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1

HEAT TRANSFER AND FLUID FLOW OVER CIRCULAR

CYLINDERS IN CROSS FLOW

M. JIBRAN HAIDER, S. NOMAN DANISH, W. A. KHAN, S.

UZAIR MEHDI & BILAL AHMED ABBASI

Department of Engineering Sciences

National University of Sciences and Technology (NUST)

Pakistan Navy Engineering College (PNEC)

Karachi

Abstract:

An extensive numerical study is conducted to determine cross flow of air (Pr=0.71)

around isothermal cylinders of circular cross-section in arrangements such as single

cylinder, inline arrays and staggered arrays. Commercial software package FLUENT

is used to solve the fluid flow and energy equations assuming the flow over the

cylinder is two dimensional, steady, viscous and incompressible bounded in a duct, as

low Reynolds number is being investigated. The width of the duct is kept 20 times the

diameter of the cylinder so that the effects of channel blockage can be avoided. The

effects of radiation are also neglected in this study. Variations in properties such as

local Nusselt number, average Nusselt number, local pressure coefficient and local

skin friction coefficient are presented around the cylinders at Reynolds number

ranging from 40 to 10,000. The results are compared with analytical, experimental

and numerical data from previous literature and are found to be in excellent

agreement. It has been found that heat transfer from a staggered array of cylinders is

slightly higher than an inline array of cylinders.

Keywords: Heat Transfer; Computational Fluid Dynamics; FLUENT; Reynolds

Number; Nusselt Number; Circular Cylinder

Corresponding Author:

M. Jibran Haider

jibranhaider@hotmail.com

9221-34812186

92-300-2508252

2

NOMENCLATURE

B.C Boundary condition

CFD Computational Fluid Dynamics

cp Specific heat at constant pressure [J/kg.K]

Cp (θ) Local pressure coefficient,

2

5.0)()()(

UppCp

D Diameter of the cylinder [m]

h (θ) Local heat transfer coefficient [W/m2.K]

k Thermal conductivity [W/m.K]

L Characteristic length = Diameter of cylinder [m]

NuL Average Nusselt number based on characteristic length,

khLNu

NuL (θ) Local Nusselt number,

kLhNuL)()(

p (θ) Local static pressure at the surface of cylinder [N/m2]

P∞ Free stream pressure [N/m2]

Pr Prandtl number,

kcp

Pr

ReL Reynolds number based on characteristic length,

LU

Re

T Temperature [K]

Tw Wall temperature [K]

T∞ Free stream temperature [K]

u x component of velocity [m/s]

U∞ Free stream velocity [m/s]

v y component of velocity [m/s]

W Width of duct [m]

Greek Symbols

θ Angular displacement measured clockwise from front stagnation point [o]

μ Dynamic viscosity [kg/m.s]

ρ Density [kg/m3]

3

1. Introduction

Investigation of heat transfer and fluid flow around cylinders has been a

popular subject because of its importance in variety of applications such as heat

exchangers, nuclear reactors, overhead cables, power generators, thermal apparatus

etc. Many researchers have analytically, experimentally and numerically determined

heat transfer and flow structures around circular cylinders placed in a bank and as

well as in isolation. A brief summary is provided below.

An extensive analytical study has been carried out by Van Der Hegge [1] to

produce a new correlation formula to determine heat transfer by natural and forced

convection from horizontal cylinders. Similarly, Refai Ahmed & Yovanovich [2] have

developed a method to determine heat transfer by forced convection from isothermal

bodies such as infinite circular cylinders, flat plates and spheres. The solution is valid

for a wide range of Reynolds and Prandtl numbers. More recently Khan et al. [3] [4]

have investigated fluid flow and heat transfer from a single circular cylinder and an

infinite circular cylinder analytically by the Von Karman –Pohlhausen method.

Correlations are obtained for heat transfer and drag coefficients which are applicable

for a wide range of Reynolds and Prandtl numbers. Effects of both isothermal and

isoflux boundary conditions are analyzed.

Meel [5] experimentally determined the circumferential heat transfer

coefficient by measuring temperature distribution on the outer surface of the cylinder.

