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OCR for page 297
Numerical and Experimental Analysis of
Propeller Wake by Using a
Surface Pane] Method and a 3Component LDV
T. Hoshino (Mitsubishi Heavy Industries, Laid., Japan)
ABSTRACT
Hydrodynamic modeling of the trailing vor
tex wake of a propeller is one of the most im
portant f actors in developing a propeller
theory. A variety of trailing vortex wake
models have been proposed hitherto. However,
details of geometrical f eatures have not been
known clearly . In the present study, f low
f ields around propeller are precisely measured
in a cavitation tunnel using a 3component
Laser Doppler Velocimeter (LDV). Based on the
experimental f inding that the pitch of the tip
vortices are smaller than that of the inboard
trailing vortex sheets, the surface panel
method with a def armed wake model of the
trailing vortices is proposed. Then, the pres
sure distributions on the blade and the f low
f ields around propeller were calculated by the
present surface panel method. A better agree
ment of pressure distributions near the hub is
observed when the hub effect is considered in
the calculations. It is shown that the calcu
lated f low f ields around propeller are in good
agreement with the measured ones. Openwater
characteristics of propeller calculated by the
present method are also in good agreement with
experimental data.
NOMENCLATURE
2a(r) Pitch of blade section = P(r)
Bi j Influence coefficient due to the j th
source panel on blade and hub surfaces
c ( r ) Chord length
Ci i Inf luence coef f icient due to the j th
doublet panel on blade and hub surfaces
Cp( Pi ) Pressure coefficient
= ( P(Pi ~PO ) /P (VA +( ri2 ) ~ /2
D Propeller diameter
e, ,e2 Local coordinates on panel
J Advance coefficient = VA /(nD)
K Number of propeller blades
KT Thrust coef f icient of propeller
= T/(pn2D4 )
K? Torque coefficient of propeller
= Q./ (p n D )
Number of chordwise wake panels
L
n
n
n
N
2
NR
o
P(Pi )
Po
P(r)
Pw (r)
Q
r
to
rh
rwh
rwT
R(P, Q)
Rick
s
SL (r)
S
Sj
t1 ~t2
T
V,
V]
vt
Normal coordinate f or blade section
Propeller rotational speed, [ rps ~
Unit vector outward normal to surface
Total number of blade and hub panels
Number of chordwise blade panels
Number of radial blade panels
Propeller center
Pressure
Static pressure at inf inity
Pitch of blade section
Pitch of trailing vortex sheet
Propeller torque
Radial coordinate f rom propeller axis
= ~y2+z2
Propeller radius = D/2
Radius of propeller hub
Radius of hub vortex
Radius of ultimate wake
Distance between field point P and
boundary point Q
Distance between the ith control point
and the jth integration point
Chordwise coordinate f or blade section
Chordwise coordinate of leading edge
Boundary surf ace
Surface of the jth panel
Tangential coordinates on panel
Propeller thrust
Cartesian coordinates in the
bladef ixed f rame
XR (r) Propeller rake
v Velocity
Vt Perturbation velocity vector tangent to
blade surface
Speed of advance
Velocity vector of relative inf low
Total velocity vector tangent to blade
surface
Wi j Inf luence coef f icient due to the j th
doublet strip on wake surface
SG (r) Pitch angle of blade section
Sw (r) Pitch angle of trailing vortex sheet
~ i j Kronecker delta
Eli Area of the ith panel
Al Potential j ump across wake surface
A¢j Discrete potential j ump in the jth
panel
Angular coordinate from generator line
of propeller = tank (y/z)
Tetsuj i Hoshino, Nagasaki Experimental Tank, Mitsubishi Heavy Industries, Ltd .
348 Bunkyou Machi, Nagasaki 852, Japan
297
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p
JO
Q
V
Subscripts
by
TE
w
+
TV
Angular coordinate of generator line
of kth blade = 2~(k1~/K
Fluid density
Perturbation velocity potential
Discrete potential in the jth panel
Angular velocity = 2nn
Gradient operator
Face and back sides of blade,
respectively ~ <=l;face, K=~;b~ck )
B Blade
D Drag
H Hub
i,j Values on panels i,j, respectively
k Value on kth blade
P Potential
Q Value on boundary point Q
r, ~Radial and tangential components,
respectively
Value on trailing edge
Wake
x,y,z Components in Cartesian coordinate,
respectively
Upper surface
Lower surface
Values on corner points ~,v of panel,
respectively
1. INTRODUCTION
In recent years, propellers with various
blade geometries such as a highly skewed pro
peller have been fitted to ships in order to
reduce the propeller induced vibration and
noise, or to improve the propulsive perfor
mance of ship. A reliable numerical method is
indispensable for the design and analysis of
such propellers.
A number of propeller design and analysis
methods based on lifting surface theories such
as Vortex Lattice Method (VLM)[1] and Quasi
Continuous Method (QCM)~2] have been develop
ed. However, the propeller lifting surface
methods are essentially based on the thin wing
theory. Therefore, they are insufficient to
predict the pressure distribution on propeller
blade, especially near the hub where the ef
fect of blade thickness and hub would be
dominant.
On the other hand, surface panel method
has been remarkably advanced in the field of
aerodynamics for the design and analysis of
threedimensional wing and bodies [3123. The
surface panel method is one of the most ad
vanced methods, because it allows precise rep
resentations of the complicated blade geometry
of the propeller such as the highly skewed pro
peller. In the past few years, the surface
panel method has been applied to marine propel
lers and also advanced turboprop problems [13
203. In most of such propeller theories, howev
er, the geometry of the trailing vortex wake
of a propeller slipstream has been treated ap
proximately because of the complexity of the
slipstream.
