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Representative terms from entire chapter:
time increment
Simulations of Forces Acting on a Cylinder in Oscillatory
Flow by Direct Calculation of the Nav~er-Stokes Equations
T. Kinoshita
Univertisty of Tokyo, Tokyo, Japan
M. Hinatsu
Ship Research Institute, Tokyo, Japan
S. Murashige
University of Tokyo, Tokyo, Japan
Abstract
Flow around floating vessels
sometimes accompanies separations, and
is unsteady in ocean waves. We select
the flow around a circular cylinder in
an oscillatory flow as a preliminary
study for it.
In this paper, the two dimensional
Navier-Stokes equations for it are
directly computed using body-fitted
coordinate, moving mesh technique, 3rd
order upwind scheme, and the MAC method
at the Reynolds number, R-=10^ and the
Keulegan-Carpenter number, K==5,7, and
10.
The computed results simulate
effect of the Kc number on the
field in excellent agreement
published data both qualitatively
quantitatively.
the
flow
with
and
Moreover, we examine the validity of
this computational scheme. The good
agreement with analytical solutions
gives reliability of it, and we
consider the effect of fineness of the
grid and the effect of the Reynolds
number minutely.
1. Introduction
The flow around a circular cylinder
in an oscillatory flow is idealization
of the flow around floating vessel like
semi-submersible in ocean waves. It
accompanies separations and is
unsteady. Hence, we must sometimes
understand mechanism of the separated
vortices to estimate hydrodynamic
forces acting on it.
Flow visualization gives a lot of
valuable qualitative information for
it. Williamson [1] oscillated a
circular cylinder in water at rest at
the Keulegan-Carpenter number, K=<60,
and the frequency parameter, ~ =255,
313
where K==UmT/D, ~ =R-/K=, R." is the
Reynolds number, R-=UmD/ v, Um denotes
the maximum velocity in a period, D the
diameter of the circular cylinder, v
the kinematic viscosity of a fluid, and
T the period of oscillation. He
related the position and strength of
vortices around it to K=. For example,
a very interesting half Karman vortex
street in the transverse direction was
observed in the range of 7
expressed under the
U=UmsinD as follows:
Cd ~ 34 J F'~sin:9 do
Cat ~ 2 UDT J F'n~,P,220 dig
condition of
(2)
Sarpkaya[6] carried out an analysis
of the experimental data of Keulegan
and Carpenter[5] as well as his own
data, and showed that the coefficients
Cat and Cm depend on not only Kc but
also R.-. In particular, it is
interesting that Cat increases and Cm
decreases with Kc in the range of about
5
computations are necessary to take into
account the three dimensional coherent
structure of the shed vortex sheets,
they remain as a next step of the
research. This paper shows two
dimensional solution to this problem.
In order to use the present
computational scheme as a practical
tool, we must determine the validity of
it. We compare the computed results
with the analytical solutions and
examine effects of total number of grid
points, minimum grid size, time
increment, and the Reynolds number on
the computed results.
2. Governing Equations
The two dimensional Navier-Stokes
equations are written in the normalized
form as follows:
Kc at + ~ ax + v a
- a0P + ~ ( a2U + a2
Kc at + u a6X + t; at =
_ dP + 1 ( a2v + a2v)
by Re ax2 au2 ( 9)
where u and v denote the velocity in
the x-direction and the velocity in the
y-direction, respectively, and p the
pressure. The velocity is normalized by
the maximum velocity of oscillation in
a period, Use, the pressure by P Um-2,
the length by the diameter of a
circular cylinder Do, and the time by
the period of the oscillation T-. A
superscript of asterisk denotes the
dimensional value.
~ cut
. . .
\,, . -. . '.'; :.'.';',. ..\'k',''\\\ V ' '
. . , ~ \ \ .
. . . . . . ~ .
Fig.1 Grid
circle. The number of grid divisions
are 120 in the circumference direction
and 50 in the radius direction. The
grid is clustered near body surface
using geometric series to obtain high
resolution. The minimum space adjacent
to the body is set to 0.005.
