analysis of fluid flow turbulence in tundishes fed by a

1. Introduction

Control of turbulence, efficiency for flotation of inclu-sions, control of nitrogen and oxygen pickup during steadyand unsteady operations and plug flows have been the prin-cipal aims of fluid flow control in tundishes. Earlier worksrelated with designs for fluid flow control made use ofweirs, dams and baffles,1—8) these works employed physicalmodeling approach to optimize flow parameters. Melt flowin multiple-strand tundishes for billet and bloom casters hasbeen studied using water modeling and mathematical simu-lations.9—14) Most of those designs lead to the employmentof dams and weirs to build a pouring box to contain the slagand control fluid turbulence. However, most of those de-signs yield slag eye openings around the ladle shroud, deadzones behind dams and weirs and a build up of slag in thepouring box which increases as the casting sequences grow.Moreover, nitrogen and oxygen pick up during unsteady(ladle and steel grade changes) and steady operations cannot be avoided using these flow control devices.

More recently flow controllers known as turbulence in-hibitors have demonstrated to be useful to avoid slag en-trapment, pick up gases and decrease of downgraded steelduring grade changes and residual steel at the end of cast-ing sequences15—20) without the inconveniences mentionedbefore. These devices are also helpful to increase flotationrate of inclusions21) through increase of plug flow and tocontrol better temperature variations during ladle change

operations.22) However, as the trend of steelmaking industryis to simplify operations and to optimize designs of equip-ments in order to decrease costs and maintain or even in-crease steel quality new approaches are required for flowcontrol in tundishes. In the present work the authors pro-pose the substitution of all tundish furniture by only theladle shroud, with an especial design, that must fulfill allfunctions of the flow control devices so far used in steelcasters.

The aims of this work involve the definition of two basicaspects of fluid flow; first to compare the fluid dynamics asinfluenced by conventional ladle shrouds (LS) with tundish-es equipped with a swirling ladle shroud (SLS) and secondthe determination of the best turbulence model to simulatemathematically the swirling effects of the entry jet on fluidflow by comparing computations with experimental mea-surements of velocity fields.

2. Experimental

Experiments were performed in a 1/2 scale model of atwo-strand slab tundish with geometric dimensions shownin Figs. 1(a) and 1(b). This model is fed by a water storingtank located above the tundish model; water from the out-lets is recycled to the tank by a pump. Both outlets haveflow meters to measure the flow rates of water out of thetundish. After reaching a steady state, for a given castingrate, a slurry of polyamide particles with a size of 20 mm

ISIJ International, Vol. 44 (2004), No. 6, pp. 1024–1032

© 2004 ISIJ 1024

Analysis of Fluid Flow Turbulence in Tundishes Fed by a SwirlingLadle Shroud

G. SOLORIO-DÍAZ, R. D. MORALES,1) J. PALAFAX-RAMOS,1) L. GARCÍA-DEMEDICES and A. RAMOS-BANDERAS

Graduated Student, Department of Metallurgy and Materials Engineering, Instituto Politecnico Nacional-ESIQIE, Apdo. Postal75-874, CP 07338, Mexico D.F.. E-mail: [email protected] 1) Department of Metallurgy and Materials Engineering,Instituto Politecnico Nacional-ESIQIE, Apdo. Postal 75-874, CP 07338, Mexico D.F., K&E Technologies S.A. de C. V. President.E-mail: [email protected], [email protected]

(Received on December 25, 2003; accepted in final form on February 27, 2004 )

A new design of a ladle shroud, obtained through water modeling, that controls turbulence of the entryjet in continuous casting tundishes is proposed. Particle Image Velocimetry (PIV) measurements indicatethat this design decreases the impact velocity on the tundish bottom to close to 1/3 of that provided by aconventional ladle shroud. This achievement is due to a swirling jet that promotes a recirculatory flow in thehorizontal planes of the tundish. The swirling effects help to dissipate the turbulence energy of the jet be-fore it impacts the tundish bottom making possible decreases of fluid velocities that impact the back andfront walls of the tundish. Turbulence models like k–e , k–w and RSM were applied to simulate the experi-mental PIV measurements of velocities in the fluid flow. Only the RSM model yielded results that agree re-markably well with the experimental determinations. These results make possible to avoid the employmentof flow control devices such as dams, weirs, turbulence inhibitors and the like in tundishes.

