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

Gaseous flows inside microfluidic devices often fall in the slip-flow regime. According to this theoretical description, the Navier-Stokes model remains applicable in bulk, while at solid walls a slip velocity boundary model shall be considered. Physically, it is well established that, to properly account for the wall curvature, the wall slip velocity must be determined by the shear stress, rather than the normal component of the velocity derivative alone, as commonly applied to planar surfaces. It follows that the numerical transcription of this type of boundary condition is generally a challenging task for standard computational fluid dynamics (CFD) techniques. This paper aims to show that the realization of the slip velocity condition on arbitrarily shaped boundaries can be accomplished in a natural way with the lattice Boltzmann method (LBM). To substantiate this conclusion, this work undertakes the following three studies. First, we examine the conditions under which the generic reflection-type boundary rules used by LBM become consistent models for the slip velocity boundary condition. This effort makes use of the second-order Chapman-Enskog expansion method, where we address both planar and curved boundaries. The analysis also clarifies the capabilities and limitations behind the considered reflection-type slip schemes. Second, we revisit the family of parabolic accurate LBM slip boundary schemes, originally formulated in [Phys. Rev. E 96, 013311 (2017)] on the basis of the multireflection framework, and discuss their characteristics when operating on curved boundaries as well as the limitations of other less accurate LBM slip boundary formulations, such as the linearly accurate slip schemes and the widely popular “kinetic-based” boundary schemes. In addition, we also discuss the numerical stability of the parabolic slip schemes previously developed, providing an heuristic strategy to improve their stable range of operation. Third, we evaluate the performance of the several slip boundary schemes debated in this paper. The numerical tests correspond to two classical 2D benchmark flow problems of slip over non-planar solid surfaces, namely: (i) the velocity profile of the cylindrical Couette flow, and (ii) the permeability of a slow rarefied gas over a periodic array of circular cylindrical obstacles. The obtained numerical results confirm the competitiveness of the LBM when equipped with slip boundary schemes of parabolic accuracy as CFD tool to simulate slippage phenomena over arbitrarily non-planar surfaces. Indeed, although operating on a simple uniform mesh discretization, the LBM yields a similar, or even superior, level of accuracy compared to state-of-the-art FEM simulations conducted on hardworking body-fitted meshes. This conclusion establishes the LBM as a very appealing CFD technique for simulating microfluidic flows in the slip-flow regime, a result that deserves further exploration in future studies.

Figure 1

Discretization of a curved wall on the uniform mesh of the LBM operating on d2Q9 lattice.

Figure 2

Cylindrical Couette flow. Panel (a): Problem geometry. Panel (b): LBM discretization. Panel (c): FEM discretization.

Figure 3

Cylindrical Couette flow velocity profiles as function of $\mathrm{Kn}=\frac{\lambda }{R_2-R_1}$, for $R_1/R_2=1/2$, ${\mathcal{C}}_1={\mathcal{C}}_2=1$. Panel (a): ${\mathrm{\Omega }}_1=0.001$ and ${\mathrm{\Omega }}_2=0$ (${\overline{u}}_{\theta }=u_{\theta }/\left({\mathrm{\Omega }}_1{R}_1\right)$ and $\overline{r}=r/R_2$). Panel (b): ${\mathrm{\Omega }}_1=0$ and ${\mathrm{\Omega }}_2=0.001$ (${\overline{u}}_{\theta }=u_{\theta }/\left({\mathrm{\Omega }}_2{R}_2\right)$ and $\overline{r}=r/R_2$).

Figure 4

Cylindrical Couette flow velocity profiles as function of $\{{\mathcal{C}}_1,{\mathcal{C}}_2\}$, for $R_1/R_2=1/2$ and $\mathrm{Kn}=\frac{\lambda }{R_2-R_1}=0.1$. Panel (a): ${\mathrm{\Omega }}_1=0.001$ and ${\mathrm{\Omega }}_2=0$ (${\overline{u}}_{\theta }=u_{\theta }/\left({\mathrm{\Omega }}_1{R}_1\right)$ and $\overline{r}=r/R_2$). Panel (b): ${\mathrm{\Omega }}_1=0$ and ${\mathrm{\Omega }}_2=0.001$ (${\overline{u}}_{\theta }=u_{\theta }/\left({\mathrm{\Omega }}_2{R}_2\right)$ and $\overline{r}=r/R_2$).

