Journal of Applied Mechanics Vol.11, pp.769-778
Numerical
on Fluid-Forces
(August 2008)
Prediction
JSCE
of Shielding
Effects
Acting on Complicated-Shaped
Object
複 雑形 状物 体 に作用 す る流 体 力の遮 蔽効 果 に関す る数値 解析
Satoru
USHIJIMA*
and
牛島 省
Nozomu
KURODA**
・黒 田 望
*Member
, Assoc. Prof. Dept. CERE, Kyoto Univ. (C-cluster, Kyoto 615-8540)
**Member
, Master Course Dept. CERE, Kyoto Univ. (C-cluster, Kyoto 615-8540)
A computational method for multiphase fields, MICS 1), was applied to estimate the
shielding effects on the fluid forces acting on a complicated-shaped object surrounded
by other objects. In the MICS, arbitrarily-shaped objects are treated with tetrahedron
elements, through which the momentum interactions between objects and fluids are accurately taken into account with a tetrahedron sub-cell method. The applicability of the
MICS was discussed with the experimental results obtained in some arrangements of the
objects which surround a target object in a flume equipped with a wave generator. As a
result, it was shown that the MICS enables us to predict reasonably the shielding effects
on the fluid forces in all cases of the present experiments.
Key Words : fluid force, shielding effect, free-surface flow, MICS
Introduction
surface flows around arbitrarily-shaped objects. Since
It is important
the fluid-solid interactions are taken into account in
this method, it is possible to estimate the fluid forces
1.
to estimate
accurately
the
fluid
forces caused by free surface flows against
a compli-
cated shaped
block.
In
acting on the complicated-shaped objects as well as
the shielding effects on it.
to take account
of
The predicted free-surface profiles and fluid forces
of fluid
with the MICS are compared with the experimental
results, which were obtained in a flume equipped with
object
such evaluations,
like a wave breaking
it is necessary
the shielding effect, which means the variation
forces due to the deformation
caused by the surrounding
such an object
Although
of the free-surface
objects, since it is rare that
exists alone in the actual conditions.
in many cases some empirical
such as the drag and lift coefficients
been utilized in the evaluation
vious that such evaluations
empirical
flows
coefficients,
Cd and CL, have
of fluid forces, it is obare inaccurate
multiphase
other
objects.
methods,
paper, a computational
method
for
to estimate
effects on the fluid forces acting on a
complicated-shaped
Numerical
2.1
fields, MICS 1), is applied
the shielding
2.
Procedures
and the shielding effects are
not taken into account.
In the present
cases of the present experiments.
since the
coefficients are usually derived in a uniform
flow with a single object
a wave generator. Some arrangements of the objects
surrounding the target object are examined and it is
shown that the MICS allows us to estimate reasonably
the fluid forces and shielding effects on them in all
object,
In contrast
which is surrounded
to the usual
by
numerical
the MICS enables us to deal with the free-
Basic
The
solid
is
multiphase
phases
the
fluids
in
Fig.1
to
the
―769―
Equations
field
is treated
collection
as
have
of
shown
different
corresponding
consisting
as
the
in
of
a mixture
of
immiscible
Fig.1.
physical
phases.
The
gas,
fluids Ħ,
incompressible
fluid
components Ħi
treatment
and
which
and
properties
This
liquid
equivalent
enables
us to deal with the complicated-shaped
objects
in-
spatially-representative
value ƒÓk(t)
as
follows:
cluded in free surface flows easily.
(6)
With this relationship, Eq.(1) is rewritten as
(7)
where
average
p and ui are volume-average
velocity
density
and mass-
component:
(8)
Fig.1
Mixture
of immiscible
and incompressible
Similarly, with the relationship uk,i = ui+uk,i, Eq.(2)
is represented as
flu-
ids
(9)
The
and
mass-conservation
Lagrangian
given
as
equations
forms
for
in
the
fluid
the
Eulerian
mixture Ħ
are
Assuming that the third term on the left-hand side of
the above equation is negligible, the following incom-
follows:
pressible condition is derived from Eqs.(7) and (9):
(10)
With the similar procedures, Eq.(4) is rewritten as
(2)
where
pk and uk,i
are density
in the
xi direction
of the
the above two equations,
is derived
and velocity
fluid-k
component
respectively.
