Simulation of Corner Solidification in a Cavity Using the Lattice Boltzmann Method

Abstract

This study investigates corner solidification in a closed cavity in which the left and bottom walls are kept at a temperature lower than its initial temperature. The liquid material in the cavity initially lies at its phase transition temperature and, due to cold boundary conditions at the left–bottom walls, solidification starts. The simulation of corner solidification was performed using a kinetic-based lattice Boltzmann method (LBM), and the tracking of the solid–liquid interface was captured through the evaluation of time. The present investigation addresses the effect of natural convection over conduction across a wide range of higher Rayleigh numbers, from 10⁶ to 10⁸. The total-enthalpy-based lattice Boltzmann method (ELBM) was used to observe the thermal profiles in the entire cavity with a two-phase interface. The isotherms reveal the relative dominance of natural convection over conduction, and the pattern of interface reveals the effective growth of the solidified layer in the cavity. To quantify the uniformity of cooling, a coefficient of variation (COV) for the thermal field was calculated in the effective solidified zone at a wide range of Ra. The results show that the value of COV increases with Ra and reduces with time. The thermal instability in the flow field is also quantified through FFT analyses.

1. Introduction

Solidification from an adjacent surface is often encountered in materials processing applications such as injection moulding. Uniformity in cooling is often required to ensure a better solidification process, which can be achieved by carefully designing the mould.

The process of melting–solidification in a closed cavity has been studied by several researchers due to its vast applications in industrial and engineering applications. Numerical study of solid–liquid phase change has been performed by applying conventional methods such as the finite difference method (FDM), finite volume method (FVM), and finite element method (FEM) [1,2,3,4,5,6]. As the solid–liquid phase change problem includes the thermal and flow behaviour in the solid and liquid regions and the tracing of the solid–liquid interface is difficult, several new techniques are available to easily solve the two-phase problem. The lattice Boltzmann method (LBM), a kinetic-based approach, is one such technique, popular for studying the solid–liquid phase change phenomena due to its ease of implementation and simple collision-streaming phenomena. The interface co-existing in the solid–liquid region can be traced as a ‘sharp’ or diffuse’ interface, which can be addressed using the immersed boundary lattice Boltzmann method (IBLBM) or enthalpy-based lattice Boltzmann method (ELBM), respectively [7,8,9,10]. ELBM is a popular technique over IBLBM for solving the solid–liquid phase change problem as it requires less grid resolution, which reduces the computational cost [11]. In ELBM, the energy equation in terms of temperature has been used for conduction- or convection-dominated phase change by several researchers, although it has required numerous iterations due to the presence of a non-linear term in the energy equation [12,13,14,15]. However, the total-enthalpy-based method introduced by Huang et al. [7] has been widely applied to solve the melting–solidification problem due to its total enthalpy term, which shows computational efficacy for the phase change problem [16,17,18]. Samanta et al. [19] reviewed the application of LBM to solid–liquid phase change problems of metals and alloys, discussing both pure LBM and hybrid LBM approaches to melting–solidification problems.

The above investigations were dedicated to one-way melting or solidification in which a single thermal gradient was present. ‘Corner melting’ or ‘corner solidification’ involves the heat absorption or rejection from two adjacent walls. In a closed cavity with the presence of opposite hot and cold walls, a dual temperature gradient is generated, ensuring faster melting or solidification for ‘corner melting’ or ‘corner solidification’, respectively. ‘Corner melting’ or ‘corner solidification’ is important in industrial applications such as material processing, additive manufacturing, continuous casting, etc., in which a better understanding of the basic metal behaviour is important during solidification. Studying corner solidification is also relevant for geological systems involving mantle convection in subduction zone involving magma flow [20].

However, limited works are available on the area of ‘corner melting’ or ‘corner solidification’ [21,22,23,24,25,26]. Recently, ‘corner melting’ of low-Prandtl-number materials was studied by Samanta et al. [18], in which they showed the effect of natural convection in the melting zone. The effect of natural convection on the melting boundary was addressed in their work with a quantitative assessment of convective instability in the melt region.

