1. Proceedings of the 6th International and 43rd National Conference on Fluid Mechanics and Fluid Power
December 15-17, 2016, MNNITA, Allahabad, U.P., India
FMFP2016-276
STEADY FLOW AROUND a PAIR of STATIONARY CIRCULAR CYLINDERS at LOW
REYNOLDS NUMBERS
Ankush Deep Kumar Rohan Singh Amit Kumar Sharma Subhankar Sen
UG student UG student UG student Assistant professor
Dept. of Mech. Engg. Dept. of Mech. Engg. Dept. of Mech. Engg. Dept. of Mech. Engg
Indian School of Mines Indian School of Mines Indian School of Mines Indian School of Mines
Dhanbad Dhanbad Dhanbad Dhanbad
826004 826004 826004 826004
ankushdeep37@mece.ism.ac.in kumarrohan@mece.ism.ac.in amitism@mece.ism.ac.in subhankars@gmail.com
Abstract
Characteristics of flow past two fixed cylinders in
tandem are investigated using a stabilised finite
element method. The governing equations are
approximately solved on a structured, non-uniform,
multi-block mesh. The diameter of upstream cylinder
(D1) is kept constant while the diameter of
downstream cylinder (D2) is varied from 0.5D1 to D1.
The flow is studied at a fixed Reynolds number 40 in
the regime of steady flow. The centre-to-centre
distance between the two cylinders is varied from 2D1
to 15D1. The flow characteristics are investigated at
the blockages 2% and 25%. As the blockage increases
the drag on the upstream cylinder and separation
angle becomes invariant to spacing as well as
diameter ratio D1/D2. The inline spacing and blockage
has significant effects on the cylinder drag as well as
cylinder surface pressure.
Keywords: Tandem circular cylinders, steady flow,
blockage, drag coefficient, pressure coefficient, separation
angle.
I. INTRODUCTION
Flow around a pair of tandem cylinders are found in several
engineering applications like chimney stacks, heat
exchanger tubes, offshore risers, etc. In certain cases, fluid
flow is affected by the presence of boundary surfaces in the
vicinity of the cylinders. Because of its vast range of
applications, studying the effects of various parameters on
flow characteristics of two fixed /vibrating cylinder setup is
of paramount significance.
Many studies have been carried out on this setup in the past.
Most of them deal with cylinders of identical shape and size
for identical circular cylinders in tandem, Singha and
Sinhamahapatra [1] explored the steady and unsteady flow
features for non-dimensional gap ratio ranging between 0.2
and 4. With increasing spacing they noted that drag on
upstream cylinder does not change appreciably while it
increases for downstream cylinder. In addition they
observed that surface pressure of upstream cylinder is
similar to an isolated cylinder.
A handful of studies explored the flow characteristics of a
pair of cylinders having different diameters .At Re=100-150,
Wang et al. [2] studied the effect of having a diameter of
upstream cylinder from 0.D1 to D1 while non-dimensional
gap size varying from 0.1 to 4.
In the current study, 2-dimensional numerical simulations
have been carried out to investigate the flow characteristics
of a pair of fixed circular cylinders in tandem. Low (2%) as
well as high (25%) blockages has been used for the
computations. The Navier-Stokes equations of motion were
solved using Finite Element Method. The diameter (D1) of
upstream cylinder diameter was fixed while the diameter of
downstream cylinder (D2) was varied from 0.5D1 to D1.
The centre-to-centre distance (S) was varied from 2D1 to
15D1. The effects of different diameter of downstream
cylinder, centre-to-centre distance, and block percentage on
drag coefficient, pressure coefficient and separation angle on
the two cylinders was studied.
II. METHODOLOGY
The steady, viscous, incompressible flow is governed by the
Navier-Stokes equations of motion. In coordinate-free form,
these equations are expressed as
. − . = 0 ...(1)
. = 0 ...(2)

2. The problem statement has been shown schematically in
Fig.1. The cylinders of diameters D1 and D2 reside in a
rectangular computational domain. The cylinders are
arranged in tandem and centre-to-centre spacing between the
cylinders is denoted by S. The origin of the Cartesian
coordinate system coincides with the centre of the upstream
cylinder. Free stream inlet, slip sidewalls, stress-free exit,
and no-slip conditions on velocity are used as boundary
conditions. The blockage, B is the ratio of diameter of
upstream cylinder and width of the rectangle, thus, =
1⁄ .
Fig.1 Domain
The governing equations (1) and (2) have been discretized
by using a stabilized finite-element formulation. This
formulation uses SUPG and PSPG stabilization terms to
provide oscillation-free solutions for velocity and pressure,
respectively [Tezduyar et al., 1992]. The computational
domain is discretized via bilinear quadrilateral elements.
Both velocity and pressure are located at all four nodes of an
element. Thus, equal order of interpolation is used for
velocity and pressure. The PSPG term takes care of equal
order interpolation for velocity and pressure. For B=2% and
S/D1=6, the number of nodes and elements in the mesh are
66887 and 66140, respectively.
III. RESULTS AND DISCUSSION
The pressure on both upward and downward portions of the
cylinder are identical therefore there is no lift generated on
the cylinder. The drag is mostly due to viscous drag.
