Similarities of Flow and Heat Transfer around a Circular Cylinder

Abstract

Modeling fluid flows is a general procedure to handle engineering problems. Here we present a systematic study of the flow and heat transfer around a circular cylinder by introducing a new representative appropriate drag coefficient concept. We demonstrate that the new modified drag coefficient may be a preferable dimensionless parameter to describe more appropriately the fluid flow physical behavior. A break in symmetry in the global structure of the entire flow field increases the difficulty of predicting heat and mass transfer behavior. A general simple drag model with high accuracy is further developed over the entire range of Reynolds numbers met in practice. In addition, we observe that there may exist an inherent relation between the drag and heat and mass transfer. A simple analogy model is established to predict heat transfer behavior from the cylinder drag data. This finding provides great insight into the underlying physical mechanism.

1. Introduction

Flow over an object is omnipresent both in nature [1,2,3] and in many engineering applications [4,5,6]. For instance, the motion of swimming and flying animals [1], growth of stalagmites [2], fall motion of hailstones [3], motion of pollutants in the atmosphere [7], complex motion of the drill string in the field of petroleum engineering [8], and flow over bridge piers, chimney stacks, offshore structures, and tower structures in civil engineering [9], aircrafts in the field of aerospace [10], nuclear fuel rods in the atomic field [5], power battery cooling structures in the field of new energy vehicles [11], heat exchanger tubes in thermal engineering [12], etc. The fluid dynamic drag [13,14,15], active and passive methods for drag reduction [16,17,18], boundary layer flow [19], flow-induced vibration [5], behavior of turbulent fluid motion [20], and instability in the wake shear layer [21,22,23] are of interest in numerous fields. Owing to its practical importance in engineering applications and theoretical significance in understanding fundamental fluid mechanics, the flow over a circular cylinder has attracted extensive study interest from both scientists and engineers. Therefore, the flow and heat transfer characteristics of circular cylinders in cross flow have been the subject of many theoretical, experimental, and numerical studies; thus, a large number of results exist and are available in the literature.

2. Literature Review

The flow over bluff bodies like spheres [24,25,26] and circular cylinders [27,28,29,30] is a classical problem in fluid mechanics. Yao et al. [9] focused on the influence of turbulence on the wind pressure and aerodynamic behavior of smooth circular cylinders. Liang and Duan [12] studied numerically the flow past a yawed circular cylinder using large eddy simulation. They pointed out that the boundary condition of the two end-plates has a significant effect on the flow behaviors in the wake. Sarıoğlu et al. [18] investigated experimentally the effect of a rod on the flow around a square cylinder at incidence. Lin et al. [21] carried out an experimental study on nonstationary three-dimensional aspects of flow past a circular cylinder and found that the flow behaviors are critically affected by the presence of a separation bubble. Ahmed and Wagner [28] focused on the vortex shedding and transition frequencies associated with flow around a circular cylinder. During an oil drilling process, complex motion of the drill string [31,32] and the settling velocity of drill cuttings in drilling fluids [33,34] have been the focus of investigations in the field of petroleum engineering. Leth-Espensen et al. [35] presented a biomass devolatilization model describing both spherical and cylindrical particles for suspension firing. Duan et al. [7] investigated the flow and heat transfer past a sphere, and they first proposed the appropriate drag coefficient to replace the inertia type definition proposed by Sir Isaac Newton.

It seems to be a common perception that the drag and customary drag coefficient would preferably have a uniform variation trend. In fact, this is a wrong perception caused by the inertia type definition. An extension research of the previous work (Reference [7]) was conducted in the present paper. The flow past a circular cylinder was investigated by introducing a new representative appropriate drag coefficient concept originally presented in Reference [7] for spheres. The Kutta-Joukowski theorem states that the force experienced by a body in a uniform stream is equal to the product of the fluid density, stream velocity, and circulation and has a direction perpendicular to the stream velocity [36]. Appling the Kutta-Joukowski theorem, the motion of flying animals, high-speed aircrafts, and cylinders of various shapes may be easily converted into a specific type of problem (Figure 1).

