Large Eddy Simulation on Flow Structure in a Dissipative Ladle Shroud and a Tundish

Abstract

Flow structures were investigated in a dissipative ladle shroud (DLS) and a tundish using Large Eddy Simulation. The numerical results were validated inside the DLS and the tundish with PIV experiments. Velocity distribution, vorticity islands and strain rate were analyzed in the DLS respectively, compared with that of a bell-shaped ladle shroud (BLS). The results showed that the three chambers of the DLS gave rise to velocity differences, fluctuating strain rates and vortices, and promoted an increase on turbulence dissipation rate; and the average velocity of outflow ranged from 0.25 to 0.5 m/s when the inlet velocity was 0.708 m/s. In the BLS, the stream flowed straightforward with relatively consistent velocity; apparent vortices were only formed in the bell end; and the outflow went down with high speed and turbulent kinetic energy. The dissipative effect of the DLS was also validated by the flow structure in the tundish. When the stream left the outlet of the DLS, it swung, got twisted and was mixed with more surrounding fluid in the tundish which decreased the mean skin friction coefficient of tundish wall and the velocity of free surface, and finally contributed to a better tundish performance.

1. Introduction

Fluid flow in the tundish has a great effect on temperature field, steel cleanness, lining wearing and device safety and finally influences the quality of molten steel and casting blank. Different flow control devices (FCDs) have been developed to control fluidynamics to enhance the tundish performance in recent years. Generally, three kinds of FCDs can be classified depending on their locations: (1) located at the outlet of tundish, for instance, stopper rods;¹⁾ (2) located in the tundish, such as turbulent inhibitors, weirs and dams;^2,3,4) (3) located at the inlet of the tundish, i.e., ladle shrouds.^{5,6,7,8,9,10,11)} The first two kinds of FCDs have earned widespread attention and gotten significant development. For the ladle shroud, it is a new research field to employ it as a flow control device for tundish operation and even replace other FCDs. As the trend of steelmaking industry is to simplify operations and to optimize designs of equipment to decrease costs and maintain or even increase steel quality, new approaches are required for flow control in tundishes. Thus, it is meaningful to get a better understand of the turbulent phenomena in the ladle shroud.

In available literature, seven types of ladle shroud can be summarized and their features are shown in Table 1. To prevent the melt stream from reoxidation by the aspiration of ambient air, a conventional ladle shroud (like a straight pipe) was initiatively designed to connect the flow between the ladle and the tundish. It is easy to make and can protect the molten steel for a certain degree, which has been widely used. The past two decades have seen several new shrouds, such as the Bell-shaped Ladle Shroud (BLS), the technique of argon injection and Swirling Ladle Shroud (SLS).

Table 1. Ladle shrouds and their features in available literature.

It is investigated that the DLS performs many advantages (shown in Table 1) compared to other shrouds which begin to draw more attention. Ken Morales-Higa⁷⁾ firstly developed a DLS and studied the flow pattern by Particle Image Velocimetry (PIV) measurement and mathematical modeling (k-ε). Solorio-Diaz⁸⁾ compared the flow field and inclusion removal in a conventional shroud and a DLS using Reynold Stress Model (RSM). However, these two numerical approaches are not very suited for studying the time evolution of unsteady flow structures, especially in the ladle shroud with high turbulence, and only steady simulations were carried out in their studies. PIV presents several limitations to characterize a turbulent flow with strong instant velocity variations and high-energy dissipation model. And the cylindrical body of LS reflects the laser light sheet very strongly in water modelling, making it hard to record the internal velocity field. Thus, mathematical modelling needs to be developed and improved to understand more details of the momentum transfer mechanisms in the shroud. Mathematical modelling can be classified into the Reynolds-averaged approach, Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS). Because of the computational cost, the Reynolds-averaged approach, typically with the two-equation (k-ε) turbulence model, has been extensively adopted. Nevertheless, it fails to provide reliable results in swirling flows and highly strained angular velocities of rotating flows.¹²⁾ The DNS is to solve directly the Navier-Stokes (N-S) equations with high computational cost. A midway method between two-equation turbulence models and DNS is LES where the N-S equations are filtered. The filtered equations are then simulated and the left smallest eddies are modeled^13,14) to get relatively accurate data with acceptable solution time.

