Consistent lattice Boltzmann modeling of low-speed isothermal flows at finite Knudsen numbers in slip-flow regime: Application to plane boundaries

The first nonequilibrium effect experienced by gaseous flows in contact with solid surfaces is the slip-flow regime. While the classical hydrodynamic description holds valid in bulk, at boundaries the fluid-wall interactions must consider slip. In comparison to the standard no-slip Dirichlet condition, the case of slip formulates as a Robin-type condition for the fluid tangential velocity. This makes its numerical modeling a challenging task, particularly in complex geometries. In this work, this issue is handled with the lattice Boltzmann method (LBM), motivated by the similarities between the closure relations of the reflection-type boundary schemes equipping the LBM equation and the slip velocity condition established by slip-flow theory. Based on this analogy, we derive, as central result, the structure of the LBM boundary closure relation that is consistent with the second-order slip velocity condition, applicable to planar walls. Subsequently, three tasks are performed. First, we clarify the limitations of existing slip velocity LBM schemes, based on discrete analogs of kinetic theory fluid-wall interaction models. Second, we present improved slip velocity LBM boundary schemes, constructed directly at discrete level, by extending the multireflection framework to the slip-flow regime. Here, two classes of slip velocity LBM boundary schemes are considered: (i) linear slip schemes, which are local but retain some calibration requirements and/or operation limitations, (ii) parabolic slip schemes, which use a two-point implementation but guarantee the consistent prescription of the intended slip velocity condition, at arbitrary plane wall discretizations, further dispensing any numerical calibration procedure. Third and final, we verify the improvements of our proposed slip velocity LBM boundary schemes against existing ones. The numerical tests evaluate the ability of the slip schemes to exactly accommodate the steady Poiseuille channel flow solution, over distinct wall slippage conditions, namely, no-slip, first-order slip, and second-order slip. The modeling of channel walls is discussed at both lattice-aligned and non-mesh-aligned configurations: the first case illustrates the numerical slip due to the incorrect modeling of slippage coefficients, whereas the second case adds the effect of spurious boundary layers created by the deficient accommodation of bulk solution. Finally, the slip-flow solutions predicted by LBM schemes are further evaluated for the Knudsen's paradox problem. As conclusion, this work establishes the parabolic accuracy of slip velocity schemes as the necessary condition for the consistent LBM modeling of the slip-flow regime.

Figure 1

Discretization of wall on LBM uniform mesh: (a) lattice-aligned wall and (b) lattice-inclined wall.

Figure 2

Different flow-lattice configurations. (a) Discretization of horizontal channel ($\theta =0$) with effective channel width given by $H=N_y-1+2{\delta }_y$, where $N_y$ is number of computational cells and ${\delta }_y$ is wall to grid node vertical distance. (b) Discretization of diagonal channel ($\theta =\pi /4$). (c) Discretization of arbitrarily inclined channel $[\theta ={tan}^{-1}\left(\frac{1}{2}\right)]$. The effective channel width is determined as $H^{\prime }=Hcos\theta$, where $H$ stands for the lattice-aligned case.

Figure 3

Poiseuille flow velocity solutions at different $\mathrm{Kn}$ numbers within slip-flow theory [Eq. (24)]. (a) No-slip velocity boundary condition ($C_1=C_2=0$). (b) First-order slip velocity boundary condition ($C_1=1.0$ and $C_2=0$ by Maxwell [15]). (c) Second-order slip velocity boundary condition ($C_1=1.1466$ and $C_2=0.9576$ by Cercignani [21]).