The flow over a circular cylinder is a classic example of flow over a bluff body and frequently serves as a benchmark to help understand flow separation and vortex shedding [37]. Extensive studies have been carried out on this classic problem in the past century. A literature survey indicates that resistance formulas for a wide range of Reynolds (Re) numbers are rarely reported, especially concerning the absence of a classical drag relationship in the entire range of Re numbers. Therefore, based on a systematic summary of previous studies, we have developed such a general empirical model for accurate prediction of the appropriate drag coefficient.

Historical experimental data for drag and heat transfer of the flow around a circular cylinder were critically examined. Analogous investigations have been reported by many researchers working in the flow and heat and mass transfer area, including internal flows [38] and external flows [7]. For external flows, Duan et al. [7] found that there exists a relation between the drag and heat transfer in spheres. For internal flows, Duan and He [38] presented an extended Reynolds analogy for slip flow heat transfer in microchannels. To the authors’ best knowledge, the internal relationship between the drag and heat transfer for cylinders has not been revealed in the existing literature. Here we present drag and heat transfer results which demonstrate that there may exist an inherent relation between the drag and heat transfer in cylinders. The appropriate drag coefficient may be associated with the Nusselt number. Thus, a simple model is proposed to predict heat transfer behavior from drag data.

Drag estimation of bodies moving through fluids is a crucial concern in engineering practice [1]. For a uniform stream past a circular cylinder, the usual definition of the drag coefficient C_D is [39]

$$C_D=\frac{F}{\frac{1}{2}\rho U_{\infty }^{2}A}=\frac{F}{\frac{1}{2}\rho U_{\infty }^2(D\cdot 1)}$$

(1)

where F represents the drag force on the circular cylinder per unit length, ρ is the fluid density, $U_{\infty }$ is the relative velocity of the fluid and the object, A is the projected area of the body in the direction of the flow, and D is the cylinder diameter.

It is well known that in a viscous cross flow, the drag acting on a circular cylinder is due to friction and inertia. Earlier efforts and significant contributions were made by Stokes [40], Oseen [41], and Lamb [42] to obtain theoretical solutions for creeping flows. Van Dyke [43] introduced the development of theoretical solutions for spheres and circular cylinders. A first approximation for the drag coefficient on a cylinder is known as the Oseen solution, and it is expressed as follows [41]

$$C_D=\frac{8\pi }{Re\left(\frac{1}{2}-\mathsf{\Gamma }-ln\frac{Re}{8}\right)}$$

(2)

where $\mathsf{\Gamma }$ = 0.577216… is Euler’s constant and Re is the Reynolds number defined by $Re=U_{\infty }D/\nu$. Tomotika and Aoi [44] derived and presented expansion formulas of the drag force at small Reynolds numbers. As a result of their efforts, a second approximation of the drag coefficient was presented [44]:

$$C_D=\frac{8\pi }{ReS}\left[1-\frac{1}{S}(S^2-\frac{1}{2}S+\frac{5}{16})\frac{R{e}^2}{32}\right]$$

(3)

where $S$ is a constant defined by $S=\frac{1}{2}-\mathsf{\Gamma }-ln\frac{Re}{8}$. Further, a third approximation of the drag coefficient is expressed as [44]

$$C_D=\frac{8\pi }{ReS}[1-\frac{1}{S}(S^2-\frac{1}{2}S+\frac{5}{16})\frac{R{e}^2}{32}-\frac{1}{S^2}(S^4-\frac{1}{3}{S}^3+\frac{7}{72}S-\frac{25}{256})\frac{R{e}^4}{{32}^2}].$$

(4)

Subsequently, further research was conducted by Proudman and Pearson [45], Kaplun [46], and Tamada et al. [47] in order to determine expansion formulas of drag force. Due to the complexity and difficulty of mathematically solving Navier-Stokes equations, it is quite difficult to extend the availability of the analytical solutions to higher Reynolds numbers. The correlation between the drag coefficient and Reynolds number can only be determined by means of experiment.

