A Fully Numerical Lifting Line Method for the
Design of Heavily Loaded Marine Propellers with
Rake and Skew
MASSACHUSETTS INSTTUTE
OFT"E0HNOLOLGY
by
Giovani Diniz
JUL 3 0 2015
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
LIBRARIES
Master of Science in Naval Architecture and Marine Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
@ Massachusetts Institute of Technology 2015. All rights reserved.
Author ......
Signature redacted
e Ament of Mechanical Engineering
May 8, 2015
:5
A
Certified by...
Signature redacted
(
Stefano Brizzolara
Research Scientist and Lecturer
Assistant Director for Research MIT Sea Grant
Thesis SuDervisor
-
redacted
Signature
- -- -- -
A ccepted by ..........-
David E. Hardt
Chairman, Departmental Committee on Graduate Students
Department of Mechanical Engineering
..-. .-s..--
-
2
A Fully Numerical Lifting Line Method for the Design of
Heavily Loaded Marine Propellers with Rake and Skew
by
Giovani Diniz
Submitted to the Department of Mechanical Engineering
on May 8, 2015, in partial fulfillment of the
requirements for the degree of
Master of Science in Naval Architecture and Marine Engineering
Abstract
This thesis aims to give a contribution to the design of heavily loaded marine propellers by numerical methods. In this work, a wake-adapted, fully numerical, lifting
line model is used to obtain the optimum circulation distribution along the propeller's
blade via variational method, presented by Coney [9]. In this context, two approaches
to the representation of the wake field are compared: the first approach utilizes Betz's
condition for moderately loaded propellers, in which the wake is aligned with the hydrodynamic pitch angle. The second approach, in which the wake is aligned with the
local velocities, utilizes Kutta's Law to create a zero-lift wake surface. A thorough
comparison of the influence of the effect of tip vortex roll-up is done. A lifting surface
method with fully aligned wake is developed and used to correct the optimum distribution of pitch and camber obtained by the new lifting line method. The resulting
geometries geometries, operating under heavily-loaded conditions, are submitted to
a preliminary analysis in a boundary element-based potential flow code to verify the
consistency of the results. This analysis confirms the better results obtained with the
fully numerical lifting line model and the variations between the approaches in terms
of circulation and pitch angle observed in the lifting line results are verified. Finally,
the performance of propeller geometries generated with the approaches studied in this
work are compared by high fidelity RANSE analysis. The CFD simulations confirm
the higher accuracy of the method in which the wake geometry is aligned with the
local velocities in terms of fulfillment of thrust requirement.
Thesis Supervisor: Stefano Brizzolara
Title: Research Scientist and Lecturer
Assistant Director for Research MIT Sea Grant
3
4
Acknowledgments
First and above all, I would like to thank wife, Sara, for being my best friend and
source of inspiration and motivation and staying by my side through it all.
I would like to thank my dear friend, Guy, for reminding me the value of true
friendship and for being my confident through these years.
Our lunch breaks are
deeply missed.
To my advisor, Professor Stefano Brizzolara, I am deeply thankful for the opportunity to work in his laboratory, working to provide me the necessary funding to
support my studies and for the endless patience and will to help, especially during
the most difficult stages of this work.
Professor Henry Marcus, advisor of the MIT International Shipping Club, was
a great professional and personal mentor who I greatly admire and thank for the
countless advices throughout these two years.
I, finally, would like to thank Professor Hermano Krebs, for a warm welcome to
MIT and his invaluable advices on how to navigate in that brave new world that MIT
was two years ago.
5
6
Contents
1
Introduction
15
2
Circulation Optimization
19
3
4
2.1
Propeller Lifting Line Model . . . . . . . . . . . . . . . . . . . . . . .
19
2.2
Induction Velocity Functions . . . . . . . . . . . . . . . . . . . . . . .
21
2.2.1
HPA Induction Functions
. . . . . . . . . . . . . . . . . . . .
21
2.2.2
LVA Induction Functions . . . . . . . . . . . . . . . . . . . . .
23
2.3
Induced Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.4
Variational Optimization . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.5
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.5.1
Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.5.2
H ub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.5.3
Induced Velocities . . . . . . . . . . . . . . . . . . . . . . . . .
30
Wake Geometry
33
3.1
HPA Wake Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.2
LVA Wake Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
3.3
H ub
38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chord and Thickness
41
4.1
Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
4.2
Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4.3
Methodology
43
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . .
.
4.4
43
45
5 Lifting Line Results
Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
5.2
Induced Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
5.3
Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
5.4
Hub Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
5.5
Rake and Skew . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
.
.
.
.
.
5.1
53
7
Lifting Surface Corrections
55
9
. . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
55
7.2
Results and Discussion
. . . . . . . . . . . . . . . . . . . . . . . .
58
.
.
Methodology
61
Results
Loading
61
8.2
Hub Radius
61
8.3
Rake . . .
63
8.4
Skew . . .
65
.
8.1
.
8
7.1
.
Blade Detail Geometry Definition
.
6
67
Verification
10 Validation
71
. . . . . . . . . . . . . . . . . . . . . . .
72
10.2 Heavily-Loaded Condition
. . . . . . . . . . . . . . . . . . . . . . .
73
.
.
10.1 Lightly-Loaded Condition
11 Conclusions
81
8
List of Figures
Design Methodology: both HPA and LVA approaches .
. . . . . .
17
2-1
Lifting line model. Bound and trailing vortices . . . . .
. . . . . .
20
2-2
Induced velocities on the lifting line
. . . . . . . . . .
. . . . . .
20
2-3
LVA scheme for the calculation of the induction function with Biot-
.
.
.
.
1-1
23
2-4
HPA discretization of lifting line . . . . . . . . . . . . .
29
2-5
HPA discretization of lifting line . . . . . . . . . . . . .
30
3-1
HPA model, shape of the trailed (fixed) vortex wake - CT = 1, J = 0.80,
.
.
.
Savart law .. . . . . . . . . . . . . . . . . . . . . . . . .
34
3-2
Local velocity alignment scheme . . . . . . . . . . . . . . . . . . . .
36
3-3
LVA model - CT
1, J = 0.80, wo = 0.1471 and KT = 0.253 . . . .
37
3-4
Comparison of wake cross-sections for HPA and LVA cases - CT
.
.
=
=
=
1,
0.80, wo = 0.1471 and KT = 0.253 . . . . . . . . . . . . . . . .
LVA model wake contraction -
.
J
3-5
.
. . . . . . . . . . . . . . . . . . . . .
wo = 0.1471 and KT = 0.253
CT =
1, J = 0.80, wo = 0.1471 and
.
K T = 0.253 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-1
Chord and thickness distributions - CT = 1, J
=
38
0.80, KT = 0.253
.
and wo = 0.1471 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
Chord and thickness distributions - CT = 0.5, J = 1.13, KT = 0.253
and wo = 0.1471 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
4-2
38
9
44
5-1
Comparison of convergence of the optimum circulation searching algorithm between different design cases (with and without rake and skew)
and solution methods (LVA/HPA).
5-2
. . . . . . . . . . . . . . . . . . .
Comparison of induced velocities with CT = 1, J = 0.80, KT = 0.253
and wo = 0.1471 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-3
48
Comparison of optimum circulation and hydrodynamic angles /3 calculated by the two methods (HPA/LVA) at increasing loading
5-4
47
. . . .
49
Comparison of results for different hub radii: optimum circulation dis3 , right
tribution (G, left graphs) and hydrodynamic pitch angle (#
graphs ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-5
51
Optimum circulation distribution - LVA method for cases with rake
and HPA/LVA no rake - CT = 1, J = 0.80, KT = 0.253 and wo = 0.1471 52
5-6
Optimum circulation distribution - LVA method for cases with skew
and HPA/LVA no rake - CT = 1, J = 0.80, KT
6-1
Blade outlines for HPA and LVA cases with CT
0.253 and wo = 0.1471
0.253 and wo = 0.1471 52
1, J = 0.80, KT =
. . . . . . . . . . . . . . . . . . . . . . . . . .
54
7-1
Lifting surface model . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
7-2
Lifting surface geometry - CT = 1, J = 0.80, KT = 0.253 and wo =
0.1471 H PA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-3
57
Comparison of case with and without lifting surface corrections for
CT = 1, J = 0.80, KT = 0.253 and wo = 0.1471 for both HPA and LVA 58
7-4
Lifting surface corrections for camber for CT = 0.5, J = 1.13, KT =
0.253 and wo = 0.1471 HPA and LVA cases . . . . . . . . . . . . . . .
7-5
Lifting surface corrections for angle of attack for CT = 1, J = 0.80,
KT = 0.253 and wo = 0.1471 HPA and LVA cases . . . . . . . . . . .
7-6
Lifting surface corrections for angle of attack for HPA cases with CT
1, J = 0.80, KT
7-7
59
=
=
0.253 and wo = 0.1471, and various hub radii . . .
Lifting surface corrections for camber for LVA cases with CT
J = 0.80, KT = 0.253 and wo = 0.1471, and various hub radii
10
=
59
60
1,
. . . .
60
8-1
CT = 0.2, J = 1.80, KT = 0.253 and wo = 0.1471
62
8-2
CT = 0.5, J = 1.13, KT = 0.253 and wo = 0.1471
62
8-3
CT = 1, J = 0.80, KT = 0.253 and wo
8-4
Blade outlines for HPA cases with CT
=
.
. . . . . . . .
.
. . . . . . . . .
64
Blade outlines for LVA cases with CT = 1, J = 0.80, KT = 0.253 and
.
wo = 0.1471, with and without rake . . . . . . . .
8-7
63
0.80, KT = 0.253 and
.
1, J
. . . . . . . . .
. . . . . . . . .
.
8-6
=
63
1, J = 0.80, KT = 0.253 and
. . . . . . . .
Blade outlines for LVA cases with CT
wo = 0.1471, and various hub radii
. . . . . . . . .
.
8-5
=
0.1471
.
wo = 0.1471, and various hub radii
=
64
Blade outlines for LVA cases with CT = 1, J = 0.80, KT = 0.253 and
65
9-1
Panel method results for lifting line
69
9-2
Pressure coefficient distribution at different radii for the two propellers
.
wo = 0.1471, with and without skew . . . . . . . .
.
designed with HPA/LVA methods . . . . . . . . . . . . . . . . . . .
9-3
Comparison of optimum circulation distribution between propellers
with and without lifting surface corrections for the two HPA/LVA approaches .....
................
............
....
10-1 Pressure distribution for a lightly-loaded propeller - CT = 0.4, J =
.
. . . . . . . . . . . . . . . . . .
10-2 Propeller geometries obtained with the HPA and LVA approaches
.
.
1.26, KT = 0.253 and wo = 0.1471
10-3 Mesh of elements for CFD analysis for HPA case using approximately
.
1.8 million cells in computational domain . . . . . . . . . . . . . . .
10-4 Convergence of simulation for the HPA and LVA geometries.....
. . . . . . . . . . . . .
.
10-5 Pressure distribution for the HPA geometry
.
10-6 Pressure distribution for the LVA geometry . . . . . . . . . . . . . .
10-7 Helicity iso-surface (40m/s2 ) of the blades of the propellers designed
.
with the HPA and LVA approaches . . . . . . . . . . . . . . . . . .
