Simulation Study on SF₆ Circuit Breaker Arc-Extinguishing Chamber Based on Lattice Boltzmann Method (LBM)

Abstract

The SF₆ circuit breaker is an essential piece of high-voltage equipment in ensuring the safe operation of the power grid. Regarding the arc-extinguishing chamber, as the most essential component, its performance is directly related to the breaking capacity of the circuit breaker. This study applies the Double Distribution Function Lattice Boltzmann Method (DDF-LBM), combined with the Smagorinsky sub-grid scale (SGS) model, to systematically simulate the dynamic breaking process of a 252 kV SF₆ arc-extinguishing chamber under 50 kA breaking current conditions. Two independent distribution functions are employed to describe the fluid field and the temperature field, respectively, thereby simulating the physical flow–heat coupling process. A dynamic simulation framework is constructed using the D2Q9 model to describe the mechanical motion of the contacts and the fluid flow. The description of contact movement is achieved by dynamically updating the geometric mesh, thereby realizing fluid–solid transformation. The research results indicate that the proposed method can simulate the pressure variation of the fluid field during the breaking process. The value of the Smagorinsky constant (C_s) exhibits a non-negligible influence on the pressure field predictions. The optimal value of C_s = 0.10 is determined through analysis, and the peak pressures at the upstream and throat measurement points reach 1.11 MPa and 1.37 MPa, respectively. Numerical simulations are conducted on the dynamic breaking process of the arc-extinguishing chamber, revealing the evolution of the pressure field upstream of the nozzle and at the throat regions. This study provides new numerical simulation methods for the investigation of SF₆ arc-extinguishing chambers and establishes a foundation for the application of the Lattice Boltzmann Method in the field of high-voltage electrical appliances.

1. Introduction

With the global energy transition and the rapid growth in electricity demands, the reliability and efficiency of high-voltage electrical equipment have become key pillars for the sustainable development of the power industry [1,2,3,4,5]. SF₆ gas has significant advantages in insulation and arc-extinguishing performance, which has attracted many researchers to conduct in-depth research on SF₆ circuit breakers [6,7,8,9,10]. The arc-extinguishing device is responsible for the current interruption and arc extinction processes, thus making the investigation of the arc-extinguishing mechanism especially significant [11]. The multi-physical interactions and chemical reactions during the current interruption process are extremely complex, making it highly difficult to predict the arc characteristics.

Investigations into arc models have been extensively conducted in both experimental and simulation domains. Rüegsegger et al. [12] analyzed the chemical composition of arc exhaust gases in SF₆ circuit breakers experimentally, revealing the complex decomposition products generated during the circuit-breaking process. Their findings provided meaningful insights for the development of models. Based on the classic Mayr model, Schavemaker and Van der Sluis [13] proposed a black box model, which can simulate the dynamic characteristics and arc current. The findings demonstrate a significant improvement in simulation accuracy. Brand and Jungblut [14] conducted a series of ion mobility experiments and identified the (9,6,4) potential. Mitchell et al. [15] proposed a two-dimensional laminar model; they applied an adaptive mesh and a controlled amount of artificial diffusion. The results indicated a strong correlation with the experiments. In order to interpret the phenomenon of shunt reactor switching transients, a practical arc model with the function of parameter estimation was established by Chang et al. [16]. Guan et al. [17] proposed an experimental method for measuring the transient gas pressure in 252-kV SF₆ circuit breakers. The experimental approach has inherent advantages regarding data accuracy, but there are unavoidable limitations in terms of high costs and measurement difficulties. Lin et al. [18] proposed a numerical model for predicting the dielectric recovery characteristic based on the streamer model. This model allows for the calculation of the temperature and electron density, significantly improving the accuracy and stability of simulations.

For numerical simulation, computational fluid dynamics (CFD) is a classic and important tool for modeling fluid flow dynamics, but it exhibits notable limitations in handling multi-physics-field coupling conditions. Therefore, magnetohydrodynamic (MHD) models on the basis of the standard CFD control equation system have been established, which couple the effects of electromagnetic fields. Zhang et al. [19] described the first attempt to propose a two-dimensional (2D) MHD model for the puffer-type arc-quenching chamber. With the development of research, a three-dimensional transient MHD model has been proposed [20]. This technical improvement facilitates the investigation of the underlying factors influencing arc stability. Furthermore, Wang et al. [21] performed experimental investigations to verify the accuracy and feasibility of the MHD-based simulation of a low-voltage circuit breaker.