Since the induced velocities on the blade
due to the helical trailing vortex wake of a
propeller are much larger than those due to
the trailing vortex wake of a wing, hydro
dynamic modeling of the trailing vortex wake
behind the propeller becomes important in
developing propeller theories. In the past,
the trailing vortex wake had been replaced by
a prescribed helical surface with a constant
pitch obtained from the undisturbed inflow or
a constant hydrodynamic pitch calculated from
the lifting line theory as described by
Hanaoka [212. In the actual propeller, the
trailing vortices leave the trailing edge of
the propeller blade and flow into the slip
stream with the local velocity at that posi
tion. Therefore, the detailed knowledge of the
velocity distributions of the propeller slip
stream would be indispensable to establish the
realistic model of the trailing vortex wake.
Due to the recent development of measuring
technique with Laser Doppler Velocimeter(LDV),
the measurements of time dependent flow fields
around propeller have been reported by many
researchers t2227~. Based on the results of
the flow field measurements, Kerwin and Lee
[1] proposed a rollup wake model which took
into consideration the contraction of the slip
stream and the rollup of the trailing vortex
sheets. However, the rollup model in which
the trailing vortices are assumed to be con
centrated into a single hub vortex and a set
of tip vortices at a certain distance behind
the blade is considered to be too simplified.
More realistic geometry of the trailing vortex
wake behind propeller has to be taken into
consideration.
In the present paper, a surface panel
method is described for analyzing the flow
fields around propeller operating in uniform
flow at first. Green's identity is applied to
obtain an integral equation with respect to
the unknown potential strength over the sur
face of the propeller blades, hub and wake.
Such method was firstly developed by Morino
for general lifting bodies [6,7J. An improve
ment on Kutta condition is added to the Morino
method. That is, the Kutta condition of equal
pressure on the upper and lower surfaces at
the trailing edge is applied in the present
study.
Next, flow fields around propeller models
operating in uniform flow are precisely
measured in a cavitation tunnel using a 3
component LDV. Based on the measured velocity
distributions of the propeller slipstream, a
deformed wake model is proposed where the con
traction of the slipstream and the variation
of the pitch of the inboard helical trailing
vortex sheets are taken into account.
Then, pressure distributions on the
propeller blade and openwater characteristics
calculated by the present method are compared
with the experimental data. Further, flow
fields around the propeller are calculated by
the surface panel method and compared with
those measured by the LDV.
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2. FORMiLATION OF PROPELLER PROBLEM
2.1 Coordinate Systems and Geometry of
Propeller
We consider a propeller rotating clockwise
with a constant angular velocity Q in an
inviscid, incompressible, irrotational flow
with a uniform axial speed VA far upstream.
The propeller consists of a finite number of
axisymmetrically arranged blades of identical
shape and a hub.
We define a Cartesian coordinate system
Oxyz with origin O fixed at the center of the
propeller, where x is measured along the down
stream axis of rotation as shown in Fig.1.
The zaxis coincides with generator line of
the first blade and the yaxis completes
righthanded coordinate system. A cylindrical
coordinate system OxrO is also introduced for
convenience. Angular coordinate ~ is measured
clockwise from the zaxis when viewed in the
direction of positive x. Radial coordinate r
is measured from the xaxis. Then, the Car
tesian coordinate system Oxyz is transformed
into the cylindrical coordinate system OxrO
by the relation
x = x, y = r sine, z = r cosO, (1)
where
r = ~ , ~ = tan~~(y/z).
Further, we introduce a helical coordinate
system (r,s,n) with pitch 2na(r). The eaxis
is measured chordwise from leading to trailing
edge of the blade section. The eaxis is
measured normal to the eaxis from face to
back side. Then, the cylindrical coordinate
system OxrO is related to the helical coor
dinate system (r,s,n) by
x = ta(r) sr n]//a(~)2+r2+xR(r),
~ = [sta(r) n/r]//a(r)2+r2,
where XR (r) is propeller rake defined by the x
coordinate of generating line at radius r.
' Vr Vz
V`~:
~_ Vy
~Vx
~Y
~x
Fig.1 Coordinate systems of propeller
Blade section of propeller is usually
defined in a way similar to that of a two
dimensional wing by the ordinates nb<(r,s) of
face and back sides along the chord where K = 1
and 2 show the face and back sides, respec
tively. Then the coordinate of a point on the
surface of the kth blade can be expressed as
x = xb<(r,s),
y = r sint~b~(r,S)+8k], ~ (3)
z = r coStebK(r~s)+Ok
where
Xb~(r,S) = La(r) Sr nbK(r~s)]t'/a(r)2+r2
+xR(r),
~(r,S) = I S+a(r) nbK(r~s)/r]/?/a(r) +r2,
= 2~(k1~/K, k = 1,2,.,K,
K = number of propeller blades.
2.2 Velocity Potential and Boundary Condition
Under the assumption of potential flow,
the flow field around a propeller is charac
terized by a perturbation velocity potential¢,
which satisfies Laplace's equation
(4)
and vanishes at infinity. We consider a bound
ary surface S. which is composed of propeller
blade surface SB, hub surface SH and wake sur
face Sw, and also unit outward normal vector n
to the surface S. Applying Green's identity,
the perturbation potential at any field point
P(x,y,z) can be written as a distribution of
source and doublet over the boundary surface
[6,73:
(2)
where
4nE¢(P) = ~Sl(Q)a3Q(~t Q')dS
at(Q) 1 dS
S anQ R(P,Q)
O for the point P inside S.
E ~ 1/2 for the point P on S.
1 for the point P outside S.
(5)
and R(P,Q) is the distance from the field
point P(x,y,z) to the boundary point Q(x',y',
z') and a /a nQ is the normal derivative to the
boundary surface at the point Q.
Kinematic boundary condition is that the
velocity normal to the blade surfaces SB and
the hub surface SH should be zero. Using rela
tive inflow velocity VI , the boundary condi
tion with respect to a moving frame fixed on
the propeller blade can be written as
eat = VI nQ = _(VA+QXE) nQ, on SB and SH,
(6)
where VA and Q are the advance and angular
velocity vectors respectively and r is the
position vector of the point P on the boundary
surface.
