An algebraic propeller model using Eulerian turbomachine theory. Wilken, Dennis Ray.

An algebraic propeller model using Eulerian turbomachine theory. Wilken, Dennis Ray.
Calhoun: The NPS Institutional Archive
Theses and Dissertations
Thesis Collection
1982
An algebraic propeller model using Eulerian
turbomachine theory.
Wilken, Dennis Ray.
Massachusetts Institute of Technology
http://hdl.handle.net/10945/20043
LiE.sARY
DUDLEY
NAVAL POSTGRADUATE SCHOOL
MONTERcY, CALIF. 93940
.
An Algebraic Propeller Model
Using Eulerian Turbomachine Theory
by
Dennis Ray Milken
B.S.E.E.
,
Purdue University
<
1974)
SUBMITTED TO THE DEPARTMEI^ OF OCEAN
ENGINEERING IN PARTIAL FULFILLMENT OF THE
DEGREES OF
OCEAN ENGINEER
AND
MASTER OF SCIENCE IN
MECHANICAL ENGINEERING
at
the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
MAY 1982
©
Dennis Ray Nil ken
1982
The author grants to M.I.T. permission to reproduce
and to distribute copies o-f this thesis document in
whole or in part.
/
'.Y
DUDLEV
NAVAL POSTGRADUATE SCHOOL
MONTEREY, CALIF. 93940
,
:
.
.
An Algebraic Propeller Model
Using Eulerian Turbc»nachine Theory
by
N
Dennis Ray Milken
o-f Ocean Engineering on May 7,
1982
partial -fulfillment o-f the requirements -for the degrees o-f
Ocean Engineer and Master o-f Science in Mechanical
Engineering.
Submitted to the Department
in
ABSTRACT
Utilizing -four -quadrant experimental data determined by Miniovich converted to the modified advance and angl e-of-advance
coefficient form, complete Karman-Knapp Circle Diagrams are
plotted for a number of 3 bl aded propellers.
Reducing the
family of curves to a rational basis,
algebraic equations are
derived to model the performance of a three bladed propeller
through all four quadrants using Eulerian turbomachine theory.
The result is a set of equations uihich can be used to generate
complete propeller thrust and torque characteristics given only
incomplete performance data suitable for use in steady state
design or dynamic simulation.
Thesis Supervisor:
Title:
Dr.
Henry M, Paynter
Professor of Mechanical Engineering
To Professor Paynter uiithout whose support and assistance
could not have completed this project.
I
To my uii-fe and children without whose love and encouragement
I
wouldn't have had any reason to.
"Inawa/f science might be described as paranoid thinicing applied to
nature: we are looking -for natural conspiracies! -for connections among
apparently disparate data."
-CARL SA6«i^
Using
Eulerian Turbomachine Theory
An Aloebraic Propeller Model
CONTENTS
Title Page
Abstract
Dedication
CHAPTER
1
1-1
1
2
3
INTRODUCTION
Primary Motivation for this Type
Method
o-f
Modeling
6
1-2 Some Current Propeller Modeling Methods
7
the Modeling Equations
9
1-3 The Objective
CHAPTER 2
o-f
DERIV^TICBM OF THE ALGEBRAIC MODEL
2-1 Fundamentals
o-f
Eulerian Turbomachines
10
2-2 Ideal, Canonical and Perfect Model Equations
for Ship Propellers
13
2-3 Determination of the Algebraic Modeling
Equations
18
CHAPTER 3
INTRODUCTION OF ADDITIONAL TERMS AND OTHER
DETAILS IN PROPELLER CHARACTERISTICS
3-1 Pitch as a Primary Determinant
40
3-2 Number of Blades and Blade Area
41
3-3 Effects of Cavitation
42
3-4 Hul /Propel er Interactions
44
3-5 Other Considerations
45
1
CHAPTER 4
1
PREDICTION OF PROPELLER PERFORMANCE USING
THE MODELING EQUATIONS
4-1 Comparison with Kfery Similar Propellers
47
4-2 Extrapolation to Other Propellers
53
4-3 Extrapolation to Missing Quadrants
54
CHAPTER 5
SUMMARY
5-1 An Overview of the Method
5-2 The Potential
APPENDIX A
-for
Further Applications
ESSEhfTIALS OF EULERIAN TURBOMACHINES
55
57
59
The Per-fect Eulerian Turbomachine
The Practical Eulerian Turbomachine
The Dissipation Function
Per Unit Variables
APPENDIX B
SELECTED EXPERIMENTALLY DETERMINED PROPELLER
DATA CONVERTED TO '4-QUADRANT' FORM
64
APPENDIX C
THE BOND GRAPH REPRESENTATION
71
LIST OF REFERENCES
74
CHAPTER
1
.
1
INTRODUCTION
1
Primary motivation for this kind of modelino method
architect or anyone interested in the modeling of
a naval
For
propellers
as a part of the design procedure leading to
integrated
ship design,
the problem long persisted where one
to estimate the performance of a
required
is
an
propeller
for
ship modeling purposes before the design has progressed to the
where one is reasonably confident as to what the
point
desired propeller should look like.
the
final
Even more frustrating
is
problem of having somehow determined the geometry of
the
propeller but not having sufficient experimental data to allow
a
comprehensive computer simulation.
Even where it has existed, considerable effort has been expen-
ded
to convert
the data into
useable form for ccwnputer simu-
lations since the practice of tabulating the results as
func-
tions of an advance coefficient has difficulty in the vicinity
of zero ship or shaft speed.
The
problems
propeller
allowances
associated
formidable since a complete
are
as
a
must
make
for complicated fluid motions which are only
just
beginning to be understood.
difficult
with modeling a machine such
to
In
model
particular, it is still quite
pass completely from propeller geometry to
detailed lift and drag forces required for specific
the
propeller
characteristics.
However,
tation
there
now exists a considerable amount of
documen-
for modeling machines y^ery much like a propeller.
For
attention
turbines and pumps have
had
the
of modelers and those who perform simulations
-for
industrial
example,
axial
decades. Taken together, propellers, turbines and pumps are in
fact,
members of a general class of machines known as
machines.
Models
for
characteristics
and design.
esis
machines
many of the large industrial
been quite successful
have
in
predicting machine
turbo-
performance
and are widely used in control system
A general model for propellers which
synthcould
incorporate that data base would have a firmer basis in experience and first principles.
1.2 Some current propeller model
no methods
present there are two common methods used by designers
At
propeller
obtain
involves
a
i
data for computer
simulations.
The
to
first
experimental determination of the characteristics of
physical model of the full size propeller followed by
con-
version of the data into the familar thrust and torque coefficient form. Such coefficients are then used in a table look-up
scheme
within the simulation program to obtain the
necessary
thrust and torque values. There are advantages to this kind of
method. First, the determination of the propeller characteristics
ensure
through experimentation (scaling
considerations
that the data used in the simulation
will
aside)
be as accu-
rate as possible. Secondly, the use of a model of the intended
propeller
acts
as a test of the geometry and reveals
at
an
early date any serious conceptual problems.
The disadvantages of such a method are fairly obvious. To test
a model
have
a propeller at an early stage in a design,
o-F
defined idea as to the range of operation of
a well
propeller and its desired geometry.
simulation,
sive
all
you must
the
To allow for a comprehen-
you must test and record the model
data
in
four quadrants of operation. Since cost is always a signi-
consideration,
ficant
the
the machining of a propeller model
and operational costs of the
set-up
sizeable investment in the design.
data
problem
of
As alluded to earlier, raw
due
The experimentally
be converted to a form that avoids such
must
a
the
to
"singularities' of the advance coefficient at zero
ship speed and zero shaft rpm.
data
represent
tests
suitable for computer simulations
not
is
and
determined
difficul-
ties.
The second method of modeling propeller characteristics involves
complex and relatively recent work in propeller
namics.
The
Cummings
C23
results.
As
undoubtedly
computer
models of Professors
are quite thorough and provide
a "complete" model,
at
Kerwin
some
hydrodyand
[13
revealing
these computer programs are
the vangard in propeller theory.
With a
firm
basis in hydrodynamic theory, the continual evolution of these
programs will ultimately achieve very accurate results. Unfortunately, these programs are imnensely long and require consi-
derable amounts of CPU time. Because of the attendant expense,
they
are not yet practical for incorporation directly into
ship simulation program.
