Effect of internal mass in the lattice Boltzmann simulation of moving solid bodies by the smoothed-profile method

A computational method for the simulation of particulate flows that can efficiently treat the particle-fluid boundary in systems containing many particles was developed based on the smoothed-profile lattice Boltzmann method (SPLBM). In our proposed method, which we call the improved SPLBM (iSPLBM), for an accurate and stable simulation of particulate flows, the hydrodynamic force on a moving solid particle is exactly formulated with consideration of the effect of internal fluid mass. To validate the accuracy and stability of iSPLBM, we conducted numerical simulations of several particulate flow systems and compared our results with those of other simulations and some experiments. In addition, we performed simulations on flotation of many lightweight particles with a wide range of particle size distribution, the results of which demonstrated the effectiveness of iSPLBM. Our proposed model is a promising method to accurately and stably simulate extensive particulate flows.

Figure 1

Schematic illustration of the $k\mathrm{th}$ rigid particle suspended in an incompressible and Newtonian fluid. In the smoothed-profile (SP) method, the original sharp interface between the particle and host fluid is replaced by a smoothed interface with a finite thickness $\xi$

Figure 2

Streamlines of steady flow past a stationary circular cylinder, which had a diameter of $D=20\mathrm{\Delta }x$ and an interface thickness of $\xi =\mathrm{\Delta }x$, and was fixed at the center of the computational domain of $800\mathrm{\Delta }x\times 800\mathrm{\Delta }x$: (a) $\mathrm{Re}=20$ and (b) $\mathrm{Re}=40$. The flows of $\left|\mathit{u}\right|<{10}^{-15}$ are not shown for visual clarity.

Figure 3

Velocity profiles of the two-dimensional fluid motion induced by the translationally oscillating circular cylinder for the case of $\mathrm{Re}=100$ and $\mathrm{KC}=5$. The velocity profiles are at the four different $x$ positions, $x=-0.6D$, 0, $0.6D$, and $1.2D$, for three different phase angles $t/T=$(a) $n+1/2$, (b) $n+7/12$, and (c) $n+11/12$, where $n$ is an integer number. The results of improved smoothed-profile lattice Boltzmann method (iSPLBM) are compared with experimental results by Dütsch et al. [23].

Figure 4

Time variation of the drag coefficient $C_{\mathrm{D}}$ of a translationally oscillating circular cylinder: (a) $\mathrm{Re}=100$ and $\mathrm{KC}=5$; (b) $\mathrm{Re}=10$ and $\mathrm{KC}=5$; (c) $\mathrm{Re}=1$ and $\mathrm{KC}=5$. The results of iSPLBM are compared with those of the smoothed-profile lattice Boltzmann method (SPLBM) and experimental results by Dütsch et al. [23] in graph (a) and the results of SPLBM in graphs (b,c).

Figure 5

Vertical (a) position and (b) velocity of a sphere sedimenting under gravity in a closed box filled with a fluid as a function of time at different Re. Conditions and parameters of our simulations are summarized in Table 2. The results of iSPLBM and SPLBM are compared with experimental results by ten Cate et al. [15].

Figure 6

Time series snapshots of five spheres floating under gravity in a closed box. The computational conditions and parameters were set to reproduce the simulation by Feng and Michaelides using their immersed-boundary lattice Boltzmann method (IBLBM) [17]. The darker (blue) colors of spheres represent larger vertical velocities.

Figure 7

Vertical (a) velocity and (b) position of spheres 1 and 2 as a function of time, and (c) trajectory of spheres 1–3. The results of iSPLBM are compared with those of the IBLBM simulation by Feng and Michaelides [17].

Figure 8

Time series snapshots of 200 spheres floating under gravity in a closed box filled with a fluid. The spheres had three different diameters of $D=10\mathrm{\Delta }x$ (blue), $15\mathrm{\Delta }x$ (white), and $20\mathrm{\Delta }x$ (red), and their numbers were set as 72, 65, and 63, respectively.

Figure 9

Probability distribution of spheres of three different sizes as a function of vertical position $z$ at the (a) initial and (b) final states.