Energy Transfer Characteristics of Surface Vortex Heat Flow Under Non-Isothermal Conditions Based on the Lattice Boltzmann Method

Abstract

During liquid drainage from intermediate vessels in various industrial processes such as continuous steel casting, aircraft fuel supply, and chemical separation, free-surface vortices commonly occur. The formation and evolution of these vortices not only entrain surface slag and gas, but also lead to deterioration of downstream product quality and abnormal equipment operation. The vortex evolution process exhibits notable three-dimensional unsteadiness, multi-scale turbulence, and dynamic gas–liquid interfacial changes, accompanied by strong coupling effects between temperature gradients and flow field structures. Traditional macroscopic numerical models show clear limitations in accurately capturing these complex physical mechanisms. To address these challenges, this study developed a mesoscopic numerical model for gas-liquid two-phase vortex flow based on the lattice Boltzmann method. The model systematically reveals the dynamic behavior during vortex evolution and the multi-field coupling mechanism with the temperature field while providing an in-depth analysis of how initial perturbation velocity regulates vortex intensity and stability. The results indicate that vortex evolution begins near the bottom drain outlet, with the tangential velocity distribution conforming to the theoretical Rankine vortex model. The vortex core velocity during the critical penetration stage is significantly higher than that during the initial depression stage. An increase in the initial perturbation velocity not only enhances vortex intensity and induces low-frequency oscillations of the vortex core but also markedly promotes the global convective heat transfer process. With regard to the temperature field, an increase in fluid temperature reduces the viscosity coefficient, thereby weakening viscous dissipation effects, which accelerates vortex development and prolongs drainage time. Meanwhile, the vortex structure—through the induction of Taylor vortices and a spiral pumping effect—drives shear mixing and radial thermal diffusion between fluid regions at different temperatures, leading to dynamic reconstruction and homogenization of the temperature field. The outcomes of this study not only provide a solid theoretical foundation for understanding the generation, evolution, and heat transfer mechanisms of vortices under industrial thermal conditions, but also offer clear engineering guidance for practical production-enabling optimized operational parameters to suppress vortices and enhance drainage efficiency.

1. Introduction

In various industrial processes such as continuous steel casting, aircraft fuel supply, nuclear power plant cooling, and chemical extraction separation, intermediate vessels serve as critical process equipment. They are responsible not only for fluid transport and transfer but also for regulating flow patterns and achieving precise temperature control. However, during actual operation, significant temperature differences exist between the internal flow field and the external environment, leading to complex heat exchange between the vessel walls and the fluid. This results in an uneven internal temperature distribution and generates unpredictable thermal disturbance forces. Under these conditions, free surface vortices frequently occur during the downward drainage process in the confined space of these vessels. These vortices represent a gas-liquid two-phase, dynamically coupled rotational flow structure formed at the free liquid surface under the combined effects of gravity, fluid properties, and external perturbations [1,2,3]. Such vortices are not merely simple fluid rotations but constitute complex flow field motions involving energy and momentum transfer. Their generation and evolution significantly alter the internal flow field structure and temperature distribution patterns within the vessel, making the temperature field—already influenced by temperature differences—more complex and difficult to control precisely [4,5,6]. More critically, due to their strong entrainment capacity, free surface vortices readily draw surface slag, oxide scale, and other light impurities toward the outlet. This not only contaminates downstream equipment and products but also, when combined with the effects of the non-uniform temperature field, collectively impacts the final temperature and purity of the fluid at the outlet [7,8,9,10]. Consequently, the chain reaction triggered by free surface vortices has become a major bottleneck constraining the improvement in key industrial indicators, including the quality of continuous cast slabs, the fuel efficiency of aircraft engines, the operational safety of nuclear power plants, and the purity of chemical products, thereby significantly impacting the overall quality and operational efficiency of modern industrial production [11,12].

To achieve precise control of fluid temperature during the drainage process, it is imperative to conduct systematic and in-depth research on the evolution mechanism of surface vortices and the dynamic distribution patterns of their velocity and temperature fields [13,14]. This requires tracing the physical origin of vortices, revealing their formation conditions, structural characteristics, and evolution paths under temperature differences, and subsequently exploring effective technical approaches to improve drainage conditions and regulate outlet fluid temperature. Such research will provide theoretical foundations and technical support for real-time vortex monitoring and active flow field control in complex industrial environments [15]. In non-isothermal flow fields, the entire evolution process of free surface vortices—from generation and development to sustained existence—involves a series of complex physical phenomena, including dynamic deformation of the free surface, coupling of multiphase media, and unsteady convective heat transfer [16,17,18]. As vortex flow fields exhibit coupled spatiotemporal multiscale characteristics, significant three-dimensional unsteadiness, and strong nonlinear turbulent features, combined with bidirectional coupling and mutual interference between velocity and temperature fields, the entire system demonstrates high dynamic complexity and physical uncertainty. Currently, there is a lack of mature theoretical models capable of characterizing these processes while remaining suitable for engineering applications. This deficiency hinders effective quantitative analysis and reliable prediction of vortex structural evolution and its thermal-flow coupling behavior, thereby constraining further improvement in temperature control precision and operational efficiency in relevant industrial processes [19,20].