8
a
1
.3
The Objective
The objective
o-f
the Modeling Equations
this thesis will be to derive a set
rela-
o-f
algebraic equations that will serve as a
simple
tively
o-f
tool
for naval architects in the ship design process. By being able
to predict accurately propeller per-formance in all
rants
operation based on Knowledge obtained from
o-f
experiments
the
naval
and in a form suitable for
computer
quad-
-four
limited
simulation,
architect can save considerable time and expense in
the simulation portion of the design.
Experimental determina-
tion of the points of minimum dissipation would be
sufficient
to allow for a comprehensive simulation for forward or
crash-
back maneuvers.
Such a set of equations may not eliminate the
need
experiment in the final stages of the design,
for actual
however they would greatly simplify the initial
Additionally,
where
experimental data exists for
propellers in only the first quadrant,
used
to
accuracy.
requirements.
particular
the equations could be
reconstruct the remaining three quadrants with
good
DERIVATION
CHAPTER 2
2.
1
o-f
the ALGEBRAIC MODEL
Fundamentals of Eulerian turbcwnachines
The
practice
data
o-f
presenting open water propeller
nondintensional
as
cients
versus
simply
advance
thrust <Kt) and torque
an absolute advance
coefficient <J)
coefficient
C3],C4]
Eulerian
turbcxnachines.
That
is,
(Kq)
coe-f-fi-
(lambda)
incorporates
which inherently restrict the machines of
tions
per-formance
or
assump-
interest
by considering only
to
high
Reynolds numbers the performance of the machine is governed by
Dimensional analysis on the
inertia and pressure forces.
pression
ex-
:
(Thrust)
F=f<'v»p,D,N,p)
yiel ds
Kt= F/PD4N2=f (^v'p/ND) =f<J)
<2.1)
Similarly for torque,
Kq= T/pD5N2=g<v^p/ND) =g<J)
<2.2)
where
Vp is the speed of advance of the propeller through the
water
and D any characteristic propeller
diameter).
performance
data
This
dimension
(usually
single parameter representation of propeller
yields the powerful result that
the
of geometrically similar propellers will,
performance
when
plotted
together, generate to good approximation a single curve.
The
same
assumptions would
,
from a fluid mechanics
stand-
point, give Euler's equation of inviscid motion. That is,
pv3eGRAD(v) = -6RAD(P)
10
(2.3)
Or,
more general
in a
to incorporate unsteady -flows,
-form
PD<v)/Dt = -6RAD<P>
<2.4)
where D/Dt is the substantive derivative.
In
conjunction with equation (2.4),
we express mass conserva-
tion by:
D<P)/Dt
where P is independent
PGRADJ^v
+
o-f
=0
position
(2.5)
and v is the vector
-fluid
velocity. Re-ference C5] chapter 3 and Re-ference 161 are use-ful
for a more detailed analysis.
A mach ine satis-fying equat i ons
experimentally,
-fied
ed
2. 4)
demonstrate
Eulerian similitude". That
will
<
is,
and (2.5) will,
what
is
as ver i-
re-ferred
to
as
geometrically similar machines
produce similar per-formance characteristics when presentin
properly chosen nondimensi onal
a
Casanady
-form.
C73
re-fers to this as a 'per-formance law'.
Based on the discussion above,
formance
done
characteristics
with
derived
small
-from
o-f
the determination
a
scale models.
the model
o-f
large turbcwnachine is
The
-full
scale
the
per-
usually
results
are
results by recognizing that at corres-
ponding points on a characteristic curve,
all
nondimensi onal
parameter ccmbinations are also equal.
However,
there are problems with scale e-ffects not evident in
11
.
such
single parameter representation.
a
In addition
geo-
to
metric dissimilarities, such as roughness and clearances,
differences are also present.
drodynamic
hy-
These include
Rey-
nold's and Froude number variations as well as cavitation
compressibility number differences.
some
Attempting to incorporate
would
of these variations into a performance law
or all
and
require that the law be multidimensional.
As an example,
one
could attempt to formulate a nondimensional representation for
thrust and torque characteristics by:
Kt=f<J,Re)
Kq=f<J,Re)
.
equivalence of the advance coefficient in
Here,
the
ction
with
the Reynolds's number would
could
one
Similarly,
lead
also form yet another
to
conjun-
similitude.
expression
for
thrust and torque
Kt=f<J,P)
Kq=f<J,P>
where P represents a nondimensional ratio for propeller pitch.
Such
a formulation could be applicable to Kaplan
controllable
turbines
or
pitch propellers where the blade pitch is varied
mechanical y
1
Multi-dimensional performance laws unfortunately lead to families
As an example,
a
a problem which single parameter laws avoid.
of curves,
Bovet C8] presents efficiency versus output at
constant head for a Kaplan turbine for various
tings.
An
formance
entail
analytical
blade
set-
treatment for a multi-dimensional
per-
terms of generating
data
would
a matrix of possible states with dimension equal
to the
law
in
12
numerical
number
o-f
nondimensional characteristic parameters. Obviously,
analysis techniques used to generate characteris-
dimensional
tic nondimensional
performance parameters are limited in their
applicability in performance simulation.
tion
To allow
incorpora-
Mould
fluid or geometrical affects into the analysis
o-f
require another approach.
That approach should produce a for-
mulation which retains the advantages of similitude yet allows
for some fluid or geometrical effects.
2.2 Ideal, canonical and perfect ship propeller equations
Figure
a propeller blade cross
shows
2.1
distance from the propeller hub
arbitrary
radial
condition
of zero nominal
dicates
speed of advance.
some
at
under
the
Figure 2.1a
in-
the tangential velocity of the blade element (Vr)
fluid velocity (Vf )
<D)
section
.
Figure 2.1b shows the lift
(L)
and
drag
and
forces with respect to an effective angle of attack a. The
magnitude
of
the effective angle of attack is a function
of
the circulation which causes an induced velocity [33.
If
we define a torque exerted by the element as Fr then,
Fr= LSin<oc)
Further,
defining
+
DCos<a)
Ff as a force perpendicular to Fr
<2.6)
in
the
direction of fluid motion (fluid pressure term),
Ff= LCosCo) - DSinCo)
13
(2.7)
1
Kfr
<a)
Fi
Qure 2
14
.
Since L and D are expressed as:
L= Cl<i4pAvXv)
<2.6)
;
D= Cd<J4pAvXv)
,
and <2.7) can be rewritten:
CdJfVr)
(2.8)
i4PAv(CdXVr - ClXV-f)
<2.9)
Fr= A^pAvCClX^f
F-f=
In per unit
(see Appendix A),
m= v(ClXq
+
(2.8)
and (2.9) become:
CdJfn)
(2.10)
p= v(ClXn - CdJCq)
(2.11)
Or using the canonical
m=
+
notation,
RgJ^q + RdSCn
(2. 12)
p= Rg3«n - RdJfq
(2. 13)
where
Rg=rq3ev
rq=rq(P)
v=SQRT(nXn + q3Cq)
Rd=rdJev
rd=rdO)
^=ARCTAN(q/n)
The correspondence with the bond graph representation shown in
Appendix B is now readily apparent.
The mechanical power can be expressed as Fr^Kfr while the -fluid
power can be expressed as
F-fX'v'-f.
15
Thus, dissipation losses can
.
be -Found to be
(2.14)
Pd= Fr3?Vr - FfXV-f
In per unit,
(2.14)
becomes
Pd= m^n -
<2. 15)
p3Sq
Substituting (2.12) and <2.13) yields
Pd= RdMvS^v
Hence,
as
would be expected,
<2. 16)
an
no power lost in dissipation
have
without
drag.
The
equations
ideal
it
i-f
would
ship propeller
could
reduce to the
generate
rather
lift
trivial
expressions
m =
Rg3?q
<2. 17a)
p = RgXn
<2. 17b)
with Rg representing an energy trans-formation modulus.
be
considered as a general
ation
o-f
separation
o-f
a number
losses.
o-f
Writing Rd
in a
more general
such losses such that at the point
loss,
propel
Rd
1
combin-
loss term that includes a
loss mechanisms such as -form
is identically zero [93,
er
16
o-f
minimum
we can de-fine
Rd can
drag
-form
and
typical
dissipation
a
PERFECT
Rd = W<P)3f<<n-q)5«<n-q)/v)
Using
-for
<2.18)
expression
general
the Kutta condition to -formulate a
the modulus Rg would suggest
Rg = rg
di-f-fering
-from
-»^
XO) 3«( <nXn) -<qXq) )/v
<2.19)
the -form proposed by Paynter by the
and (2.19)
arbitrariness in
into (2.12> and (2.13) and recognizing
WO)
and
XO)
rg
Substituting
which could represent some linear -function in n.
