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OCR for page 708
24th Symposium on Naval Hydrodynamics
Futuoka, JAPAN, 8-13 July 2002
Prediction of Slam Loads on Wedge Section using
Computational Fluid Dynamics (CFD) Techniques
Reddy D.N., Scallion T., Chengi Kuo
(Universities of Strathclyde and Glasgow, UK)
Abstract
Numerical prediction of slam loads on a wedge
shaped ship section during water entry is carried out
using Computational fluid Dynamics (CFD)
techniques by employing Finite Volume Method
(FVM) for discretization of the flow equations and
Volume Of Fluids scheme (VOF) for free-surface
capturing. Water impact of a wedge section with 30
degrees of deadrise angle is numerically simulated
while taking into account the key factors affecting
evaluation of slam loads. History of impact velocity,
pressures at distinct locations and the impact force on
whole wedge section during water impact are
predicted. The main conclusions drawn are that close
correlations with the existing experimental data are
obtained, and the effects of impact velocity
variations, domain size and three dimensionality are
significant on the numerical results for slam loads.
~ Introduction
Ships operating in open seas undergo severe motions
with increasing speeds and experience slam loads on
the hull due to continuous buffeting from the waves,
leading to loss of ship control and speed, discomfort
to crew and passengers, increased wet decks and
structural damage. Considering the several adverse
effects that slamming loads can cause to the ship, it is
essential to estimate slam loads on ship sections at
the design stage itself in order to ensure safety,
economical design and operation of the ships.
Sections at forward are more prone to slamming
impacts, and generally have either U shapes or V
shapes.
Many theoretical (Wagner (1931), Payne (1981),
Greenhow(1987), Wilson (1989), Zhao and Faltinsen
(1993, 1996), Arai et al. (1989, 1994), Sames et al.
(1999), Muzaferija et al.(2000)) and experimental
studies (Chuang (1967), Campbell et al. (1980), Zhao
et al. (1996)) on prediction of slam loads during
water entry of solid bodies have been reported, to
name a few, since the pioneering analytical work by
von Karman (1929). Comprehensive review on the
subject has also been reported by SNAME (1993).
Critical review indicates that model experiments
produced closer correlations with the ship
measurements compared to theoretical methods.
However, model experiments are prohibitively costly
and time consuming in addition to the difficulties
involved in generalizing the specific results for their
universal usage. At the same time, theoretical
methods have not reached a stage to reliably predict
slam loads on any ship plying in a seaway due to the
highly complex nature and the associated difficulties
in modeling the slamming phenomena apart from the
high computational requirements. Even though
theoretical methods cannot fully replace the need for
experiments, they certainly lead to efficient methods
of evaluating different cost effective options
available at design stage.
With no analytical solutions existing to the
complicated conserved flow equations for simulating
slamming of the ship sections considering all the
influences, the best solution options have
traditionally been through the use of numerical
methods. Numerical methods of varying vigor and
sophistication have been adopted in analyzing water
entry problems and mostly consider largely
simplified governing equations and boundary
conditions based on potential theory and sometimes
simplified further through different approaches of
reformulation. Their application was largely limited
to simple forms like two-dimensional regular and
ship shaped sections or three-dimensional regular
shapes and resulted in obtaining deeper insight into
the slamming problem rather than producing results
within engineering accuracy suitable for their
practical application. Improvements in computational
capabilities have encouraged employment of more
efficient numerical methods based on Boundary
Element (BE) and Computational Fluid Dynamics
(CFD) techniques, but they still required some form
of approximation for non-linear free surface
boundary. Zhao and Faltinsen (1993, 1996) used
boundary element techniques to predict slam loads on
wedge and ship sections accounting for splash effects
and variations in impact velocity. Their numerical
results for ship section compared favorably with the
experiments but over predicted for the case of wedge
section, and shown that the effect of three-
dimensionality is significant and also that the
numerical results were sensitive to the length of the
introduced jet part. With the emergence of several
numerical solution techniques to free surface
evolution and improved computational capabilities,
CFD methods based on Finite Volume (FV) approach
seem to be gaining ground. However, accuracy of the
numerical results during water entry of solid bodies
OCR for page 709
. . .
prlman. y requires proper modeling of the physical
slamming phenomena in addition to careful
consideration of the different key factors affecting
them.
Application of CFD techniques in Slammine
Research
To obtain better correlation of the numerical results
with the experiments, Arai et al.~1989, 1994), Sames
et al.61999), Muzaferija et al..~2000), to name a few,
employed CFD techniques to predict slamming loads
on arbitrary shaped two-dimensional sections.
Arai et al., used finite difference method for
discretization of the Euler equations and Volume of
Fluids (VOF) method for free surface evolution in
order to obtain slam loads on 2D sections assuming
constant entry velocity. He did obtain favourable
trends but the pressures did not correlate well with
those from the drop tests probably due to neglect of
variations in impact velocities. Sames et al. (1999)
and Muzaferija et al. (2000) used Finite Volume
Method (FVM) for discretization of the Navier-
Stokes equations and High Resolution Interface
Capturing (HRIC) scheme for free surface to predict
slam loads on 2D sections by considering varying re-
entry velocities. Sames et al. concluded that
prescribed vertical velocity histories significantly
affected the determination of realistic pressure levels.
Muzaferija et al. (2000) obtained slam loads on
wedge section considering three-dimensionality with
numerical results matching reasonably well with the
experiments of Zhao et al. (1996), however under the
assumption that these experiments were conducted in
restricted waters.
The present study considers numerical prediction of
slam loads consisting of total impact force and
pressures during water entry on a V-shape similar to
the case of a wedge dropped onto calm water surface.
The present numerical prediction method using CFD
techniques accounts for the key factors affecting the
accuracy of these slam loads using CFD techniques
to achieve better correlations with experiments.
