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OCR for page 607
Steady and Unsteady Characteristics of a Propeller
Operating In a NonUnifo~m Wake:
Comparisons Between Theory and Experiments
F. Genoux, R. Baubeau (Bassin d'Essais des Carines, France)
A. Bruere, M. DuPont
(Office National des Etudes et Recherches Adrospatiales, France)
ABSTRACT
The predictions of the steady and unsteady cha
racteristics of a propeller operating in a nonuni
form wake has been a task of R&D for the past de
cades, seeking to meet the increasingly demanding
requirements of acoustical discretion in the design
of propellers for ships.
The present paper exposes the latest work
conducted at the Bassin d'Essais des Carenes in
the theoretical and numerical fields to produce a
numerical code able to answer this need, as well
as the results issued from an original technology
developed at ONERA to give access to the fluctua
ting pressure field on a blade. The code issued
from these efforts is based on a linearized lifting
surface theory and is fitted for low and moderate
loadings. Its originality lies in its ability to solve
either the inverse problem or the analysis one
with the same numerical schemes. Its formulation
is adequate for the computation of both steady and
unsteady characteristics of a propeller operating
in a nonuniform incoming flow.
The convergence tests are commented to give
an idea of the robustness of the code. The numeri
cal results are compared to experimental data
available in the open litterature and to measure
ments derived from experiments conducted with
the technology of thin film pressure transducers.
INTRODUCTION
The prediction of the steady and unsteady cha
racteristics of a propeller operating in a nonuni
form wake aims to meet the increasingly deman
607
ding requirements of acoustical discretion in the
design of propellers for ships. These predictions
rely on both theoretical and numerical develop
ments able to match the designer's need for a re
liable and accurate tool. The validation of the ap
proach requires extensive and accurate experi
mental data, thus motivating highly complex and
heavy tests. From a designer standpoint, the
knowledge of steady and unsteady loadings is nee
ded to compute the levels of fluctuating forces
transmitted to ship through the shafts and the hull,
and thus to optimize the dimensioning of shaft sup
ports and bearings for fatigue. A good prediction
of radiated noise induced by the propeller as well
as the evaluation of unsteady cavitation are also
conditionned by an accurate computation of fluc
tuating pressure fields on propeller blades. Despite
the clear need for predictive tools, the progress
have been slowed for a long time due to the lack of
computational power and the release of new gene
ration computers during the two last decades have
certainly contributed to the improvments in the
area of propeller computing.
Although an extensive review of all the theore
tical and experimental works conducted on the
prediction of steady and unsteady characteristics
of propellers is out of the scope of the present
paper, it is interesting to underline the main steps
in both theories and documented experiments in
the development of tools for the computation of
unsteady loadings. The first consistent and perti
nent tools from an engineering point of view were
based on lifting line method, ranging from quasi
steady approach (1, 2, 3) to twodimensional uns
teady method (4, 5, 6~. Some refinements were
brought by combining quasisteady approach and
twodimensional unsteady method (7). The limita
OCR for page 607
tions of these methods are precisely identified  a
good review of their failures can be found in the
reference (8~. Theoretical attempts (9), using
matched asymptotic method, have been made to
solve the most obvious and severe restrictions,
such as neglecting threedimensional effects 
span, skew.
Despite the save of computational time in com
parison to heavier methods, the remaining inaccu
racies  specially for low Expanded Area Ratio
propellers  as well as the increasing computatio
nal power of computers have motivated the deve
lopment of new codes based on linearized lifting
surface theory. The reference (10) summarizes
the different steps taken in that direction for the
last twenty years. The codes developed within this
theory lead to a clear improvement of the accura
cy and reliability of the numerical results (111.
From a designer's standpoint, the use of the li
nearized lifting surface theory to solve the inver
se problem  computing a propeller geometry mee
ting propulsive requirements  is proposed in
(12), where the presence of an axisymetric body
is taken in account.
As opposed to the important number of theore
tical efforts, there are very few well documented
experiments in the open litterature. Therefore,
the possibilities of validation are limited and nu
merous numerical results are compared to the
measurements referenced in (8) and (13) publi
shed in 1968, more than twenty years ago. This
is partly due to the difficulties of gathering all the
data needed for a reliable validation. Besides the
access to the fluctuating forces transmitted to the
shaft, one has to accurately measure the flow
field feeding the propeller in its presence, and de
termine a proper procedure to deduce the effects
of the suction from the wake inhomogeneities.
Furthermore, there is no experimental data in the
open litterature that is known delivering informa
tion on fluctuating pressures at the blade surface
of a propeller. The lack of adequate sensors ful
filling the requirements for such measurements
has clearly limited the knowledge of pressure
fluctuations related to a propeller to measure
ments on adjacent walls.
