# Revisiting the Homogenized Lattice Boltzmann Method with Applications on Particulate Flows

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## Abstract

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## 1. Introduction

## 2. Modelling

#### 2.1. Drag Force

#### 2.2. Hindered Settling

## 3. Methods

#### 3.1. Homogenized Lattice Boltzmann Method

#### 3.2. Forcing Schemes

#### 3.3. Methods for Momentum Exchange

## 4. Results and Discussion of Numerical Experiments

#### 4.1. Settling Sphere

#### 4.1.1. Simulation Setup—Comparison to Literature

#### 4.1.2. Results and Discussion—Comparison to Literature

#### 4.1.3. Simulation Setup—Comparison to Correlations

#### 4.1.4. Results and Discussion—Comparison to Correlations

#### 4.1.5. Further Discussion—Onset of Unsteadiness

#### 4.2. Tubular Pinch Effect

#### 4.2.1. Simulation Setup

#### 4.2.2. Results and Discussion

#### 4.3. Hindered Settling

#### 4.3.1. Simulation Setup

#### 4.3.2. Results and Discussion

#### 4.4. Computational Efficiency

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BGK | Bhatnagar–Gross–Krook |

DNS | direct numerical simulation |

EDM | exact difference method |

EOC | experimental order of convergence |

GUO | Guo forcing |

HLBM | homogenised lattice Boltzmann method |

LBM | lattice Boltzmann method |

MEA | momentum exchange algorithm |

MEA-L | momentum exchange algorithm according to Ladd [39,40] |

MEA-W | momentum exchange algorithm according to Wen et al. [42] |

MLA | momentum loss algorithm |

MLUPps | million lattice site updates per second and per processor |

PCM | partial curve mapping |

RMSE | root mean squared error |

SCF | Shan–Chen forcing scheme |

Roman | |

A | projected area of an object in flow direction |

B | domain covered by a particle |

${C}_{D}$ | drag coefficient |

${c}_{i}$ | i-th discrete lattice velocity |

${c}_{\mathrm{s}}$ | lattice speed of sound |

d, $\widehat{d}$ | spatial dimension |

${d}_{\mathrm{B}}$ | mapping function of an object on the lattice |

${d}_{\mathrm{p}}$ | particle diameter |

D | tube diameter |

${f}_{i}$ | particle distribution function in the phase space according to the i-th lattice velocity |

${f}_{i}^{\mathrm{eq}}$ | Maxwell–Boltzmann distribution function according to the i-th lattice velocity |

$F$ | force |

${F}_{\mathrm{f}}$ | force acting on the fluid |

${F}_{\mathrm{p}}$ | force acting on the particle |

${F}^{\mathrm{B}}$ | Buoyancy force |

${F}^{\mathrm{D}}$ | drag force |

${F}^{\mathrm{G}}$ | gravitational force |

${F}^{\mathrm{H}}$ | hydrodynamic force |

g | gravitational acceleration |

I | time interval |

${I}_{\mathrm{h}}$ | discrete time interval |

${J}_{\mathrm{p}}$ | moment of inertia |

L | tube length |

${m}_{\mathrm{p}}$ | particle mass |

N, $\widehat{N}$ | resolution parameter |

${n}_{\mathrm{p}}$ | number of processes |

p | pressure |

${p}^{\mathrm{L}}$ | pressure in lattice units |

q | dimension of the velocity space of a lattice |

$r$ | distance to the center of mass of an object |

$\mathrm{Re}$ | Reynolds number |

S | speedup |

${S}_{i}$ | source term in the LBM collision step according to the i-th lattice velocity |

t | time |

${T}_{\mathrm{p}}$ | torque |

$u$ | velocity |

$\tilde{u}$ | redefined velocity in the presence of a particle |

${u}_{\mathrm{BM}}$ | hindered settling velocity according to Barnea and Mizrahi |

${u}^{\mathrm{eq}}$ | velocity used in the Maxwell–Boltzmann distribution |

${u}_{\mathrm{f}}$ | fluid velocity |

${u}_{\mathrm{O}}$ | hindered settling velocity according to Oliver |

${u}_{\mathrm{p}}$ | particle velocity |

${u}_{\mathrm{RZ}}$ | hindered settling velocity according to Richardson and Zaki |

${u}_{\mathrm{S}}$ | terminal settling velocity according to Stokes |

${u}_{\mathrm{S}-\mathrm{N}}$ | terminal settling velocity according to Schiller and Naumann |

