# Experiments on the Drag and Lift Coefficients of a Spinning Sphere

^{1}

^{2}

^{*}

## Abstract

**:**

_{c}) at each dimensionless angular speed. When 0 < Re < Re

_{c}, the lift coefficient decreases with increasing the Reynolds number, while it is constant when Re

_{c}< Re < 3500. The constant lift coefficient corresponding to different spin speeds was defined as the limit value of the lift coefficient. It is found that when 1 < Rr < 12, the limit value of the lift coefficient is 0.37, while the limit value of the lift coefficient increases with increasing dimensionless angular speed at 0 < Rr < 1. It is found that the spin increases the drag coefficient of a spinning sphere only when 0 < Rr < 10. Moreover, the performed analyses show that the drag coefficient of a spinning sphere is less than that of a non-spinning sphere when 10 < Rr < 25. Furthermore, the lift-to-drag ratio of a spinning sphere is discussed in this article.

## 1. Introduction

_{r}, where D is the sphere diameter, Ω is the angular speed, and u

_{r}is the relative velocity between the particle and fluid. Moreover, the Reynolds number is defined as Re = u

_{r}D/ν, where ν is the kinematic viscosity.

## 2. Materials and Methods

#### 2.1. Experimental Setup

^{®}3CXP, Mikrotron, Unterschleißheim, Germany), a memorizer (AQ8-CXP6D, Mikrotron, Unterschleißheim, Germany), and image processing software (Stream Pix, Norpix, Montréal, Canada). The image resolution was set to 1690 × 1710 pixels. The recording rate of the camera was set to 400 frames per second (FPS) to ensure clear images. The images were analyzed in the image processing software. Figure 1 shows the schematic view of the experimental setup.

#### 2.2. Experimental Methods

#### 2.2.1. Measuring Procedure

^{3}and 1000 kg/m

^{3}, respectively. The velocity of spinning spheres u

_{s}was in the range of 0.244 m/s to 0.434 m/s. The velocity of the fluid u = 0. The relative velocity between the particle and fluid was in the range of 0.244 m/s to 0.434 m/s. The range of angular speed Ω was from 13.956 rad/s to 223.29 rad/s. The range of dimensionless angular speed Rr was from 0.149 to 3.471. The range of the Reynolds number Re was from 610 to 3472. Table 1 presents the experimental conditions.

#### 2.2.2. Data Processing Procedure

_{s}is 0.417 m/s.

_{r}is equal to the velocity of the spinning sphere u

_{s}. The drag force is in the opposite direction of the sphere’s motion.

_{s}is the density of particles, ρ is the density of fluid. Moreover, the lift force of a spinning sphere can be obtained from the following expression:

_{LΩ}is the lift force of a spinning sphere, and C

_{LΩ}is the lift coefficient of a spinning sphere. The drag force of a spinning sphere can be mathematically expressed in the form below:

_{d}

_{Ω}is the drag force of a spinning sphere, and C

_{d}

_{Ω}is the drag coefficient of a spinning sphere.

_{s}, and β into Equations (6) and (7), the lift and drag coefficients of a spinning sphere can be calculated. It is note that u = 0.

## 3. Results and Discussion

#### 3.1. Lift Coefficient

_{c}) at each dimensionless angular speed. Until the Reynolds number is less than the critical value (i.e., 0 < Re < Re

_{c}), the lift coefficient decreases with the increase in the Reynolds number, while it is approximately constant when Re

_{c}< Re < 3500. The constant lift coefficient corresponding to different spin speeds is defined as the limit value of the lift coefficient, which is 0.37 when 1 < Rr < 12. When 0 < Rr < 1, the limit value increases with the increase in the dimensionless angular speed. Figure 7 shows the correlation among the calculated values using Equation (8), the previously measured data, and previous simulation results. It is observed that the calculated values are consistent with the previous measured data and simulation results.

_{L}

_{Ω}

_{c}and C

_{L}

_{Ω}

_{m}are the calculated and measured lift coefficients, respectively.

#### 3.2. Drag Coefficient

#### 3.2.1. Drag Coefficient of a Non-Spinning Sphere

_{d}is the total drag force of a non-spinning sphere, F

_{d}

_{ν}is the viscosity-related drag force of a non-spinning sphere, F

_{dp}is the pressure-related drag force of a non-spinning sphere, C

_{d}is the total drag coefficient of a non-spinning sphere.

_{dp}is the pressure-related drag coefficient of a non-spinning sphere. Similar to the pressure-related drag force, the viscosity-related drag force can be expressed in the form below:

_{dv}is the viscosity-related drag coefficient of a non-spinning sphere,

_{dv}and measured C

_{d}into Equation (14), the coefficient C

_{dp}can be obtained. The correlation between the pressure-related drag coefficient and the Reynolds number can be expressed as follows:

#### 3.2.2. Drag Coefficient of A Spinning Sphere

_{dΩ}/C

_{d}is equal to the ratio of the measured or simulated drag coefficient of a spinning sphere to the calculated total drag coefficient of a non-spinning sphere using Equation (16). The previous results are shown in Figure 9.

