DEVELOPMENT OF A MARINE PROPELLER DESIGN METHOD …

DEVELOPMENT OF A MARINE PROPELLER DESIGN METHOD BASED ON LIFTING LINE THEORY AND LIFTING

SURFACE CORRECTION FACTORS

By ATAUR RAHMAN

Student No.100912201

June 2015 Department of Naval Architecture and Marine Engineering Bangladesh University of Engineering and Technology

Dhaka 1000, Bangladesh

ii

DEVELOPMENT OF A MARINE PROPELLER DESIGN METHOD BASED ON LIFTING LINE THEORY AND LIFTING

SURFACE CORRECTION FACTORS

A THESIS

By ATAUR RAHMAN

Student No. 1009122001

Submitted to the Department of Naval Architecture And Marine Engineering in a

partial fulfilment of the requirements for the degree of Master of Science

in Naval Architecture and Marine Engineering

June 2015

Bangladesh University of Engineering and Technology Dhaka 1000

iii

CERTIFICATE OF APPROVAL

The thesis entitled “Development of a marine propeller design method based on lifting line theory and lifting surface correction factors” submitted by ATAUR RAHMAN holding Student no. 1009122001, session: October 2009 has been accepted as satisfactory in partial fulfilment of the requirements for the degree of Master of Science in Naval Architecture and Marine Engineering on 30th June 2015.

Board of Examiners

(I) ________________________ Dr. Md. Refayet Ullah Chairman (Supervisor)

Professor Dept. of NAME, BUET, Dhaka

(II) ________________________ Dr. Md. Shahjada Tarafder Member (Ex-Officio) Professor & Head Dept. of NAME, BUET, Dhaka

(III) ________________________ Dr. Md. Mashud Karim Member

Professor Dept. of NAME, BUET, Dhaka

(IV) ________________________ Dr. Md. Quamrul Islam Member (External)

Professor Dept. of ME, BUET, Dhaka

iv

CERTIFICATE OF RESEARCH

This is to certify that the work presented in this thesis entitled “Development of a

marine propeller design method based on lifting line theory and lifting

surface correction factors” is the outcome of the investigation carried out by the

candidate Ataur Rahman, under the supervision of Dr. Md. Refayet Ullah,

Professor, Department of Naval Architecture and Marine Engineering, Bangladesh

University of Engineering and technology, Dhaka Bangladesh.

------------------------------------ Candidate

(ATAUR RAHMAN)

------------------------------------ Supervisor

(DR. MD. REFAYET ULLAH)

v

DECLARATION

I do hereby declare that this thesis entitled “Development of a marine propeller

design method based on lifting line theory and lifting surface correction

factors” has been prepared by me and has not been submitted anywhere else for

the award of any degree or diploma or for publication.

------------------------------------ Candidate

(ATAUR RAHMAN)

vi

ACKNOWLEDGEMENT

The author wish to express his profound gratitude and heartiest thanks to Dr. Md.

Refayet Ullah, Professor, Department of Naval Architecture and Marine

Engineering, Bangladesh University of Engineering and Technology, Dhaka,

without whose continuous supervision, valuable suggestion, encouragement and

inspiration, this work would not be possible.

The author wishes further extent gratitude to Dr. Md. Quamrul Islam, Professor of

the Department of Mechanical Engineering, BUET & Dr. Md. Mashud Karim,

Professor of the Department of Naval Architecture & Marine Engineering, BUET for

their valuable comments and suggestions which improve the quality of the thesis.

With feeling of gratitude the author wish to thank Dr. Md. Shahjada Tarafder,

Professor & Head of the department of Naval Architecture & Marine Engineering,

BUET, with also all the teachers and staffs of the department for their constant

cooperation, encouragement and inspiration, and unforgettable contribution.

Full Support from his family members specially his two children Fairuz and Mahrus

sacrificed their time and love to encourage their father to complete this work. Finally

support and encourage from his father Haji Nurul Amin made him to finish this

thesis. He wish to dedicate his work, his thesis and his degree to his father,

recently who has started his journey to meet his creator in heaven.

vii

Table of Contents

Abstract ix

List of Figures x

List of Tables xi

List of Symbols xii

Chapter 1 : Introduction 1

1.1 Literature review 1

1.2 Lifting surface theory 4

1.3 Present research 6

1.3.1 Basic assumptions 6

1.3.2 Outline of research 7

Chapter 2 : Theoretical formulation 8

2.1 The velocity induced by free vortex lines 8

2.1.1 The induced velocity components determined by Biot-Savart’s Law 8

2.2 The velocity induced by radial vortex lines 13

2.3 Solution of propeller lifting line problem by vortex lattice method 16

2.3.1 Wake adapted propeller 21

2.4 Solution of propeller lifting surface problem 23

2.4.1 The reference helix 26

2.4.2 Bound vortex distribution 27

2.4.3 Vortex lattice method 29

2.4.4 Relating continuous and lattice distributions 31

2.4.5 Velocity induced by the lattice in 3-dimensional flow 33

2.4.6 Determining the camber and angle of attack 33

2.4.7 The symmetry of the velocity field 36

2.4.8 Modification of preceding results for symmetrical blades 37

viii

Chapter 3 : Design methodology 40

3.1 Design procedure 40

3.1.1 Relation between C��, KT, J�� 42

3.1.2 Propeller design using circulation theory 42

3.2 Cavitation check 49

3.3 Calculation of section chord, section lift coefficient, un-corrected section pitch and section camber 49

3.4 Calculation of lifting surface geometry 50

3.5 Calculation of corrected section pitch and section camber 51

Chapter 4 : Application of theoretical formulation for the design of a marine propeller 52

4.1 Design of marine propeller 52

4.2 Calculation of nominal mean wake 53

4.3 Calculation of blade area ratio 54

4.4 Calculation related to lifting line program 55

4.5 Cavitation check 62

4.6 Calculation of section chord, section lift coefficient and uncorrected section pitch and section camber 64

4.7 Calculation related to lifting surface program 65

4.8 Calculation of camber correction factor 69

4.9 Calculation of corrected section pitch and section camber 69

Chapter 5 : Results and discussion 70

Chapter 6 : Conclusion and recommendation 78

6.1 Conclusion 78

6.2 Further recommendation 79

References 80

Appendix – A : Program description 82

Appendix – B : Program listing, data & output 89

Appendix – C : Table of chord-load factors 130

Appendix – D : Kramer diagram 131

ix

Abstract

The marine propeller design method based on lifting line theory and lifting surface

correction factors is presented in this thesis. The blade outline is chosen in such a

way that no cavitation is occurred. The pitch and camber is determined by

satisfying the requirements for the designed load distribution. The method is an

adaptation of the vortex lattice method and used for wake adapted propeller.

A propeller with symmetrical blades and with mean lines which are symmetric

about the mid chord need no pitch correction due to lifting surface effect. In this

thesis, the developed method is applied for the propeller whose blade outline is

symmetrical about the mid-chord and the mean line chosen is parabolic which is

also symmetrical about the mid-chord. Only camber correction is done.

It is assumed that the circulation is zero at hub and tip in both lifting line and lifting

surface calculation. Also in lifting surface calculation, the circulation is zero at

Leading edge and trailing edge of the propeller blade.

For the calculation of induced velocity at any point in the flow field, the Biot-Savart

law is used. The pitch is calculated using trial and error method. The camber

distribution is corrected by using camber correction factors and compared with

published result.

x

List of Figures

Figure 2.1 Co-ordinate system and notation for helical vortices

Figure 2.2 Co-ordinate system for radial vortices

Figure 2.3 Schematic arrangement of vortex lattice with M=10, P=5

Figure 2.4 Velocity diagram – optimum lifting line

Figure 2.5 Lifting surface notation

Figure 2.6 Co-ordinate system

Figure 2.7 Expanded blade section

Figure 3.1 Ship propeller design method

Figure 3.2 Detail propeller design stages

Figure 3.3 Velocities and forces on a blade element.

Figure 4.1 Radial distribution of circulation G(x), lifting line method.

Figure 4.2 Chord-wise circulation distribution (x=0.45)



Figure 5.1 Comparison of radial variation of hydrodynamic pitch angle (βi)

Figure 5.2 Comparison of radial variation of non-dimensional circulation (G)

Figure 5.3 Comparison of radial variation of pitch-diameter ratio (P/D)

Figure 5.4 Comparison of radial variation of camber-chord ratio (f/c)

Figure 5.5 Convergence of CTi & CT values at different iteration from different

initial iteration values

Figure 5.6 Effect of thrust requirement on radial pitch-diameter ratio distribution (P/D)

Figure 5.7 Effect of thrust requirement on radial camber-chord ratio distribution (f/c)

Figure A.1 Flow chart of lifting line program

Figure A.2 Flow chart of lifting surface program

Figure D.1 Kramer diagram

xi

List of Tables

Table 4.1 Nominal mean wake calculation

Table 4.2 Blade area ratio calculation

Table 4.3 Hydrodynamic pitch angle – 1st iteration with ��= 0.77

Table 4.4 Hydrodynamic pitch angle – 2nd iteration with ��= 0.7422

Table 4.5 Hydrodynamic pitch angle – 3rd iteration with ��= 0.7297

Table 4.6 Hydrodynamic pitch angle – 4th iteration with ��= 0.73



Table 4.9 Calculation of section chord, section lift coefficient and uncorrected section pitch and section camber

Table 4.10 Radial circulation coefficient calculation

Table 4.11 Value of circulation G(x) from both methods

Table 4.12 Camber correction factor, Kc

Table 4.13 Corrected section pitch and section camber

Table 5.1 Comparison of circulation distribution by two methods

Table C.1 Chord-load factors, µnj

xii

List of Symbols

AE Expanded area of the propeller blades, m2

AP Projected area of propeller blade, m2

AE /A0 Propeller Expanded Area Ratio

A0 Disc area of the propeller= �� , m2

B.A.R Propeller blade area ratio

CD Drag coefficient�= ��.�∗�∗��∗��

� CL Lift coefficient�= �

�.�∗�∗��∗�� CT Propeller thrust loading coefficient �= �

�.�∗�∗��∗��

CTi Ideal thrust loading coefficient

Cij Fourier coefficient of circulation distribution

Cp Power coefficient �= !�.�∗�∗��∗��"

� D Propeller diameter, m

G Non-dimensional circulation = #�∗�∗$∗�

G' Non-dimensional circulation = #�∗�∗�%�

I Radial term in Fourier series for G distribution

J Chord-wise term in Fourier series for G distribution

J Propeller advance coefficient = ��&∗�� Kc Camber correction factor '= ()*+,-�&.�/012

()*+,-�&��/0123 KT Propeller thrust coefficient = �

4∗&�∗�5� KQ Propeller torque coefficient = 6

4∗&�∗�7� L Lift of the blade section, N

M No. of radial lattice elements

N No. of chord-wise lattice elements

xiii

P No. of radial control points

P Propeller pitch, m

PD Delivered Power, W

Q No. of chord-wise control points

Q Propeller torque, N.m

R Propeller Radius, m

S Non-dimensional vortex sheet strength = 8$∗�

T Propeller thrust force, N

VA Speed of advance [=VS(1-ω)], m/s

VR Resultant fluid velocity, m/s

VS Ship speed, m/s

V* Resultant fluid velocity, m/s

c Blade section chord length, m

dD Elemental drag force, N

dL Elemental lift force, N

dQ Elemental torque, N.m

dT Elemental thrust force, N

f Maximum camber of mean line

9 Non-dimensional camber '= //0;< 3

g Acceleration due to gravity, m/s2

h Depth of immersion of the propeller shaft, m

hq Slope of mean line with unit camber at point q

llll Chord length of expanded section

n Propeller revolution rate per second,1/s

xiv

pa Atmospheric pressure, N/m2

pv Vapor pressure of fluid, N/m2

p Propeller pitch, m

r Radius of control point, m

r0 Radius of helical vortex element, m

rh Propeller hub radius, m

s Distance along helical axis

u Induced velocity

> Non-dimensional induced velocity '= ��-$# 3

ua, uA Axial induced velocity, m/s

un Normal induced velocity, m/s

ur Radial induced velocity, m/s

ut, uT Tangential induced velocity, m/s

u* Displacement velocity, m/s

x Non-dimensional radius [=r/R]

xh Non-dimensional hub radius

x,y,z Cartesian co-ordinate system

z Number of propeller blades.

z,r,θ Cylindrical co-ordinate system

α Angle of attack of section relative to βi, deg

? Non-dimensional pitch correction '= @;<3

β Propeller advance angle, deg

βi Hydrodynamic pitch angle, deg

βio Hydrodynamic pitch angle at ro, deg

xv

Γ Circulation, m2/s

γ Vortex sheet strength

εi Drag-lift ratio = ;!;<� η Non-dimensional radius = -�- � η Efficiency of the propeller

ηi Ideal efficiency of the propeller

ηB Efficiency of the propeller in behind condition

λ Advance ratio = ��∗&∗��

ABC Chord load factor

ρ Density of fluid, Kg /m3

ρ Transformed radial co-ordinate

σ Transformed chord-wise co-ordinate

σ Cavitation number

D( Propeller thrust loading coefficient

ω Volumetric mean nominal wake fraction

ω Propeller rotational speed (= 2πn), 1/s

ω (x) Mean circumferential wake fraction at non-dimensional radius x

1

Chapter 1

Introduction

1.1 Literature review

Transportation across the oceans has engaged the attention of humankind since

the dawn of history. Ships started thousands of years ago as simple logs or

bundles of reeds and have developed into the huge complicated vessels of today.

Wooden sailing ships are known to have appeared by about 1500 BC and had

developed into vessels sailing around the world by about 1500 AD. For centuries,

ships were propelled either by human power (e.g. by oars) or by wind power (sails).

The development of the steam engine in the 18th Century led to attempts at using

this new source of power for ship. The first mechanical propulsion device to be

widely used in ships was the paddle wheel, consisting of a wheel rotating about a

transverse axis with radial plates or paddles to impart an astern momentum to the

water around the ship giving it a forward thrust. The paddle wheel are quite efficient

when compared with other propulsion devices but have several drawbacks

including difficulties caused by the variable immersion of the paddle wheel in the

different loading conditions of the ship, the increase in the overall breadth of the

ship fitted with side paddle wheels, the inability of the ship to maintain a steady

course when rolling and the need for slow running heavy machinery for driving the

paddle wheels. Paddle wheels were therefore gradually superseded by screw

propellers for the propulsion of oceangoing ships during the latter half of the 19th

Century.

The Archimedean screw also had been used to pump water for centuries and

proposals had been made to adapt it for ship propulsion by using it to impart

momentum to the water at the stern of a ship. The first actual use of a screw to

propel a ship appears to have been made in 1804 by the American, Colonel

Stevens. In 1828, Josef Ressel successfully used a screw propeller in an 18 m long

experimental steamship.

2

The first practical applications of screw propellers were made in 1836 by Ericsson

in America and Petit Smith in England. Petit Smith's propeller consisted of a

wooden screw of one thread and two complete turns. During trials, an accident

caused a part of the propeller to break off and this surprisingly led to an increase in

the speed of the ship. Petit Smith then improved the design of his propeller by

decreasing the width of the blades and increasing the number of threads, producing

a screw very similar to modern marine propellers. The screw propeller has since

then become the predominant propulsion device used in ships.

A screw propeller consists of a number of blades attached to a hub or boss. The

boss is fitted to the p8ropeller shaft through which the power of the propulsion

machinery of the ship is transmitted to the propeller. When this power is delivered

to the propeller, a turning moment or torque is applied making the propeller revolve

about its axis with a speed thereby producing an axial force or thrust causing the

propeller to move forward with respect to the surrounding medium (water) at a

speed of advance.

Although the screw propeller was used for ship propulsion from the beginning of the

19th Century, the first propeller theories began to be developed only some fifty

years later. To develop propeller theory for design of screw propeller it is important

to understand the terms used and to have some knowledge of certain principles of

hydrodynamics [6, 8, 14].

In the momentum theories as developed by Rankine, Greenhill and R.E. Froude for

example, the origin of the propeller thrust is explained entirely by the change in the

momentum of the fluid due to the propeller. The blade element theories, associated

with Weissbach, Redten-bacher, W. Froude, Drzewiecki and others, rest on

observed facts rather than on mathematical principles, and explain the action of the

propeller in terms of the hydrodynamic forces experienced by the radial sections

(blade elements) of which the propeller blades are composed. The momentum

theories are based on correct fundamental principles but give no indication of the

shape of the propeller.

3

The blade element theories, on the other hand, explain the effect of propeller

geometry on its performance but give the erroneous result that the ideal efficiency

of a propeller is 100 percent.

The divergence between the two groups of theories is explained by the circulation

theory (vortex theory) of propellers initially formulated by Prandtl and Betz '(1927)

and then developed by a number of others to a stage where it is not only in

agreement with experimental results but may also be used for the practical design

of propellers.

The circulation theory or vortex theory provides a more satisfactory explanation of

the hydrodynamics of propeller action than the momentum and blade element

theories. The lift provided by each propeller blade is explained in terms of the

circulation around it. The circulation theory has been used for designing propellers

for over seventy years [8, 14].

In lifting line theory, the propeller blades are replaced by straight radial vortex lines.

A free vortex sheet extends downstream from each of the lifting lines forming an

approximately helical surface. The propeller is assumed to be rotating with constant

angular velocity in an axial velocity stream, whose velocity may be a function of

radius only.

If the propeller blade is represented by a vortex line or lifting line, the effect of the

finite width of the blade is neglected [2, 13]. The variation of the induced velocity

along the chord of a blade section causes a curvature of the flow over the blade

resulting in changes to the effective camber of the blade sections and the ideal

angle of attack [4, 14].

Due to the low aspect ratio of most marine propeller blades, the use of lifting line

theory results in unacceptably large errors unless supplemented by a lifting surface

correction of some kind. The lifting line concept which is at present widely used

leads to a propeller which does not attain the design thrust for design revolutions

and speed, so corrections are made to camber. Since in many cases a thrust

4

deficiency still exists a further correction, this time to pitch and based on the

assumptions that flow curvature varies appreciably along the chord.

The modern development of the circulation theory is based on considering the

propeller blades as lifting surfaces rather than lifting lines [13]. The lifting surface

theory is used today basically to determine the camber and pitch angle at the

different radii of the propeller [10, 17, 18, 19, 23].

1.2 Lifting Surface Theory

The modern development of the circulation theory is based on considering the

propeller blades as lifting surfaces rather than lifting lines. In this case the camber

lines of propeller sections are calculated from the lifting surface equations using

blade contour and loading distribution as starting point. The discussion to substitute

the lifting-line approach by lifting-surface theories dates back to the 1950s, but the

realization of this goal was initially impossible for real ship propeller geometries due

to insufficient computing power.

The simplest mathematical model of a lifting surface theory was used by Ludwig

and Ginjel in 1944. They presented their well-known camber correction factors, with

which the result of a lifting line calculation can be corrected. Instead of a calculation

of the camber distribution over the chord, the maximum camber is calculated only.

The use of computers made it possible to solve the lifting surface equations, with

various restrictions as to the geometry of the propeller and the chord-wise loading

distribution and with mathematical simplifications. The first one, who refined the

Ludwig and Ginzel camber corrections, was Cox in 1961 [7]. He used constant

chord-wise loading distributions and showed with his calculations, the necessity of

taking into account variations in blade shapes and number of blades.

In 1961, Pien [20] reported a method to solve the lifting surface equations for the

steady case using the continuous vortex distribution. The Calculations using this

theory were performed by Cheng in 1964 and 1965 [3].

5

The method of Pien and Cheng is preferred as somewhat more reliable. Since

computers may not always be accessible to propeller designers, it is necessary to

present systematical calculations to drive a set of correction factors depending on

the most important parameters. This was done by Morgan et al in 1969. He used

Cheng’s method and varied blade area ratio, induced advance ratio, skew and

number of blades as the most important parameters, while some blade outline were

compared.

The second generation of lifting-surface methods was developed around the late

1970s when sufficient computer power became widely available. Kerwin in 1978

used a vortex-lattice method to solve the lifting surface equation, a method which

was earlier used by Falkner [5] for arbitrary wings. The lifting surface is represented

by a discrete lattice of vortices and the induced velocities are calculated over an

interval from about the quarter chord to the three-quarter chord to determine an

average camber correction factor.

Vortex-lattice methods were in the 1990s extended to rather complicated propeller

geometries, e.g. contra-rotating propellers, and unsteady propeller inflow (nominal

wake computations).

This work contains an invaluable amount of information for the designer. Some of

the conclusions drawn by Kuiper [16] have to be mentioned here :

- The span wise change in the camber chord-wise distribution is small. In

practice the use of the two-dimensional distribution is probably reasonable.

- The shape of the blade outline and the span-wise load distribution has a

significant effect on the correction factors for camber and ideal angle of attack.

- Skew has little effect on the camber, but has a large effect on the ideal angel

of attack.

The most important question of course is how do propellers designed using the

correction factors, procedure the desired thrust at the design advance ratio.