${u}_{\mathrm{sim}}$ | simulated hindered settling velocity |

${u}_{\mathrm{Steinour}}$ | hindered settling velocity according to Steinour |

${u}_{\infty}$ | maximum settling velocity |

${u}^{\mathrm{L}}$ | velocity in lattice units |

${w}_{i}$ | weighting function according to the i-th lattice velocity |

$x$ | coordinates of a lattice point |

${x}_{\mathrm{c}}$ | x-coordinate of particle center |

${y}_{\mathrm{c}}$ | y-coordinate of particle center |

${y}_{\mathrm{start}}$ | initial y-coordinate of particle center |

Greek | |

$\delta t$ | time step size in SI units |

$\delta x$ | grid spacing in SI units |

$\delta {t}^{\mathrm{L}}$ | time step size in lattice units |

$\delta {x}^{\mathrm{L}}$ | grid spacing in lattice units |

$\Delta u$ | difference between fluid velocity and the velocity in presence of an object |

${\mu}_{\mathrm{f}}$ | dynamic viscosity |

$\nu $ | kinematic viscosity |

$\rho $ | density in lattice units |

${\rho}_{\mathrm{f}}$ | fluid density |

${\rho}_{\mathrm{p}}$ | particle density |

$\tau $ | lattice relaxation time |

$\varphi $ | solid volume fraction |

$\omega $ | angular velocity |

$\Omega $ | spatial domain |

${\Omega}_{\mathrm{h}}$ | discrete approximation of the spatial domain $\Omega $ |

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**Figure 1.**Setup of the simulation according to ten Cate at al. [45].

**Figure 2.**Settling velocity in the four defined cases for different combinations of forcing and momentum exchange schemes along with the experimental results by ten Cate et al. [45].

**Figure 3.**Extract of the results for the case of $\mathrm{Re}=11.6$ with exact difference method (EDM) and momentum exchange algorithm (MEA)-W for different grid spacings along with the experimental data by ten Cate et al. [45].

**Figure 4.**Drag coefficient of correlations discussed in Section 2.1 along with the simulation results plotted against the Reynolds number, computed using the maximum settling velocity measured in the simulation.

**Figure 5.**Maximum settling velocity for ${\mu}_{\mathrm{f}}=0.005\mathrm{k}\mathrm{g}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$ over the chosen lattice velocity ${u}^{\mathrm{L}}$ (see Equation (27)).

**Figure 7.**Contours of the stream-wise vorticity for a sphere with $\mathrm{Ga}=548.97$ at $t=0.19$ s.

**Figure 9.**Results of various starting positions of the circle according to case 2 of Inamuro et al. [36].

**Figure 10.**Random distribution of spheres in the computational domain for the hindered settling simulations with a solid volume fraction of $20\%$.

**Figure 11.**Velocity field around the sphere at $t=0.23\mathrm{s}$ for the case of $\mathrm{Re}=5.29$ with a solid volume fraction of $\varphi =0.2$.

**Figure 12.**Averages over the settling velocity of the top $5\%$ of particles for various $\varphi $ and $\mathrm{Re}=49.64$. The part used for temporal averaging is depicted as dashed line.

**Figure 13.**Performance data as million lattice updates per second and core, regarding the number of simulated particles.

Case | ${\mathit{\rho}}_{f}$ in kg m${}^{-3}$ | ${\mathit{\mu}}_{f}$ in kg m${}^{-1}$s${}^{-1}$ | $\mathrm{Re}$ | $\mathit{\delta}\mathit{x}$ in m | $\mathit{\delta}\mathit{t}$ in s | ${\mathit{u}}_{\mathit{\infty}}$ in m s${}^{-1}$ (Abraham [60]) | Error in % |
---|---|---|---|---|---|---|---|

1 | 970 | $0.373$ | $1.5$ | $1.671\times {10}^{-3}$ | $3.891\times {10}^{-4}$ | $0.038$ | $10.6$ |

2 | 965 | $0.212$ | $4.1$ | $1.645\times {10}^{-3}$ | $2.41\times {10}^{-4}$ | $0.06$ | $5.0$ |

3 | 962 | $0.113$ | $11.6$ | $1.61\times {10}^{-3}$ | $1.526\times {10}^{-4}$ | $0.091$ | $4.5$ |

4 | 960 | $0.058$ | $31.9$ | $1.559\times {10}^{-3}$ | $1.01\times {10}^{-4}$ | $0.128$ | $5.3$ |

**Table 2.**Different error measures [80] for the four cases described by ten Cate et al. [45] for various combinations of momentum exchange and forcing schemes. The error in terminal velocity is given relative to the analytical values according to Abraham [60]. The results in the table refer to $N=1$.