. . . . . . . . . . .
.
10-8 Pressure field at leading edge of HPA geometry
.
10-9 Pressure field at leading edge of LVA geometry . . . . . . . . . . . .
11
10-10Pressure coefficients at blade sections at radii 0.3R and 0.8R, for the
two propellers HPA/LVA . . . . . . . . . . . . . . . . . . . . . . . . .
12
79
List of Tables
Comparison of results . . . . . . . . . . . . . . . . . . . . . . . . . .
.
9.1
68
10.1 Comparison of results from RANS analysis - CT = 0.4, J = 1.26,
. . . . . . . . . . . . . . . . . . . . .
.
KT = 0.253 and wo = 0.1471
10.2 Comparison of results from RANS analysis - CT
1, J = 0.80, KT =
. . . . . . . . . . . . . . . . . . . . . . . . .
.
0.253 and wo = 0.1471
=
72
74
13
I
14
MAP-
Chapter 1
Introduction
The lifting line model was first devised for the wing problem by Prandtl [19] and
later extended to the case of the marine propeller by Betz [2].
In his work, Betz
showed the condition for the optimum distribution of circulation along the lifting line
which Sparenberg [20] later proved to be a general result in the case of lightly loaded
propellers in inviscid, uniform flow.
Lerbs [16] extended the theory for the case of non-uniform inflow and for moderatelyloaded conditions, making use of the velocity potentials presented by Kawada [13].
Morgan [17] and Eckhardt and Morgan [11] contributed on the improvement of the
calculations of the optimum distribution of circulation and Wrench [22] introduced
asymptotic formulae for fast and accurate calculation of induced velocities.
Recent developments have continued to contribute to the improvement of lifting
line based models. Brizzolara et. al [5] revisited Lerb's original theory for contrarotating propellers incorporating modern propeller design techniques. Brizzolara [3]
explored the effects of tip-vortex roll-up and wake contraction for the analysis problem
of a propeller using a boundary element method approach.
The significance of the effects of tip-vortex roll-up and wake contraction, especially
at highly-loaded conditions, has also been explored in the works of Brizzolara et al.
[4], with attention to hub effects, and by Aran and Kinnas [1], with regards to the
adequacy of Betz's condition in the design of heavily-loaded propellers. These works
demonstrate that the shape of the optimum circulation obtained with local velocity
15
alignment may significantly differ from that obtained using the Lerbs (1952)-Wrench
(1965) method.
In the present work, a variational method, such as that presented by Coney [9],
is used to obtain the optimum circulation distribution on the propeller blade, and an
approach to the design of marine propellers is investigated focusing on the alignment
of the wake with local velocities in order to capture the effects of tip-vortex rollup and wake contraction into the optimization algorithm.
The effects of rake and
skew are also considered in this method in which induced velocities are calculated
by direct application of Biot-Savart's Law and chord and thickness distribution are
optimized using combined strength and cavitation criterion with application of the
method presented by Brizzolara et. al. [5].
The effects of rake, skew, loading and hub radius are tested and compared against
results obtained with models based on Betz's condition. Hence, in this study, two
approaches to model the lifting line are presented: the first, based on Betz's condition,
assumes a rigid alignment of the wake with the hydrodynamic pitch angle and will be
referred to as Hydrodynamic Pitch Angle Alignment approach, or HPA; the second
approach, based on the implication of Kutta's Law for a non-lifting surface, makes
sure that the wake is aligned with the local velocities. This approach will be referred
to as Local Velocity Alignment approach, or LVA.
The method developed and systematically tested in this work is that of the diagram of figure 1-1, in which the optimization of circulation is achieved through
an iterative procedure, followed by the optimization of the chord and thickness distributions and the alignment of the wake. Upon convergence, the blade geometry
characteristics is defined and lifting surface corrections are calculated and applied to
the initial geometry to obtain the final blade shape.
Present study focuses on the comparison between the results obtained using the
HPA and LVA approaches.
The former follows the current state of the art in ma-
rine propellers design by numerical methods, while the latter, features the proposed
contribution towards an improved lifting line - lifting surface design method.
These improvements aimed to extend the current methods' capabilities to ac-
16
curately design marine propellers under heavily-loaded conditions and to take into
account for the effects of rake and skew in early design stages.
Lifting Uine
FVh
and
Geo
Lake
etry
Calculate
Velocites
Induced
Indu fielocities
at
Pa Optimize Circulation
Distribution
ake
Optimize Chord
and Thickness
Distributions
Covrgence
Criteria
Definition
Surface
CLifting
Corrections
_
Propeller
Geometry
Figure 1-1: Design Methodology: both HPA and LVA approaches
The geometries obtained with both approaches are then analyzed with a boundary
element method to verify the consistency of the results and their performance are
also validated through high fidelity numerical simulations by means of a state of the
art Reynolds Averaged Navier-Stokes (RANSE) equation solver to verify compliance
with design criteria.
17
18
Chapter 2
Circulation Optimization
2.1
Propeller Lifting Line Model
The idealized lifting line model of a screw propeller represents the blades as radial lines of distributed bound vortices which provide the total required thrust and
prescribe the pitch angle of each section. In the context of inviscid, incompressible,
potential flow theory, Kelvin's Theorem is satisfied by including a series of trailed
(free) vortex lines shed from the lifting line and extending to infinity downstream of
the propeller.
In the continuous case, at each radial position r of the lifting lines, the bound
vorticity has strength F(r), and the vorticity of the shed vortex lines, Ff(r), must
satisfy 2.1.
rf (r) = dF(r)d
dr
(2.1)
In a right-handed cartesian coordinate system, with positive values of the X-axis
increasing upstream of the propeller and positive values of the Z-axis increasing towards the free-surface. Passing from the continuous theoretical model to an approximate discrete numerical model, a discreet model is idealized by a set of M discrete
vortex segments on the lifting line, as in figure 2-1 (for one single blade of the model).
Each horseshoe element, m, positioned at radius r(m), is composed of a bound
19
vortex line with piece-wise constant strength of F(m) and an upper and a lower free
vortex line shed from the edges of the bound vortex lines, with constant strength
ff(m), given by 2.2.
m
(m + 1) - IF(m)
17 (m) =
1 ,2,...,M
(2.2)
Figure 2-1: Lifting line model. Bound and trailing vortices
The lifting line segments and the free vortex lines shed from them are subject to
inflow axial velocity, Va(r), and tangential velocity, wr + Vt(r). The inflow is perturbed
by the self induced axial velocities, u,(r), and self-induced tangential velocities, ut(r),
composing the generalized velocity profile experienced by each section of the blades
shown in the diagram in figure 2-2 below.
ua
Ut
D
V
L
Vinflw
Va
T
Ft
r + Vt
r
Figure 2-2: Induced velocities on the lifting line
20
Each section experiences the generation of lift forces, L, perpendicular to the total
inflow velocities, V, and drag forces, D, parallel, to the inflow velocities. These forces
combined are projected to the global coordinate system to define the thrust, T, given
by each section, and the tangential force, F, that, multiplied by the radius, yields
the torque,
Q,
consumed by that section.
The inflow velocities prescribe the uncorrected hydrodynamic pitch angle, 3 (defined as the ratio between the inflow velocities), and the corrected hydrodynamic pitch
angle,
i (defined as the ratio between the inflow velocities and induced velocities),
as in 2.3 and 2.4, respectively.
tan /3(r)
tan Oi(r) =
2.2
Va(r)
wr + Vt(r)
Va(r) +una(r)
Va (r) + 1(r)
Wr + Ut (r) + V (r)
(2.3)
(2.4)
Induction Velocity Functions
In this work, the induction functions are defined as the velocities induced by a
vortex line of unit strength and are used in the variational optimization to obtain the
optimum circulation distribution.
The real induced velocities are the velocities induced by a vortex line with a given
circulation strength coming out as a result of the optimization procedure.
2.2.1
HPA Induction Functions
The calculation of induction functions for the hydrodynamic pitch alignment approach was made through the use of Wrench's [22] formulae, for the axial component,
ia(m, i), and tangential component, at (m, i), of the velocity induced at the lifting
line collocation point i, positioned at r(i), by a unit strength trailing vortex line m,
shed from the bound vortex located at radius r(m).
21
These velocity components are defined as 2.5 and 2.6, in the case where the collocation point's radius on the lifting line are smaller than the radius of the point which
the helicoidal free vortex line is detaching from.
ia(m, i) =
Z
4irre
6, (rn, i) =
(y - 2.z.y.yo.F1)
(2.5)
2Z2(yo.F1)
(2.6)
27rre
In the case where the collocation point's radius on the lifting line are greater than
the radius of the point which the helicoidal free vortex line is detaching from are
calculated by means of equations 2.7 and 2.8.
Z2
Ua(m,i) =
dt (m, i) =
27rr,
(2.7)
(y.yo.F2 )
(1 + 2.z.yo.F2)
47rr,
(2.8)
Where, F and F2 are functions defined as in equations 2.9 and 2.10.
F 1 =-
F
1
2zyo
F2=ui2zyo
i
I1
U
+
U
U- I
2. (
-a2
+
+
U_
1
U_
U -)]
(2.9)
(2.10)
In which, ai, a2 , U, y and yo are defined as:
a1 =
a2 =
I
24z
9y
y
(2.11)
1+hy 2
2
+2
(1 + y2)1.5
22
3y2-2
2
(1 + y2)1.5
(2.12)
U
=
YO(---y2-1
( +-y2,
_ 1)
exp
1+ y2 -
+ y02
trc
(2.14)
1
(2.15)
r, tan /3
Yo
2.2.2
(2.13)
tan 3j
LVA Induction Functions
Consider a single vortex line shed downstream of the propeller, as in figure 2-3.
The induction functions for the local velocity alignment approach are calculated by
direct application of Biot-Savart's Law, in which the velocity induced at the lifting
line collocation point i, positioned at r(i), by a unit strength trailing vortex line m,
shed from the bound vortex located at radius r(m), are defined as 2.16.
y
Figure 2-3: LVA scheme for the calculation of the induction function with Biot-Savart
law.
n(m, i) = -I Idl x S
23
(2.16)
The calculation of the induction functions via Biot Savart's Law is quintessential for
the purpose of this work in terms of including the effects of wake alignment, rake and
skew into the circulation optimization algorithm. The asymptotic formulae proposed
by Wrench [22] offer the convenience of fast computational time and accuracy in the
lightly-loaded case, but inherently neglect the alterations of the wake geometry under
heavily-loaded conditions and the induced velocities of the lifting line on itself.
This numerical way of calculating the self-induced velocities is, perhaps, the most
important element contributing to the differences (presented in the next sections) in
circulation and hydrodynamic pitch angle due to rake and skew of the propeller blade.
2.3
Induced Velocities
Induction functions are used as input parameters to the optimization algorithm,
as described in the following section, to obtain the circulation distribution along the
radius of the blade.
Once this is done, the induced velocities at the lifting line collocation point i,
positioned at r(i), due to all trailing vortex lines shed from the bound vortices are
calculated by means of equations 2.17 and 2.18 for both HPA and LVA cases.
