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)
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
M. Jibran Haider
B.C Boundary condition
CFD Computational Fluid Dynamics
cp Specific heat at constant pressure [J/kg.K]
Cp (θ) Local pressure coefficient,
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,
NuL (θ) Local Nusselt number,
p (θ) Local static pressure at the surface of cylinder [N/m2]
P∞ Free stream pressure [N/m2]
Pr Prandtl number,
ReL Reynolds number based on characteristic length,
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]
θ Angular displacement measured clockwise from front stagnation point [o]
μ Dynamic viscosity [kg/m.s]
ρ Density [kg/m3]
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  to
produce a new correlation formula to determine heat transfer by natural and forced
convection from horizontal cylinders. Similarly, Refai Ahmed & Yovanovich  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.  
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  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   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  extended the
study to three circular cylinders arranged inline. An extensive experimental study was
undertaken by Buyruk  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  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
On the other hand Wung and Chen  have utilized a finite analytic method
to determine heat transfer at various Reynolds numbers from inline and staggered tube
arrays. Buyruk  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. 
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
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 .
2. Computational Methods
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
Set boundary conditions
Initialize the solution
Iterate to obtain a solution
Check for convergence
settings and boundary
conditions or modify
meshing technique or
Check for accuracy
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
x component of conservation of momentum
y component of conservation of momentum
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
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 . 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 . 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.
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 .
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
B.C = Velocity Inlet
u = U∞
v = 0
T = T∞
B.C = Wall, No slip
u = 0
v = 0
T = Tw = 400 K
B.C = Wall
u = 0
v = 0
T = Tw = 300 K
B.C = Outflow
W = 20 D
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.
No. of Nodes
Average Nusselt No
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.
Range of Re
40 – 1000
40 – 4000
40 – 4000
and Katz 
466.0 PrRe683.0 LL
40 – 4000
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.  are also plotted. Present study is in close agreement with
all previous experimental and analytical studies.
Table 3 Values of NuL Obtained From Correlations and Our Results at 50 ≤ Re ≤ 600
ReL Vs NuL
0100 200 300 400 500 600
Knudsen and Katz (1958)
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 . Again a good
agreement is found with the previous study.
ReL Vs NuL (θ=0)
0100 200 300 400 500 600 700 800 900 1000
Kays ,Crawford &
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  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.
030 60 90 120 150 180
Krall & Eckert (1970)
Figure 6 Variation of NuL (θ) at Re=100
030 60 90 120 150 180
Krall & Eckert (1970)
Figure 7 Variation of NuL (θ) at Re=200
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 . 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.
030 60 90 120 150 180
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  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.
030 60 90 120 150 180
Thoman & Szewczyk (1969)
Figure 9 Variation of Cp (θ) at Re=200
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  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
060 120 180 240 300 360
Figure 11 Variation of NuL (θ) at Re=200 along the First Cylinder
Re=200 - Second Cylinder
060 120 180 240 300 360
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
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.
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
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
3.3.1 Local Nusselt Number
Variation of local Nusselt number is obtained and compared with the results of
Buyruk . 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
060 120 180 240 300 360
Figure 20 Variation of NuL (θ) at Re=200 along the First Cylinder
Re=200 - Second Cylinder
060 120 180 240 300 360
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
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
Figure 22 Contours of Static Pressure at Re=120
The contours of velocity at Reynolds number 40 and 500 are shown in the following
Figure 23 Contours of velocity at Re =40
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.
Table 5 Variation of Nu with Re for a Staggered Array
3.3.5 Contours of Temperature
The contours of temperature are shown in Figs 25, 26 and 27 at different
Figure 25 Contours of Temperature at Re=100
Figure 26 Contours of Temperature at Re=1000
Figure 27 Contours of Temperature at Re=5000
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
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.
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
 V. D. H. Zijnen, “Modified correlation formulae for the heat transfers by natural and
by forced convection from horizontal cylinders”, Appl. Sci. Res, Vol. 6, 1956, pp.
 G. R. Ahmed and M. Yovanovich, “Analytical method for forced convection from flat
plates, circular cylinders and spheres”, Journal of Thermophysics and Heat Transfer,
Vol. 9, No. 3, 1995, pp. 516–523.
 W. A. Khan, J. Culham, and M. Yovanovich, “Fluid flow and heat transfer from a pin
fin: analytic approach”, American Institute of Aeronautics and Astronautics, Vol. 163,
2003, pp. 1–12.
 W. A. Khan, J. Culham and M. Yovanovich, “Fluid flow around and heat transfer
from an infinite circular cylinder”, Journal of Heat Transfer, Vol. 27, 2005, pp. 785-
 D. Meel, “A method for the determination of local convective heat transfer from a
cylinder placed normal to an air stream”, Int. J. Heat Mass Transfer, Vol. 5, 1962, pp.
 T. Igarashi, “Characteristics of the flow around two circular cylinders arranged in
tandem”, JSME, Vol. 24, No. 188, 1981, pp. 323–331.
 T. Igarashi, “Characteristics of a flow around two circular cylinders of different
diameters arranged in tandem”, JSME, Vol. 25, 1982, pp. 349–357.
 T. Igarashi and K. Suzuki, “Characteristics of the flow around three circular cylinders
arranged in-line”, JSME, Vol. 27, 1984, pp. 2397–2404.
 E. Buyruk, “Heat transfer and flow structures around circular cylinders in cross flow”,
Tr. Journal of Engineering and Environmental Science, Vol. 23, 1997, pp 299–315.
 M. Mehrabian, “Heat transfer and pressure drop characteristics of cross flow of air
over a circular tube in isolation and/or in a tube bank.”, The Arabian Journal for
Science and Engineering, Vol. 32, No. 2B, 2005, pp. 365–376.
 T. Wung and C. Chen, “Finite analytic solution of convective heat transfer for tube
arrays in cross flow”, Journal of Heat Transfer, Vol. 111, 1989, pp. 633–648.
 E. Buyruk, “Numerical study of heat transfer characteristics on tandem cylinders,
inline and staggered tube banks in cross flow of air”, Int. Comm. Heat Mass Transfer,
Vol. 29, No. 3, 2002, pp. 355–366.
 K. Szczepanik, A. Ooi1, L. Aye and G. Rosengarten, “A numerical study of heat
transfer from a cylinder in cross flow”, Proceedings of 15th Australasian Fluid
Mechanics Conference, Sydney, December 13-17, 2004.
 “FLUENT 6.3 User’s Guide” , 2006.
 A. Zukauskas, “Advances in heat transfer”, Academic Press New York, 1972, pp. 93-
 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.
 R. Hilpert, “Warmeabgabe von geheizten drahten und rohren”, Forch. Geb.
Ingenieurwes, Vol. 4, 1933, pp. 215–224.
 J. Knudsen and D. Katz, “Fluid Dynamics and Heat Transfer”, McGraw Hill, New
 Kays, W. Crawford and B. Weigand, “Convective Heat and Mass Transfer”, 4th
Edition McGraw-Hill, 2005.
 K. Krall and E. Eckert, “Heat transfer to a transverse circular cylinder at low
Reynolds Numbers including rarefaction effects”, Fourth International Heat Transfer
Conference”, Vol. 3, 1970, pp. 1–11.
 M. Zdravkovich, “Flow around circular cylinders”, Vol. 1, 1997, pp. 260–273.