Fluid Mechanics of Yacht Keels Marco Trucchi - [email protected] 18 December 1996

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Abstract This thesis represents the first of a series of final projects concerning the fluid-dynamic study of yacht fin keels that will be made at the Department of Aeronautics of the Imperial College of London. The purposes of our work were: • to study what are the main characteristics of the design of yacht fin keels • to evaluate the possibility of studying the hydro-dynamics of a yacht using the a pre-existing set of computer codes for the numerical simulation of steady three-dimensional inviscid incompressible flows, without considering the free surface and using linear finite elements on unstructured tetrahedral grids • to create all the tools necessary to make parametric studies on different conventional keels and winged (or T-shaped) keels using the abovementioned codes • to make a parametric study on the influence of the addition of different tip winglets to the same conventional keel. The tools consist in computer programs, written in standard Fortran 77 language. Some of them create, on the basis of user specified shape parameters, the files containing the geometry and the mesh characteristics that are needed by the codes. The created shape is made up of a hull and a fin keel; two winglets can be attached to the keel tip. The yacht can be also heeled, up to a maximum angle of 30◦ ; the shapes attainable with the programs are satisfactorily similar to the actual. Other tools make calculations and analyze the results of the computations. The friction drag on the keel is evaluated in first approximation with an empirical method. i

Marco Trucchi - [email protected]

”Fluid Mechanics of Yacht Keels”

We have tested on three low aspect ratio rudders our codes and methods by means of the comparison of the results with experimental data taken from an external source. The computations have been performed on an inverse tapered keel of aspect ratio 1.4, attached to a classic shape hull, with heel angles from 0◦ to 30◦ and leeway angles from 0◦ to 10◦ . On the same keel many winglets of different aspect ratio, sweep angle and dihedral angle have been attached to compare their performance. Our conclusions are that, despite the approximations, the methods we have used can represent a good tool for the hydrodynamic design of a fin keel, while they are inadequate for the hull. For what concerns the winglets of the T-shaped keels we found positive effects on the lift curve slope and on the induced drag depending on the their aspect ratio and dihedral angle, while their sweep angle is almost uninfluential. Possible future developments on this subject could consist in finding a better estimate of the friction drag and in evaluating the wave drag and the unsteady behavior; the results could then be tested in the towing tank. New calculations could be made to find the influence of the addition of different tip winglets to different kinds of conventional keel, especially with different aspect ratio.

Notes Units of measure. The yacht world is still full of traditions; this, together with the fact that most of the sailing development took place in the AngloSaxon countries, brings to a heavy use, in the related works, of units of measure different from the ones of the SI. In this work we have always tried to use the metric system; some of the figures, taken from other studies, and their explanations represent an exception to this. For all the conversions to the SI units not made directly inside the text, refer to appendix B. Software. This document was prepared with the LATEX system. The set of numeric codes used for the flow simulation is the FAN system. Some long analytical calculations were made with the help of Mathematica. Our figures were prepared with Gnuplot and Tecplot. All the files related to this work are available as described in appendix C.

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Sommario Questa tesi rappresenta la prima di una serie riguardante lo studio fluidodinamico di chiglie di deriva di imbarcazioni a vela che verr`a svolta presso il Dipartimento di Aeronautica dell’Imperial College di Londra. Gli obiettivi del lavoro erano: • studiare quali sono le principali caratteristiche del progetto di una chiglia di deriva • valutare le possibilit`a di effettuare uno studio sull’idrodinamica di una imbarcazione a vela facendo uso di una preesistente serie di codici per computer, atti alla simulazione in tre dimensioni di flussi stazionari inviscidi e incompressibili, che non considerano il pelo libero dell’acqua e usano elementi finiti lineari su griglie non strutturate di tetraedri • creare tutti gli strumenti necessari per uno studio parametrico su diverse chiglie convenzionali e chiglie con alette mediante i codici sopracitati • effettuare uno studio parametrico sull’influenza dell’aggiunta di differenti alette su di una stessa chiglia convenzionale. Gli strumenti consistono in programmi per computer, scritti nel linguaggio Fortran 77. Alcuni di essi creano, sulla base di parametri di forma definiti dall’utente, i file contenenti la descrizione della geometria e delle caratteristiche della griglia di cui hanno bisogno i codici. La forma creata consiste in uno scafo con chiglia di deriva; due alette possono essere aggiunte all’estremit`a della chiglia. L’imbarcazione pu`o essere sbandata fino ad un angolo di 30◦ ; le forme ottenibili sono sufficientemente aderenti alla realt`a. Altri strumenti effettuano i calcoli necessari per analizzare i risultati dati dai codici. La resistenza d’attrito sulla chiglia `e valutata in prima approssimazione con un metodo empirico. iii

Marco Trucchi - [email protected]

”Fluid Mechanics of Yacht Keels”

Abbiamo effettuato delle prove su tre timoni di basso allungamento per verificare l’accuratezza dei metodi utilizzati, confrontando i risultati con dati di esperimenti ricavati da una fonte esterna. I nostri calcoli sono stati effettuati su di una chiglia a rastremazione inversa di allungamento 1.4, posta al di sotto di uno scafo di forma classica, con angoli di sbandamento compresi tra 0◦ e 30◦ e angoli di scarroccio compresi tra 0◦ e 10◦ . Sulla stessa chiglia sono state aggiunte alette di diverso allungamento, angolo di freccia e angolo diedro per confrontarne le prestazioni. Le conclusioni sono che, nonostante le approssimazioni, i metodi utilizzati possono rappresentare un valido strumento per il progetto idrodinamico di una chiglia di deriva, mentre sono inadeguati per uno scafo. Per quanto riguarda le alette, abbiamo trovato effetti positivi sul coefficiente angolare di portanza e sulla resistenza indotta dell’intera chiglia dipendenti dal loro allungamento e angolo diedro, mentre il loro angolo di freccia `e praticamente ininfluente. Possibili sviluppi futuri sull’argomento potrebbero consistere nel trovare una migliore stima della resistenza d’attrito, nel valutare la resistenza d’onda e il comportamento instazionario; i risultati potrebbero essere successivamente convalidati con prove in vasca idrodinamica. Nuovi calcoli potrebbero essere fatti per studiare l’influenza dell’aggiunta di alette su differenti chiglie convenzionali, specialmente di diverso allungamento.

Sommario

iv

Contents Acknowledgements

viii

Ringraziamenti

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1 Displacement yachts 1.1 Functional description . . . . . . . . . . 1.2 Yacht mechanics . . . . . . . . . . . . . 1.2.1 Velocity triangle . . . . . . . . . 1.2.2 Acting forces . . . . . . . . . . . 1.2.3 Steady motion . . . . . . . . . . . 1.2.4 Unsteady motion . . . . . . . . . 1.2.5 Points of sailing and performance 1.2.6 Polar diagrams of sails . . . . . . 1.3 Yacht requirements . . . . . . . . . . . . 1.3.1 Racing classes . . . . . . . . . . . 1.3.2 Yacht measurement rules . . . . .

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1 1 2 2 5 6 12 12 15 17 18 19

2 Hulls and fin keels 2.1 Introduction . . . . . . . . . . . . . 2.1.1 Graphic representation . . . 2.1.2 Involved parameters . . . . 2.1.3 Theorem of Buckingham . . 2.2 Kinds of resistance . . . . . . . . . 2.2.1 Wave drag . . . . . . . . . . 2.2.2 Friction drag . . . . . . . . 2.2.3 Induced drag . . . . . . . . 2.2.4 Pressure drag . . . . . . . . 2.2.5 Drag due to heel . . . . . . 2.3 Total resistance and polar diagrams

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Marco Trucchi - [email protected] 2.4 2.5

”Fluid Mechanics of Yacht Keels” . . . . . . . . . . .

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54 55 56 58 59 60 61 65 65 70 74

3 Equations of motion and their numerical solution 3.1 Navier Stokes equations for incompressible flows . . 3.1.1 Boundary conditions. . . . . . . . . . . . . . 3.2 Restrictions and approximations . . . . . . . . . . . 3.2.1 Validity of the results . . . . . . . . . . . . . 3.3 Numerical solution of steady incompressible flows . 3.3.1 Mesh generation . . . . . . . . . . . . . . . 3.3.2 Flow solution . . . . . . . . . . . . . . . . .

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77 77 79 80 83 84 84 85

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Towing tank testing . . . . . . . . . . . . . . . . Fin keel requirements . . . . . . . . . . . . . . . 2.5.1 Hydrodynamic side force and drag angle 2.5.2 Transverse stability . . . . . . . . . . . . 2.5.3 Damping of rolling oscillations . . . . . . 2.5.4 Course keeping ability . . . . . . . . . . Fluid-dynamic effects of keel shape . . . . . . . Winged keels . . . . . . . . . . . . . . . . . . . 2.7.1 Effects of the winglets . . . . . . . . . . 2.7.2 Fluid-dynamic phenomena on tip devices Design hints for a generic keel . . . . . . . . . .

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4 The FAN system 4.1 System modules and data files . . . . . . . 4.1.1 User generated files . . . . . . . . . 4.1.2 System generated files . . . . . . . 4.1.3 Using several meshes for multigrid . 4.2 Geometry data file (fan.dat) . . . . . . . 4.3 Mesh specification data file (fan.bac) . . 4.4 Boundary condition data file (fan.bco) . 4.5 Solver control data file (fan.nam) . . . . .

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89 89 91 91 92 92 95 97 98

5 Hull and keel generators 5.1 Hull and conventional keel . 5.1.1 Geometry . . . . . . 5.1.2 Mesh generation . . 5.1.3 Boundary conditions 5.1.4 Flow conditions . . . 5.2 Hull and winged keel . . . . 5.2.1 Geometry . . . . . .

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5.3

”Fluid Mechanics of Yacht Keels”

5.2.2 Mesh generation and boundary conditions Calculation of forces and moments . . . . . . . . 5.3.1 Program lift.for . . . . . . . . . . . . . 5.3.2 Program momentum.for . . . . . . . . . . 5.3.3 Profile drag estimate on the fin keel . . . . 5.3.4 Lift/Drag curves . . . . . . . . . . . . . .

6 Preliminary study 6.1 Choice of the shape . . . . . . . . . . . . 6.2 Generated meshes . . . . . . . . . . . . . 6.3 Convergence history . . . . . . . . . . . 6.4 Testing . . . . . . . . . . . . . . . . . . . 6.4.1 Geometry and mesh . . . . . . . 6.4.2 Results and comparisons with the 6.4.3 Calculation using momentum.for .

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7 Results 7.1 Conventional keel . . . . . . . . . . . . . . . . 7.1.1 Effective angle of attack . . . . . . . . 7.1.2 Lift computation on all the boat . . . . 7.2 Parametric study on winged keels . . . . . . . 7.2.1 Variation of aspect ratio . . . . . . . . 7.2.2 Variation of sweep angle . . . . . . . . 7.2.3 Variation of dihedral angle . . . . . . . 7.3 Some graphical representations and interesting Conclusions

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127 129 129 134 135 136

138 . 138 . 141 . 148 . 153 . 153 . 155 . 162 166 . 166 . 171 . 174 . 174 . 178 . 185 . 185 . 191 194

A Details of the geometry definition 197 A.1 Hull and conventional keel . . . . . . . . . . . . . . . . . . . . 197 A.2 T-shaped keel . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 B Conversion factors

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C Computer files and their availability

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D Some figures

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E Bibliography

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Contents

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Acknowledgements I wish to express my gratitude to the Imperial College of Science, Technology and Medicine of London, that allowed me to have a wonderful study experience abroad. In particular I wish to thank Professor John K. Harvey, who was my tutor for this final project, and Doctor Joaquim Peir´o, who helped me a lot with the numerical codes. Besides I wish to thank for the large help with the computers the Doctors Letty Allen, Matthew Foulkes and Rhodri Moseley; for everything concerning my stay in London Sara Payton, Kate Gibbons, John O’Leary and Gianluca Sperti.

viii

Ringraziamenti Desidero ringraziare l’Imperial College di Londra, che mi ha permesso di vivere un’affascinante esperienza di soggiorno di studio all’estero. In particolare ringrazio il professor John K. Harvey, effettivo relatore della mia tesi, e il dottor Joaquim Peir´o, che mi ha molto aiutato nella parte di studio numerico. Desidero inoltre ringraziare per i numerosi consigli di carattere informatico i dottori Letty Allen, Matthew Foulkes e Rhodri Moseley; per tutto ci`o che ha riguardato il mio soggiorno a Londra Sara Payton, Kate Gibbons, John O’Leary e Gianluca Sperti.

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List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10

Star class (F. Sweisguth, 1932) . . . . . . . . Carina (P.L. Rhodes, 1955) . . . . . . . . . Velocity triangle . . . . . . . . . . . . . . . . Forces in steady-state sailing conditions . . . Equilibrium forces in upright position . . . . All round performance . . . . . . . . . . . . Olympic triangle . . . . . . . . . . . . . . . Sail polar diagram: close-hauled conditions . Sail polar diagram: close-reaching conditions Sail polar diagram: reaching conditions . . .

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2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17

Hull representation . . . . . . . . . . . . Fin keel . . . . . . . . . . . . . . . . . . Winged or T-shaped keel . . . . . . . . . Theoretical wave drag . . . . . . . . . . Relative contributions to total resistance Optimum prismatic coefficient . . . . . . Boat drag . . . . . . . . . . . . . . . . . Relative contributions to total resistance Friction coefficient of a flat plate . . . . . Variation of Cf on a flat plate . . . . . . Origin of the vortex wake . . . . . . . . Velocity field around a finite span wing . Velocity field around a finite span wing . Mirror image . . . . . . . . . . . . . . . Boundary layer and wake . . . . . . . . . Hull polar diagram . . . . . . . . . . . . Polar diagrams for different Θ angles . .

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”Fluid Mechanics of Yacht Keels”

2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29

Effect of decrease of β . . . . . . . . . . . . . . . . Rolling oscillations on hull . . . . . . . . . . . . . . Velocity and force on a section of a rolling keel . . . Section parameters . . . . . . . . . . . . . . . . . . Experimental lift curves for different aspect ratios . Effects of keel shape on lift curve slope . . . . . . . Efficiency factor K for different tip shapes . . . . . Comparison between winged keels and straight keels End-plates on a wing . . . . . . . . . . . . . . . . . Flow over a wing tip . . . . . . . . . . . . . . . . . Cross-flow on a keel . . . . . . . . . . . . . . . . . . Marchaj’s explanation of the winglet functioning . .

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10

FAN system flowchart . . . . . . . . . . . . . . . . . . . . . Discretization process for the definition of the computational domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orientation of the curve components . . . . . . . . . . . . . Orientation of the surface components . . . . . . . . . . . . Surface region connectivity definition . . . . . . . . . . . . . Parameters for background mesh . . . . . . . . . . . . . . . Distance to point, line and triangle sources . . . . . . . . . . Function governing the spacing around a source . . . . . . . 2D mesh generated using a point source . . . . . . . . . . . Convention for the free-stream velocity angles . . . . . . . .

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93 93 94 95 96 96 97 97 99

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14

Geometry of the boundaries . . . . . . . . . . . . . . . . Geometry of the boundaries — Zoom on the hull . . . . Geometry of the boundaries — Zoom on the fin keel . . . Surfaces of the yacht . . . . . . . . . . . . . . . . . . . . Hull parameters — xy-plane . . . . . . . . . . . . . . . . Hull parameters — xz-plane . . . . . . . . . . . . . . . . Hull parameters — yz-plane . . . . . . . . . . . . . . . . Yacht from behind at Θ = 0◦ . . . . . . . . . . . . . . . . Yacht from behind at Θ = 15◦ . . . . . . . . . . . . . . . Zoom of the last figure, but with the rotation of the axes Volume distribution with heel . . . . . . . . . . . . . . . Upper midsection curve at different radius factors . . . . Some parameters for the fin keel tip . . . . . . . . . . . . Background mesh . . . . . . . . . . . . . . . . . . . . . .

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102 103 104 106 107 108 109 110 111 111 112 113 114 115

List of Figures

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”Fluid Mechanics of Yacht Keels”

5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23 5.24

Point and line sources . . . . . . . . . . . . . . . . . . . . Triangle sources . . . . . . . . . . . . . . . . . . . . . . . . Free stream angles for the keel problem . . . . . . . . . . Overview of the curve components of the T-shaped keel . . Geometry of the boundaries — Zoom on the left winglet . Geometry of the boundaries — Zoom on the right winglet Surfaces of the T-keel tip . . . . . . . . . . . . . . . . . . . Line and triangle sources on the left winglet . . . . . . . . Generic surface triangle . . . . . . . . . . . . . . . . . . . . Closed surface for the momentum theorem . . . . . . . . .

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117 118 121 123 124 125 126 128 130 134

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19

Overall view of the surface mesh . . . . . . . . . . . . . . . Two views of the surface mesh of the yacht . . . . . . . . . Front view of the surface mesh of the yacht . . . . . . . . . Cuts of the tetrahedron domain . . . . . . . . . . . . . . . View of a T-shaped keel mesh . . . . . . . . . . . . . . . . Left view of a coarser mesh (Mesh factors=2) . . . . . . . Cuts normal to z . . . . . . . . . . . . . . . . . . . . . . . Cuts normal to y . . . . . . . . . . . . . . . . . . . . . . . Cuts normal to x . . . . . . . . . . . . . . . . . . . . . . . Convergence history: CFL and multigrid . . . . . . . . . . Convergence history on work units . . . . . . . . . . . . . Geometry of the test (AR=2) . . . . . . . . . . . . . . . . Lift coefficient curves . . . . . . . . . . . . . . . . . . . . . Drag coefficient curves . . . . . . . . . . . . . . . . . . . . Maximum L/D for the different aspect ratios . . . . . . . Angle of maximum L/D for the different aspect ratios . . . Effective aspect ratio . . . . . . . . . . . . . . . . . . . . . Lift and drag coefficients calculated with the two programs Influence of the artificial viscosity . . . . . . . . . . . . . .

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142 143 144 145 146 147 148 149 149 150 151 154 157 158 159 160 161 163 164

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7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

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Marco Trucchi - [email protected] 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19

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”Fluid Mechanics of Yacht Keels” . . . . . . . . . . .

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180 181 182 183 184 186 187 188 189 190 192

A.1 Selection of support points for curve interpolation . . . . . . . 198 A.2 Support cubic for distribution of points . . . . . . . . . . . . . 200 A.3 Cubic for the segments 4 . . . . . . . . . . . . . . . . . . . . . 201

List of Figures

xiii

List of Tables 1.1

General effects of some factors on the rating value . . . . . . . 22

2.1 2.2 2.3

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5.1 5.2

Connectivity array for background mesh . . . . . . . . . . . . 115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.1 6.2 6.3 6.4 6.5

Values of the parameters chosen for the hull . . . . . Values of the parameters chosen for the fin keel . . . Values of the parameters chosen for the winglets . . . Number of nodes and elements . . . . . . . . . . . . . Comparison between our results and the experimental

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A.1 Connectivity of surface regions . . . . . . . . . . . . . . . . . . 204 A.2 Connectivity of surface regions — T-shaped keel . . . . . . . . 206

xiv

Chapter 1 Displacement yachts In this chapter we will consider the most important features of the displacement yachts, starting with a brief functional description and then giving particular importance to their motion and mechanics. Eventually we will examine the leading features of the contemporary yacht measurement rules.

1.1

Functional description

From a first point of view, according to functions, we could consider a yacht divided in four parts: Hull It contains and supports the cargo and accommodation with safety, giving the necessary buoyancy, static stability and resistance to waves. Fin Keel or Centreboard Being the lower part of the hull, it produces most of the hydrodynamic action needed for sailing. It also provides accommodation and lever for the ballast. Rudder It controls the steering of the yacht, and contributes to increase the fin action of the hull and keel. Sails They produce the driving force needed for sailing, extracting energy from the wind. This general description includes many kinds of yacht, in which the different functions can be achieved in many ways, depending mostly upon the aim of the boat and the conditions in which it will operate. Anyway it is important to remember that the same word displacement means that almost all the 1

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buoyancy is given by the Archimedes’ force, and only a small part of it is due to hydrodynamics, like in boats with hydrofoils or planing hulls. For this study, it must be noted that the characteristics described above cannot always define exactly which part of the hull should be considered the fin keel: this is due to the fact that the whole hull affects more or less the hydrodynamic behavior of the yacht. A more precise physical definition of the different parts can be given easily when the keel is designed as an appendage, like in figure 1.1, showing a Star class. This is the usual solution adopted on smaller yachts, like dinghies. Instead in figure 1.2 it is hard to say where the keel ends; all the hull contributes a lot in giving the necessary forces. In our study we will concentrate on the former configuration, so it will be possible to consider the fin keel action alone. In any case, all the studies will be made on a geometry of a keel together with a hull, in order to represent better the submerged part of the boat.

1.2

Yacht mechanics

In this section we will try to explain how a yacht works, with the description of its kinematics and dynamics. Most of the figures refer to a simple dinghy configuration, while our work will concentrate on keels of bigger boats. This does not affect the validity of the reasoning, as “big yachts, including the most sophisticated 12-Metres, are nothing but big dinghies” [14]; of course the shapes are different, but not the involved phenomena.

1.2.1

Velocity triangle

The apparent wind speed V~A , felt by a yacht, is the resultant of the vectorial sum of the true wind speed V~T and the opposite of the yacht speed V~S . See figure 1.3. V~A = V~T − V~S (1.1) The angles shown are: γ β λ

True Wind – Course Angle Apparent Wind – Course Angle Leeway Angle

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Figure 1.1: Star class (F. Sweisguth, 1932)

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Figure 1.2: Carina (P.L. Rhodes, 1955)

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Figure 1.3: Velocity triangle The leeway angle is the one between the fore-and-aft axis and the course. Yachts have the possibility of sailing in almost every direction, with the exception of the windward courses that form very small angles with the true wind (see section 1.2.5). Here comes the necessity of tacking, that is holding many successive close hauled courses, to reach a point which is exactly into the wind: the direct route is in this case impossible. When sailing to windward, the most important speed is Vmg , the speed made good to windward, that is the component of the boat speed towards the wind: Vmg = VS cos γ . (1.2)

1.2.2

Acting forces

The forces acting on a yacht can be divided in six groups: 1. gravity force 2. hydrostatic force of buoyancy on the hull 3. aerodynamic forces on any emerged item, mostly on the sails Chapter 1. Displacement yachts

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4. hydrodynamic forces on any submerged item, specially made up by side forces on the keel and the rudder 5. wave drag on the hull, due to the presence of the free surface of separation between air and water 6. inertial forces. It must be said here that for side force we intend generally the component of the fluid-dynamic force perpendicular to the far field flow direction and to the vertical axis of the boat. For the similarity of the phenomena involved it is often called lift even if, unlike on aircraft, it is not needed to balance the gravity but other forces.

1.2.3

Steady motion

The first step to study the dynamics of a yacht is to consider a situation of steady motion. We try to describe the case in which the course is constant and there are no waves on the water letting the boat oscillate. See in figure 1.4 how it can be represented in a simplified way: we study the equilibrium considering all the fluid-dynamics forces perpendicular to the yacht vertical axis. This is a fairly good approximation in most cases. Here is the explanation of the symbols used: FR FV FH FH(lat) R FV W FS FS(lat) W ∆ MP MR MY r Θ CE

Driving Force Vertical Aerodynamic Force Heeling Force Horizontal Heeling Force Water Resistance Vertical Hydrodynamic Force Side Force Horizontal Side Force Weight of the boat Displacement of the boat Pitching Moments Rolling Moments Yawing Moments Righting Arm Heel Angle Centre of Effort

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Figure 1.4: Forces in steady-state sailing conditions

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Centre of Lateral Resistance

For the moments, the second subscript is ‘A’ for Aerodynamic and ‘H’ for Hydrodynamic. The steady motion can be achieved only with the equilibrium of all forces and moments; obviously we are not considering now any inertial phenomena. We can write six equations: three of them represent the force equilibrium along three different perpendicular axes, the other three the moment equilibrium around the same axes: 1. FR =R 2. FH(lat) = FS(lat) 3. FV + W = FV W + ∆ . Using simple trigonometry, we can write: FH =

FH(lat) cos Θ

FV = FH(lat) tan Θ

FS =

FS(lat) cos Θ

FV W = FS(lat) tan Θ

and 2a. FH = FS 3a. FV = FV W 3b. ∆ = W , that are useful in writing the moment equations: 4. F1 · b1 + MP A = ∆ · c1 + MP W 5. FH · b2 + MRA = ∆ · r + MRW 6. MY A = MY W . We have assumed that the projections on the horizontal plane of the total aerodynamic and of the total hydrodynamic forces act on the same line, thus do not produce moment on the vertical axis. Many interesting thoughts can be made on this simple situation, trying to trace a logical path. Suppose we want to sail at a certain speed in a Chapter 1. Displacement yachts

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certain direction (Course in figure 1.4C). A driving force FR is needed from the sails, to balance the unavoidable drag R due to that particular motion. But the sails, for the geometry of the problem, give also a heeling force FH , that can only be balanced by the keel. So there must be a leeway angle λ to create the necessary force FS on the immersed profile. The two forces FH and FS form a couple balanced, when the yacht heels, by the displacement ∆ together with the weight of the boat W . It must be noted that both the side force FS and the heeling force FH change the drag R initially assumed: an increase in side force always produces an increase in resistance due to induced drag, while an increase in heeling force causes the angle Θ to rise, fact that can in some cases reduce the wetted area and therefore the resistance. This is an example of how much difficult it is to make a good prediction of the yacht behavior, if we consider all the parameters that interfere. Going on with the thoughts, to increase the power to carry sails of a yacht, it is clear that the righting couple ∆ · r should rise, either with an increase of ∆ or of r. Putting more ballast in the lowest possible position is one of the easiest ways to do it. Besides, when possible, the crew go as far as they can on the windward side of the boat to increase r. Centre of effort and of lateral resistance Classically, the forces are shown as if applied in two particular points: the centre of effort for the sails and the centre of lateral resistance for the hull. These points are set in the centre of gravity of the lateral surface of the considered part. It is a common opinion that it is sufficient to have the modulus, direction and application point of the force vectors to know everything about the fluid-dynamic action, for example on a rigid lift device. This is absolutely wrong, in a 3D study. The confusion comes probably from the analogy with the case of a 2D profile, in which it is always possible to reduce a system of acting forces with non-zero resultant to a single force applied in a particular point (even though this is not the customary method). The argument is treated in section 5.3.1, in which it is explained exactly what we intend for CLR in our study.

