2nd International Symposium on Seawater Drag Reduction
Busan, Korea, 23-26 MAY 2005
Design Trade-off Analysis for High Performance Ship
Hull with Air Plenums
Jin-Keun Choi, Chao-Tsung Hsiao, Georges L. Chahine
(DYNAFLOW, INC., U.S.A.)
been researched more recently in a number of
ways,
for
example,
injection
of
polymer/surfactants /fiber suspensions, microbubble injection, vorticity manipulation with riblets,
large eddy break-up devices, ventilated cavities
(or air lubrication), and active turbulence control
by using MEMS (Sellin and Moses, 1989, Gyr,
1990, Bushnell and Hefner, 1990).
In the ventilated cavity approach, the frictional
drag of a ship is reduced by covering parts of the
hull surface with an air layer or a ventilated cavity,
which, in effect, results in a decrease of the
wetted area of the ship hull (Matveev, 1999). A
similar idea of using air pressure beneath the
bottom of the ship to reduce the wetted area for
high speed ships can be found in air cushion
vehicles (ACV) and surface effect ships (SES).
These types of ships require relatively large
power to pressurize the air chamber beneath the
hull and lift the ship, which makes their
applications to larger ships difficult (Clayton and
Bishop, 1982). The ventilated cavity method is
distinguished from these methods because it
requires smaller amount of air needed only to
maintain a thin layer of ventilated cavity. The idea
of covering a hull with an air layer to reduce the
frictional drag was patented as early as in the
nineteenth century (Latorre, 1997). However, a
stable formation of such a thin cavity suffers from
interference with the ship motion, the changes of
buoyancy force, flow around the ship, and other
instabilities at gas-liquid interfaces (Kostilainen
and Salmi, 1972). Ventilated multi-hull ships use
a new class of hull forms in between the SES and
the ventilated cavity ships. That is, the volume of
air captured in the plenum of a multi-hull can vary
from a very small quantity as in a thin cavity layer
to a larger quantity as in the air chamber of a
SES. The development of a design for this new
type of ship requires an understanding on the
hydrodynamics of such hull forms, including the
behavior of the ventilated cavity air-water
interface, the interaction between the ship
generated wave and the air cavity, and the
ABSTRACT
Ship designs with captured air plenums can
provide benefits of reduced resistance and
improved seakeeping performance. In order to
design such ships, hydrodynamics computations
are needed. We have extended our boundary
element method code, 3DYNAFS©, to the study of
multi-hulls with air plenums and to predict wave
resistance in the presence of the air plenum free
surface. We also utilized an Unsteady Reynolds
Averaged Navier-Stokes (UnRANS) equation
solver to study the viscous effects around the hull
with an air plenum. We conducted parametric
studies on the air plenum and hull form
parameters. Through these systematic variations
of design parameters, we have established trends
of the total resistance, which presented very
distinct characteristics depending on the Froude
numbers.
Lastly,
we
have
successfully
demonstrated the capability of the code in dealing
with air plenums of inclined hulls. The developed
code was found a very useful tool for designs of
complex hull forms with air plenums.
INTRODUCTION
Compared to conventional displacement hulls,
hull designs with captured air plenums have
potential benefits in reduced resistance because
the wetted area of the vessel is reduced. The
impact on seakeeping performance is not known,
while it is expected that the captured air can
mitigate shock from mines. In order to evaluate
these new hull concepts, more fundamental
understanding of the hydrodynamics of such hulls
should be established first.
Resistance of a ship is conventionally
decomposed into the following three components;
the wave-making resistance, the frictional drag,
and the viscous form drag. In the past, much of
the research was focused on the reduction of the
wave-making resistance through modification of
hull forms (Wigley, 1935-36, Eggers et al., 1967,
Inui, 1980). The reduction of frictional drag has
1
domain D. By knowing either φ or ∂φ / ∂n on the
boundary S, the other quantity on S can be
obtained from (3).
To solve this equation
numerically, all surfaces are discretized into
triangular and quadrilateral panels on which linear
distributions of φ or ∂φ / ∂n are assumed. The
corresponding surface integrals are then
represented as a summation over all panels.
Writing (3) at all discrete nodal points P yields a
system of linear algebraic equations, and their
solution provides the values of the potential or its
normal derivative at all nodes. Then (3) can be
used to obtain the pressure and velocity fields at
any point in the liquid domain D.
The boundary surface S of the fluid domain
includes the vessel wetted surface, the air cavity
surface, the ocean free surface, and optionally, a
sea bottom and an inlet surface.
The boundary condition on the wetted hull
surface is an equality to the normal velocity of the
hull surface and the normal velocity of the liquid at
a point on the ship hull surface, ∂φ / ∂n :
resulting wave-making resistance, and the
geometry and location of the cavity, etc.
The objective of the present research is to
develop a ventilated cavity ship computational
fluid dynamics tool and utilize it in design of new
type of hulls with air plenums
BOUNDARY ELEMENT METHOD CODE
At DYNAFLOW, INC., we have developed a threedimensional fully non-linear boundary element
method (BEM) code, 3DYNAFS©, and utilized it for
simulations of complex 3-D free surface dynamics
such as bubbles (Chahine and Perdue, 1989,
Chahine et al., 1997), breaking waves
(Goumilevski et al., 2000), motion of floating
bodies (Kalumuck et al., 1999, Cheng et al.,
2001), and cavitation (Chahine and Hsiao, 2000).
The accuracy of the code in these applications
has been proven through comparisons with
experiments for these various applications.
Except for very thin boundary layers near the
solid boundaries where viscous effects are
important, the general features of the
hydrodynamic problem can be obtained by
assuming the flow incompressible and inviscid.