<2.18)
term
the
we can arrive at the -following
,
expressions:
m =
rgSSq *
p = rgXn
-^
Kl<P)5Cv<n-q)
<2.20)
K20) 5^v<n-q)
(2.21)
where Kl and K2 now incorporate the individual character
given
The equations above
propeller.
-for
a
PRACTICAL
re-flect
Rl
propeller
by including an
a
PERFECT propeller
(zero minimum dissipation loss) can be au^nented to
a
o-f
additional
represent
term
to
the actual
non-zero minimum dissipation loss.
m = rg^q
+
KKP) 3fv<n-q)
*
R13^n
(2.22)
p = rgXn
-»•
K2 ( P) 3C v ( n -q)
- R23^q
(2.23)
and R2 would be -functions
o-f
17
the advance coe-f icient J
-f
-for
a given propeller,
but in a simpler model
could be reduced to
constants. As will be seen later, the values
a
effect on the shape
direct
o-f
the 4-quadrant curves
the slope
given machine and in particular,
and thrust radial s. KIO) and K20)
The degree
would be identical.
on experimental
based
o-f
the model
o-f
-for
o-f
the zero torque
agreement between Ki and K2
data would be a measure
o-f
-for
advance
coe-f-f
methodology
C
12]
.
o-f
the K<P)
data. The
by Miniovich CIO] and converted to a
modi-fied
icient
-form
in order
Cll] was used
in
an
to -formulate a general model
be expanded upon in a more detailed analysis.
ence
determining
R2 and rg based on experimental
as Rl,
produced
the validity
the algebraic model ina equations
o-f
propeller characteristics requires a determination
data
a
an idealized propeller
equations <2.22) and <2,23) as a model
term as well
o-f
a given rg.
-for
2.3 Determination
Using
Rl and R2 have
o-f
abbreviated
could
that
See also Refer-
Such an analysis would include an investigation of
those factors which collectively yield the individual
charac-
ter
of a given machine yet have defineable specific
ces,
the details of which are beyond the scope of this thesis.
In
Reference Cll],
influen-
experimental results for thrust and torque
are
tabulated for eighteen three bladed propellers with vary-
ing
expanded area ratios and
propeller
Table
2.1
pitch-to-diameter
ratios.
The
designations are those used by Baker and Patterson.
is provided as a quick reference for EAR
values for each propeller.
18
and
P/D
B3 and B4 in Appendix B are plots
Figures
series
propellers.
o-f
To be useful,
must
these curves
To do so,
rationalized to a consistent basis.
data for two
rauj
o-f
the points
be
o-f
minimum dissipation Mere -found for each of the propellers (See
Appendix A>
and
region
!>
in
was fixed
B3
the
at
(quadrant
turbine operation
and in the region of normal
The mapping of
the
to those points of minimum dissipation is done over
the
(-1,-1)
at
entire q vs
That is,
third
the
curves.
the q vs
n
plane.
Hence
pattern
image''
of
the point of minimum dissipation in the
for positive torque and thrust
quadrant
first
in
plane recognizing the 'mirror
n
the curves.
into
dissipation
(quadrant
pump operation
normal
of
III)
data
that the point of minimum
such
B4
(l,i>
Those points were then used to scale figures
.
reflected
is
quadrant on the negative torque
and
thrust
scaling is reflected through
the
origin
the
rather than simply across the zero torque and thrust radial s.
Figures 2.2 and 2.3 are plots of the thrust and torque
respectively
for three propellers selected for their
curves
varying
expanded area and pitch-to-diameter ratios and are representative
the
of
form of the 4-quadrant curves for
all
of
the
propellers. With the data plotted in this form, the similarity
of
the
torque and thrust curves versus the advance angle
is
striking. In addition, the general form of the curves is quite
consistent
in
the regularity of the
specific
perturbations
which influence all of the propellers in the same manner.
19
Figure 2.2
20
Figure 2.3
21
i
1
12
Prop No.
3
4
5
6
EAR
0.5
0.5
0.5
0.5
0.5
0.5
P/D
0.6
0.8
1.0
1.2
1.4
1.6
7
8
9
10
11
12
EAR
0.8
0.8
0.8
0.8
0.8
0.8
P/D
0.6
0.8
1.0
1.2
1.4
1.6
13
14
15
16
17
18
EAR
1.1
1.1
1.1
1.1
1.1
1.1
P/D
0.6
0.8
1.0
1.2
1.4
1.6
Prop No.
Prop No.
Table 2.
continuing,
Be-fore
point
must
I
out
there
that
is
an
approximation which was used which requires an explanation. As
Figure
B2
thrust
(head)
not
Appendix 8 demonstrates,
in
in
degrees
180
measured clockwise
and
angle
generally
computed
-for
quite
-for
lines
apart.
-from
The di-f-ference
high
small.
between
spec
-fie
I
call
speed
the skew
axial
See re-ference [133.
o-f
the
curves
o-f
machine.
were
o-f
The skew
quite
1
is
angle
were
all
degree. The
o-f
losses
small,
approximated to a PERFECT propeller by
the skew angle between the two radial s -found -from the
22
The
pumps
-flow
the magnitude
Since the skew angle is
radial
angle.
the propellers used in this analysis
skew angle is a -function
angle
the
the first quadrant zero thrust
less than 3 degrees and were generally less than
amount
zero
o-f
the -first and second quadrants are obviously
the second minus 180 degrees
skew
the
the
dividing
minimum
dissipation calculations.
The scaling values for q and
As
recalculated using these adjusted angles.
then
the
and
-first
The
apart.
lated
third quadrant asymptotes
approximation was
were
n
were
a
result,
180
degrees
Justified since the calcu-
-felt
minimum dissipation angles were obtained with a
confi-
dence of plus or minus 2 degrees due to the required extrapol-
ations in the corresponding portion of the experimental
Such
approximation also simplifies the derivation of
an
modeling
equations
considerably.
2.2
lists
and
the
approximate
minimum
angles
both quadrants for six of the eighteen
along
thrust
in
angles.
The
the
propellers
with the calculated and approximated magnitudes of
and torque radial s at those
the
Table
calculated
dissipation angles
data.
the
approximate
values are shown in parenthesis.
Prop. No.
Min. Dissipation
angles
Radial Length
thrust
torque
35.2 <
34.6)
-146.0 (-145.4)
11.52(8.97)
3.92(3.80)
15.39(14.88)
25.83(22.44)
2
39.5 ( 38.5)
-142.5 (-141.5)
5.64(5.02)
4.98(4.59)
13.54(11.32)
25.65(23.88)
3
44.5 (
44.3)
-136.0 (-135.8)
4.95(4.82)
5.02(4.90)
10.54( 9.96)
5
53.5 ( 53.0)
-127.5 (-127.0)
5.06(4.78)
6.88(6.31)
9.27( 8.85)
26.99(20.92)
15
45.6 (
45.6)
-134.5 (-134.5)
6.99(6.90)
4.82(4.79)
12.10(12.04)
20.36(19.61)
6.99(6.59)
7.48(7.07)
11.96(11.47)
20.99(18.76)
1
18
58.0
(
57.8)
-122.5 (-122.3)
Table 2.2
23
19.99(18.68)
.
equations
Using
setting
and
<2.20)
rg equal
required
to
constant
(1)
-fit
and (2.21) for the PERFECT
to n, we can plot
the experimental
and solving
the
-functions
K<P)
data by setting m and p to a
data
Kl and K2 at each
-for
propeller
Since Kl and K2 must be positive at all
point.
we account for
times,
sign change in m and p in the different quadrants as
the
calculations proceed.
By plotting Kl and -K2 as functions
of
we can gain a quick qualitative
the normalized advance angle,
feel
for the similarity of Kl and K2.