The objective of the present work is to investigate the flow structure in a bell-shaped ladle shroud (BLS), a dissipative ladle shroud (DLS) and a tundish using the LES, and analyze the effects of different shrouds on flow pattern and turbulent kinetic energy dissipation phenomena. It is beneficial to have a better understanding of flow characteristics in ladle shroud and tundish so as to optimize their structures and performances.

2. Mathematical Model Development

2.1. Geometrical Description

The schemes of the BLS and the DLS are shown in Fig. 1(a). The DLS is comprised of three same straight sections, three same divergent sections and a bell-shape end. Their dimensions are listed in the Table 2. The diameters of the inlet and the outlet are 0.030 m and 0.060 m respectively. The BLS is better to be chosen to study the dissipative phenomena with the same outlet shape and area, compared with conventional ladle shroud in previous studies. As the main purpose of the current study is to investigate the effect of ladle shroud which mainly influences its surrounding fluid, it is rational to simplify the tundish structure to get high-quality refined mesh and appropriate Yplus to achieve accurate results. The structure of tundish was simplified as a rectangular tank and the computational domain was meshed into ~1750000 hexahedral cells (see Fig. 1(b)). All simulations were performed using water as the material for PIV experiments and LES modeling.

Fig. 1.

(a) Schemes of studied ladle shrouds. (b) Mesh of the model.

Table 2. Tundish parameters.

Parameters	Values
Length of the domain, mm	500.0
Height of the domain, mm	170.0
Width of the domain, mm	250.0
Submergence depth of shroud, mm	36.5
Inlet diameter of shroud d1, mm	30.0
Outlet diameter of shroud d2, mm	60.0
Water density, kg/m3	1000
Water viscosity, kg/(m·s)	0.001
Flow volume rate, L/min	30
Length of the straight section (L1), mm	50
Half length of the straight section (L2), mm	37.5

2.2. Mathematical Formulation and Procedure

In the LES model, only large scale structures are calculated. A Sub-grid scale (SGS) model is employed to characterize the dissipative effect of eddies smaller than the filter size. The governing equations are as follows:   

$$\frac{\partial u_i}{\partial x_i}=0$$(1)

  

$$\frac{\partial u_i}{\partial t}+\frac{\partial u_{i{u}_j}}{\partial x_j}=-\frac{1}{\rho }\frac{\partial p}{\partial x_i}+\frac{\partial }{\partial x_j}\left({\nu }_{eff}\right)\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)$$(2)

  

$${\nu }_{eff}={\nu }_0+{\nu }_t$$(3)

Where, the symbols p and u_i mean the pressure and filtered velocities respectively; the i and j represent the three directions in Cartesian coordinates. The residual stresses, arisen from the unresolved small eddies, are modeled using a turbulent eddy-viscosity (ν_t) in the formulas. The SGS k model need to solve the following transport equation in the present work, which involves advection, dissipation, production and viscous diffusion.¹⁵⁾   

$$\begin{array}{l}\frac{\partial k_s}{\partial t}+u_i\frac{\partial k_s}{\partial x_i}={\nu }_t|\overline{S}{|}^2-C_{\epsilon }\frac{k_s^{3/2}}{\mathrm{\Delta }}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}+\frac{\partial }{\partial x_i}\left(\left({\nu }_0+C_{kk}{K}_G^{1/2}\mathrm{\Delta }\right)\frac{\partial k_s}{\partial x_i}\right)\end{array}$$(4)

Where, Δ = (Δ_xΔ_yΔ_z)^1/3, ν_t = $C_{\nu }\mathrm{\Delta }{k}_s^{1/2}$ , S_ij = $\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)$ , |S| = $\sqrt{2S_{ij}{S}_{ij}}$ .

The C_e, C_kk and C_v can be treated as constant parameters, which were adopted as 1, 0.1, and 0.05.

The governing equations were discretized and calculated using Finite Volume Method (FVM) with 2nd-order central differencing scheme for convection terms, which is carried out in FLUENT software. Implicit Fractional Step Method (I-FSM) was employed to achieve the velocity-pressure coupling. The time integration was implemented using 2nd order implicit scheme. The time-dependent LES models were started on the base of steady state obtained by the k-ε two-equation model in the whole domain. The flow was allowed to develop for 20 second before starting collecting time statistics with a time step Δt=0.0005 second. Time statistics were collected for 50 seconds to get the mean data.

