# A Non-Equilibrium Interpolation Scheme for IB-LBM Optimized by Approximate Force

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

**Immersed boundary method and lattice Boltzmann method**

**The conventional immersed boundary–lattice Boltzmann method (IB-LBM)**

## 3. Present IB-LBM: A Non-Equilibrium Scheme and an Optimized Approximate Force

**Spread operator and interpolation operator in the non-equilibrium Scheme**

**Optimization of the proposed IB-LBM with approximate force on the IB**

**The whole process of the present IB-LBM**

## 4. Results

#### 4.1. Symmetrical Poiseuille Flow with an Immersed Boundary

_{2}norm of the velocity error is also considered:

#### 4.2. Flow Past a Fixed Circular Cylinder

#### 4.3. Sedimentation and Collision of Moving Particles

#### 4.4. A Flexible Filament Fixed at One End

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The proposed non-equilibrium scheme: (

**a**) the interpolation and spread process in two dimensions; (

**b**) the three-point interpolation in one dimension.

**Figure 4.**The comparison of the numerical solutions according to the four grid schemes and the analytic solution.

**Figure 6.**Streamline diagrams: (

**a**) four popular method results when Re = 40; (

**b**) result of present method when Re = 40 and 200.

**Figure 7.**The motion trajectory compared with paper [16]: (

**a**) motion trajectory on X-axis with time t (s); (

**b**) motion trajectory on Y-axis with time t (s).

Mesh | $\Vert {\mathit{u}}_{\mathit{n}}-{\mathit{u}}_{\mathit{a}}\Vert $ | Rate | $\Vert {\mathit{u}}_{\mathit{n}}-{\mathit{u}}_{\mathit{a}}\Vert {}_{\mathit{\infty}}$ | Rate |
---|---|---|---|---|

$20\times 20$ | 0.0955329 | 0.0635722 | ||

$40\times 40$ | 0.0480705 | 0.99085 | 0.0322156 | 0.98064 |

$80\times 80$ | 0.0245173 | 0.97135 | 0.0162343 | 0.98871 |

$160\times 160$ | 0.0123796 | 0.98584 | 0.0081475 | 0.99462 |

Grid | Grid Ratio | $\Vert {\mathit{u}}_{\mathit{n}}{\mathit{u}}_{\mathit{a}}\Vert {}_{\mathit{\infty}}$ | ${\left|\right(\Vert {\mathit{u}}_{\mathit{n}}-{\mathit{u}}_{\mathit{a}}\Vert {}_{\mathit{\infty}})}_{\mathit{n}\mathit{e}\mathit{w}}/{\left(\Vert {\mathit{u}}_{\mathit{n}}-{\mathit{u}}_{\mathit{a}}\Vert {}_{\mathit{\infty}}\right)}_{\mathit{o}\mathit{l}\mathit{d}}-1|$ |
---|---|---|---|

$10$ | 2 | 0.0651623 | |

$18$ | 1.11 | 0.0635722 | 0.02440 |

$20$ | 1 | 0.0642422 | 0.01053 |

$30$ | 0.67 | 0.0644133 | 0.00266 |

**Table 3.**Comparison of the drag coefficient, the lift coefficient, and the Strouhal number when Re = 20, 40, 100, and 200.

Reference | Tritton [42] | Calhoun [43] | Wu [19] | Qin [30] | Hu [23] | Present | |
---|---|---|---|---|---|---|---|

Re = 20 | ${\overline{C}}_{d}$ | 2.22 | 2.19 | 2.091 | 2.230 | 2.213 | 2.298 |

Re = 40 | ${\overline{C}}_{d}$ | 1.48 | 1.62 | 1.565 | 1.689 | 1.660 | 1.693 |

Re = 100 | ${\overline{C}}_{d}$ | 1.33 | 13.364 | 1.510 | 1.418 | 1.527 | |

${C}_{l}$ | 0.298 | 0.353 | 0.367 | 0.355 | |||

${S}_{t}$ | 0.175 | 0.163 | 0.169 | 0.166 | 0.176 | ||

Re = 200 | ${\overline{C}}_{d}$ | 1.77 | 1.349 | 1.493 | 1.394 | 1.495 | |

${C}_{l}$ | 0.67 | 0.718 | 0.712 | 0.723 | |||

${S}_{t}$ | 0.202 | 0.193 | 0.199 | 0.195 | 0.208 |

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Liu, B.; Shi, W.
A Non-Equilibrium Interpolation Scheme for IB-LBM Optimized by Approximate Force. *Axioms* **2023**, *12*, 298.
https://doi.org/10.3390/axioms12030298

**AMA Style**

Liu B, Shi W.
A Non-Equilibrium Interpolation Scheme for IB-LBM Optimized by Approximate Force. *Axioms*. 2023; 12(3):298.
https://doi.org/10.3390/axioms12030298

**Chicago/Turabian Style**

Liu, Bowen, and Weiping Shi.
2023. "A Non-Equilibrium Interpolation Scheme for IB-LBM Optimized by Approximate Force" *Axioms* 12, no. 3: 298.
https://doi.org/10.3390/axioms12030298