www.sut.org
Experiment of free-falling cylinders in water
Sirous Yasseri*
Safe Sight Technology, Surrey, UK
Technical Paper
doi:10.3723/ut.32.177 Underwater Technology, Vol. 32, No. 3, pp. 177–191, 2014
Received 24 February 2014; Accepted 11 August 2014
Abstract
Trajectory of objects falling into water and their landing point
and orientation are of interest for the protection of oil and
gas production equipment resting on the seabed. Falling of
small-scale model cylinders through water with low velocity
has been investigated experimentally. Several experiments
have been conducted by dropping model cylinders with the
density ratio higher than 1 into a pool. The main objectives
were to observe the trajectory and the landing point. Similar
experimental results published by other researchers are
reformulated to give common normalised landing points,
which were then used to compare with the author’s tests.
Keywords: cylinder drop experiments, offshore dropped
objects, submerged trajectory
1. Introduction
Study of falling rigid bodies through water with low
water entry speed (less than 10m/s) has great safety
significance for protection of subsea equipment
from impact. The trajectory of an object through
water governs the impact velocity, impact angle
and impact point. Many random parameters affect
such outcomes, which can only be quantified in statistical terms.
The non-linear behaviour of some shapes further
complicates the prediction of their trajectory and
the landing point. Even a simple shape such as a
cylinder or sphere dropped with the same nominal
initial conditions lands at different points. Simple
solid body motions can develop chaotic behaviour
given small perturbations to initial condition or
changes in condition of the medium (e.g. density
and current) along the object’s path.
The present paper reports the results of drop tests
of cylindrical models. The overall premise of the
experiment consists of dropping various cylindrical
* E-mail address: sirous.yasseri@gmail.com
models into water, recording their landing location
and observing their underwater trajectory over the
course of their descent. The visual observation of
trajectories was sketched and also noted in a
descriptive way. The primary purpose of the experiment was to determine where a dropped cylinder
lands. The secondary objective was the trajectory
and angle of impact, as this has some implication
on the level of damage that a dropped object can
impart.
These experiments were performed in a 2.5m
(nominal) deep pool with the assistance of a diver.
Nine cylindrical models were produced and dropped
from a fixed height (to achieve a consistent initial
velocity) and entry angle. Models were retrieved after
all nine of them were dropped and results recorded.
A number of variables govern where an object
lands if accidentally dropped. The most important
variables are:
• The object’s shape – i.e. if it is long, cylindrical,
flat or boxed shaped;
• Mass, drag and inertial coefficients;
• The inclination of the object at the time of drop;
• The initial velocity at the time of drop;
• The relative position of the centre of buoyancy
and the centre of gravity, as this causes the object
to rotate;
• Environmental condition, namely if the sea is
calm (sea surface undulation) and the current
velocity.
The chance combination of these factors causes an
object that is dropped several times from the same
point to land at different locations.
Heterogeneous state of the ocean, currents
changing with depth and time, variations of the
water density and salinity are among the important
factors that could change the trajectories. In order
to analytically replicate a specific experimental trajectory, it is necessary to have precise knowledge of
177
Yasseri. Experiment of free-falling cylinders in water
all these factors at the time of drop. Since this
information is often difficult to quantify reliably, it
appears that the flight of free-falling objects can
only be described statistically.
Objects that will be lifted offshore can be conveniently categorised into four groups:
1. Heavy objects such as blowout preventer (BOP),
Christmas tree, manifold, drill-pipe bundle and
coiled tubing lift frame;
2. Medium-weight but bulky objects, such as accumulator mud-mat, container, support for subsea
distribution unit, and completion basket;
3. Lighter-weight but bulky objects, such as lifting
frame, tie-in spool support;
4. Light-weight long objects, such as running strings
and objects less than 2 tonnes.
Accurate numerical modelling depends primarily on the ability to numerically describe complex
three-dimensional dynamic behaviour, and requires
correct accounting of all the forces acting on the
falling object. Under ideal conditions, the water column is usually represented as a semi-infinite space
with isotropic and constant properties, such as temperature, salinity and density. Under these conditions, it has been shown that an idealised cylindrical
body, free falling through the water, could reach a
number of stable or quasi-stable motion patterns
(Allen, 2006). Depending on the geometry and
distribution of mass, a range of trajectories can be
expected. Several distinct patterns can be observed,
such as straight, spiral, flip, flat and seesaw (Chu
et al., 2005). In addition, a single trajectory may
consist of a single pattern or any combination of
these trajectories.
A problem with model testing is hydrodynamic
scaling with respect to the water depth, which has
a pronounced effect on where the model lands.
The trajectory patterns of falling objects are affected
by the mass, size and shape of the models, among
other things.
the submarine pipelines against dropped objects.
DNV assumes that the landing point of an accidentally dropped object can be represented by a normal
distribution (Fig 1):
 1  x 2 
1
p(x )=
exp −    (1)
 2  δ  
2πδ
where x is the horizontal excursion and δ is the
standard deviation. DNV (2010) gives the standard
deviation as a function of weight and shape of the
falling object, and the water depth (d). The standard deviation (δ) is given by:
δ = d.tan (α)(2)
where α is the spread in the descent angle, which
can be found in Table 10 of the DNV-RP-F107
(2010) code.