Over the past five years, experimental and numerical research on free surface vortices (FSV) and air entrainment mechanisms in pump sumps/intakes has advanced rapidly, with substantial contributions from scholars worldwide. Zhang investigated the influence of bell mouth submergence depth on the spatiotemporal evolution and pressure fluctuation characteristics of FSVs, revealing that shallower submergence leads to faster vortex evolution cycles and larger pressure distortion amplitudes [21]. Li et al. developed an improved thermal multiphase lattice Boltzmann (LB) model, enabling efficient and accurate simulation of the thermal-dynamic coupling behavior in free surface vortices [22]. Yuan et al. constructed a turbulence/free-surface interaction model derived from direct numerical simulation (DNS) to predict turbulence-induced bubble entrainment. This model requires no adjustable constants and demonstrates strong robustness when applied to complex flow problems in full-scale ships [23]. Zhang et al.employing high-speed visualization and pressure testing, explored the spatiotemporal evolution and pressure fluctuation characteristics of sidewall-attached vortices (SAVs) in enclosed pump pits, identifying three distinct evolutionary structures of SAVs [24]. Deng et al. utilized linear non-modal analysis and the energy budget equation to uncover a linear mechanism wherein the free surface, at high Froude numbers, enhances the transient growth of very-large-scale motions (VLSMs) through the coupling of surface waves with shear eigenmodes (generating energy) [25]. Zhang et al. studied the effectiveness of jet flow in suppressing vortex-induced vibrations (VIV) of a near-free-surface circular cylinder, finding that shallower submergence enhances suppression. This is achieved through the synergistic effect of jet flow disrupting the wake vortices and utilizing the high-pressure region near the free surface [26]. Ali et al. utilized particle image velocimetry (PIV) to investigate the dynamic behavior of swirling flow within a 180° sharp bend, elucidating the evolution and merging mechanisms of Dean vortices under the combined action of pressure and centrifugal forces [27]. Zhang et al. investigated the evolution process and dynamic behavior of a cavitating vortex ring approaching a free surface. The study revealed that the surface drives the acceleration of the vortex ring by disrupting its pressure balance and clarified the physical principle whereby this dynamic mechanism couples with the cavitation process to form a high-speed jet [28]. Amar et al. combined experimental methods with three-dimensional numerical simulations to study unsteady surface vortices and their gas entrainment process, validating the model’s high accuracy [29].

A review of the existing literature reveals that the current research focus remains primarily on characterizing macroscopic parameters such as the vortex formation mechanism, flow field structure, and velocity distribution. While these works lay a crucial foundation for understanding the steady-state behavior of the flow, there is a lack of systematic and in-depth investigation into the nonlinear coupling mechanisms involving the temperature, pressure, and velocity fields. This coupling often exhibits strong spatiotemporal multiscale characteristics, and its inherent non-equilibrium and nonlinear feedback processes are difficult to accurately describe using simple linear superposition or one-way coupling models. At the numerical simulation level, traditional theoretical frameworks based on macroscopic conservation equations face fundamental challenges when addressing such complex multi-field coupled physics problems [30]. Under conditions involving complex shear, severe deformation, and phase change, their accuracy and robustness significantly decline, leading to substantial errors in the source term calculations for mass, momentum, and energy exchange between phases. This ultimately constrains the accurate prediction and control of transient behaviors in complex thermal-fluid systems.

The evolution of free surface vortices demonstrates highly complex interfacial morphology and flow field characteristics, necessitating specialized boundary treatment methods in numerical simulations. The requirement to incorporate real-time evolving flow fields as an integral component of the solution process substantially increases computational complexity. Capitalizing on continuous advancements in mesoscopic theory, the lattice Boltzmann method (LBM) provides an effective approach for addressing such challenges. This study employs LBM to simulate free surface vortex evolution under central drainage conditions at mesoscopic scale, revealing distribution patterns and evolutionary mechanisms of vortex velocity fields. Specifically, we employ a double-distribution-function (DDF) LBM model combined with VOF-based interface tracking to simulate both flow and thermal fields in a fully coupled manner, providing a more physically consistent representation of the heat and mass transfer during vortex evolution. Furthermore, by integrating convective heat transfer theory, we investigate the coupling mechanism between velocity and temperature fields during vortex evolution, establishing the dynamic patterns of outlet fluid temperature influenced by vortex dynamics. Unlike most prior studies that focused on isothermal or steady-state vortex behavior, this research develops a mesoscopic LBM-based model that captures the three-dimensional unsteady evolution of free-surface vortices with real-time thermal coupling. Our analysis includes an explicit investigation into how initial perturbation velocity regulates vortex intensity, core stability, and low-frequency oscillations, and how this in turn influences global convective heat transfer. Crucially, we investigate the bidirectional coupling mechanism between temperature and vortex dynamics, showing that temperature alters fluid viscosity and viscous dissipation, which feedback into vortex development and drainage time [31,32,33].

In summary, this research provides significant academic value through: (1) development of numerical models for free surface vortex fluid dynamics, (2) analysis of vortex heat and mass transfer mechanisms, and (3) exploration of vortex suppression methodologies. These contributions establish theoretical foundations and technical pathways for effective vortex control and drainage process optimization in complex thermal environments. At the practical engineering level, the research outcomes enable accurate prediction and active control of free surface vortex fields in complex environments, thereby improving fluid flow conditions at outflow regions. This advancement carries substantial practical implications for enhancing process quality and operational efficiency across multiple industrial sectors including metal metallurgy, aircraft fuel supply, and chemical separation systems.

2. Mathematical Model

The free surface vortex is a gas-liquid two-phase flow pattern widely existing in nature. Its formation process is influenced by factors such as static pressure, initial perturbation velocity, position of the container outlet, and fluid physical properties. An in-depth study of the theoretical model for vortex evolution holds significant guiding importance for understanding the heat and mass transfer characteristics in non-isothermal flow fields.

2.1. Lattice Boltzmann Model

The lattice Boltzmann equation (LBE) is a discrete form of the Boltzmann equation within a finite difference framework. It describes the macroscopic behavior of fluids by simulating the propagation and collision processes of fluid packets. This method originates from molecular dynamics and reflects macroscopic properties arising from interactions between fluid molecules in a statistical manner, such as viscosity, thermal conductivity, thermal expansion coefficient, and surface tension [34,35,36]. Due to the vast number of fluid molecules, directly solving their kinetic equations is computationally prohibitive. To reduce complexity, statistical mechanics extends the molecular scale to the packet scale, enabling representation of molecular motion characteristics while maintaining macroscopic continuity at the mesoscopic level [37,38].

The entire simulation in this study was indeed performed using the LBM, which is the sole numerical framework adopted. Within this framework, the kinetic behavior of the fluid can be characterized by the particle distribution function f(r, c, t):

$$dN=dx{\displaystyle \int f(\mathit{r},t,\mathit{c})}d\mathit{c}$$

(1)

Here, f represents the particle number density at position r, velocity c, and time t, with dN being the number of particles within the spatial volume dr and velocity interval dc. Macroscopic fluid properties can be obtained from the moments of the distribution function:

$$\rho =m{\displaystyle \int f(\mathit{r},t,\mathit{c})}d\mathit{c}$$

(2)

$$\rho \mathsf{u}=m{\displaystyle \int \mathbf{c}}f(\mathit{r},t,\mathit{c})d\mathit{c}$$

(3)

where ρ is the fluid density, u is the velocity vector, and m is the molecular mass.