For historical reasons, it is a common practice to use dynamic pressure to nondimensionalize the drag experienced by bodies. The drag coefficient can also be expressed by the following general expression [44]:

$$F=4\pi \mu U_{\infty }{\displaystyle \sum _{m=0}^{\infty }}{B}_m$$

(5)

where μ represents the viscosity. The constants B_m may be acquired via solving the following simultaneous linear algebraic equations:

$${\displaystyle \sum _{m=0}^{\infty }{B}_m}{\lambda }_{m,n}(Re)=\{\begin{array}{cc}4\hfill & (n=1),\\ 0\hfill & (n=2,3,\dots ),\end{array}$$

(6)

where ${\lambda }_{m,n}(Re)$ are functions of the Re number, and they may be expressed by

$${\lambda }_{m,n}(Re)=I_{m-n}{K}_{m-1}+I_{m+n}{K}_{m+1}+I_{m-n+1}{K}_m+I_{m+n-1}{K}_m$$

(7)

where $I_m$ and $K_m$ are the modified Bessel functions of the first and second kind, respectively. Furthermore, substituting Equation (5) into Equation (1), the general formula of C_D can be written as

$$C_D=\frac{4\pi }{Re}{\displaystyle \sum _{m=0}^{\infty }{B}_m}.$$

(8)

A literature survey shows that drag correlations for a wide range of Re numbers are rarely reported. In particular, there is no such correlation for drag in the entire range of Re numbers. The old drag coefficient diagrams were used in almost all the relevant literature and texts. When researchers verify the accuracy and reliability of the results they have obtained, they all have to use traditional drag coefficient diagrams. Therefore, a new drag coefficient diagram with rich and reliable data and a general drag model is necessary. Based on this situation, we extensively collected historical experimental data and mapped them on the latest drag coefficient-Re diagram (Figure 2). The historical experimental data collected by Schlichting [48] and other experimental results from Dryden and Hill [49], Delany and Sorensen [50], Tritton [51], Roshko [52], and Achenbach [53] for the drag coefficient of circular cylinders all fall on a single curve. The new drag coefficient diagram refers to a wide range of Re numbers and can greatly facilitate engineering applications, and it can be updated in the texts of fluid mechanics and heat transfer, since it is of great significance to teaching and scientific research.

3. Results and Discussion

3.1. Drag Force-Velocity Diagram

It is intuitive to people that the resistance of a moving body increases rapidly with velocity. For bluff-body flows, trend graphs of resistance and velocity over a broad range of Re are rarely seen in the literature, especially for flow around circular cylinders. Based on the results of previous experimental studies [48,49,50,51,52,53], a resistance-velocity diagram over a broad range of Re numbers was obtained, and it is illustrated in Figure 3. The relative velocity range is from 1.5 × 10⁻³ to 1.5 × l0³ 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 10⁶.

The classic shape of the drag coefficient curve versus Reynolds number for circular cylinders is presented in Figure 2 and also appears in every fluid mechanics text. Figure 3 shows the typical trend that the drag on a circular cylinder increases with increasing flow velocity of the fluid; however, there is a significant reduction in the customary drag coefficient C_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 × 10⁵; 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) × 10⁵, 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

Although extensive investigations on flow past circular cylinders have been conducted in pressurized wind tunnels, due to the limitations of the conditions, most of the experimental works were carried out below the critical Reynolds number Re < 4 × 10⁵. 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).

In fluid dynamics, the drag coefficient is an important dimensionless group which is utilized to quantify the drag of a body in a moving fluid. Therefore, the drag coefficient is a crucially important dimensionless parameter when calculating resistance, and we hope it can hint as to the variation trend of resistance. However, it is clear that the traditional drag coefficient does not possess this important characteristic, and its change trend with the Reynolds number may be also easily misleading. A very representative example is shown on Wikipedia [56], where the relevant statement is expressed as follows: “the drag coefficient is utilized in the resistance relation in which a lower drag coefficient indicates the body will have less aerodynamic or hydrodynamic drag”. Through the above analysis, we can see that this expression is not rigorous, since a smaller value of drag coefficient does not necessary imply that the drag acting on the body is lower. This demonstrates that the customarily used drag coefficient may not be a desirable and appropriate dimensionless parameter to describe the fluid flow physical behavior. It seems to be a common perception that the drag and customary drag coefficient would preferably have a uniform variation trend. In fact, this is a wrong perception caused by Newton’s definition, and herein lies the significance of our research work.