A series of experiments were conducted by Igarashi [6] [7] at high Reynolds Numbers

to determine pressure and drag coefficients around two circular cylinders placed in

tandem. The effects of varying the longitudinal distance between the cylinders and

their diameters were also investigated. Igarashi along with Suzuki [8] extended the

study to three circular cylinders arranged inline. An extensive experimental study was

undertaken by Buyruk [9] to determine local Nusselt Number and local pressure

coefficient around a circular cylinder for various Reynolds Numbers and blockage

ratios. This research was also extended to tube banks and the variation of local

Nusselt Number was obtained for every row with changes in longitudinal and

transverse pitches. Similarly Mehrabian [10] attempted to investigate the rate of

cooling of a cylindrical copper element by forced convection. The author has also

analyzed the uncertainty in the measurement of heat transfer characteristics of the

system.

On the other hand Wung and Chen [11] have utilized a finite analytic method

to determine heat transfer at various Reynolds numbers from inline and staggered tube

arrays. Buyruk [12] has numerically investigated heat transfer from cylinders placed

in tandem, inline tube banks and staggered tube banks. He has used a finite element

method to obtain the circumferential variation in Nusselt number for the cylinders. A

steady as well as an unsteady analysis has been undertaken by Szczepanik et al. [13]

to determine heat transfer from a cylinder in cross flow. Unsteady simulations of the

cylinder depict vortex shedding. The numerical study makes use of a k-ω turbulence

model.

4

The present study utilizes a commercial CFD software package, FLUENT

which is based on a control volume based technique to solve the governing equations

such as conservation of mass, momentum, energy and turbulence. Algebraic equations

are generated for discrete dependant variables like pressure, velocity, temperature etc,

for each control volume. Finally the discretized equations are linearized and a solution

is obtained [14].

5

2. Computational Methods

2.1. Methodology

The following methodology has been adopted in order to obtain results

through CFD simulations. It highlights the iterative procedure which must be carried

out in order to obtain an accurate set of results.

Create geometry in GAMBIT

Export mesh to FLUENT

Set solution parameters and

solver settings

Generate mesh

Set boundary conditions

Initialize the solution

Iterate to obtain a solution

Check for convergence

Modify solution

parameters, solver

settings and boundary

conditions or modify

meshing technique or

mesh density

Yes

No

Check for accuracy

Stop

Yes

No

6

2.2. Assumptions

A two dimensional analysis is performed as the length of the cylinder is kept

much greater than its diameter. The assumption that flow is incompressible is

warranted as relatively low Reynolds number is being investigated. The width of the

duct is kept much larger (20 times) than the diameter of the cylinder so that the wall

effects of the duct can be neglected. This means that the effects of boundary layer

formation on the duct boundary will not affect the flow in the vicinity of the cylinder.

2.3. Geometry and Meshing

The geometry is created in GAMBIT, which is the pre-processor for geometric

modeling and mesh generation. The rectangular computational domain is bounded by

the inlet, outlet and duct boundaries. The flow enters the domain from the inlet

boundary on the extreme left and leaves from the outlet boundary on the extreme right

for all simulations. A 2D structured mesh of non uniform grid spacing is created. The

mesh density is kept intense near the cylinder for resolving the boundary layer

accurately. The distinct points of the mesh are called nodes where all the equations

involved in the system are solved. These equations are:

Equation of continuity

0

y

v

x

u

x component of conservation of momentum

2

2

2

2

1yu

xu

x

p

y

u

v

x

u

u

t

u

y component of conservation of momentum

2

2

2

2

1yv

xv

y

p

y

v

v

x

v

u

t

v

Energy equation

Basically two slightly different techniques were utilized for mesh generation.

The first technique was applied to the mesh of a single cylinder. It involves the

generation of a block structured quadrilateral mesh around the cylinder. Figure 1

focuses on the mesh in the vicinity of a circular cylinder. It highlights the block

technique which is used for a smooth transition in mesh. A smooth transition in cell

volumes between adjacent cells is necessary as inability to do so may lead to

truncation errors.

