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International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:13 No:03 32
I J E N S IJENS © June 2013 IJENS-IJMME-5757-304913
Numerical Investigation of Laminar Flow over
a Rotating Circular Cylinder
Ressan Faris Al-Maliky Department of Mechanical Engineering Kufa University, Iraq
Corresponding author; E-mail address: Ressan_Faris@yahoo.com
Abstract-- This is the work deals with a numerical study of
laminar flow of two - dimensional, incompressible, and steady
state over rotating circular cylinder. The solution of the flow is
presented for dimensionless rotation rate varying from (1 – 6) (in
the steps of 1) at each value of Reynolds number based on
diameter of cylinder is (200, 400, 800, and 1000). Navier – Stokes
and continuity equations were solved numerically by using finite
volume technique is conducted with FLUENT version (6.2)
package program was used in present work.
Stream lines or function and vorticity
contours and pressure, lift, and skin friction coefficients results
are presented along curve length of cylinder at each value of
rotation rate and Reynolds number. The results of lift coefficient
and stream lines and vorticity contours were compared with
other previously published research that presented support the
validity of results.
Results have shown approximately increase values of
pressure, and skin friction coefficients with increasing of rotation
rate at known Reynolds number.
Index Term-- Rotating cylinder, laminar flow, skin friction
pressure lift coefficients
I. INTRODUCTION
In (CFD) computational fluid dynamics, laminar
flow past a rotating cylinder is interesting problem; it's
applications in many fields such as rockets, projectiles,
aeronautics, and marine ships.
The pressure gradient can be explained simply by
Bernoulli's principle, in which pressure and velocity are
inversely proportional. The phenomena of a rotating
cylinder's lift is know as the Magnus effect, named after a
19th century German engineer, and is related to the
circulation around an a flow field. (Rayleigh) studied the lift
of a rotating cylinder for an inviscid (frictionless) fluid, and
related lift to the circulation of a rotating cylinder by the
following formula:
L = ρ.U∞.Γ in which the circulation, Γ is given by:
Γ = 2.π.ω.R2 therefore,
L = ρ.U∞.(2.π.ω.R2)
The relationship between lift and circulation is
known as Kutta – Joukowsky relationship and applies to all
shapes, particularly to the aerodynamic shapes such as an
airplane wing.
In a laminar fluid, like air, the cylinder is subjected
to both pressure and viscous forces, and the explanation is
more complex. Studies (Smith, 1979) indicate that the
circulation does not result from the common explanation of
the air set into an opposing rotation by the friction of a no
slip wall, as this only occurs in a very thin boundary layer
next to the surface. But this motion of the fluid in the
boundary layer does affect the manner in which the flow
separates from the cylinder. Boundary layer separation is
moved back on the side of the cylinder that is moving with
the fluid, and is moved forward on the side opposing the
main stream. The wake then shifts to the side moving
against the main stream causing the flow to be deflected on
that side, and the resulting change in free stream flow
creates a force on the spinning cylinder [1].
In the present work, the asymmetrical flow was
considered, a laminar fluid which is generated by rotating a
circular cylinder in a uniform stream of fluid. There are two
basic parameters in the problem, namely, first the Reynolds
number based on the diameter of cylinder, second rotation
rate, which is a dimensionless measure of the rotation rate.
When q = 0, the motion is symmetrical about the direction
of translation and this situation has previously received a
considerable amount of attention [2].
(Watson, 1995) pointed out that the pressure field
given by Smith's asymptotic form is not single-valued and
proposed that an additional term to Jeffery's Fourier series is
necessary. However, he did not derive the force, since the
outer flow which is governed by the Navier-Stokes
equations was not obtained. The problem of flow past
rotating cylinders was considered by (Sennitskii, 1973). The
problem was studied using a boundary layer approach for
the case of a large distance between the centers of cylinders.
In the work of (Sennitskii, 1975) the first terms of an
asymptotic expansion by inverse degree of the Reynolds
number were obtained [3].