2 Numerical Method
2.1 Mathematical Model
Conservation equations of mass and momentum
(Governing Equations) and the boundary conditions
as given below are considered to simulate slamming
phenomena on an arbitrary impacting section.
Governing Equations:
Continuity equation:
I pv.n dS =0= div v =0
S
Momentum equations:
—Ipv ~7Q+ Ipvi.n AS= J(~/1 grad v-p] .n dS+ J pg dQ
dtQ 5 5( J n
Approximations: Governing equations neglect the
surface forces due to surface tension forces as they
are very small compared to the impact forces (Weber
number Wend) and do not affect the slamming
phenomena, but the same due to viscosity are
considered. Similarly, the body forces due to
centrifugal, coriolis and electro-magnetic forces do
not exist or affect the slamming phenomena and
hence are neglected, whereas body force due to
gravity (as Fn2 _ 1) is considered. Further, the
velocities of the fluid are expected to be small
compared to the velocity of sound in water (Mach
number Mw < 0.3), and it is fairly reasonable to
consider the flow to be incompressible.
Boundary conditions:
To reduce the computational requirements, the
solution domain is defined considering only half the
impacting section, since the ship and the hence the
transverse sections are traditionally symmetrical
about their central vertical plane. Additionally, the
body is considered to impact vertically in the same
direction that gravitational force acts due to which
the fluid flow can also be assumed to be symmetrical
about this plane.
Bach
( S B ~
Outlet ~ So ~ ~~
| Air |
Free Surface ( SF)
tI! fl ' I I ' t
Figure 1: Schematic diagram for computational
domain of Wedge section
Inlet (S.): Slamming impact of the body section on
to the water surface is simulated by keeping the body
stationary and letting the water at the inlet boundary
(SI) to move at the velocity of impact vats. At the
inlet boundary condition, mass flow rate is fixed
irrespective of the internal pressure as An = yin,
where n is unit normal vector along the boundary,
and vin is the instantaneous velocity of impact.
Body Boundary (SRL With the body being
stationary and impermeable, fluid velocity at the
body boundary is equal to that of the wall itself,
, which essentially means that flow exists
v = V ._77
only in tangential direction to the boundary surface vt
2
OCR for page 710
and the velocity perpendicular to the body surface Vn
is zero. Additionally, the tangential velocity of the
fluid at the wall is equal to that of the wall itself,
which follows from the fact that the viscous fluids
stick to solid (no-slip condition), i.e., the tangential
stresses Ant exist at the body boundary but the normal
viscous stresses are zero.
Free Surface (SFL TO determine the instantaneous
shape of the free surface and the forces exerted on the
fluids in contact, both kinematic and dynamic
conditions need to be fulfilled on the free surface.
Kinematic condition: No convective mass transfer
through the free surface; i.e., the fluid velocity
component normal to the free surface is equal to the
free surface velocity.
Dynamic condition: Forces acting on fluids in
contact at the free surface are in equilibrium. In the
absence of surface tension, this means that the
stresses T on both sides of the free surface are equal.
In the absence of wind, shear stresses are generally
neglected for ship flows. With the further assumption
that viscous effects on the free surface boundary are
negligible, in which case normal stresses are
neglected, the pressures on either side of the free
surface are equal, which are simply atmospheric.
Walls (Sw~The vertical boundary at far end of the
computational domain is considered to be wall where
no-slip boundary condition similar to that of body
boundary is applied i.e., the normal velocity across
the boundary is defined to be zero. Also normal
viscous stress is zero at the wall as in the case of
impermeable wall with no-slip condition.
Svmmetrv plane (Set The vertical boundary at the
left end of the computational domain Ss is considered
to be symmetry plane wall as this plane coincides
with the symmetry plane of the body section. Similar
to the case of a wall, here also the velocities normal
to the plane vn are zero. At the same time, the
tangential shear viscous stresses on the symmetry
boundary are zero, but the normal stresses Inn exist.
Outlet (SOL Outlet is open to atmosphere and there
can be fluid flow across the outlet to maintain the
atmospheric pressure. At the outlet boundary, fixed
pressure outlet boundary condition is considered
with the mass flux adjusted to satisfy continuity
equation with the direction of mass flow determined
by the pressure inside the outlet boundary is more
than the atmospheric pressure or not. The former
produces local outlet flow, whereas the latter
produces local inlet flow.
SOUTH3
SOUTH2
SOUTH1
Bod'
2.2 Numerical Solution
Computational Grid
Multi-block structured grid with common matching
interfaces has been generated for the solution domain
considering its simplicity and the ability to maintain
conservativeness and reduction in numerical
diffusion. Rectilinear grid is chosen with x-axis along
the length of the section, y and z-axes along section's
breadth and depth respectively as shown in Figure 2.
YL
. ~
L~ it'' ~ _Y
, rUTFT ;_
Figure 2: Sub-division of the Computational Domain
The sizes of computational domain (YL, ZL), grid
and its distribution are problem dependent, and are
chosen to be sufficiently large enough to obtain
numerical results independent of these factors.
Boundaries of the computational domain along the
horizontal (z=0) and vertical axes (y=0), designated
as 'Low' and 'South', are divided into two segments
(Lowl and Low2) and three segments (South!,
South2 and South3) respectively. The other two
opposite boundaries are divided in a similar way with
the number of cells on opposite sides being same to
generate structured grid. With the section's shape
remaining constant along x-axis, grid size along x-
axis is kept at one cell or CV.
Grid points along the segment containing the body
boundary are distributed uniformly along the body
girth mainly to (i) Ensure that the cell corner points
lie on the body boundary for its better representation
and (ii) Achieve smaller size of cells where the body
curvature is steep within the limits of acceptable cell
aspect ratios of 10.