The probability that the object lands within a
horizontal distance (r) of the drop point is given by
the equation:
r
P (| x | ≤ r ) = ∫ p(x )dx (3)
−r
When considering object excursion in deep water,
the spreading of long/flat objects will increase
down to a depth of approximately 180m. Below this
depth DNV assumes spreading does not increase
significantly and tends to become vertical.
Parameters used in Equation 1 account for a
nominal current (DNV, 2010). The effect of currents becomes more pronounced in deeper water
because the time it takes for an object to land on
the seabed will increase as the depth increases. This
means that any current may increase the excursion
(in one direction). At 1,000m depth, the excursion
2. Industry practice
Protection of subsea assets around a platform or
a drilling vessel is necessary for the safety of people
working on the surface; this includes environmental protection as well as limiting the commercial
loss. There were attempts in the past to gain insight
into the three questions of landing point, angle of
attack and trajectory. A few published results have
been reviewed in the present paper, and their
results are compared with the current experiments.
A major industry code of practice is the Det
Norske Veritas (DNV) recommended practice (RP)
F107 (2010), which has targeted the protection of
178
Fig 1: Landing point of falling objects according to
DNV-RP-107
Underwater Technology Vol. 32, No. 3, 2014
not too conservative. Tests that are reported in the
present paper provide some further insight.
Experimental results can be replicated using
the dynamics of objects falling through water (see
Appendix A) if the actual data are available or the
input data can be manipulated to match the experimental results. Naturally, in a blind test any similarity between experimental and theoretical results
is a matter of chance. Conversely, results of Fig 2
cannot be replicated by experiments.
If the theoretical results are conservative and
adopting them is not costly, then there is no economical reason for further refinement. If, in the
derivation of standoff distance instead of the most
probable value, the 95% confidence level was used,
then distances may not be sustainable. For a large
standoff distance, the installation vessel may need
to leave its work station to undertake an overboard
lift at a safe distance, lower the load close to the
seabed and ‘walk’ it back to the work station. This
would add substantially to the schedule, perhaps
without much benefit.
Prior to the DNV-RP-F107 (2010), practicing
engineers assumed discrete distribution of falling
objects. For this purpose, the seabed was discretised into a number of rings (generally three)
assuming a dropped cone (e.g. Yasseri, 1997). A
percentage of dropped objects were then assigned
to each ring.
Standoff distance measured from the drop point (m)
could be 10–25m for an average current velocity of
about 0.25m/s, and up to 200m for a current of
1.0m/s. Only light objects, with large surface areas,
can glide and move further away from the drop point.
Regarding the installation of subsea equipment,
it is necessary to determine how far an object would
drift if accidentally dropped. This is known as the
‘standoff point’, which identifies how far an installation vessel should be located from a subsea asset
to avoid damage should an object be accidentally
dropped overboard. For this purpose, rigid body
dynamics of falling objects are used to estimate the
distance from the drop and lifting points for various objects. Fig 2 shows the standoff distances for
typical offshore equipment from the point of drop
for three different current velocities (no wave) using
the most probable values for drag, added mass and
inertia coefficients.
This also involves calculating the hydrodynamic
loads. Data were produced for a water depth of
200m. The objects were released from a static position at the mean water level and their free-fall trajectory in the water column was tracked until they
landed on the seabed. Current loading applied
along the vertical axis, and the drift of object from
the vertical axis was monitored. The hydrodynamic
loads were calculated by hand, assuming either a
plate or cylindrical shape for the objects. These
results are useful, provided they are realistic and
Standoff distance measured from the drop point for typical objects (200m water depth)
120
0.5m/s current
110
0.75m/s current
1m/s current
100
90
80
70
60
50
40
30
20
10
0
e)
5t
1
(1
e)
2t
(4
e)
4t
(3
e)
2t
(3
e)
0t
(3
e)
4t
(2
e)
2t
(2
e)
0t
(2
e)
0t
(4
e)
8t
(3
e)
7t
(2
e)
0t
(2
e)
8t
1
t(
)
te
.5
(2
e)
0t
(2
)
(4
)
te
te
re
1
(5
ng
)
te
(5
e
e
e
e
e
p
2
le
m
m or
e
u
dl am k u
ul
us
s
4m tre
8m 8m pip
tri
er
pi × 8
ur × 8 uct
×
×
×
nt bun
pp
c
l
od no
eo g s eou
s
ct
fr
r
l
i
e
t
u
a
n
a
u
t
r
t
m
n
v
s
m
r
o
if
s
4m tm
8m 8m D
8m & s
lla
ni
an
re
st × 8 ort
M
pe g l
rs
ld
×
×
×
×
is
e
ce un cell
n
te
t p ll pi
U
is
ifo 2m 8m
hr
m 5m upp
m
m
R
bi
D
at
i
s
n
i
ou
a
r
0
0
C
u
M
S
m
a
s
2
1
t
D
M
Fr
lt
er
r2
ow
M er
&
ke
ol
oi
et
in
Bl
s
n
ne
C
sk
as spo
ta
at
ai
ai
a
B
n
t
t
m
B
o
in
on
on
C
ud
eC
C
M
Ti
e)
2t
(<
Objects
Fig 2: Standoff distance measured from the drop point (for 200m water depth)
179
Yasseri. Experiment of free-falling cylinders in water
Gilles’ 230 tests
0.4
0.3
Normalised Y axis
0.2
Model No.1
L = 15.1359cm, D = 4cm, m = 2.7cm, weight = 322.5g,
volume = 190.2028cm3, density = 1.6956g/cm3
10.380
8.052
5.725
H(cm)
–1.462
0.866
3.193
H(cm)
0.000
18.468
36.935
M(mm)
0.1
–0.4
–0.3
–0.2
– 0.1
0
0
0.1
–0.3
–0.4
Normalised X axis
Fig 4: Gilles’ (2001) 230 tests. Plot is normalised using the
water depth
Fig 3: Gilles’ (2001) model characteristics. The left-most
column indicates COM position 0, while the right-most
indicates COM position 2
3. Literature review
Free fall of cylindrical objects has received a lot of
attention in the past (e.g., Ingram, 1991; Aanesland,
1987; Colwell and Ahilan, 1992; Chu et al., 2004,
Chu et al., 2005; Chu and Fan, 2006; Chu and Ray,
2006; Chu, 2009). In the following, experimental
results available in the open literature that are relevant to the present paper are summarised.