(Ingham, 1983) obtained numerical solutions of the
two-dimensional steady incompressible Navier–Stokes
equations in terms of vorticity and stream function using
finite differences for flow past a rotating circular cylinder
for Reynolds numbers Re = 5 and 20 and dimensionless
rotation rate velocity q between 0 and 0.5. Solving the same
form of the governing equations, but expanding the range
for q, (Ingham & Tang, 1990) showed numerical results for
Re = 5 and 20 and 0 ≤ q ≤ 3. With a substantial increase in
Re, (Badr et al., 1990) studied the unsteady two-dimensional
flow past a circular cylinder which translates and rotates
starting impulsively from rest both numerically and
experimentally for 103 ≤ Re ≤ 104 and 0.5 ≤ q ≤ 3. They
solved the unsteady equations of motion in terms of
vorticity and stream function. The agreement Numerical
study of the steady-state uniform flow past a rotating
cylinder 193 between numerical and experimental results
was good except for the highest rotational velocity where
they observed three-dimensional and turbulence effects.
Choosing a moderate interval for Re, (Tang & Ingham
1991) followed with numerical solutions of the steady two-
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:13 No:03 33
I J E N S IJENS © June 2013 IJENS-IJMME-5757-304913
dimensional incompressible equations of motion for Re = 60
and 100 and 0 ≤ q ≤ 1.
They employed a scheme that avoids the
difficulties regarding the boundary conditions far from the
cylinder. Considering a moderate constant Re = 100, (Chew,
Cheng & Luo, 1995) further expanded the interval for the
dimensionless rotation rate q, such that 0 ≤ q ≤ 6. They used
a vorticity stream function formulation of the
incompressible Navier– Stokes equations. The numerical
method consisted of a hybrid vortex scheme, where the time
integration is split into two fractional steps, namely, pure
diffusion and convection. They separated the domain into
two regions: the region close to the cylinder where viscous
effects are important and the outer region where viscous
effects are neglected and potential flow is assumed. Using
the expression for the boundary-layer thickness for flow past
a flat plate, they estimated the thickness of the inner region.
Their results indicated a critical value for q about 2 where
vortex shedding ceases and the lift and the drag coefficients
tend to asymptotic values. (Nair, Sengupta & Chauhan,
1998) expanded their choices for the Reynolds number by
selecting a moderate Re = 200 with q = 0.5 and 1 and two
relatively high values of Re = 1000 and Re = 3800, with q =
3 and q = 2, respectively. They performed the numerical
study of flow past a translating and rotating circular cylinder
solving the two-dimensional unsteady Navier–Stokes
equations in terms of vorticity and stream function using a
third-order upwind scheme.
(Kang, Choi & Lee, 1999) followed with the
numerical solution of the unsteady governing equations in
the primitive variables velocity and pressure for flows with
Re = 60, 100 and 160 with 0 ≤ q ≤ 2.5. Their results showed
that vortex shedding vanishes when q increases beyond a
critical value which follows a logarithmic dependence on
the Reynolds number (e.g., the critical dimensionless
rotation rate q = 1.9 for Re = 160).
(Chou, 2000) worked in the area of high Reynolds
numbers by presenting a numerical study that included
computations falling into two categories: q ≤ 3 with Re =
103 and q ≤ 2 with Re = 104. Chou solved the unsteady two
dimensional incompressible Navier–Stokes equations
written in terms of vorticity and stream function. In contrast,
the work of (Mittal & Kumar, 2003) performed a
comprehensive numerical investigation by fixing a moderate
value of Re = 200 while considering a wide interval for the
dimensionless rotation rate of 0 ≤ q ≤ 5. They used the
finite-element method to solve the unsteady incompressible
Navier–Stokes equations in two-dimensions for the
primitive variables velocity and pressure [4].
(Dennis, 22) investigated the steady asymmetrical
flow past an elliptical cylinder using the method of series
truncation to solve the Navier-Stokes equations with the
Oseen approximation throughout the flow. He found that by
considering the asymptotic nature of the decay of vorticity
at large distances that for asymmetrical flows it is not
sufficient merely that the vorticity shall vanish far from the
cylinder but it must decay rapidly enough [2].
(Kang et al., 1999) pointed out, the simulations
may be started with arbitrary initial conditions. They
performed a numerical study with different initial
conditions, including the impulsive start-up, for Re = 100
and q = 1.0 and the same fully developed response of the
flow motion was eventually reached in all cases [4].
II. GOVERNING EQUATION and BOUNDARY
CONDITIONS
The applied system consists of a two dimensional
infinite long circular cylinder Fig. (1), having diameter D
and is rotating in a counter – clockwise direction with a
constant angular velocity ω. It is exposed to a constant free
stream velocity of U∞ at the inlet.