To enhance numerical accuracy, a double-distribution-function (DDF) model is adopted to describe the flow and thermal fields, respectively [39]. To simplify computation, the Bhatnagar–Gross–Krook (BGK) collision operator model is used. In this model, intermolecular collisions drive the system toward equilibrium, leading to the Boltzmann equation:

$${\partial }_{t}f+(\mathsf{c}\cdot \mathrm{\nabla })f=\mathrm{\Omega }\left(f\right)+\mathbf{F}$$

(4)

$${\partial }_{t}g+(\mathbf{c}\cdot \mathrm{\nabla })g=\mathrm{\Omega }(g)$$

(5)

Here, Ω(f) and Ω(g) are collision operators, F denotes the external force term, and f and g are the single-particle density and temperature distribution functions. The energy distribution function g and collision operators Ω(f) and Ω(g) are defined as follows:

$$g={\displaystyle \frac{{(\mathit{c}-\mathit{u})}^2}{2}}f$$

(6)

$$\mathrm{\Omega }(f)={\displaystyle \frac{1}{{\tau }_f}}[f^{eq}-f]$$

(7)

$$\begin{array}{l}\mathrm{\Omega }(g)={\displaystyle \frac{{(\mathit{c}-\mathit{u})}^2}{2}}\mathrm{\Omega }(f)-fZ={\displaystyle \frac{{(\mathit{c}-\mathit{u})}^2}{2}}\mathrm{\Omega }(f)-f(\mathit{c}-\mathit{u})({\partial }_t\mathit{u}+(\mathit{c}\cdot \mathrm{\nabla })\mathit{u}={\displaystyle \frac{g^{eq}-g}{{\tau }_g}}\end{array}$$

(8)

where τ_f and τ_g are the relaxation times; f^eq and g^eq are the equilibrium distribution functions for the velocity and temperature fields, respectively. The corrected distribution functions are expressed as follows:

$${\tilde{f}}_i=f_i+{\displaystyle \frac{0.5dt}{{\tau }_f}}(f_i-f_i^{eq})-0.5dt{F}_i$$

(9)

$${\tilde{g}}_i=g_i+{\displaystyle \frac{0.5dt}{{\tau }_g}}(g_i-g_i^{eq})-0.5dt{G}_i$$

(10)

Based on the Boltzmann equation, the discrete distribution functions f_i and g_i are defined as follows:

$$f_i={\displaystyle \frac{{\tau }_f}{{\tau }_f+0.5dt}}{\tilde{f}}_i+{\displaystyle \frac{0.5dt{f}_i^{eq}}{{\tau }_f+0.5dt}}+{\displaystyle \frac{0.5{\tau }_f{F}_{i}dt}{{\tau }_f+0.5dt}}$$

(11)

$$g_i={\displaystyle \frac{{\tau }_g}{{\tau }_g+0.5dt}}{\tilde{g}}_i+{\displaystyle \frac{0.5dt{g}_i^{eq}+0.5{\tau }_g{G}_{i}dt}{{\tau }_g+0.5dt}}$$

(12)

where c_i is the discrete lattice velocity in the i-th direction:

$${\mathit{e}}_i=b_l\left[\begin{array}{ccccccccccccccccccc}0& 1& -1& 0& 0& 0& 0& 1& -1& 1& -1& 0& 0& 1& -1& 1& -1& 0& 0\\ 0& 0& 0& 1& -1& 0& 0& 1& -1& 0& 0& 1& -1& -1& 1& 0& 0& 1& -1\\ 0& 0& 0& 0& 0& 1& -1& 0& 0& 1& -1& 1& -1& 0& 0& -1& 1& -1& 1\end{array}\right]$$

(13)

The D3Q19 model is adopted in this work. The equilibrium distribution functions f_i^eq and g_i^eq for the flow and thermal fields are given by:

$$f_i^{eq}(\mathit{r},t)={\omega }_i\left[{\displaystyle \frac{p_f}{b_s}}+{\rho }_0({\displaystyle \frac{3{\mathit{c}}_i\cdot \mathit{u}}{b_l^2}}+{\displaystyle \frac{4.5{({\mathit{c}}_i\cdot \mathit{u})}^2}{b_l^4}}-1.5{\displaystyle \frac{\mathit{u}\cdot \mathit{u}}{b_l^2}})\right]$$

(14)

$$g_i^{eq}(\mathit{r},t)=T{\omega }_i\left[1+{\displaystyle \frac{3{\mathit{c}}_i\cdot \mathit{u}}{b_l^2}}+{\displaystyle \frac{4.5{({\mathit{c}}_i\cdot \mathit{u})}^2}{b_l^4}}-1.5{\displaystyle \frac{\mathit{u}\cdot \mathit{u}}{b_l^2}}\right]$$

(15)

Here, $b_s=b_l/\sqrt{3}$ is the lattice sound speed, and ω_i is the weighting factor. T is the fluid temperature, and ρ₀ denotes the initial density. The fluid pressure P_f, velocity u, and temperature T are defined as follows:

$${\omega }_i=\left\{\begin{array}{c}\begin{array}{cc}0& i=0\end{array}\\ \begin{array}{cc}1/18& i=1–6\end{array}\\ \begin{array}{cc}1/36& i=7–18\end{array}\end{array}\right.$$

(16)

$$p_f=b_s^2{\displaystyle \sum _{i=0}^{18}{f}_i}$$

(17)

$${\rho }_0\mathit{u}={\displaystyle \sum _{i=0}^{18}{c}_i{f}_i}+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathit{F}dt$$

(18)

$$T={\displaystyle \sum _{i=0}^{18}{g}_i}+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}Qdt$$

(19)

where F represents the body force acting on the fluid and Q is the heat source. The terms F_i and G_i in Equations (11) and (12) are expressed as follows:

$$F_i=(1-{\displaystyle \frac{1}{2{\tau }_f}}){\omega }_i({\displaystyle \frac{{\mathit{c}}_i-\mathit{u}}{b_s^2}}+{\displaystyle \frac{({\mathit{c}}_i\cdot \mathit{u}){\mathit{c}}_i}{b_s^4}})\cdot \mathit{F}$$

(20)

$$G_i=(1-{\displaystyle \frac{1}{2{\tau }_g}}){\omega }_i\cdot Q$$

(21)

The relaxation time factors τ_f and τ_g are defined as follows:

$${\tau }_f={\displaystyle \frac{u_c{L}_c}{Re{b}_s^{2}dt}}+0.5$$

(22)

$${\tau }_g={\displaystyle \frac{{\tau }_f-0.5}{Pr}}+0.5$$

(23)

where u_c and L_c denote the characteristic velocity and length of the fluid, respectively, Re is the Reynolds number, and Pr is the Prandtl number.

