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OCR for page 571

Computation of a Nonlinear Rotational Inv~scid Flow through a
Heav~ly-Loacled Actuator Disk with a Large Hub
B. Yim
David Taylor Research Center
Bethesda, USA
Abstract
A heavily-loaded actuator disk with a large hub in an
open rotational inviscid flow is considered. With an
assumption of axisymmetry for the flow, the governing
equation is the well known Helmholtz equation for the
stream function. Similar problems have been dealt with
by many authors, but with no hub or for an annulus
with constant hub and tip radii. The additional freedom
in the boundary geometry makes the problem much more
difficult but is required to allow extensive applications to
ship propulsor hydrodynamics.
For the general solution, a nonlinear integral equation
for the stream function is formulated by use of the
Green function. The boundary conditions require both
the stream function and its normal derivative on the
boundary. However, although the former is known, the
latter has to be derived at each iteration. The solution is
obtained by successive iterative approximations
substituting the first approximation into the nonlinear
integral equation at each mesh point of the flow field.
Convergence of the solution has been shown but the
computation is quite slow. Therefore the computational
method can only be applied using high speed computer
of large memory.
For the solution at distances far down stream of the
propulsor, the Helmholtz equation can be approximated
by an ordinary differential equation, and the numerical
solution can be obtained without too much difficulty.
The velocity components and the pressure are obtained
far downstream of an actuator disk with a given set of
circulation and shear swirl distributions.
The flow near the disk is computed by an iteration
method. Any physical quantities can be obtained from
the stream functions. As an example of the applications
of this program, the effects of shear swirl flow behind a
heavily-loaded actuator disk, where the external forces
act on the flow, are computed.
Introduction
Both design and performance prediction procedures
for propellers [1] have made continuous progress. Lifting
line theory is now only used in preliminary design, and
lifting surface theory is used for the final design and
performance prediction of propellers. The theory of hull-
propeller interaction has also made progress as in the
consideration of effective wake in predicting propeller
performance. However, present practice still employs
rather crude approximations in some areas of propeller
theory. In particular, the hub effect on propeller blade
design in a shear flow has not been fully considered. The
load distribution of a propeller blade produces bound
and trailing vortices. Trailing vorticity in a shear flow is
different from that in uniform flow, producing
additional trailing vorticity which can be called the
secondary vorticity. The effect of the secondary vorticity
field on blade design and propeller performance is not
well understood. In particular the effect of onset shear
and swirl flow on the hub vortex has not been fully
investigated. For some designs, when the propeller shaft
is of small diameter and the inflow is fairly uniform,
there is less necessity to handle such things. But in the
case of a controllable pitch propeller where the shaft is
of relatively large diameter, or a submarine propeller
where the hub is part of the stern, hub effects can be
very important.
Since the boundary layer thickness of a ship at the
station where a propeller is located is sometimes the
same order of mgnitude as the propeller radius, several
types of interactions have received serious attention by
previous investigators [2,3]. In the design of wake
adapted propellers, the effective wake is used for
estimating the thrust and torque of propellers. To
estimate the effective wake from the nominal wake
velocities measured in the propeller plane in absence of
the propeller, several methods have been used. It was
either estimated empirically by multiplying the measured
circumferential mean nominal wake by a constant factor
or determined theoretically. One of the theoretical
effective wake computations for a propeller in an
axisymmetric flow field was provided by Huang and
Groves [2]. Here, the computed propeller-induced
velocities had to be used in solving the vorticity
equations. The propeller-induced velocities were
computed from potential flow theory using conventional
loading and thickness singularities. Also a combination
of this program with the conventional lifting surface
theory was considered. Experiments showed that the total
velocity profiles calculated by Huang & Groves theory
immediately upstream of the propeller were in good
agreement with measured values. However, Huang and
571

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Groves only considered the region upstream of the
propeller assuming that the energy was not influenced in
that region by the propeller. They do not consider the
flow downstream of the leading edge of the propeller,
where external forces act on the fluid. Several
axisymmetric approximations [4,51 draw attention to the
importance of the shear flow effect on the propeller
when treated as an actuator disk. Full investigation of
the shear swirl flow interaction with the blade-induced
velocity field and the secondary flow has not been
undertaken so far in propeller theory. In the present
paper, the shear swirl flow coming into the heavily-
loaded propeller with a large hub is treated as a single.-
problem rather than as the combination of separate
problems. To make this manageable, we consider it as
axisymmetric and the propeller as an actuator disk. In
addition the behavior of swirl in the wake of a propeller
in shear flow as a function of hub shape is investigated
here in an axisymmetric flow analysis. For turbomachine
design, similar problems have been considered extensively
[6]. Thus the governing equation is well known and even
exact solutions for special cases have been found.
