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OCR for page 645
A Three Dimensional Theory for the Design Problem
of Propeller Ducts In a Shear Flow
I. Falcao de Campos
(Maritime Research Institute Netherlands, The Netheriands)
ABSTRACT
A linearized theory of the threedimen
sional steady interaction between a ducted
propeller system and a radially and circum
ferentially sheared axial onset flow is pre
sented. Following duct lifting surface theory
the duct is modelled by a distribution of
pressure dipoles and sources on a reference
cylinder to represent the effects of loading
and thickness. An actuator disk model is used
to represent the effects of propeller loading.
An integral equation for the pressure distur
bance is derived which may be applied to treat
both the effects of loading and thickness. The
potential and shear interaction components of
the disturbance pressure are treated separate
ly and a computational scheme is applied to
solve the integral equation for the interac
tion pressure. The results of sample calcu
lations for the effects of duct loading in
axisymmetric and nonaxisymmetric wakes are
presented and discussed.
NOMENCLATURE
Amn ~ Ban
(cay ~ ~ )
( ex ~ er ~ e(3)
Em
F
f
G
_mn =
Gmnij,Gmnij
Gll, G12, Gal, G22
H(x)
hl,h2,h3,h4
I~,Km
k
L(~)
P(lnl )
P,P
~
Ap, Ap
Qm+l/2
q
R
Functions of shear param
eters
Shear parameters
Duct semichord
Unit vectors, Cartesian
coordinates
Unit vectors, cylindrical
coordinates
Fourier integral in the
downwash calculation
External force field
Duct camber
Kernel function
Matrix elements
Functions in degenerate
kernel function
Heaviside unit step function
Functions of shear param
eters for duct loading and
thickness
r
S
T
),U
U
U~,U2,U3,U4
u
(u,v,w)
Modified Bessel functions of
order m
And, index of radial node
i1,i2,ip,kp _ Functions of shear param
eters for propeller loading
index of radial node
Parameter in xwise Fourier
transform
Function in downwash calcu
lation
Downwash kernel function
Pressure, respectively its
xwise Fourier transform
Strength of pressure dipole,
respectively its xwise
Fourier transform
Legendre function of second
kind and half order
Strength of source distribu
tion
Radius, distance between two
joints
Transformed radius
Righthand side of
type equation
Function of source distribu
tion
Duct thickness
Undisturbed axial velocity,
respectively its modulus
Reference velocity
Parameters in analytical
defined wake
Disturbance velocity
Axial, radial and circumfer
ential components of distur
bance velocity
Fluid velocity
V(O)
TV
(x,y, z)
(x,r,8),(L,~,4)
J.A.C. Falcao de Campos, MARIN, P.O. Box 28, 6700 M Wageningen, The Netherlands
645
Influence function for ra
dial downwash velocity on
the duct
Average radial velocity in
duced on the duct
Radial velocity jump on the
duct
Cartesian coordinates
Cylindrical coordinates
OCR for page 645
&(x)
n
A
p
T
1, V
Subscripts
d
m,n
t
 Conical angle
 Expansion rate
 Dirac delta function
 Small parameter
 Normalised axial distance on
the duct
 Parameter of xwise Fourier
transform
 Fluid density
 Integral operator
 Integration volumes
 Righthand side of integral
equation
 Argument of Legendre func
tions
 Refers to duct
 Index of circumferential
harmonics
 Refers to propeller
 transverse component
Superscripts
(O)
(1)
1. INTRODUCTION
 Refers to potential, induc
tion part
 Refers to shear interaction
part
Ducted propellers are a wellestablished
means of ship propulsion. It is wellknown,
[1] that the use of a ducted propeller with an
accelerating type of duct improves the effi
ciency of the propulsor in case of heavy load
ing. Also the use of a decelerating type of
duct may be beneficial to reduce the risk of
cavitation of the propeller. A large number of
conventional duct designs, which have been
most successfully applied in practice [2][1],
are axisymmetric but the application of non
axisymmetric ducts has also been subject to
investigation both experimentally [1] and
theoretically [3][4]. These ducts have been
applied to reduce the nonuniformity of the
inflow to the propeller in the the ship's wake
leading to improved performance from the point
of view of efficiency, cavitation and vibra
tions.
For the design of ducted propellers a
number of analytical tools have become avail
able along the years. Early ducted propeller
theories [5][6] were based on linearised
annular airfoil theory for the singularity
representations of the effects of duct loading
and thickness, in combination either with an
infinite blade number model (actuator disk) or
with a finite bladed lifting line model of the
propeller. An extensive review of these the
ories was made by Weissinger and Maass in ref.
[7]. It is interesting to notice that the
truly inverse methods published to date for
designing axisymmetric propeller ducts are
based on these theories. The methods determine
the duct geometry (in the presence of a time
averaged propeller induced velocity field) for
specified duct pressure distribution [8] or
given load and thickness distributions [9].
These methods suffer from the drawback that it
is not possible to guarantee a priori that the
given pressure or load distribution will lead
to an acceptable duct geometric shape. Never
theless, those inverse methods were of great
assistance in designing famous ducted pro
peller systematic series, such as the ones
] and [10]. Often
to be modified to