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Sailing in upright position For a while let us limit our attention to a particular situation, in which the heeling angle is very small and can be neglected. The results of the reasoning affected by this hypothesis is valid in much more complicated circumstances. The case we are considering can be true with light wind, especially on dinghies, where the crew represent a considerable part of the total weight and can balance better the boat. In close-hauled conditions the heel angle is greater than in reaching, because the heeling force is bigger. It is usually better to have the yacht as upright as possible, so that the driving force is not reduced by the inclination of the total force. In figure 1.5A the total aerodynamic force FT is shown, applied to the centre of effort CE; it is split in two components in two different ways. The first is the same of figure 1.4, parallely and perpendicularly to the course direction. The second is parallely and perpendicularly to the apparent wind direction, in the aeronautical way: the total force is split in lift L and drag D. The angle of trim of the sail (or sheeting angle) δm can be adjusted to obtain the maximum driving force FR for that angle β 1 . Note also the aerodynamic drag angle εA , such as cot εA = L/D .

(1.3)

See now figure 1.5B: it shows the total hydrodynamic force RT , applied to the centre of lateral resistance CLR and split, as in figure 1.4, in resistance R and side force FS , respectively parallely and perpendicularly to the water flow direction (i.e. the course direction). Note the very important hydrodynamic drag angle εH , such as cot εH = FS /R . (1.4) In figure 1.5C both the aerodynamic and hydrodynamic forces are put together. As we have said before, to achieve steady motion the forces FT and RT must form an equilibrated system, that is they must be equal, opposite and acting in the same direction. From the geometry we can then obtain the important relation β = εA + εH . (1.5) Equation 1.5 is very important to estimate the minimum β angle attainable on a particular yacht, which is equal to the sum of the minimum drag angles. At the same speed of the boat saying that β is minimum means that γ is minimum and the course is the closest to the wind. 1

This does not mean that the sail L/D ratio is maximum, see section 1.2.6.

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Figure 1.5: Equilibrium forces in upright position

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1.2.4

”Fluid Mechanics of Yacht Keels”

Unsteady motion

The motion of a yacht can seldom be considered steady. In fact every action on the hull and on the sails provokes a variation in the motion, that changes towards a new condition of equilibrium. This can be reached with a continuous modification of the motion parameters, made possible by virtue of a feed-back between the parts. Let us try to give an example of feed-back, with the case of the acceleration after leaving the moorings, referring to figure 1.5A. At the beginning on the still boat only the aerodynamic forces act, and the true and apparent wind coincide. Therefore the initial acceleration is directed like FT , towards downwind. As soon as the boat moves2 , the water stream on the keel creates a new force that causes the boat to accelerate in a different direction. The sails give also a different force, as the apparent wind has changed. This complex succession of events brings to the condition of figure 1.5C, explained before. The cause of the unsteadiness can be: Maneuvering The crew can act on the rudder, the sails or sometimes the keel to direct the boat. Undulation of the water Waves can cause any kind of acceleration on a boat, creating forces on the hull. Rolling, pitching and yawing oscillations are common effects, negative for the sailing speed. Changeability of wind and water streams The variations can be in direction and in strength.

1.2.5

Points of sailing and performance

The direction of the yacht motion can form different angles γ with the direction of the wind. With the same true wind speed, depending on the course angle, the yacht can sail at various velocities and in different trim conditions. In figure 1.6 a polar diagram of a 12-Metre yacht is shown, for three different wind speeds. The maximum Vmg is found with the straight line parallel to 2

In this simple example we have neglected the unavoidable delay in the creation of the fluid-dynamic forces on a profile.

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Figure 1.6: All round performance

Chapter 1. Displacement yachts

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Figure 1.7: Olympic triangle the 90 degree course tangent to the curve (shown only for the 7 knot curve). The γ angle found does not coincide with the closest course angle, fact well known by sailors. In a similar way the maximum downwind speed Vdw can be found: it is not in correspondence of the running course. This explains why it is sometimes faster to make many reaching tacks instead of a single running tack, when sailing to downwind. Note also how the angle of maximum boat speed increases with wind speed, changing from about 75◦ at 7 and 12 knots to about 120◦ at 20 knots. In high winds, in fact, it is not possible to take advantage of all the available driving force of the sails when sailing close-hauled, as it would cause too much heel. Instead in reaching all the power of the sails can be used, since the FH /FR ratio is considerably smaller (see section 1.2.6). Sailing competitions It is shown in figure 1.7, as an example, the classic olympic triangle, on which many races are usually made. There are many other kinds of races, but our aim here is just to explain which can be the courses in a regatta. The classic run is A–B–C–A–B–A–B, but other possibilities are A–B–C–A–B–A–B–C–A or A–B–C–A–B–A–B–C–A–B. It must be noted that over the 60% of the time is spent in windward courses, that is the course A–B in the figure. Two reaching courses (B–C and C–A) and one running course (B–A) are the other possibilities. The yacht racing on this kind of runs will not, for example, be designed to make a good Chapter 1. Displacement yachts

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Figure 1.8: Sail polar diagram: close-hauled conditions close-reaching course, but most of the efforts will be made to improve the close-hauled situation, which is in fact often considered the most important.

1.2.6

Polar diagrams of sails

As we said before, the characteristic parameters of a yacht change a lot with the angle between the course and the wind direction. We will examine three particular cases, see figure 1.8, 1.9 and 1.10. In this three figures, the polar diagram of the sail is used to determine the condition of maximum driving force for different β angles. The parameter of the curve is the actual angle of attack α on the sail, such as α = β − λ − δm .

(1.6)

Varying α, the curve gives the lift and drag on a diagram, so that the vector from the origin to the point on the curve represents the total force on the sail. Figure 1.8 shows a yacht sailing close-hauled, with β = 35◦ and λ = 5◦ . The angle α corresponding to the maximum driving force is obtained in this situation with the dotted line, traced perpendicular to the course and tangent to the curve. The contact point corresponds to an angle of attack α of 27◦ . Chapter 1. Displacement yachts

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Figure 1.9: Sail polar diagram: close-reaching conditions In this manner it is possible to estimate with equation 1.6 the sheeting angle δm = 3◦ . In 1.9 the course is close-reaching, with β = 80◦ and λ = 2◦ . If the maximum driving force is determined like before, we find α = 28◦ , obtained with δm = 50◦ . Comparing this second situation with the first, we note that the angle α is very similar to before, but the driving force is now much bigger. In 1.10 reaching is the course, with β = 137◦ . The driving force is here bigger, but a big part of it is due to resistance. Looking at the shape of the polar and at the very big angle α = 67◦ we understand that the sail is in stall. We can now say that a good sail is very different from a good wing. A good wing is made by trying to have the maximum possible L/D ratio in every working situation; in a sail doing this is useful only for close-hauled courses. Indeed in reaching the drag contributes to the driving force at least as much as the lift, becoming even the only force in running. This is the reason why sails like spinnakers, with very large drag angles, are used while sailing to downwind. Note also the reduction of heeling force while the course angle increases, until in running it disappears. This means that the utility of the keel is reduced with big course angles. That is why in some boats it is made possible to reduce or even eliminate the wetted surface of the centreboard, by raising Chapter 1. Displacement yachts

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Figure 1.10: Sail polar diagram: reaching conditions it.

1.3

Yacht requirements

Even while designing a small part of a yacht, it is of vital importance to bear in mind the whole boat characteristics, necessary for the purpose we want to reach. While trying to improve one feature, there is an effective risk of having a significative loss in some others. Let us try to show which requisites can be considered of big importance on a generic displacement yacht: 1. Speed, in every course on which the boat will have to sail 2. Maneuverability 3. Seaworthiness, intending the ability to bear severe weather conditions with safety 4. Sea-keeping property, namely the ability to keep the motion as steady as possible in rough weather and sea conditions 5. Habitability, including also dryness of the underdeck. Chapter 1. Displacement yachts

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A list like the one shown above can apply to many craft, but the difference between one and another is the importance given to every single feature of the list, which depends on the aim of the boat. We can give many examples: on a cruiser it is probable that habitability, together with the other features that contribute to let the voyage as comfortable as possible, will be the first goal of the designer. Speed and maneuverability will still be considered important, of course, but their qualities will be affected by the primary requirements: a comfortable boat will probably be heavier and slower. On the other hand, a yacht designed to beat a speed record will have only the minimum necessary to live on it, and the environment for the crew will not be excellent. Compromise is the most important word in all cases: the skill of the designer is to combine the parameters in the best way. This is valid even if we consider just one feature at a time: see, for example, the first requirement mentioned in the list. As all the boats, with the exception of some special purpose ones, have to sail on different courses, optimizing the speed is not an easy task; every point of sailing would need a different configuration, while the actual possible changes are few and usually operable only on the sails. In a work like ours the requirements are more limited. In fact it is impossible to take directly into account features like seaworthiness or habitability while studying the fluid-dynamics of the submerged part of the hull. They will influence this study indirectly, appearing only rarely in our reasonings, but sometimes giving big help in the choices. Particular importance will now be given to the racing classes. The division in classes is fundamental, as otherwise it would be impossible to compare the performance of different boats. Some rules, applied to a yacht, give the class in which it can compete. The rules usually include formulae that take into account characteristic lengths, areas or volumes of the craft, also together with coefficients depending on the shape and materials of some parts of it.

1.3.1

Racing classes

From the constructional point of view, all racing yachts can be divided into three types: 1. Free classes 2. Development classes

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3. One-design classes They differ one from another in the degree of restriction of their designs by constructional and dimensional rules and regulations. Clearly the closest limits are imposed on the one-design classes, since their prime object is to test the competing helmsmen and crews when sailing nominally identical boats. In the remaining two types some freedom, to a greater or lesser extent, is allowed to the designers, builders, and even crews to improve the racing capabilities of their yachts. From the variations allowed in the rules, free and restricted classes contribute enormously toward technical progress, as has been demonstrated throughout the history of yachting. The more successful classes are promoted by the I.Y.R.U. as International Classes. They are fostered by special institutions whose aims are to guide their development, guard the measurement and construction rules, to organize international races, etc.

1.3.2

Yacht measurement rules

Yacht measurement rules began with no other object than of evaluating size in terms of tonnage or rating, treating the results obtained as an index of speed potentiality from which time allowance for racing might be calculated. An extension of this method was the establishment of classes, mentioned before, comprising yachts of equal measurement under the chosen system. In either case rules inevitably began to influence the shape of yachts. It must be sadly admitted that rules have sometimes inhibited the general improvement of the design and the origin of new characteristics, that were unappropriately not permitted. In fact most of the notable developments in yacht architecture were the product of faulty rules that for a moment in their history allowed to make profitable experiments in design. Despite this, today the rules are very tight and the designers have the freedom to produce only small variations, to make a boat of a certain class. For our study, it is important to know at least the basic structures of the measuring formulae3 ; it will be especially useful when talking about the keels with winglets. Furthermore we can start to understand how each part affects the global behavior by looking in the rule if it gives a penalty or a credit. 3

However, the full texts of the rules, as they appear in the respective handbooks, occupy about seventy printed pages.

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Some basic formulae of contemporary rules There are three elements of which measurement rules must be composed: 1. Length L 2. Sail area SA 3. Displacement ∆ The first two are speed-producing factors; they do not represent actual length or surface values but need little elaboration with coefficients that depend, for example, on the character of the ends of the hull or on the sail shape. The third is a speed-stopping factor, related to the wave-making resistance. We will now show, as examples, two formulae, for two different classes. International 5.5 Metre Class. The basic formula is: Ã √ √ ! L SA L + SA √ + .9 ≤ 5.5 m . 4 12 3 ∆

(1.7)

The measurements are in linear, square or cubic metres. The rule is thus composed of two additive portions, the first of which, with the heavily weighted √ 3 ∆ in the denominator, encourages a full-bodied boat with heavy displacement; the second, containing only length and sail area, encourages the lightest boat possible. In fact this formula was created to maintain a balance between the heavy yachts and the light. Without associated restrictions a rule like this would produce too great diversity between its three elements, so the flexibility of the formula is stiffened by ulterior restrictions: 1.7 m3 ≤ ∆ ≤ 2 m3 26.5 m2 ≤ SA ≤ 29 m2 1.9 m ≤ beam and, especially important for our future reasonings, draft ≤ 1.35 m

(1.8)

The maximum value of L is controlled automatically by the maximum value of ∆ and the minimum of SA . This rule offers the designer a simpler problem, at least mathematically, than many others in which the influence of the variables cannot be reduced Chapter 1. Displacement yachts

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to simple relationships. A reasonable line of development under this rule would seem to be the adoption of the maximum displacement (disposing it as economically as possible in terms of wetted surface) and to balance L and SA so that the resulting boat is not under-canvassed. For draft, the maximum allowed is a natural minimum owing to the needs of the keel, as we will investigate better in chapter 2, no compensation being offered under the rule for less than the maximum. Instead the choice of the beam is very difficult, because its effects on drag and stability are not so easy to interpret. International Cruiser-Racer Rule This rule is formulated so that the yachts are classified according to rating values of 7, 8, 9, 10.5, 12, 13.5 and 15 meters. It takes the following form: √ L + SA − F ± B ± D ± P + A ± H + C − K Pf , (1.9) Rating = 2 where the new quantities are freeboard (F ), the coefficients of beam (B), draft (D), displacement (P ), and other coefficients related to the form of the bow profile (A), the underwater profile (H), the abbreviated stern (C) and the type of the propeller fitted (Pf ). The iron keel credit (K) is given to the boats using iron instead of lead for the keel. √ Fundamentally, rating consists of L + SA . Each class has certain minimum and maximum values of the waterline length LWL and associated minimum displacement. The lower limit of the LWL gives the name to the class. The other factors are derived from a standard, but not necessarily ideal, hull, and penalties or bonuses are received by an actual yacht on the basis of the variation of its dimensions from the standard (this is the reason of the use of the ± symbol). For example, penalties will be given to the boats with draft bigger than the base dimension, credits to the ones with draft smaller than it. Table 1.1 shows the effect of some of the individual factors on the rating value; note that it is not only referred to this class, but it is general. Clearly, the factors giving credits or penalties do not depend on the particular class. It will be evident that the ability of a rule like this to control design depends on the nicest adjustment of the scales of bonuses and penalties. In a rule of similar structure to this, the formula of the Cruising Club of America, a case of failure in this adjustment became clear in 1955-56, when boats with Chapter 1. Displacement yachts

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Factor Beam Draft Displacement Sail area Freeboard Ballast ratio Propeller

”Fluid Mechanics of Yacht Keels”

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Small Penalty Credit Penalty Credit Penalty Credit Less credit

Table 1.1: General effects of some factors on the rating value extremely wide beam and shallow draft with centreboards could eliminate totally the competition of narrower, deep-keel yachts. Finally, we must admit that a big advantage of this type of rule is the ease with which the constants and scales may be modified.

Chapter 1. Displacement yachts

22

Chapter 2 Hulls and fin keels 2.1

Introduction

After having considered the general characteristics of a displacement yacht, we now go deeper inside the problems related to the main argument of our work: we examine the submerged part of the yacht, in particular the fin keel. The kinds of resistance on a hull are deeply investigated in this chapter; all the basic ideas behind the design of a keel are explained. The winged keels and their related phenomena are also presented.

2.1.1

Graphic representation

In order to represent a complex 3D shape like a hull it is not sufficient to show a orthographic projection. Figure 2.1 shows the usual way to do it. The hull is cut by four different kinds of plane. The curves found with the cuts parallel to the hull symmetry plane and to the water surface are called respectively buttocks and waterlines; the curves found with the cuts perpendicular to the preceding ones are called stations. The diagonals are found with cuts parallel to a longitudinal oblique plane. The curves are sometimes marked with numbers or letters so that it is easier to find the correspondences between the different draws. In the view with the stations, the projection from behind is put to the left of the symmetry plane, while the projection from the front is put to the right. Similarly, in the view from underneath the waterlines are shown above the symmetry plane, while the diagonals are shown under it. Sometimes the profiles of the fin keel, instead of being together with the other waterlines, are put in the lateral view capsized 23

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Figure 2.1: Hull representation on their axes. In our work we do not pretend to design an actual yacht. All the surfaces of our hull will be made on the base of not too complex mathematical functions, in order to be automatically generated by a computer program; thus the representation will not be as important as it would be in a real project.

2.1.2

Involved parameters

It is possible to describe the geometry of the submerged part of a boat with a series of few important values and features, that give the possibility of comparing the behavior of different yachts and also represent the basis with which every yacht designer must confront when considering a new shape. Hull The parameters are the total length, the waterline length LWL, the beam, the draft, the wetted area and the displacement ∆. Another variable, very useful for the wave drag, is the prismatic coefficient CP . It is defined as CP =

displaced volume LWL · maximum area of immersed sections

(2.1)

and its characteristics will be shown in section 2.2.1; basically, it measures the distribution of immersed volume along the length of the hull. If the prismatic Chapter 2. Hulls and fin keels

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coefficient is small, the hull is fine-ended; if it is big, the hull is full-ended. The fin keel is usually cut away from this calculation. The denominator is equivalent to the volume of a prism that has the cross-area of the greatest area of immersed section of the hull, and the same length as the LWL. Note that the section of maximum area does not necessarily coincide with the maximum waterline beam. The last parameter we consider is the displacement/length ratio, defined as ∆/L3 . Seen from a positive point of view, it represents the ability to carry loads and to have usable volume. From a negative point of view, it is a kind of measure of the wave drag. Usually in the related works, if the units of measure are omitted, ∆ is a force in tons and L is a length in hundreds of feet: in this case the frequent values are included between 150 ton/(0.01 ft)3 for a very light displacement boat and 450 ton/(0.01 ft)3 for a heavy one. Converting to the SI, we obtain that 1 ton/(0.01 ft)3 =0.351876 N/m3 , so the usual values are included approximately between 53 and 158 N/m3 . Other factors we do not consider directly, but that must be born in mind, are the stability or power to carry sails effectively, and the ratios between the sail area and some of the hull parameters (like the sail area/ displacement and sail area/ wetted area ratios). Fin keel Having reached a good level of development, the shapes of a sail or of a hull can usually vary only a bit, considering different yachts. This is not the case of the keel: if we examine the relative state of the art we can still notice a big variability. The involved parameters are about the same of those of an aircraft wing. We refer to figure 2.2, where the symbols mean cr ct dk Λ

Root chord Tip chord Keel draft Sweep Angle.

The principal features, in this case, are: 1. Lateral plan area 2. Aspect ratio, defined as AR = Chapter 2. Hulls and fin keels

dk 2 dk = , A cav

(2.2) 25

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Figure 2.2: Fin keel being cav the average chord. In our work, if not indicated differently, we will always consider this geometric aspect ratio. Usually it has small values, if compared to aircraft. 3. Taper ratio, defined as

ct , (2.3) cr that can sometimes, differently from classic wings, be greater than 1 (situation of inverse taper). TR =

4. Sweep angle, that is the angle between the quarter line and the perpendicular to the longitudinal axis of the hull. The sweep-back, as in figure 2.2, is positive, while the sweep-forward is negative. 5. Section profiles The effects of these parameters on the fluid-dynamic characteristics will be explained in section 2.6. Some keels can have a big torpedo-like tip (bulb). The reasons of its presence are due only to stability factors; we have already said that a ballast in needed for sailing, and sometimes the one given by a classic shape is not enough. In our work these configurations will never be considered. Note that other wing features are not usually present in a keel. A boat is most of the times designed to be symmetric, so it must not have better Chapter 2. Hulls and fin keels

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performance while making a left tack or, vice-versa, a right one. This brings to the fact that the keel itself must be symmetric in reference to the verticallongitudinal plane: it has no profile nor spar camber, no twist , no dihedral angle and its root chord is longitudinal. As stated in section 1.4, when we ask to a symmetric keel to produce a side force, it always needs a leeway angle, which creates some inconvenients for the motion of the hull, that does not go in the direction in which it would have the minimum drag but goes obliquely. Besides this, asymmetric wings have generally better characteristics. On some particular kinds of yacht, designed to go prevalently or exclusively on the same tack, asymmetric features can be found: it is for example the case of the boats made to beat a speed record. In this case it is possible that there is no leeway, even with side force. Obviously these craft can sail on the other tack too, but with a significant loss of performance. Some designers have tried to let the boat sail always without leeway, taking advantage from a device that gives the possibilty to rotate the fin keel on a vertical axis. The side force is obtained with about the same angle of attack, but the hull can go straight with less resistance. Other devices can change the shape of the profile, moving the leading and the trailing edge to create a camber. Usually the gain achieved with such kind of contrivances does not worth the design problems, and the risk of a worsening if not trimmed correctly. Keel winglets Keels can have a peculiarity: two winglets put as appendages near the tip. The reasons of the use of these T-shaped keels represent an important topic for our work, and will be explained in section 2.7. Here we only want to present their geometry. From now onwards we will use the term classic to refer to the keel without the winglets. If we know that the keels themselves can be totally different one from the other, then we can easily imagine that winglets too can be designed in many different ways. Basicly winglets can be attached in two zones: directly to the vertical part of the fin, in its deepest part, or on the ballast bulb, if present. As we said before, we will never consider the latter. The parameters of the winglets are very similar to the ones of a wing. For the reasoning made in the last paragraph, the two winglets must be symmetric as regards the hull symmetry plane, but each of them can have asymmetric characteristics; see figure 2.3. The parameters, all referred to Chapter 2. Hulls and fin keels

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Figure 2.3: Winged or T-shaped keel the winglets, are: 1. Plan area 2 2. Aspect ratio, AR = width area . Here too it is often a small number, but for reasons different to those of the vertical part.

3. Taper ratio, TR = and root chord.

cwt , cwr

where cwt and cwr are respectively the tip chord

4. Dihedral angle γ, between the quarter line of the unswept wing and the horizontal plane. In this work it is positive when downwards, as this is the most common case. 5. Sweep angle Λ, between the actual quarter line and the quarter line of the wing with dihedral angle only. Positive if backwards. 6. Section profiles 7. Angle between the root chord and the horizontal plane, positive if the leading edge is above the trailing edge. 8. Twist, that is the angle between the tip and the root chord, positive if the angle of attack of the tip section is bigger than the one of the root section. 9. Position of the connection with the vertical part; winglets are usually put in the back part of the tip, in a way such that the trailing edge of the wing root section coincides with the trailing edge of one of the lowest sections of the vertical part. Chapter 2. Hulls and fin keels

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10. Winglet root chord/keel tip chord ratio ( ccwrt ), that gives the fraction of the keel tip covered by the wing.

2.1.3

Theorem of Buckingham

In the fluid-dynamics problems it is of paramount importance to consider adimensional parameters. They can describe every phenomenon, without taking into account their dimensional quantities. The Π–products are adimensional monomials made by pure numbers and powers of quantities. Their main propriety is that they are independent: every other adimensional product is a function of them. The Π–Theorem made by Buckingham asserts that if n dimensional quantities take part in a phenomenon, the equation that governs it can be substituted by another equation with the (n − q) Π–products that they can form, where q is the number of involved basic quantities. In the case of the floating bodies, that we want to investigate, there are five dimensional quantities, and we can write: F (ρ, V, L, µ, g) = 0 .