Finite Approximations
Convection is then approximated using Hybrid
Differencing Scheme (HDS), a combination of 2nd
order Central Differencing Scheme (CDS) and the
first order Upwind Differencing Scheme (UDS) to
ensure higher stability and better accuracy by
considering the transport property of the convective
term in a better manner. The diffusive term is
discretized employing 2nd order CDS. The main point
to be noted is that, interpolation scheme of order
3
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Representative terms from entire chapter:
slam loads
higher than second makes sense only if the surface
integrals are also approximated using higher order
follllulae. Higher order schemes may reduce
numerical diffusion but give unstable solutions.
For discretizing the pressure term, the mid point rule
approximation for the surface integrals is used with
the pressure term interpolated linearly to the center of
cell faces. Source term containing the body forces
due to gravity, represented by a volume integral,
cannot be replaced by surface integrals and hence
this non-conservative term is integrated over the cell
volume by a simple second order accurate
approximation, which also becomes exact since the
fluid density p is considered to be constant.
For the present case of simulating slamming
phenomena, implicit scheme for transient terms is
considered due to its unconditional stability and
ability to obtain physically realistic and bounded
results and does not put restrictions on the size of the
time step through courant condition.
Free Surface
The shape and position of the free surface keeps
changing with time during slamming of the ships and
if the cells, which lie on the free surface boundary,
are known, implementation of the free surface
boundary conditions is straightforward. In general,
dynamic condition of the free surface is implemented
directly whereas the kinematic condition is used to
update the free surface position for each time step in
an iterative manner.
Volume of Fluid scheme (VOF) is employed for
interface capturing, as the same is considered to be
computationally more efficient and also treats the
overturning of free surface. Here, a scalar transport
equation as given below for the volume fraction 'c'
of fluid is solved in addition to the usual conservation
equations.
,9 JcdQ + |cv.n dS = 0
The values of volume fraction indicate the Guides)
present in the cell. where c=0 and 1 indicating the
, ~
cells filled completely with air and liquid
respectively, and any value between 1 and 0
indicating presence of both air and water. Physical
properties of the fluids involved in the computational
domain depend water and air, and the volume
fraction c, and are obtained through piece wise
interpolation. Limiting factors imposed on the
volume fraction c to preserve sharpness of the free
surface interface are as follows:
c=0 if C
3 Incorporation of Deceleration
effects
(ii)
It can be seen from the experimental data (Zhao et.
al., 1996) that the impact velocities are not constant
through out the slamming period, but vary depending
upon the geometrical shape and mass of the section
itself, known as 'deceleration effects'. This is mainly
due to the changes in the total force history
experienced by the section during the drop tests, i.e.,
resultant of the slamming (both hydrodynamic and
hydrostatic) impact force and the weight of the body
acting in opposite directions. Impact pressures on the
body surface are known to be functions of the
section's shape and instantaneous impact velocity. It
is therefore imperative for any numerical model to be
incorporated to consider variations in impact
velocities during slamming.
Implementation of the velocity variations during the
numerical solution is relatively straight forward if the
impact velocity profile is known in priori. However,
this may not always be available before hand and the
same may need to be estimated during simulation
itself. Deceleration effects are therefore implemented
in the present numerical method through one of the
procedures briefed below.
Assigning the instantaneous impact velocity
at every time step from the known velocity
profile with respect to impact time.
Estimating the instantaneous impact velocity
during the numerical simulation itself based
on the different instantaneous forces acting
on the impacting body.
In the first method, impact velocity profile measured
during the experiments is fitted into a polynomial as
a function of impact time. In the second method,
impact velocity of the body is estimated during the
simulation at every time step, with the initial impact
time being the instant when body touches water
surface. The first step involved in calculating
instantaneous impact velocities is to estimate the
instantaneous acceleration of the body from the
resultant vertical force (F) acting on the section. The
total resultant vertical force (F) per unit length of the
impacting body is the difference between the vertical
slamming force (S), which also includes the
hydrostatic force, and body weight (mg).
~ = m 2 = ~ - mg
Mass of the freely falling body is therefore an
additional factor influencing the impact velocity
profile with time, consequently the pressures and
hence the total slam force.
The vertical slamming force S is calculated by
integrating the instantaneous cell pressures obtained
numerically, over the whole impacting surface by
considering the corresponding impact area of each
cell in that direction, whereas the weight of the body
(mg) remains constant during the whole impact
period. From the resulting instantaneous body
acceleration, it is straightforward to get the body
velocity by multiplying the acceleration with the time
step, which is very small in the order of milliseconds
used for numerical simulation.
4 Numerical Results
To illustrate applicability of the present numerical
method to practical ship design problems on
slamming, it is necessary to validate the numerical
model based on CFD techniques through correlations
with the model experiments. This can only be
achieved by clear understanding of the different
factors influencing both the numerical and
experimental results and implementing the same
accordingly. These influencing factors broadly fall
into two categories as those
.
.
Affecting the numerical accuracy of the results
like proper representation of the boundaries,
computational domain, grid and cell sizes etc.
Representing the actual test condition like body
decelerations, position of testing tank wall
boundaries with respect to body, three-
dimensionality etc.
Details of Physical model
Experimental results as obtained by Zhao et. al.
(1996) for the water impact of the wedge section
have been used to validate the present numerical
results. The geometrical and experimental details of
the wedge section used by Zhao are given in Table 1
below.
Geometrical details of Wedge test section
Length of the section L [m]
Breadth of the section B Eml
.,
Vertical distance from Keel to knuckles D Eml
~ .
Length of measuring section Lm Em]
Weight of drop-rig including ballast [kg]
Weight of Measuring section [kg]
Initial Impact Velocity Via [m/s]
Table 1: Wedge section details (Zhao et. al., 1996)
1.000
0.500
0.l4s
0.200
241.0
14.5
~ 1SO
For better validation of the numerical results, direct
comparison with more than one parameter obtained
during the experiments as listed below is considered.
Impact velocity profiles,
Vertical impact force on the whole section, and
Pressures at certain locations on the surface.
5
4.1 Deceleration Effects
Variations in drop velocity of the body over impact
period, known as 'deceleration effects', have been
incorporated in the numerical model in order to
simulate the actual test condition of drop tests.