published in references [1
the final duct shapes needed
meet practical requirements.
Following the developments in the numeri
cal methods for the calculation of potential
flow on lifting bodies, methods for the hydro
dynamic analysis of ducted propellers evolved
to a greater degree of sophistication. More
accurate panel representations of the duct
geometry have been employed for axisymmetric
flow [11] and, more recently, for threedimen
sional flow [12][13]. These last methods for
steady threedimensional analysis have concen
trated on the complex interaction between pro
peller and duct in uniform inflow by incorpo
rating lifting surface or panel representa
tions of the propeller blades. Also complete
unsteady potential flow analysis [14] of the
ducted propeller system has been attempted.
The methods mentioned previously are
restricted to potential flows. In reality the
ducted propeller operates in the highly non
uniform flow endowed with vorticity in the
ship's wake and the interaction with this flow
is an important field of research in propulsor
design. In dealing with this problem the po
tential flow methods have retained completely
their usefulness through the introduction of
the concept of the effective onset velocity,
which is defined as the total velocity minus
the potential velocity induced by the propul
sor. The effective onset velocity has to be
computed by some (viscous or inviscid) rota
tional model for the propulsorhull interac
tion. Examples of the inviscid approach to the
computation of the effective velocity for con
ventional propellers in axisymmetric flow can
be found in [15][17].
There is a considerable amount of ref
erences in the turbomachinery literature deal
ing with the problem of solving approximate
forms of the Euler equations for determining
the inviscid disturbance flow to parallel
shear flows, as can be found in the survey
given by Hawthorne [18]. In particular, the
large shear  small disturbance approximation,
applied along the lines set in the classical
works of Karman and Tsien [19] for a lifting
line and of Lighthill [20] for a simple
source, have been used to fundamentally study
the effects of shear in the flow around aero
dynamic shapes [21][22], including the an
nular airfoil [23][24]. For an infinitely
bladed propeller, modelled by an actuator
disk, the same approach has been followed to
investigate the effects of shear in the incom
ing flow in the axisymmetric case [25][26],
plane flow [27] and threedimensional flow
646
OCR for page 645
[28]. The latter reference includes the ef
fects of shear of a radially and circumfer
entially varying axial inflow to the actuator
disk and incorporates some nonlinear terms in
the equations of motion adequate for extending
the method for heavier loadings.
For ducted propellers the effects of
shear have been investigated in [11] using a
numerical vortex panel method in axisymmetric
flow. Recently, Lee [29][30] presented a lin
earised analysis of the ducted propeller sys
tem in axisymmetric shear flow suitable for
solving the duct design problem in the pres
ence of the propeller. He showed that the
shear significantly affects not only the duct
camber and ideal angle of attack but also the
duct induced velocity at the propeller.
The objective of the present paper is to
present a threedimensional theoretical anal
ysis of the steady interaction between a
ducted propeller and a radially and circum
ferentially sheared axial inflow. Consistent
with the steady flow assumption, an infinitely
bladed propeller (actuator disk) is consi
dered. Although not strictly necessary to the
analysis, an axisymmetric loading over the
propeller disk will be assumed in this paper
for simplicity. The duct loading and thickness
may vary in the circumferential direction. The
theory may be used in the analysis of a ducted
propeller with a given duct shape but it will
be most readily applied (noniteratively) in
the inverse mode i.e. to determine the duct
section camber and angle of attack for given
thickness and loading distributions.
As in the method followed by Lee [29] in
the axisymmetric case, the analysis developed
herein approaches the solution of the problem
from the theory of the Poisson equation to
derive a system of coupled integral equations
for the disturbance pressure harmonics. How
ever, some essential differences with the
formulation used by Lee are noteworthy: First,
a single type of integral equation in the dis
turbance pressure is used throughout, encom
passing both the effects of loading and thick
ness. Second, by separating shear interaction
effects from induction effects, the method
recovers the potential formulation of the duct
lifting surface theory, see for instance [6],
[31][33], and actuator disk theory [34],
while, at the same time, the presence of any
singularities of the pressure field are re
moved from the problem for the interaction
with shear.
The paper is organized as follows: In
section 2 the theoretical analysis is pre
sented. Section 3 deals with the numerical
procedures employed so far to solve the
integral equation and compute the velocity
field. In section 4 the results of sample
calculations illustrating the effects of shear
in the velocity field due to a nonsymmetric
duct in a wake field are presented and dis
cussed. The paper closes with some remarks
regarding the basic limitations of the method
and its further development.
2. THEORETICAL ANALYSIS
2.1. Equations of Motion for General Steady
Disturbances to a Shear Flow
We start by deriving the linearised Euler
equations for the steady flow of an inviscid
and incompressible fluid in the presence of an
external force field. Anticipating the use of
sources and sinks to represent thickness ef
fects, we will assume the rate of expansion to
be zero except at the points where such singu
larities will be present.
The continuity equation for an incompres
sible fluid reads, see for instance ref. [35]:
V.~= A,
(1)
where ~ is the fluid velocity and ~ is the lo
cal rate of expansion.
The Ruler equations for the steady flow
of an incompressible ideal fluid are
(it V)4 + 9(~) = pp ,
(2)
where p is the pressure, p the fluid density
and F the external force field per unit vol
ume.
We introduce a Cartesian coordinate sys
tem (x,y,z), with unit vectors (e ,e ,e ) and
a cylindrical coordinate system with Yunii vec
tors (e ,er,e~), Fig. 1. We consider a radi
ally And circumferentially nonuniform axial
flow D(r,8), independent of the axial coordi
nate, to be disturbed baby the presence of the
ducted propeller. Let u denote the disturbance
velocity so that the fluid velocity may be
written as
](x,r,0) = U(r,8)ex + u(x,r,6) . (3)
~\~
Y
/
\
Fig. 1. Coordinate system and axial inflow.
647
x
OCR for page 645
The equations of motion (2) may be lin
earised by assuming the disturbance velocity
to be small, say of 0(~), in comparison with
the undisturbed velocity 0=0(1). Substituting
eq. (3)2 into (2) and neglecting squared terms
of 0(s ) in the disturbance velocities we ob
tain
u aU (A v)(u~ ) v(~) F (4)
since (Uex.V)(Uex) = U a(Uex)/ax = 0
We further decompose the disturbance ve
locity into its axial and transverse compo
nents
u ueX + Ut ~
ith ( ) ~ ~ With (5)
we fin that
(u. 9) (Uex) = U ax (Uex) + (Ut. v) (Uex) =
= eX(ut. MU)
and (4) may be written in the form
u aU + e~x(u~t. W) + V(~) = Fp (6)
Taking the divergence of (6) and using (1) we
obtain
92(~) + 2 ax (Ut.vu) = 9.(Fp)  U ax · (7)
We may Eliminate the transverse velocity com
ponent ut between (6) and (7) to obtain a sin
gle equation for the pressure. To do so we
first take the transverse vector component of
(6)
U ax + ~t(~) = p '
where Vt  ex x (e x V) is the gradient op
erator In the transverse plane, and substitute
(8) into (7) to obtain
V2(~) + _ :9tU.Vt(P)] =
( pt. VtU)  U ax
Equation (9) is a linear partial differ
ential equation for the pressure disturbance
which needs to be solved for given U. F and
distributions.
2.2. Ducted Propeller Model
Let us consider now the specific form of the
righthand side of eq. (9) for the external
force fields F representing the duct and pro
peller loadings and for the rate of expansion
field ~ representing the duct thickness in the
linearized theory. We note that, since ]=0(1),
from eq. (9) we will require F=O(~) and A=0(~)
Denoting by S(x,r,O) the righthand side of
eq. (9) (multiplied by p), we obtain in cylin
drical coordinates:
aFX 1 a~rFr) 1 aF
S(x,r,6) = aX + r ar + r ae
5)  U (Fr ar + rat ax)  Pu ax . (10)
Consistent with the linearization applied
to the Euler equations, we will place the
force field singularities and the rate of ex
pansion singularities (sources) producing the
disturbance velocities, on a reference surface
aligned with the undisturbed flow d. In the
present ducted propeller application we choose
the reference surface to be a cylinder of con
stant radius and chord.
We represent the duct loading by a radi
ally directed force field F=(O,F ,0) distri
buted on a cylinder of radius R and chord 2c,
extending from x=c to x=+c, Fig. 2, in the
form:
Fr = Apd(x,8)[H(x+c)H(xc)] &(rRd) , (11)
where H(x) is the Heaviside unit step func
tion, H(x)=0 if xO, and 8(x)
is the Dirac delta function. Substituting eq.
(11) into eq. (10) we obtain
S(x,r,8) =
_ Apd(x,0)[H(x+c)H(XC)]
Fir  U ar) &(rRd) + &'(rRd)]
r
l
C
648
. (12)
+c
Fig. 2. Duct geometry conventions
x
OCR for page 645
It is seen from equations (9) and (12) that
the duct loading is represented by a distribu
tion of pressure dipoles of strength Apd(X,0)
on the reference cylinder.
The duct thickness is represented by a
source distribution q(x,6) on the reference
cylinder. The rate of expansion becomes
A(x,r,0) =
= q(x,0)[H(x+c)H(xc)] &(rRd) (13)
Substitution of eq. (13) into (10) yields
S(x,r,8) =  pU aX (q(x,8)[H(x+c)
 H(xc)]) 6(rRd) . (14)
The propeller loading is assumed axisym
metric over the propeller disk. Neglecting the
radial and circumferential components, the
force field is assumed to be of the form
{=(Fx,O,O), with
Fx(x,r) = App(r) 6(xxp) , r~ ~ (22)
Ik(m)(r.a;k) = ~ m ; ; m;; ;
where I and K are the modified Bessel func
. m m
lions of order m.
We introduce the notation
a(r,8) = U ar ~ b(r,8) = U r at (23)
1. The following convention was adopted in de
fining Fourier transforms:
so
f(~) = ~ f(x)e ibex dx
_a)
with inverse f(x) = 2~ ~ f(~)ei\X dA
_co
for a continuous F.T.; f(~) = Fin
with inverse f = 1 ~ fee ins dO
n z~ O
for a discrete Fourier series.
649
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and we expand a(r,8) and b(r,8) in Fourier with
series
a(r,8) = ~ anteing
n==
a,
b(r,8) = £ bn(r)ein~ ,
n=
with
an(r) = 2~ ~ a(r,0)e ins dO ,
bnfr) = 2~ ~ b(r,8)e ins dO .
Substituting eq. (19) and eqs. (21)(23) into
the first integral of (17) and carrying out
the integration in a we obtain
(a ap + by an) R do =
a' co co
= ~ £ e ~ dk e ~ da a Ik( )(r,a;k)
_= O
co co
x ~ dX £ pa (a) P'(a;A) +
L mn  n
_00 n__m
~ n==
00
lion bmn(a) pn(a)] ~ dL ei(\ k)t (26)
_co
where Pn = dpn/da. By noting that
co
8(~k) = 2 ~ ei(Ak)( dL (27)
_
the integration in ~ can be carried out to
give the integral, the value
ao co Go a'
£ imO ~ dk eikx 2 £ ~ do Ik(m)(r,a;k)
m== = n== O
a am_n(a) pn(a;k) + in bm_n(a) pn(a;k) . (28)
The integral involving p' can be further re
duced by partial integration. Evaluating the
derivatives of the terms involving the Bessel
functions we finally obtain
~ (a at + a at) R dt
~ co
=  £ Him ~ do eiAx
m== O
x £ ~ Gmn(r~a;A) pn(a;A) da ~(29)
n=0 0
650
(24) Gmn(r,a;A) =
and
A (a) = (11ml) a (a) +
2 Km(lAlr) [Amn( a) Im( I Al a) +
Bmn(a)l\I Im_1(1\Ia)] , r>a
2 Im(lAlr) [Amn(a) Km(lAla)
Bmn(~)l\I Km_1(1\Ia)] , r
OCR for page 645
In eq. (35) we have used IK(a )(r,a;~) to de
note
m) X Im( X r) Km( X a) , ra
(36)
2.3.3. Duct Thickness
In the case of duct thickness, inserting