the incompressible
(11)
From
condition
In Eq.(11), as proposed by a CSF model 2), the surface
as
force is treated as
(3)
The
momentum
equation
for Ħ
is given
(12)
This relationship means that the surface force is transformed to the body force. Putting the third term on
the right-hand-side of Eq.(11) Di, it is written with
by
Eq.(5) as
(4)
(13)
where fi is the acceleration component of the external
force and the second term on the right-hand side of
where
Eq. (4) means the surface integration of the surface
tension Fs,i acting in the xi direction. The stress Tk,ij
is defined as
(5)
where ƒÐij,
sure,
fluid-k,
pk,uk, ek,ij
viscous
are
coefficient
the
and
Kronecker
delta,
deformation
cous coefficient
are volume-average
defined
pressure
and
vis-
as follows:
(14)
Finally, assuming that the non-linear term of
in Eq.(11) and surface tension are negligible, the conservative form of the momentum equation is derived
as
pres-
tensor
p and u
of
respectively.
Assuming
a variable ƒÓ'k
that
(t,
the
x)
in
volume
each
of Ħ
fluid
is
is
sufficiently
approximated
small,
as
(15)
its
―770―
The governing equations of the MICS consist of Eqs.
(7), (10) and (15).
2.2 Computational Method
The discretized governing equations of the fluidmixture are solved after determining the volumeaverage physical properties with the sub-cell method,
which will be described later. The three-dimensional
velocity components ui and the pressure variable p of
the discretized equations are definedon the collocated
grid points in the computational fluid cell.
The numerical procedures of the incompressible
fluid-mixture consist of three stages; prediction,
pressure-computation and correction stages. At the
prediction stage, the tentative -velocity components
are calculated at the center of the cells
ui* with a
finite-volume method. In this procedure, Eq.(15)
is discretized with the C-ISMAC method 3), which
is based on the implicit SMAC method 4) that allows
us to decrease computational time without decreasing
numerical accuracy. The equation discretized with respect to time by the C-ISMACmethod is given by
higher-order schemeto the right-hand side. The convection terms are evaluated with a fifth-order conservation FVM-QSI scheme 5) and numerical oscillations are removed by a flux-control method 5). The
C-ISMACmethod enables us to derive the simultaneous equation system easily from the implicit form of
the left-hand side of Eq.(18) as well as to preserve numerical accuracy by applying a higher-order scheme
to the explicit form on the right-hand side Eq.(18).
After solving the equation system of ui, which is
derived from the discretized equation of Eq.(18), we
obtain u*i with Eq.(17). The u*i derived at the center of the computational cell is then spatially interpolated on the cell boundary. Before this interpolation,
pressure-gradient term evaluated at the cell center is
removed from u*iin order to prevent pressure oscillation as
(19)
The cell-center velocity ui, which does not include the
pressure-gradient term, is spatially interpolated on
the cell boundaries by fb, which is a linear function in
the present study. After this procedure, the pressuregradient terms that are estimated on the cell boundaries are added to the interpolated velocity, fb(ui)
Thus, we obtain the cell-boundary velocity component ub,i as follows:
(16)
where ƒ¿
and ƒÀ
0 •… ƒ¿, ƒÀ •… 1.
are
With
the
parameters
the
following
whose
ranges
are
(20)
The
relationship,
velocity
defined
component
un+1b,i at n+
1 time-step
is
by
(17)
(21)
Eq.(16) is transformed to the following equation:
Subtracting Eq.(20) from Eq.(21), we have
(22)
where ƒÓ
the
18)
where ui becomes nearly zero when the flow field is
almost steady or the time-scale of the flow field is suf-
is
=
pn+i-pn.
incompressible
estimated
equation
condition
at
of ƒÓ
Substitution
n+1
time-step
of
given
yields
by
Eq.(22)
Eq.(10)
the
into
that
following
:
(23)
ficiently larger than the time increment At. Thus,
we can apply a simple first-order spatial discretization method to the left-hand side of Eq.(18), while
At the pressure-computation stage, Eq.(23) is
solved with the C-HSMAC method. The C-HSMAC
―771―
method
enables
boundary
us
velocity
compressible
tional
final
lar
to
obtain
the
components,
pressure
which
condition |D| •ƒ ƒÆD
cell,
the
to
where ĮD
results
those
of
of
has
been
the
C-HSMAC
is
the
the
proved
relationships
SOLA
that
in
given
or
the
the
each
While
are
method
6),
improved
method
are
the element is completely included in a single fluid
cell as shown in Fig.2 (a), ak is easily determined
from the element volume. In contrast, when the element is included in multiple fluid cells as shown in
simiit
efficiency
largely
C-HSMAC
in-
computa-
computational
is
the
method
HSMAC
represented by multiple tetrahedron elements. When
cell-
threshold.