However, the equation with a general boundary
condition has only been dealt with numerically for the
purpose of turbomachinery design. For an open propeller
without duct, the boundary conditions are different than
for turbo-machines. Wu [7] considered a similar
formulation of the problem for an actuator disk but no
one seems to have considered the wake flow field.
A simpler method of computation of the swirl flow is
first studied here and computations are performed for an
example of a useful general boundary condition which
can be used in propeller design, especially for the hub
vortex cavity analysis.
Then a full axisymmetric nonlinear numerical
problem is tackled using an iteration method. Even
though this problem is for an axisymmetric actuator
disk, this is in the shear swirl flow with a large hub of
an arbitrary shape and heavy loading. With an
application of the Green theorem and the Green function
a full treatment of the boundary conditions is attempted.
Both the boundary condition on the hub and the
discontinuity of the velocity in the slipstream behind the
propeller are properly treated. This analysis is a
continuation of a previous investigation [8] with
considerable improvements in accuracy and completeness.
Shear Swirl Flow Analysis
Although there have been many investigations of
propeller performance and propeller design, very little is
known of the flow in the propeller wake. The swirl that
is closely related to hub vortex cavitation should be
considered in the wake of a propeller whose hub length
is finite.
The image vortices inside an infinite hub are well
known. These image vortices cannot be placed on the
outside surface of the hub, or in the flow. This is
because the trailing vortices and the secondary vortices
formed near the propeller blades would not disappear
behind the propeller unless viscosity were considered, and
no other vortex can be considered in addition to them.
The problem is how these vortex lines will be rearranged
or roll up amongst each other while they are transported
downstream where the propeller hub tapers to a point
and disappears.
In experiments, the vortices around the hub roll up
and vortex cavities are often formed in the hub wake. To
understand this phenomenon, a simple model of
axisymmetric swirl flow may be useful. In general, the
common dynamic equation of incompressible inviscid
steady vortex flow can be [6] represented by
qx: = VH (1)
where
H = _ q2 + P (2)
2 e
is the total head in the assumed axisymmetric flow with
cylindrical polar coordinates (x,r,8) and the
corresponding velocity components q (u,v,w), and
vorticity components ~ (`x,cr'`~)
Considering the stream function to (x,r), the
governing equation for ~ is
a2~ ~2~ 1 B~ dH dC
+ - - r2 _ ~ {~\
Dx2 .,r2 r Br dip din ' '
C(~) = rw (4)
is the circulation distribution in the wake. H is constant
along a streamline with ~ constant and H = H(Y'). The
nominal wake is assumed to be known, i.e. the flow field
without propeller is known. Accordingly, H without the
propeller is known and designated by Ho. Using an
axisymmetric approximation, H can be written as [7]
H = Ho + QC~ (5)
where Q is the angular speed of the propeller located at
x=0 and the circulation Cat is proportional to the blade
number times the blade load distribution except for
secondary vorticity. If there is no preswirl upstream of
the propeller, the total circulation C will be equal to C1.
