# Similarities of Flow and Heat Transfer around a Circular Cylinder

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

_{D}is [39]

_{m}may be acquired via solving the following simultaneous linear algebraic equations:

_{D}can be written as

## 3. Results and Discussion

#### 3.1. Drag Force-Velocity Diagram

^{−3}to 1.5 × l0

^{3}m/s, which roughly covers all the flow conditions that exist in nature and in engineering applications, and the corresponding Reynolds number range is from 1 to 10

^{6}.

_{D}with increasing Re number (Figure 2). Obviously, the drag and the customary drag coefficient have a roughly opposite changing trend, which may be undesirable and unreasonable. As can be seen from Figure 2, with increasing Re number, the drag coefficient first decreases, then it is nearly invariant in a wide range of Re number with a rise of Re up to 2.0 × 10

^{5}; however, as a matter of fact, the drag coefficient is closely related to the Reynolds number in this regime. The most remarkable variation in C

_{D}occurs in the critical Reynolds number range (3–4) × 10

^{5}, where C

_{D}decreases from its subcritical value of 1.2 to the supercritical value of 0.2. The sudden drop in the drag coefficient marks the end of the subcritical regime and the beginning of the critical regime. In detail, this decrease in C

_{D}is a result of the transition from laminar to turbulent flow in the boundary layer [54]. After that, C

_{D}rises again in the supercritical regime, and it gradually approaches a constant value in the transcritical regime [55]. This variation trend is strange and ruleless, which further demonstrates that the customary drag coefficient may be not a proper dimensionless parameter to describe and represent the drag.

#### 3.2. The Concept of Appropriate Drag Coefficient and Its Physical Meaning

^{5}. In the supercritical and transcritical flow regimes, generally the behavior of the flow around circular cylinders is abnormally sensitive to the Reynolds number or a very small perturbation, so there are some differences in the results of different researchers (see Figure 2).

_{m}. Using the definition of the modified drag coefficient for spheres and cylinders, the general drag expressions may be restated respectively as

_{C}is desirable and reasonable in representing the drag and also shows more physical meaning. As a consequence, it is more appropriate and convenient to work out drag problems using Equation (9).

#### 3.3. General Drag Model over the Entire Range of Reynolds Numbers

_{C}. This means that the definition of the appropriate drag coefficient is more scientific and reasonable. Figure 2 demonstrates that the customary drag coefficient is a fairly complicated function of the Re number, while the appropriate drag coefficient curve is relatively very smooth (Figure 4), so it is easier to obtain a relatively simple expression to accurately describe the flow characteristics over the entire range of Re numbers.

#### 3.4. Relationship between the Drag and Heat Transfer

^{4}.

^{5}, and it may approximately hold for flow with higher Reynolds number. The rough analogy may be expressed as follows:

^{5}.

^{0.45})Nu/Pr

^{0.4}, and it is compared with the drag experimental results in Figure 5. The flow and heat transfer data in Figure 5 come from experimental studies, in which the heat transfer data were collected by Whitaker [70], and other experimental data were from Sanitjai and Goldstein [71] and Perkins and Leppert [72]. The presented model does provide a means to approximately predict the Nusselt number for the whole range of Reynolds numbers, even if no experimental results exist! In particular, this may be quite important due to the lack of information on heat transfer for high-Reynolds-number flows in the literature. In addition, the obtained results also provide a theoretical basis for the design and optimization of the shape and thermal barrier coating system of high-speed aircrafts.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

A | cross-sectional area, m^{2} |

B_{m} | constants |

C_{D} | drag coefficient |

D | cylinder, sphere diameter, m |

D_{C} | appropriate drag coefficient |

F | drag force, N |

I_{m}, K_{m} | modified Bessel function |

Nu | surface-average Nusselt number |

Pr | Prandtl number |

Re | Reynolds number, $={U}_{\infty}D/\nu $ |

S | constant |

Sh | Sherwood number |

${U}_{\infty}$ | free stream velocity, m/s |

$\mathsf{\Gamma}$ | Euler’s constant |

${\lambda}_{m,n}$ | coefficient is a function of the Reynolds number |

μ | dynamic viscosity, N·s/m^{2} |

ν | kinematic viscosity, m^{2}/s |

ρ | density, kg/m^{3} |

D | drag |

∞ | for fluid at free stream conditions |

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**Figure 1.**Flow past an object. (

**a**) Flight of an eagle; (

**b**) flow past a circular cylinder; (

**c**) flow past an airfoil.

**Figure 5.**Comparison of drag experimental results and heat transfer experimental data with the developed model (Equation (19)).

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**MDPI and ACS Style**

Ma, H.; Duan, Z.
Similarities of Flow and Heat Transfer around a Circular Cylinder. *Symmetry* **2020**, *12*, 658.
https://doi.org/10.3390/sym12040658

**AMA Style**

Ma H, Duan Z.
Similarities of Flow and Heat Transfer around a Circular Cylinder. *Symmetry*. 2020; 12(4):658.
https://doi.org/10.3390/sym12040658

**Chicago/Turabian Style**

Ma, Hao, and Zhipeng Duan.
2020. "Similarities of Flow and Heat Transfer around a Circular Cylinder" *Symmetry* 12, no. 4: 658.
https://doi.org/10.3390/sym12040658