The governing partial differential equations are the
form of continuity and Navier–Stokes or momentum
equations in two dimensions for the incompressible, steady
state, and laminar flow around a rotating circular cylinder
[5] as below:
Continuity equation:
0y
v
x
u
…(1)
x - momentum equation:
2
2
2
2
y
u
x
uν
x
P
ρ
1
y
uv
x
uu …(2)
y - momentum equation:
2
2
2
2
y
v
x
v
y
P
ρ
1
y
vv
x
vu …(3)
The boundary conditions for the flow across a
rotating circular cylinder see Fig. (1), can be written as:
at the inlet boundary: u = U∞, v = 0 at the exit boundary: p = 0
On the surface of the cylinder: u = -ω×D×sin(θ)/2,
v = -ω×D×cos(θ)/2, where 0° ≤ θ ≤ 360°.
The boundary conditions on the surface of the
cylinder can be implemented by considering wall motion:
moving wall and motion: rotational for a particular
rotational rate in FLUENT.
The above governing equations (1, 2, & 3) when
solved using the above boundary conditions yield the
primitive variables, i.e., velocity u,v, and pressure p are
calculated numerically.
III. AERODYNAMICS CHARACTERISTICS
To describe the problems must be define Reynolds
number as
μ
Dρ.URe …(4)
and dimensionless rotation rate:
2U
ω.Dq …(5)
Three relevant parameters computed from the
velocity and pressure fields are the pressure, skin friction,
and lift coefficients, which represent dimensionless
expressions of the forces that the fluid produces on the
circular cylinder, these are defined, respectively, as follows
[4]:
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I J E N S IJENS © June 2013 IJENS-IJMME-5757-304913
2p
ρ.U2
1
p-pC
…(6)
2f
ρ.U2
1
τC
…(7)
2l
ρ.U2
1
LC
…(8)
Where pressure forces act normal to the surface of
the cylinder and the shear stress acts tangential to the
surface of the cylinder.
IV. RESULTS and DISCUSSION
The numerical solution of the governing system of
partial differential equations is carried out through the
computational fluid dynamics package FLUENT version
(6.2). This computer program applies a control-volume
method to integrate the equations of motion, constructing a
set of discrete algebraic equations with conservative
properties. The segregated numerical scheme, which solves
the discretized governing equations sequentially [4].
The computational grid for the problem under
consideration is generated by using a commercial grid
generator GAMBIT and the numerical calculations are
performed in the full computational domain using FLUENT
program for varying conditions of Reynolds number and
rotation rate. In particular, the O-type shown in Fig. (2), grid
structure is created here and it consists of non-uniform
quadrilateral elements 34600 having a total of 35030 nodes
or grid points in the full computational domain. The grid
near the surface of the cylinder is sufficiently fine to resolve
the boundary layer around the cylinder [5].
The results of lift, skin friction, and pressure
coefficients have been represented from FLUENT (6.2) at
each values of Re of range (200, 400, 800, and 1000) and q
varying from 1 – 6 in the steps of 1 with angle in polar
coordinate (angular direction), as shown in Fig. (1), θ in
degrees units. Also, lines of stream function at same Re & q
ranges. While validations of the results of lift coefficient are
compared with other numerical results as shown in Tab. (1)
and give a good approach and convergence.
Lift coefficient of rotating cylinder with q at each
values of Re is represented in Tab. (2) shows Cl is increase
in negative direction, if change direction of rotating to
clockwise then lift coefficient is positive value, observe in
Tab. (2), note at each value of Re (or in each column Re is
constant) wherever increase rotation rate, will increase lift
coefficient clearly, but when compare between first column
and second until fourth each value of lift coefficient
approximately convergent or similar for same rotation rate
i.e., effect of Re is not significant on lift coefficient, in
additional to lift coefficient is greater than at no-rotate a
absolutely according to Magnus effect.
Fig. (3), shows the streamline patterns for the
various pairs of Re and q, the rotation of the cylinder is
counterclockwise while the streaming flow is from left to
right considered in this investigation. Notice that the
stagnation point lies above the cylinder, in the region where
the direction of the free stream opposes the motion induced
by the rotating cylinder. As the dimensionless rotation rate
at the surface of the cylinder increases, for a fixed Re, the
region of close streamlines around the cylinder extends far
from the wall and, as a consequence, the stagnation point
moves upwards. For the lowest q = 3, the region of close
streamlines becomes narrow and the stagnation point lies
near the upper surface of the cylinder.
