Large-Eddy Simulation of Transient Horizontal Gas ...

Large-Eddy Simulation of Transient Horizontal Gas–Liquid Flow in Continuous Casting Using Dynamic Subgrid-Scale Model Zhongqiu Liu & Baokuan Li

Metallurgical and Materials Transactions B ISSN 1073-5615 Volume 48 Number 3 Metall and Materi Trans B (2017) 48:1833-1849 DOI 10.1007/s11663-017-0947-3

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Author's personal copy Large-Eddy Simulation of Transient Horizontal Gas– Liquid Flow in Continuous Casting Using Dynamic Subgrid-Scale Model ZHONGQIU LIU and BAOKUAN LI Euler–Euler simulations of transient horizontal gas–liquid flow in a continuous-casting mold are presented. The predictions were compared with previous experimental measurements by two-channel laser Doppler velocimeter. Simulations were performed to understand the sensitivity to different turbulence closure models [k–e, shear stress transport (SST), Reynolds stress model (RSM), and large-eddy simulation (LES)] and different interfacial forces (drag, lift, virtual mass, wall lubrication, and turbulent dispersion). It was found that the LES model showed better agreement than the other turbulence models in predicting the velocity components of the liquid phase. Furthermore, an appropriate drag force coefficient model, lift force coefficient model, and virtual mass force coefficient were chosen. Meanwhile, the wall lubrication force and turbulent dispersion force did not have much effect on the current gas–liquid two-phase system. This work highlights the importance of choosing an appropriate bubble size in accordance with experiment. Finally, coupled with the optimized interfacial force models and bubble size, LES with a dynamic subgrid model was used to calculate the transient two-phase turbulent flow inside the mold. More instantaneous details of the two-phase flow characteristics in the mold were captured by LES, including multiscale vortex structures, fluctuation characteristics, and the vorticity distribution. The LES model can also be used to describe the time-averaged gas–liquid flow field, giving reasonably good agreement with mean experimental data. Thus, LES can be used effectively to study transient two-phase flow inside molds. DOI: 10.1007/s11663-017-0947-3 The Minerals, Metals & Materials Society and ASM International 2017

I.

INTRODUCTION

GAS injection techniques have been widely used in many industrial processes, e.g., steelmaking reactors (ladle, tundish, and mold) in metallurgical engineering,[1–4] nuclear reactors in nuclear engineering,[5–8] and bioreactors in chemical engineering.[9,10] In the continuous casting process, argon gas is usually employed to prevent nozzle clogging, encourage mixing, and promote flotation of nonmetallic inclusions from the molten steel by changing the flow field. Argon gas is injected into the mold through a submerged entry nozzle (SEN). Due to intense shear forces exerted by molten steel, the argon gas disintegrates into a swarm of bubbles with different diameters. Large bubbles rise toward the top surface due to buoyancy and are subsequently removed from the mold, while smaller bubbles are carried deep into the mold cavity region, becoming trapped by the solidifying shell to form blister defects. A schematic of the multiphase flow inside the mold is shown in Figure 1. Therefore, thorough understanding of the gas–liquid ZHONGQIU LIU and BAOKUAN LI are with the School of Metallurgy, Northeastern University, No. 3-11, Wenhua Road, Heping District, Shenyang, 110819, China. Contact e-mail: [email protected] Manuscript submitted November 16, 2016. Article published online March 8, 2017. METALLURGICAL AND MATERIALS TRANSACTIONS B

flow characteristics is essential to improve conventional processes and develop novel agitation methods. To develop design tools for engineering purposes, much work has been carried out in the area of computational fluid dynamics (CFD) modeling of gas–liquid flows. Two approaches are mostly used to simulate the gas–liquid flow in the mold: the Euler– Lagrange (E–L) and Euler–Euler (E–E) approach. The Lagrangian approach provides a direct physical description of the particle–fluid interaction, but it is computationally intensive and hence cannot be used for simulating systems with high dispersed-phase volume fraction, as is the case in the current study. Despite this, the E–L approach has been widely used to study transport of dispersed bubbles and inclusions in the mold. Thomas et al.[11] considered normal and tangential force balances involving ten different forces acting on a particle in the boundary layer, considering the primary dendrite arm spacing (PDAS) to study entrapment of bubbles and inclusions. Lee et al.[12] studied the effect of the thermal Marangoni force on the behavior of argon bubbles at the solidifying interface, revealing that it could play an important role in entrapment of argon bubbles. Miki et al.[13] used a dispersed-phase model (DPM) to study the mechanism of bubble entrapment and their influence on the steel flow. Liu et al.[14,15] studied transient asymmetric flow and bubble transport and entrapment inside a vertical bending caster using VOLUME 48B, JUNE 2017—1833

Author's personal copy

Fig. 1—Schematic of gas–liquid flow in continuous-casting mold.

large-eddy simulation (LES); the results showed that it is difficult for small bubbles to float upward to the top surface once they have moved downward to the curved section of the mold. In the E–E approach, dispersed bubbles are not tracked individually, but the dynamics of the dispersed phase is ensemble averaged to obtain a set of Eulerian equations, which are similar to the equations for the continuous phase. The advantage of the E–E approach is that the computational demands are much lower compared with the E–L approach, especially for systems with higher gas volume fraction. Thomas et al.[2] used an E–E model to calculate the volume fraction and velocities of argon bubbles, and their effect on the molten steel flow. Bai et al.[16] investigated the turbulent flow of molten steel and argon bubbles in the mold using the E–E approach in the CFD program CFX, solving one velocity field for the molten steel and a separate velocity field for the gas phase. The momentum equation for each phase is affected by the other phase through only the interfacial drag term. Liu et al.[17,18] studied the argon–steel two-phase flow inside a mold using a constant bubble size based on the E–E approach, incorporating the drag, lift, and virtual mass forces. Recently, Liu et al.[4] studied the polydisperse bubbly flow inside a mold using the MUltiple SIze Group (MUSIG) model, incorporating various interfacial forces including drag, lift, virtual mass, wall lubrication, and turbulent dispersion. The interaction between the dispersed argon bubbles and continuous molten steel affects the turbulence and interfacial forces (e.g., drag force, lift force, added mass force, wall lubrication force, and turbulent dispersion force) in the mold. Therefore, correct modeling of turbulence and interfacial forces is of prime importance 1834—VOLUME 48B, JUNE 2017

