Explicit and viscosity-independent immersed-boundary scheme for the lattice Boltzmann method

Viscosity independence of lattice-Boltzmann methods is a crucial issue to ensure the physical relevancy of the predicted macroscopic flows over large ranges of physical parameters. The immersed-boundary (IB) method, a powerful tool that allows one to immerse arbitrary-shaped, moving, and deformable bodies in the flow, suffers from a boundary-slip error that increases as a function of the fluid viscosity, substantially limiting its range of application. In addition, low fluid viscosities may result in spurious oscillations of the macroscopic quantities in the vicinity of the immersed boundary. In this work, it is shown mathematically that the standard IB method is indeed not able to reproduce the scaling properties of the macroscopic solution, leading to a viscosity-related error on the computed IB force. The analysis allows us to propose a simple correction of the IB scheme that is local, straightforward and does not involve additional computational time. The derived method is implemented in a two-relaxation-time D2Q9 lattice-Boltzmann solver, applied to several physical configurations, namely, the Poiseuille flow, the flow around a cylinder towed in still fluid, and the flow around a cylinder oscillating in still fluid, and compared to a noncorrected immersed-boundary method. The proposed correction leads to a major improvement of the viscosity independence of the solver over a wide range of relaxation times (from 0.5001 to 50), including the correction of the boundary-slip error and the suppression of the spurious oscillations. This improvement may considerably extend the range of application of the IB lattice-Boltzmann method, in particular providing a robust tool for the numerical analysis of physical problems involving fluids of varying viscosity interacting with solid geometries.

Figure 1

Schematic view of the IB configuration. A boundary $\mathrm{\Gamma }$ discretized by Lagrangian markers $\left\{{\mathit{X}}_{\mathit{k}}\right\}$ is immersed in the fluid domain $\mathrm{\Psi }$ discretized by the lattice nodes $\left\{{\mathit{x}}_{\mathit{i}}\right\}$. The shaded blue area represents the region where the IB forcing is applied, determined by the width of the employed kernel function (16).

Figure 2

Sketch of the physical configurations: (a) the inclined Poiseuille flow and (b) the circular cylinder towed in a periodic channel. Solid lines represent the boundaries of the computational domain, and dashed lines indicate the immersed boundaries.

Figure 3

Effect of the relaxation time on the Poiseuille flow solution: (a) standard IBM and (b) corrected IBM. ${\text{Re}}_p=0.1$ and $D_p=50\mathrm{\Delta }n$.

Figure 4

Effect of the grid resolution on the Poiseuille flow solution: (a) evolution of the flow rate $Q$ issued from the standard and corrected IB methods, for $\tau =1$ and $\tau =40$, as a function of $D_p/\mathrm{\Delta }n$; (b) evolution of $\mathrm{\Lambda }$, the relative difference between the flow rate errors of both methods, for different values of the relaxation time, as a function of $D_p/\mathrm{\Delta }n$.

Figure 5

Effect of the relaxation time on the drag force on a cylinder towed in a periodic channel, at ${\text{Re}}_c=0.01$, for the noncorrected and corrected IB methods. The cylinder diameter is set to $D_c=10\mathrm{\Delta }n$.

Figure 6

Effect of the relaxation time on the streamlines around a cylinder towed in a periodic channel, at ${\text{Re}}_c=0.01$, for the standard and corrected IB methods: (a) $\tau =1$ and (b) $\tau =30$. The cylinder diameter is set to $D_c=10\mathrm{\Delta }n$. The streamlines are plotted in the frame attached to the cylinder axis, and the relative flow is from left to right.

Figure 7

Flow past a cylinder oscillating in still fluid at ${\text{Re}}_c=3$, simulated using the noncorrected IB method for $\tau =1$: (a) selected time history of the body displacement $x_c$ and drag coefficient $C_x$, and (b) and isocontours of the nondimensional vorticity at two different instants $t_1$ and $t_2$, indicated by vertical dashed lines in (a). The diameter of the cylinder is $D_c=10\mathrm{\Delta }n$.

Figure 8

Effect of the relaxation time on the drag force exerted on a cylinder oscillating in still fluid, at ${\text{Re}}_c=3$, for the noncorrected and corrected IB methods: (a) selected time histories of the drag coefficient and (b) evolution of the RMS of the high-pass filtered drag force $\widetilde{C_x}$ as a function of the relaxation time.

Figure 9

Examination of approximation (34), for the oscillating-cylinder case: evolution of the reinterpolated momentum correction in the $x$ direction $\mathcal{I}\left[\mathcal{S}\left(G_x\right)\right]$ on the immersed boundary as a function of the polar angle $\theta$, at times (a) $t_1$ and (b) $t_2$ indicated in Fig. 7, and comparison with the approximation $\kappa G_x$. The data are issued from the same simulation as in Fig. 7.

Figure 10

Evolution of the drag force exerted on a cylinder immersed in a cross flow, as a function of the Reynolds number. The results issued from the present corrected IB method are compared to those reported in prior works of Sen et al. [30] and Tritton [31]. The cylinder diameter is $D_c=10\mathrm{\Delta }n$.