Now both H and C immediately behind the propeller are
both known functions of r and A. Thus,
IdH dH do
dip dr dr (6)
dC dC do
dip dr dr (7)
The boundary conditions at downstream infinity may be
written as follows
1 al a~
u =— ~ ~ us,,, ~ ~ O as r ~ ~ (8)
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1 a~
~ = -—a ~ ° as x ~ ~ (9)
On the hub,
r = h(x), Y' = 0 (10)
On the outer edge of the propeller and on the slip stream
Y'(r,x) = ~ (1,0) (11)
Many problems with similar governing equations have
been solved for special cases, mostly for turbomachinery.
For a propeller, there are some investigations for a
simple case [61. No investigation seems to exist for a
propeller wake with shear swirl in the presence of a finite
hub. We use non-dimensional quantities
r' = r/R, x' = x/R, q' = q/U, C' = C/(UR)
hi' = ~/(UR2), H ' = H/U2 and 1 = U/(QR)
where U is the speed at upstream infinity, R is the propeller
radius. After this substitution the prime will be omitted
as understood. Then we obtain the nondimensional
governing Equation (3).
Although the problem could be solved numerically, it
would be very complicated in the present form to obtain
an accurate solution. At first, for the purpose of
investigating hub vortex cavitation, the solution far
downstream of the hub may be very useful. Since
Bty '32~
a To, a 2 .0 asx .=
may be considered as a function of r only in Equation
(3), and the governing equation becomes an ordinary
differential equation. For special functional forms of H
and C, a closed form solution is possible. With the
general functional form of H and C, if ~(a) and
B~/8r(a) are known the solution can be obtained as an
initial value problem by a method such as the Runge
Kutta Technique. When ~ is obtained, u and C will be
known and the pressure can be obtained by
I' = Ho) - 2 {U2 + r2 (a)} (12)
or by
1 dp c2
_ _ = _
Q dr r3
Here we consider C(r) =0 outside the slipstream. Since
outside the slipstream at x~= the flow is uniform and
the pressure should be continuous, the location of the
slipstream and the pressure distribution can be uniquely
determined. The solution of the differential Equation (3)
with conditions (8) through (12) is obtained by solving
two simultaneous ordinary differential equations
do 1
dr r F1 + f(~,r)
do
dr l (13)
f(~,r) = r2 dry - C dC
with a given boundary condition at the slip stream edge
r= rs(x)
~(rs) = Nehru) (14)
p = 0 where rsto)—ru
From Equations (5) and (12)
dt s) =rsu=rs[2{HO(ru)~+ A C(ru)-C2(ru)/rs2~]l (15)
Because we assumed C(ru) = 0,
d~(r5) = rs \/2Ho(ru)
However, since rs is not known, these equations may be
solved for a range of rs values and the final value
selected such that
~(r=0) = 0 (16)
either graphically or by an iteration method. Note here
that dH/d~, dC/dt4, A, d~/dr at x=0 are all known as
functions of r(x=0). Thus, at each step of the solution
of the above simultaneous equations, from the known
value of ~ at r(x ~c=), r(x=0) can be calculated to
determine both dH/d~ and dC/dt4 at Ax =0) and
therefore, at r(x All.
Such solutions are exact at x em, under the assumed
conditions. However, the solution with (or = h) = 0 can
also be considered to be an approximate solution at any
appropriate strip of a slender hub, i.e., the hub slope is
small with respect to x.
In the present study, a sample calculation of ~ and
pressure are rgade with the following form of the
boundary conditions. At x=0, for Ho
u = (r - h)~/n/D at r < rO
D— (rO—hen
u= 1 atr~rO
v = 0 in rH < r < 1 (17)
and
C = an + air + a2r2
with constant coefficients an, al, a2 and n. Using the
above representations, the incoming velocity profile is
similar to the flow in a turbulent boundary layer, and
the circulation distribution has a radially parabolic shape
which can take on zero values at the blade hub and tip.