Introducing a cut line along the
positive part of the x-axis in the
physical plane (x,y) as shown in Fig.1,
we transform the plane (x,y) into the
computational plane (hi) whose grid
increment is set to be constant and
unity for each direction as shown in
Fig.2.
J
The continuity equation is written
in the normalized form as follows: b o d y s u r f a c e
au + an ~ o
ax an (lo)
3. Computational Procedure
3.1 Body-fitted Coordinate System and
Moving Mesh
Body-fitted coordinate system makes
it easy to compute flow around a body
of arbitrary shape, especially to treat
boundary conditions through coordinate
transformation. In this simulation,
the grid is generated as shown in Fig.1
and the system is an 'O-grid'. In this
grid, a circular cylinder of unit
diameter forms the inner boundary. The
outer boundary is a circle whose
diameter is 40 times that of the inner
I- ~ I
Fig.2 Computational plane
. -r
Actual computation is performed in this
computational plane (Gil).
As noted in the introduction, the
moving mesh technique is adopted. Now
we can consider two ways to move the
grid as follows : (1) the way to fix
the outer boundary and move only the
inner boundary, (2) the way to move the
outer boundary with the inner boundary
simultaneously. Case (1) requires
longer computational time than case
(2), because the grid must be
regenerated every moment. Hence, case
(2) is adopted in this paper. The
relations of partial differential
315
operators between the physical plane
(x,y) and the computational plane
(hi) are written as follows:
Ox ~ calf + be,,
by ~ cat + d ~ ,,
axe + ally ~ a a': + b ad + c a7,,,
+ day + e an
t - - (am + CUT) a ~ - (bar + by:) ~ ~ + a t
(11)
where t denotes the time
physical plane (x,y), ~ the
the computational plane (Gil)'
a - (T ~ JY7, . b - 77Z ~ -Jut
c ~ (~ ~ - Jan' . d - ray ~ Jo
a ~ a2 + c2, ~ - 2(ab + cd)
c ab2 + ~
d ~ am + bar, + cc + do,,
e - able + bb', + cat + dd,,
J ~ ~ (~ (y ~ ~ I ~e ~~ I
in the
time in
and
All physical quantities, i.e.
velocity and pressure, are estimated at
each intersection point of grid, which
is so-called 'regular mesh'.
3.2 Boundary Conditions
Indices (i,j) of grid points in
the computational plane(,) are shown
in Fig.2. Each boundary condition is
as follows:
(1) Body Surface (j=1)
*The velocity is set to that of a
circular cylinder.
*The first derivative of the pressure
is derived through the Navier-Stokes
equations (9) and the orthogonality of
the grid on the body surface as
follows:
a rim ~ JEK (your - Eva)
jEKC (W + do) (your - Eve)
JE (W + dV) (yours - X:U,7)
+ R] E {-y~(cu~7, + eu,,) + x`(cv7,7, rev,,)}
e
(12)
where U and V denote the motion
velocity in the x-direction and that in
the y-direction of the circular
cylinder, respectively, and E ~ zi + ~ .
316
(2) Outer Boundary (j=J,J+1)
*The velocity is set to zero.
*Using linear extrapolation, the pres-
sure is set as follows:
pa = 2p_ - pa- 2
pare = 2p~ - pa_= (13)
(3) Outside of Cut (i=-l,O,I,I+1)
*The velocity and pressure are set as
follows:
q_ ~ = qua—2
q0 = qI—1
qI = ql
q=-1 = q2 (14)
where q denotes velocity or pressure.
This condition is called 'periodic
condition'.
3.3 MAC Method and Discretization
In the present computation, the MAC
method[11] is adopted as a
computational algorithm. In this
method, pressure p is obtained by
solving the Poisson equation which is
derived by taking the divergence of the
Navier-Stokes equations as follows:
V2p —V · { (U · V U — 1 as + 1 V2£
) } Kc ~ t Re ~
15)
where u denotes the velocity vector
and £ iS written as
au av
£ ~ a Sac + a (16)
The condition of the continuity is
imposed as follows:
d£ A, _ £t (En ~ So) (17)
where it denotes the time increment,
and n the time step. The reason why
the local dilation term of the n-th
time step is left is that it prevents
the accumulation of numerical errors.