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We assume that the wake surface Sw is in
finitely thin and there is no flow and no
pressure jump across the surface Sw, while the
potential jump is allowed. The boundary condi
tion on the wake surface Sw can be written as
a++ at
= , p+ = p on Sw, (7)
and ant
where p+ are the pressures on the wake surface
Sw , and subscripts + and  denote the upper
and lower sides, respectively. For the steady
propeller problem the potential jump Al across
the wake surface is constant along an ar
bitrary streamline in the wake and can be
expressed as
¢ = ¢+ ~ on Sw.
Considering Eqs.(6) through (8), the
boundary integral equation (5) for the point P
on the blade and hub surface reduces to
2~(P)§S IS l(Q)anQ(R`p Q')dS
~iSW6~(Q')anaQ,( ~[Q'')dS
~SB+SH(VI nQ)R~p Cads on SB and SH ;
Here ~ denotes that Cauchy's principal value
should be taken and Q' is any point on the
wake surface Sw.
Eq.(9) is a Fredholm integral equation of
second kind for the velocity potential ~ and
can be solved uniquely. The resulting surface
potential distribution can be differentiated
to obtain velocities and pressures, which are
integrated to yield the total forces and
moments.
~ Upstream View
NR=12
NC=12
3. Numerical Procedure
3.1 Discretization of Propeller Blade, Hub
and Wake Surface
In order to obtain a numerical solution
for the boundary integral equation (9), the
surface of propeller blades, hub and wake is
divided into a number of small elements. In
the past application of the panel method to
the propeller problem, planar quadrilateral
panel has been used to approximate the
surface. However, the elements representing
the propeller blades must be nonplanar due to
the helical blade surface, which results in
gaps at the edge of the planar panel and
therefore numerical errors. In order to yield
a closed surface and avoid such numerical
errors, the surface of the blades, hub and
wake is approximated by a number of qua
drilateral hyperboloidal panels in the present
paper. This paneling is one of the important
features of the present method.
The discretization of a propeller is
divided into three portions, i.e., the gener
ation of blade panels, the generation of hub
panels, and the generation of wake panels. We
consider the discretization of a propeller
blade at first. In the choice of the radial
distribution of panel strips for a propeller
blade, it should be noted that the better
results could be obtained if the finer panel
strip was used in the region of rapid varia
tion of sectional properties. Therefore, we
will use the cosine spacing which concentrates
the panel strips at the hub and tip. If the
radial interval from the hub rh to the tip rO
is divided into NR small panel strips, the
radii of the corner points of each panel strip
can be expressed as follows:
where
rp = 2 (rO+rh) 2 (rOrh)cosap, (10)
l DIN +~' for ~ = 2
O for ~ = 1,
( Downstream View )
NR=12
NC=12
Fig.2 Panel arrangement for a highly skewed propeller
300
.3,~.,NR+1.
an
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In the chordwise spacing of the blade
panels, cosine spacing is considered to be the
best. Therefore, the chordwise positions of
the corner points of each panel are given by
Slav = sL(rp)~( 1cosQv), (11)
where
sir) = scoordinate of the leading edge,
c(r) = chord length of the blade,
OF = vn/Nc, v = 0,1~2,.~,Nc,
Nc = number of chordwise division.
This concentrates the panels at the leading
and trailing edges, where greater resolution
is required. A propeller blade surface is thus
discretized into NRx2Nc quadrilateral elements
per each blade.
Propeller hub is considered to be an axi
symmetric body on which the propeller blades
are mounted. The blade panels adjacent to the
hub surface are shortened or stretched to ob
tain the intersections with the hub surface.
Then the axial positions of the hub panels
meet with those of the blade panels at the
intersections. The hub portion from the lead
ing to the trailing edge is divided into some
strips equally spaced in circumferential angle
between the root blades. This generates the
panels with helical pattern on the hub. The
hub portion upstream of the leading edge of
the blade is divided into straight panels with
equal axial and circumferential spacings. On
the other hand, the hub portion downstream of
the trailing edge is helically divided with
the pitch at the root of the propeller. An ex
ample of the panel arrangement on propeller
blade and hub surface for a 5bladed highly
skewed propeller is shown in Fig.2.
Trailing vortex leaves the trailing edge
of the blade and flows into the slipstream
with the local velocity at that position.
However, the wake surface is usually ap
proximated by prescribed helical surface in
order to avoid time consuming calculation of
the slipstream velocities. In the present
paper, the surface of the trailing vortex wake
is determined based on the measured velocity
distributions of the propeller slipstream. The
wake surface is divided into NR wake strips,
which start from the trailing edges of the
blade strips. Then, each wake strip is divided
into L wake panels. The axial spacing of the
wake panel is finer near the blade and
gradually becomes coarser in the downstream.
Details of the numerical modeling of the
trailing vortex wake is shown in the following
chapter.
3.2 Linear Algebraic Equations
As mentioned above, the blade and hub sur
face is divided into N small panels Sj and the
wake surface is divided into NR X L small
panels SQ. The values of the potential and
(V~.nQ) are assumed to be constant within each
panel and equal to the values hi, A¢j and
(V~.nj) at the centroid of the panel,
respectively. Then, by satisfying Eq.(9) at
the centroid of each panel, one obtained a
system of N linear algebraic equations as
NR N
E(6ijcij)lj £~WiiLli =  £ Bij(V~ nj)(12)
for i = 1, 2, ...., N.
Here Sij is the Kronecker delta and Cij , W
and Bij are influence coefficients defined by
Cij = 2 n (R. )dSj ,
k] So a j i jk
K ~ 1 i: a ( )dSQ , (13)
k51 Q_~ 2H Stant Rink
i j k  ~ 2 HIS j R. i j k i
where Rij k and Rick are the distances from the
control point of the panel on the kth blade
and hub surfaces to the integration point on S
and SQ.
The influence coefficients Cij and Bij are
evaluated analytically in the near field [7~.
On the other hand, they are approximated by a
Taylor series expansion in the far field in
order to save computation time. The coeffi
cient Wij are also calculated by using the ex
pression for Cij. Then, Eq.(12) can be solved
numerically to yield the values of the unknown
potential fj.