The flow is then described with a velocity potential
v
v
∂φ
v r
= [V + ( Ω × r )] ⋅ n ,
b
b
∂n
r
φ , ( V = ∇φ ), satisfying the Laplace equation in
v
where V is the velocity vector of the center of
b
v
gravity of the vessel. Ω is the angular velocity of
b
the body-fixed coordinate system relative to the
space-fixed coordinate system.
rThe ocean free surface is described by
F ( x , t ) = z − ζ ( x, y, t ) = 0 . On this surface we
have two conditions. The kinematic condition
expresses that a fluid particle followed in its
motion remains at the free surface,
the liquid domain D:
∇ 2φ = 0,
in D .
(1)
The velocity potential and the pressure are related
through the unsteady Bernoulli equation:
1
∂φ
2
p + ρ ∇φ + ρ
+ ρ gz = c(t ) , (2)
2
∂t
r
dF ( x , t ) / dt = 0 .
where ρ is the liquid density, and g the
acceleration of gravity.
The boundary value problem for the potential
is solved by the boundary element method, which
is based on the Green identity, written as:
∂G ⎫
⎧ ∂φ
G −φ
⎬dS ,
∂n ⎭
⎩ ∂n
α ( P)φ ( P) = ∫∫ ⎨
S
(4)
(5)
The dynamic condition expresses the balance of
pressures at the interface:
patm + ρ gz + ρ
(3)
∂φ 1
2
+ ρ ∇φ − γ C = 0 , (6)
∂t 2
where γ is the surface tension parameter, C is
the local surface curvature, and patm is the
atmospheric pressure.
On the air plenum air-water interface, the
boundary conditions are similar to (5) and (6).
The kinematic condition (5) applies as is,
however, the dynamic condition is modified to
where α(P) is the solid angle under which the
point P sees the fluid domain D. G = 1/ MP is the
Green’s function, where M belongs to the
boundary surface S and P belongs to the fluid
2
account for the fact that the pressure inside the
cavity is pg and not patm.
MODELING OF AIR PLENUMS
In this work, we have considered the following two
approaches to handle the air plenum:
Air plenum surface as another free surface –
In addition to the ship hull and the free surface
surrounding it, the internal walls of the air plenum
are modelled and discretized. Then the free
surface of the air plenum is tracked along these
internal walls. An advantage of this approach is
that the nonlinear motion of the air plenum free
surface moving up and down along the internal
walls can be simulated directly. A disadvantage is
an increase of the problem size and computation
CPU time.
Air plenum surface as a cavity surface –
Another approach is to treat the air plenum liquid
interface surface as a cavity surface with fixed
end nodes at the edges of the ship plenum cavity.
In this case penetration of the interface edge into
the plenum and its separation from the ship nodes
are not allowed.
In this study, we have selected this last
approach because it requires less complicated
hull gridding. Two types of plenum pressure
conditions were considered: (a) constant pressure
and (b) constant air mass in the plenum. The
second condition is useful in determining the
proper air plenum pressure needed to obtain a
stable cavity. The constant air mass model uses
a polytropic pressure-volume compression law,
∂φ 1
2
pg + ρ gz + ρ
+ ρ ∇φ − γ C = 0 . (7)
∂t 2
The air plenum-water interface is predefined on
the hull surface, and a volume inside the hull
bounded, on the liquid side, by this surface is
identified as a cavity. The air-water interface is set
free to move according to the local flow.
The numerical scheme proceeds in the time
domain and the solution is obtained as a function
of time. At each time step, the matrix equation
resulting from Green’s identity is solved, and the
normal velocity at the free surface and the
potential on the solid surfaces are obtained. The
Bernoulli equation is then used to calculate
dφ / dt and update the potential at the free
surface nodes and the pressures at the solid
nodes. The free surface nodes are advanced in a
Lagrangian fashion using the previous time step
liquid velocities at these nodes.
ESTIMATION OF RESISTANCE
In the present study, we investigate the frictional
resistance and the wave-making resistance. The
total resistance coefficient, CT, can be expressed
by the following equation (Lewis, 1988):
CT =
RT
= (1 + k ) ⋅ CF ( Re ) + CW ( Fn ) , (8)
2
1
2 ρ SV
k
⎛V ⎞
pg = pg ,0 ⎜ ch,0 ⎟ .
⎝ Vch ⎠
where RT is the total resistance, S the wetted
surface area, V the ship speed, CF is the frictional
resistance coefficient, and CW is the wave-making
resistance coefficient. According to Froude’s
assumption, the frictional resistance is a function
of only the Reynolds number, Re, while the wavemaking resistance is a function of only the Froude
number, Fn. The form factor, k, which represents
the “curvature” effect of the hull, has a value
between 0.1 and 0.4 depending on the hull shape.
Here, we use a skin friction drag estimation
based on the ITTC 1957 Model-Ship Correlation
Line (Lewis, 1988). The total resistance can be
obtained by combining this CF and the wavemaking resistance obtained from 3DYNAFS©. The
wave-making resistance is calculated by
integrating the pressure over the full surface of
the hull including the air plenum surface.
(9)
Here, pg ,0 and Vch ,0 are the initial pressure and
volume in the plenum and pg and Vch are the
current values.
COMPUTATION WITH UNRANS CODE
Unsteady Reynolds-Averaged Navier-Stokes
(UnRANS) solvers have been successfully used
in propeller and ship flow problems (Wilson et al.,
1998, Gorski, 1998, Hsiao and Pauley, 1999).
Here, we employ our modified version of the
UnRANS solver, DF_UNCLE (Hsiao and Chahine,
2001) to investigate viscous effects on the ship
with an air plenum. Like the original UNCLE code
developed at Mississippi State University (Taylor,
1991, Sheng, 1994), DF_UNCLE uses artificialcompressibility method, in which a time derivative
of the pressure is added to the continuity equation
to couple it with the momentum equations
3
along each girth. The half of the ocean free
surface covers -10 m ≤ y ≤ 6 m, 0 ≤ x ≤ 6 m, and
is represented by 19x54 nodes.
resulting in a hyperbolic system. The method is
marched in pseudo time to reach a steady-state
solution with a divergence-free velocity field. To
obtain a time-accurate solution, a sub-iterative
procedure for pseudo time is performed at each
physical time step.