Observation of the experimental data (Zhao et al.,
1996) reveals that the total impact period for the
Wedge section is around 0.025 seconds with the
impact force peaking at 0.0158 seconds. Initial
impact velocity for the wedge section is 6.150 m/s
and the experimental data shows that the impact
velocity do not remain constant during the whole
impact period.
Impact of wedge section is numerically simulated
considering the deceleration effects following both
the methods i.e., using (i) known velocity profile
from the experiments (Zhao, et. al., 1996) and (ii)
estimating the instantaneous impact velocity during
simulation itself.
In the first method, impact velocity data V(t),
measured during the experiments, is fitted into a
polynomial as a function of impact time as given
below.
v(~) = 6.~s+~6.0~*` - 6398.94*r + ~02439.20*~+ 2006636.38*~4,
0.0<~<0.025 sec
In the second method, at every time step during
simulation, the impact velocity is calculated based on
the different instantaneous forces acting on the
impacting body as briefed in Section 3.
Computational Domain Size
Initial computational domain size ratios of Dy =10
and Dz =2 are chosen along y and z- axes for
generating the grid over the computational domain.
Here, Dy is the ratio of domain width YE to body half
breadth (B/2), where as Dz is the ratio of the domain
depth above deck or below keel of the body Zig to the
body depth D (Figure 2~. Computational domain size
along the x-axis is chosen to be one meter.
Grid and Cell sizes
Since, the shape of the impacting section does not
change along its length, grid size along x-axis is kept
at one cell or CV, whereas grid sizes along y and z
axes are chosen in such a way that the total number
of cells in the whole computational domain is kept at
optimum level to reduce the computational time but
produce grid independent results. Distribution of grid
points is carried out using stretching functions so that
the cells closer to the body boundary are as small as
possible apart from ensuring that the neighboring
cells at each segment boundary do not vary much in
size to achieve numerical stability during simulation.
Details of minimum cell sizes corresponding to a
particular grid size chosen initially for the
computational domain is given in Table 2.
Item | Lowl | Low2 |Southl|South2|South3| CO°maP |
Glid Cells 60 30 30 60 30 90 X12O
C~etfChlhcineg I 1.000 1.161 1.085 1.000 1.085
din cel I 4.16e-3 4.19e-3 2.57e-3 2.41e-3 2.41 e-3 ~ 406e 3
Table 2: Initial Grid and Cell Sizes for the
computational domain
Results
Convergence of the solutions is characterized by the
amount of the sum of residuals during each time step.
A preliminary study on the number of sweeps
showed that a minimum number of 15-20 sweeps
were necessary to achieve convergence. Numerical
simulations with different impact velocity profiles
have been carried out and the impact forces obtained
on the whole wedge section are shown in Figure 3.
p )11 lag I: Shim ~ lag Vetci~ Vs ~ picttm e
6 .4
ED
~ 5.6
=52
A
~ 4.8
he 4A
4D
. ~ ~ Experiments ...... constantVeL I ~ ,j
l Vay~g Vex Sodom ill Vay~g Vet (Ca'cchted)
.
O.000 ODO5 OD1O OD15
~ pacts e i: Seconds
0.020 OD25
3 )11 IDG! :Shaa ~9 brceYs ~~6ctt~ e
| a Expedients - constantVel I
12000 . . I _0ar~n 0~1 l~n~mn~ ~l, ..i~0anrnn Sol Ira~,llAm~l l
10000
~ 8000
z 6000
~ 4000
an
=
~ 2000
~ Ol i~"6 . ~ I
OD20 OD25
j . - ., ~ . a_ ~ ~,"~ - _~ . - ., U . - _ \w "_~_ - , ~
ODOO OD05 OD1O 0.015
hpactto e r: Seconds
Figure 3: Decelerations effects on Slamforce
Discussion
Impact forces obtained during constant velocity
impacts show very poor correlation with
experimental results as can be expected, with
discrepancy being more than 100%. Incorporation of
deceleration effects into the numerical model has
improved the correlation between numerical and
experimental results significantly with the forces
obtained based on instantaneous calculated impact
velocities have a closer agreement with the
experimental results compared to the case of pre-
assigned (polynomial) values. Better agreement is
6
noticed during initial stages of impact, but
differences are larger at later stages with the peak
impact forces from experiments and numerical
solutions on the measuring section being 5100 N and
6200 N respectively, even though the time instant at
which these peak values occur is nearly same. This
also is the time instant at which the flow is nearing
the knuckle or separation point. Also, the impact
velocity profile estimated during the simulation is not
matching well with the experiments.
Zhao et. al. (1996) had argued that the differences in
peak impact force on the wedge section considering
Boundary Element Methods are mainly due to the
cross flow or three-dimensional effects. This also
explains the differences in impact velocities, which
can be attributed to the fact that impacting velocities
during simulation are estimated based on the
impacting force acting on the total length of the
section, which may not be constant along its length.
Zhao had further concluded that the discrepancies in
impact force would be significant beyond the time
instant at which the flow reaches a position where the
ratio of body breadth to length crosses 0.25. Since the
present section falls in this category, it is considered
worthwhile to carry out few additional simulations on
three-dimensional sections. Further, Sames et. al.
(2000) had concluded that discrepancies can also
occur due to the effects of domain (analogous to tank
size during experiments). Before undertaking studies
on cross flow, investigations have been carried out to
obtain solutions independent of all the effects due to
different factors affecting numerical solutions like,
grid size, computational domain size etc.
4.2 Effect of Grid Size
Effect of grid size on the numerical accuracy has
been carried out by considering four different grid
sizes namely Coarse, Medium, Fine and Very fine
Grids, with the number of cells doubled successively
for the four cases along y and z axes as given in
Table 3 below. The minimum size of cells obtained
along each region for the section is given Table 4
respectively. It can be observed that the cell sizes are
the lowest near the body. Computational details for
impact of wedge section considering different grid
sizes are given in Table 5.