Case | Forcing | Momentum Exchange | RMSE | PCM | Area between Curves | Error in % Maximum Velocity |
---|---|---|---|---|---|---|

1 | EDM | MEA-W | $0.04191$ | $3.63347$ | $0.00386$ | $4.69880$ |

1 | EDM | MLA | $0.02722$ | $2.17069$ | $0.00292$ | $4.91581$ |

1 | EDM | MEA-L | $0.04187$ | $3.61223$ | $0.00386$ | $4.71524$ |

1 | GUO | MEA-W | $0.04190$ | $3.63317$ | $0.00386$ | $4.69881$ |

1 | SCF | MEA-W | $0.02616$ | $3.86421$ | $0.00323$ | $6.73646$ |

1 | SCF | MEA-L | $0.02618$ | $3.87909$ | $0.00323$ | $6.73834$ |

2 | EDM | MEA-W | $0.07019$ | $1.96464$ | $0.00703$ | $6.08454$ |

2 | EDM | MLA | $0.11885$ | $4.41254$ | $0.01115$ | $6.71691$ |

2 | EDM | MEA-L | $0.07028$ | $1.96609$ | $0.00704$ | $6.08627$ |

2 | GUO | MEA-W | $0.07019$ | $1.96473$ | $0.00703$ | $6.08455$ |

2 | SCF | MEA-W | $0.10945$ | $3.48661$ | $0.01082$ | $7.83986$ |

2 | SCF | MEA-L | $0.10952$ | $3.49102$ | $0.01082$ | $7.85094$ |

3 | EDM | MEA-W | $0.04287$ | $1.50668$ | $0.00409$ | $6.23779$ |

3 | EDM | MLA | $0.08819$ | $2.58684$ | $0.01000$ | $8.01712$ |

3 | EDM | MEA-L | $0.04289$ | $1.50718$ | $0.00409$ | $6.23545$ |

3 | GUO | MEA-W | $0.04287$ | $1.50659$ | $0.00409$ | $6.23780$ |

3 | SCF | MEA-W | $0.05442$ | $2.00959$ | $0.00585$ | $7.27864$ |

3 | SCF | MEA-L | $0.05443$ | $2.00904$ | $0.00585$ | $7.28794$ |

4 | EDM | MEA-W | $0.02578$ | $0.50550$ | $0.00271$ | $4.88694$ |

4 | EDM | MLA | $0.10395$ | $2.71061$ | $0.01302$ | $9.27789$ |

4 | EDM | MEA-L | $0.02580$ | $0.50635$ | $0.00271$ | $4.89684$ |

4 | GUO | MEA-W | $0.02578$ | $0.50551$ | $0.00271$ | $4.88695$ |

4 | SCF | MEA-W | $0.03347$ | $0.86398$ | $0.00401$ | $5.84097$ |

4 | SCF | MEA-L | $0.03349$ | $0.86637$ | $0.00402$ | $5.84066$ |

**Table 3.**Different error measures [80] for the four cases described by ten Cate et al. [45] for various combinations of momentum exchange and forcing schemes. The error in terminal velocity is given relative to the analytical values according to Abraham [60]. The results in the table refer to $N=8$.

Case | Forcing | Momentum Exchange | RMSE | PCM | Area between Curves | Error in % Maximum Velocity |
---|---|---|---|---|---|---|

