In the actual continuous casting production process, the heat transfer, fluid flow, phase change, and gas-liquid interactions of molten steel in the mold are highly complex. These processes are interconnected and mutually influential, making it extremely difficult to accurately simulate and calculate the flow behavior of molten steel in the mold. While ensuring that the calculation results are realistic and reliable, the computational process can be appropriately simplified. In this paper, the standard k-ε model was used to simulate the flow of molten steel in the slab continuous casting mold, and a three-dimensional mathematical model was established. Based on the research content involved in this study, the following assumptions were made regarding the flow and transport behavior of molten steel in the mold [11] .

1) The molten steel in the mold was treated as an incompressible fluid and considered a homogeneous phase;

2) The effects of solid-phase transformation during solidification, volume shrinkage, and heat transfer on the mold flow field were neglected;

3) Natural convection caused by density variations was neglected;

4) The effects of mold vibration, mold curvature, and mold flux on the flow field were neglected. Note that, in numerical simulation of mold flow field, surface flow velocity and liquid level fluctuation were typically regarded as core indicators to highlight the key mechanisms of molten steel flow, such as reducing slag entrapment and controlling the flotation of inclusions. The role of mold flux was mainly concentrated on heat transfer at the molten steel interface, lubrication, and inclusion adsorption, and mold flux had minimal impact on the overall flow field. Therefore, the effects of mold flux was neglected in here;

5) Heat transfer, phase change, and chemical reactions of the molten steel were neglected;

6) The continuous casting process was considered to be steady-state pouring.

For the flow of molten steel in the slab mold, based on the laws of mass conservation and momentum conservation, the fundamental governing equations included the continuity equation, the momentum equation, and the k-ε turbulence model equation. The specific formulations of each equation were as follows:

1) Continuity equation

$\frac{\partial \left(\rho v_j\right)}{\partial x_j}=0$(1)

where, ρ was the fluid density, kg/m³; v_j was the velocity in the j-direction, m/s; x_j was the distance in the j-direction, m.

2) Momentum equation

$\rho \frac{\partial \left(v_i{v}_j\right)}{\partial x_j}=-\frac{\partial p}{\partial x_i}+\frac{\partial }{\partial x_j}\left({\mu }_{\text{eff}}\frac{\partial v_i}{\partial x_j}\right)+\frac{\partial }{\partial x_i}\left({\mu }_{\text{eff}}\frac{\partial v_j}{\partial x_i}\right)+\rho g$(2)

where, v_i was the velocity in the i-direction, m/s; x_i was the distance in the i-direction, m; p was the pressure, Pa; μ_eff was the effective viscosity coefficient, kg/(m·s); g was the gravitational acceleration, m/s².

3) k-ε equations. The standard k-ε turbulence model was employed owing to its established robustness and computational efficiency, and it was successfully applied in simulating turbulent flow fields in similar industrial continuous casting molds in previous studies [12] [13] .

The turbulent viscosity was expressed as follows:

${\mu }_T=C_{\mu }\frac{k^2}{\epsilon }$(3)

k-equation:

$\frac{\partial \left(\rho k\right)}{\partial t}+\frac{\partial \left(\rho u_{j}k\right)}{\partial x_j}=\frac{\partial }{\partial x_j}\left(\left(\mu +\frac{{\mu }_T}{{\sigma }_{\epsilon }}\right)\frac{\partial k}{\partial x_j}\right)+\rho P-\rho \epsilon$(4)

ε-equation:

$\frac{\partial \left(\rho \epsilon \right)}{\partial t}+\frac{\partial \left(\rho u_j\epsilon \right)}{\partial x_j}=\frac{\partial }{\partial x_j}\left(\left(\mu +\frac{{\mu }_T}{{\sigma }_{\epsilon }}\right)\frac{\partial \epsilon }{\partial x_j}\right)+C_1\frac{\epsilon }{k}\rho P-C_2\frac{\epsilon }{k}\rho \epsilon$(5)

and

$P={\mu }_T\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}-\frac{2}{3}\frac{\partial {\mu }_m}{\partial x_m}{\delta }_{ij}\right)-\frac{2}{3}k\frac{\partial {\mu }_m}{\partial x_m}$(6)

where, μ_T was the eddy viscosity coefficient, N/m²; k was the turbulent kinetic energy, J; ε was the turbulent kinetic energy dissipation rate, W/m³; t was time, s; u_i and u_j were the energies in the i and j directions, respectively, J; μ was the dynamic viscosity of the fluid, Pa·s; P was the turbulence production term; μ_m was the dynamic viscosity, N∙s/m²; x_m was the coordinate component, m; δ_ij was the Kronecker δ symbol, which equals 1 when i = j and 0 otherwise; C₁, C₂, C_μ, σ_k, and σ_ε were empirical constants with values of 1.44, 1.92, 0.09, 1.0, and 1.3, respectively.