2.6
are
plots
the
Figures 2.4,
of Kl and -K2 for propellers
9,
and
2.5,
and
15
18
respective! y
As is readily apparent, even with
<n)
,
and
Kl
functions
in
a
K2 are y/ery similar.
response
simplified function for rg
The behavior of the
to changes in
K<F)
pitch-to-diameter
and
area ratios is also apparent and are manifest in the
expanded
amplitude variations and relative smoothness of the plots.
the basis of that similarity,
On
KO)
we form a new pair of
functions by the straightforward transformation:
Ka(P) = <K1
+
K2)/2
Kb<P) = (Kl - K2)/2
such
that
Kl equals Ka+Kb and K2
represents
an
measure
their difference.
of
average
of
equals
Ka-Kb.
the two functions while
Hence,
Kb
is
Figure 2.7 is a plot of the
Ka
a
Ka
and Kb functions for propeller number 15.
A
different approach may be to account for the general ampli-
tude differences in Kl and K2 on eitherside of the zero thrust
and torque radials and themselves.
24
Four gain parameters could
u
•H
25
IT)
(U
u
en
•H
26
O
u
3
D^
•H
Em
27
<a)
15
225
<b)
Figure 2.7
28
be
obtained
each propeller such that when Ka and
-for
maximum amplitude of KbO) will be
plotted, the
reduced. This was done
are
Kb
considerably
propeller number 15 by gain scaling
-for
the average amplitudes as outlined above. Figure 2.8 shcMus the
KbO) now approaches what may be
Ka and Kb.
result for
con-
sidered as individual propeller nuance characteristics.
Since Ka<P)
we
is essentially doubly periodic over the range of ^
approximate
form yet another function that can be used to
the entire range of F
Kl and K2 valid over
KaO)
plots of
-135
for
P
<
<
45
and
By overlaying the
.
45
F
<
<
and
225
again averaging we obtain an approximate \«F> function applic-
able
to both the thrust and torque curves and valid
for
the
that we
are
two ranges of P stated above.
To
able
retain
all
of the information available such
reconstruct
to
necessary,
we
functions
are
the original
Kl
and
K2
must define four additional
formed
intervals of -135
<
P
functions
<
45 degrees and 45
<
Those
functions.
by first separating Ka and Kb
P
<
225
as
on
the
degrees.
Labled:
We
Kal = Ka
-135
<
P
<
45
Ka2 = Ka
45
<
P
<
225 degrees
Kbl = Kb
-135
<
P
<
45
Kb2 = Kb
45
<
P
<
225 degrees.
degrees
degrees
then define the four functions that represent the
average
and differences of these functions on the two intervals of P.
29
= <Kal+Ka2)/2
K
1
= <Kal-Ka2)/2
K
2
K
=:
<Kbl + Kb2)/2
3
= (Kbl-Kb2)/2
K
4
Algebraic manipulation will
shcx«i
that:
-135 < ^ < 45 deg
Kl = K +K +K +K
3 4
45 < P < 225 deg
Kl = K -K +K -K
3
4
-135 < P < 45 deg
K2 = K +K -K -K
3 4
45 < P < 225 deg.
K2 = K -K -K +K
3
4
Our approximate KO) -function is now -found from:
12
12
12
12
KO) =
<K +K )/2
1
3
valid on the intervals
o-f
-135<F<45 and 45<^225 degrees.
the two functions Kl and K2 were the same and the identical
I-f
underlying function was responsible for the thrust and
torque
characteristics on each side of the 4-quadrant asymptote, then
K
,
K
,
would be zero. In
and K
3
1
surronary,
K<P)
is an average of
4
Kl and K2 which is then symmetrized for the
-^
and - torque and
thrust curves. Figure 2.9 shcxus the result for
number 9 on the interval -PI
To
the
of
.
<
PI.
square
K(F)
characteris-
function we can formulate an expansion
of
Two of the methods which are available involve the Walsh
functions
and the Fourier series.
waves are superimposed and
approximate simple sinusoids.
ful
P
determine some of the underlying fundamental
tics
K<P)
<
propeller
With the Walsh
in
functions,
the limit can be used to
They could be particularly use-
here since the K<F) function does have ^ery sharp minimums
at the 45 and -135 degree angles.
30
The Fourier series is some-
-135
<a)
15
Kb
-135
45
<b)
Figure 2.8
31
225
CM
0)
u
Cn
•rH
Cl4
32
.
more familar and
what
-functions,
Walsh
to the
the Fouries series can be used
the limit,
in
As a dual
used here.
ujas
to
approximate square waves.
By obtaining
-finally,
Table
potential
a
lists
2.3
propellers 9,
the first
15 and
this -family
-for
Fourier
-five
K<P)
convenient
point
approaches
have
propellers.
o-f
coe-f-f
zero
to begin the expansion.
equation
we
icien ts
for
The Fourier expansion is taken over
18.
function
angles the
by
model
KO)
-for
between the minimum dissipation angles.
the interval
given
Fourier series expansion
the
where
<2.24>
The
provides
expansion
defined
is
c
and
At those
a
is
by
equation
basic
modeling
<2.25)
By substituting
the K(P)
expression into the
equations (2.20) and (2.21), we can
determined
cctfnpare
the experimentally
thrust and torque curves to the model by
the equations over the range of -1
<
n
<
-•I
.
In
plotting
doing so, we
are assuming that the differences in Kl and K2 are sufficiently
to permit
small
their
cxnission.
Justification
of
this
assumption lies in the accuracy of the curves generated by the
model
equations
Figures
2.10
when compared to actual
and 2.11
shcHfi
experimental
the results for propeller
data.
number
18.
KO) =
c
aO
-»•
= SQRT<
SUM< a<k)Cos<kx)
a3fa
-»
b3«b
+
b(k)Sin<kx>
)
<2.24)
(2.25)
)
33
Figure
34
2. 10
Figure 2.11
35
Propel er Number
1
abc abc ab
9
15
5.20
6.27
-.26
.64
.64
k«2
-2.02 -1.06
2.28
k=s3
-1.25
.27
1.28
k=4
-.59
-.87
1.06
k=0
5.20
k=l
.59
18
•
6.27
10.31
.59
.87
-.23
-2.68 -1.15
2.92
-.57
-.19
.78
-.53
c
10.31
2.98
2.99
-3.27 -1.29
3.52
.60
-1.25
.12
1.26
.54
-.78
.01
.78
Table 2.3
To demonstrate the power
o-f
the curve resulting -from
the model
utilizing only the zeroth term
the Fourier series expansion
o-f
is plotted along with the curve plotted using the -first
Doing
terms.
so reveals that the di-f-ficulty
-fitting the
in
proper K -function lies essentially in the dissipation
o-f
the 4-quadrant curves.
the K(F>
In a sense,
both
indicate
the
torque
Additionally,
it
ple,
the
a-f-fects
are
lumped
is obvious that quite accurate results
KO) assumed
-for
the
For a loss
o-f
-fit
As more terms
can be improved in
power simulation
-for
most accurate results for thrust and torque
36
-for
turbine operation can
to be a constant.
expansion are added,
dissipation regions.
in
term.
ahead propeller operation and
the
mechanism
and thrust characteristics and
KO)
be achieved with
of
is expected since
What is interesting is that the
that a conwnon dissipation
together in the
normal
regions
-function represents a modulating dissipation term
the basic modeling equations.
plots
this
three
the
exam-
values
would require a large number
number
o-f
o-f
terms.
The limitation on
terms that can properly be added to the
lies in the residue
o-f
the
expansion
the Kl and K2 functions. Recalling that
KCP> was -formed -from an averaging process involving Kl and K2,
the -functions which represent their di-f-ferences were -formed to
allow retrieval
the original Kl and K2 data.
o-f
that these di-fferences degrade the K<P>
To the extent
approximation,
their
impact should properly be less than the last term added to the
expansion. Again
terms
areas
required
o-f
the Walsh -functions may reduce the number
,
particularly in the normal pump
the 4-quadrant diagram.
be used to construct the
is
-f
ami
1
n3Cn-^q3?q
turbine
and
The model equations can also
ar Kt and Kq versus J curves. Kt
given by equation <2.26) and Kq by equation <2.27)
simply q/n where R=
o-f
J
.
is
and THETA=Arctan J.
2
Kt = l./R3{cos (THETA)
(2.26)
2
Kq = l./R3ecos <THETA)
Here,
the
R
and
n
re-fer
<2.27)
to the appropriate values
-for
the
thrust and torque curves.