2

2

2

2

yT

xT

y

T

v

x

T

u

t

T

7

Figure 1 Mesh in the Vicinity of the Cylinder

The second technique of mesh generation is applied to inline and staggered

arrangements. The mesh for these arrangements consists of tri meshing. Due to a

relatively complex geometry generating block structured quadrilateral mesh near the

cylinders is time consuming and therefore not feasible. The mesh is shown below.

Figure 2 Mesh in the Vicinity of Four Cylinders placed in an Inline Array

2.4. Solver Settings

All numerical simulations are performed under the double precision solver as

opposed to the single precision solver. The double precision solver performs better

where pressure differences are involved and high convergence with accuracy is

demanded [14]. A pressure based solver which in previous versions of FLUENT was

referred as the segregated solver was selected, as the present study deals with an

incompressible flow. A second order upwind scheme was used to discretize the

convective terms in the momentum and energy equations. This scheme though is time

consuming but it yields an accurate solution. This high order accuracy is achieved by

a Taylor series expansion about the cell centroid [14]. A convergence criterion of 10-6

was found sufficiently accurate for this study and was applied to all residuals except

energy for which the criterion was extended to 10-9.

8

2.5. Boundary Conditions

Following boundary conditions were applied to the boundaries for all cases.

Inlet Boundary: A velocity inlet boundary condition is applied to the inlet

boundary as it is intended for incompressible flows. A uniform velocity profile is

defined normal to the inlet boundary.

Outlet Boundary: An outflow boundary condition is employed at the outlet

boundary. Its use is justified as the flow velocity and pressure at the outlet are

unknown before the solution of the problem. It works on the principle of zero

diffusion flux normal to the outflow boundary for all variables except pressure. It

merely extrapolates information from within the domain and applies to the outlet

without disturbing the upstream flow [14].

Cylinder: A wall boundary condition is selected for the isothermally heated

cylinder. The cylinder is heated to a temperature of 400 K for all simulations. In

addition a no slip condition is employed along the cylinder surface.

Duct Boundary: The fluid flow is bounded within the duct by applying the wall

boundary and no slip condition.

The mesh with dimensions and boundary conditions is shown in the figure below. It is

to be noted that the actual mesh is much finer than the one shown.

Figure 3 Meshed Computational Domain with Boundary Conditions

20 D

15 D

Inlet Boundary

B.C = Velocity Inlet

u = U∞

v = 0

T = T∞

Cylinder

B.C = Wall, No slip

u = 0

v = 0

T = Tw = 400 K

Duct Boundaries

B.C = Wall

No slip

u = 0

v = 0

T = Tw = 300 K

Outlet Boundary

B.C = Outflow

W = 20 D

x

y

9

2.6. Grid Independence Study

In order to study the effect of grid size on the results, meshes of three different

densities were created, solved and their results were analyzed. The following table

shows the details of grid sizes for flow over a single circular cylinder and the

corresponding effects on the average Nusselt number at a Reynolds number of 100.

The results were found to be grid independent beyond the “average” mesh size.

Mesh Size

No. of Nodes

Average Nusselt No

Percentage Error

1.

Coarse

4120

5.2513

--

2.

Average

14260

5.1513

1.90

3.

Fine

37220

5.1401

0.22

Table 1 Effect of Grid Size on Average Nusselt Number

3. Results and Discussion

3.1 Flow over a Single Cylinder

3.1.1 Average Nusselt Number

Table 2 represents correlations developed by various researchers relating the

average Nusselt number for a circular cylinder.

Author

Correlation

Range of Re

B.C

Zukauskas [15]

5.0

Re4493.0 LL

Nu

40 – 1000

Isothermal

Morgan [16]

471.0

Re583.0 LL

Nu

40 – 4000

Isothermal

Hilpert [17]

466.0

Re615.0 LL

Nu

40 – 4000

Isothermal

Knudsen

and Katz [18]

3

1

466.0 PrRe683.0 LL

Nu

40 – 4000

Isothermal

Table 2 Experimental Correlations of NuL for Air

The values of average Nusselt number are calculated from these correlations

and are compared with the results of present study for Reynolds number ranging from

50 – 600. The results are presented in Table 3 and Fig 5. The analytical results

obtained by Khan et al. [4] are also plotted. Present study is in close agreement with

all previous experimental and analytical studies.

