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 The Principles of

Naval Architecture Series

Propulsion

Justin E. Kerwin and Jacques B. Hadler

J. Randolph Paulling, Editor

2010

Published by

The Society of Naval Architectsand Marine Engineers

601 Pavonia Avenue Jersey City, New Jersey 07306

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 Copyright © 2010 by The Society of Naval Architects and Marine Engineers.

The opinions or assertions of the authors herein are not to be construed as official or

reflecting the v iews of SNAME or any government agency.

It is understood and agreed that nothing expressed herein is intended or shall be construed

to give any person, firm, or corporation any right, remedy, or claim against SNAME or

any of its officers or member.

Library of Congress Cataloging-in-Publication Data

Kerwin, Justin E. (Justin Elliot)

Propulsion / Justin E. Kerwin and Jacques B. Hadler.

 p. cm. — (The principles of naval architecture series)

Includes bibliographical references and index.

ISBN 978-0-939773-83-1

1. Ship propulsion. I. Hadler, Jacques B. II. Paulling, J. Randolph. III. Title.

 VM751.K47 2010623.87—dc22

2010040103

 ISBN 978-0-939773-83-1

Printed in the United States of America

First Printing, 2010

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  An Introduction to the Series

The Society of Naval Architects and Marine Engineers is experiencing remarkable changes in the Maritime Indus-

try as we enter our 115th year of service. Our mission, however, has not changed over the years . . . “an internation-ally recognized . . . technical society . . . serving the maritime industry, dedicated to advancing the art, scienceand practice of naval architecture, shipbuilding, ocean engineering, and marine engineering . . . encouraging theexchange and recording of information, sponsoring applied research . . . supporting education and enhancing the professional status and integrity of its membership.”

In the spirit of being faithful to our mission, we have written and published significant treatises on the subjectof naval architecture, marine engineering, and shipbuilding. Our most well known publication is the “Principles ofNaval Architecture.” First published in 1939, it has been revised and updated three times—in 1967, 1988, and nowin 2008. During this time, remarkable changes in the industry have taken place, especially in technology, and thesechanges have accelerated. The result has had a dramatic impact on size, speed, capacity, safety, quality, and envi-ronmental protection.

The professions of naval architecture and marine engineering have realized great technical advances. They in-clude structural design, hydrodynamics, resistance and propulsion, vibrations, materials, strength analysis usingfinite element analysis, dynamic loading and fatigue analysis, computer-aided ship design, controllability, stability,

and the use of simulation, risk analysis and virtual reality.However, with this in view, nothing remains more important than a comprehensive knowledge of “first princi-

 ples.” Using this knowledge, the Naval Architect is able to intelligently utilize the exceptional technology availableto its fullest extent in today’s global maritime industry. It is with this in mind that this entirely new 2008 treatisewas developed—“The Principles of Naval Architecture: The Series.” Recognizing the challenge of remaining rel-evant and current as technology changes, each major topical area will be published as a separate volume. Thiswill facilitate timely revisions as technology continues to change and provide for more practical use by those whoteach, learn or utilize the tools of our profession.

It is noteworthy that it took a decade to prepare this monumental work of nine volumes by sixteen authors andby a distinguished steering committee that was brought together from several countries, universities, companiesand laboratories. We are all especially indebted to the editor, Professor J. Randolph (Randy) Paulling for providingthe leadership, knowledge, and organizational ability to manage this seminal work. His dedication to this arduoustask embodies the very essence of our mission . . . “to serve the maritime industry.”

It is with this introduction that we recognize and honor all of our colleagues who contributed to this work.

 Authors:Dr. John S. Letcher Hull GeometryDr. Colin S. Moore Intact StabilityRobert D. Tagg Subdivision and Damaged StabilityProfessor Alaa Mansour and Dr. Donald Liu Strength of Ships and Ocean StructuresProfessor Lars Larsson and Dr. Hoyte C. Raven Ship Resistance and FlowProfessors Justin E. Kerwin and Jacques B. Hadler PropulsionProfessor William S. Vorus Vibration and NoiseProf. Robert S. Beck, Dr. John Dalzell (Deceased), Prof. Odd Faltinsen Motions in Waves

and Dr. Arthur M. ReedProfessor W. C. Webster and Dr. Rod Barr Controllability

Control Committee Members are:Professor Bruce Johnson, Robert G. Keane, Jr., Justin H. McCarthy, David M. Maurer, Dr. William B. Morgan,Professor J. Nicholas Newman and Dr. Owen H. Oakley, Jr.

I would also like to recognize the support staff and members who helped bring this project to fruition, espe-cially Susan Evans Grove, Publications Director, Phil Kimball, Executive Director, and Dr. Roger Compton, PastPresident.

In the new world’s global maritime industry, we must maintain leadership in our profession if we are to continueto be true to our mission. The “Principles of Naval Architecture: The Series,” is another example of the many waysour Society is meeting that challenge.

 ADMIRAL ROBERT E. KRAMEK 

  Past President (2007–2008)

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Symbol Units Description

( r  ,  ) (m, rad) 2D right-handed polar co-ordinates

(u , v , w ) m/s velocity components in the( x  , y , z ) directions

ua, u r , ut * * *

  m/s (axial, radial, tangential) in-duced velocity on a propel-ler lifting line

( x  , r  ) (m, m) coordinates of the meridi-onal plane

( x  , r  , o ) (m, m, rad) propeller coordinate system(axial, radial, azimuthal)

( x  , y ) m 2D cartesian coordinates(streamwise, vertical)

( x  , y , z ) m 3D cartesian coordinates(streamwise, spanwise, vertical)

(  ,  ) m ( x  , y ) coordinates mappedinto the   plane

a  –  parameter in the NACA a-Series of mean lines

a n – series coefficientsc m chord lengthct   /c r   – ratio of tip chord to root

chordds m shaft diameter f  ( x  ) m camber distribution (mean

line)

 f 0 m maximum camber f 0  /c – camber ratio

 f  ( k )  ig ( k ) – H 2 k iH 2 k 1 0

2i /    k

 , Sears

function g m/s2   9.81, acceleration due to

gravityh ( x  ) m cavity thicknessia( r v , r c ) – Lerbs axial induction factorit ( r v , r c ) – Lerbs tangential induction

factor

 k –2U 

c

, reduced frequency

 n rev/s rotation rate n  – index of chordwise positions n – harmonic number n – unit surface normal vector p ( x  , y ) Pa pressure field p  Pa pressure far upstream p min  Pa minimum pressure in the

flow pv  Pa vapor pressure of the fluidq m/s total velocity vector

Nomenclature

Symbol Units Description

q ( x  , y ) m/s U  u2 v2 , magni-tude of the total fluid velocity

q ( x  ) m/s magnitude of the total ve-locity on the foil surface

q p ( x  ) m/s velocity distr ibution onthe surface of a parabola

 r   m distance vector r c m circle radius in confor-

mal mapping r c  m control point radius r h m hub radius r i  m image vortex radius

 r o  m core radius of the hub vortex

 r  H  m hub radius r  L  m leading edge radius of

curvatures m spans – nondimensional chord-

wise coordinatet ( x  ) m thickness distributiont0 m maximum thicknesst0  /c – thickness ratioua r c

*   m/s induced axial velocity atradius r c

ua r c, r v*

  1/m axial horseshoe influencefunction

ua r c, r v  1/m axial velocity inducedat radius  r c  by  Z   unit-strength helices

ut r c*   m/s induced tangential ve-

locity at radius r cut r c, r v

*   1/m tangential horseshoe in-fluence function

ut r c, r v  1/m tangential velocity in-duced at radius  r c  by  Z 

unit-strength helicesuc ( x  ) m/s perturbation velocity due

to camber at ideal angle

of attackus r c, r v  1/m induced velocity along a

wake helixu t ( x  ) m/s perturbation velocity due

to thickness obtainedfrom linear theory

ut m/s tangential component ofthe velocity

utm r *   m/s circumferential meantangential velocity

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xx NOMENCLATURE

Symbol Units Descriptionuw  m axial induced velocity far

downstream

w –V  S 

V  S  V  A, wake fraction

w ( x  , t ) m/s velocity induced by thebound and shed vorticity ata point on the x axis

w *(y ) m/s downwash velocity distri-bution

wij   m/s downwash velocity at con-trol point (i , j  )

w nm,ij   1/m horseshoe influence func-tion for (i ,  j  )th control point vortex

w n,m  1/m downwash velocity in-duced by a unit horseshoe vortex

 x  m angular coordinate defined

by cos x 2

c x 

 x c  m control point positions x  m ( r  ) m rake of the midchord line x v m vortex positions x  L (y ) m leading edge versus span-

wise coordinate x T  (y ) m trailing edge versus span-

wise coordinatey  rad angular cosine spanwise

coordinate, defined by

cosy2

 S Y 

yu ( x  ) m y ordinate of 2D foil uppersurface

yl ( x  ) m y ordinate of 2D foil lowersurface

y p ( x  ) m y coordinate for a parabola z   x   iy  m complex coordinate z0 (y ) m vertical displacement of the

nose-tail line( F  x  , F y , F  z ) N force components in the ( x  ,

y , z ) directions(V a , V  r  , V t ) m/s time-averaged velocity in

the ship-fixed propeller co-

ordinate system(V  A , V  R , V T  ) m/s (axial, radial, azimuthal)effective inflow velocity

 A  m2  /s vector potentialA  –  s2  /  S , aspect ratio A n , B n  – Fourier series harmonic

coefficientsC   m/s strength of the leading edge

singularityC  ( x  ) m/s suction parameterC  A – correlation allowance co-

efficientC  D – drag coefficient

Symbol Units DescriptionC  Df  – frictional drag coefficientC  Dp  – pressure drag coefficientC  Dv – viscous drag coefficientC  L – lift coefficient

C  Lideal  – ideal lift coefficientC  M  – moment coefficient with

respect to midchordC  N   – normal force coefficientC  P  – pressure coefficient[C  P  ] min  – minimum pressure coeffi-

cient (also denoted C  P  , min )

C Qa –   Q

  V  A   R3

2

1 2

, torque

coefficient based on volu-metric mean inflow speed

C Qs –

  Q

  V  S    R

32

1 2, torque

coefficient based on shipspeed

C  S   – leading edge suction forcecoefficient

C Ts –   T 

  V  S    R2

2

1 2

, thrust

coefficient based on shipspeed

C Ta –

 A

T 

  V 2   R221

, thrust

  coefficient based on volumetric mean inflowspeed

 D m propeller diameter D  N drag per unit span D S  m full-scale propeller di-

ameter D M   m model propeller diameter E   J fluid kinetic energy F h N hub drag

 F  n –   n gD

  nD

 g

 D ,

Froude number F  N   N force normal to a flat plate F  S   N leading edge suction forceG N/m2  shear modulus of elas-

ticity

G – 2  RV 2

  , nondimen-

sional circulation H  –    */  boundary layer

shape factor H  r   – root unloading factor

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  NOMENCLATURE xxi

Symbol Units Description H t  – tip unloading factor

1 0 H 2 k, H 2 k  –    J  n ( k )   iY  n ( k ) ( n    0,1), Hankel functions of thesecond kind

 J  n ( k ), Y  n ( k ) – Bessel functions of the firstand second kind

 J  A – nD

V  A , advance coefficient

 J  S  – nD

V  S  , advance coefficient

 K  N/m2    QL S   / , calibration con-stant

 K Q –    n2 D5

Q, torque coefficient

based on rotation rate

 K T  –    n2 D4

T 

, thrust coefficientbased on rotation rate

 L  N lift (per unit span in 2D flow) L S   m length of shaft over which

 is measured M   – number of vortex panels

along the span N   – number of panels along the

chord P  m blade pitch P  B W  2  nQ , brake power P  D W delivered power P  E  W   RT V  , effective power

 P  S  W shaft power P T  W  TV  A , thrust powerQ Nm propeller torqueQ Nm brake torque R m propeller radius

 R n – V 

c0.7V  R , Reynolds number

 RT   N resistance of the hull whentowed

 Rw  m slipstream radius far down-stream

 Re – Reynolds number S  m2  /s point source strength (flow

rate) S  m2  projected areaT   N propeller thrust (or total

thrust of propeller and duct)U  m/s free-stream speedU  m/s flow speed at infinityU e  m/s boundary layer edge velocityU i m/s  U   ut (0), “free stream”

speed at the local leadingedge of a parabola

V  m/s ship speedV  m3  cavity volume

Symbol Units DescriptionV  x, y, z  m/s velocity vectorV  * m/s total inflow speedV  A  m/s speed of advanceV  A  m/s volumetric mean ad-

 vance (inflow) speed

V  R m/s  A

1/2

V 2 0.7  nD2 , re-

sultant inflow velocityV  S  m/s ship speed

V c m/s velocity on the upper foilsurface

V l m/s velocity on the lower foilsurface

V  m m/s mean foil velocity

V d m/s difference foil velocityW  Nm workW  ( x  , t ) m/s gust velocity

W o  m/s sinusoidal gust amplitude Z   – number of blades  rad angle of attack rad free-stream inclination

with respect to the  x 

axis0L (y ) rad angle of zero lift0 – normalized sinusoidal

gust amplitude

  deg arcsin r c

yc , negative an-

gular coordinate of rearstagnation point

  rad undisturbed inflow pitchangle

 c  rad wake pitch angle at ra-dius  r    r c

 i rad total inflow pitch angle v  rad wake pitch angle at ra-

dius  r    r v w  rad wake pitch angle  ( x  ) m/s vortex sheet strength (cir-

culation) per unit length  f   m/s free vortex sheet strength b m/s bound vortex sheet

strength s  m/s shed vorticity (per unit

length of wake)  rad inclination of a spanwise

 vortex with respect tothe y direction

  m boundary layer thickness * m boundary layer displace-

ment thickness k  rad blade indexing angle – open-water propeller ef-

ficiency (also denoted o )

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xxii NOMENCLATURE

Symbol Units Description

 D –  P  D

 P  E ; quasipropulsive co-

efficient

 H  –  P T 

 P  E , hull efficiency

u ( x  ), l ( x  ) n/a (upper, lower) foil surface ( x  ) m mean camber distribution ( r  ) – circulation reduction factor

(Prandtl tip factor)  –   2   / n , sinusoidal gust

wavelength ( n  is the har-monic number

  –  D M 

 D S , scale model geometry

ratio kg/(m-s) dynamic viscosity

 ( x  ) m3  /s dipole sheet strength  1/m gradient operator2  1/m2  Laplacian  m2  /s kinematic viscosity rad/s propeller angular velocity

(counter-clockwise direc-tion when looking down-stream)

  1/s vorticity vector  rad angular coordinate of a

general point on a helix ( x  ,y ) m2  /s velocity potentiali m2  /s velocity potential in the in-

terior of a foil p ( r  ) rad blade pitch angle   – 3.14159265. . .  ( x  ,y ) m2  /s stream function   kg/m3  density  ( x  ) m/s source sheet strength (flow

rate) per unit length  – cavitation number

Symbol Units Description  Pa steady state tensile

stress in the blade a Pa magnitude of the alter-

nating stress in the blade

  R  Pa the estimated level ofresidual stress from themanufacturing process

  deg tail angle   – duct loading factor w  Pa wall shear stress  ( x  ) m thickness distribution m momentum thickness ( x  ) rad  arctan(df   /dx  ), slope of

the mean line at point x c  rad projected chord m ( r  ) rad skew angle of the mid-

chord line

s  deg angular coordinate of a

stagnation pointw m momentum thickness in

the far downstream wake  ( Z  ) m mapping function   m2  /s vortex circulation (posi-

tive counter-clockwise)   ( r  ) m2  /s circulation distribution

over the span   (y ) m2  /s circulation distribution

over the span   nm m2  /s circulation of bound vor-

tex element at panel ( n , m )  o m2  /s circulation at the blade

root  –   J   / , absolute advance

coefficient  m sinusoidal gust wave

length ( z ) m2  /s complex potential

dz

d  m/s  u  iv , complex velocity

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Preface

During the 20 years that have elapsed since publication of the previous edition of  Principles of Naval Archi-

tecture, there have been remarkable advances in the art, science, and practice of the design and construction ofships and other floating structures. In that edition, the increasing use of high-speed computers was recognized andcomputational methods were incorporated or acknowledged in the individual chapters rather than being presentedin a separate chapter. Today, the electronic computer is one of the most important tools in any engineering environ-ment and the laptop computer has taken the place of the ubiquitous slide rule of an earlier generation of engineers.

 Advanced concepts and methods that were only being developed or introduced then are a part of common engi-neering practice today. These include finite element analysis, computational fluid dynamics, random process meth-ods, and numerical modeling of the hull form and components, with some or all of these merged into integrateddesign and manufacturing systems. Collectively, these give the naval architect unprecedented power and flexibilityto explore innovation in concept and design of marine systems. In order to fully utilize these tools, the modernnaval architect must possess a sound knowledge of mathematics and the other fundamental sciences that form abasic part of a modern engineering education.

In 1997, planning for the new edition of Principles of Naval Architecture was initiated by the SNAME publicationsmanager who convened a meeting of a number of interested individuals including the editors of PNA and the new

edition of Ship Design and Construction on which work had already begun. At this meeting it was agreed that PNAwould present the basis for the modern practice of naval architecture and the focus would be principles in preferenceto applications. The book should contain appropriate reference material but it was not a handbook with extensivenumerical tables and graphs. Neither was it to be an elementary or advanced textbook although it was expected to beused as regular reading material in advanced undergraduate and elementary graduate courses. It would contain thebackground and principles necessary to understand and to use intelligently the modern analytical, numerical, experi-mental, and computational tools available to the naval architect and also the fundamentals needed for the develop-ment of new tools. In essence, it would contain the material necessary to develop the understanding, insight, intuition,experience, and judgment needed for the successful practice of the profession. Following this initial meeting, a PNAControl Committee, consisting of individuals having the expertise deemed necessary to oversee and guide the writingof the new edition of PNA, was appointed. This committee, after participating in the selection of authors for the variouschapters, has continued to contribute by critically reviewing the various component parts as they are written.

In an effort of this magnitude, involving contributions from numerous widely separated authors, progress has

not been uniform and it became obvious before the halfway mark that some chapters would be completed beforeothers. In order to make the material available to the profession in a timely manner it was decided to publish eachmajor subdivision as a separate volume in the Principles of Naval Architecture Series rather than treating each asa separate chapter of a single book.

 Although the United States committed in 1975 to adopt SI units as the primary system of measurement, thetransition is not yet complete. In shipbuilding as well as other fields we still find usage of three systems of units:English or foot-pound-seconds, SI or meter-newton-seconds, and the meter-kilogram(force)-second system com-mon in engineering work on the European continent and most of the non-English speaking world prior to the adop-tion of the SI system. In the present work, we have tried to adhere to SI units as the primary system but other unitsmay be found, particularly in illustrations taken from other, older publications. The symbols and notation follow, ingeneral, the standards developed by the International Towing Tank Conference.

In recent years the analysis and design of propellers, in common with other aspects of marine hydrodynamics, hasexperienced important developments both theoretical and numerical. The purpose of the present work, therefore, isto present a comprehensive and up-to-date treatment of propeller analysis and design. After a brief introduction to

 various types of marine propulsion machinery, their nomenclature, and definitions of powers and efficiencies, the presentation goes into two- and three-dimensional airfoil theory including conformal mapping, thin and thick foilsections, pressure distributions, the design of mean lines, and thickness distributions. The treatment continues withnumerical methods including two-dimensional panel methods, source/vortex based methods, and others. A sectionon three-dimensional hydrofoil theory introduces wake vortex sheets and three-dimensional vortex lines. This isfollowed by linear lifting line and lifting surface theory with both exact and approximate solution methods.

The hydrodynamic theory of propulsors begins with the open and ducted actuator disk. Lifting line theory of propellers follows, including properties of helicoidal vortex sheets, optimum and arbitrary circulation distribu-tions, and the Lerbs induction factor method. The vortex lattice method and other computational methods aredescribed. Unsteady foil theory and wake irregularity are covered in a section on unsteady propeller forces. Thesection on cavitation describes the various types of cavitation, linear theory, partial and supercavitating foils, nu-merical methods, and effects of viscosity.

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There are sections on model testing of propellers followed by selection and design using standard series chartsand by circulation theory. Other types of propulsors such as waterjets, vertical axis propellers, overlapping pro- pellers, and surface-piercing propellers are covered. Propeller strength considerations include the origin of bladeforces and stress analysis by beam theory and finite element methods. The final section discusses ship standardiza-tion trials, their purpose, and measurement methods and instruments and concludes with the analysis of tr ial data

and derivation of the model-ship correlation allowance.

 J. R ANDOLPH P AULLING

 Editor 

xiv PREFACE

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Table of Contents

 An Introduction to the Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

 Authors’ Biography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

1 Powering of Ships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Historical Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Types of Ship Machinery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Definition of Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Propulsive Efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Two-Dimensional Hydrofoils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Foil Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.2 Conformal Mapping Essentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10

2.3.3 The Kármán-Trefftz Mapping Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112.3.4 The Kutta Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12

2.3.5 Pressure Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

2.3.6 Examples of Propellerlike Kármán-Trefftz Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

2.3.7 Lift and Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

2.3.8 Mapping Solutions for Foils of Arbitrary Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15

2.4 Linearized Theory for a Two-Dimensional Foil Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15

2.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15

2.4.2 Vortex and Source Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16

2.5 Glauert’s Solution for a Two-Dimensional Foil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

2.5.1 Example: The Flat Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

2.5.2 Example: The Parabolic Mean Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

2.6 The Design of Mean Lines: The NACA a-Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

2.7 Linearized Pressure Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20

2.8 Comparison of Pressure Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21

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iv PROPULSION

2.9 Solution of the Linearized Thickness Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21

2.9.1 Example: The Elliptical Thickness Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

  2.9.2 Example: The Parabolic Thickness Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22

2.10 Superposition of Camber, Angle of Attack, and Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .222.11 Correcting Linear Theory Near the Leading Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23

2.12 Two-Dimensional Vortex Lattice Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25

2.12.1 Constant Spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25

2.12.2 Cosine Spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25

2.12.3 Converting from  n to   ( x  ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26

2.12.4 Drag and Leading Edge Suction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26

2.12.5 Adding Foil Thickness to Vortex Lattice Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29

2.13 Two-Dimensional Panel Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30

2.13.1 Source-/Vortex-Based Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31

2.13.2 Surface Vorticity Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31

2.13.3 Perturbation Potential Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31

2.13.4 Sample Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32

2.14 The Cavitation Bucket Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33

2.15 Viscous Effects: Two-Dimensional Foil Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35

2.15.1 Coupled Inviscid/Boundary-Layer Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35

2.15.2 Measures of Boundary-Layer Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37

2.15.3 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38

2.15.4 Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39

2.15.5 Computing the Coupled Boundary Layer/Outer Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39

2.15.6 Reynolds Number Effects on Lift and Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40

2.15.7 Advanced Blade Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42

3 Three-Dimensional Hydrofoil Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44

3.1 Introductory Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44

3.2 The Strength of the Free Vortex Sheet in the Wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45

3.3 The Velocity Induced by a Three-Dimensional Vortex Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46

3.4 Velocity Induced by a Straight Vortex Segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47

3.5 Linearized Lifting-Surface Theory for a Planar Foil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .48

3.5.1 Formulation of the Linearized Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .48

3.5.2 The Linearized Boundary Condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49

3.5.3 Determining the Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49

3.5.4 Relating the Bound and Free Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50

3.6 Lift and Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51

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  PROPULSION v

  3.7 Lifting Line Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53

3.7.1 Glauert’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53

3.7.2 Vortex Lattice Solution for the Planar Lifting Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55

3.7.3 The Prandtl Lifting Line Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .593.8 Lifting Surface Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63

3.8.1 Exact Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63

3.8.2 Vortex Lattice Solution of the Linearized Planar Foil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63

4 Hydrodynamic Theory of Propulsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67

4.1 Inflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67

4.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69

4.3 Actuator Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70

4.4 Axisymmetric Euler Solver Simulation of an Actuator Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74

4.5 The Ducted Actuator Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75

4.6 Axisymmetric Euler Solver Simulation of a Ducted Actuator Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77

4.7 Propeller Lifting Line Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78

4.7.1 The Velocity Induced by Helical Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79

4.7.2 The Actuator Disk as a Particular Lifting Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81

4.8 Optimum Circulation Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82

4.8.1 Assigning The Wake Pitch Angle  w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

  4.8.2 Properties of Constant Pitch Helical Vortex Sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .84

4.8.3 The Circulation Reduction Factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86

4.8.4 Application of the Goldstein Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87

4.9 Lifting Line Theory for Arbitrary Circulation Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89

4.9.1 Lerbs Induction Factor Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89

4.10 Propeller Vortex Lattice Lifting Line Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90

4.10.1 Hub Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92

4.10.2 The Vortex Lattice Actuator Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95

4.10.3 Hub and Tip Unloading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95

4.11 Propeller Lifting-Surface Theory and Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96

4.11.1 Propeller Blade Geometry Employing Cylindrical Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96

4.11.2 Noncylindrical Blade Geometry Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .97

4.11.3 Blade Geometry Data Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .97

4.11.4 Historical Background of Propeller Lifting-Surface Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98

5 Unsteady Propeller Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101

5.1 Types of Unsteady Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101

  5.2 Basic Equations for Linearized Two-Dimensional Unsteady Foil Theory . . . . . . . . . . . . . . . . . . . . . . . . . .101

5.3 Analytical Solutions for Two-Dimensional Unsteady Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103

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vi PROPULSION

5.4 Numerical Time Domain Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.5 Wake Harmonics and Unsteady Propeller Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.6 Transverse Alternating Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108

5.7 Unsteady Three-Dimensional Computational Methods for Propellers . . . . . . . . . . . . . . . . . . . . . . . . . . . .1095.8 Unsteady Propeller Force Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109

6 Theory of Cavitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Spyros A. Kinnas

  6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.2 Noncavitating Flow—Cavitation Inception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.3 Cavity Flows—Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112

6.4 Cavitating Hydrofoils—Linearized Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .114

6.4.1 Partially Cavitating Hydrofoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.4.2 Supercavitating Hydrofoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.4.3 Analytical Solution for the Partially Cavitating Flat Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117

6.4.4 Analytical Solution for the Supercavitating Flat Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

  6.5 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.6 Leading Edge Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119

6.7 Panel Methods for Two-Dimensional and Three-Dimensional Cavity Flows . . . . . . . . . . . . . . . . . . . . . . . 120

6.8 Cavitating Propeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120

6.9 Comparisons with Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123

6.10 Effects of Viscosity on Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.11 Design in the Presence of Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124

7 Scaling Laws and Model Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125

7.2 Law of Similitude for Propellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125

7.3 Open-Water Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.4 Model Self-Propulsion Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.4.1 Wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.4.2 Augment of Resistance and Thrust Deduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.4.3 Relative Rotative Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.4.4 Hull Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .130

7.4.5 Quasi-Propulsive Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.4.6 Standard Procedure for Performance Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .130

7.5 Wake Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.6 Propeller Cavitation Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.6.1 Variable Pressure Water Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.6.2 Presentation of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134

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  PROPULSION vii

  7.6.3 Variable Pressure Circulating Water Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7.6.4 Variable Pressure Towing Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

8 Propeller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1358.2 The Design and Analysis Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135

8.3 Definition of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8.4 Preliminary Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .136

8.4.1 Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

8.4.2 Number of Revolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

8.4.3 Number of Blades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

8.4.4 Radial Load Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

8.4.5 Blade Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .138

8.4.6 Skew . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

8.4.7 Camber and Angle of Attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

8.5 Design Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139

8.6 Analysis and Optimization of the Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

8.6.1 Propeller Design by Systematic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8.6.2 Propeller Design by Circulation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

9 Waterjet Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

9.1 Hydrodynamic Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142

9.2 Inlet Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143

9.3 Pump Design and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143

9.3.1 Coupled Euler/Lifting-Surface Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

9.3.2 RANS Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

9.4 Tip Leakage Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

10 Other Propulsion Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .149

10.2 Tunnel Sterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

10.3 Vertical-Axis Propellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

10.4 Overlapping Propellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

10.5 Supercavitating Propellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

10.6 Surface Piercing Propellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .153

10.7 Controllable-Pitch Propellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .155

11 Propeller Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .156

11.2 Stresses Based on Modified Cantilever Beam Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .157

11.3 Bending Moments Due to Hydrodynamic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .157

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viii PROPULSION

11.4 Centrifugal Force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

11.4.1 Bending Moments Due to Blade Rake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .158

11.4.2 Bending Moments Due to Blade Skew . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

11.5 Strength Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15911.6 Stresses Based on Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

11.7 Minimum Blade Thickness Based on Classification Society Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

11.8 Fatigue Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

11.9 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

12 Ship Standardization Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

12.1 Purpose of Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

12.2 Preparation for Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

12.3 General Plan of Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .164

12.4 Measurement of Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

12.5 Analysis of Speed Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .165

12.6 Derivation of Model-Ship Correlation Allowance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

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 1.1 Historical Discussion  A moving ship experiencesresisting forces from the water and air that must beovercome by a thrust supplied by some mechanism. Inthe earliest days, this mechanism consisted of manu-ally operated oars, and later, of sails, and then, devicessuch as jets, paddlewheels, and propellers of many dif-ferent forms.

The earliest propulsion device to use mechanical power seems to have been of the jet type using a primemover and a pump, patents for which were granted toToogood and Hayes in Great Britain in 1661. In a jet-type propulsion device, water is drawn in by the pump anddelivered sternward as a jet at a higher velocity with the

reaction providing the thrust. Toogood and Hayes usedan Archimedian screw as the pumping device. The useof the Archimedian screw as a hydrodynamic devicehad been known from ancient times. Early applicationsof an external thrusting device to accelerate the wateralso took the form of an Archimedian screw. Thus, theorigins of modern screw propulsion and waterjets areclosely related. At the relatively low speeds of commer-cial cargo ships, the waterjet is materially less efficient. At the higher speeds of advanced marine vessels, thewaterjet is competitive, and in certain types of vesselsis supplanting the propeller. The waterjet is discussedfurther in Section 9.