By
selecting
an n and solving the modeling equations
and <2.21) for q with a given
KO)
,
<2.20)
quickly
Kt and Kq can be
determined for any value of J. Figure 2.12 is a plot of the Kt
and Kq curves for propeller number 18 using the zeroth term of
the
K<^)
expansion
and the first three terms for J
Note that the scale factors used
37
in
constructing the
>
zero.
rationa-
plots
lized
lating
Kt
o-f
thrust and torque must be considered in calcu-
and Kq so that their ranges
o-f
values
are
those
normally encountered in the literature. The scale -factors used
-for
propeller
number 18 in constructing Firgure 2.12 can
deduced from Table 2.2.
38
be
Kt
.75
>^^^
"^^
^%^
1
Term
>\
.45
3 Terms
Actual
\\
.15
1.6
,8
J
<a)
lOKq
1.2
1
Term
3
Terms
.9
—
.3
1.6
.8
J
<b)
Figure
39
2. 12
CHAPTER
3.
1
As
INTRODUCTION of ADDITIONAL TERMS and
DETAILS in PROPELLER CHARACTERISTICS
3
OTHER
Pitch as a primary determinant
the variation
be discussed in Chapter 4,
will
pitch
in
expressed as a pitch-to-diameter ratio <P/D) has a significant
From
ler.
propel-
the performance characteristics of a given
on
e-f-fect
designer's standpoint,
a
addresses
Saunders C33
obtain-
pitch-to-diameter selection primarily on the basis of
ing maximum propeller efficiency,
particularly when utilizing
chart data in the initial stages of the
series
the perspective of the modeling equations however,
is more
it
convenient to see the relationship between the relative
to diameter ratio and specific speed.
descriptions
present
meter
on
From
design.
References
C
143
pitch
and
the use of the specific speed
C
15]
para-
.
Specific speed variations have a significant effect on the
quadrant
diagram not easily seen in the propellers
4-
presented
here. Since propellers are in general '^ery high specific speed
the impact of variations of that parameter is some-
machines,
what
obscure.
ions
on
The principle effect of specific speed variat-
the 4-quadrant diagram was only touched
on
in
the
plots
of
the thrust and torque characteristics discussed ear-
lier.
It
is seen primarily in the location of
and
torque
radials of Table 2.2.
speed increases,
As the
the radials tend to rotate
the zero thrust
relative
specific
counterclockwise
while the skew angle addressed in Chapter 2 tends to decrease.
40
How variations in pitch
a
a-f-fect
the specific characteristics of
propeller is best seen in an analysis of the performance of
controllable,
<CRP)
Reference
.
is quite thorough in examining quantitatively the
1161
of
pitch propellers
reversible
impact
pitch on the efficiency and thrust and torque coefficients
of a CRP propeller.
is seen in the modeling equations as a
factor in the determination of the amplitude of
the
function. An increase in the pitch-to-diameter ratio
has
dominant
K<P)
It
two noticeable effects.
tive amplitudes
results
which
the various harmonic components
of
in
First, it tends to increase the rela-
a somewhat wider
fluctuation
KO)
in
the
of
4-
quadrant curve in the principle dissipation regions. Secondly,
it
has
a
direct affect on the zeroth term
which represents the average value of the K<P) fun-
expansion
The net effect is
ction.
(corresponding
Obviously,
an
to
a
larger
shrinking of the 4-quadrant curves
torque and
thrust
coefficients).
the 'dc' value of the expansion is dominant
since
examination of Figures 2.2 and 2.3 would indicate that the
harmonic content of K<^)
cally
all
Fourier
the
of
in quadrant
I'v'
in
the dissipation regions,
is apparently proportionally
specifiequal
for
three propellers.
3.2 Number of blades and blade area
The effect of variations in numbers of blades was not examined
in detail.
tion
of
However, as pointed out in Chapter 4, the formula-
the modeling equations was quite general
41
and utilizes
what is recognized as the actuator disk theory to obtain their
initial relationships. A comparison
o-f
4-quadrant diagrams
-for
three bladed and seven bladed propellers has not indicated any
potential problems.
the
amount
blade area expressed here
o-f
Yet Figures B4(a>
ratio.
to
Mhat is really the primary determinant is
,
B4(b>
discount any major influence
,
area
expanded
as
2.2 and 2.3 would seem even
o-f
EAR on torque
and
thrust
characteristics in general.
cavitation
3.3 E-f-fects
o-f
Cavitation
is probably the most dif-ficult aspect
Rightly so,
operation with which to deal.
a lot
o-f
propeller
o-f
time
being expended to describe the mechanics of cavita-
energy is
Referen-
tion and its influence on propeller characteristics.
ces C4],
C
16]
,
£17],
quantitatively
the
ance. Reference
C
ler
to
and
16]
CIS], and
C
19]
explore qualitatively and
perform-
role of cavitation in propeller
relates the speed of advance of a propel-
the cavitation number.
By
defining
the
cavitation
number (sigma) with equation <3.1>,
sigma=2g<h-i-hA-hW/<v'a3^*^a
<3.
1)
where g is the acceleration due to gravity,
h
propeller
hA the atmospheric
head
axis beneath the water surface,
pressure,
propeller
hV
the depth of the
the vapor pressure of water
speed of advance,
it
42
is shown
and
that (using
Ua
the
typical
.
.
values) Sigma is related to Va by the relationship of equation
<3.2)
si^a= 1070/VaXVa
How the cavitation number
with Va in Knots.
is seen in Reference C4], Chapter
of
(3.2)
16,
a-f-fects Kt
Figure 108 as a fall off
Kt and efficiency as the advance ratio decreases.
appear
and Kq
It
that the effect of changes in sigma for a higher J
smaller, also seen in Reference
C
does
is
16]
Yet nowhere is the role of cavitation included directly in the
modeling equations.
Since the intention of the modeling equa-
tions is to provide a reasonable method of determining propeller characteristics at an early stage of a ship design say for
the
purposes of simulation,
perhaps an extension to
cavitation affects is unwarranted.
include
However, we can anticipate
where such affects may be present.
Equations (2.22) and (2.23) can be modified to indicate
eral
In
gen-
relationship to cavitation number shown below.
m=rg(si^a)Xq
+
Kl(P,sigma) 5^vX(n-q)
+
Rl(sigma)3(n
(3.3)
p=rg(sigma)36n
+
K20,sigma)
- R2(sigma)3«q
(3.4)
some
the presence of cavitation is handled by
cases,
3^vX(n-q)
an
effective reduction of the propeller blade area. This could be
incorporated
dissipation
into the K(P)
functions. However,
the
greater
would alter the 4-quadrant zero thrust radial s to
43
hence,
some extent,
that Rl is also a function of sigma
Notice
case.
the Rl and R2 terms are included in this
since
the
losses will be reflected from the hydraulic to mechanial side.
Although
the potential
affects,
such
of
cavitation
assumed
a step may be beyond the scope of the
equations.
model
is there for inclusion
Again,
intent of the equations is
the
a relatively simple means to simulate propeller
provide
formance for first-cut design purposes.
that goal Mill
Any extension
to
per-
beyond
increase the complexity of the equations rapid-
ly.
3.4 Hul /propel er interactions
1
Two
1
of the familar parameters which reflect
hull/propeller
<
t)
interactions are the thrust
and the wake fraction <w)
consequence
velocity
the influence
deduction
of
factor
The wake fraction appears as
.
a
of the disparity in ship forward velocity and the
as seen by the propeller.
The wake fraction is
de-
fined as:
w =
where
Kf
<»v'-Va)/V
is the ship velocity and Va the
velocity
that
the
propeller sees.
The
thrust deduction factor
the
propeller
through
results from the requirement
to produce enough thrust to
a resistance of Rt
propel
the
of
ship
while also reducing the high pres44
sure area over the ship's stern. De-fined as:
t
= <T-Rt)/T
T is the net required thrust to propel
where
the ship
at
a
given speed. Values for both -factors are usually obtained from
over
little
modeling
vary
of
the
the approach was one of considering
the
C
16]
would indicate that
wide range of ship speeds.
In
in an open water condition and
actuator disk approach essentially taken here,
and
t
terms
hence,
w were not included or necessary.
and
t
a
equations,
propeller
for
ui
Reference
tests.
model
corrections
Owing
to
the
allowances for
complicated inflow fluid geometries due to disturbances
ahead
of the propeller have been ignored.