to capture the physics behind the phenomena occurring in the mold correctly. Researchers have tested different turbulence closures [e.g. k–e model, shear stress transport (SST), Reynolds stress model (RSM), and LES], amongst which the standard k–e model is the most widely adopted owing to its simplicity and lower computational requirements. Recently, plant observations[13–15] showed that defects in the slab are intermittent and asymmetric, suggesting that they are related to transient flow. However, most standard k–e simulations reported in literature provide time-averaged values of the turbulent flow but do not provide information about the transient flow and velocity fluctuations that are characteristic of turbulent flow phenomena. Recently, the LES approach was identified as a better way to model turbulence. It has been successfully applied to obtain the asymmetrical single-phase flow of molten steel in the mold, e.g., by Ramos-Banderas et al.,[19] Yuan et al.,[20] and Liu et al.[14,15] However, except for the study by Liu et al.,[17] relatively little work has been reported on LES modeling of multiphase flow in molds. Another critical issue in CFD modeling of gas–liquid flow in molds is proper description of the interfacial forces between the dispersed bubbles and continuous fluid. Several models for different interfacial forces have been reported in literature, a detailed account of which is provided by Tabib et al. using a bubble column[8] and recently by Liu et al. using a mold.[21] In the study by Liu et al.,[21] a sensitivity analysis of different turbulence models and interfacial force closures was carried out and various recommendations made for appropriate selection. The present work presents a sensitivity analysis of different turbulence models and interfacial force closures within the E–E framework of CFD modeling. The sensitivity to different turbulence models (standard k–e, SST, RSM, and LES) and different interfacial forces (drag, lift, added mass, wall lubrication, and turbulent dispersion force) were investigated using the experimental data of Iguchi et al., obtained from a model experiment of horizontal gas–liquid flow in a mold. The relative merits of the various turbulence models and force formulations are brought out, and some recommendations made for appropriate selection. The optimized numerical model is then used to calculate the transient two-phase turbulent flow inside the mold.

II.

HORIZONTAL GAS–LIQUID FLOW EXPERIMENT

The horizontal gas–liquid flow experiment facility was described in detail by Iguchi et al.[1] The rectangular vessel (mold) has length of 30 cm, width of 15 cm, and depth of 50 cm (with water for 40 cm). Water and air are supplied to the mold through a pipe (nozzle) using a pump and compressor, respectively. The origin of the Cartesian coordinate system (x, y, z) is located at the center of the outlet of the nozzle. The diameter of the nozzle is 0.9 cm. Water is circulated in a circuit through METALLURGICAL AND MATERIALS TRANSACTIONS B

Author's personal copy Table I. Comparison of Process Parameters Between Water Model and Actual Continuous-Casting Mold[1]

Liquid Liquid temperature Liquid density Liquid flow rate Gas Gas temperature Gas density Gas flow rate Narrow wall of the mold Wide wall of the mold SEN port size

Water Model

Actual Mold

Water 293 K (20 C) 1000 kg/m3 2.5 to 7.5 kg/min Air 293 K (20 C) 1.2 kg/m3 4 to 24 cm3/s 0.125 m 0.3 m U0.009 m

Molten steel 1833–1853 K (1560–1580 C) 7000 to 7200 kg/m3 3000 to 5000 kg/min Argon 1853 K (1580 C) 1.6 to 0.25 kg/m3 10 to 70 L/min 0.23 to 0.27 m 0.85 to 2.3 m 0.07 9 0.09 m2

a buffer tank to the mold. The water flow rate and air flow rate are controlled by different flow meters. Due to the Coanda effect, there is no attachment of the air–water two-phase jet to the wide faces for the current experimental conditions. Three dimensionless parameters for single-phase water flow (Re, Fr, and We) and one dimensionless parameter for water–air two-phase flow (Frm) were used to obtain the water flow rate and air flow rate with reference to the actual operating conditions, as presented in Table I.[1]

III.

MATHEMATICAL MODEL FORMATION

A. Euler–Euler Two-Fluid Model In the Euler–Euler two-fluid model, separate equations are required for each phase. The mass and momentum equations for phase k are generally given as follows: @ ð ak q k Þ þ r ðak qk uk Þ ¼ 0; @t

½5

uin din ; tl

½1

uin Fr ¼ pffiffiffiffiffiffiffiffi ; gdin

@ ðak qk uk Þ þ r ðak qk uk uk Þ ¼ r ðak sk Þ ak rP @t ½6 þ ak qk g þ Fk :

½2

The terms on the right-hand side of Eq. [6] represent, respectively, the stress, pressure gradient, gravity, and interfacial forces. The velocities in Eqs. [1] and [2] are defined as follows:

½3

uk ¼ u~k u0k ;