In recent years, the literature has shown that the effect of natural convection on the melt boundary is a very hot topic to researchers; flow and thermal instability analysis in the melt zone describes the effective melt behaviour over time [25,27,28]. The convective instability in the solidified layer is a concerned phenomenon that indicates the homogeneity of the solidified region. Instability analysis is performed on the solidified region to reveal the nature of the time–temperature history. Looking at the literature on solidification, it is found that relatively fewer investigations on solidification have been performed using LBM [21,22,23,25,29], and the inherent study of convective instability in the solidified layer should be the focus of further research.

In the present work, an attempt has been made to study corner solidification using the total-enthalpy-based lattice Boltzmann method at a wide range of high Rayleigh numbers. The effect of natural convection on the melting boundary was investigated Rayleigh numbers ranging from 10⁶ to 10⁸. The location of the interface was tracked over time with the variation in Rayleigh number. Thermal instability was studied in the effective solidified region, revealing temperature variations with time in the solidified zone. Statistical analysis was performed to evaluate the mean temperature and its standard deviation in the effective solidified layer with the variation in time at different Rayleigh numbers to calculate the co-efficient of variation (COV) of the thermal field in the solidified region. The value of COV in the solidified zone describes the uniformity of cooling in the solidified layers.

2. Mathematical Formulation

2.1. Model Description and Governing Equations

A closed square cavity is filled with low-Prandtl-number material, initially at its phase transition temperature (T_m). The cavity length is L while the width of the cavity is W. Two adjacent walls, the left and bottom cavity walls, are kept at a temperature (T_w) lower than the phase change temperature (T_m), while the top and right walls are insulated. The specified boundary condition of the cavity leads to solidification from the left–bottom walls. The schematic of the physical model is presented in Figure 1. As solidification starts from two adjacent left–bottom walls, the interface is moved towards the opposite corner of the closed cavity and a melt pool is observed in the top-right corner. The generation of natural convection due to gravity in the semi-liquid melt pool was observed. Dual temperature gradient presents in the entire cavity in which the temperature gradient between top and bottom walls is sharper over the temperature gradient between left and right walls as acceleration due to gravity has a greater influence in the former case. The fluid in the cavity is considered to be laminar, incompressible in nature, and natural convection is considered with Boussinesq approximation. Viscous dissipation is neglected during solidification. In the present solidification study, solidified layers grow with time, and the effective solidified area changes with time. The conventional transportation equations including the continuity equation, momentum and energy equations with source terms are given as follows [18]:

$${\displaystyle \frac{\partial {\rho }_m}{\partial t}}+\nabla .\left({\rho }_m{\mathbf{u}}_m\right)=0$$

(1)

$${\displaystyle \frac{\partial ({\rho }_m{\mathbf{u}}_m)}{\partial t}}+\nabla .\left({\rho }_m{\mathbf{u}}_m{\mathbf{u}}_m\right)=-\nabla p+\nabla .\left(\mu \nabla {\mathbf{u}}_m\right)+{\rho }_m{\mathbf{S}}_u$$

(2)

$${\displaystyle \frac{\partial \left({\rho }_m{c}_{p}T\right)}{\partial t}}+\nabla .\left({\rho }_m{c}_{p}T{\mathbf{u}}_m\right)=\nabla .\left(k\nabla T\right)+{\mathrm{S}}_{\mathrm{e}}$$

(3)

where $\rho$, u, t and p are density, velocity vector, time and pressure, respectively. T, c_p, μ and k are temperature, specific heat capacity, viscosity and thermal conductivity, respectively. Subscript ‘m’ is used for material. The source term in the momentum equation $\left({\mathbf{S}}_u\right)$ in Equation (2) can be expressed as [7,18]

$${\mathbf{S}}_u=\mathbf{g}\beta \left(T-T_{ref}\right)$$

(4)

where g and β are acceleration due to gravity and the co-efficient of thermal expansion, respectively. T_ref is the reference temperature.