The first practical steam-driven paddle ship, the

Charlotte Dundas , was designed by William Symingtonfor service on the Forth-Clyde Canal in Scotland. TheCharlotte Dundas  demonstrated the practicality ofsteam power in 1803 by towing two loaded bargesagainst a head wind that stopped all other canal boats. A few years later, in 1807, Robert Fulton constructed thefamous North River Steamboat (erroneously named theClermont by Fulton’s first biographer) for passenger ser- vice on the Hudson River in New York.

The period that followed, until about 1850, was theheyday of the side paddle wheel steamers. The first ofthese to cross the Atlantic was the American  Savan- nah , a full-rigged ship with auxiliary steam power,

which crossed in 1819. Then followed a line of now fa-miliar names, including the Canadian  Royal William ,the famous first Cunarder Britannia (in 1840), culmi-nating in the last Cunard liner to be driven by paddles,the Scotia , in 1861.

Side paddle wheels were reasonably efficient propul-sive devices because of their slow rate of turning, butthey were not ideal for seagoing ships. The immersion varied with ship displacement, and the wheels came outof the water alternatively when the ship rolled, caus-ing erratic course keeping. In addition, the wheels wereliable to damage from rough seas. From the marineengineer’s point of view, they were too slow running,

involving the use of large, heavy engines. These opera-tional weaknesses ensured their rapid decline from popularity once the screw propeller proved to be an ac-ceptable alternative. Side paddle wheels remain in usein some older vessels where shallow water prohibitsthe use of large screw propellers. Side paddles also givegood maneuvering characteristics. Paddles have alsobeen fitted at the sterns of ships, as in the well-knownriverboats on the Mississippi and other American riv-ers. Such stern-wheelers are still in use, mainly as pas-senger carriers.

Side paddle wheels were widely used on the Missis-sippi, Ohio, and other inland rivers in the first half of the

19th century (e.g., the  Natchez  and  Robert E. Lee , im-mortalized by Currier and Ives, respectively). The CivilWar and railroads marked their decline (Twain, 1907).

The development of the modern screw propeller hasa long history. The first proposal to use a screw propel-ler appears to have been made in England by Hookein 1680, followed by others in the 18th century such asDaniel Bernoulli and James Watt. The first three actualuses of marine propellers appear to be on the human- powered submarines of David Bushnell in 1776 andRobert Fulton in 1801, and the steam-driven surfaceship of Colonel Stevens in 1804. Fulton built and oper-ated the submarine Nautilus in 1800–1801. The Nauti-lus used a hand-cranked propeller when submerged and

a sail when surfaced, making it the first submersibleto use different propulsion systems for submerged andsurfaced operation. The first steam-driven propellerthat actually worked is generally attributed to ColonelStevens, who used twin screws to propel a 24-foot ves-sel on the Hudson River in 1804 (Baker, 1944).

Two significant practical applications of ship propel-lers came in 1836 by Ericsson in the United States andPettit Smith in England. The design by Smith was a singlehelicoidal screw having several revolutions that broke,resulting in a shorter screw that produced more thrust(Rouse & Ince, 1957). Ericsson’s early design consistedof vanes mounted on spokes that were attached to a hub.

The screw propeller has many advantages over the paddle wheel. It is not materially affected by normalchanges in service draft, it is well protected from dam-age either by seas or collision, it does not increase theoverall width of the ship, and it can run much fasterthan paddles and still retain as good or better efficiency.The screw propeller permits the use of smaller, lighter,faster running engines. It rapidly superseded the paddle-wheel for all oceangoing ships, the first screw- propelledsteamer to make the Atlantic crossing being the Great Britain in 1845.

The screw propeller has proved extraordinarilyadaptable in meeting the increasing demands for thrust

1

Powering of Ships

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2 PROPULSION

under increasingly arduous conditions. While other de- vices have been adopted for certain particular types ofships and kinds of service, the screw propeller has re-mained the dominant form of propulsor for ships.

 Among the more common variants of the propeller,

the use of a shroud ring or nozzle has been shown tohave considerable advantages in heavily-loaded propel-lers. The ring or nozzle is shaped to deliver a forwardthrust to the hull. The principal advantage is foundin tugs, trawlers, and icebreakers at low ship speeds,where the pull at the bollard for a given horsepowermay be increased by as much as 40% or more when com- pared with that given by an open propeller. At higherspeeds, the advantage fades, and when free running thedrag of the nozzle is detrimental.

 Another type of propulsor was used in the USS Alarm  as long ago as 1874 (Goldsworthy, 1939). Thisship carried a fixed bow gun and had to be turned toaim the gun. To keep the ship steady in a tideway, wherea rudder would be useless, a feathering paddlewheel ro-tating about a vertical axis, invented by Fowler in GreatBritain in 1870, was fitted at the stern and completelysubmerged (White, 1882). It was quite successful as ameans of maneuvering the ship, but its propulsive effi-ciency was low. The modern version of this propulsoris the Voith-Schneider propeller. Although its efficiencyis not so high as that of the orthodox propeller, and itsmaintenance is generally more costly, the advantage inmaneuverability has resulted in many applications toriver steamers, tugs, and ferries. The vertical axis pro- peller is discussed further in Section 10.3.

1.2 Types of Ship Machinery   In selecting the propel-

ling machinery for a given vessel, many factors mustbe taken into consideration, such as weight, space oc-cupied, first cost, reliability, useful life, flexibility andnoise, maintenance, fuel consumption, and its suitabil-ity for the type of propulsor to be used. It is beyond thescope of this text to consider all the machinery typesthat have been developed to meet these factors, but abrief review will not be out of place.

The reciprocating steam engine with anywhere fromone to four cylinders dominated the field of ship pro- pulsion until about 1910. Since then, it has been super-seded first by the steam turbine and more recently bythe diesel engine, and in special applications, by the

gas turbine.The first marine turbine was installed in 1894 by SirCharles Parsons in the Turbinia, which attained a speedof 34 knots. Thereafter, turbines made rapid progressand by 1906 were used to power the epoch-making bat-tleship HMS Dreadnought and the famous Atlantic liner Mauretania .

The steam turbine delivers a uniform turning ef-fort, is eminently suitable for large unit power output,and can utilize very high-pressure inlet steam over awide range of power to exhaust at very low pressures.The thermal efficiency is reasonably high, and the fuelconsumption of large steam turbine plants is as low as

200 grams of oil per kilowatt hour (kW-hr) (less than0.40 pounds [lb.] per horsepower [hp]/hour). Steam tur-bines readily accept overload, and the boilers can burnlow-quality fuels.

On the other hand, the turbine is not reversible and

its rotational speed for best economy is far in excess ofthe most efficient rotations per minute (RPM) of usual propeller types. These drawbacks make it necessary toinstall separate reversing turbines and to insert gearsbetween the turbines and the propeller shaft to reducethe propeller rotational speed to suitable values.

Most internal-combustion engines used for ship pro- pulsion are diesel1 (compression ignition) engines. Theyare built in all sizes, from those fitted in small pleasureboats to the very large types fitted in the largest con-tainerships. The biggest engines in the latter ships de- velop over 6000 kW per cylinder, giving outputs over80,000 kW in 14 cylinders (108,920 hp). The  Emma Mærsk  has 14 cylinders and 108,920 hp. There nowmay be even bigger ones. They are directly reversible,have very low fuel consumption, and are suitable forlow-quality fuel oils. An average figure for a low-speeddiesel is around 170 grams of oil per kW-hr (less than0.3 lb. per hp-hr). These large engines are directly cou- pled to the propeller. Smaller medium-speed enginesmay be used, driving the propeller through gears orelectric transmissions. As the diesel engine has grownin capacity and refinement, it has supplanted the steamturbine as the primary means of propulsion of merchantships. Only liquefied natural gas (LNG) carriers andships with nuclear propulsion plants remain beyond thereach of diesel engines.

The air that can be trapped in the cylinders forcombustion limits the torque that can be developed ineach cylinder of a diesel engine. Therefore, even whenthe engine is producing maximum torque, it producesmaximum power only at maximum RPM. Consequently,a diesel will produce power that rises rapidly with theRPM. This characteristic leads to the problem of match-ing a diesel engine and a propeller. The resistance of theship’s hull will increase with time because of aging andfouling of the hull, while the propeller thrust decreasesfor the same reasons. Therefore, over time, the loadon the engine will increase to maintain the same shipspeed. This consideration requires the designer to se-

lect propeller particulars (such as pitch) so that later,as the ship ages and fouls, the engine does not becomeoverloaded (Kresic & Haskell, 1983).

In gas turbines, the fuel is burned in compressed air,and the resulting hot gases pass through the turbine.The development of gas turbines has depended mostlyon the development of high temperature alloys and ce-ramics. Gas turbines are simple, light in weight, andgive a smooth, continuous torque. They are expensivein the quantity and quality of fuel burned, especially

1  After Rudolf Diesel, a German engineer (1858–1913).

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  PROPULSION 3

at part power. Like diesel engines, gas turbines canquickly be brought on to full load without a long warm-ing-up period. Marine gas turbines have been fitted toa small number of merchant ships. They are now usedby a majority of naval vessels, where a high power den-

sity is desired. In some applications, a large gas turbineis combined with a diesel engine of lower output, or asmaller gas turbine, with both plants connected to acommon propeller shaft by clutches and gearing. Thelower power units are used for general cruising, and thelarger gas turbine is available at little or no notice whenthere is a demand for full power.

Mechanical gearing has been used as the mostcommon means to reduce the high prime mover RPMto a suitable value for the propeller. It permits the op-eration of engine and propeller at their most economi-cal speeds with a power loss in the gears of only 1% or2%. The reduction in RPM between the prime moverand the propeller shaft can also be attained by electri-cal means.

The prime mover is directly coupled to a genera-tor, both running at the same high speed for efficientoperation and low cost, weight, and volume. In mostapplications, the generator supplies a motor directlyconnected to the propeller shaft, driving the propellerat the RPM most desirable for high propeller efficiency.This system eliminates any direct shafting betweenengine and propeller, and so gives greater freedom inlaying out the general arrangement of the ship to bestadvantage. In twin-screw ships fitted with multiple setsof alternators, considerable economy can be achievedwhen using part power, such as when a passenger ship

is cruising, by supplying both propulsion motors froma reduced number of engines. Electric drive also elimi-nates separate reversing elements and provides greatermaneuverability. These advantages are gained, how-ever, at the expense of rather high first cost and greatertransmission losses.

Nuclear reactors are widely used in submarines, in alimited number of large naval vessels, and in a class ofRussian arctic icebreakers, but are not generally con-sidered viable for merchant ships due in large part to public opposition. The reactor serves to raise steam,which is then passed to a steam turbine in the normalway. The weight and volume of fuel oil is eliminated.

The reactor can operate at ful l load almost indefinitely,which enables the ship to maintain high speed at seawithout carrying a large quantity of consumable fuel.The weight saved, however, cannot always be devotedto increase deadweight capacity, for the weight of reac-tor and shielding may approach or exceed that of theboilers and fuel for the fossil-fueled ship.

1.3 Definition of Power  The various types of marineengines are not all rated on the same basis, inasmuch asit is inconvenient or impossible to measure their poweroutput in exactly the same manner. Diesel engines areusually rated in terms of brake power ( P  B), while steamturbines are usually rated in shaft power ( P  S  ). The

International Standard (SI) unit for power is the watt(W), where 1 W  1 N-m/s. The term horsepower  is stillused, although it has two different definitions: 1  Eng-lish hp  550 ft-lb/sec  745.7 W, whereas 1 metric hp 75 kgf-m/sec   75 kgf-m/sec * 9.8067 (kg-m/sec2 )/kgf

735.5 W. Brake power   is usually measured directly at the

crankshaft coupling by means of a torsion meter or dy-namometer. It is determined by a shop test and is calcu-lated by the formula

 P  B  2  nQ B (1.1)

Where n is the rotation rate, revolutions per sec and Q B

is the brake torque, N-m.Power can also be computed using the angular rota-

tion rate of the shaft, measured in radians per second,with the simple conversion   2  n.

 Shaft power  is the power transmitted through the shaftto the propeller. For diesel-driven ships, the shaft powerwill be equal2 to the brake power for direct-connect en-gines (generally the low-speed diesel engines). For geareddiesel engines (medium- or high-speed engines), the shafthorsepower will be lower than the brake power becauseof reduction gear “losses.” For electric drive, the shaft power supplied by the motor will be lower than the brake power that the prime movers supplied to the generator be-cause of generator and motor inefficiencies. For furtherdetails on losses associated with electric drive systems,see T and R Bulletin 3-49 (SNAME, 1990).

Shaft power is usually measured aboard ship as closeto the propeller as possible by means of a torsion meter.This instrument measures the angle of twist between

two sections of the shaft, where the angle is directly proportional to the torque transmitted. For a solid, cir-cular shaft the shaft torque is

Q S  d4 G S  S 

32 S 

 L S 

(1.2)

Where d S  is the shaft diameter, m; G S  is the shear mod-ulus of elasticity of the shaft material, N/m2 ;  L S   is thelength of shaft over which torque is measured, m; and  S 

is the measured angle of twist, radians.The shear modulus G S  for steel shafts is usually taken

as 8.35  1010 N/m2 . The shaft power is then given by

 P  S  2  nQ S  (1.3)

For more precise experimental results, particularly withhollow shafting, it is customary to calibrate the shaft bysetting up the length of shafting on which the torsionmeter is to be used, subjecting it to known torque andmeasuring the angles of twist, and determining the cali-bration constant K   Q S  L S   / S  . The shaft power can then

2The brake horsepower as measured onboard ship for direct-connected diesels is lower, because of thrust-bearing friction,than the brake horsepower measured in the shop test, wherethe thrust bearing is unloaded.

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4 PROPULSION

be calculated directly from any observed angle of twistand revolutions per second as

 P  S  2  n K  S 

 L S 

(1.4)

There is some power lost in the stern tube bearing andin any shaft tunnel bearings between the stern tubeand the site of the torsion meter. The power actuallydelivered to the propeller is therefore somewhat lessthan that measured by the torsion meter. This delivered power  is given the symbol P  D .

 As the propeller advances through the water at aspeed of advance V  A , it delivers a thrust T  . The thrust power  is

 P T  TV  A (1.5)

Finally, the effective power  is the resistance of the hull, R , times the ship speed, V  .

 P  E  RV  (1.6)

1.4 Propulsive Efficiency   The efficiency of an engi-neering operation is generally defined as the ratio of theuseful work or power obtained to that expended in car-rying out the operation. In the case of a ship, the useful power obtained is that used in overcoming the resis-tance to motion at a certain speed, which is representedby the effective power P  E  .

The power expended to achieve this result is not soeasily defined. In a ship with reciprocating steam en-gines, the power developed in the cylinders themselves,the indicated power,  P  I  , was used. The overall propul-

sive efficiency in this case would be expressed by theratio P  E   / P  I  . In the case of steam turbines, it is usual tomeasure the power in terms of the shaft power deliv-ered to the shafting aft of the gearing, and the overall propulsive efficiency is

 P  P  E 

 P  S 

 (1.7)

Because of variations between T   and  R  and betweenV  A  and V  , it is difficult to define the hydrodynamic ef-ficiency of a hull-propeller combination in terms of suchan overall propulsive efficiency.

 A much more meaningful measure of efficiency of

 propulsion is the ratio of the useful power obtained, P  E ,to the power actually delivered to the propeller, P  D . This

ratio has been given the name quasipropulsive coeffi-cient , and is defined as

 D  P  E 

 P  D  (1.8)

 The shaft power  is taken as the power delivered to theshaft by the main engines aft of the gearing and thrustblock, so that the difference between  P  S  and  P  D repre-sents the power lost in friction in the shaft bearingsand stern tube. The shaft transmission efficiency  isdefined as

 S  P  D

 P  S 

  (1.9)

Thus, the propulsive efficiency is the product of thequasipropulsive coefficient  and the shaft transmis-sion efficiency

 P   D S  (1.10)

The shaft transmission loss is usually taken as about1% for ships with machinery aft and 3% for those withmachinery amidships. It must be remembered also thatwhen using the power measured by a torsion meter, theloss will depend on the position of the meter along theshaft. To approach as closely as possible to the powerdelivered to the propeller, it should be as near to thestern tube as circumstances permit. It is often assumedthat  S  1.0.

 The definition of the quasipropulsive efficiency de-scribed above has been widely used for the conventionaldisplacement ship provided with screw propellers and

is a very useful measure of the comparative propulsive performance of such ships. The effective power for theseships is based on the total hull resistance with the appro- priate appendages installed for the control and propul-sion of the ship. This definition of effective power is notso useful for high-speed vessels where different types of propulsors can be installed such as waterjets, surface- piercing propellers, or conventional screw propellers ona Z drive or with inclined shaft and struts. For these ves-sels, it is more appropriate to base the effective poweron the “bare” hull resistance, thus providing a commondefinition of quasipropulsive efficiency when comparingthe efficiencies of various propulsion alternatives. The

appendage resistance of a particular propulsor is there-fore appropriately charged to that propulsor.

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  PROPULSION 5

 2.1 Introduction  We will begin our examination ofhydrofoil and propeller flows by looking at the flowaround two-dimensional (2D) foil sections. It is impor-tant to recognize at the outset that a 2D flow is an ideal-ization. Flows around marine propellers, sailboat keelsor control surfaces are inherently three-dimensional(3D). Moreover, it is even impossible to create a truly2D flow in a wind or water tunnel. While the foil modelmay be perfectly placed between the walls of the tun-nel test section, interaction between the tunnel wallboundary layers and the foil generate 3D features thatdisturb the two-dimensionality of the flow field. Reliableexperimental measurements of 2D foil sections there-

fore require careful attention to the issue of avoidingunwanted 3D effects.Of course, 2D flows can be modeled theoretically and

are much easier to deal with than 3D flows. Moreover,the fundamental mechanism for creating lift as well asmuch of the methodology for designing optimum foilsection shapes can be explained by 2D concepts. De-sign methods for airplane wings, marine propellers, andeverything in between rely heavily on the use of system-atic foil section data. However, it is important to recog-nize that one cannot simply piece together a 3D wing or propeller in a stripwise manner from a sequence of 2Dfoil sections and expect to get an accurate answer. Wewill see later why this is true, and how 2D and 3D flows

can be properly combined. A surprisingly large number of methods exist for

 predicting the flow around foil sections, and it is im- portant to understand their advantages and disad- vantages. They can be characterized in the followingthree ways.

1. Approach: analytical or numerical2. Viscous model: potential flow (inviscid) methods,

fully viscous methods, or coupled potential flow/bound-ary layer methods

3. Precision: exact, linearized, or partially linearizedmethods

Not all combinations of these three characteristicsare possible. For example, fully viscous flows (except ina few trivial cases) must be solved numerically. Perhapsone could construct a 3D graph showing all the possi-ble combinations, but this will not be attempted here!In this Section, we will start with the method of con-formal mapping, which can easily be identified as be-ing analytical, inviscid, and exact. We will then look atinviscid, linear theory, which can either be analytical ornumerical. The principal attribute of the inviscid, linear,numerical method is that it can be readily be extendedto 3D flows.

This will be followed by a brief look at some correc-tions to linear theory, after which we will look at panelmethods, which can be categorized as numerical, invis-cid, and exact. We will then look at coupled potentialflow/boundary layer methods, which can be character-ized as a numerical, viscous, exact method.3 Finally, wewill take a brief look at results obtained by a Reynolds Averaged Navier-Stokes (RANS) code, which is fully vis-cous, numerical, and exact.4

2.2 Foil Geometry   Before we start with the develop-ment of methods to obtain the flow around a foil, we willfirst introduce the terminology used to define foil sec-tion geometry. As shown in Fig. 2.1, good foil sections

are generally slender, with a sharp (or nearly sharp)trailing edge and a rounded leading edge. The base linefor foil geometry is a line connecting the trailing edgeto the point of maximum curvature at the leading edge,and this is shown as the dashed line in the figure. This isknown as the nose-tail line, and its length is the chord ,c , of the foil.

The particular coordinate system notation used to de-scribe a foil varies widely depending on application, andone must therefore be careful when reading differenttexts or research reports. It is natural to use x  ,y as thecoordinate axes for a 2D flow, particularly if one is usingthe complex variable  z   x     iy . The nose-tail line isgenerally placed on the x  axis, but in some applications

2

Two-Dimensional Hydrofoils

3Well, more or less. Boundary layer theory involves linearizingassumptions that the boundary layer is thin, but the coupledmethod makes no assumptions that the foil is thin.4Here we go again! The foil geometry is exact, but the turbu-lence models employed in RANS codes are approximations.

Chord

LeadingEdge

TrailingEdge

Maximum Camber, f 0Maximum Thickness, t

0

f(x)

+t(x)/2

-t(x)/2

Leading Edge Radius

Nose-Tail Line

Figure 2.1 Illustration of notation for foil section geometry.

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6 PROPULSION

the x  axis is taken to be in the direction of the onset flow,in which case the nose-tail line is inclined at an angleof attack ,  , with respect to the  x   axis. Positive  x   canbe either oriented in the upstream or downstream direc-tion, but we shall use the downstream convention here.

For 3D planar foils, it is common to orient the y co-ordinate in the spanwise direction. In this case, thefoil section ordinates will be in the z direction. Finally,in the case of propeller blades, a special curvilinearcoordinate system must be adopted; we will introducethis later.

 As shown in Fig. 2.1, a foil section can be thoughtof as the combination of a mean line  , f  ( x  ), with max-imum value  f o  and a symmetrical thickness form ,t ( x  ), with maximum value to . The thickness form isadded at right angles to the mean line so that pointson the upper and lower surfaces of the foil will havecoordinates

t x 2

yu  f  x  cos 

t x 2

sin  x l  x 

t x 2

sin  x u  x 

yl  f  x  cos t x 

2

(2.1)

where   arctan(df   /dx  ) is the slope of the mean line at point x  .

The quantity f o   /c is called the camber ratio , and in

a similar manner, to   /c  is called the thickness ratio .It has been common practice to develop foil shapes byscaling generic mean line and thickness forms to theirdesired values, and combining then by using equation(2.1) to obtain the geometry of the foil surface. A majorsource of mean line and thickness form data was cre-ated by the National Advisory Committee on Aeronau-tics (NACA; now the National Aeronautics and Space Administation) in the 1930s and 1940s and assembledby Abbott and Von Doenhoff (1959). For example,Fig. 2.2 shows sample tabulations of the geometry ofthe NACA  Mean Line a   0.8  and the NACA 65A010Basic Thickness Form. Note that the tabulated mean

line has a camber ratio  f o   /c  0.0679, while the thick-ness form has a thickness ratio to   /c  0.10. Included inthe tables is some computed velocity and pressure datathat we will refer to later.

 An important geometrical characteristic of a foil isits leading edge radius,  r  L , as shown in Fig. 2.1. Whilethis quantity is, in principle, contained in the thick-ness function t ( x  ), extracting an accurate value fromsparsely tabulated data is r isky. It is therefore providedexplicitly in the NACA tables—for example, the NACA65A010 has a leading edge radius of 0.639% of the chord.If you wish to scale this thickness form to another value,all of the ordinates are simply scaled linearly. However,

the leading edge radius scales with the square of thethickness of the foil, so that a 15% thick section of thesame form would have a leading edge radius of 1.44%of the chord. We can show why this is true by consider-ing an example where we wish to generate thickness

form (2) by linearly scaling all the ordinates of thick-ness form (1)

t2 x  t1 x to  c

2to  c

1  (2.2)

 for all values of x  . Then, the derivatives dt  /dx  and d  2t  /dx  2

will also scale linearly with thickness/chord ratio. Now,at the leading edge, the radius of curvature, r  L, is

 r  L  lim x  0

2 3  2

1 dt

dx 

d

2

tdx 2

(2.3)

 evaluated at the leading edge, which we will locateat  x    0. Because the slope dt  /dx  goes to infinity at arounded leading edge, equation (2.3) becomes

 r  L  lim constant x  0

2t

0

c

3

dt

dx 

d2t

dx 2

(2.4)

 which confirms the result stated earlier.

Some attention must also be given to the details ofthe trail ing edge geometry. As we will see, the uniquesolution for the flow around a foil section operatingin an inviscid fluid requires that the trailing edge besharp. However, practical issues of manufacturingand strength make sharp trailing edges impractical.In some cases, foils are built with a square (but rela-tively thin) trailing edge, as indicated in Fig. 2.2, al-though these are sometimes rounded. An additional practical problem frequently arises in the case of foilsections for marine propellers. Organized vortex shed-ding from blunt or rounded trailing edges may occurat frequencies that coincide with vibratory modes of

the blade trailing edge region. When this happens,strong discrete acoustical tones are generated, whichare commonly referred to as singing . This problemcan sometimes be cured by modifying the trailing edgegeometry in such a way as to force flow separation onthe upper surface of the foil slightly upstream of thetrailing edge.

 An example of an “antisinging” trailing edge modifi-cation is shown in Fig. 2.3. It is important to note thatthe nose-tail line of the modified section no longer passes through the trailing edge, so that the convenientdecomposition of the geometry into a mean line andthickness form is somewhat disrupted. More complete

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  PROPULSION 7

information on this issue was presented by Michael and Jessup (2001).

The procedure for constructing foil geometry de-scribed so far is based on traditional manual drafting practices which date back at least to the early 1900s. De-fining curves by sparse point data, with the additionalrequirement of fairing into a specified radius of curva-ture leaves a lot of room for interpretation and error. In

the present world of computer-aided design software andnumerically controlled machines, foil surfaces—and ul-timately 3D propeller blades, hubs, and fillets—are bestdescribed in terms of standardized geometric “entities”such as  Non-Uniform Rational B-Splines  (NURBS)curves and surfaces. An example of the application ofNURBS technology to 2D propeller sections and com- plete 3D propeller blades was presented by Neely (1997a).

Figure 2.2 Sample of tabulated geometry and fl ow data for an NACA mean line and thickness form. (Reprinted from “Theory of Wing Sections,”

by permission of Dover Publications.)

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8 PROPULSION

Figure 2.3 An example of a trailing edge modification used to reduce singing. This particular procedure is frequently usedfor U.S. Navy and commercial applications.

BEVEL INTERSECTS SUCTION

SIDE AT XC = 0.95

 S U C T I O N  S I D

 E

NOSE-TAIL LINE

P R E SSU R E  SI D E 

-TRAILING EDGE FAIRING RADIUS

(6 TIMES THICKNESS AT KNUCKLE

BEFORE ROUNDING)

TRAILING EDGE RADIUS

(1/64)ʺ

TRAILING EDGE

(XC = 1.0)

KNUCKLE

XC = 0.95

-

tTE

2

tTE

2

 As an example, Fig. 2.4 shows a B-spline representationof a foil section. In this case, the foil, together with its sur-face curvature and normal vector, is uniquely defined bya set of 10 ( x  ,y ) coordinates representing the vertices ofthe B-spline control polygon. This is all that is needed tointroduce the shape into a computational fluid dynamicscode, construct a model, or construct the full size object.Further information on B-spline curves may be found inLetcher (2009).

2.3 Conformal Mapping 2.3.1 History  The initial development of the fieldof airfoil theory took place in the early 1900s, long be-fore the invention of the computer. Obtaining an accu-rate solution for the flow around such a complex shapeas a foil section, even in two dimensions, was thereforea formidable task. Fortunately, one analytical tech-nique, known as the method of conformal mapping, wasknown at that time, and it provided a means of deter-mining the exact  inviscid flow around a limited classof foil section shapes. This technique was first appliedby Joukowski (1910), and the set of foil geometries cre-ated by the mapping function that he developed bearshis name. A more general mapping function, which in-

cludes the Joukowski mapping as a special case, wasthen introduced by Kármán and Trefftz (1918). Whileseveral other investigators introduced different map- ping functions, the next significant development wasby Theodorsen (1931), who developed an approximateanalytical/numerical technique for obtaining the map- ping function for a foil section of arbitrary shape. The-odorsen’s work was the basis for the development of anextensive systematic series of foil sections published byNACA in the late 1930s and 1940s. Detailed accounts andreferences for this important early work may be foundin Abbott and Von Doenhoff (1959) and Durand (1963).

The old NACA section results were done, of neces-sity, by a combination of graphical and hand compu-tation. An improved conformal mapping method ofcomputing the flow around arbitrary sections, suitablefor implementation on a digital computer, was pub-lished by Brockett (1965, 1966). Brockett found, notsurprisingly, that inaccuracies existed in the earlierNACA data and his work led to the development of foilsection design charts which are used for propeller de-

sign at the present time.The theoretical basis for the method of conformalmapping is given in most advanced calculus texts (e.g.,Hildebrand, 1976), so only the essential highlightswill be developed here. One starts with the knownsolution to a simple problem—in this case the flowof a uniform stream past a circle. The circle is then“mapped” into some geometry that resembles a foilsection, and if you follow the rules carefully, the flowaround the circle wil l be transformed in such a way asto represent the correct solution for the mapped foilsection.