2.3. Boundary Conditions

1) Inlet: a constant inflow velocity (0.708 m/s) profile was applied at the nozzle inlet;

2) Outlet: a fixed pressure of 0 Pa (relative to the ambient) was applied at the outlet of the tundish;

3) Top surface and walls: free-slip boundaries with zero normal velocity were used at the top surface. All walls of the domain were considered to be no-slip, and the wall boundary domain was handled using the Werner-Wengle formulation.

2.4. Physical Modelling and Model Validation

A water model was built to validate the mathematical model with the same flow parameters (geometries and flow rate etc.). PIV was implemented to measure velocity inside the DLS and the outflow of it. A scheme of the physical experiment apparatus is shown in Fig. 2. The DLS is mainly characterized by three chambers and the inside velocity of the second chamber was chosen for measurement to verify the LES results in the shroud. The chamber was surrounded by a cubic box (filled with water) to relieve the reflection of the cylindrical body and improve the translucency. When the flow pattern reached a relatively steady state in the water model, data were collected in the interior DLS and the tundish separately.

Fig. 2.

The scheme of physical experiment apparatus.

For the velocity in the chamber, PIV vectors are shown in Figs. 3(a) and 3(c), and LES results in Figs. 3(b) and 3(d). As the wall of the chamber still reflects the laser, the velocity cannot be gotten along the wall boundary. The flow patterns of the two models show good agreement as a whole. In Figs. 3(a) and 3(b), the flow tends to straightly flow down without captured vortex and the velocity magnitude is generally around 0.7 m/s; the maximum velocity reaches about 0.75 m/s in the middle part and the velocity is relatively small along the wall (smaller than 0.5 m/s). In Figs. 3(c) and 3(d), the main stream tends to flow down from the right side of the chamber and a clockwise vortex is generated in the upper-left part of the chamber; when vortices are created, the flow vectors turn out to be more complicated and diverse which benefit the dissipation of turbulence kinetic energy. However, the location of the vortex center in LES model is a little lower than that of PIV model.

Fig. 3.

Velocity vectors in the second chamber: (a), (c): LES results; (b), (d): PIV results. (Online version in color.)

Figure 4 shows the flow field under the outlet of the DLS obtained by PIV and LES modeling. Due to the dissipative effect of the DLS, the outflow swings when it goes out from the outlet (more details are discussed in Chapter 3). The main stream flows to the left side and the right side in Figs. 4(a), 4(b) and 4(c), 4(d) respectively. It should be noted that the time step is 0.02 s in PIV measurement, while 0.0005 s in the LES modeling, which makes the PIV findings seem more stable and simple in comparison with that of numerical simulations. Due to the setting and limitation of PIV monitoring, the node space is also relatively coarse for the vectors of PIV results; it is, however, sufficient to capture the major information of the turbulence flow. The main stream flows out from the outlet at around 0.60 m/s; when the outflow is mixed with the fluid in the tundish, the stream gets distorted and its velocity deceases rapidly to less than 0.3 m/s at the bottom area. Several vortices can also be observed on both sides of the outflow owing to velocity differences among fluid elements; the velocity reaches as low as less than 0.1 m/s in the vortex regions; however, the location and dimension of the vortices are not sufficiently consistent between the vector profiles, as the flow patterns are diverse and time-dependent.

Fig. 4.

Velocity vectors under the ladle shroud: (a), (c): LES results; (b), (d): PIV results. (Online version in color.)

3. Results and Discussion

3.1. Ladle Shroud Flow Structure

3.1.1. Velocity Distribution

Velocity is the most directly perceived parameter to reflect the flow structure. Hence, velocity fields of the two shrouds were monitored at certain points, central lines and central planes (XZ plane) respectively to describe the turbulent flow.