Velocity is obtained by solving the
Navier-Stokes equations (9) using the
Euler explicit time differencing
scheme. The flow chart of computation
is shown in Fig.3.
The governing
discretized as follows:
equations are
(1) Space differencing
The terms except the convective term
are discretized by 3-points central
differencing scheme for all regions.
The convective term is discretized by
1st order upwind differencing scheme
(18) for j=2 and 3rd order upwind
differencing scheme (19) for j> 3
- ~
move body from (n)-th time
step to (n+1)-th time step
and set body surface ve-
locity U=-l,v=-l
.
compute pressure p=41 by
solving the Poisson
eq.(15)
. , ~
compute velocity until,
v~-l by solving the Navier
-Stokes eqs.(9)
! ~
+
11
Fig.3 Flow chart
U( dd`)i ~ Ui (-Ui-l+ui+l)
- luil (uil-2ui+ui+~) (18)
( d U ) (Ui-2 - 8ui l +8ui+1—Ui+2)
i 4h (19)
where h denotes the grid increment.
(2) time differencing
The Euler explicit method is used as
follows:
61' ~ (U ~—Un)
at fit (20)
3.4 Stability
The limitation of the time increment
is given by the von Neumann method, as
follows:
Kcxu (21)
where fit denotes the time increment,
~x the local grid increment and u the
local velocity.
In the case of K==5 and u=1.0, the
limitation of the time increment is
given as follows:
fit ~ 0.001 (22)
Since the van Neumann method is
valid only for linear equations, the
317
time increment must be set lower than
the above value in actual computation.
In this computation, it is set as
follows:
at ~ 0.0002 for Kc ~ 5 and 7
at ~ 0.~1 for Kc ~ to (23)
4. Computational Results
The computations are performed at
R-=10000 and K==5,7 and 10. In order to
realize asymmetric flow field,
sinusoidal motion of a circular
cylinder in the direction vertical to
the oscillation is superposed on the
oscillation at the first quarter period
of the oscillation as follows:
U -sin(2~t)
O.lxsin(4~t) for t ~ 0.0~0.25
0.0 for t > 0.2S
(24)
where U and V denotes the motion
velocity of a circular cylinder in the
x-direction and that in the y-
direction, respectively. The amplitude
and period of the superposed sinusoidal
motion are 1/lOth and half of the
oscillation, respectively.
For all computational conditions,
the circular cylinder is oscillated for
eight periods, and the results are
discussed for the data in the 8-th
period.
4.1 Hydrodynamic Forces Acting on a
Circular Cylinder
(l)In-Line Force
Published experimental data of
hydrodynamic forces are arranged by the
drag coefficient, Cat, and the inertia
coefficient, Cm, or the added mass
coefficient, C^, where Cm=l+C~ ( see
Appendix). Before describing the
computed results, we should note that
there are scatters between the data
published by different researchers as
shown in Fig.4.1. The time history of
in-line force shows a qualitatively
same pattern in each cycle, but the
magnitude and phase vary to some
degree. The differences yield the
scatters. For example, Kato et al.[12]
showed the time history of pressure
distributions, in which the magnitude
differs more than 50 per cent in some
cycle. Thus, since the value of the
coefficients depends on how to average
and estimate them, they may scatter
even in the same experimental facility.
Besides the differences of experimental
method and conditions, the reasons of
the differences of the magnitude and
phase in each cycle are considered as
follows. With progression of Rem and/or
K=, irregularity of shed vortex sheet
which has three dimensional coherent
structures[2][3] is superposed on that
which has two dimensional coherent
structures. As a result, complicated
flow field which shows strong
irregularity is formed. Hence, the
scatters are not only due to errors
but also due to essential nature of
flow field. Furthermore, the
coefficients Cot and Cm (or C^) are
considerably varied by even a little
phase shift of hydrodynamic force time
history.