The source term (S_e) in Equation (3) is written as [7,18]

$${\mathrm{S}}_{\mathrm{e}}=-\left[{\displaystyle \frac{\partial \left({\rho }_m\Delta H\right)}{\partial t}}+\nabla .\left({\rho }_m{\mathbf{u}}_m\Delta H\right)\right]$$

(5)

where ΔH is the change in latent enthalpy. For pure metal phase change, the convective term is negligible in the solidified zone [7], so Equation (4) may be written as [18]

$${\mathrm{S}}_{\mathrm{e}}=-{\displaystyle \frac{\partial \left({\rho }_m\Delta H\right)}{\partial t}}=-{\displaystyle \frac{\partial \left({\rho }_m\lambda f_l\right)}{\partial t}}$$

(6)

where λ is the latent heat of enthalpy and f_l is liquid fraction, respectively. Solid fraction in the present problem can be calculated as f_s = 1 − f_l.

To avoid the non-linearity term in Equation (3), substituting Equation (6) in Equation (3), the energy equation can be represented as [7] follows:

$${\displaystyle \frac{\partial \left({\rho }_{m}H\right)}{\partial t}}+\nabla .\left({\rho }_m{c}_{p}T{\mathbf{u}}_m\right)=\nabla .\left(k\nabla T\right)$$

(7)

The total enthalpy $H$ is expressed by sensible and latent heat transfer [16],

$$H=c_{p}T+f_l\lambda$$

(8)

The initial and boundary conditions are given as follows:

Initial condition:

$$\mathrm{at} t=0, u_m=v_m=0, \mathrm{T}={\mathrm{T}}_{\mathrm{m}} \mathrm{for} 0\le x\le L \mathrm{and} 0\le y\le W$$

(9a)

Boundary conditions:

$$\begin{array}{l}\mathrm{at} t>0,\\ u_m=v_m=0, \mathrm{T}={\mathrm{T}}_{\mathrm{w}} \mathrm{for} x=0 \mathrm{and} 0\le y\le W\\ u_m=v_m=0, \mathrm{T}={\mathrm{T}}_{\mathrm{w}} \mathrm{for} y=0 \mathrm{and} 0\le x\le L\\ u_m=v_m=0, {\displaystyle \frac{\partial \mathrm{T}}{\partial x}}=0 \mathrm{for} x=L \mathrm{and} 0\le y\le W\\ u_m=v_m=0, {\displaystyle \frac{\partial \mathrm{T}}{\partial y}}=0 \mathrm{for} y=W \mathrm{and} 0\le x\le L\end{array}$$

(9b)

The dimensionless numbers used in the present work are [18]

$${\mathrm{T}}^{*}={\displaystyle \frac{\mathrm{T}-{\mathrm{T}}_{\mathrm{w}}}{{\mathrm{T}}_{\mathrm{m}}-{\mathrm{T}}_{\mathrm{w}}}}, x^{*}={\displaystyle \frac{x}{L}}, y^{*}={\displaystyle \frac{y}{W}}$$

(10)

$$\mathrm{Prandtl} \mathrm{number}, Pr={\displaystyle \frac{\nu }{\alpha }}$$

$$\mathrm{Rayleigh} \mathrm{number}, Ra={\displaystyle \frac{\mathbf{g}\beta \Delta \mathrm{T}{W}^3}{\nu \alpha }}$$

$$\mathrm{Stephan} \mathrm{number}, \mathrm{Ste}={\displaystyle \frac{c_p\Delta \mathrm{T}}{\lambda }} \mathrm{and}$$

$$\mathrm{Fourier} \mathrm{number}, \mathrm{Fo}={\displaystyle \frac{\alpha t}{W^2}}$$

where ${\mathrm{T}}^{*}$ is dimensionless temperature; $x^{*}$ and $y^{*}$ are the dimensionless coordinates at $x$ and $y$ direction, respectively; $\nu$ is kinematic viscosity; $\alpha$ is thermal diffusivity; and $t$ is time. The asterisk mark (*) is dropped for the rest of the paper, which specifies the non-dimensional form of the variable.

