# Experimental Study on Flow Structure Characteristics of Gap Flow Boundary Layer Based on PIV

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Program

#### 2.1. Experimental System

^{2}and a length of 8.2 m. The receiving device was a stainless steel tank, which was also the water-supplying device for the whole system.

#### 2.2. Measuring Device

^{3}and a particle size of 3–6 μm.

#### 2.3. Experimental Program Design

#### 2.3.1. Test Conditions

#### 2.3.2. Arrangement of Test Sections and Test Lines

## 3. Results

#### 3.1. Calculation of Wall Friction Velocity

#### 3.2. Boundary Layer Thickness

#### 3.3. Wall Function

#### 3.3.1. Viscous Sub-Layer

#### 3.3.2. Logarithmic Layer

#### 3.3.3. Transition Layer

## 4. Discussion

## 5. Conclusions

- (1)
- In this study, two methods were used to calculate the wall friction velocity, the results of the two methods were very similar, and the maximum relative error did not exceed 5%, which indicates that using PIV to measure the velocity profile in the viscous sub-layer to solve the wall friction velocity had good precision.
- (2)
- The boundary layer thickness of the gap flow was inversely proportional to both the mean velocity of the gap flow and the gap ratio, and the ratio of the boundary layer thickness to the gap half-height was constant for all the conditions investigated in this study.
- (3)
- The boundary layer data were nondimensionalized by using the wall friction velocity and the wall position to obtain the wall function of the boundary layer of the gap flow; it was expressed as$$\left\{\right)separators="|">\begin{array}{c}{y}^{+}5.5,|{u}^{+}={y}^{+}\\ 5.5{y}^{+}26,|{u}^{+}=\frac{1}{0.071}tanh\left(0.071{y}^{+}\right)\\ {y}^{+}26,|{u}^{+}=2.78ln\left({y}^{+}\right)+3.8\end{array}$$
- (4)
- In the viscous sub-layer, the velocity gradient was proportional to both the mean velocity of the gap flow and the gap ratio, and thus, the wall friction velocity was also proportional to the mean velocity of the gap flow and the gap ratio; the experimental data in the transition region satisfied the hyperbolic tangent function well; and the logarithmic region satisfied both the logarithmic law and the law of defects, and was therefore also known as the overlap region.
- (5)
- The thickness of the logarithmic region increased with the increase in the mean velocity of the gap flow and decreased with the increase in the gap ratio. The range of the inner region of the gap flow boundary layer was $y<$0.18$\delta $ or $y<$0.13($h$/2).

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Layout of test sections and test lines: (

**a**) arrangement of test sections, ①–⑤ indicate the locations of the test sections; (

**b**) arrangement of test lines, (1)–(5) indicate the locations of the test lines in the test sections, and the blue background color indicate the flat plate.

**Figure 9.**Velocity distributions in the viscous sub-layer: (

**a**) velocity distributions in the viscous sub-layer with different gap flow mean velocities; (

**b**) velocity distributions in the viscous sub-layer with different gap ratios; (

**c**) dimensionless velocity distributions in the viscous sub-layer with different gap flow mean velocities and different gap ratios.

**Figure 10.**Velocity distributions in the logarithmic layer: (

**a**) velocity profile normalized using the inner velocity scale; (

**b**) velocity profile normalized using the outer velocity scale.

Gap Ratio $\mathit{\beta}$ | Plate Height $\mathit{a}/\mathrm{mm}$ | Square Pipe Height $\mathit{b}/\mathrm{mm}$ | Square Pipe Width (Gap Width) $\mathit{c}/\mathrm{mm}$ | Gap Height $\mathit{h}/\mathrm{mm}$ |
---|---|---|---|---|

0.6 | 30 | 50 | 50 | 20 |

0.7 | 35 | 50 | 50 | 15 |

0.8 | 40 | 50 | 50 | 10 |

Gap Ratio $\mathit{\beta}$ | Mean Velocity of Gap Flow ${\mathit{u}}_{\mathit{m}/m/s}$ | Reynolds Number of Gap Flow $\mathit{R}\mathit{e}$ |
---|---|---|