Then, at x = 0
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Jr 1 (r - h)l/n + 2 h(r - h)l/n + l r
(or) = u(r)rdr = D 1 /n + 2 1 /n + 1 r = rH ' s =
J rH
at rH < r ~ rO
(or) = intro) + 2 - 2 at r ~ rO R
d = D {(r-h)l/n+l + h(r-h)l/n} at rH < r ~ rO
d = D {(rO-h)l/nr} at r > rO
or
Also assumed at x=0, from Equation (12)
dr 2 dr {U2 + r2 C2(~)) + 13 C2
dH ~ {(r - h)l/n ~ (r - h)l/n - l
+ an + air + a2r2 (a + 2a r)
at rH < r < rO (19)
aO+a~r+a2r2
= (al + 2a2r) at r ~ rO
The pressure distribution induced by the particular shear
swirl flow considered here through Equation (17) is given
in Figure 1. Even if the circulation near the hub is zero,
the pressure rapidly decreases near the axis at x loo. If
the pressure is lower than the vapor pressure Pv, a cavity
may be formed and will behave like a solid boundary. If
a streamline has ~=0 and P= Pv, this will be a cavity
boundary. In Figure 1, the axial and the tangential
velocity components, u, w and the pressure p/g are given
at each radial position for three assumed slip stream
radii at x em, rS=0.98, rs=0.95 and rs=0.915. ~=0 at
r=0 is achieved only in one case rs=0.915. In the other
two cases, ~=0 for r>0 when the hub is infinitely long.
Even when the radius of an infinitely long hub decreases,
the quantities u, w and - p/g increase near the hub.
These quantities downstream of the actuator disk near
r = 0 increase very rapidly to oo. The boundary values of
u and w at x=0 are shown as broken lines.
Full Axisymmetric Solution
Now we try to solve Equation (3) in its full
axisymmetric form. We consider
to =—r2 + rt4
in Equation (3). Then it changes to
V214 - ~ = - g (21 )
574
1.0
one
0.95
0.915
0.8
. 0.6
0.4
O.
at x=o
)at~ if/ /
\

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where d is the delta function
and
G(r,x; Q,4) = G(e.~; r,x)
G(r,x; Q,4) ~ O as PQ
Ge(r,x; Q.~) ~ O as PQ
Such a function has been obtained by Wu [7] and is
represented by the 2nd order Legendre function which
has a logarithmic singularity at PQ .0. From Equations
(21) and (24)
GV24,-~V2G = - Gg + (or - Q)(X - t)/r
Integrating this equation in the space bounded by the Besides, at ~ = x
hub boundary and planes at x= x~ and XN we obtain
Ji (GV2~-~V2G) pdedt = - J:GgQdQd4+~(r,x)
V D where
Using the Green theorem
From Equation (21)
am 1 am 1 or
On r an - 2 an on r = rH (27)
From Equations (26) and (27)
am 1 altar ax ar I ax 1 ar
an = r ar ~ an - an at / at -—a (28)
where a / a is the slope of the hub with respect to x and
—do= dx, —do= -dr (29)
an an
~ = (l in_ e ~
Q 2 t=xL
in = ( ~ ~~)
tptr,x) = - JIG an ~ an A') pdg + JiGgtp,4)pdedk When we insert Equations (28~-~30) into Equation (25)
Q D we obtain
-Jo
is
GV aa9' pdg
where Q is along the contour at x =xL and XN and the hub
boundary, n is the inward normal to the fluid, Qs is along
the slip stream where at4/8n is discontinuous as much as
~ a ala n while ~ and a G./ a n are continuous. Along the
contour Q. the boundary conditions are given as where
~ = 0 on the hub r = rH
or
~ = _ H
The normal velocity on the hub is zero, or
am
= 0
at
where t is the unit vector along the tangent to the hub.