The non-dimensionless forms of Equations (1)–(3) are given following [13,30]

$$\nabla .{\mathbf{u}}^{*}=0$$

(11)

$${\displaystyle \frac{\partial {\mathbf{u}}^{*}}{\partial t*}}+\nabla .\left({\mathbf{u}}^{*}{\mathbf{u}}^{*}\right)=-\nabla p^{*}+\nabla .\left(Pr\nabla {\mathbf{u}}^{*}\right)+PrRa\beta {\mathrm{T}}^{*}$$

(12)

$${\displaystyle \frac{\partial {\mathrm{T}}^{*}}{\partial t*}}+\nabla .\left({\mathbf{u}}^{*}{\mathrm{T}}^{*}\right)=\nabla .\left(\nabla {\mathrm{T}}^{*}\right)-{\displaystyle \frac{1}{Ste}}{\displaystyle \frac{\partial f_l}{\partial t^{*}}}$$

(13)

The non-dimensional boundary conditions are [30]

$$\mathrm{IC}: \mathrm{at} t^{*}=0\hspace{1em}\hspace{1em}\hspace{1em}{u}^{*}=0, v^{*}=0, {\mathrm{T}}^{*}=1 \mathrm{for} 0\le x^{*}\le 1 \mathrm{and} 0\le y^{*}\le 1$$

(14a)

$$\begin{array}{l}\mathrm{BC}: \mathrm{at} t^{*}>0\\ u^{*}=0, \hspace{1em}\hspace{1em}\hspace{1em}{v}^{*}=0 \hspace{1em}\hspace{1em}{\mathrm{T}}^{*}=0 \hspace{1em}\hspace{1em}\hspace{1em}\mathrm{for} x^{*}=0\\ u^{*}=0, \hspace{1em}\hspace{1em}\hspace{1em}{v}^{*}=0 \hspace{1em}\hspace{1em}{\displaystyle \frac{\partial T^{*}}{\partial x^{*}}}=0 \hspace{1em}\hspace{1em}\mathrm{for} x^{*}=1\\ u^{*}=0, \hspace{1em}\hspace{1em}\hspace{1em}{v}^{*}=0 \hspace{1em}\hspace{1em}{\mathrm{T}}^{*}=0 \hspace{1em}\hspace{1em}\hspace{1em}\mathrm{for} y^{*}=0\\ u^{*}=1, \hspace{1em}\hspace{1em}\hspace{1em}{v}^{*}=0 \hspace{1em}\hspace{1em}{\displaystyle \frac{\partial T^{*}}{\partial y^{*}}}=0 \hspace{1em}\hspace{1em}\mathrm{for} y^{*}=1\end{array}$$

(14b)

Subscript ‘m’ has been omitted in the non-dimensional governing Equations (11)–(13) and boundary conditions from Equations (14a) to (14b). The asterisk ‘*’ is omitted from the next section and, for simplicity, corresponding variables are used without the asterisk.

2.2. Lattice Boltzmann Model (LBM)

The double distribution function (DDF)-based lattice Boltzmann method (LBM) is used to solve the momentum and energy equation [18]. The estimation of the velocity field was analysed using density distribution functions at the mesoscopic level while the calculation of the energy field followed the density distribution of total enthalpy. The present LB model initially solves the LB equation in terms of velocity and enthalpy, and temperature is calculated from the total enthalpy in the computational domain. A single relaxation-based LB equation is used for the formulation of corner solidification problem. The LB equations are represented in the present form after discretisation in time and discrete velocity space. The density distribution function for flow can be expressed as [7,16,18]

$${\mathrm{f}}_{\mathrm{i}}(x+{\mathbf{e}}_{\mathrm{i}}\Delta t,t+\Delta t)={\mathrm{f}}_{\mathrm{i}}(x,t)-{\displaystyle \frac{1}{{\mathrm{\tau }}_{\mathrm{f}}}}\left[{\mathrm{f}}_{\mathrm{i}}(x,t)-{\mathrm{f}}_{\mathrm{i}}^{\mathrm{e}\mathrm{q}}(x,t)\right]+\Delta t{\mathrm{F}}_{\mathrm{i}}$$

(15)

where f is velocity distribution function, ${\mathbf{e}}_{\mathrm{i}}$ is the discrete velocity in direction i, time step is Δt, ${\mathrm{\tau }}_{\mathrm{f}}$ represents the dimensionless relaxation time for flow field, and ${\mathrm{F}}_{\mathrm{i}}$ is the force term.