The boundary conditions set in this article were described in relation to the following aspects.

1) Tundish nozzle inlet: Defined as a velocity inlet, its velocity was derived from the casting speed, nozzle dimensions, and mold dimensions based on mass conservation. The turbulent kinetic energy and turbulent dissipation rate were determined by Equations (7) and (8), respectively.

The values of k and ε were as follows:

$k=0.01v_{\text{inlet}}^2$(7)

$\epsilon =\frac{2k^{1.5}}{d_{\text{nozzle}}}$(8)

where, v_inlet was the inlet velocity of the nozzle, m/s; d_nozzle was the hydraulic diameter of the nozzle, m.

2) Symmetry plane: The velocity component perpendicular to the symmetry plane and the gradients of all physical quantities in the direction normal to the symmetry plane were zero.

3) Free surface: The same boundary conditions as those for the symmetry plane were applied, while the effects of surface tension and the mold powder layer were neglected.

4) Wall surfaces: Both the mold and nozzle walls were treated as no-slip solid walls. The velocity component perpendicular to the surface and the normal components of other physical quantities were zero. The flow field near the wall region was treated using the standard wall function.

5) Outlet: The velocity at the lower outlet of the mold section was defined as the exit velocity from the mold, which corresponded to the drawing speed of the solidified shell out of the mold during continuous casting.

The process parameters used in the simulation calculations are detailed in Table 1 . The Submerged Entry Nozzle (SEN) structure employed featured a double-port design with a concave bottom and downward-facing outlets.

During the continuous casting process, the molten steel in the tundish is poured into the mold through a Submerged Entry Nozzle (SEN). The immersion depth of the nozzle determines the state of the upward reflux flow, which in turn affects the temperature of the free liquid surface, the movement of the liquid slag, and the melting performance of the mold flux. A reasonable immersion depth can effectively prevent slag entrapment and improve the quality of the cast slab [16] . Therefore, it was necessary to investigate the influence of the SEN immersion depth on the flow field in the mold. When the exit area of the submerged entry nozzle, the nozzle outlet inclination angle, and the casting speed were 40 mm × 55 mm, 12˚, and 1.6 m/min, respectively, the flow fields in the mold were studied at immersion depths of 110 mm, 120 mm, and 130 mm. The distribution of the flow field in the longitudinal central section of the mold is shown in Figure 4 . It can be observed that when the immersion depth was 110 mm, the molten steel flowing out of the nozzle exit formed an asymmetric flow field. The molten stream on one side had difficulty in reaching the narrow face of the mold, while the stream on the other side could reach it. This was likely due to significant energy dissipation during the molten steel flow and uneven static pressure distribution after the steel exited the nozzle [17] . Studies have shown [18] that a shallow immersion depth can easily lead to an asymmetric flow field in the mold, causing unreasonable liquid surface fluctuations and uneven temperature distribution inside the mold, which are major causes of surface cracks and slag entrapment in casting slab. Those issues should be avoided in production. When the immersion depths were 120 mm and 130 mm, the flow fields formed after the molten steel exiting the nozzle were similar and exhibited quasi-symmetric. The flow fields on both sides of the nozzle in the mold showed symmetry, ensuring uniform temperature distribution of the molten steel in the nozzle exit region, thereby promoting uniformity of the initial shell and contributing to improved cast slab quality. From the above analysis, it can be concluded that an immersion depth of 110 mm was not conducive to improving cast slab quality.