6

1.3 Present Research

In the present work, we consider the solution of the both lifting line and lifting

surface problem for a propeller with arbitrary blade outline, pitch distribution and

circulation distribution operating in an axially directed velocity field. It is assumed

that the radial circulation distribution is known from lifting line solution and that the

blade surface is to be formed from a known mean line type by determining the

camber and the pitch at each radius. These two parameters are to be determined

by the requirement that the desired radial circulation distribution is obtained with the

sections operating at their ideal angle of attack. The chord-wise circulation

distribution will then be determined by two conditions such as the boundary

condition that the flow be tangent to the blade surface and the Kutta condition.

In the present approach, no restrictive assumptions need to be made as to the

circulation distribution or blade outline. The procedure adapted for lifting line and

lifting surface theory is similar to the method of Kerwin [13]. The continuous

distribution of radial and helical vortices is replaced by a lattice of discrete vortex

lines. The lattice can be considered as formed from a number of horseshoe vortex

elements of constant strength. The velocity induced at an arbitrary point in space

by each lattice element can be determined by integration according to the law of

Biot-Savart. By determining the velocity at a number of control points on the blade

surface at the mid-points of the lattice a set of linear equations may be formed

relating the strengths of the lattice elements to the shape of the blade surface.

1.3.1 Basic Assumptions

The fluid is assumed to be frictionless and incompressible. The inflow velocity, is

assumed to be axial and a function of radius only.

For the solution of lifting surface problem, the free vortex system is assumed to lie

on a helical surface whose pitch is determined from lifting line theory with the same

radial load distribution. The blade surface is assumed to be approximately on the

helical reference surface.

7

The problem is linearized to the extent that the boundary condition is applied on

the helical surface rather than on the blade itself and the induced velocities are

assumed to be small relative to the resultant inflow. It is assumed that the Kutta

condition holds, i.e. that the bound circulation is zero at the trailing edge. It is also

assumed that the bound circulation is zero at the blade tip and at the hub.

1.3.2 Outline of Research

As the present work is concerned with the development of a marine propeller

design method based on lifting line theory and lifting surface correction factors, first

of all, a design methodology is proposed and discussed in Chapter 3. The major

portion of the design procedure involves lifting line and lifting surface theory

application. The theoretical formulation related to the velocity induced by free

vortex lines and radial vortex lines are derived in Chapter 2. Solution of propeller

lifting line problem and propeller lifting surface problem are also formulated here. In

Chapter 4, the derived theoretical formulations are applied for the design of a wake

adapted propeller with symmetrical blade outline. Numerical results related to the

design of the propeller are presented and compared for validity in chapter 5.

8

Chapter 2

Theoretical Formulation

2.1 The velocity induced by free vortex lines

The free vortex lines generated by a propeller is assumed to be of true helical

shape i.e. their radius and pitch remains constant, and there will be z number of

vortices of equal strength symmetrically located around the circumference. The

axial extent of the set of vortices are

i) Finite, in the case of vortex segment lying on the blade surface and

ii) Semi-infinite, in the case of free vortex system extending downstream

from the trailing edge of the blade.

The velocity induced by a vortex line of arbitrary shape may be expressed in terms

of an integral taken along the vortex line by means of Biot-Savart’s law.

In a vortex lattice approximation to the lifting line and lifting surface problem, the

velocity induced at an arbitrary point in space by a segment of a helical vortex line

must be determined. Since this is a three-dimensional problem, the Biot-Savart

integrals would appear to provide the best way of obtaining the induced velocities.

In the case of finite interval, the integration may be performed by numerical

methods. In the semi-infinite case, numerical integration may be used up to a

sufficient large distance downstream.

2.1.1 The induced velocity components determined by Biot-Savart’s Law

As shown in Figure 2.1, a left handed Cartesian co-ordinate system is located with

the Z axis along the propeller axis of rotation with positive direction downstream.

The Y axis passes through the control point, i.e., the point in space where the

velocity is to be determined. A cylindrical system (z, r, θ) is oriented so that the line

z=0, θ=0 in the cylindrical system corresponds to the Y-axis in the Cartesian

system.

9

Figure 2.1 Coordinate system and notation for helical vortices

10

There will be z helical vortices (one from each blade) which have the following

properties:

a) The vortices all start with the same axial coordinate z0, radial coordinate r0,

but with different angular co-ordinates θ= θp, p=1, 2, 3 ….. z

b) The vortices are of constant radius r0, and constant pitch angle βio

Biot-Savart’s Law may be written

�� = �� … … (2.1)

Where, Γ = Vortex strength (m2/sec)

��= Vector distance from vortex element to control point (m)

��= Vector element of distance along the vortex (m)

��= Vector induced velocity (m/sec)

The distance S has the following x, y and z components:

�� = �−�� + ��, � − �� + ��, −!� − ��"#�$%�& … (2.2)

Where θ is the angular coordinate measured from θp as shown in Figure 2.1

The vortex element dl is

�� = �� + ��, −�� + ��, "#�$%'&�� … (2.3)

The cross product ��()��( is as follows

��)�� = �� * +�� ,�� -�� (� + ��) −��(� + ��) "#�$%'−��(� + ��) � − �� (� + ��) −!� − ��"#�$%'*

11

=��0!�� + �� − "#�$%'�� − �� + �� − �� ++��&,��"#�$%'�−�� + �� + �� + ��& + !�� + ��,−�� + �� + ��1 … (2.4)

and the scalar quantity S3 is

S3 = ��4 + ��4 − 2��6 ��θ + ��+(!� + �� tan$%')4&3/4 … (2.5)

Substituting Eqs. (2.2 - 2.5) in Eq. (2.1) and summing over the z blades gives the

following expressions for the axial, tangential and radial velocity components.

�< = �= = �.?@�� ∑ B �C�DB �−�� + �� + ��&�� … (2.6)

�E = �F = �.?@�� ∑ B � [−"#�$%'. 0� − �� + ��1C�DB + �� + ��(!� +�� tan$%')H�� … (2.7)

�? = �I = �.?@�� ∑ B � [(��"#�$%' + !�). � �� + ��C�DB − �� tan $%' �� +��H�� … (2.8)

The above equations, after due changes in nomenclature, are in agreement with

Ullah [22] and Glover [9] formula. Furthermore, in the special case when one of the

helix starting angles, θp , as well as the axial starting point, x0 , are zero, these

expressions agree with those given by Ullah [22] and Glover [9]. This latter case

corresponds to the velocity components at a blade in propeller lifting line theory.

Equations (2.6) to (2.8) can be made non-dimensional in terms of the following

variables

J = ��

K = =@? … … (2.9)

��( = ��?L�

12

The non-dimensional induced velocity components ��( can then be written as

�<�� = J �∑ BM�N

C�DB �−J + � �� + ��&�� … (2.10)

�E�� = J�∑ 1P32 �− tan $� . 01 − J cos�� + �U�1 + sin�� + �U� 0K +WU=1

J� tan $� 1& �� … ... (2.11)

�?�� = J2 �∑ 1P32 X−tan $� . �� + �U� + Y�"#�$� + KJZ � �� + �U�[WU=1 ��

… … (2.12)

where the denominator in each of the integrals above is

P�N = �(K + J� tan$%')4 + 1 + J4 − 2J� �� + ��&3/4 … (2.13)

In general, the velocity component normal to a particular boundary is to be

determined. Let (l, m, n) be the (x, y, z) components of a unit vector normal to the

surface. The non-dimensional normal velocity is then given by

\]�� = �. �<�� + ^. �?�� + �. �E�� … (2.14)

The velocity component to a true helical surface can be determined by substituting

the component of unit normal in Eq. (2.14). Choosing the positive direction for the

normal to be directed upstream, i.e. in the direction in which a propeller would

normally be developing thrust, there follows,

� = −cos $% ^ = 0 � = sin $% \]�� = −�<�� cos $% +�E��sin $%

Where, βi is the pitch angle of the helix at the control point radius, r

13

2.2 The velocity induced by radial vortex lines

The velocity induced by a straight radial vortex segment of constant strength can

be obtained by integration using Biot-Savart’s Law.

The notation to be used is shown in Figure 2.2. A set of z radial vortex lines are

located at angles θp and extend from r1 to r2.

Figure 2.2 Coordinate system for radial vortices

14

The component of the vector element of Vortex lines �� are

�� = ��, ��6 ��,0& … … (2.15)

and the distance from the vortex element to the control point is

�� = �−��, � − ��6 ��, −!�& … … (2.16)

The cross – product ��)�� is as follows

��)�� = �� * +�� ,��( -��(sin �� cos �� 0−�� sin �� − �� cos �� −!�* . = ��0−!�� , !��, ��1 … … (2.17)

And the scalar quantity S3 is

S3 = ��4 + ��4 − 2�� +!�4&3/4 … … (2.18)

Substituting the quantities into the expression for Biot-Savart’s Law Eq. (2.1), the

following expressions for the velocity components are obtained

�< = �= = �� ∑ ?`%]ab?@( )�=�DB?4?B

�? = 0 ... ... (2.19)

�E = �F = �� ∑ c=@d'`ab?@( )�=�DB?4?B

These can be expressed in terms of the non-dimensional quantities

J = ?@? K = =@? (2.20)

��( = ��?L�

15

resulting in the following expressions

��< =∑ ��=�DB � ef�/NeNeg … … (2.21)

��E = −K ∑ � ��=�DB � ef�/NeNeg

where, h�N = �K4 + 1 + J4 − 2J� ��&�N … (2.22)

Equations (2.21) can be integrated to give the following

��< =∑ ��.=�DB ij … … (2.23)

��E =−K ∑ � ��=�DB . ij

where, ij = ecd'`ab�kNl`%]ab�fg/NmegeN K4 + ��4�� ≠ 0 … (2.24)

ij = cB4�eld'`ab�NmegeN K4 + ��4�� ≠ 0 … (2.25)

The latter form corresponds to the case when the vortex segment coincides with

the Y axis, at which point the velocity is Zero as can be seen from Eq. (2.21)

The velocity normal to a helical surface with pitch angle βi at a radius r is

�]�� = −�<��. 6 �$% +�E��. ��$%

which in this case can be written as

�]�� = ∑ ��. 6 �$% + K6 ��. ��$%&=�DB i� … (2.26)

16

2.3 Solution of Propeller Lifting line problem by vortex lattice methods

The solution required for the problem is to find the radial distribution of circulation to

produce a free vortex sheet of true helical shape in homogeneous flow, i.e. the

optimum propeller. By computing the velocity induced by each element of the lattice

at a number of control points on the lifting line, a set of linear equations results

relating the strength of the individual vortices to the resultant slope of the flow at the

control points. In general, the velocity induced at some point on a propeller blade

will be due to both the free vortex system and the bound vortices. However, in

lifting line theory where the blades have been replaced by straight, radial bound

vortices only the free vortex system need be considered. This is because the

resultant velocity induced anywhere on one lifting line by a symmetrically arranged

set of lifting lines of equal strength is zero.

To obtain the radial distribution of circulation by a lattice method the free vortex

sheet is replaced by a finite number of helical line vortices as shown schematically

in Figure 2.3. By computing the velocity induced by each element of the lattice at a

number of control points on the lifting line, a set of linear equations result relating

the strength of the individual vortices to the resultant slope of the flow at the control

points.

The usual non-dimensional circulation which is defined as

op = �4�qrs … … (2.27)

where, ᴦ = Strength of bound vortex at radius r (m2/Sec)

r = Radius of the vortex element under consideration (m)

R = Propeller Radius (m)

Vs = Speed of ship (m/sec)

17

In the present work, it is more convenient to use u* as the non-dimensionalizing

velocity so that G will be independent of loading

o = �4�qt∗ … … (2.28)

Where u* = displacement velocity, defined in Figure 2.4 (m/sec)

Figure 2.3 Schematic arrangement of vortex lattice with M=10, P=5

18

Figure 2.4 Velocity diagram – Optimum lifting line propeller

19

To proceed with the specific formulation of the problem, it is first assumed that the

strength of the bound vortex representing each blade is given by an I-term Fourier

sine series

o(v) = �(w)4�qt∗ = ∑ #%x%DB ��v … … (2.29)

Where G is the non-dimensional bound vortex strength and ρ is a new variable

which is zero at the hub radius rh and π at the tip. The variables ρ and x are related

by

y = B4 (1 + yz) −B4 (1 − yz)� �v v = � �cB[BlF{c4FBcF{ H … … (2.30)

The vortex distribution given by Eq. (2.29) is automatically zero at the hub and tip

for any values of the coefficient ai.

The vortex lattice arrangement is shown schematically in Figure 2.3. The interval

from r = rh to r = R is divided into M equal spaces and the radius to the inner end of

the m th space is called rom. The continuous bound vortex distribution G(r) is

replaced by a stepped distribution whose value is equal to that of the continuous

distribution at the mid-point of each interval.

o| = o[B4 }(��)|lB + (��)|~H,(1 ≤ ^ ≤ � − 1) oB = o[B4 }(��)4 +�z~H,(^ = 1) … (2.31)

o� = o[B4 }� + (��)� ~H,(^ = �)

The free vortex lines originate at (ro)m where the value of Gm changes. Calling the

free vortex at (ro)m , o|�� there follow

o|�� = o| −o|cB … … (2.32)

20

This can be made to hold for m = 1, 2……..M+1 by defining the non-existent vortex

segments

o� = o�lB = 0 … … (2.33)

It should be noted that the same result could be obtained by noting that the

strength of the continuous free vortex sheet at a radius r is (dG/dr) and replacing

the derivative of G by the first order central difference.

The free vortex lines can be considered as replacing a continuous vortex sheet

which extends ½ space on either side of the free vortex. The only exception is at

both ends, where in the continuous case, the sheet must end at the hub and blade

tip. It would therefore seem reasonable to move the end vortices in 1/8 space so

that they would be located approximately in the region which would actually be

occupied by the sheet. In this case, the free vortices are at the following radii :

(��)| = �z + (?c?{)(|cB)� 2 ≤ ^ ≤ �

(��)B = �z + B� . (qc?{)�

(��)�lB = 1 −B� . (qc?{)� … (2.34)

The velocity is to be computed at P control points located at radii r1, r2 ………….rp

midway between free vortex elements. There is no restriction on how many of the

available control point positions are to be used.

The non-dimensional velocity component induced at rp by a set of semi-infinite

helical vortices originating from each blade with radius rom are

(��<)|� = (�<)|�. ��?b�� = 4Fb(t�)�bt∗��

(��E)|� = (�E)|�. ��?b�� = 4Fb(t�)�bt∗�� … (2.35)

(��])|� = (�])|�. ��?b�� = 4Fb(t�)�bt∗��

21

Where �� is the non-dimensional velocity as defined in Chapter 2.1, U is the

dimensional velocity and the subscriptions a, t and n axial, tangential and normal

components.

The requirement that the relative flow at the lifting line be of constant pitch can be

seen from Figure 2.4 to be

�∗ = t�� = t��N �� = t�� = 6 ��"#�" … (2.36)

expressed in terms of either the tangential, axial or normal components. These

relations make use of the known result that the resultant induced velocity is normal

to the helical surface formed by the free vortex system.

2.3.1 Wake adapted propeller

The preceding development can be extended very easily to the case where the

pitch of the free vortex system is arbitrary, and the axial inflow velocity Va is a

prescribed function of radius.

It is assumed that the pitch angle of the free vortex system βi (r) and geometric

inflow angle β(r) = tancB � r��.?� is known and that the non-dimensional circulation G

is to be determined.

In this case it will be necessary to compute the normal velocity component, since

the resultant velocity is not necessarily normal to the free vortex sheets.

In this case, the boundary condition may be written as follows:

(�])� = B4Fb∑ (�])|��lB|DB .∑ #%x%DB (�|∗ sin �. v| −�|cB∗ sin �. v|cB) =��∗ . (cos $%)� … … (2.37)

23

2.4 Solution of Propeller Lifting Surface problem

Now we consider the problem of determining the camber and pitch correction for a

propeller with a prescribed blade outline, mean line type and radial load distribution.

The pitch and camber corrections are determined by the requirement that the

prescribed radial load distribution be obtained with the sections operating at their

ideal angle of attack.

The nomenclature used in this section is basically the same as in the lifting line

case except that an extra dimension must be added due to the chord wise load

distribution.

As shown in Figures 2.5 and 2.6, an (x’, y’, z’) Cartesian coordinate system is fixed

on the propeller with the Z’ axis axial and the Y’ axis passing through the tip of the

index blade. The X’ axis completes the left handed system. A cylindrical system (z’,

r’, θ’) corresponds to the (x’, y’, z’) system with θ’=0 on the Y’ axis and the positive

θ’ clockwise when loading in the positive Z’ direction.

A moveable Cartesian system (x, y, z) and a corresponding cylindrical system (z,r,

θ) is oriented with the Z axis axial and Y-axis (on θ =0 line) passing through a

particular control point on the index blade.

There are PxQ control points on the index blade where p= 1, 2 …..P indicates

radial positions and q = 1, 2…….. Q indicates chord wise position. It should be

mentioned that all pairs of coordinates or subscripts referring to radial and chord

wise directions are given adjacent alphabetic symbols with the higher symbol

(alphabetically) referring to the chord wise direction.

There will be P x Q possible positions for the moveable system and the notation ypq

for example means the Y-axis of the moveable system corresponding to the pq-th

control point. Following the notation, the quantities θpq and (zo)pq are the

displacement of the moveable system measured from the fixed system.

24

A non-dimensional radius is defined as x=r/R where R is the radius of the propeller.

To distinguish the radius of a control point from that of a helical vortex line (on the

end of a bound vortex segment) the latter is given a zero subscript. The non-

dimensional quantities

J = ?@?

K = =?

Figure 2.5 Lifting Surface notation

25

Figure 2.6 Coordinate System

26

Finally a curvilinear system is defined at any radius by the intersection of an axial

cylinder with the reference of helical surface. The origin is taken at the mid-chord

line of the blade whose angular coordinate in the (z’, r’, θ’) system in �. The S axis

is along the helix with the positive direction towards the trailing edge. The n-axis is

perpendicular to S and lies on the cylindrical surface with the positive direction

upstream as shown in Figure 2.6. If the cylindrical surface is expanded and viewed

from the propeller axis out towards the tip, a blade section results as shown in

Figure 2.7. The chord length of the expanded section is l(r), consequently,

s = - l/2 corresponds to the leading edge and

s = + l/2 corresponds to the trailing edge.

The angle of attack of the section relative to the reference helix is α and the

maximum camber measured from the nose-tail line is given the symbol f.

2.4.1 The reference Helix

The blade surface is assumed to be approximately on a helical surface whose pitch

at any given radius is determined by the angle of relative flow according to lifting

line theory with the same radial load distribution. In the present work it is assumed

that the reference helix passes through the y’-axis. As the helix is of constant pitch,

any radial line will be contained in the surface. It is further assumed that the bound

vortex segments are radial, and that the axial distance between a control point and

a vortex element is the same as the helical surface were of constant pitch

corresponding to the pitch at the control point radius.

27

Figure 2.7 Expanded blade section

2.4.2 Bound Vortex Distribution

The bound circulation distribution over the blade surface will be expressed by a

trigonometric series in the variables ρ and σ which are related to r and s by

� = B4 (� + �z) −B4 (� − �z) cos v .

� = − 4 cos � … … (2.41)

from which there follows

r =rh when ρ = 0

r = R when ρ = π

r = -l/2 when σ = 0

r = +l/2 when σ = π … … (2.42)

28

The vortex sheet strength γ can be converted to a non-dimensional quantity s by

dividing by the displacement velocity U* as defined in the preceding section. It is

assumed that S can be represented by a series of the form

S(ρ, σ) = �� X∑ C�� sin iρ. cot �4��DB + ∑ ∑ C� sin iρ sin jσ¢ DB��DB [ … (2.43)

The second part is a Fourier sine series, which has the property that S=0 along the

edge of the blade for any value of constant Cij

The first term goes to zero all along the trailing edge, but tends to infinity at the

leading edge. For a fixed value of ρ this is the chord wise circulation distribution of

a flat plate at a small angle of attack in two dimensional flow. According to

linearized two dimensional thin airfoil theory the chord-wise circulation distribution

of any mean line can be obtained by superimposing the flat plate distribution and a

general distribution which is zero at both the trailing and leading edge. The angle of

attack for which the coefficient of the “flat plate” term is zero is called the “ideal

angle of attack”.