Distinct from traditional simulations, the Lattice Boltzmann Method (LBM) is increasingly being recognized as a potential mesoscale approach by researchers. The LBM solves the macroscopic conservation equations by simulating the mesoscopic kinetics of a simplified particle system. This method has attracted extensive attention in the field of fluid mechanics [22,23], as well as in the plasma-jet field [24]. In thermal fluid simulations, a simplified thermal LBM model has been proposed. It achieves an effective balance between computational efficiency and acceptable accuracy [25]. The Double Distribution Function (DDF)-LBM method is capable of simulating argon arc plasmas [26]. In the field of circuit breakers, there are limited investigations so far, but the above studies have indicated its non-negligible potential. Sun et al. [27] applied the LBM to develop an arc model. This method has been proven to effectively simulate the arc process via experiments, and their findings provide a new perspective for the operation of circuit breakers. Cui et al. [28] analyzed the pressure characteristic distribution during the interruption process of a 252 kV SF₆ circuit breaker. The researchers concluded that the results showed strong consistency with experimental tests, and they noted considerable potential for further investigation. Capturing transient pressure fluctuations in a SF₆ circuit breaker during a 50 kA dynamic breaking process remains a significant challenge. To overcome this, investigating pressure evolution from a mesoscopic perspective is fundamental to understanding the macroscopic breaking performance. Based on the above, this study aims to develop a numerical simulation method for the dynamic breaking process of the SF₆ arc-extinguishing chamber based on the Double Distribution Function Lattice Boltzmann Method (DDF-LBM). By solving the distribution functions of the fluid and temperature fields, the challenge of coupling multiple physical fields is effectively addressed. Considering the transient interruption process of the circuit breaker and the high velocity of the gas flowing through the narrow nozzle, the system exhibits strong turbulence. This study introduces the Smagorinsky sub-grid scale (SGS) model to improve the simulation accuracy by adjusting the relaxation time. Furthermore, this article systematically analyzes the influence of the Smagorinsky constant C_s on the simulation results, providing a basis for the selection of C_s values. This work uses the LBM method to simulate the pressure characteristics during the breaking process of the SF₆ circuit breaker arc-extinguishing chamber, providing direct technical support for the engineering application of the LBM method in high-voltage electrical appliances.

2. Numerical Methods

2.1. LBE Evolution Equation and DDF Model

This section sequentially introduces the basic details of the mathematical methods used for the purpose of this study. The core numerical approach is based on the Lattice Bhatnagar–Gross–Krook (LBGK) model [22]. This model constitutes a discrete velocity formulation of the continuous Boltzmann equation, which governs the evolution of the particle distribution function f (c, x, t) in phase space.

$${\displaystyle \frac{\partial f}{\partial t}}+c·\nabla f+{\displaystyle \frac{F}{m}}·{\nabla }_{c}f=\mathsf{\Omega }$$

(1)

where c is the particle velocity, F stands for the external force, m is the mass of a particle, ∇_c refers to the velocity gradient operator, and Ω is the collision operator.

While the core of the conventional LBM addresses mass and momentum conservation, accurately simulating an arc-extinguishing chamber requires the addition of an energy conservation model to capture the intense heating from the arc. The Double Distribution Function (DDF) Lattice Boltzmann Method (LBM) serves as the core numerical framework to simulate the mass and heat transfer of SF₆ gas. It enables the efficient and stable simulation of the fluid dynamics and heat transfer processes in the arc-extinguishing chamber. The procedure of this mesoscopic method begins with discretizing the velocity, space, and time. Specifically, the standard two-dimensional (2D) nine-velocity (D2Q9) model is used to simulate the flow by employing nine discrete velocity directions (Figure 1). Figure 1 shows that only nine possible directions (i = 0 to 8) for the movement and collision of fluid particles exist in the LBM, which correspond to the nine points in the square grid. The expression of the discrete velocity vector e_i is

$$e_i=\left\{\begin{array}{cc}\left(0,0\right)& i=0\\ \left[\cos {\displaystyle \frac{\left(i-1\right)\pi }{2}},\sin {\displaystyle \frac{\left(i-1\right)\pi }{2}}\right]& i=1,2,3,4\\ \sqrt{2}\left[\cos {\displaystyle \frac{\left(2i-1\right)\pi }{4}},\sin {\displaystyle \frac{\left(2i-1\right)\pi }{4}}\right]& i=5,6,7,8\end{array}\right\}$$