That is,
am am ar am ax
_=_ + _= 0 (26)
at ar at ax at
However,
(30)
(25) JiG ( 6) d do _JG am {1 ( de )2t do
D Q
- JGA a { 1 + ( do ) ~ do - iG~QdQ + f(r,x) (31 )
Is XN
f(r,x) =J( 2 - ap 2 ) Q do +l am 2 do do
Q=rH Q=rH
+l am (a - 2 ) do -; G am do (32)
t=XL t=XL
al' am ar am ax
= +
an ar an ax an
Since 14~ .0 as Amp, the line integral at ~ loo disappears.
Note the effect of slipstream is first included here among
solutions solved by the similar technique. Although ~ is
continuous throughout the flow including the slipstream,
we know that the velocity or the derivatives of ~ may
not be continuous across the slipstream especially when
the circulation distribution is not continuous. This may
cause the line integral along the slipstream to be non-
negligible. Numerically this gives a considerable
complexity because neither the shape of the slipstream
nor the velocity discontinuity is known a priori.
However, in each iteration this could be known
approximately. Equation (31) is a complicated nonlinear
integro-differential equation for ~ because g is a
complicated nonlinear function of A, and ~g is not
known on the boundary. Wu formulated the problem for
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a simple case of no shear and no hub without slipstream,
only with the heavily-loaded actuator disk. He obtained
the Green function and suggested solving the nonlinear
integral equation by an iteration technique, showing as
an example the case with a given load distribution on the
propeller disk but did not obtain numerical results. Later
Greenberg [10] performed the numerical computations
for the same problem in a somewhat different manner
with a uniform loading and a nonuniform loading, and
demonstrated the convergence of the iteration with the
same Green function obtained by Wu [7].
The present problem includes a large hub and
incoming shear flow with pre-swirl in addition to the
heavily-loaded propeller disk. Because of this complexity,
there exists a line integral term, in addition to the area
integral that was handled by Wu and Greenberg et al.
The line integral includes the normal derivative of the
unknown which changes the integral equation to an
integro-differential equation.
When an approximate solution ~ is assumed
everywhere, Bt4/8n on the boundary will be known
accordingly, and H. C and their derivatives along the
streamlines can also be determined. Therefore when these
quanitities are inserted in Equations (31) and (32) the
iterated solution can be obtained. If the continuation of
this process converges to a solution, it will be the desired
one if the solution exists.
Computation of tY
The first order solution may be considered with a
straight slipstream through the blade tip (r= 1). As a first
approximation the streamlines are assumed to be
constant along
r =—(rT - rH(x)} m= 1, 2, ·~. mO (33)
The stream functions at x= XL is given as Fir) from
which we obtain the shear distribution
1 al
r or
-, or vice versa.
At x = A, g can be calculated from the given A, H
and C at x = A. Then along each streamline A, H. C,
dH/d~ and dC/d~ are constant. Therefore when ~ is
known at any field point H. C, dH/d`Y, and dC/d~ will
be known automatically. Thus g will be known and we
can evaluate the area integral of Equation (31). The
mesh is created with lines (33) and the vertical lines
x = + Ax (34)
with lix intervals.
At each mesh point, the value of G is precalculated
by a good approximation [10]. Then
Jru
AH
GgQdg (35)
is first calculated at each vertical line by Simpson's rule.
The logarithmic singularity of G in the numerical
integration of Equation (35) and in the slipstream line
integral is treated as follows:
when x=t, and Q ~r(/O)
G ~ - 1 1 log (r-e)2
4n r
r2
Though the integration of the log function does not
produce any singular behavior, the numerical treatment
requires care.
J GgQde =J { GgQ + 2g° log ~r-e~ } do
rH rH
+ 2 r [(rH—r){log~rH—rig—1 }—(ru—r)(log~ru—ret—1~]
where gO is the value of go at Q = r, the singular point.
Then the values of Equation (35) at ~ = + Ax are
integrated by Simpson's rule to produce the area integral.
Since ~ on the boundary is given from the boundary
condition, the function f(r,x) in Equation (32) is
determined and it does not change by iteration. However
3~/8g on the hub r=rH, a~/at on x=x and ~ at
x = XN are not known a priori. The line integral terms
have to be iterated by calculating 3~/ap and 3~/~t
from the first approximated ~ distribution.