Chapter 2. Hulls and fin keels

(2.4)

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Symbol Unit of measure

Meaning

ρ

[ Kg m3 ]

Density of the water

V

[m s]

Flow speed

L

m

µ

[ Kg ms ]

Viscosity coefficient

g

[m s2 ]

Acceleration of gravity

Reference length

Table 2.1: The symbols and their relative units of measure are shown in table 2.1. The basic quantities are three: length, time and mass. While the choice of some of the quantities is obvious, two things must be noted. The first is the presence of the acceleration of gravity g. Its importance comes from the fact that it appears in the formulae related to the behavior of the water waves. For example, the wave speed in deep waters VW is given by: s

VW =

gλ , 2π

(2.5)

where λ is the wave length. The second is the absence of the pressure. This is due to the incompressibility of the water, since phenomena involving the pressure are related to the fluid compressibility present at flow velocities comparable to the sound speed in the same fluid (i.e. at high Mach numbers). The dynamic pressure 1/2ρV 2 is often used to obtain the pressure dimensions. With the help of the theorem we find that the number of Π–products is 5 − 3 = 2. The classic choice is to take the Reynolds number and the Froude number. Reynolds number It is defined as Re =

Chapter 2. Hulls and fin keels

ρV L VL = , µ ν

(2.6)

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where ν is the kinematic viscosity, such as ν=

µ . ρ

Its unit of measure is [m2 /s]. L and V are respectively a characteristic length and velocity. Clearly, the factors that let the Reynolds number raise are the speed and the length, while the kinematic viscosity lets it lower. The Reynolds number roughly denotes the ratio between the inertia forces and the viscosity forces. A low value means that the viscous effects are not negligible, and indeed they have a big importance for the flow behavior. While it increases, the viscous effects lose importance, compared to the inertial ones. It is very important that the scale of the length is chosen correctly: a classic example of this is given by the flow inside and outside the boundary layer, explained in section 3.2. We say that two phenomena are in fluid-dynamic similarity when they have, besides a geometrical similarity, the same Reynolds numbers. This fact can be explained clearly with the Navier-Stokes equations (section 3.1). Let us see how much the Reynolds number can be in a problem like ours: a flow over a fin keel. The characteristic length is in this case the mean chord, the velocity is the free stream velocity. As an example, consider a mean chord of 1 m and a speed of 4 m/s in water at 15◦ C (ν = 1.141 10−6 m/s2 ). We find Re =

4×1 = 3.51 106 . −6 1.141 10

(2.7)

Here is a very important observation: despite the big difference between a yacht and an aircraft, both can have a very similar Reynolds number. The kinematic viscosity for the air at 15◦ C is 14.6 10−6 m/s2 ; an airplane having a mean chord of 2 m and flying at a speed of 25.6 m/s has exactly the same Reynolds of the above yacht. On aircraft many efforts have been made in research, while on yachts this has not happened yet. Fortunately, many results taken from aeronautical studies can be applied successfully to sailing, especially when the involved shapes are similar, like in keels or rudders. Froude number It is defined as

Chapter 2. Hulls and fin keels

V Fr = √ . gL

(2.8)

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Again L and V are a characteristic length and velocity. This number is strictly related to the wave drag, as we will see better in section 2.2.1. Basicly, it represent in some way the development of waves along the hull; it roughly represents the ratio between the inertia forces and the gravitational forces. At the moment we just want to see which can be its magnitude in our problem. Note that in practice,√in the related works, it is usually preferred to use the dimensional term V / L instead of the Froude number. Consider the same yacht of before, in the same conditions. Usually the reference length is here different, and coincides with the waterline length: suppose it is 8 m, so this hull and keel configuration can represent quite well the one studied later in this project. Speed is (like before) 4 m/s. We find Fr = √

4 = 0.452 . 9.81 × 8

(2.9)

There is no corrispondence with the aircraft in this case. The only vague similitude we can make is between the Froude and the Mach numbers, even if the waves that develop on a free surface are very different from the ones on a wing flying at velocities comparable with the sound speed.

2.2

Kinds of resistance

It is very useful to try to separate the contributions to the drag given by each different factor. It must be noted that it is not possible to find this division in the reality, because all the physical effects are strictly joint one to the other and, for example, an improvement in one kind of resistance can bring to a worsening in another. It is, anyway, the best method we can use. We will examine the following factors: • Wave drag • Friction drag • Induced drag or drag due to lift • Pressure drag • Drag due to heel

Chapter 2. Hulls and fin keels

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2.2.1

”Fluid Mechanics of Yacht Keels”

Wave drag

The wave drag represents, especially at higher speeds, the most important contribute to the total resistance. This kind of resistance is given by the creation of a system of waves by the hull. If the hull does not produce an evident wave system, a bigger part of the driving force is available to balance other forms of resistance, and the boat accelerates considerably. This is the case of the planing hulls, which is totally different from the displacement yachts we are considering. Qualitatively we can assess that the bigger the bow and stern waves, the higher the wave resistance. The dimension of these waves depend exclusively on the shape of the hull and on the speed. The bow wave can be seen as a rise of the water due to the rise of the pressure around the stagnation zone; the stern wave as a compensation of the ‘hole’ in the water left by the boat while moving. At low speed the bow and stern waves are independent and their length is considerably smaller than the waterline length. While the speed increases, the amplitude and the length of both the waves rise, and they start to interfere. The theory applied to the front wave only brings to the result that the wave drag is proportional to V 4 . The actual behavior is different, as there is a contribute due to the interference between the bow and the stern wave. The contribute is sinusoidal, as while increasing the speed the situations of constructive and destructive interference alternate; see figure 2.4, where the sinusoidal contribution has been magnified. Using the theory of the waves in deep waters, we find relations like the equation 2.5. But as the waves move at the same speed of the boat, we can say that the wave length is λ=

2πV 2 g

(2.10)

and, using equation 2.8, its relation to the Froude number is 2πV 2 λ = = 2πFr2 . L gL

(2.11)

Now we want to use a new parameter: the number of waves n present on the side of the hull. Clearly, it is n = Lλ , so from equation 2.11 we obtain the important 1 (2.12) Fr = √ 2πn Chapter 2. Hulls and fin keels

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Wave drag

Actual wave drag part prop. to V^4

Froude number

Figure 2.4: Theoretical wave drag (the scale of the sinusoid is exaggerated) and

s

gL . (2.13) 2πn The concept we introduced before is now evident: the Froude number depends tightly on the number of waves. If n decreases, Fr increases, i.e. the yacht has accelerated. For the displacement yachts the situation with one only wave between the bow and the stern represents a limit. From that velocity onwards, the drag has a dizzy increment. Equation 2.13, making the substitutions, becomes for n=1 s √ 9.81L V = = 1.25 L in SI units, (2.14) 2π1 which is √ a formula known by all the sailors. It is often written as V = 1.34 L, using the knots for the speed and the√feet for the length. That is the reason of the common use of the term V / L: one can directly compare it to the limit value (1.25 or 1.34, depending on the units of measure). The speed given by equation 2.14 is not unsurpassable, but, especially for heavy displacement hulls, at it the drag increment becomes so big that it is really difficult to go faster. Thus practically the maximum speed depends V =

Chapter 2. Hulls and fin keels

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Figure 2.5: Relative contributions to total resistance mostly on the waterline length: that is why it is so important in the class rules (see section 1.3.2). Figure 2.5 shows the total drag increment on yachts of different displacement/ length ratio, defined in section 2.1.2 and here measured in ton/ (0.01 ft)3 . The total drag includes kinds of resistance we will consider later, but for the moment it is only important to know that the wave drag is by far the biggest contribution at high speed. On the light displacement yachts the increment of resistance is less evident than on the heavy ones, for which there is a steep barrier. The number of waves on the hull, together with the Froude number, can therefore be taken as an indicator for the kind of motion (clearly, for a given craft). See table 2.2. The wave resistance depends on the leeway angle, too. The drag is clearly bigger if the hull moves laterally, not in the fore and aft direction. With a leeway angle, the front surface on which the water stagnates giving rise to the bow wave is less ‘hydrodynamic’. This contribution is less important than one can imagine, especially if compared to the other sorts of resistance due Chapter 2. Hulls and fin keels

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n

Fr

√ m V √ ] [ L s

V √ √ [ kn ] L ft

Kind of motion

5 3 2 1 0.75 0.15

0.17 0.25 0.30 0.40 0.46 1.03

0.56 0.72 0.88 1.25 1.44 3.23

0.60 0.77 0.94 1.34 1.55 3.46

slow medium fast fast forced planing

Table 2.2: to leeway, related to the output of side force. In particular, the side force on the keel causes a variation in the wave pattern, that can bring to a worsening of the total wave drag. Influence of prismatic coefficient The fullness or finess of hull ends relative to midship sections has a considerable effect upon the wave patterns generated due to hull motion, in particular on the position and the height of the bow and the stern crests, and also on the dynamic lift produced on the hull bottom. This, in turn, affects to a large extent the wave-making resistance. In order to picture the fullness of the ends relative to the largest section of the hull, the so called prismatic coefficient CP , already mentioned in section 2.1.2, is employed. Optimum prismatic coefficients √ range from 0.50 to 0.70, depending on the Froude number (or the V / L ratio): see figure 2.6. From that sketch it can be argued that √ conventional heavy displacement yachts, which sail at lower values of V / L for the reasons explained before, should have fineended hulls, while fast planing boats should be full-ended. A distinction can be made also regarding the weather conditions in which the boat is supposed to sail most of the time. A yacht designed for light winds should have CP in a range of 0.50–0.53; conversely, sailboats designed for strong winds should have a higher CP , even more than 0.65. This behavior is contrary to what might perhaps be expected, and was not generally recognized for a long time.

2.2.2

Friction drag

The phenomena related to the viscosity drag are so many and so complex that in this work they can only be treated briefly. Elementary concepts like Chapter 2. Hulls and fin keels

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Figure 2.6: Optimum prismatic coefficient laminar and turbulent boundary layer will be taken for granted. The friction drag in a boat is a considerable part of the total drag only at low speeds, without leeway and especially on light craft. The fact that that the contribution of viscous drag is smaller at high velocities is clearly visible in figure 2.7 and 2.8, both relative to the same boat. The reason of this is to be found in the fact that friction drag roughly increases proportionally to V 2 , while the wave drag to V 4 or even to higher powers of the velocity. The two figures refer to a light displacement yacht, thus the increment due to the wavemaking resistance is not exorbitant like it can be on other craft. Nevertheless we can see how the ratio between the friction and the total drag changes from 1 at very low speeds and becomes about 0.4 for a speed of 7 knots. If the boat had the possibility of planing, then the wave-making resistance would decrease at speeds higher than that and friction would become again a substantial part; but it is not the case we want to examine. If a angle of leeway is present, another big contribute to the total resistance is given by the induced drag, as we will see in section 2.2.3. Thus the influence of the friction can be even less than the one of figure 2.8.

Chapter 2. Hulls and fin keels

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Figure 2.7: Boat drag Skin friction coefficient The skin friction Df of a flat plate, or a surface with slight curvature, can be calculated according to the familiar formula 1 Df = ρV 2 Cf A , 2

(2.15)

where ρ is the density of the fluid, V is the velocity of flow, Cf is the skin friction coefficient for the appropriate Reynolds Number and A is the wetted area. As might be expected, the friction coefficient which enters into equation 2.15 is not constant but is largely controlled by the character of the flow in the boundary layer. Reynolds’ discovery and further contributions made by Rayleigh, Prandtl, Blasius and others, made it clear that the flow character depends upon the relative predominance of inertial and viscous forces, as represented by the value of Reynolds number (Re) (discussed in section 2.1.3), the inertial forces favouring turbulent flow (higher Re), while the viscous forces promote laminar flow (lower Re). The main lines in figure 2.9 are the ones marked with laminar flow, turbulent flow and transition. They come out from a big number of experiments and represent the relationship between the skin friction coefficients Cf and Reynolds Number for the flow over the two sides of a flat plate in the corresponding boundary layer conditions. For the two solid curves it would be Chapter 2. Hulls and fin keels

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Figure 2.8: Relative contributions to total resistance better to say fully laminar and turbulent flow, because they represent the Cf respectively in the case of laminar and turbulent flow over all the flat plate surface. Note that the transition cannot be defined as precisely as the other two situations, due to the dispersion of the data, so the dash-dotted curves represent the borders of the zone in which is likely that the transition occurs. In order to appreciate the practical significance of transition, it is instructive to consider a situation in which the boundary layer is laminar on the forward part of the body, turbulent on the after part, with the dividing transition point between the two and shifting with every change of Reynolds number. To give an example, we may find that at certain Reynolds number, in other words at a certain boat speed, the transition occurs somewhere along the hull, some distance from the bow. Experimental evidence enables us to assume that transition is likely to occur when the critical Reynolds number Recr of about 5 105 is reached. Thus, if the boat speed is V = 2 m/s and we neglect the pressure gradients, the distance L at which transition is expected to develop will be L=

Recr ν 5 105 × 1.141 10−6 = = 0.285 m . V 2

(2.16)

If the speed of the boat increases, the transition point will gradually be shifted towards the bow. This is anyway a pessimistic evaluation, as the critical Reynolds number can be even ten times greater, and a negative pressure gradient can help a lot in mantaining the flow laminar. The results shown are also useful in situations different from the ones of the flat plate. They are particularly good on surfaces with a slight curvature. Prediction of skin friction in those cases requires some empirical knowledge about factors such as surface roughness, pressure gradient, surface flexibility, Chapter 2. Hulls and fin keels

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Figure 2.9: Friction coefficient of a flat plate

Chapter 2. Hulls and fin keels

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Figure 2.10: Variation of Cf of a smooth flat plate with Re and position of mean transition point T.P. etc., which may delay or promote flow change from laminar to turbulent. For a smooth, flat plank the transition occurs in the range of Recr = 3 to 5 105 ; for hull or foil when flow is affected by favourable pressure gradient, this range is shifted towards higher values of Re. If the critical value of Reynolds number Recr is assumed to be 5.0 105 , then at Reynolds number of 15.0 105 the flat plate of length L would be expected to have laminar flow over the forward third of its length and turbulent flow over the remaining two thirds. The friction coefficient Cf could be estimated as sum of one-third of the laminar flow coefficient for the relevant Reynolds number of 5.0 105 and two-thirds of the turbulent flow value for Re = 1.5 106 . The set of curves in figure 2.10 show the variation of friction coefficient Cf of a smooth plate with Reynolds number and position of mean transition point behind the leading edge. It can be inferred from it that transition is of some importance in estimating friction drag since this rather evasive phenomenon of transition is largely responsible for uncertainties while translating model experiments into full-scale prediction. On the reduction of skin friction. It seems from what we have said that the best method to reduce the friction drag consists in trying to reduce Chapter 2. Hulls and fin keels

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the wetted area and to keep the flow laminar as much as possible. The reduction of wetted area has so much influence on the governability of the whole boat that it is usually very difficult to try to optimize it considering also its influence on the dynamic behavior of the craft. If we observe the development of the racing yachts in the last fifty years we can notice a big reduction in the value of the wetted area. But it is undeniable that on the older boats it was easier to maneuver and keeping the course. It is, as we have said many times, always a matter of compromise: nowadays the designers have the speed as first goal, rather than governability. A negative pressure gradient contributes to keep the flow laminar. Usually on the profiles it is likely that the boundary layer is laminar until the point of minimum pressure. But it is not always good to try to extend the laminar zone as much as possible, because other determining phenomena take part: they are briefly explained in section 2.2.4. Influence of roughness To preserve laminar flow and minimum frictional reistance it is particularly important for the leading edge to be smooth. Even slight roughnesses on the bow or leading edge of the centerboard or rudder may cause turbulent flow, while some roughness away from these edges is less damaging. Since, however, it is the forward portions of hull or centerboard that most frequently suffer damage, it is necessary to give special attention to them when overhauling the boat. The next point to consider is the influence of surface imperfections or roughness in the turbulent flow zone, normally created when sailing at average speeds. Although turbulent flow is characterized by velocity fluctuation in all directions inside the boundary layer, there always remains a sublayer at the very surface of the hull in the form of a thin laminar film. This covers slight roughness and makes the wetted area hydrodynamically smooth as long as the protuberances of the rough areas are sufficiently encased within the laminar film. The thickness of this sublayer decreases with the rise in the value of Re, and a speed is reached at which the grain of the roughness begins to emerge from the laminar film, with a consequent increase in resistance. The already considered figure 2.9 gives the curves, prepared by Schlichting, of the influence of various degrees of surface roughness on the skin friction coefficient Cf at different values of Re. It is now necessary to define the term “a hydrodynamically smooth surface” in relation to turbulent flow. Above Chapter 2. Hulls and fin keels

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Finished and polished surface 5 10−4 mm Smooth marine paint 0.05 mm Galvanized metal (average) 0.15 mm Ordinary wood 0.5 mm Barnacle growth (average) 5 mm Table 2.3: curve A are drawn curves a, b, c, d, e, f, and g. representing the coefficients of friction for different grain sizes (or degrees of roughness). Numbers beside the curves give the ratio L/k, i.e. the length of wetted area, L, to the height of the grains, k, which enables us to apply the result from this graphic to yachts of different lengths. For example, taking the International Canoe, and assuming L = 5.2 m, V = 5 m/s, the appropriate Reynolds number can be calculated: VL 5 × 5.2 Re = = = 2.28 106 . −6 ν 1.141 10 Now, taking the roughness of the immersed hull (or the grain size) as 0.5 mm, then L/k = 5.2/(.5 10−3 ) = 10400 ∼ = 104 . Thus from the graphic the skin friction coefficient, when Re = 2.28 107 and L/k = 104 (curve c), is Cf = 0.005 (point 3). Table 2.3 gives the degree of roughness k for different types of surface. In the example shown, the coefficient Cf is calculated for smooth bare wood. By covering it with marine paint, k will be reduced to 0.05 mm, and then, by a similar calculation, curve e will produce a coefficient of skin friction Cf = 0.003. Yet such a surface is still not hydrodynamically smooth, and the coefficient Cf could be brought down to the point 2 on curve A by polishing to reduce the roughness to the permissible level, to what we could call critic roughness kcr . This can be put in relation to the Reynolds number by L Re = 100 , (2.17) kcr found empirically from the graphic. In our case we find that k should not exceed 0.02 mm, if the surface is to be hydrndynamically smooth. Point 1 on the ‘transition curve’ below curve A indicates that laminar flow can occur on certain parts of the hull. Going back to equation 2.17, and remembering equation 2.6, we find that kcr = 100 Chapter 2. Hulls and fin keels

ν . V

(2.18) 43

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Basically then, the permissible roughness depends on the boat speed. As the speed increases, the required standard of smoothness becomes more stringent, especially for planing and skimming boats. For nonplaning boats the required smoothness is about 0.05 mm, and this is not very difficult to achieve using marine paints or varnishes.

Finally, there is another incredible way of reducing skin friction. The transition from laminar flow to a turbulent boundary layer, with its increase in resistance, is connected with the appearance of oscillations, that is instability of flow within the boundary layer. It is the rigid surface of the hull which promotes this. Max Kramer [12] developed a method for repressing this dynamic instability for greater Reynolds numbers by imitating the structure of a dolphin’s skin. The skin of a dolphin and of other fast swimming creatures acts as a damper to absorb oscillatory energy from the boundary layer and convert this energy into heat. The same mechanism on the synthetic skin was obtained with two rubber layers separated by stubs, with the remaining space filled with damping fluid. The results were really fascinating: in some cases the flow could be laminar on the 80 per cent of the surface, on a geometry that showed fully turbulent flow with a rigid skin. The friction drag could be reduced even of 60 per cent. Unfortunately since Kramer’s tests only a few experimenters have been able to measure a reduction in friction drag using flabby skins.

2.2.3

Induced drag

We have already said that the immersed part of the boat produces in most of the courses a side force, thus it can be considered very similar to a vertical half wing. Even if we do not consider the viscosity effects at all, on a finite span wing there is always a resistance due to the lift, usually called induced drag. As we will see, it would be better to call it “drag due to the aspect ratio”, as it does not exist on a inifinite span wing. This part is fundamental for the study we want to make, as this is the only type of drag that we can measure directly with the inviscid code we are going to use. It is now necessary to define two classic adimensional quantities: the lift

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coefficient CL and the drag coefficient CD : L CL = 1 2 ρV A 2 D CD = 1 2 . ρV A 2

(2.19) (2.20)

L, D and A are respectively the lift, drag and plan area of the wing; ρ and V the density and velocity of the flow at infinity. As usual, L and D are defined as the perpendicular and parallel to the flow components of the fluid-dynamic force. The Prandtl theory about the finite span wings gives a relation between the induced drag coefficient and the lift coefficient: CL2 . (2.21) e π AR In the formula e is the Ostwald coefficient (that depends on the wing shape) and AR is the aspect ratio. It is valid for wings of high aspect ratio and small sweep angle. Note that the aspect ratio defined in equation 2.2 is valid for a keel: for a wing instead of the keel draft dk we must put the total span length. We can try to explain the reasons of the induced drag existence. At the trailing edge of a finite span wing there is a vortex sheet, in which the whirls tend to roll up in two bigger tip vortexes that persist even far behind the wing. See figure 2.11 to understand the origin of this vortex layer. On the two surfaces of the wing there is a cross-flow, that is a flow with a component parallel to the greater wing dimension, due to the difference of pressure between the upper surface and the lower. The cross-flow is absent on the symmetry plane streamlines and becames stronger going towards the tips; its trasversal component is towards the tips on the lower surface (which is in overpressure conditions), and vice-versa on the upper surface (which is in depression conditions). At the trailing edge the upper and lower flows reunite, so a vorticity appears. The main effect of the vorticity is to change the local angle of attack of the flow on the sections of the wing. In figure 2.12 this can be clearly seen. The tip vortexes induce a downwash velocity, that reduce the effective angle of attack. Considering the downwash w small if compared to the velocity of the far field V0 , we can write: w , (2.22) αi ∼ = V0 CDi =

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Figure 2.11: Origin of the vortex wake

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Figure 2.12: Velocity field around a finite span wing where αi is the induced angle of attack. This means that in section 1 αi is less than in section 2, where the downwash is bigger. The induced angle of attack causes the induced drag: see figure 2.13. As we are not considering the viscosity, the lift on a foil is perpendicular to the flow direction. But the flow forms with the foil chord an effective angle of attack αeff = α − αi , (2.23) so the lift is rotated backwards of the angle αi , as regards the situation without induced velocity. Remembering that the decomposition of the fluiddynamic force in lift and drag depends on the far-field flow direction, we can see that the lift of a section has in reality a drag component. The Prandtl theory assesses that to obtain the minimum induced drag the downwash should be constant along the span width. Thus the induced angle of attack should be constant too. This situation is reached with an ellptic lift distribution, which can be achieved either with an elliptic planform or with a particular distribution of camber and twist along the span. The first goal of the Ostwald factor e in equation 2.21 is to correct the Chapter 2. Hulls and fin keels

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Figure 2.13: Velocity field around a finite span wing value of the induced drag depending on the lift distribution. If we do not consider the pressure drag (section 2.2.4), for the elliptic load e = 1, while in all the other cases it is less than 1. The sweep angle and the taper ratio also influence the induced drag. In all cases a sweep-back or a sweep-forward angle provoke an increase in it, and the same happens for a taper which lets the planform be different from an elliptic shape. On high aspect ratio wings these negative effects are considerable, while they lose importance with the decrease of aspect ratio. Anyway combining together the sweep angle and the taper ratio properly it is always possible to obtain a near-elliptical lift distribution, so that the induced drag is almost as low as the one of the unswept elliptic foil. Induced drag on a yacht Most of the reasonings made are valid for a yacht. In particular, it must be noted that the aspect ratio of a fin keel is usually low. This is due to the fact that a certain lateral area is needed to achieve enough side force at reasonable leeway angles, and to improve the course-keeping ability. But the draft has a Chapter 2. Hulls and fin keels

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limit, given either by the class rules (see section rulesformulae), or by other necessities such as the one of sailing in shallow waters. An exception is given by some yachts, designed for example to sail only in the open ocean, that can even have a draft of 8 meters, with aspect ratios of 10 or more. Fortunately the lift distribution of low aspect ratio wings tends always to be elliptical, whichever is the planform shape. So this one can be chosen thinking about other effects than the induced drag. The hull too produces a part of the side force needed. It behaves like a bad wing, with very low aspect ratio. Thus, even if the side force it produces is very small, its induced drag can be considerable. Equation 2.21 is valid only for high aspect ratios, thus it is not directly applicable to our problem. The behavior of induced drag is anyway similar, as it is proportional to the square of the lift coefficient. But instead of the geometric aspect ratio and the Ostwald factor it is better to use the effective aspect ratio AReff : CL2 CL2 CDi = = . (2.24) π AReff K π AR In equation 2.24 we have also introduced the efficiency factor K, that puts in relation the effective and the actual aspect ratio: AReff = K · AR .