The computed results of Cat and Can
are shown in Figs.4.1 and 2 with
experimental data of Sarpkaya[6] and
Tanaka et al.[13] and the analytical
solutions[8][9] which are rewritten by
equations (5) m(8) by noting Cm=l+C~.
These figures show that these
computational results simulate very
well the effect of the Kc number on the
drag coefficient Cat and the added mass
coefficient Can.
C d
25
2.0
15
10
05 \`
0.0
Exp. by Sarpkaya
>I
o Computed /
o
/ /~ ,/'
/ /
/~(Theory by Bessho
/ G>"
" ~' Theory by Wang
O ~
Fig.4.1 Drag coefficient, Cat
C a
1.?
1.OI
08t
06:
04:
11 7
. . ~
0.0
~0.2
-04
~ ~ ~ Theory by Wang
O \ \ Theory by Bessho
\? `(
\ \%
\ O \
Exp. by Sari\
o Computed
\
-06 1 , , , , 1 1
4 6 8 10 12 14 16
K c
Fig.4.2 Added mass coefficient, Ca
Next, in order to examine that the
present computation simulates unsteady
phenomena accurately every moment, time
histories of in-line force are shown in
Figs.5.1,2 and 3.
F in = F in, ~ p Um'2D ~2 ~
. ...
2. 5 F
computed
2. an r- ________, MOrison ':
/ \
, ~
C) ~ -
O 0, 5
, ~
~ n _ / ~ \
/ /, \ ~ I
n s r— /
I '' \\
- 1. C / , \'
I ~ ~
- /
I/
- 2 . n ~ ! , , , , ! , , . , ' , , , , ! , , , , !
An, to 1 n 2. ~ 3. C ¢. 0 5. 0 S. On 7 n
Phase o= 2 ~ t -, T
Fig.5.1 Time history of in-line force
(K==5)
F in = F in ( D U ~-2 D ·/ 2 )
. . ,
2. ~ I ' ' '
— computed
1.5 ~ ---- Morison
0 1.^
c)
O A, 5
u —
w
-
fJ '~>'
`) " n 7 \
~ r ~ | \ ~ _
~ c.s ~ ,1 'it' ~
H -
~ / ~ _
- !. ~ ~~
—2 . ~ ' ' ' ' I ~ ' , , , , ! , , , , 1 , , , , ~ , . . .
n !. ^ 2. C 3. n ~ n s c s 0 7. n
ase o= 2;~t T
4 6 8 lO 12 14 16 Fig.5.2 Time history of in-line force
K c (K==7)
\'`
i no'
a) -
c,) 2.^ -
S~
o
.
a)
. -
,' i \\
,! \ ~
/' '
. .
~ -` /,
_ I ~ /
H _, n ~ /,'
~ \ ~ ,
~ I,
,
2 . ~ x, , ! ~ , , , , I , . .
~ ^ . ^ ?. ~ ~ ^
Phase
318
Fig.5.3 Time history of in-line force
(K==lO)
Almost no experimental data of time
histories of in-line force have been
published, but we can get the
approximation for them by substituting
the experimental data of the
coefficients into Cat and Cm in the
Morison equation (1). The broken lines
in these figures denote the
approximation. Here it should be noted
that the Morison equation assumes an
odd-harmonic in-line force, neglects
higher frequency components and cannot
express the experimental data in the
range of K==8m 25. The first
assumption means that the flow field
repeats every half period of the
oscillation. At K== 5 and 7, the
computed results are in very good
agreement with the approximation by the
Morison equation as shown in Figs.5.1
and 2. While at Kc=lO, they are not in
so good agreement with them as at K==5
and 7 and are not odd-harmonic
definitely as shown in Fig.5.3. This
matter is discussed in Chapter 4.2 more
minutely.
(2)Transverse Force
The comparison of the computed
results of the maximum value of
transverse force, F=~.m~x with
experimental data of it[6] is shown in
Fig.6.