10

Table 3 Values of NuL Obtained From Correlations and Our Results at 50 ≤ Re ≤ 600

ReL Vs NuL

3

4

5

6

7

8

9

10

11

12

13

0100 200 300 400 500 600

ReL

NuL

Zukauskas (1972)

Morgan (1975)

Hilpert (1933)

Knudsen and Katz (1958)

Khan (2005)

Present Study

Figure 4 Comparison of NuL Vs ReL

It can be clearly seen from Fig 4 that as the Reynolds number increases,

Nusselt number also increases. The increase in Reynolds number is brought about

only by an increase in the free stream velocity as all the other parameters are kept

constant. The increased velocity will increase the average heat transfer coefficient

around the cylinder which eventually increases the average Nusselt number.

3.1.2 Local Nusselt Number

Figure 5 shows the plot of Nusselt number at the stagnation point and is

compared with the results given by Kays, Crawford and Weigand [19]. Again a good

agreement is found with the previous study.

Reynolds

Number

Zukauskas

[15]

Morgan [16]

Hilpert [17]

Knudsen and

Katz [18]

Khan [4]

Present

Study

Experimental

Experimental

Experimental

Experimental

Analytical

Numerical

50

3.18

3.68

3.81

3.83

3.7

3.82

100

4.49

5.10

5.26

5.29

5.2

5.15

200

6.35

7.07

7.26

7.31

7.4

7.12

300

7.78

8.56

8.77

8.83

9.0

8.59

400

8.99

9.80

10.03

10.10

10.3

9.97

500

10.05

10.89

11.13

11.20

11.5

11.17

600

11.01

11.86

12.12

12.20

12.5

12.28

11

ReL Vs NuL (θ=0)

5

10

15

20

25

30

35

0100 200 300 400 500 600 700 800 900 1000

ReL

NuL (θ=0)

Kays ,Crawford &

Weigand

Present Study

Figure 5 Comparison of ReL Vs NuL (θ=0)

The variation of local Nusselt number along the cylinder is presented in Figs 6

and 7 in comparison with the results of Krall and Eckert [20] for Reynolds numbers

100 and 200. Krall and Eckert kept the same boundary condition of no slip and

uniform wall temperature on the cylinder as is done in the present study.

Re=100

0

2

4

6

8

10

12

030 60 90 120 150 180

θ

NuL (θ)

Krall & Eckert (1970)

Present Study

Figure 6 Variation of NuL (θ) at Re=100

Re=200

0

2

4

6

8

10

12

14

16

030 60 90 120 150 180

θ

NuL (θ)

Krall & Eckert (1970)

Present Study

Figure 7 Variation of NuL (θ) at Re=200

12

Again the results are in good agreement. In both the graphs presented above the

values obtained by Krall and Eckert are slightly higher than the present study. The

possible reason of deviation may be related to a higher blockage factor in the study of

former authors. Present study is based on a blockage factor of 0.05.

3.1.3 Local Pressure Coefficient

The analytical results of local pressure coefficient along the surface of the

cylinder have been provided by Zdravkovich [21]. He has reported the results of

Kawaguti and Apelt. Figure 8 provides a comparison of those results with present

study at Reynolds number of 40.

Re=40

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

030 60 90 120 150 180

θ

Cp (θ)

Apelt (1961)

Kawaguti (1953)

Present Study

Figure 8 Variation of Cp (θ) at Re=40

An excellent agreement with the analytical results is observed. Experimental

study over cylinders at such low Reynolds number yields a greater percentage of

error. Therefore the significance of analytical and numerical study at low Reynolds

number is much more feasible. Similarly Zdravkovich [20] has reported the results of

Thoman & Szewczyk who carried out a computational study of flow over a circular

cylinder. The results at a Reynolds number of 200 are compared with the study and

found to be very close.

Re=200

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

030 60 90 120 150 180

θ

Cp (θ)

Thoman & Szewczyk (1969)

Present Study

Figure 9 Variation of Cp (θ) at Re=200

13

3.2 Flow over an Inline Array of Cylinders

The next stage of analysis was to simulate flow over circular cylinders placed

in an inline configuration as shown in Fig 10. Results were first obtained for 4

cylinders placed in a 2 x 2 array for which the longitudinal and transverse distances

between the cylinders were kept at 2 times the diameter of the cylinder. Later flow

over 25 cylinders placed in a 5 x 5 array was simulated.