Figure 5 shows ten-second instantaneous velocity histories at three points (p1, p2 and p3, located at the center, 1/2 and 1/4 of the outlet radius respectively). In the BLS, the velocity of p1 fluctuates between 0.5 m/s and 0.8 m/s, and averages at 0.673 m/s when the inlet velocity is 0.708 m/s; the ten seconds see apparent fluctuation for p2, and its velocity ranges from 0.1 m/s to 0.8 m/s and reaches the average at 0.394 m/s; the average velocity of p3 is the lowest and amounts to 0.185 m/s, which displays a gap of −0.5 m/s compared with that of P1. In the DLS, the average velocities of the three points seem to be a little even and range from 0.25 m/s to 0.5 m/s; the velocity of p1 shows larger fluctuations (from 0.1 m/s to 0.65 m/s) and the standard deviation is 0.087 (0.033 for the BLS). Meanwhile, the standard deviations of velocity of p2 and p3 are 0.132 and 0.107 respectively in the DLS which are larger than that of the BLS (0.108 and 0.051 respectively). These indicate that the outflow velocity of DLS gets divided with larger fluctuations rather than focuses at center with higher velocity and less fluctuation, and the average values of the DLS points tend to be more even, which can decrease liquid level fluctuations and impact forces of ladle shroud on the tundish bottom.

Fig. 5.

Instantaneous velocity histories at three points of the two LS outlets: (a) BLS; (b) DLS. (Online version in color.)

The turbulent flow is much more complex to be described in detail and velocity varies with time stochastically. Therefore, five typical velocity profiles (denoted as mode 1 to 5) are chosen to show the flow characteristics at the centerlines of the two shrouds, which are shown in Fig. 6. The velocity magnitude increases as the liquid flows down and seems to be consistent before the bell-shaped end in the BLS. A drop of from 0.85 m/s to nearly 0.65 m/s can be seen at the bell part. In the DLS (see Fig. 6(b)), the inlet velocity stands at 0.708 m/s and slightly goes up at the initial straight section; the velocity firstly declines in the divergent section and ascends at the following convergent section of the first chamber, and increases again in the following straight section. This phenomenon is recurrent and the velocity fluctuates between 0.35 m/s and 0.85 m/s. The velocity fluctuations and difference promotes the rate at which turbulence kinetic energy is converted into thermal energy (ε). With the dissipative effect of the shroud and the divergent outlet, the outflow gets divided and the velocity decreases to a range of 0.27 m/s to 0.49 m/s, much lower than that of the BLS.

Fig. 6.

Typical velocity profiles at the shroud centerlines. (a) BLS. (b) DLS. (Online version in color.)

Figure 7 shows four typical velocity vector profiles at the central XZ plane for the BLS and the DLS. The velocity continuously rises in the straight pipe section and bias flows occur at the bell part in the BLS. The main stream flows to the left side, expands, to the right side, and focuses with time. However, the main stream is limited at the center and vortices are generated around them at the bell-shaped end. In the DLS, the velocity decreases in the chambers and increases in the straight sections, and many vortices are formed in the shroud. Similarly, bias flows take place and go out from the outlet in many directions. The main stream expands to the whole outlet and its velocity becomes smaller and more even, compared with that of the outflow in the BLS.

Fig. 7.

Four typical velocity vector profiles at the central XZ plane. (a) BLS. (b) DLS. (Online version in color.)

The three chambers represent the main feature of the DLS, which change the flow structure of the shroud. A typical flow pattern in one of the chamber is shown at the central plane XZ in Fig. 8(a), where the color map is the same to that of Fig. 7. At a start of an arbitrary time of 0 s, a spindly clockwise vortex is generated near the left side wall at the divergent section and changed into a round vortex at the tip after 0.1 second; during this time, the stream mainly flows down from the right side of the chamber and gets hindered by the left vortex. After another about 0.1 s, an anticlockwise vortex is generated at the right tip and extruded into slender shape by the stream at 0.286 s. The turbulent flow is much more diverse than the flow patterns above; stream swings and vortices are frequently generated, which promotes a decrease of velocity magnitude and an increase of turbulent fluctuations. Figure 8(b) shows the velocity vector of Reynolds Stress Model simulation in the published work,⁸⁾ which cannot capture the vortices and fluctuations of the fluid flow like LES.

Fig. 8.

(a) A typical flow pattern in the chamber at the central plane XZ of the DLS. (b) Velocity vector of Reynolds Stress Model simulation⁸⁾ (c) ink dispersion in the first chamber. (Online version in color.)