M
Ua(i)
=
F(m)1Ta(m, i)
(2.17)
i)
(2.18)
m=1
M
ut(i) = )
F(m)'t(m,
m=1
The total inflow velocities, V(m), at each radius, r(m), are thus the vector sum of
all axial and tangential velocities experienced by the section. Its modulus is, hence,
defined by equation 2.19.
V(m) =
[Va(m) + ua(m)]2 + [Vt(m) + wrc(m) + ut(m)]2
24
(2.19)
2.4
Variational Optimization
In this study, the variation method proposed by Coney [9] is used to find the
optimum circulation distribution on the lifting line in which the propeller's thrust
and torque are calculated from the elemental lift force, L, calculated on each bound
vorticity segment by the Kutta's 2.20.
L = pIP x V
(2.20)
Developed thrust, T, and absorbed torque, Q, are obtained by numerical integration
of the lift forces generated at each section, as defined in equations 2.21 and 2.22,
obtained when decomposing the expression for V into its components.
M
T
=
pZ E
[(Vt(m) + wrc(m) + ut(m))] I7(m)Ar
(2.21)
M=1
M
2V (Va(m)
- pZ
+ ua m)) c(m)Cdv(m)Ar]
M=1-
M
Q
[(V1(m) + wrc(m) + ut(m))] ]F(m)r(m)Ar
pZ
(2.22)
m=1
M
[2 V (Va(m)
M=1
+ ua(m)) c(m)Cd(m)r(m)Ar
-
-
- pZ
In accordance with the calculus of variations, an auxiliary function, H, is defined,
as in 2.23, by adding to the torque to be minimized the adjoint term which results
from the multiplication of the so called Lagrange's multiplier A times the difference
between the calculated thrust T and the required value T,.
H = Q + A(T - T)
25
(2.23)
Taking the expressions of the partial derivatives of H, with respect to the unknowns
Fi and Lagrange multiplier, one obtains the system of equations that permits to find
the optimum circulation distribution to maximize efficiency (minimizing torque) while
respecting the constraint of the required thrust.
OH
=H0
ari
=Va(i).r(i)Ar
M
+ 1
[F(m) ii (i, m)r(m)Ar + F(m)Pa(m, i)r(i)Ar]
M=1
(2.24)
+ A [Vt (i) + wr(i)] Ar
M
[F(m)'it(i, m)Ar + F(m)it (m, i)Ar],
+ A1
1
m=
for i = 1, 2, ... , M
OH
OA
0
Tr
(2.25)
M[M
+ pz E
Vt(m) + wr(m) + I3 F(n)t (n, 0)Ar F(m)Ar
m=1
.
n=1
The unknowns in 2.24 and 2.25 are the M strength values of the bound vortices,
1(m), plus the scalar value of the Lagrange multiplier, A, composing a non-linear
system of equations to be solved.
2.24 and 2.25 are non-linear system which is linearized by rewriting these expressions assuming an initial value for the Lagrange multiplier, A, and the tangential
induced velocities, u*. Then the expressions for the partial derivatives of the auxiliary
function H become 2.26 and 2.27.
=
Va(i).r(i)Ar
M
[F(m)ia(i, m)r(m)Ar + F(m)Pa(m, i)r(i)Ar]
+
m=1
(2.26)
+ A [Vt(i) + wr(i')] Ar
+
I
[F(m)ft(i, m)Ar + r(m)lit(m, i)Ar]
,
0
M=1
for i =1, 2,..., M
26
M
T, = pz 13 [Vt(m) + Wr(m) + ztAr + u*(m)] F(m)Ar
(2.27)
M=1
Viscous drag is considered in the optimization algorithm. This force is defined as
in equation 2.28, in which, c(r) and CD,(r) are the chords and two-dimensional drag
coefficient of the sections, respectively.
1
2
One can then substitute T for (T - F,) in 2.23 and carry on the partial derivatives
to account for a deduction in thrust, making 2.27 become 2.29.
M
Tr
pz E3 [V,(m) + wr(m) + ftAr + u* (m)] F(m)Ar
m=1 M(2.29)
+ Ipz E
VW*(m) [Va(m) + u*(m) ] c(m)Cd(m)Ar
M=1
The solution of this linear system (i.e. the values of ri and A) is used to obtain
the new values of u* by equation 2.18 and to update the value of A.
This step is
repeated typically for not more than 6 iterations to reach convergence.
Once the values of A and u* are calculated, the system is recalculated, and so
forth, until convergence.
During these iterations, however, the wake is kept frozen and no changes are made
to the induction functions. So, it is necessary to perform an outer iteration algorithm
that, starting from the initial wake geometry, converges to the final aligned wake,
which gives the final induced velocities.
This procedure is important to ensure the convergence of the method, especially at
higher loading conditions (i.e. CT >1.0), and essential to avoid tip loaded solutions,
as argued by Coney [9].
2.5
Results and Discussion
The operation conditions for all cases presented hereinafter, are given in form of the
thrust coefficient, CT, the advance ratio, J, mean wake coefficient, wo, and propeller
27
thrust coefficient, KT, defined in 2.30 - 2.32, respectively, where R is the propeller's
radius and VA = Vs(1 - wo).
p=R
C
o
pr R2 y2
(2.30)
VA
J = N.D
KT =
(2.31)
Tr
TD
pN2D4
(2.32)
The application cases considered in this work regard a 5-bladed, z
=
5, with a
diameter D = 5[m].
2.5.1
Discretization
The discretization of the lifting line and the wake trailing vortex lines is assumed
to be uniform, as all Ar in the formulae are not functions of the radius.
Figure 2-4 (CT = 1, J = 0.80, KT = 0.253 and wo = 0.1471) shows the results
obtained from a convergence study on the number of elements.
The distribution
of the optimum circulation and of the hydrodynamic angle, 1i along the radius for
M = 15, 20, 25 where there is virtually no difference between the results of the
method, and, therefore, unless otherwise mentioned, M = 15 elements will be used
in all results presented in the rest of the study.
In spite of the equivalence of the formulae presented here with the case of a
cosine spacing discretization, showed by Coney [9], the cosine spacing applied in
this work caused erratic behavior of the numerical method, significantly affecting the
convergence and accuracy of the results, which was not experienced otherwise.
This problem was related to sudden jumps in values of induction functions, for
both HPA and LVA approaches, due to the reduced distances between trailing vortex
lines at the tip.
In order to use a cosine distribution, the expression of Biot-Savart kernel, used
to calculate the induction functions for the LVA approach, have to be changed to
28
0.016
Circulation
,75,
Hydrodynamic
-e-- M = 15
-~M=20
e-
0.014
Pitch Angle
-e-M = 15
70
M = 25
M=
25
66
0.012
60
55
00
45
0.008
40
-
0.006
35
0.2
04
03
05
06
07
0.2
09
08
03
04
05
0
07
0.8
09
1
Figure 2-4: HPA discretization of lifting line
a Rosenhead-Moore kernel (i.e., replacing 1/(IS11)
3
by 1/(JII112+62) , in equation
2.16). This workaround solved the convergence problem but introduced an additional
parameter 3 on which the circulation results are highly dependent.
Therefore, the uniform spacing of lifting line and wake trailing vortices was adopted
through this work, and the results of the method are restricted to a linear spacing on
the lifting line but also highly insensitive to the number of elements, M, utilized to
discretize it.
2.5.2
Hub
The presence of the hub incurs in additional terms to be considered in the optimization algorithm, as indicated in the work of Coney [9]. On one hand, the induction
functions must account for the influence of the image of lifting and trailing vortex
lines, and on the other hand, the expression of forces must account for an additional
hub drag caused by and dependent on the finite circulation at the root of the blade.
The induction functions are calculated using the same formulae utilized for the
wake trailing vortices, 2.5-2.8, for the HPA approach, and 2.16, for the LVA. Hence,
the induced velocities, including the hub, become 2.33 and 2.34.
M
Ua(i) =
, r(m) (fa (m, i) + 9a(ma, i))
m=1
29
(2.33)
M
Ut(j)
=
(2.34)
F(m) (it (m, i) + ft(m
E
m=1
According to Wang [21], a Rankine vortex structure is the appropriate model to
represent the hub. The integration of the pressure field due to the Rankine vortex
structure yields to the additional hub drag, 2.35, proportional to the square of the
circulation at the hub and a function of the hub vortex core radius, ro.
Fh=
167r
(2.35)
log-+ 31 F
ro
If the hub is to be considered in the optimization, equation 2.35 must be included
in 2.21 of thrust for the optimization algorithm, in similar manner as the viscous drag
on each blade has been previously considered.
2.5.3
Induced Velocities
In order to validate the induction functions calculated with the Biot-Savart Law,
defined in 2.16, a hydrodynamic pitch aligned wake (HPA wake geometry) was utilized
to calculate the circulation distributions using Wrench's asymptotic formulae and the
direct application of Biot-Savart's Law.
Circulation
HP
0.015
0 014
iot
0
var-
Hydrodynamic
Pitch Angle
e- HPA-wrenci,
HPA-B t Savart
HPA~~fl~
70 -A
-
-
0013
65
--
0
-
0012
0011
-
0.01
45
-
0009
40
0 008
-
35
0007
0 006
02
-
13
3
03
04
05
06
07
08
09
1
02
03
04
05
06
Figure 2-5: HPA discretization of lifting line
30
07
08
09
As shown in figure 2-5, the results are exactly the same for a propeller with CT = 1,
J = 0.80, KT = 0.253 and wo = 0.1471. So it is concluded that in the HPA approach
the calculation of the induction functions are interchangeable.
Although the method to calculate the induction functions in the HPA cases does
not influence the results, all HPA cases will use Wrench's asymptotic formulae in
this work, unless otherwise mentioned, in order to maintain consistency with current
state of the art referred to in this work and to explicitly present the differences
found between the method currently used to design marine propellers and the method
proposed in this work.
31
I
32
Chapter 3
Wake Geometry
The geometry of the free vortex lines shed downstream of the lifting line are not
known a priori and, in general, should be calculated as part of the solution of the
design problem.
Betz [2] showed that the optimum distribution of circulation was
obtained when the pitch angle of the helicoidal wake trailing vortices is that equal to
the hydrodynamic pitch angle calculated at the lifting line.
Betz's condition is the predominant assumption in all modern propeller design
codes 1 and is used by most authors and propeller designers. This assumption, also
used by Coney [9], is replicated here in the HPA approach to highlight the differences
between this and the LVA method proposed in this work.
3.1
HPA Wake Geometry
Betz's condition, 3.1, has been extensively utilized and studied to model the wake
field and has proven to be suitable to design propellers operating in moderatelyloaded conditions. The helicoidal shape of the free vortex lines are generated keeping
a constant pitch (the corrected hydrodynamic pitch, 3) along the free vortex lines
shed downstream of the blade, according to Lerbs [16].
1such as MIT-PLL,
by Kerwin [14], OpenPROP, by Brenden Epps [12] among others
33
1
tan
tan31
7-1w(r)
(3.1)
1 -- wo
Figure 3-1 shows the lifting line model created using Betz's condition to generate
the propeller wake geometry.