2.2. LBM–N–S Relationship

The Boltzmann equation at the mesoscopic scale forms the theoretical foundation of the LBM. It describes the kinetic behavior of fluid molecules under specific assumptions, providing a bridge between microscopic dynamics and macroscopic fluid motion [40]. To establish the correspondence between the LBM and the Navier–Stokes equations, the Lattice Boltzmann equation is expanded using a Taylor series.

$$dt{D}_i{f}_i(\mathit{r},t)+{\displaystyle \frac{1}{2}}d{t}^2{D}_i^2{f}_i(\mathit{r},t)=-{\displaystyle \frac{1}{\tau }}\left[f_i(\mathit{r},t)-f_i^{eq}(\mathit{r},t)\right]$$

(24)

By applying the Chapman–Enskog multi-scale expansion, multiple time and spatial scales are introduced:

$$t=\epsilon t_1+{\epsilon }^2{t}_2+\dots +{\epsilon }^{n_b}{t}_{n_b}$$

(25)

$$\mathit{r}=\epsilon {\mathit{r}}_1+{\epsilon }^2{r}_2+\dots +{\epsilon }^{n_b}{\mathit{r}}_{n_b}$$

(26)

The distribution function is then expanded as follows:

$$f_i=f_i^{(0)}+\epsilon f_i^{(1)}+{\epsilon }^2{f}_i^{(2)}+\dots +{\epsilon }^{n_b}{f}_i^{(n_b)}$$

(27)

Substituting the above expansions into the Taylor-expanded Lattice Boltzmann equation and collecting terms by order of ε, the zeroth and first moments yield the macroscopic conservation equations.

$${\displaystyle \frac{\partial \rho }{{\partial }_t}}+\mathrm{\nabla }\cdot (\rho \mathit{u})=0$$

(28)

$${\displaystyle \frac{\partial \rho \mathit{u}}{{\partial }_t}}+{\mathrm{\nabla }}_1\cdot (\rho \mathit{u}\cdot \mathit{u})=-\mathrm{\nabla }\cdot p+\mathrm{\nabla }\cdot \sigma +O({\mathit{u}}^3)$$

(29)

$${\sigma }_{ij}=\rho \upsilon ({\displaystyle \frac{\partial {\mathit{u}}_i}{\partial {\mathit{r}}_j}}+{\displaystyle \frac{\partial {\mathit{u}}_j}{\partial {\mathit{r}}_i}})$$

(30)

$$\upsilon =(\tau -0.5)b_l^{2}dt={\displaystyle \frac{1}{3}}(\tau -0.5){\displaystyle \frac{d{\mathit{r}}^2}{dt}}$$

(31)

Hence, the LBM can recover the macroscopic mass, momentum, and energy conservation equations from simple discrete evolution equations at the mesoscopic level.

2.3. Boundary Conditions and Immersed Boundary Method

Boundary conditions are crucial for numerical stability and computational accuracy. In complex geometries, traditional bounce-back boundary schemes may fail to maintain momentum conservation. Therefore, the immersed boundary mthod (IBM) is adopted in this study to handle the coupling between the solid wall and the fluid. The velocity and temperature update equations at fluid nodes are expressed as follows:

$$\begin{array}{cc}{\mathit{u}}^{*}={\displaystyle \frac{1}{{\rho }_0}}{\displaystyle \sum _{i=0}^{18}{\mathit{c}}_i{f}_i}& d\mathit{u}={\displaystyle \frac{\mathit{F}dt}{2{\rho }_0}}\end{array}$$

(32)

$$\begin{array}{cc}{T}^{*}={\displaystyle \frac{1}{{\rho }_0}}{\displaystyle \sum _{i=0}^{18}{g}_i}& dT={\displaystyle \frac{Qdt}{2}}\end{array}$$

(33)

In the IBM framework, the solid boundary is represented by Lagrangian points rb, while the fluid domain uses a Eulerian grid. Information transfer between the two is realized through a discrete delta function:

$$\mathit{F}({\mathit{r}}_f^k)={\displaystyle \sum _{j=1}^{n_b}\mathit{F}({\mathit{r}}_b^j)}D({\mathit{r}}_f^k-{\mathit{r}}_b^j)\mathrm{\Delta }{s}_j$$

(34)

$$Q({\mathit{r}}_f^k)={\displaystyle \sum _{j=1}^{n_b}Q({\mathit{r}}_b^j)}D({\mathit{r}}_f^k-{\mathit{r}}_b^j)\mathrm{\Delta }{s}_j$$

(35)

Here, F and Q represent the momentum and heat source terms, respectively; the subscripts b and f denote the solid boundary and fluid particles; Δs_j is the surface area of the j-th Lagrangian point; n_b is the total number of Lagrangian points; and D(r_f − r_b) is the discrete delta function used to smooth the physical quantities across the fluid–solid interface:

$$D({\mathit{r}}_f-{\mathit{r}}_b)={\displaystyle \frac{1}{h_l^3}}d\left({\displaystyle \frac{X_f-X_b}{h_l}}\right)d\left({\displaystyle \frac{Y_f-Y_b}{h_l}}\right)d\left({\displaystyle \frac{Z_f-Z_b}{h_l}}\right)$$

