. The bubblediameter db is accepted to be a constant in this study.

B. Turbulence Models

In the current study, the turbulence is treated differ-ently for different phases. Considering the low densityand the flow rate of gas phase, the turbulence ofdispersed gas bubble is calculated with a simple zeroequation model. For the continuous liquid phase, boththe unsteady SAS model and LES are used.

1. Scale-adaptive simulationThe SAS concept is based on the introduction of the

von Karman length-scale (Lvk) into the turbulence-scaleequation. Additionally acquired information allows theSAS model to dynamically adjust itself resolving theLES-like structures in the unsteady regions of the flowfield. Recently, Menter and Egorov[12] proposed that theLvk term can be transformed and implemented into anyscale-defining equation, e.g., employing the SSTtwo-equation system to form the SST–SAS turbulencemodel. The governing equations of the SST–SAS modeldiffer from those of the SST–RANS model[37] throughintroducing an additional scale-adaptive source termQSAS in the transport equation for the specific turbu-lence dissipation x (also referred as turbulence eddyfrequency). The equations for the turbulent kineticenergy k and the specific turbulence dissipation x can bedefined as

@ðqlalkÞ@t

þr � alqlulkð Þ

¼ alPk � alqclkxþr � al ll þlT;lrk

� �rk

� �½7�

@ðqlalxÞ@t

þr � alqlulxð Þ

¼ alxkPk 1� f1 þ f2

L

Lvk

� �2" #

�alqlx2 cl � c1=4l f3� �

þ alQSAS þr � allT;lrU

rx

� �

þal2qlrU

1

xrk � rx� k

x2rx � rx

� �;

½8�

where Pk is the turbulent production term dependingon the rate of strain tensor S as follows:

Pk ¼ lT;lS2; with Sj j ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi2SijSij

p;

Sij ¼1

2

@ul;i@xj

þ @ul;j@xi

� �;

½9�

where L is the length-scale of the modeled turbulence,

L ¼ffiffiffik

p=ðc1=4l � xÞ; ½10�

where Lvk is the von Karman length-scale.

Lvk ¼ maxjSu0lj j ;CS1CSASD

� �; with

u0lj j ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXðiÞ

@2ul;i

@xj@xj

� �2vuut :

½11�

The additional source term QSAS is defined by

QSAS ¼ max qf2jS2 L

Lvk

� �2

�CS1 �2qlkrU

�max1

x2rxrx;

1

k2rkrk

� �; 0

( ):

½12�

The values of the constants in above equations aref1 = 0.8, f2 = 3.51, f3 = 0.0288, rF = 2/3,j = 0.41, cl = 0.09, and CS1 = 2. The SGS filterwidth D is equal to the cubic root of the controlvolume.The equilibrium eddy viscosity of the SAS model can

be defined by

lT;l ¼ ql1

CSAS� Lvk

� �2

Sj j; ½13�

where the model constant CSAS is chosen to be 0.131.This formula has a similar structure as one for theSmagorinsky eddy viscosity used for the LES describedin the next section.

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2. Large eddy simulationIn the LES model, the key element is the SGS model

which determines the effect of the unresolved turbulentscales. The model proposed by Smagorinsky[38] is usedto calculate the turbulent viscosity lT,l.

lT;l ¼ ql CSDð Þ2 �S�� ��; ½14�

where the superscript ‘‘–’’ denotes the first filtering;CS is the Smagorinsky constant. It was found thatthe best results for a wide range of flows areobtained for CS � 0.1. However, employing a con-stant parameter is the most critical shortcoming ofthis simplified model.