(2)

The continuous Boltzmann equation is then transformed into an easily computable discrete lattice Boltzmann equation (LBE) via the Bhatnagar–Gross–Krook (BGK) approximation, which simplifies the collision operator into a relaxation process governed by a single relaxation time. In the BGK model, the particle distribution function is assumed to relax toward its local equilibrium at a rate determined by a single relaxation time. The resulting discrete evolution equation is expressed as

$$f_i\left(x+e_i∆t,t+∆t\right)=f_i\left(x,t\right)+{\displaystyle \frac{1}{{\tau }_f}}\left[{f_i}^{eq}-f_i\right]+∆t·F_i$$

(3)

where f_i is the flow field distribution function in direction i, τ_f is the flow field relaxation time, e_i is the discrete velocity, f_i^eq is the equilibrium distribution, and F_i represents external forces. The DDF model adopted in this study uses two independent distribution functions: f (for mass and momentum) and g (for temperature). Although both functions evolve according to the BGK collision operator towards their local equilibrium states, they are governed by distinct relaxation times (τ_f and energy field relaxation time τ_g) to account for their physical processes, such as viscous dissipation in the flow and heat conduction in the temperature field. The equation used to describe the evolution process of the energy distribution function from the current state g_k (x, t) to the next moment g_k (x + e_kΔt, t + Δt) is

$$g_k\left(x+e_k∆t,t+∆t\right)=g_k\left(x,t\right)+{\displaystyle \frac{1}{{\tau }_g}}\left[{g_k}^{eq}-g_k\right]+∆t·Q_k$$

(4)

where Q_k denotes the discrete energy source term; it is used to inject the macroscopic volumetric heat generated by the arc into the evolution process of the microscopic distribution function in an appropriate mathematical form. In the loaded case, arc heating is represented by a prescribed equivalent volumetric heat source. The macroscopic source term is defined as

$$Q\left(x,t\right)=Q_0{f}_t\left(t\right)\varphi \left(x\right)$$

(5)

where Q₀ is the reference source magnitude, f_t(t) is the time-dependent activation function, and ф(x) is the normalized spatial distribution function of the arc-heating region. In the present implementation, the source starts at t = 15 ms, reaches its maximum at t = 25 ms, and then decreases linearly to zero over the following 10 ms.

$$f_t\left(t\right)=\{\begin{array}{c}0, t<15 \mathrm{ms},\\ {\displaystyle \frac{t-15}{10}}, 15 \mathrm{ms}\le t\le 25 \mathrm{ms},\\ 1-{\displaystyle \frac{t-25}{10}}, 25 \mathrm{ms}<t\le 35 \mathrm{ms},\\ 0, t>35 \mathrm{ms}.\end{array}$$

(6)

The heat source is applied in the inter-contact/nozzle throat region, as highlighted in Figure 2. In the numerical implementation, spatial deposition is prescribed as a throat-centered Gaussian-type heating region with its center located near the nozzle throat. The source width in both directions is controlled by the same characteristic spread parameter used in code implementation. The peak source magnitude is determined from the equivalent arc power divided by the prescribed arc volume, yielding Q₀ ≈ 1.06 × 10¹¹ W/m³. To avoid non-physical overheating, the source magnitude is further limited by an upper bound of 5 × 10¹¹ W/m³ in the numerical implementation. The discrete source term Q_k in the DDF-LBM energy equation is constructed from this prescribed macroscopic heat source. Therefore, the loaded case results should be interpreted as the thermo-fluid response of the interrupter under an imposed equivalent arc-heating condition. The thermal balance in a circuit breaker arc is governed by the difference between the Joule heat generated by the high current flowing through the plasma resistance and the radiant heat energy radiated from the high-temperature plasma to the surroundings. The equilibrium distribution function is expressed as

$$f_i^{eq}=\rho {\omega }_i\left[1+{\displaystyle \frac{\left(e_i\cdot \mathit{u}\right)}{{\theta }_s^2}}+{\displaystyle \frac{(e_i\cdot \mathit{u}{)}^2}{2{\theta }_s^4}}-{\displaystyle \frac{{\mathit{u}}^2}{2{\theta }_s^2}}\right]$$