Unfortunately, no unified tool based on the li
nearized lifting surface theory was available to
allow the computation of both steady and unsteady
characteristics of a propeller operating in a non
uniform flow, within the solving of the inverse
problem or of the direct one. The present paper
exposes the latest work conducted at Bassin
d'Essais des Carenes de Paris in the theoretical
and numerical fields to develop a numerical code
able to eliminate some of the previously mentioned
restrictions, as well as the experimental results
obtained in its facilities from an original technolo
gy developed at the Office National d'Etudes et de
Recherches Aerospatiales aimed at giving access
to the fluctuating pressure field on a blade. The
numerical results are compared to experimental
data found in the open litterature and to the ones
obtained at Bassin d'Essais des Cardnes. The expe
riments are documented as carefully as possible to
allow comparison.
MODEL AND SOLUTION PROCEDURE
The numerical code is based on a linearized lif
ting surface theory and is fitted for low and mode
rate loadings. The starting core was developed in
the late seventies and limited to the solving of the
inverse problem (12~. Implemented at Bassin
d'Essais des Cardnes, the code has gone through
many evolutions and is now stabilized in its matu
re form.
Its originality lies in its ability to solve the in
verse problem  determination of pitch and camber
laws for a given shaft power  as well as the di
rect analysis  computation of thrust and torque
for a given geometry  within the same formula
tion. Such a feature allows the validation of the
code used in its inverse mode  the important
mode for the designer  by checking the accuracy
of the results computed in its direct mode on geo
metries of reference propellers.
The code has been extended to permit the calcu
lation of unsteady forces due to the interactions of
a propeller with a nonuniform steady incoming
flow. This calculation remains possible in both in
verse and direct modes, thus allowing skew opti
misation in the design process.
The Figure 1 shows the geometry of the pro
blem. The propeller, which is represented by its
geometry H. operates behind an axisymetric body
C whose geometry is given by the equation of its
meridian. The body's advance speed,
608
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Vs. is supposed constant and the single screw pro
poller rotates at a constant rotation speed n. The
plane of reference used to describe the propeller
geometry is named [I. The helicoids emitted by the
blades are noted I;.
In both inverse and direct modes, the diameter,
D, of the propeller, its number of blades, Z. the
spanwise laws of skew, rake, maximum thickness
and chord length, as well as the chordwise law of
thickness are given. The spanwise pitch and cam
ber laws and the chordwise camber laws are
known in the direct mode and are unknown in the
inverse mode. The effective wake is supposed to
be known in the propeller plane.
righthanded propeller /~/ Y
~ ~\l
ar V x V
(4) ~(1 ) ~
I _~
~ (3) (2)
Fig. 1 Geometry of the problem
(1) Propeller H
(2) Helicoids £
(3) propeller plane r]
(4) Body C
The geometry of the propeller H is described in
the system of polar coordinates of axis the body
axis. Following ITTC standards, one defines a right
hand orthogonal system of Cartesian coordinates
with the origin O coinciding with the centre of the
propeller. The longitudinal axis x coincides with
the body axis, positive downstream; the trans
verse axis, positive part; the third axis z positive
upward. One uses a cylindrical system with origin
O and longitudinal axis x.
The effective wake field is described by the
three components, Vr, Via, Vx, of the velocity
vector, Or, 0), in the propeller plane, written in
cylindrical coordinates. The three components are
known by their harmonica! amplitudes, Aj k' and
phases(p; k' for k varying from 0 to infinity:
Vj = Vs ~ I, Aj, Or ~ cos [k ~ +(Pj, k]
k=0 (1
The fluid is assumed to be incompressible and
the flow irrotational. No presence of cavitation is
considered within this work. Therefore, the abso
lute velocity field derives from a potential A,
which satisfies the Laplace equation:
Ad>=0 (2)
Steady case
In the steady case, the only amplitudes of ve
locity components that are not equal to zero are
Ar 0 and Ax o The phases Hi 0, are equal to zero.
The absolute potential ~ can be split in two
terms:
Her, E, x) =Vsx+~(r', E', x) `3'
where the first term takes in account the body
advance velocity and the second term is the rela
tive velocity potential written in the polar coordi
nate system, (O. r', 8', x), rotating with the pro
peller.
The boundary conditions are written on the body
C and on the propeller, H. including its hub. If n
is the normal to the boundary, these conditions can
be written as:
 on the body C :
Vt n\si.n
2n n R
where As is the advance ratio of the propeller,
 on the propeller H:
at
,
= al ~ (,8 B) ^\/ ~ ~ AX, o(r)\s ~
(5)
where hi and hi+ are the positions of back and
face of the blade sections at the reduced radius, it,
609
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Vs. is supposed constant and the single screw pro
peller rotates at a constant rotation speed n. The
plane of reference used to describe the propeller
geometry is named n. The helicoids emitted by the
blades are noted A.
In both inverse and direct modes, the diameter,
D, of the propeller, its number of blades, Z. the
spanwise laws of skew, rake, maximum thickness
and chord length, as well as the chordwise law of
thickness are given. The spanwise pitch and cam
ber laws and the chordwise camber laws are
known in the direct mode and are unknown in the
inverse mode. The effective wake is supposed to
be known in the propeller plane.
righthanded propeller y)
2
. dC
(4) (1) :12)
Fig. 1 Geometry of the problem
(1) Propeller H
(2) Helicoids ~
(3) propeller plane rat
(4) Body C
The geometry of the propeller H is described in
the system of polar coordinates of axis the body
axis. Following ITTC standards, one defines a right
hand orthogonal system of Cartesian coordinates
with the origin O coinciding with the centre of the
propeller. The longitudinal axis x coincides with
the body axis, positive downstream; the trans
verse axis, positive part; the third axis z positive
upward. One uses a cylindrical system with origin
O and longitudinal axis x.
