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OCR for page 760
Hull Vibration Excitation by Propeller Sources:
a Link between Hydrodynamics and Marine Acoustics
R. Kinns,
(Dept. of Marine Technology, University of Newcastle, U.K.)
N. Peake and 0. Rath Spivack
(DAMTP, University of Cambridge, Cambridge, U.K.)
Abstract
Acoustic boundary element models are used to solve
the Helmholtz equation, in order to explore the na-
ture of fluctuating hull pressures due to propeller
sources, when the wavelength of underwater sound
is comparable to hull dimensions. Sources are rep- sources.
resented by stationary monopoles and dipoles near a Considerable effort has been devoted to the pre-
rigid hull. Results for submerged and floating bodies diction of hull pressure fluctuations near ship pro-
are described. The hullsurface is assumed to tee rigid pelters, due to cavitating and non-cavitating pro-
throughout. pelter sources. The principal focus has been on peri-
A simple ellipsoidal representation of submerged and odic pressure fluctuations at multiples of bpf. Above
floating bodies is used first to aid understanding of cavitation inception, these arise primarily from cyclic
how hull pressure distributions are affected by the lo- cavitation due to rotation of the propeller in a spa-
cation and frequency of propeller sources. The sea tially non-uniform wake field, but the effects of fluc-
surface is represented using image technioues. to en- tuating forces, rotating steady forces and blade thick-
ness can all be significant. There is increasing suc-
cess in predicting random fluctuations by appropriate
scaling of data from cavitation tunnels and towing
tanks. Progress in prediction of periodic and random
pressure fluctuations near the propeller, has, however
outpaced the capability to predict hull vibration ex-
citation.
which can be particularly obtrusive, as well as sin-
gle frequency components at multiples of propeller
blade passing frequency (bpf). Satisfactory vibration
prediction requires estimation of the distribution of
fluctuating pressure over the whole of the hull sur-
face, not just fluctuating pressure near the propeller
sure zero pressure there when the body is floating.
Solid boundary factors are used to indicate the prin-
cipal effects of source location, frequency and the sea
surface. Finally, results from an independent model
of a cruise liner hull are used to illustrate the com-
bined effects of diffraction, interference and flotation
on hull forces, with explicit modelling of the free sur-
face.
It has been known for many years that procedures
which focus on a limited area near the propellers can
. give poor accuracy in prediction of hull vibration ex-
intrO]UCtlOn citation. For example Cox, Vorus, Breslin and Rood
(1978) show how significant forces due to cavitation
Increasing emphasis on prediction of hull vibration can extend more than 15 propeller diameters forward
due to propeller sources stems from the need to of the propeller itself. The results of that work are
meet demanding requirements for passenger comfort. described by Breslin and Andersen (1994), who point
These now focus on broadband random vibration, out that the surface of the sea can have a critical ef-
OCR for page 761
feet, which is not reproduced correctly in cavitation
tunnels. It is, nevertheless, still common to limit at-
tention to an area that extends only a few diameters
forward of the propeller position and to assume that
only hull forces on the same side of the ship as the
propeller are significant in a twin-screw ship. The
potential errors in these approximations are further
increased by the effects of compressibility.
It has almost always been assumed that there is no
need to take account of the compressibility of wa-
ter in prediction of hull vibration excitation at low
multiples of bpf. The effects of retarded time are in
fact significant at even low frequencies. The reason is
that the wavelength of underwater sound is typically
half the hull length at bpf and comparable to the hull
beam at only four times bpf. This has various related
effects. Firstly, variations of phase with distance from
the source lead to interference between forces on dif-
ferent parts of the hull. Secondly, hull diffraction
depends on frequency, so that the amplitude of pres-
sure on the hull due to a source of constant strength
varies with frequency. Both effects are present for
submerged as well as floating bodies. Thirdly, the
quenching effect of the free surface changes with fre-
quency, which has a particularly marked influence on
hull forces due to sources that are near the stern
waterline of a surface ship. None of these features
are reproduced correctly in a numerical analysis that
solves Laplace's, rather than the Helmholtz equation.
The hull vibration excitation problem has at least as
much in common with the prediction of underwater
radiated noise (Ross, 1987) as prediction of hull pres-
sures in the immediate vicinity of a propeller.
, . . . . .. . . . . .
The work described in this paper has its origins in the
design of cruise liner propulsion systems for low vi-
bration. It has been demonstrated at full scale (Kinns
& Bloor, 2000) that the choice of propeller rotation
direction can have a very marked effect on cavitation-
induced vibration. The selection of rotation direction
can influence both the nature of cavitation patterns
and the spatial location of the principal sources. If
reduced source strength is associated with increased
distance of the source from the sea surface, then the
hull excitation might be reduced by a smaller amount
that expected from observation of cavitation sources.