(2.25)

Clearly, the higher the efficiency factor, the lower the induced drag. This is how we will consider the induced resistance in our work. The effective aspect ratio, or the efficiency factor, take into account not only the non-elliptic shape of the lift distribution and the error of a theory valid for high aspect ratio, but also the effect of the so-called mirror image. The fluid-dynamic behavior of the immersed part of a yacht, if the surface of the water is sufficiently flat, is like the one of a body composed by the part itself and its mirror image, made in reference to the water surface. The distance S shown in figure 2.14 can roughly represent the intensity of the induced drag, in the sense that the bigger it is, the smaller the induced drag is. With the mirror image the effective aspect ratio is, like for S, equal to the twice of the geometric (K=2); in the first definition of AR of equation 2.2, the numerator increases fourfold while the denominator twofold, being doubled both dk and A. Unfortunately the water surface is nothing like a flat plate. Thus the efficiency factor can be much less than two. Besides this, in our work we will Chapter 2. Hulls and fin keels

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Figure 2.14: Mirror image

Figure 2.15: Boundary layer and wake consider only the calculations on the keel, even if the hull is present. The perfect mirror image given by the computer code doubles the effective aspect ratio of all the immersed part, and not of the keel alone. So in our case too K will be smaller than the theoretical.

2.2.4

Pressure drag

If we do not consider the viscosity, we do not find any kind of resistance on a wing with no angle of attack. It is the D’Alembert paradox: in the part of the wing near the trailing edge there is a total retrieval of pressure, that counterbalances perfectly the overpressure near the leading edge. Theoretically this is valid for every kind of force on every object, even wings with angle of attack; in fact the lift can be present only with viscosity, because the vorticity that produces it comes from the detachment of the starting vortex, which occurs for the presence of the viscosity. Without lift the induced drag is not present, either. But this is not the reality. Due to viscosity effects, the boundary layer cannot be adherent to the surface until the trailing edge, and a separation occurs somewhere before it (figure 2.15). After the separation the pressure is constant and equal to the one of the point of separation, thus smaller than in the inviscid case. The balance is broken, and the wing feels a force of resistance: the pressure drag. Chapter 2. Hulls and fin keels

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The turbulent boundary layer behaves differently from the laminar for what concerns the separation. The layers near the surface of the former have more energy, so the flow can stay attached longer. We can then understand that the transition on a profile from laminar to turbulent boundary layer is necessary, otherwise the small friction drag achieved with the presence of a big laminar zone becomes unuseful for the rise in pressure drag due to a laminar, and therefore anticipated, separation. The pressure drag is strictly related to viscosity phenomena, but its quantity does not depend much on the Reynolds number. A big difference can be present only between the cases of laminar separation and of turbulent separation. The pressure drag depends also on the angle of attack; usually in the experimental data the pressure and the friction drag are put together to show their dependence on it. As we will see better, the increment of these two kinds of drag together is approximately proportional to the square of the lift coefficient, and thus roughly to the square of the angle of attack. This proportionality must not be confused with the one of the induced drag, even if its behavior is similar, as the latter has a totally different origin, as explained in section 2.2.3. Here comes clear the second purpose of the Ostwald coefficient e of equation 2.21: to consider the square dependence to the lift coefficient of the pressure drag. This effect reduces further e. Depending on the shape of the foil and its attitude (incidence relative to the flow), the skin friction and the pressure drag may sometimes change drastically and one of them may completely overshadow the other. The most evident example of this is the one of a flat plate put parallely and perpendicularly to the flow. In the first case the drag is almost entirely the result of skin friction and the wake is negligible; the friction coefficient Cf is in the order of 0.004–0.008. Whereas in the second there is entirely pressure drag distinguished by a conspicuous wake; the pressure drag coefficient CD is of the order of 1.9 (2D flow). It means that pressure drag can be 250-500 times greater than friction drag. But this is not the case of the keels we are going to study, as their usual shapes are such that the pressure drag is very small, at least at the leeway angles at which craft sail.

2.2.5

Drag due to heel

A last kind of resistance on a sailing yacht is the drag due to heel. We have said that an angle of heel is unavoidable on craft without moving ballast, as Chapter 2. Hulls and fin keels

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effect of the component of the aerodynamic force perpendicular to the course. The experimental results, such those shown in figure 2.17, demonstrate how big can be the rise in the hydrodynamic resistances caused by the heeling force. With a heel angle of 20◦ we can note an increment in resistance of about 4% in comparison to the upright situation. With 30◦ heel, the increment rises to about 15%. Excessive heeling affects not only speed but also the ability to sail to windward. The hydrodynamic drag angle εA can increase strongly with the heel angle. We will se that the heeling angle can also have positive effects on the yacht performance. Usually there is a critical angle, beyond which it is better to reduce the side force, for example taking down the sail area, in order to reduce the heeling force, even if it seems that a part of the driving force is wasted. In such cases the rise in driving force would be less than the one in heel resistance. The situation explained is tipical of the heavy displacement yachts. On the contrary, on lighter yachts a moderate amount of artificially induced heel, within the limits of 20–30◦ , can even improve performance. When sailing in a light wind it is actually advisable to shift the crew to leeward to the centerline to reduce the wetted area and the friction resistance.

2.3

Total resistance and polar diagrams

It is very useful to show the side force and the total hull resistance on the same diagram. In the resulting polar diagram, in the same way of those of the sails, the vector traced from the origin to the points of the curve represents the actual total force, if the units of measure are the same for the two axis. The total resistance coefficient CD is given by CD = CD0 (Re, Fr, Θ) + k(AR) CL2

(2.26)

where we can recognize all the single kinds of resistance described earlier. In fact the friction depends mostly on the Reynolds number, the wave drag on the Froude number, the heel drag on the angle of heel Θ; the second addendum is the induced drag. We must always remember that all the types of resistance are more or less connected, so it is not possible to separe totally the different contributes. Equation 2.26 explain us that the polar diagram, for a given boat and speed, with a certain angle of heel, is represented by a parabola. In figure 2.16 Chapter 2. Hulls and fin keels

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Figure 2.16: Hull polar diagram the forces on a 6-Metre hull at a speed V = 3 m/s with Θ = 20◦ are shown. Here the scales are different for the two axis. The tangent to the curve represents the maximum L/D ratio, thus the minimum hydrodynamic angle εH . It is showed here for the first time a common problem: the values of side force and resistance of the real sailing conditions are definitely smaller than the ones of minimum εH . Not all the resources of the hull are therefore exploited, because of the already mentioned necessity of having a certain lateral area with a draft limit, that brings to sail at low CL values. Figure 2.17 shows the effects of the heel angle on the polar of a yacht with LWL=7.2 m. The resistance with heel is greater than without it, as we said before, but it should be noted that the hull when heeled develops side force even when leeway is negative. When the angle of leeway is zero, but the heel is 20◦ , the side force is quite considerable, nearly 30the yacht. In such a case the heeled hull functions in effect as a hydrofoil, and somehow reduces the drift necessary to achieve sufficient lift. Up to certain limits, then, the negative effects of heeling which appear as an additional resistance, are compensated by the reduced leeway.

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Figure 2.17: Polar diagrams for different Θ angles

2.4

Towing tank testing

The goal of the towing tank tests is to evaluate the lift and drag forces on the actual hull, making measurements on a model. If the geometry is the same, as it is between model and reality, we would only need to evalutate the drag coefficient CD and the lift coefficient CL on the model. In fact these would be the same of the actual yacht, provided that the Reynolds and the Froude numbers coincide in the two cases (see section 3.1). Unfortunately it is not so easy to obtain on the model the same adimensional numbers of the reality. Suppose we make a model with dimensions equal to one tenth of the real hull. To have the same Reynolds, if the fluid has the same kinematic viscosity, the velocity on the model should be ten time greater.√But to have the same Froude, as g is constant, the velocity should be 1/ 10 times the real one, so about one third of it. William Froude tried to solve this problem for the drag, suggesting to keep on the model only the same Froude number to evaluate the wave drag, and then to estimate the friction with calculations taken from experimental data on a flat plate, on which no wave drag is present, at the same Reynolds number.

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The drag coefficient is thus seen as composed of two separate parts: CD (Fr, Re) = CD (Fr) + CD (Re) .

(2.27)

On the model with the same Froude number of the reality the total CD is measured and, subtracted from it the CD (Re), the CD (Fr) is found, which is the same for the actual hull. This, summed to the CD (Re) evaluated for the actual hull, gives the actual CD . The results of this method are quite good and generally this is the way in which the tests are made. Clearly, the errors in the evaluation of the resistance come from the fact that in reality it is not possible to separate totally the two kinds of resistance, because one influences the other. Also, being the model at a considerably lower Reynolds number, on it the laminar boundary layer is more extended; to avoid this, generators of turbulence must be put near the bow or leading edge, changing sometimes substantially the flow patterns. There are similar problems concerning the separation. To reduce the relative importance of these errors the only way is to use the biggest possible models.

2.5

Fin keel requirements

The most prominent feature of a sailing yacht hull is the keel, which can be considered one of the elements present with the greater variability of shape on the yacht. This means that designers still have not deeply understood its way of functioning. One of the reasons of this lack of development is to be found in the excessive trust that naval architects initially had in the direct application of the aerodynamic theories to the problems of sailing. Clearly there are a lot of analogies between the two subjects, but the importance is to understand where the theories inherited from the aeronautical field can be applied and where not. The functions of a keel are: 1. To generate a hydrodynamic side force at the minimum additional drag, at a reasonable leeway angle 2. To provide a location for ballast in low position below the waterline (large transverse stability) 3. To provide good damping against rolling oscillations 4. To secure good directional stability and balance (course keeping ability) Chapter 2. Hulls and fin keels

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5. To support the boat when on the ground. About the last function, only few things can be said. The importance is not to forget that most keels must resist to such situation, thus an appropriate structural calculus is to be made on them. The other functions are very important for our work, and we will examine them one by one.

2.5.1

Hydrodynamic side force and drag angle

The first characteristic of the list can be considered the most important. The side force does not differ from the lift of an aircraft wing, and it is needed to balance its opposite aerodynamic side (heeling) force produced by the sails. It is clear that, being the side force a purpose and the resistance an always unwanted phenomenon, the keel should be designed as to provide the maximum possible FS /R ratio, thus the minimum hydrodynamic drag angle εH . The kinds of resistance present on the keel alone are all the ones explained in section 2.2 but the wave drag, which is a phenomenon due to the presence of the free surface and thus not involving the keel. The action of the keel, instead, change sometimes considerably the flow patterns near the surface, so the wave drag on the hull depends on it. Remembering equation 1.5, εH is directly involved in the determination of the angle β between the course and the apparent wind. If, with a design improvement, we manage to lower the value of εH , then β decreases. If we imagine to start from the situation of figure 1.3 on page 5 and to improve our keel in the sense just noted, the lowering of β can be exploited in two ways. The first, of figure 2.18A, is to reduce the angle γ, sailing at the same speed VS . This means that the course sailed gets closer to the true wind direction. The second, of figure 2.18B, is to keep the same course (same angle γ), but to go faster. In both cases the speed made good to windward Vmg increases. It will be necessary to draw a new polar diagram relative to the all-round performance, like the one in figure 1.6 on page 13, to find the new maximum Vmg . The close-hauled conditions are important to evaluate the efficiency of the keel, but not the only ones. The need of side force will be less in the other courses, even maybe null in running. In the latter course it would be better to have a very small keel, just the minimum to obtain the features number 3 and 4. As in our work we consider only fixed lift devices, and not for example centerboards that can be raised, we must remember that the keel will have to work in very different conditions of production of lift. In practice the small Chapter 2. Hulls and fin keels

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Figure 2.18: Effect of decrease of β hydrodynamic angle εH will be necessary in all the range of lift coefficients, between zero and the maximum value. As the lift coefficient depends on the angle of attack, which depends on the leeway angle, the research for the small εH should be done in all the range of leeway angles, between zero and the maximum value. The reasonings just made are the basis from which we can start to try to answer to the difficult question: “How can we compare different keels?”. We want now to explain what we intend for “a reasonable leeway angle”. We have already seen in section 2.3 that it happens that a yacht never sails at the leeway angle at which the L/D ratio is maximum. In that discussion we were talking about total resistance, which is very different from the resistance of the keel alone we are considering now. Anyway, all that is valid for the whole hull, it is valid for the keel, in terms of requirements. It is important that the leeway of maximum L/D ratio is not too big, otherwise the yacht sails always in a condition far from the best. If the aim is to sail at small leeway angles, then the curve CL –angle of attack should be as steep as possible: in this way even with small leeway angles the hull can produce forces big enough to balance the heeling force. Big leeway angles do not create problems for the keel, that could bear angles of attack even double than the usual ones, but for the hull, for which they create bigger wave drag and especially bigger induced drag, as its aspect ratio is small. The first step to design a keel is to try to understand in which conditions Chapter 2. Hulls and fin keels

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the yacht will have to sail, and how much time compared to the total in each of them. For example, boats designed to race on the olympic triangle (figure 1.7) will be trimmed to have a high speed made good to windward, as they will have to sail most of the time tacking to windward. The choice of the keel will be guided by these conditions, which can be used to weight its performance. Concluding, all the things said in this section can be summarized in these assertions regarding the hydro-dynamic of the fin keel: • The curve L/D-angle of attack must be steep, reaching a high maximum value at a small angle of attack • The curve L-angle of attack must be steep It is hard to say what “steep” means in an absolute way. On the other hand, these assertions are perfect to compare two different keels, and represent our criterion of valutation.

2.5.2

Transverse stability

Being the lower part of the hull, the keel represents the best place in which the ballast can be put. The necessity of the ballast can be clearly seen in figure 1.4: the boat heels because of the heeling force on the sail, and the side force on the keel balances it, producing a rolling moment. This moment can be balanced with the couple weight–buoyancy: the keel thus represents a good righting lever. Once that the displacement has been decided, the designer should try to put the center of gravity of the boat in the lowest possible position. That is why keels are usually made of iron or lead. The large transverse stability that can be achieved with this contrivance brings to the possibility of carrying bigger sails at the same heel angle, or of sailing less heeled with the same sails. Both mean improvement in the speed: there is a close analogy with the case of two dinghies, one with light and the the other with heavy crew. With high winds the heavy crew can keep the boat more upright, and thus go faster because in that way they can exploit better the sail force. On the other hand, if they have the possibility, they can decide to use bigger sails, that causes them to navigate with the same heel angle of the light crew with smaller sails, but clearly faster.

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Figure 2.19: Rolling oscillations on hull

Figure 2.20: Velocity and force on a section of a rolling keel

2.5.3

Damping of rolling oscillations

The rolling oscillations induced by the action of wind and wave forces represent a negative effect on the yacht resistance. In figure 2.19 we can see the situation of a hull rolling at a angular velocity ω. The effect of rolling is that on the keel there is no more a flow of constant direction, but it depends on the distance r from the centre of the roation, more or less like on a propeller. The velocity w perpendicular to the keel is then: w(r) = ω · r

(2.28)

The velocity w(r) is summed to the velocity V0 of the flow, causing a variation of angle of attack, rising as we move away from the centre of rolling rotation. In figure 2.20 a section of a rolling keel without an initial angle of attack is shown. The angle of attack due to the roll produces a force F 1 , which is always opposite to the transversal motion and subtracts energy from it, 1

As it is a force of a single section, it would be better to say force per unit of length.

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damping the rolling oscillation. The sections far from the centre of rotation give the maximum contribute, for the fact that they have a bigger angle of attack of the flow (thus a bigger force), and they produce more damping moment for the longer arm. The only risk is that, if ω and r are big (i.e. fast oscillation and deep appendage), the tip can stall, reducing notably its action. In the last paragraph we have reasoned as if everything was steady, which is not true: being an oscillation, it is clear that the angular velocity ω is not constant. The drag and lift computation becomes very complex, for the presence of vortex shedding phenomena and delays in the production of forces. In particular it must be remembered that the lift attainable in an unsteady case can be far greater than in a steady one; besides, greater angles of attack can be born without stalling. Both characteristics are positive for damping. Anyway, what we have learned from the steady study can be considered as a first simplified approach to the design made thinking at roll damping. We can assess: • The keel should be fast in developing the fluid-dynamic force • The unsteady resultant force should be applied in the lowest possible position (i.e. low centre of lateral resistance) • The curve CL -angle of attack should be steep, to create big forces even for small velocities w due to rolling.

2.5.4

Course keeping ability

It is very difficult to design a keel so that it secures course keeping ability, that is the directional stability and balance, specially in rough seas. A design made without thinking about this would bring to an efficient boat in steady conditions, but not usable in reality. The most important parameter involved is the lateral area. Its value should be always greater than what the steady fluid-dynamic calculation alone would bring to. Attention should be paid not to create a boat so “stiff” on its course to become hard to keep while maneuvering. In a study like ours it will not be possible to consider directly the course keeping ability. The positive characteristics concerning it that our design will have will come only from the similarity with other actual boats. Chapter 2. Hulls and fin keels

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Figure 2.21: Section parameters Unfortunately, it is impossible to design the keel in such a way that all the abovementioned functions can be maximized to one’s full satisfaction. Th reason is that there is a conflicting interdependence of the keel design factors which determine the resistance, stability, damping and course keeping ability of the boat. Moreover, it is not possible to maximize two interdependent variables at the same time. The keel design depends on weighting the significance of the abovementioned functions. Usually, in the case of contemporary racing and even cruising boats, the greatest emphasis is put on the functions 1 and 2, at the expense of the others, especially the 4th. One of the reasons is that the first two functions directly determine the speed performance, while the others much more indirectly.

2.6

Fluid-dynamic effects of keel shape

The first step to make when designing something is to understand what are its main purposes. For a keel, it is what we have done many times in this work, particularly in section 2.5. The second step is to examine how the changes in the characteristics of the object affect its behavior. This is the aim of this section. Basically we will examine the effects of the parameters of section 2.1.2 on the fluid-dynamic behavior of the keel. 2D effects. First of all we want to consider the 2D effects, i.e. the effects due to the shape of the sections of the keel. Here is a list of the changeable parameters: for each of them there are specified the effects of an increase of the parameter itself. Everything is referred to figure 2.21. Thickness/chord ratio (t/c) The stall angle, the pressure drag and the volume for the ballast increase. Chapter 2. Hulls and fin keels

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Maximum thickness position (m) The profile drag without angle of attack and the L/D ratio decrease, the ballast is nearer to the stern. Radius of leading edge (r) The L/D ratio, the profile drag without angle of attack and the stall angle increase. All the effects mentioned in the list are more or less related to viscosity. The code we are going to use is inviscid, so no one of these will be shown by our calculations. The profiles we are going to use are of the NACA 00XY series. They are commonly used either for keels or for rudders, and a huge quantity of experimental data on them is available from aeronautical studies. This series includes symmetric profiles, with the position of the maximum thickness set at 30% of the chord; they are defined by analytical functions. The ‘XY’ digits represent the percentage thickness/ chord ratio of the profile, thus the NACA 0012 profile has t/c=12%. In section A.1 it is shown how these profiles have been inserted in the geometry of the keel. 3D effects. We have already seen in section 2.2.3 how the 3D parameters, especially the aspect ratio, act in changing the value of the induced drag, which is also strictly related to the lift (equation 2.21). That is why now we want to show their influence on the lift curve slope CL0 of equation 2.29. The aspect ratio is the most important parameter in this case too. Knowing the lift curve slope c0L∞ of the 2D profile, that is an infinite aspect ratio wing, it is possible to estimate CL0 with: CL0 =

c0L∞ dCL . = c0L∞ dα 1 + πAR

(2.29)

The 2D theory gives c0L∞ = 2π, and the experimental data confirm this [1]. Equation 2.29 is valid for big aspect ratios. When the aspect ratio is very small the lift-curve is not linear but concave. On the other hand with small angles of attack the curve can still be considered linear. This is exactly the case of the keel we consider, with small aspect ratio at small angles of attack: equation 2.29 is not valid, but the important fact is, anyway, that the smaller is the AR, the smaller is the lift curve slope. Figure 2.22 shows the experimental data on aerofoils of the same section (G¨ottingen 389) but of different AR.

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Figure 2.22: Experimental lift curves for different aspect ratios For small aspect ratios, the huge cross-flow on the greater part of the wing is the reason of the non-linearity of the lift curve and of the inaccuracy of the theory on induced drag; between the hypothesis of the Prandtl theory we find that the sections are considered like 2D profiles, i.e. each of them acts independently of its neighbouring sections except for the induced downwash: the strong cross-flow impedes this. In these cases characteristics like the tip shape can have big influence on the behavior of the wing. The maximum attainable CL is not much affected by the aspect ratio. Because of this, together with what we have just said about the lift curve slope, the stall angle increases with the lowering of the aspect ratio. This does not interest much the design of classic keels, because the usual leeway angles are small if compared to the stall angles. On winglets of T-shaped keels (section 2.7) and on rudders the situation is different, and this remark is very important. The effects on lift curve slope of sweep-back angle Λ and of taper ratio, together with the ones of aspect ratio, are shown in figure 2.23. It is very important to note that all the effects are not so considerable when the aspect ratio is small, as all the curves unite in one only. In all cases, a sweep angle gives a negative contribute, while the best taper ratio Chapter 2. Hulls and fin keels

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Figure 2.23: Effects of keel shape on lift curve slope changes depending on the value of the sweep angle. It must be remembered that a similar behavior has been noted for the induced drag in section 2.2.3. The situation at angles of attack near the stall is influenced a lot by the sweep angle and taper ratio. Experimental evidence indicates that a foil with sweep-back or more taper stalls first at the tip, while a foil with sweep-forward or less taper stalls first at the root. But, as we have already mentioned many times, keels usually work at very small incidence, far from stall, so these phenomena can be almost forgotten in the classic keel design. The only thing that must be noted is that the centre of lateral resistance, as it can be understood intuitively from the stall behavior, is moved away from the root by the presence of a sweep-back or a big taper, and viceversa. The last two paragraphs bring us to the fundamental conclusion that in practice on keels with low aspect ratio (less than about 2.5) the other shape parameters do not have much influence. The keel shape itself can be chosen to satisfy other requirements as well as the fluid-dynamic ones. This is necessary to understand the reasons of the choice of the keel shape on which we will make our calculations.

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2.7

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Winged keels

T-shaped keels, or winged keels, have already been described in section 2.1.2. Keel design has often been a matter of fashion in the past, and probably most of the reasons that let many designers choose the winglets were of this kind. To give an example, we can remember that T-shaped keels came to the fore after the victory of Australia II in the 1983 America’s Cup contest. Its keel design was considered revolutionary, having inverse taper and tip winglets. Until some years ago it was very common to see keels like the Australia II’s on racing or even on cruising yachts. Instead today it is far more probable to see tip bulbs with attached winglets or tip bulbs shaped as to obtain the same advantages given by the winglets. In reality both the basic ideas of Australia II’s keel were not new. About the winglets, Lilienthal had already applied them in his drawings of hangliders in 1894, taking the idea from the shape of the tip feathers of soaring birds. Lanchester, who was the first to understand the connection between the presence of the wing tip vortexes and the induced drag, patented in 1897 the ‘capping planes’, sort of profiled end-plates. Many years later, but long before 1983, winglets were successfully mounted on aeroplane wings and end-plates were used when testing 2D profiles in wind tunnels. Even if some attempts have been made to use end-plates in keel design, we will consider directly only the winglets, which have had much more success.