The Kutta condition is applied to obtain
the values of the unknown potential jump Alj
on the wake surface. An equal pressure Kutta
condition is introduced in the present study.
A detailed formulation of the numerical Kutta
condition is shown in APPENDIX.
3.3 Velocity and Pressure on the Surface
The velocity and pressure distributions on
the blade and hub surfaces can be evaluated
directly by taking the gradient of the in
fluence coefficients for the velocity poten
tial Eq.~13~. However, it takes too much com
putation time because the influence coeffi
cients for the induced velocity must be newly
calculated. On the other hand, the velocity
and pressure on the surface can be obtained
also by differentiating the velocity potential
over the body surface which is already known.
This method takes much shorter computation
time than the former but numerical differen
tiation is used to introduce some numerical
errors.
In the present paper, the latter method is
adopted to calculate the velocity and pressure
distributions on the body surface. The numeri
cal differentiation was conducted as follows
[12~. The distributions of the velocity poten
tial ~ are approximated by a quadratic equa
tion passing through the potentials at the
centroids of three adjacent panels as
= at2 + bt + c,
(14)
where t is the surface distance and a, b, and
c are the coefficients of the quadratic equa
tion. Then, the derivatives of the potential
301
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along the tangent directions tl and t2 to the 3.4 Field Point Velocity
panel surface can be expressed as
ttl ~ at = 2alt1+b1,  (15)
~t2 ~ eat = 2a2t2+b2.
2
respectively.
Next, we take the e1 axis in the direction
of to and the e2 axis in the direction perpen
dicular to t1 in the plane composed of t1 and
t2 as shown in Fig.3. Denoting the unit vec
tors in the directions of the el, e2, andt2
axis by el , e2, and t2 , respectively, the
derivatives of the potential along the e1 and
e2 axes can be expressed as
te1 ~ Den (tl, l
at (t2(t2el)ltl ~ (16)
f~2 = ae2 (t2 e2)
Then the perturbation velocity tangent to the
body surface can be obtained by
Vt = filet + fe2e2, (17)
Adding the tangential component of the rela
tive inflow velocity VI, we obtain the total
velocity tangent to the body surface as
Vt = VI_(VI n)n+vt
= [(VI el)+¢e1 den + [(VI e2)+le2]e2, (18)
where
n = elxe2.
The pressure on the body surface can be
expressed by Bernoulli's equation as
P(Pi) = PO+ 2P[ VI I Vt I ], (19)
where
pa = static pressure at infinity,
p = density of water.
The pressure is finally expressed in terms of
the nondimensional pressure coefficient Cp(Pi)
, which is defined as
C ( ) P(Pi) Po (20)
where
WO = VIVA 2 + ~ rQ) 2 .
t2e2
/ t2~e2 \
1
_ e' ' \  t',e'
P2
Fig.3 Local coordinate system on a panel
302
The induced velocities at the field point
P outside the closed surface S can be evalu
ated by taking the gradient of the velocity
potential ~ as follows:
v(P) = Vpt(P)
An SB+SH¢(Q) PanQ(R`p Q')dS
+4~Sw6~(Q')9pana '(R~P[q',)dS
4~ SB+SH (VI nQ)VP(R(P Q) )dS, (21)
Then, Eq.(21) can be approximated by
N NR N
i j1 TjVPCij +j=£16ljVPWijj£1 (VI nj )VPBij,
(22)
VPCij = Is Vpan (R j )dSj],
VpW i i = k Zl Q[~4~§SQVPan'(Ri to )dSt], (23)
VpBii = k~lt4~i7Sj9P(Rijk)dSj].
Here, the influence coefficients VpCij , VpBij
and VpWij can be evaluated analytically in a
manner similar to the determination of the in
fluence coefficients for the potential t83.
3.5 Thrust and Torque of Propeller
The total forces and moments acting on a
propeller can be obtained by integrating the
pressures over the blade and hub surfaces.
Denoting the components of the outward normal
vector ni by (nxi' nyi, nzi), the potential
components of the thrust Tp and torque Qp of
the propeller can be expressed as
N
Tp = K ~ P(Pi)nXi6Si'
i=
N
Qp = K ~ P(pi)(nyi zinzi yi)6si'
i=1
(24)
where
Nisi = area of panel,
(Xi, yi, zi ) = coordinates of the point Pi .
With the skin friction coefficient C f(P i) ~
the viscous components of the thrust TD and
torque QD of the propeller can be written as
TO = 2 pK ~ Cf(Pi)VtXilVtil6Si'
1 N
(25)
QD = 2 pK. £ Of (Pi )(Vtz i ·YiVtyi ·Zi ) IVt i I AS i
1 = 1
where
(Vtxi,vtyi ~ Vtzi) = components of the tangen
tial velocity Vt i at the point P. i
OCR for page 297
Then we obtain the total thrust and torque
of the propeller as
T = Tp + TD, Q = QP + QD. (26)
Finally, advance, thrust and torque coeffi
cients are defined as
VA T
where
J = D' KT = my, KQ = my. (27)
n = propeller rotational speed,
D = propeller diameter.
4. MEASUREMENTS OF FLOW FIELD AROUND PROP~.T.RR
4.1 3component LDV System
The LDV system used in the present study
is a five beam, twocolor, 3component type
with a 3watt Argonion laser as shown in
Fig.4. This LDV system allows simultaneous
measurement of three components of time depen
dent velocities around a propeller [253.
In order to measure the time dependent
velocity at the specified field point, one
propeller revolution is divided into 256 an
gular positions and the velocity data are com
bined with the present angular positions of
the propeller. In the present measurements,
total of 5120 data are collected for each
velocity component and rearranged according to
each angular position. Then, mean values and
standard deviations of the velocity are ob
tained at each blade angular position.
The water in the tunnel is filtered to
10 ~ m particle size and then seeded with 4 Am
metallic coated particle which is best for the
present 3component LDV system because of its
high reflection index and adequate size.