(14) and (21) into the righthand side of eq.
(17), we obtain
S(L,a,¢) do =
v
_ P ~ eimb J do J da a
~ m= = O
J dk IK(m)(r,a;k)eik(X~~) ~(aR~
_=
a:{[H(~+c)H(~c)] J U(a,¢) q(L,¢)e ¢> ~ .
(37)
Introducing the function
Tm(L,a) = 2~ J U(a,¢) q(t,¢)e i ~ do , (38)
and carrying out the integration in a we have
for the integral (37)
2p ~ e Rd J dX IK( )(r,Rd;~)
Go
x ~ dL ei\(X a) dad {Tm(L,Rd)[H(~+c)H(~c)]} .
a (39)
In the previous expression the integration in
~ may be performed by introducing the Fourier
expansion
Tm(A) = J Tm(L,Rd)e ~ dL (40)
c
to yield
S(L,a,¢) do = 2 ~ Tim ~ dX eii
v m= =
x ipX Rd IK( )(r,Rd;~) Tm(~)
(41)
2.3.4. Propeller Loading
With the propeller loading given by eq.
(15), the righthand side of eq. (17) is re
duced in the same way as the previous cases.
The integral is
~ R
v
=  J dL JP da a J do £ eim(~~~)
= O O m=
I dk IK(m)(r,a;k)eik(X~~) Ap (a) (x )
_= (42)
Since the load distribution is assumed inde
pendent of +, the integration in ~ gives
In .
J e lmt df = an 8mO , (43)
where ~ is the Kronecker delta, and, hence,
the integral becomes
R
v
~ aim J dk JP da IK( )(r,a;k)
m== = O
alp (a) J eik(x~) (x ) d:
= 2 £ aim J do Minx
m== _
i~x R ,^`
x iX e P JP IK`U'(r,a;~) App(a) a da. (44)
2.3.5. Integral equation
Gathering the previous results expressed
by eqs. (29), (35), (41) and (44), by equating
the corresponding harmonics of the left and
righthand sides of eq. (17) we may write
(symbolically) the integral equation in the
form
~ ~ ~
Pm T Pn = Em '
with the operator T defined by
(45)
ao co
Pn ~ J Gmn(r~a;~) pn(a;~) da (46)
n=0 0
The function ~ takes different forms for
the effects of dumct loading, thickness and
propeller loading. In the case of duct loading
it is given by
651
OCR for page 645
~m(r;A) =
= Rd IK( )(r,Rd;~) APd (a) +
m
a,
2Rd IK )(r,Rd;~) ~ am_n(Rd) APd (I)
n== n
(47)
It is easily seen from the properties of
the modified Bessel functions that ~m(r;~)
is discontinuous at the duct reference cyl
inder r=R . The discontinuity amounts to the
(Fourier transformed) pressure jump on the
duct, i.e. the strength of the pressure dipole
Id ( A)
m
In the case of duct thickness, eq. (41),
the function takes the form
(r;~) = ipA Rd IK( )(r,Rd;~) Tm(~) · (48)
It is also seen that this function is
continuous at r=R but has discontinuous first
derivatives at that radius.
Finally, in the case of propeller load
ing, the function Am is
(r;A) =
ink Ret Ink
= id e ~ [~ IK`V'(r,~;~) App(~) ~ do, (49)
which is a continuous function of the radius
with continuous first derivatives.
The eq. (45) constitutes an infinite sys
tem of onedimensional integral equations,
with a discontinuous kernel, for the distur
bance pressure harmonics. At this stage it
should be remarked that, as shown in the
classical work in shear flow problems [19],
[20], the advantage of having introduced the
xwise Fourier Transform is to reduce the
threedimensional problem to a set of de
coupled twodimensional problems, each one for
a single value of the parameter ~ in eq. (45).
However, it should be noted that, in the case
of a circumferentially sheared inflow, the
circumferential pressure harmonics do not
decouple. In fact, they are coupled through
the harmonics of the shear parameters, as
shown by the form of the integral operator in
eq. (46).
We would like also to stress that the
integral operator T is a function of the shear
parameters a and b only, being independent of
duct loading, thickness or propeller loading,
which affect only the righthand side of eq.
(45).
2.4. Separation of Potential and Interaction
Effects
The numerical approach to the solution of
the integral equation (45) has to be done with
care. First, the kernel G is discontinuous
at ran, as given in (32). This may not consti
tute a major problem since a number of numer
ical techniques are available to handle this
type of kernel, as it may be found, for in
stance, in ref. [36]. Second, the righthand
side of eq. (45) is either discontinuous at
r=Rd, as in the case of duct loading, or has
the discontinuity in its first derivative, as
in the case of duct thickness. The integral
equation (45) is of the second kind and this
means that the solution will inherit the dis
continuous behaviour at r=Rd. This fact has
major consequences for the computation of the
velocity field because, as shown in the next
section, differentiation of the pressure in
the radial direction is required to derive the
radial velocity component from the radial mo
mentum equation.
In the pursuit of an accurate numerical
solution of the integral equation (45) and of
an accurate computation of the associated ve
locity field, it is quite natural to distin
guish two contributions to the pressure field,
respectively the potential pressure p(O),
which satisfies the Poisson equation (9) in
the absence of shear, ~tU=O, and the interac
tion pressure p(1) which satisfies the remain
ing part of the equation (9). For the trans
formed equation (45) in Fourier space this
means that
p = p(O) + p(1)
with
p(O) = ¢(0)
m m
and
(50)
(51)
Pm )  T pml) = ¢(1) + T ¢(0) (52)
In eqs. (51) and (52) we have decomposed
the righthand side into a shearindependent
term 9(0) and a sheardependent term 9(1). In
the case of duct loading we have m
Em ) = Rd IK( )(r'Rd;~) Ad (I) (53)
m
and
a)
.(1) = 2Rd IK(m)(r,Rd;A) ~ am_n( d) Pd
n== n
(54)
In the cases of duct thickness and pro
peller loading the sheardependent term ~ml)
is inexistent and Amy) is given by eqs. (48)
and (49), respectively.
652
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The potential flow problem eq. (51) for
the duct loading and thickness leads to the
formulation of the duct lifting surface the
ory. Solution of the loading (vortex distri
bution) and thickness (source distribution)
problems has been given in refs. [6], [31],
[32], where expressions can be found for the
evaluation of the radial velocity (downwash)
at the duct reference cylinder. Expressions
for the duct induced axial and tangential ve
locities are also given in ref. [6]. We will
discuss briefly the solution of this problem
in the section devoted to the computation of
the velocity field.
The solution of the shear interaction
problem eq. (52) requires the computation in
the righthand side of eq. (52) of the term
T M(°). A straightforward computation allows
this term to be expressed in terms of func
tions which can be easily evaluated numeri
cally. The results of these computations for
all cases are given in the Appendix A. It may
be readily verified using the expressions of
the Appendix A that the righthand side of
eq. (52) in the cases of duct loading and
thickness is continuous and has continuous
first derivatives at r=Rd, a property which is
shared by the solution pml) of the shear in
teraction problem.
2.5. Velocity Field
In accordance with the decomposition of
the pressure field into its potential and in
teraction parts, we write for the disturbance
velocity
~ (0) (1)
u = u + u
(55)
where u(O) is the velocity associated with the
potential pressure p(O), which we call the in
duction velocity, and u(1) is the shear inter
action velocity (0)
By definition u satisfies eq. (1) and
the momentum equation
U aaX + V(P(O)) = Fp, (56)
while u(1) is a solenoidal velocity field
V U(1) o
satisfying
(57)
U aaX + eXP(ut°) + utl)).~7U] + V(~)) = 0
(58)
If the pressure fields p(O) and p(1) are
known, respectively from eq. (51) and the so
lution of eq. (52), the velocity fields u(O)
and u(1) can be evaluated by integrating eqs.
(56) and (58). We note that, in using (58) for
the determination of the interaction velocity,
the potential part of the solution affects the
radial and tangential components of the inter
action velocity only through the interaction
pressure p(1). For the axial velocity com
ponent, however, there is an additional term
u(O). VU coupling the two problems and this
requires the calculation of the induction ve
locity utO) first.
Integrating eq. (56) from x== to x, we
obtain for the induction velocity components
u(O) = _ p(O) 1 ~ x dE, (59)
v( ) =  U ~ ar ( p ) d: + U ~ p dE, (60)
W(°) 1 ~X 1 a (P(°)) d: (61)
In writing eq. (59) to (61)
we have made
u(O) and the
use of the fact that the velocity
disturbance pressure p(O) vanish at infinity
upstream, x=~.
In the cases of duct loading and thick
ness, F =0 and the evaluation of the axial and
tangential velocity components u(O) and u(1)
do not pose particular problems. They can be
computed from the inverse Fourier transform of
~m°), attention being paid to the calculation
on both sides of the duct reference cylinder
r=R _0. However, in the evaluation of the ra
dia] velocity component from eq. (60) special
care has to be taken in carrying out the re
quired differentiation and integration at
r=Rd. Such evaluation constitutes the subject
of duct lifting surface theory dealing with
the evaluation of the Cauchy singularity in
the kernel function for the radial downwash.
The solution of this problem can be found in
refs. [6], [31][33] and will not be treated
in detail here. In Appendix B the relevant
expressions are collected for the case of duct
loading, together with an outline of their
derivation from the present formulation.
In the case of propeller loading F =0 and
with F given by eq. (15), we obtain [or the
axial coXmponent
u(O) =  Pu(O) + ~U H(xxp), r
OCR for page 645
u(1) = _ p(1) _ a ~x(v(O)+v(l)) do
 b ~ (w(O)+w(1)) do (63)
_co
', ( 1 )
=
~ u ~ a (p(l)) dE, (64)
_
w =  U ~ r ae ( p ) d: ' (65)
_a)
with the definition (23) of a and b. Again, in
deriving (63) to (65) we have made use of the
fact that the interaction pressure and veloc
ities vanish at infinity upstream. In eqs.
(63)(65), which hold for the cases of loading
and thickness we have been considering, v(1)
and w(1) have to be evaluated first from the
interaction pressure; u(1) is then computed
from eq. (63) with known values of v(O), v(1),
w(O) and w(1)
2.6. Duct Boundary Conditions
We describe the duct surface by speci
fying the deviations of the outer and inner
surfaces from the reference cylinder due to
the conical angle, camber and thickness dis
tributions of the duct sections, Fig. 2. The
outer and inner surfaces may be given by the
expression
r(x,0) = Rd ~ a(~)(xc) + f(x,0) + 2 t(x,8) ,
c
OCR for page 645
3. NUMERICAL SOLUTION PROCEDURE
For the purposes of numerical analysis a
dimensionless form of the equations is used.
The duct radius Rd is taken as reference
length, velocities are made dimensionless by a
reference velocity DO, taken here as the (fi
nite) asymptotic value of the velocity U at
large radii, and pressures are made dimension
less by pU 2.
To solve numerically the integral equa
tion (52) we first apply a transformation to
the radial coordinate r=r(r) which maps the
integration interval (0,) onto the interval
(0,1). Various possibilities exist to choose
such transformation, but we have applied a
simple exponential transformation defined by
r = 1  exp(ar) , (79)
with the constant a=ln(1/2) chosen as to map
the duct radius r=1 to r=0.5. The integral
equation is solved by a quadrature method us
ing the trapezoidal rule. Introducing the
equidistant nodes
r; = (i1)/(NR1) , i=l,,NR , (80)
the integral in eq. (46) evaluated at the node
i is approximated by (omitting the parameter
for simplicity)
~ mn(ri'~) Pn(~) do =
= 2 Gill ~ wiJ G12j Pn
J
(2)
2 G21i ~ wij G22j Pnj ' (81)
with
ill Km( I HI ri ) ~ G21i = Im( I HI ri )
G12j = [A n(ri ) Im( 1 Al r; ) +
Bmn(rj ) I \1 Im_l( I A rj ) ] (dr/dr)
G22j = [Amn(rj ) Km(  )~ rj )
Bmn(rj ) 1 x1 Km_l( l Ad rj ) ] (dr/dr)j
(82)
w(.J) and wi2) being the weights of trape
zoidal rule. For the first N+1 harmonics of
the disturbance pressure, the discretized form
of the lefthand side of equation (52) reads
with
N NR =
( Pn )mi = ~ N jI1 Gmnij Pn; , (83)
G .. = G .. for man or id
mn~J man