2.7.1

Effects of the winglets

In this part we will refer to concepts explained in section 2.5, trying to connect the effects of tip devices to the fin keel requirements. The main effects of the tip winglets are: 1. increase of the lift force 2. decrease of the induced drag 3. increase of the friction drag 4. lowering of the centre of gravity 5. lowering of the centre of lateral resistance 6. difference in the unsteady behavior Chapter 2. Hulls and fin keels

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Figure 2.24: Efficiency factor K for different tip shapes A first attempt to explain how these effects are created will be made in section 2.7.2; now we only want to discuss the improvements or worsenings that the winglets can bring. The effects 1, 2, 4, 5 are positive, the 3rd is negative, while the 6th can be both. The increase of the lift force means that the slope of the lift curve is more steep; it is like if the aspect ratio had increased, if we look at equation 2.29. In this case if we want to obtain the same lift force of the keel without tip devices we have to reduce the leeway, which is a positive fact mainly for the reduction of both the induced and the wave drag of the hull alone, as we have already mentioned. The second effect brings to an increase of the value of the coefficient K in equation 2.24 (page 49) and 2.25. Here again it is like if the aspect ratio had increased. We consider this effect more important than the first, as it can contribute directly to an effective reduction of the hydrodynamic drag angle. The performance of different shapes of winglets can be found with not Chapter 2. Hulls and fin keels

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Figure 2.25: Comparison between winged keels and straight keels easy theoretical aerodynamic calculations. This kind of analysis, together with some experiments, has been made by Cone [6]. Some of his results are shown in figure 2.24. Note that the closed tips of the figure are not solid so the free-stream air may pass through. The increment of effective aspect ratio can be huge in some cases, like for the forms G, H, L and N. The forms M and N let us understand that it is important that the dimension perpendicular to the wing plan is big. The forms that are more similar to the ones attainable with the addition of winglets on a keel are the B and the G. Garrett [8] makes an important observation about this apparent improvement. He assesses that it would always be better, when possible, to use a straight keel instead of a winged one, even when not considering viscosity. The two kinds are compared in figure 2.25. On the x-axis there is the ratio between the winglet span b and the total draft d, while on the y-axis there is the ratio between the effective and the real draft. The effective draft comes directly from the effective aspect ratio already defined. The solid line represents the behavior of the winged keel; as just stated, it behaves like if the aspect ratio, thus the draft, had increased, so the curve rises. The dashdotted line represents the behavior of a classic (straight) keel originated from Chapter 2. Hulls and fin keels

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the winged one by rotating the winglets downwards; in this case the effective draft is equal to the real, given by the rotation by dclassic = d + b/2. As it can be seen, the obtained classic keel is always better than the corresponding winged one. The last result is also strictly related to the third effect. Clearly the addition of the winglets increases the amount of wetted surface, increasing the friction drag. Besides, due to the junctures between the vertical part and the tip devices, the drag has an ulterior rise for interference. If compared to the T-shaped one, the so called ‘originated classic keel’ mentioned before, besides having bigger reduction of induced drag and no problem of interference, has also only about half of the increase of wetted surface. Here we come to the point: why are winged keels so widespread, if we have seen that the classic ones are better? The answer is the same that brings to the design of low aspect ratio keels, even if they are worse: the draft has a limit given, as we have already said, by the formulae of the racing rules or by the fact that the yacht should also be able to sail in shallow waters (see section 2.2.3). There are no limits of other kind; we could think about structural limits, but for the usual aspect ratios they are by far not reached, neither for breaking or for too much flexibility, thus they do not intervene. Putting the winglets is therefore an easy way to obtain a virtual increase in the aspect ratio without increasing the draft. Or, if a dihedral angle is present, to obtain a big virtual increase in the aspect ratio with a small increase of the real one, and thus of the draft. Considering the first three effects together, it appears that to obtain an effective fluid-dynamic improvement the design of the winglets must be accurate. Starting from a classic configuration and afterwards putting the winglets, an increase in the steepness of the lift curve is very probable, while an effective reduction in the total drag, or better a rise in the L/D ratio, is not sure and sometimes hard to achieve. Even if we cannot note a fluid-dynamic improvement on the winged keel alone, the effects 4 and 5 can justify the success of this kind of design. For the fact that a considerable part of the volume is now concentrated near the tip, and being the winglets of the same material of the rest of the keel (usually steel or lead), the centre of gravity is notably lowered. The power to carry sail, or transverse stability, increases, and there can be an effective gain in the performance. From a first point of view the lowering of the centre of lateral resistance could seem to be negative. The arm of the couple formed by the side force of the fin keel and the heel force of the sail rises, so the Chapter 2. Hulls and fin keels

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heel rises too. Besides it is likely that the lift distribution varies more from the optimum elliptic and the induced drag becomes bigger. This is a crucial point and will be examined better in section 2.8; summarizing, the loss in induced drag due to the effect we are considering is defeated by the gain in wave drag on the hull, which is deeply related to the keel lift and the distance of its point of application from the free surface (free surface effect). For what concerns the unsteady behavior, the question is still open. None of the authors of the bibliography gives much information about this topic, which can represent a challenge for the people who will make other projects starting from what we have done in this. In particular, it could be interesting to calculate how the winglets affect the speed in creating the fluid-dynamic forces, both side force and drag, when the angle of attack changes or with rolling and yawing oscillations, and how these effects change the course keeping ability. The intuition can bring to think that pitching oscillations and heaving cause drag on the winglets, due to stall or vortex shedding. In fact to avoid the stall usually winglets have low aspect ratio, so that the stalling angle of attack on them is greater. Curiously Garrett [8] mentions some studies that assess that a negative effect of this kind is present only when the waves come aft, while there is a driving force when the waves come fore. Considering the global performance of a yacht, the general belief of nowadays about the winglets is that they can actually help while sailing to windward and with large and moderate angles of leeway, thus at higher speeds. When sailing to downwind or at low speeds the increment of wetted area is the main effect and the friction worsens the performance2 . Looking again at equation 2.24 we can understand the reason for which at large angles of leeway the improvement due to the winglets is greater. If we start from a classic keel with low AR, we see that any tip device which increases the effective AR has a large effect on the induced drag: the more heavily the keels are loaded (higher CL thus higher angle of leeway), the greater the gain. The yacht Australia II had many other design improvements besides the winglets, which therefore cannot be considered the main factor that brought to its victory of the America’s Cup, especially because the races were made with low wind speed. Finally, it must be noted that a winged keel gives problems when putting the boat on the ground and generally while sailing, because it tends to hook fishing nets, algas and all sorts of floating debris. Besides it is more vulnerable 2

Remember that friction is the major part of drag at low speed; see for example figure 2.8 on page 39).

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Figure 2.26: End-plates on a wing and its costs of manufacturing are higher than a conventional one, thus the cruising in shallow water becomes risky. All these negative effects can be bearable on a racing yacht, but maybe not on a cruising yacht.

2.7.2

Fluid-dynamic phenomena on tip devices

Behind the functioning of the tip devices there are two main ideas. The first is applicable to the end-plates. By using them the development of a trailing vortex round the keel tip could be prevented (figure 2.26). In that ideal case the flow would tend to become like the two dimensional, with constant lift distribution as if the aspect ratio was infinite. Consequently, the kinetic energy deposited in the trailing vortex would no longer be dissipated and, as a result, the induced drag might be eliminated. In practice the end-plate can be considered as a barrier that makes it difficult for the flow to get around the keel tips; the bigger the barrier, the more effective it is. Some experiments proved that the end-plates increased the effective aspect ratio, but were very sensible to the misalignments with the local flow direction and increased a lot the wetted area and thus the friction drag. The sensibility to the misalignments brings to the possibility of designing the end-plate for one particular sailing condition only. The second idea is applicable to the winglets. Their aim is to extract some of the large amount of kinetic energy contained within the trailing vortex system, both reducing it and improving the keel performance. We have already explained the origin of the trailing edge vorticity in section Chapter 2. Hulls and fin keels

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Figure 2.27: Flow over a wing tip 2.2.3, giving pariticular importance to the cross-flow on the wing surfaces (figure 2.11 on page 46). The cross-flow is maximum near the tip: this is clear in figure 2.27, taken from a photograph and lately elaborated. All the streamlines that pass from the leeward to the windward side, to use yachting terms, tend to roll up in a single trailing vortex. On a keel yacht, the cross-flow near the tip and the trailing vortex is particularly strong for two reasons: first, the small aspect ratio, that causes a remarkable induced drag; second, the particular lift distribution. A hint to explain the second cause can be found in the Prandtl theory, which assesses that the intensity of the vorticity at the trailing edge of a wing at a particular distance from the wing axis is proportional to the variation of section lift at the same distance along the span. To avoid the free surface effect (section 2.8) the keel is usually made to have the load distributed in the lowest possible position; this causes a strong variation of section lift near the tip, because the lift itself has to pass from high values to zero in a short distance. The vorticity of the vortex layer of figure 2.11 is thus concentrated near the tip not only after the vortexes have rolled up, but also just after the trailing edge, even if in a lesser way. When the angle of leeway is large, the local flow on the suction side of a yacht keel can form an angle αi with the far-field flow direction of even 40◦ . The situation is shown in figure 2.28. Clearly, the higher the angle αi , the stronger the tip vortex. Marchaj [14] tries to explain how a winglet works in figure 2.29. The vectors of forces and velocities in figure 2.29A refer to a small section (in terms of width) of the winglet at a certain spanwise location from the windward side of the keel. The velocity Vi induced by the tip vortex when vectorially added to the flow velocity VS ahead of the keel (which is approximately equal to the boat velocity) gives the magnitude of the resultant local velocity VR , as well as its direction indicated in figure 2.28. The direction of VR is such that, in effect, the winglet operates at an angle of incidence αi which will be different for the section close to the keel winglet junction compared with the Chapter 2. Hulls and fin keels

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Figure 2.28: Cross-flow on a keel section close to the tip of the winglet. The resultant force FT produced on the winglet gives a small net driving component Dr in the direction of the boat motion. Its magnitude will, of course, depend on the winglet efficiency i.e. its L/D ratio. Observe in figure 2.29B that a similar result in terms of driving component Dr is obtained on the winglet mounted on the opposite, leeward, side of the keel. Marchaj considers the principle on which the winglets’ functioning is based totally different from the one of the end-plates. In our opinion, before considering the results of our work, we do not agree totally with him. We think that the winglets, just like the end-plates, can in reality act as ‘fences’ for the flow, preventing it from going around the tip. Clearly the efficiency of the winglets in this sense is much lower than the one of the end-plates, especially for the lower keel tip fraction covered. Finally, to mention another idea on this subject, Garrett [8] says that the function of the winglets is to move the trailing vortex from the keel tip to the winglet tips. Being increased in this virtual way the distance of the true vortex core from its mirror image, the effective aspect ratio is increased and the induced drag reduced.

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Figure 2.29: Marchaj’s explanation of the winglet functioning

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2.8

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Design hints for a generic keel

In this section we want to summarize some of the already mentioned ideas related to the keel design; some new and very important reasonings will also be made. We have seen in section 2.5 which are the requirements for a fin keel. In particular, on page 55 the required functions are listed: we will refer again to the items of that list. When designing a keel, the first step is to consider on which courses the yacht will have to sail most of the time. If racing on a Olympic triangle, for example, the boat will be in close-hauled conditions for the 60% of the time. When sailing to windward, like in that case, all the functions of the list are important. On the other hand when sailing to downward only the functions 3 and 4 are useful. Thus a first deduction is that a bigger and deeper keel is needed when the yacht has to sail mostly in close-hauled conditions. On the other courses, in fact, a big keel has negative effects. The wind speed has also a big importance. In higher winds the yacht sails at higher angles of leeway and thus higher CL . This affects the optimization of the L/D ratio: its maximum can be at higher CL to obtain while sailing the same efficiency of a low wind case; it must be born in mind that the yacht usually sails at CL s much smaller than the value of maximum efficiency, as a big lateral area is needed to secure a good course keeping ability. But the main point is here another: the relative importance of the keel resistance on the total resistance decreases as the wind increases. The reason of this is clear: with high winds the boat speed is high, thus the wave drag represents the bigger part of the total resistance and all the kinds of resistance of the keel have a secondary importance. The choice will be therefore made on a bigger keel, provided that the yacht will also have to sail in close-hauled conditions. The classic aerodynamic theory would bring the design to a fin keel with constant profile, elliptic planform, maximum possible aspect ratio and no sweep angle. The elliptic planform is difficult to achieve on a ballast keel; it is more convenient to use a trapezoidal shape with a taper ratio of 0.33, which gives a good approximation of elliptic load. But with low aspect ratio the load tends always to be elliptical, whatever is the planform. This brings to try to put the ballast in a low position, losing just a little in the hydrodynamic performance. To obtain this, it is possible to use many contrivances: the inverse taper ratio, the thickening of the profile towards the tip, the tip bulbs and the tip winglets. All these expedients but the last would cause Chapter 2. Hulls and fin keels

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a sure worsening in the aerodynamic behavior of a wing, but for the keel it depends on other factors, mainly the presence of the free surface. The free surface effect is the variation in the wave drag on the hull due to the side force on the keel. The pressure distribution on the sides of the hull and the keel is asymmetrical, and this causes disturbance on the water surface. The waves created by the fin keel interfere with the waves due to displacement. The magnitude of this effect depends on the distance between the zone on which the side force is created and the free surface. Clearly the greater the distance, the lesser the effect; the centre of lateral resistance CLR, being a kind of measure of it, becomes very important. If the yacht heels, the distance between the CLR and the surface decreases substantially; this means that if we find a way to lower the former, the gain can be considerable. A sweep-back angle causes a good lowering of the CLR: this explains why it is so common on the keels. Keeping the taper ratio constant, for every heel angle there is an optimum sweep-angle that minimizes the drag. While with a sweep angle the wave drag decreases, in fact, it must be remembered that the induced drag increases, especially if the aspect ratio is not low. A large number of experiments has demonstrated that without heel the best sweep angle is zero. This means that the free surface effect depends a lot on the heel angle, and thus on boats like dinghies other factors should be considered more important. Another fact brings to try to lower the CLR: at higher speeds it is very difficult to obtain side force near the keel root, as the two sides can communicate and every pressure difference can be cancelled by the waves. One of the best ways to increase the side force near the keel tip is to use the inverse taper, even if it worsens the induced drag. The inverse taper can bring to a remarkable improvement of the surface effects; being these reduced, a sweep angle is no more needed, indeed it is better to design a straight keel to avoid an ulterior rise in the induced drag. The only reason of the presence of a sweep-back on this kind of keels is that it avoids the hooking of fishing nets, algas and all sorts of floating items, problem already seen as present on winged keels. Inverse tapered keels are not so common as one could imagine knowing their positive effects, maybe because they look weird. One of the most important reasons of Australia II’s success was probably this feature, more than the presence of the winglets.

The conclusions of all these reasonings, especially according to Garrett Chapter 2. Hulls and fin keels

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in [8], are: • The boats that sail without heel angle (like the not ballasted dinghies) should have high aspect ratio fin keels with no sweep angle, and should be tapered so to achieve a near-elliptic lift distribution. • When the fin keel causes a change in the free surface wave pattern, it should be with inverse taper and no sweep. If the last condition is a problem, a small sweep angle is acceptable to let the leading edge be less inclined towards the bow. The taper should be higher if the aspect ratio is small. • If the draft must be limited, it can be possible to improve the windward course performance by adding tip winglets. The design must be accurate, as a decrease in the wave and induced drag is coupled to an increase in the friction drag. • When sailing to downwind the main factor is the wetted surface, which should be the minimum possible, indipendently on the shape and compatibly with the loss in course keeping ability. • If the inverse taper is not desirable, a sweep-back angle can be used, larger if the aspect ratio is small.

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Chapter 3 Equations of motion and their numerical solution In our work all the experimental studies have been made using numerical codes. The codes belong to the FAN system described in chapter 4 and have been kindly supplied by Dr. J. Peir´o. They have been used without any modification; nevertheless it was very important to know their functioning principles in order to understand them better: the parameters that the programs need as input are many and complex, and if inserted incorrectly can bring to a waste of efforts or even to wrong results. The aim of this chapter is to present the equations that govern the fluiddynamic phenomena involved on a yacht hull and the way they can be solved numerically. We will start from the complete equations, which would describe accurately the problem, and then we will apply the simplifications needed to let the computation possible with the available means.

3.1

Navier Stokes equations for incompressible flows

The equations of interest are in our case the 3D incompressible Navier Stokes equations. They can be obtained imposing the mass and the momentum conservation on an infinitesimal volume of the fluid. We obtain: ∇0 · V~ 0 = 0

77

(3.1)

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

∂ V~ 0 ρ  0 + (V~ 0 · ∇0 )V~ 0  = −∇0 p0 + µ∇02 V~ 0 + F~ 0 , ∂t

(3.2)

where V~ 0 is the velocity vector, ρ the density, p0 the pressure, µ the dynamic viscosity and F~ 0 a generic other force (in our case, the gravity force). The prime symbol (0 ) has been and will be used to denote that the quantity or the differential operator is dimensional; this is necessary not to create confusion with the corresponding non dimensional quantity or operator. Making numerical calculations it is far more convenient to use the non dimensional form of the equations, which we will refer to if not indicated differently: ∇ · V~ ∂ V~ + (V~ · ∇)V~ ∂t

= 0 = −∇ψ +

(3.3) 1 2~ ∇V , Re

(3.4)

where Re is the Reynolds number and the pressure ψ is the static pressure p with the hydrostatic component extracted: ψ = p + zFr−2 ,

(3.5)

being Fr the Froude number. The hydrostatic term comes from the consideration of the gravity force, that is commonly neglected in aeronautical studies. All lengths and velocities have been adimensionalized by a characteristic length L and a velocity Vref ; the pressure is adimensionalized by 2 ρVref . Equation 3.3 is usually called the continuity equation and represents the mass conservation, while equation 3.4 is obtained from the momentum conservation, it is vectorial and represents 3 scalar relations, one for every direction in space. Together they form a system of 4 equations with 4 unknowns: these are the pressure p = p(x, y, z, t) and the 3 components of the velocity vector V~ = V~ (x, y, z, t), that are u(x, y, z, t), v(x, y, z, t) and w(x, y, z, t), respectively parallel to the x, y and z axis. The first member of equation 3.4 represents the inertia forces, being in fact the Lagrangian derivative of the velocity, thus the substantial acceleration: ∂ V~ ∂ V~ ∂ V~ ∂ V~ ∂ V~ dV~ = +u +v +w = + (V~ · ∇)V~ . dt ∂t ∂x ∂y ∂z ∂t Chapter 3. Equations of motion and their numerical solution

(3.6)

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The second member of the same equation represents the pressure forces with the first addendum and the viscous forces with the second. It is also possible to impose the energy conservation, finding another equation. But in this incompressible case the energy equation is unuseful to describe the motion, being the mentioned equations sufficient to solve the system. It would be necessary in problems of heat transfer; but even in that case its only interference with the motion would be made through the variation of the kinematic viscosity ν (and thus on Re) due to the variation of temperature, and not directly on the variables. From the equations it is clear what we have said about towing tank testing: provided that the geometries are similar, two different flows are in fluiddynamic similarity if their Reynolds and Froude numbers are the same. In fact the equations, being nondimensional, become exactly the same.

3.1.1

Boundary conditions.

In a problem there are three kinds of boundary condition. The first is the hull surface, which behaves like a wall on which all the components of the velocity are null: this means that u = v = w = 0 on it. The second is given bt the far-field surfaces, which represents the flow at an infinite distance from the boat; on them the velocity must be set equal to the free stream velocity V~∞ . The third is the most difficult to represent: the free surface, thus the surface of separation between air and water. When the effects of surface tension and viscosity are neglected, its boundary condition consist of two equations. The first, the dynamic condition, states simply that the pressure acting on the free surface is constant, or equal to atmospheric as is for normal boat waves. The second, the kinematic condition, states that the free surface is a material surface, or in other words, once a fluid particle is on the free surface, it forever remains on it. Following [7], if z = β(x, y, t) is the free surface location, this second condition becomes: dβ ∂β ∂β ∂β =w= +u +v . dt ∂t ∂x ∂y

(3.7)

Equation 3.7 only permits solutions where β is single valued and thus does not permit bow wave breaking phenomena, not easy to formulate numerically. The difficulties encountered while adding the free surface discretization in a pre-existing code are many: being, clearly, the form of the surface unknown Chapter 3. Equations of motion and their numerical solution

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at the beginning of the computation, the mesh must have the possibility of moving and stretching towards the final and actual position of the surface.

3.2

Restrictions and approximations

Computational Fluid Dynamics (CFD) can today give good results about flow behavior with the solution of the Navier Stokes equations. The study we want to make has however a very complex geometry, thus the solution of the complete equations would be impossible with the computers made available to us for our purpose. Besides, we have to be satisfied with the code we can use, which is however already powerful; we have not the possibility of modifying the source code, so the only possible changes for us are the variations of the input parameters commonly requested to the program user. The important fact is to comprehend which data obtained from the code we can use should be considered reliable and which not. To do this, we consider the limitations one at a time. 1. The code can only compute the steady state solution. This means that the first term of equation 3.4, namely the temporal derivative, is 0. This brings to a heavy simplification in the solving algorithm, but limits our attention excluding some topics that could be very interesting, like for example the unsteady flow over a winged keel. 2. The free surface cannot be represented like we have explained before, but only as a wall or a symmetry plane, with no undulation. This is far from the reality of the problem, as the waves that are created near the hull in every sailing situation cause the wave drag, which is fundamental for the total drag calculation. Besides, they influence in some way the flow patterns all over the hull, in particular near the surface. The configuration we are going to study is composed by a hull and a fin keel, and we will examine all the results about it. In spite of this, we already know that the data about the hull will be affected by considerable errors, and thus they will not be reliable. On the other hand the calculations on the keel will be far more precise, thank to its bigger distance from the surface, even if always affected by the same kind of errors, so difficult to estimate. The keel tip, that we are going to study in particular, will be even less affected.

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As we said before, the introduction of the free surface in a pre-existing code is really problematic: this can represent a great challenge for the students who are going to continue the study we have begun. The fact that the upper surface is flat, together with the incompressibility of the water, causes the gravity term (zFr−2 in the ψ definition) to be uninfluential on the flow motion. The pressure gradient due to hydrostatic forces is constant, so the hydrostatic pressure depends only on the value of the z coordinate. Neglecting that term we will not find any buoyancy force; anyway this could be found with easy calculations on the geometry and is unuseful for our purposes. What is more important is that for the rest the calculations are not affected by the absence of that term. This is very important for our work as the code we use has no gravity term, as it was designed mainly for aeronautical applications. 3. While the FAN system can solve the steady incompressible Navier Stokes equations, we will only have the possibility of computing an inviscid flow. This is due to the fact that the geometry we consider is complex, and a study which includes viscosity needs much more nodal points than an inviscid one. It is important to understand why this happens. From now onwards the characteristic length for the equations, and thus for the Reynolds number, will be the average keel chord, as it was already in section 2.1.3. The Reynolds numbers of common flows that interest the aeronautical field are very high; on the boats, even if the speeds are lower, for the different characteristics of the fluid the Reynolds numbers are similar: in equation 2.7 on page 31 we found a value of 3.51 106 in a probable sailing situation. We have already mentioned that the Reynolds number represents fairly the ratio between the inertia and the viscous forces. This reasoning derives directly from the equations of motion, taking the characteristic length and velocity as typical quantities. With the high Reynolds numbers of the situations we are considering it is easy to understand that the viscous term can be neglected. Equation 3.4, considering equation 3.6 and without the viscous and the gravity terms becomes: dV~ = −∇p . dt

(3.8)

Equation 3.8 corresponds in the dimensional form to the famous Euler equaChapter 3. Equations of motion and their numerical solution

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tion:

dV~ 0 = −∇0 p0 . (3.9) dt0 It is valid both for compressible and incompressible flows: the difference between the two cases stands in fact in the continuity equation. With the Euler equation it is no more possible to provide the no-slip wall conditions, which are in fact a consequence of the action of viscosity. Thus the first boundary condition considered before becomes a condition of impermeability: the component of the velocity normal to the wall must be zero. If ~n is the versor normal to the wall pointing inside the body, the condition can be written as V~ · ~n = 0. It is then clear that the Euler equation cannot represent properly the flow in proximity of the walls. There the viscous and the inertia forces are comparable; the region in which this happens is the boundary layer. Inside the boundary layer the velocity gradient is high and Euler is not valid; outside it Euler is valid, and the boundary condition of impermeability apply not directly on the wall, but on the limit outside the boundary layer1 . Making reasonings on the orders of magnitude of the different terms of the dimensional equation 3.2, we find, using the characteristic quantities: ρ

V∞02 L V∞0 viscous force ≈ µ . δ inertia force ≈ ρ

The only new quantity is the boundary layer thickness δ. This was the great intuition of Prandtl, who used in the boundary layer L as length scale for the inertia forces and δ as length scale for the viscous forces. As the two kinds of forces are comparable in the boundary layer, we can write: ρ

V∞02 V0 ≈µ ∞ L δ

and then, using the definition of Reynolds number, 1 δ ≈√ . L Re

(3.10)

1

In reality the correct condition includes a flow not tangent to the boundary layer, but that leaves it.