The radial circulation distribution is obtained by integrating γ over the chord at a

particular radius

Γ(v) = � ¤(v, �)�� `¥ �� …. … (2.44)

or in terms of non-dimensional quantities

o(v) = B�M � �(v, �)�� `¥ �� … … (2.45)

where G is the non-dimensional circulation defined in the preceding section as

o = ��Mt∗ Substituting Eq. (2.43) for s in Eq. (2.45) and integrating gives the result

o(v) = ∑ (26%' +6%B) sin �vx%DB … … (2.46)

29

If we now require that a particular radial load distribution o(v) is to be obtained in

the sections operating at their ideal angle of attack, there follows that 6%'=0, and

that 6%Bare the known Fourier coefficients of the radial circulation distribution. The

remaining coefficients

6%¦ i = 1, 2, ………….. I

j = 2, 3, …………J

which do not contribute to the radial load distribution are to be determined by the

boundary condition on the blade surface. For later use it will be convenient to

define

§¦(v) = ∑ C� sin iρx%DB … … (2.47)

So that the Eq. (2.43) becomes

�(v, �) = �̈© ∑ §¦(v) sin ª�«¦DB … … (2.48)

provided the angle of attack at each radius is ideal.

2.4.3 Vortex Lattice

The continuous bound vortex sheet is to be approximated by a finite number of

radial bound vortex segments each with constant strength. At the ends of each

segment a free vortex of the same strength must be shed forming a “horseshoe”

vortex system as shown in Figures 2.5 and 2.6. Naturally, parts of the free vortex

system originating from bound vortices at the same and immediately adjacent radii

coincide. Although this fact will be useful for computational process, each

horseshoe system will be considered logically to be an independent unit.

30

The lattice arrangement is obtained by dividing the interval between the hub and

the blade tip into M equal spaces. The free vortices are shed at radii

(��)| = (qc?{)(|cB)� + �z … … (2.49)

except at the ends, where they are moved in 1/8 space towards the interior of the

blade (as in the lifting line case).

There are N radial vortex elements between any two adjacent values of r0. These

will be centered at

�| = B4 [(��)| + (��)|lBH … … (2.50)

and will be located by dividing the chord length at rm into N equal panels with the

bound vortex at the mid-point of each panel.

The chord-wise position relative to the mid-chord line is given by

�|] = �4¬ (2� − − 1) … … (2.51)

and the angular coordinate measured clockwise from the y’ axis is

�|] = �¬M (�� )�F� (2� − − 1) … (2.52)

Control points are located at the midpoints of the panels formed by the horseshoe

elements. In general, there will be many more horseshoe elements than control

points, and it is completely arbitrary which of the possible control point arrangement

are to be used. However, to simplify the computations somewhat, it will be

assumed that the chord-wise arrangement of control points will be the same at

each radial position used. The number of chord wise control points is given by the

expression

® =¬c4l¯gc¯N¯g … … (2.53)

31

Where, ζ1 is the number of radial vortex elements between each control point and

ζ2 is the number of unused control point positions between the leading edge and

the first control point. If Eq. (2.53) is a fraction, only the integer part is to be retained.

The control point angles are then given by

��° = b¬M . (��)bFb . [2}�B(± − 1) + �4 + 1~ − H … (2.54)

There are a total of P radial positions used, and are subject only to the restriction

that P≤ M. the total number of control points is PXQ.

2.4.4 Relating continuous and lattice distributions

Let Gmn be the non-dimensional strength of the bound vortex located at θmn and

centered at rm. The strengths of the individual elements are first of all subject to the

requirement that the radial load distribution be the same as in the continuous case.

∑ o|] = (§B)|¬]DB … … (2.55)

The remaining N-1 requirements will be that the lattice and continuous distributions

induce the same velocity at each of the N-1 possible control point positions in two-

dimensional flow.

From thin airfoil theory, the non-dimensional velocity induced at the qth control

point by the N vortices at a particular radius rm is

(t�)�²t∗ =¬M� ∑ ��4(]c°)cB¬]DB … … (2.56)

where un is the dimensional velocity normal to the vortex sheet.

The velocity induced at the same point by the continuous distribution can be shown

to be

32

(t�)�²t∗ = 4M� ∑ §¦ cos(ª + 1)�°«¦D� … (2.57)

where, �° = coscB X¬c4°¬ [ ,0 ≤ �° ≤ ³ … (2.58)

Equating (2.58) and (2.57) for each value of q the following equation is obtained.

∑ ��4(]c°)cB¬]DB = 4¬∑ §¦ cos(ª + 1)�°«¦D�

q = 1, 2 ……………… N-1 … (2.59)

which combined with Eq. (2.55) results in a set of N linear equations for the

unknown Gmn.

Let the solution of this set of equations be expressed in the form

o|] =∑ ´]¦§¦|«¦DB … … (2.60)

The chord load factors ´]¦ are constants which can be computed once and for all.

Values of µ are given by Falkner [5]. Values of ´]¦ correct upto 6 decimal places,

were computed for

N = 2, 4, 6, 8

and J = 0, 1, 2 ……………. N-1

using IBM 650 and these result appear in Appendix -C

33

2.4.5 Velocity induced by the lattice in 3-Dimensional flow

Let �|]�° be the normal component of the non-dimensional velocity induced by the

complete horseshoe system o|] at the control point at rp , θpq

The subscript n for ‘normal’ will be omitted in this section since only the normal

component will be considered.

As � is related to the dimensional velocity u by

�|]�° =�|]�° ��?b�µ¶ …. … (2.61)

which can also be expressed in terms of the non-dimensional circulation

�|]�° = t��b²t∗ 4Fb·µ¶ … … (2.62)

This velocity can be computed by a procedure which is outlined in Appendix – A.

2.4.6 Determining the Camber and Angle of Attack

As was mentioned previously, it is assumed that the blade surface is to be formed

such that its expanded sections may all be derived from a single mean line by

suitably selecting the camber-chord length ratio (f/l) and the angle of attack α at

each radius. The angle of attack is to be measured from the induced inflow angle βi

determined from the lifting line theory. It is also assumed that the magnitude of the

resultant inflow velocity is the same as in the lifting line case, namely, V*.

The value of f/l and α at each radius are determined by the boundary condition that

the flow be tangent to the mean line at each control point. The slope of the mean

line relative to βi at a particular chord-wise station is

¸� −ℎ° . �º��

34

where, hq is the slope of the mean line with unit camber ratio.

The boundary condition can be written as

¸� −ℎ° . �º�� = − Brb∗ �∑ ∑ �|]�°¬]DB�|DB & − ($% − $)� … (2.63)

assuming that the induced angles are very small.

Introducing Eq. (2.62) and noting that

($% − $) ≅ t∗ ��r∗ … … (2.64)

there follows,

¸� −ℎ° . �º�� =− Brb∗ ��∗ (cos $%)� + B4Fb∑ �|∗ ∑ �|]�°¬]DB�|DB o|]m (2.65)

It is now convenient to express u*/v* in terms of the lift coefficient of the section.

From Kutta-Joukowski’s Law

�¼ = v½∗Γ�� … … (2.66)

where dL is the lift force acting on an element of bound vortex of radius dr and ρ is

the fluid mass density.

The lift coefficient is

6¾ = ¾gNw(r∗)N.? = 4�r∗ =¿4��À̈ Á . �t∗r∗� … (2.67)

Replacing G by b1 in the Eq. (2.67) and combining with Eqs. (2.65) and (2.60) there

follows

35

ÂbÃÄ −ℎ° . �Å̈�bÃÄ =− (/M)b4�(Æg)b �(cos $%)� + B4Fb∑ �|� ∑ �|]�°¬]DB�|DB ∑ ´]¦«¦DB §¦|m

… …. (2.68)

where ζmn is a factor which takes into account that u* may be a function of radius

and is defined by

ζ|� = t�∗tb∗ = (��)�c(��)�(��)bc(��)b . �?�?b m … (2.69)

The quantities on the left in Eq. (2.68) are the angle of attack and the camber ratio

per unit lift coefficient and are given by the symbol

¸ = ÂÃÄ … … (2.70)

Ç = º/ÃÄ … … (2.71)

In the two dimensional flow, these are constants which depends only on the type of

mean line. The ratio of camber required in three dimensional to that required for an

equal lift coefficient in two dimensional flow is the camber correction factor as

defined in current propeller design methods. However, a similar definition cannot be

used for the pitch correction since the ideal angle of attack of many mean lines in

two dimensional flow is zero.

Equation (2.68) written for each control point represents a set of linear equations

for ¸ , Ç and the coefficients of the non-lift producing part of the circulation

distribution.

Rearranging the Eq. (2.68) to put the unknowns on the left and introducing Eq.

(2.47)

36

È��(Æg)bFb�©̈�b É ¸ − È��(Æg)bFbz²�©̈�b É Ç + ∑ ζ|� ∑ �|]�°¬]DB�|DB ∑ ´]¦«¦D4 ∑ 6%¦x%DB sin �v|

= −2y�(cos $%)� −∑ ζ|�∑ �|]�°¬]DB ´]B�|DB ∑ 6%Bx%DB sin �v| p = 1, 2, 3 ……………… P

q = 1, 2, 3 ………………..Q …. … (2.72)

if the number of radial terms I in the Fourier series for the circulation distribution is

equal to the number of radial control point positions P, and if the number of chord-

wise terms J is one less than Q, the number of unknowns will be

2Ê + i(Ë − 1) = 2Ê + Ê(® − 2) = Ê® … … (2.73)

which equals to the number of equations. The reason that J=Q-1 is that the first

term of the series is determined in advance by specifying the radial load distribution.

Consequently, there must be at least two chord-wise control points in order to

determine a pitch and camber correction.

2.4.7 The symmetry of the velocity field

In the special case when both the blade outline and the mean line are symmetrical

about the Y'-axis, the important simplification results from the symmetry of the

integrals determining �|]�° As a result, it can be shown that within the limitations

of the assumptions outlined in the Chapter 1, a propeller with symmetrical blades

has no pitch correction due to lifting surface effect.

37

2.4.8 Modification of preceding results for symmetrical blades

The development as per done will now be modified to take advantage of these

results. The continuous vortex strength in Eq. (2.43) is re-written as follows

S(ρ, σ) = �� X∑ ∑ C� sin iρ sin(2j − 1)σ¢ DB��DB [ … (2.74)

which is symmetrical about the mid-chord. Control points will be distributed only

over the downstream half of the chord, and in particular cannot be located at the

mid-chord, since this will result in matrix A in Eq. (2.78) being singular.

It is also convenient to define N as the number of chord-wise lattice elements on

each side of the mid-chord, so that the total number is 2N.

The angular co-ordinates of the bound vortex elements are given by the expression

�|] = �4¬M (��)�F� (®] − 2 − 1) …. (2.75)

which replaces Eq. (2.52). The number of chord-wise control points Q is still given

by Eq. (2.53) since N has been re-defined.

However, the expression for the control point angles Eq. (2.54) is now as follows

��° = b¬M (��)bFb (ζB(± − 1) +ζ4 + 1) … (2.76)

38

The final set of equations are practically containing the same as in Eqs. (2.70) and

(2.71) except that the terms containing the pitch correction ¸ are no longer present.

4³(§B)�y�ℎ°Ç − ∑ ζ|� ∑ �|]�°4¬]DB ∑ ´]¦«¦D4�|DB ∑ 6%¦ x%DB sin �v|

= 2y�(cos$%)� +∑ ζ|� ∑ �|]�°4¬]DB ´]B�|DB ∑ 6%Bx%DB sin �v|

… … (2.77)

The set of equation can be written as matrix notation

ÍÎ¾ . )¾ =ÏÎ … … (2.78)

Where,

ÍÎ¾ =ÐÑÒÑÓ (Æg)bFbz²�©̈�b ……… Õ- = (U − 1)® + ±� = U −∑ ζ|�∑ �|]�°4¬]DB ´]¦�|DB sin �v| …Õ - = (U − 1)® + ±� = Ê + (� − 1)(Ë − 1) + (ª − 1)

ÏÎ = 2y�(cos$%)� +∑ ζ|� ∑ �|]�°4¬]DB ´]B�|DB ∑ 6%Bx%DB sin �v|

…… - = (U − 1)® + ±

)¾ = Ö4³Ç�%¦ …… � = U ≤ Ê…… � = Ê + (� − 1)(Ë − 1) + (ª − 1)

39

As,

�ºd�t]d'??×dE× =�ºd�4M = ÃÄ�� (for parabolic camber line)

�ºd�d'??×dE× = ØÃ �ºd�L]d'??×dE× �ºd�3M = ØÃ �ºd�4M

Therefore camber correction factor

ØÃ = �ÅÙ��©�ÅÙ�N© = �ÅÙ�ÚÄÛÜ = 4³Ç as Ç = �ÅÙ�ÃÄ

Therefore

)¾ = ÕØÃ �%¦ …… � = U ≤ Ê…… � = Ê + (� − 1)(Ë − 1) + (ª − 1)

40

Chapter 3

Design Methodology

3.1 Design Procedure

The design procedure followed for the design of a wake adapted propeller is

provided in Figure 3.1 and Figure 3.2 below.

Figure 2.1 : Ship propeller design method

Figure 3.1 : Ship propeller design method

Known data : Thrust to be produced, diameter, no. of blades, shaft revolutions, ship speed, wake variations and blade area ratio

Calculate : Advance ratio, thrust loading coefficient and ideal thrust loading coefficient

Calculate : Ideal efficiency using KRAMER DIAGRAM

Calculate : Hydrodynamic pitch angles without induced velocities

Calculate : Development of compute program for the calculation of optimum hydrodynamic pitch angles considering induced velocities through iterations using Lifting Line Theory.

The induced velocities will be calculated using Biot-Savart Law. The computer program will also include calculation of thrust, torque, thrust coefficient, power coefficient and delivered power.

Calculate : B.A.R considering cavitation

Calculate : Development of computer program for the calculation of section pitch and section camber considering correction factors using Lifting Surface Theory.

Calculate : Section chord, section lift coefficient, uncorrected section pitch and section camber

41

Data :

Figure 3.2 : Detail propeller design stages

Vs, ω(x), D, n, c(x), T, z

CT, J, CTi

Kramer Diagram

ηi : Ideal Efficiency

Hydro dynamic pitch angles - tan β - optimum tan βi(trial and error through iteration)

(Lifting line calculation)

T, Q, CT, CP, PD, AE/A0 (cavitation consideration)

Design of Blade section CL(c/D), f/c, P/D

Final Blade geometry f/c, P/D

Lifting surface correction to f/c, P/D

42

3.1.1 Relation between ÝÞß, KT, àáß

For the present design procedure propeller diameter, blade number and initial

chord distribution is taken as known parameter. For a propeller disk area, A0 and

speed of advance VA producing thrust T, the thrust loading coefficient of the

propeller is defined as

6âã = âgNwã@räN … … (3.1)

For a propeller diameter D and speed of rotation, n producing thrust, T the thrust

coefficient of the propeller is defined as

Øâ = âw]NMÛ … … (3.2)

Also for a propeller diameter, D, speed of rotation, n and speed of advance, VA, the

advance coefficient of the propeller is defined as

Ërã = rä]M … … (3.3)

6âã = âgNwã@räN = âw]NMÛ . �]NMN�räN = Øâ . �� . B«åäN = �� . Îæ«åäN … (3.4)

3.1.2 Propeller design using Circulation theory

The circulation theory of propeller is used to design a propeller in detail to obtain a

prescribed distribution of loading along the radius, a pitch distribution matching the

mean circumferential wake at each radius.

As with the design methods using methodological series data, several methods

exist for designing propeller with circulation theory. All these methods are

fundamentally the same but have difference only in details. One such method is

considered here.

43

It is assumed that the following quantities are known possibly through a preliminary

design calculation using methodical series data

- The ship speed, Vs, and the corresponding thrust, T

- The propeller diameter, D

- The number of blades, z

- An estimated blade area ratio AE / A0

- The depth of immersion of the shaft axis, h

- The propeller revolution rate, n

For a “Wake adapted propeller”, the mean effective circumferential wake factor, ω(x)

as a function of non-dimensional radius, x = r/R must also be known.

The volumetric mean nominal wake fraction ω is given by

� = � �(F).FFgç{gN(BcF{N) … … (3.5)

xh being the non-dimensional hub radius.

The mean velocity of advance of the propeller is therefore

VA = VS (1-ω) … … (3.6)

The next step is to determine the thrust loading coefficient

6âã = âgNwã@räN … … (3.7)

Where A0 = (π/4) D2, is the disc area of the propeller

The advance coefficient

Ërã = rä]M … … (3.8)

44

To estimate the ideal thrust loading coefficient Cè� (i.e. for an inviscid fluid), the

calculation of advance ratio, λ and average drag-lift ratio, ϵ� are required.

ë = rä�]M … … (3.9)

ì% = tan ¤ = Ã©ÃÄ = �.�C . ãíã@ − 0.02 … (3.10)

and thus obtained,

6â% = ÃæäBc4îï� … ... (3.11)

Next it is necessary to estimate the ideal efficiency ηi of the propeller. This is

usually done with the help of the Kramer diagram, which gives ηi as a function of Cè� , λ and z.

The hydrodynamic pitch angle to be determined at the various radii x,

tan $% = ��e� . X Bc�Bc�(F)[�Û … … (3.12)

according to Van Manen when

tan $ = rð[Bc�(F)H�]MF … … (3.13)

After determining the hydrodynamic pitch angle at the different radii, one may now

calculate the radial distribution of the ideal thrust coefficient.

Îæ�F = Ú.ÚÄ© � ÜçñåscòæåðmN.«åsN.C� �� … … (3.14)

Similarly the radial distribution of the ideal torque coefficient

45

Îó�F = F.Ú.ÚÄ© �}Bc�(F)~lôäåð mN.C� �� … … (3.15)

Integrating then yields a value of the ideal thrust coefficient,

Øâ% =� Îæ�F . �yBF{ … … (3.16)

Similarly, the ideal torque coefficient,

Øõ% =� Îó�F . �yBF{ … … (3.17)

The ideal efficiency,

J% = «åð(Bc�)4� . Îæ�Îó� … … (3.18)

Now the ideal thrust loading coefficient can be calculated as

6â% = �� . Îæ�«åäN … … (3.19)

This value of 6â% must be equal to the initial value of 6â% calculated from the Eq.

(3.11). If the two values of 6â% are not in agreement, the value of J% must be altered

and the calculation is to be repeated until the initial value of 6â% from Eq. (3.11) and

the final value obtained from Eq. (3.19) are in sufficiently close agreement .

Eckhardt and Morgan [4] suggest that the number of iteration to bring this about

can be redone by using the following empirical relation :

J%(Ø + 1) = e�(Î)BlÚæ�(@)öÚæ�(÷)øÚæ�(@) … … (3.20)

46

When J%(Ø) and J%(Ø + 1) are the value of J% for the K-th and (K+1)–th iterations, 6â%(0) is the desired value of 6â% and 6â% (K) is the value obtained in the K-th

iteration.

One may determine the radial distribution of the viscous thrust coefficient,

ÎæF = Ú.ÚÄ© � ÜçñåscòæåðmN.«åðN .�BcÚ©ÚÄ ��mC� �� … … (3.21)

Similarly the radial distribution of the viscous torque coefficient,

ÎóF = F.Ú.ÚÄ© �}Bc�(F)~lôäåð mN.�BlÚ©ÚÄ �� mC� �� … (3.22)

Integrating then yields a value of the viscous thrust coefficient,

Øâ = � ÎæF . �yBF{ … … (3.23)

Similarly, the viscous torque coefficient,

Øõ =� ÎóF . �yBF{ … … (3.24)

The viscous efficiency,

J = «åð(Bc�)4� . ÎæÎó … … (3.25)

Now the viscous thrust loading coefficient can be calculated as

6â = �� . Îæ«åäN … … (3.26)

and this should agree sufficiently well with the initial value at the start of the design

calculation. If the value of 6â obtained from Eq. (3.7) differs initially from the value

obtained from Eq. (3.26), the value of 6â must be altered and the entire calculation

47

starting with an initial estimate of J% repeated until a satisfactory is obtained

between initial and the final value of 6â. After this one may determine the power

coefficient.

The power coefficient is defined as

6j = j©gNwã@rä� … … (3.27)

Where the delivered power ÊM is obtained as

ÊM = 2³�® … … (3.28)

® =Øõv�4Pù … … (3.29)

Also, ú = Øâv�4P� … … (3.30)

The delivered power ÊM must match the design delivered power of the ship

propulsion plant; otherwise the ship speed and the corresponding propeller thrust

must be changed and the calculation repeated until the calculated 6j matches the

delivered power available.

The efficiency of the designed propeller in the behind condition is obtained as

Jû = ÃæÃü … … (3.31)

From the blade section velocity diagram in figure 3.3, it can be seen that

B4 . Lärð = ��.��(��c�)�� [1 − �(y)H … … (3.32)

B4 . Lærð = �� .��(��c�)�� [1 − �(y)H … … (3.33)

B4 . Lärð = B�� B4 Lærð� … … (3.34)

rýrð = ��(��c�)�� [1 − �(y)H … … (3.35)

48

Figure 3.3 Velocities and forces on a blade element

The non-dimensional circulation is defined as

op = ��Mrð =�� − 1�. o. [1 − �(y)H … (3.36)

o = ��ML∗ = ��Mrð . rðL∗ … … (3.37)

rðL∗ = �� − 1�. [1 − �(y)H … … (3.38)

Once the radial distribution of the hydrodynamic pitch angle for a specific ideal

thrust loading coefficient has been detected, the value of the product Ã.ÃÄM at various

radii can be obtained,

Ã.ÃÄM = 4�� [Bc�(F)Hlôäåð= 4��åýåð

… … (3.39)

using the relation L = ρ VR Γ … … (3.40)

49

3.2 Cavitation check

For warship propellers with special sections [8], the thrust loading coefficient,

þd = 0.013 + 0.5284��.�q + 0.3285��.�q4 − 1.0204��.�q3 … (3.41)

(0.11 ≤ ��.�q ≤ 0.43) which is the upper limit, and where

��.�q = �älwz�c��gNwr@.�ýN and þd = æäbgNwr@.�ýN … (3.42)

If the þd of the designed propeller is below that which is upper limit value, then the

propeller is non-caviting.