3.1. Gilles’ tests
Gilles’ tests (Gilles, 2001; Chu et al., 2005) consisted of dropping circular cylinders into the water
where each drop was recorded by underwater cameras from two viewpoints. The controlled parameters for each drop were: centre of mass (COM)
position, initial velocity, drop angle and the ratio
of cylinder’s length to diameter.
Log-normal distribution fitting of Gilles’ tests
0.5
Distance between landing and drop point (r in m)
–1
–0.5
Expon. (Gilles’ tests)
0
0.5
1
1.5
Standard normal variable
Fig 5: Gilles’ (2001) results in log-normal probably distribution plot
180
0.4
–0.2
Model No.3
L = 9.1199, D = 4cm, m = 1.47cm, weight = 215.3g,
volume = 114.6037cm3, density = 1.8786g/cm3
6.662
5.592
4.521
H(cm)
–1.368
–0.297
0.774
H(cm)
0.000
6.847
13.694
M(mm)
0.05
0.3
–0.1
Model No.2
L = 12.0726cm, D = 4cm, m = 1.7cm, weight = 254.2g,
volume = 151.709cm3, density = 1.6756g/cm3
8.450
6.609
4.768
H(cm)
–1.546
0.277
2.119
H(cm)
0.000
12.145
24.290
M(mm)
Gilles’ tests
0.2
2
2.5
3.0
y = 0.0948epx(0.8505r)
R2 = 0.9521
Underwater Technology Vol. 32, No. 3, 2014
Ray’s models (2006)
L
(cm)
D
(cm)
Mass
(g)
COM
(cm)
Density
(g/cm3)
Number
of tests
31.85
4.83
563.4
13.75
2.224
8
27.94
4.02
473
14.2
2.224
12
31.75
5.18
831
15.85
1.754
11
31.75
5.18
808
16.10
1.754
11
Bomb
COM
L
D
Shell
COM
D
L
Cylinder
COM
L
D
Capsule
COM
L
D
Fig 6: Shapes used by Ray (2006)
correlation was found between the landing point
and the drop angle, although vertical drops were
more likely to land closer to the point of drop.
3.2. Ray’s tests
Ray (2006) used four polyester resin shapes for his
experiments. He reported 42 experiments using
various shapes as shown in Fig 6. Dimensions of
Ray’s model are also noted in Fig 6.
The cylinder is the most predictable and stable
shape of all (Ray, 2006). Out of a total of 11 test runs,
Ray’s tests (42 points)
0.5
0.4
0.3
0.2
Normalised Y axis
Gilles used three cylindrical shapes which had a
circular diameter of 4cm; however, the lengths were
about 15cm, 12cm and 9cm, respectively. The bodies were constructed of rigid plastic with aluminiumcapped ends. Inside each was a threaded bolt,
running lengthwise across the model, and an internal weight. The internal weight was used to vary the
COM (see Fig 3). The drop angles were 15°, 30°, 45°,
60° and 75°. This range produced velocities whose
horizontal and vertical components varied in magnitude, and allowed for comparison of trajectory
sensitivity with the varying velocity components.
Gilles (2001) reported results for 230 tests.
Gilles (2001) used five COM positions. For positive COM cases, the COM was located below the
centre of buoyancy. For negative cases, the COM
was located above the centre of buoyancy prior to
release. Fig 4 shows Gilles’ results in normalised
form. The x and y coordinates were normalised by
dividing them by the pool’s depth. Circles of 10%,
20%, 30% and 40% of water depth are superimposed on Fig 4.
According to Gilles’ experiment, 80% of objects
landed within 20% of the water depth. Almost 99%
of objects landed within 40% of the water depth,
which clearly indicates a non-normal distribution,
contrary to the DNV assumption. Fig 5 shows these
results on a lognormal probability plot. Appendix B
shows how to determine the lognormal distribution
parameters using the experimental data. Directionality seen in Fig 4 is owing to the experimental
setup. All tests were done at one corner of the pool.
