# Numerical Study of Baroclinic Acoustic Streaming Phenomenon for Various Flow Parameters

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Geometries

#### 2.1. Rectangular Channel Geometry

#### 2.2. Pt-100 Thermometer Geometry

## 3. Mathematical Model and Numerical Implementation

#### 3.1. Numerical Calculations

#### 3.2. Implementation of the Wall Osculations

#### 3.3. Definition of Metrics of Total, Fluctuating and Streaming Velocities

**u**over 10 cycles of the acoustic wave:

#### 3.4. Nusselt Number Calculation

## 4. Results and Discussion

#### 4.1. Impact of the Third Dimension on the Streaming

#### 4.2. Impact of Streaming Flow on Heat Transport

#### 4.3. Application of Streaming Phenomenon in Pt-100 Resistance Thermometer

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Investigated geometries. (

**a**) Rectangular channel geometry. 1—moving left wall, 2—bottom wall, 3—top wall, 4—right wall, L—channel length, H—channel height. (

**b**) Geometry of the Pt-100 thermometer filled with air, the dimensions are in mm. The 0.9 mm end walls are modelled as oscillatory boundary conditions with the frequency $f=20$ kHz.

**Figure 2.**Comparison of the x and y components of streaming velocity ${u}_{mx}$ and ${u}_{my}$ for various mesh resolution with the results from [4]; (

**Left**): Plots of ${u}_{mx}$ along H of the resonator for $x=0.75$ L; (

**Right**): Plots of ${u}_{my}$ along H of the resonator for $x=0.5$ L.

**Figure 3.**Comparison of the current results with [4] for various temperature difference between the top and bottom wall; (

**Left**): plots of ${u}_{mx}$ along height of the domain for $x=0.75$ L; (

**Right**): plots of ${u}_{my}$ for $x=0.5$ L.

**Figure 4.**Comparison of the x-component of instantaneous total velocity

**u**, along horizontal mid-plane of the resonator at four different instants: $\omega t=(0,\pi /2,\pi ,3/2\pi $), obtained in the current study with the results from [4], case 1A*.

**Figure 5.**Streamlines, magnitude, and vectors of the streaming velocity ${\mathrm{U}}_{\mathrm{m}}$ for f = 20 kHz, cases 1A*, 1B* and 1C*. (

**a**) Case 1A* (isothermal flow). (

**b**) Case 1B*, $\Delta T=20$ K. (

**c**) Case 1C*, $\Delta T=60$ K.

**Figure 6.**Streamlines depicting streaming velocity ${\mathrm{U}}_{\mathrm{m}}$ for cases 2A–2C (f = 40 kHz). (

**a**) Case 2A (isothermal flow). (

**b**) Case 2B, $\Delta T=20$ K. (

**c**) Case 2C, $\Delta T=60$ K.

**Figure 7.**Streamlines depicting streaming velocity ${\mathrm{U}}_{\mathrm{m}}$ for Cases 3A–3C (f = 80 kHz). (

**a**) Case 3A (isothermal flow). (

**b**) Case 3B, $\Delta T=20$ K. (

**c**) Case 3C, $\Delta T=60$ K.

**Figure 8.**Changes of scalar metrics of: (

**a**) streaming velocity ${\overline{u}}_{m}$; (

**b**) total velocity $\overline{u}$; (

**c**) fluctuation velocity ${\overline{u}}^{\prime}$; (

**d**) ratio of streaming velocity to fluctuation velocity ${\overline{u}}_{m}/{\overline{u}}^{\prime}$, in the function of $\Delta T$ and frequency f.

**Figure 9.**(

**Left**): comparison of x component of streaming velocity ${u}_{mx}$ for cases 1C*, 2C, and 3C along vertical line for $x=0.75$ L, $0<y<H$; (

**Right**): comparison of y component of streaming velocity ${u}_{my}$ for cases 1C*, 2C, and 3C along vertical line for $x=0.5$ L, $0<y<H$.