KEY WORDS: tundish; ladle shroud; turbulence dissipation; PIV; mathematical models.

and a density of 1 030 kg/m3 was fed through a syringe inthe upper part of the ladle shroud. By following the tracksof these particles, using a Particle Image Velocimetry (PIV)equipment, velocity fields of water were recorded. Particletracks were recorded by a Coupled Charged Device (CCD)equipped with a lens of depth field. All signals areprocessed in a PC; using the cross correlation techniqueand Fast Fourier Transforms they are converted into fluidvelocities.23—25) Figure 2 shows a scheme of the experi-mental setup. Two ladle shrouds were investigated the firstone is of a conventional (LS) design and its dimensions areshown in Fig. 3(a) the second is a swirling ladle shroud(SLS) which consists of central pipe with three intermedi-ate chambers that work as a brake of fluid velocity and anupper blade whose function is to start a swirling motion ofthe fluid all through the shroud length up to its tip. The tipof the shroud has bell shape to reinforce the braking effecton the fluid which flows into the tundish. Water flow ratewas 5.83104 m3/s corresponding to 3.8 tons of liquidsteel/min in the current tundish, according to the Froudecriterion. This flow was maintained constant for this work.

3. Theory of Turbulence Models

The most commonly used turbulence model is that de-vised by Jones and Launder28) and known as k–e , it hasmany advantages; its concept is simple, is implemented invery commercial codes and it has demonstrated capabilityto simulate correctly many industrial processes like com-bustion,27) fluid flow in tundishes15,16) and multiphaseflows28) among many other applications. Nevertheless, itfails to provide reliable results of swirling flows and highlystrained angular velocities of rotating flows.30) Since we aredealing here with a complex swirling flow two other turbu-lence models were tried; the k–w model of Wilcox29) andthe Reynolds Stress model (RSM) which uses additionalequations to calculate the Reynolds stresses of the flow.30)

Next lines summarize the main characteristics of each tur-bulence model.

3.1. k–ee Model

This model, the k–w and the RMS models belong to agroup of turbulence models known as Reynolds Average

ISIJ International, Vol. 44 (2004), No. 6

1025 © 2004 ISIJ

Fig. 1. The geometric dimensions of the tundish (m).

Fig. 2. scheme of the particle image velocimetry equipment employed in the experiments of physical modeling.

Fig. 3. Geometric dimensions of the experimental shrouds; (a)conventional ladle shroud (LS); (b) swirling ladle shroud(SLS).

Navier–Stokes (RANS) that provide time averaged flowvariables. These variables can be dependent on time forcases of unsteady state flows. The k–e and k–w models usethe hypothesis of the isotropic eddy-viscosity which ismodeled through the flow fields of the turbulent kinetic en-ergy and the specific dissipation rate. All simulations in thiswork are performed under steady state conditions, accord-ingly the transport equations of the k–e model are,Equation of continuity

..............................(1)

Momentum equation

....(2)

the closure of this system of partial differential equations isobtained through the equations shown in Table 1 which in-cludes the transport equations for the turbulent kinetic ener-gy, k, its dissipation rate, e , the turbulent viscosity, mT

(m effmmT) and empirical constants.

3.2. k–ww Model

This is an empirical model based also on model transportequations for the turbulence kinetic energy k and the specif-ic dissipation rate w , which can also be thought as the ratioof e to k . Naturally continuity and momentum transfer bal-ances are expressed also by Eqs. (1) and (2) and the closurefor the system of equations is provided by equations oftransport for the turbulent kinetic energy, k, and its specificdissipation rate, w , the turbulent viscosity model and theempirical equations presented in Table 2 to close the sys-tem.