Let us start with the flow around a circle. We knowthat in a 2D ideal flow, the superposition of a uniform

free stream and a dipole (whose axis is oriented inopposition to the direction of the free stream) willresult in a dividing streamline whose form is circu-lar. We also know that this is not the most generalsolution to the problem, because we can additionallysuperimpose the flow created by a point vortex of ar-bitrary strength located at the center of the circle. Thesolution is therefore not unique, but this problem willbe addressed later when we look at the resulting flowaround a foil.

To facilitate the subsequent mapping process, we willwrite down the solution for a circle of radius  r  c whose

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  PROPULSION 9

center is located at an arbitrary point ( x c ,yc ) in the x   y plane, as shown in Fig. 2.5. The circle will be required tointersect the positive x  axis at the point x   a , so that theradius of the circle must be

 r c  x c a2 yc

2 (2.5)

We will see later that in order to obtain physically plausible foil shapes, the point x   a must either be inthe interior of the circle or lie on its boundary. This sim- ply requires that x c  0. Finally, the uniform free-stream velocity will be of speed U   and will be inclined at anangle  with respect to the x  axis.

With these definitions, the velocity components (u , v )in the x  and y directions are

u x, y U  cos U  cos2 sin  

2  r 

 r c

 r 

2

v x, y U  sin U  sin2 cos  

2  r 

 r c

 r 

2

  (2.6)

where r  and  are polar coordinates with origin at thecenter of the circle , so that

 x  x c  r  cosy y

c

 r  sin  (2.7)

Note that we are following a strict right-handedcoordinate system, so that positive angles and posi-tive tangential velocities are in a counterclockwisedirection. A vortex of positive strength,  , thereforeinduces a velocity which is in the negative x  directionon the top of the circle and a  positive x  direction atthe bottom.

Figure 2.5 shows the result in the special case wherethe circulation,  , has been set to zero, and the resultingflow pattern is clearly symmetrical about a line inclinedat the angle of attack—which in this case was selectedto be 10 degrees. If, instead, we set the circulation equal

to a value of    7.778695, the flow pattern shown inFig. 2.6 results.

Clearly, the flow is no longer symmetrical, and thetwo stagnation points on the circle have both moveddown. The angular coordinates of the stagnation pointson the circle can be obtained directly from equation(2.6) by setting r    r c and solving for the tangential com- ponent of the velocity

ut v cos u sin

2U  sin   

2  r c

  (2.8)

Figure 2.4 An example of a complete geometrical description of a foilsection using a fourth order uniform B-spline. The symbols connectedwith dashed lies represent the B-spline control polygon, which com-pletely defines the shape of the foil. The resulting foil surface evaluatedfrom the B-spline is shown as the continuous curve. The knots are thepoints on the foil surface where the piecewise continuous polynomialsegments are joined. The upper curve shows an enlargement of theleading edge region. The complete foil is shown in the lower curve.

Figure 2.5 Flow around a circle with zero circulation. The center of the circleis located at x  0.3, y  0.4. The circle passes through x  a  1.0.The fl ow angle of attack is 10 degrees.

X

      Y

-2 -1 0 1 2

-2

-1

0

1

2

(x,y)

θ

a-aβ

r 

r c

xc,y

c

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10 PROPULSION

If we set u t  0 in equation (2.8) and denote the angu-lar coordinates of the stagnation points as s , we obtain

sins

  

4  r cU   (2.9)

 For the example shown in Fig. 2.6, substituting

 r c

1.32 + 0.42

1.3602,    7.778695, U   1.0, and

  10 degrees into equation (2.9), we obtain

sins 0.45510 : s 17.1deg,  142.9deg  (2.10)

In this special case, we see that we have carefully se-lected  in such a way as to move the rear stagnation pointexactly to the point a on the x  axis, since s   , where

   arcsinyc

 r c  (2.11)

 2.3.2 Conformal Mapping Essentials  Conformalmapping is a useful technique for solving 2D ideal fluid problems because of the analogy between the proper-ties of an analytic function of a complex variable and

the governing equations of a fluid. We know that the flowof an ideal fluid in two dimensions can be representedeither by a scalar function  ( x  ,y ) known as the velocity potential , or by a scalar function   ( x  ,y ) known as thestream function . To be a legitimate ideal fluid flow, bothmust satisfy Laplace’s equation. The fluid velocities canthen be obtained from either, as follows.

 x 

 

yu

v

y

 

 x 

  (2.12)

 Now let us suppose that the physical x  ,y coordinatesof the fluid flow are the real and imaginary parts of acomplex variable z   x   iy . We can construct a com- plex potential   ( z ) by assigning the real part to bethe velocity potential and the imaginary part to be the

stream function

 z  x, y i  x, y  (2.13)

 As the real and imaginary parts of   each satisfyLaplace’s equation,   is an analytic function.5  In addi-tion, the derivative of  has the convenient property ofbeing the conjugate of the “real” fluid velocity, u  iv . Aneasy way to show this is to compute d  /dz by taking theincrement dz in the x  direction

d

dz

 x 

 x 

 

 x 

u iv

i   (2.14)

 where the second line of equation (2.14) follows directlyfrom equation (2.12). If you are not happy with this ap- proach, try taking the increment dz in the iy direction,and you will get the identical result. This has to be true,because  is analytic and its derivative must thereforebe unique.

We now introduce a  mapping function    ( z ), withreal part   and imaginary part  . We can interpret the z plane and the   graphically as two different maps. Forexample, if the z plane is the representation of the flowaround a circle (shown in Figs. 2.5 and 2.6), then each pair of x  ,y coordinates on the surface of the circle, oron any one of the flow streamlines, will map to a cor-

responding point   ,  in the   plane, depending on the particular mapping function   (z). This idea may makemore sense if you take an advanced look at Fig. 2.7. Thefancy looking foil shape was, indeed, mapped from acircle.

While it is easy to confirm that the circle has beenmapped into a more useful foil shape, how do we knowthat the fluid velocities and streamlines in the   planeare valid? The answer is that if  ( z ) and the mappingfunction   ( z ) are both analytic, then  (  ) is also ana-lytic. It therefore represents a valid 2D fluid flow, butit may not necessarily be one that we want. However,if the dividing streamline produces a shape that we

accept, then the only remaining flow property thatwe need to verify is whether or not the flow at largedistances from the foil approaches a uniform streamof speed U  and angle of attack  . We will ensure the proper far-field behavior if the mapping function isconstructed in such a way that     z in the limit as  zgoes to infinity.

Figure 2.6 Flow around a circle with circulation. The center of the circle islocated at x  0.3, y  0.4. The circle passes through x  a  1.0.Note that the rear stagnation point has moved to x   a .

X

      Y

-2 -1 0 1 2

-2

-1

0

1

2

(x,y)

θ

a-aβ

r 

r c

xc,y

c

5Remember, an analytic function is one that is single valuedand whose derivative is uniquely defined (i.e., the value ofits derivative is independent of the path taken to obtain thelimiting value of   /  z ).

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  PROPULSION 11

 Finally, the complex velocity in the   plane can sim- ply be obtained from the complex velocity in the z plane

d 

d 

d 

d  z

d 

d  z

d 

d  z

u iv u iv   z

  (2.15)

 Even though we introduced the concept of the com-

 plex potential,  , we do not actually need it. From equa-tion (2.15), all we need to get the velocity field around thefoil is the velocity around the circle and the derivativeof the mapping function. And, of course, we need themapping function itself to find the location of the actual point in the   plane where this velocity occurs.

Evaluating expressions involving complex variableshas been greatly facilitated by the availability of com- puter languages that permit the declaration of complexdata types. In addition, commercial graphics pack-ages designed specifically to handle output from com- putational fluid dynamics (CFD) codes can be used togenerate high-quality graphs of flow streamlines, color

contours of velocities and pressures, and flow vectors.When applied to conformal mapping solutions, there is practically no limit to the resolution of flow details. Theinformation was all there in Joukowski’s time, but themeans to view it was not! The flow figures in this sec-tion were all generated by a procedure of this type de- veloped by Kerwin (2001).

 2.3.3 The Kármán-Trefftz Mapping Function  TheKármán-Trefftz transformation maps a point z to a point  using the following relationship

  a z a   z a 

 z a    z a   (2.16)

where  and a are given real constants, whose purposewe will discover shortly. The derivative of the mappingfunction, which we will need to transform the velocitiesfrom the z plane to the   plane can be obtained directlyfrom equation (2.16)

 4 2a2   z a 1  z a 1  z a    z a  2

d 

dz  (2.17)

We can see immediately from equation (2.16) that when   1 the mapping function reduces to     z , so this pro-duces an exact photocopy of the original flow! Note also,that when z  a ,     a. Since we want to stretch out thecircle, useful values of  will therefore be greater than 1.0.

Finally, from equation (2.17), the derivative of themapping function is zero when z  a . These are calledcritical points  in the mapping function, meaning thatstrange things are likely to happen there. Most difficultconcepts of higher mathematics can best be understoodby observing the behavior of small bugs. Suppose a bugis walking along the perimeter of the circle in the z plane,starting at some point  z  below the point a . The bug’sfriend starts walking along the perimeter of the foil in the  plane starting at the mapped point   ( z ). The magnitudeand direction of the movement of the second bug is re-lated to that of the first bug by the derivative of the map- ping function. If d   /dz is nonzero, the relative progress ofboth bugs will be smooth and continuous. But when thefirst bug gets to the point a , the second bug stops dead inits tracks, while the first bug continues smoothly. After point a , the derivative of the mapping function changessign, so the second bug reverses its direction. Thus, a

sharp corner is produced, as is evident in Fig. 2.7.The included angle of the corner (or tail angle in this

case) depends on the way in which d   /dz  approacheszero. While we will not prove it here, the tail angle   (indegrees) and the exponent    in the mapping functionare simply related

  1802     2    

180  (2.18)

so that the tail angle corresponding to      1.86111 is25 degrees, which is the value specified for the foil shownin Fig. 2.7. Note that if   2 in equation (2.18) the result-ing tail angle is zero, (i.e., a cusped trailing edge results).In that case, the mapping function in equation (2.16)

reduces to a much simpler form which can be recognizedas the more familiar Joukowski transformation

a2

 z  z   (2.19)

 Finally, if   1, the tail angle is    180 degrees, orin other words, the sharp corner has disappeared. Sincewe saw earlier that     1 results in no change to theoriginal circle, this result is expected. Thus, we see thatthe permissible range of  is between (1,2). In fact, since practical foil sections have tail angles that are generallyless than 30 degrees, the corresponding range of    isroughly from (1.8,2.0).

Figure 2.7 Flow around a Kármán-Trefftz foil derived from the fl ow arounda circle shown in Fig. 2.6 with a specified tail angle of    25 degrees.

X

      Y

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Karman-Trefftz Section: xc=-0.3 y

c=0.3   τ=25 degrees

Angle of Attack=10 degrees.

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12 PROPULSION

If a rounded leading edge is desired, then the circlemust pass outside of  z    a . On the other hand, wecan construct a foil with a sharp leading and trailingedge by placing the center of the circle on the imaginaryaxis, so that a circle passing through  z    a will also pass through z  a . In this case, the upper and lower

contours of the foil can be shown to consist of circulararcs. In the limit of small camber and thickness, thesebecome the same as parabolic arcs.

 2.3.4 The Kutta Condition  We can see from equa-tion (2.6) that the solution for the potential flow arounda circle is not unique, but contains an arbitrary valueof the circulation,  . If we were only interested in this particular flow, it would be logical to conclude, fromsymmetry, that the only physically rational value forthe circulation would be zero. On the other hand, if thecylinder were rotating about its axis, viscous forcesacting in a real fluid might be expected to induce a cir-culation in the direction of rotation. This actually hap-

 pens in the case of exposed propeller shafts which areinclined relative to the inflow. In this case, a transverseforce called the  Magnus effect will be present. This isdescribed, for example, in Thwaites (1960) who gaveseveral examples including the Flettner Rotor Ship thatcrossed the Atlantic Ocean in 1922 propelled by two vertical-axis rotating cylinders replacing the masts andsails. Thwaites also described the use of fluid jets ori-ented tangent to the surface of a cylinder or airfoil to al-ter the circulation. However, these are not of interest inthe present discussion, where the flow around a circle issimply an artificial means of developing the flow arounda realistic foil shape.

Figure 2.8 shows the local flow near the trailingedge for the Kármán-Trefftz foil shown in Fig. 2.5. Theflow in the left figure shows what happens when thecirculation around the circle is set to zero. The flow onthe right figure shows the case where the circulation isadjusted to produce a stagnation point at the point a

on the x  axis, as shown in Fig. 2.6. In the former case,there is flow around a sharp corner, which from equa-tion (2.15) will result in infinite velocities at that pointsince d   /dz is zero. On the other hand, the flow in theright figure seems to leave the trailing edge smoothly.If we again examine equation (2.15), we see that theexpression for the velocity is indeterminate, with bothnumerator and denominator vanishing at z  a . It canbe shown from a local expansion of the numerator anddenominator in the neighborhood of  z  a  that thereis actually a stagnation point there provided that thetail angle    0. If the trailing edge is cusped (   o ),the velocity is finite, with a value equal to the compo-

nent of the inflow that is tangent to the direction of thetrailing edge.Kutta’s hypothesis was that in a real fluid, the flow

 pattern shown in the left of Fig. 2.8 is physically impos-sible, and that the circulation will adjust itself untilthe flow leaves the trailing edge smoothly. His conclu-sion was based, in part, on a very simple but clever ex- periment carried out by Prandtl in the Kaiser WilhelmInstitute in Göttingen around 1910. A model foil sectionwas set up vertically, protruding through the free sur-face of a small tank. Fine aluminum dust was sprinkledon the free surface, and the model was started up fromrest. The resulting flow pattern was then photographed,

Figure 2.8 Flow near the trailing edge. The figure on the lef t is for zero circulation. Note the fl ow around the sharp trailing edge and the presence of astagnation point on the upper surface. The figure on the right shows the result of adjusting the circulation to provide smooth fl ow at the trailing edge.

X

      Y

1.5 1.75 2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Zero Circulation

Stagnation Point

X

      Y

1.6 1.7 1.8 1.9 2 2.1 2.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Karman-Trefftz Section: xc=-0.3 y

c=0.4   τ=25 degrees

Angle of Attack=10 degrees. Circulation,  Γ=-7.778

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  PROPULSION 13

as shown in Fig. 2.9. The photograph clearly shows theformation of a vortex at the trailing edge which is thenshed into the flow. Because Kelvin’s theorem states thatthe total circulation must remain unchanged, a vortexof equal but opposite sign develops around the foil.Thus, the adjustment of circulation is not arbitrary butis directly related to the initial formation of vortex inthe vicinity of the sharp trailing edge. While this pro-cess is initiated by fluid viscosity, once the vortex hasbeen shed, the flow around the foil acts as though it isessentially inviscid.

This basis for setting the circulation is known as theKutta condition, and it is universally applied when invis-cid flow theory is used to solve both 2D and 3D lifting problems. However, it is important to keep in mind thatthe Kutta condition is an idealization of an extremelycomplex real fluid problem. It works amazingly wellmuch of the time, but it is not an exact solution to the problem. We will see later how good it really is!

In the case of the present conformal mapping methodof solution, we simply set the position of the rear stag-nation point to s   . The required circulation, fromequation (2.9) is,

   4  r cU  sin    (2.20)

 2.3.5 Pressure Distributions  The distribution of pressure on the upper and lower surfaces of a hydrofoilis of interest in the determination of lift and drag forces,cavitation inception, and in the study of boundary layerbehavior. The pressure field in the neighborhood of thefoil is of interest in studying the interaction betweenmultiple foils, and in the interaction between foils andadjacent boundaries. The pressure at an arbitrary pointcan be related to the pressure at a point far upstreamfrom Bernoulli’s equation,

1

2

1

2 p    U 2  p   q2

 where q is the magnitude of the total fluid velocity at the point in question,

q u2 v2

 and (u ,v ) are the components of fluid velocity obtained

from equation (2.15). The quantity  p  is the pressurefar upstream, taken at the same hydrostatic level. Anondimensional pressure coefficient can be formed bydividing the difference between the local and upstream pressure by the upstream dynamic pressure

 p  p   q

U  1

2

C  P   1

2  U 2

 Note that at a stagnation point, q  0, so that the pres-sure coefficient becomes C  P   1.0. A pressure coefficientof zero indicates that the local velocity is equal in mag-nitude to the free-stream velocity, U  , while a negative

 pressure coefficient implies a local velocity that exceedsfree stream. While this is the universally accepted con- vention for defining the nondimensional pressure, manyauthors plot the negative of the pressure coefficient. Inthat case, a stagnation point will be plotted with a valueof C  P    1.0.

 2.3.6 Examples of Propellerlike Kármán-Trefftz Sections  Figures 2.10 to 2.14 show the foil sections, pressure contours, and stream traces for two foil sec-tions operating at an angle of attack,  , of 0 and 10 de-grees. The mapping parameters are identified on each plot, and the contour levels for the pressure coefficientare the same for all graphs in order to permit directcomparison. Figure 2.10 shows a “skeleton” section witha sharp leading (and trailing) edge at an angle of attackof zero. Note that the pressure contours and streamtraces are symmetrical about the midchord, and thatthe dividing streamline therefore passes smoothly overthe upper and lower surfaces of the leading edge.

Figure 2.11 shows the same section at an angle ofattack of 10 degrees. The dividing streamline now im- pacts the foil on the lower surface slightly downstream

Figure 2.9 Early fl ow visualization photograph showing the developmentof a starting vortex (Prandtl & Tietjens, 1934). (Reprinted by permission ofDover Publications.)

Figure 2.10 Skeleton at zero angle of attack.

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14 PROPULSION

of the leading edge. The flow around the sharp leading

edge from the lower to the upper surface produces alocal region of high velocity and hence low pressure.While the highest pressure coefficient contour (lowest pressure) is shown as 4.0, it is actually infinite right atthe leading edge.

Figure 2.12 shows a section generated with the samemapping parameters as in Figure 2.10 except that  x chas been moved from zero to  0.05, thus producing arounded leading edge. The angle of attack is zero inthis case, and the flow pattern is no longer symmet-ric about the midchord. However, the dividing stream-line impacts the foil right at the leading edge and passes smoothly over the upper and lower foil surface.

Figure 2.13 is the same foil, but at an angle of attack of10 degrees. The dividing streamline impacts the foil onthe lower surface, as in Fig. 2.11, but the high velocityregion near the leading edge is less extreme. Finally,Fig. 2.14 shows a close-up of the leading-edge regionfor this case.

We will see that the effect of foil geometry andangle attack on the detailed flow around the leading

edge is of extreme importance in propeller design,

and the analytical results shown here are provided asa preview.

 2.3.7 Lift and Drag  Determining the overall liftand drag on a 2D foil section in inviscid flow is incrediblysimple. The force (per unit of span) directed at right an-gles to the oncoming flow of speed U  is termed lift andcan be shown to be

 L   U     (2.21)

while the force acting in the direction of the oncomingflow is termed drag is zero. Equation (2.21) is known as Kutta-Joukowski’s Law .6

 We can easily verify that equation (2.21) is correct

for the flow around a circle by integrating the y and  x components of the pressure acting on its surface. With-out loss of generality, let us assume that the circle is

Figure 2.11 Skeleton at 10 degrees—leading edge close up.

Figure 2.12 Kármán-Trefftz section with a rounded leading edge at zeroangle of attack.

Figure 2.13 Foil shown in Fig. 2.12 at an angle of attack of 10 degrees.

6 The negative sign in the equation is a consequence of choos-ing the positive direction for x  to be downstream and using aright-handed convention for positive  .

Figure 2.14 Close-up of leading-edge region of the foil shown in Fig. 2.13.

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  PROPULSION 15

 centered at the origin, and that the angle of attack iszero. In this case, the velocity on the surface of the cir-cle, from equation (2.8), is

ut 2U  sin    

2  r c

  (2.22)

 As before, we can write down the pressure fromBernoulli’s equation

1

2 p  p   U 2 ut 

2  (2.23)

and the lift is the integral of the y  component of the pressure around the circle

 2 

0

 L     p  p  sin  r cd   (2.24)

By substituting equations (2.22) and (2.23) into equa-tion (2.24), and recognizing that only the term containingsin2  survives the integration, one can readily recoverequation (2.21).

In a similar way, we can write down the integral fordrag

 2 

0

 D     p  v  p 

 cos  r cd  (2.25)

and show that all terms are zero.We will now resort to “fuzzy math” and argue that

equation (2.21) must apply to any foil shape. The argu-ment is that we could have calculated the lift force onthe circle from an application of the momentum theo-rem around a control volume consisting of a circular path at some large radius r    r c . The result must bethe same as the one obtained from pressure integration

around the foil. But if this is true, the result must alsoapply to any foil shape, because the conformal mappingfunction used to create it requires that the flow fieldaround the circle and around the foil become the sameat large values of r  .

 2.3.8 Mapping Solutions for Foils of Arbitrary Shape  Closed form mapping functions are obviouslylimited in the types of shapes that they can produce.While some further extensions to the Kármán-Trefftzmapping function were developed, this approach waslargely abandoned by the 1930s. Then, in 1931, The-odorsen (1931) published a method by which one couldstart  with the foil geometry and develop the mapping

function that would map it back to a circle. This wasdone by assuming a series expansion for the mappingfunction and solving numerically for a finite number ofterms in the series. The method was therefore approxi-mate and extremely time-consuming in the precomputerera. Nevertheless, extensive application of this methodled to the development of the NACA series of wing sec-tions, including the sample foil section shown in Fig. 2.2.

 An improved version of Theodorsen’s method, suit-able for implementation on a digital computer, was de- veloped by Brockett (1966). Brockett found, as noted in2.3.1, that inaccuracies existed in the tabulated geom-etry and pressure distributions for some of the earlier

NACA data. Brocket’s modified NACA-66 thicknessform was developed at that time and has been usedextensively for propeller sections.

By the mid 1970s, conformal mapping solutions hadgiven way to panel methods, which we will discuss later.

This happened for three reasons:

1. Conformal mapping methods cannot be extendedto 3D flow, while panel methods can.

2. Both methods involve numerical approximationwhen applied to foils of a given geometry, and imple-mentation and convergence checking is more straight-forward with a panel method.

3. Panel methods can be extended to include viscousboundary layer effects.

2.4 Linearized Theory for a Two-Dimensional Foil Section

 2.4.1 Problem Formulation  In this section, wewill review the classical l inearized theory for 2D foilsin inviscid flow. The problem will be simplified bymaking the assumptions that the thickness and cam-ber of the foil section is small and that the angle of at-tack is also small. The flow field will be considered asthe superposition of a uniform oncoming flow of speedU   and angle of attack  and a  perturbation velocity field  caused by the presence of the foil. We will usethe symbols u ,v  to denote the perturbation velocity,so that the total fluid velocity in the x  direction will beU  cos   u , while the component in the y directionwill be U  sin   v .

The reader should be warned that for analytical de- velopments, it is more efficient to have the origin x   0at the foil midchord, whereas for practical foil geometry,

it is more common to have x   0 represent the leadingedge. The reader should take care to correctly interpreteach equation, but the author will warn the reader eachtime the coordinate system is redefined.

The exact kinematic boundary condition is that theresultant fluid velocity must be tangent to the foil onboth the upper and lower surface

on y yu

U  sin v

U  cos u

U  sin v

U  cos u

dyl

dx 

dyu

dx 

on y yl  (2.26)

 However, since we are looking for the linearized so-lution, three simplifications can be made. First of all,since  is small, cos   1 and sin    . But if the cam-ber and thickness of the foil is also small, the perturba-tion velocities can be expected to be small compared tothe inflow.7 Finally, since the slope of the mean line,  ,

7  Actually, this assumption is not uniformly val id, since the perturbation velocity wil l not be small in the case of the flowaround a sharp leading edge, nor is it small close to the stag-nation point at a rounded leading edge. We will see later thatlinear theory will be locally invalid in those regions.

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16 PROPULSION

is also small, the coordinates of the upper and lowersurfaces of the foil shown in equation (2.1) will beapproximately,

yu

 x   f  x  t x 

2

yl x   f  x  t x 

2

(2.27)

 Introducing these approximations into equation(2.26), we obtain the following

df  x dx 

dt x dx 

1

2

U  v  x U 

on y = 0

df  x dx 

dt x dx 

1

2

U  v  x U 

on y = 0

dyl

dx 

dyu

dx 

 (2.28)

 Note that the boundary condition is applied on the liney    0 rather than on the actual foil surface, which isconsistent with the linearizing assumptions made sofar. This result can be derived in a more formal wayby carefully expanding the geometry and flow field interms of a small parameter, but this is a lot of work andis unnecessary to obtain the correct linear result. Thenotation v and v means that the perturbation velocityis to be evaluated just above and just below the  x  axis.Now, if we take half of the sum and the difference of thetwo equations above, we obtain

df  x 

dx 

  v x  v  x 

2U 

dt x dx 

  v x  v

  x U 

  (2.29)

 We now see that the linearized foil problem hasbeen conveniently decomposed into two parts. Themean value of the vertical perturbation velocity alongthe  x   axis is determined by the slope of the camberdistribution f  ( x  ) and the angle of attack,  , measuredin  radians . The jump in vertical velocity across the x   axis is directly related to the slope of the thick-ness distribution, t ( x  ). This is the key to the solutionof the problem, because we can generate the desired

even and odd behavior of v ( x  ) by distributing vort icesand sources along the x  axis between the leading andtrailing edge of the foil, as will be shown in the nextsection.

 2.4.2 Vortex and Source Distributions  The veloc-ity field of a point vortex of strength located at a point  on the x  axis is

u x, y  

2 

y

 x   2 y2

v x, y    

2 

 x   

 x   2 y2

  (2.30)

 while the corresponding velocity field for a point sourceof strength S  is,

u x, y   S 

2 

 x   

 x   2 y2

v x, y   S 

2 

y

 x   2 y2

  (2.31)

 We next define a vortex sheet  as a continuous dis-tribution of vortices with strength    per unit length.The velocity field of a vortex sheet distributed between x   c  /2 to x   c  /2 will be

c / 2

2c / 2

 

c / 2

2c / 2

u x, y

v x, y

1

2 

  y x   2 y2

d 

d 1

2 

   x    x   2 y2

(2.32)

 It is instructive to look at the velocity field in thespecial cases where the vortex strength is constantover the interval. This result will also be useful laterwhen we look at panel methods. As shown by Katz andPlotkin (1991), in this case  comes outside the integral,and equation (2.32) can be integrated analytically, giv-ing the result

 x  c   22 y2

 x  c   22 y2lnv x, y

   

2 

tan1 tan1y

 x  c   2

y

 x  c   2u x, y

   

2  (2.33)

 The reader should be aware that the evaluation ofthe arctangent such as in equation (2.33) must be donecarefully to ensure that the resulting angle is in the cor-rect quadrant. This requires that both the numeratorand denominator of the argument to the arctangent beknown—not just the resulting quotient.8

 Figure 2.15 shows the velocity field obtained fromequation (2.33) for points along the y  axis in the casewhere the vortex sheet strength has been set to     1.Note that a jump in horizontal velocity exists across thesheet, and that the value of the velocity jump is equal tothe strength of the sheet. This fundamental property of

a vortex sheet follows directly from an application ofStokes theorem to a small circulation contour spanningthe sheet, as shown in Fig. 2.16.

  dx  u dx  0 udx  0

  u u  (2.34)

8 Most programming languages provide intrinsic  functionsfor the arctangent (such as ATAN2 in Fortran95) that requirethat the numerator and denominator be supplied separately.In more precise mathematical terms, ATAN2 (y , x  ) (or equiva-lent) returns the principal value of the argument of the com- plex number z   x   iy .

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  PROPULSION 17

 Even though Figure 2.15 was computed for a uniform

distribution of   ( x  ) between x 1 and x 2 , the local behaviorof the u component of velocity close to the vortex sheetwould be the same for any continously varying distri-bution. On the other hand, the v component of velocitydepends on   ( x  ), but is continuous across the sheet.Figure 2.17 shows the v component of velocity along the x  axis, again for the case where     1.

We can develop similar expressions for the velocityfield of a uniform strength source sheet. If we let thestrength of the source sheet be   per unit length, the velocity field of a source sheet extending from x   c  /2to x   c  /2 will be

c / 2

2c / 2

 

c / 2

2c / 2

u x, y

v x, y

1

2 

   x   

 x   2

y2

d 

d 1

2 

  y x   2 y2

  (2.35)

 Again, if we specify that the source strength is con-stant, equation (2.35) can be integrated, so give the result

(2.36)lnu x, y    x  c   22 y2

 x  c   22 y2

 

4 

v( x, y) tan1   x  c   2

y

 

2  tan1

  x  c  2

y

 Figure 2.18 shows the v  component of the velocity

obtained from equation (2.36) evaluated just above and just below the x  axis for a value of    1. The jump in the vertical velocity is equal to the value of the source sheetstrength, which follows directly from a consideration ofmass conservation.

  v v (2.37)

Returning to equation (2.29), we now see that, withinthe assumptions of linear theory, a foil can be representedby a distribution of sources and vortices along the x  axis.The strength of the source distribution,   ( x  ) is known di-rectly from the slope of the thickness distribution

  x  U 

  dt x dx 

(2.38)

 while [since v x , y 0 v v

2] the vortex sheet dis-

tribution must satisfy the relationship

c / 2

2c / 2

1

2    x   

  c   d  U 

df 

df  x    (2.39)

Figure 2.15 Vertical distribution of the u  velocity at the midchord of aconstant strength vortex panel of strength    1.

x1

  x2

u

Figure 2.16 Illustration of the circulation path used to show that the jump inu velocity is equal to the vortex sheet strength,   .