In preliminary summary, the instantaneous velocity histories of monitored points, centerlines and planes indicate that the outlet velocity of the DLS fluctuates more severe at a relatively low speed compared with that of the BLS. Many vortices are formed both in the chambers and at the outlets which promote turbulent fluctuations and a decrease of average velocity from 0.708 m/s to a range of 0.25 m/s to 0.5 m/s.

3.1.2. Deformation and Turbulent Energy Dissipation

The deformation and energy dissipation are important properties of the turbulent flow, which can be used to characterize the flow structure in the ladle shroud. Deformation tensor describes how fluid elements deform as a result of fluid motion, which is expressed as:   

$$D_{ij}=\frac{\partial u_i}{\partial x_j}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)+\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}-\frac{\partial u_j}{\partial x_i}\right)$$(5)

Where S_ij = $\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)$ , and ς_ij = $\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}-\frac{\partial u_j}{\partial x_i}\right)$ ; S_ij is the symmetric part of the deformation tensor, which represents the rate of shear strain tensor for incompressible fluid; ς_ij is the anti-symmetric part and the vorticity tensor. The relationship between dissipation rate of kinetic energy and the strain rate fluctuation can be expressed as:   

$$\epsilon =2\nu \overline{s_{ij}{s}_{ij}}$$(6)

Where s_ij represents the fluctuating strain rate and ν is the fluid kinematic viscosity. It indicates that the strain rate fluctuations are parallel to ε. Strain rate profiles (denoted as mode 1 to 5) are shown at the two shroud centerlines in Fig. 9, where various differences can be seen in the two shrouds. In the BLS, the strain rate ranges from 5 to 25 s⁻¹ in the straight section, and rises to nearly 80 s⁻¹ in the bell-shaped section. In the DLS, strain rate fluctuates between 5 and 140 s⁻¹, a big range; the strain rate tends to increase in the first chamber and the first two straight pipe sections and fluctuates in the following section. Thus, the strain rate of the DLS is larger and fluctuates more severe than that of the BLS, which encourages more vortices and larger turbulence dissipation rates and finally makes the outlet velocity decrease and fluctuate.

Fig. 9.

Typical Strain Rate profiles at the shroud centerlines. (a) BLS. (b) DLS. (Online version in color.)

Vorticity is a measure of the rotation of a fluid element as it moves in the flow field, and is defined as the curl of the velocity vector:   

$$\zeta =\nabla \times u_i$$(7)

The dissipation rate of kinetic energy is proportional to the mean square fluctuations of the vorticity vector, which can be seen in the following equation.   

$$\epsilon =\nu \overline{{\varsigma }_i{\varsigma }_i}$$(8)

Thus, the turbulence energy dissipation is also represented by the vortex magnitude in the fluid. Figure 10 shows the vorticity field and the evolution tracks of typical vortices of the DLS within an arbitrary time of 0.22 s in the DLS. Fluid elements only tend to rotate along the wall boundary and the magnitude ranges from 0 to 300 s⁻¹ when the flow goes downward in the BLS; apparent vortices are formed in the bell-shaped tip, which can also be seen from Fig. 7. In the DLS, however, more and larger islands of vorticity are created. A typical dissipation and evolution behavior of vortices can be described as follows: at an arbitrary time of 0.00 s, a vortex (surrounded by an oval in Fig. 10) is generated at the divergent section of the first chamber; after 0.07 s, the vortex moves to the convergent section and is enlarged, and the vorticity magnitude is decreased as well; during the following time, the vortex dissipates into many small vortices and flows down; meanwhile, a new vortex is formed at the divergent section at 0.15 s. These phenomena are recurrent and similar in other chambers, and turbulent kinetic energy gets dissipated in this course.

Fig. 10.

Vorticity fields of the BLS and DLS and typical evolution tracks of vortices in the DLS. (Online version in color.)