Gilles (2001) grouped his tests into six series,
with the differentiating factor between the series
being the drop angle. However, no discernible
0.1
–0.5 –0.4 –0.3 –0.2 –0.1
0
0
0.1
0.2
0.3
0.4
0.5
–0.1
–0.2
–0.3
–0.4
–0.5
Normalised X axis
Capsule
Shell
Bomb
Cylinder
Fig 7: Landing point for Ray’s (2006) tests (42 points)
181
Yasseri. Experiment of free-falling cylinders in water
Model
Shape
Dimensions (cm)
No. of
tests
Mass – 1692.0g
Density – 1.335g/cm3
Model
1
6.5cm
COM
13.0cm
D: 13.0
Scale: N/A
Distance from
COM to COV = 0
13
D: 14.9
H: 13.3
Scale: N/A
Distance from
COM to COV = 0
9
Mass – 2815.0g
Density – 1.722g/cm3
13.3cm
Model
2
COM
5.5cm
14.9cm
7.45cm
COM
Mass – 1145.0g
Density – 1.615g/cm3
7.0cm
6.2cm
COM
1.6cm
15.0cm
Model
3
7.5cm
COM
D: 15.0
H: 7.0
Scale:
COM to COV =
+0.373
along Z axis
15
Mass – 813.0g
Density – 1.388g/cm3
COM
3.8cm
6.3cm
3.2cm
16.0cm
Model
4
8.2cm
COM
13.3cm
5.5cm
7.8cm
L: 16.0
W(Back): 7.8
W(Front): 13.3
H(Back): 6.3
H(Front): 3.8
Scale: 1/6
COM to COV = 0
14
Fig 8: Details of Allen’s (2006) models
nine of the shapes exhibited an almost vertical
trajectory pattern from the point of water entry to
the landing point. Models were launched orthogonal to the water’s surface with initial velocities
ranging from 28m/s to 67m/s. The upper bound
of velocity in offshore applications, when an object
hits the surface of the water, is about 15m/s.
182
Fig 7 shows the landing points for all Ray’s 31 exp­eriments normalised using the water depth. Owing
to vertical entry, all impact points for cylinders are
within 10.5% of water depth. Despite the high velocity and nominal vertical entry, impact points are
somewhat away from the projection of entry point on
the pool’s bottom. Similar observations were made
Underwater Technology Vol. 32, No. 3, 2014
by Bushnell (2001), where he fired missiles at 380m/s
into a pond from a nominal vertical direction.
3.3. Allen’s tests
Allen (2006) experimented with small samples in a
2.5m-deep pool using four shapes. These shapes
are shown in Fig 8. The model included a sphere
that is symmetric and has equal weight distribution
about its three axes.
Fig 9 shows the landing point for all shapes in
Allen’s tests. Again, the same pattern emerges, i.e.
dispersion clustering within 20% of the depth.
Some of the hemi-spherical shapes landed beyond
this limit. However, no excursion extended beyond
50% of the depth, and the largest excursion was for
flat-shape objects. Allen’s (2006) results also follow
a lognormal distribution.
Bushnell (2001) fired missiles into a pond at
high velocity (about 300m/s) at angles of attack
between 1 and 3.5°. In addition, the test specimen
had tail fins, some of which broke during the test.
In Bushnell’s test, most missiles hit the pond floor
within a circle of 0.2% of the water depth. Bushnell
experimented with missiles at a high velocity through
the water where flow separation creates a cavity of
air around the body. That cavity is also called cavitation, which remains in the water long after the
missile has passed to influence the trajectory.
4. Current model tests
COM location, into a 2.5m-deep pool at low velocity, similar to Gilles’ models (2001) (see Fig 10). The
entry of each model into the water was recorded by
the surface experimenter. A mat with gridlines was
placed on the pool’s bottom to facilitate the recording of the landing points. All models were numbered as well as marked with bright colours for
ease of detection.
The testing was conducted by two experimenters. One experimenter remained at the surface and
was responsible for launching the models and recording all test data. The other experimenter was inside
the pool with breathing apparatus, and reported
the landing point and the observed trajectory to the
surface experimenter for recoding. No test was discarded. However, prior to the commencement of
the actual testing, several trial runs were attempted
in order to eliminate all likely problems. This trial
run also provided a procedure for performing the
tests. No underwater cameras were used to record
the trajectory of the dropped cylinders.
The controlled parameters were the cylinders’
physical parameters (length-to-diameter ratio,
COM) and initial drop conditions (initial velocity,
and drop angle). The models were based on the
assumption that a 9m tubular is dropped in 180m
of water, thus producing a 20:1 ratio. The depth of
the pool was ~2.5m. From this ratio, 12cm was
­chosen for the length of the model. Two additional
models, 9cm and 15cm in length, were also used
The author’s experiments consisted of dropping
three cylindrical models, of various lengths and
L
δ
Allen’s tests (51 points)
COV COM
0.4
d
C o G of extra mass
0.3
Normalised Y axis
0.2
x
Fig 10: The current model
0.1
– 0.4
– 0.3
– 0.2
– 0.1
0
0
0.1
0.2
0.3
0.4
Φ
– 0.1
–0.2
Weight
Entry point
Distance to
COM
–0.3
–0.4
Normalised X axis
Sphere-Model 1
(Fig 8)
Hemisphere-Model 2
(Fig 8)
Model 3 (Fig 8)
Model 4 (Fig 8)
Fig 9: Allen’s (2006) tests. Plot is normalised using the water
depth
~ 2.5m
depth
Fig 11: Schematic of experimental setup
183
Yasseri. Experiment of free-falling cylinders in water
for the sensitivity analysis. The outer radius of all
the models was 4cm.
The models were constructed of rigid plastic
with a cap at both ends. Fig 11 shows a schematic of
the experimental setup. A mass was inserted inside
the model (Fig 11) to vary the COM for each model.