γ(x)

u+

u-

dx

Figure 2.17 Horizontal distribution of the v  velocity along a constant strengthvortex panel of strength    1.

x1

  x2

v(x)

Figure 2.18 Horizontal distribution of the v  velocity along a constant strengthsource panel of strength    1.

x1

  x2

v(x)

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18 PROPULSION

The symbol “c” superimposed on the integral signindicates that the form of this integral is known as a“Cauchy principal value integral,” as will be discussedin the next section.

This decomposition of foil geometry, velocity fields, and

singularity distributions has revealed a very important re-sult. According to linear theory, the vortex sheet distribu-tion, and hence the total circulation, is unaffected by foilthickness, since it depends only on the mean line shapeand the angle of attack. This means that the lift of a foilsection is unaffected by its thickness. Now, the exact con-formal mapping procedure developed in the previous sec-tion shows that lift increases with foil thickness, but onlyslightly . So, there is no contradiction, as linear theory isonly supposed to be valid for small values of thickness. Wewill see later that viscous effects tend to reduce the amountof lift that a foil produces as thickness is increased. So, insome sense, linear theory is more exact than exact theory!We will return to this fascinating tale later.

To complete the formulation of the linear problem,we must introduce the Kutta condition. Since the jumpin velocity between the upper and lower surface of thefoil is directly related to the vortex sheet strength, it issufficient to specify that   (c  /2)  0. If this were not true,there would be flow around the sharp trailing edge.

2.5 Glauert’s Solution for a Two-Dimensional Foil  Inthis section, we will summarize the relationship betweenthe shape of a mean line and its bound vortex distribu-tion following the approach of Glauert (1947). A distri-bution of bound circulation   ( x  ) over the chord inducesa velocity field v ( x  ) which must satisfy the linearizedboundary condition developed earlier in equation (2.39)

df 

dx  v x  U    (2.40)

 Glauert assumed that the unknown circulation   ( x  )could be approximated by a series in a transformed  x coordinate, x ~,

 x  c

2cos x   (2.41)

 Note that at the leading edge, x   c  /2, x ~  0, whileat the trailing edge, x   c  /2, x ~    . The value of x ~ at themidchord is    /2. The series has the following form:

a n

sin nx a0

1 cos x 

sin x   n1 ˜ 

 ˜   x  2U    (2.42)

  All terms in equation (2.42) vanish at the trailing edgein order to satisfy the Kutta condition. Since the sineterms also vanish at the leading edge, they will not beable to generate an infinite velocity, which may be pres-ent there. The first term in the series has therefore beenincluded to provide for this singular behavior at the lead-ing edge. This first term is actually the solution for a flat plate at unit angle of attack obtained from the Joukowskitransformation, after introducing the approximation thatsin    . It goes without saying that it helps to know theanswer before starting to solve the problem!

With the series for the circulation defined, we cannow calculate the total lift force on the section fromKutta-Joukowski’s law,

 L   U      U   x dx 

c  2

c  2

  cU 2  a0   a1

 

2

  (2.43)

 Equation (2.43) can be expressed in nondimensionalform in terms of the usual lift coefficient

C  L  2 a 0   a 1

 L

1

2  U 2c

  (2.44)

 We will next develop an expression for the distribu-tion of vertical velocity, v , over the chord induced by thebound vortices

v x    d c1

2  

c  2

c  2

 ( )

 x   

  (2.45)

 Note that the integral in equation (2.45) is singular,since the integrand goes to infinity when x     .

To evaluate the integral, one must evaluate theCauchy principal value, which is defined as

 f  x,  d   lim   f  x,  d     f  x,  d 0c  2

c  2

 

 x 

c  2

 

c  2

 x 

c  (2.46)

 What equation (2.46) says, in simple terms, is that if theintegral goes to  on one side of the point x     and goesto  at the same rate on the other side, the two will can-cel out, leaving a finite result. The detailed steps in carry-

ing out the integration of equation (2.45) may be found inGlauert (1947) and the final result is amazingly simple:

v x U 

a0   a

 n cos nx 

 n1

  (2.47)

 Solving equation (2.40) for df   /dx   and substitutingequation (2.47) for v , we obtain the desired relationshipbetween the shape of the mean line and the series co-efficients for the chordwise distribution of the boundcirculation

a0

 

0

df dx 

1

    dx 

a n cos nx dx 

 

0 

2 

df dx 

  (2.48)

 A particularly important result is obtained by solvingequation (2.48) for the angle of attack for which the a0

coefficient vanishes

ideal

 1

 

 n

0

 

df 

dx  dx  (2.49)

This is known as the ideal angle of attack , and it is particularly important in hydrofoil and propeller designbecause it relates to cavitation inception at the leadingedge. For any shape of mean line, one angle of attack

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  PROPULSION 19

exists for which the velocity is finite at the leading edge.From the symmetry of equation (2.49), we see that theideal angle of attack is zero for any mean line that issymmetrical about the midchord. Combining equations(2.48), (2.49), and (2.50) gives an alternate form for the

lift coefficientC  L  2 

ideal C 

 L

ideal

(2.50)

 where C Lideal  1  is the ideal lift coefficient, which is

the lift coefficient when the angle of attack of the foilequals ideal angle of attack.

 2.5.1 Example: The Flat Plate  For a flat plate atangle of attack  , we can see immediately from theGlauert results that a0   and a n  0 for n  0. The liftcoefficient is then found to be C  L  2   and the boundcirculation distribution over the chord is

  x   2U 

sin x 

1  cos x   (2.51)

 This result, together with some other cases that wewill deal with next, are plotted in Fig. 2.19. In this figure,all of the mean lines have been scaled to produce a liftcoefficient of C  L  1.0. In the case of a flat plate, the angleof attack has therefore been set to   1/(2  ) radians.

 2.5.2 Example: The Parabolic Mean Line  The equa-tion of a parabolic mean line with maximum camber f 0 is

 f  x   f o

 x 

c    21

2

  (2.52)

 so that the slope is

df  8 f o x 

c2 dx 

(2.53)

but because x   c  /2cos x ~, the slope can be written as

df    f o 4

ccos x 

dx   (2.54)

Therefore, we can again solve for the Glauert coef-ficients of the circulation very easily:

a n

0 for n>1

1

  a

0   dx  0

 

0

df 

dx 

cos2 xdx  4a1 c

4 f 0   f 0

c

2

  

 

0

(2.55)

 The lift coefficient is then given by the expression

C  L

 2 a0

 a1

 2 a  4  f 

0

c  (2.56)

 and the circulation distribution becomes

8U  f 

0

c  x  2U  sin x 

sin x 

1  cos x   (2.57)

The solution for the parabolic camber line thereforeconsists of the sum of two parts: a li ft and circulationdistribution proportional to the angle of attack anda lift and circulation distribution proportional to thecamber ratio. This is true for any mean line, exceptthat in the general case, the lift due to angle of attackis proportional to the difference between the angle ofattack and the ideal angle of attack (  ideal ). The lat-ter is zero for the parabolic mean line due to its symme-try about the midchord. The result plotted in Fig. 2.19

is for a parabolic mean line operating with a lift coef-ficient of C  L    1.0 at its ideal angle of attack, whichis zero.

2.6 The Design of Mean Lines: The NACA a-Series  Froma cavitation point of view, the ideal camber line is onethat produces a constant pressure difference over thechord. In this way, a fixed amount of lift is generatedwith the minimum reduction in local pressure. As thelocal pressure jump is directly proportional to thebound vortex strength, such a camber line has a con-stant circulation over the chord. Unfortunately, thistype of camber line does not perform to expectation,since the abrupt change in circulation at the trailing

edge produces an adverse pressure gradient whichseparates the boundary layer. One must therefore beless greedy and accept a load distribution that is con-stant up to some percentage of the chord, and then al-low the circulation to decrease linearly to zero at thetrailing edge. A series of such mean lines was devel-oped by the NACA and this work is presented by Abbottand Von Doenhoff (1959). This series is known as thea-series, where the parameter “a” denotes the fractionof the chord over which the circulation is constant. Theoriginal NACA development of these mean l ines, whichdates back to 1939, was to achieve laminar flow wingsections. The use of these mean lines in hydrofoil and

Figure 2.19 Comparison of shape and vortex sheet strength for a fl at plate,parabolic mean line, NACA a  1.0, and NACA a  0.8, all with unitlift coefficient.

X

      Y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

Flat Plate

NACA a=1.0

Parabolic

NACA a=0.8

X

      V    o    r      t    e    x    s      h    e    e      t    s      t    r    e    n    g      t      h  ,       γ      (    x      )

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Flat Plate

Parabolic

NACA a=1.0

NACA a=0.8

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20 PROPULSION

 propeller applications to delay cavitation inception wasa later development.

These shapes could, in principle, be developed fromthe formulas developed in the preceding section by ex- panding the desired circulation distribution in a sine se-

ries. However, a large number of sine series terms wouldbe necessary for a converged solution, so it is better tointegrate equation (2.39) directly. As   ( x  ) consists onlyof constant and linear segments, the integration can becarried out analytically. The resulting expression forthe shape of the mean line for any value of the param-eter “a” and lift coefficient, C  L , is

1

1 a  a ln1

2

 x 

c

 f  x c

C 

 L

2 a  1  a

 x 

c

2

ln1

1  x 

c

2

1  x 

c

2

1  x 

c2

1

4

ln  x 

c

 x 

c

2

1  x 

c  g h

 x 

c

1

4

 (2.58)

where

1

2ln a  g

1

1 a

1

4

1

4a2

1 a2 ln1 a1

2h

1

1 a

1

4  1 a2

 g

 Note that these equations assume coordinates with x   0 at the leading edge and x   c at the trail ing edge.

Except for the NACA a  1.0 mean line, this series ofmean lines is not symmetrical about the midchord. Theideal angles of attack are therefore nonzero, and maybe found from the following equation. Reverting backto coordinates with x   c  /2, x ~  0 at the leading edgeand x   c  /2, x ~    at the trailing edge, we have

ideal

  df    C  Lh1

    dx  2 a  1

 

0

dx  (2.59)

 Experience has shown that the best compromisebetween maximum extent of constant circulation andavoidance of boundary layer separation corresponds to

a choice of a  0.8. The tabulated characteristics of themean line, taken from Abbott and Von Doenhoff (1959),are given in Fig. 2.2.

2.7 Linearized Pressure Coefficient  The distribution of pressure on the upper and lower surfaces of a hydrofoil isof interest both in the determination of cavitation incep-tion and in the study of boundary layer behavior. We sawin the preceding section on conformal mapping methodsthat the pressure at an arbitrary point can be related to the pressure at a point far upstream from Bernoulli’s equation

 p    q21

2 p

   U 2

1

2  (2.60)

 where q is the magnitude of the total fluid velocity at the point in question

q  U  cos u2 U  sin v2   (2.61)

and  p  is the pressure far upstream, taken at the same

hydrostatic level. A nondimensional pressure coefficientcan be formed by dividing the difference between the localand upstream pressure by the upstream dynamic pressure

C  P 

   1  p  p

  U 21

2

2q

U    (2.62)

  As the disturbance velocities (u ,v ) are assumed to besmall compared with the free-stream velocity in lineartheory, and because cos   1 and sin    ,

q 1  2  2

2

U 

u

U 

q2

U 

u

U 

v2

U 

v

U 2   1  2   (2.63)

 so that the pressure coefficient can be approximated by

C  P 

  2 u

U   (2.64)

This is known as the linearized pressure coefficient ,which is valid only where the disturbance velocities aresmall compared to free stream. In particular, at a stag-nation point where q  0, the exact pressure coefficientbecomes 1, while the linearized pressure coefficientgives an erroneous value of 2!

For a linearized 2D hydrofoil without thickness, theu component of the disturbance velocity at points justabove and below the foil is u     /2. Thus, the linear-

ized pressure coefficient and the local vortex sheetstrength are directly related, with

C  P 

U 

   (2.65)

 on the upper surface and

C  P 

U 

   (2.66)

 on the lower surface.Cavitation inception can be investigated by compar-

ing the minimum value of the pressure coefficient on thefoil surface to the value of the cavitation index 

 p  pv 

12  U 2

  (2.67)

 where pv is the vapor pressure of the fluid at the operat-ing temperature of the foil. Comparing the definitions of  and C  P , it is evident that if C  P     , then p   p v. Sup- pose that a foil is operating at a fixed angle of attack at a value of the cavitation index sufficiently high to ensurethat the pressure is well above the vapor pressure every-where. It is therefore safe to assume that no cavitationwill be present at this stage. Now reduce the cavitationnumber, either by reducing p or increasing U  . The point

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  PROPULSION 21

on the foil surface with the minimum pressure coeffi-cient, (C  P  ) min , will reach vapor pressure first, and thiswill occur when  (C  P  ) min    . At this point, equilibriumcan exist between liquid and vapor, so that in principlefluid can evaporate to form a cavity.

The physics of this process is actually very compli-cated, and it turns out that the actual pressure at whicha cavity forms may be below the vapor pressure and willdepend on the presence of cavitation nuclei in the fluid.These may be microscopic free air bubbles or impuritiesin the fluid or on the surface of the foil. If there is an abun-dance of free air bubbles, as is generally the case near thesea surface, cavitation will occur at a pressure very closeto vapor pressure. On the other hand, under laboratoryconditions in which the water may be too pure, cavita-tion may not start until the pressure is substantially be-low vapor pressure. This was responsible for erroneouscavitation inception predictions in the past, before theimportance of air content was understood.

2.8 Comparison of Pressure Distributions  Because the vortex sheet strength   ( x  )/U  and the linearized pressurecoefficient is equivalent, we now have all the necessaryequations to compare the shape and pressure distributionsfor a flat plate, a parabolic camber line, the NACA a  1.0mean line, and the NACA a  0.8 mean line. We will com- pare them at a lift coefficient of 1, with all three mean linesoperating at their ideal angles of attack. Figure 2.19 showsthe shape (including angle of attack) of the four sections inquestion. Note that the slope of the flat plate and parabolicmean line is the same at the three-quarter chord, whichis an interesting result that we will come back to later. Itis also evident that the slope of the NACA a  0.8 mean

line is also about the same at the three-quarter chord, andthat the combination of ideal angle of attack and meanline slope makes the back half of the parabolic and NACAa  0.8 mean lines look about the same.

The NACA a  1.0 mean line looks strange becauseit looks more or less the same as the parabolic meanline, but with much less camber, yet it is supposed tohave the same lift coefficient. The logarithmic form ofthe latter makes a difference, and we can see that inthe enlargement of the first 10% of the chord shown inFig. 2.20. Even at this large scale, however, there is noevidence of the logarithmically infinite slope at the end. As indicated earlier, lift predicted for the NACA a  1.0

is not achieved in a real fluid, so our first impressiongained from Fig. 2.19 is to some extent correct.2.9 Solution of the Linearized Thickness Problem  We

will now turn to the solution of the thickness problem.Equation (2.38) gives us the source strength,   ( x  ) di-rectly in terms of the slope of the thickness form, whileequation (2.35) gives us the velocity at any point ( x  ,y ).Combining these equations, and setting y  0, gives usthe equation for the distribution of horizontal perturba-tion velocity due to thickness

uc

  dt  d d 

 x   

1

U 

2 

c /2

c /2  (2.68)

 2.9.1 Example: The Elliptical Thickness Form  Thethickness distribution for an elliptical section is

 x 2

c   2t x  t

01   (2.69)

 where the origin is taken at the midchord, so that the

leading edge is at x   c  /2 and the trailing edge is at x   c  /2. Transforming the chordwise variable as before

 x  cos x c

2  (2.70)

 the thickness function becomes

t x  t0   1  cos2 x  t0 sin x   (2.71)

 and its slope is

dt

dx  t

0cos x  t

0

cos x 

2

csin x 

dx 

dx    (2.72)

 The integral for the velocity then becomes

sin d c

2

cos t

0c   2 sin 1

2  

0

 u

U    c

2

ccos x   cos 

  (2.73)

t0

 c

u

U 

0

 

ccos x  cos 

cos    t0

cd

 

(2.74)

 Linear theory therefore yields the very simple result inthis case that the nondimensional horizontal disturbance

Figure 2.20 Enlargement of Fig. 2.19 showing the difference between anNACA a  1.0 and parabolic mean line near the leading edge.

X

      Y

0 0.025 0.05 0.075 0.10

0.01

0.02

0.03

Parabolic

NACA a=1.0

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22 PROPULSION

 velocity, u  /U  , is constant over the chord, with a valueequal to the thickness/chord ratio of the elliptical section.It turns out that this result is exact at the midchord, and very nearly correct over most of the chord. However, lin-ear theory has a serious flaw in that no stagnation point

results at the leading and trailing edge. Of course, the as-sumption of small slopes is not valid at the ends, so thebreakdown of linear theory in these regions is inevitable.

 2.9.2 Example: The Parabolic Thickness FormThe parabolic thickness form has the same shape as a parabolic mean line, except that it is symmetrical abouty  0. This is sometimes referred to as a bi-convex foil.The shape of this thickness form, and its slope are

t x  t0

 x 

c   21

2

  (2.75)

c2

8t0 x 

dx 

dt   (2.76)

 and the horizontal disturbance velocity is

1

2 

u

U 

c  2

c  2c

8t0 

 x   d 

 c 2

4t0c  2

c  2c

 

c2

 

 x   d 

  (2.77)

 The above Cauchy principal value integral is one ofa series of such integrals whose evaluation is given by Van Dyke (1955)

uU  4t0

  c  2 x 

c  2 x  x log c

 c2   (2.78)

In this case, the velocity is logarithmically infiniteat the leading and trailing edge, so linear theory failsonce again to produce a stagnation point! However,the logarithmic singularity is very local, so the resultis quite accurate over most of the chord. This result is plotted in Fig. 2.21, together with the result for an ellipti-cal thickness form.

Note that the maximum velocity occurs at the mid-

chord and has a value4

 u  U  t0 / c 1.27324 t0 / c . An

elliptical thickness form with the same thickness/chord

ratio would therefore have a lower value of (C  P  ) min andwould therefore be better from the point of view of cavi-tation inception.

2.10 Superposition of Camber, Angle of Attack, and

Thickness  We can combine mean lines and thicknessforms to produce a wide range of section shapes. Thelinearized perturbation velocity can be determined sim- ply by adding the perturbation velocity due to thickness,camber at the ideal angle of attack, and flat plate load-ing due to departure from the ideal angle of attack

u x  ut x  u

c x  U 

ideal   c  2  x 

c  2  x   (2.79)

 The linearized pressure coefficient can also be deter-mined by superimposing these three effects by equation(2.64), which is reproduced here

C  P   2 u

U   (2.80)

For example, by adding a parabolic mean line witha camber ratio of  f 0  /c   0.05 to a parabolic thicknessform with thickness ratio t0  /c  0.10, we obtain a sectionwith a flat bottom and parabolic top. This is known asan ogival section,9 which was commonly used for ship propellers in the past, and is still used for many quantity produced propellers for small vessels.

 Assuming the angle of attack is the ideal angle of attackof the parabolic mean line,  ideal  0 in this case, the cir-

culation [equation (2.58)] becomes  x  8U   f 0 sin x c .

The disturbance velocity due to camber at the midchord( x ~     /2) is

 f 0c

sin x   4  0.05  1.0  0.200uc 1      4U  2  U 

 on the upper surface, and  0.200 on the lower surface.The velocity due to thickness, from equation (2.78), willbe u t   /U   0.127 on both the upper and the lower surface.Hence, on the upper surface

0.327u

U   C P  0.654

9 Actually, the upper contour of an ogival section is a circu-lar arc, but this approaches a parabolic arc as the thicknessbecomes small.

Figure 2.21 Shape and velocity distribution for elliptical and parabolicthickness forms from linear theory. The thickness/chord ratio, t  o /c ,  0.1.The vertical scale of the thickness form plots has been enlarged for clarity.

Chordwise Position, x/c

      S      h    a    p    e  ,

      t      (    x      )    a    n      d

    v    e      l    o    c      i      t    y  ,    u      /      U

-0.5 -0.25 0 0.25 0.5

-0.1

-0.05

0

0.05

0.1

0.15

Parabolic

u/U Parabola

u/U Ellipse

Elliptical

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  PROPULSION 23

 while on the lower surface

0.073u

U   C P  0.146

2.11 Correcting Linear Theory Near the Leading Edge

We saw in the preceding sections that linear theory can-not predict the local behavior of the flow near a roundleading edge because the assumption of small slopes isclearly violated. While this does not affect the overall lift,any attempt to predict pressure distributions (and cavi-tation inception) near the leading edge will clearly fail.However, as the problem is local, a relatively simple cor-rection to linear theory can be used to overcome this dif-ficulty. This problem was first solved by Lighthill (1951). Amore recent mathematical treatment of this problem maybe found in Van Dyke (1975). An improved formulationof Lighthill’s method was introduced by Scherer (1997),who also cited earlier work by Brockett in 1965, who dis-covered a 1942 publication (in German) by Riegels. Thederivation presented here is based, in part, on class notes prepared by Robert J. Van Houten at MIT in 1982.

Figure 2.22 shows the velocity distribution near theleading edge of an elliptical thickness form obtainedboth by linear and exact theory. Linear theory gives thecorrect answer at the midchord, regardless of thicknessratio, but fails to predict the stagnation point at the lead-ing edge. On the other hand, as the thickness ratio isreduced, the region of discrepancy between exact andlinear theory becomes more local. If the foil is thin, lin-ear theory can be expected to provide the correct globalresult, but it must be supplemented by a local solutionin order to be correct at the leading edge. The technique

of combining a global and local flow solution is known

formally as the method of matched asymptotic expan-sions . However, we will follow a more informal path here.

We saw earlier that the leading edge radius of a foil, r  L , scales with the square of the thickness/chord ratio.If we are concerned with the local flow in the leadingedge region, the maximum thickness of the foil occursat a point that is far away from the region of interest. Infact, if we consult our resident small bug, as far as it isconcerned, the foil extends to infinity in the x  direction.

The relevant length scale for the local problem is there-fore the leading edge radius. As shown in Figure 2.23, ashape which does this is a parabola (turned sideways).We can find the equation for the desired parabola easilyby starting with the equation of a circle of radius r  L withcenter on the x  axis at a distance r  L back from the lead-ing edge, and examining the limit for x    r 

y p

 x  r  L

2  r  L

2 2 y p x  2 r 

 L x  x 2 2 r 

 L x 

 x y

 p

2 r  L

2

 Note that in this section, the coordinates are definedwith x   0 at the leading edge and x   c at the trailing edge.The velocity distribution on the surface of a parabola

in a uniform stream U i can be found by conformal map- ping to be10

 x 

 x  r  L  2q P  U i (2.82)

10 The procedure is to start with the potentia l solution forthe flow approaching an infinite flat wall (sometimes called“corner flow”) and mapping the flat wall into a parabola. Thederivation wil l not be presented here.

Figure 2.22 Comparison of surface velocity distributions for an ellipticalthickness form with t  o /c  0.1 and t  o /c  0.2 obtained from an exactsolution and from linear theory.

Distance From Leading Edge, x/c

      S    u    r      f    a    c    e      V    e      l    o    c      i      t    y  ,    q      /      U

0 0.1 0.2 0.3 0.4 0.50.6

0.8

1

1.2

Linear Theory (q/U=1+to /c)

Exact Solution: to /c=0.2

Exact Solution: to /c=0.1

Exact solution has stagnation point (q=0) at x=0

Figure 2.23 Local representation of the leading edge region of a foil by aparabola with matching curvature at x  0. This is sometimes referred to asan “osculating parabola.”

x

    y

r L

Osculating Parabola

Circle

(2.81)

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24 PROPULSION

and this is plotted in Fig. 2.24. We used U i in equation(2.82) rather than the foil free-stream velocity, U  , be-cause the local leading edge flow is really buried in theglobal flow field. The only remaining task, therefore, isto assign the proper value to U i .

Let us define ut ( x  ) as the perturbation velocity due tothickness obtained from linear theory. The total surface velocity according to linear theory is thenq ( x  ) U   ut ( x  ).

In the limit of x   c , the linear theory result becomesq (0) U   ut (0). On the other hand, in the limit of x    r Lthe local leading edge solution becomes q ( x  )  U i . Thus,the “free stream” in the local leading edge solution mustapproach U  i  U   ut (0), and the complete expression forthe surface velocity then becomes

 x 

 x  r  L

   2q x  U  u

t x    (2.83)

 The Lighthill correction can be extended to includethe effects of camber and angle of attack. Define uc ( x  )as the perturbation velocity due to camber at the idealangle of attack. With x   0 at the leading edge, equation(2.79) can be written as

c  x 

 x q x  U  u

t x  u

c x   I 

ideal   (2.84)

 Multiplying equation (2.84) by the same factor repre-senting the local leading edge flow gives the result

1 q x    u

t  x    u

c  x 

 x  r  L

  2

 x 

U U U (2.85)

c  x 

 x  r  L

  2 

ideal

 Note that square root infinity in equation (2.84) hasnow been canceled and that the velocity at the leadingedge has the finite value

ideal

q0 2

 r  L

  c

2

 r  L

  c

C  L

C  L

2 

ideal

U (2.86)

 The velocity predicted from equation (2.86) can becompared with NACA tabulated results obtained fromTheodorsen’s conformal mapping method for a varietyof foil types. This is done in Table 2.1, and it is clear thatthe Lighthill correction works very well.

In addition to correcting the velocity right at the lead-ing edge, we can also use Lighthill’s rule to modify the velocity and pressure distribution from linear theoryover the whole forward part of the foil. However, if wewere to apply equation (2.85) to an elliptical thicknessform, we would find that the result would be worse atthe midchord. For example, we know that the exact value of the surface velocity at x   /c  0.5 for an ellipti-

cal thickness form with a thickness/chord ratio of 20%is q  /U   1.2, and that results in a pressure coefficientof C  P    0.44. We would also get the same result withlinear theory. However, if we apply equation (2.85), wewould get

C  P 

0.385

 r  L

0.5 0.02to

c

2

q  U  1.1770.5

0.5  r l  2

1   to

c

  (2.87)

 As suggested by Scherer (1997), a variant of theoriginal Lighthill formula [equation (2.85)] solves this problem, and can be derived as follows. If we take thederivative of equation (2.81),

dy

dx 

 r  L

2 x    (2.88)

 and form the quantity

1

1 2

2

dy

dx 

 x 

 x  r 

 L

  (2.89)

Figure 2.24 Surface velocity distribution near the leading edge of a semi-infinite parabola.

Distance from leading edge, x/r L

      S    u    r      f    a    c    e    v    e      l    o    c      i      t    y  ,    q      /      U

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Table 2.1 Velocity q / U  at the Leading Edge for Various ThicknessForms at Unit Lift Coefficient

Section Type   rL /  c   Lighthill Theodorsen

NACA 16-0006 0.00176 5.37 5.47

NACA 16-0012 0.00703 2.68 2.62

NACA 16-0021 0.02156 1.53 1.57

NACA 63A006 0.00265 4.37 4.56

NACA 0006 0.00400 3.56 3.99

Next Page

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44 PROPULSION

P

Vu

Vl

Vm

+Vd

-Vd γ

δ

Figure 3.2 Velocity diagram in the tangent plane.

3.1 Introductory Concepts  We saw in the precedingsection that a 2D foil could be represented in linear the-ory by a vortex and source distribution, and that the lif ton the section was due entirely to the former. We alsosaw that linear theory yielded surprisingly accurateresults in comparison to exact theory, particularly forfoil sections that were relatively thin. In this section, wewill therefore extend the concept of a thin, lifting 2Dfoil section to three dimensions. Such an idealization istermed a lifting surface .

We will start by considering a surface of vanishingthickness, but otherwise arbitrary shape, as illustratedin Fig. 3.1. We will further assume that this lifting sur-

face is placed in a steady, irrotational flow field, and thatthe fluid, as in the case of 2D flow, may be regarded asincompressible and inviscid.

The fluid velocities at an arbitrary point P  on the lift-ing surface must, of course, be tangent to the surface.However, the velocities at corresponding points on theupper and lower sides of the surface need not be equalin either magnitude or direction. The boundary condi-tion simply requires that they be coplanar.

Let us denote the velocity on the upper surface as V uand the velocity on the lower surface as V l. These canbe viewed in a plane tangent to the surface at point P  , as shown in Fig. 3.2, and represented as the vectorsum of a mean velocity, V  m, and a difference velocity, V d.

Therefore,

V l V  m V d

V u V  m

V d(3.1)

 where

V  m 1

2V u V l

V d 1

2V u V l

(3.2)

  A nonzero difference velocity implies the presence ofa vortex sheet whose strength at the point P  is17

  n V d ,    2 2V d  (3.3)

and whose direction is normal to the plane formed bythe surface normal vector, n, and the difference veloc-ity vector, V d. Equation (3.3) may be verified simply by

calculating the circulation around a small contour, asillustrated in Fig. 3.3.We now define  as the angle between the mean flow

and the vorticity vector at point P  , and proceed to calcu-late the pressure jump across the lifting surface usingBernoulli’s equation

  p  pl  pu 1

2  V 2 V 2lu

(3.4)

  A simple application of the law of cosines relates theupper and lower velocities to the mean and difference velocities as follows

V 2 V 2 V 2  2V  mV d cos    2 l m   d

V 2

V 2

V 2

 2V  mV d cos    2 u m d

(3.5)

 which may then be combined with equations (3.3) and(3.4) to give the result

  p   V  m  sin   (3.6)

3

Three-Dimensional Hydrofoil Theory

Figure 3.1 A lifting surface.