When the mesh (33) is set from the beginning, the
iterated solution ~ will change at the given mesh points
at each iteration and will converge to a solution if the
solution exists. There, H. C and the derivatives which
are only functions of ~ have to be interpolated
numerically at each point from the given values of Hi),
C(~), etc. at x= xL. Since the values of to on the
boundary are already known from the boundary
conditions, they do not have to be calculated on the
boundary. This fact is very convenient because the
boundary integral/the line integral is more singular than
the area integral. However the values of the line integral
at the points other than at the boundary have to be
calculated and need special care.
Iteration and Convergence
Iteration techniques for nonlinear equations are
familiar to those who use high speed computers.
However, since the present problem may be considered
to involve simultaneous equations with an extremely large
number of variables, it is rather impractical to use the
conventional Newton Raphson method. Because it might
be rather simpler and more economic to consider a naive
first order iteration without modification of values, this
approach was tested first. However in this case
sometimes the solution seemed to oscillate after a certain
course of convergence, so that a simple modified quasi
Newton Raphson method was considered [8].
The convergence using the simple modified quasi
Newton Raphson method was found to be very sensitive
576

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to the computational error. Whenever any error exists in
the computation the solution does not converge. This has
been studied in detail in a previous paper [8]. However,
even though it just converges it is not necessarily the
right solution. Besides the question of uniqueness, the
solution has to satisfy all of the boundary conditions. In
fact it converged without line integrals along the lines
x=x and the slipstream, giving a wrong solution. It
also converged to an unreasonable solution when
unreasonable values were given as the boundary
condition. Therefore not only the convergence but also
satisfaction of boundary conditions must be checked.
Propeller Characteristics
Now the axisymmetric stream function is obtained
everywhere, all the physical quantities can be computed.
These are used for the characteristics of an infinite
bladed propeller.
Well known formulae for propeller characteristics are
given as follows: The thrust coefficient
T neQZR4 = 4A J w1(r,° + ) { A - w2(r,0 + ) } rdr
rH
where 2w~r equals the circulation distribution F
set of data obtained in a water tunnel experiment. The
first example has already been used for an approximate
analysis as shown in Figure 1. The full axisymmetric
analysis with the boundary conditions given in Equation
(17) are now tested for the hub geometry shown in
Figure 2. The efficiency changes corresponding to the
inflow velocity distributions shown in Figure 3 are shown
in Figure 4. These changes are compared with the results
calculated by simple momentum theory without
considering the thrust deduction. It is obvious from the
CQ equation that CQ will decrease when u deceases, thus
increasing the efficiency. However, the reduction in u
may be as the result of increment in the friction and/or
form drag of the ship. The second example corresponds
to one of the sets of test data of Huang and Groves as
shown in Table 1 with the hub geometry shown in Figure
5. The velocity distribution is taken from the measured
longitudinal velocity component at the station
x=xL= -0.482 as shown in Figure 6.
XjX~
corresponding to the propeller blade circulation ~ with a
Z bladed propeller such that
km ZE = Foo/K = 2wlr
where K is the Goldstein factor and w2 is the ~
component of the velocity at the actuator disk. The
torque coefficient
CQ = QR5 = 4A2 J u(r,0) w~(r,0) r2dr
rH
The efficiency
The pitch angle
71 = A CT/CQ
= cot- ~ (rr7C(r)/A)
where rlC(r) is the radial distribution of the propeller
efficiency
r1c(r) = { 1 - r W2(r,0 + )}/u(r,0)
The other conventional thrust coefficients CTS=
T/(e/2 U2nR2) has the relation with CT, CT=O.SA2CTS.
Numerical Examples and Discussions
For checking the numerical accuracy and solution
behavior for parameter changes, two examples were
numerically tested. One example uses the boundary
conditions given in Equation (17) and the other uses a
~ 1.0
1 1 1 1 ~
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4
Fig. 2. Hub geometry 1.