The contours of positive and negative vorticity are
presented in Fig. (5), the positive vorticity is generated
mostly in the lower half of the surface of the cylinder while
the negative vorticity is generated mostly in the upper half.
For the dimensionless rotation rates of q = 3 and 4, a zone of
relatively high vorticity stretches out beyond the region
neighboring the rotating cylinder for 0° ≤ θ ≤ 90°,
resembling "tongues" of vorticity. Increasing q, the rotating
cylinder drags the vorticity so the ‘tongues’ disappear and
the contours of positive and negative vorticity appear
wrapped around each other within a narrow region close to
the surface. Based on the velocity and pressure fields
obtained from the simulations for the various Re and q
considered [4].
While Figs. (3, 5) shown stream lines and vorticity
contours are represented to purpose of comparison with
Figs. (4, 6) in other numerical results (J. C. Padrino, et al) in
same conditions of the flow in which give same behavior
and convergence in shape approximately excepting some
small differences due to different in number of mesh nodes,
iteration loop to arrive convergence values, and levels of
contour. In additional to lift coefficient is greater than it's
value at no-rotating cylinder.
Figs. (7, 8, 9, and 10) are shown pressure
coefficients of flow past rotating cylinder along curve length
or angle from front as in Fig. (1) at various values of Re, q,
in which observe wherever increasing rotation rate, pressure
coefficient will increase without looking to Reynolds
number.
Each curve differ than one to other while maximum
values of pressure coefficient in four figures are -43, -40.7, -
34.8, and -34.6 appear at (225 – 240) degree & q = 6, i.e.,
cylinder front approximately at Reynolds number are: 200,
400, 800, and 1000 respectively. Maximum value of
pressure coefficient are convergent approximately, this lead
to vary in Re from 200 to 1000 has small effect or not
significant on pressure coefficient in same time value of
rotation rate change from 1 – 6.
Further published researches don’t deal with skin
friction of flow past rotating cylinder and don’t meet paper
discuss prediction to estimate skin friction numerically past
rotating cylinder just past stationary cylinder such as (E.
Achenbach, 1968) [7] investigated skin friction at 0° ≤ θ ≤
360° over stationary cylinder at 6×104 ˂ Re ˂ 5×106 with
smooth surface experimentally and defined three states of
the flow: the subcritical, critical, and supercritical then
specified separation angle in each region.
In this present Figs. (11, 12, 13, and 14) shown
skin friction coefficient along curve length or angle from
point on cylinder surface to origin in Fig. (1) at known
Reynolds number & rotation rate, observe in these figures
mostly (not along curve length) increase values of skin
friction with increasing of rotation rate of each case i.e., Re
is known and constant except at Re = 200, skin friction at q
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I J E N S IJENS © June 2013 IJENS-IJMME-5757-304913
= 5 has maximum value greater than it's at q = 6 happen at
point (15°, 1.156) i.e., at back of cylinder, while another
figure it's maximum values at point: (315°, 1.112), (30°,
0.667), and (30°, 0.596) at q = 6, Re = 400, 800, and 1000
respectively.
Commonly each curve of skin friction has five
vertex or apex in various location, as behavior of curve half
is repeat after θ = 180°, but reversely in shape and differ in
values of another half i.e., shape of skin friction coefficient
curve in upper surface same as lower surface.
V. CONCLUSIONS
1. comparison of the numerical results with other
published researches for lift coefficient and stream
lines and vorticity contours therefore give good
agreement.
2. increase lift coefficient with increasing rotation rate
and effect of Reynolds number is not significant or
small change on lift coefficient.
3. increasing rotation rate, pressure coefficient will
increase without look to Reynolds number i.e.,
variation in Reynolds number from 200 to 1000 has
small effect or not significant on pressure
coefficient in same time value of rotation rate
change from 1 – 6.
4. mostly increase values of skin friction coefficient
with increasing of rotation rate at Reynolds number
is equal 400, 800, and 1000 except at Re = 200.