Re ¼

We ¼

ql din u2in ; r

uin Frm ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðgQg Þ=ðdin uin Þ

½4

where Re is the Reynolds number, Fr is the Froude number, We is the Weber number, and Frm is the modified Froude number. uin is the liquid flow velocity at the nozzle inlet, and din is the inner diameter of the nozzle. Once the water and air flow reached steady state, photographs of the water–air two-phase jets were taken using a camera. The velocity components of the water phase in the x, y, and z directions are u, v, and w, respectively. The mean velocities of the u and v components along the centerline of the z = 0 plane were measured by two-channel laser Doppler velocimetry. The uncertainty in these measurements was less than 3 pct. The farthest axial position from the origin, x = 21 cm, was chosen because the flow field upstream of this position was hardly affected by the flow issuing out of the nozzle placed at the opposite wall. More details can be found in the report by Iguchi et al.[1]

METALLURGICAL AND MATERIALS TRANSACTIONS B

½7

where uk is the part of the velocity for phase k that will be resolved in the numerical simulations, u~k is the instantaneous velocity, and u0k is the part unresolved by the numerical simulation. Most models derive Eqs. [1] and [2] through ensemble averaging, where u~k and u0k represent the mean and fluctuating velocity, respectively. When Eqs. [1] and [2] are obtained through a filtering operation through a box-type filter function,[15] these terms are, respectively, the grid-scale and subgrid-scale (SGS) velocities. In LES, large eddies are resolved directly, while small eddies are modeled. The stress term for phase k is described as follows: T 2 ½8 sk ¼ leff;k ruk þ ðruk Þ Iðr uk Þ : 3 The effective viscosity of the liquid phase, leff,l, is composed of three contributions: the molecular viscosity, the turbulent viscosity, and an extra term due to bubble-induced turbulence: leff;l ¼ lL;l þ lT;l þ lBI;l :

½9

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Author's personal copy The effective viscosity of the gas phase is based on the effective liquid viscosity as proposed by Jakobsen et al.[22]: qg leff;g ¼ leff;l : ½10 ql The model proposed by Sato et al.[23] was used to take account of turbulence induced by bubble movement, using the expression ½11 lBI;l ¼ ql Cl;BI ag dg ug ul with a model constant Cl,BI, which is equal to 0.6. B. Turbulence Closure Models 1. Standard k–e model When the standard k–e model is used, the turbulent viscosity (lT,l) is calculated as k2 : e

lT;l @ql al x þ r ðal ql ul xÞ ¼ r al ll þ rx @t rx;SST 1 @k @x x þ al c Pk ql bSST : 2ql al ð1 F2 Þ rx;SST x @xj @xj k ½18 The blending functions F1 and F2 are used to govern the crossover point between the k–e and k–x model, being given by ! pffiffiffi 500ll k 2 ; ½19 ; F1 ¼ tanhðU1 Þ; U1 ¼ max 0:09xdb ql xd2b F2 ¼ tanh U42 ; "

! # pffiffiffi 500ll 4ql k k U2 ¼ min max ; ; ; 0:09xdb ql xd2b dx þ rx;SST d2b

½12

½20

The turbulent kinetic energy k and turbulent dissipation e are calculated as lT;l @ðql al kÞ þ r ðal ql ul kÞ ¼ r al rk þ al ðG ql eÞ; @t rk

where rk,SST, rx,SST, c, and bSST are the model constants. More detailed descriptions of these model constants can be found in Menter.[24]

lT;l ¼ Cl ql

½13 lT;l @ðql al eÞ þ r ðal ql ul eÞ ¼ r al re @t re e e2 þ al Ce1 G Ce2 ql ; k k

3. Reynolds stress model (RSM) When the RSM is used to describe anisotropic turbulence, the stress term of phase k is described as h i

sk ¼ leff;k ruk þ ðruk ÞT þ r ql u0i u0j : ½21

½14

with the standard model constants Cl = 0.09, rk = 1.00, re = 1.30, Ce1 = 1.44, and Ce2 = 1.92. The term G represents the production of turbulent kinetic energy, described by G ¼ sl : rul :

½15

2. Shear stress transport (SST) model The SST model is a hybrid version of the k–e and k–x models with a specific blending function. The shear-induced turbulent viscosity lT,l is given by lT;l ¼

qal k ; maxðal x; SF1 Þ

S¼

pffiffiffiffiffiffiffiffiffiffiffiffiffi 2Sij Sij :

½16

The ensemble-averaged transport equations of the SST model are given as lT;l @ql al k þ r ðal ql ul kÞ ¼ r al ll þ rk @t rk;SST þ al Pk ql bSST kx; ½17

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Fig. 2—Grid and boundary conditions used in simulations.


Author's personal copy term Pij, the pressure strain term /ij, and the dissipation term wij, respectively defined as i @ h 0 0 0 DT;ij ¼ ql ui uj uk þ pðdkj u0i þ dik u0j Þ ; ½23 @xk @uj @ui Pij ¼ ql u0i u0k u0j u0k ; @xk @xk

½24

@u0i @u0j þ /ij ¼ p ; @xj @xi

½25

@u0i @u0j ; @xk @xk

½26

wij ¼ 2ll

where d is the Kronecker factor. 4. Large-eddy simulation (LES) model In the LES model, the key element is the subgrid-scale model, which determines the effect of unresolved turbulent scales. In the current work, the model proposed by Smagorinsky[25] is used to calculate the turbulent viscosity lT,l: lT;l ¼ ql ðCS DÞ2 S~; ½27 where CS is the Smagorinsky constant. S~ is the characteristic filtered rate of strain, and D = (DiDjDk)1/3 is the filter width. A CS value of around 0.1 has been found to yield the best results for a wide range of flows.[8,9,17,21,26] However, CS is not a universal constant, which is the most serious shortcoming of this simple model. In view of the uncertainty in specifying the constant CS, Germano et al.[27] proposed a dynamic subgrid model in which CS is not chosen arbitrarily but rather computed as 1 Lij Mij ; 2 Mij Mij