17The quantity    is frequently referred to as the “vorticity,”even though it is really the vortex sheet strength. Vorticity,strictly speaking, is the curl of the velocity vector, and a vor-tex sheet is the limit of a thin layer of fluid containing vortic-ity as the thickness of the layer goes to zero and the vorticitygoes to infinity.

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46 PROPULSION

First, define a curvilinear s1  coordinate which beginsat a point s L at the leading edge and is everywhere tan-gent to the mean velocity vector V  m. This passes thetrailing edge at some point sT   and proceeds into thewake. Next, define an s2  coordinate with an origin atan arbitrary point in the wake, which lies in the wakeand is orthogonal to the s1  coordinate. Let sW   be theintersection of the s1  and s2 coordinates. Thus, s L  andsT  are functions of the parameter sW  . Finally, define s E 

as the s2  coordinate of a point just outside the wake.We now see that the selected circulation contour starts

 just upstream of s L , passes over the hydrofoil to a point just above sT  , follows down into the wake keeping justabove the s1 coordinate, makes a left turn at the point sW  ,and proceeds over the wake to the point s E  . The return tripis similar, except that the contour remains below the wakeand the hydrofoil until it reaches the starting point at s L .The contour therefore consists of two almost closed loopsthat are connected by a pair of parallel curves. The circu-lation around the forward loop, from Stokes theorem, is

  1s2 sW   S T 

 S  L

 bs1; s2 sW ds1  (3.7)

The circulation around the portion of the contourconsisting of the two parallel connecting paths is zero,because it is everywhere tangent to the vorticity vector.Therefore, to keep the total circulation zero

  2s2 sW   S  E 

 S W 

  f s2 ds2   1s2 sW   (3.8)

where 2  is the circulation around the second loop inthe wake. The final result is obtained by differentiatingequation (3.8) with respect to sW 

 S  E 

 S W 

d  1dsW 

d

dsW 

  f s2ds2

  f sW  (3.9)

 Thus we see that the strength of the free vorticity isrelated to the spanwise derivative of the bound circu-lation around the hydrofoil. However, the differentialds2 must be taken in the wake, not on the hydrofoil. Ifthe free vortex lines were to move straight back (which

will be assumed subsequently in linearized hydrofoiltheory), then the spanwise increment in the wake andon the hydrofoil would be the same, and this distinctionwould be unimportant. However, in the more generalcase in which deformation of the free vortex wake isallowed, equation (3.9) is an exact result.

3.3 The Velocity Induced by a Three-Dimensional Vortex

Line  As a first step in the solution of the lifting surface problem just formulated, we need to be able to computethe velocity field induced by 3D vortex sheets. As thesesheets can be thought of as being composed of elemen-tary vortex lines, we can first determine their individual velocity field, and then obtain the velocity induced by theentire vortex sheet by integration. The expression for the velocity induced by an arbitrary 3D vortex line is knownas Biot-Savart’s law .

We will start the derivation by considering a flow inwhich the vorticity is confined to a volumeV  within thefluid. We wish to find the velocity V   at a general point P  ( x  ,y , z ) as illustrated in Fig. 3.8.

 An element of vorticity withinV  is shown at the point

Q (  , ,  ), and the distance vector r to the field point P  is

 r   x   i y  j   z   k

 r   x   2 y 2  z  2

  Any vector field whose divergence is zero everywhere

and whose curl is nonzero in a portion of the field can beexpressed as the curl of a vector field whose divergenceis zero (see, for example, Newman, 1977, p. 115). Inter- preting the former as the velocity, V , and the latter asthe vector potential  A,

V   A,  A 0 (3.10)

 The vorticity is V . Introducing equation(3.10) and using a particular vector identity, we canexpress the vorticity in terms of the vector potential A

V   A  A  A

Figure 3.7 Circulation path used to determine the strength of the free vortic-ity in the wake.

ω

Q(ξ,η,ζ)

r   V

P(x,y,z)

 υFigure 3.8 Notation for velocity,V , at point P(x,y,z)  induced by a volumedistribution of vorticity ( ,, ) contained in volumeV .

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  PROPULSION 47

 Because the divergence of A is zero, this reduces to

2 A   (3.11)

The solution of equation (3.11), which is a vector formof Poisson’s equation, is

 A V 

1

4 

dV 

 r (3.12)

  At this point, we can verify directly that the diver-gence of A is zero

v  

 S   A

1

4 

1

4 

dV 

 r 

 ndS 

 r 

 The last integral can be seen to be zero as a conse-quence of Kelvin’s theorem, which requires that vortexlines not end in the fluid.

We can now roll out the volume into a long, thin vor-tex tube, which in the limit becomes a vortex line. Asshown in Fig. 3.9, dl is a differential element of lengthalong the vortex line and s  is a unit vector tangent tothe vortex. The volume integral in equation (3.12) thenbecomes a line integral

  

4 

sdl

 r  A

 

 and the velocities may then be obtained by taking thecurl of A

  V   A    

4 

 

  

4  r   dl

 r 3

s  r dl

s(3.13)

 The latter form of equation (3.13) is known as Biot-Savart’s law. The last step can be verified by workingout each component. For example, the x  component is

i j k

s

 r 

 x 

y

y

 z

 z

s x 

 r 

sy

 r 

s z

 r 

sy

 r 

s z

 r   i  …

(3.14)

 Remembering that  r   is a function of ( x  ,y , z ) but s  isnot, the x  component then becomes

s z sy s z sy

y

 z

1

 r 

1

 r 

y  r 3  r 3

 z  (3.15)

 Going through the same operation with the x  compo-nent of the last form of equation (3.13)

sy z   s zy

i

i

 j k

s x    sy   s z

 x      y  z  

s  r 

 r 3  r 31

 r 3

(3.16)

 we see that this is the same as equation (3.15), thus veri-fying equation (3.13).

In summary, the velocity field of a concentrated vor-

tex line of strength G is,

V    dl  

4 

s  r 

 r 3(3.17)

 which can be written out in component form as follows

sy z s zy u   dl

  

4    r 3

s z x    s x  z  v   dl

  

4    r 3

s x y sy x   w   dl

  

4    r 3

(3.18)

 Equation (3.18) is particularly useful in deriving ex- pressions for the velocity induced by particular shapesof vortex lines, as we will see later.

Some comments should be made about the limits ofthe integral in Biot-Savart’s law. From Kelvin’s theorem,a vortex line must have constant strength and cannotend in the interior of the fluid. If we are really solvingfor the velocity field of a concentrated vortex, then theintegral must be taken over a closed path. However, westarted out this section with the observation that a vor-tex sheet could be considered to be made up of individual vortex filaments representing bound and free vorticity. Inthis case, the strengths of the vortex filaments can vary

along their length, provided that the variation of boundand free vorticity is always set in such a way that Kelvin’stheorem is satisfied. The velocity induced by one compo-nent of the vortex distribution can still be obtained fromBiot-Savart’s law, but the strength of the vortex in equa-tion (3.17) will have to be moved inside the integral, andthe contour of integration will not necessarily be closed.

3.4 Velocity Induced by a Straight Vortex Segment  The velocity field of a straight vortex segment serves as asimple illustration of the application of Biot-Savart’s law.However, it is also a very useful result because the nu-merical solution of more complicated geometries can beobtained by discretizing the vortex sheet into a lattice of

Figure 3.9 Development of a vortex line. On the left is a volume distributionof vorticity . In the middle, the volume has been put through a pasta ma-chine to form a noodle with cross-section area da. On the right, the noodlehas been turned into angel’s hair, with zero cross-sectional area and infinitevorticity, but with the total circulation kept fixed.

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48 PROPULSION

straight concentrated vortex elements. The result to be

derived here therefore serves as the influence functionfor such a computational scheme.

We can simplify the analysis by choosing a coordi-nate system such that the vortex segment coincideswith the x  axis and the field point, P  , lies on the y axis,as shown in Fig. 3.10. This is not really a restriction,as one can always make a coordinate transformationto do this, and the resulting velocity vector can thenbe transformed back to the original global coordinatesystem.

The vortex extends along the x  axis from x 1 to x 2 . Inthis case

s 1, 0, 0 and r   , y, 0

 and from equation (3.17) we can see immediately thatu  v  0, so we only need to develop the expression for w

wy   

4 

 x 2

 x 1

 

yd 

2 y23 2

  

4    2 y2

  x2

 x1

 x 2b

  

4 

 x 1c

  (3.19)

Where 2 x 2 y 2b  and 1 x 2 y 2c . This resultcan also be expressed in terms of the two angles 1 and2 , which are i llustrated in Fig. 3.10

wy    4 y

cos 2  cos 1

 Two limiting cases are of particular interest. For aninfinitely long vortex, 1  2  0 so that

wy    

2 y

 which is the correct result for a 2D vortex. For a semi-infinite vortex extending from x   0 to infinity, 1     /2and 2  0 so that the velocity is half that of an infinite vortex. This result is useful in lifting line theory whichwe will be looking at later.

The complete velocity field is shown in Fig. 3.11 fora vortex extending along the  x   axis from  1 to  1.The variation in velocity with y is shown for severalfixed values of  x  . In this plot, the velocity has beennondimensionalized by the factor   /2 y  so that theresults can be interpreted as the ratio of the veloc-ity induced by the vortex segment to that induced byan infinite vortex of the same strength located the

same distance away. Therefore, for | x  |  1 the resultapproaches a value of unity as y becomes small. For| x  |   1, the result approaches zero for small y . Forlarge y distances away from the vortex segment, thenondimensional velocity becomes independent of  x and decays as 1/y . The dimensional velocity thereforedecays as 1/y2 .

3.5 Linearized Lifting-Surface Theory for a Planar Foil3.5.1 Formulation of the Linearized Problem  We

will now consider the case of a hydrofoil of zero thick-ness, whose surface lies very nearly in a plane that isaligned with the oncoming flow. Such a surface, for ex-ample, might be exactly flat but inclined at a small angle

of attack with respect to the flow. More generally, how-ever, the surface may have some arbitrary distributionof camber and angle of attack. The only restrictions arethat the resulting deviation of the surface from the ref-erence plane be small and that the slopes of the surfacebe everywhere small.

Figure 3.12 illustrates the particular notation for this problem. A Cartesian coordinate system is orientedwith the positive  x   axis in the direction of a uniformonset flow of magnitude U  . The y  axis is normal to U 

and the ( x  ,y ) plane serves as the reference surface. Thetips of the hydrofoil are located at y  s  /2, so that itstotal span is s .

Radial Distance, y

     N    o    r    m    a     l     i    z    e     d     V    e     l    o    c     i     t    y ,

     2     π    y    w     /      Γ

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x=0.0

x=0.5

x=1.0

x=2.0

x=1.5

-1   1   x

y(x,y)

Figure 3.11 Normalized velocity, 2 yw (x ,y )/  , induced by a straight vortexsegment.

x1   x

2

y

P(0,y,o)

bc

ξ

dl

θ1   θ

2

r 

Γ

0Figure 3.10 Notation for a straight line vortex segment using a local coor-dinate system with the x  axis coincident with the vortex and the field point,P, located on the y  axis.

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  PROPULSION 49

 The outline, or planform, of the hydrofoil is describedby the curves x    x  L (y ) and x    x T  (y ), denoting respec-tively the projection of the leading and trailing edges onthe ( x  ,y ) plane.

 A section of the surface at spanwise position y  isshown in Fig. 3.13. The notation is the same as for a 2Dfoil, as illustrated in Fig. 2.1, except that the angle ofattack,  (y ), and the vertical displacement of the nose–tail line,  z0 (y ), must be considered as part of the 3Dgeometry. In addition, the vertical coordinate is now z ,rather than y . The angle of attack of the nose-tail linerelative to the oncoming flow is  (y ) and the distribu-

tion of camber is denoted by the function  f  ( x  ,y ). Thecamber is measured in a direction normal to the nose–tail line, and its maximum value at any spanwise stationis f 0 (y ). When we add thickness later, it will similarly bedenoted by the function t ( x  ,y ) with a maximum valuet0 (y ) at each spanwise location. The thickness func-tion will then be added at right angles to the camberline, with t /2 projecting on each side. Finally, the chordlength c (y ) is the length from the leading to the trail ingedge measured along the nose–tail line.

If the angle of attack is small, we can assume that cos  1 so that the chord length is essentially the same asits projection on the ( x  ,y ) plane

cy  x T y  x  Ly

 Similarly, as the angle of attack has been assumed tobe small, we can write the z coordinate of a point on thesurface as follows

 zs x, y  z0y y x  f  x, y

 where z0 (y ) is the elevation of the nose–tail l ine at x   0.3.5.2 The Linearized Boundary Condition  The

exact boundary condition is that the normal componentof the total fluid velocity vanish at all points on the hy-drofoil surface

V  · n  0 on z  z S  x, y

 If the slopes of the surface are small, the unit normal

 vector can be approximated as

i  n   j  1 k z

s

 x 

 zs

y

 The fluid velocity can be expressed as the sum of theoncoming flow U  and a disturbance velocity field withcomponents (u ,v ,w ). The total velocity is therefore

V  U  ui vj  wk

 and its dot product with the normal vector is

 zs

yV  n U  u v w

 zs

 x (3.20)

 If the camber and angle of attack is small, we can ex- pect that the disturbance velocities will be small com- pared with the oncoming flow. We can therefore eliminatethe products of small quantities in equation (3.20) to ob-tain the final form of the linearized boundary condition

w  U    on z  0 zs

 x (3.21)

 Note that the boundary condition is satisfied on thereference plane z  0 rather than on the actual surface,which is the same approximation as was made for lin-earized 2D theory.

It is important to note that equation (3.21) does not in-

 volve the slope of the surface in the spanwise direction,and actually looks just like the boundary condition for2D flow. This is not the result of any assumption that thespanwise slopes are smaller than the chordwise ones,but follows from the assumption that the predominant velocity is in the chordwise direction.

3.5.3 Determining the Velocity  The next step isto determine the vertical component of the disturbance velocity induced by the bound and free vortices repre-senting the hydrofoil and its wake. If the disturbance ve-locities are small, we can assume that the mean inflow isequal in magnitude and direction to the oncoming flow.This means, in particular, that the bound vorticity will be

XY

Z

u

w

v

xL(y)

xT

(y)

-s/2

s/2

γb

γf 

γf 

U

c(y)

Figure 3.12 Notation for a planar hydrofoil.

xt

xl

z

f(x,y)α

z0

zs(x,y)

n(x,y)

Figure 3.13 Cut-through foil section at fixed spanwise location, y .

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50 PROPULSION

oriented in the y direction, and that the free vorticity willbe oriented in the x  direction. The vortex sheets are pro- jected onto the ( xy ) plane, and the trailing vortex wakewill consist only of semi-infinite straight vortices extend-ing downstream in the x  direction from the trailing edge.

It is important to remark at this point that this as-sumption regarding the wake does not conform to realityeven for hydrofoils that have small angles of attack andcamber. The planar-free vortex sheet we are assuming isactually unstable, and its outboard edges tend to roll upand form concentrated tip vortices. While this is a matterof concern in many applications (as we shall see subse-quently in looking at marine propellers), it is fortunatelynot of great importance to the problem at hand. The rea-son is that the velocity induced at points on the liftingsurface by an element of free vorticity in the distant wakeis both small and insensitive to its precise position.

 An expression for the velocity induced at a point( x  ,y ,0) on the projection of the hydrofoil surface by thebound vorticity b can be obtained directly from equa-tion (3.18)

wb x , y, 0    b ,  x   

 

s  2

s  2 

 x T 

 x  Lc   d d

1

4    r 3(3.22)

 Here   and  are dummy variables corresponding to x  and y . The integral over the span is in the direction ofthe elementary bound vortex lines, as in equation (3.17),while the integration in the chordwise direction of thecontinuous vortex sheet is equivalent to the total circu-lation of a concentrated vortex line. As the denomina-tor in equation (3.22) vanishes when  x     and y   ,the integral is singular. The singularity is of the Cauchy principal value type, which we saw earlier in the solu-tion of the 2D problem.

The exact distance vector,  r , from an element of vorticity on the hydrofoil surface located at ( , , s ) toanother point on the surface ( x  ,y , zs ) where we want tocalculate the velocity has a magnitude

 x   2 y 2  zs  2 r 

 but because linear theory projects everything onto the( xy ) plane, this reduces to

 x   2 y 2 r 

 The velocity induced by the free vorticity can be de-

 veloped from equation (3.17) in the same way, giving theresult

w f   x , y, 0     f  , y

 

s  2

s  2 

 x  Lc   d d

1

4    r 3(3.23)

 Note that in this case the upper limit of the    inte-gration is not the trailing edge, but extends to infinitydownstream.

3.5.4 Relating the Bound and Free Vorticity  Therelationship between the bound and free vorticity can beobtained using the same approach as was used in Section3.2. However, things are now simpler, because the s1 and

s2 coordinates in Fig. 3.7 are now just the x  and y coordi-nates. If we chose a circulation path as shown in Fig. 3.14,where the two connected loops almost touch the hydrofoilat the point ( x  ,y ), it is evident from Stokes’ theorem that

 x 

 x  Ly 

s  2

y

 b , yd      f  x , d 0 (3.24)

 Differentiating equation (3.24) with respect to y ,keeping in mind that the lower limit of the first integralis a function of y , we obtain an expression for the free vorticity at any point within the hydrofoil,

 x 

 x  L

  f  x , y  b x  L, y dx  Ldy

 

d y

 b  , y (3.25)

 The first term in equation (3.25) can be interpreted asthe bound vorticity that runs into the leading edge beingturned 90 degrees to become free vorticity. For example,if a hydrofoil had constant bound vortex strength overits surface, the second term in equation (3.25) would bezero, and the only source of free vorticity on the hydro-foil would be due to the first term.

If the point x  moves downstream of the trail ing edge,equation (3.24) becomes

 x T y

 x  Ly

s  2

y b , yd     f  x , d 0

 

(3.26)

 which looks almost the same, except that the upper limit ofthe first integral is now also a function of y . Differentiationwith respect to y therefore results in one additional term

 x T y

 x  Ly  f y  b x  L, y   d

dx  Ldy

 

y

 b , y

 b x T , y d  

dy

dx T 

dy

(3.27)

 Thus, the free vorticity in the wake is independent of x 

and depends only on the spanwise derivative of the totalbound circulation around the hydrofoil. This result agreeswith the more general result given in equation (3.9).

Figure 3.14 Circulation contours to get free vorticity on the foil.

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  PROPULSION 51

 The formulation of the linearized planar hydrofoil problem is now complete. The boundary condition givenin equation (3.21) relates the geometry and angle ofattack of the hydrofoil to the vertical component of theinduced velocity. The latter is related to the bound and

free vorticity by equations (3.22) and (3.23). Finally, thefree vorticity is related to the bound vorticity by equa-tions (3.25) and (3.27). Thus, the relationship betweenthe hydrofoil geometry and its bound vortex distributionis established. Finally, the distribution of pressure jumpacross the lifting surface (which is commonly referred toas the load distribution ) is directly related to the bound vorticity from equation (3.6).

If we are given the geometry and angle of attack, wecan solve for the load distribution using the above equa-tions. As the unknown bound vortex distribution appearsinside an integral in this case, we must solve an integralequation. This is known as the analysis problem .

On the other hand, if we are given the load distribu-tion, the quantities inside the integrals are known, andwe can obtain the slope of the hydrofoil section directlyby integration. One more integration is then required toget the actual shape of the hydrofoil from the slope of itssurface. This is known as the design problem .

While this is simple enough in principle, the problemis complicated by the fact that the integrals in eithercase are singular, and no closed form solution existseven for this linearized problem. One must therefore re-sort to numerical procedures, or to a combination of an-alytical and numerical techniques to solve the problem.Before we get into this, we will develop expressions forthe overall lift and drag forces on a hydrofoil, which can

be done analytically.3.6 Lift and Drag  The resultant force per unit span

on a section at spanwise location y can be resolved intocomponents F  z (y ) in the z direction and F  x  (y ) in the x  di-rection. The force in the z direction, being at right anglesto the direction of the oncoming flow, as we saw earlierin looking at 2D flow, is termed lift , while the force in the x  direction is termed drag . In a real fluid, the drag forceconsists of a contribution due to viscous stresses on thesurface of the hydrofoil and a contribution due to the pres-ence of the trailing vortex system. The latter is termed in-duced drag and is the only component of drag consideredin our present inviscid analysis. In the case of 2D flow,

there is no free vorticity, and hence, no induced drag.We can derive the force acting on a section by apply-ing the momentum theorem to a control volume of in-finitesimal spanwise extent dy extending to infinity inthe x  and z directions, as shown in Fig. 3.15. The forcein the z direction is

 S 

 F  zyy     wV  ndS  (3.28)

 where the integral is taken over all surfaces of the con-trol volume.

The contributions of the top, bottom, and front surfacesto equation (3.28) can be seen to be zero, because the ve-locity at large distances decays at a faster rate than the

area increases. The contribution of the sides is also zerodue to the fact that w is an even function of z while V  n isodd. This leaves the flux of momentum through the aft sur-face. As this is far from the hydrofoil, the velocity inducedby the bound vorticity goes to zero. The only induced ve-locity is that due to the free vortices, which has no compo-nent in the x  direction. Thus, the momentum flux is

 F  zyy   U w, y, zdzy (3.29)

 Because we are infinitely far downstream, the velocity induced by the free vortices appears as that

due to a sheet of vortices of infinite extent in the  x direction

s  2

s  2w, y, z

1

2 

 

  d

  y 2  z2

y (3.30)

 Combining equations (3.29) and (3.30) and reversingthe order of integration gives the result

s  2

s  2 

 F  zy   U 

2 

 

  dzd

 y 2  z2

y (3.31)

 The integral in equation (3.31) is

dz  tan1

 y 2

 z2

y   z

y

(3.32)

 which is simple enough, except that we have to be care-ful in evaluating the limits. As z→ the inverse tangentbecomes    /2 depending on the sign of  z  and y   .The safe way is to break up the spanwise integral intotwo intervals, depending on the sign of y  

 

y

s  2

  U  F  zy   dtan1

2 

 

 z

y

 

s  2

yd

tan1

  U 

2 

 

 z

y

(3.33)

Figure 3.15 Control volume for momentum analysis for lift.

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52 PROPULSION

which leads us to the final result

 F  zy   U 

2  y   s   2

  s   2   y   U   y

  U 

2

(3.34)

 because   (s  /2)     (s  /2)  0. Thus, the total lift forceon a section is the same as that which would result if thedistribution of bound circulation over the chord wereconcentrated in a single vortex of strength  (y ).

We cannot use the same control volume to determinethe drag, because  x  directed momentum is convectedacross the sides of the control volume, and we wouldneed to know more about the details of the flow to cal-culate it. However, we can determine the total induceddrag by equating the work done by the drag force whenadvancing the hydrofoil a unit distance to the increasein kinetic energy in the fluid. For this purpose, we canmake use of Green’s formula

 S 

 E   

2  dS 

 n(3.35)

 to determine the kinetic energy,  E  , in the fluid regionbounded by the surface S  . In equation (3.35),  is the ve-locity potential and n is a unit normal vector directed out-ward from the control volume, as illustrated in Fig. 3.16.

The contribution to the integral in equation (3.35)from all the surfaces except for those cut by the free vortex wake is zero as the outer boundaries move toinfinity. On the inner surface, the normal derivative ofthe velocity potential is w (,y ,0) on the upper portion,

and w (,y ,0) on the lower portion. The jump in potential(u  l ) is equal to the circulation   (y ) around the hy-drofoil at spanwise position y .

The kinetic energy imparted to the fluid as the foil ad- vances a unit distance in the x  direction is the induceddrag , F  x 

induced. This, in turn, can be equated to the total

kinetic energy between two planes far downstream sep-arated by unit x  distance,

   yw, y, 0dy

 F  x 

induced

s  2

s  2

  

2

l uw, y, 0dys  2

s  2

  

2

 

s  2

s  2

  

2

 

s  2

s  2lw, y, 0dy

  

2

uw, y, 0dy

(3.36)

 We can relate this to the velocity field near thehydrofoil as follows. Suppose that the total bound circu-lation were concentrated on a single vortex line coinci-dent with the y axis, as shown in Fig. 3.17. The velocityw (0,y ,0) induced by the free vortices would be half the value induced at infinity, as shown earlier. Defining adownwash velocity18

w*y  w0, y, 0

 equation (3.36) becomes

 F  x 

induced

  

s  2

s  2  yw*

ydy (3.37)

 The total induced drag force is therefore the same asthat which would result if the resultant force on eachspanwise section were normal to the induced inflow velocity, V *(y ), as shown in Fig. 3.18. Here, V *(y ) is theresultant of U  and w*(y ), and F  (y ) is the resultant of thelift F  z (y ) and the induced drag F  x  (y ).

XY

Z

u

w

v

-s/2

s/2  γ

f =-dΓ /dy

U

C

C

Γ(y)

w*(y)

Figure 3.17 Concentration of bound vorticity along a lifting line.

18“Downwash” is the nomenclature used by the aerodynamiccommunity, because when the lift on an airplane wing is up-ward, the resulting w* is downward. Note that herein, w* isdefined positive upward.Figure 3.16 Control volume for kinetic energy far downstream.

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54 PROPULSION

The last step in equation (3.43) makes use of Glau-ert’s integral, which we saw before in the solution of the2D foil problem

 

0

c I  ny   d  cos n

cos y cos 

sin nysin y

(3.44)

 Figure 3.19 shows the first four terms of the Glauertseries for the spanwise distribution of circulation, while

Fig. 3.20 shows the resulting induced velocity. Note thatthe leading term produces a constant velocity of  1 overthe span, while the higher terms produce progressively

larger and more oscillatory velocity distributions. Notethat the velocity given in equation (3.43) is indetermi-nate at the tips, where sin y  0, but that it can easily beevaluated, giving the result that w* / U    n2a n . The factthat the velocity at the tips induced by each term in the

series grows quadratically with n has important practi-cal consequences, which we will discuss later.

 At this point , the total force in the z direction can befound

s  2

s  2

 

s  2

s  2

 F  z

  

2

 F  zydy   U     ydy

 

 n1 

 

0

  U 2s2  a n sin ny sinydy

  U 2s2a1 

(3.45)

 which is seen to depend only on the leading term in the

assumed series for the spanwise distribution of circula-tion. The remaining terms serve to redistribute the liftover the span, but do not affect the total.

The total induced drag force can now be computedfrom the formula obtained in the previous section

s  2

s  2

 F  x       w*y  ydy (3.46)

 which can be accomplished by substituting equations(3.39) and (3.43) into (3.46). As this involves the productof two series, two summation indices are required. Not-ing that all but the diagonal terms in the product of thetwo series vanish on integration, (0,  ), the steps neces-

sary to obtain the final result are as follows

 

0

 

 n1 F  x    

U    na n sin ny

sin y

 

 m1

2Uss

2sin ydy a m sin my

 

 n1

 

 m1 

 

0

  U 2s2 na n sin ny   a m sin mydy

 

 n1

   U 2s2 

2  na

 n

(3.47)

 It is instructive to extract the leading coefficient inthe circulation series and to express it in terms of thetotal lift force F  z from equation (3.45)

 n2

2

1  n 

2  U 2s2

2

  F  z

 n2

  U 2s2a2 F  x 

  

21

 

 n1

a n

a1

2

a n

a1

(3.48)

Spanwise position, y/s

     C     i    r    c    u     l    a     t     i    o    n ,

      Γ     (    y     )     /     2     U    s

-0.5 -0.25 0 0.25 0.5-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1 a1

a2

a3

a4

Figure 3.19 Plot of first four terms of Glauert’s circulation series.

Spanwise position, y/s

     D    o    w    n    w    a    s     h    v    e     l    o    c     i     t    y ,

    w     *

 (    y     )     /     U

-0.5 -0.25 0 0.25 0.5-20

-15

-10

-5

0

5

10

15

20

a1

a2

a3a

4

Figure 3.20 Plot of velocity induced by first four terms of Glauert’s circula-tion series.

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  PROPULSION 55

 Thus, we see that the reciprocal of the term in brack-ets is a form of efficiency which is maximized whena n  0 for  n   1. The presence of higher terms in thecirculation series does not change the lift , but increasesthe drag. The optimum spanwise distribution of circula-

tion is therefore one in which the lift is distributed as siny, or in physical coordinates, as an ellipse.

For a fixed spanwise distribution of lift, equation(3.48) shows that induced drag is directly proportionalto the square of the total lift, inversely proportional tothe square of the speed, and inversely proportional tothe square of the span.

This result is frequently presented in terms of lift anddrag coefficients based on planform area. This requiresthe introduction of a nondimensional parameterA  calledaspect ratio , which is the ratio of the span squared tothe area, S , of the hydrofoil

A  s2

S 

(3.49)

 Defining the total lift and induced drag coefficients as,

C  L  F  z

  U 2S 1

2

C  D   F  x 

  U 2S 1

2

(3.50)

the nondimensional form of equation (3.48) becomes

 n2

C  D  1 

 

a n

a1

2

 n

2C  L

 A 

(3.51)

 While equation (3.51) is more concise, it can lead tothe erroneous conclusion that increasing aspect ratioalways reduces induced drag. It does reduce induceddrag if the increase in aspect ratio is achieved by in-creasing the span. However, if it is achieved by keep-

ing the span fixed and reducing the chord, equation(3.48) shows that the drag is the same. The confusionis caused by the fact that if the area is reduced, thelift coefficient must be increased in order to obtain thesame lift. Therefore, in this case both the li ft and dragcoefficients increase, but the dimensional value of thedrag remains the same.