1 | EDM | MEA-W | $0.03241$ | $4.35586$ | $0.00422$ | $8.24923$ |

1 | EDM | MLA | $0.03399$ | $2.84034$ | $0.00448$ | $8.27399$ |

1 | EDM | MEA-L | $0.03241$ | $4.35589$ | $0.00422$ | $8.24930$ |

1 | GUO | MEA-W | $0.03241$ | $4.35586$ | $0.00422$ | $8.24923$ |

1 | SCF | MEA-W | $0.03442$ | $4.41524$ | $0.00457$ | $8.49026$ |

1 | SCF | MEA-L | $0.03442$ | $4.41527$ | $0.00457$ | $8.49039$ |

2 | EDM | MEA-W | $0.03224$ | $1.11747$ | $0.00347$ | $4.32345$ |

2 | EDM | MLA | $0.05467$ | $1.03172$ | $0.00550$ | $4.44263$ |

2 | EDM | MEA-L | $0.03224$ | $1.11747$ | $0.00347$ | $4.32358$ |

2 | GUO | MEA-W | $0.03224$ | $1.11748$ | $0.00347$ | $4.32345$ |

2 | SCF | MEA-W | $0.03326$ | $1.10431$ | $0.00359$ | $4.59350$ |

2 | SCF | MEA-L | $0.03326$ | $1.10431$ | $0.00359$ | $4.59360$ |

3 | EDM | MEA-W | $0.02349$ | $0.64709$ | $0.00162$ | $4.43872$ |

3 | EDM | MLA | $0.06483$ | $1.49952$ | $0.00563$ | $4.83740$ |

3 | EDM | MEA-L | $0.02349$ | $0.64703$ | $0.00162$ | $4.43877$ |

3 | GUO | MEA-W | $0.02349$ | $0.64709$ | $0.00162$ | $4.43872$ |

3 | SCF | MEA-W | $0.02079$ | $0.76920$ | $0.00152$ | $4.67580$ |

3 | SCF | MEA-L | $0.02079$ | $0.76898$ | $0.00152$ | $4.67593$ |

4 | EDM | MEA-W | $0.08260$ | $1.06138$ | $0.00223$ | $4.96350$ |

4 | EDM | MLA | $0.08638$ | $1.67762$ | $0.00927$ | $6.09874$ |

4 | EDM | MEA-L | $0.08259$ | $1.06147$ | $0.00223$ | $4.96360$ |

4 | GUO | MEA-W | $0.08260$ | $1.06139$ | $0.00223$ | $4.96350$ |

4 | SCF | MEA-W | $0.08877$ | $1.03403$ | $0.00251$ | $5.16140$ |

4 | SCF | MEA-L | $0.08875$ | $1.03418$ | $0.00251$ | $5.16147$ |

N | Forcing | Momentum Exchange | RMSE (Mean) | PCM (Mean) | Area between Curves (Mean) | Error in % Maximum (Mean) |
---|---|---|---|---|---|---|

1 | EDM | MEA-W | $0.04519$ | $1.90257$ | $0.00442$ | $5.47702$ |

1 | EDM | MLA | $0.08455$ | $2.97017$ | $0.00927$ | $7.23193$ |

1 | EDM | MEA-L | $0.04521$ | $1.89796$ | $0.00443$ | $5.48345$ |

1 | GUO | MEA-W | $0.04519$ | $1.90250$ | $0.00442$ | $5.47703$ |

1 | SCF | MEA-W | $0.05588$ | $2.55610$ | $0.00598$ | $6.92398$ |

1 | SCF | MEA-L | $0.05590$ | $2.56138$ | $0.00598$ | $6.92947$ |

8 | EDM | MEA-W | $0.04268$ | $1.79545$ | $0.00288$ | $5.49373$ |

8 | EDM | MLA | $0.05997$ | $1.76230$ | $0.00622$ | $5.91319$ |

8 | EDM | MEA-L | $0.04268$ | $1.79546$ | $0.00288$ | $5.49381$ |

8 | GUO | MEA-W | $0.04268$ | $1.79545$ | $0.00288$ | $5.49373$ |

8 | SCF | MEA-W | $0.04431$ | $1.83069$ | $0.00305$ | $5.73024$ |

8 | SCF | MEA-L | $0.04430$ | $1.83069$ | $0.00305$ | $5.73035$ |

**Table 5.**Mean experimental order of convergence (EOC) for different error and similarity measures regarding $N=8$. Results are averaged over all values for a given combination of forcing and momentum exchange scheme as well as all cases.

Forcing | Momentum Exchange | EOC (RMSE) | EOC (RMSE) | EOC (Area between Curves) | EOC (Area between Curves) |
---|---|---|---|---|---|

EDM | MEA-W | $1.79$ | $1.81$ | $2.06$ | $1.39$ |

EDM | MLA | $1.37$ | $2.00$ | $1.49$ | $1.66$ |

EDM | MEA-L | $1.79$ | $1.81$ | $2.05$ | $1.36$ |

GUO | MEA-W | $1.79$ | $1.81$ | $2.06$ | $1.39$ |

SCF | MEA-W | $1.57$ | $1.03$ | $1.72$ | $1.55$ |

SCF | MEA-L | $1.57$ | $1.03$ | $1.72$ | $1.55$ |

**Table 6.**Deviation of of the drag coefficients calculated by homogenized lattice Boltzmann method (HLBM) simulations to different drag correlations.