½28

~i u~j u~bi u~bj ; Lij ¼ ud

½29

Fig. 3—Comparison between simulated and experimental profiles with different turbulence models for different gas flow rates (a) Qg = 0 cm3/s, (b) Qg = 4 cm3/s, and (c) Qg = 12 cm3/s.

b~S c d ^ 2 q S Mij ¼ 2D ~ij S~S~ij :

½30

The individual Reynolds stresses u0i u0j are computed via an extra transport equation:

The concept of the dynamic procedure is to apply a second filter (called the test filter) to the equations of ^ is equal to twice the motion. The new filter width D grid filter width D. Both filters produce a resolved flow field. The difference between the two resolved fields is the contribution of small scales with size in between the grid filter and test filter. This is a standard procedure, and details can be found elsewhere.[27,28]

CS ¼

@ðql al u0i u0j Þ þ r al ql ul u0i u0j ¼ al DT;ij þ Pij þ /ij þ wij : @t ½22 The four terms on the right-hand side are the turbulent diffusion term DT,ij, the stress production METALLURGICAL AND MATERIALS TRANSACTIONS B


Author's personal copy Table II.

Drag Force Coefficient Models

Authors Ishii and Zuber

Model [32]

CD ¼

[33]

Schiller and Naumann

Zhang and Vanderheyden[30]

24 Reb

0:15Re0:687 b

Correlation q jul ug jdb Reb ¼ l l

1þ , 8 >

b < 24=Re 24 ; 0:44 1 þ 0:15Re0:687 CD ¼ max Re b b > : 0:44

l

if Reb 1 if 1 Reb 1000 if 1000 Reb 2 105

24 ffi CD ¼ 0:44 þ Re þ 1þp6ffiffiffiffiffi Re b b

3 C FD ¼ ag ql D ug ul ðug ul Þ; 4 db

½32

where CD is the drag force coefficient. 2. Lift force Due to the horizontal velocity gradient, bubbles rising in a liquid are subjected to a lateral lift force. Following the formulation of Drew and Lahey,[29] this force can be correlated to the relative velocity and the local liquid velocity as ½33 FL ¼ ag ql Cl ug ul r ul ; where CL is the lift force coefficient. 3. Virtual mass force The virtual mass force accounts for relative acceleration, i.e., the additional work performed by bubbles in accelerating the surrounding liquid. The acceleration of the liquid is taken into account through the virtual mass force, given by Dug Dul FVM ¼ ag ql CVM ; ½34 Dt Dt where CVM is the virtual mass force coefficient.

Fig. 4—Comparison between simulated and experimental profiles when using different drag force coefficient models for different gas flow rates (a) Qg = 4 cm3/s and (b) Qg = 12 cm3/s.

C. Closure models for interfacial forces The total interfacial forces can be categorized into five main terms: drag force, lift force, virtual mass force, wall lubrication force, and turbulent dispersion force: Fk ¼ Flg ¼ Fgl ¼ FD þ FL þ FVM þ FWL þ FTD ; ½31 where Flg denotes the momentum transfer from gas to liquid phase, and vice versa for Fgl. 1. Drag force The interphase momentum transfer between gas and liquid by drag forces is given by 1838—VOLUME 48B, JUNE 2017

4. Wall lubrication force Under certain circumstances, the dispersed phase is observed to concentrate in a region close to, but not immediately adjacent to, the wall. This effect may be modeled by the wall lubrication force, which tends to push the dispersed phase away from the wall, which can be modeled as 2 FWL ¼ CWL ag ql ul ug nw ;

½35

where CWL is the wall lubrication force coefficient, and ~ nw is the outward vector normal to the wall. 5. Turbulent dispersion force The turbulent dispersion force results in additional dispersion of phases from regions of high to low volume fraction due to turbulent fluctuations. The turbulent dispersion force can be calculated as FTD ¼ CTD CD

mt;g ral rag ; rt;g al ag

½36


Author's personal copy where CTD, mt,g, and rt,g are the turbulent dispersion coefficient, turbulent kinematic viscosity for the gas phase, and the turbulent Schmidt number of the gas phase, respectively. By default, the turbulent Schmidt number rt,g = 0.9 is adopted.

Fig. 5—Comparison between simulated and experimental profiles with different lift force coefficient models for different gas flow rates (a) Qg = 4 cm3/s and (b) Qg = 12 cm3/s.

D. Numerical Details In this work, the commercial CFD package CFX-14.5 combined with the CFX Command Language (CCL) was used to simulate the horizontal gas–liquid flow in the continuous-casting mold. To compare the simulation results with the water model experimental measurements, the geometry, material properties, and boundary conditions were set to correspond to the water model of Iguchi et al.[1] At the inlet, a mass flow rate boundary condition was used, and the volume ratio of gas to liquid was obtained from the respective mass flow rate of the water experiment. As the diameter of the injected bubbles is unknown, a monodisperse bubble size of 1 mm was specified; the sensitivity to bubble size is discussed in the next section. At the outlet, a mass flow rate boundary condition was used. The top surface of the mold cavity is modeled using a degassing boundary condition, where dispersed bubbles are permitted to escape, but liquid is not. Along the walls, wall functions were applied for the k–e model and SST model, while no-slip boundary conditions were adopted for the RSM model and LES model. Considering the requirement of the Euler–Euler approach and the limitation of D/db ‡ 1.2,[7] and in order to capture enough large-eddy structure characteristics to analyze the two-phase dynamic state distribution in the mold, the domain was divided into various finite volumes from 8 9 104 to 8 9 105 for db = 5 mm to db = 0.5 mm. Boundary conditions and a typical grid layout are shown in Figure 2. In all the simulations, a bounded third-order accurate QUICK scheme was used for discretization of the convective terms. Converged solution was considered to be reached when the residuals of all variables were less than 104. All simulations were transient with flow simulated for 100 seconds and data monitored during the calculation. Also, data were time averaged over the last 50 seconds. However, the LES approach must be run for a sufficiently long time to obtain a stable statistical flow field. To save calculation time, this calculation was first carried out using the standard k–e turbulence model to obtain a steady flow