Thus, in this work, we propose the use of a new, more representative drag coefficient quantity, namely, the modified drag coefficient that is defined in the following manner [7].

$$D_C=Re{C}_D=\frac{F}{\frac{1}{2}\mu U_{\infty }}$$

(9)

By comparison with Equation (1), it is clearly observed that the above definition of the appropriate drag coefficient is simpler and easier to use. Furthermore, by the use of Equation (8), the appropriate drag coefficient can also be expressed as follows:

$$D_C=Re{C}_D=4\pi {\displaystyle \sum _{m=0}^{\infty }{B}_m}.$$

(10)

From the above formula, it is noted that the new drag coefficient is directly related to the constants B_m. Using the definition of the modified drag coefficient for spheres and cylinders, the general drag expressions may be restated respectively as

$$F_{D,spheres}=\frac{D_C}{8}\pi \mu D{U}_{\infty },$$

(11)

$$F_{D,cylinders}=\frac{D_C}{2}\mu \cdot 1\cdot U_{\infty }.$$

(12)

A striking similarity in the equation structure emerges between the drag expressions in terms of the modified drag coefficient and the linear Stokes drag expression for spheres [57]. The linear Stokes drag of spheres at low Reynolds numbers may be expressed as

$$F_{D,spheres}=3\pi \mu D{U}_{\infty }.$$

(13)

The Stokes drag for spheres is also referred to as the Stokes-Einstein-Sutherland equation, and it was developed based on the analytical solution of Navier-Stokes equations by Sir George Gabriel Stokes [40].

Based on the discussion above, we observe that the proposed dimensionless group D_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

Historical experimental drag data for circular cylinders were processed and presented according to the definition of the modified drag coefficient. Figure 4 demonstrates the appropriate drag coefficient of circular cylinders as a function of the Re number. It can be seen that the modified drag coefficient rises with increasing Re number as the drag force itself does. From Figure 3 and Figure 4, it is observed that the drag and the modified drag coefficient have roughly the same variation trend with increasing Re number. Figure 4 is more reasonable and intuitive in reflecting the original physical behavior and natural tendency. Therefore, Figure 4 does serve as a significant visual aid in drag analysis and optimum structural design. Furthermore, the curve is relatively smooth and there does not exist a minimum for D_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.

Due to the absence of a classical drag relationship over the entire range of Re numbers, based on the systematic summary of previous research, according to the definition of the appropriate drag coefficient, an extensive empirical model of the appropriate drag coefficient with great accuracy was developed by means of a weighted least square fit in the entire Re number range. The simple model obtained is expressed as follows:

$$D_C=1.38R{e}^{0.95}+7.72R{e}^{0.31}+1.82.$$

(14)

As can be seen from Figure 2 and Figure 4, the modified drag coefficient curve is smoother than the traditional drag coefficient curve. The model developed is in good agreement with the analytical solution and almost all experimental data. Using the appropriate drag coefficient definition to solve the problems of flow past an object, the change in resistance of the object with the velocity of the fluid can be intuitively reflected by the relation between the appropriate drag coefficient and Re number in the graph. The above discussion demonstrates that the new modified drag coefficient may be a preferable dimensionless parameter to describe fluid flow physical behavior so that fluid flow problems can be solved in a simple and intuitive manner.

3.4. Relationship between the Drag and Heat Transfer

For convection heat transfer from cylinders, Tomotika et al. obtained the Oseen solution, and they proposed a mean Nusselt number correlation which is valid at low Reynolds numbers and of very limited utility [58]. Lately, Khan et al. [59] performed an analytical study on heat transfer from cylinders and proposed the corresponding relationship. The analytical solution of the average Nusselt number obtained for both thermal boundary conditions is expressed as follows:

$$\frac{Nu}{R{e}^{1/2}P{r}^{1/3}}=\{\begin{array}{l}0.593\mathrm{for}\mathrm{constant}\mathrm{wall}\mathrm{temperature}\hfill \\ 0.632\mathrm{for}\mathrm{constant}\mathrm{heat}\mathrm{flux}\hfill \end{array}.$$

(15)

A large number of experimental studies on convection heat transfer from cylinders were performed by researchers. Based on experimental data, Kramers [60] suggested a mean Nusselt number correlation for 0.1 ≤ Re ≤ 10⁴.

$$Nu=0.42P{r}^{0.2}+0.57P{r}^{1/3}R{e}^{0.5}$$

(16)