C-HSMAC
method
satisfy
in
a
and
of
1).
The
given
by
Fig.2 (b), a fluid cell is divided into multiple sub-cells
and ak is determined from the number of sub-cells included in the element, which are shown as gray cells
in Fig.2 (b). The accuracy of the sub-cell method can
be improved using the smaller sub-cells.
(24)
(25)
(26)
where the superscript k stands for the iteration stepnumber of the C-HSMAC method. The initial values
(a) Tetrahedron element
included in a fluid cell
of ukb,i and pk are ub,i given by Eq.(20) and Pn respectively.
The
discretization
linear
of
equation
system
BiCGSTAB
using
D|•ƒƒÆD
2.3
above
is
in
the
to
determined
cal
the
is
all
solved
iterative
with
the
completed
when
cells.
2.4
|
in
in
each
the
the
of
of the
cell
is fixed
T-Type
Solid Model
A solid object in the flow is numerically represented
using T-type solid model. The object surface model,
values
for
Sub-cell method (thick grid lines stand for
fluid cell and thin lines in (b) indicate subcells )
computation
is
derivation
physical
value ƒÕ
The
in
computational
which
which
Fig.2
simultaneous
Method
tions,
the
yields
equations
satisfied
shown
be
7).
three
Sub-Cell
As
of ƒÓ,
method
the
Eq.(24)
(b) Tetrahedron element
in multiple fluid cells
is
is
governing
mixture
of
computational
based
space,
cell
the
the
on
fluids
Since
Eulerian
with
model" , which represents an object with multiple
sphere elements, T-type model is advantageous in re-
grid
volume-average
estimated
need
cell.
the
created with a CAD software as shown in Fig.3 (a),
is divided into multiple tetrahedron elements shown
in Fig.3 (b). Compared with the "sphere-connected
equa-
physithe
spect that its approximations of volume, mass and inertia tensors are more accurate. The contact spheres,
shown in Fig.3 (c), are used only in the collision de-
following
equation:
tection between objects.
(27)
where ƒÕg
liquid
and ƒÕl
phases
object—k.
phases
the
shown
volume
values
of
is
in
the
and ƒÕbk
fraction
a computational
fraction
fraction ƒ¿k
as
physical
respectively
The
in
are
fluid
solid
part
approximated
Fig.2.
of
As
the
cell
is
the
is
that
liquid
is
later,
a
of
by
solid
the
solid
f
of Solid Objects
rotational motions. The basic equation of the translational motion is given by
and
(28)
The
where Mb is the mass of an object,
method,
object
Movements
The movements of the T-type solid model are calculated with the basic equations for translational and
and
by ƒ¿k.
sub-cell
a
gas
and
given
defined
with
shown
in
2.5
is
v is the velocity
vector of its center and dot means time differentiation.
―772―
where ƒÖ
is
tensor
in
tation
matrix
on
the
the
the
the
basic
object.
instead
of
cation
the
and
Eqs.(28)
N
is
The
are
velocity
attitude
and
rotation
2.6
angular
of
the
of
the
is
object,
object
a ro-
to
quaternions
matrix
are
is
imposed
related
with
the
inertia
R
torque
performed
of
I
processes
multiplication
and
the
external
numerical
actually
attitude
vector,
R.
The
determined
loform
(29).
Fluid Forces Acting on Objects
The fluid forces acting on the objects are calculated
with the pressure and viscous terms obtained from the
computational results of Eq.(15):
(a) Surface model
(30)
where
FCk
acting
on
FCki•E
is
with
body ƒÐk
the
sub-cell
the
F
on
of
the
the
C
shown
as
solid
FCk
in
hand
in Eq.(29)
moments
from
xi
a
in
fluid
component
is
is
calculated
fluid
cell
fraction
C
given
from
AC,
ak
cell
by
Eq.
density
calculated
of
with
method.
right
N
of
volume
T-type
the
its
vector
Fcki
volume
of
torque
and
and
summation
(b) Polygon model
force
component
the
the
In
fluid
object-k
The
(30)
the
rGC •~
object
in
center
model,
as
all
fluid
side
of
shown
cells
is
in
the
Eq.(28).
is obtained
Fck,
xc ,k to
the
the
vector
Similarly,
from
where
Fig.4,
force
the
rGC
center
the
summation
is
of
the
vector
a fluid
cell
Fig.4.
