Diffuse-reflection boundary conditions for a thermal lattice Boltzmann model in two dimensions: Evidence of temperature jump and slip velocity in microchannels

We discuss the implementation of diffuse reflection boundary conditions in a thermal lattice Boltzmann model for which the upwind finite difference scheme is used to solve the set of evolution equations recovered after discretization of the velocity space. Simulation of heat transport between two parallel walls at rest shows evidence of temperature jumps at the walls that increase with Knudsen number. When the walls move in opposite directions with speeds $\pm u_W$, fluid velocity slip is observed at the walls, together with temperature jumps.

Figure 1

Temperature dependence of the nondimensionalized average speed, Eq. (15) for $\mathbf{u}=0$. The continuous line shows the theoretical result of the kinetic theory, Eq. (13), and is provided for comparison.

Figure 2

The square lattice used for finite difference LB simulation of channel flow: ●, bulk nodes; 엯, boundary nodes; ◻, ghost nodes outside the walls. Walls are situated at half lattice spacing between boundary nodes and ghost nodes. Periodic boundary conditions are used in the vertical direction.

Figure 3

Diffuse reflection procedure: ◻, ghost nodes; 엯, boundary nodes; ∎, wall points where the distribution functions $f_{ki}$ $(i=1,2,8)$ follow the Maxwellian distribution law.

Figure 4

Heat transport between two parallel walls at rest—stationary profiles of various nondimensionalized quantities, for $n_{\mathit{\text{initial}}}=2\times {10}^7$, ${\theta }_{\mathit{WL}}=0.50$, ${\theta }_{\mathit{WR}}=1.50$: (a) density, (b) temperature, (c) pressure, (d), Knudsen number, (e) average speed $\overline{c}$, and (f) spurious velocity $u_x$. Results recovered using the constant relaxation time carry the symbol $\tau$ while results recovered using the density dependent relaxation time defined by Eq. (19) carry the symbol $\Lambda$.

Figure 5

Heat transport between two parallel walls at rest: nondimensionalized temperature profiles recovered in the stationary state using the density dependent relaxation time $\tau$ given by Eq. (19), for ${\theta }_{\mathit{WL}}=0.90$, ${\theta }_{\mathit{WR}}=1.10$ and three values of the initial fluid density $n$.

Figure 6

Heat transport between two parallel walls at rest: Knudsen number dependence of temperature jumps at the left and right walls, recovered using both versions of the relaxation time. Results recovered using the constant relaxation time carry the symbol $\tau$ while results recovered using the variable relaxation time defined by Eq. (19) carry the symbol $\Lambda$. Wall temperatures are (a) ${\theta }_{\mathit{WL}}=0.90$, ${\theta }_{\mathit{WR}}=1.10$; (b) ${\theta }_{\mathit{WL}}=0.50$, ${\theta }_{\mathit{WR}}=1.50$.

Figure 7

Couette flow: (a) velocity profiles and (b) temperature profiles recovered using both versions of the relaxation time for wall temperatures ${\theta }_{\mathit{WR}}={\theta }_{\mathit{WL}}=1.00$ and velocities $u_{\mathit{WR}}=-u_{\mathit{WL}}=0.1$. Results recovered using the constant relaxation time carry the symbol $\tau$ while results recovered using the variable relaxation time defined by Eq. (19) carry the symbol $\Lambda$.

Figure 8

Couette flow: effect of Knudsen number on the temperature jump (a) and velocity slip (b) for $u_W=0.5$ and three values of the wall temperature. Results recovered using the constant relaxation time carry the symbol $\tau$ while results recovered using the variable relaxation time defined by Eq. (19) carry the symbol $\Lambda$.