numerical investigation of laminar flow over a … · laminar flow of two - moved back on the side...

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: [email protected]

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-

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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]:



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



= 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, [email protected], 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



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∞



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



(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.



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



(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



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



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



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


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




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


numerical investigation of laminar flow over a … · laminar flow of two - moved back on the side...

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