Table III. Lift Force Coefficient Models Model Tomiyama

[31]

Saffman and Mei[34,35]

Interfacial Force Coefficient Correlation 8 b jdb if Eo > < = y 1C Wd 1 ¼ CW ðEoÞ max 0; CWD WC b P1 > > y : ; yW C Wd WC b


Correlation CW1 ¼ 0:01; CW2 ¼ 0:05 yW £ (CW2/CW1)Ædb CWC = 10 CWD = 6.8 P = 1.7


Author's personal copy Table V. Turbulent Dispersion Force Coefficient Models Author(s)

Model [41]

Lopez de Bertodano

FTD lg

Burns et al.[42] (Favre-averaged drag model)

FTD lg

= CTDqlkal

m rag ral ¼ CTD CD rt;g al ag t;g

Correlation CD = 0.2 CD = 1.0, rt,g = 0.9

better agreement with experimental data. For all the models, note that the peak value of the v profile was overpredicted near the nozzle inlet. This could be due to the fact that buoyancy of air bubbles was not predicted accurately, because constant bubble size was considered in all the simulations. The effect of air bubble size on the gas–liquid flow is discussed in the next section. Therefore, the drag force coefficient model of Zhang and Vanderheyden,[30] which gave better predictions for both u and v, was used in further simulations.

Fig. 9—Comparison between simulated and experimental profiles with different bubble sizes for different gas flow rates (a) Qg = 4 cm3/s and (b) Qg = 12 cm3/s.

was used to calculate the lift force, the virtual mass force was considered with CVM = 0.5, and neither the wall lubrication force nor turbulent dispersion force model was considered. Figure 4 shows the effect of using these different drag force coefficient models on the axial and vertical mean velocity components ( u and v) along the x axis, comparing the predicted results with the measured data from the water model experiments. It can be seen that all the drag force coefficient models could predict the u profile at low gas flow rate (Qg = 4 cm3/s) quite well. However, none of the models could capture the peak value of the v profile measured in experiments. Anyhow, the predictions of all three models were acceptable at 5 to 25 cm away from the nozzle inlet. For the higher gas flow rate (Qg = 12 cm3/s), the drag force coefficient model of Zhang and Vanderheyden[30] was found to give 1842—VOLUME 48B, JUNE 2017

2. Effect of lift force The lift force is proportional to the continuous phase density. Hence, it is mainly significant when the dispersed phase density is either less than, or of the same order of magnitude as, the continuous phase density. Also, it is proportional to the continuous phase shear rate. Hence, it is most significant in shear layers whose width is comparable to the dispersed phase mean diameter. To study the effect of the lift force, the lift force coefficient models reported by Tomiyama et al.,[31] Saffman[34] and Mei et al.,[35] and Legendre and Magnaudet,[36] and CL = 0.5 were chosen, as shown in Table II. Meanwhile, the Zhang and Vanderheyden drag force coefficient model[30] was used to calculate the drag force, the virtual mass force was considered with CVM = 0.5, and neither the wall lubrication force nor turbulent dispersion force model was considered. Figure 5 shows a comparison of the predicted axial and vertical mean velocity components ( u and v) along the x axis versus experimental data. From all the figures, it can be observed that, when the lift force is included, gas bubbles gradually move out towards the narrow face of the mold and become larger. According to these results, when the lift force is not considered, there is a large error between the simulation predictions and measurement data for the water model, especially for high gas flow rate. Therefore, the lift force is critical for successful prediction of the two-phase flow in the mold and should not be ignored. For all the lift force coefficient models, it is seen that the deviation in the results at higher gas flow rate is significant compared with that observed at low gas flow rate. All the results show that, when the Legendre and Magnaudet model[36] was used, the predicted mean velocity components agreed well with the experimental data, so this model of CL was used in the current work (Table III). 3. Effect of virtual mass force The virtual mass force is proportional to the continuous phase density, hence being significant when the dispersed phase density is less than the continuous METALLURGICAL AND MATERIALS TRANSACTIONS B


Fig. 10—Predicted air volume fraction inside the mold: averaged using the SST model (a) and instantaneous using the LES model (b).

phase density. Also, by its nature, it is only significant in the presence of large accelerations. The virtual mass force was neglected in many previous simulations in accordance with the observations made by Thakre and Joshi,[37] Sokolichin and Eigenberger,[38] and Tabib et al.[8] Figure 6 shows the effect of the virtual mass force on the axial and vertical mean velocity components ( u and v) along the x axis in the mold, comparing the numerical predictions of the current model with different virtual mass force coefficients from CVM = 0.1 to CVM = 1.0 with the experimental data. Meanwhile, the Zhang and Vanderheyden drag force coefficient model[30] was used to calculate the drag force, the Legendre and Magnaudet lift force coefficient model[36] was considered, and neither the wall lubrication force nor turbulent dispersion force model was considered. From the results for low gas flow rate (Qg = 4 cm3/s), note that the predictions of u and v for all the models are almost equal. For the higher gas flow rate (Qg = 12 cm3/s), significant deviation is seen in the results for the different CVM values at higher gas flow rate compared with that observed at low gas flow rate. However, this deviation is small compared with the condition of no virtual mass force, so the virtual mass force can be neglected in the current two-phase flow conditions. However, in previous work by the authors,[21] the virtual mass force had a significant influence on the bubble Sauter mean diameter and could not be ignored when modeling a polydisperse bubbly flow in a continuous-casting mold. In the current work, when the virtual mass force coefficient CVM = 0.5 was considered, the predicted axial and METALLURGICAL AND MATERIALS TRANSACTIONS B