The effective wake field is described by the
three components, V r, Via, Vx, of the velocity
vector, Or, 8), in the propeller plane, written in
cylindrical coordinates. The three components are
known by their harmonica! amplitudes, Aj k' and
phases(pj k' for k varying from 0 to infinity:
t
Vj = Vs ~ >, Al, k(r ) cos [k ~ +(Pi, k])
k_0 t1 )
The fluid is assumed to be incompressible and
the flow irrotational. No presence of cavitation is
considered within this work. Therefore, the abso
lute velocity field derives from a potential A,
which satisfies the Laplace equation:
~ ~ = 0 (2)
Steady case
In the steady case, the only amplitudes of ve
locity components that are not equal to zero are
Ar 0 and Ax o. The phases Hi 0, are equal to zero.
The absolute potential ~ can be split in two
terms:
O(r, 8, x) = Vs x+~(r', B', x) `3'
where the first term takes in account the body
advance velocity and the second term is the rela
tive velocity potential written in the polar coordi
nate system, (O. r', B', x), rotating with the pro
peller.
The boundary conditions are written on the body
C and on the propeller, H. including its hub. If n
is the normal to the boundary, these conditions can
be written as:
 on the body C :
(4)
where As is the advance ratio of the propeller,
 on the propeller H:
at,+
r _
5~ !1 = ae + (A B) 3/ ~ ~ Ax' o(r)\s
(5)
where q~ and q~ are the positions of back and
face of the blade sections at the reduced radius, in,
610
OCR for page 607
duced by the interactions between the propeller
and the hull. It can be split into two terms:
 q1 strength of the source due to the sources
distributed on the projected propeller H',
 q2 strength of the source due to the doublets
distributed on the projected propeller H' and the
helicoids E.
The intensity of sources located on the projec
ted propeller H' is directly related to the thick
ness law of the blade profile at the considered ra
dius, according to the relation:
14~+  n)N
~ q an M
_
2~nR M R (10)
It should be underlined that it is not necessary
to compute the strength of the source induced by
the flow arond the body without the propeller.
The use of equation (8) and the KuttaJoukowski
condition allow to derive the equation relating
camber, pitch and source strength:
Vs
rl) R
A r, of r )
O AM
AX,o(r)
11 qP V [I ] OM dSP
1  qP y ~3 nM dSp
4 a R2 R V :~ . nM dSp
I1 ~ 6¢Tr. Ed R V ~rlP.PMl . Be dSp
4 ~ Jr ~ R2 L ~3 J
Ha+ + art)
~ an AM
2 R
,
+ (By)\/ ~ +Ax,O(r)\s M (11)
In the direct problem, the unknowns 6, and
5tTr Ed can be directly computed from the known
values of pitch and camber.
In the inverse problem, the designer chooses
the normalized circulation law and the perfor
mances  either thrust or torque to be by the pro
peller. Thus, the circulation law is defined with an
unknown multiplicative constant Fmax
In both modes, the discretization of the equation
(1 1 ) produces a linear system with a predomi
nently diagonal matrix. The resolution of the sys
tem does not raise any particular difficulty. The
forces are computed using the Joukowski theorem.
In the inverse mode, the shockfree entrance
condition supresses the suction force at the lea
ding edge of profiles. The integration of forces and
moments produces a second degree equation with
the unknown Fmax Reference (12) details the ap
proach.
In the direct mode, the leading edge suction
force has to be taken in account and is calculated
with the method described in (14) and (15~.
The suction effects due to the potential effects
of the propeller on the body can be computed by
integrating the efforts on its surface. These ef
forts are directly calculated using Lagally theo
rem.
At last, the linearization of the equations with
respect to As allows to compute the performances
at offdesign conditions close to the design point.
Unsteady case
In the presence of nonuniformities in the flow
feeding the propeller, the wake field velocity vec
tor can be split into two parts:
Vj(r, D)  Vs Al, O(r) + Vj(r, 8) (12)
with:
00
~
Vj(r' D) = Vs At, Al, k(r)COS[k ~ +9j, k]
k1
611
(13)
OCR for page 607
~
where V' are the three components of the ve
locity fluctuations encountered by the propeller
blades during the rotation.
Besides, it is assumed that the geometry of the
helicoids, I, is not affected by the inhomogeneities
of the incoming flow. Therefore, the solution of
the potential is split into three terms:
~
O = vs x + ~(r'' D'? x)+ ¢(rl, D', x) (14)
With the mentioned assumption, the lineariza
tion of the problem eases considerably its solving,
for the two first terms are solution of thesteady
problem. As a further simplification, the unsteady
interactions between the propeller and the body
are neglected and the indetermination between
sources and doublets is solved by assuming that
the fluctuations of potential are only related to the
doublets.