In view of the uncertainty in specifying the constantCS, Germano et al.[39] proposed a dynamic SGS model.Then, Lilly[40] proposed a modification of Germano’smodel, in which the CS is not arbitrarily chosen, but iscomputed by

CS ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiLijMij

2M2ij

s½15�

Lij ¼ �duiuj þ �ui �uj ½16�

Mij ¼ D2 �S��� ��� �Sij � D2 d�S�� �� �Sij; ½17�

where the Lij and Mij are the assumed Gaussian vari-ables. The concept of the dynamic procedure is toapply a second filter (called the test filter) to the equa-tions of motion. The superscript ‘‘^’’ denotes the sec-

ondary filtering. The new filter width D is twice thewidth of the initial grid filter D. Both filters produce aresolved flow field. The difference between themreflects the contribution of the small scales being in-be-tween the filter sizes. The details of this filtering proce-dure can be seen elsewhere.[39,40]

C. Interfacial Forces

In the Euler–Euler model, the interfacial momentumtransfer exhibits a dominant effect in the multiphasemomentum equations. There is still no agreement in thecommunity on the universal closures to be used. In thecurrent study, the drag force FD, lift force FL, virtualmass force FVM, and turbulent dispersion force FTD areconsidered.

Fm ¼ Flg ¼ �Fgl ¼ FD þ FL þ FVM þ FTD; ½18�

where Flg denotes the momentum transfer terms fromthe gas phase to liquid and vice versa for Fgl.

FD ¼ � 3

4agql

CD

dbug � ul�� ��ðug � ulÞ ½19�

FL ¼ agqlCLðug � ulÞ � r � ul ½20�

FVM ¼ agqlCVMDug

Dt�Dul

Dt

� �½21�

FTD ¼ CTDCDmt;grt;g

ralal

�ragag

� �; ½22�

where CD is the drag force coefficient, which can beevaluated by correlation of several distinct Reynoldsnumber ranges for the individual bubbles as proposedby Ishii and Zuber.[41] The lift force coefficient CL isset to 0.5 based on the study of Drew and Lahey.[42]

The virtual mass force coefficient CVM is taken to be0.5 for spherical bubbles. According to standard modelvalues, the turbulent dispersion coefficient of the gasphase CTD,g = 1 and the turbulence Schmidt numberof the gas phase rt,g = 0.9 are adopted here. The tur-bulence dispersion force is not employed within theLES due to the high turbulent structure resolution,which allows naturally for track dispersion mecha-nisms of the gas phase in the liquid flow.

III. EXPERIMENTAL DATA AND SIMULATIONSETUP

A. Experimental Data

The experimental trials were carried out for thehorizontal air–water flow in a well-designed apparatus,schematically shown in Figure 1.[1] The mold has arectangular cross section of 0.3 m90.15 m filled withwater up to a height of 0.4 m. The mixed water and airare supplied into the mold through a nozzle with adiameter of 9 mm. Water circulates in the circuitthrough a buffer tank to the mold. The water flow rateand air flow rate are controlled by corresponding flowmeters. In order to control the level of the free surface,

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an overflow pipe is mounted at the right side of themold. Measurements are performed along the mold’scenterline at the nozzle inlet level (red line in Figure 1)for the water flow rate of 5 l/min. Once the water and airflows reach a quasi-steady-state regime, the water–airtwo-phase flow is captured using a video camera. The

horizontal and vertical velocity components of the liquidphase are simultaneously measured by a two-channellaser Doppler velocimetry. Air bubbles are detected inthe size range of 0.3 to 5 mm. More experimental detailscan be found in the study of Iguchi and Kasai.[1]

B. Simulation Setup

Numerical simulations are performed employing theCFD code ANSYS-CFX-14.5 combined with the CFXcommand language (CCL). The calculation domain(geometry), material properties, and boundary condi-tions were set corresponding to the water model asreported in Iguchi et al.[1] A mass flow rate boundarycondition was used at the inlet, and the volume ratio ofgas was estimated based on the water experiment. Thebubbles are treated as monodispersed with a constantdiameter of 3 mm, which corresponds to experimentobservations. The top surface of the mold cavity ismodeled as a degassing boundary condition, whichmeans an outlet for the dispersed gas phase. Along thewalls, a no-slip boundary condition is applied for thecontinuous phase and a free slip condition for thedispersed gas bubbles. At the outlet, a water mass flow isfixed assuming that no gas can go with the outflow.A second-order, center-differencing scheme was used

for the spatial discretization; while a first-order, fullyimplicit, backward-differencing scheme was employedfor the time discretization. The SIMPLEC algorithmwas employed for the pressure–velocity coupling. Thetime step of all simulations is 0.001 second. The choiceof the time step is determined by the criterion that themaximum Courant–Friedrichs–Levy (CFL) numbermust be less than one (i.e., Dt 6 D/|u|). Converged

Fig. 1—Schematic of Iguchi’s experimental apparatus, reprintedfrom Ref. [1] (Color figure online).