Acoustic boundary element modelling techniques
have allowed the effect of source location to be ex-
plored for both ellipsoidal bodies and real hull shapes
(Bloor, 2002~. In that work, the surface of the sea
was modelled throughout as a pressure release sur-
face. The hull surface was assumed to be rigid in
most of the exploration, so that only the hull shape,
rather than its dynamic response which depends on
internal structure, was modelled. Following initial
investigation of the modelling approach using simpli-
fied shapes (Bloor & Kinns, 2000), the effect of source
position and frequency has been studied in depth
for a typical cruise liner hull shape (Kinns & Bloor,
2002a). The work has been extended to include the
effects of dipole sources (Kinns & Bloor, 2002b),
which can represent fluctuating propeller forces, fi-
nite blade thickness and the effects of fluid flow, in
addition to monopole sources that represent the prin-
cipal effects of unsteady cavitation.
Two approaches can be used to determine the effects
of the free surface of the sea. If the surface is as-
Mach number is not replicated in model-scale tests sumed to be flat, then it is possible to use image
that replicate Fioude number. If a 1/s scale model is techniques, whereby the hull is reflected in the sur-
used, then the wavelength ~ is ~ too large in rela- face and the underwater sources are reproduced with
tion to hull dimensions. The effects of compressibil-
ity are therefore underestimated. Thus, the effects
of the free surface and the relative phases of pres-
sure on different parts of the hull are not the same at
full and model scales, for a given multiple of bpf, or
any other component in the frequency spectrum. It
also means that the acoustic far fields of higher order
opposite sign above the surface. This ensures that the
pressure is zero everywhere on the notional sea sur-
face. This is the approach used by Vorus (1974) for
solution of Laplace's equation. It is used here to solve
the Helmholtz equation. An alternative approach is
to model the sea surface explicitly, which allows the
effects of surface gravity waves to be explored. This
sources, such as axial dipoles that represent fluctuat- approach was favoured by Bloor (2002~.
ing thrust, are more significant at full than at model In this paper, we demonstrate first that results ob-
scale. tained using boundary element models agree closely
2
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Representative terms from entire chapter:
hull forces
with those derived from analytical expressions for the
submerged sphere. We then show how fluid compress-
ibility influences fluctuating pressure, solid boundary
factor and hull force distributions on ellipsoidal bod-
ies, where the sources are monopoles and dipoles close
to the body. The body can be either submerged, or
floating. We explore dependence on frequency, by
varying the wavelength in relation to hull dimensions
over ranges of practical interest. Finally, we show
how results for a floating ellipsoid are related to those
for a real cruise liner hull shape, using results in Kinns
& Bloor (2002a,b).
Formulation of the problem and
fundamental equations
We consider here the calculation of the pressure field
(and related quantities) generated by an acoustic
source ensonifying a submerged body. The assump-
tion is made that the source is harmonic, which al-
lows calculations to be carried out in the frequency
domain.
The governing equation for this problem is the
Helmholtz equation, which can be written as:
V2~(r) + k24(r) = 0,
where <;b is the field (e.g. the acoustic pressure) and
k = w/c is the wavenumber, with w the frequency of
the source and c the speed of sound in the fluid.
The field is related to surface pressure (see e.g. Morse
and Feshbach, 1953) by the integral:
otr)= ~ Gin dS—/~o~'ndS, (2)
where G is the Green's function, S the scatterer sur-
face, and n is the outward normal on the surface.
To solve equation 2 appropriate boundary conditions
must be imposed. We have considered a scatterer
with a perfectly reflecting surface, and therefore used
Neumann boundary conditions with the derivative of
the field vanishing on the surface:
~ = 0 ~
which reduces equation 2 to:
Is In
(3)
The time harmonic dependence can be reintro-
duced explicitly by using the full form of the field
¢(r) exp—(ixt), to take compressibility into account.
Numerical solution using bound-
ary elements
Here we summarise the numerical treatment of the
boundary integral equation 2 to find the surface fields
(the field or its normal derivative for hard or pres-
sure release surfaces respectively) on a surface, for an
arbitrary source. The integral equation can be dis-
cretized, giving rise to a large matrix system which
can be inverted using a 'black box' inversion routines
such as NaG or Lapack routines.
Let us consider a medium with axes x, y, z where z is
the vertical axis directed upwards, x is the 'direction
of propagation' or the axis of the scatterer, and y is
the transverse horizontal coordinate. The surface is
denoted by S(x, y, z).