(c) Contact spheres
Fig.3
The
basic
equation,
equation
T-type
of the
can be written
solid model
rotational
motion,
as the following
Euler
Fig.4
form:
(29)
―773―
Fluid force acting
in a fluid cell
on a tetrahedron
element
3.
Application
of prediction
3.1
Experiments
The fluid forces acting
method
on the target
object
block
as much
as possible
in case-S.
sur-
rounded by multiple objects were measured in a wavegenerator
flume, in order to understand
the shielding
effects on the fluid forces.
The schematic
view of the flume equipped
wave generator
movements
is shown
in Fig.5.
with a
The horizontal
of the vertical plate on the left side of the
flume can be controlled
by a PC, so that it can gen-
erate the wave flows above the box fixed on right side
of the flume. The target
object and the surrounding
objects are placed on the box as detailed later.
The lengths L1 and L2 of the flume are both 0.7m.
The width of the flume B is 0.19m.
The height of
Fig. 5 Experimental flume (plane and side views)
the box in the flume hb is 0.1m and the width of the
flume is 0.19m.
0.104m
The initial water depth ho was set at
and the water level is same as the height of
the bottom
edge of the target
object.
water depth due to the generated
The maximum
wave was about 147
mm at the point 100 mm apart from the front edge of
the box.
The target
object,
wave breaking
as shown in Fig.6, is a four-leg
block, whose height is about
56mm
and specific density is around 2.14. The target object
was supported
by a steel plate on which four strain
gages are fixed to measure the streamwise
of the fluid forces acting on the object.
hand, the other objects surrounding
component
On the other
the target block
are six-leg blocks as shown in Fig.7. The side length
Fig.
6
Target
four-leg
block with
Fig.7
Six-leg
supporting
d shown in Fig.7 is about 20mm.
Three
cases of experiments
the different arrangements
were carried
out with
of the surrounding
blocks,
as shown in Fig.8. In case-F, three six-leg blocks are
placed in front of the target
four-leg block. Similarly,
three six-leg blocks are fixed behind the target
block
in case-B. In case-S, two six-leg blocks are placed next
to the target
block as shown in Fig.8. The positions
of the surrounding
distances
The
of the block edges from the side walls, b1
and b2 are 10mm
distance
blocks are shown in Fig.9.
and 15mm
respectively,
while the
from the front end of the box, d1 and d2 are
100mm and 50mm
surrounding
respectively.
The attitudes
blocks in case-S are different
F and case-B, so that they can approach
of the
from casethe target
―774―
block
plate
3.3
Comparison
between
experiments
and
predictions
The computational
results obtained
with only a sin-
gle target block are shown in Fig.10 and Fig.11, which
are taken from our previous paper 8). It has been confirmed that
the agreements
of the time histories
for
the fluid forces shown in Fig.11, acting in the streamwise direction, are satisfactory
surrounding
Fig.8
Arrangement
Fig.9
3.2
Positions
Conditions
In
the
tional
cells
tial
conditions
The
unsteady
ment •¢t
The
with
up
with
while
elements
number
same
1.0 •~
10-2
the
1.0 •~
10-5m2
blocks
in
experiments.
the
Fig.10
Predicted free-surface profiles around a
single block, t = 1.1 (s)
Fig.11
Time
histories
single
block
blocks
the
those
for
125
a
water
/sec.,
volume
region.
the
The
ini-
experiments.
with
time
incre-
and
target
The
fluid
block
respectively.
block
The
set
at
The
are
is
152
same
set
rep-
node
consists
tetrahedron
cell.
were
created
to
and
six-leg
air
computations
were
generator
elements
nodes.
each
blocks
mesh
The
surrounding
in
of
the
and
tetrahedron
78
50 computa-
second.
elements.
the
air
proceed
models
416
38 •~
three-dimensional
as
software
and
was
of
in the
including
tetrahedron
points,
cosity
are
CAD
resented
244
up
m,
numerical
the
the
set
0.25
140 •~
computations
=
blocks
of the surrounding
predictions,
are
1.4 •~ 0.19 •~
objects.
