Implementation of diffuse-reflection boundary conditions using lattice Boltzmann models based on half-space Gauss-Laguerre quadratures

The Gauss-Laguerre quadrature method is used on the Cartesian semiaxes in the momentum space to construct a family of lattice Boltzmann models. When all quadrature orders $Q_x$, $Q_y$, $Q_z$ equal $N+1$, the Laguerre lattice Boltzmann model $\mathrm{LLB}(Q_x,Q_y,Q_z)$ exactly recovers all moments up to order $N$ of the Maxwell-Boltzmann equilibrium distribution function $f^{\left(\mathrm{eq}\right)}$, calculated over any Cartesian octant of the three-dimensional momentum space. Results of Couette flow simulations at $\mathrm{Kn}=0.1$, $0.5$, $1.0$ and in the ballistic regime are reported. Specific microfluidic effects (velocity slip, temperature jump, longitudinal heat flux) are well captured up to $\mathrm{Kn}=0.5$, as demonstrated by comparison to direct simulation Monte Carlo results. Excellent agreement with analytic results is obtained in the ballistic regime.

Figure 1

Comparison between the stationary profiles (longitudinal fluid velocity $u_y$, temperature $T$, transversal heat flux $q_z$, longitudinal heat flux $q_y$) across the channel in Couette flow at $\mathrm{Kn}=0.1$ and $\mathrm{Kn}=0.5$ obtained with the model $\mathrm{LLB}(7,7,7)$ and the corresponding DSMC results [57, 58].