0.6 | 1 | 28,435 |

1.5 | 42,652 | |

2 | 56,870 | |

0.7 | 1 | 22,967 |

1.5 | 34,450 | |

2 | 45,933 | |

0.8 | 1 | 16,587 |

1.5 | 24,880 | |

2 | 33,174 |

$\mathit{\beta}$ = 0.6 | |||

${u}_{m}$ =1 m/s | ${u}_{m}$ = 1.5 m/s | ${u}_{m}$ = 2 m/s | |

Method 1 | 0.0568 | 0.0820 | 0.1056 |

Method 2 | 0.0549 | 0.0781 | 0.1004 |

Relative error | 3.35% | 4.76% | 4.92% |

$\mathit{\beta}$ = 0.7 | |||

${u}_{m}$ = 1 m/s | ${u}_{m}$ = 1.5 m/s | ${u}_{m}$ = 2 m/s | |

Method 1 | 0.0577 | 0.0836 | 0.1084 |

Method 2 | 0.0567 | 0.0806 | 0.1035 |

Relative error | 1.73% | 3.59% | 4.52% |

$\mathit{\beta}$ = 0.8 | |||

${u}_{m}$ = 1 m/s | ${u}_{m}$ = 1.5 m/s | ${u}_{m}$ = 2 m/s | |

Method 1 | 0.0580 | 0.0838 | 0.1088 |

Method 2 | 0.0577 | 0.0817 | 0.1049 |

Relative error | 0.52% | 2.51% | 3.58% |

${\mathit{u}}_{\mathit{m}}$ | 1 m/s | 1.5 m/s | 2 m/s | ||
---|---|---|---|---|---|

${\u2206\mathit{y}}^{+}\u2215\frac{{\u2206\mathit{y}}^{+}}{{\mathit{\delta}}^{+}}$ | |||||

$\mathit{\beta}$ | |||||

0.6 | 59/0.126 | 94/0.143 | 129/0.154 | ||

0.7 | 41/0.114 | 66/0.131 | 95/0.148 | ||

0.8 | 18/0.074 | 40/0.117 | 55/0.127 |

${\mathit{u}}_{\mathit{m}}$ | 1 m/s | 1.5 m/s | 2 m/s | ||
---|---|---|---|---|---|

K | |||||

$\mathit{\beta}$ | |||||

0.6 | 0.0708 | 0.0695 | 0.0716 | ||

0.7 | 0.0717 | 0.0704 | 0.0711 | ||

0.8 | 0.0715 | 0.0713 | 0.0712 |

Types of Boundary Layers | Ranges of Viscous Influence |
---|---|

Gap flow boundary layer | $0.18\mathsf{\delta}$$\mathrm{or}0.13(\mathrm{h}$/2) |

Round pipe flow [32,33] | $0.1\mathrm{R}$$,0.12\mathrm{R}$ |

Flat plate boundary layer [34,35] | $0.2\mathsf{\delta}$$,0.3\mathsf{\delta}$ |

Open flow boundary layer [36,37,38,39,40] | $\left(0.1\text{\u2013}0.25\right)\mathsf{\delta}$$,\left(0.15\text{\u2013}0.5\right)\mathrm{h}$ |

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**MDPI and ACS Style**

Sun, L.; Sun, X.; Li, Y.; Wang, C.
Experimental Study on Flow Structure Characteristics of Gap Flow Boundary Layer Based on PIV. *Water* **2023**, *15*, 3989.
https://doi.org/10.3390/w15223989

**AMA Style**

Sun L, Sun X, Li Y, Wang C.
Experimental Study on Flow Structure Characteristics of Gap Flow Boundary Layer Based on PIV. *Water*. 2023; 15(22):3989.
https://doi.org/10.3390/w15223989

**Chicago/Turabian Style**

Sun, Lei, Xihuan Sun, Yongye Li, and Cheng Wang.
2023. "Experimental Study on Flow Structure Characteristics of Gap Flow Boundary Layer Based on PIV" *Water* 15, no. 22: 3989.
https://doi.org/10.3390/w15223989