5. in each case skin friction coefficient has five vertex
in various location, as behavior of curve half is
repeat after θ = 180°, but shape of skin friction
coefficient in upper surface same as lower surface
VI. SCOPE of the STUDY and LIMITATIONS
Rotating circular cylinder application play important rule in
missiles and projectiles where it's rotation add lift force as
well as original lift force when existence cross wind that
lead to increase in range, stability, and performance just as
decrease drag force in aerodynamics fields. In automotive
design, good aerodynamic consideration aims for the least
drag to achieve efficiency, and also to optimize negative lift
particularly in motor sport. Similarly such effort has been
proven to tremendously save the fuel cost in the aviation
industry.
Other engineering applications of cylinder like
structures such as air flow past a group of buildings or
bundle of pipes in a chemical plant, where require reduce
wind force on their side.
In heat transfer, coolant flow past tubes in a heat
exchanger, sea water flow past columns of a marine
structure, twin chimney stacks.
The flow around a rotating cylinder involves
complex transport phenomenon because of many factors
such as the effect of cylinder rotation on the production of
lift force and moment. There are two parameters that
influence this flow problem: The Reynolds number, and the
rotation rate of the cylinder is non-dimensionalized
quantities, the first Reynolds No. is limited flow model i.e.,
laminar or turbulent, while the second represented relative
velocity of uniform flow and rotational cylinder.
In this paper low Reynolds No. is considered to
generate laminar, steady, incompressible flow, no slip
without average roughness surface of cylinder. All these
factors are limited this work and any cases outside this field
are not satisfying assumptions of model.
REFERENCES [1] John Middendorf, "CFD Modeling of Wind Tunnel Flow over
Rotating Cylinder", Computation Fluid Dynamics, Professors Tracie
Barber/Eddie Leonardi, May 30, 2003. [2] D. B, Ingham and T. Tang, "A Numerical Investigation into the
Steady Flow Past a Rotating Circular Cylinder at Low and
Intermediate Reynolds Numbers", Reprinted from Journal Of Computational Physics Vol. 87, No.1, New York and London, March
1990.
[3] Surattana Sungnul and Nikolay Moshkin, "Numerical Simulation of Steady Viscous Flow past Two Rotating Circular Cylinders",
Suranaree J. Sci. Technol. 13(3):219-233, May 30, 2006.
[4] J. C. Padrino and D. D. Joseph, "Numerical study of the steady-state uniform flow past a rotating cylinder", J. Fluid Mech. (2006), Vol.
557, pp. 191–223, 2006 Cambridge University Press.
[5] Varun Sharma and Amit Kumar Dhiman, "Heat Transfer from a Rotating Circular Cylinder in the Steady Regime: Effects of Prandtl
Number", Indian Institute of Technology Roorkee, Roorkee – 247
667, dhimuamit@rediffmail.com, India.
[6] Sanjay Mittal, S. & Bhaskar Kumar, "Flow Past a Rotating Cylinder",
J. Fluid Mech. Vol. 476, 303–334, Cambridge University Press,
United Kingdom, 2003. [7] E. Achenbach, "Distribution of Local Pressure and Skin Friction
around a Circular Cylinder in Cross-Flow up to Re = 5×106", J. Fluid Mech., Vol.34, pp.625-639, 1968.
NOMENCLATURE
Latin
symbols Description
L Lift force (N)
q rotation rate
Re Reynolds number
r radius in polar coordinate
U∞ free velocity of fluid (m/s)
p∞ pressure as the radial coordinate r goes to
infinity (N/m2)
p local pressure (N/m2)
Cp pressure coefficient
Cf skin friction coefficient
D diameter of the cylinder
R Radius of the cylinder
Cl lift coefficient
Uco free velocity of air (m/s)
u, v velocity component in x, y – direction
respectively (m/s)
x Cartesian coordinate in horizontal direction
(m)
y Cartesian coordinate in vertical direction (m)
Greek
symbols Description
τ shear stress (N/m2)
α angle of attack
Γ circulation (m2/s)
ρ density of the air (kg/m3)
θ angle in polar coordinate (degree)
ω angular velocity (rad/s)
µ dynamic viscosity of the fluid (kg/m.s)
v kinematic viscosity of the fluid (m2/s)
symbols Abbreviations
CFD Computational Fluid Dynamic
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:13 No:03 36
I J E N S IJENS © June 2013 IJENS-IJMME-5757-304913
Fig. 1. laminar flow over rotating cylinder has diameter is D
Table I
Numerical lift coefficient of the steady state laminar flow past a rotating cylinder
No. Re q Present study J. C. Padrino & D. D.