3.3 Calculation of section chord, section lift coefficient, un-corrected

section pitch and section camber

As the section chord is taken as known parameter and if the cavitation criteria fulfils

then there is no problem. Otherwise the section chord is to be adjusted. Section lift coefficient, CL can be calculated from the Eq. (3.39).

The uncorrected section pitch can be calculated as

Êt]d'??×dE× = 2³� tan $% … … (3.43)

So, �jM�t]d'??×dE× = ³y tan $% … … (3.44)

as section angle of attack α = 0

The chosen mean camber line is taken as of parabolic shape [1, 11, 15, 21]. Now for the parabolic mean line [14]

6¾ = 2³¸ + 4³ ºd … … (3.45)

As α = 0, the camber chord ratio is calculated as

�ºd�t]d'??×dE× = ÃÄ�� … … (3.46)

50

3.4 Calculation of lifting surface geometry

The lifting surface program is developed based on theoretical formulation

developed in Chapter 2.4. By taking radial load distribution and hydrodynamic pitch

angle from lifting line program, the necessary calculation is performed using lifting

surface theory and section pitch and camber correction factors are calculated. As

the blade outline is assumed symmetrical and the section thickness symmetrical

about the mid chord, there is no pitch correction. Only the section camber

correction is calculated. The correction factor is calculated as Kc. Also chord-wise

circulation distribution is calculated and later integrated to get the radial circulation

distribution. This radial circulation distribution is checked with lifting line circulation

distribution.

The vortex sheet strength γ is converted to non-dimensional quantity S by dividing

by the displacement velocity u* as defined in Chapter 2.

It is assumed that S is represented by the series

S(ρ, σ) = �� X∑ ∑ C� sin iρ sin jσ¢ DB��DB [ … … (3.47)

For the present calculation, I = 3 and J = 2

Later the non-dimensional quantity G, is calculated as

o(v) = B�M � �(v, �)�� `¥ �� … … (3.48)

For parabolic mean line,

Ç(y) = Ç|<F �1 − �4FÃ �4m … … (3.49)

So that the slope is

ºF =−�Fº��çÃN … … (3.50)

51

But since

y = − d4 cos � … … (3.51)

`¥ = d4 sin � … … (3.52)

The slope can be written as

ºF = 4 º��çd cos � … … (3.53)

At L. E x = - c/2, σ = 0

At T. E x = + c/2, σ = π

For

Ç|<F� = 1.0and �Ç�y = −1.3333 σ = 109.47 0

Also for

Ç|<F� = 1.0and �Ç�y = −2.66666 σ = 131.805 0

3.5 Calculation of corrected section pitch and section camber

As it is assumed that the blade section outline is symmetrical and also the

thickness form is symmetrical about the mid chord, there will be no section pitch

correction. Only section camber correction is to be taken into account for

calculation of section camber. The corrected section camber is

�ºd�Ã'??×dE× = Ød �ºd�t]d'??×dE× … … (3.54)

where, Kc is the section camber correction factor, calculated using lifting surface

theory.

52

Chapter 4

Application of theoretical formulation for the design of a marine propeller

4.1 Design of Marine Propeller

Based on the theoretical formulation developed in Chapter 2 and related design

methodology proposed in Chapter 3, the design of a marine propeller is performed

here which uses the data of US Navy combatant DTMB 5415 ship [12].

Propeller Data

Thrust, T = 237321 lb = 107649 Kg

Ship Speed, Vs = 21.1 knots = 10.8538 m/s

No. of blades, z = 4

Propeller Diameter, D = 21.1 ft = 6.4329 m

Density of water, ρ = 104.49 kg sec2 / m4

Radius (non- dimensional)

x

Wake distribution

ω(x)

Chord distribution

c(x) / D

0.2 0.424 0.220

0.3 0.344 0.249

0.4 0.288 0.271

0.5 0.249 0.284

0.6 0.221 0.289

0.7 0.200 0.284

0.8 0.185 0.260

0.9 0.174 0.209

1.0 0.161 0.000

53

4.2 Calculation of nominal mean wake

Table 4.1: Nominal mean wake calculation

Radius ,

x

Wake Distribution,

ω(x) x . ω(x) S.M.

Product of the

function

0.2 0.424 0.0848 1 0.0848

0.3 0.344 0.1032 4 0.4128

0.4 0.288 0.1152 2 0.2304

0.5 0.249 0.1245 4 0.4980

0.6 0.221 0.1326 2 0.2652

0.7 0.200 0.1400 4 0.5600

0.8 0.185 0.1480 2 0.2960

0.9 0.174 0.1566 4 0.6264

1.0 0.161 0.1610 1 0.1610

∑ = 3.1346

x.ω(x)dxBF{ = 13 ∗ 0.1 ∗ 3.1346 = 0.1045

12 (1 − yz4) = 12 (1 − 0.24) = 0.48 Therefore, Nominal mean wake,

� =� x.ω(x)dxBF{B4 (1 − yz4) = 0.10450.48 = 0.2177

54

4.3 Calculation of Blade Area Ratio

Table 4.2: Blade area ratio calculation

Radius

x

Chord Distribution,

c(x) / D S.M. Product of the function

0.2 0.220 1 0.220

0.3 0.249 4 0.996

0.4 0.271 2 0.542

0.5 0.284 4 1.136

0.6 0.289 2 0.578

0.7 0.284 4 1.136

0.8 0.260 2 0.520

0.9 0.209 4 0.836

1.0 0.000 1 0.000

∑ = 5.964

c(y)D . dxB

F{ = 13 ∗ 0.1 ∗ 5.964 = 0.1988

ÍfÍ� = 2!³ c(y)D . dxB

F{ = 2 ∗ 4³ ∗ 0.1988 = 0.5062

55

4.4 Calculation related to lifting line program

Í� =³4 ∗ (6.4329)4 = 32.5015^4

½ã = ½̀ (1 − �) = 10.8538 ∗ (1 − 0.2177) = 8.4909^/� 6â = úB4 vÍ�½ã4 = 107649B4 ∗ 104.49 ∗ 32.5015 ∗ (8.4909)4 = 0.8793

ì% = tan ¤ = 6M6¾ =0.4! . ÍfÍ� − 0.02 = 0.44 ∗ 0.5062 − 0.02 = 0.0306

ë = ½ã³�P = 8.4909³ ∗ 1.6933 ∗ 6.4329 = 0.2481

6â% = 6â1 − 2ëì% = 0.87931 − 2 ∗ 0.2481 ∗ 0.0306 = 0.8929

From the Kramer diagram,

For z =4, ë = 0.2481 and 6â% = 0.8929

The ideal efficiency, J%= 0.77

Hydrodynamic pitch angle – 1 st iteration with ��= 0.77

tan $ = rð[Bc�(F)H�]MF = B�.�ù3�∗[Bc�(F)H�∗B.��33∗�.�34�∗F = 0.3172 X[Bc�(F)HF [ Optimum tan $% = ��e� . X Bc�Bc�(F)[�Û =�(F)e�

Where �(y) = tan $ X Bc�Bc�(F)[�Û

56

Table 4.3: Hydrodynamic pitch angle – 1 st iteration with ��= 0.77

Radius

x 1-ω(x) tan$

Pitch

angle, $

(deg)

1-ω � 1 − �1 − �(y)m�Û F(x) tan$%

Hydrodynamic

Pitch angle, $% (deg)

0.2 0.576 0.9135 42.4129 0.7823 1.2581 1.1493 1.4926 56.1790

0.3 0.656 0.6936 34.7456 0.7823 1.1412 0.7915 1.0280 45.7897

0.4 0.712 0.5646 29.4498 0.7823 1.0732 0.6059 0.7869 38.1998

0.5 0.751 0.4764 25.4748 0.7823 1.0311 0.4913 0.6380 32.5372

0.6 0.779 0.4118 22.3834 0.7823 1.0032 0.4131 0.5365 28.2154

0.7 0.800 0.3625 19.9263 0.7823 0.9834 0.3565 0.4630 24.8423

0.8 0.815 0.3231 17.9081 0.7823 0.9698 0.3134 0.4070 22.1452

0.9 0.826 0.2911 16.2313 0.7823 0.9601 0.2795 0.3630 19.9494

1.0 0.839 0.2661 14.9028 0.7823 0.9489 0.2525 0.3280 18.1570

6â%(1) = 0.7259 (1st iteration)

6â%(0) = 0.8929 (required)

The value of J% for the next iteration is

J%(2) = J%(1)1 +Ãæ�(�)cÃæ�(B)ùÃæ�(�)= 0.771 + �.��4�c�.�4ù�ù∗�.��4�

= 0.7422

57

Table 4.4 Hydrodynamic pitch angle – 2 nd iteration with ��= 0.7422

Radius

x F(x) tan$% Hydrodynamic Pitch

angle, $%(deg)

0.2 1.1493 1.5485 57.1463

0.3 0.7915 1.0665 46.8420

0.4 0.6059 0.8164 39.2279

0.5 0.4913 0.6619 33.4998

0.6 0.4131 0.5566 29.1020

0.7 0.3565 0.4803 25.6550

0.8 0.3134 0.4222 22.8905

0.9 0.2795 0.3766 20.6347

1.0 0.2525 0.3402 18.7901

6â%(2) = 0.8412 (2nd iteration)

6â%(0) = 0.8929 (required)

For J%(1) = 0.77, 6â%(1) = 0.7259

For J%(2) = 0.7422, 6â%(2) = 0.8412

By linear interpolation for the required 6â%(0) = 0.8929,

J%(3) = 0.7297

58

Table 4.5 Hydrodynamic pitch angle – 3 rd iteration with ��= 0.7297

Radius

x F(x) �� Hydrodynamic Pitch

angle, ��(deg)

0.2 1.1493 1.5751 57.5882

0.3 0.7915 1.0847 47.3273

0.4 0.6059 0.8304 39.7055

0.5 0.4913 0.6732 33.9492

0.6 0.4131 0.5662 29.5174

0.7 0.3565 0.4885 26.0369

0.8 0.3134 0.4295 23.2412

0.9 0.2795 0.3830 20.9577

1.0 0.2525 0.3461 19.0889

6â%(3) = 0.8947 (3rd iteration)

6â%(0) = 0.8929 (required)


J%(4) = J%(3)1 +Ãæ�(�)cÃæ�(3)ùÃæ�(�)= 0.72971 + �.��4�c�.��ù∗�.��4�

= 0.73

59

Table 4.6: Hydrodynamic pitch angle – 4 th iteration with ��= 0.73

Radius


angle, ��(deg)

0.2 1.1493 1.5744 57.5775

0.3 0.7915 1.0843 47.3155

0.4 0.6059 0.8300 39.6939

0.5 0.4913 0.6729 33.9382

0.6 0.4131 0.5659 29.5073

0.7 0.3565 0.4883 26.0276

0.8 0.3134 0.4293 23.2327

0.9 0.2795 0.3829 20.9499

1.0 0.2525 0.3459 19.0816

6â%(4) = 0.8931 (4th iteration)

6â%(0) = 0.8929 (required)

Including viscous effect

6â(4) = 0.8639

6â(0) = 0.8793 (required)

The calculation under predicts by 1.75 %. So If the 6â is increased by 1.75%, then

6â(0) = 0.9085

60


J%(5) = J%(4)1 +Ãæ(�)cÃæ(�)ùÃæ(�)= 0.731 + �.��ùc�.��3Bù∗�.��ù

= 0.7275

Table 4.7 Hydrodynamic pitch angle – 5 th iteration with ��= 0.7275

Radius

x F(x) ��

Hydrodynamic Pitch

angle, ��(deg)

0.2 1.1493 1.5798 57.6664

0.3 0.7915 1.0880 47.4135

0.4 0.6059 0.8329 39.7905

0.5 0.4913 0.6753 34.0293

0.6 0.4131 0.5679 29.5916

0.7 0.3565 0.4900 26.1052

0.8 0.3134 0.4308 23.3040

0.9 0.2795 0.3842 21.0156

1.0 0.2525 0.3471 19.1424

6â%(5) = 0.9042 (5th iteration)

6â%(0) = 0.9085 (required)


J%(6) = J%(5)1 +Ãæ�(�)cÃæ�(ù)ùÃæ�(�)= 0.72751 + �.��ùc�.��4ù∗�.��ù

= 0.7268

61

Table 4.8: Hydrodynamic pitch angle – 6 th iteration with ��= 0.7268

Radius


angle, ��(deg)

0.2 1.1493 1.5813 57.6914

0.3 0.7915 1.0891 47.4409

0.4 0.6059 0.8337 39.8176

0.5 0.4913 0.6759 34.0549

0.6 0.4131 0.5684 29.6153

0.7 0.3565 0.4905 26.1270

0.8 0.3134 0.4312 23.3241

0.9 0.2795 0.3845 21.0340

1.0 0.2525 0.3474 19.1595

6â%(6) = 0.9072 (6th iteration)

6â%(0) = 0.9085 (required)

Difference = - 0.14 %

Including viscous effect

6â(6) = 0.8772

6â(��±��) = 0.8793

Difference = - 0.24 %

Hence values of 6th iteration is accepted.

62

4.5 Cavitation check

For warship propellers with special sections,

þd = 0.013 + 0.5284��.�q + 0.3285��.�q4 − 1.0204��.�q3 (0.11 ≤ ��.�q ≤ 0.43) ��.�q = �älwz�c��gNwr@.�ýN

Here,

pA = 101.325 kPa

ρ = 1025 kg/m3

g = 9.81 m/s2

h = 4.5 m

pv = 1.704 kPa

V0.7R = V2A + (0.7πnD)2

Now B4v½�.�q4 = 667.5517-Ê#

Uã + vℎ� −U� = 144.8696-Ê#

So,

��.�q = Uã + vℎ� −U�B4v½�.�q4 = 0.2170

Cavitation Number, þd = 0.1327

Therefore 0.1327 is the upper limit of þd for warships with special sections

63

Now for the present case

þd = æäbgNwr@.�ýN

Here,

T = 1053.689 KN

Ap = (1.067- 0.229 P/D) AD

P/D = π x tan βi = 0.4905

AD =(AE/A0) (π/4) D2 = 16.4523 m2

Ap = 15.7066 m2

So

þd = 0.1005

As the þd = 0.1005 is lower than þd = 0.1327 (upper limit) at σ0.7R=0.217,

The propeller is non-cavitating.

64

4.6 Calculation of section chord, section lift coefficient and uncorrected

section pitch and section camber

�jM�t]d'??×dE× = ³y tan $% and

�Ç��t]d'??×dE× = 6¾4³

The calculation is presented in the table below

Table 4.9 Calculation of section chord, section lift coefficient and

uncorrected section pitch and section camber

Radius

x

Hydrodynamic

Pitch Angle,

βi (deg)

uncorrected

Pitch

Diameter

ratio, (P/D)

c.CL/D

Chord

Distribution

c/D

Lift

Coefficient

CL

Uncorrected

Section

camber, f/c

0.2 57.6914 0.9936 0.0000 0.2200 0.0000 0.0000

0.3 47.4401 1.0264 0.0952 0.2490 0.3823 0.0304

0.4 39.8164 1.0476 0.1085 0.2710 0.4004 0.0319

0.5 34.0524 1.0616 0.1066 0.2840 0.3754 0.0299

0.6 29.6131 1.0714 0.0979 0.2890 0.3388 0.0270

0.7 26.1282 1.0787 0.0854 0.2840 0.3007 0.0239

0.8 23.3193 1.0834 0.0697 0.2600 0.2681 0.0213

0.9 21.0349 1.0873 0.0493 0.2090 0.2359 0.0188

1.0 19.1579 1.0914 0.0000 0.0000 0.0000 0.0000

65

4.7 Calculation related to lifting surface program

In lifting surface program, radial circulation coefficient is required as input. The

necessary calculation is presented in table 4.10

Table 4.10 Radial circulation coefficient calculation

x=0.45 v = coscB(0.375) = 67.98�

x=0.65 v = coscB(−0.125) = 97.18�

x=0.85 v = coscB(−0.625) = 128.68�

v = coscB �1 + yz − 2y1 − yz m o(v) = �6%¦ sin �v3%DB

Radius x ρ (deg) Circulation, G( ρ)

0.45 67.98 0.0835

0.65 97.18 0.1105

0.85 128.68 0.0985 0.0835 =ÍB sin 67.98� + Í4 sin 135.96� + Í3 sin 203.94�0.1105 =ÍB sin 97.18� + Í4 sin 194.36� + Í3 sin 291.54�0.0985 =ÍB sin 128.68� + Í4 sin 257.36� + Í3 sin 386.04� 0.0835 = 0.9271ÍB + 0.6952Í4 − 0.4057Í30.1105 = 0.9922ÍB − 0.2480Í4 − 0.9302Í30.0985 = 0.7806ÍB − 0.9758Í4 + 0.4390Í3 0.08285 = 0.9199ÍB + 0.6898Í4 − 0.4025Í30.10244 = 0.9199ÍB − 0.2299Í4 − 0.8624Í3 0.01959 = 0 − 0.9197Í4 − 0.4599Í3 Í4 = −0.213 − 0.500054Í3 0.02071 = 0.2299ÍB + 0.1724Í4 − 0.1006Í30.07682 = 0.6898ÍB − 0.1724Í4 − 0.6467Í3 0.09753 = 0.9197ÍB + 0 − 0.7473Í3ÍB = 0.10605 + 0.8125Í3 0.0985 = 0.10356 + 1.561191Í3 Í3 = −0.0032411 = 63B Í4 = −0.0196793 = 64BÍB = −0.1034166 = 6BB

66

The Radial Circulation distribution G(x) obtained by the lifting line method is plotted

and presented in Figure 4.1.

Figure 4.1 Radial Distribution of Circulation G(x) Lifting line method

From the figure, it is found that the maximum circulation is available between the

radiuses 0.6 to 0.8

The vortex sheet strength S(ρ,σ) is calculated at non dimensional radius x= 0.45, x

= 0.65 and x = 0.85 at chord wise positions from 0 degree to 180 degree by lifting

surface method. Integrating the value of S(ρ,σ), we get the circulation G(ρ). The

results obtained from the above mentioned case is plotted in Figure 4.2 to 4.4.

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.2 0.4 0.6 0.8 1

G (

Circ

ulat

ion)

x (Non-dimensional radius)

67

Fig 4.2 : Chord-wise Circulation Distribution (x=0.45)


0.00

0.05

0.10

0.15

0.20

0.25

0 20 40 60 80 100 120 140 160 180

S (

Non

-dim

ensi

onal

vor

tex

she

et

stre

ngth

)

σ (Chordwise position in degree)

0.00

0.05

0.10

0.15

0.20

0.25

0 20 40 60 80 100 120 140 160 180

S (

Non

-dim

ensi

onal

vor

tex

she

et

stre

ngth

)


68


Circulation G(x) calculated by both lifting line and lifting surface method are

compared in Table 4.11 below.

Table 4.11 Value of Circulation G(x) from both methods

Radius, x ρ (deg) Circulation, G

(lifting line)

Circulation, G

(lifting surface)

0.45 67.98 0.083505 0.084085

0.65 97.18 0.110505 0.111272

0.85 128.68 0.098511 0.099196

0.00

0.05

0.10

0.15

0.20

0.25

0 20 40 60 80 100 120 140 160 180

S (

Non

-dim

ensi

onal

vor

tex

she

et

stre

ngth

)


69

4.8 Calculation of Camber correction factor

The camber correction factor, Kc is calculated from the Lifting Surface program and

presented in Table 4.12

Table 4.12 Camber correction factor, K c

Radius x Camber Correction factor Kc

0.45 1.253

0.65 1.330

0.85 1.382

4.9 Calculation of corrected section pitch and section camber

The corrected section pitch and section camber is calculated and presented in

Table 4.13

Table 4.13 Corrected section pitch and section camber

Radius

X

Uncorrected

Pitch Diameter

ratio, (P/D)

Corrected

Pitch Diameter

ratio, (P/D)

Uncorrected

section

camber, f/c

Camber

Correction

factor, Kc

Corrected

section

camber, f/c

0.2 0.9936 0.9936 0.0000 1.130 0.0000

0.3 1.0264 1.0264 0.0304 1.180 0.0359

0.4 1.0476 1.0476 0.0319 1.230 0.0392

0.5 1.0616 1.0616 0.0299 1.270 0.0380

0.6 1.0714 1.0714 0.0270 1.315 0.0355

0.7 1.0787 1.0787 0.0239 1.345 0.0321

0.8 1.0834 1.0834 0.0213 1.360 0.0290

0.9 1.0873 1.0873 0.0188 1.380 0.0259

1.0 1.0914 1.0914 0.0000 1.380 0.0000

70

Chapter - 5

Results and Discussion

Based on the theoretical formulation developed in Chapter 2 and related design

methodology proposed in Chapter 3, the design method calculations of the marine

propeller are performed in Chapter 4 which use the data of US NAVY Combatant

DTMB 5415 Ship [12].