Fig. 2—Predicted air volume fractions in the center plane by different models of SST: (a) 3 mm, (d) 1 mm; LES: (b) 3 mm, (e) 1 mm; SAS: (c)3 mm, (f) 1 mm (Color figure online).

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 50B, FEBRUARY 2019—547

solution is reached when the residuals of all the variablesare less than 10�4. The flow was simulated for thephysical time of 100 seconds, and the data of watervelocities were monitored during the calculation process.The data were sampled and time-averaged over the last50 seconds.

IV. RESULTS AND DISCUSSION

A. Instantaneous Flow Characteristics

Snapshots of the instantaneous distribution of the gasphase volume fraction are shown in Figure 2 obtainedrespectively with the SST, LES and SAS turbulencemodels. They are presented for different bubble diam-eter of 1 mm and 3 mm and are sampled at the centerplane of the simulation domain. The volume fractionfield illustrates the dynamics of the formed bubbleplume showing a typical fan pattern. For the RANSapproach (SST model), the air distribution is morehomogenous and steady, as shown in Figures 2(a) and(d). The pulsating motion of the gas phase cannot be

captured although the turbulent dispersion force isconsidered. The reason is that a typical URANS modelis applied without taking into account the turbulentlength-scale effect. The pulsating motions of air gas canbe captured well by LES and SAS methods along theouter front of the jet. The pulsating movement isimportant to the transport and entrapment of thebubbles and nonmetallic inclusions in the actual castingmold. The corresponding distribution pattern of the gasvolume fraction in Figure 2 shows a significant devia-tion for two bubble diameters. One can see that thesmaller bubbles can travel across the mold further thanthe bigger ones, and their dispersion is broader as well.Concluding, the bubble size is an important factor forthe bubble transport and its correct prediction.In order to visualize the characteristic three-dimen-

sional eddy structure computed by the SAS and LESmodels, various iso-surfaces of the Q-criterion[43] aregiven in Figure 3 calculated using following relation:

Q ¼ 1

2ðX2 � S2Þ; ½23�

Fig. 3—Instantaneous iso-surfaces of Q-criterion by LES: (a) 50 1/s2, (b) 100 1/s2, (c) 500 1/s2; by SAS: (d) 50 1/s2, (e) 100 1/s2, and (f) 500 1/s2

(Color figure online).

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Fig. 4—Comparison between the simulated and experimental profiles for different gas-flow rates (a) Qg = 0 cm3/s, (b) Qg = 4 cm3/s, and (c)Qg = 12 cm3/s (Color figure online).

Fig. 5—Predicted averaged air volume fractions for different gas-flow rates (a) Qg = 4 cm3/s and (b) Qg = 12 cm3/s.

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 50B, FEBRUARY 2019—549

where X is the vorticity tensor.Both results of the SAS and LES approaches are

obtained on the same grid and using the same time step.The strong mixing zone along the liquid jet can beclearly seen from both simulations. Of cause, moresophisticated turbulent structures are captured by theLES model. Compared to conventional URANS model,the SAS model resolves more unsteady structures due tothe introduction of the von Karman lengths scale Lvk. Itis a direct result of the term-by-term modeling of theexact transport equation for the integral length-scaleintroduced by Rotta as discussed elsewhere.[12] The Lvk

allows the SAS model to adjust to the already resolvedscales in the flow field, thereby avoiding the excessivedamping introduced by the conventional URANS.