`1y The field can be expressed as the sum of incident
and scattered components. Denoting the incident
field with Hi, and assuming Neumann boundary con-
ditions, i.e. a rigid surface, the governing boundary
integral is given by:
—4'~' ~ ~ ' ~ogr'jdr' (4)
iLrJ = Berg— /
JS
where r = (x, y, z) and r' = S(x', y', z'), say. G(r, r')
is the free space Green's function, given by:
eiklr—r'I
G(r r')=
By taking the limit as: r ~ rS we obtain:
Oitrs)=
In most cases of interest in the project the surface can
be expressed as a function of x and a radial angle
it. In order to treat this equation numerically it is
therefore convenient to convert the integrations to
be with respect to x, 6. We thus obtain:
jirlctrs) = otrS)—,/ ,/ ~ 6~ )~(
The incident fields due to a monopole and dipole re-
spectively are:
ikr
hi = A—,
r
and
Vi =—D l
i ~ eikr
_ COSOL-
kr r '
where or is the angle between the dipole axis and r,
and A and D are source strengths (see for example
Kinns & Bloor, 2002b).
In addition to the total pressure on the hull surface, it
is interesting to calculate the Solid Boundary Factor
(SBF), which can vary considerably over the curved
hull surfaces. In particular its magnitude can fluctu-
ate widely where high multiple scattering occurs.
Another quantity which is of practical interest to ship
builders is the total force on the hull, which can be
calculated by integrating the pressure over the sur-
face. The cumulative force, which is the integral over
part of the surface for increasing longitudinal extent
forward of the stern, shows how excitation of different
parts of the hull contribute to the total.
Validation for the submerged
sphere
The numerical code which implements the boundary
elements solution described in the previous section
has been validated by comparison with an analytical
solution.
Analytical solutions exist for simple bodies ensonified
by a monopole. In particular, we have used the an-
alytical solution (e.g. Shelton and James, 1997) for
the total pressure on a rigid spherical shell due to a
monopole source. The expression for the total pres-
sure at a point (r, §) on the surface of the shell can
be written as
car, 6) = Pier, 6) + User, §), (20)
where the subscripts refer to incident and scattered
term respectively. If the monopole source is located
at (Ro, 0), each term can be expressed as an infinite
2.5~
sum of Bessel and Hankel functions and their deriva-
tives, which depend on the size of the sphere a and
on the position and frequency of the source through
aim the arguments ha and kRo.
The surface pressure on a perfectly reflecting sphere
with diameter d has been calculated with a monopole
`19y source located at a distance d/2 from the surface
of the sphere for a range of frequencies, using both
the analytical and the numerical Boundary Element
code. The amplitude and the real and imaginary part
of the field have been compared and show excellent
agreement. Figure (1) for example shows the ana-
lytical and numerical results for the real part of the
surface pressure ~ = d/4. Other cases show similar
agreement.
Surface pressure (roar pan) on sphere for monoDole. `=d/4
I 1
—analytical ~
—bou~daryebment I
1.5
:.~.5 .
o
EL O
-
-1.S
Hi Jit \~
1
0 4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Distance along source direction
Figure 1
Results for a submerged ellipsoid
Having validated the numerical code, we then applied
it to the case of a generic axisymmetric ellipsoid with
a ratio 1:5 of diameter to length. Discretization of
the hull surface was carried out as described earlier,
using 32 points in the radial dimension, and 60 along
the length of the hull, i.e. a total of 1920 surface
elements.
We present below results for different types of source,
located at varying distances from the stern on the
principal axis of the ellipsoid, and for a range of fre-
5
quenches.
Monopole excitation
We have calculated fields due to a monopole source
located at 0.25d for four different wavelengths: ~ =
8d,4d,2d,d.
2
1.8t
1.6t
1.4i
1.2
0.8
0.6
0.4
0.2
Solid boundary factor for monopole at d/4
_ , , , , , .
—X=Bd
—- A=4d
k=2d
X=1d
~~ ,'4
~~ __ _ _:;~~,<
t
0 1 2 3 4 5 6 7 8 9 10
O-
Figure 2
The results are shown in Figure 2, where the SBF
is plotted against the distance along the x-axis, and
has been calculated along the line on the top surface
of the ellipsoid defined by the intersection with the
plane [y—Z]-
SoLd boundary factor for monopoly at distance d
1.8
I1.6
1.4 1
1
0.8
0.6
0.4
0.2
_—X=&d
—;~=4d
X=2d
- - X=1d
O-
0 1
2
~ . ~
i~ ~
1{
. ll
is
3 4 5 6 7 8 9 10
Figure 3
The same set of results has also been obtained for the
case where the position of the source is further from
the ellipsoid, at x = d. These are shown in Figure 3.