Figure 3-1: HPA model, shape of the trailed (fixed) vortex wake - CT = 1, J
wo = 0.1471 and KT = 0.253
3.2
0.80,
LVA Wake Geometry
A more generalized condition to be assumed on the wake trailing vortices is that
each vortex line should not generate any lift forces. Therefore, in accordance with
Kutta's Law, each segment of the wake must be aligned with the total local velocities
to generate zero-lift forces, as in equation 3.2.
L = p IF x V = 0
(3.2)
The velocity field is composed of the inflow and induced velocities calculated by
means of direct application of Biot-Savart's Law as in 3.3.
U =
_,
4H
JIS-113
34
(3.3)
In the cartesian coordinate system, the velocity components are as in 3.4, where
UX'Y'z are the three components of the induced velocities and 0 is the angle between
the Z and Y coordinates of the control points.
V = (Va + ux)z + (V - w.r. sin 0 + uy)J + (w.r. cos 0 + u,)k
(3.4)
The M + 1 trailing vortices are subdivided into a discrete number N of segments,
and consequently, N + 1 nodes. The coordinate of each node is represented as Xi (M)
and the length of each segment, 6'.
The alignment scheme consists of an iterative procedure. At first, the total velocity
at the midpoint of the segment, 0(m), is calculated and used to determine the new
position of the next point on the vortex line, X'
i+1
(m), by means of 3.5.
.
This points, however, may not be same as the next point in the vortex line, Xi+ 1
Hence, the equivalent displacement between the points, 6 = X'1 (m) - Xi+1 (M) is
calculated and subsequently applied to all points downstream of the starting row (see
figure 3-2(a)).
Once all points of all trailing vortex lines are corrected, the procedure moves on
to calculate the induced velocities in the second row of elements (see figure 3-2(b))
and apply the equivalent displacements to all points downstream of the second row,
and so forth all nodes composing the wake geometry are shifted.
- i+1
X'
(m) = Xi(m) + 6s fi~m
I
for: m
1,2,...,M
and: i
1,2,..N
P()
11(3.5)
In simple words, the algorithm aligns the wake elements with the local velocities
and adjusts the shape of the remaining row of points to match the shape of the parent
row, one row at a time.
In this manner, the variations obtained from the initial geometry are propagated
downstream in a controlled fashion which allows the convergence of the solution. The
35
rii+1\X.1
ri
V,
(M)
X'.I (mk
(a) Step 1
VI.1
X,+ 1
+M
r,
X,+2 (M)
rt
(b) Step 2
Figure 3-2: Local velocity alignment scheme
36
wake geometry generated with this scheme is shown in figure 3-3, is the same propeller
of figure 3-2, i.e. with CT = 1, J = 0.80, wo = 0.1471 and KT = 0.253.
Ati
Figure 3-3: LVA model - CT = 1, J = 0.80, wo = 0.1471 and KT = 0.253
A view of the cross-sections of the wake, for both the HPA and LVA cases, are
shown in figure 3-4 in which the development of the tip vortex roll-up can be clearly
seen starting at about a quarter turn downstream of the lifting line and becoming
fully developed (third curve from right to left) after one complete turn of the wake.
It is worth noting the sections displayed are not evenly spaced nor their heights
reflect the real Z-coordinates of the points actually used for the computation. Each
shape represents a row of points the points in the wake grid. This representation is
ideal for the visualization the shape of the roll-up development but tend to hide the
wake contraction.
The real wake shape reveals the contraction of the wake between consecutive
complete turns of the wake downstream of the blade and it is clearly seen in figure
3-5. At the second complete turn of the wake, a contraction of nearly 4% in diameter
is obtained for a propeller with CT = 1, J = 0.80, wo = 0.1471 and KT = 0.253.
37
Wake Cross-Sections - HPA
3
2.5
'K
-
2
1.5
-
1
0.5
-10
12
-6
-8
-4
-2
0
-4
-2
0
x
Wake Cross-Sections - LVA
3
2.5
1.5
-
2
1 --10
'12
-6
-8
Figure 3-4: Comparison of wake cross-sections for HPA and LVA cases - CT
=
1,
J = 0.80, wo = 0.1471 and KT = 0.253
Figure 3-5: LVA model wake contraction
KT
=
3.3
-
CT
1, J
=
0.80, wo
0.1471 and
0.253
Hub
The propeller hub is modeled by an image vortex method ideally extending infinitely
downstream of the lifting line, while in practice it is approximated with a sufficiently
long extension, i.e., comprising the extent of the wake trailing vortex lines in the
proposed numerical model.
Kerwin and Leopold [15] modeled the image vortex lattice with hub trailing vortex
38
lines positioned at a radii, re, given by equation 3.6, in which, rt, are the radii of the
free vortices at the wake of the propeller.
2
=
-r
L
rt
(3.6)
The authors also showed that the hub image of vortex lines, positioned as above,
should be aligned with the pitch angle given by 3.7 in order to satisfy the impervious
condition of the hub's surface, i.e., null normal velocities at the hub radius.
tan Ov = tan ##r
(3.7)
In the LVA approach, however, the wake lines are not aligned with the hydrodynamic pitch angle, so the alignment of the hub images is performed with the local
pitch angle of each segment in an attempt to hold the condition above as close as
possible to its true meaning.
An alternative approach to avoid this problem would be to model the hub by a
distribution of potential flow singularities (sources or vortices) on the hub surface
through, similarly to what it is done in boundary element methods.
Coney [9] proposed this other method and showed the equivalence of results between the two. In this work, the image vortex method was utilized but the panel grid
should be considered for future development in order to improve the convergence of
the methods featuring larger hub radii, or hubs with variable radii.
39
40
Chapter 4
Chord and Thickness
The method utilized in this work to optimize the chord and thickness distribution
follows the work of Brizzolara et. al.
[5].
In that work, the determination chord
and thickness distributions is performed based on a veery efficient combination of
strength and cavitation criteria proposed by Connolly [10] and Castagneto e Maioli
[8], respectively.
The aim of using these criteria simultaneously is to define chord
and thickness distributions that both provide enough structural strength to the blade
while avoiding the development of cavitation.
4.1
Strength
The semiempirical method proposed by Connolly utilizes the blade bending moment
stresses, -r, and radial centrifugal stresses, a7, defined as in 4.1 and 4.2, respectively,
to define the state of stresses at the propeller's blade.
In 4.1, A 1 and A 2 are tabulated coefficients which are functions of radial position of
the section, R is the propeller's radius, T and
Q
are the developed thrust and torque,
and, the product of the chord and the square of the section's maximum thickness,
ct 2 , represents the section inertia modulus.
2
-,ct =A
R.K
Z
27rR
A1 P T +A2- Q4.1
R
41
(4.1
In 4.2, i and 4 represent the rake and rake angle, respectively, C1 and C2, are
tabulated values, and n is the propeller's rotational speed.
Oc =
n2.R 2
1010
C +
i RC2
t C
s
~Tons
in 2 1
)
(4.2)
The total radial stress is assumed as the sum of both bending moment and centrifugal stresses and is restricted to a fraction 1/K.Krob of the yield stress of the
blade's material as in 4.3.
In this equation, K, is the safety factor and
Krob
is a
calibration factor to weigh the strength over the cavitation criterion, according to
Brizzolara et. al [3].
9c +
, 5
aam =
(4.3)
Ks.Krob
Replacing 4.3 into 4.4, one obtains
[R.K
ct2 =A1
z
4.2
2 1rR
P
T+A2- Q]
R
(4.4)
Gam -
O'c
Cavitation
For cavitation inception, Castagneto e Maioli use a simplified calculation of the
minimum pressure coefficients, C7""n, on each section of the blade and compare it
with the cavitation number, uo, and a safety factor for cavitation Kp, as in 4.5.
Kpor
(4.5)
=CP"a"
The minimum pressure coefficient is estimated by the semiempirical formula 4.6,
in which the h values are tabulated and CLf and CL, are the portions of the lift
coefficient due to camber and angle of attack, respectively.
C
=P
+ h1CLf + h 2 -C + h3CLa
42
1
(4-6)
-
--- -
Defining p = CLf/(CLf)CL),
x
1/V./_ and knowing that CL
CLf + CL,
equation 4.6 can be rewritten as 4.7:
x 3 (h2 V/t 2-+ [(hip + h3(1
4.3
-
p)) CLC + ( -
/1 + Ko-o)
0
(4.7)
Methodology
Since the strength and cavitation criterion depend on interrelated variables, the
chord and thickness optimization problem is solved iteratively. Initially, the centrifu.
gal stresses are assumed null, and equation 4.4 is solved for the section's modulus ct2
Equation 4.7 is solved for x, which defines the section chord, c, and, with the section
modulus, the thickness, t.
The centrifugal stresses are then calculated by means of 4.2 and are included in
equation 4.4, and so forth until the convergence of the method. Brizzolara et. al. [3]
also note the necessity of constraining the minimum tip thickness (around 3%) and
thickness over chord ratio at the root of the blade (no more than 15%) and it is up
to a design criteria to define the proportion of lift generated due to camber and angle
of attack, p, and the safety factors, K, and Kp, and the tuning factor, Krob.
4.4
Results and Discussion
Using the methodology presented above, the chord and thickness distributions for
a propeller under lightly-loaded condition (CT = 1, J = 0.80, KT = 0.253 and
wo = 0.1471, and Kp = 0.75, K, = 5 and K,,b = 1.2, using the HPA and LVA
approaches) is presented in figure 4-1.
While in figure 4-2 are displayed the results for a propeller operating in heavilyloaded conditions (CT
=
0.5, J = 1.13, KT = 0.253 and wo = 0.1471 and Kp = 0.75,
K, = 5 and Krob = 1.2, using the HPA and LVA approaches).
43
0.
Chord-Olameter Ratio
2
Thickness-Chord Ratio
025
-e- HPAI
-,LVA
0. 21
0.2
-
02
0. 19
0 15
-
0. 18
17
0.1
0. 16
0. 15
005
-
0.
13
0.
02
03
04
05
0.6
07
0,8
0.9
02
I
0.3
0.4
05
0.6
0.7
0.8
0.9
Figure 4-1: Chord and thickness distributions - CT = 1, J = 0.80, KT = 0.253 and
wo = 0.1471
Chord-Diameter Ratio
0.14
Thickness-Chord Ratio
0.26
1I9HPAI
A LVAI
0.13
-LVA
0.2
0.12
0.11
r
01
.
0.09
015
0.08
005
0.07
0'
0.2
0.3
04
05
0.6
07
0.8
09
1
02
03
0.4
05
06
xr
07
08
09
1
Figure 4-2: Chord and thickness distributions - CT = 0.5, J = 1.13, KT =-0.253 and
wo = 0.1471
The first case presents significantly large chords at the tip of the propeller, highlighting the effect of the cavitation inception criteria, constraining the minimum chords
at sections under high velocity inflow. In the second case, the propeller operates at a
lightly loaded condition. In this case, the chords at the tips are greatly reduced and
the root thicknesses are significantly larger, highlighting the predominant strength
criteria constraining the section modulus at the root.
44
-
- .........
.
