A Comparison Between the Interpolated Bounce-Back Scheme and the Immersed Boundary Method to Treat Solid Boundary Conditions for Laminar Flows in the Lattice Boltzmann Framework

Abstract

In this paper, the interpolated bounce-back scheme and the immersed boundary method are compared in order to handle solid boundary conditions in the lattice Boltzmann method. These two approaches are numerically investigated in two test cases: a rigid fixed cylinder invested by an incoming viscous fluid and an oscillating cylinder in a calm viscous fluid. Findings in terms of velocity profiles in several cross sections are shown. Differences and similarities between the two methods are discussed, by emphasizing pros and cons in terms of stability and computational effort of the numerical algorithm.

Appendix

In order to prevent momentum and pressure leakages, the immersed body should be represented by a sufficiently large [16, 26] number of Lagrangian IB points. Therefore, we performed and report here a preliminary analysis on the effect of the number of IB points for the test case of Sect. 5.1 (flow over a rigid cylinder). The drag coefficient $C_d$ experienced by the cylinder at $\mathcal {R}e=10$ is computed with $350 \times 220$ lattice points by progressively refining the number of Lagrangian IB points representing the cylinder surface, thus reducing the solid mesh spacing $\Delta S$. The drag coefficient is computed as

$$\begin{aligned} C_d = \frac{F_x}{\bar{\rho }V^2R}, \end{aligned}$$

(15)

where $F_x$ is the horizontal component of the total force acting upon the cylinder and $\bar{\rho }$ is the average density. The force is computed according to Eq. 13 over all the $\varvec{X}_w$ idealizing the cylinder. The relative error $\varepsilon$ is defined as

$$\begin{aligned} \varepsilon = \frac{C_d^{\mathrm {c}}-C_d^{\mathrm {r}}}{C_d^{\mathrm {r}}}, \end{aligned}$$

(16)

where $C_d^{\mathrm {c}}$ is the drag coefficient computed for a given number of IB points and the reference value $C_d^{\mathrm {r}}$ is computed for a very fine solid mesh consisting of 1024 IB points, which corresponds to a mesh size $\Delta S=0.06$. In Fig. 10, the relative error is plotted against the solid mesh spacing $\Delta S$. It is possible to observe that the error practically vanishes for $\Delta S \le 0.5$, which is therefore the value used in the numerical analyzes reported in the present paper. Specifically, a further refining over such value corresponds to a plateau of the curve. Finally, it is worth to notice that the relative error assumes negative values, since $C_d^{\mathrm {c}} < C_d^{\mathrm {r}}$. Such behavior has to be addressed to the fact that the lower the number of IB points, the more permeable the cylinder surface is, thus the resultant drag coefficient reduces as well.

Fig. 10

Relative error in the computation of $C_d$ at $\mathcal {R}e=10$ for different values of the solid mesh spacing $\Delta S$ (red curve). The blue line represents $\varepsilon =0$ (Color figure online)