3.3. Reynolds Stress Model (RSM)

This model abandons the isotropic eddy-viscosity hy-pothesis and closes the RANS equations by solving trans-port equations for the Reynolds stresses, together with anequation for the dissipation rate. The exact transport equa-tions for the Reynolds stresses can be expressed as,

..................................(3)

where Cij is convection term, DT,ij is turbulent diffusion,DL,ij is molecular diffusion, Pij is stress production, F ij is

pressure strain, e ij is dissipation and Fij is production termby system rotation. The turbulent diffusion, pressure strainand dissipation terms need to be modeled. Then DT,ij isevaluated using the gradient diffusion model of Daly andHarlow.31) The pressure-strain term, F ij, is modeled accord-ing to the proposals of Gibson and Launder32) andLaunder.33) The dissipation term, e ij, is evaluated throughthe Sarkar model34) as,

..........................(4)

where YM2reM2t is and additional dissipation term. The

turbulent Mach number Mt is defined as

.................................(5)

where “a” is the speed of sound and k is the turbulent kinet-ic energy. The kinetic energy for modeling a specific termis obtained through the trace of the Reynolds stress tensor:

.................................(6)

the scalar dissipation rate of the kinetic energy, e , is calcu-lated through an equation similar to that for the k–e modelgiven in Table 1. The turbulent viscosity is also calculatedthrough an equation that is also similar to that for the k–emodel and is given in Table 1.

k u ui i 1

2

Mk

at 2

ε δ ρεij ji Y

2

3( )M

2ρ ε εΩk j m ikm i m jkm

ij

u u u u

F

( )

p

u

x

u

x

u

x

u

xi

j

j

i

i

k

j

k

ij ij

∂∂

∂∂

µ∂∂

∂∂

ε

2

Φ

∂

∂µ

∂∂

ρ∂∂

∂∂x x

u u u uu

xu u

u

x

D Pk k

i j i jk

j ki

k

ij ij

( )

L,

∂∂

ρ∂

∂ρ δ δ

xu u u

xu u u p u u

C Dk

k i jk

i j k kj i ik j

ij ij

( ) [ ( )]

T,

∂∂

ρ∂∂

∂∂

µ∂∂

∂∂x

u uP

x x

u

u

u

xji j

i j

i

j

j

i

( ) eff

∂∂

ρx

uj

j( )0


© 2004 ISIJ 1026

Table 1. The k–e Model of turbulence

Table 2. The k–w Model of turbulence

3.4. Boundary Conditions

3.4.1. k–e ModelTo deal with the modeling of near wall region the wall

functions by Launder and Spalding are employed.35) Non-slipping conditions were applied as boundary conditions toall solid surfaces of the tundish.

At the entry jet and at the outlets the flow profiles wereassumed to be flat and calculated by

UinQ/Anozzle ................................(7)

The inlet values for k and e at the inlet were calculated withthe following equations:

kin0.01U 2in .................................(8)

e in2k in3/2/Dnozzle ..............................(9)

Gradients of the turbulent kinetic energy and its dissipationrate are zero at the symmetry planes and at the free bathsurface.

3.4.2. k–w ModelThe wall boundary conditions for the velocity field are

treated in the same way as in the k–e model including thenear wall approach through empirical functions. The samecan be said about the inlets and outputs of fluid in thetundish. Regarding the wall function for w the followingexpression was applied29);

..............................(10)

where b0* is an empirical constant which value is presentedin Table 2, k is the Von Kármán constant and y is the per-pendicular distance from the wall surface. Surface rough-ness was not considered in this work. Gradients of k and wat the symmetric planes including the bath surface are zero.Boundary conditions at the inlet and the outlets of thistundish for k and w are similar to the previous turbulencemodel.