Changes with frequency are small until the wave-
length falls towards the diameter of the ellipsoid.
Then, diffraction causes a substantial increase in SBF
near the bow. A surprising feature of the results is
the local minimum in SBF just forward of the stern,
which is almost independent of frequency in the range
considered. We intend to explore this further.
The SBF results are only influenced slightly by the
separation of the source and the stern of the ellipsoid.
This applies also to results for the axial and trans-
verse dipoles, which have the directivity indicated by
equation 19.
Dipole excitation
Calculations have been carried out also for transverse
and axial dipoles. In the case of an axial dipole the
distribution of the fields (and consequently of the
Solid Boundary Factor) on the hull is axisymetric,
as was the case for a monopole source. If the source
is a transverse dipole, this axisymmetry is lost. We
illustrate this in Figure 4, where the distribution of
the acoustic pressure due to an axial and transverse
dipole with ~ = d at distance d from the hull are com-
pared. On the grey scale, lighter grey corresponds to
higher values of the pressure.
Surtace pressure due to transverse dipole
Surtace pressure due to axial dipole
Figure 4
6
B
The SBF distributions due to a transverse dipole
at distance d from the hull, with its axis along the
z—axis, are shown in Figure 5, where the SBF is plot-
ted for the same chosen contour as before and for the
same four frequencies.
Solid boundary factor for trensvase dipole at distance d
Figure 5
1.8
1.6
1.4
1.2
0.E
0.e
O.~.
O.;
o
I~ ~ =~
Figure 6
Results for a floating ellipsoid
Our numerical code can also be applied to the case of
a floating body, by implementing the image method
(see for example Breslin 1994) to represent the source.
We present here results for a half-submerged ellip-
soid ensonified by a monopole, and show the effect of
changing frequency and source depth.
The monopole source is placed below the ellipsoid at
typical propeller position, at d/10 and d/5 forward of
stern waterline.
Figures 6 is equivalent to 5, but for the case of an SBFforsourccto~rdd/1O.aideD~dl1ObeowFoatinchuI'
axial dipole.
Again, changes with frequency are small until the
wavelength is reduced to about the the diameter of
the ellipsoid. There are, however, substantial dif-
ferences between the SBF distributions for the two
dipole orientations and the monopole. In the case
of the transverse ellipsoid, the SBF remains almost
constant over most of the length. It rises increasingly
sharply toward the bow when the wavelength is re-
duced below twice the diameter. In the case of the
axial dipole, the SBF changes with frequency nearer
the stern when the wavelength is similarly reduced.
1.',~
,.;
0.~
0.(
0.4
0.2
7
ash
Figure 7
`+-1~
For the case where the source is at d/10, we only
show results for a source depth of d/10 below the
hull surface (Figure 7~. In the other case, we show
results for two different depths, namely d/10 and d/5
(Figures 8 and 9~.
2.5 .
2
.c
~ As8d
-—A=4d
A=2d
A=1d i/:
! I
0 1 2 3 4 5 6 7 8 9 10
distant along hull
Figure 8
f 7g2°~°°—-— )
a.
distance along hull
Figure 9
The results for the floating ellipsoid show the pow-
erful effect of the free surface of the sea in changing
the dependence of SBF on frequency. As expected,
the SBF is close to 2 for locations on the hull near
the source, at any frequency in the range of interest
where the source to hull separation is also a small
fraction of a wavelength. The change with frequency
elsewhere on the hull surface can be very large, espe-
cially towards the bow, showing why it is important
to include the effects of a finite speed of sound in
surface ship analysis.
Cumulative hull forces can be calculated simply by
summation of forces on hull elements up to a given
station forward of the stern, taking account of the
surface orientation and dealing with the in-phase and
out-of-phase components separately. The cumulative
force magnitude can be derived from the two com-
ponents at the specified summation limit. This can
also be done for port and starboard sides separately,
when the source is positioned to one side of the hull,
in order to explore whether the force is dominated
by hull excitation on the same side as the assumed
propeller source.
Figure 10 shows how the cumulative vertical force
changes with distance along the ellipsoid, when the
source is d/5 forward of the stern, d/10 below the
hull and has frequencies such that the underwater
sound wavelength is 8, 4, 2 and 1 times the ellipsoid
diameter. The effects of increasing SBF are countered
by interference between forces on different parts of
the hull surface. Thus, the cumulative force oscillates
with larger relative amplitude as frequency increases,
and decays more slowly relative to its mean, while
the total force tends to fall.