Figure 4

Accuracy $\left|\mathcal{E}\right({\tilde{u}}_x\left)\right|$ as function of relaxation parameter $\mathrm{\Lambda }$. Note, the meaning of $\mathrm{\Lambda }$ may be either of a free parameter (MRT/TRT) or a viscosity-dependent one (REG/BGK) (see Table 2). All cases refer to horizontal channel $(\mathrm{\Theta }=1)$ with midway walls $({\delta }_y=\frac{1}{2})$, resolved by $N=5$ grid nodes along width, i.e., channel width $H=N-1+2{\delta }_y=5$ (simulation units). (a) No-slip flow regime ($\mathrm{Kn}=0$) where BB, CK, and LI slip schemes are exact at $\mathrm{\Lambda }=\frac{3}{2}\frac{{\delta }_y^2}{2}=\frac{3}{16}$. (b) First-order slip-flow regime ($\mathrm{Kn}=0.1, C_1=1$, and $C_2=0$ [15]) where BB, CK, and LI slip schemes are exact at $\mathrm{\Lambda }=\frac{3}{2}(\frac{{\delta }_y^2}{2}+C_1\lambda H)=\frac{33}{16}, \mathrm{\Lambda }=\frac{3}{2}\frac{{\delta }_y^2}{2}=\frac{3}{16}$, and $\mathrm{\Lambda }=\frac{3}{2}(\frac{{\delta }_y^2}{2}+C_1\lambda {\delta }_y)=\frac{9}{16}$, respectively. (c) Second-order slip-flow regime ($\mathrm{Kn}=0.1, C_1=1.1466$, and $C_2=0.9576$ [21]) where BB, CK, and LI slip schemes are exact at $\mathrm{\Lambda }=\frac{3}{2}(\frac{{\delta }_y^2}{2}+C_1\lambda H+C_2{\lambda }^2)\simeq 2.70, \mathrm{\Lambda }=\frac{3}{2}(\frac{{\delta }_y^2}{2}+C_2{\lambda }^2)\simeq 0.55$, and $\mathrm{\Lambda }=\frac{3}{2}(\frac{{\delta }_y^2}{2}+C_1\lambda {\delta }_y+C_2{\lambda }^2)\simeq 0.98$, respectively. The MR slip schemes (not shown here) are exact $\forall$ slip-flow regimes and $\forall \mathrm{\Lambda }$.

Figure 5

Accuracy $\left|\mathcal{E}\right({\tilde{u}}_x\left)\right|$ as function of grid resolution $H$. All cases refer to horizontal channel $(\mathrm{\Theta }=1)$ with midway walls $({\delta }_y=\frac{1}{2})$. Convergence rates are obtained through a linear regression, with the numerical values of slopes summarized in Table 9. Unless otherwise stated, the solutions are obtained for the fixed relaxation combination $\mathrm{\Lambda }=\frac{1}{4}$. (a) No-slip flow regime ($\mathrm{Kn}=0$). (b) First-order slip-flow regime ($\mathrm{Kn}=0.1, C_1=1$, and $C_2=0$ [15]). (c) Second-order slip-flow regime ($\mathrm{Kn}=0.1, C_1=1.1466$, and $C_2=0.9576$ [21]). The MR slip schemes (not shown here) are exact for all slip-flow regimes, $\forall \mathrm{\Lambda }$ values and $\forall H$; hence, concept of grid convergence does not apply.

Figure 6

Poiseuille flow velocity solutions at $\mathrm{Kn}=0.1$ within slip-flow theory. Continuous black line represents exact analytical solution [Eq. (32)]. Markers denote the LBM nodal point solutions, which are predicted by different slip boundary schemes. All cases refer to inclined channel $[\theta ={tan}^{-1}\left(\frac{1}{2}\right)]$, resolved by $N_y=6$ grid nodes along its width. (a) No-slip velocity boundary condition ($C_1=C_2=0$). (b) First-order slip velocity boundary condition ($C_1=1.0$ and $C_2=0$ by Maxwell [15]). (c) Second-order slip velocity boundary condition ($C_1=1.1466$ and $C_2=0.9576$ by Cercignani [21]).

Figure 7

Error profiles, from Fig. 6, of streamwise velocity component $\mathrm{\Delta }{\tilde{u}}_{x^{\prime }}={\tilde{u}}_{x^{\prime }}-{\tilde{u}}_{x^{\prime }}^{\left(\mathrm{exact}\right)}$, where ${\tilde{u}}_{x^{\prime }}=j_q/j_q^{\left(\max \right)}$ and the analytical solution ${\tilde{u}}_{x^{\prime }}^{\left(\mathrm{exact}\right)}=j_q^{\left(\mathrm{exact}\right)}/j_q^{\left(\max \right)}$, with $j_q^{\left(\mathrm{exact}\right)}$ given by Eq. (32) and $j_q^{\left(\max \right)}=F_q{H}_q^2/\left(8\nu \right)$. The mean error $\langle \mathrm{\Delta }{\tilde{u}}_{x^{\prime }}\rangle$ is quantified in Table 11. Panel (a): No-slip velocity boundary condition ($C_1=C_2=0$). Panel (b): First-order slip velocity boundary condition ($C_1=1.0$ and $C_2=0$ by Maxwell [15]). Panel (c): Second-order slip velocity boundary condition ($C_1=1.1466$ and $C_2=0.9576$ by Cercignani [21]).

