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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 three-dimen-
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 non-axisymmetric 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 x-wise Fourier
transform
Function in downwash calcu
lation
Downwash kernel function
Pressure, respectively its
x-wise Fourier transform
Strength of pressure dipole,
respectively its x-wise
Fourier transform
Legendre function of second
kind and half order
Strength of source distribu
tion
Radius, distance between two
joints
Transformed radius
Right-hand 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 646
&(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 x-wise Fourier
transform
- Fluid density
- Integral operator
- Integration volumes
- Right-hand 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 well-established
means of ship propulsion. It is well-known,
[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 non-uniformity 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 three-dimen-
sional flow [12]-[13]. These last methods for
steady three-dimensional 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 propulsor-hull 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 three-dimensional flow
646
OCR for page 647
[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 non-linear 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 three-dimensional 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 (non-iteratively) 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 non-symmetric
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 non-uniform 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 648
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 (U-t.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
right-hand 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 right-hand 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(x-c)] &(r-Rd) , (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(X-C)]
Fir- - U ar) &(r-Rd) + &'(r-Rd)]
r
l
-C
648
. (12)
+c
Fig. 2. Duct geometry conventions
x
OCR for page 649
OCR for page 650
OCR for page 652
OCR for page 653
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(x-c)] &(r-Rd) (13)
Substitution of eq. (13) into (10) yields
S(x,r,8) = - pU aX (q(x,8)[H(x+c)
- H(x-c)]) 6(r-Rd) . (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(x-xp) , r
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 m-n- - -n
_00 n__m
~ n=-=
00
lion bm-n(a) pn(a)] ~ dL ei(\ k)t (26)
_co
where Pn = dpn/da. By noting that
co
8(~-k) = 2 ~ ei(A-k)( 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) = (1-1ml) 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 651
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 right-hand 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~~) ~(a-R~
_=
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 right-hand 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
right-hand 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
~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 one-dimensional 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
x-wise Fourier Transform is to reduce the
three-dimensional problem to a set of de-
coupled two-dimensional 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 right-hand 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 right-hand
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 right-hand side into a shear-independent
term 9(0) and a shear-dependent 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 shear-dependent term ~ml)
is inexistent and Amy) is given by eqs. (48)
and (49), respectively.
652
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 right-hand 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 right-hand 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 + e-XP(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(x-xp), r