(36)

where (X, Y, Z) denote the coordinates in the Eulerian and Lagrangian systems, respectively. Substituting Equations (35) and (36) into Equations (37) and (38) yields:

$$d\mathit{u}({\mathit{r}}_f^k)={\displaystyle \sum _{j=1}^{n_b}d\mathit{u}({\mathit{r}}_b^j)}D({\mathit{r}}_f^k-{\mathit{r}}_b^j)\mathrm{\Delta }{s}_j$$

(37)

$$dT({\mathit{r}}_f^k)={\displaystyle \sum _{j=1}^{n_b}dT({\mathit{r}}_b^j)}D({\mathit{r}}_f^k-{\mathit{r}}_b^j)\mathrm{\Delta }{s}_j$$

(38)

where:

$$d(l)=\left\{\begin{array}{cc}\hfill \left(3-2\left|l\right|+\sqrt{1+4\left|l\right|-4l^2}\right)/8& \left|l\right|\le 1\hfill \\ \hfill \left(5-2\left|l\right|+\sqrt{-7+12\left|l\right|-4l^2}\right)/8& 1<\left|l\right|\le 2\hfill \\ 0& \left|l\right|>2\hfill \end{array}\right.$$

(39)

The no-slip wall boundary condition is applied to stationary boundaries. For fluid particles reaching the wall boundary, a bounce-back scheme is imposed, such that:

$$\mathit{u}({\mathit{r}}_b^j)={\displaystyle \sum _{k=1}^{n_f}\mathit{u}({\mathit{r}}_f^k)}D({\mathit{r}}_f^k-{\mathit{r}}_b^j)h_l^3$$

(40)

$$T({\mathit{r}}_b^j)={\displaystyle \sum _{k=1}^{n_f}T({\mathit{r}}_f^k)}D({\mathit{r}}_f^k-{\mathit{r}}_b^j)h_l^3$$

(41)

Wall temperature conditions are enforced via a thermal bounce-back scheme. For isothermal walls, the temperature distribution functions are set to the equilibrium values at the wall temperature. For adiabatic walls, the temperature gradient is set to zero by applying a symmetric boundary condition.

3. Implementation of Free Surface Vortex Numerical Model

3.1. Vortex Geometry and Numerical Model

This study establishes a numerical model for vortex simulation, with its specific structure and spatial arrangement illustrated in Figure 1a. The model centers around a cylindrical container connected to a vertical drainage pipe at the center of its base. The container has a total height of 0.5 m and a diameter of 0.4 m, ensuring adequate flow development while maintaining a manageable computational domain. The bottom drainage pipe is 0.1 m long with an inner diameter of 0.03 m; this slender pipe structure plays a crucial role in establishing stable discharge flow during the drainage process. The lower red region in the container is initialized as the water phase, while the upper blue region is designated as the continuous gas phase. A predefined tangential velocity component is imposed within the water phase region as the initial perturbation velocity field, effectively inducing initial rotational momentum in the liquid. This strategy significantly enhances both the physical realism of the numerical simulation and the controllability of the initial flow field development. Figure 1b displays the numerical model of the vortex within the confined physical space, with its corresponding mesh scheme shown in Figure 1c. All numerical simulations in this study were conducted using the commercial software XFlow, which is based on the LBM. The LBM computation employs an adaptive lattice, where the base spatial resolution is uniformly set to 3 mm. Two automatic refinement levels are activated dynamically in regions exhibiting strong gradients, including the vortex core, shear layer, and free-surface interface. The time step is internally controlled according to LBM stability constraints, ensuring that the Courant number remains below 0.2 throughout the simulation.

To describe the dynamic behavior of the gas-liquid interface, a multiphase flow model based on the VOF method is selected. The free surface is captured using a VOF-like interface-tracking scheme within the LBM, in which the gas–liquid interface is reconstructed from the distribution functions without explicit mesh reconstruction [41,42]. The VOF and LBM are coupled in a unified framework where the interface is advected by the LBM velocity field. This model tracks the deformation, breakup, and coalescence of the free surface. Furthermore, to represent the vortex structures and energy transport mechanisms generated during vortex evolution, a turbulence model was coupled to the simulation, enabling a more realistic representation of the complete dynamic characteristics of the vortex from generation to dissipation.

3.2. Boundary and Initial Conditions

To address the flow characteristics of the swirling field, this study divides the computational domain into three types of physical boundaries: inlet, outlet, and walls. The inlet boundary is set at the gas injection zone on the top of the model, employing a pressure inlet condition to establish a stable gas-liquid interaction environment at the free surface. The outlet boundary is located at the end of the slender bottom drainage pipe, primarily responsible for liquid discharge and exhibiting gas entrainment when the swirling flow is fully developed. A pressure outlet condition with a no-backflow assumption is adopted to suppress reverse pressure gradients and backflow disturbances, accurately describing the free outflow behavior of the gas-liquid two-phase flow while ensuring numerical stability and physical rationality. All wall surfaces are treated with a no-slip boundary condition, supplemented by wall functions to resolve near-wall flow, thereby establishing a closed and physically sound numerical simulation system for swirling flows. Based on the established swirling flow model, mesh system, and boundary conditions, a pressure-based incompressible solver is selected for numerical simulation. This solver is suitable for pressure-dominated flows at low Mach numbers, particularly for physical scenarios involving rotation and coupled transport at the gas-liquid interface.

The coupling solution calculation flowchart of the LBM-LES method is shown in Figure 2. A transient LBM solver is employed to capture the unsteady evolution characteristics of the free-surface swirling flow throughout its entire process from generation to dissipation. Regarding thermal boundaries: Thermal Inlet: The gas phase at the top is set to ambient temperature. Thermal Outlet: The outflow is assumed to be thermally fully developed, with a zero-temperature-gradient condition. Initial Temperature: the liquid phase is initialized with a stratified temperature profile [43,44,45,46]. Both the momentum equations and turbulence transport equations are discretized using the second-order upwind scheme, which combines numerical stability with high resolution and low numerical dissipation, enabling accurate capture of key structural evolutions in the swirling field, such as shear layers and vortex core regions. In summary, the numerical setup adopted in this study balances computational accuracy, stability, and efficiency, providing a reliable simulation foundation for revealing the mechanisms of cross-scale heat transfer and flow dynamics in swirling flows.