Thus, the only boundary condition remaining ap
plies to the projected propeller surface, H':
Vet · n _ O
In R (15)
is:
The integral equation associated to the problem
~ R
1
~
V r(r' Ok)
~
V~(r, Ok)
~
Vx(r' Ok)/
1 ~
6¢P R V: Pl nMdSP
lIOR2 Lm3J
SAP R V Up.PM1 nM dSp
~3 ~
() (1 6)
The angle Ok is the sum of the angle due to the
propeller rotation and the phase shift from blade
to blade. Thus, it is time dependent:
Ok2n ant (k~)
( 1 7)
The KuttaJoukowski is implicitly satisfied at
the trailing edge of the blades but, in the unsteady
case, the value of the potential jump, 6¢, is not
uniform on the helicoids £ and is not constant in
time at the trailing edges of the blades.
To solve the problem raised by the timedepen
dency of the potential, the following procedure has
been implemented:
 the value of the doublet associated to the jump
~
of potential 6¢ at the trailing edge at a given time
is obtained from the integral equation (16),
 the doublet element associated to the potential
jump is convected downstream on the helicoid at a
speed equal to the steady component of the local
atinfinity velocity,
 the computation is actually conducted on the
first blade only, due to the bladetoblade periodi
city of the solution. Nevertheless, the other blades
are taken in account in the calculation,
 the computation, which can be seen as a tran
sient approach, is stopped when the periodicity of
the unsteady circulation is achieved.
Numerical procedure
The mesh used to solve the problems is based on
a collocation method described in (14) and (15~.
This method allows to use nonplanar surface ele
ments which are required to get accurate results
in the case of highly skewed propellers.
In the unsteady case, the calculation of the uns
teady pressures is required. Thus, the time deri
vative of the potential is deduced from the analy
tical computation of the potential. Special care is
to be taken for the doublet locations in the vicinity
of the trailing edge. Therefore, second degree dou
blets have been used to insure the continuity of
doublet intensity and of its first derivative at the
borders of the panels adjacent to the trailing edge.
Moreover, the timestep is chooser according to
the Shannon rule to allow consistent harmonica!
analysis. The numerical tests show that high en
ough discretizations in space and time give good
stability of the results.
6~2
OCR for page 607
COMPARISON TO E}OSWELL EXPERIMENTS
This section presents and comments the nume
rical results when simulating the experiments re
forenced in (8) and (13~. These experiments
consisted in measuring the characteristics of a se
ries of threeblade propellers in open water condi
tions and behind two wake screens  a threecycle
wake screen and a fourcycle screen. Four propel
lers were designed with the same diameter,
thrust, speed of advance and rotational speed. The
first three propellers have different Expanded
Aspect Ratio: 0.3 for the Propeller NRSDC N°
4132, 0.6 for Propeller NRSDC N° 4118 and 1.2
for Propeller NRSDC N° 4133. The fourth propel
ler, Propeller NRSDC N° 4143 has a highly skew
of 120° and an Expanded Aspect Ratio of 0.6. In
the trials behind the wake screens, besides the
monitoring of fluctuating resulting forces and mo
ments, the measurements included averaged
thrust and torque measurements and the wake
surveys.
The computations were made on an ALLIANT FX
80 machine equiped with 8 processors and 256
Mbytes of RAM. Both open water and behind wake
generators conditions were numerically tested. In
the open water tests, the design point plus two
offdesign points were computed. The mesh used in
all cases was 1 1 points spanwise and 15 points
chordwise. The durations of each case were in the
order of 5 mn CPU for the steady cases and 3 mn
CPU for the unsteady cases.
Figures 2, 3, 4 and 5 show the comparison bet
ween computational and experimental results in
the open water case. The thrust and torque coeffi
cients are plotted versus the advance coefficient.
These variables are defined as:
KTo= T
p n2 D4
KQ0= Q
p n2 D5
JO= VO
n D (18)
The analysis of the curves shows a satisfactory
agreement between computation and measurement
for the four tested propellers. The worst results
are found with the smallest EAR propeller while
the highly skewed propeller's performances are
well predicted. For all propellers, the computed
slopes of thrust and torque coefficients curves
versus advance coefficient are very close to the
experimental ones on a reasonably wide range of
advance coefficient.
Figures 6 and 7 show the comparisons between
computations and experiments in the unsteady
cases. As in reference (13), only the nonskewed
propellers have been represented. The fluctuating
force and torque coefficients are defined as:
KTi _ Tj, z
p n2 D4
for i = x, y, z
KQi = i' Z for i = x, y, z

p n2 D3
(1 9)
~
~
where Tj z and Q; z, i = x, y, z are respecti
vely the axial, horizontal and vertical components
of the force and moment fluctuating at the blade
rate frequency of the propeller.
The results obtained in the two experimental
configurations  threecycle and fourcycle wake
screens are gathered on these two figures. The
threecycle screen generates the axial fluctuating
force and torque, while the fourcycle screen ge
nerates the fluctuating transverse forces and ben
ding moments.
With figure 6, one can see that the axial forces
are well predicted. On the other hand, the trans
verse forces seem to be overestimated by the
computation although the trend of the coefficients
with EAR is respected. Figure 7 shows that the
predictions of fluctuating moments are good in am
plitude as well as in trend versus EAR.