A total of nine models with three different lengths
and three centres of mass for each length were produced. Owing to symmetry, by turning the models
around, the COM can be below or above the centre
of buoyancy.
After dropping all nine models, the experimenter
inside the pool reported the results and recovered
the models. The surface experimenter then signalled
his readiness for the next nine tests. This cycle then
was repeated until all the tests for a given drop
angle were concluded.
The initial orientation of the drop can be set
between 0° and 90°, with 0° being a vertical initial
orientation and 90° being horizontal. Drop angles
were 15°, 30°, 45°, 60° and 75°. This produces a situation where the cylinder is heading directly downward throughout the event. If an object dropped
vertically even with no initial rotation, the object will
still tend to acquire a horizontal velocity component
as it falls through the water. This results in the object
impacting the seabed at a random angle and location. Several trajectory patterns (straight, spiral, flip,
flat, seesaw and combination) were observed during
the experiment, but the four patterns as shown in
Fig 12 were more frequent. The COM position had
the largest influence on the trajectory of cylinders.
The drop point scatter plot is shown in Fig 13.
When the COMs and buoyancy coincided, objects
Arc trajectory
–1.5
–1
0
–0.5
0
0.5
Straight trajectory
1
1.5
–1.5
–1
–1.5
–1
–1.5
–1
–2.5
–2.5
X or Y
X or Y
Early flip
Late flip
–0.5
1
1.5
0.5
1
1.5
–1.5
–2
0
0.5
–1
–2
0
0.5
1
1.5
–1.5
–1.5
Water depth (Z)
–1
–1
–0.5
0
–1
–1.5
–2
–2
–2.5
–2.5
X or Y
0
–0.5
–0.5
Water depth (Z)
0
–0.5
Water depth (Z)
Water depth (Z)
–0.5
0
–0.5
X or Y
Fig 12: More frequent trajectory patterns of falling cylinders (see also Chu et al., 2005)
184
Underwater Technology Vol. 32, No. 3, 2014
Landing points of 324 model cylinders dropped
into 2.5m of water
0.4
0.3
Normalised Y axis
0.2
0.1
– 0.4
– 0.3
– 0.2
– 0.1
0
0
0.1
0.2
0.3
0.4
– 0.1
– 0.2
– 0.3
– 0.4
Normalised X axis
Fig 13: A plot of the landing point of all author’s 324 tests
fell near the drop point for more than 90% of
the time, while a small distance between COM and
the centre of volume (COV) caused the variability.
When the centre of buoyancy is ahead of the COM,
the model flips while descending and experienced the largest excursion from the drop point.
When the drop angle is close to 90o and the centre
of buoyancy and COM coincide, then the descent is
mostly vertical with only a slight slant. When the
buoyancy centre is above the mass centre, models
approach the floor at acute angles between 120°
and 180°. The trajectory patterns of each model
type are affected by the size and shape of the models. The four generalised patterns were not shared
among the various shapes, but rather they were
quite consistent within each shape type.
Fig 14 shows the distance between the landing
point and drop point in a lognormal probability
plot. It can be seen that the lognormal distribution describes the probability of the landing spot
fairly well. Fig 15 shows results from another set
of experiments, where a cylinder with a length
of 10cm and diameter of 12cm, and a cube of
15cm × 15cm × 15cm were dropped using the
same experimental setup. These 140 points also
follow a lognormal distbution similar to Fig 14.
Fig 16 shows a few of the observed trajectories of
the ­f alling cylinder and cube in the second set of
experiments.
Probability plot of landing points for the current tests (324 points)
–3
–2.5
–2
– 1.5
–1
– 0.5
0
0.5
1
1.5
2
2.5
3
Distance between landing point and the drop point
0.5
Experimental data
Expon. (Experimental data)
0.05
Standard normal variable
y = 0.1033exp(0.8147r)
R2 = 0.9828
Fig 14: Author’s results on a lognormal probability plot (324 points)
185
Yasseri. Experiment of free-falling cylinders in water
5. Computational approach
The motion of three-dimensional rigid bodies
freely falling through water has been addressed by
many researchers. Appendix A outlines the formulation necessary to track such motion. MATLAB,
commercial software, was used to solve these equations. Owing to a rigid body assumption, tracking
of one point gives enough data to plot the position
of the falling object (Fig 17). Fig 18 shows some
examples of trajectories using different values of
control variables.
These equations replicate the experimental
results fairly well. The method is applied to cylindrical bodies initially released below the free surface in calm water. Depending on the initial body
orientation, body aspect ratio and mass distribution, it is possible to identify key characteristic patterns of the body motion, which compare
remarkably with observations in the tank-drop
tests. The present study has a significant implication to the manoeuvring of underwater vehicles.
Abelev et al. (2007) presented experimental results
of falling heavy cylindrical shapes, which are in
line with the author’s results.
clear that any satisfactory explanation of this complex phenomenon must include a description of
the instantaneous fluid forces experienced by the
falling object, as well as the inertial and gravitational effects that are present.
6. Concluding remarks
Not all falling objects travel straight downward
under the influence of gravity, even in the air.