(7)

$$g_i^{eq}=T{\omega }_i\left[1+{\displaystyle \frac{\left(e_i\cdot \mathit{u}\right)}{{\theta }_s^2}}+{\displaystyle \frac{(e_i\cdot \mathit{u}{)}^2}{2{\theta }_s^4}}-{\displaystyle \frac{{\mathit{u}}^2}{2{\theta }_s^2}}\right]$$

(8)

where u is the macroscopic velocity; ω_i denotes the weight coefficients (standard D2Q9 model: 4/9 for the center, 1/9 for axial directions, 1/36 for diagonal directions); θ_s is the lattice speed of sound; and e_i is the discrete velocity vector. In the DDF-LBM, the equilibrium distributions f_i^eq and g_i^eq are directly computed from macroscopic quantities (density ρ, velocity u, and temperature T) using low-order expansions of the Maxwell–Boltzmann distribution. By applying the Chapman–Enskog expansion to f_i and g_k, the LBE recovers the macroscopic continuity equation (${\displaystyle \frac{\partial \rho }{\partial t}}+\nabla ·\left(\rho u\right)=0$), Navier–Stokes momentum equation (${\displaystyle \frac{\partial \left(\rho u\right)}{\partial t}}+\nabla ·\left(\rho uu\right)=-\nabla \mathrm{P}+\nabla ·\left(\tau \right)+F$), and convection–diffusion energy equation (${\displaystyle \frac{\partial T}{\partial t}}+\mathbf{u}\cdot \nabla T=\alpha {\nabla }^{2}T+Q$).

2.2. Macroscopic Reconstruction

The macroscopic physical quantities of density ρ, velocity u, and temperature T are reconstructed by the following equations:

$$\rho =\sum _i{f}_i$$

(9)

$$\rho u=\sum _i{f}_i{e}_i$$

(10)

$$T=\sum _k{g}_k$$

(11)

The coupling of the DDF-LBM is primarily reflected in the feedback and driving mechanisms of key parameters. For instance, the flow velocity u is substituted into the equilibrium distribution function g_i^eq, and the thermal field counteracts the fluid field from dynamically calculating τ_f and ρ, thus achieving comprehensive thermal–flow coupling. It should be noted that the standard LBM is inherently constrained to low-Mach-number and low-compression regimes. The SF₆ arc-extinguishing process involves intense localized heating that induces severe thermal expansion and density variations. In this study, the localized high-compression effects can be reasonably approximated without violating the core Chapman–Enskog expansion stability to ensure the validity of the DDF-LBM framework.

2.3. Dynamic Physical Properties

In Equations (3) and (4), the relaxation times τ_f and τ_g are significant parameters that control the viscosity ν and thermal diffusion coefficient α, enabling them to adapt to the real-time changes in the arc-extinguishing chamber environment.

$${\tau }_f={\displaystyle \frac{\nu }{{c_s}^2∆t}}+0.5$$

(12)

$${\tau }_g={\displaystyle \frac{\alpha }{{c_s}^2∆t}}+0.5$$

(13)

where c_s is the lattice sound velocity, equal to ${\displaystyle \frac{1}{\sqrt{3}}}$. The established LBM framework enables the real-time verification of the viscosity and thermal diffusivity within the arc-extinguishing chamber by detecting changes in ρ and T that affect the relaxation time τ_f and τ_g values, thereby ensuring physical fidelity. Since the ideal gas law is not applicable to SF₆ under high-temperature and high-pressure conditions, the transportation coefficient β is incorporated to modify its behavior for non-ideality:

$$P=\rho RT/\beta$$

(14)

where R is the gas constant. Using this equation, the changes in the temperature field are fed back in real time to the pressure gradient, thereby driving the evolution of the airflow field. The coefficient β serves as a reference mapping factor to reconcile the real gas behavior with the lattice isothermal EOS. In this study, β is expressed as a constant based on the initial reference state to simplify the model:

$$\beta ={\displaystyle \frac{R{T}_{ref}}{{c_s}^2}}$$

(15)

where T_ref is the reference temperature. This ensures that the lattice pressure evolution is consistent with the physical pressure at the reference temperature, while allowing the model to effectively capture macroscopic pressure wave propagation.