3.3. Model Validation

To ensure the accuracy and reproducibility of the swirling flow numerical simulation results, this study selected Odgaard’s theoretical vortex model and the experimental data from SooHwang as validation benchmarks for systematic comparative analysis [47,48], with the results shown in Figure 3. The figure displays the tangential velocity distribution on a horizontal cross-section at height H = 0.01 m. It can be observed that in the region near the central rotation axis, the tangential velocity increases rapidly with radius, reaching a peak at a certain critical radius; as the radius further increases, the velocity gradually decays, overall exhibiting the characteristic structure of a classical Rankine vortex, namely the combination of rigid rotation and free vortex patterns. This trend shows strong agreement with the experimental data from SooHwang’s tidal power plant vortex simulations while also aligning with the theoretical predictions of Odgaard’s model. This validation not only qualitatively confirms the reliability of the present numerical model in capturing the flow field structural characteristics during vortex evolution but also, through quantitative comparison, highlights the model’s strong representational capability in handling the coupling mechanism between free and forced vortices. Consequently, it establishes a rigorously validated numerical simulation foundation for subsequent in-depth research on complex swirling flow problems involving cross-scale energy transfer and unsteady interactions between vortex structures.

4. Numerical Results and Analysis

4.1. Formation Mechanism of Two-Phase Vortex Flow Transport

To elucidate the dynamic mechanism of the vortex flow field, this section presents a numerical simulation of the vortex evolution process under ambient temperature conditions. Figure 4a–c show the cloud maps of the gas–liquid two-phase distribution at different time instants. It can be observed that a typical funnel-shaped vortex structure gradually develops above the bottom outlet and extends upward toward the free surface over time. A distinct interfacial boundary layer emerges at the gas–liquid interface, exhibiting dynamic evolution characterized by the progressive formation of surface depression and central gas entrainment. Hence, Figure 4d,e display the two-dimensional streamline distributions, revealing the synergistic interaction between tangential and radial velocities during vortex development. The streamlines exhibit nearly concentric circular patterns far from the vortex core, whereas near the vortex center, they transition into spiral contraction structures due to the enhancement in axial velocity. This observation further confirms the kinetic transition from a forced vortex to a free vortex during the evolution process. Overall, these results not only visualize the macroscopic morphology of vortex evolution but also, through the topological structure of streamlines, uncover the dynamic processes of momentum transport and energy conversion within the vortex. This provides a reliable visualized basis for subsequent quantitative analysis of vortex stability and energy dissipation mechanisms.

During the vortex evolution process shown in Figure 4, as the fluid is discharged through the bottom outlet, a pronounced concave deformation gradually forms at the center of the free surface, indicating that the flow has entered the air-entraining vortex stage (corresponding to Figure 4b). At this stage, an air core emerges within the vortex center and remains stably suspended inside the liquid, continuously extending downstream. Eventually, it penetrates the entire flow field and reaches the outlet, marking the establishment of a fully developed air-entrainment state. Notably, once the vortex becomes completely developed (as illustrated in Figure 4c), the flow pattern at the outlet undergoes a significant transformation: the liquid switches to a wall-adhering flow mode, forming an annular liquid film along the inner wall of the outlet pipe, while the central region of the pipe is occupied by a stable air core. This unique flow structure not only reduces the effective flow area of the outlet pipe—resulting in a noticeable decrease in drain rate—but also enhances the momentum and energy exchange efficiency across the gas-liquid interface due to the enlarged interfacial area and increased relative velocity between phases.

Figure 4d–f illustrate the evolution of the free-surface flow field. As the drainage proceeds, the initially quiescent surface streamlines begin to converge toward the outlet center, exhibiting a typical inward-spiraling pattern. The streamline density increases significantly near the vortex core region. This phenomenon indicates that, as the fluid continues to discharge, the decreasing radius of rotation leads to an increase in tangential velocity, causing the centripetal force to gradually overcome the centrifugal force generated by rotation, thereby driving the fluid toward the central axis. During this process, the vortex continuously entrains the upper-layer fluid downstream through a helical suction mechanism, resulting in a three-dimensional helical flow structure characterized by strong coupling between circumferential rotation and axial downward motion.

When the outlet is located at the geometric center of the container, the vortex exhibits strict axisymmetric characteristics, and the tangential velocity distribution is symmetric about the central axis. To investigate the evolution of flow-field dynamics during vortex development, two key stages are selected: the initial surface depression and the critical vortex penetration stage, as shown in Figure 5. Tangential velocity distributions are extracted at three representative heights—H₁ = 0.05 m, H₂ = 0.10 m, and H₃ = 0.15 m, as shown in Figure 5. The left side of the axis corresponds to the tangential velocity during the initial depression stage, while the right side shows the distribution at the critical penetration stage. Analysis reveals that the tangential velocity profiles at both stages exhibit the typical Rankine combined vortex structure: a solid-body rotation region near the core where velocity increases linearly with radius, and an outer region following the decay law of a free vortex. As the height decreases from H₃ to H₁, the tangential velocity peak increases significantly within the same stage, indicating a clear axial gradient in vortex intensity. Furthermore, at the same height, the tangential velocity during the critical penetration stage is markedly higher than that in the initial depression stage, revealing a continuous angular momentum concentration toward the vortex core during its evolution. As the vortex develops into the penetration stage, the rotational kinetic energy further intensifies, and the vorticity concentration effect in the core region becomes more pronounced, leading to higher tangential velocities at equivalent radial positions. This systematic analysis validates the theoretical model of vortex evolution under symmetric drainage conditions and provides quantitative insights into the mechanisms governing the development of vortex instability.

Based on the aforementioned observations and analysis, the following dynamic mechanism can be deduced. Driven by the initial perturbation velocity, the vortex structure first generates near the bottom outlet and subsequently develops and propagates axially upstream. As the vortex evolves, the tangential velocity of fluid particles near the core region initially increases sharply with radius before decaying—specifically, reaching a peak at a characteristic radius that defines the vortex’s core action region. In contrast, regions farther from the vortex center experience minimal rotational influence, maintaining consistently low tangential velocities without significant change. Furthermore, the radius of the free surface vortex is not constant but exhibits notable dynamic contraction during development. During initial formation, the vortex operates with a larger radius and a more dispersed rotational region. As the vortex progresses to the penetration stage, its radius significantly decreases, demonstrating redistribution of angular momentum both axially and radially toward the center. This reflects the continuous focusing of vortex energy into the core region throughout its evolution.