vertical mean velocity components ( u and v) along the x axis agreed well with the measured data. So, the value of the virtual mass force coefficient was chosen constant at 0.5 in this work. 4. Effect of wall lubrication force The wall lubrication force is usually modeled in conjunction with the wall lift force. In situations where the lift force pushes bubbles towards the narrow wall, the wall lubrication force acts in the opposite direction to ensure that bubbles accumulate a short distance away from the narrow wall. However, the wall lubrication force was neglected in all previous simulations in the mold. Figure 7 shows the effect of the wall lubrication force on the axial and vertical mean velocity components ( u and v) along the x axis in the mold, comparing the predictions using two wall lubrication force coefficient models proposed by Antal et al.[39] and Frank et al.[40] with the experimental data; the details of these wall lubrication force coefficient models are presented in Table IV. Meanwhile, the Zhang and Vanderheyden drag force coefficient model[30] was used to calculate the drag force, the Legendre and Magnaudet lift force coefficient model[36] was considered, the virtual mass force was considered with CVM = 0.5, and the turbulent dispersion force model was not included. From these results, in accordance with previous studies, it can be seen that the wall lubrication force had no clear effect on the results. Clearly, the wall lubrication force has a small influence on the current two-phase flow system in the mold. Therefore, the wall lubrication force was neglected in the current model (Table IV).


Author's personal copy drag force. The effect is to move bubbles from areas of high to low concentration. Hence, this effect will usually be important in turbulent flow with significant interfacial drag force. In the current work, simulations were carried out with CTD = 0.2 for the turbulent dispersion force model proposed by Lopez de Bertodano[41] and CTD = 1.0 for the Favre-averaged drag force model reported by Burns et al.[42] Meanwhile, the Zhang and Vanderheyden drag force coefficient model[30] was used to calculate the drag force, the Legendre and Magnaudet lift force coefficient model[36] was considered, and the virtual mass force was considered with CVM = 0.5, while the wall lubrication force model was not included. Figure 8 presents a comparison of the predicted results when using these different turbulent dispersion force coefficient models versus the measured data for different gas flow rates. It can be seen that the turbulent dispersion force models had no clear effect on the predictions of the axial and vertical mean velocity components ( u and v) along the x axis in the mold. Clearly, the turbulent dispersion force has a small influence on the current two-phase flow system in the mold. Therefore, the turbulent dispersion force was also neglected in the present model (Table V). C. Effect of Bubble Size

Fig. 11—Predicted water velocity vector inside the mold: averaged using the SST model (a) and instantaneous using the LES model (b).

Fig. 12—Predicted air volume fraction distribution along the x axis of the mold.

5. Effect of turbulent dispersion force The turbulent dispersion force results in additional dispersion of phases from regions of high to low volume fraction due to turbulent fluctuations. This is caused by the combined action of turbulent eddies and interfacial 1844—VOLUME 48B, JUNE 2017

The turbulent flow of molten steel inside the mold depends greatly on the amount and size of injected argon bubbles. Accurate determination of the mean bubble diameter is crucial, as the bubble size influences the momentum transfer through the interfacial area concentrations and interfacial forces; For example, bubbles of different sizes are subject to lateral migration due to forces acting in the lateral direction, i.e., the lift force, which is different from the main drag force direction. Furthermore, the bubble lift force is found to change sign as the bubble size varies, as studied by Tabib et al.[8] Figure 9 shows the effect of bubble size on the axial and vertical mean velocity components ( u and v) along the x axis in the mold based on the LES turbulence model. For all cases, the Zhang and Vanderheyden drag force coefficient model[30] was used to calculate the drag force, the Legendre and Magnaudet lift force coefficient model[36] was considered, and the virtual mass force was considered with CVM = 0.5, while neither the wall lubrication force model nor turbulent dispersion force model was included. The bubble sizes considered are 0.5, 1, 2, 4, and 5 mm. From these results, it can be seen that, when the bubble size is smaller than 2 mm, there is large error between the simulation predictions and measurement data for the water model, especially for high gas flow rate. Bubble size of 4 and 5 mm was found to give axial and vertical mean velocity components close to the experimental values for the different gas flow rates. By analyzing photographs of the water–air two-phase jet in the water model experiment, Iguchi et al.[1] found that the maximum bubble diameter was about 5 mm, which is suitable for the current model. However, owing to the fundamental importance of the bubble size to the gas–liquid flow, the predictions for the bubble size METALLURGICAL AND MATERIALS TRANSACTIONS B


Fig. 13—Time history plots of axial water velocity at three different points along the x axis of the mold (a) x = 2 cm, (b) x = 7 cm, and (c) x = 20 cm.

