Lattice Boltzmann model for incompressible axisymmetric thermal flows through porous media

The present work proposes a simple lattice Boltzmann model for incompressible axisymmetric thermal flows through porous media. By incorporating forces and source terms into the lattice Boltzmann equation, the incompressible Navier-Stokes equations are recovered through the Chapman-Enskog expansion. It is found that the added terms are just the extra terms in the governing equations for the axisymmetric thermal flows through porous media compared with the Navier-Stokes equations. Four numerical simulations are performed to validate this model. Good agreement is obtained between the present work and the analytic solutions and/or the results of previous studies. This proves its efficacy and simplicity regarding other methods. Also, this approach provides guidance for problems with more physical phenomena and complicated force forms.

Figure 1

Schematic diagram of the first test case.

Figure 2

A comparison of the axial velocity (a) and temperature (b) profiles for different external forces and wall temperatures of the first test as obtained from the present LBE model simulation (solid line) versus the analytical solution (symbols).

Figure 3

Schematic diagram of the second test case.

Figure 4

A comparison of the axial velocity (a) and temperature (b) profiles for different permeability and wall temperatures of the second test as obtained from the present LBE model simulation (solid line) versus the analytical solution (symbols).

Figure 5

A comparison of the axial velocity (a) and temperature (b) profiles for different external forces and wall temperatures of the second test as obtained from the present LBE model simulation (solid line) versus the analytical solution (symbols) in the presence of inertial effect.

Figure 6

Schematic diagram of the third case.

Figure 7

A comparison of the axial velocity (a) and temperature (b) profiles for different external forces and wall temperatures of the third test as obtained from the present LBE model simulation (solid line) versus the analytical solution (symbols).

Figure 8

A comparison of the axial velocity (a) and temperature (b) profiles for different external forces and wall temperatures of the third test as obtained from the present LBE model simulation (solid line) versus the analytical solution (symbols) in the presence of inertial effect.

Figure 9

Schematic diagram of the fourth case [32].

Figure 10

A comparison of the velocity (a) and temperature (b) profiles for $\varepsilon =0.4,\mathrm{Da}={10}^{-3},\mathrm{Rp}=0.5$, and $\mathrm{Re}=100$ and different wall temperatures of the fourth test as obtained from the present LBE model simulation (solid line) versus the results of Rong et al. [32] (circles).