Ftranmax~
3.0:
2 . 0
~ . O
0 . 5
O.
o
o o
3 4 5 10 50 100 Kc
Fig.6 Maximum value of Transverse
force, F=~.m~x
The experiment was not performed at
R-=10000, but the computed results seem
to show dependence of it on Kc and R."
fairly well. The experiment shows that
the ratio of the frequency of
fluctuation of transverse force to the
frequency of the oscillation of the
cylinder is two in the region of the Kc
number set in this computation. Time
histories of transverse force in
Figs.7.1,2 and 3 shows that.
s — \ ,
\
~ - I
A ~ __ !
a)
~ '
V,
SO
\ / _ \
Fig.7.1 Time history of transverse
force (K==5)
1 tran = F tran ~ p U m 2 D' 2
!. O I,, ,, ,,
a)
C)
So O.v
o
a) -c. ~
U.
a'
~ - ! .
~Q
5 - _,
ED
Exp. by Sarpkaya i. n !. n
O Computed
.,,,,, ., .. ~ 1 ' ' ' ' ~ ' ' '
. s ~ I \
A I \
2. 0 3. 0 k. O
Phase
Fig.7.2 Time history of transverse
force (K==7)
F tran s F tran ~ p U ~2 D '' 2 ~
(. n , ,,, ,,,, I,,,, ,,,, i,,
3. 0 ~
0) 2. C —
O ,.c ~ 1
Lo - ,
an.. ~ \ ~ \
`¢ -2. ~ ~ \ i
E-' - V
-3. ~ —
_ it. . o ~ I , , ! . , ~ i , ' , , , , ! , . , , ~
^ ^ !. n 2. C 3. ~ (. n 5. ~ 5. n ?. n
0 = 2 Ir ~ T
I \
An,
11
-
-
Phase
Fig.7.3 Time history of transverse
force (Kc=lO)
319
We consider that a transverse force
depends on positions of vortices
strongly. Time history of transverse
force is a good measure of periodicity
of the flow field.
4.2 Time Evolution of Vorticity
Time evolution of vorticity is shown
in Figs.8.1 and 2 where the time
increment is set to 0.1, contour pitch
2.0, a solid line denotes a
counterclockwise vortex, and a broken
line a clockwise vortex. The circular
cylinder is oscillated from side to
side along the x-axis, a black circle
in each figure shows the position of an
oscillating circular cylinder, and it
is located on the left end at t=7.0.
Williamson[1] classified aspects of
vortices around the circular cylinder
into some groups by Kc for R." of the
order 103, and in the range of K=<13
they are summarized as follows:
(a) K=<7
Vortices which are generated in a
half period become a pair with vortices
which are generated in the last half
period. He called it 'pairing of at-
tached vortices', although one of the
pair is not attached but shed.
(b) 7
(O) t=7.0
·>
(I) t=7.}
(6) t=7.6
Fig.8.1 Vorticity contour, pitch=2.0, K==5
solid line: counterclockwise, broken line: clockwise
321
~ rim ~~ f f )
. - ; r .
A
(3)t=7.3
D
D'
.
(4)t=7.4
MA
B_A ~
at\
B'
, I]
J
//
,.. ,, ;... ,...
, ;;,. i,. ;,
'1 ~
D
I !/
.
· . /
In_-,
=. ~
. - , =.. · ' ,.
B' I \
N ~ \ ~
(8jt=7.8 ~=
A-
~ . .....
.~ .
`:; .
D
,,
~ me:
i....
.
. .
"N
Fig.8.2 Vorticity contour, pitch=2.0, Kc=lO
·. \.,
solid line: counterclockwise, broken line: clockwise
322
., .. ~ ~
I/ —
\ ., ~
<.