**Figure 10.**Velocity magnitude and vectors of streaming on the 2D cross-sections ($xy$ planes) of the 3D resonator for $z=(0.5,0.375,0.125)$ W. (

**a**) $xy$ plane at $z=0.5$ W. (

**b**) $xy$ plane at $z=0.375$ W. (

**c**) $xy$ plane at $z=0.125$ W.

**Figure 11.**Velocity magnitude and vectors of streaming on the 2D cross-sections ($xz$ planes) of the 3D resonator for $y=(0.5,0.375,0.125)$ H. (

**a**) $xz$ plane at $y=0.5$ H. (

**b**) $xz$ plane at $y=0.375$ H. (

**c**) $xz$ plane at $y=0.125$ H.

**Figure 12.**Change of the scalar metric of the streaming flow along the width (

**left**) and height (

**right**) of the resonator.

**Figure 13.**Mean temperature and isotherms for the non-isothermal cases. (

**a**) Case 1B*, $\Delta T=$ 20 K, $f=$ 20 kHz. (

**b**) Case 1C*, $\Delta T=$ 60 K, $f=$ 20 kHz. (

**c**) Case 2B $\Delta T=$ 20 K, $f=$ 40 kHz. (

**d**) Case 2C $\Delta T=$ 60 K, $f=$ 40 kHz. (

**e**) Case 3B, $\Delta T=$ 20 K, $f=$ 80 kHz. (

**f**) Case 3C, $\Delta T=$ 60 K, $f=$ 80 kHz.

**Figure 15.**From the top to bottom: changes of the streaming vertical and horizontal components of velocity, streaming velocity magnitude, and mean temperature along the thermometer channel taken at the height $y=0.825$ H.

**Figure 16.**Comparison of the vertical temperature profiles of the flow with and without streaming for 3 locations $L=(7.5,88.5,165.5)$ mm. (

**Left**): temperature profiles; (

**Right**): temperature difference between flow with and without streaming.

Case | f, kHz | $\mathbf{\Delta}\mathit{T}$, K | L, mm | H, mm | Number of Cells |
---|---|---|---|---|---|

1A | 4500 | ||||

1B | 20 | 60 | 8.825 | 0.64 | 18,000 |

1C | 72,000 | ||||

1A* | 0 | ||||

1B* | 20 | 20 | 8.825 | 0.64 | 72,000 |

1C* | 60 | ||||

2A | 0 | ||||

2B | 40 | 20 | 4.413 | 0.48 | 27,000 |

2C | 60 | ||||

3A | 0 | ||||

3B | 80 | 20 | 2.206 | 0.32 | 27,000 |

3C | 60 |

Case | $\Delta \mathit{T}$, K | f, kHz | ${\mathit{Nu}}_{\mathit{top}}$ | ${\mathit{Nu}}_{\mathit{bot}}$ |
---|---|---|---|---|

1B* | 20 | 20 | 1.321 | 1.258 |

1C* | 60 | 20 | 1.239 | 1.221 |

2B | 20 | 40 | 0.856 | 1.305 |

2C | 60 | 40 | 1.015 | 1.241 |

3B | 20 | 80 | 0.640 | 0.941 |

3C | 60 | 80 | 0.752 | 0.887 |

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**MDPI and ACS Style**

Baran, B.; Machaj, K.; Malecha, Z.; Tomczuk, K.
Numerical Study of Baroclinic Acoustic Streaming Phenomenon for Various Flow Parameters. *Energies* **2022**, *15*, 854.
https://doi.org/10.3390/en15030854

**AMA Style**

Baran B, Machaj K, Malecha Z, Tomczuk K.
Numerical Study of Baroclinic Acoustic Streaming Phenomenon for Various Flow Parameters. *Energies*. 2022; 15(3):854.
https://doi.org/10.3390/en15030854

**Chicago/Turabian Style**

Baran, Błażej, Krystian Machaj, Ziemowit Malecha, and Krzysztof Tomczuk.
2022. "Numerical Study of Baroclinic Acoustic Streaming Phenomenon for Various Flow Parameters" *Energies* 15, no. 3: 854.
https://doi.org/10.3390/en15030854