Joseph [4]
Mittal & Kumar [6]
1. 200 3 -10.278 -10.34 -10.366
2. 200 4 -17.43 -17.582 -17.598
3. 200 5 -27.14 -27.0287 -27.055
4. 400 4 -17.388 -18.0567 ---
5. 400 5 -27.635 -27.0112 ---
6. 400 6 -31.018 -33.7691 ---
7. 1000 3 -9.914 -10.6005 ---
Fig. 2. the O-type grid structure mesh
θ x
y
r ω
U∞
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Table II lift coefficient with rotation rate & Reynolds No.
lift coefficient Cl
q Re = 200 Re = 400 Re = 800 Re = 1000
1 -2.217 -2.014 -1.798 -1.77
2 -5.406 -5.254 -4.896 -5.174
3 -10.278 -10.231 -10.051 -9.914
4 -17.43 -17.388 -16.86 -16.452
5 -27.14 -27.635 -22.5 -23.463
6 -32.594 -31.081 -30.522 -26.333
Fig. 3. Stream lines for various pairs of Re and q [4].
q q
q q
q q
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:13 No:03 38
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(a) Re = 200, q = 4 (b) Re = 200, q = 5
(c) Re = 400, q = 4 (d) Re = 400, q = 5
(e) Re = 400, q = 6 (f) Re = 1000, q = 3
Fig. 4. Stream lines for various pairs of Re and q for present study.
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I J E N S IJENS © June 2013 IJENS-IJMME-5757-304913
Fig. 5. Vorticity contours for various pairs of Re and q. The negative vorticity is shown as dashed lines. The rotation of the cylinder is counterclockwise
while the streaming flow is from left to right [4].
(a) Re = 200, q = 4 (b) Re = 200, q = 5
q q
q q
q q
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(c) Re = 400, q = 4 (d) Re = 400, q = 5
(e) Re = 400, q = 6 (f) Re = 1000, q = 3
Fig. 6. Vorticity contours for various pairs of Re and q for present study
Re = 200
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
5
0 45 90 135 180 225 270 315 360
Cp
q = 1
q = 2
q = 3
q = 4
q = 5
q = 6
θ
Fig. 7. Pressure coefficient vs. angular position at Re = 200
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I J E N S IJENS © June 2013 IJENS-IJMME-5757-304913
Re = 400
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
5
0 45 90 135 180 225 270 315 360q
Cp q = 1
q = 2
q = 3
q = 4
q = 5
q = 6
Fig. 8. pressure coefficient vs. angular position at Re = 400
Re = 800
-40
-35
-30
-25
-20
-15
-10
-5
0
5
0 45 90 135 180 225 270 315 360q
Cp q = 1
q = 2
q = 3
q = 4
q = 5
q = 6
Fig. 9. pressure coefficient vs. angular position at Re = 800
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I J E N S IJENS © June 2013 IJENS-IJMME-5757-304913
Re = 1000
-40
-35
-30
-25
-20
-15
-10
-5
0
5
0 45 90 135 180 225 270 315 360
q
Cp q = 1
q = 2
q = 3
q = 4
q = 5
q = 6
Fig. 10. Pressure coefficient vs. angular position at Re = 1000
Re = 200
0
0.2
0.4
0.6
0.8
1
1.2
0 45 90 135 180 225 270 315 360
q
Cf
q=1
q = 2
q = 3
q = 4
q = 5
q = 6
Fig. 11. Skin friction coefficient vs. angular position at Re = 200
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Re = 400
0
0.2
0.4
0.6
0.8
1
1.2
0 45 90 135 180 225 270 315 360q
Cf
q=1
q = 2
q = 3
q = 4
q = 5
q = 6
Fig. 12. Skin friction coefficient vs. angular position at Re = 400
Re = 800
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 45 90 135 180 225 270 315 360q
Cf
q=1
q = 2
q = 3
q = 4
q = 5
q = 6
Fig. 13. Skin friction coefficient vs. angular position at Re = 800
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Re = 1000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 45 90 135 180 225 270 315 360q
Cf
q = 1
q = 2
q = 3
q = 4
q = 5
q = 6
Fig. 14. Skin friction coefficient vs. angular position at Re = 1000
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