_{dΩ}/C

_{d}varies from 1 to 1.6. However, the drag coefficient of a spinning sphere is less than the total drag coefficient of a non-spinning sphere for 10 < Rr < 25. When 0 < Rr < 2.5, the ratio C

_{dΩ}/C

_{d}increases as Rr increases. Beyond this range, the ratio C

_{dΩ}/C

_{d}decreases as Rr increases. To interpret this phenomenon, two empirical correlations were developed to simulate the effect of particle’s spin on the drag coefficient based on the present experimental data and the results of Barkla and Auchterlonie [4]. When 0 < Rr < 2.5 and 0 < Re < 3500, the correlation can be expressed as follows:

#### 3.3. Lift-to-Drag Ratio

_{ld}) can be expressed in the form below:

_{ld}increases with increasing dimensionless angular speed at a certain Reynolds number. At 100 < Re < 3500 and 0 < Rr < 0.85, k

_{Ld}increases with increasing dimensionless angular speed at a certain Reynolds number and increases with increasing Reynolds number at a certain angular speed. At 100 < Re < 3500 and 0.85 < Rr < 6, k

_{ld}increases with increasing Reynolds number at a certain angular speed and can be approximated as a constant value with increasing dimensionless angular speed at a certain Reynolds number. The present equations were preliminarily verified by the previous results.

## 4. Conclusions

- (1)
- The obtained experimental data reveal that the lift coefficient is related to the Reynolds number and dimensional angular speed. There is a critical Reynolds number (Re
_{c}) at each dimensionless angular speed. When 0 < Re < Re_{c}, the lift coefficient decreases as the Reynolds number increases, while it is constant when Re_{c}< Re < 3500. The constant lift coefficient corresponding to different spin speeds was defined as the limit value of the lift coefficient. This coefficient is 0.37 when 1 < Rr < 12, while the limit value of the lift coefficient increases with the increase in dimensionless angular speed for 0 < Rr < 1. - (2)
- Compared with the total drag coefficient of a non-spinning sphere at a certain dimensionless angular speed, the drag coefficient of a spinning sphere is higher when 1 < Rr < 10. When 1 < Rr < 10, the ratio of the drag coefficient of a spinning sphere to the total drag coefficient of a non-spinning sphere C
_{dΩ}/C_{d}is between 1 and 1.6. However, the drag coefficient of a spinning sphere is less than the total drag coefficient of a non-spinning sphere when 10 < Rr < 25. The ratio C_{dΩ}/C_{d}increases with increasing dimensionless angular speed in the range of 0 < Rr < 2.5, while the ratio C_{dΩ}/C_{d}decreases with an increase in Rr beyond this range. To interpret this phenomenon, two empirical correlations were developed to describe the effect of particle spin on the drag coefficient based on the experimental data and the results from the literature. - (3)
- When 0 < Re < 100, the lift-to-drag ratio of a spinning sphere k
_{ld}increases with increasing dimensionless angular speed at a certain Reynolds number. At 100 < Re < 3500 and 0 < Rr < 0.85, k_{Ld}increases with increasing dimensionless angular speed at a certain Reynolds number and increases with increasing Reynolds number at a certain angular speed. At 100 < Re < 3500 and 0.85 < Rr < 6, k_{ld}increases with increasing Reynolds number at a certain angular speed and can be approximated as a constant value with increasing dimensionless angular speed at a certain Reynolds number.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 8.**Variation in the drag coefficient against the Reynolds number [30].