E
CCI
1.40
1.20
1.00 1
n Rn -l
. _ 1
0.60 _ \
0.40 _
0.20 _
· compu ted
Bes sho
O. ~ ~`
a)
| - !. ~ ~ \ /~
Wang
Kc
Fig.9.1 Comparison of drag coefficient,
Cat between computed results and
analytical solutions
Cm
2.10 _
2.05 _
2.00 -
1.95 _
1.90 _
1.85 _
1.80 _
~ · compel
0.00 1.00 2.00 3.00 4.00 5.00
Kc
Fig.9.2 Comparison of inertia coef-
ficient, Cm between computed
results and analytical so-
lutions
U - sin(2~t)
V; ~ O.Olxsin(4~t) for t - 0.0~0.25
~ O.O for t ~ 0.2S
(25)
In equation (25), the amplitude of
superposed sinusoidal motion in the y-
direction is set to 1/lOth of equation
(24) and the period the same as
equation (24). There are no
differences between the two results
qualitatively. Time histories of in-
line force are shown in Fig.10. Here
it should be noted that it is
difficult to get the same disturbance
in different water tanks, even though
same experiments are performed. This
is one of the reasons why experimental
data scatter. Although it is almost
impossible to express the disturbance
in the tank numerically, we consider
F in = F in / (p Us'2D'/2 )
I,,0,,,,,.,,,,,,,,,.,,,,,,.,,~.,
~ computed ( di s t xo . 1 ) ,~\
3 0 t ~ , computed // \
r ~~~~~-~~ Morlson 1/ \ \
{, /. — — ' ",\
\' /
J
. 1,,,, 1 ., . . 1, . . .
!. ~ 2. 0 3. 0 4. 0 5.0 6.0 7. ~
Phase 0= 2 ~ t ·/T -
Fig.10 Effect of initial disturbance
that the difference between computed
results with the different initial
disturbances as shown in Fig.10 may be
obtained under different experimental
conditions.
(3) Effect of total number of grid
points on Cat, Cam, and flow field
The grid used in the present
computation is determined by the
total number and the minimum
increment. Resolution of flow field by
finer grid is higher, while stability
of the solution is lower if the same
time increment is used. Since we
cannot compute so long, minimum time
increment is limited. Hence, we cannot
make the grid infinitely fine. Table 1
shows that Cut and Can computed by the
grid 120x 50 and 140x 60 are in the
range of scatters of experimental data,
but that those by the grid lOOx 30 are
clearly not in the range.
Table 1 Effect of total number of grid
points on Cat and C"
(K==5, R.=104)
r loo x 30 120 X50 140 X60
Cd 1.031 0.480 0.664
Ca 0.649 0.795 0.809
Figure 11 shows that the computed
results by the grid 120 x50 and 140 x60
qualitatively agree with experimental
data very well, but that those by the
grid lOOx 30 do not agree with them.
323
\\\\// ~ ::
\~\~\`'t'\~\\\~'\~Al Il l l Ail ,l I/ // l,
(8)t=7.8
(a) 100x 30
1 1 - ~ /~"f ',,' -/'~,1
(8)t=1.8
(b) 120x 50
(8)t=1.8
~11~
Cal
(c) 140x 60
Grid near the cylinder surface Vorticity contour
Fig.11 Effect of total number of grid points on flow field
(K==5, R.=10^, t=7.8)
324
(4) Effect of minimum grid increment on
Cat and C"
We can do the same consideration as
the total number of grid poets. Finer
grid increment near the surface of the
cylinder makes the resolution of
boundary layer higher, but the
stability of the solution lower if the
same time increment is used. Table 2
shows that the coarser and finer
minimum increments than that used in
the present computation do not give
good results of Cut and Cam.
Table 2 Effect of minimum grid
increment on Cal and C"
(K==5, R-=104)
x=0.003 ~ x=0.005 ~ x=0.007
0.703 0.480 0.703
~ 0.469 0.795 0.469
(5) Effect of time increment on Cut and
C - .
Through the consideration of the
stability, finer time increment makes
the stability of the solution higher
and give a solution closer to the true
one. But finer time increment makes
time derivative of the residual of the
continuity equation larger and the
results may not suffice for the mass
conservation law. Table 3 shows that
Cat and Can using ~t=O.OOO1 are not in
the range of scatters of experimental
data.