2.4. Smagorinsky Sub-Grid Scale (SGS) Model

Large eddy simulation (LES) is a turbulence simulation method. In order to capture the turbulence within the arc-extinguishing chamber, an explicit Smagorinsky sub-grid scale (SGS) model is implemented by dynamically updating the local relaxation time in this study. Therefore, the Reynolds number (Re), as an important indicator, is calculated during the simulation process to detect turbulence. Based on the Smagorinsky model, the eddy viscosity correction method is applied to transform the dissipative effect of turbulence into an effective viscosity ν_eff that varies in space and time. It is expressed as

$${\nu }_{eff}=\nu +{\nu }_t$$

(16)

where ν and ν_t indicate the molecular viscosity and the turbulent eddy viscosity, respectively. The value of ν_t is related to the strain rate tensor (S), which is calculated using a formulation derived from the Smagorinsky model:

$${\nu }_t={\left(C_S∆\right)}^2\left|\mathit{S}\right|$$

(17)

where C_S is the Smagorinsky constant, determined as 0.1; Δ is the lattice filter width, which is equal to 1.0. S is used to describe the deformation rate of the fluid, indicating the symmetrical part of the gradient tensor of the fluid velocity field. The definition is

$$\mathit{S}=\mathbf{0.5}{(\nabla \mathit{u}+\left(\nabla \mathit{u}\right)}^T)$$

(18)

Therefore, the local strain rate tensor is synchronously calculated based on the current macroscopic velocity u. Since the viscosity is directly related to the relaxation time in the LBM, the effective relaxation time τ_eff is

$${\tau }_{eff}={\displaystyle \frac{{\nu }_{eff}}{{c_s}^2∆t}}+0.5$$

(19)

Then, injecting Equation (16) into Equation (19), we obtain

$${\tau }_{eff}={\displaystyle \frac{3{\nu }_t}{∆t}}+\left(0.5+{\displaystyle \frac{3\nu }{∆t}}\right)={\displaystyle \frac{3{\nu }_t}{∆t}}+{\tau }_f$$

(20)

In the collision process, the value of τ_eff directly governs the evolution of f_i, thereby allowing the turbulence model to indirectly influence the dynamic physical properties and the energy field g_i.

3. Modeling Methodology

3.1. Geometric Model

The arc-extinguishing chamber of a 252 kV SF₆ circuit breaker with a breaking current of 50 kA is simulated in this study. Referencing [27], a simplified geometric model (shown in Figure 2) with dimensions of 220 mm × 72 mm is manually constructed to represent the physical arc chamber domain. The blank area is the calculation area, the moving and stationary contacts are displayed as grey rectangles, and the insulated wall is three-sectioned (with annotations in Figure 2). The areas AB and EF are gas inlets, and CD, MN, KL, OP, are gas outlets. To ensure numerical accuracy and resolve the steep pressure gradients near the nozzle throat, a global grid refinement factor of 2 is applied, resulting in a fine mesh of 440 × 144 grids in total. This results in uniform physical lattice spacing of Δx = 0.5 mm. The physical dimensions are scaled by a refinement factor to convert millimeter units into grid indices.

In SF₆ circuit breaker simulations, the geometric structure evolves in real time with the separation motion of the contacts. Considering the relative motion in the arc-extinguishing chamber, this simulation models the breaking process by changing the length of the stationary contacts (from 220 mm to 0). This movement directly changes the boundary shape and flow channels of the fluid domain. In the LBM, the numerical labels 0 and 1 are implemented to distinguish between fluid regions and solid boundaries, respectively. Initialized as an (N_x × N_y) NumPy array, the computational domain is filled with a default value of 0. Solid structures, including the stationary insulating walls and moving contacts, are generated through array slicing and geometric operations. The stationary contacts are modeled as two rectangular blocks located within a specific range of y coordinates, and the x position is updated over time based on a specified motion curve (in Figure 3) for the simulation of the breaking process of the circuit breaker. In addition, Figure 3 provides the pressure variation curve of the gas inlet over time.