3.7.2 Vortex Lattice Solution for the Planar Lifting Line  We saw in the previous section that Glauert’s an-alytical solution for the 2D foil could be replicated withhigh precision by a discrete VLM. It is therefore reason-able to expect that the same success can be achievedwith a vortex lattice lifting line method. In both cases,the motivation is obviously not to solve these particular problems, but to “tune up” the vortex lattice techniqueso that it can be applied to more complicated problemsfor which there is no analytical solution.

 As illustrated in Figure 3.21, the span of the liftingline is divided into M  panels, which may or may not beequally spaced, and which may be inset a given distancefrom each tip.19  The continuous distribution of circu-lation over the span is considered to be replaced by astepped distribution that is constant within each panel.The value of the circulation in each panel is equal to the value of the continuous distribution at some selected

19

The optimum tip inset is not at all obvious at this point, butit will be addressed later.

Continuous Circulation Distribution, Γ(y)

Γ1

Γ2

ΓM

yc(1)   y

c(2)   y

c(M)

yv(1) y

v(2)   y

v(M+1)

Tip Inset

Control Points

Concentrated Free Vortes Lines

Figure 3.21 Notation for a vortex lattice lifting line. In this case, there are eight uniformlyspaced panels, with a quarter panel inset at each end.

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56 PROPULSION

 value of the y coordinate within each panel. The induced velocity will be computed at a set of control points . Thecoordinate of the control point in the n th panel is yc ( n ),and the corresponding circulation is   ( n ).

Because the circulation is piecewise constant, the

free vortex sheet is replaced by a set of concentrated vor-tex lines shed from each panel boundary, with strengthequal to the difference in bound vortex strength acrossthe boundary. This is equivalent to replacing the contin-uous vortex distribution with a set of discrete horseshoe vortices, each consisting of a bound vortex segment andtwo concentrated tip vortices. The y coordinate of the panel boundaries, which are then the coordinates of thefree vortices, will be denoted as yv ( n ). If there are  M  panels, there will be M   1 free vortices.

The velocity field of this discrete set of concentrated vortices can be computed very easi ly at points on thelift ing line, because the singular integral encounteredin the continuous case is replaced by the summation

  

 m1

w*yc n  w*      mw n, m n (3.52)

 where w n,m  is the velocity induced at the control pointyc ( n ) by a unit horseshoe vortex surrounding the pointyc ( m ). As the bound vortex segment of the horseshoedoes not induce any velocity on the lifting line itself, theinfluence function w n,m  consists of the contribution oftwo semi-infinite trailing vortices of opposite sign

w n, m

1

4 yv m yc n1

4 yv m  1 yc n

  (3.53)

However, it is clear that the resulting velocity will notbe accurate for all values of y . In particular, the velocitywill become  as yc is moved past any of the vortexcoordinates yv . Nevertheless, our intuition says that theresult might be accurate at points that are more or lessmidway between the vortices.

Our intuition is correct, as illustrated in Fig. 3.22,which shows the distribution of induced velocity w*(y )for an elliptically loaded lifting line using 10 equallyspaced panels inset one quarter panel from each tip.The velocity has been computed at a large number of

 points within each panel, and one can clearly see the ve-locity tending to  near each of the panel boundaries.The velocity can obviously not be calculated exactly onthe panel boundaries, so what is shown in the graph is asequence of straight lines connecting the closest pointscomputed on each side.

 Also shown in Fig. 3.22 is the exact solution for theinduced velocity, which in this case is simply a constant value w*(y )/U    1. The numerical solution does not lookat all like this, but if you look closely, you can see that thenumerical values are correct at the midpoints of each ofthe intervals. We would therefore get the right answer if wechose the midpoints of each interval as the control points.

The lift and induced drag can now be written as sumsof the elementary lift and drag forces on each panel

 M 

 n1

 F  z   U      nyv n  1  yv n (3.54)

 M 

 n1

 F  x      w* n   nyv n 1 y n n (3.55)

 Equation (3.52) can represent the solution to twodifferent types of problems. The first is the design problem, where the circulation distribution  (y ), andhence the total li ft, is given. We can use equation (3.52)directly to evaluate w*, and we can then use equation(3.55) to obtain the induced drag. We will also see laterthat the downwash velocity is an important ingredientin establishing the spanwise distribution of angle at-tack required to achieve the design circulation. Thesecond is the analysis problem where we are given thespanwise distribution of downwash, w*, and we wishto determine the circulation distribution. If we writedown equation (3.52) for  M   different control pointsyc  (1). . .yc ( n ). . .yc ( M  ), we obtain a set of simultaneousequations where w n,m is the coefficient matrix, w* is theright-hand side, and  is the unknown. Once  is found,we can obtain both the l ift and the drag from equations(3.54) and (3.55).

The remaining question is how to determine the op-timum arrangement of vortex and control points. Whilemuch theoretical work in this area has been done, rightnow we will use a cut and dry approach. This is facili-tated by a simple  FORTRAN95 program called  HVLL ,

Spanwise position, y/s

     D    o    w    n    w    a

    s     h    v    e     l    o    c     i     t    y ,

    w     * /     U

0 0.1 0.2 0.3 0.4 0.5-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

Vortex Positions

Control Point Positions

Figure 3.22 Spanwise distribution of velocity induced by a vortex lattice.The spacing is uniform with 10 panels and 25% tip inset. Due to symmetr y,only half the span is shown.

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  PROPULSION 57

which calculates both the exact and the numerical values of the induced velocity, total lift, and total in-duced drag for a circulation distribution defined by anyspecified number of Glauert coefficients a j  . The design  problem is first exercised by calculating the numerical

approximation to the downwash induced by the speci-fied circulation distribution. The analysis option is thenexercised by calculating the numerical approximationto the circulation starting from the exact downwashassociated with the originally specified circulation dis-tribution. In both cases, the total lift and induced dragcan be computed and compared with the exact values.Thus, the accuracy of a given lattice arrangement andthe convergence of the method with increasing numbersof panels can be studied.

The simplest arrangement consists of equally spaced panels with no tip inset and with the control points atthe midpoint of each panel. This scheme will be demon-

strated for the simple case of elliptical loading, wherethe exact downwash is a constant. The results for M   8 panels is shown in Fig. 3.23. Here we see that the pre-dicted circulation distribution has the correct shape,but is uniformly too high. The numerical result for thedownwash is quite good in the middle of the span, butgets worse at the tips.

Table 3.1 shows the effect of number of panelson the computed forces. For example, if the circula-tion is specified, the error in predicted lift is 1.3%with 8 panels and reduces to 0.1% with 64 panels.However, the error in drag is much greater, rang-ing from 10.1% with 8 panels to 1.3% with 64 panels.

On the other hand, if the downwash is specified, thecomputed lift and drag is in error by about the sameamount, ranging from 12.5% with 8 panels to 1.6%with 64 panels.

While this might not seem too bad, it is easy to get

much better results without any extra computing effort.The problem with the tip panel is that the strength of thefree vortex sheet in the continuous case has a squareroot singularity at the tips, which is not approximatedwell in the present arrangement.

Figure 3.24 and Table 3.2 shows what happens if thetip panels are inset by one quarter of a panel width. Nowthe induced velocity in the tip panel is much better (butstill not as good as for the rest of the panels), and theerror in forces is around 1% for 8 panels, and 0.1% or lessfor 64 panels. One can explore the result of changingthe tip inset and find values that will either make the

Figure 3.23 Comparison of vortex lattice and exact results for an elliptically loadedlifting line with a1  1.0. The solution was obtained with eight panels using uniformspacing with zero tip inset.

Spanwise Position, y/c

     G     i    r    c    u     l    a     t     i    o    n ,

      Γ     /     U    s

     D    o    w    n    w    a    s     h ,

    w     * /     U

-0.5 -0.25 0 0.25 0.50

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

W-NUM

G-NUM

G-Exact

W-Exact

Table 3.1. Convergence of Vortex Lattice Lifting Line with ConstantSpacing and 0% Tip Inset

Constant Spacing—Zero Tip Inset

Percent Errors in Vortex Lattice Predictions for  F  z, F 

 x, and  F 

 x /( F 

 z)2

Panels Given ( y) Given w*( y)

  8 1.3 7.6 10.1 12.5 12.5 11.1

16 0.5 4.2 5.1 6.3 6.3 5.9

32 0.2 2.3 2.6 3.1 3.1 3.0

64 0.1 1.2 1.3 1.6 1.6 1.5

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58 PROPULSION

induced velocity at the tip, the total lift, or the induceddrag correct. However, no single value will be best forall three. Thus, a tip inset of one quarter panel is con-sidered to be the best. A proof that a quarter panel insetis correct in the analogous situation of the square rootsingularity at the leading edge of a 2D flat plate was pub-

lished by James (1972). Another possible spacing arrangement is motivated

by the change in variables used by Glauert in the solu-tion of the lifting l ine problem. We saw that this arrange-ment worked very well for the 2D problem. In this case,the vortices and control points are spaced equally in theangular coordinate y . This arrangement is called cosinespacing, and the equations for yv ( n ) and yc ( n ) can befound in the code. In this case, no tip inset is required.Proofs that this arrangement is correct may be found inLan (1974) and Stark (1970).

Figure 3.25 and Table 3.3 shows what happens whencosine spacing is used with eight panels, but where thecontrol points are located midway between the vorti-ces, as is the case with constant spacing. Figure 3.26 isan illustration of how a fast computer can make up fora certain amount of human stupidity. Using 64 panels,

the predicted circulation looks quite good, although it isstill a little high. The downwash is again accurate overa lot of the midspan, but the results at the tips are evenmore of a disaster. Increasing the number of elementslocalizes the problem, but the computed values at the tipare still way off. Despite this, the total forces seem to beconverging with an error of around 2 % with 64 elements.

Fortunately this is not the real cosine spacing, and itis included as a cautionary tale for numerical hackers. In real cosine spacing, the control points are mapped withthe same cosine transformation as the vortices (Table 3.4).They are therefore not in the middles of the intervals, butare biased towards the tips. Figure 3.27 results shows that

this arrangement is extremely accurate, even with eight panels. Note, in particular, that the lift and drag obtainedfrom the circulation found by specifying w* is exact, andthat the ratio of drag to lift squared, F x  / F z

2 is exact for anynumber of panels.

 All of the examples considered so far are for ellip-tical loading. The remaining two figures show the re-sults of adding an additional coefficient with a valueof a3     0.2 to the Glauert series for the circulation.This unloads the tips (which may be desired to delaytip vortex cavitation inception), producing large upwardinduced velocities in the tip region and increase in the

Spanwise Position, y/c

     G     i    r    c    u     l    a     t     i    o    n ,

      Γ     /     U    s

     D    o    w    n    w    a    s     h ,

    w     * /     U

-0.5 -0.25 0 0.25 0.50

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

G-Exact

W-Exact

G-NUM

W-NUM

Figure 3.24 Comparison of vortex lattice and exact results for an elliptically loadedlifting line with a1  1.0. The solution was obtained with eight panels using uniformspacing with 25% tip inset.

Table 3.2. Convergence of Vortex Lat tice Lifting Line with ConstantSpacing and 25% Tip Inset

Constant Spacing—25% Tip Inset

Percent Errors in Vortex Lattice Predictions for  F  z, F 

 x, and  F 

 x /( F 

 z)2

Panels Given ( y) Given w*( y)

  8 1.0 1.6   0.4 0.3 0.3   0.3

16 0.4 0.6   0.1 0.1 0.1   0.1

32 0.1 0.2   0.0 0.0 0.0 0.0

64 0.0 0.1   0.0 0.0 0.0 0.0

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  PROPULSION 59

induced drag. Figure 3.28 shows the results obtained us-ing “good” cosine spacing with eight panels. The resultsare extremely close to the exact value, although a smalldiscrepancy is visible in the graph. Figure 3.29 showsthe same case calculated with 32 panels. The resultsnow appear to be right on top of the exact results. Inaddition, the increased number of panels provides muchbetter resolution of the behavior of the circulation anddownwash near the tips.

3.7.3 The Prandtl Lifting Line Equation  Our discus-sion of lifting line theory so far has addressed the ques-tion of relating the spanwise distribution of circulationto the downwash, lift, and induced drag. In addition, wehave found the spanwise distribution of circulation, whichminimizes the induced drag.

Lifting line theory, by itself, does not provide anyway of determining the lift generated by a particularfoil shape, since the details of the flow over the actual

surface are completely lost in the idealization of a lift-ing line. While the lifting surface equations developedearlier will provide the means to solve this problem, asimpler alternative exists if the aspect ratio of the foil(the ratio of the span to the mean chord) is high.

This idea was originated by Prandtl (Prandtl & Tietjens,1934), who reasoned that if the aspect ratio is sufficientlyhigh, the foil section at a given spanwise position acts asthough it were in a 2D flow (remember the near-sightedbug), but with the inflow velocity altered by the downwash velocity obtained from lifting line theory. The solution tothe problem of analyzing the flow around a given foil thenrequires the solution of two coupled problems: a local 2D problem at each spanwise position and a global 3D liftingline problem. This idea was formalized many years later bythe theory of matched asymptotic expansions where the so-lution to the wing problem could be found in terms of an ex- pansion in inverse powers of the aspect ratio. The matchedasymptotic solution may be found in Van Dyke (1975), butwe will only present Prandtl’s original method here.

First recall that the sectional lift coefficient, C  L , is

C  Ly  F  zy 2  y

Ucy  U 2cy

1

2

(3.56)

 where c (y ) is the local chord as illustrated in Fig. 3.12. Tokeep things as simple as possible for the moment, let us as-sume that the foil sections have no camber. Then, if the flowwere 2D, the lift coefficient at spanwise positiony would be

C  Ly  2y2  yUcy (3.57)

Spanwise Position, y/c

     G     i    r    c    u     l    a     t     i    o    n ,

      Γ     /     U    s

     D    o    w    n    w    a    s     h ,

    w     * /     U

-0.5 -0.25 0 0.25 0.50

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

W-EXACT

G-EXACT

G-NUM

W-NUM

Figure 3.25 Comparison of vortex lattice and exact results for an elliptically loaded lift-ing line with a1  1.0. The solution was obtained with eight panels using cosine spacingwith central control points.

Table 3.3. Convergence of Vortex Lattice Lifting Line with CosineSpacing and Central Control Points

Constant Spacing—Central Control Points

Percent Errors in Vortex Lattice Predictions for  F  z, F 

 x, and  F 

 x /( F 

 z)2

Panels Given ( y) Given w*( y)

  8 1.1 12.1 14.0 15.2 15.2 13.2

16 0.3   6.9   7.4   7.7   7.7   7.1

32 0.1   3.6   3.8   3.9   3.9   3.7

64 0.0   1.9   1.9   1.9   1.9   1.9

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60 PROPULSION

Following Prandtl’s theory, equation (3.57) can bemodified to account for 3D effects by reducing the angleof attack by the induced angle

C  Ly  2 y iy2  yUcy

 2y w*y  U 

(3.58)

 Both the circulation   and the downwash w*  canbe expressed in terms of the coefficients in Glauert’sexpansion, from equations (3.39) and (3.43). Equation(3.58) then becomes

C  Ly  

 n1

y  

 n1

a n sin ny4s

cy

2   na n sin ny

sin y

(3.59)

 Equation (3.59) must hold for any spanwise positiony along the foil. Given a distribution of chord length c (y )and angle of attack  (y ), we can find the first  M  coef-ficients in the Glauert expansion for the circulation bysatisfying equation (3.59) at M  spanwise positions. Thesolution will presumably become more accurate as M  is

increased. Another alternative is to go back to equation (3.58)

and use a VLM to solve for discrete values of thecirculation. Because equation (3.52) gives us the down-wash, w* at the  n th panel as a summation over the  M  panels, we obtain the following set of simultaneousequations for the M  unknown vortex strengths  n

C  Lyc n  2 2   nUc n

 n

 n  1, … M 

 

 M 

 m1

   mw n,m

1

U  (3.60)

 Both methods work well, but are approximations,as the results depend either on the number of terms re-tained in the Glauert series or on the number of panelsused in the vortex lattice.

 An exact solution to equation (3.59) can be obtainedby inspection in the special case that the chord distribu-tion is elliptical and the angle of attack is constant. Ifwe define c0 as the chord length at the midspan, we canwrite the chord length distribution as

cy c0

2y

s1

2

(3.61)

Table 3.4. Convergence of Vortex Lattice Lifting Line with CosineSpacing and Cosine Control Points

Cosine Spacing—Cosine Control Points

Percent Errors in Vortex Lattice Predictions for  F  z, F 

 x, and  F 

 x /( F 

 z)2

Panels Given ( y) Given w*( y)

  8 0.6 1.3 0.0 0.0 0.0 0.0

16 0.2 0.3 0.0 0.0 0.0 0.0

32 0.0 0.1   0.0 0.0 0.0 0.0

64 0.0 0.0 0.0 0.0 0.0 0.0

Spanwise Position, y/c

     G     i    r    c    u     l    a     t     i    o    n ,

      Γ     /     U    s

     D    o    w    n    w    a    s     h ,

    w     * /     U

-0.5 -0.25 0 0.25 0.50

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

W-EXACT

W-NUM

G-EXACT

G-NUM

Figure 3.26 Comparison of vortex lattice and exact results for an elliptically loadedlifting line with a1  1.0. The solution was obtained with 64 panels using cosine spac-ing with central control points.

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  PROPULSION 61

 which has a projected area S     c0s  /4 and an aspect ra-tio A   s2  /S   4s  /(  c0 ). Before introducing this chordlength distribution in (3.59), we must transform it intothe y variable using equation (3.38)

cy c0 sin y  (3.62)

Equation (3.59) then becomes

C  Ly  A   2   na nsin ny

sin y

 n1

 

 n1

a n   sin nysin y

(3.63)

Figure 3.28 Comparison of vortex lattice and exact results for a tip-unloaded liftingline with a1  1.0 and a3 0.2. The solution was obtained with eight panels usingcosine spacing with cosine control points.

Spanwise Position, y/c

     G     i    r    c    u     l    a     t     i    o    n ,

      Γ     /     U    s

     D    o    w

    n    w    a    s     h ,

    w     * /     U

-0.5 -0.25 0 0.25 0.50

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

W-NUM

G-NUM

G-Exact

W-Exact

Figure 3.27 Comparison of vortex lattice and exact results for an elliptically loadedlifting line with a1  1.0. The solution was obtained with eight panels using cosinespacing with cosine control points.

Spanwise Position, y/c

     G     i    r    c    u     l    a     t     i    o    n ,

      Γ     /     U    s

     D    o    w    n    w    a    s     h ,

    w     * /     U

-0.5 -0.25 0 0.25 0.50

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

W-EXACT

W-NUM

G-NUM

G-EXACT

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62 PROPULSION

but this equality can only hold if the circulation distri-bution is elliptical (i.e., if a n  0 for n  1). In this case,the local lift coefficient, C  L (y ) and the total lift coeffi-cient C  A a1 are equal, and equation (3.63) reduces to

C  Ly C  L 2

1  2  A (3.64)

 This remarkably simple formula captures the essen-tial role of aspect ratio controlling the rate of change oflift with angle of attack. As the aspect ratio approachesinfinity, the lift slope approaches the 2D value of 2  . Asthe aspect ratio becomes small, the lift slope approacheszero. This result is plotted in Fig. 3.30, together with ac-curate numerical results obtained from lifting surfacetheory and with results obtained from the theory ofmatched asymptotic expansions (VanDyke, 1975).

 An amazing attribute of Prandtl’s theory as appliedto an elliptical wing is how well it works even for lowaspect ratios. Of course, if you look closely at Fig. 3.30,

 you can see that Prandtl’s theory always over-predictsthe lift, and that the percent error increases with de-creasing aspect ratio. Another important observation isthat, even at an aspect ratio of A  8, the lift slope issubstantially below the 2D value of 2  .

The three curves labeled 2nd approx. , 3rd approx. ,and  modified 3rd approx  . are a sequence of solutionsobtained from the theory of matched asymptotic expan-sions. The first of these looks almost like Prandtl’s re-sult, namely

C  Ly C  L 21  2  A   (3.65)

which is a little more accurate for high aspect ratios, butfalls apart for low aspect ratios. Note that it predicts thata foil with an aspect ratio of 2 will have zero lift at allangles of attack! The higher order matched asymptoticapproximations remain accurate for progressively lower values of aspect ratio.

Figure 3.31 shows the application of Prandtl’s equa-tion to determine the effect of planform taper on circu-lation distribution. As expected, the circulation near the

Figure 3.30 Lift slope, dC L/d , of an elliptic wing as a function of aspect ratio,A. (From Van Dyke, 1975; reprinted by permission of Elsevier Publications.)

Figure 3.29 Comparison of vortex lattice and exact results for a tip-unloaded liftingline with a1  1.0 and a3 0.2. The solution was obtained with 32 panels usingcosine spacing with cosine control points.

Spanwise Position, y/c

     G     i    r    c    u     l    a     t     i    o    n ,

      Γ     /     U    s

     D    o    w    n    w    a    s     h ,

    w     * /     U

-0.5 -0.25 0 0.25 0.50

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

G-Exact

W-Exact

G-NUM

W-NUM

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  PROPULSION 63

tips decreases (and the circulation at the root increases)as the ratio of tip chord to root chord, ct   /c r  , is decreased.

Before leaving our discussion of Prandtl’s lifting lineequation, let us consider what happens if the foil sec-tions have camber. As the coupling between 3D liftingline theory and a local 2D flow is based on the total liftat each spanwise section, it does not matter whether

the lift is generated by angle of attack, camber, or somecombination of the two. We can therefore generalizeequation (3.57) by including the 2D angle of zero lift ofthe local section, 0 L (y )

 2 y 0 Ly2  yUcy

C  Ly (3.66)

 For a section with positive camber, the angle of zerolift is generally negative, thus increasing the lift in ac-cordance with equation (3.66). All we have to do is re- place  (y ) with  (y) 0 L (y ) in equations (3.58) and(3.59) to treat the general case of cambered sections. Itis also easy to include real fluid effects by replacing the2D lift slope of 2

  and the theoretical angle of zero lift

with experimentally determined values. In this way, theresults of 2D experiments can be applied to 3D flows, provided that the aspect ratio is high.

For the special case of an elliptical foil, the span-wise distribution of circulation will be elliptical if (y )  0 L (y ) is constant over the span. This can beachieved, for example, by having both the angle of at-tack and the zero lift angle constant over the span, orby some combination of the two whose difference isconstant. In the former case, the spanwise distributionof lift will remain elliptical if the angle of attack of the

entire foil is changed by a constant amount (say due tosome different operating condition). However, in thelatter case, a constant increment in angle of attack willintroduce a spanwise variation in the quantity  (y ) 0 L (y ). In that case, elliptical loading will only be gener-

ated at one particular angle of attack.3.8 Lifting Surface Results

3.8.1 Exact Results  The solution of the linearized problem of a planar foil involves the solution of a singu-lar integral equation whose main ingredients are givenin equations (3.22), (3.23), and (3.25). We would expectthat an analytical solution could be found in the simplecase of a rectangular planform and with zero camber, yet this is unfortunately not the case. Tuck (1991) devel-oped highly accurate numerical solutions for this caseby a combined analytical/numerical approach whichinvolved an extrapolation of the error obtained by dif-ferent levels of discretization. In particular, Tuck foundthat the lift slope of a square (aspect ratioA  1.0) foil is

C  L

 1.460227 (3.67)

 with a confidence of “about” 7 figures. Obviously, thisdegree of accuracy is of no practical value, but it is im- portant to have exact solutions for specific cases to testthe accuracy of numerical methods. For example, if youare examining the convergence of a numerical methodas a function of panel density, you might be misled if the“exact” value that you are aiming for is even slightly off.

 A large number of investigators have published val-ues for the lift slope of a flat, circular wing (a flying man-hole cover) over the time period from around 1938–1974.

Their values range from 1.7596 to 1.8144, with severalagreeing on a value of 1.790. None of these are closedform analytic solutions and some of the differences canbe attributed to insufficient numbers of terms used inseries expansions. But in 1986, Hauptman and Miloh ob-tained an exact solution based on a series expansion ofellipsoidal harmonics. In particular, they were able toderive the following simple equation for the lift slope ofa circular wing

 1.79075032C  L

8  2(3.68)

 and also obtained a somewhat more complicated equa-tion for the lift slope of any elliptical planform.

To our knowledge, no other exact solutions exist.However, these two results are extremely valuable in validating the VLM, which we will explore in the nextsection.

3.8.2 Vortex Lattice Solution of the Linearized Planar Foil  We would obviously not have gone to allthe trouble of developing the vortex lattice solution forthe 2D foil and for the planar lifting line if we had notanticipated putting these two together to solve the liftingsurface problem. This can be done very simply in thecase of a rectangular foil, as shown in Fig. 3.32.

Spanwise position, y/s

     C     i    r    c    u     l    a     t     i    o    n ,

      Γ     /     2     U    s

-0.5 -0.25 0 0.25 0.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Rectangle: ct /c

r =1.0

Triangle: ct /c

r =0.0

Tapered: ct /c

r =1/3

Figure 3.31 Effect of planform shape on spanwise distribution of circulationobtained from Prandtl’s lifting line equation. The foils all have an aspectratio of A  4 and are at unit angle of attack.

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64 PROPULSION

 As in 2D flow, the chord of the foi l is divided into N cosine spaced panels, with concentrated vortex linesand control points spaced in accordance with equa-tion (2.92). Similarly, the span of the foil is dividedinto  M   cosine spaced panels as was done in the de- velopment of vortex lat tice lift ing line theory. Eachconcentrated bound vortex element of strength    nm

is combined with a pair of free vortices of strength  extending downstream to infinity, thus forminga  horseshoe vortex   element. If we let i, j   denote thechordwise and spanwise indices of a control point,then the vertical velocity at a particular control pointwill be

 N 

 n1

 M 

 m1

wij  

   nmw nm,ij  (3.69)

 where w nm,ij  is the horseshoe influence function , whichis defined as the vertical velocity at the (i,j  )th control point induced by a horseshoe vortex element of unitstrength centered at the ( n,m )th grid point. Each horse-shoe consists of three straight vortex segments, whose velocity at any control point position can be obtainedfrom equation (3.19).

From the linearized boundary condition

w

U   zs

 x 

 we can form a set of N    M  simultaneous equations forthe N    M  unknown vortex strengths   nm . Once the circu-lation strengths are known, we can obtain the spanwisedistribution and total value of lift using the same equa-tions as in vortex lattice lifting line theory.

Figure 3.33 shows the vortex lattice arrangement for afoil with aspect ratio A   2.0 using a relatively fine grid,with 32 spanwise and 16 chordwise panels. Note that the panels near the leading and trailing edges at the tip areextremely small.

Table 3.5 shows the computed lift slope for a foil withaspect ratio A  1.0 using a systematically refined gridranging from N   1 to N   128 panels in the chordwisedirection and M   4 to M   128 panels in the spanwisedirection. The finest grid in the study required the so-

Starboard Tip

     L    e    a     d     i    n    g     E     d    g    e

     T    r    a     i     l     i    n    g     E     d    g    e

Vortex Lattice Grid

Rectangular Foil: A=2

32 Spanwise Panels

16 Chordwise Panels

     S    p    a    n    w     i    s

    e    v    o    r     t    e    x

Chordwise vortex

Control points

Figure 3.33 Vortex lattice grid for a rectangular foil with aspect ratio A  2.In this example, there are 32 spanwise and 16 chordwise panels. The plot onthe upper right is an enlargement of the starboard tip near the trailing edge.

Table 3.5. Convergence of Vortex Lattice Calculation for Rectangular Foil with Aspect Ratio 1.0

M/N 1 4 8 16 32 64 128

  4 1.428988 1.458702 1.459285 1.459351 1.459357 1.459357 1.459357

  8 1.428994 1.459262 1.460010 1.460097 1.460105 1.460106 1.460106

  16 1.428994 1.459306 1.460085 1.460196 1.460209 1.460210 1.460210

  32 1.428994 1.459309 1.460091 1.460206 1.460222 1.460224 1.460224

  64 1.428994 1.459309 1.460091 1.460207 1.460224 1.460226 1.460226

128 1.428994 1.459309 1.460091 1.460207 1.460224 1.460226 1.460227

Tabulated values of dC L/d . Each row shows convergence with number of chordwise vortices. Each column shows convergence with number of spanwise panels.

C

C

      C♦ 

♦     

♦     

O

n=1   2   3 N

m=1

2

3

4

M

n,m

i,j

w(n,m,i,j)

trailing edge

Horseshoe element

Control point

⇒ ∞

Figure 3.32 Notation for a vortex lat tice solution for a rectangular foil.

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  PROPULSION 65

lution of a set of 8192 simultaneous equations!20  Notethat this result agrees to seven significant figures withTuck’s result. But note also, that a very coarse 4  8 grid produces a result which is in error by only 0.07%.

The next step in complexity is to develop a vortex lat-tice scheme for foils with arbitrary planforms, includ-ing ones with curved leading and trailing edges. Thenatural model problem for this case is the circular foil,

since we know the exact solution.Early vortex lattice schemes maintained a Cartesiangrid, with bound vortex elements oriented in the y direc-tion and trailing vortex elements oriented in the x  direction.This meant that as the chord length changed with spanwise position, the vortex lattice grid had to have abrupt stepsbetween each chordwise panel. A better arrangement is toadapt the grid to the planform, as shown in Fig. 3.34. The vortex lines that follow the general shape of the leading andtrailing edges are no longer necessarily in the y directionand are therefore not necessarily bound vortices .