Figure 10 Four Cylinders Placed in an Inline Arrangement

3.2.1 Local Nusselt Number

Distribution of local Nusselt number has been obtained for the inline

configuration. The results are compared with that of Buyruk [12] for a Reynolds

number of 200 and shown in Figs 11 and 12 for upstream and downstream cylinders.

A very good agreement is observed for both the cylinders.

Re=200 - First Cylinder

0

2

4

6

8

10

12

14

16

060 120 180 240 300 360

θ

NuL (θ)

Buyruk (2002)

Present Study

Figure 11 Variation of NuL (θ) at Re=200 along the First Cylinder

Second

Cylinder

First

Cylinder

14

Re=200 - Second Cylinder

0

1

2

3

4

5

6

7

8

9

10

060 120 180 240 300 360

θ

NuL (θ)

Buyruk (2002)

Present Study

Figure 12 Variation of NuL (θ) at Re=200 along the Second Cylinder

3.2.2 Contours of Static Pressure and Velocity

The contours of static pressure are shown in Fig 13 at a Reynolds number of

200 around the cylinders. The shifting of the front stagnation points on the front

cylinders (red region) and the rear cylinders (green region) is evident. This is due to

the venturi effect created between the two rows of cylinders which creates suction and

shifts the front stagnation point.

Figure 13 Contours of Static Pressure at Re=200

15

Velocity contours are also shown below at a Reynolds number of 40 and 200.

Figure 14 Contours of Velocity at Re=40

Figure 15 Contours of Velocity at Re=200

The increase in velocity between the cylinders as Reynolds number increases

is shown by the red region. The wakes generated by the cylinders at the front are

disturbed due to the presence of rear cylinders. The disturbance is much greater at a

Reynolds number of 200 than at 40.

3.2.3 Average Nusselt Number

The results of average Nusselt Number were obtained for the inline array of 5

by 5 and are presented in Table 4. The longitudinal and transverse distances were kept

at 2.5 times the diameter of the cylinder.

16

Reynolds

Number

Nu

100

3.005

500

6.907

1000

11.277

2000

18.064

3000

23.453

4000

28.192

5000

33.175

6000

38.235

7000

43.581

8000

49.150

9000

54.433

10000

59.462

Table 4 Variation of Nu with Re for a 5 x 5 Inline Array

3.2.4 Contours of Temperature

The contours of temperature are shown below for various Reynolds numbers.

Figure 16 Contours of Temperature at Re=100

Figure 17 Contours of Temperature at Re=1000

17

Figure 18 Contours of Temperature at Re=5000

The plots above show the decrease in the thermal boundary layer as Reynolds

number increases from 100 to 1000 and finally to 5000. So the temperature gradient at

a higher Reynolds number is very steep which gives a better heat transfer. It is also

evident that the diffusion of temperature contours occurs much early downstream of

the cylinders at a lower Reynolds numbers. Lastly, the symmetry of temperature

contours can be observed about the central row. It is to be noted that there is no shift

in the stagnation points of the cylinders present in the central row while every other

cylinder experiences some change in stagnation point.

3.3 Flow over a Staggered Array of Cylinders

The flow was also simulated over a staggered array of 3 cylinders as shown in

Fig 19 and then for 23 cylinders. The longitudinal and transverse pitches for the three

cylinders are kept 2.

Figure 19 Three Cylinders Placed in a Staggered Arrangement

First

Cylinder

Second

Cylinder

18

3.3.1 Local Nusselt Number

Variation of local Nusselt number is obtained and compared with the results of

Buyruk [12]. The comparison is shown in Figs 20 and 21 at a Reynolds number of

200 for the first and second cylinders.