3.4.3. The RSM ModelAt walls, explicit boundary conditions are applied using

log-law expressions disregarding convection and diffusionin the transport equations for the stresses in Eq. (3). Using alocal coordinate approach for tangential, t , normal, h andangular directions, l , the Reynolds stresses at the wall adja-cent cells to a wall are computed from

...............(11)

All simulations were performed for water with physicalproperties of viscosity of 0.001 Pa-s and a density of1 000 kg/m3. Calculating velocity fields of water allows adirect comparison with the experimental flow fields deter-mined using the PIV technology. This approach will be use-ful to define which of the three turbulence models is capa-ble to emulate more closely the actual experimental deter-minations of water flow in the tundish model.

The computational algorithm employed for all mathe-matical simulations is that known as SIMPLER.36)

Computation convergence was obtained when the residualsof the output variables reached values lower than 1105.All mathematical simulations were performed by two PC’sat the Laboratory of Mathematical Simulation of MaterialsProcessing and Fluid Dynamics, IPN-ESIQIE. All resultswere stored in a CD-ROM for further analysis and futureelaboration of reports.

4. Results and Discussion

Just for comparing the effects of the LS and the SLS onfluid flow the k–e model was used and the results areshown in Figs. 4(a) and 4(b) respectively for the symmet-ric-transversal plane in the entry box. At a glance is evidentthat the SLS decreases dramatically the fluid velocities andconsequently flow turbulence in the pouring box. Using theLS, the fluid impacts the tundish bottom with a velocity ashigh as 0.70 m/s, the jet core drags fluid volumes which areclose to it and after impacting fluid velocities are as high as0.35–0.40 m/s along the tundish bottom. Fluid flow recircu-lations are formed in the longitudinal and transversalplanes. With the SLS, the jet impacts the tundish bottomwith smaller maximum velocities of 0.25 m/s, almost onethird of that obtained using the LS. Recirculating flows inthe and transversal planes are, consequently, less developedand with smaller magnitudes of velocity vectors as can beseen by comparing Fig. 4(b) with Fig. 4(a). From these re-

u

k

u u

kλ τ η

2

0 655 0 255. .

u

k

u

kjτ

2 2

1 098 0 247. , . ,

ωβ κ

k

y

1 2

0

/

*


1027 © 2004 ISIJ

Fig. 4. Velocity fields at the symmetry-transversal plane predict-ed by the k–e model using, (a) conventional ladle shroud,(b) swirling ladle shroud.

sults is clear that the SLS provides the capacity to decrease,very appreciable, fluid flow turbulence.

The next step for the presentation of these results is relat-ed with the capability of different turbulence models to pre-dict flow patterns using the SLS. Comparing Fig. 5(a) (k–emodel), Fig. 5(b) (k–w model) and 5(c) (RSM model), allfor the SLS shroud, is evident that there are appreciable dif-ferences among the results predicted by these turbulencemodels. The k–e model, as mentioned before, predicts re-circulating flows completely developed involving the fullliquid level in the tundish, on one side, in the other; theentry jet is straight until the impact point on the tundishbottom. Predictions made with the k–w model indicate thepresence of two recirculating flows at each side of the entryjet; however, each flow seems to be composed of smaller re-circulations close to the tundish bottom and close to thebath free surface. Moreover, the velocity vectors ascendingalong the tundish walls reach the top of the liquid level inthe present case while results predicted by the k–e modeldo not show ascending vectors until that position. In otherwords the shearing stresses effects of the entry jet on thesurrounding fluid are higher using the k–w model thanusing the k–e model. It is also noteworthy to observe athickening effect of the entry jet by velocity field spreadingjust below the SLS tip. This shows the capability of thek–w model to predict shear flow spreading rates fromround jets as in the present case.29) Now, using the RSMmodel velocity fields similar to those predicted by the k–wmodel are obtained. However, is clear that RSM predicts

not only a thickening of the entry jet but also its twisting,which is a consequence of the swirling flow effects yieldingnon-isotropy conditions of Reynolds stresses, promoted bythe SLS.