GRID
SIZE
Coarse
Med.
Fine
Very
Fine i
WEDGE SECTION: GRID DISTRIBUTION
Lowl~ow2~outh~South2| South3
15 8 8 15 8
.
30 15 15 30 15
60 30 30 60 30 ~-
l20 60 60 120 60
Near body
(Whole Domain)
15X15 (23X31)
30X30 (45X60)
-60X60 (9OX120)
120X120 (180X240)
Table 3: Grid Sizes for computational domain
Grid Size
Coarse
Medium
Fine ~
Very Fine .
Lowl
3.00
l 47
0.72
0.36
Wedge Section: Minimum Cell Size [mm]
Low2
10.79
4.52
2.02
1.19
Southl
10.92
6.3
2.78
1.31
South2
7.00
3.60
1.69
0.85
South3
101.50
5.08
2.71
5.28
Whole
Domain
300
1.47
0.72
0.36
Table 4: Min. ell siz es corn to differ rent G. id sizes
Grid Size
Coarse
Medium
.
Fine
Very fine
Wedge: Computation time on
Pentium II, 300 MHz, 64 Mb RAM machine
l No. of CVs:
I Near body
| (Comp. Dom.)
1
225 ( 713)
900 ( 2700)
l
3600 (10800)
14400 (43200)
T Min. l
I Cell |
I Size |
I im] |
.623e-3
~.81 le-3 |
12.406e-3
Tl.203e-3 1
No.of
Time
Steps
(nt)
54
112
222
456
IMin.
|time step
l size dt
| |secl
e4
1 .25e4
4.70e-S
|2.80e-5
ICPU
Time
I [has:
| mind
10 o4
0:15
1:25
Table 5: Computation time for different grid sizes
Numerical results consisting different slamming
parameters like impact velocity profile, pressures at
distinct locations and the impact force on the whole
wedge section are presented in Figure 4. Observation
of time series of impact velocity profiles vertical
impact force and pressures clearly indicate that effect
of grid size is not significant and a grid size of 60X60
near body (90X120 in the whole domain) is
considered to be sufficient to capture all the flow
features for the impact of the wedge section.
(A) WEDGE Summing Velocity Vs Impact time
1 6.4 r
1 .~ 6.0 ~
1 ~s'6t
h Grid :15 X 15 r
1 ~ 4.8 ~ - Grid :30 X30
I ~ I I Grid :60 X 60
05 4~4 ~ Grid 120X 120
4.0
1 o.ooo
~ ' - ~
1
0.005 0.010 0.015 0.020 0.025 1
Impact bme in Seconds |
l
I (B) WEDGE: Slamming force Vs Impact time
7000
6000
1 ~40001 ~ ~ 1
3000 1 ~ ~ x _ ~ \
in, 2000 -1 _~ 1 =Grid 60 X60
1 E 1000 1 '' 1 Grid: 120 X 120 1 1
1 ~ 1~ 1
I ~ 0-r I
1 0.000 0.005 0.010 0.015 0.020 0.025
I Impact time in Seconds
(C) WEDGE: Pressure distribution over Breadth
( at impact time t ~ 0.0202 seconds )
0.00 0.05 0.10 0.15 0.20 0.25
Section Breads in m
Figure 4:Effect of Grid size on Slamming parameters
7
4.3 Influence of Domain boundaries
The main aim of these investigations is to identify the
minimum domain size based on the impacting body
dimensions, which yield domain independent
numerical solutions. Domain sizes in both the
directions, i.e., along breadth (YL) and depth (ZL) of
the impacting body, have been varied systematically
to study its effects on the numerical results. Details
of the systematic variation of the domain size ratios
DB (= YL/(B/2)) and DO (= ZL / D) as considered are
given in Table 6. Actual dimensions of the
computational domain sizes will be dependent on the
physical size of the section. Simulations are carried
out considering independent variation of the domain
size ratios (DB, DD) to have a better understanding of
their effect on the slam loads.
Wedge section: Domain size along y and Taxis
YL 1m1 DB ~ Y. /(B/2) ZI Iml
,
0.75 3 0.290
1.25 5 0.580
2.50 10 0.870
3.75 15 1.160-
- DD=ZI /D
2
4
. 6
8
Table 6:Domain sizes along y and z-axes
Numerical results containing (i) impact velocity
variations and (ii) impact force on the section are
presented in Figure 5 and Figure 6 for the domain
variations along breadth and depth respectively.
3)! eDGE:Sims ~'Vetcg`Vs Attire
6.0
5.6
~ i.:
.R 4.8
D ~ g
.- 4.0
6.4 -
~ pacts e ~ Seconds
¢~11 EDGI:S~m s'b~ceVs &;acttill6
~ pact ~ e :~ S econds
Figure 5: Effect of Domain size along section 's
breadth (Y-axis)
'}' IDO1'Sbs ~ it'Tetcl~g ~ pectin ~
6A
. . . . _-
^,.L ~
.~ ~ ~
{D.
ODOO 0~05 0.010 OD15 0~20 OD2S
~P3 t~e~SeCO:6S
{till IDGI:Sh. ~ ii' Arc ~ pecitic ~
6000
solo
logo
; -
~ 3000
,, 2000
.~, 1000
Van
I
_~
I=' -' -
~ 2L~ - 6 ZL~ - 8
9'
O D20 O D2S
.
O TOO O DOS O [10 O [15
~ PaCt~ e ~ S8CO3dS
Figure 6: Effect of Domain size along section 's depth
(z-axis)
These results (Figure 5 and Figure 6) clearly indicate
that, domain independent results are obtained when
the domain ratios DB and DD are greater than 10 and
4 respectively. All further simulations on these
sections are thus carried out keeping these limitations
in mind to obtain domain independent results.