We will therefore call them spanwise vortex linesand will have to be careful when computing forces. FromStokes’ theorem, the total circulation around the foil at a

 particular spanwise location yc is the sum of the circula-tions of the spanwise elements, no matter how they are in-clined. Suppose we let  be the inclination of a particularspanwise vortex with respect to the y direction, the force per unit length on the vortex is   U  cos( ). However, thelength of the spanwise vortex is  y  /cos( ), where  y  isthe width of the panel in the spanwise direction. So, theforce per unit span is still equal to   U  , regardless of theinclination of the vortex  ,  . In addition, Kelvin’s theorem

is satisfied if the vortex system is built from horseshoeelements, originating from a spanwise vortex element.

The only trick in dealing with a circular planform isto provide for a finite chord at the tip in order to pre- vent having all the spanwise vortices coming togetherat a single point. This can be done rationally by solv-ing for the tip chord such that the area of the approxi-mate quadrilateral tip panel is equal to the area of the

actual circular arc segment. This is shown in Fig. 3.34.Figure 3.35 shows a much finer grid, with 64 panels overthe span and 32 panels over the chord. Figure 3.36 shows

20 Advantage was taken of port–starboard symmetry. Other-wise there would have been 16,384 equat ions.

Streamwise coordinate, x/s

     S    p    a    n    w     i    s    e    c    o    o    r     d     i    n    a     t    e ,

    y     /    s

-0.5 0 0.5-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Leading Edge   Trailing edge

Spanwise vortex lines   Chordwise vortex lines

Control points

Figure 3.34 Vortex lattice grid for a circular foil with an 8  8 grid.Streamwise coordinate, x/s

     S    p    a    n    w     i    s    e    c    o    o    r     d     i    n    a     t    e ,    y     /    s

-0.5 0 0.5-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Leading Edge   Trailing edge

Spanwise vortex lines   Chordwise vortex lines

Control points

Figure 3.35 Vortex lattice grid for a circular foil with 64 spanwise and32 chordwise panels.

Streamwise coordinate, x/s

     S    p    a    n    w     i    s    e    c    o    o    r     d     i    n    a     t    e

 ,    y     /    s

-0.1 0 0.10.3

0.35

0.4

0.45

0.5

Spanwise vortex lines

Chordwise vortex lines

Control points

Tip

Figure 3.36 Enlargement of the tip region of the vortex lattice grid for acircular foil with 64 spanwise and 32 chordwise panels.

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66 PROPULSION

an enlargement of the tip region, where the finite, equalarea tip chord is barely visible.

The computed lift coefficients for these two casesare C  L  1.782 for the 8  8 grid and C  L  1.790 for the64  32 grid. Recall that the exact solution for a flat cir-cular foil is C  L  1.791, so it is clear that the VLM workswell for nonrectangular planforms.

Extension of the VLM to swept and tapered planformsis relatively simple. Figure 3.37 shows the vortex latticegrid for a tapered foil whose leading edge is swept back45 degrees. One particular horseshoe vortex element

is highlighted in Fig. 3.37. The influence function forthis element again consists of the contribution of threestraight vortex segments.

Figure 3.38 shows vortex lattice grids for three foil planforms with constant chord and varying amountsof sweep. Figure 3.39 shows the effect of sweep on thespanwise distribution of circulation,  (y ), for thesethree foils with zero camber and constant angle of at-tack. The foils have a constant chord length of c  0.2s ,

and therefore have an aspect ratio of A  5.0. The foilwith zero sweep has an (almost) elliptical circulationdistribution. The swept back foil has substantially in-creased circulation at the tip and decreased circulationat the root. On the other hand, the foil with forwardsweep has reduced circulation at the tips and increasedcirculation at the root. This means that if one wantedto have a swept back foil with an elliptical distributionof circulation over the span, the angle of attack of thetip sections would have to be reduced, compared withthose at the root. The converse would be true for a foilwith forward sweep. Note also that both forward andaft sweep result in a reduction of the total lift com- pared with a foil with zero sweep. The results shown inFig. 3.39 were obtained with a vortex lattice with 128 panels over the span in order to resolve the very abruptchange in slope of the spanwise distribution of circula-tion at the midspan.

Figure 3.38 Vortex lattice grids for rectangular foils with zero and 45degree sweep. The chord is c /s  0.2. The grid consists of 32 spanwiseand 16 chordwise panels.

Spanwise position, y/s

     C     i    r    c    u     l    a     t     i    o    n     d     i    s     t    r     i     b    u     t     i    o    n ,      Γ

     /      Γ     (    a    v    g     )

-0.5 -0.25 0 0.25 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Zero Sweep

Swept Back 45o

Swept Forward 45o

Figure 3.39 The effect of sweep on the spanwise circulation distribution.

U

Figure 3.37 Vortex lattice grid for a swept, tapered foil. The root chord isc r /s  0.5 and the tip chord is c t   0.2. The leading edge is swept back45 degrees. The grid consists of 16 spanwise and 8 chordwise panels.One particular horseshoe element is highlighted.

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  PROPULSION 67

4.1 Inflow   Propellers generally operate in the verycomplex flow field that exists at the stern of a ship. Thisflow may be highly turbulent and spatially nonuniform.To add to the difficulty, this flow may be altered signifi-cantly by the presence of the propeller. While the veloc-ity distribution in the plane of the propeller is partiallydue to the potential flow around the ship, its origin islargely viscous. Unlike a purely potential flow, the vor-ticity in this flow field interacts with itself as it is ac-celerated by the action of the propeller. This means thatthe total velocity at a point near the propeller is not sim- ply the linear superposition of the inflow (in the absenceof the propeller) and the velocity induced by the propel-ler, but includes an additional interactive component. A full numerical simulation of the combined flow prob-lem requires massive computational resources, and the validity of the outcome is limited by present empiricalmodeling of turbulence. It is therefore a practical neces-sity to employ a simpler flow model for most propellerdesign and analysis applications.

Rather than including this complex interaction in thesolution of the propeller flow problem, it is traditionallyassumed that the specified inflow is an effective inflow ,which is defined in a coordinate system fixed on theship as the total time-averaged velocity in the presenceof the propeller minus the time-average potential flowvelocity field induced by the propeller itself  . The nomi-

 nal inflow is defined as the flow that would be presentin the absence of the propeller. If there is no vorticityin the inflow field, this definition reduces to the usualresult that the total velocity is the linear superpositionof the inflow in the absence of the propeller, and the

 velocity induced by the propeller. Thus, in this case, thenominal and effective inflows are the same. For now, wewill focus on the inflow itself.

In order to represent a given inflow field, we firstdefine a ship-fixed cylindrical coordinate system withthe  x   axis (positive downstream) coincident with theaxis of rotation of the propeller, as shown in Fig. 4.1.The origin of the coordinate is in the  plane of the pro- peller  , which serves as the reference point for all axialdimensions of the propeller blade surfaces. The radialcoordinate is denoted by r  , and the angular coordinateby  o , which is measured in a clockwise (right-handed)sense when looking downstream with o  0 being at12 o’clock. As the inflow is periodic in

o , the three com-

 ponents of the time-averaged velocity in the ship-fixedcylindrical coordinate system, V a , V  r  , V t , can be ex- pressed as a Fourier series. The harmonic coefficientsfor each component, A n , B n , are functions of position inthe meridional plane, ( x  , r  ), and nondimensionalized bythe ship speed, V  S 

(4.1)

 Figures 4.2, 4.3, and 4.4 show a typical axial inflowfield as might be measured in a towing tank or possiblycomputed by a viscous flow solver.21 Figure 4.2 shows acontour plot of the axial velocity over the propeller disk,

and the presence of the hull boundary layer is evident atthe top of the disk. This particular hull has a skeg, so theregion of moderately low velocity extends to the bottomof the disk. Velocities close to free-stream can be seenat the outermost radius at angular positions of roughly30 degrees on each side of the bottom of the disk.

V a x , r , o  V  S  Aa x , r   Aa x , r  cos no

 n10  n

V  r 

 x , r , o

  V  S 

 A r  x , r   A r  x , r  cos no

 n10  n

V t x , r , o  V  S  At x , r   At x , r  cos no

 n10  n

 Ba x , r  sin no

 n1 n

 B r  x , r  sin no

 n1 n

 Bt x , r  sin no

 n1 n

 4

Hydrodynamic Theory of Propulsors

21 Because the hull is symmetrical, the velocity field should besymmetrical about the vertical centerplane. However, in thiscase, slight asymmetry is apparent due to model constructionor measurement errors.Figure 4.1 Propeller coordinate system and velocity notation.

Y

Z

X

VA

ω

ua

ur

ut

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68 PROPULSION

Figure 4.3 shows the variation in axial velocity with an-gular position for a set of radii. Finally, Fig. 4.4 showsthe cosine coefficients,  A n , for one particular radius.Note that the inflow harmonic coefficients decrease rap-idly with increasing harmonic numbers. While it wouldseem from Fig. 4.4 that only the first few harmonics areof importance, we will see later that the apparently in- visible higher harmonics are actually responsible for

 producing unsteady propeller shaft and bearing forces. A similar diagram could be made for a series of

axial positions over the extent of the propeller. In the

 past, it has generally been assumed that the var iationin inflow is small enough that that the inflow can beassumed to be a function only of radial and angular position—thus greatly simplifying the ana lysis of the propeller flow. However, current propel ler design/analysis methods can accept a fully 3D inflow, and thisis important for special applications such as podded propulsors or waterjet pumps.

We next introduce another cylindrical coordinatesystem that rotates with the propeller. The  x  and r  co-ordinates are the same as before, but   represents the

Z

     Y

-1 -0.5 0 0.5 1-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

V

0.950

0.925

0.900

0.875

0.850

0.825

0.800

0.775

0.750

0.725

0.700

0.675

0.650

0.625

0.600

0.575

0.550

0.525

0.500

0.475

0.450

Figure 4.2 Contours of axial velocity for a t ypical single-screw ship.

Ship-Fixed Angular Coordinate  θ0

     A    x     i    a     l     I    n     f     l    o    w     V    e

     l    o    c     i     t    y     V

    a     /     V

    s

0 100 200 3000.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0.280

0.440

0.600

0.680

0.760

0.920

Radius r/R

Figure 4.3 Plot of the angular variation in axial velocity at several radii. Figure 4.4 Bar graph of the cosine harmonics of the wake field at one radius.Harmonic Number, N

     A    x     i    a     l     V    e     l     i    c     i     t    y     C    o    s     i    n    e     H    a    r    m    o    n     i    c ,

     A    n

0 2 4 6 8 10-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

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  PROPULSION 69

angular coordinate of an arbitrary point relative tothe angular coordinate of a reference point on the  keyblade of the rotating propeller. If a propeller is rotatingwith angular velocity  in a counterclockwise directionwhen looking downstream,22  the relationship between

the fixed and rotating coordinate system is

o t (4.2)

If the inflow is nonuniform with respect to o in theship-fixed coordinate system, an observer rotating withthe propeller will see a time-varying velocity. Undernormal operating conditions, the response of a propel-ler blade to each harmonic of the inflow is essentiallylinear. Thus, each inflow harmonic gives rise to an un-steady blade force at a frequency, n , related to the in-flow harmonic number, n . The steady, or time-average,force is therefore determined almost entirely by thezero’th harmonic. One should be cautioned that this as-sumption may not be valid if massive amounts of bladeflow separation or cavitation is present.

To determine steady propeller forces, we have there-fore simplified the problem considerably. We now simplyhave a given radial distribution of axial, radial, and tan-gential effective inflow velocity

V  A x , r   V  S  Aa0, r  A 0

V  R x , r   V  S  A r 0, r 0

V T  x , r   V  S  At0, r 0

  (4.3)

  Again, if the propeller inflow is to be approximatedby its value at the plane of the propeller, the velocitiesgiven in equation (4.3) are a function of radius only.

Finally, a useful inflow quantity is the advancespeed, V 

_ A , which is a weighted average of V  A ( r  ) over

the propeller disk. This term, which originates fromself- propulsion model testing, is the experimentally de-termined speed of advance of a propeller operating inuniform flow where either the thrust or torque matchthe values obtained in a test where the same propelleris operating in conjunction with the hull. This defini-tion is very clear in the context of a model test, but can-not be rigorously defined except as the outcome of anelaborate numerical simulation of the hull and propel-ler. However, a volumetric mean advance speed can becomputed very easily as

 rV  A r dr  R

 r h

2

 R2  r 2h

V  A (4.4)

 where  R  is the radius of the propeller and  r h  is theradius of the propeller hub. Alternatively, equation(4.4) can be modified to produce a thrust-weighted ortorque-weighted average velocity assuming that these

quantities are known. It is therefore important for thereader to keep in mind that the precise definition of V 

_ A

depends on the problem being considered. And finally,the overbar symbol, V 

_ A , which is used here to distin-

guish an averaged quantity from a point quantity, V  A ( r  ),

is generally omitted in propeller literature.4.2 Notation  Figure 4.1 shows a right-handed pro-

 peller placed in the rotating coordinate system de-scribed in the preceding section, together with theinflow V  A ( r  ). If present, the tangential inflow, V T ( r  ),would be positive by right-hand rule with respect to the x  axis. The propeller has a maximum radius  R  (or di-ameter D ) and is located in the vicinity of the origin ofthe coordinate system.23  The propeller has  Z  identicalblades that are symmetrically placed on a hub, whichin turn is attached to a shaft. The hub and shaft can bethought of as an arbitrary axisymmetric body alignedwith the x  axis, but it is frequently idealized as a cylin-der of radius r 

 H  , as shown in Fig. 4.1 or ignored entirely

in preliminary hydrodynamic analyses.The propeller induces a velocity field with Carte-

sian components (u ,v ,w ) or cylindrical components(ua ,u r  ,ut ) in the axial, radial, and circumferential di-rections, respectively. The total velocity field is thenthe superposition of the propeller advance speed, ro-tational speed, and induced velocity field and has thefollowing components in the axial, radial, and circum-ferential directions

 r  V T  r  ut r V  R r  u r  r V  A r  ua r    (4.5)

The propeller produces a thrust force T  in the nega-tive x  direction and absorbs a torque Q about the x  axis,

with a positive value following a right-handed conven-tion. These can be nondimensionalized either on the ba-sis of the ship speed , V  S 

C T  S 

T 

1

2  V 2  R2 S 

C Q S 

Q

1

2  V 2  R3 S 

(4.6)

 or on the volumetric mean inflow speed , V _ A , as defined

in equation (4.4)

C T  A

T 1

2  V 2  R2 A

C Q A

Q

1

2  V 2  R3 A

  (4.7)

23 Variations in placement of the propeller in the  x  directionwill simply add a constant to the rake , while variations in theangular placement of the y axis wil l add a constant to theskew .

22 This is commonly referred to as a right-handed propeller,which rotates clockwise when looking upstream. This is gen-erally the assumed direction of rotation when developingequations for propeller flow.

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70 PROPULSION

 Alternatively, the forces may be nondimensional-ized with respect to a nominal rotational velocity,  nD ,

where n 2 

 is the number of propeller revolutions per

second K 

T 

   n2 D4

T 

 K Q    n2 D5

Q  (4.8)

 The kinematics of the flow depends only on the ratioof the inflow and rotational velocities, and this is cus-tomarily expressed as an advance coefficient

 J  S V  S 

 nD  (4.9)

 or

 J  A V  A

 nD  (4.10)

 By introducing equation (4.9) into equation (4.6), wesee that the two systems of force coefficients are relatedas follows

C T  S 

 K T 8

 J 2 S  

C Q S 

 K Q16

 J 2 S  

  (4.11)

with a similar relationship for C T 

 A

 and C  Q

 A

. In most cases, the K T  , K Q system is preferred, as the pro-

 peller rotational speed, n , can be precisely measured bothin the laboratory and on shipboard, and the coefficients

remain finite in the case of static thrust, when V  A  0. Onthe other hand,  K T  and  K Q become infinite at one pointduring a crashback maneuver,24 when the shaft changesits direction of rotation, while C T  and C Q are finite duringcrashback until the ship comes to a complete stop. How-ever, the C T  , C Q system also has a more direct relationshipto propeller efficiency. Thus, both systems are useful andhave therefore been retained over the years.

The  power coefficient  based on torque and shipspeed is

QC  P 

Q S 

1

2  V 3  R2 S 

C Q S  

 J  S 

(4.12)

 and the hydrodynamic efficiency of the propeller is

s TV  S 

Q

C T  S 

C  P Q

 S 

C T  S 

 K T   J  S 

2  K Q

 J  S 

C Q S 

 (4.13)

 with a similar relationship in terms of V _ A .

4.3 Actuator Disk   We will first consider the simplest possible idealization of a propeller: the so-called actuator

disk , which was first introduced by Rankine (1865) andFroude (1889). The physical propeller is replaced by a per-meable disk of radius R , with vanishing thickness in the x  direction. The disk introduces a uniform jump in total pressure,  p0 , to the fluid passing through the disk, which

tends to accelerate the fluid in the positive axial directionand thus results in a thrust force in the negative x  direc-tion. No tangential velocity, or swirl ,25 is introduced by thedisk, and as a consequence of the principle of conservationof angular momentum, there is no torque supplied.

It is difficult at this point to relate this theoretical de- vice to a real propeller. However, we will ultimately seethat the actuator disk is really the limit, in an ideal fluid,of a propeller with an infinite number of blades, zerochord length, and infinite rotational speed.26 But rightnow, we will develop expressions for the thrust and ef-ficiency of an actuator disk based on conservation ofmomentum and energy.

The actuator disk is assumed to be operating in anunbounded, inviscid fluid, with a uniform axial inflow velocity, V  A , and uniform static pressure,  p , far up-stream. As the flow is axisymmetric and without swirl,we are left with axial and radial velocities, u ( x  , r  ), v ( x  , r  )as a function of x  and r  only.

Because the flow is inviscid, the total pressure,  p0 ,in accordance with Bernoulli’s equation, is constantalong any streamline, except for those that pass throughthe disk, where a total pressure rise of  p0 occurs. Fardownstream of the disk, we can expect that flow quan-tities will be independent of  x  , with the tip streamlineachieving some limiting radius Rw . The axial perturba-tion velocity will be a constant uw for r    Rw and will be

zero for r    Rw . The radial velocity will be zero for all ra-dii vw  0, and the static pressure will be independent ofradius, with a value equaling the upstream value p  p .

 As no fluid is created within the disk and the axialand radial velocities are continuous, the increase in to-tal pressure  p0 is felt entirely as an increase in static pressure,  p . This pressure rise can be found by writingBernoulli’s equation between a point far upstream and a point far downstream along any stream tube that passesthrough the disk

 p    V 2  p  p 1

2   V  A uw

21

2

 p  puw V  A uw

2

 A

(4.14)

  As   p is independent of radius, equation (4.14) verifiesthe important assumption that the axial perturbation

24 This is where the ship reverses the direction of rotation ofits propeller(s) while moving at normal ahead speed until theship comes to a complete stop and starts moving backward.

25 Swirl is defined as the product of the radius, r  , and the tan-

gential velocity, ut, and is commonly used in turbomachineryliterature.26 Actually, other limiting processes involving counter-rotating propellers or propeller/stator combinations can also be shownto lead to the actuator disk.

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  PROPULSION 71

 velocity in the slipstream far downstream is indepen-dent of radius, uw  constant.

The total thrust on the actuator disk, T  , can be writ-ten down immediately as

T   p   R2

  uw     R2

V  A uw

2 (4.15)

 Introducing the definition of thrust coefficient fromequation (4.6), we obtain a compact nondimensionalform of equation (4.15) that only involves the ratio of theslipstream velocity to the advance speed

C T   2T 

1

2   R2V 2 A

uw

V  A1

uw

V  A

1

2(4.16)

 We can also obtain an independent expression for thethrust based on conservation of momentum, and thiswill enable us to obtain information about the velocity

field at the disk. The general vector form of the momen-tum equation is

 F    S  pndS    

 S V V  ndS   (4.17)

which states that the flux in momentum of the fluid passing through an arbitrary control volume is equal tothe sum of the pressure forces acting on the boundaryof the volume and the total body force, F  , acting withinthe volume. In this case, the resultant body force is thetotal thrust, T  , which acts in the negative  x  direction.Thus, the x  component of the momentum equation canbe written as

T    S  pn x dS    

 S V  A ua  n x dS   (4.18)

We will dispose of the pressure integral first, as it for-tunately will turn out to be zero! Referring to Fig. 4.5, wechoose a control volume whose outer surface corresponds

to the stream surface which passes through the tip of thedisk, and whose upstream and downstream boundariesare sufficiently far from the disk for the velocity and pres-sure to have reached their limiting values. Thus, at theupstream boundary, the control volume radius is Ru , the

 pressure is po , and the velocity is V  A . At the downstreamboundary, the radius is Rw , the pressure is again po , andthe velocity is V  A  uw . Conservation of mass requires that

 R2V  A  R2V  A uwwu (4.19)

The net pressure force acting on the two ends is there-fore   po R

2  R2u w  in the positive x  direction. Determiningthe x  component of the pressure force acting on the outersurface presents a problem because we do not know thedetails of its shape or pressure distribution. To overcomethis problem, we will examine another control volumewhose inner boundary matches the outer boundary ofthe present control volume. The outer boundary will bea stream surface whose upstream and downstream radii, R, R

u d are large enough for the velocity to have returnedto the free-stream value V  A . As the new control volumelies outside the propeller slipstream, the velocity at boththe upstream and downstream faces is V  A . Conservationof mass requires that the two faces have equal area, andas the pressures are equal there is no net pressure force. Again, conservation of mass requires that

 R2  R2  R2  R2u d u d  (4.20)

so that the x  component of the pressure force acting on theouter boundary is   po R

2  R2u w  in the negative x  direction. As there is no momentum flux out of and no body forcewithin the outer control volume, the net integral of pressuremust be zero. Therefore, the pressure force on the innerboundary of the outer control volume is   po R

2  R2u w  in the positive x  direction, which must be equal and opposite tothe force on the outer surface of the inner control volume.Now we see that this force just balances the difference in pressure force on the ends of the control volume, thus prov-ing the assertion that the net pressure force is zero.

We need to be a little careful in developing the ex- pression for the mass flow rate m  , as the velocities in the plane of the disk cannot be assumed to be independentof radius. We first introduce the following notation forthe axial perturbation velocity at the disk

u* r  u0, r a (4.21)

which will also be used later in developing propeller lift-ing line theory. Now consider a differential stream tubeof radius dr  at the disk. The mass flow rate through thestream tube is

dm   V  A u* r 2  rdr a   (4.22)

and the total thrust is

T   uw aV  A u* r 2  rdr  R

0

  uwV  A u*  R2a

(4.23)Figure 4.5 Control volume for actuator disk momentum calculation. Thestream tube contraction has been exaggerated for clarity.

VA

VA

VA+u

w

p∞

p∞

Ru

Rd

Ru

∞

Rd

∞

VA

  p∞

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72 PROPULSION

where we define the mass-averaged axial perturbation velocity at the disk as

 R

0au* r 2  rdr au*

1

  R2   (4.24)

 Comparing equations (4.15) and (4.23), we see thatthe mass-averaged axial perturbation velocity at thedisk is exactly half of the perturbation velocity fardownstream

au* 2

uw (4.25)

 If we introduce equation (4.25) into equation (4.16),we obtain a very useful expression that relates the axial velocity at the disk to the thrust coefficient

au*

2V  A

 1 1 C T (4.26)

 For small values of thrust coefficient, C T   1, equa-

tion (4.26) becomesau*

4V  A

C T (4.27)

 Most textbook presentations of actuator disk theory,for simplicity, do not distinguish between averaged flowquantities and their actual radial distribution, and there-fore leave the impression that the perturbation velocity atthe disk is independent of radius. In fact, detailed compu-tations show that the axial perturbation velocity is nearlyconstant over most of the disk but decreases somewhatas the radius approaches the radius of the disk. Outsideof the disk, the induced velocity is initially negative, buttends to zero at larger radii. This can be seen from Fig. 4.6,

which shows the results of a numerical computation usinga procedure that will be described shortly.

We can now write down an exact expression for theradius of the slipstream far downstream, Rw , by apply-ing conservation of mass

a w  V  A u*  R2   V  A uw  R2 (4.28)

and rearranging

1 1 C T 

2 1 C T 

uw

V  A1

au*

V  A1

 Rw

 R

  (4.29)

 Thus, we see that as the thrust loading increases, theultimate slipstream radius,  R

w , decreases. In the limit

of static thrust,27  when the advance speed is zero, thethrust coefficient becomes infinite, and the slipstreamradius reaches an asymptotic limit of Rw   R  1   2, asshown in Fig. 4.7.

In this case, Schmidt and Sparenberg (1977) have shownthat the tip stream tube has a logarithmic spiral behaviorin the immediate vicinity of the tip, passes upstream, and

27 This is sometimes referred to as bollard pull, signifyingthat the ship is being held stationary at a dock while poweris applied to the propeller. It is also a condition encounteredinitially when a ship starts from rest.

Total axial velocity (VA

+ua(0,r))/V

A

     R    a     d     i    u    s    r     /     R

0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.30

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 4.6 Numerical computation of radial distribution of total axial veloc -ity in the plane of the actuator disk. The jump in velocity should theoreticallyoccur right at r /R  1.0 but is “smeared out” over about 2% of the radiusdue to the finite grid spacing used in the simulation.

10-1

100

101

102

103

Thrust Coefficient, CT

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Limit of Static Thrust

Figure 4.7 Ultimate slipstream radius as a function of thrust coefficient, C  T  ,from equation (4.29).

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  PROPULSION 73

then doubles back to pass through the disk at a radius ofapproximately 0.97 R , as shown in Fig. 4.8. Furthermore,stream tubes just outside cross the disk twice. Fluid par-ticles traversing these stream tubes experience no net to-

tal pressure rise, so that their pressure and velocity fardownstream return to free-stream values. In spite of thiscomplex flow behavior near the tip, equation (4.29) is stilltheoretically exact. However, one should remember thatan exact solution to the actuator disk problem does notimply that it is a physically realizable flow.

The efficiency of the actuator disk can be defined asthe ratio of the useful work done by the device to thetotal energy supplied to it. As shown in Fig. 4.9, in afixed coordinate system, the disk will be moving in the

negative x  direction with speed V  A . In a time increment t , the disk moves   x   V  A t in the negative x  direction,and the output work will be W   T   x  . At the same time,the total amount of kinetic energy imparted to the fluid

will be increased by some amount, which we will call E  . Conservation of energy then requires that the totalenergy input be (T   x    E  ). The efficiency will then be

T  x  T 

T  x  E T  E    x (4.30)

 We now need to obtain an expression for  E  . Duringthe time interval  t , a fluid particle in the slipstream fardownstream will have moved a distance

 x w

V  A uwt    x V  A uw

V  A(4.31)

 relative to the disk . The increase in kinetic energy im- parted to the fluid is therefore the kinetic energy con-tained in a cylindrical volume of fluid of radius  Rw andaxial length   x w,

 E     u21

2w     R2

w    x V  A uw

V  A(4.32)

Introducing equation (4.28), we obtain

 E 

 x 

V  A  u2   R21

2w

V  A u* a(4.33)

0.1

0.2

0.3

0.4   0.5   0.6   0.7   0.8

0.9

1.0

0.99

1.0

10

x

y

0.50.0−0.5−1.0

Figure 4.8 Stream tubes near the tip of an actuator disk in static thrust (Schmidt andSparenberg, 1977). Note that the tip stream tube (labeled 1.0) initially goes upstream.

Figure 4.9 Control volume for actuator disk energy balance.

δxw

δx

VA

uw

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74 PROPULSION

Using equation (4.23), this can be written explicitlyin terms of thrust

uw

V  A

 E 

 x 

1

2

au*

V  AT  T  (4.34)

 The efficiency equation (4.30) can now be put in termsof the thrust coefficient, which is the desired result

1 1 C T 

2 (4.35)

 Equation (4.35) is exact, despite the fact that the simplemomentum/energy analysis presented here cannot quan-tify the complex local flow near the disk. This importantequation shows how efficiency reduces with increas-ing thrust coefficient, and serves as an upper bound onefficiency for “real” propulsors. This result is plotted inFig. 4.10, together with the results for the more generalcase which we will consider later.  Swirl is a term origi-nating in the turbomachinery literature and is defined as( radius )  (tangential velocity ). For a classical actuatordisk, the only external force acting on the fluid is in theaxial direction so that no tangential velocities are present.

4.4 Axisymmetric Euler Solver Simulation of an Actua-tor Disk   The results developed so far are for the sim- plest possible case of an actuator disk with uniformtotal pressure rise with no addition of swirl. While onecan treat the general case, we will defer this until wehave developed the fundamentals of lifting line theory.We will then be able to show that the actuator disk canbe recovered as a special limit of lifting line theory.

However, it would seem that the basic understanding ofthe flow created by an actuator disk would be enhancedif we could somehow see the actual distribution of ve-locity and pressure, which is something that was not

available to Rankine and Froude. We will therefore leapahead in time and look at the complete flow using a nu-merical Euler solver.