The equilibrium density distribution function ${\mathrm{f}}_{\mathrm{i}}^{\mathrm{e}\mathrm{q}}$ can be written as [7]

$${\displaystyle f_i^{eq}={\omega }_i\mathit{\rho }\left[1+{\displaystyle \frac{{\mathbf{e}}_i.\mathbf{u}}{c_s^2}}+{\displaystyle \frac{\mathbf{u}\mathbf{u}:({\mathbf{e}}_i{\mathbf{e}}_i-c_s^2\mathbf{I})}{2c_s^4}}\right]}$$

(16)

where $\mathrm{\omega }$ is weighting factor, ${\mathrm{c}}_{\mathrm{s}}$ is lattice speed, and I is unit vector.

In Equation (15), ${\mathrm{F}}_{\mathrm{i}}$ is the body force term, and the discretised body force term is given as [7,18]

$${\displaystyle F_i={\omega }_i\left(1-{\displaystyle \frac{1}{2{\tau }_f}}\right)\left({\displaystyle \frac{{\mathbf{e}}_i-\mathbf{u}}{c_s^2}}+{\displaystyle \frac{{\mathbf{e}}_i.\mathbf{u}}{c_s^4}}{\mathbf{e}}_i\right)\left(\mathit{\rho }.{\mathbf{S}}_{\mathit{u}}\right)}$$

(17)

where, in Equation (17), ${\mathbf{S}}_u$ depicts the source term. The macroscopic density and velocity can be expressed as [18]

$$\rho ={\displaystyle \sum _0^{\mathrm{n}}{\mathrm{f}}_{\mathrm{i}}}$$

(18)

$$\rho \mathbf{u}={\displaystyle \sum _0^{\mathrm{n}}{\mathbf{e}}_{\mathrm{i}}}{\mathrm{f}}_{\mathrm{i}}+\mathrm{F}\Delta t$$

(19)

Similarly, the distribution function for energy in terms of enthalpy is given as [7]

$${\mathrm{g}}_{\mathrm{i}}(x+{\mathbf{e}}_{\mathrm{i}}\Delta t,t+\Delta t)={\mathrm{g}}_{\mathrm{i}}(x,t)-{\displaystyle \frac{1}{{\mathrm{\tau }}_{\mathrm{e}}}}\left[{\mathrm{g}}_{\mathrm{i}}(x,t)-{\mathrm{g}}_{\mathrm{i}}^{\mathrm{e}\mathrm{q}}(x,t)\right]$$

(20)

where $\mathrm{g}$ is the energy distribution function and ${\mathrm{\tau }}_{\mathrm{e}}$ represents the dimensionless relaxation time for the energy field.

The equilibrium distribution function ${\mathrm{g}}_{\mathrm{i}}^{\mathrm{e}\mathrm{q}}$ is given as [18]

$${\mathrm{g}}_{\mathrm{i}}^{\mathrm{e}\mathrm{q}}=\left\{\begin{array}{l}H-c_p\mathrm{T}+{\mathrm{\omega }}_{\mathrm{i}}{c}_p\mathrm{T}\left(1-{\displaystyle \frac{u_m^2}{2c_s^2}}\right)\begin{array}{ccc}& \mathrm{i}=0& \end{array}\\ {\mathrm{\omega }}_{\mathrm{i}}{c}_p\mathrm{T}\left[1+{\displaystyle \frac{{\mathit{e}}_i.{\mathbf{u}}_m}{c_s^2}}+{\displaystyle \frac{{\mathit{e}}_{\mathrm{i}}.{\mathbf{u}}_m}{2c_s^4}}-{\displaystyle \frac{{\mathbf{u}}_m^2}{2c_s^2}}\right]\begin{array}{ccc}& \mathrm{i}\ne 0& \end{array}\end{array}\right.$$

(21)

Summing up, all distribution functions estimate the total enthalpy, which can be given as follows:

$${\displaystyle \sum _0^{\mathrm{n}}{\mathrm{g}}_{\mathrm{i}}}=H$$

(22)