VALIDATION OF THE METHODS
We have validated 3DYNAFS© for wave resistance
problems using the classical case of a Wigley hull
of length 6.1 m, studied the effect of the size of
the modeled free surface, and of grid resolution
for the Froude number 0.25.
Through this
validation study, we concluded that we may use
the following conditions and grid parameters:
• The half hull discretized with 21x11 grid
• The ocean free surface domain size of 0 ≤ x ≤
6 m, -10 m ≤ y ≤ 6 m with 19x54 grid.
• A linearized free surface condition on the
ocean free surface can be used while
maintaining nonlinear free surface conditions
on the plenum/water interface.
Under these conditions, the force in the
direction of the inflow, obtained by integrating
pressure over the hull, resulted in a predicted
wave resistance of 12 N. The components of this
unsteady wave resistance force show similar
behavior in time as reported by Celebi (2000).
Figure 1: Fisheye view of the grid for the truncated
Wigley hull (21x11 nodes on half body) and the ocean
free surface (19x54 nodes on half domain).
The flow field around the truncated Wigley
hull without air plenum at speed 1.93 m/s (Froude
number 0.25) is computed as shown in Figure 2.
For a similar computation with an air plenum, the
air plenum free surface is specified on the bottom
of the hull in a nearly rectangular region, -1.22 m
≤ y ≤ 1.22 m, -0.11 ≤ x ≤ 0.11 m. The resulting
deformation of the air plenum surface is shown in
Figure 3. In these two computations with and
without the air plenum, the inflow velocity is
gradually accelerated from zero to reaches full
speed at 2.0 s. The frictional resistance is
estimated from the wetted area of the hull, which
is 4.3 m2 originally and reduced to 3.5 m2 if the air
plenum exists. The estimated frictional resistance
of the hull with wetted bottom is 23.7 N, while that
of the hull with air plenum is 19.0 N. The
predicted wave-making resistances of both hulls
with and without the air plenum are almost the
same at 8 N.
BASELINE HULL FORM
As a first step to varying the hull form
systematically, we used a truncated Wigley hull as
a baseline hull form. The hull form can be
expressed by the following equations with the xaxis pointing to the starboard, the y-axis pointing
to the bow, and the z-axis pointing up:
2
⎡ ⎛ y ⎞2 ⎤ ⎡ ⎛ z ⎞ ⎤
x
= ⎢1 − ⎜
⎟ ⎥ ⋅ ⎢1 − ⎜ ⎟ ⎥ , (10)
B / 2 ⎢⎣ ⎝ L / 2 ⎠ ⎥⎦ ⎢ ⎝ D1 ⎠ ⎥
⎣
⎦
z
0.01
0.009
0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0
-0.001
-0.002
-0.003
-0.004
-0.005
-0.006
-0.007
-0.008
-0.009
-0.01
for − D ≤ z ≤ 0 , where B , L , and D are
respectively the beam, the length, and the draft of
the hull, D1 ( D1 > D ) is the draft of the original
Wigley hull, and z = 0 is the undisturbed ocean
surface. The dimensions used are B =0.61 m,
L =6.1 m, D1 =0.38125 m, and D =0.2 m.
The grids for the truncated Wigley hull and the
ocean free surface are shown in Figure 1. We are
exploiting port-starboard symmetry of the
problem, and the half of the hull is discretized with
21 nodes in the length direction and 11 nodes
Figure 2: Free surface wave around the truncated
Wigley hull without air plenum at Froude number 0.25.
4
surface toward the downstream end. This is due
to the ocean free surface wave profile that
induces a wavy pressure distribution along the
hull (also indicated by the green and blue contour
lines on the side of the hull). This influence of the
wave profile along the hull on the cavity interface
has important implications for practical designs.
The resistance, illustrated in Figure 5, shows no
appreciable difference of wave-making resistance
between the long and short plenum cases at this
Froude number. On the other hand, the longer
plenum results obviously in greater reduction of
the frictional resistance.
Z
H:21x11, FS:19x54 uniform, Quad panels
Tmin,Tmax,TstepFactor = 1000 100 0.5
Cubic spline for dphids on FS
With Geometric solid angles, Averaging on phi, W=100
Freed edge downstream, Zero slope perimeter
Damping near FS edge, Yout=2, CDout=0.8
2
Ramping up with exp(-10.0 t )
X
Y
P
Air Plenum :
50% ship length at midship
Constant chamber air mass
Initial chamber volume = 1.0 m3
Kgas=1.0
Initial Pair = 103192 Pa
Linearized free surface, Truncated Wigley, Fn=0.250
103200
103180
103160
103140
103120
103100
103080
103060
103040
103020
103000
102980
102960
102940
102920
102900
102880
102860
102840
102820
102800
Vertical dimension 10 times magnified
Time History of Air Plenum Volume and Pressure
Figure 3: Fisheye view of the truncated Wigley hull with
air plenum at Froude number 0.25. Converged air
plenum interface for a 50% ship-length plenum. Inflow
is from the right to the left. Notice that the pressure
scale is very fine around the hydrostatic pressure at the
depth of the ship bottom. The vertical Z scale of the
interface shape is magnified by 10 to highlight the
interface deformation.
103200
Volume Change F
Plenum Pressure F
Volume Change M
Plenum Pressure M
Volume Change A
Plenum Pressure A
Volume Change [m3]
0.003
AIR PLENUM SIZE AND LOCATION
0.002
103100
0.001
103050
The air plenum volume change is relative to the initial volume.