${\mathit{\mu}}_{f}$ in kg m${}^{-1}{s}^{-1}$ | ${\mathit{C}}_{D}$ (Simulated) | Stokes | Abraham | Schiller and Naumann | |||
---|---|---|---|---|---|---|---|

Re | Error in % | Re | Error in % | Re | Error in % | ||

$2.0\times {10}^{-2}$ | $86.6942$ | $0.26$ | $-7.72$ | $0.23$ | $-24.93$ | $0.24$ | $-17.33$ |

$1.5\times {10}^{-2}$ | $56.7155$ | $0.45$ | $7.33$ | $0.40$ | $-17.98$ | $0.42$ | $-8.42$ |

$1.0\times {10}^{-2}$ | $29.6648$ | $1.02$ | $26.31$ | $0.84$ | $-14.14$ | $0.90$ | $-2.68$ |

$5.0\times {10}^{-3}$ | $10.5937$ | $4.09$ | $80.42$ | $2.90$ | $-9.38$ | $3.08$ | $2.74$ |

$2.5\times {10}^{-3}$ | $4.5797$ | $16.35$ | $212.00$ | $9.18$ | $-1.61$ | $9.57$ | $6.94$ |

$1.25\times {10}^{-3}$ | $2.2965$ | $65.40$ | $525.82$ | $26.57$ | $3.29$ | $26.83$ | $5.34$ |

$6.25\times {10}^{-4}$ | $1.2539$ | $261.60$ | $1266.86$ | $70.48$ | $-0.79$ | $69.50$ | $-3.52$ |

$3.125\times {10}^{-4}$ | $0.7339$ | $1046.40$ | $3100.03$ | $173.68$ | $-11.84$ | $170.80$ | $-14.74$ |

$1.5625\times {10}^{-4}$ | $0.5213$ | $4185.60$ | $8992.70$ | $404.05$ | $-15.27$ | $406.40$ | $-14.28$ |

$7.8125\times {10}^{-5}$ | $0.4543$ | 16,742.40 | 31,595.60 | $900.47$ | $-8.31$ | $948.67$ | $1.76$ |

**Table 7.**Maximum settling velocity for ${\mu}_{\mathrm{f}}=0.005\mathrm{k}\mathrm{g}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$ for a chosen lattice velocity ${u}^{\mathrm{L}}$ (see Equation (27)). Additionally, given is the deviation to the velocity obtained with ${u}^{\mathrm{L}}=0.0025$ and the change of maximum velocity to the value obtained with the next smaller ${u}^{\mathrm{L}}$.

${\mathit{u}}^{L}$ | $0.0025$ | $0.005$ | $0.01$ | $0.02$ | $0.04$ | $0.08$ | $0.16$ |
---|---|---|---|---|---|---|---|

${u}_{\infty}$ in $\mathrm{m}{\mathrm{s}}^{-1}$ | $0.02975$ | $0.03002$ | $0.03042$ | $0.03102$ | $0.031981$ | $0.03375$ | $0.03768$ |

Deviation in % | 0 | $0.89$ | $2.24$ | $4.25$ | $7.50$ | $13.45$ | $26.64$ |

Change in % | - | $0.89$ | $1.33$ | $1.97$ | $3.11$ | $5.54$ | $11.6$ |

Case | $\mathit{\tau}$ | $\mathbf{\Delta}\mathit{p}$ | D | L | ${\mathit{d}}_{p}/\mathit{D}$ | ${\mathit{y}}_{c}/\mathit{D}$ | Error in % |
---|---|---|---|---|---|---|---|

Inamuro et al. case 2 | $1.4$ | $8.167\times {10}^{-4}$ | 200 | 200 | $0.25$ | $0.2642$ | $-3.33\%$ |

Inamuro et al. case 6 | $0.757$ | $2.337\times {10}^{-4}$ | 200 | 200 | $0.25$ | $0.2548$ | $-5.83\%$ |

Inamuro et al. case 11 | $1.4$ | $1.633\times {10}^{-3}$ | 200 | 400 | $0.25$ | $0.2726$ | $-4.35\%$ |

Inamuro et al. case 14 | $0.95$ | $3.207\times {10}^{-3}$ | 100 | 400 | $0.5$ | $0.3590$ | $-5.48\%$ |

Li et al. | $0.75$ | $2.670\times {10}^{-4}$ | 100 | 400 | $0.25$ | $0.2756$ | $-4.10\%$ |

**Table 9.**Simulated hindered settling velocity regarding different $\mathrm{Re}$ and $\varphi $ in comparison to different correlations.

