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Procedia Engineering 34 ( 2012 ) 110 – 115

1877-7058 © 2012 Published by Elsevier Ltd.

doi: 10.1016/j.proeng.2012.04.020

9

th

Conference of the International Sports Engineering Association (ISEA)

Analysis of Flow around a Flying Pipe

Katsuya Hirata

a

*, Yosuke Kida

a

, Hirochika Tanigawa

b

, Jiro Funaki

a

a

Department of Mechanical Engineering, Doshisha University, Kyoto 610-0321, Japan

b

Department of Mechanical Engineering, Maizuru National College of Technology, Maizuru 625-8511, Japan

Accepted 02 March 2012

Abstract

The present aim is to reveal the flow past a rotating pipe which is immersed parallel to the mainstream. At first, we

conduct field observations of a flying pipe using a pair of high-speed video cameras, together with motion analyses

based on their recorded images, which quantitatively reveal both paths and angular velocities of the flying pipe. In

addition, we conduct numerical simulations by a finite difference method, whose results suggest that the pipe-rotation

effect becomes remarkable for a rotation parameter

:

! 0.4.

© 2012 Published by Elsevier Ltd.

Keywords: Projectile sports; aerodynamics; flying disc; motion analysis; numerical simulation

1. Introduction

The present study concerns the aerodynamics of a flying pipe in rotation, with which we might be

familiar as “X-zylo [1].” Some flying pipes could fly faster than such a similar projectile as a flying disc

or “Frisbee,” and farther than a football. The flying disc is very popular not only as a toy, but also as an

instrument of such games as Ultimate, Disc golf, Guts, Crosbee, 500, Disc dog, Flying disc freestyle,

Fricket, Double disc court, Friskee, Durango boot, Flutterguts, Kan-jam and so on. At present, games

using the flying pipe have not been established well, and it’s only application is for a toy. However, the

flying pipe could have the potential for a projectile-sports instrument, as well as the flying disc.

From an aerodynamic point of view, both the flying pipe and the flying disc are regarded as bluff

bodies, namely, non-streamline-shaped bodies. However, the flow past such a bluff body at higher

Reynolds numbers has been important at various practical aspects in aeronautical and mechanical

*Katsuya Hirata. Tel.: +81-774-656461; fax: +81-774-656830.

E-mail address: khirata@mail.doshisha.ac.jp.

Available online at www.sciencedirect.com

111

Katsuya Hirata et al. / Procedia Engineering 34 ( 2012 ) 110 – 115

engineering fields as well as sport fields, it is one of rather recent topics in the long history of fluid

mechanics. Among bluff-body problems, there have been less research concerning three-dimensional

ones. As three-dimensional basic studies, it seems appropriate to consider axisymmetric bodies with

simple geometries like a sphere and a disc, whose knowledge are required in the analyses of many flying

or suspended objects in fluid. But, even the flow past such a simple axisymmetric body has not been

revealed enough in comparison with such a two-dimensional body as a circular cylinder, despite of wide

ranges of it’s applicabilities. A pipe or a tube is another simple axisymmetric bluff body. And, researches

on the flow past a pipe have been still less active than a sphere or a disc, although they are useful in many

industrial fields, such as the designs for combustors, ventilator nacelles, screw casings, streamers and

flow meters, as well as flying toys. Although there have existed few researches on the flow past a pipe,

we can find several researches on the flow past a ring, a torus or a washer, which is not in rotation but

stationary [2] í [8]. Recently, the flow has attracted our attention in the context of a new-concept wind-

mill design [9].

The present aim is to reveal the flow past a rotating pipe which is immersed parallel to the mainstream. At

first, we conduct field observations of a flying pipe in rotation using a pair of high-speed video cameras,

together with motion analyses based on their recorded images, which quantitatively show both paths and

angular velocities of the flying pipe. In addition, we conduct numerical simulations by a finite difference

method based on the MAC scheme.

2. Method

2.1. Model and Parameters

Figure 1a shows the present flying-pipe model: namely, a rotating pipe in uniform flow. In addition,

Fig. 1a also shows the present coordinate system, which is a cylindrical one (r,

T

, z) with its origin O at

the front centre.