4.2 Definition of Field Point Velocity
We define the velocity components in the
x, y and zaxis directions of the flow
around a propeller to be vx, vy, vz as shown
in Fig.1. Then, the velocity components in the
r and 0directions vr and vet can be expressed
as:
LDV optical system
 Laser =
~ ~ 3d mensional
Photo multipliers . . traverser
green ,
Display
Traverse 1. and
controller  printer
1t ~t l
ital Mini computer
Memory:512kB
HardJ flexible
Blade pulses/rev. Disk devices
Fig.4 3component LDV system
Table 1 Principal particulars of propeller
models
Propeller
Diameter of Propeller ( mm )
Pitch Ratio at 0.7R
Expanded Area Ratio
Bass Rado
Number of Blades
Blade Thickness Fraction
Rake Ande (deg.)
A
250.0
0.8000
0.6500
0.1800
5
0.0500
8.0
B
__50.0
1.000d
0.6500
0.1800
0.0500
8.0
. C
250.0
1.2000
0.6500
0.1800
5
0.0500
8.0
Table 2 Conditions for flow measurement by LDV
Advance Ratio' I
0.4Q, 0.56, 0.72

0.581 0.701 0.90
Ohio, 0.84, 1n8
PI .
A
8
C .
X/rO = 0.?5 O 0.25 0 5 10
20 r/rO=
~1.1
, 1.0 095
° 98 0.85
_ 0.7
· Measuring Point
Fig.5 Measuring positions around propeller
Vr = vy sinO+vz cos6, ~ (28)
vie = vy cosOvz sine,
If the velocity measurements were conducted in
a vertical plane, the ycomponent vy is iden
tical but opposite in sign with the tangential
component vie, and the zcomponent vz is iden
tical with the radial component v r as follows :
vx = vx,
Vr = Vz,
via = vy.
(29)
In a uniform flow' the flow field around
propeller is axisymmetrical and oscillating
with the blade frequency. If the propeller
rotates with a constant angular velocity, the
time dependent velocity measurements at a cer
tain radius correspond to the measurements at
the same radius for many different angular
positions of the measuring points at a fixed
propeller position. Hence, the velocity
measurements with respect to a certain propel
ler position give the instantaneous velocity
distribution of the propeller at a certain
time.
4.3 Propeller Models and Measuring Conditions
Velocity measurements of the propeller
slipstream were conducted in a uniform flow
for three propeller models which are five
bladed and different in pitch. Principal par
ticulars of propeller are shown in Table 1.
303
OCR for page 297
In the LDVmeasurements, propeller rota
tional speed was kept constant of n=20 rps and
advance speed of the propeller VA was changed
to vary the advance coefficient J. Measuring
conditions for each propeller are shown in
Table 2. Measuring positions of the flow field
around the propeller by LDV were taken
upstream and downstream of the propeller as
shown in Fig.5.
4.4 Results of Measurements of Flow Field
around Propeller
As an example of the results of the LDV
measurements, circumferential variations of
three components of the velocities around the
propeller B at the advance coefficient of
J=0.70 are shown in Figs.6  8. It is shown
that each velocity component is periodically
fluctuating with the blade frequency.
Fig.6 shows the velocity fluctuations
measured at various radial positions just
upstream of the propeller ( x/rO=0.25 ). The
variations of the axial and tangential
velocities vx, via are observed at inner radii.
Fig.7 shows the velocity fluctuations at
axial position of x/rO=0.25 just behind the
propeller. The sudden change of the radial
velocity component shows the velocity jump
across the trailing vortex sheet. The slope of
the velocity jump of the radial component at
inner radii ~ r/rO < 0.5 ~ is opposite to that
at outer radii ( r/rO>0.8 ). This shows that
the strength of the trailing vortex changes
its sign between the inner and outer radii.
Propeller B ( x/rO = 0.25)
4.0 i
3.0
2.0 ~ VX/VI
1.0 =~==Z
 QO ma_
~  1.0 VT/VA
20 ' ' ! ~
O 90 180 270 360
O (dim)
4.0
3.0
2.0
1.0
is0.0
1.0
2.0
4.0
3.0
2.0
~1.0
as0.0
1.0
2.0
4.0
3.0
2.0
1.0
of0.0
~ 1.0
2.Oo
Propeller B ( x / rig = 0.25 )
r/r. = 0.90
4.0 _
3.0 _.
2.0 ~ VX/VA
in; 1.0 _
pi o ~V,/V
20  ' ~'
O 90 180 270 360
O (de`)
4.0
:# 3.0 _ r/rB = 0.70
~0
1.0
0.0
~  1.0 _
2.0 )
. _ .
_
Vx/V~
1 ~ : .  ~ _
V'/\l~
1 ~ ~
Vr/V,
1 _
I I I I
90 180 270 360
O (den.)
r/rO = 0.30
0 90
4.0 _  _
:~ 3.0 _ r/r.= 0.50
~ 2.0 VX/VA
 1.0
~ on ~
~  1.0  Vr/VA
20 ~_ I
' O 90 180 270 360
8 (661.)
4.0 _
3.0
2.0 _
_ 1.0
Jo o.o
~ 1.0
_,n
VX /VA
_
_ V/V
180 270 360
O (don)
Fig.6 Circumferential variations of veloci
ties upstream of propeller B
~ J=0.70, x/rO=0.25 ~
_ r/rO = 0.70
r/rO = 0.50 Velocity Defect
VT/VA V0I0C;tY JUmP
I '. ' '
O 90 1 80 270 360 _ _ _
O (den.) 4.0 O (den.)
_ r/rO = 0.30 : ~3.0  r/rO = 0.90
~ 1.0~=~,
__~ ~;F 00 I ye~10~
_~.~ ~ 90 180 270 360
O (den)
4.0
3.0
 2.0
1.0
~ 00
~ 1.0
2.0
O  Propeller B ( x/r. = 0.25 )
0 _ r/r. ~ 1.00
0 vX/v~
0 ~
O ~0~=
O. Vr/V~
O ~ I 1 1
O 90 180 270 360
O (de`)
AD
3.0
;S 12 oO ~ =
;sy~lOo :'' ~
ia 20
n on 180 2~0 360
r/rO = 0.95 Velocity JUmP due to Tip Vortex
r/rB = 0.20
___ _
90 180 270 360
~ (de`)
v
O ~ Vo/VA V`/VA
~=~C:=
AXE
l l l l
90 180
~ (de`)
270 360
4.0
3.O
2.0
1.0
0.0
1.0
2.0
90 180
O (deg.)