G mii = 1  Gmmii '
655
mnlJ
2 Glli G12; WiJ , ji . (84)
With the mapping (79) and the equidistant
node distribution an equal number of nodes are
placed inside and outside the duct. The map
ping ensures a larger concentration of nodes
near the axis r=O, where the Bessel functions
K are rapidly varying. To achieve an accept
a~le accuracy near r=0 and at large radii ex
ponential scaling of the Bessel functions is
applied. The integrals appearing on the right
hand side of equation (52) (see Appendix A)
are evaluated by trapezoidal integration using
the same set of nodes (80).
The system of equations is solved in
sequence for the set of values of the param
eter ~ coinciding with the nodes of Laguerre
integration in the interval (0,=) which is
used for inverting the pressure xwise Fourier
Transform. A LU factorization of the matrix is
carried out for each value of X, and then used
to obtain the solution vectors for the differ
ent righthand sides (47), (48) and (49).
With the interaction pressure obtained
from the inversion of the double Fourier
Transform (20) at a specified number of axial
planes, the interaction velocity field is com
puted with the eqs. (63)(65). The integrals
are computed by trapezoidal integration. Cen
tral differencing is applied to evaluate the
radial and circumferential derivatives.
4. RESULTS OF SAMPLE CALCULATIONS
To illustrate the effects of shear a num
ber of sample calculations were carried out
for an analytically defined wake field. Here
we will only discuss the results concerning
the effects of duct loading on the velocity
field.
The wake field chosen is a sinusoidal
perturbation superposed to the axisymmetric
wake field used by Lee in ref. [30]. It is
defined by the expression
U(r,8) = 1  U1 e 2 (1  U. r2 sing U O)
(85)
OCR for page 645
With U1=0.72, U2=1.70 and U3=0 we recover the
axisymmetric wake of Lee. For the wake field
(85) the functions Amn and Bmn approach zero
at the axis sufficiently fast to ensure a zero
value of the element G22j at the axis, j=1.
In all the calculations a chord/diameter
ratio c/R =0.5 has been chosen. In general the
duct loading and thickness distributions are
specified by a sum of chordwise modes which
are used in the solution of the duct lifting
surface problem. In the present application we
have chosen the NACA a=0.8 chordwise loading,
which is constant from the leading edge to 0.8
of the chord and decreases linearly to zero
from 0.8 of the chord to the trailing edge.
The Kutta condition of zero loading at the
trailing edge is automatically satisfied by
this load distribution. The duct loading is
expressed in terms of the duct section lift
coefficient defined by
ZZUO ~Zc' _1 d (86)
Since the theory is linear, the distur
bance pressure and velocities are proportional
to the loading coefficient CL, and the results
hold for an arbitrary loading. The results are
presented for a considerably high duct loading
and a strongly sheared inflow which may be
considered as representatives of a typical
ducted propeller application. In such cases
the disturbance velocity is no longer small in
comparison with the velocity of the incoming
flow. Of course the assumption of small
disturbance velocities does not hold in such
cases and nonlinear effects will certainly
affect (in an unknown manner) the accuracy of
the predictions. In any event, the results are
easily scaled to smaller loadings.
To examine the convergence of the numeri
cal solution of the integral equation a compu
tation for axisymmetric flow was carried out
first for a section lift coefficient CL=0.9.
Fig. 3 shows the convergence of the interac
tion pressure at the duct inlet plane x=0.5
with the number of radial nodes. The conver
gence of the Fourier inversion procedure was
checked by performing the calculations with 7
and 15 nodes of the Laguerre integration, the
results remaining unchanged. It can be seen
that with a number of 65 nodes the solution
coincides with the solution with 129 nodes
except near the axis r=0. It must be remarked
that none of these solutions accurately sat
isfy the condition of a zero radial derivative
at the axis, as shown in the enlarged view of
the region near the axis in Fig. 4, although
with increasing number of nodes the numerical
error decreases. This was to be expected
since, with the present quadrature solution
method, the condition is not enforced explic
itly and, therefore, the accuracy of the
solution will depend on the discretization
error of the integrals appearing both on the
°°5°~""1""1""1""1""~""1""1""1""1""~
0.045
0.040
OD35
Cat
to
=) 0.030 ,
Q

