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ABSTRACT
In this work, a simple gas kinetic scheme is presented for solving the 3D incompressible thermal flow problems. In the scheme, the macroscopic-governing equations are discretized by finite volume method, and the numerical fluxes at cell interface are reconstructed by the local solution of the Boltzmann equation. To compute the numerical fluxes, two equilibrium distribution functions are introduced. One is the sphere function for calculating the fluxes of mass and momentum equations, and the other is the D3Q6 discrete velocity model for evaluating the flux of energy equation. Using the difference of equilibrium distribution functions at the cell interface and its surrounding points to approximate the nonequilibrium distribution function, and at the same time considering the incompressible limit, the numerical fluxes of macroscopic governing equations at the cell interface can be given explicitly and concisely. Numerical results showed that the present scheme can predict accurately the thermal flow properties at a wide range of the Rayleigh numbers.
Nomenclature
| c | = | peculiar velocity of particles |
| cs | = | sound speed |
| D | = | space dimension |
| = | discrete particle velocity | |
| f | = | density distribution function |
| feq | = | equilibrium state for f |
| fE | = | buoyancy force |
| Fn | = | numerical flux vector |
| g | = | gravity acceleration |
| gS | = | sphere function |
| = | temperature distribution function | |
| = | equilibrium state for | |
| I | = | unit tensor |
| L | = | characteristic length |
| Ma | = | Mach number |
| n | = | unit normal vector |
| Nf | = | number of the faces |
| Nu | = | Nusselt number |
| p | = | pressure |
| Pr | = | Prandtl number |
| Q | = | source term |
| Ra | = | Rayleigh number |
| Si | = | area of the ith interface |
| t | = | time |
| T | = | temperature |
| Tm | = | average temperature |
| u | = | velocity vector |
| u0 | = | reference velocity |
| Vc | = | characteristic thermal velocity |
| W | = | conservative variable vector |
| β | = | thermal expansion coefficient |
| Δ t | = | streaming time step |
| κ | = | thermal diffusivity |
| μ | = | dynamic viscosity |
| ν | = | kinematic viscosity |
| = | particle velocity | |
| ρ | = | density |
| = | collision time scale for | |
| = | collision time scale for f | |
| = | moment vector | |
| = | volume of cell I | |
| Subscripts | = | |
| I | = | cell index |
| 1, 2, 3 | = | coordinate component |
| α | = | particle index |
| Superscripts | = | |
| face | = | cell interface |
| L | = | left side of cell interface |
| R | = | right side of cell interface |
| sph | = | spherical surface |
Nomenclature
| c | = | peculiar velocity of particles |
| cs | = | sound speed |
| D | = | space dimension |
| = | discrete particle velocity | |
| f | = | density distribution function |
| feq | = | equilibrium state for f |
| fE | = | buoyancy force |
| Fn | = | numerical flux vector |
| g | = | gravity acceleration |
| gS | = | sphere function |
| = | temperature distribution function | |
| = | equilibrium state for | |
| I | = | unit tensor |
| L | = | characteristic length |
| Ma | = | Mach number |
| n | = | unit normal vector |
| Nf | = | number of the faces |
| Nu | = | Nusselt number |
| p | = | pressure |
| Pr | = | Prandtl number |
| Q | = | source term |
| Ra | = | Rayleigh number |
| Si | = | area of the ith interface |
| t | = | time |
| T | = | temperature |
| Tm | = | average temperature |
| u | = | velocity vector |
| u0 | = | reference velocity |
| Vc | = | characteristic thermal velocity |
| W | = | conservative variable vector |
| β | = | thermal expansion coefficient |
| Δ t | = | streaming time step |
| κ | = | thermal diffusivity |
| μ | = | dynamic viscosity |
| ν | = | kinematic viscosity |
| = | particle velocity | |
| ρ | = | density |
| = | collision time scale for | |
| = | collision time scale for f | |
| = | moment vector | |
| = | volume of cell I | |
| Subscripts | = | |
| I | = | cell index |
| 1, 2, 3 | = | coordinate component |
| α | = | particle index |
| Superscripts | = | |
| face | = | cell interface |
| L | = | left side of cell interface |
| R | = | right side of cell interface |
| sph | = | spherical surface |