$\mathrm{Re}$ | $\mathit{\varphi}$ | u_{sim} in m s${}^{-1}$ | Error in % (u_{RZ}, u_{S}) | Error in % (u_{O}, u_{S}) | Error in % (u_{BM}, u_{S}) | Error in % (u_{RZ}, u_{S-N}) | Error in % (u_{O}, u_{S-N}) | Error in % (u_{BM}, u_{S-N}) |
---|---|---|---|---|---|---|---|---|

$0.53$ | $0.05$ | $-0.00071$ | $-23.38$ | $-10.17$ | $-3.97$ | $-15.90$ | $-1.46$ | $5.35$ |

$0.53$ | $0.1$ | $-0.00055$ | $-24.69$ | $-16.87$ | $-7.46$ | $-17.28$ | $-8.81$ | $1.51$ |

$0.53$ | $0.15$ | $-0.00045$ | $-20.36$ | $-19.03$ | $-6.15$ | $-12.46$ | $-11.18$ | $2.95$ |

$0.53$ | $0.2$ | $-0.00037$ | $-15.63$ | $-21.35$ | $-5.47$ | $-7.19$ | $-13.73$ | $3.70$ |

$0.53$ | $0.25$ | $-0.00031$ | $-5.06$ | $-18.20$ | $0.80$ | $4.52$ | $-10.27$ | $10.57$ |

$5.29$ | $0.05$ | $-0.00691$ | $-46.58$ | $-34.75$ | $-30.24$ | $-20.82$ | $-4.00$ | $2.63$ |

$5.29$ | $0.1$ | $-0.00551$ | $-48.15$ | $-37.75$ | $-30.71$ | $-22.56$ | $-8.41$ | $1.95$ |

$5.29$ | $0.15$ | $-0.00444$ | $-48.68$ | $-40.61$ | $-31.16$ | $-22.72$ | $-12.61$ | $1.29$ |

$5.29$ | $0.2$ | $-0.00367$ | $-47.08$ | $-41.07$ | $-29.17$ | $-19.62$ | $-13.29$ | $4.22$ |

$5.29$ | $0.25$ | $-0.00295$ | $-46.26$ | $-41.77$ | $-28.24$ | $-17.62$ | $-14.32$ | $5.58$ |

$49.46$ | $0.05$ | $-0.06962$ | $-74.81$ | $-67.71$ | $-65.47$ | $-18.17$ | $3.13$ | $10.26$ |

$49.46$ | $0.1$ | $-0.05656$ | $-76.34$ | $-68.63$ | $-65.08$ | $-21.76$ | $0.19$ | $11.53$ |

$49.46$ | $0.15$ | $-0.04757$ | $-76.80$ | $-68.71$ | $-63.73$ | $-21.83$ | $-0.06$ | $15.84$ |

$49.46$ | $0.2$ | $-0.03889$ | $-77.69$ | $-69.34$ | $-63.14$ | $-23.30$ | $-2.07$ | $17.71$ |

$49.46$ | $0.25$ | $-0.03300$ | $-77.49$ | $-68.01$ | $-60.58$ | $-20.96$ | $2.17$ | $25.90$ |

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**MDPI and ACS Style**

Trunk, R.; Weckerle, T.; Hafen, N.; Thäter, G.; Nirschl, H.; Krause, M.J.
Revisiting the Homogenized Lattice Boltzmann Method with Applications on Particulate Flows. *Computation* **2021**, *9*, 11.
https://doi.org/10.3390/computation9020011

**AMA Style**

Trunk R, Weckerle T, Hafen N, Thäter G, Nirschl H, Krause MJ.
Revisiting the Homogenized Lattice Boltzmann Method with Applications on Particulate Flows. *Computation*. 2021; 9(2):11.
https://doi.org/10.3390/computation9020011

**Chicago/Turabian Style**

Trunk, Robin, Timo Weckerle, Nicolas Hafen, Gudrun Thäter, Hermann Nirschl, and Mathias J. Krause.
2021. "Revisiting the Homogenized Lattice Boltzmann Method with Applications on Particulate Flows" *Computation* 9, no. 2: 11.
https://doi.org/10.3390/computation9020011