In our design calculations number of blades, propeller diameter, ship speed, thrust

to be delivered, radial wake variation and section chord length variation are taken

as known data as same as stated by Kamal & Sallah [12]. Only the methodology of

design is different.

i. Initially this design method takes thrust to be produced by the propeller as a

known data and thrust developed by the designed propeller is found from the

program as output parameter. The validity of the method can be checked from

comparison of the two data. In this thesis, the thrust value is also checked with

the reference [12] data which is 107649 kg and from the program output the

value is 1053.71 kN or 107449 Kg, which is only 0.19% less from the desired

value. So the thrust computed by this design method agrees well with the thrust

predicted by Kamal & Sallah [12].

ii. The calculated chord wise circulation distribution by lifting surface theory is then

integrated to get the radial variation of the resultant circulation, which is

compared with the values developed from lifting line theory. Table 5.1 shows

the comparison of circulation distribution between lifting line and lifting surface

methods. The variation of the circulation distribution calculated by both methods

is agreed with each other.

71

Table 5.1: Comparison of circulation distribution by two methods

x ρ (deg) Circulation, G from lifting line method

Circulation, G by lifting surface method

Difference

0.45 67.98 0.083505 0.084085 + 0.6945%

0.65 97.18 0.110505 0.111272 + 0.6941%

0.85 128.68 0.098511 0.099196 + 0.6953%

iii. The radial variation of hydrodynamic pitch angle (βi) computed by the present

method is compared with that predicted by Kamal & Sallah [12] and presented

in Figure 5.1.

Figure 5.1 Comparison of radial variation of Hydrodynamic pitch angle (βi)

10

20

30

40

50

60

70

0.2 0.4 0.6 0.8 1

βi

(H

ydro

-dyn

am

ic p

itch

an

gle

in

de

gre

e)

x (Non-dimensional Radius)

βi (Present work)

βi ([12])

72

From the Figure 5.1, it is observed that the calculated value of hydrodynamic

pitch angle (βi) at hub differs slightly with the predicted value but it fully agree

with that value at the tip. The difference in section hydrodynamic pitch angle (βi)

is due to different boundary condition of loading.

iv. The radial variation of Circulation (G) computed by the present method is

compared with that predicted by Kamal & Sallah [12] and presented in Figure

5.2.

Figure 5.2 Comparison of radial variation of Non-dimensional Circulation (G)

From the Figure 5.2, it is observed that the calculated value of Circulation (G) at

hub differs slightly with the predicted value but it fully agree with that value at

the tip. The difference in section Circulation (G) is due to different boundary

condition of loading.

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.2 0.4 0.6 0.8 1

G (

Non

-dim

ensi

onal

Circ

ulat

ion)


G (Present work)

G ([12])

73

v. The radial variation of Pitch-Diameter ratio (P/D) computed by the present


in Figure 5.3.

Figure 5.3 Comparison of radial variation of Pitch-Diameter ratio (P/D)

From the Figure 5.3, it is observed that the calculated value of Pitch-Diameter ratio

(P/D) at hub fully agree with the predicted value but it differs slightly with that value

at the tip. The difference in section Pitch-Diameter ratio (P/D) is due to the choice

of different mean line. In the present design, parabolic mean lines is used where in

reference [12] NACA mean lines is used.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.10

1.20

0.2 0.4 0.6 0.8 1

P/D

(P

itch-

Dia

met

er r

atio

)


P/D (Present work)

P/D ([12])

74

vi. The radial variation of Camber-Chord ratio (f/c) computed by the present


in Figure 5.4.

Figure 5.4 Comparison of radial variation of Camber-Chord ratio (f/c)

From the Figure 5.4, it is observed that the calculated Camber-Chord ratio (f/c)

value at hub differs slightly with the predicted value but it fully agree with that value

at the tip. The difference in section Camber-Chord ratio (f/c) is due to the choice of

different mean line. In the present design, parabolic mean lines is used where in

reference [12] NACA mean lines is used.

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.2 0.4 0.6 0.8 1

f/c

(C

am

be

r-C

ho

rd r

ati

o)


f/c (Present work)

f/c ([12])

75

vii. Initially the ideal efficiency (ηi) is predicted by using Kramer’s Diagram (taken

as ηi=0.77) and several iterations are done to find out final CT. The same CT

value is obtained from any suitable arbitrary prediction of efficiency (ηi), this

time it is taken as ηi=0.70. The comparison of the two method is presented in

Figure 5.5

Fig 5.5 Convergence of CTi & CT values at different iteration from different initial

iteration values

From the Figure 5.5, it is observe that for different initial value of ideal efficiency the

CTi values differs initially but after some iterations both CTi values & CT values

converge to a same definite value. So the iterations can be started from any initial

value of ideal efficiency.

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1 2 3 4 5 6 7

Cтi

& C

тva

lue

Iteration number

Convergence of Cтi & Cт value at different iteration from different

initial effeciency value

Cтi at assumed ηi =0.7

Cт at assumed ηi =0.7

Cтi at assumed ηi =0.77

Cт at assumed ηi =0.77

76

viii. The Pitch-Diameter ratio distribution is calculated for the designed thrust to be

produced and it is compared with the same distribution calculated by varying

±10% of the designed thrust. The result of the above procedure is presented in

Figure 5.6.

Fig 5.6: Effect of thrust requirement on radial Pitch-Diameter ratio distribution (P/D)

From the Figure 5.6, It is observed that the increase of thrust value results the

increase of Pitch-Diameter ratio and the decrease of the thrust value results the

decrease of the Pitch-Diameter ratio and it is very usual.

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

P/D

(P

itch-

Dia

mte

r ra

tion)

x (Non dimensional Radius)

Radial Pitch ratio (P/D) distribution

P/D at 10% increase of Design Thrust

P/D at Design Thrust T

P/D at 10% decrease of Design Thrust

77

ix. The Camber-Chord ratio distribution is calculated for the designed thrust to be

produced and it is compared with the same distribution calculated by varying

±10% of the designed thrust. The result of the above procedure is presented in

Figure 5.7.

Fig 5.7: Effect of thrust requirement on radial Camber-Chord ratio distribution (f/c)

From the Figure 5.7, It is observed that the increase of thrust value results the

increase of Camber-Chord ratio and the decrease of the thrust value represents

the decrease of the Camber-Chord ratio and it is very usual.

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

f/c (

Cam

ber-

Cho

rd r

atio

)


Radial Camber-Chord distribution (f/c)

f/c at 10% increase of Design Thrust

f/c at Design Thrust T

f/c at 10% decrease of Design Thrust

78

Chapter - 6

Conclusion and recommendation

6.1 Conclusion

The development of a design method of a wake adapted marine propeller using

lifting line theory with lifting surface correction factors has been presented in this

thesis. From the above results and discussions, following conclusions can be

drawn:

• The present method based on lifting line theory corrected with lifting surface

theory is prospective for marine propeller design.

• The present design method adopts wake distribution easily.

• The computed radial variations of circulation, hydrodynamic pitch angle, Pitch

ratio (P/D), camber ratio (f/c) agree well with those predicted by other

researchers.

• The pitch ratios at different radial positions increase with the increase in thrust

requirement. Similarly the camber ratios at different radial positions increase

with the increase in thrust requirement.

79

6.2 Further Recommendation

Areas for future research could be concentrated as follows:

• The basic assumption of the program was symmetrical shape section of the

blade, and no rake or skew was considered, so no pitch correction was

required. The present lifting surface program might be modified to take into

account variations in skew and rake to get the pitch corrections.

• The program might be further modified to accommodate finer lattice spacing

with an increase in the number of chord-wise control points in order to obtain

additional terms in the Fourier series for the circulation distribution. Then a

more accurate value of pitch and camber correction factors may be obtained.

• This program was developed for high speed crafts using parabolic shape

sections in all radial positions. The program may be modified by introducing

NACA shape sections in all radial positions or mixed parabolic and NACA

shape sections in different radial positions.

80

Reference

1. Avci, A. G. & Korkurt, E., 2011, “Wake adapted propeller design application to Navy ships”, INT-NAM 2011, p465-475.

2. Celik, F. and Guner, M., 2006, “An improved lifting line method for the design of marine propellers”, Marine technology, Volume 43, No. 2.

3. Cheng, H. M., 1964-1965 “Hydrodynamic aspects of propeller design based on lifting surface theory”, Part 1 & 2, David Taylor model basin reports, 1802 & 1803,

4. Eckhardt, M.K. and Morgan, W. B., 1955, “A propeller design method”, Transaction of the Society of Naval Architects and Marine Engineers (SNAME), U.S.A., Volume 63.

5. Falkner, V. M., 1947, “The solution of lifting-plane problems by vortex-lattice theory”, Aeronautical Research Council, R&M no. 2591.

6. Geoffrey, G. C. and Morgan, W. B., 1972, “The use of theory in propeller design”, SNAME, Chesapeake section.

7. Geoffrey, G. C., 1961, “Correction to the camber of constant pitch propeller”, RINA (page 227-243).

8. Ghose, J.P. and Gokarn, R.P., 2004, “Basic ship propulsion”, Allied Publishers Pvt. Ltd., India.

9. Glover, E. J., 1972, “A design method for the heavily loaded marine propeller”, Transaction RINA.

10. Harley, E. E., 1965, “A comparison of the lifting surface corrections Calculated by different methods for three propeller designs”, Hydrodynamics laboratory, Department of Navy, U.S.A., Report No-2049.

11. Hill, J. G., 1949, “The design of propellers”, Transaction SNAME, p143-171.

12. Kamal, I. Z. M. & Sallah, M. S. B. M., 2011, “Wake adapted propeller design based on lifting line and lifting surface theory”, MIMET.

13. Kerwin, J. E., 1961, “The solution of propeller lifting surface problems by vortex lattice methods”, MIT, Ph. D. thesis.

14. Kerwin, J. E., 2007, “Hydrofoils and propellers”, MIT, U.S.A.

15. Krishna, Y. G., 2015, “Design of fixed pitch propeller and water jet propulsion system for a frigate”, Indian journal of science and technology, Vol.7 (S7), 76-84.

81

16. Kuiper, G. 1971, “Some remarks on lifting surface theory”, ISP.

17. Lerbs, H.W., 1955, “Propeller pitch correction arising from lifting surface effect”, DTMB, Washington D.C., U.S.A., Report No. 942.

18. Morgan, W.B., Silovic, V. and Denny, S. B., 1968, “Propeller lifting surface corrections”, Transaction of the Society of Naval Architects and Marine Engineers (SNAME), U.S.A., Volume 76.

19. Parra, C., 2013, “Numerical investigation of the hydrodynamic performances of marine propeller”, University of Galati, Gdynia, Master’s thesis.

20. Pien, P. C., 1961, “The calculation of marine propellers based on lifting surface theory”, Journal of Ship Research, Vol 5, No.2.

21. Robert, F. K., 1973, “High speed propeller design”, SNAME, Spring meeting, Lake Buena Vista, April 2-4

22. Ullah, M. R., 1984, “The application of three-dimensional wing theory to the analysis of marine propeller performance”, Ph.D. thesis, University of Newcastle Upon Tyne, UK

23. Wilkins, J. R., 2012, “Propeller design optimization for tunnel bow thrusters in the bollard pull condition”, MIT, U.S.A., Masters of Science in mechanical engineering thesis.

82

Appendix - A

Program Description

Computer program were prepared to obtain numerical solutions of the following two

problems :

a. Determine the non-dimensional radial circulation distribution for a lifting line

propeller within a prescribed distribution of tanβ & tanβi

b. Determine the camber correction for a propeller within symmetrical blade

outline and a mean-line which is symmetrical about the mid-chord.

The principal source language was FORTRAN, each of the programs known as

lifting line program and lifting surface program consists of a number of specially

prepared subroutines. In some cases the same subroutines used for both programs.

Brief description of the main program and the principal sub routines will be given in

the following sections listing of the source programs are given in Appendix - B

Lifting line Program

A flow chart for the execution of the lifting line program is provided in Fig A1 below.

Fig A1 : Flow chart of lifting line program

MAINPROGRAM

SINTN CMA SOLVEFREPIN

83

SINTN = It is a function subroutine, performs integration.

CMA = It is a function sub routine.

PIN = It is a function subroutine, performs parabolic integrations.

FRE = This subroutine calculates the velocity induced by the free vortex line.

SOLVE = This subroutine solves a set algebraic equations.

Input data line preparations

The data file consists of 6 lines,

Line 1 : NCORD = 1 for viscous fluid, 2 for viscous fluid

NBL = Number of blades

DIA = Diameter of propeller (m)

DIAH = Propeller hub diameter (m)

BAR = Blade area ration

Vs = Slip speed (m/s)

RPS = Propeller rotational speed (rps)

Line 2 : XR (I), I = 1,9

9 values of non-dimensional radius

Line 3 : XWAKE (I) , I = 1,9

9 values of wake function at XR(I)

Line 4 : XTBI (I), I=1,9

9 values of hydrodynamic pitch angle (deg) at XR(I)

Line 5 : NST = Number of lattice spaces along lifting line

NCP = Number of control points along lifting line

Line 6 : NSC(I), I = 1, NCP

List of values of lattice containing the control point

84

Output file preparation

The main output is presented in the tabular form. In this respect two blades are presented.

Table 1

Col.1 : XR (I), I = 1,9


Col.2 : UA/VS (I), I = 1,9

9 values of non-dimensional axial induced velocity

Col.3 : UT/VS (I), I = 1,9

9 values of non-dimensional tangential axial velocity

Col.4 : US/VS (I), I = 1,9

9 values of non-dimensional induced velocity, u*

Table 2

Col.1 : XR (I), I = 1,9


Col.2 : XWAKE (I), I = 1,9

9 values of wake fraction at XR (I)

Col.3 : GVS (I), I = 1,9

9 values of non-dimensional circulation based on ship speed, Vs.

Col.4 : GUS (I), I = 1,9

9 values of non-dimensional circulation based on induced velocity, u*.

Col.5 : BETI (I), I = 1,9

9 values of section hydrodynamic pitch angle, deg

Col.6 : VR/VS (I), I = 1,9

9 values of non-dimensional section resultant inflow velocity

85

Col.7 : CCL/D (I), I = 1,9

9 values of section lift length coefficient.

Col.8 : DKT (I), I = 1,9

9 values of section Thrust coefficient.

Col.9 : DKQ (I), I = 1,9

9 values of section torque coefficient.

Additional output

WAKE = Mean wake fraction

KT = Propeller thrust coefficient

KQ = Propeller torque coefficient

EFF = Propeller Efficient

CT = Propeller thrust loading coefficient

CP = Propeller power coefficient

PD = Delivered power

T = Propeller thrust

Q = Propeller torque

EFFB = Efficiency of the designed propeller in the behind condition.

86

Lifting Surface Program

A flow chart for the execution of lifting surface program is provided in the fig A2 below

FigureA2 : Flow chart of lifting surface program

CMA It is a function of subroutine

PIN It is a function of subroutine performs parabolic interpolations

CAMBER This subroutine calculates the camber correction factors

HELIX This subroutine calculates the velocity induced by the free vortex lines on the blade and also beyond the blades upto 6 revolutions.

BOUND This subroutine calculates the velocity induced by the bound vortex lines

SOLVE This subroutine solves a set of algebraic equations.

SINTN It is a function of subroutine, performs integrations.

CMA PIN CAMBER

MAINPROGRAM

HELIX BOUND SINTNSOLVE

87

Input the file preparations

The data file consists of 9 lines

Line 1 : X(N), N = 1,9

9 values of non-dimensional radius.

Line 2 : XCHORD(N), N = 1,9

9 values of C/D at X(N)

Line 3 : XTBI (N), N = 1,9

9 values of hydrodynamic pitch angle, tanβi at X(N).

Line 4 : XTB(N), N = 1,9

9 values of advance angle tanβ at X(N).

Line 5 : KTEST = 1

MT = 8

NT = 3

NPT = 3

NZ1 =1

NZ 2 = 0

NG = 3 = Z = Number of blades

ALAM = 0.2423 = λi = x tanβi at X = 0.7

RH = 0.2 = Xh = non-dimensional hub ratio

GNZL = 0.08

Line 6 : MC (N), N = 1, 8

MC (1) = 3 = X(3) = 0.45

MC (2) = 5 = X(5) = 0.65

MC (3) = 7 = X(7) = 0.85

88

Line 7 : ((HMU(N,J), N=1,NT) J=1,JT)

Chord load factor, µ

HMU(1,1) = µ11= 0.109375

HMU(2,1) = µ21= 0.182292

HMU(3,1) = µ31= 0.208330

HMU(1,3) = µ13= 0.189887

HMU(2,3) = µ23= -0.002532

HMU(3,3) = µ33= -0.187355

Line 8 : H(N), N = 1, NQT

Camber slope, hq

H(1) = -1.333333

H(2) = -2.666666

Line 9 : XGAM (N), N = 1, 8

Circulation coefficient, An

XGAM (1) = +0.1034166

XGAM (2) = -0.0196793

XGAM (3) = -0.0032411

Output file preparation

The main output is presented in the tabular form

Table

Col 1 : XR (I), I = 1,3


Col 2 : Kc (I), I = 1,3

3 values of camber correction factor, Kc

89

Appendix - B

Program Listing, Data and Output

C*********************************************************************************************** C C "LIFTING LINE PROGRAM FOR MARINE PROPELLER CALCULATION" C C*********************************************************************************************** C JUNE 2015 C****************************************************************** C C BEGINING OF THE MAIN PROGRAME C DIMENSION XR(9), XWAKE(9), XTBI(9), NSC(4), XTB(9),XWAKEI(9) DIMENSION THSS(27), THS(28), BETI(9), VRVS(9), UAVS(9), UTVS(9) DIMENSION USVS(9),CCLD(9), DKT(9), DKQ(9) DIMENSION RZ(25), TANBZ(25),TBETAZ(25), R(24), RHO(25), BG(4) DIMENSION AG(4,4), UG(4,25), ZETA(4,25), CG(4), GAMMA(9), CIRC(9) DIMENSION TANBI(24), TBETA(24), COSBI(24) OPEN (UNIT=11, FILE='PA7.DAT') OPEN (UNIT=22, FILE='PA7.OUT') OPEN (UNIT=33, FILE='PA71.OUT') C DATA THSS/2*0.0625,3*0.125,6*0.25,16*0.5/ C C READ FROM DATA FILE C READ(11,1) NCOND,NBL,DIA,DIAH,BAR,VS,RPS READ(11,2) (XR(I), I=1,9) READ(11,2) (XWAKE(I), I=1,9) READ(11,2) (XTBI(I), I=1,9) READ(11,3) NST, NCP READ(11,4) (NSC(I), I=1,NCP) 1 FORMAT(5X, 2I2, 5F7.4) 2 FORMAT(5X, 9F7.4) 3 FORMAT(5X, 2I2) 4 FORMAT(5X, 4I2) WRITE(33,1) NCOND,NBL,DIA,DIAH,BAR,VS,RPS WRITE(33,2) (XR(I),I=1,9) WRITE(33,2) (XWAKE(I),I=1,9) WRITE(33,2) (XTBI(I),I=1,9) WRITE(33,3) NST,NCP WRITE(33,4) (NSC(I),I=1,NCP) C C CALCULATION - 1 C TPI=8.0*ATAN(1.0) RDS=57.2958 THS(1)=0.0 DO 5 I=2,28