Jet swirling from the SLS shroud is actually detected ex-perimentally by PIV measurements as is shown by compar-ing Figs. 6(a) and 6(b) for the conventional and the SLSshroud, respectively. As seen in Fig. 6(a) the conventionalLS yields a straight jet that impacts the tundish bottomwhile, as shown in Fig. 6(b), the SLS yields a twisted-swirling jet. Since neither the k–e model nor the k–wmodel predict swirling flows under the present experimen-tal conditions we can anticipate that the RSM model of tur-bulence is actually more suitable for flow simulation ofwater in models and, eventually of liquid steel in tundishes,using the SLS as is seen by comparing Fig. 6(b) with Fig.5(c).

Since the fluid flow patterns in the entry box have a de-finitively strong influence on the rest of the fluid flowthroughout the fluid volume, in the tundish, is interesting tocontinue with the prediction performance of the three tur-bulence models. On this line, Figs. 7(a), 7(b) and 7(c) showthe velocity fields of the fluid at the entry-longitudinalplane predicted using the k–e , k–w and RSM models, re-spectively. In all cases high velocity gradients are observedbetween the entry jet and the surrounding flow. The firstmodel predicts “S” shaped fluid streams toward the outletsand the flow is non-symmetrical. In the right end-wall thereis a clear recirculating flow while in the left end-wall the


© 2004 ISIJ 1028

Fig. 5. Velocity fields at the symmetry-transversal plane predicted mathematically using different models of the turbu-lence, (a) k–e model, (b) k–w and (c) RSM model.

fluid goes directly to impact with that wall coming backlater toward the outlet. The fluid that surrounds the SLS isclearly dragged by the entry jet. Velocity field predictedwith the k–w model indicates also non-symmetrical charac-

teristics; the fluid surrounding the SLS, different to the pre-vious case, is unevenly sheared at both sides of the SLS:The recirculating flows in the outlet box of both end wallsof the tundish are also different. Most interesting is the flowpredicted by the RSM model which, as mentioned before,predicts adequately the swirling motion of the fluid. Allpredictions using this model indicate a flow close to besymmetrical, although not completely symmetrical, thefluid that surrounds the SLS is sheared following hemi-cir-cles at both sides of the SLS. After the entry box the flowobserves plug characteristics toward the outlet boxes. It isappropriated to underline here that the computed results in-dicate non-symmetrical flows at each side of either the LSor SLS shroud in vertical-transversal or vertical-longitudi-nal planes. Most of CFD researchers compute fluid flowsusing only a half or even a fourth of tundishes and coppermolds assuming complete symmetry. Erroneously they ne-glect the fluctuating nature of turbulence, which has proved,theoretically and experimentally37) that small changes, fromfluctuations, of flow variables may promote radical changesof fluid flow patterns in metallurgical vessels.

Thus as is seen here all three models yield very differentpredictions about the velocity fields of the fluid inside ofthis tundish, although, all of them completely agree in thefact that the SLS is able to decrease very considerably thefluid velocity from the entry jet that impacts the tundishbottom. Therefore, since in principle the RSM model yieldsapparently better agreement with the PIV measurements,the following results are focused to verify that hypothesis asis explained in the next lines.

Comparisons of velocity fields between experimental andmathematical simulations using the RSM model of turbu-lence at the vertical-symmetric planes of this tundish areshown in Figs. 8(a)–8(d) corresponding to the positions in-dicated in the central scheme. Figure 8(a) shows the experi-mentally determined velocity field of a plane at position 1and Fig. 8(b) shows the corresponding prediction by theRSM model. The existence of a recirculating “eye” markedwith number “1” is observed in the lower bottom corner ofFigure 8a and the predictions indicate the formation of this“eye” at approximately the same position as is seen in the


1029 © 2004 ISIJ

Fig. 6. Swirling motion of the entry jet determined trough PIVmeasurements; (a) conventional ladle shroud and (b) SLSdesign.