Table 3 Effect of time increment on Cat
and Can
(K==5, R-=104)
;
~t=0.00041
1 1 --1
Cd
Ca
; .
0.814 0.480
0.668 0.795
0.654
0.739
(6) Effect of the Reynolds number on Cat
and Can
The experimental data by Sarpkaya[6]
and Tanaka et al.[13] indicate that Cat
decreases and Can does not vary with
progression of the Rue number. Figures
12.1 and 2 show that the computed
results do not simulate the tendency.
3
CC
1.80 .
1.60
1.40 _
1.20 _
1 r,n _
lo
lo
O Computed
Exp. by Tanaka
et al
0 0
x 104
1.50 2.00 2.50
R.
Fig.12.1 Effect of the Reynolds number
on drag coefficint, Cut
C,-.
l.oor
r, A r. L
_ . ~ ~
0.60 _
0.40 _
0 . 20 _
O Computed
x104
0.50 1.00 1.50 2.00 2.50
O O
Exp. by Sarpkaya
R.
Fig.12.2 Effect of the Reynolds number
on added mass coefficint, Can
Why can we get very good solutions at
R-=10^ ? We can consider the reason as
follows. As noted earlier, the big
vortices generated every half period
are predominant for the flow field.
Hence, in the computation at R-=10^, we
can consider that 3rd order upwind
scheme controls the transition to
turbulence near the surface of the
cylinder and the flow field outside it,
including predominant vortices, is
correctly estimated. For this reason,
we can obtain very good results at
R-=104. Then we need to consider the
effect of numerical dissipation on the
solutions. At the higher Reynolds
number, the local effective Reynolds
number near the surface of the cylinder
becomes too high for 3rd order upwind
scheme to control the transition to
turbulence there, and, as a result, the
numerical dissipation makes the
Reynolds number lower. he need to
consider the cause of some errors in
more detail to estimate the dependence
of the Reynolds number on the flow
field.
For more precise estimate of
hydrodynamic forces Acting upon a
circular cylinder in an oscillatory
flow, the following is hereafter
required;
*observations of phenomena for long
periods
*simulation of three dimensional coher-
ent structures of the flow field.
In this paper, all computation is
performed using FACOM VP-100, and it
takes about 70 minutes for 8 periods of
the oscillation under the condition of
the time increment ~t=0.0002 which is
40000 time steps.
6 Conclusions
(1) Solutions for the flow around an
oscillating circular cylinder at
R-=10000 and K==5,7 and 10 are obtained
by the direct calculation of the
Navier-Stokes equations using body-
fitted coordinates system, moving mesh
technique, and the MAC method. The
results are in excellent agreement with
published experimental data quantita-
tively.
(2) The computed flow patterns show the
'pairing of attached vortices' and
'transverse street' at this range of
K=, where Cat increases and Cm decreases
drastically. The change of aspects of
vortices is very similar to
Williamson's observation at R." of the
order 103.
(3) Time histories of the computed in-
line force are in good agreement with
those which are given by the Morison
equation and published experimental
data of Cat and Cam at K==5 and 7. They
show that this computation simulates
unsteady phenomena accurately every
moment.
(4) Time history of the computed in-
line force is not odd-harmonic at all
at Kc=lO.
(5) The reliability of the present
computation is confirmed by the
comparison of it with the analytical
solutions. On the other hand, the
computed results vary with total number
of grid points, minimum grid increment,
and time increment. But we cannot
determine the best ones before getting
solutions and comparing them with
experimental data. In order to
estimate The effect of the Reynolds
number on the flow, we need to consider
the cause of some errors in more
detail.
(6) For more precise estimate of
hydrodynamic forces around an
oscillating circular cylinder,
structures of the flow field must be
observed for long period and regularity
of it must be investigated. Since
three dimensional coherent structures
of the flow field may affect the
regularity, we need three dimensional
simulation to discuss them more
completely.
The authors acknowledge some useful
comments by Prof.Y.Ikeda of the
University of Osaka Prefecture and
Dr.Y.Kodama and Dr.T.Hino of Ship
Research Institute.