When the immersion depths of the Submerged Entry Nozzle (SEN) were 120 mm and 130 mm, respectively, the velocity distributions of the molten steel at the free surface of the mold were shown in Figure 5 . As the immersion depth of the SEN increased from 120 mm to 130 mm, the maximum velocity of the molten steel at the free surface increased from 0.45 m/s to 0.48 m/s. This may be attributed to the enhanced shear effect of the upward reflux on the steel-slag interface, which increased the probability of slag entrainment [19] . The fluctuation conditions of the molten steel at the free surface under the two immersion depths are shown in Figure 6 . It can be observed that in the region near the narrow face of the mold, the fluctuations of the molten steel at the free surface were predominantly upward, and as the SEN immersion depth increased from 120 mm to 130 mm, the fluctuation height increased from 12.71 mm to 20.29 mm. In the region near the SEN, the fluctuations were predominantly downward, and as the SEN immersion depth increased from 120 mm to 130 mm, the fluctuation height increased from −5.21 mm to −5.72 mm. Based on the above results, it can be concluded that neither reducing nor increasing the immersion depth of the SEN alone can meet the requirements of actual continuous casting production. Other measures were needed to optimize the flow field inside the mold.

The flow field in the slab mold is closely related to the outlet area of the Submerged Entry Nozzle (SEN). If the outlet area of the nozzle is either too large or too small, it can easily lead to an unbalanced flow of molten steel at the outlet. A properly sized nozzle outlet area can appropriately increase the flow resistance of the molten steel at the outlet, which helps the molten steel flow out from both sides of the nozzle to form a more balanced flow and thereby optimizes the flow field [20] . When the immersion depth of the submerged entry nozzle, the nozzle outlet inclination angle, and the casting speed were 120 mm, 12˚, and 1.6 m/min, respectively, the flow fields in the slab mold were studied under three different nozzle outlet area conditions, namely 30 mm × 50 mm, 40 mm × 55 mm, and 50 mm × 60 mm. The distribution of the flow field in the central longitudinal section of the mold is shown in Figure 7 . It can be observed that under the conditions of 30 mm × 50 mm and 40 mm × 55 mm nozzle outlet areas, the flow field distributions were similar and exhibited a generally symmetrical pattern. However, when the nozzle outlet area was 50 mm × 60 mm, after the molten steel flowed out from the nozzle outlet, it began to rise before reaching the narrow face of the mold, and the symmetry of the flow field decreased. The upward deflection of the left-side flow stream became more pronounced. This was mainly because the excessively large nozzle outlet area caused an imbalance in the outflow of molten steel from the two sides of the nozzle [21] .

Under the conditions of nozzle outlet areas of 30 mm × 50 mm and 40 mm × 55 mm, the velocity distribution of molten steel at the free surface of the mold is shown in Figure 8 . The maximum velocities in both cases were 0.45 m/s, which were higher than the required liquid surface velocity for production. The surface fluctuation conditions under these two nozzle outlet areas were shown in Figure 9 , with fluctuation ranges of −7.00 mm to 13.03 mm and −5.21 mm to 12.71 mm, respectively. It can be concluded that reducing the nozzle outlet area on the basis of 40 mm × 55 mm led to an intensification of both positive and negative surface fluctuations. This was mainly because a smaller nozzle outlet area increased the pressure inside the nozzle, thereby increasing the kinetic energy of the molten steel at the nozzle exit, which in turn resulted in higher surface velocity and greater fluctuation. Based on the above results, it was manifested that neither reducing nor increasing the outlet area of the Submerged Entry Nozzle (SEN) can meet the requirements of actual continuous casting production. Other methods must be employed to optimize the flow field within the mold.

During the continuous casting process, changes in the casting speed determine the variation in the steel throughput. Under the same conditions, a lower casting speed results in a smaller velocity of the molten steel stream at the nozzle outlet, which can easily lead to biased flow, affecting the flow field inside the mold and causing defects such as longitudinal cracks on the surface of the cast slab. Conversely, an excessively high casting speed increases the velocity of the molten steel stream exiting the nozzle, thereby increasing the impact pressure on the narrow face of the mold and enhancing the upward reflux. This leads to excessive fluctuation of the liquid surface and can also drive the liquid slag layer near the narrow face toward the center of the mold, resulting in an uneven distribution of the slag layer thickness and promoting the formation of surface defects on the cast slab. An appropriate casting speed ensures effective flow of the molten steel at the free surface, which helps enhance temperature compensation and uniform distribution of the molten steel at the free surface within the mold, thereby reducing the occurrence of casting defects. Therefore, it was necessary to consider the influence of casting speed on the mold flow field. When the immersion depth of the Submerged Entry Nozzle (SEN), the nozzle outlet area, and the nozzle outlet inclination angle were 120 mm, 40 mm × 55 mm, and 12˚, the changes in the mold flow field under casting speeds of 1.5 m/min, 1.6 m/min, and 1.7 m/min were studied. The flow field in the longitudinal central section of the mold is shown in Figure 10 . It can be observed that under the same conditions, when the casting speed varied, the basic characteristics of the flow field inside the mold did not change significantly, and the overall distribution of the flow field remained largely similar.