12
10
2;
~ x=8d
— A=4d
A=2d
=1d
~ _
1 2 3 4 5 6 7 8 9 10
distance song hut
Figure 10
The principal effects come from interference between
the source and its negative image above the sea sur-
8
face. In effect, this is related to the ratio of the pres- the analysis at 12 and 24 Hz, but almost identical
sure due to a vertical dipole with its elements sepa- results can be obtained using other extents and el-
rated by twice the source immersion, to that due to ement distributions that ensure convergence for the
the submerged source in isolation. selected values of A. The boundary element model
The pressure ratio is close to unity for locations near for analysis at 12 and 24 Hz (Kinns & Bloor 2002a,b)
to the submerged source, but varies greatly with fre- uses 600 elements to describe the hull, with an addi-
quency elsewhere when the wavelength is not large in tional 1,176 elements to describe the sea surface. The
relation to ellipsoid dimensions. Significant changes hull elements are distributed between 25 longitudinal
with frequency are still apparent when the wave- segments, with 24 elements around the hull in each
length is twice the length of the ellipsoid. It is only
when the wavelength is very large indeed in rela-
tion to hull dimensions that the SBF distribution ap-
proaches the solution obtained using Laplace's equa-
tion.
These results for the ellipsoid show why there are
such large changes in SBF distributions for surface
ship hulls with increasing frequency, as described by
Kinns and Bloor (2002a). The following cumulative
force distributions are derived using their model.
Results for a cruise liner hull
A numerical description of a real cruise liner hull was
used in Kinns & Bloor (2002a,b) to illustrate applica-
tion of the modelling techniques to a real hull shape.
The same hull description is used here. The hull is
typical of modern twin-screw cruise liners having a
single skeg. It has a nominal waterline length of 251
metres and a beam of 32.2 metres. The draught is
8.3 metres in level trim.
Results are presented here for frequencies of 12, 24
and 48 Hz. These are the maximum values of one,
two and four times propeller blade passing frequency
(bpf) in the ship selected for detailed analysis. The
underwater sound wavelength ~ is respectively 125,
62.5 and 1.25 metres. The beam is then close to 0.25N Figure 11
0.5A and ~ respectively. The principal difference be- Figure 12 shows the selected locations of sources in
tween the numerical models for the floating ellipsoid the plane of the propeller disc, which is 8.4 metres
and the cruise liner is that the sea surface was mod- forward of the stern waterline. Figure 11 also shows
elled explicitly in the latter case, while image moth- hull sections near to the propeller disc. The monopole
oafs were used for the ellipsoid. source is on the periphery of the propeller disc, 40° in-
The sea surface is defined for a specified extent ei- board of top dead centre. This is close to the position
ther side of the hull, as far forward as the bow. The of maximum cavitation volume for the selected ship
surface extends by the same distance in the stern di- with inward rotating propellers. The dipole sources
rection. An extent of 300 metres has been used for are positioned at the hub of the propeller, on the star-
segment. Figure 11 shows the surface element distri-
bution on the hull itself, for the starboard side only.
The hull, but not the assumed excitation, is sym-
metric. For clarity, elements are shown for alternate
longitudinal stations, specified according to distance
from the stern waterline.
The sea surface extent is reduced to 100 metres for
analysis at 48 Hz, where ~ is only 31.25 metres. Also
the numbers of hull and sea surface elements are in-
creased to 984 and 1,560, because the element sizes
would otherwise be too large in relation to A.
3 6 9 12
Tranevsras co~oroinatc (metres)
l
15 18
_35-
·-~--&1_
_tZ8 _
.._..174—
_Z31 -
·-~--318_
_433 _
·540~
—7&0 _
·· O··~35m
_~438_
· O ll02 _
9
board side only. Sources on the port side are set to Results for the dipole can be presented as the ratio
zero. of the magnitude of the cumulative hull force to the
dipole force. The results are then valid for any ship
having the same geometry and relative source posi-
tion, regardless of scale.
O
-,-3
64
-12
_ ~
0 3
I
+ )
15 18
6 9 12
Tranever" coordinate (metros)
Sechon
a S.em
5~
als.'m
S~
at 10.4
Par
be. -
.4~pd
.
ban
+ U$30-
_~_
Figure 12
0 50 100 150 200
Distance from stern waterilne (mobas)
_ 48 Hz source
.~. 24Hz source
12Hz source
t.0 _
o' .