Reference
C
16]
also addresses the pitch
dependency
deduction factor and gives general ranges of
thrust
ratios of
-^
poration
of the thrust deduction and wake fractions
detai
1
1
to -1.
of
the
for P/D
t
References C4] and C20] address the incorin
more
.
3.5 Other considerations
As
formulated,
many
of
the modeling equatins simply do
the fine details of propeller design.
however, including blade rake, skew-back,
edges,
details
hub
fairing
which
not
Most of
the requirements
45
these
the shaping of blade
and blade profiles are in large
go beyond
address
for
the
measure
initial
stages
isics
o-f
Once the 'macroscopic' character-
propeller design.
o-f
type is determined,
a generic propeller
cavitation
and
blade loading requirements
speci-fic
examined but only in the detail
or
can
o-f
be
tests
by actual model
allcMAied
in
propeller
-frcwn
the begin-
a complex computer program specializing
by
the impact
hydrodynamics.
A tacit assumption which has been incorporated
ning
is that the nondimensional
tainable
advance
-from
per unit quantity q is
,
the dimensional values
Va and in turn,
There is, hcwuever
,
a term called the axial
directly accounted
speed
o-f
the -flow rate through the propeller.
which is included in the momentum theory
not
propeller
o-f
ob-
-for
in-flow -factor a C4]
-for
propeller
the calculations.
in
It
action
has
been
assumed to be a quanti-f iabl e constant.
The
e-f-ficiency
the behavior
calculation
o-f
o-f
o-f
a given
the K<P)
propeller and its relationship
function has not been
the e-f-ficiency
o-f
a given
explored.
machine is
to
The
straight-
forward, but the qualitative relationship of the efficiency to
pitch-to-diameter
or
expanded area ratio
equations is yet to be determined.
dissipation
fran
is
lish
or
in
a
modeling
That the points of minimum
points of best efficiency can
experimental data or inferred from other
accepted,
the
be
determined
considerations
but the use of the modeling equations to estab-
'best' efficiency machine
given
specific
requirements is a topic that begs further work.
46
operating
CHAPTER
4.
PREDICTION o-f PROPELLER PERFORMANCE
MODELING EQUATIONS
4
THE
Comparison with very similar propellers
1
only a few
Ni th
the eighteen propellers available
o-f
igated in detail regarding the speci-fic
theless
o-f
USING
the -family
o-f
propellers.
peller numbers 9,
As seen by a comparison
and 18,
15,
o-f
o-f
overall. The
plots
o-f
e-f-fect
Appendix B.
o-f
to
1.0
1.6
the Kl and K2 -functions in the dissipa-
regions while slightly increasing the average
tion
EAR
-function retains the
P/D ratios
o-f
pro-
o-f
Intuitively at least, the higher EAR
0.8 to 1.1.
to smooth out
KO)
the
same general shape over the range
appears
-functions, never-
is possible to extend the results to the remainder
it
and EAR'S
KO)
invest-
magnitude
EAR is more easily seen in the raw data
In
Figure B4(b)
has its most pronounced
e-f-fect
-for
example,
at ^ equals
-i-PO
the higher
and
-90
degrees. This is also shown in Figures 2.2, 2.3, 2.4, and 2.5.
A point by point comparison will also indicate that the variation
EAR has little
in
e-f-fect
on the thrust
other
than at those two particular points
o-f
curve.
Seen more clearly in Figure B4<a),
it
Figures 2.2 through 2.5.
what
obvious
turbine region
the
4-quadrant
is rein-forced in
The expanded area ratio has as some-
larger e-ffect on the Kl curve (torque),
the normal
characteristics
o-f
particularly in
the 4-quadrant curve. This is not
Figure B4<b) where the plot
the raw data
would
indicate that EAR has essentially no e-ffect outside the
prin-
in
ciple dissipation regions.
47
o-f
The
e-f-fect
o-f
Beginning with the raw data
obvious.
see the
e-f-fect
ination
o-f
An exam-
contrary to intuition, the closer to the origin
corresponding
the higher the relative magnitudes
Ct and Cq
2.5 and 2.6 reveal
nearly
we can
Figure B3(a),
the analysis.
of P/D ratio throughout
that a given point lies,
Figures
o-f
Figures 2.2 and 2.3 show this in-fluence quite well.
Note however,
the
more
changes in pitch-to-diameter ratio is much
coe-f-f
icients.
comparison
A
o-f
o-f
that the in-fluence of P/D ratio is
the same for both the thrust and
torque
characteris-
tics. Table 2.3 points out that the various principle frequen-
cies of
the K<P)
average
'dc'
function change in amplitude as well as
The substantial variations in
term.
the
peak
the
amplitudes of the K<P) function are also evidenced in the
of Appendix B.
data
are intriguing.
B3(a)
in
The apparent critical points
raw
Figure
in
The increase in P/D ratio is reflected
the relative specific speeds of the various propellers
is
the
demonstrated
in
the gradual
and
counterclockwise rotation
zero thrust and torque radial s.
Those apparent
of
critical
points must lie along the pivot axis of that rotation but
the
reason for the correspondence isn't clear.
By examining Tables 2.2 and 2.3 and Figures 2.2 through 2.6 as
the figures of Appendix B,
well
as
with
\fery
rating
the
is possible to
reasonable accuracy the appropriate values
points
and the fundamental values of
propellers of the data base.
thrust
it
With
that
KO)
infer
of
for all
knowledge,
and torque plots could be generated and compared
Figures 2.2 and 2.3.
48
the
of
the
with
By
plotting the higher harmonic amplitudes versus the
a ccwnparison can be
harmonic amplitude,
zeroth
made between three
o-f
propellers examined in detail regarding the K<P) -function,
the
Figure
the amplitudes of
shows
4.1
propellers number 9,
would
ratios
area
suggest
particular
a
for
By extrapolating along the
and 18.
15,
for a propeller whose pitch-to-diameter
curves
second
and
-first
against the zeroth harmonic amplitude
plotted
harmonics
the
and
expanded
zeroth
harmonic
amplitude, we can approximate the values of the Fourier series
expansion terms for the first two harmonics.
As
propeller number
example,
an
has
1
a
pitch-to-diameter
ratio of 0.6 and an expanded area ratio of 0.5.
zeroth
amplitude is equal
harmonic
to four,
Assuming
the
first
two
the
harmonic amplitudes can be inferred by extending the curves to
the
value
appropriate
modeling
series
against
the
curves.
Figure
zeroth
harmonic.
with those amplitudes of
equations
Fourier
the
of
expansion
terms,
experimentally
a
Using
respective
the
comparison can
determined
thrust
the
made
be
and
4.2 shcxus the result for the positive
torque
thrust
curve.
Owing
to the influence of
zero,
some
curve.
values of
is expected in that region of
error
However,
the
the higher harmonics near J
as
K<^)
Figure 4.2 indicates,
expansion terms
yield
the
equals
thrust
even extrapolated
quite
reasonable
results
for
the
of
the
curves.
The
value of the zeroth harmonic (first term of
the
Fourier
series expansion) has a substantial
normal
pump and
49
turbine
portions
influence in
the
1
8
5
Zeroth Harmonic Amplitude
<a)
Zeroth Harmonic Amplitude
<b)
Fi
gure 4
50
.
Actual
Model
Figure 4.2
51
neighborhood
higher
order
equals
zero
equals zero and acts with
J
o-f
neglected
the
terms to produce part of the error near
values
Figure 4.2.
o-f
With more
the
study
J
the
o-f
trends in the harmonic amplitudes with pitch-to-diameter ratio
better approximations
expanded area ratio,
and
and torque curves
experimental
the
o-f
data.
error in Figure 4.2 is in the plot
The scaling which
was
produce the rationalized data depends on the values
at the point
A
o-f
to
and q
n
o-f
those
are
points,
icient y so to cause errors much like those of Figure 4.2.
1
comparison
weakness
o-f
the curves
Figure 4.2 also points
out
o-f
the sharper peaks in the curve
number
o-f
reproduce
terms
o-f
a Fourier series expansion
the experimental
simulation
quadrant
large
is required
to
data in the -fourth quadrant dissi-
pation region. A piecewise continuous approximation
-fourth
to
gener-
by the modeling equations indicates that a -fairly
ated
a
expansion methods in -forming the K(F) approxi-
The absence
mation.
o-f
the Fourier series representation as opposed
o-f
other possible
the
done
Those scale -factors
to errors in the location
sensitive
quite
su-f -f
minimum dissipation.
o-f
the thrust
similar propellers will be possible.
o-f
Another important source
o-f
o-f
would provide improved
o-f
K<F)
in
results
-for
a
requiring better correlation to experimental
during say, transient analysis.