0
We present the effects of the air plenum size and
location for two Froude numbers: Fn=0.25 and
0.50. For Fn=0.25, these four air plenum interface
dimensions and locations are considered:
• Length: 0.2 L, width: 0.5 B, starting location:
at 0.25 L from the bow (noted 3F)
• Length: 0.2 L, width: 0.5 B, starting location:
at 0.40 L from the bow (noted 3M)
• Length: 0.2 L, width: 0.5 B, starting location:
at 0.55 L from the bow (noted 3A)
• Length: 0.5 L, width: 0.5 B, starting location:
at 0.25 L from the bow (noted 9M)
The plenum pressure was varied based on a
constant air mass in an initial chamber volume of
1.0 m3. The wave profiles of the first three cases
were very similar, which resulted in essentially the
same wave resistances for all three cases. In
Figure 4, the chamber volumes and pressures are
compared. It is interesting to notice that the air
plenum located near the bow has the greatest
time variations in both quantities, and that the one
located in the middle has the least variations.
This is related to the intensity of the local pressure
variations around the air plenum, which strongly
depends on the location.
The converged air plenum interface shape for
the plenum with 50% ship length is shown in
Figure 3. It is observed that the air plenum bulges
out near the cavity leading edge and forms a wavy
103150
-0.001
0
10
20
Plenum Pressure [Pa]
0.004
103000
102950
40
30
Time [s]
Figure 4: Comparison of the air plenum pressure
changes and the volume changes for the three air
plenum locations; near the bow (F), at the mid-ship (M),
and near the stern (A).
40
Resistance [N]
30
20
Wetted, Total
Wetted, Friction
Wetted, Wave-making
15%L plenum, Total
15%L plenum, Friction
15%L plenum, Wave-making
50%L plenum, Total
50%L plenum, Friction
50%L plenum, Wave-making
10
0
0
10
20
30
40
Time [s]
Figure 5: Comparison of the resistance components for
short (20% ship length) and long (50% ship length) air
plenums.
For the Fn=0.50, two medium size air plenums
were also considered in addition to the four
variations discussed above.
5
•
The wave profiles along the hull are
compared in Figure 6. The wave heights are very
similar near the bow for all the cases studied, but
are different downstream of the air plenum. The
three cases with lower wave resistance (3F, 6F,
9M) show flatter wave profiles downstream of the
forward plenum edge. A location of the air
plenum near the high bow wave appears to result
in a mitigation of the wave height downstream.
Length: 0.35 L, width: 0.5 B, starting location:
at 0.25 L from the bow (noted 6F)
• Length: 0.35 L, width: 0.5 B, starting location:
at 0.40 L from the bow (noted 6A)
The resulting resistance and lift forces are
tabulated in Table 1. As expected, the total
resistance decreases as the air plenum covers
more area of the hull. One interesting observation
is that the wave-making resistance is appreciably
affected by the size and location of the air plenum.
This trend, found only at the higher Froude
number, is very different from that at Fn=0.25.
The air plenums located forward, i.e., 3F and 6F,
have advantages in wave-making resistance over
other locations, showing lower wave resistance in
addition to reduced frictional resistance. In the
case of the short forward air plenum (3F), the
reduction in wave resistance, 6 N, is almost as
large as frictional resistance reduction, 7 N.
These findings suggest that a well designed air
plenum can benefit both wave resistance and
frictional drag.
0.2
3 rows forward (-5.4 N)
3 rows middle (+0.8 N)
3 rows aft (+0.6 N)
6 rows forward (-7.1 N)
6 rows aft (+0.0 N)
9 rows (-6.9 N)
No cavity
0.15
Height
0.1
0.05
0
ity
av
tc
Af
y
vit
ca
le
dd
Mi
-0.05
Table 1: Resistance and lift force acting on the
truncated Wigley hull with variations of air plenum size
and location for Fn = 0.50. All forces are in N.
Air plenum
Wetted bottom
With an
air
plenum
Resistance
Total
Viscous Wave
-0.1
181
85
97
-766
169
78
91
-748
3M
175
78
97
-696
3A
175
78
97
-691
6F
163
72
91
-685
6A
170
72
98
-793
9M
158
68
90
-669
-3
-2
-1
0
1
2
3
Longitudinal position
Lift
3F
F
y
vit
ca
rd
a
orw
Figure 6: Comparison of wave profiles along the hull for
Froude number 0.50. The forces in the parentheses
represent the increase of decrease relative to the
wetted bottom case.
LENGTH TO BEAM RATIO
The hull length to beam ratio, L/B, is one of the
most important design parameters. We have
studied the effect of this ratio by varying the ship
length and beam while keeping its depth and
displaced volume the same. The geometry of the
air plenum edge and volume are scaled according
to the length-to-beam ratio. Here we focus mainly
on the wave resistance, which is affected by the
interaction of ship generated free surface wave
and the plenum air/water interface.
The
importance of such interaction was already
identified in previous studies (Amromin, 2000).
We considered L/B ratios of 2.5, 5, 7.5, and 10.
The wave profiles along the hull are
compared in Figure 7 with the abscissa
normalized by the ship length. As expected, the
lower L/B results in larger wave height, and thus
higher wave resistance. However, the frictional
resistance is greater for higher L/B ratios (larger
wetted surface area), and this increase in
frictional resistance exceeds the decrease in the
During these numerical simulations, the ship
was fixed in space no matter how large a force or
moment it experienced1. As expected, the ship
tends to sink due to the bottom suction effect at
high speed, illustrated by the negative lift forces.
From the table, we can see that the air plenum
reduces this sinkage force, and the larger the air
plenum the less the sinkage force. The greatest
change in the sinkage force is found for Case 9M,
where it is reduced about 13%. This is another
advantage of an air plenum since at high speed
sinkage contributes to increased ship resistance.
1
In this study, we did not utilize the Fluid Structure Interaction
©
(FSI) module of 3DYNAFS , which takes into account the ship’s
response to the hydrodynamic forces.
6
wave-making resistance. Consequently, a high
L/B ratio hull has a higher total resistance.
