# Fluid–Structure Interaction Simulation of a Coriolis Mass Flowmeter Using a Lattice Boltzmann Method

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Fluid Domain

#### 2.1.1. Navier–Stokes Equations

#### 2.1.2. Lattice Boltzmann Method

#### 2.1.3. Moving Boundary Methods

#### 2.2. Structural Domain

#### 2.2.1. Navier–Cauchy Equation

#### 2.2.2. Direct Methods

#### 2.3. Fluid-Structure Interaction

#### 2.3.1. Coupling Conditions

#### Kinematic Condition

#### Dynamic Condition

#### Geometric Condition

#### 2.3.2. Segregated Approaches

#### 2.3.3. Implementation

- The OpenLB instance calculates the hydrodynamic forces acting on the boundary for each solid node according to Equation (17).
- The hydrodynamic forces are communicated and collected from each worker to the master process.
- The master process maps the collected boundary forces to the finite element grid by integrating the force on each finite element mesh point.
- The mapped boundary forces are written into an Elmer input deck file (.sif).
- Elmer is restarted by the master process using the input deck file (.sif) and a related restart file (.dat).
- The Elmer instance is closed after the displacement velocity and the deformed mesh is written to disk as an unstructured mesh file (.vtu) and a new Elmer restart file (.dat) is created.
- The master process reads the mesh file (.vtu) and uses the built-in OpenLB voxelizer, which decides whether a point is outside or inside the fluid domain and allows the later distance calculation.
- The master process maps the displacement velocity of the FEM grid to the LBM link intersection points ${\mathit{x}}_{w}$ by a linear interpolation procedure and distributes the information to each worker process.
- The OpenLB instance reconstructs the particle distribution functions for the fresh nodes by using the extrapolation refill algorithm (see Equation (16)).
- The collide and stream algorithm is executed (see Equation (5)).
- After the streaming step is executed, the unknown particle distribution function is calculated by the curved boundary approach using the mapped displacement velocity (see Equation (15)).

## 3. Setup of the Coriolis Mass Flowmeter Test Case

#### 3.1. Boundary Conditions and Initial Conditions

#### 3.1.1. Structural Domain

#### 3.1.2. Fluid Domain

#### 3.1.3. Coupling Conditions

#### 3.2. Mesh Generation

#### 3.2.1. Structural Domain

#### 3.2.2. Fluid Domain

## 4. Results of the Coriolis Mass Flowmeter Test Case

#### 4.1. Modal Analysis

#### 4.2. Phase Shift Calculation

## 5. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 7.**Connection of the critical mesh regions. (

**a**) Node plate—measuring pipes. (

**b**) Sensor—measuring pipes. (

**c**) Exciter—measuring pipes.

**Figure 9.**Geometry displacement due to excitation mode (mode 2) and Coriolis twist mode (mode 8). (

**a**) Mode 2, front view; (

**b**) Mode 2, top view; (

**c**) Mode 8, front view; (

**d**) Mode 8, top view.

**Figure 10.**Structural response at frequency ${f}_{exc}=50.00\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$.

**Figure 11.**Structural response at frequency ${f}_{exc}=104.28\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$.

Structural Properties | Fluid Properties | ||
---|---|---|---|

${\rho}^{s}$ | $7870\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$ | ${\rho}^{f}$ | $998\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$ |

E | $210\phantom{\rule{0.166667em}{0ex}}\mathrm{GPa}$ | ${\eta}^{f}$ | $0.207\phantom{\rule{0.166667em}{0ex}}\mathrm{Pas}$ |

${\nu}^{s}$ | $0.3$ |

Region | $\Delta {\mathit{x}}^{\mathit{s}}$ in m |
---|---|

Outer housing | 0.035 |

Body | 0.030 |

Sensors and exciter | 0.005 |

Measuring pipes | 0.010 |

Node plates | 0.005 |

**Table 3.**Lattice Boltzmann method (LBM) discretization parameters for the both investigated mass flows.

Mass Flow in $\frac{\mathbf{kg}}{\mathbf{h}}$ | $\Delta {\mathit{x}}^{\mathit{f}}$ in m | $\Delta {\mathit{t}}^{\mathit{f}}$ in s | ${\mathbf{Ma}}^{\mathbf{LB}}$ |
---|---|---|---|