Figure 6 illustrates the spatial distributions of the three velocity components—Vx, Vy, and Vz—at two cross-sectional heights, 0.05 m and 0.15 m above the bottom, during the air-entrainment stage of the vortex. Comparative analysis indicates that although the overall variation trends of the velocity components are similar at different heights, there exist notable local discrepancies near the vortex core. In the vicinity of the core region, all three velocity components exhibit sharp variations, reflecting the coupled action of vortex rotation and axial suction. In contrast, in regions far from the vortex center, both the radial (Vx) and tangential (Vy) components approach zero, suggesting that the influence of the vortex is spatially localized.

From the perspective of variation along the axial direction, the Vx component shows minimal difference between the two heights, indicating that the radial velocity distribution is relatively uniform along the vertical axis. The Vy component exhibits the most pronounced change, showing a clear axial gradient—its magnitude increases significantly as the height decreases, revealing a non-uniform distribution of rotational intensity along the vortex axis. The Vz component displays intermediate variation but demonstrates a distinct axial acceleration trend. Comprehensive analysis shows that the vortex evolution originates from the bottom central region, where both tangential and axial velocities increase progressively as the drainage continues. The closer to the bottom, the stronger the velocity intensification effect becomes. This phenomenon elucidates the physical mechanism of the vortex during the air-entrainment stage: the intense suction near the outlet continuously draws in fluid from the surrounding region around the vortex core, forming a high-energy, steep-gradient flow structure localized near the bottom. This reflects the spatial localization and axial attenuation characteristics of the suction-induced flow field within the vortex.

Figure 7 present the radial distribution characteristics of tangential velocity in the flow field under different initial disturbance velocities. At the early stage of drainage, the circumferential flow dominated by inertial forces exhibits an alternating positive–negative velocity distribution symmetric about the central axis. The overall velocity variation is smooth, and no distinct extrema are observed. As the free-surface vortex forms and develops, the flow energy gradually concentrates toward the center, leading to high-speed rotation within a very small radius, where a pronounced velocity peak region emerges.

Comparative analysis of the tangential velocity peak positions at different evolution stages reveals a noticeable offset between the vortex core and the geometric outlet center, indicating the inherent dynamic instability of the vortex structure in high-speed rotating flows. The spatial position of the vortex core undergoes low-frequency random oscillations over time. Further investigation shows that the amplitude of this oscillation is positively correlated with the initial disturbance velocity—a larger disturbance velocity results in a greater vortex-core deviation amplitude, accompanied by intensified pressure fluctuations near the outlet. This pressure-pulsation effect, induced by the unsteady oscillation of the vortex, not only affects the stability of the vortex structure, but may also have a significant influence on the downstream flow patterns and energy dissipation mechanisms, thereby providing critical insight into the processes of energy transport and instability during vortex evolution.

Figure 8 illustrates the complex vortex system composed of upward-stacked Taylor vortices and penetrating vortices near the central outlet region. The formation of a strong axial flow at the outlet represents a dynamic response to angular momentum conservation when the fluid migrates inward from the sidewalls under the influence of gravity. The results indicate that the interaction between the axially dominated vortex flow and the circumferentially driven Taylor vortices enhances the shear effect among fluid layers, increases the mixing efficiency, and consequently strengthens both the global flow-field mixing intensity and the convective heat transfer performance. In addition, due to the static pressure imbalance induced by gravity during the drainage process, the outlet fluid experiences high-velocity discharge. Combined with the tangential inertial forces triggered by initial disturbances, the fluid is driven to move downward along a helical trajectory, ultimately forming a self-organized swirling structure. This finding provides essential theoretical insight into the multi-scale vortex coupling mechanisms in rotating flow fields and offers valuable guidance for the optimization of engineering fluid transport processes.

During the drainage process, regions closer to the central outlet exhibit significantly enhanced angular velocity and energy gradients, leading to a simultaneous increase in tangential and axial velocities in the flow field above the outlet, forming a high-speed suction effect. As the liquid level continuously drops and gravitational potential energy can no longer maintain adequate fluid supply from the periphery to the outlet, the energy balance of the flow field is disrupted. The liquid surface undergoes depression evolution and eventually forms a penetrating air core structure that occupies the physical space of the outflow passage. After the vortex fully penetrates, the outlet flow rate sharply decreases. At this stage, the high tangential velocity in the core region intensifies the fluid’s circumferential rotational kinetic energy, promoting the transport of centrally accumulated energy to the surroundings through the generation and diffusion of local vortices. This flow field reconstruction process, driven by energy gradients and dominated by vortex dissipation, reveals the dynamic characteristics of energy redistribution in rotating flow fields. It also provides a crucial theoretical basis for enhancing mixing and optimizing mass transfer through flow field control in engineering applications, demonstrating significant application value particularly in complex flow systems involving coupled abrupt energy gradient changes and rotational instability.

4.2. Effect of Temperature on Flow Field Evolution

In practical industrial operations, the temperature of the flow field often undergoes significant variations due to the combined effects of reaction heat exchange, ambient temperature fluctuations, and process control adjustments. The key thermophysical properties of the fluid are strongly temperature-dependent. During the formation and evolution of a free-surface vortex, the fluid motion is governed by the coupled action of multiple forces, including gravity, viscous forces, and surface tension. Temperature variations, by altering the thermophysical parameters of the fluid, further influence the vortex stability, structural morphology, and energy dissipation characteristics. Therefore, a systematic investigation into the regulatory mechanisms by which temperature variations affect the vortex flow structure and dynamic behavior is of great theoretical and practical significance. On one hand, it deepens the understanding of the vortex evolution mechanism under multi-physical-field coupling conditions; on the other hand, it provides explicit engineering guidance for flow control and heat-transfer optimization in industrial systems.