90

THS(I)=THS(I-1)+THSS(I-1) 5 CONTINUE DO 51 I=1,28 THS(I)=THS(I)*(0.5*TPI) 51 CONTINUE DO 11 I=1,9 XWAKEI(I)=XWAKE(I)*XR(I) 11 CONTINUE WAKEI=SINTN(XR,XWAKEI,9) WAKE=WAKEI/(0.5*(1.0-XR(1)*XR(1))) DO 6 I=1,9 XTBI(I)=TAN(XTBI(I)/RDS) XTB(I)=(VS*(1.0-XWAKE(I)))/(0.5*TPI*RPS*DIA*XR(I)) 6 CONTINUE C C CALCULATION - 2 C RH=XR(1) MAX=NST+1 DO 7 N=1,9 XTBI(N)=XTBI(N)*XR(N) 7 XTB(N)=XTB(N)*XR(N) AMT=NST DELM=(1.0-RH)/AMT HDELM=0.5*DELM AM=RH-HDELM RZ(1)=RH+0.25*HDELM TEMP=RZ(1) YP=PIN(TEMP,XR,XTBI,9) TANBZ(1)=YP/RZ(1) DO 8 M=1,NST R(M)=AM+DELM AM=R(M) TEMP=AM RHO(M)=CMA(TEMP,RH) RZ(M+1)=R(M)+HDELM IF(M-NST)19,10,19 10 RZ(M+1)=RZ(M+1)-0.25*HDELM 19 TEMP=RZ(M+1) YP=PIN(TEMP,XR,XTBI,9) TANBZ(M+1)=YP/RZ(M+1) YP=PIN(TEMP,XR,XTB,9) TBETAZ(M+1)=YP/RZ(M+1) TEMP=R(M) YP=PIN(TEMP,XR,XTBI,9) TANBI(M)=YP/R(M) YP=PIN(TEMP,XR,XTB,9) TBETA(M)=YP/R(M) COSBI(M)=1.0/SQRT(1.0+TANBI(M)**2) 8 CONTINUE DO 9 I=1,NCP MS=NSC(I)

91

TDEL=R(MS)*(TANBI(MS)-TBETA(MS)) TBI=TANBI(MS) CBI=COSBI(MS) BG(I)=2.0*R(MS)*CBI DO 9 M=1, MAX ZETA(I,M)=RZ(M)*(TANBZ(M)-TBETAZ(M))/TDEL TBZ=TANBZ(M) XT=RZ(M) XF=R(MS) CALL FRE(XT,XF,TBZ,TBI,CBI,NBL,THS,UN) UG(I,M)=2.0*TPI*R(MS)*UN 9 CONTINUE C C CALCULATION - 3 C DO 18 I=1,NCP DO 18 K=1,NCP AG(I,K)=0.0 18 CONTINUE RHO(MAX)=0.0 DO 24 I=1,NCP AI=I SN1=0.0 DO 24 M=1,MAX IF(M-1)21,21,22 21 J=M GOTO 23 22 J=M-1 23 SN2=SIN(AI*RHO(M)) DO 20 K=1,NCP AG(K,I)=AG(K,I)+UG(K,M)*(SN2*ZETA(K,M)-SN1*ZETA(K,J)) 20 CONTINUE SN1=SN2 24 CONTINUE CALL SOLVE (AG,BG,NCP,CG) DO 25 M=1,9 GAMMA(M)=0.0 TEMP=XR(M) XTBI(M)=XTBI(M)/TEMP XTB(M)=XTB(M)/TEMP TEMP1=CMA(TEMP,RH) DO 26 I=1,NCP AI=I GAMMA(M)= GAMMA(M)+SIN(AI*TEMP1)*CG(I) 26 CONTINUE USVS(M)=(XTBI(M)/XTB(M)-1.0)*(1.0-XWAKE(M)) CIRC(M)=GAMMA(M)*USVS(M) 25 CONTINUE C C CALCULATION - 4 C VSJ=VS/(RPS*DIA)

92

VAJ=VSJ*(1.0-WAKE) CDL=(0.4/NBL)*BAR-0.02 IF(NCOND.EQ.1) CDL=0.0 DO 34 J=1,9 T1=ATAN(XTBI(J)) T2=ATAN(XTB(J)) VAVS=1.0-XWAKE(J) VTVS=(TPI*0.5*XR(J))/VSJ VRVS(J)=VAVS*COS(T1-T2)/SIN(T2) UTVS(J)=VRVS(J)*SIN(T1)*TAN(T1-T2) UAVS(J)=TAN(T1)*(VAVS/TAN(T2)-UTVS(J))-VAVS VAT=VAVS+UAVS(J) VTT=VTVS-UTVS(J) BETI(J)=ATAN(VAT/VTT) VRVS(J)=VAT/(SIN(BETI(J))) CCLD(J)=TPI*CIRC(J)/VRVS(J) DKT(J)=(CCLD(J)*(VTT*VTT)*(VSJ*VSJ)*(1.0-CDL*TAN(BETI(J))) 1 *NBL)/(4.0*COS(BETI(J))) DKQ(J)=(XR(J)*CCLD(J)*(VAT*VAT)*(1.0+CDL/TAN(BETI(J))) 1 *NBL)/(8.0*SIN(BETI(J))) 34 CONTINUE 38 TKT=SINTN(XR,DKT,9) TCT=(8.0/(0.5*TPI))*(TKT/(VAJ*VAJ)) TKQ=SINTN(XR,DKQ,9) C C WRITE TO OUTPUT FILE C WRITE(22,35) 35 FORMAT(2X,'TABLE:PROPELLER PERFORMANCE CHARACTERISTICS', 1 /7X,'(LIFTING LINE PROGRAM)', 2 /2X,27(1H-)) WRITE(22,36) NBL,DIA,DIAH,BAR,VS,WAKE,RPS 36 FORMAT(/2X,'INPUT DATA'/2X,10(1H-)/2X, 'NBL=',I1, 1', DIA=',F7.4,'M, DIAH=',F7.4,'M, BAR=',F7.4/2X, 2'VS=',F7.4,'M/S, WAKE=',F7.4,', RPS=',F7.4,'R.P.S.') WRITE (22,27) WRITE (22,28) WRITE (22,29) WRITE (22,30) DO 31 J=1,9 WRITE (22,32) XR(J), UAVS(J), UTVS(J), USVS(J) 31 CONTINUE 27 FORMAT (/2X,'OUTPUT RESULTS'/2X,14(1H-)) 28 FORMAT (10X,'INDUCED VELOCITY') 29 FORMAT (2X,25(1H-)) 30 FORMAT (2X,'XR',7X,'UA/VS',3X,'UT/VS',3X,'US/VS') 32 FORMAT (4(2X,F6.4)) WRITE (22,40)VSJ,VAJ,CDL WRITE (22,42) DO 43 L=1,9 BETI(L)=BETI(L)*57.2958 WRITE (22,44) XR(L),XWAKE(L),CIRC(L),GAMMA(L),BETI(L),VRVS(L),

93

1 CCLD(L), DKT(L), DKQ(L) 43 CONTINUE WRITE (22,45) TKT,TCT,TKQ 40 FORMAT (/2X, 'VSJ=',F8.4,10X, 'VAJ=',F8.4,10X,'CDL=',F8.4) 42 FORMAT (2X, 62(1H-)/2X,'XR',8X,'XWAKE',3X,'G/VS',4X,'G/US', 1 3X,'BETAI',3X,'VR/VS',3X,'CCL/D',5X,'DKT',5X,'DKQ') 44 FORMAT (2X,F6.4,8(1X,F7.4)) 45 FORMAT (2X,'KT=',F10.5,2X'CT=',F10.5,2X,'KQ=',F10.5) IF(NCOND.EQ.1) GO TO 99 CONS=1.025*(RPS*RPS)*(DIA**4.0) THRUST=CONS*TKT TORQUE=CONS*DIA*TKQ POWDEL=TPI*RPS*TORQUE POWCO=POWDEL/(0.5*1.025*((TPI/8.0)*DIA*DIA)* 1 ((VS*(1.0-WAKE))**3.0)) EFFB=TCT/POWCO WRITE(22,46) THRUST,TORQUE,POWDEL,POWCO,EFFB 46 FORMAT(/2X,'THRUST=',F10.2,2X,'KN' 1 /2X,'TORQUE=',F10.2,2X,'KN.m' 2 /2X,'DELIVERED POWER=',F10.2,2X,'KW' 3 /2X,'POWER COEFFICIENT=',F10.6 4 /2X,'EFFICIENCY(BEHIND)=',F10.6) 99 STOP END C C END OF MAIN PROGRAM C C SUB - 1 C FUNCTION SINTN(X,Y,N) DIMENSION X(9), Y(9) SINTN=0.0 T=0.0 DX1=X(2)-X(1) DD1=(Y(2)-Y(1))/DX1 IF(N-2)1,2,3 2 SINTN=(Y(2)+Y(1))*DX1/2.0 GOTO 1 3 DO 4 I=3,N DX2=X(I)-X(I-1) DD2=(Y(I)-Y(I-1))/DX2 D1=(DD2-DD1)/(DX1+DX2) D2=(DD1+D1*DX1)/2.0 D3=D1/3.0 ST=((D3*DX1-D2)*DX1+Y(I-1))*DX1 SINTN=SINTN+(T+ST)/2.0 T=((D3*DX2+D2)*DX2+Y(I-1))*DX2 DX1=DX2 DD1=DD2 IF(I-3)4,5,4 5 SINTN=SINTN+SINTN 4 CONTINUE

94

SINTN=SINTN+T 1 RETURN END C C SUB - 2 C FUNCTION PIN(TP,AX,AY,NO) DIMENSION AX(9), AY(9) ARA(Q1,Q2,Q3,Q4,Q5,Q6,Q7)= 1 Q5*(Q4-Q2)*(Q4-Q3)/((Q1-Q2)*(Q1-Q3))+ 2 Q6*(Q4-Q1)*(Q4-Q3)/((Q2-Q1)*(Q2-Q3))+ 3 Q7*(Q4-Q1)*(Q4-Q2)/((Q3-Q1)*(Q3-Q2)) IF(TP-AX(1))2,1,3 1 PIN=AY(1) GOTO 9 2 PIN=ARA(AX(1),AX(2),AX(3),TP,AY(1),AY(2),AY(3)) GOTO 9 3 IF(TP-AX(2))2,4,5 4 PIN=AY(2) GOTO 9 5 DO 6 I=3,NO M=I IF(TP-AX(I))8,7,6 7 PIN=AY(I) GOTO 9 6 CONTINUE 8 PIN=ARA(AX(M-2),AX(M-1),AX(M),TP,AY(M-2),AY(M-1),AY(M)) 9 RETURN END C C SUB - 3 C FUNCTION CMA(TP,RH) PI=4.0*ATAN(1.0) IF (TP-0.999)2,2,1 1 CMA=PI GOTO 6 2 CN=(1.0+RH-2.0*TP)/(1.0-RH) IF(ABS(CN)-0.00001)3,3,4 3 CMA=0.5*PI GOTO 6 4 CTN=SQRT(1.0-CN**2)/CN CMA=ATAN(CTN) IF(CTN)5,6,6 5 CMA=CMA+PI 6 RETURN END C C SUB - 4 C SUBROUTINE FRE(XT,XF,TBZ,TBI,CBI,NBL,THS,UN) DIMENSION THS(28),UX(5),UY(5),UZ(5)

95

TPI=8.0*ATAN(1.0) RDS=57.2958 ZF=0.0 TF=0.0 F1=XF*SIN(TF/RDS) F2=XF*COS(TF/RDS) E3=TBZ UXF=0.0 UYF=0.0 UZF=0.0 DO 12 NB=1,NBL PHIZ=TPI*(NB-1)/NBL DO 13 J=1,27 DTH=(THS(J+1)-THS(J))/4.0 K=1 IF(NB.GT.1) K=2 L=0 DO 14 M=1,5,K L=L+1 TE1=THS(J)+DTH*(M-1) TE=PHIZ+TE1 E1=SIN(TE) E2=COS(TE) E4=ZF-XT*TE1*E3 A=XF*XF+XT*XT-2.0*XT*F1*E1-2.0*XT*F2*E2+E4*E4 A=A**1.5 A=1.0/(2.0*TPI*A) B=A*XT*E3*1.0 UX1=-E1*E4/E3-F2+XT*E2 UX(L)=B*UX1 UY1=F1-XT*E1-(1.0/E3)*E2*E4 UY(L)=B*UY1 UZ1=(1.0/E3)*E2*(F2-XT*E2)+(F1-XT*E1)*E1/E3 UZ(L)=B*UZ1 14 CONTINUE IF(NB.GT.1) GOTO 15 UXF=UXF+(DTH/3.0)*(UX(1)+UX(5)+2.0*UX(3)+4.0*(UX(2)+UX(4))) UYF=UYF+(DTH/3.0)*(UY(1)+UY(5)+2.0*UY(3)+4.0*(UY(2)+UY(4))) UZF=UZF+(DTH/3.0)*(UZ(1)+UZ(5)+2.0*UZ(3)+4.0*(UZ(2)+UZ(4))) GOTO 13 15 UXF=UXF+(DTH*2.0/3.0)*(UX(1)+UX(3)+4.0*UX(2)) UYF=UYF+(DTH*2.0/3.0)*(UY(1)+UY(3)+4.0*UY(2)) UZF=UZF+(DTH*2.0/3.0)*(UZ(1)+UZ(3)+4.0*UZ(2)) 13 CONTINUE 12 CONTINUE UA=-UZF UT=-UXF UR=UYF UN=-UA*CBI+UT*(TBI/CBI) RETURN END C

96

C SUB - 5 C SUBROUTINE SOLVE(A3,B3,M,X3) DIMENSION A3(4,4),B3(4),X3(4) C THIS SUBROUTINE SOLVES A SET OF ALGEBRIC EQUATIONS C A3(I,J)*B3(J)=X3(I), I=1,2,3.......M C MM=M-1 DO 1 K=1,MM KP=K+1 L=K DO 2 I=KP,M IF(ABS(A3(I,K)).LE.ABS(A3(L,K))) GOTO 2 L=I 2 CONTINUE IF(L-K)5,5,3 3 DO 4 J=K,M AJ=A3(K,J) A3(K,J)=A3(L,J) 4 A3(L,J)=AJ AB=B3(K) B3(K)=B3(L) B3(L)=AB C ELEMINATION PROCESS 5 DO 1 I=KP,M FAC=A3(I,K)/A3(K,K) A3(I,K)=0.0 DO 6 J=KP,M 6 A3(I,J)=A3(I,J)-FAC*A3(K,J) 1 B3(I)=B3(I)-FAC*B3(K) C SOLUTION AND BACK SUBSTITUTION X3(M)=B3(M)/A3(M,M) I=M-1 7 I2=I+1 SUM=0.0 DO 8 J=I2,M 8 SUM=SUM+A3(I,J)*X3(J) X3(I)=(B3(I)-SUM)/A3(I,I) I=I-1 IF(I)9,9,7 9 RETURN END

97

Iteration 1

Input

010406.432901.286600.810010.853801.6933

00.200000.300000.400000.500000.600000.700000.800000.900001.0000

00.424000.344000.288000.249000.221000.200000.185000.174000.1610

56.179045.788238.199232.535828.213624.844122.141419.950918.1543

2404

04101622

Output

TABLE: PROPELLER PERFORMANCE CHARACTERISTICS

(LIFTING LINE PROGRAM)

-----------------------------------------------------------------------------------

INPUT DATA

-------------------

NBL=4, DIA= 6.4329M, DIAH= 1.2866M, BAR= .8100

VS=10.8538M/S, WAKE= .2176, RPS= 1.6933R.P.S.

OUTPUT RESULTS

------------------------------

INDUCED VELOCITY

------------------------------

XR UA/VS UT/VS US/VS

.2000 .1131 .1689 .3652

.3000 .1538 .1581 .3163

.4000 .1732 .1363 .2804

.5000 .1810 .1155 .2547

.6000 .1832 .0983 .2359

.7000 .1827 .0846 .2218

.8000 .1813 .0738 .2113

.9000 .1803 .0654 .2040

1.0000 .1759 .0577 .1948

98

VSJ= .9964 VAJ= .7796 CDL= .0000

-----------------------------------------------------------------------

XR XWAKE G/VS G/US BETAI VR/VS CCL/D DKT DKQ

.2000 .4240 .0000 .0000 56.1790 .8295 .0000 .0000 .0000

.3000 .3440 .0149 .0471 45.7882 1.1298 .0829 .0732 .0114

.4000 .2880 .0206 .0734 38.1992 1.4314 .0903 .1444 .0229

.5000 .2490 .0237 .0932 32.5358 1.7329 .0860 .2163 .0347

.6000 .2210 .0250 .1061 28.2136 2.0353 .0773 .2801 .0454

.7000 .2000 .0247 .1112 24.8441 2.3389 .0663 .3267 .0533

.8000 .1850 .0225 .1065 22.1414 2.6435 .0535 .3437 .0563

.9000 .1740 .0177 .0865 19.9509 2.9491 .0376 .3052 .0502

1.0000 .1610 .0000 .0000 18.1543 3.2574 .0000 .0000 .0000

KT= .17324 CT= .72590 KQ= .02813

99

Iteration 2

Input

010406.432901.286600.810010.853801.6933

00.200000.300000.400000.500000.600000.700000.800000.900001.0000

00.424000.344000.288000.249000.221000.200000.185000.174000.1610

57.146346.841139.226733.497229.099825.656322.885720.635518.7886

2404

04101622

Output



------------------------------------------------------------------------------------

INPUT DATA

-------------------


VS=10.8538M/S, WAKE= .2176, RPS= 1.6933R.P.S.

OUTPUT RESULTS

----------------------------

INDUCED VELOCITY

------------------------------


.2000 .1179 .1825 .4005

.3000 .1650 .1760 .3527

.4000 .1906 .1556 .3176

.5000 .2033 .1345 .2923

.6000 .2091 .1164 .2739

.7000 .2113 .1015 .2601

.8000 .2120 .0895 .2497

.9000 .2125 .0800 .2426

1.0000 .2094 .0712 .2336

100

VSJ= .9964 VAJ= .7796 CDL= .0000

-----------------------------------------------------------------------


.2000 .4240 .0000 .0000 57.1463 .8260 .0000 .0000 .0000

.3000 .3440 .0163 .0461 46.8411 1.1255 .0908 .0781 .0126

.4000 .2880 .0231 .0729 39.2267 1.4272 .1019 .1596 .0263

.5000 .2490 .0273 .0933 33.4972 1.7291 .0991 .2452 .0409

.6000 .2210 .0292 .1066 29.0998 2.0318 .0903 .3235 .0544

.7000 .2000 .0291 .1119 25.6563 2.3358 .0783 .3824 .0648

.8000 .1850 .0268 .1071 22.8857 2.6407 .0637 .4060 .0691

.9000 .1740 .0211 .0869 20.6355 2.9466 .0450 .3627 .0619

1.0000 .1610 .0000 .0000 18.7886 3.2551 .0000 .0000 .0000

KT= .20076 CT= .84120 KQ= .03383

101

Iteration 3

Input

010406.432901.286600.810010.853801.6933

00.200000.300000.400000.500000.600000.700000.800000.900001.0000

00.424000.344000.288000.249000.221000.200000.185000.174000.1610

57.588247.326439.704233.946629.515226.038123.236520.958519.0873

2404

04101622

Output



-----------------------------------------------------------------------------------

INPUT DATA

-------------------


VS=10.8538M/S, WAKE= .2176, RPS= 1.6933R.P.S.

OUTPUT RESULTS

---------------------------

INDUCED VELOCITY

--------------------------------


.2000 .1199 .1888 .4172

.3000 .1700 .1844 .3700

.4000 .1984 .1647 .3352

.5000 .2135 .1437 .3102

.6000 .2211 .1252 .2920

.7000 .2246 .1097 .2783

.8000 .2263 .0971 .2680

.9000 .2275 .0871 .2609

1.0000 .2251 .0779 .2520

102

VSJ= .9964 VAJ= .7796 CDL= .0000

------------------------------------------------------------------------


.2000 .4240 .0000 .0000 57.5882 .8243 .0000 .0000 .0000

.3000 .3440 .0169 .0456 47.3264 1.1234 .0944 .0802 .0131

.4000 .2880 .0243 .0726 39.7042 1.4251 .1073 .1664 .0278

.5000 .2490 .0289 .0932 33.9466 1.7271 .1052 .2584 .0438

.6000 .2210 .0312 .1068 29.5152 2.0300 .0965 .3435 .0588

.7000 .2000 .0312 .1122 26.0381 2.3342 .0840 .4084 .0703

.8000 .1850 .0288 .1074 23.2365 2.6393 .0685 .4354 .0753

.9000 .1740 .0227 .0871 20.9585 2.9453 .0485 .3898 .0677

1.0000 .1610 .0000 .0000 19.0873 3.2539 .0000 .0000 .0000

KT= .21354 CT= .89475 KQ= .03660

103

Iteration 4 - (Non-Viscous Solution)

Input

010406.432901.286600.810010.853801.6933

00.200000.300000.400000.500000.600000.700000.800000.900001.0000

00.424000.344000.288000.249000.221000.200000.185000.174000.1610

57.577547.314739.692733.935729.505126.028923.227920.950719.0800

2404

04101622

Output



---------------------------------------------------------------------------------

INPUT DATA

-------------------


VS=10.8538M/S, WAKE= .2176, RPS= 1.6933R.P.S.