Fig. 7. Velocity fields at the symmetry-longitudinal plane predicted mathematically using different models of the turbu-lence. (a) k–e model, (b) k–w model and (c) RSM model.

point marked by number “1” in Fig. 8(b). Experimental de-terminations indicate the existence of velocity vectors descending from the upper left corner as is seen in pointmarked with number “2” in Fig. 8(a) which has its equiva-lent in the point marked with number “2” in Fig. 8(b). Inpoint marked with point “3”, Fig. 8(a), velocity vectors areoriented toward the upper right corner and the same can beseen in the RSM model predictions in Fig. 8(b), point “3”.Experimental results and mathematical predictions of ve-locity fields in the outlet box (plane 2 in the central scheme)are shown in Figs. 8(c) and 8(d), respectively. Again ob-serving the agreement between points marked with num-bers “1”, “2”, and “3”, regarding the velocity field, in thesetwo Figures we can affirm that indeed the RSM model ofturbulence is able to predict very well the experimental de-terminations.

In the horizontal planes it was observed, through the PIVmeasurements, that the fluid flow is still more complex dueto the horizontally swirling-rotating entry jet. Figures 9(a)and 9(b) show the experimental and mathematically simu-lated velocity fields at a horizontal plane respectively locat-ed at half the liquid level in the tundish, as is indicated inthe central scheme. Point marked with number “1” in Fig.9(a) (this view involves only about 2/3 of the tundish width)shows an “eye” of the recirculating flow promoted by theswirling entry jet which is also observed in the predicted re-sults with number “1” in Fig. 9(b) (this view involves thecomplete tundish width) located slightly closer to the frontwall of the tundish. Points marked with number “2” in Figs.9(a) and 9(b) agree since both show that the fluid flows up-

stream as a part of a large recirculating flow. Finally pointsmarked with number “3” in Figs. 9(a) and 9(b) agree clear-ly in the downstream orientation of the velocity vectors.Further comparisons between experimental determinationsand mathematical simulations are observed in Figs. 9(c)and 9(d) respectively for a horizontal plane located close tothe tundish bottom (outlet box) as is indicated in the centralscheme (plane 2). Points marked with number “1” in Figs.9(c) and 9(d) show the same downstream orientation of thevelocity vectors, besides, points marked with number “2” inboth Figures show the remarkable agreement about the ex-istence of a shearing flow between two streams of fluidmeeting in that point. Points marked with number “3” showthe same downstream flow oriented toward the half heightof the end wall. In summary, the mathematical predictionsare in remarkably good agreement with the experimentalmeasurements corroborating that the RSM turbulencemodel is recommendable to simulate swirling flows in atundish.

There are still various aspects that should be studied tohave a complete assessment about the performance of theSLS as a reliable fluid control device including tracer stud-ies, flotation of inclusions and metal-slag interactions.These aspects will be matter of future publications in thefield. However, so far this work has demonstrated the feasi-bility for decreasing turbulence, which is a first requirementof flow control. Moreover, there is still a theoretical matterleft in order to explain which, is the reason of why RSMmodel is more capable than k–e and k–w models to simu-late these types of flows. The explanation can be that, dif-


© 2004 ISIJ 1030

Fig. 8. Measured and mathematically predicted velocity fields using the RSM model at symmetry longitudinal planes;(a) PIV measurements at plane 1, (b) mathematical predictions at plane 1, (c) PIV measurement at plane 2 and (d)mathematical predictions at plane 2.

ferent to the eddy-viscosity models, the RSM model takesinto account transport and diffusion phenomena ofReynolds stresses accounting for the effects of streamlinecurvature, swirl, rotation and rapid changes of strain rate,these factors are not considered in the former models.

5. Conclusions

A new flow control device for steel flow in tundishes(SLS) has been designed using water modeling, PIV mea-surements and mathematical simulations using three modelof turbulence, and the main conclusions derived from thisstudy are as follows:

(1) The SLS decreases the impact velocity of the fluidon the tundish flow and consequently decreases the turbu-lence of the entering jet. Impact velocity using the SLS isabout 1/3 of that when a conventional ladle shroud is em-ployed.