Sometimes the falling objects follow complicated
downward trajectories as they fall. It is therefore
Landing points for dropped short cylinder and
a cube of 150cm × 150cm × 150cm
Fig 16: A few of the observed trajectories for the second
set of tests
0.4
0.3
Trajectory of a cylindrical object with heavy nose
10
0
0.1
– 0.4
– 0.3
– 0.2
–0.1
0
0.1
0.2
0.3
– 0.1
– 0.2
Normalised X axis
Drop point
–20
–30
–40
–50
–70
– 0.4
Cylinder
0.4
–60
– 0.3
Box-shape
Fig 15: Scatter plot for two shapes: a short cylinder with
length/diameter < 1 and a small cube (140 points)
186
Nose position
–10
0
Water depth (m)
Normalised Y axis
0.2
–80
–10
0
10
20
30
40
50
60
70
Horizontal position (m)
Fig 17: Tracking the orientation of a falling object with
position of COM lower than COV
80
Underwater Technology Vol. 32, No. 3, 2014
Fig 18: A few calculated trajectories for a falling cylinder with different orientation, initial velocity
and hydrodynamic conditions
The entry of a solid into water or other liquids
gives rise to a sequence of complex events. The
­evolution of the air cavity behind the falling body
after the initial impact significantly influences the
dynamics and trajectory of low-speed projectiles
(see e.g., Chu and Fan, 2006; Chu and Ray, 2006).
The effect of air cavity dynamics is more pronounced in large body water entry. Leaves, tree
seeds and paper cards falling in air are spectacular
examples of time-dependent fluid dynamics at
intermediate Reynolds numbers, at which both
inertial and viscous effects are important. The trajectory of a falling card typically appears to be very
complex, with the card either oscillating from side
to side (fluttering) or rotating and drifting sideways (tumbling).
It is acknowledged that there are deficiencies in
representing the landing point using a statistical
distribution. Fitting a distribution to a set of data
only tells us how the dispersion would look. The
data behind the DNV recommendation (2010)
could not be located, which makes the comparisons between the DNV recommendation and the
current experiment difficult. In addition, the lognormal curve may not represent the effect of all
input parameters adequately. It was chosen as
the first approximation with an overall goal of
developing a stochastic model evaluation. Other
distributions may be examined and could potentially represent the experimentally measured data
better.
Real objects of any significance are generally
longer than 6m, but still much smaller than the
water depth for deepwater locations. Thus, the ratio
of water depth to the object length is quite high.
Hence, replicating it in a model test requires a very
small model if the pool is not very deep. It is desirable that the choice of scale is representative of the
offshore industry’s ratio of water depth to object
length, but at the same time it should be large
enough to cover typical offshore tubular sizes. The
ratio of water depth to the tubular length, for a
187
Yasseri. Experiment of free-falling cylinders in water
Entry angle
Wave slope
Drift owing to object hitting the wave at
different location
Fig 19: The effect of undulating sea surface on the trajectory
and drifting
dropped tubular model of 12m in length in a water
depth of 240m, is 20. Therefore, for a pool depth of
2.5m, the model length should be around 120mm.
Ingram (1991) surmises there is a depth beyond
which an object would fall flat, implying there is a
­single pattern in deep water. However, no single
pattern was observed in these tests.
Pulling together all tests reported here, the following observations can be made about the landing location of free-falling cylinders:
• About 50% of the time, objects land within 10%
of the water depth.
• About 80% of the time, objects land within 20%
of the water depth.
• About 90% of the time, objects land within 30%
of water depth.
• About 95% of the time, objects land within 40%
of water depth.
• About 98% of the time, objects land within 50%
of water depth.
These tests were performed in pools where the water
surface only experiences small ripples. However,
there is significantly more undulation in the ocean
surface. Such undulation will cause more deviation
to the object and help it to travel further (Fig 19).
References
Aanesland V. (1987). Numerical and experimental investigation of accidentally falling drilling pipes. OTC 5494,
19th Offshore Technology Conference, April 1987,
Houston, USA.
Abelev AV, Valent PJ and Holland KT. (2007). Behaviour of
a large cylinder in free fall through water. IEEE Journal of
Ocean Engineering 32: 10–20.
Allen CR. (2006). Mine drop experiment II with operational
mines shapes (MIDEX II). MS Thesis, Naval Postgraduate
School, Monterey, CA, USA, 306pp. Available at http://
faculty.nps.edu/pcchu/web_paper/thesis/06Mar_Allen.
pdf, last accessed <14 September 2014>.
188
Bushnell JM. (2001). Tail separation and density effects on
the underwater trajectory of the JDAM. MS Thesis, Naval
Postgraduate School, Monterey, CA, USA, 88pp. Available
at http://calhoun.nps.edu/public/handle/10945/4406,
last accessed <14 September 2014>.
Chu PC. (2009). Mine impact burial prediction from one
to three dimensions. ASME Journal of Applied Mechanics
Review 62.
Chu PC and Fan CW. (2006). Prediction of falling cylinder
through air-water-sediment columns. ASME Journal of
Applied Mechanics Review 73: 300–314.
Chu PC and Ray G. (2006). Prediction of high speed rigid
body manoeuvring in air-water-sediment columns. In:
Rahman M and Brebbia CA (eds.). Advances in Fluid
Mechanics V. Southampton, UK: WIT Press, 123–132.
Chu PC, Fan CW, Evans AD and Gilles AF. (2004). Triple
coordinate transforms for prediction of falling cylinder
through the water column. Journal of Applied Mechanics
71: 292–298.
Chu PC, Gilles AF and Fan CW. (2005). Experiment of
falling cylinder through the water column. Experimental
Thermal and Fluid Science 29: 555–568.