4.4 Effect of Viscosity and Gravity
Viscosity effects on the water entry problems are
analysed by considering the coefficient of viscosity
of water as l .O l e-6 m2/sec. Gravitational forces were
introduced into the numerical model in the form of
body force acting vertically down along the negative
z-axis. Simulations including gravity, gravity and
viscosity together were carried out as shown in
Figure 7. Effects due to viscosity and gravity are
found to be minimal probably due to low fluid
velocities and size of the section. Greenhow (1989)
also concluded that gravitational forces are important
only when impact time t > V/2g, where g is
acceleration due to gravity.
11 cDa ~ · a ~e ~ ing Sbxc- V. ~ proceed -
~ ~n
anon
500
400
B 300
.. 2 0 0
~ 100
;
~ O
0 . Jr N 0 V is c 08 icy, N 0 C ra vibes j A_ |
w id] V ~sc08ily
o ~/ W ith G savvy
. / . W ith C m viny a n d V is c o ~ ity | l
o.ooo o.oos o.olo o.o.s ~ ^~~
~ pretty · ~ Abscond.
0.020 0.025
Figure 7: Effect of Gravity and Viscosity on impact
force of Wedge section
8
It can be observed that numerically obtained value
for peak impact force on the measuring length of
wedge section (0.2m), which is independent of grid
and domain sizes is around 5600 N. whereas the
same from the experiments is around 5100N. Also
the impact velocities from the numerical solution are
lower than those obtained from the experiments
especially at later stages of impact, indicating that the
numerically estimated impact forces are higher,
which is also the case. To understand this
discrepancy and the effects of cross flow on the
impact forces, further studies are carried out
considering the three dimensional wedge section.
4.5 Effect of Cross flow
To study the effects of Cross Flow along the sections
length (x-axis) on the slamming force during drop
tests, impact of three-dimensional wedge section as
used during the experiments (Zhao et al. (1996~) is
considered. For this purpose, the computational
domain for the wedge section is extended along the
longitudinal direction (A ~ beyond the section's
length (L/2) as shown in Figure 8. In this regard,
domain size ratio Do along x-direction is defined as
the ratio of domain width XL. to body half Length
(L/2~. Few parametric studies are carried out to
identify the minimum domain and grid size ODIN
along x-axis, which yield domain and grid
independent numerical solutions.
~ z ~
i L/2 7.L
D
..... ....
Body ZL X
Vin /~ ~ ~ ~ -
Figure 8: Extent of Computational domain
along section 's length (x-axis)
Grid Generation
The Wedge section used in drop tests has constant
sectional shape along its length with the measuring
section located exactly at the center. The impacting
section is symmetrical about both x-y and y-z planes.
By exploiting this double symmetry of the impacting
section, the computational domain is discretized
considering only one quarter of the section to reduce
the total number of CVs in the whole computational
domain leading to reduced computation time.
Grid and Domain sizes alone length
Grid and domain sizes along the x-direction (length)
are varied systematically considering the grid size of
40X40 (near body) along y and z-axes due to
limitations in computational facility. It was also
observed that results considering this grid size are not
significantly different from using grid of size 60X60.
Details of the systematic variations of grid and
domain sizes (D~) along x-axis are given in Table 7
and Table 8 along with the actual dimensions of the
domain, which naturally varies with the length of the
impacting body (L/2 = 0.5 m).
3D Wedge section: 3D Grid size along x-axis
Grid Grid Grid in Min.Cell
Size Near body whole Domain Size iml
Coarse SX40X40 1 SX60X80 3.608e-3
Fine 1 OX40X40 20X60X80 3.608e-3
Table 7: Grid sizes along length
3D Wedge section: 3D Domain size along x-axis
Details of Grid and Domain
sizes along y and z axes
Grid Size (Near Body) 5X40X40
Grid Size (Whole Domain) I OX60X80
.
Ye (DB) 2.500m(10)
Zip (DD) 0.870m (8) .
Case | XL DL
1 ~ 0.75 1.5
2 ~ 1.50 3
3 1 2.50 5
4 ~ 5.00 10
Table 8: Computational domain sizes along length
Computational time
Details of computational time required for
undertaking the simulations on Wedge section using
3D grid are given in Table 9.
Grid
Size
Coarse
Fine
No. of CVs:
Near body
(WholeDomain)
SX40X40
(lSX60X80)
10X40X40
(20X60X80)
3D Wedge section: Computation time on
Pentium II, 300 MHz, 64 Mb RAM machine
.
Min.
Cell Size
lml
3.608e-3
3.608e-3
No. of
Time
Steps (nt)
152
152
l
Min.time
step size
dt |sec
8.9e-S
8.9e-S
CPU
time
h ml
10:17
14:25
Table 9: Details of computation time for different 3D
grid sizes of Wedge section
Slamming impact forces obtained for different grid
and domain sizes along x-axis are shown in Figure 9
and Figure 10 respectively.
Figure 9: Time series of slammingforces for
different grid sizes along x-axis
9
91 ~DO~s8~— ~ ~g db=:e Ve ~ D—Otto
7000
6000
4000 ~
3000 . f ._~.~XL~ ~ 10 `_
D 2000- Jo - XL~- 5 me_ ~
j 10 0 0 7 _XL /L 1 .5
O
o.ooo 0.005 0.010 0.015 0.020 0.025
~ ~—ate" · ~ s-cood.
Figure 10: Time series of slammingforces for
different domain sizes along x-axis
Results from Figure 9 indicate that the effect of grid
size along x-axis is not significant, understandably
due to the fact that the section shape does not change
along the section length. Results from Figure l O
indicate that a domain size ratio of Do= 5 along
section's length is enough to obtain domain
independent results.
The slamming impact forces obtained on each
transverse strip along the length are presented in
Figure 11. This figure clearly shows the effect of
cross flow, with the total impact force on the section
reducing as we move from center to the edge (section
1 to section 5~. In other words, as the immersed
sections breadth to length ratio increases, considering
the length to be equal section's position along length
from the edge, the effect of cross flow becomes more
significant in reducing the impact force on the whole
section.