BASSIN D'ESSAIS DES CARENES EXPERIMENTS
In the prediction of unsteady efforts, one of the
expected advantage of the linearized lifting surfa
ce methods over lifting line theories is to give ac
cess to the fluctuating force amplitudes as well to
the fluctuating pressure field with a better accu
racy. This last hope is a requiroment for a better
computation of sheet cavitation in nonuniform
613
OCR for page 607
flow conditions. The lack of experimental data al
lowing the validation motivated new experiments
consisting in equiping a propeller with pressure
thin film transducers.
Thin film pressure transducer technology
The objective of the thin film pressure transdu
cers is to give accurate measurements of the fluc
tuating pressures at specified points located on the
surface of a propeller blade with the smallest al
terations of the flow around the profiles. Besides,
the sensitive part of the captor has to be as small
as possible. To meet these requirements, a new
kind of transducers thin film pressure transdu
cers  have been developed (16~.
Principle
The pressure gauge uses capacitive variations.
The sensitive element is made of a flexible dielec
tric foil metallised on both sides, thus forming a
capacitor (see figure 8~.
Pressure Fluctuations
L 5 2 6
7 1 `,~ it/
4~3
Or..;
~ ., ~.~.~.~ ~
 3
Fig. 8 View of pressure thin film transducer
(1 ) Sensitive electrodes
(2) Glue layers
(3) Profile
(4) Dielectric sheets
(5) Guard ring electrodes
(6) Preamplifier
(7) Polarisation source
The thickness of the dielectric foil varies under
the action of a pressure fluctuation. That produces
a relative variation of the capacitance which is li
nearly proportional to the pressure fluctuation.
The coefficient relating the pressure fluctuations
to the capacitance variations is directly related to
the mechanic characteritics of the foil  Elastic
modulus and Poisson coefficient. The variation of
capacitance is transformed into a voltage varia
tion using a polarization source V connected to the
first electrode of the sensitive element and to the
preamplifier connected to the second electrode.
Thus, the output voltage Vs is proportional to the
pressure fluctuation.
In order to reduce the leakage effects on the
sensor sensitivity, guard rings are mounted on
both sides of the sensitive connection, which links
the sensitive electrode to the input of the pream
plifier. The guard rings are driven by the output
voltage of the preamplifier whose gain is one.
Therefore, no current flows through the sensitive
electrode into guard electrode capacitor and the
leakage capacitors are completely cancelled.
The basic transducer is composed of three 12.5
,um thick dielectric sheets made of Kapton:
 the first sheet contains the connections, the
upper and lower sensitive electrodes, the upper
guard ring,
 the second sheet isolates electrically the
lower sensitive electrode and the lower guard,
 the third sheet contains the lower part of the
guard ring.
The electrodes and the connections are manu
factured by metal vacuum deposit which is 0.2 Am
thick. The three sheets are bonded together and at
the profile surface with 3 to 5 Am glue layers.
On 0 0.30 m propellers, 3 to 6 transducers can
be mounted on the same blade in the vicinity of
high curvature area such as leading edge. The inte
gration on the blade itself is done using bonding
technique and does not require special equipment.
The preamplifiers are located as close as possible
to the transducer, in the blade root. From the
preamplifier exit, the signal is sent through co
axial cables to a rotating electronics feeding a slip
ring. The same ring is used to bring the DC power
to the preamplifiers and the probes. Underwater
experiments require special water leakproof cau
tion. A fourth 50 ,um dielectric sheet made of
Kapton metalled on one side is bonded on the three
other ones. A special epoxy paint is coated all
over the blade surface. Experience shows that the
life time of the transducers is more than a month,
6~4
OCR for page 607
even in deep water.
Performances
The main specifications of thin film pressure
transducers, given under a 100 V polarization
level, are:
 sensitivity:
 frenquency range:
 defectivity at 1 kHz:
 sensitive standard area:
 thickness':
 temperature range accepted:
2.5 10 8 V/Pa
1 to105Hz
2 Pa/Hz1 /2
4 mm x 6 mm
170 ,um
0°C to 60°C
The microelectronics volume required to mount
the microelectronics associated to three preampli
fiers is less than 10 mm x 7 mm x 1.7 mm
Experimental program
The experimental program focused on the mea
surement of fluctuating pressures on the blades of
a skewed sevenblade propeller operating in the
wake of a submarine model. It also included the
wake survey by Laser Doppler Velocimeter. The
LDV used at Bassin d'Essais des Carenes gives ac
cess to the three components of the velocity field
in the propeller plane.
The propeller was equiped with six thin film
pressure transducers, three on the back and three
on the face. They were located midchord at three
different radii, respectively 0.5R, 0.7R and O.9R.
The model was mounted under the plateform of the
towing tank Bassin 111 (length 220 m) and driven at
constant speeds. All the presented tests were done
at the same advance coefficient, Js, equal to 0.63.