3.2. Initialization and Simulation Workflow

The initial conditions for the arc-extinguishing chamber are defined by the physical properties of the SF₆ gas under standard operating conditions, with pressure of 0.7 MPa and a temperature of 300 K. The key thermophysical properties of SF₆ are as follows: the specific gas constant R_SF6 = 56.9 J/(kg·K), the kinematic viscosity ν = 2.56 × 10⁻⁶ m²/s, the adiabatic sound speed c_s = 137.03 m/s, and the thermal diffusivity α = 1.0 × 10⁻⁴ m²/s. To bridge the physical domain and the lattice domain in the LBM framework, dimensional scaling factors are employed for accurate mapping while ensuring numerical stability. The physical parameters are converted to lattice units via scaling factors:

$$L_r={\displaystyle \frac{L_{phys}}{L_{lu}}}$$

(21)

$$u_r={\displaystyle \frac{c_s}{{\theta }_s}}$$

(22)

L_r is the length scaling factor; it converts physical lengths (L_phys) to lattice grid units (L_lu). u_r is the velocity scaling factor, and the value of the lattice sound speed θ_s is constant, equal to 1/√3. In addition, the velocity scale multiplier U_r is set to bridge the physical and lattice domains; it is expressed as

$$U_r={\displaystyle \frac{U_{phy}}{u_{latt}}}$$

(23)

where U_phy denotes the characteristic physical velocity, and u_latt is the corresponding velocity in lattice units. In this study, u_latt is determined as 0.1. This strategy ensures that the value of u_latt throughout the computational domain remains significantly lower than θ_s. Consequently, the maximum local lattice Mach number (M_a = |u_latt|/c_s,latt) is effectively suppressed, ensuring that the simulation remains within the stable, weakly compressible regime. The time-scaling factor is calculated based on Equations (21) and (22):

$$t_r={\displaystyle \frac{L_r}{u_r}}$$

(24)

Density conversion is expressed as

$${\rho }_r={\displaystyle \frac{{\rho }^{\prime }}{\rho }}$$

(25)

where ρ is the lattice unit (=1), and ρ′ is the physical density, derived from the equation ${\rho }^{\prime }={\displaystyle \frac{p_0}{R_{{SF}_6}T}}$.

3.3. Boundary Conditions

This simulation employs the DDF approach, which governs the momentum and energy fields via two distinct distribution functions. To ensure physical fidelity, boundary conditions must be consistently defined for both fields. In the context of circuit breaker analysis, specifically for current interruption and arc quenching, various boundary conditions are considered along the flow path to meet the requisite physical constraints. Accordingly, the following sections will address the boundary conditions for the fluid and energy fields separately, following the flow path.

For the pressure boundary, a time-dependent Zou–He pressure boundary is implemented at the inlet to simulate the compression of SF₆ gas by the puffer cylinder. This provides the primary driving force for the gas blast via the equation $P=\rho {c_s}^2$. The Mei–Luo–Shyy (MLS) method is implemented to simulate the motion of the contacts. It dynamically updates the moving boundary and ensures proper momentum exchange between the solid contacts and the fluid field. The no-slip bounce-back boundary condition is used for the solid wall area; it reverses the momentum of fluid particles at the boundary, resulting in zero macroscopic velocity at the wall. At the outlet zone, the gradient extrapolation boundary condition is applied. It simulates the state downstream of the arc-extinguishing chamber leading to the atmosphere and allows the pressure waves and gas flow to exit the domain without non-physical reflections.

The energy boundary condition manages the dissipation of arc-generated heat and governs the energy exchange between the hot plasma and the surrounding cold gas. To simulate the cooling mechanism of the circuit breaker, an isothermal boundary is applied at the inlet, maintaining a constant ambient temperature of 300 K to represent the injection of fresh, cold SF₆ gas. Conversely, the nozzle walls and electrode surfaces are modeled as adiabatic boundaries, reflecting the assumption that heat conduction into the solid materials is negligible during the transient 50 ms quenching process. It is worth mentioning that, although the arc region is not a spatial boundary, the arc effect is represented by a prescribed equivalent volumetric heat source coupled to the energy field. The mathematical definition of this source term is provided in Section 2.1.

4. Results and Analysis

The simulation results of the dynamic fluid field in the SF₆ circuit breaker arc-extinguishing chamber are presented in this section. The LBM was adopted as the numerical methodology to simulate the complex interruption process. The explicit SGS model was employed to capture the turbulence effect to improve the simulation’s stability and the accuracy of the transient compressible flow. Sensitivity tests of the Smagorinsky constant C_S were also conducted to investigate the influence on the magnitude of the sub-grid-scale (SGS) viscosity.