Chapter 5
Lifting Line Results
The results obtained using the HPA and LVA approaches are presented here in
order to highlight the capabilities of the novel LVA approach and draw preliminary
conclusions from the comparison of the results (namely, circulation and hydrodynamic
pitch angle) obtained with the two approaches.
This will be done via sensitivity analysis of the results to a systematic variation of
loading conditions (i.e. thrust coefficient), hub radius and rake and skew distributions.
The study is intended to correlate these parameters with the effects of tip vortex
roll-up and wake contraction, eventually considering how the optimum circulation is
affected by them, in a series of propeller design cases.
At this point, it is worth noting that the totality of current lifting line propeller
design methods (based on Lerb's or Wrench's definition of induced velocities) can
not consider skew and rake and they are eventually recovered in the lifting surface
correction stage of the design.
This is another major advantage of the proposed
method, which consider both these important geometry modification right at the
first step of the design when the optimum circulation distribution is computed.
5.1
Convergence
The convergence of the iterative search for optimum circulation distribution (Chapter 2) is nearly monotonic for all cases considered, as can be seen in figure 5-1. The
45
criteria for convergence is imposed as the relative error between consecutive iterations
constrained to a maximum tolerance of 0.1%.
For HPA cases, even at heavily-loaded conditions, the method required no more
than 4 iterations to achieve convergence with specified tolerance. LVA cases, on the
other hand, whilst also nearly monotonic, achieved convergence in no more then 10
iterations for the same imposed tolerance level.
5.2
Induced Velocities
Figure 5-2 displays the induced velocities calculated at the lifting line of the test
case propeller with CT
=
1, J = 0.80, KT = 0.253 and wo = 0.1471, comparing:
induced velocities using HPA and LVA cases (see figure 5-2(a)); the solution between
a skewed propeller and one without skew using the LVA approach (see figure 5-2(b));
and, the solution between a raked propeller and one without rake using the LVA
approach (see figure 5-2(c)).
From these results, one can infer the strong dependence of the results on the
assumption utilized to describe the wake geometry (i.e., the differences between HPA
and LVA approaches) and the inclusion of both rake and skew in the calculation of
the induced velocities.
5.3
Loading
The comparison of the results obtained between the HPA and the LVA approaches
in terms of the loading of blade shows significant differences between them in both
the distributions of circulation and hydrodynamic pitch angle.
As a general rule, including the effects of tip vortex roll-up and wake contraction
into the optimization algorithm (LVA approach), compared to the HPA approach,
yields circulation distributions that are greater at the root and smaller at, and close,
to the tip.
In terms of the hydrodynamic pitch angle even more significant differences between
46
Relative Gamma Error
Relative Gama Error
2
[%)
[%I
0 _________________________________
--------
-1
-
-2
2
---
-4
3
-6
.8
-1
1.2
1
1.8
1.6
14
2
2.2
26
24
2
1
3
28
4
3
Relative TamBtal Error [%)
Reative TntBtal Error
5
6
7
5
6
7
[%]
-2
2
-3
.4
-6
.5
14
1.2
1
2
18
16
22
28
26
24
3
4
iterhton
3
1
(b) LVA
(a) HPA
Relative Gamma Error [%)
0
-4
-
Rolative Gamma Error [%]
2
-6
-/
3
-1
12
14
I
2
3
4
5
6
7
a
1
9
15
2
25
3
35
4
45
5
4
45
5
Relative Tanietal Error [%J
Relative TanBtIal Error J%I
1
0
0-
-2
-2
3
3
1
15
2
25
(d) LVA
(c) LVA 20' rake
3
35
500
skew
Figure 5-1: Comparison of convergence of the optimum circulation searching algorithm between different design cases (with and without rake and skew) and solution
methods (LVA/HPA).
the methods are observed. Neglecting tip vortex roll-up and wake contraction seems
to underestimate the necessary pitch at nearly every section of the blade to produce
the necessary thrust.
An important aspect of these results is that, as the load on the blades increases,
the negligible differences between the lightly-loaded case, (Ct = 0.2, figure 5-3(a)),
47
Axial
-
3
-
Induced
Velocit es
-+- HPA
-HPA
|--LVAJ
-0.9
IN-
03
-,-LVA
-0.8
2,5
0.2
Tangential induced Velocities
07
0.4
0.5
06
07
00
14
09
2
0.3
04
05
06
0.7
0.8
09
08
09
0
09
(a) Comparison of induced velocities between HPA and LVA approaches
Tangential Induced Velocities
Axial Induced Velocities
-
3
-e- LVA
- -LVA-skew 52-
-e- LVA
-s-- LVA-skew
506
0 5
2
15
.5
05
0
0.2
03
0.5
0.4
0.6
Xf
0.7
0.8
0.9
1
02
03
04
05
07
06
XT
(b) Comparison of induced vel ocities with and without skew
Axial Induced Velocities
-e- LVA
---LVA-rake
-
3
201
Induced Velocities
--- LVA-rake 206
-
1-
0
-
-14
-
15
-
-1.6
-
-
2
Tangential
LVA
---
S-1
02
03
0.4
05
06
07
08
0.9
1
802
03
04
05
06
07
1
(c) Comparison of induced velocities with and without rake
=
become apparent at the heavily-loaded condition (Ct
1, figure 5-3(c)).
=
1, J
0.80, KT = 0.253
Figure 5-2: Comparison of induced velocities with CT
and wo = 0.1471
=
This is an essential fact and a validation criteria of the LVA approach, once it is
known (and proved theoretically) that the velocity profile at lightly-loaded conditions
should satisfy Betz's condition and, hence, the fact that there is virtually no difference
48
Hydrodynamic Pitch Angle
Circulation
+HPA
F--ALVA
-e-HPA
I
0.014
70
LVA _
-
0016
65
60
0012
55
-
0,01
50
45
0008
40
0006
351
02
002
1
-
0004
03
04
0.5
0.6
Xr
07
08
09
1
0.3
O03
05
0.4
04
05
06
06
xr
09
09
00
0.7
0.8
0.7
(a) CT = 0.2, J = 1.80, KT = 0.253 and wo = 0.1471
HPA
-e-A-LVA
--HPA
-A-LVA
Go
-
Hydrodynamic Pitch Angle
Circulation
0025
55
-50
0.02
S45
40
0015
35
F
07
08 05-
30
25
0.01
02
0
3
04
0.5
06
07
08
I
0.2
0.9
0.3
0.5
04
06
07
08
09
1
(b) CT = 0.5, J = 1.13, KT = 0.253 and wo = 0.1471
Hydrodynamic Pitch Angle
Circulation
SHPA
SLVA
-e-HPA
-ALVA
5
50
0.03
45
a40
0025
35
30
0.02
25
20
0012
0.2
03
04
05
06
07
0.8
(c) CT = 1, J =
0.2
09
0.80, KT
=
03
03
04
0.4
0.253 and wo
00
05
=
0006
07
0,7
00
08
05
0,9
1
1
0.1471
Figure 5-3: Comparison of optimum circulation and hydrodynamic angles 3i calculated by the two methods (HPA/LVA) at increasing loading
between the methods seen in 5-3(a) is reassuring of the coherence of the results
obtained with the LVA approach.
49
5.4
Hub Radius
The geometry of the propeller may also feature a wide range of hub radii. Figures
5-4(a) and 5-4(b) display results using the HPA and LVA approaches, respectively,
for several non-dimensional hub radii
(xrh
= 0.2, 0.275, 0.35).
The results show significant variation in terms of circulation distribution for both
HPA and LVA approaches, and nearly no difference in terms of hydrodynamic pitch
angle.
The difference of the optimum circulation, as the next sections will show,
determines the distribution of lift on the blades and will define the distributions of
camber and angle of attack along the span of the blade and generate considerably
different geometries.
In addition, the proportion of lift generated by either camber or angle of attack is
directly related to cavitation inception, as discussed in Chapter 4, and highly impacts
the efficiency of the propeller, especially under heavily-loaded conditions, so these are
important differences to be accounted for.
5.5
Rake and Skew
Following a systematic variation of the rake and skew distributions and using the
heavily loaded case, CT= 1, J
=
0.80, KT = 0.253 and wo = 0.1471, the results
obtained in terms of circulation and hydrodynamic pitch angle are presented in figure
5-5 and 5-6.
Figure 5-5 shows the comparison between the results of a propeller with no rake,
given by the HPA and LVA approaches and two propellers with 4D = 10 and 20 [deg].
The rake of the blades tends to decrease the circulation at lower sections and increase
it at higher sections, closer to the tip. The presence of rake, however, does not seem
to affect the hydrodynamic pitch angle distribution.
Figure 5-6 shows how increased skew angle Os = 300 and 50' tend to increase the
load at the root and relieve the tip of the propeller, and, at the same time, increase
the pitch at the root and reduce it at the tip.
50
Circulation
0.ooo
Hydrodynamic
= 0.2
rh - 0.275
- -rh = 0.35
Pitch Angle
-e- rh = 0.2
A rh = 0.275
e rh = 0.35
-e -rh
.5
0.032
-
0.034
50
003
45
0.028
-
40
CD 0.026
35
0.024
30
0.022
25
002
20
0.018
0.016
0.: 2
0.3
04
005
(a)
06
CT
07
= 1, J
0,8
5
02
09
0.3
0.80, KT = 0.253 and wo
04
05
00
07
08
09
0.1471 - HPA
Circulation
Hydrodynamic Pitch Angle
-e- rh= 0.2
1 rh = 0.275
-- rh - 0.35
-e- rh = 0.2
-rh - 0.275 _
-- e- rh = 0.35
55
-
0035
45
35
-
0025
-
-0
0.03
-
30
25
C
2
0.3
04
0.5
06
0.7
38
-
0.02
02
09
0.3
04
05
06
08
07
0.9
(b) CT = 1, J = 0.80, KT = 0.253 and wo = 0.1471 - LVA
)
Figure 5-4: Comparison of results for different hub radii: optimum circulation distribution (G, left graphs) and hydrodynamic pitch angle (0j, right graphs
In both figures above, the results using the HPA approach were included to highlight the significant variations between the conventional method currently used in
industry and academia. These results reiterate the importance of accounting for the
effects of rake and skew distributions into the circulation optimization and provide
reasonable, consistent results offer interesting insights for further investigation on this
matter.
51
Circulation
. + HPA
-e- HPA
-,LVA
-e-- LVA - rake19
-- LVA - rake 20
0.035
Hydrodynernic Pitch Angle
60
55
-a-LVA
-
-'
0
e-
LVA - rake 10
--
LVA - rake
2
0
1
45
003
40
0.025
35
30
002
25
0.015
20
0.01
1
02
03
0.4
0.5
06
07
0.8
15 1
02
0.9
03
04
05
0M
07
08
1
0.9
Figure 5-5: Optimum circulation distribution - LVA method for cases with rake and
HPA/LVA no rake - CT = 1, J = 0.80, KT = 0.253 and wo = 0.1471
1
0035
-e-
HPA
---
LVA
e
Circulation
1
1
Hydrodynamic Pitch Angle
60
1
55
LVA - skew 30
- -LVA - kew 501
50
-e--e-
HPA
LVA
LVA - skew
. +
LVA - skew
3
0
5
0
1
-
0503
45
40
03 0025
35
-
30
25
0 02 -~
20
(77
15
02
03
04
05
00
07
08
0.2
0.9
03
0.4
0.5
0.6
0.7
0,8
0.9
1
Figure 5-6: Optimum circulation distribution - LVA method for cases with skew and
HPA/LVA no rake - CT = 1, J = 0.80, KT = 0.253 and wo = 0.1471
52
Chapter 6
Blade Detail Geometry Definition
Once convergence is achieved, the initial geometry of the propeller, that is, the
initial distribution of camber, angle of attack and geometric pitch angle, is obtained.