No. | D (mm) | u_{s} (m/s) | Ω (rad/s) | Rr | Re |
---|---|---|---|---|---|

1 | 2.5 | 0.262 | 38.380 | 0.366 | 655.000 |

2 | 2.5 | 0.244 | 59.890 | 0.614 | 610.000 |

3 | 4 | 0.374 | 13.956 | 0.149 | 1496.000 |

4 | 4 | 0.361 | 20.350 | 0.225 | 1444.000 |

5 | 4 | 0.311 | 65.130 | 0.838 | 1244.000 |

6 | 4 | 0.315 | 77.920 | 0.989 | 1260.000 |

7 | 6 | 0.417 | 34.889 | 0.502 | 2502.000 |

8 | 6 | 0.386 | 223.290 | 3.471 | 2316.000 |

9 | 6 | 0.385 | 90.250 | 1.406 | 2310.000 |

10 | 6 | 0.379 | 90.250 | 1.429 | 2274.000 |

11 | 6 | 0.376 | 185.140 | 2.954 | 2256.000 |

12 | 6 | 0.376 | 90.250 | 1.440 | 2256.000 |

13 | 6 | 0.374 | 87.220 | 1.399 | 2244.000 |

14 | 6 | 0.366 | 195.380 | 3.203 | 2196.000 |

15 | 6 | 0.366 | 163.280 | 2.677 | 2196.000 |

16 | 6 | 0.359 | 177.930 | 2.974 | 2154.000 |

17 | 6 | 0.358 | 167.470 | 2.807 | 2148.000 |

18 | 6 | 0.353 | 186.660 | 3.173 | 2118.000 |

19 | 6 | 0.345 | 115.336 | 2.006 | 2070.000 |

20 | 8 | 0.434 | 103.980 | 1.917 | 3472.000 |

21 | 8 | 0.428 | 104.670 | 1.956 | 3424.000 |

22 | 8 | 0.419 | 98.640 | 1.883 | 3352.000 |

_{s}is the velocity of spinning spheres, Ω is the angular speed, Rr is the dimensionless angular speed, Re is the Reynolds number.

No. | D (mm) | u_{s} (m/s) | Ω (rad/s) | Rr | Re | C_{lΩ} | C_{dΩ} |
---|---|---|---|---|---|---|---|

1 | 2.5 | 0.262 | 38.380 | 0.366 | 655.000 | 0.212 | 0.627 |

2 | 2.5 | 0.244 | 59.890 | 0.614 | 610.000 | 0.349 | 0.677 |

3 | 4 | 0.374 | 13.956 | 0.149 | 1496.000 | 0.132 | 0.502 |

4 | 4 | 0.361 | 20.350 | 0.225 | 1444.000 | 0.098 | 0.550 |

5 | 4 | 0.311 | 65.130 | 0.838 | 1244.000 | 0.364 | 0.657 |

6 | 4 | 0.315 | 77.920 | 0.989 | 1260.000 | 0.339 | 0.648 |

7 | 6 | 0.417 | 34.889 | 0.502 | 2502.000 | 0.261 | 0.570 |

8 | 6 | 0.386 | 223.290 | 3.471 | 2316.000 | 0.318 | 0.657 |

9 | 6 | 0.385 | 90.250 | 1.406 | 2310.000 | 0.379 | 0.629 |

10 | 6 | 0.379 | 90.250 | 1.429 | 2274.000 | 0.378 | 0.657 |

11 | 6 | 0.376 | 185.140 | 2.954 | 2256.000 | 0.331 | 0.697 |

12 | 6 | 0.376 | 90.250 | 1.440 | 2256.000 | 0.376 | 0.674 |

13 | 6 | 0.374 | 87.220 | 1.399 | 2244.000 | 0.393 | 0.672 |

14 | 6 | 0.366 | 195.380 | 3.203 | 2196.000 | 0.402 | 0.708 |

15 | 6 | 0.366 | 163.280 | 2.677 | 2196.000 | 0.325 | 0.745 |

16 | 6 | 0.359 | 177.930 | 2.974 | 2154.000 | 0.356 | 0.768 |

17 | 6 | 0.358 | 167.470 | 2.807 | 2148.000 | 0.329 | 0.783 |

18 | 6 | 0.353 | 186.660 | 3.173 | 2118.000 | 0.371 | 0.792 |

19 | 6 | 0.345 | 115.336 | 2.006 | 2070.000 | 0.449 | 0.800 |

20 | 8 | 0.434 | 103.980 | 1.917 | 3472.000 | 0.358 | 0.682 |

21 | 8 | 0.428 | 104.670 | 1.956 | 3424.000 | 0.353 | 0.709 |

22 | 8 | 0.419 | 98.640 | 1.883 | 3352.000 | 0.353 | 0.750 |

_{s}is the velocity of spinning spheres, Ω is the angular speed, Rr is the dimensionless angular speed, Re is the Reynolds number,

**C**is the lift coefficient of a spinning sphere,

_{lΩ}**C**is the drag coefficient of a spinning sphere.

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**MDPI and ACS Style**

Zhou, S.; Zhang, G.; Xu, X. Experiments on the Drag and Lift Coefficients of a Spinning Sphere. *Water* **2022**, *14*, 2593.
https://doi.org/10.3390/w14172593

**AMA Style**

Zhou S, Zhang G, Xu X. Experiments on the Drag and Lift Coefficients of a Spinning Sphere. *Water*. 2022; 14(17):2593.
https://doi.org/10.3390/w14172593

**Chicago/Turabian Style**

Zhou, Shuang, Genguang Zhang, and Xiaoyang Xu. 2022. "Experiments on the Drag and Lift Coefficients of a Spinning Sphere" *Water* 14, no. 17: 2593.
https://doi.org/10.3390/w14172593