distribution become very important for understanding the hydrodynamics of the casting mold, which requires further work. D. Instantaneous Horizontal Gas–Liquid Flow Characteristics The flow structures present within various reactors are instantaneous phenomena, which cannot be captured when averaging procedures are used to quantify flow properties. In this context, LES can be an effective tool to study instantaneous flow profiles, but to date, few attempts have been made to understand the two-phase flow structures inside the mold. Therefore, it was considered desirable to make a systematic attempt in this work. Coupled with the optimized interfacial force models and bubble size (5 mm), the LES was used to calculate the transient two-phase turbulent flow inside the mold. The instantaneous (LES model) and averaged (SST model) flow profiles (air volume fraction and water velocity vector) at the center plane of the mold are shown in Figures 10 and 11. It can be seen that the LES could capture more details of the two-phase flow structures, including multiscale vortex structures and fluctuation characteristics. Figure 12 shows the predictions for the METALLURGICAL AND MATERIALS TRANSACTIONS B

air volume fraction along the x axis of the mold when using the different turbulence models. From the averaged air volume fraction predicted using the SST model, it can be seen that higher gas hold-up regions are observed near the nozzle inlet, indicating that the impingement depth of the air bubbles is steady and small. However, from the photographs of the water–air two-phase jet obtained from the water model experiment of Iguchi et al.,[1] it can be seen that the impingement depth of the air bubbles showed pulsatile change and was closer to the opposite narrow wall of the mold. This phenomenon can be captured by the LES model, as shown in Figures 10 and 12. Three time history plots of the axial water velocity (u) at three different points (x = 2, 7, and 20 cm away from the nozzle inlet, respectively) along the x axis in the mold are shown in Figure 13(a) through (c) between 30 and 60 seconds. All data were sampled every 0.001 seconds from the simulation results. From these figures, it can be seen that the LES captures the transient velocity fluctuation behavior observed in previous experiments by Iguchi et al.,[1] Miki and Takeuchi,[13] Yuan et al.,[20] Thomas et al.,[43] and Timmel et al.[44] Note that the current model captures the experimental observations and measured mean velocities reasonably well. The value of u decreases monotonically from x = 2 to 20 cm along the x axis. This plot clearly indicates that strong high-frequency fluctuations exist in the two-phase jet. The fluctuation frequency and amplitude are low at x = 2 cm, located close to the nozzle inlet. The reason may be that the air volume fraction in this region is steady, as shown in Figures 10 and 12. According to the results obtained at the monitoring point at x = 7 cm (Figure 13(b)), the fluctuation frequency and amplitude are much larger, with maximum amplitude of u of about 130 cm/s. Compared with the results in Figures 10 and 12, it is seen that the transient value and fluctuation of the air volume fraction at x = 7 cm are larger than at x = 2 cm, indicating that the fluctuation behavior of the water velocity in this region is mainly induced by the changing air volume fraction. Therefore, the air volume fraction has an important impact on the momentum transfer between the air bubbles and water. The results obtained at the monitoring point at x = 20 cm (Figure 13(c)) show that the fluctuation amplitude is still large, but the air volume fraction is almost equal to zero, as shown in Figure 12. The reason may be that the inertial force of the water jet has not been dissipated, which could induce continuous oscillation of the water jet. It is well known that swirling jet flow occurs in continuous-casting molds, which can trigger instabilities of the molten steel flow. Therefore, it is necessary to analyze the vortex distribution inside the mold. The vorticity represents the orientation and angular velocity of the local rotation and is therefore often used as a criterion for the existence of vortices, and also to describe their creation, transformation, and extinction. For better understanding of the two-phase turbulent jet flow inside the mold, it is of interest to examine the vorticity distribution and its role in vortex phenomena. Vorticity is a measure of the rotation of fluid elements VOLUME 48B, JUNE 2017—1845


Fig. 14—Instantaneous isosurfaces of various vorticity magnitudes inside the mold at 100 seconds (a) 50/s, (b) 40/s, (c) 30/s, (d) 20/s, (e) 10/s, and (f) 5/s.

(velocity swirling strength) in the flow field, namely the curl of the velocity field, being twice the rotating speed of the fluid: X ¼ r~ u ¼ 2x0 ;

½37

where X is the vorticity, ~ u is the velocity vector of the liquid, and x¢ is its angular speed of rotation. Figure 14 shows isosurfaces for various vorticity magnitudes from 50 to 5/s inside the mold at 100 seconds. Vorticity is present in any vortex and often provides a simpler representation because of the rotational nature of vortices. Areas with high vorticity magnitude are located at the core of the two-phase jet. The maximum value of vorticity is located closer to the opposite narrow wall of the mold, as shown in Figure 14(a). The reason may be due to the 1846—VOLUME 48B, JUNE 2017

impingement of the turbulent jet flow on the narrow wall. Due to the effect of bubble buoyancy on the water flow, areas with vorticity magnitude greater than 5/s are mainly located at the upper recirculation zone of the mold, as shown in all the figures. Many pronounced large-scale vortex structures can be clearly seen inside the mold, containing various small-scale vortices between them. These results also show that the vorticity distribution is asymmetric in the mold, corresponding to the asymmetrical flow obtained in previous studies. All these observations suggest that the two-phase turbulent flow field is transient and random; the two-phase flow pattern in the mold is expected to be asymmetrical. Based on the above comparison of the water model and predicted results, it is clear that the current model can predict the transient twoMETALLURGICAL AND MATERIALS TRANSACTIONS B


Fig. 15—Time-averaged flow field in the mold (a) 2-D and (b) 3-D.