0.8
0.6
0.4
0.2
The head distribution from Equations (2) and (5) is
approximated from the nominal wake values because not
enough information is available for the head. The blade
circulation given in the Table is converted by the relation
zr ~ r00 = 2wlr
taking the Goldstein factor K= 1 because for the
7-bladed propeller k is very close to 1. With this
approximation, the thrust coefficient CTS, using the r
values of Table 1, is very close to the measured value
and the value from the lifting surface computation.
The velocity components at the propeller plane (x =0)
and a little downstream of the plane (x=0.192) are
shown in Figure 6. These values are taken at the iteration
where the average relative error is less than 0.005~Vo with
20 iterations. A simple approximation assuming that
u = nominal wake + F<~,/(2rtan+) from actuator disk
theory is also plotted in Figure 6. The three calculated
577

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1 x=x1
1.0 ~
0.9 L
0.8:
0.7 g
0.6:
0.5 g
0.4
0.3 1 1 1 1 1
0.4 0.6 0.8 1.0
u
Fig. 3. Shear distribution at x= -0.8.
1 n
_ _ PRESENT THEORY
0 8 _—=~/ n=2 0.8
MOMENTUM
0.6 - THEORY n=4
0.4
///
///
n=2 /3 /4/ 1 1 1 ~ 1 1 1 1 1 =\
/ // - 0.5 - 0.3 - 0.1 0 0.1 0.3 0.5 0.7
Fig. 5. Hub geometry 11.
1 .2 _
1.0 _
0.6 _
1.0
0.8
0.6
0.4
0.2
0.0
//
MEASURED AT / /L X=0
WITH PROPELLER 7/ AWL X—0. 192
0 2 NOM NAL WAKE
~ 0 I // \ + ?CO/~2rtan l)
0O0 2 4 6 8 10
CT—10-2 0 Z: ~ ~ ~ ~
2 0 2 0.4 0.6 0.8 1.0
Fig. 4. Efficiency-thrust coefficient relation.
Table 1. Tested propeller data.
CTS = 0.356, HA = 1.268
r/R
0.211
0.250
0.300
0.400
0.500
0.600
0.700
0.800
0.900
0.950
1.000
1.050
1.100
1.200
Nominal
Wake
0.387
0.417
0.454
0.520
0.579
0.631
0.677
0.720
0.763
0.785
0.806
0.826
0.845
0.880
Fig. 6. Axial velocity components near the propeller disk.
r
curves are very close to each other. The efficiency
computed with the boundary condition given by the
measured longitudinal velocity at x= -0.480 IS 1.15
which is close to the value 1.17 from actuator disk
theory.
0 0023 If uniform inflow is assumed, the propeller efficiency
0.0051 with the same value of F`,0 in open water is 0.82 this
0.0107 means the approximate corresponding wake fraction Wf is
0.0150 0.287 from 1.15 (1-wf)=0.82. It is well known that the
0.0172 overall propulsive efficiency also depends upon the thrust
O. 0140 deduction which reduces the efficiency. The numerical
0 0089 results show that when F00 =0 at the blade tip, the effect
0.0055 of the slipstream line integral is negligible. This is
0.0000 reasonable because it has been analytically proven [11]
0 0000 that both u and w are proportional to F0O on the
0.0000 propeller plane when hub Is absent.
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Since the governing equation (31) holds in a shear
swirl flow and no explicit vortex distribution is used for
the solution, it is not clear here how the conventional
bound and trailing vortices plus secondary vortices [12]
are involved. However, this could be interpreted using
the vortex ring method [13].
The numerical program with iterations adopted in the
present paper is mainly for the purpose of checking the
feasibility of the application of the iteration method to a
nonlinear problem with the reasonable convergent results.
The convergence is quite stable with the change of mesh
size and the degree of interpolation (when a Lagrangian
interpolation is used).