270 360
Fig.7 Circumferential variations of velocities downstream of propeller B ( J=0.70, x/rO=0.25 )
304
OCR for page 297
4.0
3.0
2.0
1.0
~ 0.0
;~c 1.0
_ a n
Propeller B ( x / r0 = too )
r/r8= 0.70
VX/VA
V'/V~
at_
~UT/VA
Lou ~1 ~
0 90 180 270 360
O (de&)
4.0
3.1
;, 21
1.1
~ &1 _~
~  1.0 _
2.0 1 I
4.0 _
3.0
~0
1.D
~ O.C _
  1.t
4.1
3.1
2.1
1 1
~ al
  1.1
;b
[1
~'T;~
180 270 360
(dew)
I _ Propeller B ( x / r0 = 1.00 )
_ r/r.= 0.95
~ vX/v~
1 ~
1 i_
~Vr/V~
1 I I I ... I
0 90 180 270 360
O (de`)
4.0
30 _ rare = 0.90
2.0 Vx/VA Velocity lump due to nip Vortex
:~_,20
;a 20
0 90 180 270
~ (de`)
r/rO = 0.30 ~ 4t  r/rO = 0.85
VX/VA . V8~ by Jump due to Tip V0~8x
~ ~`1.C
~ o.c
VT/VA~ 1a
L.U ~~ 90 ~~ 180 270 ~ 360  2.a
~ (46~)
_ r/r. = 0.20
~ VX/VA
_ Vr/V~
' ' ' '
90 180 270 360
O (de&)
4.0
3.0
z0
~ 1.0
o.o .
~  1.0
a _2.0!
Fig.8 Circumferential variations of velocities
Further, this corresponds to the opposite
slope of the radial circulation distribution
at each radius. The strong variations in the
radial velocity component at r/rO=0.95 seem to
be due to strong tip vortices. Therefore, the
tip vortices are considered to be located near
this radius just behind the propeller. The
velocity defects in the axial and tangential
velocity components observed at the position
where the velocity jump of radial component
occurred correspond to the viscous wake of the
boundary layer on the propeller blades.
Fig.8 shows the velocity fluctuations at
axial position of x/rO =1.00 downstream of the
propeller. Same tendency is observed on the
axial, tangential and radial velocity
components. The velocity defects in the axial
and tangential components, however, become
small. This shows the diffusion of the viscous
wake of the blades. The trailing vortex
sheets are still observed as the velocity
jumps of radial component. The tip vortices
are considered to be located between r/rO=0.85
and r/rO=O.90, because the velocity change of
axial component is opposite between those two
radial positions. Same tendency was observed
for the other operating conditions and the
other propellers.
Velocity distributions around the propel
ler B at the advanced ratio of J=0.70 are
shown in Fig.9 as the form of the equi
velocity contour curves of axial component and
the velocity vectors of cross components
plan parallel to the propeller plane.
.. , l . .... .l
270 360
4.0
3.D

. 2.0
1.0
0.0
~ 1 0
2.0
90 180
(de`)
Van /V
90 180
O (do`)
270 3BO
downstream of propeller B ( J=0.70, x/rO=l.OO )
figure is observed from the downstream side of
the propeller. Since the propeller is right
handed, the tip vortices are rotating in the
anticlockwise direction as shown in Fig.9.
Radial position of the center of the tip vor
tices moves from the blade tip to the inner
radii along the downstream direction. This
means the contraction of the slipstream.
Trailing vortex sheets are also observed in
these figures. Angular position of the tip
vortex is larger than that of the trailing
vortex sheet and the difference of the angular
position is increasing along the downstream
direction. This means that the pitch of the
tip vortex is smaller than that of the trail
ing vortex sheet.
4.5 Hydrodynamic Pitch
Using the circumferentially averaged axial
and tangential velocities vx, Van, hydrodynamic
pitch angle of the trailing vortex sheets in
the propeller slipstream can be calculated by
v
= tan~1 ( Rev )' (30)
,, .._ _
VX = mean axial velocity.
v = mean tangential velocity.
Radial distributions of the hydrodynamic
pitch angle just behind the propeller for
three kinds of propellers are shown in Fig.10,
In a comparing with the geometrical pitch angles as
This follows :
305
OCR for page 297
Vx/V~ = ~Tip Vortex
Trailing
Vortex Sheet
VX / VA =_~
Trailing //~tl:t
Vortex ~ ~1.`
Tip Vortex ~
 C
_ ~
Propeller B
x/r'  0.25
x/ rO = 1 00
Trailing Tin VnrteY '
~ . 
Trailing .
Vortex ? ','t ",; ~~
Sheet arm=
. ·...
3
. ~.
. 
Fig.9a Velocity distributions downstream of propeller B ~ J=0.70 )
306
OCR for page 297
Propeller B
~x/rn=2.00 z \A
f ~
~ Vortex Sheet
Fig.9b Velocity distributions downstream of propeller B ~ J=0.70 )
Propeller A Propeller B Propeller C
x/r'= 0.25
7n
tar
x/rc = 0.25
blarks
II J = 0.72
J ~6 Pitch Angle
_ _o 1 = 0.40
. _
60 _
50 _ ~50 _
~ O ~bit
,,'43 \~9w 30 _
'6 ~
3u _
20 _
10 _
\~
x/rD= 0.25
Marks
Geometrical Pitch Ang1e
I{} J = 0.90
~J = 0 70 Hydrodynamic
Pitch Angle
O J = 0.50
_ ~
"l to 0.2 0.4 0.6 0.8 1.0 1~2 al 1.0 0.2 0 4 0.6 0.8 1.0 1.2
r/r.
on
10
n
 9a
r/r.