52
data
4.2 Extrapolation to other propellers
the modeling equations are quite
As developed in this thesis,
and although they
general
cular set
o-f
three
they *r# equally Appli'
aded propellers,
b1
cable
to other types
sents
2-quadrant data
propellers as well.
o-f
-for
parti-
their establishment to a
otAie
a
-few
seven
Hecker [21] prebl
propellers
aded
including a hybrid. Plotting the experimental data in the same
as with the three bladed machines,
manner
the similarity
o-f
-form
o-f
to
.
o-f
the experimental
that generated by the modeling equations using
only
the various parameters was quite
suc-
approximate
cess-ful
earl-
the curves to those presented
ier. Although not shcxun here, a comparison
data
one is struck with
values
-for
the principle areas
Although
dissipation regions as expected,
o-f
error were
the
the comparison did not indi-
cate
any serious problems with the applicabilty
ling
equations.
A
in
more detailed comparison
the
o-f
the
o-f
moderesults
obtained from the modeling equations to much di-f-ferent propellers is yet to be done.
4-quadrant
Considering
experimental data
-for
the scarcity
a large
variety
o-f
types
and the limitations inherent in this -form
tion,
such a ccMnparison wasn't possible. However,
o-f
the equations to date would tend to en-force
-for
their wide ranging applicability.
53
o-f
complete
propeller
presenta-
o-f
the success
the
argument
4.3 Extrapolation to missino quadrants
of the advantages
One
o-f
the modeling equations lies in their
ability to reconstruct essentially the entire 4-quadrant diagram using only partial
information.
Particularly when suffic-
mini-
ient information is available to determine the point of
mum dissipation, a "universal" K<^) function can
be scaled to
fit the rationalized data and the appropriate values of
and
^v'p
ND at the point of minimum dissipation substituted to generate
the nondimensional
either
Ct and Cq values or
values of thrust and torque directly.
If
only the first quad-
rant point of minimum dissipation is known,
can
be
approximated with little loss of
particularly
model
the dimensional
the third quadrant
accuracy.
This
is
useful where first quadrant data exists but
the
from which the data was taken is no longer available.
Extending the concept slightly, a comparison of available data
on
the rating points of various propellers could form a
base
from
which the approximate rating point of
machine could be inferred.
mental
with
specific
Thus, if only ^ery limited experi-
results are available,
constructed
a
data
the 4-quadrant curve could
some confidence by comparing
the
modeling
equations results with whatever information is available.
54
be
SUMMARY
CHAPTER 5
An overview
o-f
the method
The application
o-f
Eulerian turbcwnachine theory to the
5.
1
simulation
and
ing
history.
success-ful
o-f
long
and
While the particular method used here
more recent,
somewhat
pump/turbine systems has a
model-
this thesis represents an initial
is
at-
tempt to apply the methodology to high specific speed extended
Using
machines.
o-f
the
the model
que,
general.
in
equations were able
mechanisms which contribute to
tor-
overall
losses of the system can be seen to ''reflect'
through
the
transformation of rotational mechanical
energy. The plotting of the
fluid
to
the
i.e.,
energy
and
under-
behavior
out a common dissipation mechanism for thrust and
point
the
importantly,
area
this procedure was able
dissipation mechanisms of propellers
more
Perhaps
ratios as a basis,
a -family
-for
expanded
some of the important fundamental
reveal
lying
experimental data
bladed propellers with varying
three
pitch-to-diameter
to
y/ery ccwnplete
transl at ional
to
KO) function
for different
propellers showed their strong correlation in form, a correlation which could in a more thorough examination be seen in the
relative
magnitudes of the components of the
expansions.
Certainly,
Fourier
series
the zeroth component of the expansion
represents an important parameter since a quite reasonable fit
can
be
made
over much of the four
quadrants
utilizing
it
solely in the determination of the model 4-quadrant diagram.
This
method
also introduces to the simulation
55
of
propeller
.
characteristics
a
procedure
values
di-f-ficulty with singular
reduces
o-f
avoids
characteristics
relatively
lend
they
to only two,
simple and basically
th^nselves to computer
the
o-f
the advance coefficient.
both
complete
which
of
It
are
The model equations
reasonably well behaved and quite smooth.
are
most
number of required curves to represent
the
propeller
which
straightforward.
manipulation
Thus,
little
with
dif f icul ty
From
the standpoint of the ship designer,
ship
dynamics
initial
ly
which
include a realistic
the simulation
propeller
of
the
in
stages of the design would be much easier and certain-
more cost effective.
A generic set of equations with
the
specific values of a given propeller rating point incorporated
in
them would be sufficient to provide reasonable results
the initial
steady
design stages which are oriented more toward
near
state ahead and astern operation than a thorough simu-
lation over all
is
in
determined,
quadrants.
Once the geometry of the propeller
the equations can be updated to
provide
in-
creased accuracy. That accuracy depends on the number of terms
of
the expansion one is willing to incorporate into the simu-
lation program.
necessary
test
until
Thus,
a
scale model of the propeller is
the need arises for a
self-propelled
not
model
or a detailed investigation is required into the cavita-
tion or vibration characteristics of the propeller.
56
5.2 The potential
Since
-further applications
-for
thesis represents an initial examination
this
applicability
o-f
propellers,
Eulerian turbcwnachine theory to
little has been addressed regarding a quantitative
o-f
various
propeller
types.
It
ccxnparison
was demonstrated
data
bladed
base
-for
this analysis could be applied
propellers as well but only to the extent
the
that
results obtained for the three bladed propellers which
the
the
oi
-formed
to
seven
a
quick
o-f
qualitative look for comparison purposes.
There
is a great deal
that can be done to enhance the useful-
ness of the modeling equations.
K<P)
particular, values for the
In
function could be tabulated based on the
examination
of
available propeller data for various combinations of governing
parameters.
series
For
example,
the first few terms of the Fourier
expansions could be tabulated for given
pitch-to-dia-
meter and expanded area ratio combinations and used in a table
look-up scheme such that a ship designer wishing to simulate a
given
propeller
point
and
would need only select a
combination of P/D and EAR
rating
particular
The
ratios.
computer
program could then select the necessary coefficients and
ceed with the computations.
Or,
if
a
pro-
comparative study of the
rating points of various well designed propellers used in ship
propulsion
lished
such
were to be conducted,
a
data base could be estab-
that a ship designer would need
only
generic type of propeller to begin a simulation.
select
a
The computer
would determine the appropriate rating point and corresponding
57
coe-f -f
icients to initialize the program.
The
-form
useful
the modeling equations make
o-f
applications to
-for
propellers <CRPP)
.
system
control
use-ful
reversible
controllable,
Not only are they convenient
simulations, but they could
-form
synthesis.
particularly
them
-for
pitch
conducting
the necessary -framework
system
Such a control
-for
could
a
be
automated propulsion systems particularly involving
in
CRP propel ers.
1
There
are
involving
small
other
high
harbor
propellers
craft
K^ery
applications
including
low speed screws like those used
thrust,
with
uses
potential
tugs.
or
high
The
skew
applications
rates
or
propellers are also worthy of investigation.
in
to
exotic
perhaps
hybrid
Yet another
use
involves an examination of supercavi tat ing propellers and they
modeling
for simulation.
With the promising results obtained
here, follow-on efforts could be quite rewarding.
58
Essentials
Appendix A
o-f
Eulerian Turbomachines
a 'black-box' representation,
In
machine
modeled as in Figure Al
be
can
parameters are torque (t),
velocity
<'v')
,
propeller like any turbo-
a
rate
o-f
sha-ft
and fluid flow rate <F)
where
,
fluid
an
IDEAL
rotation (N)
Ne define
.
various
the
propeller as one that, in the process of power conversion from
mechanical
rotation to axial fluid flow,
experiences no los-
ses. Such a propeller may have little connection with reality,
however the analysis of such a machine can provide qualitative
insight into the mechanics of the energy transfer
more
comfortable
process.
model would include the effects of
A
various
types of losses into a PRACTICAL propeller model diagrammed as
in
Figure A2.