References
1
326
. Williamson, C.H.K., "Sinusoidal flow
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"Mechanics of Wave Forces on Offshore
Structures", Van Nostrand Reinhold
Company (1981)
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internal friction of fluids on the
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oscillating viscous flows", J.Fluid
Mech. Vol.32, p.35 (1968)
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by Oseen's scheme(4th report, two-
dimensional oscillating flow without
uniform velocity", J. the Society of
the Naval Architects of Japan. Vol.
161, p.42 (1987)
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order accurate difference solution
of vortex generation from a circular
cylinder in an oscillatory flow",
J. Comput. Phys. Vol.69. No.2, p.362
(1987)
11. Harlow, F.H., and Welch, J.E.,
"Numerical calculation of time de-
pendent viscous incompressible flow
of fluid with free surface", The
Physics of Fluids. Vol.8. No.12,
p.2182 (1965)
12. Kato, S., and Ohmatsu, S., "Hydro-
dynamic forces and pressure distribu-
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Appendix Relations Between Flow Around
an Oscillating Circular
Cylinder in a Fluid at Rest
and Flow Around a Circular
Cylinder Fixed in an
Oscillating Flow (All
quantities are dimensionally
defined)
(a) Flow around an Oscillating Circular
Cylinder in a Fluid at Rest
Assuming that a circular cylinder
oscillates
governing
follows:
dU + au,
dt at
in the x-direction, the
equations are written as
+ u'aU. + tray
_p apt + v(a u2
at + u ax. + v all'
~—ad P. + v( a V2
a y, 2
, +a2v
ayes
(26)
au + av2 ~ 0 (27)
where U=U(t) denotes the velocity of
the circular cylinder, a superscript of
prime the value in the coordinate
system which moves with the circular
cylinder, and v the kinematic
viscosity.
Boundary conditions are written as
follows:
u' = - U at r —
v' = 0
u' = 0
v' = 0
at r ~ a
where r denotes the distance from the
center of the circular cylinder and a
the radius of the circular cylinder.
(b) Flow around a Circular Cylinder
Fixed in an Oscillatory Flow
Assuming that the ambient flow
oscillates in the x-direction, the
governing equations are written as
follows:
ant + U0U + viny
eat + UbV+ any
_ _ asp + v( ~ 2 + a 2)
P a y ( a X2 a y2 )
(29)
(30)
where U=U(t) denotes the velocity of
the oscillatory flow.
Boundary conditions are written as
follows:
u = —U
{v = 0 at r -
u = 0
(v = 0
at r = a
(31)
In infinitely far field from the
circular cylinder, the velocity is
written as
u = - U (t)
v = 0
(32)
Thus, equation (29) in the far field
becomes the following equations:
dt P ad (33.a)
~g (33.b)
where Ping denotes the pressure in the
far field, i.e. the pressure in the
uniformly oscillatory flow. Equation
(33.b) represents that the pressure in
the oscillatory flow, Pins, is a
function of only x and t, i.e. Ping =
P,~(x,t).
The pressure is divided into two
parts as follows:
(28) p = Pawn + Pa,-= (34)
327
where P=i-= denotes the pressure due to
the disturbance caused by a circular
cylinder. Using equations (33) and
(34), equation
follows:
d t + ad t + U ad X + V Ha y
(29) is rewritten as
1 apsis at
p ~ ~ + v ( 2
at +ua~ +vag
J
+ a 2)
an
1 aPaist
_ _.
P dY
( a 2v
Arc
a2v
at
Equation (26) is the same form as
equation (25). The only difference
between the flow(a) and the flow(b) is
that the pressure includes the
additional term Pans whose gradient is
required to accelerate the undisturbed
flow in the flow(b). Integrated around
the cylinder, Pawn gives rise to an
inertia force, which is pa a 2) ~U/0 t
per unit length. The inertia
coefficient, Cm, which is defined in
the flow(b), is related to the added
mass coefficient, Can, which is defined
in flow(a) as follows:
(35) Can = 1 +- Cam (36)
3~