Fig.10 Comparison of radial distributions of hydrodynamic pitch
( Effect of propeller loading )
~ t l (P(r)
where
P(r) = geometrical pitch distribution.
In spite of difference of the operating condi
tion of the propeller, the hydrodynamic pitch
angles Qw are nearly constant and slightly
larger than the geometrical pitch angles FIG.
Same tendency is kept for the three propellers
which are different in pitch. This is the
reason why the geometrical pitch was intro
duced for the pitch of the ultimate trailing
vortex wake in QCM [23.
The hydrodynamic pitch Pw (r) of the
propeller slipstream can be obtained by
PW(r) = 2nr tan6W. (32)
7n~
60 _
50
_`
bit
40
c,
30
20 _
10 _
O
0.0
Marks
Geometrical Pitch Angle
I1 = 1.08
~I = 0 84 HydrodYnarnic
Lo J ~ 0.60
b _o
0.2 0.4 0.6 0.8 1.0 1.2
r/r.
angle of propeller slipstream
An example of radial distributions of the
(31) hydrodynamic pitch of the slipstream is shown
in Figs.ll  13 for the propeller B. It is
known that the hydrodynamic pitch increases as
the distance from the propeller increases.
Amount of increase in the hydrodynamic pitch
becomes larger as the advance ratio decreases.
This shows the tendency opposed to the conven
tional wake model, in which the hydrodynamic
pitch of the slipstream had been assumed to be
proportional to the advance coefficient [21].
Increase in the hydrodynamic pitch would be
due to the contraction of the slipstream along
the downstream direction. Pitch of the tip
vortices is obtained from the axial variations
of the angular positions of the center of the
tip vortices and also plotted in Figs.ll  13.
Pitch of the tip vortices is considerably
smaller than that of the inboard helical
trailing vortex sheets.
307
OCR for page 297
Propeller B
t
a
0.6 _
1.0
0.8 _
1~ Propeller Tip
~^
0.4 _
0.2 _
nil
Jan 0.50
Hydrodynamic Pitch
Dox/r' 0.25
 x/r' 0.50
to x/r'. 1.00
 v  x/r.~ 2.00
\~\\`
~ ' 46%
~ __W
Fig.ll Variations of hydrodynamic pitch down Fig.14
stream of propeller B ( J=O.SO )
Propeller B
or
new
.
_ 0.6 _
0.4 _
0.2 _
00 '
0.0 0.5 1.0
Pw/D
I, Tip Vortex Hydrodynamic Pitch
,L§L PropelbrTiP ~ x/r' 0.25
L I`V _
, I x/r' 0.50
ox/r'= 1.00
v  x/r'. 2.00
Fig.12 Variations of hydrodynamic pitch down
stream of propeller B ( J=0.70 )
Propeller B
= o.so
; ~Vortex Hydrodynamic Pitch
11 ~x/r=0.25
_~] 0
1.0
n8
0.4
n2
n.o
)

.3 _
.2 _
Ol 1.0 ~5
Propeller lip
Fig.13 Variations of hydrodynamic pitch down
stream of propeller B ( J=O.90 ~
___~ x/r'= 0.50
__o x/r'= 1.00
~v x/ro" 2.00
2.0
The present LDVmeasurement shows that the
helical trailing vortex sheets behind propel
ler are not always concentrated into a set of
tip vortices and a single hub vortex as shown
in Fig.14. The rollup model of the trailing
vortex sheets would be too simplified. In or
der to construct a more realistic model of the
trailing vortex sheets, it is necessary to
take into account the increase in pitch of the
Blade
Sex
_.~._
Trailing Vortex Sheet
~_,/ Up Vortex
A model of vortex pattern of propel
ler ~ from 16th ITTC reportt28] ~
fit
Transition Wake
Ultimate Wake
1 '
ran
ran
, ~
 t
Fig.15 Model of propeller slipstream
inboard helical trailing vortex sheets and the
decrease in pitch of the tip vortices near
outer edge of the slipstream.
4.6 Numerical Modeling of Trailing Vortex Sheet
A linear wake model of the trailing vor
tex sheets which was based on the geometrical
pitch of blades and ignored the contraction of
slipstream was used in the previous papert203.
Based on the measured velocity distributions
of the propeller slipstream, a new wake model
of the helical trailing vortex sheets is
considered. In the present study, the trailing
vortex wake is divided into two parts, transi
tion wake region and ultimate wake region as
shown in Fig.15. Contraction and variations of
pitch of the trailing vortex sheets are con
sidered in the transition wake region. On the
other hand, radial positions and pitch of the
trailing vortex sheets are kept constant in
the ultimate wake region.
Contraction of the propeller slipstream is
considered at first. If the radial interval
from the hub vortex radius r = rw h to the tip
vortex radius r = rw T in the ultimate wake is
divided into NR small panel strips by the
cosine spacing, the radii of each panel strip
can be expressed as follows:
308
OCR for page 297
Propeller B
1.0 _
0.9 _
0.8 _
0.] _
0.6 _
0.5 .0
Blade Tip Measured Fitted
Cat
By_
O J=0.50
, ~____J  0.70
= D.90
o
I I ~,
0.5 1.0 1.5 2.0
x/r.
Fig.16 Comparison of contraction of slipstream rw
Table 3 Principal particulars of DTRC propel
ler models
Propeller Number
Diameter of Propeller
Pitch Ratio at 0.7R (mm)
Expanded Area Ratio
Bass Ratio
Number of Blades

Blade Thickness Fraction
Skew Angle (deg.)
Rake Angle (deg.)
Blade Section
Design Advanced Coefficient
P.4679
610
1.572
0.755
0.300
3
0.099
51
a
NACA

1.077
.
P.4718
610
0.888
0.440
0.300
3
0.069
25
O
NACA
0.751
T/r 0 = 0.887  O . 12 5s, ( 36)
Linear Wake Model where
s = slip ratio = 1  J/p,
p = pitch ratio at 0.7 radius.