To investigate the influence of temperature on the vortex flow field evolution mechanism, Figure 9 displays the vorticity distribution characteristics in isothermal flow fields under different temperature conditions. The figure reveals that as the flow field temperature gradually increases, the vorticity intensity in the region above the outlet significantly strengthens, and the vortex core structure becomes more concentrated. This indicates that temperature changes substantially alter the tangential velocity distribution during vortex evolution, thereby systematically affecting the vorticity distribution and transport behavior in the surrounding area. From a physical mechanism perspective, vortex formation essentially results from a dynamic balance between centrifugal forces generated by the fluid under combined initial perturbation and gravity. At lower temperatures, the fluid possesses a higher viscosity coefficient, and the stronger viscous dissipation effect significantly hinders the circumferential and axial movement of fluid parcels, thereby delaying the vortex formation process. As temperature increases, weakened intermolecular forces lead to reduced viscosity, diminishing the resistance that viscosity imposes against centrifugal forces. This allows the fluid to more readily organize effective rotational motion under the constraint of angular momentum conservation, significantly accelerating the vortex evolution process. Consequently, elevated temperature, by reducing fluid viscosity, effectively promotes fluid momentum transfer and rotational organization capability, making it easier to form structurally stable free surface vortices during drainage. Simultaneously, as vortex development becomes more thorough and the gas entrainment stage prolongs, the total drainage time correspondingly increases. This discovery not only reveals the intrinsic relationship between temperature, physical properties, and vortex dynamics but also provides a theoretical basis for optimizing drainage efficiency through temperature regulation in industrial processes.

To elucidate the heat transfer mechanisms and their influence on fluid mixing behavior in swirling flow fields, this study obtained temperature isosurface distributions at three characteristic temperatures (331 K, 328 K, and 320 K) during vortex evolution through numerical simulation, as shown in Figure 10. During the initial vortex formation stage, the flow field exhibits a clear thermal stratification structure: due to thermal buoyancy effects, the highest temperature region concentrates near the liquid surface, while the bottom region maintains the lowest temperature due to proximity to heat dissipation boundaries, establishing a stable temperature gradient between them. As the flow progresses to the vortex air-entrainment stage, surface-level high-temperature fluid is rapidly directed toward and discharged through the outlet under strong vortex suction. Simultaneously, mid-level temperature fluid experiences a significant retention effect due to the coupled action of Coriolis and centrifugal forces generated by the vortex velocity field, maintaining the longest residence time within the container. The heat transfer mechanism during this stage primarily manifests as intense mixing between upper high-temperature fluid and mid/lower-level fluid under strong shear action, thereby achieving rapid vertical heat transport.

When the vortex develops into the gas extraction stage, the accumulated low-temperature fluid at the bottom begins to participate in the discharge process. However, residual low-temperature fluid remains observable in peripheral areas distant from the outlet. These retained low-temperature fluids gradually integrate into the mainstream temperature field through progressive mixing and thermal diffusion with the intermediate transition zone fluid, ultimately completing the dynamic reconstruction of the temperature distribution throughout the entire container. This spatiotemporal evolution sequence of the temperature field not only visually demonstrates the crucial regulatory role of vortex structures in fluid heat transfer processes—by organizing specific flow patterns to achieve directional transport and redistribution of fluid from different temperature zones—but also reveals the complex mixing mechanisms experienced by fluid layers at different temperatures during vortex evolution. This includes the synergistic effects of physical processes such as shear mixing, retention effects, and progressive diffusion, providing important visual evidence for deepening the understanding of heat-mass coupled transport characteristics in rotating flow fields.

Figure 11 presents the trajectories of fluid particles and their corresponding temperature distributions (represented by particle color) at various heights during three distinct evolutionary stages: the vortex air-entrainment stage, the gas extraction stage, and the late drainage stage. Under the coupled mechanism of vortex suction and rotation, upper-layer high-temperature fluid particles are continuously drawn into the outlet. As the drainage process advances and the liquid surface gradually descends, the temperature of fluid particles near the outlet undergoes significant changes, reflecting the strongly non-equilibrium characteristics of local heat transport. Particularly after the vortex fully penetrates, the axial velocity in the flow field weakens to some extent. Meanwhile, dominated by centrifugal force, fluid particles in the near-bottom central region begin to spread radially outward from the center. This motion behavior effectively disrupts the original thermal boundary layer structure and diminishes the stratified characteristics of the temperature distribution, thereby systematically enhancing the convective heat transfer efficiency throughout the entire flow field and achieving multi-scale coordinated heat transport in both radial and axial directions.

5. Conclusions

This study established a gas-liquid two-phase vortex numerical model based on the lattice Boltzmann method to investigate the evolutionary dynamics of vortices under central drainage conditions and their complex coupling mechanisms with the temperature field. The main research findings can be summarized in the following three aspects.

(1)

Revealed vortex evolutionary dynamics and flow field structural characteristics. The vortex flow field strictly follows axisymmetric characteristics, with its tangential velocity distribution exhibiting the classical Rankine combined vortex structure. Vortex energy continuously concentrates toward the core during evolution, manifested as dynamic contraction of the vortex core radius and significantly higher peak tangential velocity during the critical penetration stage compared to the initial depression stage. This reflects the redistribution and concentration effect of angular momentum during vortex development.

(2)

Quantified the crucial regulatory role of initial perturbation on vortex behavior. The initial perturbation velocity is a key parameter controlling vortex intensity and stability. Increased perturbation velocity directly enhances vortex intensity and promotes earlier transition to fully developed stage. Furthermore, high perturbation conditions induce spatial instability of the vortex core position, directly leading to intensified pressure fluctuations near the outlet, posing potential risks to downstream flow field stability. This provides an important basis for vibration and noise control in engineering applications.

(3)

The bidirectional coupling mechanism between the temperature field and vortex evolution has been systematically elucidated. This study reveals that temperature regulates the physical pathway of vortex evolution by altering the thermophysical properties of the fluid. Under low-temperature conditions, the higher fluid viscosity induces stronger viscous dissipation, which suppresses the rotational motion of fluid microclusters and consequently delays the formation and development of the vortex. Conversely, as the temperature increases, the viscosity decreases, enabling the fluid to organize more effective rotational motion under the constraint of angular momentum conservation, thereby accelerating the vortex evolution process and prolonging the drainage duration.