The value of temperature is calculated from enthalpy in the following manner [16]:

$$\mathrm{T}=\left\{\begin{array}{l}{\displaystyle \frac{H}{c_p}}\begin{array}{cccc}& & & \mathrm{T}<{\mathrm{T}}_{\mathrm{w}}\end{array}\\ {\mathrm{T}}_w+{\displaystyle \frac{H-H_{\mathrm{w}}}{H_{\mathrm{m}}-H_{\mathrm{w}}}}\left({\mathrm{T}}_l-{\mathrm{T}}_s\right)\begin{array}{c}\hspace{1em}{\mathrm{T}}_{\mathrm{w}}\le \mathrm{T}\le {\mathrm{T}}_{\mathrm{m}}\end{array}\\ {\mathrm{T}}_{\mathrm{m}}+\left(H-H_{\mathrm{m}}\right)/c_p\begin{array}{cc}& \mathrm{T}>{\mathrm{T}}_{\mathrm{m}}\end{array}\end{array}\right.$$

(23)

where ${\mathrm{T}}_l$ and ${\mathrm{T}}_s$ are the liquidus and solidus temperatures of the material, respectively, in the present work for pure metal ${\mathrm{T}}_l={\mathrm{T}}_s={\mathrm{T}}_{\mathrm{m}}$. Liquidus enthalpy and solidus enthalpy are expressed as $H_{\mathrm{m}}$ and $H_{\mathrm{w}}$, respectively.

The macroscopic parameters kinematic viscosity (ν) and thermal diffusivity (α) are related to mesoscopic relaxation time parameters according to Chapman–Enskog expansion as [7]

$$\mathrm{\nu }={\mathrm{c}}_{\mathrm{s}}^2\left({\mathrm{\tau }}_{\mathrm{f}}-0.5\right)\Delta t$$

(24)

$$\mathrm{\alpha }={\mathrm{c}}_{\mathrm{s}}^2\left({\mathrm{\tau }}_{\mathrm{e}}-0.5\right)\Delta t$$

(25)

Setting of relaxation parameters is very crucial to avoid numerical instability for phase change problems. The relaxation parameter for velocity distribution was chosen to be 0.52 for the present work, which shows a stable solution. The convection effect was induced through the gβ term in the Rayleigh number. All other lattice parameters were calculated from the dimensionless numbers assuming c_p = 1 and lattice space Δx = 1 [16,18].

The bounce-back method was used in the present work to implement the physical boundary conditions in LBM following the published works [16,18,30].

3. Results and Discussion

Corner solidification of low-Prandtl-number material was performed using the total-enthalpy-based lattice Boltzmann method. The computational domain is subdivided into 181 × 181 lattice grids, and the grid spacing (Δx = Δy) is taken as 1. For the present numerical simulation, the time step was chosen to be Δt = 1. A D2Q9 lattice stencil is used in which one centre node and eight neighbour nodes are connected with the lattice linkage. The temperature distribution in the entire cavity was observed with the evaluation of time. The interface between solid and liquid regions is traced, and the influence of natural convection on the melting boundary is observed. The study is performed at a wide range of higher Rayleigh numbers (Ra), 10⁶ to 10⁸, maintaining the Prandtl number (Pr) at 0.02 and Stephan number (Ste) at 0.01. Thermal instability in the solidified region was investigated through performing a time–temperature history in the solidified layer. Two locations are fixed in the effective solidified zone, (0.2, 0.2) and (0.2, 0.8), and the temperature fluctuations with time are observed. Finally, the time–temperature was quantified using the Fast Fourier Transform (FFT) algorithm and the intensity amplitude of temperature with frequency was observed. The coefficient of variation (COV) of the temperature in the solidified region was calculated to estimate the cooling homogeneity in the corner solidification problem.

3.1. Grid Independency Test and Validation

A FORTRAN code is developed to simulate the present corner solidification problem. A grid independency test was performed using different grid values.

Table 1. Average Nusselt number on heating walls using different grid resolutions.