3.2. Tundish Flow Analysis

The purpose of designing new ladle shroud is to control the fluid flow in the tundish, which is also studied under these two ladle shrouds. For this, instantaneous velocity contours of tundish with the BLS are shown in Fig. 11. The main stream impacts at the bottom and is divided into substreams when the flow goes down from the BLS outlet; large-scale eddies are generated and then dissipated into small-scale eddies. The flow seems to be quasi-periodic: the stream mainly flows to the left side of the tundish at an arbitrary time of 0 s, and is evenly divided at 0.7 s, and then turns to the right side at the following 0.7 s. Nevertheless, the main stream is relatively focused and slightly twisted. It directly impacts at the bottom with high velocity and also is prone to resulting in larger liquid level fluctuations and a larger slag eye, which will be discussed further.

Fig. 11.

Instantaneous velocity contours of tundish under the BLS. (Online version in color.)

In the tundish with the DLS, the flow structure shows more complex characteristics. Figure 12 shows the quasi-periodic velocity contours of the tundish: the stream initially goes straight, flows to the left side at 0.43 s and is twisted at 0.84 s; the flow turns to the right side and gets twisted again at the following 1.24 s and 1.64 s; then, the stream flows to the left again. The outlet stream of the shroud gets twisted and is mixed well with more surrounding fluid; and finally impacts at the bottom with a relatively low speed. This will enlarge the plug volume and improve the mixing in the tundish.^7,21)

Fig. 12.

Instantaneous velocity contours of tundish under the DLS. (Online version in color.)

It is unprocurable to measure tundish lining wear by any of the models developed in this study; however, it is possible to associate the areas of lining wear to the wall skin friction coefficient, which is directly proportional to shear stress between steel flow and tundish wall. Figure 13 shows the mean skin friction coefficient (its unit is 1) distribution of tundish wall under the two shrouds of 50 s average (considering the limited calculating time, the contour is not totally symmetric as expected, but enough to capture the main features). The maximum coefficient presents at the bottom wall, and reaches 1.2 for the BLS and 0.5 for the DLS respectively. On the sidewall, the maximum friction coefficient decrease from 0.5 to 0.3 when the BLS is replaced by the DLS. Thus, the tundish wall is effectively protected from being worn with respect to fluid flow under the DLS.

Fig. 13.

Mean skin friction coefficient distribution of tundish wall under the two shrouds. (a) BLS. (b) DLS. (Online version in color.)

As only one phase (water) is considered in the LES model, tundish slag cannot be studied. Nevertheless, the velocity of free surface can work as an index to assess the situation of liquid level. As shown in Fig. 14(a), the mean velocity of free surface ranges from 0.01 m/s to 0.13 m/s and peaks at the left side of the shroud with the BLS. When the DLS is used, the maximum value decreases almost 40% and reaches 0.08 m/s (see Fig. 14(b)); the velocity of the free surface appears to be more even, which can reduce the possibility of slag entrapment. This can be illustrated from the flow pattern in the tundish (see Figs. 11 and 12). The outflow of the DLS mixes with more fluid and the velocity of the backflow to the free surface becomes smaller, which promotes a relatively quiet liquid level.

Fig. 14.

Mean velocity of free surface with the two shrouds. (a) BLS. (b) DLS. (Online version in color.)

4. Conclusions

Large Eddy Simulation was employed to study the flow structure of ladle shrouds and a tundish. From the results, the following conclusions can be drawn:

(1) Compared with the flow pattern in the BLS, the velocity decreases and vortices are formed in the three chambers in the DLS which promotes higher velocity fluctuations and turbulence dissipation rate. The outflow leaves out the shroud at an average velocity of 0.25 to 0.5 m/s when the inlet velocity is 0.708 m/s in the DLS.

(2) Larger and more voticity islands are formed in the DLS, which gives origin to higher dissipation rate of turbulent energy when the BLS is replaced by the DLS. Vortices are formed and dissipated into smaller ones as the stream flows down in the DLS.

(3) The strain rate is larger and fluctuates more severe in the DLS, which encourages larger turbulence dissipation rates.

(4) In the tundish under the DLS, the flow structure shows more complex characteristics and seems to be quasi-periodic. The outflow of the shroud gets twisted and is mixed with more fluid, which contributes to a decrease on mean skin friction of tundish wall and free surface velocity.

Acknowledgements

The first author, Jiangshan Zhang, would like to thank Dr. Chuanbo Ji and Dr. Liyuan Sun for their constant encouragement and help. This research is supported by National Science Foundation of China (No. 51304016), which is highly acknowledged as well.

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