The circulations, F(r), and chords, c(r), are used to define the lifting coefficient of
each section, by means of equation 6.1.
_21'(r)
CL(r)
r(r)
V(r)c(r)
(6-1)
The lift generated at each section is due to both camber and angle of attack. In
order to ensure a sufficient margin to face cavitation, it is good practice to assume
that a proportion p = 90% of the total lift is generated by camber and the remaining
lift by angle of attack.
Hence, the angle of attack can be defined by equation 6.2, in which the slope of
the CL - a curve of the foil is assumed constant and equal to 2.
C (r)=(-p)
2.7r
(6.2)
The maximum camber of the section is defined by equation 6.3, in which Kt and Kf
are the coefficients defined by Castagneto and Maioli [8], and t(r) are the maximum
thicknesses of each section.
53
f (r)
c(r)p.cLi(r)
=
(6.3)
(Kf (1 + Kt('))
In this work, a NACA16 thickness distribution and mean line a = 0.8 are used and
the geometry of each foil that defines the sections of the propeller in the HPA case
with CT
=
1, J = 0.80, KT = 0.253 and wo = 0.1471 are presented in figure 6-1.
Camber-Chord Ratio
Angle of Attack
0.55
0.028
F-e-HPA
-a-LVA
0026
0.024
045
0022
04
0.02
0.35
0018
03
0016
0.25
02
0.3
04
05
0.6
0.7
08
02
0.9
012
03
0.4
Figure 6-1: Blade outlines for HPA and LVA cases with CT
0.0
=
06
0.7
0.8
09
1, J = 0.80, KT = 0.253
and wo = 0.1471
In general, the solution of HPA cases seems to present significantly higher camberchord ratios and angle of attack values, compared compared to the LVA approach,
especially in proximity of the hub.
54
Chapter 7
Lifting Surface Corrections
7.1
Methodology
Having defined the initial geometry of the blades, corrections on angle of attack
and maximum section camber can be made by modeling the geometry of the propeller
by zero thickness lifting surfaces, as proposed by Morgan [18]. In this model, shown
in figure 7-1, horseshoe elements are positioned along the chord of the blade and free
vortex lines are shed downstream, similar to the lifting line model. Assuming a zerothickness blade section, the boundary condition at each radius is the impermeability
of the surface, and, hence, equation 7.1, must be satisfied.
a(r) +
Of (r, xc)U
OXCe
)
-
[
V
(r, Xc)
U
(r)]
V
(7.1)
In 7.1, xc correspond to the chord-wise coordinate a point in section at radius r,
Un is the normal components of the velocities induced by lifting surface at that point
and U is the normal components of the velocities induced by the lifting line.
Defining the non-dimensional circulation as G = F/7rVaD, each horseshoe element
positioned along the chord has non-dimensional strength Gr(r, 6) (where 6 is the
polar angle of that point), which, if integrated along the chord of each section, must
be equal to the total circulation given by the lifting line, as in equation 7.2.
55
Figure 7-1: Lifting surface model
rOt
(7.2)
G,(r, 0)d9
G(r, 9) =]
In order to satisfy Kelvin's theorem, the strength of the circulation of the free
vortex lines, Gf (r, 0), are given by equation 7.3.
Gf (r, 9) = -
dr
dr
(7.3)
dr
Substituting equation 7.2 into 7.3, one obtains:
-
ddr - Gr(r, 01) d r - Gr(r, Ot) dr
(7.4)
+
Gf (r, 9)
From which the induced velocities are then calculated by means of direct application
of Biot Savart's Law, as in 7.5.
il(P) =
GC(r, )
2
_. ddr +
S11 3
56
Gf (r, 9) S_.
2S|1
ddr
3
(7.5)
After some algebraic manipulations, one can obtain the expression for the normalized induced velocities from the lifting surface in equation 7.6, where U/V are the
induced velocities as calculated from the lifting line.
z.
G,(r,0) SX- ddr
+
S-n
The new mean line of each section is reconstructed through the alignment of each
II SV1
Al
Vr Vr
segment with the total inflow velocity (including the velocities calculated by the lifting
surface) by means of equations 7.7 and 7.8, respectively.
f(r,xc) +xca(r)=
-~c[U
dO
(7.8)
The geometry of the lifting surface, shown in the example of figure 7-2, is used to
perform the given calculations for section camber and angle of attack.
Figure 7-2: Lifting surface geometry - CT
HPA
1, J =0.80, KT =0.253 and w0
57
0.1471
Results and Discussion
7.2
The comparison of the angle of attack and camber distributions between a case
corrected for lifting surface and an uncorrected one is displayed in figure 9-3(a), for
the HPA approach, and in figure 9-3(b), for the LVA approach.
This comparison shows the significant increase in angle of attack and a modest
variation in camber defined by the lifting surface corrections. As shown in figure 7-3
the difference in angle of attack for a heavily-loaded case is extremely important.
Angle of Attack [dag]
Camber-Chord
00241
LS
Ratio [m]
-HPA
-7-
-HP
-
LS
022A
-
002
4
0016
-
0018
0 0142
0 012
0.01
C2
0000
0.2
03
0.4
05
0.6
07
0.8
0,9
1
04
0.3
0.5
06
0,7
08
0,9
(a) Comparison of angle of attack distribution between corrected and uncorrected HPA cases
,Angle of
,
7
Attack [deg]
Camber-Chord
Ratio [m]
-e- LVA
-
F-- -LV-A
--- LVA
- LS
0.022-
6
LVA - LS
0.-2-
002
0.018
4
0.016
3
0.014
0.012
001
02
0.3
0.4
0.5
0.6
0.7
0.8
0.008
1
0.2
0.9
0.3
014
05
06
017
08
089
(b) Comparison of angle of attack distribution between corrected and uncorrected LVA cases
Figure 7-3: Comparison of case with and without lifting surface corrections for CT
1, J = 0.80, KT = 0.253 and wo = 0.1471 for both HPA and LVA
58
The differences between the HPA and LVA cases are also significant, especially in
terms of angle of attack. As seen in figure 7-4, the difference between the results of the
corrections for camber are somewhat minor and follow the same trend but displaced
slightly higher for the HPA case. The corrections for angle of attack, however, show
the opposite, with the LVA results being much higher than the HPA results, as seen
in figure 7-5.
I .1o'4
Camber Correction-Chard
Ratio
.
Camber-Chord Ratio [m]
0022
F --- - -, A-
0.02
0
0.018
0016
0014
-3
0,012
0 010.2
03
04
05
06
07
08
09
0.2
0
Figure 7-4: Lifting surface corrections for camber for
04
3
CT
0.5
0.6
07
08
0.9
= 0.5, J = 1.13, KT = 0.253
and wo = 0.1471 HPA and LVA cases
Angle of Attack (deg]
Angle of Attack Correction [deg]
-HPA
--- LVAI
-- LVA
5
5
3
40
0.2
-
-
4
03
0.4
05
0.6
07
08
10,2
09
0.3
0.4
05
06
Figure 7-5: Lifting surface corrections for angle of attack for CT
KT = 0.253 and wo = 0.1471 HPA and LVA cases
59
017
08
0.9
1, J = 0.80,
The lifting surface corrections are also influenced by the hub radius of the design.
For a propeller with CT = 1, J = 0.80, KT = 0.253 and wo = 0.1471, the variation
of angle of attack with hub radius is shown in figure 7-6 in the HPA case, and the
variation of camber in LVA case, shown in figure 7-7.
Angle of Attack [deg)
Angle of Attack Correction [deg)
~A-
-
5
rh - .2
rh - 0.275
rh -0.35
5
-
-9--e-
-+-
5-a-rh
e-
rh rh =
0.2
0.275,
0.5
4.5
4.5
4
3.5
0,
3.5
3
2.5
2.5
2
2
1.5
1.5
0.512
02
47
093
04
05
0.6
07
08
0.8
1
03
02
0.4
05
07
0.6
08
Figure 7-6: Lifting surface corrections for angle of attack for HPA cases with CT
J = 0.80, KT = 0.253 and wo = 0.1471, and various hub radii
10
10,a
--n
Camber-Chord
Camber Correction-Chord Ratio
Ratio
1
09
=
1,
[m]
0
0.024
08 0.275
- -rh- .35
-02
-e-
0.022
rh :0275
rh -0.35
002
-04
0.018
-06
0.016
'1
0,014
-08
0.012
001
-1 2
02
0.008
03
0.4
05
0.6
07
08
09
2
03
0.4
05
0.6
07
08
0.9
Figure 7-7: Lifting surface corrections for camber for LVA cases with CT = 1, J =
0.80, KT = 0.253 and wo = 0.1471, and various hub radii
Significant changes were obtained with various radii especially for the distribution
of camber. The angle of attack is also affected but to a minor extent showing localized
variations at the root of the blade.
60
Chapter 8
Results
8.1
Loading
The variation between the blade outlines generated can be explained using the same
reasoning used for the lifting line results. In terms of loading, the differences between
the methods are negligible for lightly-loaded conditions, when the LVA method nearly
replicates the behavior of the HPA method.
These differences, however, become significant for the heavily-loaded condition,
in which the effects of tip vortex roll-up and wake contraction greatly affect the
circulation distribution, and, as a consequence, the distribution of camber and angle
of attack of the blade.
The comparison between the blade outlines found for various loading conditions
of the blades are done for: a lightly-loaded case with CT= 0.2, J = 1.80, KT
0.253
and wo = 0.1471 (see figure 8-1); a moderately-loaded case with CT= 0.5, J
1.13,
KT
=
0.253 and wo = 0.1471 (see figure 8-2); and, a heavily-loaded case with CT
J = 0.80, KT
8.2
=
=
1,
0.253 and wo = 0.1471 (see figure 8-3).
Hub Radius
When it comes to the radius of the hub, the blade outline, for both HPA and LVA
cases, has a comprehensive variation.
As the lift generating portion of the blade
61
Carnber-Chord Rato [m]
Angle of Attack [deg]
1
+HP A]
- -LVA
|e-HPA
j a-LVA-
0,017
0.9
0.016
0.8
0.015
0.014
0.7
0.013
0.6
0.012
0.011
0.5
0.01
0.009
07
0..2
03
0.4
0.5
0.6
0.7
0.8
0.008
0.9
Figure 8-1: CT = 0.2, J = 1.80, KT
-
0.4
0.3
02
=
04
0.0
0.7
0.8
0.9
0.253 and wo = 0.1471
Camber-Chord Ratio (m]
Angle of Attack [deg]
3
0.5
-e- HPA
+ HPA
-A- LVA
-A- LVA
0.02
1-
2.5
0.018
0.016
0,014
1.5
0.01
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-
0,012
1 114
0.002
0.2
0.9
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 8-2: CT = 0.5, J = 1.13, KT = 0.253 and wo = 0.1471
becomes smaller, both camber and angle of attack tend to increase at the root of the
blade and tend to recover the values found for smaller hub radii, at higher sections of
the blade. This is seen from the results in figures 8-4 and 8-5, for the HPA and LVA
cases, respectively, with CT = 1, J = 0.80, KT = 0.253 and wo = 0.1471 and hub
radii of rh = 0.2, 0.275 and 0.35.