The losses can be calculated by equation Al,
Pd = T^N - PMV
where
represents the power
Pd
<A1)
dissipation.
The
amount
of
varies with the advance ratio such that two mini-
dissipation
mum will occur,
one in the range of normal pump operation and
the other approximately 180 degrees away in the four
quadrant
diagram. Using the positive semi-definite dissipation function
described
by
Paynter C9],
the equation for calculating
power dissipation from the experimental
al
form)
the
data <in nondimension-
is found to be simply:
6<^) = 2ltCqVp - CtJAp
59
<A2)
"v*
Figure Al
LOSSES
<>
p
u
N
Fiqure A2
60
dimensional
same results will be obtained using either a
The
or
a nondimensional
By
plotting
angle
the dissipation
to the arctangent
o-f
J,
at which
function.
the dissipation
o-f
the losses will
angle
o-f
square root
the
equal
-formulation
versus
the
the quadratic nature
result in a nearly linear graph such that
o-f
the
the minimum dissipation occurs is easily
dis-
cernabl e.
Figure A3 shows the behavior
o-f
the advance
ingly enough,
occurs
sion
scaling
o-f
-for
the dissipation as a
normal
o-f
(Interest-
"del',
the smaller
turbine operation.) The exten-
PERFECT Eulerian turbomachine
a
o-f
comes
to eliminate the appropriate amount
-for
-function
1.
propeller number
the two minimum values
that the expression
to
icient
the region
in
to
coe-f-f
o-f
the general
o-f
proper
-from
dissipation so
dissipation -function
goes
zero at the prescribed operating point (best e-f-ficiency or
minimum dissipation). The PRACTICAL Eulerian turbomachine then
supplements
this dissipation function by an amount
equal
to
the minimum loss.
Per
as
Unit variables as used throughout this paper are
the
nondimensional
predetermined
ratio of a parameter
base value.
For a pump,
value
defined
some
to
those base values are
taken to be at the point of best efficency or point of minimum
dissipation.
obviously
the
Mhen
presenting such data in a per
reference
operating point must
reconstruct the original experiment.
61
unit
be
form,
known
to
<
u
•H
fa
62
.
per unit variables has significant
Using
they allcxM free manipulation
o-f
easily
o-f
handled numbers such as
the base values.
the
derivation
dimensional
1
the per
selection
and -1 by proper
-feel
-for
their behavior.
propeller data which provided the
basis
-for
the propeller equations was already in a
o-f
-form
o-f
Thus the per unit values give a qualita-
tive as well as quantitative
Since
-^
o-f
neighborhood
can be made to range in the
variables
First,
the applicable equations with-
Secondly, the values
out regard to dimensionality.
unit
advantages.
such that the physical
dimensions
o-f
the
non-
Minio-
vich's propellers was no longer readily apparent, the analysis
was
done in a non-dimensional -form to retain its
generality.
This may prompt one to conclude that the computational results
are somewhat arbitrary,
and D to derive all
the paper,
however one need only select
,
N,
the appropriate values. As utilized within
the per unit values are -found by:
p=P/Po
'v'p,
t=T/To
,
n=N/No
63
,
q=F/Fo
,
etc
Selected Experimentally Determined Propeller
Data Converted to ^4-Quadrant^ Form
Appendix B
data
The
by Miniovich CIO] and converted
presented
modified
advance
quadrant
form
coe-f-f
coefficient)
(torque
shown in Figure Bl.
constant
in polar form
coefficient)
utilizing
the
the
to
4-
and
Cq
convention
This is possible by recognizing that at a
thrust or torque (which the Karman-Knapp Circle dia-
utilizes),
gram
icien t -form Cll] was converted
plotting Ct (thrust
by
to
defining relations used
the
by
Baker
and
Patterson shc^n below:
Ct » T / pD2(^Jp2+N2D2)
Cq = Q / PDSCv'pS+NaDZ)
a = Arc tan
»v»p/ND
can be rewritten in per unit form as:
R =
1
/ SQRT( ABS(Ct)
)
for the thrust curve
R =
1
/ SQRT( ABS(Cq)
)
for the torque curve
and
a = Arctan q/n
and
As
a
result,
-1.5<J<-0.667
the
and
overlap
of
.667< 1/J<
1
64
.5
for both.
the
data
in
the ranges of
act as a partial
check
by
\
plotting on top
o-f
each other.
This kind of representation
o-f
the data yields in two closed curves the ccxnp-lete characterisof the propeller.
tics
method
Wylie and Streeter [223 address
this
applied to solution of transient problems using
as
An additional
digital computer.
a
advantage is the avoidance of
the problems with singularities where ever propeller speed
of
advance or shaft rpm approaches zero. See also Reference [233.
The curves are
speed,
fic
axial
describes
also
easily recognizible as typical for high speciflow pumps.
See Reference C24] where Knapp
the 4-quadrant representation
pumps
of
and
and outlines a procedure by which the representation
turbines
could be used to deduce the transient behavior. Figure B2 is a
sketch
of
the
flow vs speed plane
indicating
various
the
regions of operation. The zero head and torque radial s are the
asymptotes to which the thrust and torque curves approach
for
a given machine.
Two series are shown here.
of
Figures B3<a) and B3(b) are
thrust and torque for propellers 1,3,
and
B4<b)
15.
The
plots
and 6. Figures B4<a)
are thrust and torque curves for propellers 3,9 and
plots
reveal
the effect of variation
diameter ratio and expanded area ratio.
65
in
pitch-to-
Figure Bl
II
<+)
Discharge
.Dissipation
DISSIPATION
NORMAL
PUMP
(
->
Speed
<+)
DISSIPATION
DISSIPATIOr^'^
Zero Head
Zero Torque
NORMAL
TURBINE
III
Speed
<-)Discharge
Figure 02
66
l\>
Figure BSa
67
Figure B3b
<68
Figure B4a
69
Figure B4b
70
Appendix C
Paynter
C9]
The Bond Graph Representation
presents the canonical bond graph
representation
for an Eulerian turbomachine as:
GY
RG
Rq
Rui
It
is also represented as a modulated qyrator series adjoined
to a shunt resistor:
Rs
In
the -first case,
the system equations are easily -found
to
be:
mssRgJfq-t-RwJ^n
<DI)
p=Rg3fn-Rq3Cq
<D2>
using the pump sign convention.
71
In
the equations are -found to be
the second case,
m = Rg
X
q + Rs X
<n-q)
<D3)
p — Rg
3C
n
- Rs
<n-q)
<D4)
By converting <D3)
3f
to canonical
and <D4)
it
-form,
is
evident
v where
that Rs can be expressed as K<P)
3C
V = SORT
)
<D5)
qXq)/v3{v
<D6)
(
nJfn
+
qXq
and
Rg = R6
A great deal
Rs
3«
<n3^n
+
simplification is
a-f-forded
by making the simple approximation
quality
KJ^v
o-f
+
with little loss in
o-f
where K absorbs the individual character
Rg =
o-f
a
n
and Rs =
particular
machine.
As a more widely applicable model,
be
general
dissipations could
adjoined to the l-junctions to account for -fluid and mech-
anical
losses, i.e.
Rdq
Rdn
72
The equations in matrix form:
m
A
Rs+Rdn
Rg-Rs
<D7)
Rg-Rs
Rs-Rdq
<D8)
more complete model Mould leave the dissipation elements as
•functions
individual
of
the advance coefficient to absorb part
behavior of a machine and smooth out the
of
the
require-
ments in the Rs term to fit experimental data.
The
tion
reader interested in pursuing the bond graph
is
representa-
directed to Reference C25] for an 'entry level'
proach to bond graphs and to Reference C26] for a more
ced treatment.
73
ap-
advan-
LIST
o-f
REFERENCES
"A Design Theory -for Superand Leopo1d,R.,
CI] Kerwin,J.E.,
72, 1964
cavitating Propellers', Transactions SN(ii#1E, 'v'ol
.
"Numerical Prediction o-f Propeller Characteristics", Journal of Ship Research, SNAME Mar. 1973
[23 Cunwnings,D.E.
,
Vol.
"Hydrodynamics in Ship Design
C3] Saunders,H.E.
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,
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o-f
Engineering
Thesis
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Wilken
An algebraic propeller model using
Eulerian turbomachine
theory.
thesW58823
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3 2768 001 95812 7
DUDLEY KNOX LIBRARY
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