Colwill RD and Ahilan RV. (1992). Reliability analysis of the
behaviour of dropped objects. OTC 6918, 24th Offshore
Technology Conference, April 1992, Houston, USA.
Det Norske Veritas (DNV). (2010). Risk assessment of pipeline
protection. Recommended practice No. DNV-RP- F107.
Olso, Norway: DNV, 45pp.
Evans A. (2002). Hydrodynamics of mine impact burial. MS
Thesis, Naval Postgraduate School, Monterey, CA, 2002.
Available at http://calhoun.nps.edu/handle/10945/
5310?show=full, last accessed <14 September 2014>.
Friedman A, Zilman G and Miloh T. (2003). Dynamics of a
body falling in waves. In: Proceedings of 18th International Workshop of Water, Waves and Floating Bodies
(IWWWFB), Ann Arbour, MI, USA.
Gilles A. (2001). Mine Drop Experiment (MIDEX). MS Thesis, Naval Postgraduate School, Monterey, CA, USA, 446pp.
Available at http://faculty.nps.edu/pcchu/web_paper/
thesis/gilles.pdf, last accessed <14 September 2014>.
Ingram JG. (1991). Experimental modelling and risk assessment of dropped offshore tubulars. MS Thesis, School of
Industrial and Manufacturing Science, Cranfield Institute of Technology, Cranfield, UK.
Kim Y, Lin KY and Yue KD. (2002). Motion dynamics of
three-dimensional bodies falling through water. In:
Rainey RCT and Lee SF (eds.). Proceedings of 17th
International Workshop of Water, Waves and Floating
Bodies (IWWWFB). London: WS Atkins Consultants,
81–84. Available at http://www.iwwwfb.org/Abstracts/
iwwwfb17/iwwwfb17_21.pdf, last accessed <14 September
2014>.
Ray G. (2006). Bomb strike experiments for mine countermeasure. MS Thesis, Naval Postgraduate School, Monterey, CA, USA, 197pp. Available at http://faculty.nps.edu/
pcchu/web_paper/thesis/06Mar_Ray.pdf, last accessed
<14 September 2014>.
Von Mises R. (1959). Theory of Flight, 1st edition. New York:
Dover Publications Inc, 629pp.
Wierzbicki T and Yue DK. (1986). Impact damage of Challenger crew compartment. Journal of Spacecraft and Rockets
23: 646–654.
Yasseri S. (1997). Risk-based decision analysis for the protection of marine pipelines from dropped objects. In: Proceedings of 7th International Offshore and Polar Engineering
Conference, Honolulu, USA, 251–256.
Underwater Technology Vol. 32, No. 3, 2014
Appendix A
jm
x
Rigid body dynamics of a cylinder falling
through water
There are various methods of describing the motion
of a rigid body; see for example Von Mises (1959),
Chu et al. (2005), Wierzbicki and Yue (1986), Kim
et al. (2002) and Friedman et al. (2003). In the following, equations which are needed to track the
dynamics of motion of three-dimensional rigid bodies falling through water (Chu et al., 2005) are given.
Fig 20 shows a right circular symmetric cylinder
whose centres of mass (X) and volume (B) are on the
cylinder’s longitudinal axis. Parameters (L, d, x̄)
denote the cylinder’s length, diameter and the distance between the two points (X, B). The positive
case is when the centre of mass (COM) is lower
than the centre of volume (COV), i.e. x̄ is positive
(Evans, 2002).
Three orthogonal right-handed coordinate systems (Fig 21) are used describe the dynamics of a
falling cylinder through the water which are:
E-coordinate, M-coordinate and F-coordinate systems. The body and buoyancy forces and their
moments are given in the E-coordinate system, the
hydrodynamic forces (drag, lift and inertia forces)
and their moments are given in the F-coordinate,
and the cylinder’s moments of gyration are given
in the M-coordinate. The E-coordinate system is
fixed to the Earth’s surface with horizontal sides as
x and y axes are parallel with the Earth’s surface
(along the two sides of the pool), and vertical direction as z axis (upward positive, Fig 20).
Suppose the cylinder is falling into the water.
The cylinder rotates around its main axis (r 1) with
an angle ψ1 and an angular velocity of Ω. Its position
d
L
2
L
2
Fig 20: M-coordinate with the COM as the origin X (im, jm); x̄
is the distance between the COV (B) and COM (X); L and d
are the cylinder’s length and diameter (Chu et al., 2005)
is represented by the COM, and its orientation is
represented by two angles: ψ2 and ψ3 (Fig 3). Here,
ψ2 is the angle between the r 1 axis and the horizontal plane, and ψ3 is the angle between the projection of the main axis in the (x, y) plane and the x
axis. The angle (ψ2 + π/2) is usually called attitude.
The relative coordinate is rigidly connected with
the cylinder. The origin (O) of the relative coordinate system coincides with the COM; the r 1 axis is
along the central line of the cylinder; the r 2 axis
is perpendicular to the plane constructed by r 1 axis
and z-axis (r 1-z plane); and the r 3 -axis lies in the
(r 1-z) plane and is perpendicular to r 1 axis. The
axes (x, y, z) and (r 1, r 2, r 3) follows the right-hand
rule.
Let V* = (V*1, V*2, V*3 ) the three components of the
velocity of COM, i.e. the origin velocity of the coordinate system (r 1, r 2, r 3). The geometric centre (GC)
is located at (x̄, 0, 0), if for GC below COM, x̄ > 0 and
for GC above COM, x̄ < 0. The relative coordinate
system (r 1, r 2, r 3) is obtained by one translation and
two rotations ψ2 and ψ3 of the E-coordinate system.