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This result is of paramount importance: if the Reynolds number is of the order of 106 , then the δ is about 1/1000 of L. The boundary layer is therefore very thin, it is just a skin that surrounds the bodies. Another important result derives from the Prandtl intuition: the pressure is constant on a direction transversal to the boundary layer. We can now understand why it is impossible with our means to compute a viscid flow. The boundary layer, having strong gradients inside, needs many nodal points in CFD to be represented correctly. But being very thin for the high Reynolds number, it would need decidedly a lesser scale of distribution of nodal points in comparison with the zone outside it. The geometry we consider is complex, thus even without the boundary layer we require more than 35 000 points and 190 000 tetrahedra to solve the Euler flow: the program uses about 40 Megabytes of hard disk memory to store all the information related to one kind of geometry, and this is almost all the disk space we have available. The computation with also the viscosity, and thus the boundary layer, is neither to be proposed; with some efforts we could obtain a solution for a Reynolds number of about 1 000, which however would be totally unrealistic and therefore unuseful. Anyway the results that we can find solving the inviscid equations can be good. The fact that the boundary layer is very thin means that the geometric error we make is negligible. But the most important thing is that the pressure values we find on our walls, which is in reality, for what we said, the pressure just outside the boundary layer, is equal to the actual pressure on the walls, for the constancy of the pressure on the transversal direction. Finally, the fact that the boundary layer can be laminar or turbulent does not affect our calculations in any way.

3.2.1

Validity of the results

We want to summarize in this section what we have said about the equations we will use, and to discuss what kind of results can be found solving them. In our work we are going to solve with the computer the non dimensional steady incompressible Euler equations (

∇ · V~ (V~ · ∇)V~

= 0 = −∇p ,

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with the boundary conditions (

V~ · ~n = 0 V~ = V~∞

at wall at infinity

The free surface is represented as a flat wall, thus the wave drag cannot be computed. The viscosity is not considered, therefore no friction drag can be found; the same happens for the pressure drag, that is dependent on separation, again due to viscosity effects. What we can find with sufficient accuracy is the pressure distribution on the surfaces. Even neglecting the separation is not so serious as separation usually occurs, on wing-like devices, near the trailing edge. Integrating the pressures we can compute the lift (or side force) and the induced drag, besides the moments of the forces acting on the keel. Lift calculations are usually performed in this way, so we can be optimist on their accuracy, once we know that the solver works properly. On the other hand, we have less information on the accuracy attainable for the induced drag computation, so we will have to be careful and try to test in some way those results: to find out if they are more or less correct represents one of our main purposes, so many reasonings on this subject will be made during the whole study.

3.3

Numerical solution of steady incompressible flows

We want to explain briefly in this section the numerical method for the solution of the three dimensional Navier Stokes equations of incompressible flow employed in the FAN system, the set of computer programs presented in chapter 4. The method is based upon the use of finite elements, with unstructured tetrahedral grids [17].

3.3.1

Mesh generation

The spatial domain is subdivided into an unstructured assembly of tetrahedral elements. The discretization is achieved, for domains of arbitrary geometrical complexity, by employing a grid generator which uses a form of the advancing front concept in which nodes and elements are simultaneously created. This process begins with the use of the advancing front method Chapter 3. Equations of motion and their numerical solution

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to triangulate the computational boundaries and then the three dimensional form of the method is used to create the tetrahedral elements within the computational domain. User specified grid parameter distribution functions control the shape and size of the generated elements.

3.3.2

Flow solution

Many efforts in the aeronautical field have been made to find algorithms for the solution of the compressible flow equations on complex shape configurations, so the unstructured grid methods have been extensively developed and have now reached a high level of sophistication, both in terms of grid generation procedures and flow solver accuracy. On the other hand, a comparatively small amount of effort has been devoted to the development of methods for the solution of the incompressible flow equations, especially with general tetrahedral grids, presumably because of the stability difficulties associated with central difference type schemes for this class of problems. The main difference between the incompressible and the compressible flow modeling is in the continuity equation. We have seen that equation 3.1 represents the continuity equation for the incompressible case. With the compressibility we would find, in the dimensional form: ∂ρ + ∇0 · (ρV~ 0 ) = 0 , 0 ∂t

(3.11)

which forms with the other equation a set of hyperbolic equations, much more easier to solve than the incompressible set. The idea of the solver is to compute the steady state solution of the incompressible flow equations applying techniques which were developed originally for the solution of the compressible Navier Stokes equations. To achieve this, the artificial compressibility concept of Chorin, conceived in 1967, has been employed. Basicly his idea consists in modifying the continuity equation, here in the non dimensional form, in this way: ∂p + c2 (∇ · V~ ) = 0 . ∂t

(3.12)

The parameter c is not physical and should be optimized to let the study converge quickly. Equation 3.12, together with equation 3.4, do not represent the incompressible situation until the steady state is reached: when this Chapter 3. Equations of motion and their numerical solution

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happens the temporal derivatives become 0 and the equations become correct. Therefore it is a time dependent method, in which the transient evolution does not represent the reality. The name “artificial compressibility” comes from the analogy that may be drawn between the above equation and the compressible flow equations of a fluid whose equation of state is given by p = c2 ρ .

(3.13)

Thus ρ is an artificial density and c may be referred to as an artificial sound speed. Writing the equation as 1 ∂p + ∇ · V~ = 0 c2 ∂t

(3.14)

it can be guessed that it should be c À 1, so it is like solving ∇ · V~ with small perturbations. But it is always necessary to make some attempts to set it correctly. A large value of it leads presumably to a more accurate solution, but needs very small time steps to ensure stability; viceversa for a small value. In this implementation the choice of c is made with c2 = min(csta, csr · |V~ |2 ) ,

(3.15)

where csta and csr are parameters given by the user. The flow variables are normalized so that the modulus of the free stream velocity is unity. The finite element method (FEM) is employed. The numerical solution is obtained by applying a Galerkin method in space to the modified equations leading to a coupled set of ordinary differential equations in time. After having created the tetrahedral mesh, all the nodes (vertices) and elements are numbered. If we call ~ = [p, u, v, w]T U the solution vector of the equations, than basically the standard finite ele~ ∗ , which is ment approach seeks a piecewise linear approximate solution U constructed in the form ~∗ = U ~ J (t)NJ (x, y, z) U

J = 1, 2, . . . , Mnode

(3.16)

where Mnode denotes the number of nodes in the grid, NJ (x, y, z) is the piecewise linear finite element function associated with node J of the grid ~ J (t) is the value of the approximate solution at node J at time t. and U The nodal values are determined from a standard Galerkin method, which Chapter 3. Equations of motion and their numerical solution

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is based upon a weak formulation of the problem, with the shape function themselves used as the weighting functions. We will not go deeper inside this problem. It is possible to demonstrate that the application of the Galerkin method to the inviscid form of the equations leads to a scheme which is central difference in character. Thus an artificial viscosity, which is designed to maintain the second order accuracy of the method, is added to stabilize the solution. The artificial viscosity is of high order and can be controlled by a user-specified constant. The steady state solution of the equation set found with the Galerkin method is achieved by advancing the system using an explicit multi-stage time marching scheme. It is possible to choose the number of stages in the Runge-Kutta time integration, to a maximum of 5. The actual time step ∆t is limited by the Courant number (CFL), given by the user. This basically states that the fastest waves in the system may not be allowed to propagate farther than the smallest mesh spacing over the course of a time step. The 1D linear equation of convection can help us understand the utility of the Courant number: ∂u ∂u +a =0 (3.17) ∂t ∂x it is a ∆t CFL = . (3.18) ∆x In the equations a is the constant wave propagation speed and ∆x is the mesh spacing. Given a certain CFL, the time step is defined as ∆t ≤

CFL ∆x . a

(3.19)

This means that the time step must be smaller if the mesh is finer, or the wave velocity is higher; the code we use has the possibility of choose a different time step for each node, according to the local satisfaction of the stability criterion (local time stepping). The CFL depends on the time marching scheme. We will use the classic 4 stage Runge Kutta scheme, for which the theory for the √ 1D linear equation gives CFL = 2 2 ∼ = 2.8. In the code, to use a condition similar to that of equation 3.19, ∆x must be replaced by a value representing the spacing of all the elements adjacent to the considered node and a by an expression representing the local wave velocity, so the situation is much more complicated. This is the reason why effectively the maximum usable CFL Chapter 3. Equations of motion and their numerical solution

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number is smaller than the theoretical mentioned. The allowable Courant number may be made larger by smoothing the residuals at each stage. Each residual is replaced by an average of itself and the neighboring residuals. An effective method to accelerate convergence is the multigrid method. The basic idea of the multigrid method is to use the residuals computed on a given grid to drive a time marching scheme on a coarse grid. The corrections to the unknowns computed on the coarser grid are then added to the fine grid solution. The time marching scheme on the coarse grid may itself be accelerated by the use of an even coarser grid and, in this way, the multigrid concept can be extended to incorporate any number of coarse grids. The only restriction is that the coarsest mesh must still represent the geometry in an adequate way. Advancing the solution on the coarse grid is computationally relatively inexpensive, because of the reduced number of unknowns and the larger allowable time step size. With this code the user, with the same grid generator, creates first an unstructured fine grid, then a sequence of nonnested coarse grids redefining the grid control functions. An approach with a fully unstructured multigrid algorithm like this needs appropriate procedures to transfer information between the different meshes. Finally, it must be said that all the reasonings made in this section remain valid for the incompressible Euler equations, i.e. the above equations without the viscous term. In fact in reality all the calculations we are going to make are inviscid.

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Chapter 4 The FAN system The FAN system is a set of computer programs for the numerical simulation of the incompressible three dimensional Navier Stokes equations using unstructured tetrahedral meshes. It was made in 1995 by the Computational Dynamics Research Ltd., at the Innovation Centre of the University College of Swansea. The techniques implemented, based on finite elements, can be applied to a variety of geometric configurations and design conditions, and have been already shown in section 3.3. In this section we will describe how the various modules of the FAN system function, the input data files they need and the output data files they generate; most of the information is taken from the system manual [5].

4.1

System modules and data files

The system is composed of four main modules, that are four files executable on the UNIX operating system. They were all written in standard Fortran 77. The modules are: • ST a generator of surface triangulations • VT a three dimensional mesh generator • PP a flow solver pre-processor • FS a multigrid incompressible Navier Stokes flow solver. The modules are run independently, with the only interaction between the different modules occurring via the files that they use for input and output. 89

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Figure 4.1: FAN system flowchart

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The FAN system employs a fixed convention for naming the data files that the different modules use. To each flow simulation is assigned a problem name, in these examples fan, and a three character extension is added to the problem name, e.g. fan.dat, to indicate the type of data being used. The links between the main modules and the data file names are illustrated schematically in the flowchart included in figure 4.1. We will now describe only the files that will be useful for our thesis.

4.1.1

User generated files

• fan.dat contains the definition of the curve components, surface components, curve segments and surface regions required for the geometrical description of the boundary of the computational domain to be discretized. • fan.bac contains the definition of the background mesh and the source location which provides the spatial distribution of the mesh parameters. • fan.bco assigns flags associated to different boundary condition types for the curves and surfaces which form the boundary of the computational domain and which will be passed to the flow solver. • fan.nam is the Fortran Namelist used to specify the flow conditions and coefficients (if they differ from the built-in defaults) required by the flow solver. All these files are written in ASCII format and are very important for our study. A big part of it, in fact, will be related to the creation of these four user-generated files; therefore they will be described in dedicated sections. Refer again to figure 4.1 to see to which modules they form input.

4.1.2

System generated files

• fan.fro stores the information about the triangulation of the surface in the three dimensional space and the parametric space. • fan.gri holds the description of the generated tetrahedral mesh • fan.un1 stores the nodal values of the primitive flow variables (pressure and velocity) Chapter 4. The FAN system

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• fan.rsd contains the history of the convergence of the L-2 norm of the residuals of the flow variables. • fan.rot contains the information needed when periodic boundary conditions are imposed (thus not in our case) • fan.sol contains all the information required by the flow solver. • fan.RST, fan.RVT, fan.unk are auxiliary file for restarting the modules, when the calculations are split in different sessions. The first four files will be fundamental for the calculations we will make; the others will be unuseful for that purpose, but necessary to run the modules.

4.1.3

Using several meshes for multigrid

If the computation of the steady state flow solution is to be accelerated by the use of multigrid, a set of tetrahedral meshes is required. Each mesh is obtained by suitably modifying the spatial distribution of mesh parameters. Although the description of the geometry of the computational domain is the same for all the meshes, the user is asked to produce one geometry definition file (.dat) for each mesh, giving to each file a different name. The modules ST and VT create then respectively a set of .fro and a set of .gri files, that form after the pre-processing of module PP a single .sol file to be used by the flow solver FS. Therefore the .sol file contains all the information related to all the meshes.

4.2

Geometry data file (fan.dat)

This file contains the geometrical description of the boundary of the three dimensional computational domain. The discretization process is shown in figure 4.2. The first step is to define the boundary edges, by means of curve components and curve segments. The edges define on the surface components, described below, the geometry of the surface regions, which constitute the boundary faces. These must form a closed surface that contains all the 3D domain. Note that in this file, like in all the other input files, the data must be inserted following a particular syntax, explained on the manual [5]. For sake of brevity the syntax will not be reported here, but clearly the Fortran Chapter 4. The FAN system

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Figure 4.2: Discretization process for the definition of the computational domain

Figure 4.3: Orientation of the curve components programs we have written follow it exactly to create automatically the FAN input files. Here is the description of the single objects: Curve components. They are defined by means of an ordered set of nu points. The curve component is the curvature continuous cubic spline which is interpolated through these points. The orientation of the curve —the direction of its tangent— is obtained from the order in which its defining points are given (see figure 4.3); this orientation is arbitrarily assigned. The user gives the coordinates of all the points of all the curves, identifying each of these with a number. Surface components They are defined by means of a rectangular network of points. Similarly to the curve components, the surface component interpolates these points with a planar Ferguson bi-cubic spline. In the network it is possible to define two parametric coordinates (v, w). The surface will have nv points in the v-direction and nw in the w-direction. The points are input sequentially and numbered as depicted in figure 4.4. The first nv points in the sequence form the first curve in the v-direction (the curve w = 0), the second group of nv points forms the curve w = 1, etc., up to nw groups, with the last group representing the curve w = nw − 1. The normal to the surface Chapter 4. The FAN system

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Figure 4.4: Orientation of the surface components at any point is defined as the vector product of the tangent vectors in the v-direction and the w-direction at that point of the surface. Here again the user gives the coordinates of all the points of all the surfaces, giving to each of these an ID number. Curve segments A curve segment is associated to a curve component and is the portion of it that belongs to the boundary of the computational domain. A segment can also result from a rigid motion of a portion of a component. The user gives the correspondences between the segments and the components and the eventual values to define the rigid motions. Surface regions A surface region is associated to a surface component. The region to be triangulated on a surface component is defined by a closed loop of oriented curve segments. The orientation of the loop of curves must agree with the defined orientation of the surface (see figure 4.5). The curve segments are ordered by the module ST sequentially, so that the surface area computed, according to the previous convention for the surface normal, is positive: in the figure such area is painted in grey. The user gives the correspondence between the regions and the components and defines the loops of curve segments for each region. Similarly to the curve segments, the surface regions can also result from a rigid motion of a portion of a surface component. Chapter 4. The FAN system

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Figure 4.5: Surface region connectivity definition At this point it is important to emphasize that the 3D point coordinates and vector components employed in the FAN system to represent the geometry of the computational domain and the generated mesh are referred to a cartesian frame of reference that follows the right-hand rule. While defining the geometry we must be careful to let all the normals point inside the domain, and not outwards, otherwise the module VT will produce an error, trying to generate the mesh outside the computational domain.

4.3

Mesh specification data file (fan.bac)

This file contains the spatial distribution of mesh parameters specified by means of: Background mesh It is a mesh of tetrahedral elements in which the mesh parameters are specified as nodal values. Referring to figure 4.6, the user specifies for each node the coordinates, the cartesian components of the vectors α~i (i = 1, 3) and the corresponding spacing δi (i = 1, 3). The element connectivities are also to be given. Distribution of sources The spacing, equal in all directions, at a point is defined as an exponential function of the distance to point, line and triangle Chapter 4. The FAN system

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Figure 4.6: Parameters for background mesh

Figure 4.7: Distance to point, line and triangle sources sources. The distance is measured for the three kinds of source like in figure 4.7. In practice each point of the line or triangle sources behaves like a point source, being, as explained later, the minimum distance the one which rules. The dependence of the spacing δ on the distance x is: (

δ(x) =

δ1 e|

δ1 x−xc D−xc

| log 2

if x ≤ xc if x > xc

(4.1)

and it is plotted in figure 4.8. Each source ensures a maximum spacing of δ1 up to a distance of xc , then a spacing that grows exponentially with the distance, assuming the value 2δ1 at a distance D. The user gives the source coordinates (1, 2 or 3, depending on the kind of source) and for each of the coordinates the values of the three parameters δ1 , xc and D. In the line (or triangle) source the values of the three parameters change linearly Chapter 4. The FAN system

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Figure 4.8: Function governing the spacing around a source

Figure 4.9: 2D mesh generated using a point source between the 2 (or 3) values inputted by the user for the extremes of the line (or triangle). Finally, as the various sources interact, in each point of the domain the spacing will be the minimum between the maximums allowed by the sources. In figure 4.9 it is shown a mesh obtained with the use of a point source.

4.4

Boundary condition data file (fan.bco)

In this file the user specifies the kind of each boundary.

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Curve segments. The flags for the curve segments determine whether the velocity tangency condition at wall nodes is applied or not. This correction should not be applied if the variation in direction of the normals on surfaces adjacent to the curve —or point— is large. These points are singular, and must be flagged by the user. The possibilities are that all the points of the curve are singular, none, the first and the last, only the first or only the last. Each point needs to be flagged only once, when it belongs to different curves. Surface regions. The surface region boundaries can be of four types, of which only three interest us, being one related to the periodic surfaces we will never make use of. The first is the wall: each surface of the boat hull is of this kind. Making an inviscid calculation the walls represent a condition of impermeability, thus the velocity must be tangent to them (see section 3.2). The second is the symmetry surface. Theoretically it is like the wall, because each wall represents a kind of “mirror” for any other object in the field. Numerically the two definitions differ a bit in the imposition of the velocity near the surface. Being the water surface flat in our approximation, we should consider it a symmetry surface for what we said in section 2.2.3 about the mirror image. Anyway for that surface we will try both the kinds, and see if the calculations are much affected by this. The third and last kind is the far field. It is valid both for the inflow and the outflow and there the velocity is set equal to the far field value given by the user in the .nam file.

4.5

Solver control data file (fan.nam)

This fourth file provided by the user contains a set of flow conditions and algorithmic constants for the flow solver. The names of the parameters are shown in typewriter fonts. First of all, the user specifies the modulus of the free-stream velocity umod= |V~∞ | and the angles α and β (alpha and beta in degrees) shown in figure 4.10. The pressure of the far field is automatically set to 1 (p∞ = 1.0). Most of the other parameters have been already presented in this chapter, and are: • ncycl: total number of timesteps (multigrid cycles) • cfl: value of the CFL number Chapter 4. The FAN system

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Figure 4.10: Convention for the free-stream velocity angles • diss2: dissipation coefficient (finest grid) • diss1: dissipation coefficient (coarser grids) • dissp: factor used to determine the dissipation coefficient for the continuity equation, i.e. dissp·diss1 or dissp·diss2 • meshc: number of the coarsest mesh to be used. The abovementioned parameters have been modified during our study, while the following have been maintained equal to the ones suggested by Dr. J. Peir´o: • nstage=4: number of stages (max 5) in the Runge-Kutta time integration • ndis(i): logical variable that indicates whether the dissipation terms are computed or not at the i-th stage of the Runge-Kutta time integration; in our case it is true for the first two stages • csta=0.3 and csr=1.0: artificial compressibilty parameters, described in equation 3.15 • relax=1.0: wall relaxation factor multiplying the value of ∆V~ at the nodes on wall type surfaces1 1

Set to 1.0 means the full correction is made, i.e. the corrected velocity is made fully

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• nsmth=0: number of residual smoothing iteration • low=.false.: being false, means that a high (second) order solution is used • reynolds=0.0: Reynolds number per unit length (conventionally 0.0 for the inviscid computation) • lg=1: multigrid cycle type: 1=V-cycle, > 1=W-cycle • nite0=1: number of pre-smoothing iterations • nite1=1: number of iterations in the current mesh • nite2=0: number of post-smoothing iterations • ncyci=1000: number of cycles for interpolation from coarse to fine mesh

tangent to the wall. A different value could help the flow to develop smoothly in the initial stages of the timestepping procedure.

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100

Chapter 5 Hull and keel generators In this chapter we want to explain the functions related to keel.for and tkeel.for, two programs that we have written in standard Fortran 77. Basically they are two tools that create automatically the input files needed by the FAN system, in order to represent the submerged part of a yacht. For the first program this consists of a hull and a conventional fin keel, which is put under the hull like an appendage. For the second program the geometry is similar, but the fin keel has in addition two winglets on the tip. The programs are fully parametric, so all the dimensions and angles can be changed by the user to vary the shape of the boat, while other parameters control the mesh generation. Most of the parameters are requested while the program runs, but for sake of simplicity some others, which require modifications rarely, are set in special sections of the program lists: to modify these the programs have to be re-compiled each time. We have decided not to put the lists of the programs (respectively 1400 and 2100 lines) inside this thesis. Instead of the lists, we will try to describe their functioning satisfactorily; a big part of the description can be found in appendix A. Anyway, the programs and all the computer files created during this work are available as explained in appendix C. The Fortran programs are fully commented, so they can be modified quite easily for other purposes. We will now explain separately how each program creates each file, always referring to chapter 4, in which the FAN system is presented. The second program explanation will require less space, as it will exploit many of the things said for the first.