~ 0.8 -
:P
O 0.6-
_
~ 0.4-
=>
0.2-
o.o .
v ~ ~ r I ~
0 50 100 150 200
Distance from stern watarilna (matr - )
_ Vowel dim
- I Tnnewrso dlpob
Axial dipole
Figure 14
Figure 14 shows the cumulative vertical force ratio
distributions for 12 Hz sources at the propeller hub.
The force in the vertical direction tend to be larger
than in other directions, because of the shape of the
hull. The force due to the vertical dipole reaches a
maximum at about 15 metres forward of the stern,
while the maximum is only reached between about 60
Figure 13 and 80 metres forward of the stern for the transverse
Figure 13 shows the cumulative vertical force on the and axial dipoles.
hull at frequencies of 12, 24 and 48 Hz for a monopole
source having a maximum rate of change of mass
flux equal to 1 n4 kg/sec2 The effects of the ch~n~?s
in SBF and interference between forces on different
parts of the hull surface can be seen clearly, with the
initial peak moving aft as frequency increases. The
scaling of results with ship size is discussed by Kinns
and Bloor (2002a), who also show how forces change
with source position and are distributed between port
and starboard sides. These results reflect the charac-
teristics described for the floating ellipsoid.
10
The distribution for the axial dipole shows the ef-
fect of the inverted pressure field aft of propeller hub.
There is a low secondary maximum at the propeller
position, which is only a few percent of the overall
maximum. The effects of the sea surface at the stern
reduce its value substantially. The cumulative force
ratio falls with increasing integration extent before
rising again, so that the ratio for a distance up to
about 12 metres forward is close to zero.
Conclusions
1.0 -
o
° 0.8-
° 0.6-
s
o
~ 0.2-
3
of ~
o
+vo~ ~
I Traneve~ d—e
- ~ -bold dada
J ~
50 100 150 200
Distance from stern waistline (mebas)
Figure 15
Figure 15 shows the effect of increasing the hub
source frequency from 12 to 24 Hz. The cumulative
ratios for the dipole sources are similar at 12 and 24
Hz up to about 12 metres forward, but then change
with frequency. The effects of phase interference are
more marked at 24 Hz, so that the cumulative force
ratios for the axial and transverse dipoles reach max-
imum values nearer the stern. The changes in the
cumulative hull force distribution due to the verti-
cal dipole show the increased influence of its acoustic
field at 24 Hz. The maximum cumulative force is now
reached at about 60 metres forward. These changes
are caused primarily by the increased dipole acous-
tic pressures for the same near-field pressure mag-
nitudes, compounded by diffraction and free surface
effects that vary with frequency.
These results echo an important analysis by Cher-
tock (1965), who showed that the ratio of the force
on the outside of a submarine hull to the force trans-
mitted by the tailshaft is almost constant, regardless
of hull shape and the precise propeller force distribu-
tion. Chertock's analysis did not include the effects
of a finite speed of sound, nor was there any need to
represent the remote sea surface. The force ratio is
much larger for a cruise liner because the propeller is
under, rather than behind, the hull. Also, the present
results are influenced significantly by the longitudinal
location of the propeller relative to the stern water-
line.
Acoustic boundary element models have been used to
explore the nature of fluctuating hull pressures due to
propeller sources. The Helmholtz equation has been
solved, so that the effects of a finite speed of sound
are included. For the present study, these sources
were represented by stationary monopoles, represent-
ing the principal effects of cavitation, and by station-
ary dipole sources in different directions, which repre-
sent fluctuating forces at the propeller. The hull sur-
face was assumed to be rigid, with a pressure-release
surface to represent the surface of the sea.
In the first boundary element model, we used a simple
ellipsoidal representation of submerged and floating
bodies to support understanding of how hull pres-
sure distributions are affected by the location and
frequency of propeller sources. The model was ver-
ified by comparing results for a submerged sphere
with analytical solutions that are available for that
case. The sea surface was represented using image
techniques, to ensure zero pressure there. This ap-
proach has been used previously to obtain solutions
using Laplace's equation.
We have concentrated on calculation of solid bound-
ary factors (SBFs), which represent the effect of the
rigid hull and pressure-release surface on the pressure
that would be measured on a virtual hull surface for
the same source in an unbounded sea. In the case
of the submerged ellipsoid, having an aspect ratio of
5:1, we found significant diffraction effects when the
wavelength was reduced towards the minor diameter,
for a source lying aft of the ellipsoid along its longitu-
dinal axis. These caused significant departures from
the solution obtainable using Laplace's equation. At
wavelengths that are not much larger than the length,
these are compounded by interference between forces
on different parts of the hull surface. Results have
been presented for monopoles, longitudinal dipoles
and axial dipoles.