4.1. Validation Strategy

In this section, the simulation results of the generated model are compared with the experimental results [17]. To verify the correctness of the numerical implementation and the boundary condition treatment, simulations were first conducted under no-load conditions, where arc-induced heating and the thermo-fluid coupling effects are not considered.

Table 1. Comparison of maximum pressure upstream of nozzle and throat under no-load breaking conditions.

Maximum Pressure (MPa)
Upstream		Throat
Test	LBM	Test	LBM
1.05	1.13	1.00	1.10

shows the pressure characteristics of the peak pressure upstream of the nozzle and nozzle throat measurement points. The mean absolute percentage error is calculated as 8.81%. This deviation is acceptable for validating the feasibility of the simulation model. Figure 4 shows the simulation results of the LBM upstream of the nozzle and the nozzle throat of the arc-extinguishing chamber during the no-load interruption process. It can be seen that the initial pressure of the system remains stable, with the pressure at both observation points maintained at approximately 0.7 MPa (absolute pressure). Then, the upstream pressure exhibits a relatively smooth and monotonous upward trend, reaching a global maximum (1.13 MPa) at about 25 ms, while the pressure at the throat increases more significantly and exhibits obvious fluctuations at the peak value (1.10 MPa). After this, the contact distance further increases, and the gas in the jet is fully released, causing the upstream pressure to decrease and eventually return to the base pressure state. The numerical simulation results capture the key physical phenomena during the no-load breaking process, including the initial contact separation, the post-nozzle opening, and the late nozzle-opening stages. This process indicates that the basic flow solver and moving boundary treatment are correctly implemented.

4.2. Effect of Smagorinsky Constant on Pressure Evolution

Within the LBM framework, the effect of C_s is manifested in the effective relaxation time, thus governing the resulting numerical dissipation. To systematically investigate the influence of the C_s value on the pressure response in the arc-extinguishing chamber, this study designed five sets of comparative simulation cases, with the C_s values set at 0.06, 0.08, 0.10, and 0.12 and a scenario without SGS (corresponding to C_s = 0). The quantitative comparison of the pressure responses under different C_s values is summarized in

Table 2. Pressure response characteristics under different C_s values.

Cs	Upstream Peak Pressure (MPa)	Throat Peak Pressure (MPa)
0 (No SGS)	0.97	1.40
0.06	0.82	1.26
0.08	0.90	1.32
0.1	1.11	1.37
0.12	0.891	0.92

. The peak pressure in the no-SGS case (C_s = 0) is 1.40 MPa at the throat and 0.97 at the upstream measurement points. As the C_s value increases from 0.06 to 0.10, the peak pressure at the throat exhibits an incremental trend (from 1.26 MPa to 1.37 MPa), while, as the C_s value increases from 0.10 to 0.12, the throat peak pressure declines to 0.92 MPa; for the upstream measurement point, it measures 0.82 MPa at C_s = 0.06, increases to 0.90 MPa at C_s = 0.08, reaches a maximum of 1.11 MPa at C_s = 0.10, and drops to 0.89 MPa at C_s = 0.12. According to the experimental findings presented in [17], the simulation correlates better with the actual situation at C_s values of 0 and 0.1. When C_s is too small, the turbulent viscosity becomes insufficient, resulting in inadequate energy transfer efficiency. As C_s = 0.10, an optimal balance is achieved, which effectively simulates turbulent fluctuations without causing excessive energy dissipation.

Figure 5 provides the pressure variation under different C_s values at both the upstream and throat measurement points during the process of breaking. Close inspection of the case without considering the SGS model indicates that the pressure evolution experiences intense oscillations and multi-peak conditions. Without SGS, the turbulent fluctuations in the flow field remain under-resolved, which contradicts physical reality and leads to excessive pressure accumulation. In contrast, the case with C_s = 0.1 (highlighted by the solid line) exhibits a single dominant pressure rise stage, which is more consistent with the test results.