The resistance and lift obtained from the
above wetted bottom cases and corresponding
runs with air plenum are shown in Table 2. For
the runs with an air plenum, the air plenum is
located in the middle of the bottom and has
dimensions of 0.5 L x 0.5 B. In all the L/B cases
studied, the wave resistance remains more or less
the same regardless of the presence of an air
plenum. That is, the reduction of the total
resistance is purely from the reduction in the
viscous drag at this Froude number regardless of
L/B. The amount of reduction in viscous drag is
slightly larger for higher L/B because higher L/B
hulls have larger wetted surface area. It is
interesting that lift with air plenums is lower than in
the corresponding wetted bottom cases for L/B ≤
5.0. This means more sinkage is expected for
small L/B hulls if an air plenum exists.
number 0.50. Examples of the converged wave
patterns on the free surface are shown in Figure
8. The corresponding hydrodynamic forces acting
on the hull are shown in Table 3. At this high
Froude number, the wave resistance is found to
be slightly affected by the existence of the air
plenum. This effect is, however, negligible
compared to the larger reduction in viscous drag.
Figure 8: Free surface waves for wetted bottom hulls
with L/B=5 (left) and 10 (right) for Froude number 0.50.
Wave height is magnified by 2.
0.15
Table 3: Resistance and lift of L/B variations for Froude
number 0.50. All forces are in N.
L/B=2.5
L/B=5
L/B=10
0.1
z [m]
L/B
4.9
0.05
7.5
0
-0.05
10.0
-0.5
-0.25
0
0.25
Air
plenum
wetted
cavity
wetted
cavity
wetted
cavity
Total
255
243
191
178
181
162
Resistance
Viscous
55
50
65
51
85
68
Wave
200
200
126
127
97
94
Lift
-1369
-493
-959
156
-766
77
0.5
y/L
Figure 7: Comparison of wave profiles along the hull for
Froude number 0.25. The abscissa is normalized by
the ship length.
Table 2: Resistance and lift or L/B variations for Froude
number 0.25. All forces are in N.
L/B
2.5
4.9
10.0
Air
plenum
without
with
without
with
without
with
Total
25
23
28
24
32
27
Resistance
Viscous
Wave
11
14
8
15
16
12
12
12
24
8
19
8
Lift
-403
-526
-270
-339
-164
-160
Figure 9: Wave and total resistance coefficients vs. L/B
for fully wetted truncated hull and in the presence of an
air plenum.
The above various hulls, with and without an
air plenum, were simulated also for the Froude
7
resistance is also observed. It is obvious that the
benefit of the air plenum is consistent throughout
the range of spacings we have studied. The lift
forces are compared in Figure 13. The predicted
lift forces indicate that the sinkage force is
decreased greatly when the air plenums exist in
all three spacing cases, which illustrate another
benefit of the air plenums.
The resistance coefficients normalized by
2
1
for the fully wetted surface and in the
ρ
SV
2
presence of an air plenum are plotted in Figure 9.
It is observed that for Fn = 0.25 the reduction of
the total resistance is slightly enhanced for L/B
near 6 or 7 due to the additional reduction of the
wave resistance. This suggests that there might
be an optimum L/B for this Froude number.
Air Plenum Volume Change (relative to the initial volume)
0.02
140000
3
Volume Change [m ]
To address twin hull configurations, the truncated
Wigley hull with L/B=10 is used as a demi-hull.
Three spacings (1B, 2B and 3B) between the two
demi-hulls are considered. Air plenums of 50%
ship length and 50% demi-hull beam are
considered.
A numerical solution of such a twin hull with
the spacing 2B is shown in Figure 10, where the
free surface wave interference pattern can be
observed. In this simulation with air plenum, the
air plenum pressure is set to a constant, 103,192
Pa, which corresponds to the hydrostatic pressure
at the bottom of the hull at rest. The time history
of the air plenum volume change is shown in
Figure 11. The air cavity grows as the ship
accelerates from rest, and then stabilizes around
0.02 m3, once the ship has reached the desired
speed. The predicted wave resistance indicates
that, at this Froude number, this resistance is little
affected by the presence of an air plenum.
160000
Plenum Pressure [Pa]
TWIN HULLS
0.03
0.01
Volume Change
Plenum Pressure
0
-0.01
120000
100000
0
5
10
15
20
25
30
80000
35
Time [s]
Figure 11: Time history of the air plenum volume. Air
plenum pressure is set to a constant value of 103,192
Pa.
250
Total
200
Resistance [N]
Total (single hull)
150
Wave-making
100
Wave-making
(single hull)
50
0
Wetted bottom : solid lines
With air plenum : dashed
0
2
4
6
s/B
Figure 12: Resistance of the twin hulls with the spacing
1B, 2B, and 3B for Froude number 0.50.
0
Wetted bottom : solid lines
With air plenum : dashed
Lift [N]
-500
Figure 10: Fish-eye view of the converged free surface
shapes with air plenum at Froude number 0.25.
-1000
The predicted free surface waves around the
twin hulls for Fn = 0.50 are much higher than
those of the lower Froude number. The predicted
resistances at Fn = 0.50 are plotted in Figure 12.
While the reduction in the total resistance is
obvious, a small decrease in the wave-making
-1500
0
1
2
3
4
s/B
Figure 13: Lift force of the twin hulls of the spacing 1B,
2B, and 3B for Froude number 0.50.
8
VISCOUS EFFECTS
BEHAVIOR
ON
AIR
domain boundary conditions, two planes of
symmetry are used to specify the boundary
conditions at the plane of symmetry and at the
ocean free surface. For the later, we preliminarily
use a symmetry plane “double body”
approximation (zero Froude number limit).
To study the viscous effects, we conducted
UnRANS computations for four different Reynolds
numbers, Re = 1.2x106, 2.4x106, 4.8x106, and
1.2x107. Steady state solution in each case is
achieved when the integrated forces converged
within four digits. Comparison of the dimensional
total drag forces is shown in Figure 15. A good
agreement in the dimensional total drag force can
be observed when the form factor k in the
empirical equation (8) is chosen to be 0.15.
PLENUM
This section describes our preliminary study of
viscous effects on the air plenum behavior using
the unsteady RANS solver, DF_UNCLE.