Figure 5

Cylindrical Couette flow with ${\mathrm{\Omega }}_1=0.001$ and ${\mathrm{\Omega }}_2=0$. Accuracy $|{\mathcal{E}}_u|$ as function of grid resolution $H$ (note, $H:=R_2-R_1$), where $R_1/R_2=1/2$ and $\mathrm{\Lambda }=3/16$ are fixed. Convergence rates are quantified in Table 4. Panel (a): No-slip flow regime ($\mathrm{Kn}=0$). Panel (b): Moderate slip-flow regime ($\mathrm{Kn}=0.01$, ${\mathcal{C}}_1={\mathcal{C}}_2=1$). Panel (c): Large slip-flow regime ($\mathrm{Kn}=0.1$, ${\mathcal{C}}_1={\mathcal{C}}_2=1$).

Figure 6

Same as Fig. 5, but with ${\mathrm{\Omega }}_1=0$ and ${\mathrm{\Omega }}_2=0.001$. Panel (a): No-slip flow regime ($\mathrm{Kn}=0$). Panel (b): Moderate slip-flow regime ($\mathrm{Kn}=0.01$, ${\mathcal{C}}_1={\mathcal{C}}_2=1$). Panel (c): Large slip-flow regime ($\mathrm{Kn}=0.1$, ${\mathcal{C}}_1={\mathcal{C}}_2=1$).

Figure 7

Cylindrical Couette flow with ${\mathrm{\Omega }}_1=0.001$ and ${\mathrm{\Omega }}_2=0$. Accuracy $|{\mathcal{E}}_u|$ as function of $\mathrm{\Lambda }$, with $H=15$ fixed (note, $H:=R_2-R_1$) and $R_1/R_2=1/2$. Panel (a): No-slip flow regime ($\mathrm{Kn}=0$). Panel (b): Moderate slip-flow regime ($\mathrm{Kn}=0.01$, ${\mathcal{C}}_1={\mathcal{C}}_2=1$). Panel (c): Large slip-flow regime ($\mathrm{Kn}=0.1$, ${\mathcal{C}}_1={\mathcal{C}}_2=1$).

Figure 8

Same as Fig. 7 but with ${\mathrm{\Omega }}_1=0$ and ${\mathrm{\Omega }}_2=0.001$. Panel (a): No-slip flow regime ($\mathrm{Kn}=0$). Panel (b): Moderate slip-flow regime ($\mathrm{Kn}=0.01$, ${\mathcal{C}}_1={\mathcal{C}}_2=1$). Panel (c): Large slip-flow regime ($\mathrm{Kn}=0.1$, ${\mathcal{C}}_1={\mathcal{C}}_2=1$).

Figure 9

Cylindrical Couette flow accuracy $|{\mathcal{E}}_u|$ at $\mathrm{Kn}=0.1$ as function of slippage coefficients, with $\mathrm{\Lambda }=3/16$, $H=15$ (note, $H:=R_2-R_1$), and $R_1/R_2=1/2$ fixed. Panel (a): ${\mathrm{\Omega }}_1=0.001$ and ${\mathrm{\Omega }}_2=0$ with ${\mathcal{C}}_1=1$ and ${\mathcal{C}}_2\in [0,100]$. Panel (b): ${\mathrm{\Omega }}_1=0$ and ${\mathrm{\Omega }}_2=0.001$ with ${\mathcal{C}}_1\in [0,100]$ and ${\mathcal{C}}_2=1$.

Figure 10

Periodic array of solid cylindrical obstacles. Panel (a): Problem geometry. Panel (b): LBM discretization. Panel (c): FEM discretization.

Figure 11

Permeability of a creeping flow across a periodic array of cylindrical obstacles at low volume fraction $p=0.2$. Accuracy $|{\mathcal{E}}_{k_{\mathrm{eff}}}|$ as function of $\mathrm{\Lambda }$, with $H=33$ fixed [$H$ is defined on Fig. 10]. Panel (a): No-slip flow regime ($\mathrm{Kn}=0$). Panel (b): Large slip-flow regime [$\mathrm{Kn}=0.1$, $\mathcal{C}=1$)].

Figure 12

Permeability of a creeping flow across a periodic array of cylindrical obstacles at low volume fraction $p=0.2$, developing in large slip-flow regime ($\mathrm{Kn}=0.1$). Accuracy $|{\mathcal{E}}_{k_{\mathrm{eff}}}|$ as function of slippage coefficient $\mathcal{C}$, with $\mathrm{\Lambda }=3/16$ and $H=33$ fixed [$H$ is defined on Fig. 10].