62
Angle of Attack [deg]
7
Camber-Chord Ratio [m]
0.024
-e-
-H HPA
-A-LVA
-~i~
HPA|
-4-LVA
0.022
6
0.02
0.018
R4
0.016
0.014
.
0.
3
0,012
0.01
1
0.2
0.3
0.4
05
0.6
07
0.8
0.9
0
0.3
04
0.5
06
07
0.9
0.8
Figure 8-3: CT = 1, J = 0.80, KT = 0.253 and wo = 0.1471
Camber-Chord Ratio [m]
Angle of Attack [deg]
5.5
-e- rh = 0.2
-A- rh = 0.275
erh - 0.35
-
--9
-A-9
0.024
rh =0. 2
rh = 0.275
rh - .3
0.022
-
4.5
3.5
-
-p
0.02
-
4
0.016
3-
0.014
2.5 -0.012
-
2
0.01
1.5
0.2
0.3
0.4
0 5
0.6
0.7
0.8
0.0080.
0.9
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 8-4: Blade outlines for HPA cases with CT = 1, J = 0.80, KT = 0.253 and
wo = 0.1471, and various hub radii
8.3
Rake
The presence of rake significantly impacts on the blade outline of the propeller. As
seen in figure 8-6, the raked geometry presents camber and angle of attack distributions that are considerably smaller at root sections and higher at the tip.
This difference in geometry is quintessential for the proper operation of each foil
under the wake of the ship.
An overestimated angle of attack may lead to face
63
Angle of Attack [deg]
Camber-Chord Ratio [m]
+-rh
-A-- rh
-e- rh
-
0.026
7
-e- rh =
-A- rh -e-rh -
-0.2
0.275
= 0.35
0024
0.2
0.275
0.35
6
0.022
5
0.02
0,010
D.4
0.016
0.014
3
0.012
2
0.3
0.4
0.5
0.6
0.7
02
0.9
0.8
-
0.00
0.2
03
0.4
0.5
0.6
07
1
09
0.8
Figure 8-5: Blade outlines for LVA cases with CT = 1, J = 0.80, KT = 0.253 and
wo = 0.1471, and various hub radii
cavitation, whereas an underestimated camber and angle of attack may cause underperformance and dissatisfaction of design requirements.
Angle of Attack [deg]
7
Camber-Chord Ratio [m]
0.024
-8
-A-
F--
LVA
LVA . *k201
LVA
I-,&-- LVA - rk20
0.022
6
0.02
5
0.018
4
0.016
3
0.014
0.012
0.01
0.2
0.3
0.4
0.5
0.6
Xr
0.7
0.8
0,00.
0.9
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 8-6: Blade outlines for LVA cases with CT = 1, J = 0.80, KT = 0.253 and
wo = 0.1471, with and without rake
64
1
8.4
Skew
In figure 8-7, the comparison between the blade outline of propellers with and
without skew, both solved with the LVA approach with CT = 1, J
=
0.80, KT = 0.253
and wo = 0.1471, is displayed where it is observed that skewed geometries tend to
increase both camber and angle of attack at lower sections of the blade.
Angle of Attack [deg]
Camber-Chord Ratio [m]
0.035
LVA
-A,- LVA - skrn
-e-
,a-- LVA -
sk50
0.03
7
0.025
002
4
0.015
3
0.01
02
0.3
0.4
05
0.6
0.7
0.8
0005
1
0.9
0
2
0.3
0.4
0.5
0.
Figure 8-7: Blade outlines for LVA cases with CT = 1, J = 0.80, KT
wo = 0.1471, with and without skew
65
07,
=
08
09
0.253 and
66
Chapter 9
Verification
To verify the consistency of results obtained by the two different design approaches,
an analysis of the two different propeller geometries was performed by a low order
boundary element method with flow-aligned wake capability [3], extensively validated
over different types of propellers (e.g., [6] [7]).
The analysis aimed to assess the fulfillment of the design requirements by the
fully numerical lifting line model and, at the same time, to assess if the higher fidelity
method which allows for the exact geometric features of the propeller (in particular
camber, thickness distribution along the chord and hub walls effect) could predict the
same performance variations between the HPA and LVA cases.
The results of this analysis are summarized in table 9.1, presenting the thrust
and torque coefficients and the efficiency of the propellers obtained for both the HPA
and LVA cases, and, for each case, including or not lifting surface corrections in the
methodology to derive the propeller's geometry characteristics.
The data indicates a decrease in efficiency while including the effects of wake
alignment in the optimization procedure. The LVA approach presented about 1.2%
in reduction of efficiency compared to the HPA.
This decrease in efficiency is likely to be related to the effects of the tip-vortex
roll-up once both geometries were subjected to the same design assumptions (with
the exception of the geometry of the wake). Furthermore, the sections of the blades
seem to be working equivalently well in terms of alignment of the local velocities,
67
as seen in figure 9-2, due to the pitch and camber corrections of the lifting surface
model.
Table 9.1: Comparison of results
Method
KT
10.KQ
ij
Lifting line
HPA
0.2532
0.4064
0.8000
LVA
0.2532
0.4105
0.7950
Panel method
HPA no LS
0.2529
0.4497
0.7161
HPA LS
0.2657
0.4780
0.7079
LVA no LS
0.2472
0.4453
0.7068
LVA LS
0.2589
0.4721
0.6981
The pressure coefficient distribution along the chords of the blade sections whose
geometry was obtained excluding lifting surface corrections shows a common trend
experienced by both geometries. This indicates their operation to be comparable, at
least from what can be inferred with a potential flow based analysis, and leads to the
conclusion that the effects of the wake alignment with the local velocities plays an
important role in the operation of heavily-loaded propellers.
The panel method analysis also showed significant differences in terms of circulation and induced velocities between the HPA and LVA designs. The variation in
thrust coefficient between HPA and LVA predicted by the panel method is nearly 3%.
The closest thrust to the design value is achieved by the propeller LVA with lifting
surface corrections which confirms the closer fidelity of the fully numeric lifting line
model with flow-adapted vortical wake.
In addition, the comparison between the effects of lifting surface correction in the
final performance of the propeller was analyzed. As the data shows, the total thrust
produced by propellers submitted to lifting surface correction procedure is generally
higher in comparison to those which were not submitted to corrections.
68
I
Jill'
II'"
The increase in thrust is mainly due to the increase of camber and, to a lesser
extent, to the angle of attack distributions added by the lifting surface method. This
is observed in figures 9-1(a) and 9-1(b), and it is in accordance with the initial design
philosophy adopted to have a larger margin on the possibility of face cavitation by
assigning a 10% lift contribution to angle of attack.
13
0.035
1.25
0.03
0-025
1.2
0.02
1.15
0.015
1
0.01
1
-HPA-NoLS
0DD5
-
HPA-noLS
--
HP-LS
1.05
HPA-LS
0.2
--
04
0.2
0.8
0.6
0.4
0.8
0-6
r/R
r/f
(a) Comparison of camber distributions due (b) Comparison of geometric pitch distributions due to lifting surface corrections
to lifting surface corrections
Figure 9-1: Panel method results for lifting line
The lifting surface corrections, obtained with the presented method which considers the hub effects, tend to increase the angle of attack of the root sections and
decrease their maximum camber. The result of this correction appears to be correct
as the pressure distribution for the profiles at the blade are relatively smooth, as
presented in 9-2.
.m -eJi
(
in. m
OM
04
0,2
9'
0
42
-DA
44
4's
01
0.2
0
04
06
06
0
& 00
1
0
01
(a)
02
01
00 4
(b)
00
0.0
6 00
1
(c)
Figure 9-2: Pressure coefficient distribution at different radii for the two propellers
designed with HPA/LVA methods
69
The increase in thrust is also observed in the comparison of the circulation distributions, as seen in figures 9-3(a) and 9-3(b), between the HPA and LVA approaches.
SR
-
IPAmL8
4
45
4
.
2
02 0
r 0'$
019
04
O-S
&a
26
r~~
(b4oprsnofcme1itibtosdet
4itn
2
I
A
12
03
04
01
a4
V7
01
ufc
orctosfrLAapoc
01
(a) Comparison of circulation distributions due
to lifting surface corrections for HPA approach
Figure 9-3: Comparison of optimum circulation distribution between propellers with
and without lifting surface corrections for the two HPA/LVA approaches
For the HPA approach, in which lifting surface corrections were performed, however, the relative error is increased to 5.7% in comparison with the case in which no
corrections were made.
The panel method analysis not only confirms the decrease in efficiency obtained
using the LVA approach but it also indicates the importance of considering the effects
of tip-vortex roll-up and wake contraction and established the importance of including
lifting surface corrections to in the design procedure.
The version of the panel method analysis, however, is also affected by the assumptions on the geometry of the wake to perform its calculations. Therefore, in order to
assess the advantage of employing one approach over the other, a fully viscous analysis
is essential to simulate the conditions experienced by the propellers in operation.
70
Chapter 10
Validation
The validation of the results obtained with the proposed methods was done using
an analysis with a Reynolds-Averaged Navier-Stokes Equation (RANSE) solver in
Star-CCM+ and applied to two cases. The CFD simulation mimics an open water
test on the propeller in full scale. Realizable k-epsilon turbulence model with two
layer wall function have been used.
The first case analyzed was that of a propeller running in moderately-loaded conditions, and, the second case, of a propeller operating under heavily-loaded conditions.
This is done because, as it was shown, under lightly-loaded conditions, the geometries generated by the HPA and the LVA approaches is virtually the same. So,
analyzing a propeller operating in this condition is a means to confirm that the overall
design methodology presented throughout this work is capable to design a geometry
that in fact satisfies the design criteria specified, i.e., provides the required thrust a
the given rotational speed and advance ratio.
Once the adequacy of the method is confirmed, two geometries are generated for a
propeller operating at heavily-loaded conditions, using the HPA and LVA approaches,
and their performance is compared. In this manner, the difference between the results
can be correlated with confidence to the approach used to design, i.e., either the HPA
or the LVA approach.
71
10.1
Lightly-Loaded Condition
For the lightly loaded condition, a propeller with z = 5, D = 5[m], CT = 0.4,
J = 1.26, KT = 0.253 and wo = 0.1471 was utilized to assess the overall design
methodology utilized in this work. The results shown in table 10.1 show a very good
adherence of the results with the requirement.
Table 10.1: Comparison of results from RANS analysis - CT = 0.4, J = 1.26, KT =
0.253 and wo = 0.1471
T [kN]
Requirement
310
HPA/LVA
311
Q
KT
10.KQ
-
0.2532
-
407
0.2526
0.6610
[kN-m]
r7
-
-
0.4866
At this initial stage of development, an error in overall thrust provided on less
than 0.5% is a unequivocal confirmation of the overall adequacy of the methodology
utilized in this work to design marine propellers. The pressure field displayed in figure
10-1 shows the good alignment of sections with local velocities.