Q 0.025
a, 0.020
Lo 0.015
m
OD11
OD05 ~1 ~ ~ ~ !, , 1 I 1 ., I I 1 1
ODOR
ODD O. 20 0.40 0.60 0.80 1.00 170 1.40 DO 1.BO 2.00
,,~L~
. \+
x
+ NR=32
x KR=65
NO
x
x
x
x
AX i;
r/Rd
Fig. 3. Convergence of interaction pressure at
x=0.5 with the number of radial
nodes. Axisymmetric flow.
c/Rd=0.5, a=0.8, CL=O.9.
+  32
X N165
029
OD40
_
OCR for page 645
left and righthand sides of the integral
equation. It should also be noted that the
numerical solution exhibits the expected de
gree of smoothness at the duct radius r=1. We
insist in the importance of obtaining a smooth
pressure distribution since the velocity field
is derived from it by numerical differentia
tion.
The radial component of the interaction
velocity on different planes is shown in Fig.
5 for the solution with 65 nodes. The inter
action radial velocity is rather small. In
Fig. 6 the axial velocity distribution due to
the effect of the duct loading inside and out
side the duct is shown for this case. The ve
locity distribution of the oncoming flow is
also shown in this figure. The induction ve
locity is simply the potential flow velocity
in uniform flow divided by the local inflow
velocity. This of course produces the large
velocities close to the symmetry axis. The
interaction velocity is negative and corrects
to a certain extent this extreme behaviour.
Nevertheless, for this strongly sheared inflow
and, in contrast with the uniform flow case,
the present method predicts in shear flow an
increase to the axis of the disturbance veloc
ity due to the duct. Again near the axis the
interaction velocity and thus, the disturbance
velocity due to the duct is influenced by the
local error in the pressure distribution.
As a second application we considered a
nonaxisymmetric wake field defined by (85)
with U1=0.72, U2=1.70, U3=2.0 and U4=0.5. The
0.030
n non
O 0.010