Figure 4.11 shows the flow streamlines and contoursof axial velocity for an actuator disk obtained from the MTFLOW  (Multipassage Through Flow design/analysis) program developed by Drela (1997). The computationaldomain extends four disk radii upstream and down-stream, and three disk radii outward. The computa-tional grid contains 240 elements in the axial directionand 120 elements in the radial direction, making a totalof 28,800 cells. The Euler solution is obtained iterativelyby adjusting the the streamtube cross-sectional area insuch a way as to satisfy the equations of conservation of

mass, momentum, and energy across the boundaries ofeach cell. The jump in total pressure across the actuatordisk is introduced as an imposed field quantity along therow of cells closest to x   0 extending from r   / R  0.02 to r   / R  1 using a procedure developed by Kerwin (2003).

The grid is not shown in Fig. 4.11 as it would com- pletely mask the velocity contours. However, Fig. 4.12shows a close-up of the flow near the disk, togetherwith the grid. The contraction of the tip streamtube isevident, along with the growth in axial velocity withdistance downstream. It is also evident that the axial velocity grows to its ultimate downstream value muchmore rapidly at the outer radii than at the inner radii. Fi-

nally, note that there is a region of axial velocity that islower than free-stream just outside the tip streamtube.These characteristics are similar to those present for a“real” propeller. Figure 4.13 shows a similar close-up ofthe pressure coefficient. The uniform jump in pressureacross the disk is evident, and the return to free-stream pressure upstream and downstream is just what onewould expect from Bernoulli’s equation.

Figure 4.14 shows the variation of axial velocity and pressure coefficient at a fixed radius of r   / R  0.5. Thetriangular symbols show the pressure jump, which inthis case was  p  /2   0.553. The computed pressuredistribution tends toward these two values except that

Figure 4.10 Efficiency as a function of thrust coefficient for the generalcase of an actuator disk with swirl. The curve for J  0 corresponds toequation (4.35).

Thrust Coefficient CT

     E     f     f     i    c     i    e    n    c    y

0 2 4 6 8 1020

30

40

50

60

70

80

90

100

Efficiency of Optimum Actuator Disk with Swirl

No Swirl: J = 0

J = 2

J = 1

J = 0.5

X

     R

-4 -3 .5 -3 -2 .5 -2 -1 .5 -1 -0.5 0 0.5 1 1 .5 2 2 .5 3 3. 5 40

1

2

3

Figure 4.11 Computed contours of axial velocity over the complete com-putational domain.

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  PROPULSION 75

the jump in pressure is spread out slightly in the x  direc-tion and is shifted slightly. This is a consequence of thefinite grid size in the Euler solution. In spite of this, thecomputed efficiency, thrust, and power coefficient and velocity far downstream all agree within 1% of the valuefrom actuator disk theory in this example.

4.5 The Ducted Actuator Disk   An important exten-sion of classical actuator disk theory to include the ef-fect of an idealized duct was developed by Oosterveldand published in van Manen and Oosterveld (1966),Oosterveld (1968), and van Manen, Oosterveld, and

Witte (1966). A portion of the tip streamtube is con-sidered to be a zero-thickness duct, which acts as themean-line of an annular airfoil. A duct loading factor, , is introduced such that the total thrust generated bythe propeller and duct is defined as T  , while the thrust provided by the actuator disk itself is T  . The thrust pro- vided by the duct is therefore (1    )T  , which can be pos-itive (thrust) or negative (drag) depending on whether  is less than or greater than unity. The axial extent andshape of the duct does not have to be specified, as wasthe case with the actuator disk itself.

Figure 4.12 Computed streamlines and contours of axial velocity in the vicinity of the actuator disk.

XFigure 4.13 Computed streamlines and contours of pressure coefficient in the vicinity of the actuator disk.

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76 PROPULSION

The resulting changes to the classical actuatordisk equations are relatively small. As equation (4.15)gives the thrust on the disc alone, T  only needs to bereplaced by   T  . Equation (4.23) remains unchanged be-cause it comes from a momentum balance upstreamand downstream which includes the force on both theactuator disk and the duct. Equating these two expres-sions for duct results in the following modification to

equation (4.26)

au*

2 V  A

1 2  1  C T  (4.36)

while the induced velocity far downstream is now pro- portional to   CT 

uw

V  A 1   1  C T  (4.37)

 The radius of the slipstream,  Rw, given in equation(4.29) now becomes

1 1  C T 

2  1  C T 

 Rw

 R(4.38)

Finally, equation (4.35), which gives the efficiency asa function of thrust coefficient, becomes

1 1  C T 

2(4.39)

 Note that these equations all reduce to the classi-cal actuator disk results when there is no thrust on theduct (   1).

Figure 4.15 shows the effect of the duct loading factor,  , on the velocity at the disk, the radius of the slipstreamfar downstream, and the efficiency for a ducted actuatordisk with a total thrust coefficent of 1.0 obtained fromequations 4.36–4.39. When    1, the velocity at the diskis greater than if there were no duct, while the converse istrue when    1. The former is termed an accelerating ductwhich has its origins in the Kort Nozzle (Kort, 1934) devel-

oped in the 1930s, while the latter is termed a deceleratingduct or pumpjet . The efficiency of a ducted actuator diskincreases with decreasing values of   , indicating that anideal accelerating ducted propeller would be more efficientthan a propeller without a duct. In fact, equation (4.39)would suggest that in the limit of  → 0, a ducted actuatordisk will have 100% efficiency for any thrust coefficient!This may seem puzzling at first unless you also look at thedependence of slipstream radius on   , where the ratio ofslipstream radius to disk radius goes to infinity as  → 0.

 A more realistic interpretation of this result is to con-sider the downstream radius of the slipstream fixed. As   is decreased, the disk radius becomes smaller, and

the velocity through the disk increases. This suggeststhat an accelerating duct therefore provides a means ofimproving efficiency when the propeller radius is lim-ited, and this has been confirmed by van Manen andOosterveld (1966) and others. However, there are sev-eral fundamental reasons that limit the amount that  can be reduced in an actual ducted propeller design.

• The increase in frictional drag on the propeller bladesand duct offsets the gain in ideal efficiency.• Flow diffusion within the duct may result in flowseparation.• Cavitation inception is more likely due to the increasedflow speed at the propeller.

Figure 4.14 Computed axial velocity, V  x /V  A, and pressure coefficient, C  P  , as a functionthe axial coordinate, x , at a fixed radius r /R   0.5.

x/R

     V    x

     /     V    a

     C    p

-2 -1 0 1 2 3 41

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Vx

Cp

± ΔCp /2

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  PROPULSION 77

  As a result, practica l values of   for an accelerat-ing duct are generally in the range from 0.8 to 1.0.For the same reasons, a decelerating duct, whichwould not appear to have any merit from the pointof view of actuator disk theory, may be an effectivechoice in some situations, particularly for high-speed

applications.

4.6 Axisymmetric Euler Solver Simulation of a Ducted

 Actuator Disk   The same axisymmetric Euler solver thatwas used to simulate an actuator disk in Figs. 4.13 and4.14 can be used to represent an actual duct. As shownin Kerwin (2003), the actuator disk can be modeled asa lifting line propeller combined with a lifting line sta-tor. The stator is designed to cancel the tangential veloc-ity generated by the propeller so that the combinationacts like an actuator disk. This equivalence is describedmore fully in Section 4.7.2.

Figure 4.16 shows the streamlines and swirl contoursfor a particular case. As swirl is only present betweenthe rotor and stator, and has a uniform value, this canbe identified by the red region extending from the hubto the duct.

 A comparison between efficiency computed by theEuler/lifting line code and the ideal efficiency from equa-tion (4.39) for three sample ducts is given in Table 4.1.The ducts are all generated from a modified NACA66thickness form with a thickness/chord ratio of 10%, and

an NACA a  0.8 mean line with camber ratios as tabu-lated. The duct chord length is equal to 1.0 and the ductangle of attack is as tabulated. A critical challenge of the

 τ=Tp /T

     V    e     l    o    c     i     t     i    e    s    a    n     d    s     l     i    p    s     t    r    e    a    m    r    a

     d     i    u    s

     E     f     f     i    c     i    e    n    c    y ,

     η

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

0.77

0.78

0.79

0.8

0.81

0.82

0.83

0.84

0.85

0.86

0.87

0.88

(VA

+ua

*)/V

A

(VA

+uw

)/VA

Rw /R

η

Accelerating   Decelerating

Figure 4.15 The effect of the duct loading factor,  , on the total velocity at the disk, thetotal velocity downstream, the radius of the slipstream far downstream, and the efficiency.

Figure 4.16 Streamlines and contours of swirl for an ideal postswirl ductedpropulsor. C  T    0.6769 and    0.740.

Table 4.1 Comparison of Efficiency Predicted by an Euler/Lifting LineCode and by Actuator Disk Theory for Three Different Duct Shapes

 d   (  f  o /  c) d   C T        A

5.0 0.06   0.6769 0.740 90.7 89.9

7.5 0.06   0.5570 0.904 90.7 89.8

7.5 0.00 1.0585 0.951 83.2 82.9

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78 PROPULSION

numerical calculation is to obtain accurate values of theinviscid thrust on the duct. This requires some care inensuring that the small leading edge region of the ductis properly resolved. The results are quite close, with adiscrepancy of less than 1% in all three cases. The lastcase, where the duct thrust is relatively low, agreedwithin 0.3% of the actuator disk value. Note the sig-nificant difference in slipstream radius as a function of

distance downstream of the duct for the case shown inFig. 4.16, where    0.740, and in Fig. 4.17, where    0.951.In the first example, the value of   is low, indicating anaccelerating duct, and the slipstream radius expands af-ter leaving the duct trailing edge. In the second example,the value of   is closer to unity, and the slipstream radiuscontracts slightly downstream of the duct. These resultsare in qualitative agreement with the actuator disk re-sults plotted in Fig. 4.15.

4.7 Propeller Lifting Line Theory   We will now takeone step closer to the real world and develop a lifting linerepresentation of a propeller. As shown in Fig. 4.18, each

 propeller blade can be considered as a lifting surfacewith some distribution of bound and free vortex sheetstrength. We then consider the limiting case of vanish-ing chord length, which is represented in the right-hand portion of the figure. As in the case of the planar foil,

the bound vortex sheet reduces to a single concentrated vortex of strength  ( r  ) on each blade. As all blades willhave the same circulation distribution in circumferen-tially uniform flow, we can select one blade (or liftingline) and designate it as the key blade .

The strength of the free vorticity in the wake may befound from the relationships developed in Section 3.2.In this case, the curvilinear coordinate, s1 , will be moreor less helical in form as i llustrated in Fig. 4.18, but mayalso contract (as we saw with the actuator disk) as it progresses downstream. However, if we make the as-sumption that the free vortex lines are convected down-stream with a constant radius, the free vortex sheetstrength, as in the case of the planar lifting line, canbe obtained directly from the derivative of the spanwise(in this case, radial) distribution of circulation

d  

dr  

 f  r  (4.40)

 where the vector direction of positive  f  points down-stream along the helix, by right-hand rule.

We can develop expressions for the forces actingat radius  r  on the key lifting line from a local appli-cation of Kutta-Joukowski’s law. Figure 4.19 shows acombined velocity and force diagram. The axial andtangential induced velocities due to the helical free vortex system, u* r , u* r ta

, combine with the effective

inflow velocity components V  A ( r  ), V T  ( r  ) and the pro- peller rotational speed   r  to produce an inflow veloc-ity V * oriented at an angle i with respect to the planeof rotation

V  A r  u* r 2   r  V T  r  u* r 2a tV * r  (4.41)

 i r  tan1

t r  V T  r  u* r aV  A r  u* r 

(4.42)

XFigure 4.17 Streamlines and contours of swirl for a postswirl ducted propul-sor. C  T    1.0585 and    0.951.

Figure 4.19 Velocity and force diagram at a particular radial position ona lifting line.

Figure 4.18 Illustration of the concept of a lifting line propeller as a limitof vanishing chord length. The radial distribution of blade circulation, (r ),remains the same so that the strength of the trailing vortex sheet,   f (r ) isunchanged.

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  PROPULSION 79

 The quantity i    is therefore analogous to the in-duced angle in wing lifting line theory.

The inviscid (Kutta-Joukowski) force per unit radiuson the vortex, F i ( r  ) is therefore

 F i r    V 

*

 r    r  (4.43)and is directed at right angles to V  *.

 It is relatively simple to include the effect of v iscousdrag at this stage by adding a force  F v ( r  ) acting in adirection parallel to V * . This force may be estimatedon the basis of an experimentally determined, or the-oretically calculated, 2D sectional drag coefficientC  Dv ( r  ). This means, of course, that the section chordlengths c ( r  ) must be specified. The viscous drag forcewill then be

1 F v r 

2   V * r  2 c r C  Dv r  (4.44)

 These forces can then be resolved into componentsin the axial and tangential direction, integrated over theradius and summed over the number of blades to pro-duce the total propeller thrust and torque.

V *   cos i V *2 cC  Dv sin i dr 1

2T   Z  

 R

 r h

(4.45)

V *   sin i V *2 cC  Dv cos i  rdr 1

2Q   Z  

 R

 r h

(4.46)

 Note that V *cos i is simply the total tangential veloc-ity acting at the lifting line, (  r   V T   u

*t), and that V *

sin i

 is the axial velocity,V  A

ua

*.

 4.7.1 The Velocity Induced by Helical VorticesThe velocity induced at radius  r c  on the  key  liftingline by a set of Z   unit strength helical vortices shedat radius  r v  (with the vector direction of the circu-lation pointed downstream) can be expressed as anintegral using the law of Biot-Savart as developed inequation (3.17).

d

2 r v r c cos  k3  2

 

0u

a r 

c, r 

v

Z

 k1  r v tan w2  r v  r c2 2

 r v  r v  r ccos  k1

4 (4.47)

Z 

 k1 

0ut r c, r v

1

4 

2 r v r c cos  k3  2

 r v tan w2  r v  r c2 2

 r v sin  k r v tan w r c  r v cos  k

(4.48)

In the above expressions,  w  is the pitch angle ofthe helix at r v. According to linear theory, w    , butwe will leave w unspecified at the moment in order tofacilitate subsequent refinements to the theory. The variable of integration,  , is the angular coordinate ofa general point on the helix shed from the key blade.

The corresponding angular coordinate of a point onthe  k th blade is found by adding the blade indexingangle

 k 2   k 1,...Z Z 

 k 1(4.49)

The total induced velocity on the lifting line can nowbe obtained by integrating the contributions of the heli-cal vortices over the radius

 R

 r h

ua r c, r vdr v  r v

 r ua r c *

 

 R

 r h

ut r c, r vdr v  r v

 r ut r c *

(4.50)

 However, equations (4.47) and (4.48) cannot be eval-uated analytically, so one must resort to some form ofnumerical solution. On the other hand, if the limits ofintegration are changed from 0, to ,, they wouldthen represent the velocities induced along a radialline by a set of helical vortices extending to infinityin both directions. This would be equivalent to the ve-locity induced infinitely far downstream by the free vort icity shed from the original lifting line. Since theintegrands in equations (4.47) and (4.48) are even withrespect to the variable of integration,  ,28 the velocitiesinfinitely far downstream are double their values at thelifting line.

Now imagine a helicoidal coordinate system, withone coordinate along the helix, one coordinate radial,and the third coordinate normal to the first two. Far

downstream, the flow will be independent of the heli-cal coordinate. Thus, the flow is 2D in terms of the tworemaining helicoidal coordinates. The potential prob-lem for this type of flow was solved independently byKawada (1933) and Lerbs (1952), and a derivation maybe found in Appendix 1 of Lerbs (1952).

The resulting potential can be expressed in terms ofinfinite sums of modified Bessel functions. While directevaluation of these functions might be as time consum-ing as numerical integration of the Biot-Savart law re-sult, fortunately highly accurate asymptotic formulasfor the sums of Bessel functions exist. This enabledWrench (1957) to develop the following closed form ap-

 proximations to the induced velocities.For r c   r v :

ua r c, r v   Z 

4  r cy 2Z yy0 F 1

ut r c, r v   Z 

2

2  r cy0 F 1

(4.51)

28 Provided that the order of summation over the blade index, k , is reversed when  is replaced by  .

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80 PROPULSION

For r c   r v :

ua  r c, r v yy0 F 2Z 

2

2  r c

ut  r c, r v y1 2Z y0 F 24  r c

Z (4.52)

 where

1 ln1 y21.5

3y2 2 1

U  1

U  exp

Z 

y0 1 y2 1 y 1 y2 1

0

1 y2 1 y2 0

y  r c

 r v tan w

y0 1

tan w

1 lnU 

1 U 1 y21.5

3y2 2

 F 1 U 

1 U 

1

24Z 

1

2Z y0

0.25

01 y2

1 y2

09y2 2

1 y21.5

0

 F 2 12Z y0

1U  1

124Z 

09y2

21 y21.5

0

0.25

01 y2

1 y2

 

(4.53)

 Figure 4.20 shows the results of equations (4.51) and(4.52) for a particular case. As one would expect, the velocity tends to   as the control point radius ap- proaches the vortex radius. When the control point iswell inside the vortex, the axial velocity approaches a

constant value, which increases with decreasing pitchangle of the helix. When the control point is outsidethe vortex, the axial velocity approaches zero rapidlywith increasing radius and is relatively insensitive to pitch angle.

The reverse is true with the tangential velocity. In-side the vortex, the tangential velocity approaches zerowith decreasing radius. Outside the vortex, the tangen-tial velocity appears to reduce slowly with increasingradius. The tangential velocity is relatively insensitiveto pitch angle, except in the immediate vicinity of the vortex. This is in contrast to the axial velocity, whichis extremely sensitive to pitch angle when the control point is inside the vortex.

The limit of infinite number of blades is of partic-ular importance, as this will yield an axisymmetricflow that we can relate to the actuator disk. As theblade number is increased, the quantity U   in equa-tion (4.53) approaches zero if  r 

c   r 

v , but approaches

infinity if  r c    r  v . As a result,  F 1  in equation (4.53)approaches zero if  r c   r v , and  F 2 approaches zero if r c   r v . Equations (4.51) and (4.52) then reduce to thesimple expressions

For r  c   r v :

ut  r c, r v 0

ua

 r c, r v   Z 

4  r v tan w (4.54)

For r c   r v :

ua  r c, r v 0

ua  r t, r v   Z 

4  r c

(4.55)

Figure 4.21 shows the effects of the number of bladeson the axial and tangential velocity including the infiniteblade limit obtained from equations (4.54) and (4.55).

The singular behavior of the induced velocities issimilar to that of a straight vortex, where we know thatthe velocity behaves as 1/( r c   r v ). It is therefore useful

Control Point radius, rc /R

     A    x     i    a     l     I    n     d    u    c    e     d     V    e     l    o    c     i     t    y  ,    u    a

     /     V     A

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1. 4 1 .5-10

0

10

20

βw=10Deg

20

30

6010

60

Control Point radius, rc /R

     T    a    n    g    e    n     t     i    a     l     I    n     d    u    c    e     d     V    e     l    o    c     i     t    y  ,    u     t     /     V     A

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5-20

-10

0

10

10

60

60

Figure 4.20 Normalized velocity (u a/V  A, u t  /V  A) induced on a lifting line at radiusr  c /R  by a set of semi-infinite helical vortices of strength  2 RV  A  originating atr  v    1.0. The number of blades in this case is Z  5. Results are shown for pitchangles  w  10, 20, 30, 40, 50, 60 degrees.

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  PROPULSION 81

to factor out the singular part, leaving a regular func-

tion that depends on the geometry of the helix. Lerbs(1952) defined an induction factor  as the ratio of the velocity induced by  Z   helical vortices to the velocityinduced by a semi-infinite straight vortex of the samestrength

ia r c, r v ua r c, r v

1

4  r c  r v

ut r c, r v1

4  r c  r v

it r c, r v (4.56)

  As the radius of the vortex, r v , approaches the radius

of the control point, r c , the velocity induced by the heli-cal vortices will approach the value induced by a semi-infinite vortex oriented in a direction tangent to thehelix at its starting point on the lifting line. Therefore,as r c → r v , we find that

cos  i

4  r c  r vua r c, r v

sin  i

4  r c  r vut r c, r v

  (4.57)

 Comparing equation (4.57) with equation (4.56), wesee that in the limit as  r c  →  r v , the axial and tangen-tial induction factors become, respectively, cos i  andsin i . Therefore, the induction factors remain finite asthe vortex point and control point coincide, while theactual velocity tends to infinity. The reason for the mi-nus sign in the definition of the axial induction factoris strictly for the convenience of making the inductionfactors positive.

 An example of the axial induction factor is shown inFig. 4.22.

With the helical vortex influence functions known,we are now ready to tackle the evaluation of the sin-gular integrals for the induced velocities given in equa-tion (4.50). However, before we do this, we will revisitactuator disk theory—this time from the point of view

of a lifting line vortex model. 4.7.2 The Actuator Disk as a Particular Lifting

 Line  It turns out that we can construct an actuatordisk using the concepts of propeller lifting line theory.29

Consider the limiting case of an infinite bladed lift-ing line with zero hub radius and uniform circulation ( r  )     over the radius. In the limiting process, wewill keep the product of the number of blades, and thecirculation per blade, Z  constant. Furthermore, as we

29 This idea was pursued independently by Morgan (2009) and possibly by others.

Control Point radius,  rc /R

     T    a    n    g    e    n     t     i    a     l     I    n     d

    u    c    e     d     V    e     l    o    c     i     t    y

     ,

    u     t     /     V     A

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5-5

0

5

10

2

7

2

7

∞

Control Point radius,  rc /R

     A    x     i    a     l     I    n     d    u    c

    e     d     V    e     l    o    c     i     t    y

     ,

    u    a

     /     V     A

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5-5

0

5

10

2

7

2

7

∞

Figure 4.21 Effect of blade number on the velocity induced on a lifting line at radiusr  c  by a set of semi-infinite helical vortices originating at r  v   1.0. The pitch angle is  w  30 degrees. Results are also shown for an infinite number of blades from equa-tions (4.54) and (4.55). The total circulation, Z , is kept constant as the blade numberis varied and matches the value used for the five -bladed propeller shown in Fig. 4.20.

Control Point radius, rc /R

     A    x     i    a     l     I    n

     d    u    c     t     i    o    n     f    a    c     t    o    r  ,     i    a

0.5 0.75 1 1.25 1.50

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Βw=10 Deg

20

30

40

5060

rc /R

     i      a

0.95 0.975 1 1.025 1.050.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Βw

=10 Deg

60

Figure 4.22 Axial induction factors for a five-bladed propeller derived fromFig. 4.20. The enlarged plot shows the local behavior near r  c /r  v  1. Theanalytical limit of i  a  cos  w  is plotted as square symbols on the graph.

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82 PROPULSION

saw that an actuator disk generates zero tangential in-duced velocity (and hence absorbs zero shaft torque),we will need to adopt one of the following models:

1. We can assume that the rotational speed,  is high,so that u*

t   r  . This is equivalent to taking the limit of

zero advance coefficient, J  A .2. We can construct an ideal counter-rotating propel-

ler by superimposing two identical, infinite bladed lift-ing lines rotating in opposite directions. In this case, theaxial induced velocities from each lifting line compo-nent will add, while the tangential component will can-cel. The quantity Z  will be divided equally between thetwo components, but the sign of  will be opposite. Withthe counter-rotating model, the specification of advancecoefficient is arbitrary.

We will next show that either model will recover theresults obtained previously by momentum/energy con-siderations. We begin by writing down the relationshipbetween thrust and circulation, using equation (4.45).Setting the viscous drag to zero, and making the as-sumption that u*

t   r 

T     V * cos  iZ   dr     r Z   dr    Z    R2  

2

 R

0 

 R

0

(4.58)

 which we can also express as a thrust coefficient

C T  2

  Z    R2

Z   

2

1  V  A   R22    V 2 A

  (4.59)

 We next obtain an expression for the axial induced velocity, ua * . Because the circulation is constant, therewill be a concentrated helical tip vortex of strength shed from each blade, and from equation (4.54), thesewill induce a constant axial velocity

ua

*   Z   

4  R tan w R  (4.60)

 and zero tangential velocity. But there will also be aconcentrated hub vortex of strength - shed from eachblade at r   0. From equation (4.55), these will inducezero axial velocity. The tangential velocity induced bythe hub vortex will not be zero (unless we use the coun-ter-rotating option), but we have assumed, no matter

what, that it will be negligible.If we make the linearizing assumption that the tan-

gent of the pitch angle of the tip vortices, w is equal tothat of the undisturbed flow, tan  , we can eliminate Z in equation (4.60) using equation (4.59)

4  R

 V  AC T  1

V  A

 R

V  A

1

4

C T 

V  A

ua

* 2

  (4.61)

This agrees with the actuator disk result in the limitof vanishing C T  where u*

 a  /V  A  1, as derived previouslyin equation (4.27). However, we can obtain a more ac-curate result if we recognize that the pitch angle of the

tip vortices, w , is not the undisturbed angle  , but thatit, at least initially, is the angle i , where

tan  i  R

V  A ua

*

  (4.62)

 On setting w    i , we can rewrite equation (4.60) as

ua

*   Z       R

4  R V  A ua

*  (4.63)

4  RV  A uaua Z   R* *   (4.64)

and use equation (4.59) to eliminate Z 

Z    V 2C T  A

  (4.65)

 to obtain the result

4

V 2C T  A

V  A uaua * *

  (4.66)

 Finally, we can put equation (4.66) in standard qua-dratic form and solve it for u*

 a  /V  A

0V  A 4

ua C T 

2

V  A

ua

* *

  (4.67)

2

ua*

V  A

1 1 C T  (4.68)

 which is exactly the actuator disk result given in equa-tion (4.26). This is remarkable, in a way, because the

actuator disk result includes the contraction of the slip-stream (but is unaware of tip vortices), while the pres-ent result models the tip vortices as constant radiushelical lines with constant (although suitably adjusted) pitch angle. We will see later how this relates to the so-called moderately loaded theory of propellers.

We can now easily derive the expression for effi-ciency. The input power is

Q    r V  A uaZ   dr * R

0

 Z   V  A ua R21

2*

(4.69)

 while the output power is

1

2TV  A   V  AZ    R

2

  (4.70)

 So the efficiency,  is the ratio of the two

TV  A 1 2

Q1

ua*

V  A

1 1 C T 

  (4.71)

 This is also the exact actuator disk result given inequation (4.35).

4.8 Optimum Circulation Distributions  We would liketo obtain the radial distribution of circulation,  ( r  ),which will minimize the torque, Q , for a prescribed

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  PROPULSION 83

thrust, T  . The diameter, advance coefficient, blade num-ber, and effective inflow are specified.30 Although otherconsiderations, such as the inception of tip vortex cavi-tation, may require us to depart from this optimum, thisis generally a logical starting point in the design process

(Fig. 4.23).This problem can be solved using the method of cal-culus of variations and one can find detailed accounts ofthis approach in Yim (1976), Coney (1989), and Breslinand Andersen (1994). Although the resulting equationscan be solved rapidly on current computers, they are non-linear, intricate in appearance, and provide little direct physical insight.

On the other hand, an earlier approach developed byBetz (1919) for a propeller in uniform inflow, V  S  , and laterextended by Lerbs (1952), is relatively simple to deriveand is physically intuitative. Suppose we have a distri-bution of circulation which is optimum and which gen-erates the desired thrust. Now suppose that we perturb

this circulation by adding an increment of circulation

If  ( r  ) is truly optimum, then * must be independentof radius. Otherwise, circulation could be decreased ata radius where * is low, and increased by a correspond-ing amount at a different radius where *  is high. Theresult would be a reduction in torque for a fixed thrust,

thus contradicting the assertion that  ( r  ) is optimum.So far, this is essentially a physically based statement ofthe variational principle.

However, if we attempted to express equation (4.72) interms of circulation and induced velocity influence coeffi-cients, the resulting expressions would be far from simple.This is because the increment of circulation introduced ata particular radius, r  , not only changes the force locally,but alters the force over the entire radius as a result of itsaltering of the induced velocity distribution. Betz (1919)overcame this obstacle by employing a principle devel-oped by Munk (see Durand, 1963) that states that the totalforce on a lifting surface is unchanged if an element ofbound circulation is displaced in a streamwise direction.

Munk’s theorem follows from the principle that theforce on a lifting surface can be obtained solely from afar-field momentum/energy analysis (as we did for the planar lifting surface), and that the far-field flow de- pends only on the strength of the trailing vorticity. This,in turn, is unaffected by a streamwise displacement ofthe bound vorticity.

Betz therefore added the increment of bound vortic-ity far downstream, so that there would be no interac-tion between the added circulation and the flow at the propeller lifting line. On the other hand, the local forceacting on the added element of bound circulation mustinclude the effect of the doubling of the induced veloci-

ties far downstream

T    r  2ut r *   r 

V  S 

 2ua r      r  r Q    *  (4.73)

 If we assume that u* t ( r  )    r  and that u*

 a ( r  ) V  S  ,we can perform some algebraic manipulations

(4.74)

*  r 

V  S 

TV  S 

Q

 r 

V  S  V  S  2ua

*

 r  2ut

*

V  S 

2ua

  V  S 

ua

2*

 r  2ut    r ut

2**

*

0 0.5 1 1.5

Advance Coefficient, Js

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

Inviscid

L/Dv = 50

L/Dv = 25

CT = 1.0

Actuator Disk

Figure 4.23 Efficiency versus advance coefficient for a five-bladed propellerwith optimum radial distribution of circulation in uniform fl ow. Results are givenfor inviscid fl ow and for viscous fl ow with sectional lif t/drag ratios of 25 and50. The actuator disk result is shown as the symbol plotted at  J  s  0.0.

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