Pressure coefficient and drag coefficient varies strongly as a
function of spacing between cylinders and blockage.
Flow characteristics at 2% Blockage
The drag coefficient of upstream cylinder does not change
appreciably with gap size as well as diameter ratio. The
results are in close agreement with that reported in [1]. Drag
coefficient of downstream cylinder is considerably smaller
than the first cylinder and increases with increase in centre
to centre distance as well as diameter of the rear cylinder. It
can be attributed to the variation in separation point and
suction created on the forward face of the downstream
cylinder and the rear face of upstream cylinder by the
vortices in the gap [1]. The drag is negative on rear cylinder
when S/D1=2 and D1/D2=2.
Cp on stagnation point of rear cylinder does not vary
appreciably with change in diameter ratio. It, however,
increases with increase in spacing. It is negative for S/D1≤
5. On base point, Cp always remains negative. Its magnitude
decreases with increasing gap size S/D1 up to 5, and then
starts increasing. The rate of change in value with increasing
S is higher when diameter ratio is larger.
Fig.2 Cp values for D1=D2 at S/D1=2 for varying angles
at 2% blockage
Flow Characteristics at 25% Blockage
The mean drag on the upstream cylinder increases
significantly as the blockage is ascended from 2% to 25%,
which is expected. As the gap increases, the drag
coefficients for different diameter ratios become same and
do not change with gap size.
The value of the drag coefficient of the downstream cylinder
decreases with decrease in the cylinder size.
Unlike low blockage, the Cp on the stagnation point of first
cylinder increase continuously with increasing gap size for
all diameter ratios in 25% blockage. Cp at base end of this
cylinder follow similar trend as their Cp on stagnation point
counterparts, but here the magnitude continuously decreases.
The pressure coefficient of the front end of the downstream
cylinders show the same trend for both blockages, but the
range of values is greater for 25% blockage.
Fig. 3 Cp values for D1=D2 at 25% B and S/D1=2

3. Fig.5 Cd1 values for different diameter ratios at 25%
blockage
Pressure distribution on the upstream cylinder shows steady
state behaviour. It does not change appreciably with change
in S and diameter ratio in low B. However, at 25% B its
stagnation pressure increases considerably. Cp of
downstream cylinder remains negative for at all surface for
⁄ ≤ 6 in low blockage. Cp at leading edge increases and
that of base point decreases with increasing S and B.
Fig. 6 Cd2 values for different diameter ratios at 25%
blockage
Separation Angle
Blockage of 2%
At blockage of 2% the value of separation angle for
upstream cylinder is approximately constant with maximum
value at S/D1=5 for all diameter ratios. The constant nature
can be attributed to the free slip condition and low blockage.
The average value 53.4° is found to be same with the
separation angle of an isolated cylinder 53.3° [4]
For downstream cylinder, appreciable changes are observed
with the separation angle decreasing upto S/D1=5, which is
in agreement with reported result [1]. The separation points
then move upstream with increase in spacing. Hence the
wake of the rear cylinder initially shrinks and then grows as
spacing is increased. The dip in the value of separation angle
in the initially is maximum for D1/D2=2.
Blockage of 25%
As the blockage is increased, the separation angle of
upstream cylinder is not affected by change in size of rear
cylinder also.
The separation points on the downstream cylinder move
upstream as the size of it is increased. For all diameter ratios
the values of separation angles shows slight increase with
increase in spacing.
Fig.7 Separation angles at 25% blockage for upstream
cylinder
IV.CONCLUSIONS
FEM based numerical simulation are performed for steady
flow around tandem cylinders at Re=40. Both, cylinder
spacing and diameter of downstream cylinder are varied and
flow characteristics, such as, Cd and Cp and separation angles
have been reported. At low blockage, drag coefficient of
upstream cylinder changes negligibly with varying S/D1.
For high blockage the drag coefficient of upstream cylinder
remains almost constant and becomes constant after S/D1≥
5. For downstream cylinder the variation in drag is more
pronounced. A small negative drag for the rear cylinder is
(Fig.6). For this case Cp is greater at base point than that at
stagnation point. More uniform variations in Cp values is
observed for high B case than their low blockage
counterparts. The separation angle for upstream cylinder is
not affected with spacing and size of rear cylinder at high
blockage.
REFERENCES
1. S. Singha, K. P. Sinhamahapatra, High Resolution
Numerical Simulation of Low Reynolds Number
Incompressible Flow About Two Cylinders in Tandem, J.
Fluids Eng 132(1), 011101 (Dec 15, 2009) .
2. Y. T. Wang , Z. M. Yan , H. M. Wang, Numerical
simulation of low-Reynolds number flows past two tandem
cylinders of different diameters Water Science and
Engineering , 6(4) 433-445,(2013).
3. Tezduyar, T.E., Mittal, S., Ray, S.E. and Shih, R..
Incompressible flow computations with stabilized bilinear
and linear equal order interpolation velocity– pressure
elements. Comput. Methods Appl. Mech. Engrg., 95: 221–
242(1992).
4. Zdravkovich M. M. 1997, flow around Circular
Cylinders, Volume 1: fundamentals, Oxford University
Press, New York.