. _
° 0.000

._
0.010
n non
\/ \
~` ,,
.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Radius r/Rd
Fig. 5. Interaction radial velocity due to
duct loading. Axisymmetric flow.
c/Rd=0.5, a=0.8, CL=O.9.
wake field is shown in Fig. 7. At all radii
the velocity is lowest at O=0. (say the upper
part of the propeller plane) and highest at
O=180 deg. (the lower part of the propeller
plane). To examine the capability of the duct
to change this incoming velocity, two types of
duct loading were considered: an Axisymmetric
loading with the section lift coefficient
CL=0.9, as in the previous Axisymmetric case,
and an asymmetric loading defined by CL =
CLo + ACL cos 0, with ACL = 0.15CLo; this cor
responds to a 30% variation of duct section
loading around the circumference, the highest
loading being placed where the incoming veloc
ities are lowest and vice versa. The same
chorddiameter ratio c/R =0.5 and the same
chordwise load distributidon are assumed. The
integral equation was solved using a number of
65 radial nodes, for 7 nodes of the Laguerre
integration. A number of 3 circumferential
harmonics was sufficient to carry out the so
lution in the present case. The induction,
interaction and the total disturbance axial
velocities at x=0 inside the duct are shown,
respectively in Figs. 8, 9 and 10 for the duct
with Axisymmetric loading. Similar results are
shown in Figs 11, 12 and 13 for the case of
asymmetric duct loading.
In the case of Axisymmetric loading oppose
site tendencies are found for the inner radii
up to about 0.5 and the outer radii. For the
outer radii the total disturbance velocities
are lower at O=180 deg., where the incoming
 xlRd = +0.3
x/Rd= 0.0 ,'
x/Rd = 0.3 /
\ ''
',,, . it\
0.1
n
~ "
/ "
. . .
/ ~
1.0
0.9
0.8
0.7
0.6

O 05

0.4
c' 0.3
o
> 0.2
ct
0.1
0.0

"'

UIUO/
.. /
U/UO ~
 Total=lnd.+lnt.
Induction
 Axisymmetric Wake
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Radius r/Rd
Fig. 6. Axial velocity due to duct loading at
x=O. Axisymmetric flow.
c/Rd=0.5, a=0~8, CL=O.9.
657
OCR for page 645
velocity is large while for the inner radii
large velocities are found at that angular po
sition. This is caused by the large positive
interaction velocity calculated for the small
er radii, see Fig. 9 for r/Rd=0.30. The domi
nant term in eq. (63) is the second term which
couples the negative radial velocities with a
large positive value of the shear parameter a
(note that for this wake field this parameter
is larger than on the base axisymmetric wake).
For the outer radii the interaction velocities
are very small, and the total disturbance ve
locities are dominated by the induction part.
In the case of asymmetric loading the
induction part shows a more pronounced vari
ation, Fig. 11, as expected from the duct load
variation. The level of interaction velocities
decreases considerably at the inner radii, as
shown in Fig. 12, especially at the angular
position O=180 deg. where the local duct load
ing is lowest. The interaction velocities at
the outer radii remain small. Still an in
crease of the total disturbance velocity to
the axis is found in this case, Fig. 13.
Clearly, an important result of practical in
terest is the better capability of the second
duct with asymmetric loading to make the in
coming flow more uniform.
out 1
0.7 l
0.6
o 1
= 0.5
~ 1
1 ~
1.0
0.9
Go 0.8