Figure 8

Error profiles, from Fig. 6, of transversal velocity component $\mathrm{\Delta }{\tilde{u}}_{y^{\prime }}={\tilde{u}}_{y^{\prime }}-{\tilde{u}}_{y^{\prime }}^{\left(\mathrm{exact}\right)}$, where analytical solution is ${\tilde{u}}_{y^{\prime }}^{\left(\mathrm{exact}\right)}=0$. The mean error $\langle \mathrm{\Delta }{\tilde{u}}_{y^{\prime }}\rangle$ is quantified in Table 11. Panel (a): No-slip velocity boundary condition ($C_1=C_2=0$). Panel (b): First-order slip velocity boundary condition ($C_1=1.0$ and $C_2=0$ by Maxwell [15]). Panel (c): Second-order slip velocity boundary condition ($C_1=1.1466$ and $C_2=0.9576$ by Cercignani [21]).

Figure 9

Error profiles of flow streamlines angle $\theta ={tan}^{-1}\left({\tilde{u}}_y/{\tilde{u}}_x\right)$ (measured with respect to the fixed, lattice-aligned frame $(x,y)$) and determined as $\mathrm{\Delta }\theta =\theta /{\theta }^{\left(\mathrm{exact}\right)}-1$, where analytical solution is given by ${\theta }^{\left(\mathrm{exact}\right)}={tan}^{-1}\left(1/2\right)$. The mean error $\langle \mathrm{\Delta }\theta \rangle$ is quantified in Table 11. Panel (a): No-slip velocity boundary condition ($C_1=C_2=0$). Panel (b): First-order slip velocity boundary condition ($C_1=1.0$ and $C_2=0$ by Maxwell [15]). Panel (c): Second-order slip velocity boundary condition ($C_1=1.1466$ and $C_2=0.9576$ by Cercignani [21]).

Figure 10

Accuracy $\left|\mathcal{E}\right({\tilde{u}}_{x^{\prime }}\left)\right|$ as function of relaxation parameter $\mathrm{\Lambda }$. All cases refer to inclined channel $[\theta ={tan}^{-1}\left(\frac{1}{2}\right)]$ resolved by $N_y=6$ grid nodes its along width. (a) No-slip flow regime ($\mathrm{Kn}=0$). (b) First-order slip-flow regime ($\mathrm{Kn}=0.1, C_1=1$ [15]). (c) Second-order slip-flow regime ($\mathrm{Kn}=0.1, C_1=1.1466$, and $C_2=0.9576$ [21]). The MR slip schemes (not shown here) are exact for all slip-flow regimes and for all $\mathrm{\Lambda }$ values.

Figure 11

Accuracy $\left|\mathcal{E}\right({\tilde{u}}_x\left)\right|$ as function of grid resolution $H$, with $\mathrm{\Lambda }$ fixed and given by Table 10, with ${\delta }_y=\frac{1}{2}$. Convergence rates are quantified in Table 12. (a) No-slip flow regime ($\mathrm{Kn}=0$). (b) First-order slip-flow regime ($\mathrm{Kn}=0.1, C_1=1$ [15]). (c) Second-order slip-flow regime ($\mathrm{Kn}=0.1, C_1=1.1466$, and $C_2=0.9576$ [21]). The MR slip schemes (not shown here) are exact for all slip-flow regimes, $\forall \mathrm{\Lambda }$ values and $\forall H$; hence, the concept of grid convergence does not apply.

Figure 12

Normalized mass flow rate [Eq. (33)] as function of $\mathrm{Kn}$ for the three slip-flow conditions: (i) no slip ($C_1=C_2=0$), (ii) first-order slip ($C_1=1.0$ and $C_2=0$ by Maxwell [15]), (iii) second-order slip ($C_1=1.1466$ and $C_2=0.9576$ by Cercignani [21]), and (iv) second-order slip $[C_1\left(\mathrm{Kn}\right)$ and $C_2\left(\mathrm{Kn}\right)$ as in Table 1 by Wang et al. 20].