Lattice Boltzmann model for a steady radiative transfer equation

A complete lattice Boltzmann model (LBM) is proposed for the steady radiative transfer equation (RTE). The RTE can be regarded as a pure convection equation with a source term. To derive the expressions for the equilibrium distribution function and the relaxation time, an artificial isotropic diffusion term is introduced to form a convection-diffusion equation. When the dimensionless relaxation time has a value of 0.5, the lattice Boltzmann equation (LBE) is exactly applicable to the original steady RTE. We also perform a multiscale analysis based on the Chapman-Enskog expansion to recover the macroscopic RTE from the mesoscopic LBE. The D2Q9 model is used to solve the LBE, and the numerical results obtained by the LBM are comparable to the results obtained by other methods or analytical solutions, which demonstrates that the proposed model is highly accurate and stable in simulating multidimensional radiative transfer. In addition, we find that the convergence rate of the LBM depends on the transport properties of RTE: for diffusion-dominated RTE with a large optical thickness, the LBM shows a second-order convergence rate in space, while for convection-dominated RTE with a small optical thickness, a lower convergence rate is observed.

Figure 1

D2Q9 lattice scheme used for a 2D geometry in a lattice Boltzmann model.

Figure 2

Radiative heat flux along the $x$ direction in a purely absorbing medium with extinction coefficients $\beta =0.1$, 1, and 10. (a) Comparison with the FEM, and (b) comparison with the zone method.

Figure 3

The centerline incident radiation along the $y$ direction in an isotropically scattering medium with emissivities $\varepsilon =0.1$, 0.5, and 1.

Figure 4

Radiative heat flux along the $x$ direction in an isotropically scattering medium with emissivities $\varepsilon =0.1$, 0.5, and 1.

Figure 5

Radiative heat flux along the $x$ direction in an anisotropically scattering medium with scattering albedos $\omega =0$, 0.5, and 0.9.

Figure 6

Grid independence test for results from the LBM.

Figure 7

Global relative errors of the lattice Boltzmann model with different lattice steps.

Figure 8

Radiative intensity distributions obtained by the Galerkin FEM, an analytical solution, and the LBM (from left to right) for various absorbing coefficients. (a) ${\kappa }_{\mathrm{a}}=0.1{\mathrm{m}}^{-1}$, (b) ${\kappa }_{\mathrm{a}}=1{\mathrm{m}}^{-1}$, and (c) ${\kappa }_{\mathrm{a}}=10{\mathrm{m}}^{-1}$.

Figure 9

Temperature distributions for various conduction-radiation parameters. (a) $Pl=0$, (b) $Pl=0.01$, (c) $Pl=0.1$, and (d) $Pl=1$.