=

 0.7
o 0.6
a)
> 0.5
Cal
._
X
0.4
0.3
0.2
n 1
1 ~r/Rd = 0.83 ~r/Rd = 1.00
)( r/Rd =0.61 ~r/Rd =0.71
r/Rd=0.42 )( r/Rd= 0.51
~r/Rd = 0.00 (9 r/Rd = 0.30
o.o
30 60 90 120 150 180 210 240 270 300 330360
Angular Position (deg.)
Fig. 7. Axial velocity of nonaxisymmetric
wake field. Eq. (85) with U1=0.72,
U2=1.7, U3=2, U4=0.5.
0.45: 1
~ r/Rd = 0.61
() r/Rd = 0.51
[S r/Rd = 0.30
0.40
~ =~0.35
~ 1 _ ~ )
O0.4
0.2
0.1
0.0  _
. _
o
a)
ct 0.30
. _
x
in:
n25
~TT~ I I ___
0 30 60 90 120 150 180 210 240 270 300 330 360 0 30
Angular Position (deg.)
r/Rd = 0.96
r/Rd = 0.91
r/Rd = 0.83
r/Rd = 0.71
\\
,;
. . . . .
60 90 120 150 180 210 240 270 300 330 360
Angular Position (deg.)
Fig. 8. Induction part of the axial velocity due to duct loading at x=O.
Nonaxisymmetric wake. Axisymmetric duct loading. c/Rd=0.5, a=0.8, CL=0.9.
658
OCR for page 645
5. SUMMARY AND CONCLUSION
A linearized analysis of the threedimen
sional steady interaction between a ducted
propeller system, modelled by a suitably cho
sen external force field and source distribu
tion, and a nonaxisymmetric sheared axial
onset flow has been given. The basic assump
tion of the theory is that the disturbance
velocities should remain small compared with
the velocity of the onset flow. Consistent
with this assumption, linearised boundary con
ditions are applied on the duct surface. As a
consequence of the linearization the effects
of duct loading and thickness and propeller
loading can be treated separately and super
posed.
For the threedimensional problem a for
mulation in terms of the pressure seems an
obvious choice and a linear integral equation
has been derived for the disturbance pressure,
which can be applied to describe both loading
and thickness effects. By separating the po
tential pressure from the interaction pres
sure, the integral equation governing the in
teraction part may be in principle solved with
great accuracy using suitable numerical proce
dures. The solution of the potential part and,
more specifically, the computation of the cor
responding induced velocity field can be ob
tained from the results of duct lifting sur
face theory.
As it stands, the analysis may be applied
to solve the design problem of a duct in the
ship's nominal axial wake field in the pres
ence of the propeller. This requires the spec
ification of the duct loading and thickness
distributions, both circumferentially and
0.34
on
o2l
~ 0.1 1
. _
0 0.0
>
Cal 0. 1
in:
1
0.2
0.3
n4
/ ~
/ \
/ ~ r/Rd = 0.61 \
An/ (9 r/Rd = 0.51 \
[S r/Rd = 0.30
, . . . . .
0 30 60 90 120 150 180 210 240 270 300 330 360
Angular Position (deg.)
chordwise, and the propeller radial load dis
tribution. The computation of the disturbance
velocities on the duct reference cylinder en
ables the determination of the duct camber and
conical angle distributions. Due to the non
linear character of the interaction thrust
forces between duct and propeller some itera
tion will be required to meet given duct and
propeller thrust values.
A computational scheme has been developed
to solve the integral equation and evaluate
the velocity field. The scheme has been ap
plied to investigate the influence of shear on
the velocity field due to the effects of duct
loading for axisymmetric and nonaxisymmetric
wake fields. Relatively low computational
costs are associated with the utilization of
the scheme when compared with a more direct
numerical approach to the solution of the
problem.
Although the present results have not yet
been validated by a proper comparison with
other calculation methods or experimental
data, the following conclusions may be drawn
from the results of the sample calculations
presented in this paper:
 The effects of shear on the disturbance ve
locity inside the duct are found to be
rather significant for the strongly sheared
wake fields used in the calculations. Both
for the axisymmetric wake and the nonaxi
symmetric wake fields, the axial disturbance
velocity distribution considerably differs
from the distribution in uniform flow.
 For the axisymmetric case a disturbance ve
locity distribution increasing to the sym
metry axis is found as a result of the ef
fect of shear. Also for the nonaxisymmetric
it< r/Rd = 0.96
r/Rd = 0.91
r/Rd = 0.83
r/Rd = 0.71
O o.oo
.~
o
>
._
0.05
0 30
. . . . .
60 90 120 150 180 210 240 270 300 330 360
Angular Position (deg.)
Fig. 9. Shear interaction part of the axial velocity due to duct loading at x=O.
Nonaxisymmetric wake. Axisymmetric duct loading. c/Rd=0.5, a=0.8, CL=0.9.
659
OCR for page 645
wake the effects of shear considerably af
fect the disturbance velocity distribution
both radially and circumferentially at the
inner radii up to r/Rd=0.5. For the outer
radii the effects of shear become much
smaller.
 A large effect on the disturbance velocities
on the inner radii arising from the effects
of shear, is found if the duct circumferen
tial load distribution is changed.
The foregoing results certainly indicate
the need for pursuing the development of a
model for the interaction between the propel
ler, duct and wake field, if the design of
wakeadapted ducts using design criteria for
flow rectification is aimed at. At present,
work on this linearised model will concentrate
on the application of the method to duct de
sign including the effect of duct thickness
and propeller loading. In addition, steps to
validate the numerical results of the model
will be undertaken.
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3. Turbal, V.K., "Theoretical Solution of
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°9
owl
0.7 l
o 0.6.

~ 0.5
o
> 0.4
Ct
x
6 0.3
0.2
0.1
~'
/ \
~ r/Rd = 0.61
C) r/Rd = 0.51
r/Rd = 0.30 .
o
4. George, M.F., "Linearised Theory Applied
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1987, pp. 93122.
0.50: 1
0.451
0.40
>, 0.35
.~
0 1
o

0.30
Ct
. _
~:
0.25
n20
r/Rd = 0.96
~ r/Rd = 0.91
C) r/Rd = 0.83
r/Rd = 0.71
. . . . . . . .
0 30 60 90 120 150 180 210 240 270 300 330 360
Angular Position (deg.)
. . . . . . . . . . . .
0 30 60 90 120 150 180 210 240 270 300 330 360
Angular Position (deg.)
Fig. 10. Total disturbance axial velocity due to duct loading at x=O.
Nonaxisymmetric wake. Axisymmetric duct loading. c/Rd=0.5, a=0.8, CL=0.9.
660
OCR for page 645
13. Kinnas, S.A. and Coney, W.B., "A Syste
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for the Prediction of Unsteady Hydrodynamic
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Finite Number of Blades," Proceedings of the
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15. Huang, T.T., Wang, H.T., Santelli, N. and
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651673.
17. Dyne, G., "A Note on the Design of Wake
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Quarterly of Applied Mathematics, Vol. 3,
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21. Weissinger, J., "Linearisierte Profil
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Mechanica 10, 1970, pp. 207228.
0.7 l
0.6 t
o
0.5
.~
O 0.4
X 03
0.2
0.1
0.0
22. Weissinger, J., "Linearisierte Profil
theorie bei ungleichformiger Anstrdmung. Teil
II: Schlanke Profile," Acta Mechanica 13,
1972, pp. 133154.
23. Weissinger, J. and Overlach, B., "Grund
lagen zu einer Theorie des Ringflugels in
axialsymmetrischer Scherstromung," ZAMM 55,
1975, pp. 413421.
24. Overlach, B., "Linearisierte Theorie der
axialsymmetrischen Stromung um Ringflugel bei
ungleichformiger Anstromung," Dissertation,
1974, Karlsruhe.
25. Goodman, T.R., "Momentum Theory of a Pro
peller in a Shear Flow," Journal of Ship Re
search, Vol. 23, No . 4, Dec. 1979, pp . 242
252.
26. Falcao de Campos, J.A.C. and Van Gent,
W., "Effective Wake of an Open Propeller in
Axisymmetric Shear Flow," Netherlands Ship
Model Basin Report No. 500303SR, May 1981.
27. Van der Vegt, J.J.W., "Actuator Disk in a
TwoDimensional NonUniform Flow," Interna
tional Shipbuilding Progress, Vol. 30, No.
348, Aug. 1983, pp. 158178.
28. Van Gent, W., "A Model of Propeller
Ship Wake Interaction," Proceedings of the
International Symposium on Propeller and
Cavitation, 1986, Wuxi, China.
29. Lee, H., "Ducted Ship Propellers in Ra
dially Sheared Flows," Ph.D. Thesis, Stevens
Institute of Technology, 1985.
30. Lee, H., "Effects of Radially Sheared
Inflow on the Design of Propeller Ducts,"
Third International Symposium on Practical
Design of Ships and Mobile Units, 1987,
Trondheim, Norway.
31. Weissinger, J., "Zur Aerodynamik des
Ringflugels I. Die Drukverteilung dunner, fast
drehsymmetrischer Ringflugel in Unterschall
stromung," D.V.L. Bericht Nr. 2, 1955.
0.45 .
r/Rd = 0.61
r/Rd = 0.51
 Cl r/Rd = 0.30
~.<
0.40
o
, 0.35
.~
o
>
ct 0.30
x
0.25
r/Rd = 0.96
r/Rd = 0.91
r/Rd = 0.83
[1 r/Rd=0.71
_ ~ 1 1 1 1
0 30 60 90 120 150 180 210 240 270 300 330 360
Angular Position (deg.)
, . . . . .
0 30 60 90 120 150 180 210 240 270 300 330 360
Angular Position (deg.)
Fig. 11. Induction part of the axial velocity due to duct loading at x=O.
Nonaxisymmetric wake. Asymmetric duct loading. c/Rd=0.5, a=0.8, CLo=0.9.
661
OCR for page 645
32. Weissinger, J., "Zur Aerodynamik des
Ringflugels III. Der Einfluss der Profil
dicke," D.V.L. Bericht Nr. 42, 1957.
33. Ordway, D.E., Sluyter, M.,M. and
Sonnerup, B.O.U., "ThreeDimensional Theory of
DuctedPropellers," TARTR602, Aug. 1960,
Therm Advanced Research, Ithaca 1, New York.
34. Hough, G,.R., and Ordway, D.E., "The
Generalized Actuator Disk," Developments in
Theoretical and Applied Mechanics, Vol. II,
Pergamon Press, Oxford [etc.], 1965, pp.
317336.
0.5
0.4
0.3
o
, 0.1
.~
o
0
Ct
FAX
0.2
0.0
0. 1
no
._
0.3
0.4
rn m
/ \
r/Rd = 0.61
() r/Rd = 0.51
r/Rd = 0.30
0 30 60 90 120 150 180 210 240 270 300 330 360
Angular Position (deg.)
35. Batchelor, G.K., An Introduction to Fluid
Dynamics, Cambridge University Press,