OUTPUT RESULTS

-----------------------------

INDUCED VELOCITY

-------------------------------


.2000 .1198 .1886 .4168

.3000 .1699 .1842 .3696

.4000 .1982 .1645 .3348

.5000 .2132 .1435 .3098

.6000 .2208 .1250 .2915

.7000 .2243 .1095 .2778

.8000 .2259 .0970 .2675

.9000 .2272 .0870 .2605

1.0000 .2247 .0777 .2516

104

VSJ= .9964 VAJ= .7796 CDL= .0000

-----------------------------------------------------------------------


.2000 .4240 .0000 .0000 57.5775 .8243 .0000 .0000 .0000

.3000 .3440 .0169 .0456 47.3147 1.1235 .0943 .0801 .0131

.4000 .2880 .0243 .0726 39.6927 1.4252 .1071 .1662 .0278

.5000 .2490 .0289 .0932 33.9357 1.7272 .1050 .2581 .0437

.6000 .2210 .0311 .1068 29.5051 2.0301 .0963 .3431 .0587

.7000 .2000 .0312 .1122 26.0289 2.3342 .0839 .4078 .0702

.8000 .1850 .0287 .1074 23.2279 2.6393 .0684 .4346 .0751

.9000 .1740 .0227 .0871 20.9507 2.9454 .0484 .3892 .0675

1.0000 .1610 .0000 .0000 19.0800 3.2539 .0000 .0000 .0000

KT= .21322 CT= .89342 KQ= .03653

105

Iteration 4 - (Viscous Solution)

Input

020406.432901.286600.810010.853801.6933

00.200000.300000.400000.500000.600000.700000.800000.900001.0000

00.424000.344000.288000.249000.221000.200000.185000.174000.1610

57.577547.314739.692733.935729.505126.028923.227920.950719.0800

2404

04101622

Output



----------------------------------------------------------------------------------

INPUT DATA

------------------


VS=10.8538M/S, WAKE= .2176, RPS= 1.6933R.P.S.

OUTPUT RESULTS

----------------------------

INDUCED VELOCITY

------------------------------


.2000 .1198 .1886 .4168

.3000 .1699 .1842 .3696

.4000 .1982 .1645 .3348

.5000 .2132 .1435 .3098

.6000 .2208 .1250 .2915

.7000 .2243 .1095 .2778

.8000 .2259 .0970 .2675

.9000 .2272 .0870 .2605

1.0000 .2247 .0777 .2516

106

VSJ= .9964 VAJ= .7796 CDL= .0610

-----------------------------------------------------------------------


.2000 .4240 .0000 .0000 57.5775 .8243 .0000 .0000 .0000

.3000 .3440 .0169 .0456 47.3147 1.1235 .0943 .0748 .0139

.4000 .2880 .0243 .0726 39.6927 1.4252 .1071 .1578 .0298

.5000 .2490 .0289 .0932 33.9357 1.7272 .1050 .2475 .0477

.6000 .2210 .0311 .1068 29.5051 2.0301 .0963 .3312 .0650

.7000 .2000 .0312 .1122 26.0289 2.3342 .0839 .3956 .0790

.8000 .1850 .0287 .1074 23.2279 2.6393 .0684 .4232 .0858

.9000 .1740 .0227 .0871 20.9507 2.9454 .0484 .3801 .0783

1.0000 .1610 .0000 .0000 19.0800 3.2539 .0000 .0000 .0000

KT= .20619 CT= .86395 KQ= .04100

THRUST= 1037.72 KN

TORQUE= 1327.43 KN.m

DELIVERED POWER= 14122.90 KW

POWER COEFFICIENT= 1.384647

EFFICIENCY(BEHIND)= .623951

107

Iteration 5

Input

010406.432901.286600.810010.853801.6933

00.200000.300000.400000.500000.600000.700000.800000.900001.0000

00.424000.344000.288000.249000.221000.200000.185000.174000.1610

57.666447.412639.789334.026829.589426.106423.299321.016419.1408

2404

04101622

Output



-----------------------------------------------------------------------------------

INPUT DATA

--------------------


VS=10.8538M/S, WAKE= .2176, RPS= 1.6933R.P.S.

OUTPUT RESULTS

-----------------------------

INDUCED VELOCITY

-------------------------------------


.2000 .1202 .1899 .4202

.3000 .1708 .1859 .3731

.4000 .1998 .1664 .3384

.5000 .2153 .1453 .3134

.6000 .2232 .1268 .2952

.7000 .2270 .1112 .2815

.8000 .2288 .0985 .2712

.9000 .2302 .0884 .2642

1.0000 .2279 .0791 .2553

108

VSJ= .9964 VAJ= .7796 CDL= .0000

------------------------------------------------------------------------


.2000 .4240 .0000 .0000 57.6664 .8240 .0000 .0000 .0000

.3000 .3440 .0170 .0455 47.4126 1.1231 .0950 .0805 .0132

.4000 .2880 .0245 .0725 39.7893 1.4247 .1082 .1676 .0281

.5000 .2490 .0292 .0932 34.0268 1.7268 .1063 .2607 .0443

.6000 .2210 .0315 .1068 29.5894 2.0297 .0976 .3471 .0596

.7000 .2000 .0316 .1122 26.1064 2.3339 .0851 .4130 .0714

.8000 .1850 .0291 .1074 23.2993 2.6390 .0694 .4406 .0764

.9000 .1740 .0230 .0871 21.0164 2.9451 .0491 .3947 .0687

1.0000 .1610 .0000 .0000 19.1408 3.2537 .0000 .0000 .0000

KT= .21581 CT= .90427 KQ= .03711

109

Iteration 6 - (Non-Viscous Solution)

Input

010406.432901.286600.810010.853801.6933

00.200000.300000.400000.500000.600000.700000.800000.900001.0000

00.424000.344000.288000.249000.221000.200000.185000.174000.1610

57.691447.440139.816434.052429.613126.128223.319321.034919.1579

2404

04101622

Output



------------------------------------------------------------------------------------

INPUT DATA

-------------------


VS=10.8538M/S, WAKE= .2176, RPS= 1.6933R.P.S.

OUTPUT RESULTS

-----------------------------

INDUCED VELOCITY

-------------------------------


.2000 .1203 .1902 .4211

.3000 .1711 .1864 .3741

.4000 .2002 .1669 .3394

.5000 .2158 .1459 .3144

.6000 .2239 .1273 .2962

.7000 .2278 .1117 .2826

.8000 .2296 .0990 .2723

.9000 .2311 .0889 .2652

1.0000 .2287 .0795 .2564

110

VSJ= .9964 VAJ= .7796 CDL= .0000

--------------------------------------------------------------


.2000 .4240 .0000 .0000 57.6914 .8239 .0000 .0000 .0000

.3000 .3440 .0170 .0455 47.4401 1.1229 .0952 .0806 .0133

.4000 .2880 .0246 .0725 39.8164 1.4246 .1085 .1680 .0282

.5000 .2490 .0293 .0932 34.0524 1.7266 .1066 .2614 .0445

.6000 .2210 .0316 .1068 29.6131 2.0296 .0979 .3482 .0598

.7000 .2000 .0317 .1122 26.1282 2.3338 .0854 .4145 .0717

.8000 .1850 .0293 .1074 23.3193 2.6389 .0697 .4422 .0768

.9000 .1740 .0231 .0871 21.0349 2.9450 .0493 .3962 .0691

1.0000 .1610 .0000 .0000 19.1579 3.2536 .0000 .0000 .0000

KT= .21653 CT= .90730 KQ= .03727

111

Iteration 6 (Viscous Solution)

Input

020406.432901.286600.810010.853801.6933

00.200000.300000.400000.500000.600000.700000.800000.900001.0000

00.424000.344000.288000.249000.221000.200000.185000.174000.1610

57.691447.440139.816434.052429.613126.128223.319321.034919.1579

2404

04101622

Output



----------------------------------------------------------------------------------

INPUT DATA

--------------------


VS=10.8538M/S, WAKE= .2176, RPS= 1.6933R.P.S.

OUTPUT RESULTS

-----------------------------

INDUCED VELOCITY

-------------------------------


.2000 .1203 .1902 .4211

.3000 .1711 .1864 .3741

.4000 .2002 .1669 .3394

.5000 .2158 .1459 .3144

.6000 .2239 .1273 .2962

.7000 .2278 .1117 .2826

.8000 .2296 .0990 .2723

.9000 .2311 .0889 .2652

1.0000 .2287 .0795 .2564

112

VSJ= .9964 VAJ= .7796 CDL= .0610

--------------------------------------------------------------


.2000 .4240 .0000 .0000 57.6914 .8239 .0000 .0000 .0000

.3000 .3440 .0170 .0455 47.4401 1.1229 .0952 .0753 .0140

.4000 .2880 .0246 .0725 39.8164 1.4246 .1085 .1594 .0303

.5000 .2490 .0293 .0932 34.0524 1.7266 .1066 .2507 .0485

.6000 .2210 .0316 .1068 29.6131 2.0296 .0979 .3362 .0662

.7000 .2000 .0317 .1122 26.1282 2.3338 .0854 .4021 .0806

.8000 .1850 .0293 .1074 23.3193 2.6389 .0697 .4306 .0877

.9000 .1740 .0231 .0871 21.0349 2.9450 .0493 .3869 .0800

1.0000 .1610 .0000 .0000 19.1579 3.2536 .0000 .0000 .0000

KT= .20936 CT= .87727 KQ= .04180

THRUST = 1053.71 KN

TORQUE = 1353.49 KN.m

DELIVERED POWER = 14400.17 KW

POWER COEFFICIENT = 1.411831

EFFICIENCY (BEHIND) = .621369

113

C ********************************************************************************************** C "LIFTING SURFACE PROGRAM FOR MARINE PROPELLER CALCULATION" C ********************************************************************************************** C JUNE 2015 C *************************************************************** C * BEGINING OF THE MAIN PROGRAME C DIMENSION X(9), XCHORD(9), XTBI(9), XTB(9), MC(3) DIMENSION CHORD(8), TANBI(8), TBETA(8), D(3), HMU(6,2), H(2) DIMENSION XGAM(3), COSBI(8), RHO(8) DIMENSION R(8), RZ(9), XRHO(9),TANBZ(9),COEFZ(3,2) C COMMON/COM1/MT,NZ1,NZ2,MC,NPT,NQT,NBOTH,NG,JT COMMON/COM2/CHORD,COSBI,ANT,R,TANBZ,RZ,TANBI COMMON/COM3/RHO,XGAM,TBETA,HMU,H,COEFZ C OPEN (UNIT=11, FILE='PB2.DAT') OPEN (UNIT=22, FILE='PB2.OUT') OPEN (UNIT=33, FILE='PB21.OUT') C C * READ FROM DATA FILE C READ(11,101) (X(N),N=1,9) READ(11,101) (XCHORD(N), N=1,9) READ(11,101) (XTBI(N), N=1,9) READ(11,101) (XTB(N), N=1,9) READ(11,102) MT,NT,NPT,NZ1,NZ2,NG,RH READ(11,103) (MC(N), N=1,3) C 101 FORMAT(9F8.6) 102 FORMAT(6I4,F8.6) 103 FORMAT(3I4) C WRITE(33,101) (X(N),N=1,9) WRITE(33,101) (XCHORD(N), N=1,9) WRITE(33,101) (XTBI(N), N=1,9) WRITE(33,101) (XTB(N), N=1,9) WRITE(33,102) MT,NT,NPT,NZ1,NZ2,NG,RH WRITE(33,103) (MC(N), N=1,3) C C * CALCULATION - 1 C NQT=(NT+NZ1-NZ2-2)/NZ1 JT=NQT NBOTH=NT+NT RH=X(1) C READ(11,104) ((HMU(N,J),N=1,NT),J=1,JT) READ(11,105) (H(N), N=1,NQT) READ(11,106) (XGAM(N), N=1,3) C 104 FORMAT(6F10.7)

114

105 FORMAT(2F10.7) 106 FORMAT(8F10.7) C WRITE(33,104) ((HMU(N,J),N=1,NT),J=1,JT) WRITE(33,105) (H(N), N=1,NQT) WRITE(33,106) (XGAM(N), N=1,3) C DO 51 N=1,3 COEFZ(N,1)=XGAM(N) COEFZ(N,2)=0.0 51 CONTINUE C DO 15 N=1,9 TEMP=X(N) XRHO(N)=CMA(TEMP,RH) XTBI(N)=XTBI(N)*X(N) XTB(N)=XTB(N)*X(N) 15 CONTINUE C ANT=NT AMT=MT DELM=(1.0-RH)/AMT HDELM=0.5*DELM AM=RH-HDELM RZ(1)=RH+0.25*HDELM TEMP=RZ(1) Y=PIN(TEMP,X,XTBI,9) TANBZ=Y/RZ(1) C DO 9 M=1,MT R(M)=AM+DELM AM=R(M) TEMP=AM RHO(M)=CMA(TEMP,RH) RZ(M+1)=R(M)+HDELM IF(M-MT)19,10,19 10 RZ(M+1)=RZ(M+1)-0.25*HDELM 19 TEMP=RZ(M+1) Y=PIN(TEMP,X,XTBI,9) TANBZ(M+1)=Y/RZ(M+1) TEMP=R(M) Y=PIN(TEMP,X,XTBI,9) TANBI(M)=Y/R(M) Y=PIN(TEMP,X,XTB,9) TBETA(M)=Y/R(M) TEMP=RHO(M) Y=PIN(TEMP,XRHO,XCHORD,9) CHORD(M)=Y COSBI(M)=1.0/SQRT(1.0+TANBI(M)**2) 9 CONTINUE C DO 6 N=1,NPT

115

M=MC(N) D(N)=0.0 DO 6 I=1,NPT AI=I D(N)=D(N)+XGAM(I)*SIN(AI*RHO(M)) 6 CONTINUE C DO 30 N=1,NT K=NBOTH-N+1 DO 30 J=1,JT HMU(K,J)=HMU(N,J) 30 CONTINUE C WRITE(22,115) WRITE(22,107)MT,NT,NPT,NQT,NZ1,NZ2,NG,RH WRITE(22,108)(R(N),N=1,MT) WRITE(22,109)(CHORD(N),N=1,MT) WRITE(22,110)(TANBI(N),N=1,MT) WRITE(22,111)(TBETA(N),N=1,MT) WRITE(22,112)(D(N),N=1,NPT) WRITE(22,113)((HMU(N,J),N=1,NT),J=1,JT) WRITE(22,114)(H(N),N=1,NQT) C 107 FORMAT(/2X,'MT( NO. OF RADIAL LATTICE SPACES)= ',I2, + /2X,'NT( NO. OF CHORDWISE LATTICE SPACES)=',I2, + /2X,'NPT( NO. OF RADIAL CONTROL POINTS)=',I2, + /2X,'NQT( NO. OF CHORDWISE CONTROL POINTS)=',I2, + /2X,'NZ1( NO. OF RADIAL VORTEX ELEMENTS BETWEEN EACH' + ,' CONTROL POINTS)=',I2, + /2X,'NZ2( NO. OF UNUSED CONTROL POINTS BETWEEN LEADING' + ,' EDGE AND FIRST CONTROL POINT)=',I2, + /2X,'NG( NO. OF BLADES)=',I2, + /2X,'RH( NON DIMENSIONAL HUB RADIUS)=',F8.6) 108 FORMAT(/2X,' X ',8F8.5) 109 FORMAT(/2X,'CHORD=',8F8.5) 110 FORMAT(/2X,'TANBI=',8F8.5) 111 FORMAT(/2X,'TBETA=',8F8.5) 112 FORMAT(/5X,'GAMMA=',3F10.6) 113 FORMAT(/5X,'HMU=',6F10.6) 114 FORMAT(/5X,'H=',2F10.6) 115 FORMAT(/2X,'OUTPUT (LIFTING SURFACE PROGRAM)', 1 /2X,30(1H-)) CALL CAMBER STOP END C C * END OF MAIN PROGRAM C C * SUB - 1 C FUNCTION CMA(TP,RH) PI=4.0*ATAN(1.0)

116

IF (TP-0.999)2,2,1 1 CMA=PI GOTO 6 2 CN=(1.0+RH-2.0*TP)/(1.0-RH) IF(ABS(CN)-0.00001)3,3,4 3 CMA=0.5*PI GOTO 6 4 CTN=SQRT(1.0-CN**2)/CN CMA=ATAN(CTN) IF(CTN)5,6,6 5 CMA=CMA+PI 6 RETURN END C C * SUB - 2 C FUNCTION PIN(TP,AX,AY,NO) DIMENSION AX(9), AY(9) ARA(Q1,Q2,Q3,Q4,Q5,Q6,Q7)= 1 Q5*(Q4-Q2)*(Q4-Q3)/((Q1-Q2)*(Q1-Q3))+ 2 Q6*(Q4-Q1)*(Q4-Q3)/((Q2-Q1)*(Q2-Q3))+ 3 Q7*(Q4-Q1)*(Q4-Q2)/((Q3-Q1)*(Q3-Q2)) IF(TP-AX(1))2,1,3 1 PIN=AY(1) GOTO 9 2 PIN=ARA(AX(1),AX(2),AX(3),TP,AY(1),AY(2),AY(3)) GOTO 9 3 IF(TP-AX(2))2,4,5 4 PIN=AY(2) GOTO 9 5 DO 6 I=3,NO M=I IF(TP-AX(I))8,7,6 7 PIN=AY(I) GOTO 9 6 CONTINUE 8 PIN=ARA(AX(M-2),AX(M-1),AX(M),TP,AY(M-2),AY(M-1),AY(M)) 9 RETURN END C C * SUB - 3 C SUBROUTINE CAMBER C DIMENSION CHORD(8),COSBI(8),R(8),MC(3),TIL(8),THETA(8,2) DIMENSION TANBI(8),RZ(9),P(12),PSI(12),TANBZ(9) DIMENSION THSS(27),THS(28),WN(12),U(24,3,2) DIMENSION RHO(8),XGAM(3),TBETA(8),HMU(6,2),H(2) DIMENSION GAMMA(8),SNRHO(3),PSIB(12),NFLIP(12) DIMENSION A(6,6),B(6),F(6),BUG(3),ANS(6),COEFZ(3,2),ANS3(6) DIMENSION XX(3),GX(6),XCHORD(3),SIG(7),SRSI(7) C

117

COMMON/COM1/MT,NZ1,NZ2,MC,NPT,NQT,NBOTH,NG,JT COMMON/COM2/CHORD,COSBI,ANT,R,TANBZ,RZ,TANBI COMMON/COM3/RHO,XGAM,TBETA,HMU,H,COEFZ C DATA THSS/2*0.0625,3*0.125, 6*0.25,16*0.5/ C PI=4.0*ATAN(1.0) THS(1)=0.0 DO 886 I=2,28 THS(I)=THS(I-1)+THSS(I-1) 886 CONTINUE DO 888 I=1,28 THS(I)=THS(I)*PI 888 CONTINUE C C * CALCULATION - 1 C NIP=2*(NZ1-NZ2-1) NTT=NBOTH+NBOTH NTM1=NTT-1 NP=1 DO 1 M=1,MT TIL(M)=CHORD(M)*COSBI(M)/(2.0*ANT*R(M)) IF(M-MC(NP)) 1,2,1 2 DO 3 NQ=1,NQT TEMP=2.0*NZ1*NQ-NIP THETA(NP,NQ)=TIL(M)*TEMP 3 CONTINUE NP=NP+1 1 CONTINUE C K101=NPT*NQT DO 4 K=1,K101 B(K)=0.0 F(K)=0.0 DO 4 L=1,K101 A(K,L)=0.0 4 CONTINUE N=1 DO 5 NU=1,NTM1,2 TEMP=NBOTH-NU PSI(NU)=TIL(1)*TEMP PSIB(N)=PSI(NU) PSI(NU+1)=PSI(NU) N=N+1 5 CONTINUE TBZ=TANBZ(1) DO 6 N=1,NTT P(N)=PSI(N)+PSI(NTT) 6 CONTINUE C ******************************************** DO 38 NP=1,NPT

118

MS=MC(NP) ETA=RZ(1)/R(MS) TBI=TANBI(MS) CBI=COSBI(MS) DO 38 NQ=1,NQT PSIT=PSI(1) THETAC=THETA(NP,NQ) XT=RZ(1) XF=R(MS) TBZ=TANBZ(1) CALL HELIX(XT,XF,TBZ,TBI,CBI,PSIT,THETAC,NG,THS,PSIB,WN) M=2 DO 38 N=1,NBOTH U(N,NP,NQ)=WN(M) M=M+2 38 CONTINUE C C ********************************************1 C DO 14 M=1,MT RB1=RZ(M) RB2=RZ(M+1) GAMMA(M)=0.0 C DO 21 I=1,NPT AI=I SNRHO(I)=SIN(AI*RHO(M)) GAMMA(M)=GAMMA(M)+SNRHO(I)*XGAM(I) 21 CONTINUE C N=1 C DO 22 NU=1,NTM1,2 TEMP=NBOTH-NU PSI(NU)=TIL(M)*TEMP PSIB(N)=PSI(NU) IF(M-MT)23,24,23 23 PSI(NU+1)=TIL(M+1)*TEMP GO TO 25 24 PSI(NU+1)=PSI(NU) 25 IF(PSI(NU)-PSI(NU+1))26,27,27 26 NFLIP(NU)=0 NFLIP(NU+1)=8 AM=PSI(NU) PSI(NU)=PSI(NU+1) PSI(NU+1)=AM GO TO 84 27 NFLIP(NU)=8 NFLIP(NU+1)=0 84 N=N+1 22 CONTINUE C