4.3. Dynamic Pressure Response

In this section, the pressure characteristics in the arc-extinguishing chamber during an interruption are simulated through the proposed DDF-LBM-based model. As described in Section 2 and Section 3, the dynamic pressure response of the arc-extinguishing chamber was simulated for the loaded 50 kA case, in which the arc effect was represented by a prescribed equivalent volumetric heat source introduced into the energy equation. The source was activated from 15.0 ms to 35.0 ms, corresponding to the assumed arc-burning interval. Figure 6 presents the temporal variation in the pressure increment at the upstream location and the throat location during the process of breaking with a Smagorinsky constant C_S of 0.1. Generally, the pressure responses at both measurement points exhibit typical unsteady characteristics, including an initial slow increase, a subsequent rapid rise to the peak, and a later decaying oscillation, while noticeable differences are observed in terms of the peak pressure amplitude and dynamic response characteristics. It can be seen that the pressure at both the upstream and throat locations marginally increases with nearly identical magnitudes at the initial 10 ms. Figure 7 shows the spatial distribution of the pressure field at 10 ms, 20 ms, and 30 ms. Close inspection of Figure 7 shows that the pressure wave has not yet propagated on a large scale at 10 ms. This indicates that the flow remains in a quasi-steady state and is mainly governed by the inlet pressure condition. In this stage, the contacts have only recently begun to separate, and the effective flow area of the nozzle is limited. Subsequently, the nozzle gradually opens, where a rapid rise in pressure is observed. The peak pressure reaches 1.35 MPa at 32 ms at the throat and 1.1 MPa at the upstream observation point at 25 ms. This observation indicates the formation of a critical throttling condition, and the 7 ms difference between the two measurement points reflects the process of pressure wave propagation from the upstream to the throat position. It can be explained by the fact that the thermal expansion effect dominates, overwhelming the Bernoulli effect. Figure 7 indicates a high-pressure area that appears in the central region and is spreading to both sides at 20 ms, and the visible interference patterns and complex wave structures can be observed. In the 30 ms snapshot of the fluid flow, the pressure distribution becomes further complicated and diffuses downstream. The throat forms a minimum effective flow area, limiting the mass flow rate; moreover, with the continuous gas supply from upstream, the throat becomes the controlling section, leading to strong local compression and a sharp increase in pressure. It can be concluded that the pressure response is not simply determined by the inlet conditions but is jointly governed by local geometric constraints and unsteady compression effects. Then, in the later stage of opening, the stationary contact no longer blocks the main nozzle, reducing the flow resistance and enabling more effective gas discharge. The chamber begins to relieve the overall pressure, and the pressure gradually decays with reduced fluctuations.

5. Conclusions

This study focuses on the arc-extinguishing chambers of SF₆ circuit breakers. It employs the Lattice Boltzmann Method (LBM) coupled with an explicit Smagorinsky sub-grid scale (SGS) model to simulate the fluid field and investigate the pressure variation characteristics inside the arc-extinguishing chamber. This method provides a new technical option for the numerical simulation of high-voltage electrical equipment, and the following main conclusions can be drawn.

• This method employs the D2Q9 dual distribution function model to simultaneously solve for the fluid field and temperature field, enabling it to capture the thermodynamic coupling effects during the arc extinction process. The explicit SGS model is applied to simulate turbulence and control the numerical dissipation intensity.
• The simulation results indicate the significance of the application of the SGS model in simulating turbulence. The Smagorinsky constant value C_s shows a significant and non-monotonic effect on the pressure response of the arc-extinguishing chamber. The optimal C_s value is determined as 0.10 according to the simulation results. The strong dependence of the pressure response on C_s also indicates that the accuracy of numerical simulation is crucial in predicting the interruption performance of circuit breakers. The optimal selection of the C_s value has been proposed, filling the research gap in this field.
• To capture the pressure characteristics, two measurement points were arranged, located at the upstream of the nozzle and throat sections. The peak pressure at these two measurement points under the condition of C_s = 0.10 reached 1.11 MPa (at the upstream) and 1.37 MPa (at the throat). Based on the simulation results, the pressure field within the arc-extinguishing chamber can be divided into four stages: the initial stage (t < 20 ms), the rising stage (20 ms ≤ t < 25 ms), the peak stage (25 ms ≤ t < 35 ms), and the decay stage (t ≥ 35 ms).
• This study adopted a simplified 2D representation of the arc chamber in the simulations, introducing certain limitations. The proposed model provides a scientific basis for the application of the LBM in the simulation of plasma in arc-extinguishing chambers and offers useful suggestions for the selection of turbulence model parameters. This study advances the application of the LBM in the field of high-voltage electrical appliances.