Fig. 16—Time-averaged air volume fraction profiles along the mold at y = 0 (x axis) and y = 1 cm.

phase flow behavior in the mold with reasonable success. E. Time-Averaged Horizontal Gas–Liquid Flow Field In the current work, time-averaged velocities from the LES were calculated according to N 1X ui ðx; t; n0 Þ; N!1 N n0

ui ðx; tÞ ¼ lim


½38

where ui is the average velocity and N is the computational time step. The averaging was started at time step n0. The time averages for each case were taken over 50 seconds. The time-averaged horizontal gas–liquid flow field in the mold is shown in Figure 16, for water flow rate of 5 L/min and gas flow rate of 12 cm3/s. In the average water flow pattern profiles along the center plane of the mold (Figure 15(a)), two large vortexes are seen in the lower recirculation zone of the mold. Due to the buoyancy of air bubbles, the water jet escaping from the nozzle inlet (x = 0 cm) is lifted in the process of forward movement. Then, a large vortex forms in the upper recirculation zone when the water jet impinges on the narrow wall of the mold. Figure 15(b) shows the three-dimensional (3-D) two-phase flow pattern inside the mold. An interesting phenomenon is observed: looking from the narrow wall, it can be seen that two vortexes with opposite directions are formed on both sides of the gas phase, which is induced by the movement of gas bubbles. A more quantitative comparison for different gas flow rates is presented in Figure 16, which shows the average air volume fraction profiles along the mold at y = 0 (x axis) and y = 1 cm. It can be seen that the air volume fraction in different regions of the mold naturally increases with increasing air gas injection at the inlet. In case of lower water flow rate, bubbles were found floating closer to the nozzle inlet, while in case of higher water flow rate, bubbles penetrated deeper into the mold and closer to its narrow wall. These


Author's personal copy variation tendencies are consistent with the previous water model experimental results of Iguchi et al.[1] and Liu et al.[4,18] Therefore, the LES model can also be used to describe the time-averaged gas–liquid flow field, giving reasonably good agreement with mean experimental data.

V.

CONCLUSIONS

CFD simulations of horizontal gas–liquid flow in a continuous-casting mold were performed to study the sensitivity to different turbulence models (k–e, SST, RSM, and LES) and different interfacial forces (drag, lift, virtual mass, wall lubrication, and turbulent dispersion), compared with measurements of the velocity fields by two-channel laser Doppler velocimeter. Then, the transient gas–liquid flow in the mold was successfully simulated. The following conclusions can be drawn: At low gas flow rate, the simulations using the LES and SST models allowed better prediction of the u profile. None of the models could capture the peak value of the v profile measured experimentally. For higher gas flow rate, the LES and SST models could predict the change trend of the u profile better. All the models gave an unrealistically high v profile near the nozzle inlet. On the whole, the LES model showed better agreement with experiments compared with the other three RANSbased models in all three cases. For various gas flow rates, the drag force coefficient model of Zhang and Vanderheyden was found to give better agreement with experimental data. The CL value proposed by Legendre and Magnaudet was found to give better predictions. Although inclusion of the virtual mass force only resulted in small improvements of the velocity component predictions, a virtual mass force with coefficient CVM = 0.5 was considered. In accordance with previous studies, the wall lubrication force and turbulent dispersion force had no clear significant effect on the results. Accurate determination of the mean bubble diameter is crucial, as the bubble size influences the momentum transfer through the interfacial area concentrations and interfacial forces. Bubble sizes of 4 and 5 mm were found to give axial and vertical mean velocity components close to experimental values for the different gas flow rates. In summary, the LES model could capture more instantaneous details of the two-phase flow characteristics in the mold quite well, including multiscale vortex structures, fluctuation characteristics, and the vorticity distribution. It can also be used to describe the time-averaged gas–liquid flow field.

ACKNOWLEDGMENTS The work reported in this paper was funded by the National Natural Science Foundation of China (Grant No. 51604070). 1848—VOLUME 48B, JUNE 2017

NOMENCLATURE CD CL CVM CS CTD CWL Cl,BI Cx1, Cx2 db DT,ij Eo F1, F2 Fr Fk, F1g, Fgl FD FL FVM FWL FTD g Gk k ~ nw P Pij Pk Q Re S t u We yx

Drag coefficient (dimensionless) Lift force model constant (dimensionless) Virtual mass force model constant (dimensionless) Smagorinsky constant (dimensionless) Turbulent dispersion model constant (dimensionless) Wall lubrication force model constant (dimensionless) Sato and Sekiguchi model constant (dimensionless) Wall lubrication constants (dimensionless) Bubble diameter (m) Turbulent diffusion term of Reynolds stress model Eo¨tvo¨s number (dimensionless) Blending functions (dimensionless) Froude number (dimensionless) Interfacial forces between the two phases (N/m3) Drag force (N/m3) Lift force (N/m3) Virtual mass force (N/m3) Wall lubrication force (N/m3) Turbulent dispersion force (N/m3) Gravity acceleration vector (m/s2) Rate of production of turbulent kinetic energy (dimensionless) Turbulent kinetic energy (m2/s2) Outward vector normal to the wall (dimensionless) Static pressure (N/m2) Stress production term of Reynolds stress model Production rate of turbulence (dimensionless) Volume flow rate (m3/s) Reynolds number (dimensionless) Strain rate tensor (1/s) Physical time (s) Velocity (m/s) Weber number (dimensionless) Distance from the wall boundary (m)

GREEK LETTERS a b q e leff lL lT t lBI r rt,g

Volume fraction (dimensionless) Constant (dimensionless) Density (kg/m3) Turbulent kinetic energy dissipation (m2/s3) Effective viscosity (N s/m2) Molecular viscosity (N s/m2) Turbulent viscosity (N s/m2) Kinematic viscosity (m2/s) Bubble-induced turbulence viscosity (N s/m2) Surface tension (N/m) Turbulent Schmidt number (dimensionless) METALLURGICAL AND MATERIALS TRANSACTIONS B

Author's personal copy s d /ij wij x x¢ X D

Stress (N/m2) Kronecker factor (dimensionless) Pressure strain term of Reynolds stress model Dissipation term of Reynolds stress model Turbulent frequency (1/s) Angular speed of liquid rotation (rad/s) Vorticity (1/s) Filter width (m)

SUBSCRIPTS b g k l

Bubble Gas phase Index of gas/liquid phase, or turbulent kinetic energy Liquid phase REFERENCES

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