~ 1 ~ ~ Z
Tip Vortex Lx
Fig.17 Comparison between linear and deformed
wake models
rW~.1 = 2 (rWTrWh) 2 (rWT~rWh)cOs(~p' (33)
Radial positions of the trailing vortex sheet
at the trailing edge of the propeller blade
must coincide with those of the panel strips
on the blade surface given by Eq.~10). Then
variations of the radial positions of the
trailing vortex panel strips in the transition
wake region can be approximated by a polyno
mial expression as
rtp = rp(rprwp) f r ( A) ~ (34)
where
fry) = ~ +1.01351.92052+1.228430.32144,
XXTE
=
' XFXTE '
xTE = xcoordinate at the trailing edge of
blade.
XF = xcoordinate of the point where the
ultimate wake region starts.
Based on the measured results, the radius
of the tip vortices in the ultimate wake
region can be expressed as a function of slip
ratio s as follows:
The radius of the hub vortex and the axial
coordinate of the starting point of the ul
timate wake are kept constant as
rwh/rO = 0.1, xF/rO = 2.0. (37)
Variations of the radial positions of the
center of the tip vortices calculated by the
above equations are shown in Fig.16, comparing
with those obtained from the results of the
flow measurements by LDV. It can be said that
the contraction of the slipstream is approxi
mated well by using the present formulae.
Variations of the pitch distributions in
the transition and ultimate wake regions can
be also expressed in the similar manner as the
contraction of the propeller slipstream. The
deformed wake model based on the measurements
of the flow field around the propeller are
compared with the conventional linear wake
model in Fig.17. Large deformation of the tip
vortices can be observed in the new wake
model, which is similar to the wake model
based on the measured wake pitch shown by
Jessup [27~.
5.NUMERICAL EXAMPLES
5.1 Pressure Distribution on Blade
In order to evaluate the accuracy and the
applicability of the present panel method,
David Taylor Research Center ~ DTRC ) propel
ler models P.4718 and P.4679 were selected,
since very precise measurements of blade sur
face pressure were conducted by Jessupt29,30~.
Both propellers are threebladed and different
in skew. Principal particulars of the propel
lers are shown in Table 3.
5.1.1 Effect of Iterative Cotta Condition
In the present calculation, the propeller
blade surface was divided into 240 hyper
boloidal quadrilateral panels per blade
(NR=12, NC=10) and the hub surface was ap
proximated by 155 panels per 1/3 sector for
both propellers. Panel arrangements of the
propellers are illustrated in Fig.18.
309
OCR for page 297
DTRC P.4679 ( upstream view )
NR=12
NClO
DTRC P.4718 ( Upstream View )
NR=12
NC" 10
Fig.18 Panel arrangements for DTRC propeller models
a First Solution  0 NC = 6
a Second & Third Iterative Solutions a NC10
0 NC = 14
0.3 0.3
DTRC P4718 r/ro = 0.7 DTRC P4718 r/ro = 0.7
&  3 0 2 Suction Side NR12 &  3 0 2  Suction Side NR = 12
Il' on kit ~I,! of
. I I ' ' ~'T
O.
0.3r
0.2
IO 3
I ~^0.1
1
110.0
1
0.1 _
02 , . . .
0.0 0.2 0.4 0.6 0.8 1.0
DTRC P4679 r/r' = 0.7
NR=12
~ Suction Side NC=10
1 ~
\b
\\
D /:
Pressure Side '`
Fraction of Chord
Fig. 19 Comparison of chordwise pressure
distributions at 0.7 radius
~ ef f ect of Kutta condition ~
The chordwise pressure distributions on
the blade at 0.7 radius were calculated at
several steps of the iterative Kutta condi
tion. Fig. 19 shows comparison between the
f irst, second and third solutions . The f irst
solution which corresponds to the application
of the Morino Kutta condition gives large dis
crepancy of the pressure on the upper and
lower sides at the trailing edge, while the
second and third solutions give almost equal
pressure at the trailing edge by the applica
tion of the iterative Kutta condition. It can
be pointed out that the convergence of the
present iterative Kutta condition is very fast.
0.1
0.3
0.2
I red
O 3
G _ ~0.1
1
11
0.
Cal
1
DTRC P4679 r/ro = 0.7
2  Suction Side NR = 12
O.t ~
0.1: Pressure Side
_ ..,
0.0 0.2 0.4 0.6 0.8 1.0
Fraction of Chord
Fig. 20 Comparison of chordwise pressure
distributions at 0.7 radius
( ef feet of panel size )
5.1.2 Effect of Panel Size
Comparative calculations f or three kinds
of discretized models were conducted to inves
tigate the ef feet of panel size . The propeller
blade was replaced with 144 (Nc =6), 240
(Nc=lO), and 336 (N c=14) panels per blade
respectively and the hub was approximated by
125, 155, and 185 panels per 1/3 sector
respectively .
The chordwise pressure distribution at 0.7
radius were also calculated and compared as
shown in Fig. 20. Results Of NO =6 are fairly
different from those of Nc=lO and 14. There
310
OCR for page 297
Panel Method
with Hub
 Panel Method O ~ Experiment
Without Hub ( Jessup )
0.4
Suction Side r/ro = 0.5
0.3
I
OCR for page 297
x/rO = 0.25
Measured by LDV Calculation with Deformed Wake Calculation with Unear Wake
z z z
1 ~1
~ ~ {''~~t
~ .
~ 1 ~ _1
Fig.23 Comparison of axial components of velocities upstream of propeller B
J=0.70, x/r =0.25
x/rO = 0.50
Measured by LDV Calculation with Deformed Wake Calculation with Linear Wake
z z z
~2
~2 am/=
W1~:
Fig.24 Comparison of axial components of velocities downstream of propeller B
~ J=0.70, x/r =0.50 )
Measured by LDV
In' "
x/rO = 0.50
Calculation with Deformed Wake Calculation with Linear Wake
z z
I'.
it,
_y
Fig.25 Comparison of cross components of velocities downstream of propeller B
( J=0.70, x/r =0.50 )
312
v,