Much stronger dependence on frequency is observed
when the source is below the floating ellipsoid, with
the sea surface in its plane of symmetry. Then, the
sea surface has the effect of transforming a monopole
into a vertical dipole, causing interference between
the pressure fields due to the submerged monopole
11
and its negative image above the sea surface. This
causes SBF distributions to depend on frequency in
the range where the wavelength is not much larger
than the hull length, compounding the effects Of Cox, B D, Vorus, W.S., Breslin, J.P. and
diffraction and interference that are present for a Rood, E P. "Recent theoretical and experi-
submerged body. For the surface ship, this causes
departures from SBF distributions calculated using
Laplace's equation at frequencies that are well below
typical propeller blade passing frequency.
Finally, we used an independent boundary element
model, with explicit representation of the sea surface,
to show how hull forces depend on the nature and
frequency of submerged sources for a cruise liner hull
with twin screws. Features in the results are clearly
related to those for the floating ellipsoid.
Acknowledgements
The work described in this paper forms part of on-
going research to improve understanding of hull ex-
citation and underwater sound radiation due to pro-
pellers. The authors are grateful to BAE SYSTEMS,
QinetiQ, Dstl and PRO Princess Cruises for their
support and encouragement. They are also grateful
to Meyer Werft of Papenburg, Germany, for provid-
ing data that describe the hull of a modern cruise
liner.
References
Bloor, C.D., "A Study of the Acoustic Pressures on
a Ship's Hull due to its Propellers", Cambridge Uni-
versity PhD thesis, 2002.
Bloor, C.D. and Kinns, R., "Development of Acous-
tic Boundary Element Models for the Prediction of
Fluctuating Hull Forces due to Propeller Cavitation",
Proceedings of NCT'50, Newcastle, April 2000, pp.
247-262.
Breslin, J.P.
and Andersen, P.
Hydrodynamics of Ship Propellers, Cambridge
University Press, 1994
Chertock, G., "Forces on a Submarine Hull Induced
12
by the Propeller", Journal of Ship Research, Septem-
ber 1965, pp 122-130.
mental developments in the prediction of pro-
peller induced vibration forces on nearby bound-
aries", Proceedings of Twelfth Symposium on Naval
Hydrodynamics, 1978, pp 278-299.
Kinns, R. and Bloor, C.D., "The Effect of Shaft
Rotation Direction on Cavitation-Induced Vibration
in Twin-Screw Ships", Proceedings of NCT'5O, New-
castle, April 200O, pp. 231-246.
Kinns, R. and Bloor, C.D., "Fluctuating Hull Forces
due to Propeller Cavitation", to be published in
RINA Transactions 2002.
7
Kinns, R. and Bloor, C.D., "Hull Vibration Excita-
tion due to Monopole and Dipole Propeller Sources",
to be published in Journal of Sound and Vibration,
2002.
Morse, P.M. and Feshbach, H., "Methods of Theoret-
ical Physics", McGraw-Hill, New York, 1953.
Ross, D., "Mechanics of Underwater Noise", Penin-
sula Publishing, Los Altos, California, 1987.
Shelton, E.A. and James, J.H., "Theoretical Acous-
tics of Underwater Structures", Imperial College
Press, London, 1997.
Vorus, W.S., "A Method for Analysing the Propeller-
induced Vibratory Forces Acting on the Surface of
a Ship Stern" Transactions SNAME Vol.82 1974
, . . .
pp.186-210.
DISCUSSION
H.B.Clausen,G.M.Keith,and
U. M. Rasmussen
0degaard & Danneskiold-Samsee A/S, Denmark
We congratulate the authors on this thorough and
detailed investigation, and we thank them and
the organisers of the 24th symposium for the
opportunity to contribute to this discussion. It
has often, and for a long time, been taken for
granted that compressibility plays no significant
role in the calculation of propeller-induced hull-
pressure fluctuations, and a systematic
evaluation of its effects is very welcome.
The authors are very persuasive about the
importance of compressibility in the accurate
calculation of solid boundary factors at distances
from the propeller comparable to the acoustic
wavelength of the pressure pulsation. From their
analysis it can not be doubted that frequency
dependent diffraction effects have a significant
influence on the hull-pressure fluctuations at
these distances. We agree with all the authors'
conclusions regarding the effects of
compressibility on hull-pressure fluctuations.
However, in order to evaluate compressibility
effects on the vibration response of the ship,
analyses such as those presented by the authors
must be viewed in the light of vibration analyses
of the ship's structure, which are usually
expressed in terms of its structural vibration
eigenmodes. It is the interaction of the
distribution of the magnitude of the local
pressure fluctuations with the eigenmode shapes,
rather than the cumulative hull forces that
determines the vibration response of the ship. A
relative estimate of the strength of the local
pressure fluctuations may be acquired from the
article by differentiating the cumulative force
distributions, given for example in figure 13.