Re=200 - First Cylinder

0

2

4

6

8

10

12

14

060 120 180 240 300 360

θ

NuL (θ)

Buyruk (2002)

Present Results

Figure 20 Variation of NuL (θ) at Re=200 along the First Cylinder

Re=200 - Second Cylinder

0

2

4

6

8

10

12

14

16

18

060 120 180 240 300 360

θ

NuL (θ)

Buyruk (2002)

Present Study

Figure 21 Variation of NuL (θ) at Re=200 along the Second Cylinder

A close examination of the plot reveals that the values of local Nusselt number

obtained by Buyruk for the second cylinder are not exactly symmetrical over the

upper and lower surfaces. The configuration of the cylinders is such that flow should

be symmetrical for the second cylinder. On the other hand our results are exactly

symmetrical.

19

3.3.3 Contours of Static Pressure and Velocity

The contours of static pressure are shown in Fig 22 at a Reynolds number of

120. The shifting of the front stagnation points is clearly visible on the two cylinders

at the front. As expected symmetrical pressure contours are obtained for the second

cylinder.

Figure 22 Contours of Static Pressure at Re=120

The contours of velocity at Reynolds number 40 and 500 are shown in the following

figures.

Figure 23 Contours of velocity at Re =40

20

Figure 24 Contours of Velocity at Re=500

It can be clearly seen that at a higher Reynolds number separation of the

boundary layer from all three cylinders occurs early. Therefore the wakes created in

Fig 24 are much greater than those in Fig 23.

3.3.4 Average Nusselt Number

Flow is simulated over a staggered array of 23 cylinders. Longitudinal and

transverse pitches are kept at 2.5 each. The results of average Nusselt numbers are

obtained for various Reynolds Numbers.

Reynolds

Number

Nu

100

3.470

500

9.941

1000

14.875

2000

22.346

3000

28.661

4000

34.495

5000

40.432

6000

46.784

7000

54.304

8000

61.821

9000

68.310

10000

74.671

Table 5 Variation of Nu with Re for a Staggered Array

21

3.3.5 Contours of Temperature

The contours of temperature are shown in Figs 25, 26 and 27 at different

Reynolds numbers.

Figure 25 Contours of Temperature at Re=100

Figure 26 Contours of Temperature at Re=1000

Figure 27 Contours of Temperature at Re=5000

22

As the Reynolds number increases from 100 to 5000 the thickness of the

thermal boundary layer decreases significantly. Therefore the temperature gradient at

Reynolds number 5000 is much greater than that at Reynolds numbers 1000 or 100.

This high temperature gradient is responsible for the increased heat transfer as

Reynolds number increases. As in the case of an inline array, the symmetry of

temperature contours can also be observed for a staggered array about the central row.

Figure 28 shows a comparison of the Nusselt number for inline and staggered arrays.

NuL Vs ReL

0

10

20

30

40

50

60

70

80

100

500

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

ReL

NuL

Inline Array

Staggered Array

Figure 28 Variation of NuL Vs ReL for Inline and Staggered Arrays

It can be concluded that the heat transfer from a staggered array is higher than

that from an inline array when subjected to the same Reynolds number. This is also

confirmed by the temperature contours of the two arrangements as the diffusion of

contours is more intense in the staggered array as compared to the inline array.

Conclusions

Numerical study has been undertaken to analyze heat transfer and flow

characteristics past a single cylinder, inline array and staggered array at various

Reynolds numbers. It can be concluded from the results that:

1. Heat transfer from a staggered array of cylinders is slightly higher than

that from an inline array of cylinders.

2. The simulated results of local Nusselt number, average Nusselt number

and local pressure coefficient from circular cylinders are in well agreement

with the analytical, experimental and numerical results available in

existing literature.

23

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[13] K. Szczepanik, A. Ooi1, L. Aye and G. Rosengarten, “A numerical study of heat

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Mechanics Conference, Sydney, December 13-17, 2004.

[14] “FLUENT 6.3 User’s Guide” , 2006.

[15] A. Zukauskas, “Advances in heat transfer”, Academic Press New York, 1972, pp. 93-

160.

[16] V. Morgan, “The overall convective heat transfer from smooth circular cylinders”,

Advances in Heat Transfer, Vol.11, Academic Press, New York, 1975, pp. 199–264.

[17] R. Hilpert, “Warmeabgabe von geheizten drahten und rohren”, Forch. Geb.

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24

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