Pressure signals were monitored, processed and
stored. A fast Fourier analysis was applied after
several successive acquisitions to improve the si
gnal to noise ratio and to check the stationnarity
of the signal. With a power spectaltype analysis,
the pressure levels for the harmonics at the Shaft
Rate frequency were extracted.
Tables 1, 11 and 111 give all the informations rela
tive to the geometries of both the propeller BA No
2515 and the model. The symbols used are the
ones recommended by the ITTC. The back and face
' including the waterproof protection
coordinates are given in the usual frame in per
centage of the projected chord length on the blade
mean line.
The harmonica! content measured by LDV is
given in Tables IV, V and Vl, which contain the ele
ven first harmonica! half amplitudes and phases
for ten dimensionalized radii ranging from 0.0265
m to 0.1165 m and for the axial, orthoradial and
radial components of the velocity vector. The
coefficients Aj k introduced in formula (13) are
obtained for k ranging from 1 to 11 by multiplying
by 0.002 the values read in the tables. The coeffi
cients Ax 0 are given in Table 1.
The experimental measurements are presented
in Table Vll. The eight first harmonics at the Shaft
Rate frequency are given for three model speeds
that were selected during the tests. The reference
is 1 pPa at O dB.
Numerical results
The computations were conducted with the code
tested on the Boswell experiments. Figure 9 shows
the results obtained in the steady case (harmonic
O of the wake only) and the unsteady case (com
plete wake) when the propeller operates in the
model wake. The reference advance coefficient is
taken at the design point, i.e. Js equal to 0.63.
There is a good agreement between calculations
and measurements.
Figures 10 and 1 1 give the amplitudes of the
harmonics of fluctuating forces and moments ac
ting on one blade. No dynamical balance allowed a
comparison to experiments.
The pressure calculated versus time on the
mesh points, analysed with a fast Fourier algo
rithm at the Shaft Rate frequency, and finally in
terpolated at the locations of the transducers.
Table Vlil gives the power pressure levels in dB
computed at the model speed of 1 m/s.
Discussion
The fluctuating pressure induced by the flow in
homogeneities can be written as:
615
OCR for page 607
oo
P= 2, Pk ei2nknt
k= 1 (20)
~
where the coefficients Pk are complex numbers
containing amplitude and phase informations for
the harmonica! order k. The power spectrum ana
lysis gives the coefficients ok versus k, realated
~
to the complex Pk according to:
* 1/2
~ ~
Ok = Pk Pk
2 (2 1 )
~ * ~
where Pk is the conjugate of Pk.
One introduces a pressure coefficient at the
harmonica! order k defined as:
Kp,k  20 109 Ok  40 109 Vs (22)
where the coefficients ok are the values co
ming from either the experiment Table Vll  or
the computation  Table Vlil.
Assuming that the wake harmonica! content is
not affected by Reynolds effects, one finds that
the pressure coefficients should remain constant
except the harmonic 1 which is affected by Froude
effects. Unfortunately, the analysis of the experi
mental data shows that the coefficients Kp i are
far from being constant. These discrepancies can
be attributed to the weak signaltonoise ratio ob
tained at the velocity of 1 m/s besides the
Reynolds and Froude effects affecting the wake
content. Nevertheless, it should be underlined that
experiments conducted on that model geometry in
various test facilities  wind tunnel and towing
tank  did not show a strong influence of the
Reynolds number on the wake harmonica! content.
For these reasons, only values of pressure
coefficients obtained at the two highest model
speeds have been averaged. This operation was
also applied to the harmonic SRI despite the pre
ceeding remark. Figures 12 to 17 show the compa
risons between the calculated and averaged pres
sure coefficients. The analysis of the results
underlines a good agreement between computation
and experiment. The only problems encountered
are mainly for the harmonic 1 and, on figure 17,
for the harmonic 5. The discrepancies for harmo
nic 1 can be associated to the absence of correc
tion for the immersion variations in the averaging
of pressure coefficients measured at different ve
locities. For the second problem, there is no clear
explanation: the experiment as well as the compu
tation can be suspected at this stage.
Nevertheless, it should be underlined that these
results are quite encouraging. The differences bet
ween calculations and experiments are in most
cases less than 3 dB. Such results are accurate
enough to allow noise radiation prevision and give
credit to the results obtained for the amplitude of
fluctuating forces and moments.
CONCLUSIONS
The code that was developed at Bassin d'Essais
des Carenes offers a good reliability, partly due to
the use of new collocation techniques insuring the
consistency of the method. The comparisons to ex
periments found in the open litterature and to
measurements realized at Bassin are quite rewar
ding, both in steady and unsteady operating condi
tions.
Nevertheless, as a dampening to this optimism,
it should be remembered that a difference of 6 dB
on levels is associated to a ratio of 100% on the
linear values. Further experimental tests are un
derway to increase the base of experimental data.
Specifically, measurements will be carried out in
the vicinity of the leading edge and LDV acquisi
tions have already been realized in a plane adjan
cent to a propeller operating in a nonuniform
flow. Such measurements are required to confirm
the absence of propeller influence on the wake
harmonica! content.
As future possible developements, the method
has the potential to take in account the unsteadi
ness of the flow due to the presence of low
frequency turbulent structures in the wake.