Wedge :Slamming Force Vs Impacttime
3000 - .......... .. ~
2 ~ 0 0 ~ # \
E Jr S action 1
E 1000 · ~ Section 2
of Jo Section 3
SOO ~ Section 4
O ~ ~ S action 5
0 0.005 0.01 0.015 0.02 0.025
Im pact tim ~ in secede
Figure 11: Slammingforces on different transverse
sections of Wedge section - Section I is at center,
Section 5 is nearest to the edge along section length
edge compared to the sections at the center, and the
same can be observed in Figure 13 considering the
pressure distribution at a particular time instant.
Further, this cross flow observed initially at the edges
slowly spreads towards the center with the increasing
impact time as the flow is nearing knuckles upwards.
Free surface elevations obtained for the section 1 at
center are shown in Figure 14.
.. . .
Figure 12: Flow distribution along the length of
Wedge section at t=0.292 sec
. .
Figure 13: Pressure distribution along the length of
Wedge section at t=0.292 see
it'
/~L · 0 '49 see
~1
TIC Is
/~ - ~ - 0.110 sec
- 0.044 see
Figure 14: Free Surface elevations at different time
instants during impact of Wedge section
Comparison of 2D and 3D results
Comparison of the results obtained for the Wedge
section considering two-dimensional and three-
dimensional grid is shown in Figure 15.
Flow patterns along the impacting wedge section are
shown in Figure 12. It can be observed that, the fluid
initially flowing upwards along positive z-direction
tends to move away from the center to the edges of The maximum impact force on the measuring section
the section longitudinally (along x-axis), clearly reduces from 5600 ~N/0.2m] in 2D case to
indicating the 'Cross flow'. This cross flow reduces 5200tN/0.2m] in 3D case. This comparison shows
the vertical component of the impacting velocity interesting features about the effects of cross flow or
thereby reducing the local impact pressures on the three-dimensionality as summarized below.
section. In other words, reduction in the impact force
on the section occurs due to the cross pow of the
fluid from center to the edges of the section's length.
This reduction is greater for the sections nearer to the
· Numerically calculated impact velocity profile
matches very closely with the experimentally
measured values
10
.
.
Vertical Impact force on the measuring section
and also the whole section in 3D case is less than
those obtained from the 2D case, clearly
indicating the effect of cross flow and three
dimensionality.
Vertical Impact forces obtained from 2D and 3D
cases match well during initial stages of impact
but differ during later stages of impact with the
3D case predicting smaller magnitudes
indicating occurrence of cross flow.
. . .
O.000 o.oos 0.010 0.015
~ p.ct~ ~ ~ atonal
7000
6000
~ A A ~
VVV ' _
4000
0 3000
2000
#
1000
o
O .00
0.020 0.025
~ it\
O —2D C ad: 40X40 1
3D C ndt lOXdOX40 |
0.005 0.010 0.015 0.020 0.025
Figure 15: Effect of Three-dimensionality on
slamming parameters of Wedge section'
Comparison with the Experiments and other
Theoretical data
Results obtained using the present improved
numerical method consists of (i) impact velocities
during drop tests, (ii) pressure distributions along the
boundary of the impacting body and (iii) Slamming
impact force on the measuring section and (iv) free
surface evolution at different time instants during the
impact. All the results except impact velocity profiles
are compared with the Boundary Element Method
(Zhao et al., 1996) and the CFD method (Muzaferija
et al., 2000~. A simultaneous comparison of all the
results is made with the experimental results also as
obtained by Zhao. Both Zhao and Muzaferija use the
experimentally determined impact velocity profiles
for their numerical simulation whereas the present
CFD method estimates the same during simulation
itself.
Zhao et al. carried out simulations using two
dimensional grid and corrected the impact forces on
Wedge section based on the Meyerhoff (1970) results
to account for the cross flow and three
dimensionality. Muzaferija et al. carried out initial
simulations using 2D grid and accounted for cross
flow by undertaking further simulations using 3D
grid. The present numerical calculations also
consider both 2D and 3D grids for simulating the
water impact of Wedge section.
Comparison of results on Wedge section from all the
methods is shown from Figure 16 to Figure 19.
Figure 16 and Figure 17 show comparison of vertical
impact velocity and forces on the section
respectively. Pressure distributions on the surface and
at locations of the section and different time instants
are shown in Figure 18 and Figure 19 respectively.
W ZDC Z s 8 ~e ~ iz~g V.bc~ty Ve be pecttl.
6 .:
6.~ .
5.6
:^
S.2 -
4.8 -
.4
.o .
0.000
Exp. (Zhao etaL, 199
_ C ~ D (P resent)
0.005 0.010 0.015
~ pacts e n 5 econds
0.020 0.025
Figure 16: Comparison of impact velocities on
Wedge section
| 11 ~DC ·'81~ - ~ ins lb=- Va ~ pectt~ -
7000
.._ +' - '- ~`
6000. ..~-'~ ~
solo ~~' ~ \`,
~ 4000 ~~
.~; 3 0 0 0 . `~ ~ _ Hi.
,11~ 2000 ~ `` Exp. (Zhao stat, 1996)
·~ BEM (Zhao etaL, 1996)
1000 . - _
{' CFD ~ uza~ ctaL ,2000
0, ~ ~ CFD (Present)
0.000 o.oos u.ulu u.u;~ .
b~pacttile h Seconde
-
.020 0.02s
Figure 17: Comparison of slammingforces on
Wedge section obtainedform different methods
Comparison of slamming forces on wedge 30 section
indicates that the results obtained by the present
numerical method correlate well with the
experiments Figure 17. Zhao (1996) obtained better
correlation by correcting the impact forces for cross
flow and three-dimensionality. Muzaferija et al.
obtained still better correlations for the peak
slamming force with slight deviations with the
experimental values are observed during the initial
and final stages of impact. It may be however be
remarked here that he achieved these results by
considering a particular situation where the domain
boundaries are restricted with domain size ratio (Dx)
along length being 0.25. The results obtained using
the present method did not encounter this situation.