Let the position vector (P) be represented by PE and
z(k )
kf
ψ2
jm
jf
km
y (j)
o•
if
o•
V2
o
im
X
B
Vr
im
ψ3
V1
x (i )
Fig 21: Cylinder orientation and relative coordinate system
189
Yasseri. Experiment of free-falling cylinders in water
PB in the Earth and relative coordinate systems. PE
and PB are connected by:
cos ψ3 − sin ψ3 0


PE =  sin ψ3 cos ψ3 0


 0

0
1


xm∗ 
 cos ψ2 0 sin ψ2 
 


 PB +  ym∗  (A1)
 0
1
0
 


z∗ 
− sin ψ 0 cos ψ 
2
2
 m 


where (xm*, ym*, zm*) represent the position of COM in
the Earth’s coordinate system.
The motion of a solid object falling through a
fluid is governed by two principles: (1) momentum
balance and (2) moment of momentum balance.
Let Vw = (Vw1, Vw2, Vw3) be the water velocity, and (ω1,
ω2, ω3) be the components of the angular velocity,
referring to the direction of the relative coordinate
system. Variables are made non-dimensional by:
t=
g ∗
L ∗
V∗
t ,=
ω ,V =
(A2)
L
g
gL
where g is the gravitational acceleration, and L
the length of the cylinder. The non-dimensional
momentum equations for COM are given by (Van
Mises, 1959):
ρ − ρw
dV1
F∗
sin ψ2 + 1 (A3)
+ ω2V 3 − ω3V 2 =
ρ
ρg ∀
dt
F∗
dV 2
+ ω3V1 = 2 (A4)
dt
ρg ∀
F∗
ρ − ρw
dV 3
− ω2V1 =
cos ψ2 + 3 (A5)
dt
ρ
ρg ∀
where ∀ is the volume of the cylinder, ρ is the cylinder
density, ρw is the water density and (F 1*, F 2*, F 3*) are the
components of fluid drag. The non-dimensional
equations of the moment of momentum for circular symmetric cylinder are:
LM 1∗
dΩ J 3 − J 2
(A6)
ω2 ω3 =
+
dt
J1
g J1
d ω2
LM 2∗ (A7)
x ∗ ∀(ρ − ρw )L
=−
cos ψ2 +
dt
J2
g J2
d ω3 LM 3∗
(A8)
=
dt
g J3
*
where x̄ is the distance between COM and GC, and
(M1*, M 2*, M 3*) are the components of the moment
190
owing to drag (J1, J 2, J 3). The three moments of
gyration are:
J 1 = ∫ (r22 + r32 )dm ∗ , J 2 = ∫ (r32 + r12 )dm ∗
and J 3 = ∫ (r12 + r22 )dm ∗
(A9)
The orientation of the cylinder (ψ2, ψ3) is determined by:
d ψ2
dψ
= ω2 , cosψ2 3 = ω3 (A10)
dt
dt
The eight non-dimensional nonlinear equations
(A2 to A5), (A6 to A8) and (A10) are the fundamental equations for determining the cylinder
movement in the water.
All the equations were solved using the 4th
order Runge-Kutta scheme for integration with
time in MATLAB. At any instant, the hydrodynamic loads were computed using a panel method.
Appendix B
Probability plotting
A method that may be used to visualise distributions and estimate parameters is probability
plotting, also referred to as linear least-squares
regression or regression on ordered statistics. This
technique involves finding a probability and data
scale that plots the cumulative distribution function (CDF) of a hypothesised distribution as a
straight line. The corresponding linearity of the
CDF for the sample data provides a measure of the
goodness-of-fit of the hypothesised distribution.
The n test results (ri, r 2,…, rn) were arranged in
increasing order; the mth value is marked on the vertical axis and corresponds to a cumulative probability
of p = m/(N + 1), marked on the horizontal axis. Then
the value of standardised normal variable for each
p was calculated using the normal distribution table.
The distance between the drop point and the
landing point were ordered in terms of increasing
magnitude in a table, and the cumulative frequency
of each ratio were determined, which is equal to the
cumulative probability of p. For each p, the value for
the corresponding standard normally distributed
variable is then calculated and noted in that table.
Samples of (Zi , ri) are plotted on the logarithmic
normal distribution graph as shown in Fig 14.
The straight line equation that best fits the samples can be written as:
y = a e bx(B1)
where a and b are constants.
Underwater Technology Vol. 32, No. 3, 2014
Taking logarithm of both sides of this equation
results in:
lny = lnα + bx(B2)
The standard normal variable becomes:
s=
ln ri −λ
or lnri = λ + ζs (B3)
ζ
Equating two equations gives the ordinate intercept of the straight line and its slope, i.e.:
λ = lnα & b = ζ(B4)
Then the mean value of landing points (r̄ ) is:
r̄ = rE = exp(λ + 0.5ζ2)(B5)
From Fig 14, it can be seen that they can be
approximated by the following straight line:
y = 0.1033e 0.8147x(B6)
λ = ln(0.1033) = −2.27012;
ζ = b = 0.8147(B7)
r̄ = rE = exp(−2.27012 + 0.5 × 0.81472)
= 0.1439 (14.39% of the water depth).
(B8)
191