Besides, prediction of fluctuating sheet cavitation
seems to be feasible because of the confidence in
the accuracy of computed pressure field.
616
OCR for page 607
FE~NCES
(1 ) Lewis, F. M., Tachmindji, A. J., "Propeller
Forces Exciting Hull Vibrations," Transactions of
SHAME, Vol. 62, 1954
(2 ) Ritger, P. D., Breslin, J. P., "A Theory for
the QuasiSteady and Unsteady Thrust and Torque
of a Propeller in a Ship Wake," Experimental
Towing Tank Report 686, Jul 1958, Stevens
Institute of Technology, Hoboken, New Jersey,
USA
(3 ~ MacCarthy, J. H., "On the Calculation of
Thrust and torque Fluctuations of Propellers in
Nonuniform Wake Flow," Report 1533, Oct 1961,
Naval Ship Research and Development Center,
Washington, D.C., USA
(4) Lewis, F. M., "Propeller Vibration Forces,"
Transactions of SNAME, Vol. 71, 1963
(5) Sears, W. R., "Some Aspects of
Nonstationnary Airfoil Theory and its Pactical
Application," Journal of Aeronautical Sciences,
Vol. 8, N° 3, Jan. 1941
(6) Sevik, M., "Measurements of Unsteady
Thrust in Turbomachinery," American Society of
h/lechanical Engineers, Paper 64FE15, Mar.
1 964
(7) Krohn, J. K., "Numerical and Experimental
Investigations on the Dependence of Transverse
Force and Bending Moment Fluctuations on the
Blade Area Ratio of FiveBladed Ship Propellers,"
proceedings of the Fourth Symposium on Naval
Hydrodynamics, Office of Naval Research, Aug.
1 962
(8 ~ Denny, S.B., "Cavitation and OpenWater
Performances Tests of a Series of Propellers
Designed by LiftingSurface Methods," Report
2878, Sept. 1968, Naval Ship Research and
Development Center, Washington, D.C., USA.
(9) Guermond, J.L., ~Thdorie Asymptotique
Instationnaire de la ligne Portante. Application a
l'Helice Marine en Sillage Non Uniformed, These de
Docteurlng~nieur, 1985, University Pierre et
Marie Curie, Paris, France
(10) Kerwin, J. E., Marine Propellers," [annual
Review of Fluid Mechanics, Vol. 18, 1986, pp.
367403.
(11 ) Kerwin, J. E., "Prediction of Steady and
Unsteady Marine Propeller Performance by
Numerical LiftingSurface Theory,. Transactions
of the Society of Naval Architects and Marine
Engineers, Vol. 86, 1978, pp. 218253.
(12) Luu, T. S., Dulieu, A., "Calcul de l'Helice
Fonctionnant en Arribre d'un Corps ~ Sym~trie
Axiale," Proceedings of Association Technique
Maritime et Adronautique, 1977, pp. 301317.
(13) Boswell, R.J., Miller, M.L., "Unsteady
Propeller LoadingMeasurement, Correlation with
Theory, and Parametric Study, Report 2625,
Oct. 1968, Naval Ship Research and Development
Center, Washington, D.C., USA
( 1 4 ) Guermond, J.L., "About Collocation Methods
for Marine Propeller Design," Proceeding of
Propeller '88 Symposium, Paper N°8, Sept. 88
(15) Guermond, J.L., "Collocation methods and
liftingsurfaces,. European Journal of Mechanics,
B/Fluids, Vol 8, n°4, 1989, pp. 283305
(16) Portat, M., Bruere, A., Godefroy, J.C.,
Helias F., "ON ERA developed thin film transducers
and their applications," Proceedings of Sensors
Symposium, Jan. 1982
617
OCR for page 607
Fig. 15 Propeller BA N° 2515
0.7R Face Midchord
200
A
._
Cal
._
a)
o
A
._
._
o
Q
y
a)
._
._
ID
o
C'
100
o
200
100
O
200
180
160
140
120
100
Harmonic Order
Fig. 16 Propeller BA N° 2515
0.9R Back Midchord
5
Harmonic Orde
Fig. 17 Propeller BA N° 2515
0.9R Face  Midchord
ld43] ~
~ ~' ~~l ~ d ~
~  ~ ~ ~ ~ ~ ~ 
 . ~ ~ ~ ~ ~ ~ ~ ~
1 2 3
.