(2) Fluid from the entering jet, using the SLS, pro-motes a swirling motion that is transmitted along thetundish and through the horizontal planes of the tundish.

(3) By comparing the mathematical simulations withthe PIV measurements it is found that the k–e and k–w tur-bulence models are unable to predict the complex flows ex-perimentally observed. However, the Reynolds StressModel predicts acceptably well the experimental determina-tions of velocity fields of the fluid.

Acknowledgements

The authors give the thanks to the institutions IPN,CoNaCyt, SNI and COFAA for their support to the Groupof Mathematical Simulation of Materials Processing andFluid Dynamics at IPN-ESIQIE.

Nomenclature

Dnozzle: Bore size of ladle shroudfb: A pseudo-constant which is a function of another

one named Xw in the k–w model, see Table 2fb*: A pseudo-constant, which is a function of Xk in

the k–w model, see Table 2k: Turbulent kinetic energyl: Eddy length

P: PressureQ: Flow rate of liquidui: Time averaged fluid velocity in direction “i”ui: Turbulent fluctuation of fluid velocirty in direction

“i”

Greek symbolsa : A constant in the k–w model equal to 13/25 b : A pseudo-constant given in Table 2

b0: A constant in the k–w model equal to 9/125b*: A pseudo-constant in the k–w as function of b0*

and the pseudo-constant fb* in the k–w modeld ij: Delta of KrocnekerXw: A pseudo-constant in the k–w model


1031 © 2004 ISIJ

Fig. 9. Measured and mathematically predicted velocity fields (using the RSM model) at symmetry-horizontal planes. (a)PIV measurements at plane 1; (b) mathematical predictions at plane 1 ; (c) PIV measurements at plane 2; (d)mathematical predictions at plane 2.

Xk: Gradient of specific dissipation rate as a functionof the scalar gradient of the kinetic turbulence en-ergy in the k–w model, see Table 2

e : Dissipation rate of kinetic energy W k: Angular velocityw : Specific dissipation rate of turbulent energyr : Density of fluids : A constant equal to 1/2 in the k–w model

s*: A constant equal to 1/2 in the k–w models k: A constant in the k–e model in the turbulent ki-

netic energy equation, see Table 1se: A constant in the dissipation rate of the kinetic

turbulent energy in the k–e model, see Table 1.m : Fluid viscosity

m eff: Effective viscosity of fluidv: Dynamic viscosity of fluid, m /r

REFERENCES

1) A. McLean, L. J. Heaslip and I. D. Sommerville: ContinuousCasting, Vol. 1, ISS Warrendale, PA, (1983), 67.

2) S. Tanaka, M. Lye, M. Salcudean and R. I. L. Guthrie: 24th Ann.Conf. of Metallurgists, CIM, Montreal, (1985), 142.

3) Y. Sahai and R. Ahuja: Ironmaking Steelmaking, 13 (1986), 241.4) C. S. Damle and Y. Sahai: ISIJ Int., 35, (1995), 163.5) J. J. Chen: Proc. Steelmaking Conf., Vol. 78, ISS. Warrendale, PA,

(1995), 593.6) Y. Sahai and T. Emi: ISIJ Int., 36 (1996), 1166.7) A. K. Sinha and Vassilicos: Ironmaking Steelmaking, 25 (1998), 387.8) L. J. Heaslip and J. Schade: Iron Steelmaker, 26 (1999), 33.9) J. Knoepe and J. Mastervich: Proc. Steelmaking Conf., Vol. 69, ISS,

Warrendale, PA, (1986), 777.10) M. L. Lowry and Y. Sahai: Proc. Steelmaking Conf., Vol. 72, ISS,

Warrendale, PA, (1989), 71.11) K. M. Godiwala, S. K. Sinha and C. S. Sivaramkrishnan: Proc.