119

DO 8 NU=2,NTM1,2 IF(PSI(NU)-PSI(NU+1))9,8,8 9 AM=PSI(NU) NF=NFLIP(NU)-1 PSI(NU)=PSI(NU+1) NFLIP(NU)=NFLIP(NU+1)+1 PSI(NU+1)=AM NFLIP(NU+1)=NF 8 CONTINUE C TBZ=TANBZ(M+1) C DO 7 N=1,NTT P(N)=PSI(N)+PSI(NTT) 7 CONTINUE C C***********************************************2 C DO 14 NP=1,NPT J1=(NP-1)*NQT MS=MC(NP) ETA=RZ(M+1)/R(MS) TBI=TANBI(MS) CBI=COSBI(MS) ZETA=((TANBI(M)-TBETA(M))/(TBI-TBETA(MS)))*(R(M)/R(MS)) ALAM=R(MS)*TBI IF(M-1)80,80,81 80 BUG(NP)=2.0*R(MS)*COSBI(MS) 81 DO 14 NQ=1,NQT K=J1+NQ IF(M-MS)83,82,83 82 A(K,NP)=GAMMA(MS)*R(MS)*H(NQ)/CHORD(MS) B(K)=B(K)+BUG(NP) 83 F(K)=0.0 C DO 41 N=1,NBOTH U(N+16,NP,NQ)=U(N,NP,NQ) 41 CONTINUE PSIT=PSI(1) THETAC=THETA(NP,NQ) XT=RZ(M+1) XF=R(MS) TBZ=TANBZ(M+1) CALL HELIX(XT,XF,TBZ,TBI,CBI,PSIT,THETAC,NG,THS,PSIB,WN) C DO 36 NU=1,NTT N=NFLIP(NU)+(NU+1)/2 U(N,NP,NQ)=WN(NU) 36 CONTINUE C C***********************************************3 C

120

DO 37 N=1,NBOTH ANGLE=PSIB(N)-THETA(NP,NQ) CALL BOUND(RB1,RB2,ETA,ALAM,ANGLE,NG,UB) W=UB+(U(N+8,NP,NQ)-U(N+16,NP,NQ)) IF(JT-2)99,40,40 C 40 DO 35 I=1,NPT C DO 35 J=2,JT L=NPT+(I-1)*(JT-1)+J-1 A(K,L)=A(K,L)-SNRHO(I)*W*HMU(N,J)*ABS(ZETA) 35 CONTINUE C 99 F(K)=F(K)+W*HMU(N,1) 37 CONTINUE C B(K)=B(K)+F(K)*GAMMA(M)*ABS(ZETA) 14 CONTINUE WRITE(22,207) WRITE(22,203)((A(K,L),L=1,K101),K=1,K101) WRITE(22,208) WRITE(22,203)(B(K),K=1,K101) C CALL SOLVE(A,B,6,ANS3) C WRITE(22,209) WRITE(22,203)(ANS3(K),K=1,K101) C 203 FORMAT(6E13.5) 207 FORMAT(/33X,'COEFFICIENT MATRIX A(K,L)'//) 208 FORMAT(/28X,'RIGHT HAND SIDE B(K)'//) 209 FORMAT(/28X,'SOLUTION MATRIX X(L)'//) C N=1 DO 90 K=1,NPT MS=MC(K) ANS(N)=R(MS) ANS(N+1)=ANS3(K) N=N+2 90 CONTINUE C J=2*NPT WRITE(22,210) WRITE(22,211)(ANS(N),N=1,J) K=NPT+1 C DO 92 I=1,NPT C DO 94 J=2,JT COEFZ(I,J)=ANS3(K) K=K+1 94 CONTINUE

121

C 92 CONTINUE C WRITE(22,212) WRITE(22,213)((COEFZ(I,J),J=1,2),I=1,3) C 210 FORMAT(/27X,'RADIUS CAMBER FACTOR KC') 211 FORMAT(27X,F5.3,8X,F7.3) 212 FORMAT(/27X,'CIRCULATION DISTRIBUTION COEFFICIENTS C(I,J)') 213 FORMAT(27X,2E15.5) C C******************************************************************4 C XH=0.2 XX(1)=R(3) XX(2)=R(5) XX(3)=R(7) XCHORD(1)=CHORD(3) XCHORD(2)=CHORD(5) XCHORD(3)=CHORD(7) GX(1)=COEFZ(1,1) GX(2)=COEFZ(2,1) GX(3)=COEFZ(3,1) GX(4)=COEFZ(1,2) GX(5)=COEFZ(2,2) GX(6)=COEFZ(3,2) DO 101 K=1,3 RHOO=ACOS((1.0+XH-2.0*XX(K))/(1.0-XH)) RHOD=RHOO*(180.0/PI) GRHO=0.0 DO 102 I=1,3 GRHO=GRHO+GX(I)*SIN(I*RHOO) 102 CONTINUE WRITE(22,103)XX(K),RHOD,GRHO 103 FORMAT(/2X,'X=',F10.6,2X,'RHO= ',F10.6,2X,'GRHO= ',F10.6) DO 106 M=1,7 SIGMA=(PI/6.0)*(M-1) SIG(M)=SIGMA SIGMAD=SIGMA*(180.0/PI) GRS=0.0 L=0 DO 104 J=1,2 DO 105 I=1,3 L=L+1 GRS=GRS+GX(L)*SIN(I*RHOO)*SIN(J*SIGMA) 105 CONTINUE 104 CONTINUE SRS=(4.0/XCHORD(K))*GRS DSDIG=(XCHORD(K)/2.0)*SIN(SIGMA) SRSI(M)=SRS*DSDIG WRITE(22,107)RHOD,SIGMAD,SRSI(M) 106 CONTINUE

122

107 FORMAT(2X,'RHO= ',F10.5,2X,'SIGMA= ',F10.5,2X,'SRSI=',F10.5) GR=SINTN(SIG,SRSI,7)/PI WRITE(22,108)GR 108 FORMAT(2X,'GR=',F10.6) 101 CONTINUE RETURN END C C * SUB - 4 C SUBROUTINE HELIX(XT,XF,TBZ,TBI,CBI,PSIT,THETAC,NBL,THS,PSIB,UN) DIMENSION THS(28),UX(5),UY(5),UZ(5),PSIB(12),UN(12) TPI=8.0*ATAN(1.0) ZF=XF*THETAC*TBI TF=THETAC F1=XF*SIN(TF) F2=XF*COS(TF) E3=TBZ UXF=0.0 UYF=0.0 UZF=0.0 C C * OFF-BLADE CALCULATION C DO 12 NB=1,NBL PHIZ=TPI*(NB-1)/NBL DO 13 J=1,27 DTH=(THS(J+1)-THS(J))/4.0 K=1 IF(NB.GT.1) K=2 L=0 DO 14 M=1,5,K L=L+1 TE1=PSIT+THS(J)+DTH*(M-1) TE=PHIZ+TE1 E1=SIN(TE) E2=COS(TE) E4=ZF-XT*TE1*E3 AA=XF*XF+XT*XT-2.0*XT*F1*E1-2.0*XT*F2*E2+E4*E4 AA=AA**1.5 AA=XF/AA BB=AA*XT*E3 UX1=-E1*E4/E3-F2+XT*E2 UY1=F1-XT*E1-(1.0/E3)*E2*E4 UX(L)=BB*UX1 UY(L)=BB*UY1 UZ1=(1.0/E3)*E2*(F2-XT*E2)+(F1-XT*E1)*E1/E3 UZ(L)=BB*UZ1 14 CONTINUE IF(NB.GT.1) GOTO 15 UXF=UXF+(DTH/3.0)*(UX(1)+UX(5)+2.0*UX(3)+4.0*(UX(2)+UX(4))) UYF=UYF+(DTH/3.0)*(UY(1)+UY(5)+2.0*UY(3)+4.0*(UY(2)+UY(4)))

123

UZF=UZF+(DTH/3.0)*(UZ(1)+UZ(5)+2.0*UZ(3)+4.0*(UZ(2)+UZ(4))) GOTO 13 15 UXF=UXF+(DTH*2.0/3.0)*(UX(1)+UX(3)+4.0*UX(2)) UYF=UYF+(DTH*2.0/3.0)*(UY(1)+UY(3)+4.0*UY(2)) UZF=UZF+(DTH*2.0/3.0)*(UZ(1)+UZ(3)+4.0*UZ(2)) 13 CONTINUE 12 CONTINUE UA=-UZF UT=-UXF UR=UYF UN2=-UA*CBI+UT*(TBI/CBI) C C * ON-BLADE CALCULATION C UN(1)=UN2 UN(2)=UN2 DTH1=PSIB(1)-PSIB(2) DTH2=DTH1 DO 16 J=3,11,2 DO 17 NB=1,NBL PHIZ=TPI*(NB-1)/NBL DTH=DTH1/4.0 K=1 IF(NB.GT.1) K=2 L=0 DO 18 M=1,5,K L=L+1 TE1=PSIT-DTH2+DTH*(M-1) TE=PHIZ+TE1 E1=SIN(TE) E2=COS(TE) E4=ZF-XT*TE1*E3 AA=XF*XF+XT*XT-2.0*XT*F1*E1-2.0*XT*F2*E2+E4*E4 AA=AA**1.5 AA=XF/AA BB=AA*XT*E3 UX1=-E1*E4/E3-F2+XT*E2 UX(L)=BB*UX1 UY1=F1-XT*E1-(1.0/E3)*E2*E4 UY(L)=BB*UY1 UZ1=(1.0/E3)*E2*(F2-XT*E2)+(F1-XT*E1)*E1/E3 UZ(L)=BB*UZ1 18 CONTINUE IF(NB.GT.1) GOTO 19 UXF=UXF+(DTH/3.0)*(UX(1)+UX(5)+2.0*UX(3)+4.0*(UX(2)+UX(4))) UYF=UYF+(DTH/3.0)*(UY(1)+UY(5)+2.0*UY(3)+4.0*(UY(2)+UY(4))) UZF=UZF+(DTH/3.0)*(UZ(1)+UZ(5)+2.0*UZ(3)+4.0*(UZ(2)+UZ(4))) GOTO 17 19 UXF=UXF+(DTH*2.0/3.0)*(UX(1)+UX(3)+4.0*UX(2)) UYF=UYF+(DTH*2.0/3.0)*(UY(1)+UY(3)+4.0*UY(2)) UZF=UZF+(DTH*2.0/3.0)*(UZ(1)+UZ(3)+4.0*UZ(2)) 17 CONTINUE

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UA=-UZF UT=-UXF UR=UYF UN2=-UA*CBI+UT*(TBI/CBI) UN(J)=UN2 UN(J+1)=UN2 DTH2=DTH2+DTH1 16 CONTINUE RETURN END C C * SUB -5 C SUBROUTINE BOUND(RB1,RB2,ETA,ALAM,ANGLE,NG,UB) C G=NG DELBL=6.2831853/G S=0.0 R=RB2/ETA RLAM=R/SQRT(R**2+ALAM**2) A=R**2+(ALAM*ANGLE)**2 PHI=ANGLE C DO 1 N=1,NG T=0.0 CP=COS(PHI) SP=SIN(PHI) B=-2.0*R*CP C=ALAM**2*ANGLE*CP+R**2*SP X=RB1 D=B**2-4.0*A C DO 2 I=1,2 IF(ABS(D)-0.0001)3,3,4 4 Y=-2.0*(2.0*X+B)/(D*SQRT(A+B*X+X**2)) GO TO 5 3 Y=-1.0/(2.0*(X-0.5*B)**2) 5 IF(I-1)6,7,6 7 T=T-Y X=RB2 GO TO 2 6 T=T+Y 2 CONTINUE C S=S+T*C PHI=PHI+DELBL 1 CONTINUE C UB=S*RLAM RETURN END C

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C * SUB -6 C SUBROUTINE SOLVE(A3,B3,M,X3) DIMENSION A3(6,6),B3(6),X3(6) C * THIS SUBROUTINE SOLVES A SET OF ALGEBRIC EQUATIONS C * A3(I,J)*X3(J)=B3(I), I=1,2,3.......M MM=M-1 DO 1 K=1,MM KP=K+1 L=K DO 2 I=KP,M IF(ABS(A3(I,K)).LE.ABS(A3(L,K))) GOTO 2 L=I 2 CONTINUE IF(L-K)5,5,3 3 DO 4 J=K,M AJ=A3(K,J) A3(K,J)=A3(L,J) 4 A3(L,J)=AJ AB=B3(K) B3(K)=B3(L) B3(L)=AB C * ELEMINATION PROCESS 5 DO 1 I=KP,M FAC=A3(I,K)/A3(K,K) A3(I,K)=0.0 DO 6 J=KP,M 6 A3(I,J)=A3(I,J)-FAC*A3(K,J) 1 B3(I)=B3(I)-FAC*B3(K) C * SOLUTION AND BACK SUBSTITUTION X3(M)=B3(M)/A3(M,M) I=M-1 7 I2=I+1 SUM=0.0 DO 8 J=I2,M 8 SUM=SUM+A3(I,J)*X3(J) X3(I)=(B3(I)-SUM)/A3(I,I) I=I-1 IF(I)9,9,7 9 RETURN END C C SUB - 7 C FUNCTION SINTN(X,Y,N) DIMENSION X(11), Y(11) SINTN=0.0 T=0.0 DX1=X(2)-X(1) DD1=(Y(2)-Y(1))/DX1 IF(N-2)1,2,3 2 SINTN=(Y(2)+Y(1))*DX1/2.0

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GOTO 1 3 DO 4 I=3,N DX2=X(I)-X(I-1) DD2=(Y(I)-Y(I-1))/DX2 D1=(DD2-DD1)/(DX1+DX2) D2=(DD1+D1*DX1)/2.0 D3=D1/3.0 ST=((D3*DX1-D2)*DX1+Y(I-1))*DX1 SINTN=SINTN+(T+ST)/2.0 T=((D3*DX2+D2)*DX2+Y(I-1))*DX2 DX1=DX2 DD1=DD2 IF(I-3)4,5,4 5 SINTN=SINTN+SINTN 4 CONTINUE SINTN=SINTN+T 1 RETURN END C

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INPUT (LIFTING SURFACE PROGRAM)

0.2000000.3000000.4000000.5000000.6000000.7000000.8000000.9000001.000000

0.2200000.2490000.2710000.2840000.2890000.2840000.2600000.2090000.000000

1.5813001.0890000.8337000.6758000.5684000.4905000.4311000.3846000.347400

0.9135000.6936000.5646000.4764000.4118000.3625000.3231000.2911000.266100

0008000300030001000000040.200000

000300050007

00.109375000.182292000.208330000.1898870-0.0025320-0.1873550

-1.3333330-2.6666660

00.1034166-0.0196793-0.0032411

OUTPUT (LIFTING SURFACE PROGRAM)

MT (NO. OF RADIAL LATTICE SPACES) = 8

NT (NO. OF CHORDWISE LATTICE SPACES) = 3

NPT (NO. OF RADIAL CONTROL POINTS) = 3

NQT (NO. OF CHORDWISE CONTROL POINTS) = 2

NZ1 (NO. OF RADIAL VORTEX ELEMENTS BETWEEN EACH CONTROL

POINTS) = 1

NZ2 (NO. OF UNUSED CONTROL POINTS BETWEEN LEADING EDGE AND

FIRST CONTROL POINT) = 0

NG (NO. OF BLADES) = 4

RH (NON DIMENSIONAL HUB RADIUS)= .200000

X .25000 .35000 .45000 .55000 .65000 .75000 .85000 .95000

CHORD= .23737 .26003 .27829 .28738 .28777 .27474 .23920 .15999

TANBI= 1.28775 .94442 .74663 .61751 .52661 .45895 .40652 .36502

TBETA= .78537 .62261 .51710 .44195 .38568 .34181 .30634 .27786

GAMMA= .083502 .110501 .098509

HMU= .109375 .182292 .208330 .189887 -.002532 -.187355

H= -1.333333 -2.666666

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COEFFICIENT MATRIX A(K,L)

-.18003E+00 .00000E+00 .00000E+00 -.52872E+01 -.43739E+01 .30313E+01

-.36006E+00 .00000E+00 .00000E+00 -.49529E+01 -.42353E+01 .29756E+01

.00000E+00 -.33279E+00 .00000E+00 -.78165E+01 .19511E+01 .90462E+01

.00000E+00 -.66559E+00 .00000E+00 -.71427E+01 .17007E+01 .89308E+01

.00000E+00 .00000E+00 -.46674E+00 -.98781E+01 .13130E+02 -.62672E+01

.00000E+00 .00000E+00 -.93347E+00 -.88402E+01 .11796E+02 -.49407E+01

RIGHT HAND SIDE B(K)

-.23541E+00 -.45996E+00 -.48212E+00 -.92093E+00 -.73110E+00 -.13672E+01

SOLUTION MATRIX X(L)

.12532E+01 .13299E+01 .13823E+01 .44996E-02 -.30738E-02 .18116E-03

RADIUS CAMBER FACTOR KC

.450 1.253

.650 1.330

.850 1.382

CIRCULATION DISTRIBUTION COEFFICIENTS C(I,J)

.10342E+00 .44996E-02

-.19679E-01 -.30738E-02

-.32411E-02 .18116E-03

X= .450000 RHO= 67.975680 GRHO= .083502

RHO= 67.97568 SIGMA= .00000 SRSI= .00000

RHO= 67.97568 SIGMA= 30.00000 SRSI= .04345

RHO= 67.97568 SIGMA= 60.00000 SRSI= .12819

RHO= 67.97568 SIGMA= 90.00000 SRSI= .16700

RHO= 67.97568 SIGMA= 120.00000 SRSI= .12231

RHO= 67.97568 SIGMA= 150.00000 SRSI= .04005

RHO= 67.97568 SIGMA= 180.00000 SRSI= .00000

GR= .084082

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X= .650000 RHO= 97.180760 GRHO= .110501

RHO= 97.18076 SIGMA= .00000 SRSI= .00000

RHO= 97.18076 SIGMA= 30.00000 SRSI= .05963

RHO= 97.18076 SIGMA= 60.00000 SRSI= .17334

RHO= 97.18076 SIGMA= 90.00000 SRSI= .22100

RHO= 97.18076 SIGMA= 120.00000 SRSI= .15816

RHO= 97.18076 SIGMA= 150.00000 SRSI= .05087

RHO= 97.18076 SIGMA= 180.00000 SRSI= .00000

GR= .111269

X= .850000 RHO= 128.682200 GRHO= .098509

RHO= 128.68220 SIGMA= .00000 SRSI= .00000

RHO= 128.68220 SIGMA= 30.00000 SRSI= .05496

RHO= 128.68220 SIGMA= 60.00000 SRSI= .15765

RHO= 128.68220 SIGMA= 90.00000 SRSI= .19702

RHO= 128.68220 SIGMA= 120.00000 SRSI= .13788

RHO= 128.68220 SIGMA= 150.00000 SRSI= .04355

RHO= 128.68220 SIGMA= 180.00000 SRSI= .00000

GR= .099193

130

Appendix - C

Table of chord-load factors

Table C.1: Chord-load factors, µ nj

N j=1 j=2 j=3 j=4 j=5 j=6 j=7

N=2 1 0.5

2 0.5

N=4

1 0.195312 0.292969

2 0.304688 0.152344

3 0.304688 -0.152344

4 0.195312 -0.292969

N=6

1 0.109375 0.182292 0.189887 0.134187

2 0.182292 0.182292 -0.002532 -0.184823

3 0.208333 0.069444 -0.187355 -0.131896

4 0.208333 -0.069444 -0.187355 0.131896

5 0.182292 -0.182292 -0.002532 0.184823

6 0.109375 -0.182292 0.189887 -0.134187

N=8

1 0.072007 0.126010 0.146058 0.129591 0.080010 0.010426 -0.052892

2 0.123367 0.154209 0.068072 -0.069119 -0.154856 -0.124451 0.004068

3 0.147079 0.110310 -0.065429 -0.153981 -0.054423 0.118564 0.147254

4 0.157547 0.039387 -0.148701 -0.076562 0.129269 0.108879 -0.098430

5 0.157547 -0.039387 -0.148701 0.076562 0.129269 -0.108879 -0.098430

6 0.147079 -0.110310 -0.065429 0.153981 -0.154423 -0.118564 0.147254

7 0.123367 -0.154209 0.068072 0.069119 -0.154856 0.124451 0.004068

8 0.072007 -0.126010 0.146058 -0.129591 -0.129591 0.080010 -0.052892

131

Appendix - D

Kramer diagram

Figure D.1 : Kramer diagram

DEVELOPMENT OF A MARINE PROPELLER DESIGN METHOD …

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