The correlation by Kramers is recommended due to its simplicity. In addition, Fand [61] analyzed the results of other investigators for liquids and air and correlated previous researchers’ and his experimental data by an equation as follows:

$$Nu=(0.35+0.34R{e}^{0.5}+0.15R{e}^{0.58})P{r}^{0.3}.$$

(17)

Owing to the similarity of governing differential equations for heat and mass transfer, analogy has been developed as a useful tool [62]. Based on the classical analogical approach, some analogy studies were carried out on the flow of fluids through a fluidized bed [63], turbulent flow in circular pipes [64,65], fully developed turbulent flow of power law fluids [66], slip flow heat transfer in microchannels [38], drag-reduced turbulent channel flow [67], and material evaporation behavior [68]. It is an efficient and applicable approach to predict heat and mass transfer coefficients from hydrodynamic results, especially for complex engineering problems. The change of the thermal field for different Reynolds numbers can be predicted from that of flow [69], particularly for the heat transfer behavior of the wake behind a heated body. This means the role played by the heat flux has a similar nature to that of the drag. It was found that there may exist a strong similarity between the drag and heat and mass transfer. It is expected that one transport process may be associated with another transport process, allowing one to be determined if the other is known. For example, an analogy exists between the diffusion of heat and electrical charge. This part focuses on an analogy between the appropriate drag coefficient and the mean Nusselt number.

In general, engineers may be interested in the availability of simple models which can be applied to extended studies. Through in-depth comparisons and analyses of heat transfer correlations and the new drag model, based on comprehensive consideration of simplicity, ease of use, and high accuracy, an analogy between drag and heat transfer for cylinders was developed. The analogy is applicable for a wide range of Reynolds numbers, 0.1 ≤ Re ≤ 10⁵, and it may approximately hold for flow with higher Reynolds number. The rough analogy may be expressed as follows:

$$\frac{D_C}{7.5+2.5R{e}^{0.45}}\cong Nu\cong Sh.$$

(18)

It is well known that the thermal field and concentration field can be related to the velocity field through Re. It is established that a relationship exists between momentum transfer and heat and mass diffusivities via Equation (18). The key engineering parameters, the Nusselt number and the Sherwood number, can be predicted using the presented analogy in the calculation and analysis of heat and mass transfer for convenience due to the simplicity of this expression, but there may be some loss in accuracy. Here, Equation (18) is valid for gases in which Pr is near unity in the heat transfer process. Therefore, in order to extend the availability of the simple analogy, the Prandtl number effect was considered and added in a more exact analogy. The exact analogy was developed for a wide range of Reynolds numbers, 0.1 ≤ Re ≤ 10⁵.

$$\frac{D_C}{7.5+2.5R{e}^{0.45}}\cong \frac{Nu}{P{r}^{0.4}}$$

(19)

The above model for circular cylinders is presented as (7.5 + 2.5Re^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.

An inherent relation between the drag and heat and mass transfer was obtained by analogy with hydrodynamic drag data. The strong analogy further proves that the new modified drag coefficient may be a proper dimensionless parameter to describe and represent the drag, and it provides great insight into the mechanism of different transport phenomena.

4. Conclusions

The flow over regular-shaped bodies like spheres and circular cylinders represents a classical and conventional problem in hydromechanics. It was found that the customary drag coefficient may not be a proper dimensionless parameter to describe and represent the drag for flow past bluff bodies.

In the present work, it was demonstrated that a new modified drag coefficient may be a preferable dimensionless parameter to describe fluid flow physical behavior and reflect the real variation trend of drag force. A general simple drag model with fundamental accuracy was developed and is universally valid for all Reynolds number regimes. It is convenient to predict the hydrodynamic drag on cylinders utilizing the general simple model.

It was observed that there may exist a strong similarity between the drag and heat and mass transfer. It was established in this paper that a relation exists between the drag and heat transfer around a circular cylinder. This may be quite crucial due to the lack of information on the Nusselt number and the Sherwood number for most bluff bodies in the existing literature. In this context, the presented model does offer a method for predicting the Nusselt number and the Sherwood number for flow over other body shapes. The proposed simple means may provide great insight into the design and optimization of the shape and thermal barrier coating system of high-speed aircrafts.

In future research work, we will further extend analogies to other body shapes employing the appropriate drag coefficient concept.