Grid Size	Average Nusselt Number at Left–Bottom Walls
101 × 101	3.581
151 × 151	3.620
181 × 181	3.622
201 × 201	3.622

shows the result of the grid independency test to calculate the average Nusselt number for the corner melting problem. It is seen from Table 1 that different grid resolutions (101 × 101, 151 × 151, 181 × 181, and 201 × 201) are used to calculate the average Nusselt number at the left–bottom walls of a corner melting problem. The deviation between 101 × 101 and 151 × 151 grids is less than 0.05%, while comparing 181 × 181 and 201 × 201 grids, the calculated Nusselt number is quite similar. In the present work, 181 × 181 grids are chosen for further calculation.

The present FORTRAN code is validated with benchmark data from [31], in which left-heated melting in a square enclosure was carried out at Pr = 0.02 and Ra = 2.5 × 10⁴. In a left-heated square cavity melting, the average Nu varies over time, as seen in Figure 2. Figure 2 shows a good result of the present code, which is used for further simulation for the rest of the work. The code is further validated by experimental and numerical research carried out by Wolff and Viskanta [32], which is shown in Figure 3. Figure 3a–c show a velocity map in the liquid region at different time intervals at Ra = 1.6 × 10⁵ and Fo = 0.57, 3.92 and 10.1, respectively, for a left-heated melting problem in a cavity with an aspect ratio of 0.75. The left column in Figure 3 depicts the velocity mapping at the liquid zone by Wolff and Viskanta [32] while the right column indicates the present code result. It is observed from the figures that the present LB code effectively captures the flow circulation in the liquid zone.

The code is validated against other published works [33,34] for a natural-convection-dominated square cavity problem.

Table 2. Average Nusselt number on heating wall of the square cavity.

	Ra = 103	Ra = 104	Ra = 105
Basak et al. [33]	4.1563	6.2476	9.391
Du et al. [34]	4.1655	6.2676	9.352
Present Work	4.1584	6.2362	9.357

shows the calculated average Nu for different Ra, which agree favourably with data from the literature.

3.2. Effect of Rayleigh Number

The temporal evolution of isotherms with interface location is shown in Figure 4, Figure 5 and Figure 6. Figure 4 describes the isotherms in the entire cavity with time at Ra = 10⁶. The interface position with time changes with time, and it is observed that conduction initially dominates, so no distortion can be found on the interface. As time progresses, convection begins to dominate over conduction and the interface becomes distorted, which is found after Fo*Ste = 0.003. While heat is rejected from the left and bottom walls, solidification starts from the left and bottom walls and the solidified area grows with elapsed time. The isotherms have been found in the entire cavity, which depicts the heat accumulation due to insulation of the top–right walls.

Similarly, Figure 5 describes the isotherm patterns at a higher Ra of 10⁷. The interface position with time is identified in Figure 5. The convection effect is greater compared to the earlier case; as Ra increases, free convection shows a wavier nature in the isotherms. It is seen from Figure 5 that, near the cold walls, the solidification front is practically vertical, and when the heat transfer regime changes from conduction to convection, a progressively inclining slope appears away from the cold walls.

At Ra = 10⁸, the solidified zone grows faster compared to the earlier cases. It is seen from Figure 6 that heat rejection is more pronounced near the bottom wall compared to the left wall as at high Ra the buoyancy effect is more effective in the bottom wall rather than the left wall. The temperature gradient between the top and bottom walls is greater than the thermal gradient between the left and right walls. Gravitational acceleration due to gravity acts downward on the cavity, which influences the free convection between the top and bottom walls. It is observed from Figure 4, Figure 5 and Figure 6 that with an increase in the Ra, the effective melt zone reduces faster. The findings demonstrate that, for the same dimensionless period, the solidification front advances towards the cavity, suggesting that, at a higher Ra of 10⁸, convection has a dominant effect.

3.3. Thermal Instability in Solidified Zone

To study the thermal instability in the solidified zone, a time–temperature history was observed under conditions where convection dominates over conduction. Two different positions were chosen in the growing solidified zone: (0.2, 0.2) and (0.2, 0.8). The former is located near the corner position of the left–bottom walls while the latter is located adjacent to the top wall. As the effective solidified area changes with time, an observation of transient temperature changes was performed at the selected location to investigate the thermal fluctuation in the solidified area. The study was performed at different Ra, as shown in Figure 7, Figure 8 and Figure 9. The time–temperature history was analysed at several time intervals to observe the change in temperature in the growing solidified area.