of computations
numerical
of
of surrounding
in case that there is no
of fluid forces
acting
on a
of
sub-cell
kinematic
1.0 •~
density
as
that
10-6
of
of
vis-
On the
and
files obtained
the
in Fig.12,
the
seen that
other
in case-F,
Fig.13
the wave
the existence
―775―
hand,
and
the predicted
case-S
Fig.14,
and case-B
distorted
blocks.
pro-
are shown
respectively.
flows are largely
of the surrounding
free surface
It can be
due to
Fig.12
(a) t = 0.9 (s)
(a) t = 0.9 (s)
(b) t = 1.0 (s)
(b) t = 1.0 (s)
(c) t = 1.1 (s)
(c) t = 1.1 (s)
(d) t = 1.2 (s)
(d) t = 1.2 (s)
Fig.13
Predicted results of free-surface flows
Predicted results of free-surface flows
(case-S)
(case-F)
―776―
In case-F, as shown in Fig.12, the wave flows are
trapped by the three six-leg blocks in front of the
target block. Thus the free surface level in front of
the target block in Fig.12 (c), which was taken at t =
1.1 sec., is smaller than that of the single target block
shown in Fig.10. It can be seen that the shielding
effects of three six-legblocksare so large that only the
weakened wave flows that pass through them collide
with the target block.
The time histories of the fluid forcesagainst the tar(a) t = 0.90 (s)
(b) t = 1.05 (s)
(c) t = 1.20 (s)
(d) t = 1.35 (s)
Fig.14
Predicted results of free-surface flows
(case-B)
get block in case-F are shown in Fig.15. It is obvious
that the peak value of the fluid forces is about 1/3
compared with the maximum value shown in Fig.10.
It is also seen that the experimental values and predictions shownin Fig.15 are in good agreement. Conclusively, the shielding effects in case-F are reproduced
well in the present computational method.
On the other hand, in case-Sshown in Fig.13, since
there are no obstacles in front of the target block, the
waveflowsacting on it are almost same as those in the
case of the single block as shown in Fig.10. However,
the flowspassing through the side of the target block
are trapped by two six-leg blocks next to the target
one. This fact makes the velocity and pressure distributions around the target block different from the
single block as shown in Fig.10. Thus, the maximum
value of the fluid forces shown in Fig.16 is slightly
smaller than that in Fig.11. The shielding effect is
certainly admitted in case-S, while the influence is
smaller than case-F.
Similarly to case-S, there are no obstacles in front
of the target block in case-B,as shownin Fig.14. Nevertheless, it can be thought that the flow conditions
are differentfrom the single block in the followingtwo
points: 1) the gradient of water level and the fluid velocity become smaller due to the six-legblocksbehind
the target one, and 2) the reflectedwaves occur due to
the blocksbehind. As shownin Fig.17, the peak value
of the fluid forces is smaller than that in Fig.11 due
to the above effect 1). In addition, it is demonstrated
that the negative fluid force occurs in the experimental results in Fig.17 followingthe positive one. The
negative fluid force is caused by the reflected wave
flows due to the effect 2). This negative fluid force
is also predicted in Fig.17, while it is underestimated
compared with the experiments.
―777―
The
tent
shielding
effect
in case-B,
reflected
is also
meanwhile
waves
admitted
to some
the fluid forces
are additionally
caused
ex-
due to the
by the
blocks
behind.
4.
Conclusion
The computational
method
MICS was applied
to
estimate the shielding effects on the fluid forces acting
on a complicated-shaped
objects.
The predicted
object surrounded
forces for three arrangements
jects,
Fig.15
Time
histories
of fluid forces
in case-F
by other
free surface profiles and fluid
of the surrounding
case-F, S and B, were discussed
ob-
with the ex-
perimental results. It has been shown that the MICS
enables us to predict reasonably the shielding effects
on the fluid forces in all three cases, which seems to
be difficult to predict
with the usual computational
methods.
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D.B.Kothe,
and C.Zemach.A
continuum method for modeling surface tension.
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pp.21-30,
Fig.16
Time
histories
of fluid forces in case-S
Fig.17
Time
histories
of fluid forces
in case-B
2002.
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I.Nezu. Fifth-order conservative scheme with flux
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