Quite clearly, the strongest local pressure
fluctuations occur close to the propeller, and it is
our experience that this region of farce
magnitude local pressure
forcing alone
determines the vibration response of the ship.
In the frequency range in which compressibility
begins to play a noteworthy role, the vibration
response becomes highly localised. In this case,
the vibrational response away from the propeller
is dominated by the structural transmission of the
forcing from the large pressure fluctuations near
the propeller. The structural transmission is
considerably more efficient than the spherically
divergent hydrodynamic transmission.
Our comments are based on our experience with
vibration analysis of ships. In general it is
difficult to make statements about the relevance
or otherwise of compressibility on fluid-structure
interaction problems without conducting a
detailed analysis of the structural vibration
response. Evaluation of the coupled problem is,
of course, contingent on being able to predict the
compressible pressure fluctuations, and in this
respect, the authors have made a significant
contribution to the debate.
AUTHORS' REPLY
We thank Henrik Clausen and his colleagues for
their constructive observations. Our aim has
been to develop an analysis tool that allows the
effect of compressibility to be explored, in terms
of both its influence on hull vibration excitation
and on underwater sound radiation from sources
in the presence of the hull surface. The analysis
is designed to expose effects at low to medium
frequencies, where the wavelength of underwater
sound is comparable to hull dimensions.
Our consideration of cumulative forces, for
different integration extents, was designed to
show how large the area of integration should be
to capture hull excitation in its entirety. We
think that previous emphasis on maximum hull
pressure has sometimes distorted design of
propulsion arrangements for minimum hull
vibration, because it places too much emphasis
on dimensions such as clearance between the
propeller and the hull, when the real emphasis
should be on reduction of acoustic source
strengths. We hope that our approach will allow
a clearer distinction between different effects in
the future, and thereby lead to improved design
optimization.
We agree that the actual effect of the complete
hull pressure distribution on vibration can only
be determined by considering the nature of the
eigenmodes that govern hull response at a given
frequency. Sometimes, these eigenmodes will
exhibit antipodes near the position of maximum
hull pressure, as well as being close to
resonance. This is the situation of greatest
concern to the structural designer, where
maximum hull pressure will tend to determine
the vibration characteristics of the ship. Our
analysis suggests that the hull pressure at
relatively large distance from the propeller may
sometimes have a significant influence on
vibration, especially if the maximum near the
propeller is close to a node in the principal
vibration eigenmode. We agree that the
significance of compressibility in these cases can
only be determined for a given ship design using
a coupled analysis of the structural vibration
response.
DISCUSSION
Stephane Cordier
Bassin d'Essais des Carenes, France
Our experience shows that resultant forces are
strongly dependent on the nonstationary nature
of the source position (propeller rotation). Can
you elaborate on the possibility of your code to
model moving sources?
AUTHORS' REPLY
We elected to look at simple stationary sources
first, in order to identify the underlying nature of
hull force distributions and their dependence on
source frequency. We are intending to
generalize the code to include moving sources.
DISCUSSION
Merle Lucie
Bassin d'Essais des Carenes, France
Could you give some comments on the
dimensional results of the floating hull case
compared to the cruiser liner hull case?
AUTHORS' REPLY
The graph showing the cumulative vertical force
for the floating hull case (Figure 10 in the paper)
was derived for a monopole source with
amplitude 4~ kg/sec2. The minor diameter of the
ellipsoid was 2 metres and the vertical scale
represents the cumulative force amplitude in N
for that case. The cumulative force for any
ellipsoid diameter d (metres) can be obtained by
multiplying the hull forces in Figure 10 by did,
for a specified value of Lid. Forces are directly
proportional to source strength.
The comparison with results for the cruise liner
hull in Figure 13 must be limited to the
qualitative trend of these curves for different
wavelengths, because the hull shape and source
location parameters are different. In particular,
the source is beneath the keel in Figure 10 while
it is offset to starboard for the results in Figures
12 to 14. Also, the keel depth of the ellipsoid is
large in relation to that of the cruise liner.
In a related paper (Kinns and Bloor, 2002b), hull
forces for the cruise liner have been presented in
the non-dimensional form: FIMob as a function
of xib, where Fis the cumulative hull force, b is
the beam, Mo is the monopole source strength
and x is the distance forward from the stern
waterline. The non-dimensional hull force due
to a dipole is FIF`I, where F`' is the dipole source
strength.