Ahsohte ftrusw, (Pa
(b) Back pressure distribution
(a) Face pressure distribution
Figure 10-1: Pressure distribution for a lightly-loaded propeller - CT = 0.4, J = 1.26,
KT = 0.253 and wo = 0.1471
In this case, a very low open-water efficiency is achieved with the method. However,
in this situation, the rotational speed of the propeller (83.25 rpm) is very slow for
the given advance ratio and yields the very small chords of the blades, as it is seen
72
in figure 10-1. A real propeller would likely be operating in much faster rotational
speeds and would likely have larger chords, and thus possibly higher efficiency.
In these models, it is also worth noting that the shape of the hub is constructed
with an arbitrary sinusoidal shape to simulate a real hub geometry. The geometric
parameters are also extrapolated to define the geometry at the tip of the propeller
and at the root. In addition, in a real design, the odd shape of the blade at the tip
should be smoothed to avoid sharp edges, which induce localized vorticity.
10.2
Heavily-Loaded Condition
Lastly, the geometries of the propeller are generated for each approach, as shown in
figure 10-7(a), for the HPA approach, and in figure 10-10(b), for the LVA approach.
(b) LVA Propeller Geometry
(a) HPA Propeller Geometry
Figure 10-2: Propeller geometries obtained with the HPA and LVA approaches
The discretization of the geometry and the computational domain into a mesh of
finite volume elements is shown in figure 10-3 for the geometry generated with the
HPA approach, in which nearly 1.8 million cells were utilized.
The geometries generated by the HPA and LVA approaches are simulated in a
free-running condition using the discretized meshes generated. In this calculations,
73
Figure 10-3: Mesh of elements for CFD analysis for HPA case using approximately
1.8 million cells in computational domain
a rotating reference frame system instead of a rotating mesh in order to simplify
calculations and reduce processing time. This assumes to a steady solution, however,
in which time-dependent effects, such as induced vibrations are neglected. The results
of the steady state analysis of both the HPA and LVA geometries are shown in table
10.2.
Table 10.2: Comparison of results from RANS analysis - CT = 1, J = 0.80, KT
0.253 and wo = 0.1471
Q
r7
10.KQ
-
0.2532
-
669
645
0.2234
0.4190
0.6794
787
759
0.2552
0.4920
0.6602
T [kNl
Requirement
775
HPA
LVA
[kN-m]
74
-
KT
-
The viscous simulation shows unequivocally the higher accuracy of the LVA approach in designing heavily-loaded propellers. While the geometry created using the
HPA approach provided a thrust nearly 12% lower than the required, the error given
by the geometry given by the LVA approach is only 0.8% above the required.
Note that a comparison of efficiency would not be appropriate at this point as
the thrust provided by the HPA method is far from the required, which means its
actual operation point in-behind the hull would not be the one simulated. To obtain
a reasonable comparison, the propeller provided with the HPA approach would have
to be simulated with at higher rotational speed (one which would provide similar
thrust to the LVA). The efficiency at that point would be then compared to the LVA
method
The convergence of the simulation was achieved for both cases within 1200 iterations with tolerance of 0.5% on the values of thrust coefficient for both simulations,
as shown in figure 10-4.
I-
J5~
2
0
M00
4M0
flO
01
12C
0
#61on
Figure 10-4: Convergence of simulation for the HPA and LVA geometries
The pressure distribution on the blades between of two methods shows the increased
blade face pressure, figure 10-6(a), with the LVA geometry compared to the HPA, in
75
figure 10-5(a), whereas the back pressure present more subtle differences, as seen in
figures 10-6(b) and 10-5(b) for the LVA and HPA geometries, respectively.
In both geometries, the effect of the hub on the pressure distribution can be seen,
especially at the back of the blades.
In these areas, the pressure drop is not as
significant as the remaining areas of the blade.
(b) Back pressure distribution
(a) Face pressure distribution
Figure 10-5: Pressure distribution for the HPA geometry
(b) Back pressure distribution
(a) Face pressure distribution
Figure 10-6: Pressure distribution for the LVA geometry
Furthermore, at the tip, the drop in pressure representing the development of
the tip-vortex is more pronounced in the HPA case compared to the LVA case, as
2
can be seen from the comparison of helicity iso-surface (40m/s ) between the two
approaches, as shown in figure 10-10
Last, the significant differences in terms angle of attack obtained between the geometries obtained with the two approaches is observed in the pressure distribution at
76
(a) HPA
(b) LVA
2
Figure 10-7: Helicity iso-surface (40m/s ) of the blades of the propellers designed
with the HPA and LVA approaches
the leading edge of the blades. The LVA geometry presents pressure field more consistent with a correct alignment with inflow velocities compared to the HPA geometry,
77
as can be inferred from figures 10-8 and 10-9.
Figure 10-8: Pressure field at leading edge of HPA geometry
Figure 10-9: Pressure field at leading edge of LVA geometry
The pressure coefficient distributions are calculated at the face and back of sections
of the propellers located at 0.8.R. It is clearly seen that the both HPA and LVA
methods present a good alignment with the local velocities.
The LVA approach, however, presents lower back pressure and higher face pressure,
in accordance with the significantly higher overall thrust developed.
The plot of the pressure coefficient distribution along the chord of the section also
indicates the good alignment with the local velocities confirming the good results
obtained with the LVA approach thus far. Both leading and trailing edges of the sec-
78
tions experience vanishing pressure, which confirms the adequacy of the methodology
utilized.
2Y
2,0
600000-
K
....
.......
-400000-
-400000
J
L
18000
-100.06
L
,
-03'5
-03
-0I
02
0 00 0305 02 05 0 005 04
Dmto5101AI).0
-01 -0 05
0
(a) HPA blade section at xrh
Fo1
XY
=
0.8R
500000.
0 ,1 0
IJ0.00000-
10.001
-04
-03s
-00
-025
d0
-01
015
0
-005
O0,soo 10101
005
5if
(b) LVA blade section at xrh
01
05
02
05s
03
5
04
0.8R
Figure 10-10: Pressure coefficients at blade sections at radii 0.3R and 0.8R, for the
two propellers HPA/LVA
79
80
.
...
Chapter 11
Conclusions
In this work, a systematic methodology is proposed to design wake adapted propellers under heavily-loaded conditions. By comparing a propeller geometry created
with a local velocity alignment approach (LVA) with that created with a conventional
approach based on Betz' condition (HPA), it is shown that neglecting the effects of
tip-vortex roll-up and wake contraction in the circulation optimization is detrimental
to the overall accuracy of the design in terms of fulfillment of design requirements.
At first, the results of the lifting line obtained with the HPA method compared
with the LVA method indicate a significant decrease in efficiency when the effects
of the aligned wake are considered through the fully numerical method (LVA). This
observed trend goes in the right direction, i.e. it tends to bring the efficiency estimated
by the lifting line down to more realistic values. Furthermore, the results obtained
with the two approaches are virtually the same under moderately- and lightly-loaded
conditions, which proves the consistency and theoretical validity of the proposed LVA
approach.
The extensive variation of hub radii showed the robustness of the method. The
possibility of including the effects of rake and skew of the blades into the the circulation optimization demonstrated the extended capabilities of this approach to
overcome the limitations of the conventional methodology.
In addition, it was shown that lifting surface corrections are necessary to adjust
camber and angle of attack distributions. A comparison between geometries to which
81
these corrections were made with those to which they were not demonstrated the
importance of those corrections in the design of the propeller to satisfy design criteria.
When corrections are not performed, the final geometry is considerably under pitched,
as it was confirmed by the following preliminary panel method verification.
This analysis, based on boundary element method, showed that, only when the
proposed fully numerical lifting line method is combined with lifting surface corrections, the resulted geometry is able to satisfy the initial design thrust requirements.
The comparison of circulation distribution confirmed the general trend of the
LVA approach to load the root of the blades and relieve the load at the tip, and
also showed (through the calculation of pressure coefficients distributions along the
chord of various sections along the span of the blade) the adequacy of the methods
in providing the correct camber and angle of attack to meet design criteria.
Finally, the numerical viscous analysis confirmed the higher accuracy of the LVA
approach over the HPA approach for propellers operating under heavily-loaded conditions. In the example case of a heavily loaded propeller, the total thrust provided
by the geometry designed using the HPA approach was about 12% lower than the
requirement, while the geometry designed by the LVA approach was only 0.8% above
design criteria. The latter getting the closest result to the design value, also in terms
of radial load distribution (especially close to the tip and the hub).
82
Index
Ct - Thrust coefficient, 28
F, - Viscous drag, 27
J - Propeller advance coefficient, 28
G - non-dimensional bound circulation, 56
KT - Propeller thrust coefficient, 28
Gf - non-dimensional free vortex circula-
L - Lift force, 34
tion, 56
V - total velocity, 34
H - Auxiliary function, 25
6s - Length of wake segment , 35
K - strength safety factor, 42
1 - propeller efficiency, 34
Kp - cavitation safety factor, 42
Or - bending moment stress, 41
Kf - Camber section coefficient, 54
X' - Position of ith wake element , 35
Krob - tuning factor, 42
u - Induced velocity, 35
Kt - Thickness section coefficient, 54
cL - Lift coefficient, 53
Q
rh - Hub radius, 39
T - Propeller thrust, 25
rt - Bound vortex radius, 39
U - normal lifting line ind. velocity, 55
r, - Free vortex radius, 39
U, - normal lifting surf. ind. velocity, 55
Ux - X-component o induced velocity, 35
V - total velocity, 24
UY - Y-component o induced velocity, 35
Va inflow axial velocity, 20
u, - Z-component o induced velocity, 35
Vt inflow tangential velocity, 20
w - wake fraction, 34
F - bound vortex circulation, 19
wo - mean wake fraction, 34
Ff free vortex circulation, 19
z - Number of blades , 22
<D - rake angle, 42
CL, - lift coefficient due to angle of attack,
a - Angle of attack, 53
43
'ta - axial induction velocity function, 22
CLf - lift coefficient due to camber, 43
h _ hub axial induction velocity func-
Cp - pressure coefficient, 42
Fh -
- Propeller torque, 25
tion, 30
hub drag force, 30
ff - tangential induction velocity function,
83
22
?ith
- hub tangential induction velocity function, 30
-
uncorrected hydrodynamic pitch an-
gle, 34
3
-
uncorrected hydrodynamic pitch an-
gle, 21
i
-
corrected hydrodynamic pitch angle,
21
f3
-
corrected hydrodynamic pitch angle,
34
w propeller rotational speed, 20
a, - centrifugal stress, 42
F - Force on section, 25
u - vector induction velocity function, 23
c - section chord, 41
f
- section camber, 54
i - rake, 42
p - camber-angle of attack lift ratio, 43
t - section thickness, 41
ua - axial induced velocity, 24
ua induced axial velocity, 20
ut - tangential induced velocity, 24
ut induced tangential velocity, 20
84
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