In order to compute the flow around a bottomtruncated Wigley hull with a Navier-Stokes solver,
we have generated a multi-block structured grid
for a half ship. The computational domain is
constructed by locating all the far-field boundaries
at least three ship length away from the ship hull
surfaces. Two surface grids, an H-type grid with
61x31 grid points and an O-type grid with 61x21
grid points are generated respectively for the side
and the bottom surfaces of the ship. The volume
grid is partially shown in Figure 14 and has a total
of 0.5 million grid points and is composed of six
H-H type grid blocks and one O-H type block grid
located under the ship bottom. To resolve the
boundary layer, the grid is clustered near the ship
surface and gradually stretched out to the far field.
The first grid point above the ship surface is
3.00E+01
Emprical Equation with k=0.15
Navier-Stokes Computation
Total Drag Force (N)
2.50E+01
adjusted to satisfy y + ≅ 1 for each Reynolds
number tested so that the turbulence model can
be properly applied.
2.00E+01
1.50E+01
1.00E+01
5.00E+00
0.00E+00
0.0E+00
2.0E+06
4.0E+06
6.0E+06
8.0E+06
1.0E+07
1.2E+07
1.4E+07
Reynolds Number
Figure 15: Comparison of the total drag force between
the Navier-Stokes computations and the empirical
equation (8) for different Reynolds numbers.
In order to study drag reduction due to the air
plenum, a portion of the ship bottom surface (0.5
L x 0.5 B, corresponding to approximately 20% of
whole hull surface area) is prescribed as a cavity
area at which free surface boundary conditions
are applied. In this unsteady simulation of the
cavity development, the converged steady state
solution for the fully wetted ship is used as an
initial condition. Since the current Navier-Stokes
solver is based on the artificial-compressibility
method (Chorin, 1967), a Newton iterative
procedure is performed in each physical time step
in order to satisfy the continuity equation. The
time-accurate solution is accepted when the
maximum velocity divergence is less than 10-3.
Figure 16 shows the evolution of the cavity at
different time steps for Re = 1.2x107 and Fn = 0.25.
We observe that oscillation waves form near the
leading edge of the cavity and propagate
downstream. The developing cavity only slightly
alters the pressure distribution on the ship side
surfaces and thus does not modify much the
Figure 14: The multi-block structured grid generated for
a half of the domain.
All boundary conditions are treated in an
implicit manner. For the physical boundaries, free
stream velocities and pressures are specify at the
inflow and far–field boundaries and the method of
characteristic (MOC) (Merkle and Tsai, 1986) is
applied at the outflow boundary with the free
stream pressure specified. On the solid surfaces,
no-slip flow and zero normal pressure gradient
conditions are used. On the cavity surface, the
free surface boundary conditions described earlier
are applied. To complete the specification of
9
viscous pressure drag. Figure 17 shows the
dynamic pressure distribution on the hull for the
The
dimensionless time, T ≡ tL / U = 1.2.
presence of the cavity significantly reduces the
skin friction to result in overall drag reduction.
Figure 18 shows the comparison of the total drag
force for ships with and without a cavity at
different Reynolds numbers. Percentages of drag
reduction are shown in Figure 19. It is important
to note that although the cavity area is 20% of the
total hull surface area, the drag reduction is about
14% which is less than 20%. That is because the
cavity influences the flow field nearby and leads to
an increase in the skin friction on the hull surface
around it. Figure 20 shows the comparison of the
skin friction coefficient on the ship bottom surface
for ships with and without cavity. It is observed
that the skin friction on the surface area near the
cavity is increased.
T=0.8
Figure 17: The dynamic pressure distribution on the
hull for dimensionless time T = 1.2.
3.00E+01
Without Air Plenum
With Air Plenum
Total Drag Floce (N)
2.50E+01
T=1.0
2.00E+01
1.50E+01
1.00E+01
5.00E+00
0.00E+00
0.0E+00
2.0E+06
4.0E+06
6.0E+06
8.0E+06
1.0E+07
1.2E+07
1.4E+07
Reynolds Number
Figure 18: The comparison of the total drag force for
ships with and without cavity at different Reynolds
numbers.
T=1.2
T=1.4
20
D r ag R e d u ctio n P er ce n tag e
18
Figure 16: The time evolution of the cavity at different
time steps for Re = 1.2x107 and Fn = 0.25.
16
14
12
10
8
6
4
2
0
0.0E+00
EFFECT OF AIR PLENUM PRESSURE
2.0E+06
4.0E+06
6.0E+06
8.0E+06
1.0E+07
1.2E+07
1.4E+07
Reynolds Number
In this section, we consider the influence of nonoptimal air plenum pressures on the cavity
behavior. To do so, the air plenum pressure was
set to various values. The considered air plenum
was in the middle of the ship bottom, half ship
length long and half beam wide. The inflow was
slowly ramped up and constant air pressures of pg
= 103,000 Pa, 103,100 Pa, 103,200 Pa, and
103,300 Pa were imposed. The converged air
plenum shapes are shown in Figure 21 for the
three cases. The numerical method had no
problem converging to a steady solution. For low
air plenum pressures, the cavity/water interface
intrudes in the plenum. For higher pressures, it
extrudes into the water.
Figure 19: The percentage of drag reduction with air
plenum applied.
Figure 20: The comparison of the total drag force for
ships with and without cavity for Re =1.2x107 and Fn =
0.25.
10
air plenum with a length 50% of the ship length
and a width 50% of the beam was placed in the
middle of the ship flat bottom. Figure 24 shows
the air/liquid interfaces looking from the stern.
One can see that the cavity surface tends to
remain horizontal. The resulting wave resistances
with and without the air plenum are compared in
Figure 25. The wave resistance of the heeled hull
with an air plenum is slightly higher than that
without the air plenum.
However, the total
resistance with the air plenum is lower than that
without air plenum due to the larger frictional
resistance advantage.
The air/water interface shapes along the
center plane of the ship are compared in Figure
22. The middle part of the cavity surfaces for the
four conditions are more or less evenly spaced
with an approximate maximum interval of 0.01 m.