Governing geometric parameters in non-dimensional forms are a reduced pipe’s thickness t/d and a

reduced pipe’s length l/d, where the model’s dimensions d, l and t denote the mean diameter, the length

and the thickness of the pipe, respectively. t/d and l/d are defined as follows.

and

where d

o

and d

i

denote the outer and inner diameters of the model, respectively.

A governing kinetic parameters in non-dimensional forms are the Reynolds number Re and a rotating

parameter

:

. Re is defined by

where U

and

Q

denote the mean flow velocity of a uniform mainstream and the kinetic viscosity of

fluid, respectively. If we regard d instead of t as a characteristic length scale, we define another Reynolds

number Re(d) based on d, given by

:

is defined by

.

,

,

,

(1)

io

io

dd

dd

d

t

io

2

dd

l

d

l

(3)

Q

tU

Re

f

(2)

(5)

f

U/

d

::

2

o

(4)

Q

dU

dRe

f

)(

112 Katsuya Hirata et al. / Procedia Engineering 34 ( 2012 ) 110 – 115

where

:

denotes the angular velocity of pipe’s rotation. Then,

:

represents the ratio of pipe’s rotating

velocity at the outer diameter to flow velocity of the mainstream.

The base pressure p

b

measured at the centre of pipe’s back face and the predominant frequency f of p

b

are non-dimensionalised as a base suction coefficient

C

pb

and the Strouhal number St, respectively. Their

definitions are as follows.

and

where p

and

U

denote the static pressure in the mainstream and the density of fluid, respectively.

2.2. Field Observation

Figure 1b shows the present experimental apparatus for field observation. The motion of a model (No.

1 in the figure), which is thrown by a player, is recorded by a pair of high-speed video cameras (No. 2).

Two cameras are synchronised with each other by a trigger-pulse generator (No. 3). Two personal

computers (No. 4) are connected to the cameras by IEEE1394, in order to initialise/monitor the cameras

and storage/analyse the recorded data. For calibration of the present stereo system, we use four colour

corns (No. 5).

2.3.

Numerical Analysis

In many actual situations, most of the flow at Re < 10

6

could be usually regarded as incompressible

and viscous. So, we consider the incompressible full Navier-Stokes equations for the present numerical

analyses. We approximately solve the equations using the MAC method in a finite-difference scheme, a

third-order-upwind difference method in spatial discretisation of convective terms, a second-order-central

difference method in spatial discretisation of the other terms, and the Euler explicit method in a time

marching.

As a spatial grid, we use a regular cylindrical grid with unequal spacing, as shown in Fig. 3. The grid

numbers in the r,

T

and z directions are 150, 42 and 105, respectively. The minimum grid size

'

r

min

,

'T

and

'

z

min

are 0.14t, 0.05S rad and 0.20t, respectively. Computational-domain sizes in the r and z

directions are 58t and 26t, respectively. The former is equal to 10d/2, and the latter is equal to 22l. Such

computational parameters as the grid and computational-domain sizes are determined by many

preliminary trials, to achieve negligible influences upon results.

O

r

T

U

l

d

i

d

d

o

z

r

O

v

T

3

2

1

5

4

Fig. 1.(a) Flying-pipe model: a rotating pipe in uniform flow, together with coordinates; (b) Experimental apparatus for field

observation. 1, Model; 2, high-speed video cameras; 3, trigger-pulse generator; 4, PC’s; 5, colour corns for calibration

,

(6)

(7)

f

U

ft

St

2

b

b

2

1

f

f

U

pp

C

p

U

113

Katsuya Hirata et al. / Procedia Engineering 34 ( 2012 ) 110 – 115

The boundary conditions on the pipe’s surfaces are viscid. On the outer boundaries of the

computational domain, we suppose the Dirichlet condition; that is, v

r

= 0, v

T

= 0 and v

z

= U

.

At a time step

'

t = 1.0×10

-4

t/ U

, we proceed with these time-manching computations. During the

computations, we monitor the value of

C

pb

, to judge whether the total computation time is enough or not

for fully-saturated conditions.