Three different time intervals are used to investigate the temperature fluctuation at particular points, as shown in Figure 7a,b for Fo = 0.4, Figure 7c,d for Fo = 0.5, and Figure 6e,f for Fo = 0.6. Figure 7 reveals that the initial temperature fluctuations near the corner position are sharper than the position adjacent to the top wall due to the dual temperature gradient present in the corner position.

Figure 8a–c describe the time–temperature history at three time intervals at Ra = 10⁷. Initially, the temperature fluctuates more, and with an increase in time, it reduces for all time intervals: Fo = 0.4, Fo = 0.5, and Fo = 0.6. Thermal instability slightly increases with an increase in Ra from 10⁶ to Ra = 10⁷. Thermal fluctuation is clearly visible in Figure 9 at Ra = 10⁸. Instability in temperature was quantified using Fast Fourier Transform (FFT) analysis. Figure 10 depicts the frequency analysis for thermal instability, which is generated at Ra = 10⁸. Figure 10a,b present the results of the frequency analysis at a time interval of Fo = 0.5 while Figure 10c,d express results at Fo = 0.6. It is seen from Figure 10 that the peak frequency was observed at a much lower value.

As time increases, peak frequency increases from f* = 0.0174 to f* = 0.0188 with the increment of the effective solidified area, which is shown in Figure 10a–c. Similarly, near the top wall peak, the frequency increases from f* = 0.019 to f* = 0.021 with the expansion of the effective solidified area. FFT analysis confirms that there is only a single frequency present at different locations in the solidified zone.

3.4. Coefficient of Variation (COV) in Solidified Zone

To investigate the uniform cooling in the square cavity, statistical analysis was performed after thermal instability analysis. The co-efficient of variation (COV) was analysed in the effective solidified zone at different Ra.

The mean temperature $(\overline{\mathrm{T}})$ in the solidified area $(\mathrm{A})$ is calculated as follows:

$$\overline{\mathrm{T}}={\displaystyle \frac{1}{\mathrm{A}}}{\displaystyle \underset{\mathrm{A}}{\iint }\mathrm{T}(x,y)\mathrm{d}\mathrm{A}}$$

(26)

The standard deviation $(\mathrm{\sigma })$ is calculated as follows:

$$\mathrm{\sigma }={\displaystyle \frac{1}{\mathrm{A}}}{\displaystyle \underset{\mathrm{A}}{\iint }{\left(\mathrm{T}(x,y)-\overline{\mathrm{T}}\right)}^2\mathrm{d}\mathrm{A}}$$

(27)

The coefficient of variation (COV) of the temperature field may be expressed as the ratio of mean temperature and standard deviation of temperature in the solidified area.

$$\mathrm{C}\mathrm{O}\mathrm{V}={\displaystyle \frac{\overline{\mathrm{T}}}{\mathrm{\sigma }}}$$

(28)

The evolution of COV at different Ra is depicted in Figure 11. It is observed that the level of initial COV varies from 95 to 205 as Ra changes from 10⁶ to 10⁸. Figure 11 reveals that the COV initially decreases sharply and then increases slightly for all Ra. The lower value of COV implies more homogeneity in the thermal field in the solidified part. Towards the end of solidification, the level of homogeneity achieved is comparable with a value of around 75.

4. Conclusions

In this study, using the total-enthalpy-based lattice Boltzmann method, corner solidification was examined. The effect of natural convection on the solidification front with the progression of time has been presented in the Rayleigh number range between 10⁶ and 10⁸. The study shows that solidification is initially dominated by conduction, and then as time passes, convection becomes more significant. The solidification front advances more quickly due to convection as Ra increases.

Analyses of thermal instability were performed in the effective solidified area using FFT analysis. The peak frequency in the solidified zone is observed, which increases with time in the growing solidified area. Furthermore, the coefficient of variation for the temperature field was analysed in the solidified zone. The results show that as time elapses, the solidified zone reaches a level of constant uniformity of the thermal field at all Ra.