B. Time-Averaged Results

In order to get a quantitative comparison with theexperimental data, the time-averaged results are ana-lyzed in this section. The experimental measurements ofthe time-averaged axial velocity �u and vertical velocity �vare taken along the centerline at the nozzle level, as

indicated in Figure 1. The applied water flow rate is setto 5 l/min, while the air flow rate is varied from 0 to12 cm3/s. The simulation results for both models areaveraged for 50 seconds starting from the time instantt = 50 seconds to quantitatively compare with theexperimental data.Figure 4 shows the profiles of the mean axial and

vertical velocities for different turbulence models undervarious gas-flow rates. For the single-phase water jet,shown in Figure 4(a), it can be seen that good quanti-tative agreement with the experimental data is obtainedby both models. The results of SAS simulation haveslightly higher values compared with the LES in bothprofiles. At the low gas-flow rate: Qg = 4 cm3/s, asshown in Figure 4(b), both simulations give a betterprofile prediction for �u than that for �v. The measuredpeak value of �v profile is located at 2 cm away from thenozzle inlet; however, it may be noticed that none of themodels could capture it well. The reason most probablyis due to the small gas volume fraction predicted by bothmodels, which cannot provide enough lift force for theliquid jet, as shown in Figure 5(a). The predictedtime-averaged peak value and the distribution of gas

Fig. 6—Predicted averaged liquid’s effective viscosities for different gas-flow rates (a) Qg = 0 cm3/s, (b) Qg = 4 cm3/s, and (c) Qg = 12 cm3/s.

550—VOLUME 50B, FEBRUARY 2019 METALLURGICAL AND MATERIALS TRANSACTIONS B

volume fraction by SAS model are similar to thoseobtained with the LES. The prediction of SAS modelgives an acceptable result at a distance ranging from 7 to25 cm away from the nozzle. Figure 4(c) shows theprofiles of �u and �v for the higher gas-flow rateQg = 12 cm3/s. It can be seen that both simulationscan predict the experimental trend of �u profile withacceptable deviation level. For �v profile, both SAS andLES results overpredict the �v magnitudes close to theinlet. This effect can be attributed to the predicted highergas volume fraction near the SEN, as shown inFigure 5(b), at about 4.1 pct, which can lift the jet. Itshould be especially stressed here that according to theexperimental measurements, the peak value of �v close tothe SEN is larger at the lower gas-flow rates. Despitebeing really confusing, the reliability of the experimentaldata is respected and accepted here. This effect should besubjected for further study.

Figure 6 presents the comparison of the turbulenteffective viscosity predicted respectively by the SST,SAS, and LES turbulence models for different gas-flowrates. It can be seen that the predicted maximumeffective viscosity by SST model is about 1.7 timeslarger than that by SAS; the latter allows avoiding the

excessive damping introduced by the conventionalURANS approach. However, compared with LES,larger viscosity values were obtained; this variance isattributed to the eddy viscosity computing method seenin Eqs. [13] and [14]. The eddy viscosity ratio W can bedefined by

W ¼lSAST;l

lLEST;l

¼ Lvk

CSCSASD

� �2

� CS1CSASDCSCSASD

� �2

¼ CS1

CS

� �2

¼ 2

CS

� �2

>1:

½24�

Therefore, the predicted eddy viscosity by the SAS islarger than that by the LES model. In addition, theliquid’s effective viscosity increases with the increasinggas injection rate. Especially at high gas rate ofQg = 12 cm3/s, a significant peak value is found atabout 9 cm away from the nozzle. Unfortunately, acomparison with the experimental data is not possible,since the effective viscosity represents a model variableonly and it is hard to relate it with any physicalquantity.

Fig. 7—Time history plots of the axial velocity u at point 1 by (a) LES and (b) SAS.

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C. Spectral Analysis

In turbulent flows, a wide range of the length-scalesexists, limited by the dimensions of the flow domain andby the diffusive scales of molecular viscosity. Therebythe spectral analysis is a valuable tool to investigate thedetails of turbulence. Since the SAS model is animproved URANS formulation based on the vonKarman turbulent scale, it allows the formation of aturbulent spectrum under the unstable flow conditions.Thereby the power spectrum analysis is performed forboth SAS and LES models and presented in thissection. Firstly, the transient axial velocity u and verticalvelocity v at Point 1 in the jet region (x = 5 mm,y = 0 mm, z = 0 mm) are monitored during the cal-culating process. Taking the high gas-flow rate(Qg = 12 cm3/s) as an example, the transient historiesof axial velocity u and vertical velocity v are respectivelyshown in Figures 7 and 8. The period of 5 secondspresented in these figures corresponds to 5000 samplepoints for the simulations. It can be seen that both SASand LES reflect the transient fluctuation behavior, butwith different amplitudes of the fluctuations. Thestronger amplitude of the fluctuations was found inthe LES results. Good agreements with the averagedexperimental data for axial velocity u have been