Cambridge, 1967.
36. Baker, C.T.H., The Numerical Treatment of
Integral Equations, Clarendon Press, Oxford,
1977, pp. 375.
37. Erdelyi, A., ea., Tables of Integral
Transforms, Vol. 1, McGrawHill, New York,
1954, pp. 106.
O o.oo
.~
o
ct
<: 0.05
r/Rd = 0.96
r/Rd = 0.91
C) r/Rd = 0.83
r/Rd = 0.71
. . . . . . . . . . .
O 30 60 90 120 150 180 210 240 270 300 330 360
Angular Position (deg.)
Fig. 12. Shear interaction part of the axial velocity due to duct loading at x=O.
Nonaxisymmetric wake. Asymmetric duct loading. c/Rd=0.5, a=0.8, CLo=O.9.
0.
o.8
0.7
o 0.6

~ 0.5
o
> 0.4
Ct
._
x
0.3
0.2
0.1
\ ~
/
~ r/Rd = 0.61
() r/Rd = 0.51
r/Rd = 0.30
o.so
n45
n~n
o

>~ 0.35
o
~D
0.30
0.25
0.20
) ~ '
5 ~
_]
. . . . . . . . . . .
O 30 60 90 120 150 180 210 240 270 300 330 360
Angular Position (deg.)
Fig. 13. Total disturbance axial velocity due to duct loading at x=O.
0 30 60 90 120 150 180 210 240 270 300 330 360
Angular Position (deg.)
Nonaxisymmetric wake. Asymmetric duct loading. c/Rd=O.S, a=0.8, CLo=O.9.
662
OCR for page 645
APPENDIX A
ShearDependent Pressure Disturbance
Term T ~n°)
Duct Loading
The term is
w
T ~n° ) = Rd ~ G( )(r;~) APd ( \)
n= mn n
with
Gd )(r;~) = 21XlKn(l~lRd) [Km(l~lr) h1(r) +
mn
Im(lAlr) h2(r)]
21Al In(lklRd)
+
{m(l~lr) h4(Rd) '
rRd ~ (89)
and
(93)
G12(r) Amn(r) Im(l~lr) +
Bmn(r) 1\l Im 1(l~lr) , (94)
G22(r) = Amn(r) Km(l~lr)
Bmn(r) ~ Km 1(l~lr)
Duct Thickness
T ~n°) = ipX Rd
where
Gd )(r;~) = 2 Kn(~Rd) [Km(~r) h1(r) +
Im(~r) h2(r)] +
2 In(~Rd) Im(~r) h4(Rd) '
rRd . (98)
Propeller Loading
(O) ~ p (L)
m
with
~ +
(99)
G( )(r;~) = 2 Km(~r) i1(r) +
m
2 Im(lAlr) i2(r) + 2 Im(l~lr)
x ip(Rp) ~ G22(a) Ko( 1 ~l a) da,
p
r
OCR for page 645
DISCUSSION
All H. Nayfeh
Virginia Polytechnic Institute and State University, USA
The convergence problem near the axis may be alleviated if one uses
an analytical rather than a numerical solution there. Such a procedure
was used by Nayfeh, Kaiser, and coworkers in the seventies to treat
acoustic waves propagating in circular ducts. The results were
published in the AIAA Journal, Journal of Sound and Vibration, and
the Journal of the Acoustical Society of America.
AUTHORS' REPLY
The author thanks Prof. Nayfeh for pointing out the possibility of
using an analytical solution in the vicinity of the axis. Indeed, see
for instance references [1] and [2]; the form of the solution near the
axis is determined by the potential part of the solution, even in the
absence of a uniform flow core and certainly in the case of bounded
shear parameters a and b at the axis. In this case, the analytical
solution for each circumferential pressure harmonic behaves like the
modified Bessel function Im multiplied by a constant. Its asymptotic
behavior for small arguments can then be used to determine the
constant by collocation of the integral equation (45) at a point off the
center line but sufficiently close to it.
[1] Eversman, W., Effect of Boundary Layer on the Transmission
and Attenuation of Sound in an Acoustically Treated Circular Duct,"
Journal of the Acoustical Society of America, Vol. 49, No. 5, 1971,
pp. 13721380.
[a] Nayfeh, A. H., Kaiser, J. E. and Telionis, D. P., Acoustics of
Aircraft EngineDuct Systems," AIAA Journal, Boll 13, No. 2, 1975,
pp. 130153.
DISCUSSION
William B. Morgan
David Taylor Research Center, USA
This is a very interesting paper on the design problem of ducted
propellers. I have a question concerning the optimum dueled
propeller in a shear flow. Can you say anything about how to
calculate the optimum ducted propeller considering both the optimum
propeller and the optimum duct shape in a shear flow?
AUTHORS' REPLY
Dr. Morgan raises the question of how to optimize both the propeller
and the duct in a shear flow. Although the interaction with shear
covers only a particular aspect of the propulsorhull interaction, the
question may be addressed without considering in detail the shear
producing mechanism which is the presence of the ship's hull with its
boundary layer and wake. From an untheoretical point of view, it
would be interesting to know for a given sheared inflow velocity field
what are the load distributions on the duct reference cylinder and on
the propeller disk which minimize the kinetic energy of the fluid left
far behind the ducted propeller system and which satisfy a given
thrust constrains". Such optimization would lead to the d,ucted
propeller loading for maximum recover of the kinetic energy of the
fluid present in the sheared incoming flow. The formulation of such
optimization problem has not been attempted. From a more practical
point of view, the present model may be conceivably used to
determine the duct shape for maximum attainable rectification of the
inflow to the propeller, for given propeller radial load distribution in
a given wake field. This is a desirable feature from the standpoint of
avoiding cavitation on the propeller blades which may also lead to
efficiency improvement.
665
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