20,000 | $4.056\xb7{10}^{-3}$ | $1.177\xb7{10}^{-4}$ | $8.660\xb7{10}^{-3}$ |

40,000 | $4.056\xb7{10}^{-3}$ | $5.885\xb7{10}^{-5}$ | $8.660\xb7{10}^{-3}$ |

Mode | ${\mathit{\omega}}^{2}$ in ${\mathbf{Hz}}^{2}$ | f in $\mathbf{Hz}$ | Physical Meaning |
---|---|---|---|

1 | $2.92\xb7{10}^{5}$ | 86.02 | |

2 | $\mathbf{4.29}\xb7{\mathbf{10}}^{\mathbf{5}}$ | 104.28 | excitation mode |

3 | $6.01\xb7{10}^{5}$ | 123.42 | |

4 | $9.50\xb7{10}^{5}$ | 155.14 | |

5 | $1.11\xb7{10}^{6}$ | 167.65 | |

6 | $1.48\xb7{10}^{6}$ | 193.85 | |

7 | $2.25\xb7{10}^{6}$ | 238.87 | |

8 | $\mathbf{3.03}\xb7{\mathbf{10}}^{\mathbf{6}}$ | 277.26 | Coriolis twist mode |

9 | $6.40\xb7{10}^{6}$ | 402.60 | |

10 | $8.05\xb7{10}^{6}$ | 451.44 |

**Table 5.**Excitation and Coriolis twist frequency for water- and air-filled tubes in comparison to measurement data.

Simulation | Measurement | Error in % | |
---|---|---|---|

${f}_{exc,air}$ | 104.28 | 101.00 | 3.24 |

${f}_{exc,water}$ | 83.94 | 81.41 | 3.11 |

${f}_{Coriolis,air}$ | 277.26 | 249.00 | 11.35 |

${f}_{Coriolis,water}$ | 222.92 | 205.00 | 8.74 |

Mass Flow in $\frac{\mathbf{kg}}{\mathbf{h}}$ | ${\mathit{\varphi}}_{\mathbf{sim}}$ in $\mathbf{Mrad}$ | ${\mathit{\varphi}}_{\mathbf{ref}}$ in $\mathbf{Mrad}$ | Error in % | Coupling Steps |
---|---|---|---|---|

20,000 | - | 0.62 | Instable | 51 |

20,000 | - | 0.62 | Instable | 101 |

20,000 | 0.59 | 0.62 | 4.7 | 202 |

40,000 | 1.18 | 1.23 | 4.1 | 202 |

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**MDPI and ACS Style**

Haussmann, M.; Reinshaus, P.; Simonis, S.; Nirschl, H.; Krause, M.J.
Fluid–Structure Interaction Simulation of a Coriolis Mass Flowmeter Using a Lattice Boltzmann Method. *Fluids* **2021**, *6*, 167.
https://doi.org/10.3390/fluids6040167

**AMA Style**

Haussmann M, Reinshaus P, Simonis S, Nirschl H, Krause MJ.
Fluid–Structure Interaction Simulation of a Coriolis Mass Flowmeter Using a Lattice Boltzmann Method. *Fluids*. 2021; 6(4):167.
https://doi.org/10.3390/fluids6040167

**Chicago/Turabian Style**

Haussmann, Marc, Peter Reinshaus, Stephan Simonis, Hermann Nirschl, and Mathias J. Krause.
2021. "Fluid–Structure Interaction Simulation of a Coriolis Mass Flowmeter Using a Lattice Boltzmann Method" *Fluids* 6, no. 4: 167.
https://doi.org/10.3390/fluids6040167