Because the present problem neglects viscosity, the
effect of viscosity on the shear flow cannot be computed.
The effect of shear on the propeller can be computed
from the measured velocities or the computed velocities
by the method of reference [2]. Therefore the present
program could be most effectively used to find the
propeller shear interference at or behind the propeller
plane with the measured upstream velocities or the
effective wake computed by the Huang and Groves
method. Since experiments show that the propeller does
not affect the flow beyond about two propeller diameters
upstream [2] of the propeller the nominal wake at a
station two propeller diameters upstream of the propeller
plane may be effectively used to compute total flow
using Equations (2), (5) and (3). If there is no shear or
swirl upstream of the propeller, the problem becomes
much simpler because the area integral outside the
domain bounded by the slipstream, the propeller plane,
and the hub disappears. However, the line integrals along
the hub and the slipstream (when FOO (tip) ¢0) are needed
in addition to the area integral.
The present program may be further extended to
evaluate ducted propellers and multistage propellers such
as contra-rotating propellers in the shear swirl flow. In
fact for the problem of contra-rotating propellers in
shear swirl flow, the present computer program can be
applied without too much change.
Acknowledgments
The investigation reported herein was supported by
the David Taylor Research Center's Independent
Research Program. The author would like to thank Dr.
T.T. Huang, Mr. Justin McCarthy, Dr. William B.
Morgan, and Dr. Milton Martin for their continuous
encouragement.
References
1. Kerwin, J.E. and C.S. Lee, "Prediction of Steady
and Unsteady Marine Propeller Performance by
Numerical Lifting Surface Theory," Trans. SNAME,
Vol. 86, 1978.
2. Huang, T.T. and N.C. Groves, "Effective Wake
Theory and Experiment," Proceedings of 13th
Symposium of Naval Hydrodynamics, ONR, 1980.
3. Nagamatsu, T. and T. Sasajima, "Effect of
Propeller Suction on Wake," Jour. of SNJE, Vol.
137, pp. 58-63.
c
4. Dyne, Gilbert, "A Note on the Design of Wake-
Adapted Propellers," J.S.R., Vol. 24, No. 4, Dec.
1980, pp. 227-231.
Goodman, Theodore R., "Momentum Theory of a
Propeller in a Shear Flow," J.S.R., Vol. 23, No. 4,
Dec. 1979, pp. 242-252.
6. Horlock, J.H., "Actuator Disk Theory,
Discontinuities in Thermo-Fluid Dynamics,"
McGraw-Hill, Inc., 1978.
7. Wu, T.Y., "Flow Through a Heavily Loaded
Actuator Disk," Schiffstechnik, Bd. 9, Heft 47,
1962, pp. 134-138.
8. Yim, B., "An Iteration Method for a Nonlinear
Shear Flow Through a Heavily-Loaded Actuator
Disk With a Large Hub," Proceedings of the
International Conference on Numerical Methods in
Engineering; Theory and Application, Edited by
G.N. Pande & J. Middleton, Martinus Nijhoff
Publishers, 1987.
9. Courant, R., "Method of Mathematical Physics,"
Vol. II, Interscience Pub. New York, 1862.
10. Greenberg, M.D., "Nonlinear Actuator Disk
Theory," Zeitschr, Flugwissensch, Bd. 20, Heft 3,
1972.
11. Morgan, W.B. and Wrench, J.W. Jr., "Some
Computational Aspects of Propeller Design,"
Method in Mathematical Physics, Vol. 4, Academic
Press Inc., New York, pp. 301-331, 1965.
12. Yim, B., "A Note With Secondary Vortex Caused
by a Thin Foil in a Nonuniform Flow," J.S.R., Vol.
32, Nov. 1988.
13. Cox, B.D. "Vortex Ring Solutions of Axisymmetric
Propeller Flow Problems," MIT, Dept. of Naval
Arch. & Marine Eng., Rept 68-13, June 1968.
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