This is expected because the water head due the
gas pressure differences of 100 Pa corresponds
to 0.01 m. The predicted wave resistances are
compared in Figure 23. The wave resistance of
the case with pg = 103,100 Pa is the smallest of
all. Noticing that this corresponds to the least
deformed cavity interface, we may conclude that
the wave resistance increases as the cavity
surface deforms the most. This suggests that the
pressure in the air plenum should be such that the
cavity neither intrudes nor extrudes too much from
the flat bottom surface of the hull.
Figure 22: Crosscut geometry of the air plenum on the
center plane of the ship for four different imposed air
plenum pressures.
Figure 21: Converged air plenum geometry with
various air plenum pressures; from top left, pg = 103,000
Pa, 103,100 Pa, 103,200 Pa, and 103,300 Pa. Zcoordinate (vertical axis) is magnified 10 times.
AIR PLENUM BEHAVIOR WITH HEELED HULL
A successful design of a hull with air plenums will
require analysis of the plenum behavior in waves
and during maneuvering. As a preliminary step
towards full study of sea keeping performance, air
plenum behaviors with an inclined hull were
considered.
The truncated Wigley hull used above was
subjected to heels to starboard of 5o and 10o with
and without an air plenum. A full grid without any
symmetry was used in these simulations. A
gradual heeling was introduced at the beginning
of the simulation. The desired heeling was
reached after 1.25 s and the hull was fixed at that
posture until the end of the simulation. The inflow
was also gradually accelerated and reaches the
full speed (Fn=0.25) near 15 s.
Simulation in the presence of the air plenum
was performed under the same conditions. The
Figure 23: The effect of air plenum pressure on the
wave-making resistance.
When the hull with the air plenum (length 50%
the ship length and width 50% of the beam) was
subjected to the same 10o heel, the free surface
and air plenum geometries became as shown in
Figure 26. The pressure in the air plenum was
set to 103,100 Pa, and the air plenum surface
converged to a shape close to horizontal in the
middle. The free surface elevations along the hull
are compared in Figure 27. Waves on the port
side are stronger than those on the starboard
11
case with 3o trim by stern is shown in Figure 29.
The ship waves just behind the stern are
observed to be stronger than those near the bow,
which is expected because the hull has more
submerged volume near the stern.
side. The presence of the air plenum modifies the
wave very little. The effect of air plenum on the
resistance of a 10o heeled hull is shown in Figure
28. The wave-making resistance is not affected
by the air plenum in this case, and as a result, the
total resistance has the full benefit of reduction of
the frictional resistance.
Figure 26: The converged free surface and air plenum
o
geometry (looking from the stern). 10 heel. The hull is
made transparent for a better view of the cavity surface.
Figure 24: Steady state free surface and the air plenum
geometry (looking from the stern). 5o heel. The hull is
made transparent for a better view of the cavity surface.
Free Surface Elevation along the Hull (Fn=0.25)
0.1
0 o Heel Wetted bottom
10 o Heel Wetted bottom Port
10 o Heel Wetted bottom Starboard
10 o Heel with plenum Port
10 o Heel with plenum Starboard
Z [m]
0.05
0
-0.05
-3
-2
-1
0
1
2
3
Y [m]
Figure 27: Comparison of the free surface elevation
along the hull with and without the air plenum. Fn =
0.25, 10o heel.
Figure 25: Comparison of the resistance components
with and without air plenum. Fn = 0.25, 5o heel.
We also studied the effect of trim on the
truncated Wigley hull (with and without an air
plenum) subjected to 2o and 3o trims by bow and
by stern.
The port-starboard symmetry was
utilized in these cases. As in the case of heeling,
the trim was gradually increased at the beginning
of the simulation. The desired trim was reached
at 1.25 s and then the hull was fixed at that
posture until the end of simulation. This angular
adjustment of the hull was achieved much earlier
than the gradual acceleration of the inflow which
reaches the full speed (Fn= 0.25) near 15 s. As
an example of the free surface wave pattern, the
converged free surface for the wetted bottom
Figure 28: Comparison of the resistance components
with and without the air plenum. Fn = 0.25, 10o heel.
12
of this pressure over the air chamber walls is
equal to the integration of the same pressure on
the cavity surface. The resultant has a horizontal
component, which contributes toward the
resistance, and is equal to the product of the air
pressure, pg, and the horizontal projection of the
cavity area. As a result of this horizontal force,
the resistance will increase if the air pressure is
greater than the average hydrostatic pressure
plus other dynamic pressures.
In Figure 32, three converged air plenum
geometries obtained with different air pressures
are compared. As expected, the air plenum
surface always converges to a horizontal surface
at different depths, which is deeper with higher air
plenum pressure. The air plenum volumes are
compared in Figure 33, and the total resistance is
compared in Figure 34. The larger the plenum
pressure, the larger the plenum volume increase
and the larger the resistance. By lowering the air
plenum pressure to 102,600 Pa, we can actually
reduce the total resistance close to zero if we can
maintain the cavity surface as shown at the top of
Figure 32.
AIR PLENUM BEHAVIOR WITH TRIM
The wave profiles along the hull are compared in
Figure 30. The higher wave profile for trim by bow
is consistent with the fact that the wave resistance
of bow trim is higher than that for the even keel.
Figure 29: Converged free surface elevation of the fully
wetted bottom truncated Wigley hull with 3o trim by
stern. Froude number 0.25.
Free Surface Elevation along the Hull (Fn=0.25)
0.1
0 o Trim Wetted bottom
3 o Trim by stern Wetted bottom
3 o Trim by stern with plenum
3 o Trim by bow Wetted bottom
3 o Trim by bow with plenum
Z [m]
0.05
0
-0.05
-3
-2
-1
0
1
2
3
Y [m]
Figure 30: Comparison of free surface elevation along
the hull for the three trim conditions with and without the
air plenum. Froude number 0.25.