3. Results and discussion

3.1. Field observation

Figure 2a shows an example of field observations; namely, an instantaneous image obtained by a high-

speed video camera. The model flies from the right to the left. We can see that an orbit of the model is

approximated to be rather a horizontal straight line, not an obvious parabolic curve.

Table 1 summarises typical dimensions and observed data of the model, together with non-dimensional

geometric and kinetic parameters for the field observation. The non-dimensional geometric parameters

are as follows: t/d = 0.01 and l/d = 0.5. And, the obtained non-dimensional kinetic parameters are as

follows: Re

| 500 (Re(d) | 50000) and

:

| 0.15.

3.2. Numerical simulation

Table 2 summarises non-dimensional geometric and kinetic parameters of the model for numerical

analysis. Re is fixed to 1.0×10

2

, and the tested range of the the rotation parameter

:

is 0 í 1.2. We

should note that the model is not in rotation for

:

= 0.

Figure 3a shows time histories of the base suction coefficient

C

pb

for several values of

:

. At first, we

see the result for

:

= 0.0 (in non-rotational motion). At the time enough after the start-up of computation

(at

W

U

/t t 100),

C

pb

becomes periodic being independent of initial conditions. As will be shown in flow

visualisation such as Fig. 5, this periodicity is related with an alternate shedding of ring-like vortices from

the model’s inside and outside. Next, we see the results for

:

z 0.0 (in rotational motion). We can

confirm the same periodicity as that for

:

= 0.0. However, we can also confirm some effects of

:

from

a quantitative point of view: that is, the time histories are not identical one another for different values of

:

. So, we discuss the

:

effects, next.

We examine some

:

effects in Figs. 3b, 4a and 4b. At first, Fig. 3b shows a time-mean base suction

coefficient (

C

pb

)

mean

versus

:

. We can see that (

C

pb

)

mean

monotonically increases with increasing

:

.

More specifically, for

:

Ń

! 0.4, (

C

pb

)

mean

becomes obviously larger than that for

:

= 0.0. For reference,

Fig. 3b also shows a formula 0.274 + (

:

)

2

by a dashed line, and another formula 0.274 + (C

BS

:

)

2

with

an empirical coefficient C

BS

= 0.15 by chained line. In the former, we consider resultant velocity of the

mainstream and the model’s rotation. Much larger (

C

pb

)

mean

by this theory suggests a nonlinearity of the

flow. The latter is a modified one using C

BS

.

Model

r

O

T

r

z

O

26t (= 22l)

58t (= 10 )

d

2

Orbit

Model

Fig. 2. (a) Computational grid; (b) An example of field observation; an instantaneous image by a high-speed video camera. A model

flies from the right to the left

114 Katsuya Hirata et al. / Procedia Engineering 34 ( 2012 ) 110 – 115

Table 2 Model’s geometric/kinetic parameters

for numerical analysis.

Reduced thickness t/d 0.16

Reduced length l/d 0.2

Reynolds number Re 100

(Reynolds number Re(d) based on d) (600)

Rotation parameter 0 1.2

(Pipe-edge-velocity ratio)

:

*

Table 1 Model’s dimensions and geometric/kinetic

parameters for field observation.

Diameter d 0.1 m

Thickness t 0.001 m

Length l 0.05 m

Reduced thickness t/d 0.01

Reduced length l/d 0.5

Translation distance per a rotation 1.8 m

Angular velocity 30 rad/s

Translation speed 8.0 m/s

(30 km/h)

Reynolds number Re 500

(Reynolds number Re(d) based on d) (50000)

Rotation parameter 0.15

(Pipe-edge-velocity ratio)

:

U

d

2

:

=

:

/U

*

Second, Fig. 4a shows a non-dimensional base-suction amplitude (

C

pb

)

max

(

C

pb

)

min

versus

:

. We

can see that (

C

pb

)

max

(

C

pb

)

min

tends to keep a constant of 0.08, being independent of

:

, in contrast to

Fig. 3b.

