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ABSTRACT
A double distribution function lattice Boltzmann method (LBM) with multirelaxation time is implemented to simulate the Rayleigh–Benard convection melting of a typical low-melting-point metal in a rectangular cavity. Typical cases frequently encountered in practice with constant heat flux/constant temperature boundary conditions are parametrically investigated, with corresponding dimensionless results outlined; the influence of inclination angle of the cavity is also clarified. The computational speed of the current LBM would reach about 40 times faster than that of conventional finite volume method as performed by commercial software Fluent. The obtained results would be valuable for guiding practical thermal design.
Nomenclature
| c | = | lattice speed (m/s) |
| cp | = | specific heat capacity (J/kg · K) |
| cs | = | sound speed in lattice space (m/s) |
| = | propagation velocity of the ith discrete particle (m/s) | |
| = | body force (N/m3) | |
| Fo | = | Fourier number |
| f | = | density distribution function |
| fl | = | liquid fraction |
| = | gravitational acceleration (m/s2) | |
| g | = | enthalpy distribution function |
| H | = | total enthalpy (J/kg) |
| Hc | = | cavity height (m) |
| ΔH | = | fusion latent heat (J/kg) |
| I | = | unit matrix |
| k | = | thermal conductivity (W/m · K) |
| Lc | = | characteristic length (m) |
| M | = | orthogonal transformation matrix |
| Ma | = | Mach number |
| m | = | distribution function in moment space |
| Nu | = | Nusselt number |
| = | unit normal vector on liquid–solid interface | |
| P | = | dimensionless pressure |
| Pr | = | Prandtl number |
| p | = | pressure (N/m2) |
| = | heat flux (W/m2) | |
| Ra | = | Rayleigh number |
| S | = | diagonal matrix |
| Ste | = | Stefan number |
| ST | = | dimensionless subcooling |
| T | = | temperature (K) |
| t | = | time (s) |
| = | dimensionless velocity vector | |
| U0 | = | characteristic velocity for natural convection (m/s) |
| = | (u, v), velocity vector (m/s) | |
| = | dimensionless movement velocity of solid–liquid interface | |
| = | movement velocity of solid–liquid interface (m/s) | |
| W | = | cavity width (m) |
| wi | = | weighted coefficient of the ith discrete particle |
| X, Y | = | dimensionless rectangular coordinates |
| x, y | = | rectangular coordinates (m) |
| Greek letters | = | |
| α | = | thermal diffusivity (m2/s) |
| β | = | volume thermal expansivity (1/K) |
| γ | = | aspect ratio of the cavity |
| ε | = | small disturbance value |
| Θ | = | dimensionless temperature |
| θ | = | inclination angle (°) |
| υ | = | kinetic viscosity (m2/s) |
| ρ | = | mass density (kg/m3) |
| φ | = | melting fraction |
| τ | = | dimensionless relaxation time |
| Subscripts | = | |
| b | = | bottom wall |
| C | = | cold wall |
| eq | = | equilibrium distribution |
| H | = | hot wall, constant heat flux boundary condition |
| i | = | = 0, 1, 2 … 8, discrete particle index |
| l | = | liquid phase |
| la | = | lattice space |
| m | = | melting point |
| j | = | index of lattice node on cavity bottom |
| T | = | constant temperature boundary condition |
| r | = | reference value |
| s | = | solid phase |
Nomenclature
| c | = | lattice speed (m/s) |
| cp | = | specific heat capacity (J/kg · K) |
| cs | = | sound speed in lattice space (m/s) |
| = | propagation velocity of the ith discrete particle (m/s) | |
| = | body force (N/m3) | |
| Fo | = | Fourier number |
| f | = | density distribution function |
| fl | = | liquid fraction |
| = | gravitational acceleration (m/s2) | |
| g | = | enthalpy distribution function |
| H | = | total enthalpy (J/kg) |
| Hc | = | cavity height (m) |
| ΔH | = | fusion latent heat (J/kg) |
| I | = | unit matrix |
| k | = | thermal conductivity (W/m · K) |
| Lc | = | characteristic length (m) |
| M | = | orthogonal transformation matrix |
| Ma | = | Mach number |
| m | = | distribution function in moment space |
| Nu | = | Nusselt number |
| = | unit normal vector on liquid–solid interface | |
| P | = | dimensionless pressure |
| Pr | = | Prandtl number |
| p | = | pressure (N/m2) |
| = | heat flux (W/m2) | |
| Ra | = | Rayleigh number |
| S | = | diagonal matrix |
| Ste | = | Stefan number |
| ST | = | dimensionless subcooling |
| T | = | temperature (K) |
| t | = | time (s) |
| = | dimensionless velocity vector | |
| U0 | = | characteristic velocity for natural convection (m/s) |
| = | (u, v), velocity vector (m/s) | |
| = | dimensionless movement velocity of solid–liquid interface | |
| = | movement velocity of solid–liquid interface (m/s) | |
| W | = | cavity width (m) |
| wi | = | weighted coefficient of the ith discrete particle |
| X, Y | = | dimensionless rectangular coordinates |
| x, y | = | rectangular coordinates (m) |
| Greek letters | = | |
| α | = | thermal diffusivity (m2/s) |
| β | = | volume thermal expansivity (1/K) |
| γ | = | aspect ratio of the cavity |
| ε | = | small disturbance value |
| Θ | = | dimensionless temperature |
| θ | = | inclination angle (°) |
| υ | = | kinetic viscosity (m2/s) |
| ρ | = | mass density (kg/m3) |
| φ | = | melting fraction |
| τ | = | dimensionless relaxation time |
| Subscripts | = | |
| b | = | bottom wall |
| C | = | cold wall |
| eq | = | equilibrium distribution |
| H | = | hot wall, constant heat flux boundary condition |
| i | = | = 0, 1, 2 … 8, discrete particle index |
| l | = | liquid phase |
| la | = | lattice space |
| m | = | melting point |
| j | = | index of lattice node on cavity bottom |
| T | = | constant temperature boundary condition |
| r | = | reference value |
| s | = | solid phase |