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Z

FLOW

Y

28 X

12

26

9

17

7

10

4 14

13

16

27 8

8

3

24

18

11

5 12 19

6 15

Figure 5.1: Geometry of the boundaries

5.1 5.1.1

Hull and conventional keel Geometry

Let us start with the description of the geometry data file keel.dat. The 29 curve segments and the position of the 13 surface regions are shown in the figures 5.1, 5.2 and 5.3. In practice the boundaries of the 3D domain form a big parallelepipedon; on its upper face there is the submerged part of the yacht, like if it had been printed from outside. The convention for the identification numbers is that for the curve segments they are written in sans serif type style (e.g. 16), with an arrow indicating the orientation. The limits of each curve are indicated with small circles only on the keel tip; in all the other cases the limits are in the intersections between the different Chapter 5. Hull and keel generators

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Z 26 Y X

12 21

9

22

25

10 20

11

29

3

13

23

27

Figure 5.2: Geometry of the boundaries — Zoom on the hull

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Z

22 Y

Origin X

7 1 23

1 2 2 3

4 6

5

Figure 5.3: Geometry of the boundaries — Zoom on the fin keel

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curves, so there cannot be any doubt. The curves in background are dashed. The identification numbers of the surfaces are instead written in italic type style (e.g. 12) with a box around. The box filled in gray means that the corresponding surface is in background. For example the surface 8 represents the ‘floor’ of the domain. The origin of the cartesian axes is set in the junction between the keel leading edge and the hull. The x-axis is longitudinal and goes from the bow to the stern, the z-axis is vertical and the y-axis forms a dextrorse tern with the first two, thus it is directed towards the right of the boat (see figure 5.3). In the previous sketches the grids of the surfaces were not represented, so only the skeleton of the boat was shown. In figure 5.4 there is an overall view of the hull surfaces from a similar angulation. No rigid motion is used to create any curve segment starting from a curve component or any surface region starting from a surface component. Thus the first ones correspond directly to the second ones. For the details of the geometry definition, see appendix A; many interesting tricks used to create the shapes described in this chapter are shown there. User defined parameters The reference length for all the geometry is the chord of the keel root: its length is 1 and goes from the origin (0,0,0) to the point (1,0,0). All the other dimensions derive from the parameters that the user enters. Hull parameters. The hull parameters that can be inserted by the user while running the program, referred to the figures 5.5, 5.6 and 5.7, are: • waterline length LWL • maximum width • percentage position of the maximum width on the waterline length • angles γ, β1 , β2 , α1 , α2 , α3 , α4 • z coordinate of the water surface (hull draft) • radius factor (rf ) • angle of heel Θ Chapter 5. Hull and keel generators

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Z

Y X

Figure 5.4: Surfaces of the yacht

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Y

Z

X

LWL

Max width distance Distance of root section centre

β1

β2

Max width

Figure 5.5: Hull parameters — xy-plane

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Z

Y

hull draft

α1 α3

X

α2 α4

Figure 5.6: Hull parameters — xz-plane

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Z

X

30

Y

o

γ

Figure 5.7: Hull parameters — yz-plane For usual dimensions, the angle γ should not exceed 80◦ , otherwise the spline curve representing the midsection tends to be wrong. Heel angle and radius factor. In figure 5.8 the domain seen from behind the yacht is shown. To achieve the heel rotation we have decided to rotate all the parallelepipedon, instead of rotating the yacht. The reason is simple: the definition of the surfaces that form a cuboid are much less complicated than the ones that form a yacht. These are exactly in the same position of before the rotation, in reference to the axis. Only the surfaces 10 and 13, that are the surfaces of the hull nearest to the water surface, change their shape. They are the only ones that are to be modified, and this is a great simplification; this is the reason why we split in two each lateral surface of the hull: only the upper part changes, while the lower (surfaces 9 and 11) remains the same. The fixed angle of 30◦ in figure 5.7 represents the maximum heel angle. In fact when Θ = 30◦ the surface 13 becomes simply a line and the two curves 20 and 29 coincide. A maximum angle of 30◦ should satisfy the demands of most of the hypothetical users. The situation at a heel angle of 15◦ is shown in figures 5.9 and 5.10. Note Chapter 5. Hull and keel generators

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Z

X

Y

Figure 5.8: Yacht from behind at Θ = 0◦ the rotation of the parallelepipedon in the first figure, and of the axes in the second. The z-axis still represent the vertical of the boat, but it is no more perpendicular to the water surface. The rotation can be made only in one direction, the one shown in the figures. The keel rotates towards the left (clearly for a viewer looking from stern to bow), so the sails heel to the right: in practice our yacht can sail only on left tacks, but being symmetric this is enough. The flow, when a leeway angle is present, will always come from the right side, i.e. the free stream velocity v (along the y-axis) will be negative. As we have already said, the calculations we consider more correct and useful for our purposes are the ones on the keel. The hull is important because it deflects the flow, which arrives on the keel very differently from the free stream. Anyway we considered important to simulate the heel in a good way. In first approximation when a boat heels the submerged volume remains constant, as the same buoyancy force — the Archimedes’ push— is needed. In general using our system of reference, fixed on the boat and not on the water, the distribution of the submerged volumes lets the water surface rotate around different axes, depending both on the shape and on the heel angle. This means that the geometry of the submerged part changes heavily and in a way difficult to estimate: it is not an easy task to give a good representation of it. We wanted to avoid this problem making the geometry so that the rotation of the water surface was made, for the usual heel angles, around the same axis, and precisely the intersection between the yacht symmetry plane and the plane of the water surface when Θ = 0◦ ; it is in practice the longitudinal axis of the boat at the water level. The condition under which this Chapter 5. Hull and keel generators

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Z

X

Y

Figure 5.9: Yacht from behind at Θ = 15◦ Z

X Y

Figure 5.10: Zoom of the last figure, but with the rotation of the axes Chapter 5. Hull and keel generators

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Figure 5.11: Volume distribution with heel happens is shown in figure 5.11, where the rotation axis is represented by the point A. The left area filled in grey represents the boat volume that emerges with the presence of a heel angle Θ, while the right area the boat volume that sinks. The condition is that the two volumes must be equal for every possible heel angle. We deduce that the surfaces which are the boundaries to those volumes must be symmetric in reference to the axis A. This, together with the fact that the yacht has a longitudinal symmetry plane, lets the condition be: all the surfaces that can be wetted by a heel rotation must be symmetric in reference to the axis A. To achieve this, we decided to create the midsection curve as an arc of circumference with radius r. To obtain the symmetry requested, it is not necessary that the centre C of the circumference is the point A: the importance is that such centre is on the horizontal plane, i.e. the horizontal line in figure 5.11. The radius factor rf is then defined as: rf =

1 2

f . (boat width)

(5.1)

The different cases are shown in figure 5.12, where C is the centre of the circumference used to create the left curve. The angle of 60◦ includes the part of the curve that can be affected by a heel rotation, which in fact cannot be greater than 30◦ . If rf is 1, the centre of the circumference is the point A; if greater than 1 the curve becomes less round, viceversa if lesser. Varying the parameter rf together with all the others, it is possible to create reasonable midsection curves; the part that sinks when heeling is the less similar to an actual yacht, but the error should not be particularly big, due to the fact that the hull waves are not represented. Besides it is easy to find the intersection between the water plane and the hull surface for any heel angle. Eventually Chapter 5. Hull and keel generators

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Figure 5.12: Upper midsection curve at different radius factors the surfaces 13 and 10 are made as to follow the midsection curve created with this system. Fin keel parameters. Referring to section 2.1.2 and to figure 5.13, the fin keel parameters the user can change are: • percentage position of the middle point of its root section on the waterline length • XY digits for NACA 00XY profile at the root • XY digits for NACA 00XY profile at the tip before the rounding • aspect ratio • sweep angle • taper ratio • ratio between the rounded part length rl and the keel draft • ratios f r/lnl and rr/lnl

5.1.2

Mesh generation

The program keel.for creates automatically the .bac file, which, as we have seen in section 4.3, regulates the spatial distribution of the mesh elements. The user can insert three parameters, that we called mesh factors: the global, Chapter 5. Hull and keel generators

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Figure 5.13: Some parameters for the fin keel tip the hull and the keel mesh factors. After many attempts, made manually by changing the program, the mesh generation is now made so that if all the three mesh factors are 1, the number of tetrahedra is optimum for our possibilities, giving a good representation of the domain without wasting disk space and time while computing. With this contrivance it is very easy to create different meshes for a multigrid computation: while the geometry (and thus the .dat file) remains the same for all the meshes, the element size specification (.bac file) can be modified cleverly by inserting different mesh factors. Besides, if the FAN system runs on a machine with different characteristics from the one we have used, it is possible to fully exploit its means by toggling the mesh factors, for example giving to them values lesser than 1 if the machine is more powerful, or more disk space is available. Background mesh For an initial mesh in a flow simulation, the background mesh is kept as simple as possible. Usually it specifies a constant spacing equal to the desired spacing at the far-field boundaries. Here we define the background mesh as a cube —big enough to cover the computational domain— split into six tetrahedra, as depicted in figure 5.14. Their connectivity array is shown in table 5.1. The global mesh factor mg regulates the constant spacing δ in this way: δ = 1.4 · mg . (5.2)

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Figure 5.14: Background mesh

Element 1 2 3 4 5 6

1 1 1 2 2 2

Nodes 2 4 2 8 6 8 3 4 7 4 7 8

8 6 5 7 8 6

Table 5.1: Connectivity array for background mesh

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This means that for the finest mesh (mg=1) we have used a spacing of 1.4. This could seem too big, remembering that the keel root chord is 1. But the spacing is effectively 1.4 only far from the yacht, as near it many sources regulate in a more efficient way the spacing. The background spacing is independent of the dimensions of the yacht. Source distribution The source distribution concentrate more elements in the regions of the domain where more resolution is needed. Basicly in our problem a higher resolution is needed near all the yacht, in particular near the fin keel that we are going to study approfonditely. Also the zones in which there are big velocity gradients need a smaller spacing. We refer to equation 4.1 on page 96 and to figure 4.8 for the definition of the three variables (δ1 , xc and D) related to the sources. Our actual definition of the sources comes from a large number of attempts, made manually to obtain satisfactory results in terms of discretization. Here again if the hull mesh factor mh and the keel mesh factor mk are set to the value 1, the distribution is optimized. The spacing in proximity of the hull and the keel is made in some ways dependent on the yacht dimensions, besides, obviously, on the mesh factors. On [18] we can see that the average value for the ratio D/xc is 2. We usually have preferred to use a smaller ratio; this means that in our case the effect of a source fades more rapidly while moving away from it. The line sources on the keel are an exception, being their D/xc ratio about 3.7. It would be unuseful to put the expressions, which depend on some dimensions of the yacht, that the program uses to calculate the values of the three source parameters for every source. The expressions can be easily found in the list of the program, in the subroutine for the creation of the .bac file, and in case modified without any problem. We now want to describe briefly where the sources are put, and the order of magnitude of their spacing and influence sphere. Many examples of meshes generated with this distribution of sources may be found in all the rest of the project. Point sources. Two point sources have been placed in the bow and in the 1 1 stern. For them the value of δ1 is about 45 of the waterline length, xc 18 and 1 D 15 . They are indicated with P1 and P2 in figure 5.15. To have an idea of the value of the three parameter, we can say that they are about 0.17, 0.45 Chapter 5. Hull and keel generators

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P1

P2

L8 L9

Z

L1

L2

X Y

L3-L4

L5

L6-L7

Figure 5.15: Point and line sources and 0.53 for the problem on which we will concentrate our efforts. It must always born in mind that the reference length is the length of the keel root chord. Line sources. There are 9 line sources: they are indicated with an ‘L’ and numbered in figure 5.15. The most important can be considered the ones that follow the edges of the keel (number 1 to 7); in those regions there are 1 very high velocity gradients. The spacing is .035, so about 30 of the chord; xc is 0.09 and D 0.33, so these source spread their influence a lot. L1, L2 and L5 are placed exactly on the keel edges (thus on the curve segments), while two line sources are put on each cubic of the rounding: even being outside the domain, because inside the keel, the important fact is that their influence is felt inside the domain. This is also the case of L8 and L9, put inside the hull near its longitudinal curves. Their parameters are about five time greater than those of the keel, except for D which is, relatively, smaller, i.e. their influence fades faster. Triangle sources. The triangle sources are shown in figure 5.16, indicated with a ‘T’. T1 and T2 are put inside the keel, to cover with their influence all the keel surface. The spacing is 0.045, and xc is 1.5 times the maximum keel thickness of the corresponding section; the ratio D/xc is 1.6. Chapter 5. Hull and keel generators

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T5

T6

T7 T8 Z

T2

T4

X Y

T1

T3

Figure 5.16: Triangle sources T3 and T4 are put behind the keel so to distribute a big number of points in the wake. There in fact we expect to find the tip vortex, which is strictly related to the induced drag that we want to compute. The spacing is about 2.5 times the one of T1 and T2, and the influence is the same. The longitudinal dimension of these triangles is of the order of the average keel chord. The triangles are placed on the yacht symmetry plane; it is probable that the wake passes through their influence zone even if a flow incidence angle is present on the keel, for two reasons. The first is the sidewash of the keel, i.e. the flow tends to leave the keel parallely to its plane: it corresponds to the downwash of an aeroplane wing, consequence of the action-reaction principle applied to the fluid-dynamic lift. The second is that their influence zone is quite wide, transversally; anyway, the mesh generation must be able to compute flows with different leeway angles, so it could not be tuned for one particular condition. T5 and T6 are placed inside the hull, and spread their influence on the left lateral surface of the hull. T7 and T8 do the same on the right side. Their extremes are shown in the figure, for the left side only: the two central extremes are set in the points of maximum width of the keel root section and the water level hull section, while the other two points are in the bow and in the stern. The spacing is about 1.33 times the spacing of the 2 point sources, but the influence fades faster than theirs.

Chapter 5. Hull and keel generators

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It must be noted that the mesh factors affect only the spacing δ1 , and not the position or the influence radius of the sources. The hull mesh factor mh multiplies the spacing of the 2 point, 2 line and 4 triangle sources placed on the hull. The keel mesh factor mk does the same with all the other sources, thus the 7 lines and 2 triangles placed on the keel and the 2 triangles of the keel wake.

5.1.3

Boundary conditions

The program keel.for creates automatically also the file .bco, in which the kind of each boundary is specified. Curve segments. The only curve segments that need to be flagged as composed by singular points are the ones representing the trailing edge of the keel, thus the segments 3 and 5. All their points are singular, because the normals of the two surfaces that are adjacent to them have a big difference in direction. No other point will be considered singular. This coincides with the hints given by [18]. Surface regions. All the surfaces of the yacht are considered walls, while all the surfaces of the parallelepipedon but the upper are considered of the farfield kind. The only doubt can be present about the surfaces 3 and 12, that represent the water surface. We have already said in this thesis that generally the water surface can represent a ‘mirror’ for the submerged surfaces only at low Froude numbers, otherwise the waves are too big to be neglected. Fortunately, being the water surface of our computation absolutely flat, we have tryed to put on surfaces 3 and 12 the flags of symmetry and of wall, and to make a test. We measured the lift and drag, as we will explain we can do, with a leeway angle of 5◦ , both with no heel and with a heel of 30◦ . We found no appreciable difference, so we argued that it does not matter which of the two kinds is imposed. We then imposed those surfaces to be walls.

5.1.4

Flow conditions

In all this chapter we have explained how the 3D domain for a numerical computation can be created automatically by the program we have written. Once that the .dat, .bac and .bco files have been made, in case for more than one mesh, and the modules ST, VT and PP of the FAN system have Chapter 5. Hull and keel generators

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run with those files as input, the code is ready to start the computation. In the .sol file, in fact, all the information about the domain is stored; we only need to provide the .nam Fortran namelist file , contatining variables to set the flow conditions and to control the flow solver (see section 4.5). The values are not created by the program keel.for, but they can be modified easily with the use of a text editor Free-stream velocity First of all, we set the modulus of the free-stream velocity (umod) to 1. Not being present the Reynolds nor the Froude number, the value we give to that variable is uninfluential for the kind of flow. The variations of pressure and thus the forces and moments acting on the boat are proportional to the square of the free-stream velocity, but as we will use only non-dimensional parameters the results we will consider will be independent of it. In figure 4.10 on page 99 the convention of the FAN system for the freestream velocity angles is shown. Being 1 the modulus of the velocity, we can split the vector in the components parallel to the cartesian axes: V~∞ = (cos α cos β)~ı + (cos α sin β)~ + (sin α)~k . (5.3) Now, in our case the definition of the angles is different: see figure 5.17. The two angles that the user can choose are the heel angle Θ and the leeway angle λ. Being all the ‘towing tank’ inclined of the angle Θ, the free-stream flow will not be parallel to the x-direction if an angle λ is present, because it has to be parallel to the water surface. If we split again the free-stream velocity vector in the cartesian components, but this time using the angles Θ and λ, we obtain: V~∞ = (cos λ)~ı + (− sin λ cos Θ)~ + (− sin λ sin Θ)~k . (5.4) Comparing the equations 5.3 and 5.4, respectively in their component along z and x, we find: sin α = − sin λ sin Θ cos α cos β = cos λ ,

(5.5) (5.6)

from which we can find the expressions to obtain the two angles α and β we have to insert in the .nam file, paying attention to put the correct signs: α = arcsin(− sin λ sin Θ) cos λ β = − arccos( ), cos α Chapter 5. Hull and keel generators

(5.7) (5.8) 120

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Z

Y Flow

X θ

λ

Figure 5.17: Free stream angles for the keel problem

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Variable ncycl cfl diss1 diss2 dissp

”Fluid Mechanics of Yacht Keels”

Value 400 1.0 0.4 0.6 1.0

Table 5.2: For the geometry, both α and β are always negative. The small Fortran program abeta.for does these simple calculations automatically: it is useful when editing the .nam file, to obtain quickly the values that are to be inserted. Other variables. As we have said in section 4.2, only some of the parameters of the .nam file have been changed during our study, besides, obviously, the ones related to the free-stream velocity. As first attempts the values of table 5.2 have been inserted. The total number of timesteps was unknown at the beginning of the series of computations. In the first attempt we have put 400, but at a future time we will set it to the minimum necessary to obtain a sufficient reduction of the pressure residuals. For what concerns the CFL number, we will try to increase it to the maximum allowed by the code for our problem. The dissipation coefficients, instead, have been suggested by Prof. J. Peir´o. Finally the last parameter, meshf, depends on the number of meshes used for the multigrid. We will try different configurations, with 1, 2 or 3 meshes, and we will decide which is the best for us.

5.2

Hull and winged keel

The program tkeel.for has been created modifying and enlarging the program keel.for, so we will refer in this section to much information already given about that. Some of the information will also be given in appendix A.

5.2.1

Geometry

Basically the figures 5.1, 5.2 and 5.3 are still valid, with these exception:

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Z Y X

Figure 5.18: Overview of the curve components of the T-shaped keel • the curve 29 becomes the curve 51, because of the fact that it disappears when the heel angle is 30◦ , so it must have the last ID number • the surface 13 becomes the surface 23, for the same reason The situation with the winglets is shown in figure 5.18, where the curves in background are represented like the lines in foreground. The shape of the vertical part has been changed, also to demonstrate that the program can effectively create different geometries. The position of the winglets in this example is a bit uncommon, because they are set quite far from the keel tip, mainly for representation reasons; anyway it is possible to find these shapes in the reality, especially when a dihedral angle is present. To attach the winglets we have modified also a part of the lateral surface of the vertical part, so that there is a zone of smooth connection between the parts. The idea has come up seeing some photographs of winged keels of this kind. Also, it would have been almost impossible to find the coordinates of the intersection curve between the lateral surface and the winglet surfaces, Chapter 5. Hull and keel generators

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Z

Y X

29 30

13

31

35

36 32 37

14

34

33

15 39

16

38

17 40

Figure 5.19: Geometry of the boundaries — Zoom on the left winglet because of their complexity. Instead we have used some tricks to obtain the desired geometry; it is like if there was a ‘patch’ near the winglet attachment. The new curve components and surface regions are shown, with the usual convention, in the figures 5.19 and 5.20, with different view angles to let them be as clear as possible; it is better to look always at the axes not to lose the orientation. The new surface grids are shown in figure 5.21; the errors in the connection zone are due to the program we are using for the figures, and not to actual geometric errors. For the details of the definition of the new curves and surfaces, refer to appendix A. Chapter 5. Hull and keel generators

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Z

X Y

41

42

31

18 45

46

48

33 43

20

19

21 44

47

49 22 50

Figure 5.20: Geometry of the boundaries — Zoom on the right winglet

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Z Y X

Figure 5.21: Surfaces of the T-keel tip

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User defined parameters Besides all the parameters that can be inserted in the keel.for program, in tkeel.for the user can specify some of the variables that were presented in section 2.1.2 on page 27, and in particular: • percentage of keel tip chord covered by the winglets • XY digits for the NACA 00XY profile of the winglet • winglet aspect ratio • winglet sweep angle • winglet taper ratio • winglet dihedral angle The trailing edges of the winglets are joint to the vertical part always in correspondence of its trailing edge. The profile is the same for all the winglet sections.

5.2.2

Mesh generation and boundary conditions

Mesh generation. The mesh is created in the same way of the problem of the conventional keel, plus the addition of some sources. In figure 5.22 the new sources are shown, for the left winglet only. The distribution is specular for the right winglet. On each side there are 3 new line sources, put on the edges of the winglet, and 4 new triangle sources. Two triangles cover the winglet, two its wake. The criterion for the choice of the source parameters is the same adopted on the vertical part, made exception for the longitudinal extension of the triangles of the wake, that is fixed equal to the one of the triangles T3 and T4 of figure 5.16. The spacing of all the sources related to the winglets is regulated by the winglet mesh factor mw, in the same way in which the keel mesh factor regulates the sources related to the vertical part of the keel. Boundary conditions. The boundary conditions of the problem without winglets are still valid. Besides, all the points belonging to the trailing edges of the winglets —the curves 38 and 48— are to be flagged as singular; no other new point is singular. All the new surfaces are walls. Chapter 5. Hull and keel generators

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Z Y X

T10

L12

L10 T9

T12

T11 L11

Figure 5.22: Line and triangle sources on the left winglet

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5.3

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Calculation of forces and moments

The codes of the FAN system give the output files described in section 4.1.2. In this section we want to show two of the tools we created to examine those files and to extract and analyze the data from them. Besides we want to show an empirical method to evaluate the friction drag on the fin keels. All the nodes, surface triangles and tetrahedra have an identification number inside the codes. The values of the four variables (the pressure and the three components of the velocity) for every node are put ordered in the .un1 file. Both the .fro and the .gri file store the three coordinates of each node. Besides the first stores the three ID numbers of the nodes that form each triangle, and the second stores the four ID numbers of the nodes that form each tetrahedron. We have created two programs in Fortran 77 language to calculate the hydrodynamic forces acting on the yacht. They operate in two different ways: the first is lift.for and calculates the forces and moments using the values of the pressure on the surface nodes; the second is momentum.for and uses the momentum theorem to calculate only the forces, using in practice the values of all the four variables on the surfaces of a parallelepipedon that includes the part of the yacht on which the calculation is made.

5.3.1

Program lift.for

The first thing that the program does is to read the data from the .fro and the .un1 files in order to get all the information required. This consists in knowing all the coordinates and the pressure of the nodes of each triangle, plus the number of the surface to which each triangle belongs. To make correct calculations the program subtracts from the values of the pressures the value of the pressure of the free stream (that is 1). This would create problems as we are not integrating the pressure on a closed surface; thus even if the pressure was constant and equal to the free stream value everywhere in the domain, we would find a false force. For example in the case of figure 5.3, integrating a constant pressure on the surfaces and 2, thus on the fin keel surfaces, we would find a force parallel to the z-axis.

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Figure 5.23: Generic surface triangle Force calculation The situation for the generic i-th triangle is shown in figure 5.23. The three nodes A, B and C are ordered so that it is possible to know the direction of the ~ × AC. ~ normal pointed outside the domain making the vectorial product AB If the normal points outside the domain of the flow, it means that it points inside the yacht, thus its direction corresponds to the one of the force F~i . The area of the triangle can be expressed by: Atriangle =

~ × AC| ~ |AB . 2

(5.9)

Being the finite elements linear, the average pressure acting on the triangle will be the average of the three pressures of the vertexes. Thus the generic force F acting on the triangle is given by: ~ × AC ~ p1 + p2 + p3 AB · , F~i = 3 2

(5.10)

that is the average pressure multiplied by the area. Chapter 5. Hull and keel generators

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The user can choose on which surfaces the calculation will be made; the program makes the vectorial summation on all the triangles of the selected ~ surfaces and finds the resultant R: N

~ = R

triangles X

F~i

(5.11)

i=1

Thus a single free force is found with this calculation. To find the drag and lift (or better, side force) the program projects the resultant R on different directions: see figure 1.4. The drag is found projecting the resultant on the direction of the free stream velocity defined in equation 5.3. Being the modulus of such velocity equal to 1, the velocity vector coincides with its versor and to find the drag it is sufficient to make the scalar product: ~ · V~∞ . D=R (5.12) For what concerns the side force, after many reasonings we have decided to consider it exactly like we usually consider the lift in aeronautics. On an aeroplane the lift is the component of the aerodynamic force perpendicular to the drag and parallel to the aircraft symmetry plane. On the yacht such symmetry plane corresponds to a plane perpendicular to the z-axis of our system, which is in practice a plane parallel to the deck. The versor of the direction of the lift, that we call ~l = (lx , ly , lz ) is then perpendicular to the drag (thus to the free stream velocity) and its z-component lz is 0. Solving the system of equations, with lx and ly as unknowns: (

~l · V~∞ = 0 lx2 + ly2 = 1

(5.13)

we find the versor ~l as: v u u ~l = ( −t

v u

2 2 u V∞y V∞y t1 − , − ,0) , 2 +V2 2 +V2 V∞x V∞x ∞y ∞y

(5.14)

where the signs have already been chosen so that the lift force is positive when the leeway angles are positive, i.e. the flow comes from the right of the boat having the y-component negative. The lift is then easily found projecting the total resultant on the lift versor: ~ · ~l . L=R Chapter 5. Hull and keel generators

(5.15) 131

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The versor ~c of the third component of the force, which is in aeronautics the cross-wind force and here we will call C, is found with the vectorial product: ~c = ~l × V~∞ . (5.16) The orientation is such that this third versor is directed towards the sails; note that it is parallel to the z-axis only when the yacht is unheeled. Like before, the third force C is found projecting the resultant on the corresponding versor: ~ · ~c . C=R (5.17) The output of the program, related to the forces, is given in the form of force coefficients. See for example the equations 2.19 and 2.20 to see how they are defined with dimensional terms. Being all the quantities in our calculations non dimensional, for us the generic force coefficient CF depends on the generic calculated force F in this way: CF =

F 1 2 V 2 ∞

A

.