Thirdly, Fig. 4b shows the Strouhal number St versus

:

. As well as Fig. 3b, St monotonically

increases with increasing

:

. More specifically, for

:

Ń

! 0.4, St becomes obviously larger than that for

:

= 0.0. For reference, Fig. 4b also shows a formula 0.113(1 + (

:

)

2

) by a dashed line, and another formula

0.113(1 + (C

ST

:

)

2

) with an empirical coefficient C

ST

= 0.1 by a chained line. Both the former and the

latter are defined as well as Fig. 3b. Again, much larger St by the former suggests a nonlinearity of the

flow.

0

0.2

0.4

0.6

050

150 200

rev0.0

rev0.1

rev0.3

rev0.6

rev0.9

rev1.2

:

= 0.0

:

= 0.1

:

= 0.3

:

= 0.6

:

= 0.9

:

= 1.2

W

U

/t

0 50 150 200

0.6

0.4

0.2

0

C

pb

0.26

0.28

0.3

0.32

0 0.2 0.4 0.6 0.8 1 1.2

(C

pb

)

mean

:

0.274+

(C

:

)

2

*

BS

0.274+

(

:

)

2

*

Fig. 3. (a) Time history of base suction for Re = 100; (b) Time-mean base suction versus rotation parameter

:

for Re = 100

:

0.02

0.04

0.06

0.08

0.1

0 0.4 0.8 1.2

(C

pb

)

max

(C

pb

)

min

0.112

0.113

0.114

0.115

0.116

0 0.2 0.4 0.6 0.8 1 1.2

St

:

0. 113(1+

(C

:

)

2

)

*

ST

0. 113(1+

(

:

)

2

)

*

Fig. 4. (a) Base-suction amplitude versus rotation parameter

:

for Re = 100; (b) number versus rotation parameter

:

for Re =

100. ) Strouhal

Finally, Fig. 5 shows typical visualised flows and reveal velocity vectors and pressure distribution at

:

= 0.0, 0.4, 0.8 and 1.2, respectively. We can see that all those flows are characterised by alternative

Table 1. Model’s dimensions and observed data, together

with geometric and kinetic parameters for field observation

Table 2. Model’s geometric and kinetic parameters for

numerical analysis.

115

Katsuya Hirata et al. / Procedia Engineering 34 ( 2012 ) 110 – 115

shedding of ring-like vortices from the model’s inside and outside, and that they are very similar one

another despite of different values of

:

C

p

0.7

-1.05

C

p

0.7

-1.05

C

p

0.7

-1.05

C

p

0.7

-1.05

Fig. 5. Velocity vectors and pressure distribution, (a) for Re = 100 and

:

= 0.0 at

W

U

/t = 151, (b) for Re = 100 and

:

= 0.4 at

W

U

/t = 15 (c) for Re = 100 and

:

= 0.8 at

W

U

/t = 150, and (d) for Re = 100 and

:

= 1.2 at

W

U

/t = 149

4. Conclusions

In order to reveal the flow past a rotating pipe which is immersed parallel to the mainstream, we

conduct field observations of a flying pipe in rotation using a pair of high-speed video cameras, together

with motion analyses based on the recorded images, which quantitatively show both paths and angular

velocities of the flying pipe, and the values of main kinetic parameters. In addition, we conduct numerical

simulations by a finite difference method based on the MAC scheme, whose results suggest that the pipe-

rotation effect becomes remarkable for a rotation parameter

:

! 0.4.

References

[1] http//www.x-zylo.com

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[3] K. Izumi and M. Takamoto, Experimental Study of the Wake behind the Ring, Trans. TERC, 6 (1982), 25-34.

[4] G. J. Sheard, M. C. Thompson, K. Hourigan and T. Leweke, The evolution of a subharmonic mode in a vortex street, Journal of

Fluid Mechanics, 534 (2005), 23 í 38.

[5] K. Hirata, J. Funaki and N. Tuno, On the Critical Geometry of a Ring in Flow, Trans. JSME, Ser. B, 67 (2001), 3101í3109. (in

Japanese).

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