Figure 31: Comparison of resistance components
between the wetted bottom and the air plenum case. 3o
trim by stern. pg = 103,100 Pa.
An air plenum of length 50% the ship length is
placed in the middle of the ship, and is set to a
constant air pressure of 103,100 Pa.
The
resistance components for the hull with trim by
stern with and without the air plenum are
compared in Figure 31. The air plenum here
increases the wave resistance, but the decrease
of resistance due to the reduced wetted area
exceeds the increase of the wave resistance.
When the hull with an air plenum is subjected
to trim, the pressure in the air plenum significantly
affects the resistance. In this case, the integration
Figure 32: Converged air plenum surface shapes for 3o
trim by stern cases with air plenum pressures 102,600
Pa, 103,100 Pa, and 103,300 Pa (from top to bottom).
13
converged air plenum geometry, which is nearly
horizontal in the middle, is shown in Figure 35.
Figure 36 shows the converged air plenum
shape for 2o trim by bow. The air plenum has
converged to this shape in a very stable way
following an initial phase of adjustment as shown
in Figure 37. The resistance components in
Figure 38 also show that the resistance values
converge to their final values in a stable way. In
the case of a 2o trim by bow, the air plenum
increases the wave resistance slightly but still
reduces the total resistance.
Figure 33: The air plenum volume changes for 3o trim
by stern cases with air plenum pressures 102,600 Pa,
103,100 Pa, and 103,300 Pa.
Figure 37: Time history of the air plenum volume
change for 2o trim by bow. pg = 103,100 Pa.
Figure 34: The total resistance for 3o trim by stern
cases with air plenum pressures 102,600 Pa, 103,100
Pa, and 103,300 Pa.
Figure 35: Air plenum geometry for 3o trim by bow.
Inflow is from the right to the left. pg = 103,100 Pa.
Figure 38: Time history of the resistance components
for 2o trim by bow. pg = 103,100 Pa.
CONCLUSIONS
Figure 36: Air plenum geometry for 2o trim by bow.
Inflow is from the right to the left. pg = 103,100 Pa.
In order to predict ventilated cavity behavior and
hydrodynamic performance of hulls with air
plenum, we have expanded and used our
boundary element method code, 3DYNAFS©. The
developed code can predict the interaction
between the ocean free surface and the air
The air plenum of length equal to half the ship
length and of a constant air pressure of 103,100
was located at the center of the bottom, and was
used to make runs with 3o trim by bow. The
14
Chahine, G. L., Duraiswami, R., and Kalumuck,
K.M., “Boundary element method for calculating
2D and 3D underwater explosion bubble behavior
including fluid structure interaction effects”,
NSWC Tech. Report, NSWCDD/TR-93/52, 1997.
plenum / water interface. We have varied the hull
form
and
the
air
plenum
parameters
systematically,
and
have
studied
the
hydrodynamic performance of each combination.
We have also studied the effect of viscosity on the
air plenum behavior by using the unsteady RANS
solver, DF_UNCLE. Major new findings from this
study are listed below:
• For a given ship and air plenum, the positive
influence of the air plenum depends strongly
on the Froude number.
• At Froude number 0.25, the effect of air
plenum on the wave profile and accordingly
the wave-making resistance is negligible.
However, the free surface wave around the
ship affects the air plenum behavior to
different degrees depending on the location
and size of the air plenum.
• At Froude number 0.50, there are appreciable
effects of the air plenum on the wave-making
resistance. If the air plenum is located close
to the bow wave region, reductions in the
wave-making resistance are possible.
• For twin hulls at Froude number 0.50, the air
plenum reduces not only the frictional
resistance but also the wave-making
resistance for all spacings studied.
• Viscous flow solutions indicated that viscous
drag estimated from the empirical method
provides
acceptable
results especially
suitable for the initial design stages.
• The cavity behavior was found to be more or
less the same over the range of Reynolds
numbers studied between 1.2 and 12 million.
• The converged steady state air plenum
surface tends to stay horizontal even when
the hull is inclined.
Chahine, G.L. and Hsiao, C.-T., “Modeling 3D
Unsteady Sheet Cavity Using a Coupled
UnRANS-BEM Code”, 23rd ONR Naval
Hydrodynamics Symposium, Val de Reuil,
France, Sep. 2000.
Cheng, J.-Y., Chahine, G.L. and Kalumuck, K.M.,
“Computations of hydrodynamic characteristics of
a floating amphibious vehicle using BEM”,
BETECH2001, Florida, 2001.
Chorin, A.J., “A numerical method for solving
incompressible viscous flow problems,” Journal of
Computational Physics, Vol.2, pp.12-26, 1967.
Clayton, B.R. and Bishop, R.E.D., Mechanics of
Marine Vehicles, E. & F.N. Spon Ltd., London,
Great Britain, 1982.
Eggers, K., Sharma, S.D., and Ward, L.W., “An
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Determining the Wavemaking Characteristics of a
Ship Form”, Trans. SNAME, Vol. 75, 1967.
Gorski, J. J., “Pressure Field Analysis of a
Propeller with Unsteady Loading and Sheet
Cavitation”, Proceedings, AIAA paper, 1998.
Hsiao, C.-T., Pauley, L. L. “Numerical
Computation of Tip Vortex Flow Generated by a
Marine Propeller”, Journal of Fluids Engineering,
Vol. 121, pp. 638-645, 1999.
Goumilevski, A., Cheng, J, and Chahine, G.,
“Wave breaking on a sloping beach: comparisons
between experiments and simulations”, Proc. 14th
ASCE Engr. Mech. Conf., Austin, TX, May 2000.
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ACKNOWLEDGEMENT
This work was conducted at DYNAFLOW, INC. in
Jessup, MD, USA, (www.dynaflow-inc.com). We
would like to acknowledge the support of the US
Office of Naval Research under an SBIR Phase I
Award No. N00014-03-M-0394 monitored by Dr.
Paul Rispin. This support is greatly appreciated.
16