This can probably be due to the fact that he had
concluded that domain independent results are
achievable with domain size ratios of DO = 3 and DB
= 10. But it was observed during the present studies
11
that the domain size along depth is very sensitive and
the same need to be increased further to 4 or 6 to
achieve domain independent results. This fact could
also be verified considering the 2D results, where the
peak slamming force from the present method is
5600 (N/0.2m] (Figure 15), whereas the same
obtained by Muzaferija is around 6000 tN/0.2m].
Incidentally, this value of 6000 tN/0.2m] is similar to
that obtained for a domain ratio of 2 along depth
(Figure 6~. Basically, as the section is flatter, the
domain ratio has to be larger. It is thus believed that,
domain size restriction could perhaps be the reason
for his results to be slightly deviating from the
experimental results.
UL}ll EDGE:Pr.~or'VaD't;Deci~ isp~ct~.
{Platdep~ ~ OD125s )
8
6
=~ l 1~
43 0 2
O_
~ O ~
me
~ O
~ ~ Exp.(2hao,19961 ~ 8E11 2hao,19961
A
_ ~ CFD (S3:es,20001 CFD presents
.000 0.005 OD10 OD15
h pactec e ~ S econds
) ~ I D G I: P r Acute Va:i~tiDe ~ ~ pact
IP2etde~tt 0.0375s )
OD20 OD25
. 1 ~ E P lZb2o~l996) ~ BEE 12hao,19961
.-
. ~ ~ CFD ISa~es,2000)—CFD $resent)
0.00 OD1 OD1 OD2
~ pacts e ~ Seconds
O D2 0.03
(C )'1 IDS I: Pn8BSU~ Vln~t60 v its ~ pacttiB 6
( P 3 atd.pti r 0.0625 11t )
Exp.~2hao,1996) ~ DEt' (2hao,1996)
- ~ | ~ CFD iSa~es,20001 - CFD present
0.020 0.02S
once o.oos OD10 OD1S
pretty e ~ Seconds
) ~ ~ D G I: P rhesus V8D1UDB v is ~ pact tie 6
( P. atlepth ~ 0.0875 ~ )
8
6
a
ODOO O.OOS OD10 OD1S
~pactts e ~ Seconds
10 ·
8
E:p.i2hao,1996) ~ Bell 12hao,l996
CFD iSa~es,20001 CFD Resents
O D20 0.02S
) ~ IDS I: P 28SU~ VasatiDe v ith ~ p8Ctti'
(P5 ~td6pth a 01318 ~ )
~ E:p.¢hao,1996~ ~ BE11 12hao,1996~
6. · CFD (Sa~es,20001 —CFD Resend-
~ .
2. .
O-
O D 00
O DOS O D10 0 D1S
~ pacts e ~ Seconds
0.020 0 D2S
Figure 18: Comparison of Slam pressures histories
at different locations on the surface of Wedge section
obtainedirom different methods
(A) WEDGE: Pressure distribution along She surface
~ at Impact time ~ ~ 0.00435 "coeds )
6
{¢ 5;
3~} 4
~ oc 3
c 2
O- . _ , .
0.000 0.200 0.400 0.600
Nondim~nabnal Depth d S - Sian
_ _
~ Exp. (Zhao et al.,1996)
· BEM (Zhao et al., 1996)
CFD (Muzaferija et al., 2000
CFD (Present)
o.eoo Boon
(B) WEDGE: Pressure dIstrlbutIon along the surface
( at Impact time ~ ~ 0.0158 seconds )
Exp. (Zhao et al.,1996) · BEM (Zhao et al., 1996)
CFD (Muzafenja et al., 2000 ) CF D (present) 7~ K
e
7
6
·r^ s
t.84
lo 2
1
O
0.00~) 0200 0.400 o.eoo
NandImendonal Depth d Season
0.800 1.000
6
4
83- 3
's2
O.XO 0200
(C) WEDGE: Pressure distribution along the surface
( at Impact time ~ ~ 0.0202 seconds )
I ~ Exp. (Zhao et al.,1996) · BEM (Zhao et al, 1996)
CFD (Muzafenja et al., 20X) —CFD (Present)
1 ~
0.400 o.eoo 0.800 1.000
No~lm~i~ Depth°'Sectbn
Figure 19: Comparison of Slam pressures along the
surface of Wedge section obtainedfrom different
methods at different time instants
5 Conclusions
Numerical solution method using CFD techniques
considers slamming of two dimensional wedge
section produce reasonably accurate results for
different slamming parameters suitable for design
purposes. Finite volume discretization techniques
considering simple rectilinear grid deals with the
slamming phenomena quite effectively. Application
of volume of fluids method has captured the free
surface evolution including breaking and overturning
effectively. The present numerical method also
considers the key factors affecting the slamming of
wedge section including varying impact velocities,
domain size and cross flow. Based on the present
studies the following conclusions are drawn.
There are two main conclusions: Firstly, close
correlations of the present numerical results with the
experiments have been obtained considering the
histories of both impact velocity and slam loads for
the water entry of wedge section during drop tests.
Secondly, Effect of impact velocity variations and
computational domain size, cross flow and three-
dimensionality on the numerical accuracy of the
results is significant.
12
Acknowledgements
This work is a part of PhD study, which has been
financially supported by Lloyds Register of Shipping
(LRS), London, and contribution of LRS is gratefully
acknowledged. Special thanks are due to Dr. Susan
E. Rutherford and Dr. Paul C. Westlake of LRS for
their constant support throughout the project. We
also thank Prof. Brian Baxter for his invaluable
comments and suggestions. The opinions expressed
herein are those of the authors.
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13