Steelmaking Conf., Vol. 77, ISS, Warrendale, PA, (1994), 703.12) M. L. Lowry and Y. Sahai: Iron Steelmaker, 19 (1992), 81.13) S. Joo, J.W. Han and R. I. L Guthrie: Metall. Trans. B, 24B (1993),

779.14) D. Mazumdar, G. Yamanoglu and R. I. L. Guthrie: Steel Res., 68

(1997), 293.15) S. López-Ramírez, J. Palafox-Ramos, R. D. Morales, M. A. Barrón-

Meza and M. Velásquez-Toledo: Steel Res., 69 (1998), No. 10–11,423.

16) S. López-Ramírez, R. D. Morales and J. A. Romero-Serrano: Num.Heat Transfer, A, 37 (2000), 68.

17) R. D. Morales, J. Palafox-Ramos, S. López-Ramirez, M. A.Domínguez-Crespo, C. Rincón, D. Salazar and A. Dainton: Proc.Steelmaking Conf., Vol. 81, ISS, Warrendale, PA, (1998), 325.

18) A. Vargas-Zamora, R. D. Morales, J. Palafox-Ramos and L. Garcia-Demedices: Int. J. Heat Mass Transfer, 46 (2003), 3029.

19) Aguilar-Corona, R. D. Morales, M. Díaz-Cruz, J. Palafox-Ramos andH. Rodríguez-Hernández: Steel Res., 73 (2002), 433.

20) L. García-Demedices, R. D. Morales, S. López-Ramírez, J. de J.Barreto-Sandoval, J. Palafox-Ramos and M. Diaz-Cruz: Steel Res.,72 (2001), 346.

21) R. D. Morales, J. de J. Barreto-Sandoval, S. Lopez-Ramirez. J.Palafox-Ramos and D. Zacharias: Metall. Mater. Trans. B, 31B(2000), 1505.

22) J. Palafox-Ramos, J. de J. Barreto-Sandoval, S. López-Ramírez andR. D. Morales: Ironmaking Steelmaking, 28 (2001), 101.

23) R. Sánchez-Pérez, R. D. Morales, M. Díaz-Cruz, O. Olivares-Xometel and M. Díaz-Cruz: ISIJ Int., 43 (2003), 637.

24) M. Raffel, C. Willert and J. Kompenhans: Particle ImageVelocimetry, Springer-Verlag, Berlin, New York, (1998), 160.

25) H. J. Odenthal, I. Lemanowicz, R. Gorissen and H. Pfeiffer: Metall.Mater. Trans. B, 33B (2002), 163.

26) W. P. Jones and B. E. Launder: Int. J. Heat Mass Transfer, 15 (1972),301.

27) F. C. Lockwood and C. A. Romo-Millanes: J. Int. Energy, 65 (1992),144.

28) A. Ramos-Banderas, R. D. Morales, L. García-Demedices and M.Diaz-Cruz: ISIJ Int., 43 (2003), 653.

29) D. C. Wilcox: Turbulence Modeling for CFD, DCW Industries, LaCañada CA, (2000), 119.

30) B. E. Launder: Int. J. Heat Fluid Flow, 10 (1989), 282.31) B. J. Daly and F. H. Harlow: Phys. Fluids, 13 (1970), 2634.32) M. M. Gibson and B. E. Launder: J. Fluid Mech., 86 (1978), 491.33) B. E. Launder: Int. J. Numerical Methods, 9 (1989), 963.34) S. Sarkar and L. Balakrishnan: ICASE Report 90-18, NASA CR

182002, (1990).35) B. E. Launder and D. B. Spalding: Comp. Methods Appl. Mech.

Eng., 3 (1974), 269.36) T. J Chung: Computational Fluid Dynamics, Cambridge University

Press, New York, London, (2002), 106.37) S. B. Pope: Models of Turbulence, Cambridge University Press, New

York, London, (2000), 65.


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