observed in both models, but they do not fit well forvertical velocity v, which has been discussed in theSection IV–B.The transfer of energy is analyzed through the power

spectrum densities (PSD) of axial velocity u taken atpoint 1, as shown in Figures 9 through 11. A fastFourier transform (FFT) of velocity signal using theWelch method with nonoverlapping sections, and aHanning window was performed. The frequency domainis resolved from 0.2 to 500 Hz. The lowest frequency islimited by the total data collection time during thesimulation of 5 seconds. The highest frequency is half ofthe signal sampling rate of 1000 Hz. For the sin-gle-phase water jet, as shown in Figure 9, it is seen thatthe PSD is very similar between two cases; the results ofLES show higher values compared with the SAS. Theclassical � 5/3 law is maintained properly for bothturbulent models according to the single-phase turbu-lent-flow calculations proposed by Van Cauwenbergeet al.[44] When the air gas is injected, as shown inFigures 10 and 11, the predictions for the SAS modelare in good agreement with the LES in the low-fre-quency region. Hence, the large-scale eddies’ magnitudescan be well captured by the current SAS model. Theclassical � 5/3 PSD slope is violated,[45] since the decay

Fig. 8—Time history plots of the vertical velocity v at point 1 by (a) LES and (b) SAS.

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is too strong for the high frequencies above 70 Hz. Dueto the larger eddy viscosity, the PSD of SAS is moredamped at the high-frequency spectrum.

V. CONCLUSIONS

The primary goal of this study was to assess theapplication of the SAS turbulence model in modelingthe transient two-phase flow in the continuous-castingprocess. The predicted results by the SAS model showgood agreement with the results of the experiment andthe LES model, both for the time-averaged velocity fieldand the transient-flow characteristics. The conclusionsare summarized as follows:The introduction of the von Karman length-scale into

the turbulence-scale equation allows the SAS model todynamically adjust to the resolved turbulent structures,avoiding the excessive damping introduced by theconventional URANS models. The LES-like pulsatingbehavior and large-scale eddy magnitudes at theunsteady regimes of the multiphase turbulent flow arecaptured.The SAS model shows a wide turbulent spectrum. The

classical � 5/3 law of power spectrum densities (PSD) ofaxial velocity is obeyed properly for the single-phaseflow. For the two-phase flow, the predictions are in agood agreement with the LES in the low-frequencyregion. The classical � 5/3 PSD slope is violated for themultiphase flow, while strong decay for the frequencies> 70 Hz is detected. The PSD is underestimated by SASmodel at high frequencies.In conclusion, the SAS model offers an attractive

alternative to the existing LES approach or to the otherhybrid RANS/LES models for the strong unsteadyflows. It can be a promising option worthy of adoptionin conjunction with the Euler–Euler model to predict thebubbly flow.

ACKNOWLEDGMENTS

This work was financially supported by the Funda-mental Research Funds for the Central Universitiesof China (No. N162504009), the National NaturalScience Foundation of China (Nos. 51604070 and51574068) and the China Scholarship Council (No.201706085027). The financial supports by theRHI-Magnesita AG; the Austrian Federal Ministry ofEconomy, Family, and Youth; and the NationalFoundation for Research, Technology, and Develop-ment within the framework of the Christian DopplerLaboratory for Advanced Process Simulation ofSolidification and Melting are gratefully acknowl-edged.

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Fig. 10—Power spectrum densities of axial velocity u at point 1under Qg = 4 cm3/s (Color figure online).

Fig. 9—Power spectrum densities of axial velocity u at point 1 underQg = 0 cm3/s (Color figure online).

Fig. 11—Power spectrum densities of axial velocity u at point 1under Qg = 12 cm3/s (Color figure online).

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