4 5 6
Harmonic Order
622
~ Kp computed
g Kp averaged
HI Kp computed
1~11 Kp averaged
HI Kp computed
88 Kp averaged
7 8
OCR for page 607
Table I  Propeller BA NO 2515 and model geometries
Propeller BA NO 2515 geometry
Diameter D = 0.235 m
Hub diameter d = 0,047 m
Number of blades Z = 7
Rig ht handed
£ C/D H/D US iT 100*Re/C as/d x Ax ~ o
0.200 0.1603 0.2960 0.0000 0.000 4.7203 0.0105 0.4183
0.250 0.1702 0.3801 0.8673 0.0019 3.8411 0.0334 0.4289
0.300 0.1799 0.4698 2.7244 0.0060 3.1369 0.0587 0.4376
0.350 0.1900 0.5515 4.7791 0.0110 2.5618 0.0805 0.4513
0.400 0.2007 0.6158 7.1098 0.0172 2.0907 0.0950 0.4714
0.450 0.2110 0.6617 9.7462 0.0248 1.7157 0.1006 0.4883
0.500 0.2195 0.6924 12.6561 0.0333 1.4238 0.0974 0.5016
0.550 0.2257 0.7108 15.7396 0.0423 1.1871 0.0891 0.5220
0.600 0.2300 0.7198 18.8603 0.0513 0.9809 0.0818 0.5566
0.650 0.2323 0.7209 22.0084 0.0603 0.7993 0.0772 0.6033
0.700 0.2322 0.7150 25.2566 0.0693 0.6431 0.0736 0.6364
0.750 0.2284 0.7019 28.6280 0.0781 0.5086 0.0691 0.6513
0.800 0.2189 0.6808 32.0666 0.0863 0.3967 0.0638 0.6754
0.850 0.2009 0.6490 35.3733 0.0931 0.3188 0.0629 0.7074
0.900 0.1702 0.6014 38.7550 0.0976 0.2470 0.0479 0.7315
0.950 0.1274 0.5306 42.4303 0.0993 0.2136 0.0082 0.7535
0.975 0.0931 0.4826 44.0882 0.0980 0.2200 0.0055 0.7657
1.000 0.0429 0.4270 45.4417 0.0946 0.2466 0.0299 0.7774
is the reduced radius, r/R
C is the chord length
H is the geometrical pitch,
US is the skew angle expressed in a,
iT is the total axial displacement. Rake is considered positive downstream,
Re is the curvature radius of the blade at the leading edge,
ds/dx is the slope of the mean camber line versus the profile mean line at the leading edge
Model geometry
Model length LM= 6.9 m
Model Midsection diameter DM= 0.625 m
The meridian equation is given as:
if O < x ~ 0.045914 LM, then r / RM= 5.7532*x
if 0.045914 < x < 0.40171, then r / RM= Yaft((x0.005439)*0.6/0.39627)
if 0.40171 < x < 0.73582, then r / RM= 1.
if 0.73582 < x < 1., then r / RM= Yfore((x0.33955)*0.6/0.66045)
where x is the longitudinal axis described in the paper and Yaft and Yfore are functions of x defined by:
Yaft = x*sqrt(23.992791.82582*x+161.51997*x;~140.33319*x;~+46.64635*x4)
Yfore = x*sgrt(6.44272+12.57402*x68.57971 *x2+81 ~ 66653*x3+32.1 0355*x4)
623
OCR for page 607
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~ CO CD (D (D ~n
a)
Q ~ ~ ~CO Ct)
cn ~CD CD CD CO
 C ~, ~ ~ ~
o
E
(n

~n
a) x a
~ c.) ~ ~c'
D ~D
C~
._
u~
~...
Z wr000
630
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DISCUSSION
Hajime Maruo
University of California at Santa Barbara, USA
The most important feature of the unsteady propeller characteristics
is the change of efficiency when the propeller is operating in the non
uniform wake. It is expressed by the relative relative efficiency. It
is widely recognized that the relative rotative efficiency is lightly
higher than one. Another feature of unsteady characteristics is the
phase shift of the fluctuating thrust and torque. This is due to the
term associated with B¢/3t in the pressure equation. In the present
paper, data concerning the above quantity are not given. The
computation of a propeller characteristics, when the propeller is
operating in circumferentially varying wake, has been carried out by
us several years ago (lSth ONR Symposium, Hamburg, 1984).
According to our experience, the simple linearized lifting surface
theory is not able to provide results which show satisfactory
agreement with experiment, and the nonlinear deformation of the
trailing vortex sheet in the slip stream must be taken into account.
AUTHORS' REPLY
The following Table gives the values of the steady components Tio
and Qio' i = x, y' z resulting from the interactions between the
propeller N° 2525 and the nonuniform wake at the nominal design
point Is equal to 0.63. Pig. 9 shows the comparison between the
axial thrust and torque TO and Qxo' and the thrust and the torque
computed in the uniform wake which nevertheless takes in account
the radial gradient of axial velocities.
V
Ti o
10*Qi,o 0.170
x
0.142 0.0019
Y z
0.0021
0.0133 0.0257
In our opinion, the relative relative efficiency is not affected as much
by the circumferential nonuniformities as by the radial gradient of
axial velocity field. In the ideal case of a purely axisymmetric body
ending with a large aft conicity angle, the relative rotative efficiency
would be substantial while, in the absence of fins, rudders, ..., the
unsteady periodical efforts would remain null.
In regard to the question relative to phase shift, it appeared that
phases are not important to know with accuracy in terms of their
practical meaning. Therefore, the results are not given.
At last, a theory mixing linearized lifting surface and nonlinear
deformation of vortex sheddings, as suggested by the discussion, does
not seem consistent. Moreover, the theory developed in the present
paper gives numerical results in good agreement with the
experimental measures available at the present time, thus, no
extension in the suggested direction has been considered.
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