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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
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