The velocity distribution of the free-surface molten steel under different casting speeds is shown in Figure 11 . Under the casting speeds of 1.5 m/min, 1.6 m/min, and 1.7 m/min, the maximum flow velocities were 0.41 m/s, 0.45 m/s, and 0.39 m/s, respectively. The fluctuation conditions of the free-surface molten steel under different casting speeds are shown in Figure 12 . When the casting speed was 1.5 m/min, the fluctuation range of the free surface was from −5.75 mm to 16.71 mm; at 1.6 m/min, it was from −5.21 mm to 12.71 mm; and at 1.7 m/min, it was from −4.60 mm to 15.15 mm. It can be observed that as the casting speed increased or decreased, the positive fluctuation of the free surface became significantly stronger, while the negative fluctuation decreased with increasing casting speed. Increasing the casting speed led to enhanced turbulent kinetic energy of the upward return flow at the mold top, which in turn increased the impact intensity of the upward return flow on the steel-slag interface, thereby intensifying the fluctuations at the interface. Excessive fluctuation of the free surface was not conducive to maintaining the stability of the steel-slag interface and can easily lead to defects such as molten steel exposure and slag entrapment, thereby deteriorating the quality of the cast slab. When the casting speed was reduced, as can be seen from Figure 12 , the fluctuation on one side of the free surface was weaker while on the other side was stronger. This may be due to the fact that the turbulent kinetic energy of the upward return flow was stronger on one side and weaker on the other after the molten steel flowed out from the nozzle, resulting in a slight flow deviation. Therefore, taking into consideration the velocity and fluctuation conditions of the free surface, a casting speed of 1.6 m/min was deemed appropriate.

The change in the inclination angle of the nozzle outlet has a significant impact on the morphology of the flow field inside the mold. When the inclination angle of the nozzle outlet is relatively small, the distance that the molten steel flows from the nozzle outlet to the narrow face of the mold is shortened, and the velocity attenuation of the molten steel is slower. After the jet impacts the narrow face, a relatively strong upward return flow is formed. When the inclination angle of the nozzle outlet increases, the jet direction of the molten steel flowing from the nozzle outlet deflects downward toward the lower part of the mold, the distance that the molten steel travels to reach the narrow face is increased, and due to the influence of resistance and gravity, the turbulent kinetic energy of the upward return flow decreases. Therefore, it was necessary to study the influence of the inclination angle of nozzle outlet on the flow field in the mold. When the immersion depth of the Submerged Entry Nozzle (SEN), the nozzle outlet area, and the casting speed were 120 mm, 40 mm × 55 mm, and 1.6 m/min, respectively, the changes in the mold flow field under nozzle outlet inclination angles of 12˚, 13˚, 14˚, and 15˚ were investigated. The flow field at the central longitudinal section of the mold is shown in Figure 13 . It can be seen that under the same conditions, when the inclination angle varied, the basic characteristics of the flow field inside the mold did not change significantly, and the distribution of the flow field remained largely similar.

The velocity distribution of the free-surface molten steel under different inclination angles is shown in Figure 14 . It can be observed that as the inclination angle increased, the maximum velocity of the free-surface molten steel showed a decreasing trend, with values of 0.45 m/s, 0.41 m/s, 0.41 m/s, and 0.40 m/s, respectively. When the inclination angle was relatively small, molten steel flowing out of the nozzle impacted the narrow face of the mold and generated a strong upward return flow. The high turbulence intensity of this upward return flow led to a greater velocity of the free-surface molten steel. The fluctuation of the free liquid surface under different nozzle outlet inclination angles is shown in Figure 15 . It can be seen that as the inclination angle increased, the molten steel surface fluctuation decreased. This was because a smaller inclination angle resulted in greater turbulence intensity of the upward return flow, which intensified the fluctuation of the liquid surface. When the nozzle outlet inclination angle was 15˚, the fluctuation range of the free liquid surface was between −4.00 mm and 3.00 mm, which fell within a reasonable range. Based on the above analysis, under the same conditions, the inclination angle of 15˚ can achieve a reasonable mold flow field.