(5.18)

Moment calculation While for the force calculation it was unuseful to know on the triangle the position of the point P in which the resultant F~i of the pressure forces is applied, for the moment calculation it is important. The calculation of the position of P is not trivial. The program uses the auxiliary axes ξ and ψ (respectively of versors m ~ and ~n) with origin in the point A, like in figure 5.23. the ξ-axis is directed towards C, while the ψ-axis is perpendicular to the first and lies in the plane of the triangle. We have made integrals involving the pressure p(ξ, ψ), which varies linearly on the triangle, and found the coordinates of P in the auxiliary axes for a generic triangle: R

Area p(ξ, ψ) ξ dξ dψ |Fi | R p(ξ, ψ) ψ dξ dψ . = Area |Fi |

ξP =

(5.19)

ψP

(5.20)

~ in global coordinates is then The vector AP ~ = ξP m AP ~ + ψP ~n , Chapter 5. Hull and keel generators

(5.21) 132

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so we can find the moment m ~ i of the force Fi around the origin by writing: ~ × Fi = (~ ~ ) × Fi . m ~ i = OP ri + AP

(5.22)

The total moment M around the origin is given by the summation of all the moments m ~ i: N triangles X ~ = M m ~i (5.23) i=1

~ around the origin could now be projected on any The total moment M axis. We have preferred to consider the moments around the three axes of the yacht, independent of the flow direction and the heel angle, i.e. the axis of our computational domain; these, in fact, would be the more appropriate when calculating the moment equilibrium of the whole yacht. Like for the force coefficients, it is more convenient to use the moment coefficients than the actual moments: the formula is in this case CM =

M , Acav

1 2 V 2 ∞

(5.24)

where cav is the average chord. Centre of lateral resistance ~ to the centre of lateral resistance CLR, the moIf we apply the resultant R ~ creates around the axes x and z are equal to the components ments that R of the total moment on the same axes. The CLR is defined to be on the yacht symmetry plane, thus it is yCLR = 0. The other coordinates of the CLR are therefore found in this way: Mz Fy Mx = − Fy

xCLR =

(5.25)

zCLR

(5.26)

The forces Fx and Fz , in fact, do not give any contribute to the moments Mz and Mx if they are applied on the symmetry plane. But they give their contribute to the moment My . It is therefore possible to fully describe the forces and moments acting on ~ the moment My and the x and z coordinates the yacht giving the resultant R, Chapter 5. Hull and keel generators

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Figure 5.24: Closed surface for the momentum theorem of the centre of lateral resistance CLR. These are in fact the quantities on which we will make our reasonings; with this contrivance it becomes unuseful to give the moments Mz and Mx . The CLR is useful because it gives immediately a measure of the height at which the lift is produced.

5.3.2

Program momentum.for

This program uses a different method to evaluate the forces on the yacht: it uses the momentum theorem. It is far more complex than the other one, but as it has been in practice unuseful during our study we have decided to describe it less deeply. On the other hand, it can be very useful for other purposes at a future time, because of what we have discovered while making the tests on our methods; for now we want just to explain briefly how it works. All the information needed is read from the files .un1 and .gri, and precisely all the coordinates and the unknown values of the four nodes of each tetrahedra of the domain. The program can apply the momentum theorem to the fluid contained in the closed surface shown in figure 5.24. The dimensions and the position of the parallelepipedon are asked to the user, so that instead of the two keel lateral surfaces there can be any surface in the upper part of the domain. We can write that the force acting on the

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surfaces of the keel is: F~ =

X

pA~n +

X

V~ (V~ · ~n)A .

(5.27)

In this equation the versor ~n is the normal versor to the surfaces that points inside the domain. The summation is extended to all the portions of the surfaces but the ones on which we want to evaluate the forces; A is the area of the portion considered. To summarize, the program makes six cuts, one for each face of the parallelepipedon. The cuts are made in the tetrahedra, finding triangles and quadrangles; each of the quadrangles is transformed into two triangles. On every cut the triangles that actually belong to the faces are selected: the summations shown in equation 5.27 are extended to all the triangles selected in this way. The values of the unknowns at the vertexes of the triangles are found by interpolation from the values at the vertexes of the tetrahedra. On every triangle the value of the quantity (V~ or p) is considered constant and equal to the average of the values at the vertexes. No moment calculation can be made using this method, and the force results are given exactly in the same way of the program keel.for.

5.3.3

Profile drag estimate on the fin keel

We have seen that the friction drag can represent a big percentage of the total drag of a yacht, especially at low speed. That reasoning was made considering also the wave drag, so important for the floating bodies. As we are not considering the wave drag, the profile drag (thus the friction and the pressure drag together) becomes even more important. The addition of the winglets on a conventional keel studied with an inviscid code, for example, should bring to a sure improvement in the fluid-dynamic characteristics of the yacht; on the contrary there can be a general worsening, if we consider the profile drag. We therefore want to use an empirical method

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that helps us in our reasonings. Of course we will have to remember that the errors can be very big, with such an approximate method. The estimate we will use comes from the experimental data of [1], where the four digit NACA profiles are fully studied. Only the 2D profiles are studied there, so apply those results to low aspect ratio keels, on which the cross-flow is conspicuous, will bring to sure errors. This estimate, in fact, could be more accurate if each section of the keel could be considered like a part of an infinite aspect ratio wing; but this is not our case, as the transversal velocity component is not negligible. Fortunately the data does not depend much on the thickness of the profile; even if we have used different four digit symmetrical profiles (NACA 0016 to 0018 for the vertical part, NACA 0014 for the winglets), we will use for them the same estimate. The Reynolds number can be taken equal to about 3 106 , like in equation 2.7. We will use the next formula, also used for a similar problem in [14]: Cd = Cd0 + kCl2 = 0.0085 + 0.017 Cl2 .

(5.28)

The Cd and the Cl represent respectively the drag coefficient and the lift coefficient of the infinite aspect ratio profile, thus they are defined in a different way of the CD and CL of a finite wing. But, as we consider the same estimate for all the keel whatever is the profile, the Cd of the formula corresponds to the CD we are searching for, and the Cl we have to insert corresponds to the CL of all the keel.

5.3.4

Lift/Drag curves

Having defined in some way the profile drag, it is possible to evaluate the efficiency of the keel, thus its L/D ratio (or better in non dimensional terms the CL /CD ratio). From the equations 2.26 and 5.28 we can write that the total drag coefficient is: CD = CD0 +

CL2 CL2 = CD0 + kCL2 + . eπAR πAReff

(5.29)

From our calculation we hope to extract the induced drag coefficient CDi , thus the last addendum of the last member of equation 5.29. We can then find the effective aspect ratio AReff = CL2 /(πCDi ) , so together with the profile drag estimate of the last section we have everything to compute the Chapter 5. Hull and keel generators

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drag. We could also write the drag in the form with the Ostwald coefficient. Equalizing the two expression for the drag dependent on the lift square we can find: AReff /AR e= . (5.30) 1 + πkAReff Being the behavior of the drag the one of equation 5.29, the following results concerning the maximum L/D ratio can be found: q

CL(L/D)

max

=

eπAR CD0

(5.31)

eπAR . 4CD0

(5.32)

s

(L/D)max =

Finally, if the lift curve is linear and the lift curve slope is CL0 , from equation 5.31 we can find the angle of maximum L/D: q

α(L/D)max =

Chapter 5. Hull and keel generators

eπAR CD0 CL0

.

(5.33)

137

Chapter 6 Preliminary study We want to show in this chapter the characteristics of the hull and the keels on which we made all the calculations, with and without the winglets. The geometries and the meshes was created with the programs shown in chapter 5. After having chosen the parameters requested by those tools, we created different meshes and studied the convergence of the method, in order to use in the numerous future computations the best configuration. Afterwards modifying the program keel.for we generated a simple geometry, on which we could make tests and compare our results with the experimental data that we had available, trying also to find a good estimate for the friction drag that the codes cannot compute. We also tested the differences between the two programs for the force calculation.

6.1

Choice of the shape

Following what we had found and described in section 2.8, we opted for an inverse tapered keel of aspect ratio 1.4. All the parameters that we had to insert in the program keel.for, explained in chapter 5, are shown in table 6.1 for the hull and in table 6.2 for the fin keel. The only parameter that is not fixed, for what concerns the hull and the conventional keel, is the angle of heel. In fact we studied such conventional configuration with different heel angles at different leeway angles, which however influence only the free stream variables and not the geometry. As we wanted to make a parametric study on the influence of the addition of different winglets on the same keel, the same parameters were also inserted in the program tkeel.for, together with the others that were related to the 138

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Hull parameter waterline length LWL maximum width position of the max width on LWL γ β1 , β2 α1 , α2 , α3 , α4 z coordinate of the water surface (hull draft) radius factor (rf ) angle of heel Θ

Value 8 2.2 60% 40◦ 23◦ , 45◦ 22◦ ,35◦ , 8◦ , 5◦ 1.2 0.9 variable

Table 6.1: Values of the parameters chosen for the hull

Fin keel parameter position of the root middle point on LWL root profile profile at the tip before the rounding aspect ratio sweep angle taper ratio rl / keel draft f r/lnl, rr/lnl

Value 55% NACA 0016 NACA 0018 1.4 2◦ 1.54 0.15 0.15, 0.15

Table 6.2: Values of the parameters chosen for the fin keel

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Winglet parameter part of keel tip chord covered by the winglets profile of the winglet winglet aspect ratio winglet sweep angle winglet taper ratio winglet dihedral angle

Value 60% NACA 0014 variable variable 0.5 variable

Table 6.3: Values of the parameters chosen for the winglets winglets, shown in table 6.3. Among these, the not fixed parameters were the aspect ratio, the sweep angle and the dihedral angle. Parameter farfield. Among the parameters that can be changed directly into the lists of the program, we payed particular attention to farfield, described in appendix A on page 199. Basically it regulates the dimension of the parallelepipedon of the domain: a small value of it could bring to wrong calculation, because the domain is not big enough, while a big value would bring to waste the efforts, because big part of the domain would have the same conditions of the far-field. The initial value of 4 that we put copying from the manual [18] is too big: we made some tests and found that it can be reduced without varying the solution up to a value of 1.5. This is the value that we used in all our calculations. In reality as the tetrahedra near the far-field surfaces have a big spacing, the reduction in the number of total elements —thus nodes— is not so evident: see in fact for example figure 6.1 or figure 6.4, in which it can be seen that most of the nodes are near the yacht and only a few are far. The geometries obtained with the values of the tables are not shown directly, but they can be seen in the figures in which we show the correspondent generated meshes: see section 6.2 and, more generally, all the figures put in the next chapters until the end. The fact that a hull created with the use of our programs can represent in a good way an actual shape is confirmed by the the value of the already described prismatic coefficient and displacement/ length ratio, whose values in the discretization belong to the real yacht’s ranges.

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Prismatic coefficient. The volume of the hull can be calculated by subtracting from the volume of the parallelepipedon, that can be easily found, the volume of the 3D domain, which is given from the FAN module that makes the triangulations (ST), and afterwards subtracting the volume of the fin keel, that can be estimated manually. Also the maximum area of the immersed sections can be easily found manually. For our shape we found, using equation 2.1, CP =

displaced volume 7.5 = = 0.54 , LWL · max area of immersed sections 8 · 1.75

(6.1)

which is inside the range of optimum values shown in figure 2.6. Displacement/length ratio. Referring to section 2.1.2, and using the conversion factor of page 30 we find that the displacement/ length ratio of our yacht is: ∆ = 143.7N/m3 = 408.4ton/(0.01 ft)3 , 3 LWL

(6.2)

that corresponds to a heavy displacement yacht.

6.2

Generated meshes

In the next pages there are many figures that show the meshes that the program generates on the geometries created by inputting in keel.for and tkeel.for the values of the parameters displayed at the beginning of this chapter. They are the consequence of the big efforts fully described in chapter 5, and thus they do not require many more comments. The figures from 6.1 to 6.5 show the meshes generated by putting values of 1 to all the mesh factors. Figure 6.1 is an overall view of the surface mesh; the different spacing of the zones must be noted. Figures 6.2 and 6.3 show the orthographic projection of the yacht surface mesh. To show quickly how the tetrahedra are distributed we have made two cuts in the 3D domain, one longitudinal and one transversal: see figure 6.4. It can be clearly seen the extension of the influence of the sources; in particular the triangle sources of the wake are visible in the longitudinal cut. Finally it is possible to see the generated mesh for a T-shaped keel in figure 6.5.

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Chapter 6. Preliminary study Figure 6.1: Overall view of the surface mesh

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Figure 6.2: Two views of the surface mesh of the yacht Chapter 6. Preliminary study

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Z

Y

X

Figure 6.3: Front view of the surface mesh of the yacht

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Figure 6.4: Cuts of the tetrahedron domain Chapter 6. Preliminary study

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Z Y

X

Figure 6.5: View of a T-shaped keel mesh

Nodes Tetrahedra Triangles Surface nodes

Conventional keel mf =1 mf =2 31406 5577 163250 27663 12526 3238 6265 1621

T-shaped keel mf =1 39600 199025 14805 7373

Table 6.4: Number of nodes and elements

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Z

Y

X

Figure 6.6: Left view of a coarser mesh (Mesh factors=2) Number of nodes and elements. See the first two columns of table 6.4 to see how many elements and nodes are created in the problem of the conventional keel putting all the mesh factors mf respectively 1 and 2, for Θ = 0◦ . In the case of the T-shaped keel all the numbers are higher, but they depend on the shape of the winglets, especially on their aspect ratio. See the last column of table 6.4 to see the number of elements and nodes for a T-shaped keel with winglet aspect ratio 1, Θ = 0◦ and all the mesh factors 1. An example of a mesh created with mf = 2 is shown in figure 6.6: there can be seen the differences with the mf = 1 mesh of figure 6.2. Program profile.for. Between the Fortran programs we have created for this thesis we want to describe quickly profile.for, that makes cuts to the mesh surfaces, giving an output like the ones of figure 6.9, 6.8 and 6.7. The user can specify on which surfaces the cut will be made, the number and the position of the cuts; the cuts can be made only with planes parallel to the coordinates. The goal was to make a representation similar to the one shown in figure 2.1, especially giving the coordinates on superimposed grids. The result is not as good as if we had made the cuts on the actual geometry: in the figures the imperfections due to the triangulation can be seen. The method used is similar to the one used by momentum.for, but clearly the cuts are Chapter 6. Preliminary study

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1.5 ’zcuts.dat’

1

0.5

0

-0.5

-1

-1.5 -4

-3

-2

-1

0

1

2

3

4

5

Figure 6.7: Cuts normal to z made on the triangles instead of the tetrahedra. The program can also draw the pressure profiles and the velocity profiles found cutting the surfaces. The output is written on a file, so the profiles can be very useful for subsequent comparisons or calculations. We do not describe how the program works, because to study the pressure we used the powerful package Tecplot; however this program could be needed for other purposes related to the subject.

6.3

Convergence history

As we knew from the beginning that we had to make a big amount of calculations on different shapes, it was important to try to let the codes converge as quickly as possible. We therefore made some tests to evaluate which configuration could be considered the best for us. We took the problem of the conventional keel, with no heel and a leeway angle of 5◦ . The codes started the computation with the free-stream conditions in all the nodes of the domain. We decided that a sufficient accuracy could be reached with the reduction of the pressure residual L2 norm to 1/1000 of the initial value; the pressure residual is usually taken as the most important as it is strictly related to the equation of continuity. The initial value of the L2 norm is about 2.0, so we wanted to reach a value of about 2.0 10−3 . Cf l number. The first attempts were made changing the CFL number, using a single grid generated with mf = 1. The CFL had been setted to 1.0, following the hints given by the manual. Trying to raise it, we found that Chapter 6. Preliminary study

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1.5 ’ycuts.dat’ 1

0.5

0

-0.5

-1

-1.5

-2 -4

-3

-2

-1

0

1

2

3

4

5

Figure 6.8: Cuts normal to y 1.5 ’xcuts.dat’ 1

0.5

0

-0.5

-1

-1.5

-2 -1.5

-1

-0.5

0

0.5

1

1.5

Figure 6.9: Cuts normal to x

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10 CFL 1.0 CFL 1.6

Pressure L2 residuals

1

0.1

0.01

0.001 50

100

150 Cycles

200

250

300

10 1 mesh 2 meshes 3 meshes

Pressure L2 residuals

1

0.1

0.01

0.001 50

100

150 Cycles

200

250

Chapter 6. Figure Preliminary study 6.10: Convergence history: CFL and multigrid

300

150

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10 1 mesh 2 meshes

Pressure L2 residuals

1

0.1

0.01

0.001 0

50

100

150

200

250 Work Units

300

350

400

450

500

Figure 6.11: Convergence history on work units

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the calculation did not converge for a value of 1.7; thus we decided to take a value of 1.6. The upper part in figure 6.10 shows the different corvengence history for the values of the CFL of 1.0 and 1.6. The convergence is quite slow, but it can be seen that after a slower start the higher CFL converges more rapidly. As the convergence is with a good approximation linear in the logarithmic diagram, we can estimate that to reach a value of the residuals of 2.0 10−3 we would have to make about 1000 cycles with CFL = 1.0 and 650 cycles with CFL = 1.6. The ratio 1000/650 gives 1.55: a value similar to 1.6, as we expected. Multigrid. The second step was to perform some multigrid computations. Using the mesh factors it is easy to create different meshes; we made a computation using 2 grids (found with mf = 1 and mf = 2) and another using 3 grids (found with mf = 1, mf = 2 and mf = 4). The convergence history is shown in the lower part of figure 6.10. Using 2 meshes the requested reduction of the residuals is reached in 300 cycles, while with 3 meshes it is never reached, because it stops at a value of 5.0 10− 3 after 150 cycles. Made curious by this fact, we realized that the other residuals had stopped their lowering after about only 100 cycles. The reason of the stopping of the residuals with 3 meshes is to be found in the fact that the mesh factor of 4 used is too big, because the corresponding spacing of the triangles cannot give a correct representation of the surfaces. The codes are powerful because the different grids can be not-nested, but in this case they have problems of intergrid transfers and do not converge any more after some cycles. The manual in fact suggest to start to build the coarsest mesh as the minimum necessary to represent the geometry, and then to find the finer grids dividing the spacing by a factor of 2 each time. We made the contrary because we wanted to exploit the available means at their maximum, so we tuned the mesh factors as to create the most complex possible mesh when having a value of 1. We therefore decided to use 2 grids. But the reducion in the number of cycles is not as good as represented in figure 6.10, where the comparison is made on multigrid cycles. A multigrid timestep requires more calculations than a single grid; while this required about 25 seconds of CPU time1 , the 2 grid timestep required about 41 seconds (and the 3 grid timestep about 48 seconds). To make a correct comparison we must transform the multigrid 1

Note that while making the calculations a second of CPU time corresponded to much more real time, as the machines worked in multitasking.

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cycles in work units, thus in single grid timesteps: we multiply the multigrid cycles by 41/25. The real comparison is then shown in figure 6.11. It can be seen that to achieve the desired reduction about 500 work units are necessary to the 2 grids, so the saving is only 150 cycles. Anyway with all the calculations that we made we can estimate to have saved more than 100 hours of CPU time by the use of 2 grids instead of 1. Note also that about 50 work units could be saved by starting the computation from the final values of the unknowns found with different free-stream conditions, but clearly on the same mesh. Anyway we decided not to employ this method, that would give problems of other kind. Finally, the convergence speed depends on the coefficients of artificial viscosity. We did not change them to make tries related to the convergence, but only for the reason explained in section 6.4.3. We realized by those experiments that higher values of such coefficients let the computation converge more rapidly, with a sacrifice in the accuracy as the situation moves away from the inviscid case. On the contrary smaller values bring to very slow convergence, or even to the unstabilization of the method. We then decided to keep the values suggested at the beginning.

6.4

Testing

The code has already been tested a lot by other people, but we felt necessary to perform some tests ourselves. We therefore searched for some experimental data on low aspect ratio wings in incompressible flows, as we have seen that the phenomena on such lift devices can be very different from the ones of the high aspect ratio wings. After a research we found a table on Principles of Naval Architecture, copyrighted by the Society of Naval Architects and Marine Engineers about experimental data on low aspect ratio rudders.

6.4.1

Geometry and mesh

Between the various configurations shown on the table we found, we chose the ones with aspect ratios 1, 2 and 3, sweep angle −8◦ , squared tip shape, taper ratio 0.45, constant section shape NACA 0015, Reynolds number 2.5 106 . Our geometry is shown in figure 6.12 for the aspect ratio equal to 2. It must be noted that on the table they always consider the aspect ratio of the Chapter 6. Preliminary study

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Z

X Y

Figure 6.12: Geometry of the test (AR=2) rudder together with its mirror image, that corresponds to an AR=1 with our system. In their experiments the rudders were tested against a groundboard with a gap of 1/200 of the root chord, that we have not considered at all. It was important to use the same criteria we had employed for the keel generator, so to demonstrate that the methods we were going to use on the yachts were precise. Thus to create the geometry and the mesh we have modified the program keel.for, creating testa.for. Basically the geometry is the same of figure 5.1 and 5.3. The surfaces representing the side of the hull have been eliminated, thus all the surfaces have been renumbered. To create the square tip a flat surface has been added, just like for the tips of the winglets (see for example figure 5.21). No heel angle is necessary. The mesh is regulated by 3 line sources that follow the edges of the rudder, 2 triangle sources put inside and 2 others put in the wake, just like the fin keel part of keel.for. We used the multigrid method with 2 grids. The finer grid, when the AR was 2, had 18126 nodes and 96344 tetrahedra.

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6.4.2

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Results and comparisons with the experimental data

For the three rudders we had the data related to: the lift coefficient curve slope at an angle of attack of 0◦ , the CL at 10◦ and 20◦ , the CLmax , the stall angle, the (L/D)max and the angle of attack for the (L/D)max , the L/D at the stall angle. We cannot make any observation on the stall: it is a phenomenon that we cannot consider with our means, thus all the data related to it did not help us; all the other data was instead useful. On each rudder we decided to make 5 computations, respectively at angles of attack of 0◦ , 5◦ , 10◦ , 15◦ , and 20◦ . All the calculations on the surfaces shown in this section have been made with the program lift.for; in section 6.4.3 there are some results we got with momentum.for. Table 6.5 summarizes the results we have found, that are also shown in the figures 6.13 to 6.17. The lift curve slope at the origin is very precise in all the three cases. The precision in the lift coefficient decreases while moving away from the origin; at 20◦ for the AR=1 and 2 the difference from the experimental values is big. The curves are concave instead of being convex, as it usually happens at low ARs (see figure 6.13). This is however not so important, because the leeway angles practically never exceed the 10◦ . The computed drag curves are very similar to parabolas, as we expected: the L2 norm of the differences between the measured values and the quadratic interpolation is of the order of 10−4 . This is also clear in figure 6.14, where the values we found are well fitted by the quadratic curves; in this case we had no direct corrispondence with the experimental data. The parabolas are found searching for the curves CDi = a + bα2 that minimize the L2 norm; the value of b can be found in table 6.5. Using the profile drag estimate of section 5.3.3 and the formulas of section 5.3.4 we found the maximum L/D and the angle of maximum L/D. Knowing that such angle was small, we have used only the CL and CDi results included between 0◦ and 10◦ . The results are in all the cases pessimistic for what concerns the maximum attainable L/D (figure 6.15). The angles at which the maximum is found is always smaller than the reality (figure 6.16). The correspondence is in general quite bad. Fortunately at least the relative results are not distorted; in other words the influence of the AR in our calculations is the same of the real: increasing the aspect ratio the maximum L/D increases, while the angle of maximum L/D decreases. This could be Chapter 6. Preliminary study

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AR=1

AR=2

AR=3

³

´

dCL dα α=0◦ dCDi d(α2 ) CL @ 10◦ CL @ 20◦

AReff L/Dmax α for L/D ³ ´ max dCL dα α=0◦ dCDi d(α2 ) CL @ 10◦ CL @ 20◦

AReff L/Dmax α for L/D ³ ´ max dCL dα α=0◦ dCDi d(α2 ) CL @ 10◦ CL @ 20◦

AReff L/Dmax α for L/Dmax

”Fluid Mechanics of Yacht Keels”

Our results 0.0224

Experiments 0.023

0.000304 0.221 0.411 0.51 6.75 5.2◦ 0.0396

— 0.25 0.59 — 8.0 8.0◦ 0.040

0.000399 0.394 0.736 1.23 10.3 4.5◦ 0.0570

— 0.42 0.91 — 12.2 6.0◦ 0.056

0.000435 0.566 1.07 2.33 13.8 4.2◦

— 0.56 1.09 — 15.8 4.6◦

Table 6.5: Comparison between our results and the experimental

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1.2

1

AR=1 AR=2 AR=3 Experimental

lift coefficient CL

0.8

0.6

0.4

0.2

0 0

5

10 angle of attack [degrees]

15

20

Figure 6.13: Lift coefficient curves useful for the comparison of different keels. What is more important for us, besides the lift curve slope, is the effective aspect ratio AReff , found with both the lift curve and the induced drag curve as explained in section 5.3.4. The AReff is smaller than the actual AR ; in figure 6.17 it is drawn together with the bisecting line, to see how much it is smaller.

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0.18

0.16

Induced drag coefficient

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0 0

5

10 angle of attack [degrees]

15

20

Figure 6.14: Drag coefficient curves

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16

14

Our results Experimental

12

Maximum L/D

10

8

6

4

2

0 0

0.5

1

1.5

2 Aspect ratio

2.5

3

3.5

4

Figure 6.15: Maximum L/D for the different aspect ratios

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10

Angle of maximum L/D

8

Our results Experimental

6

4

2

0 0

0.5

1

1.5

2 Aspect ratio

2.5

3

3.5

4

Figure 6.16: Angle of maximum L/D for the different aspect ratios

Chapter 6. Preliminary study

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3

2.5

Effective aspect ratio

2

1.5

1

0.5

0 0

1

2 Aspect ratio

3

4

Figure 6.17: Effective aspect ratio

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6.4.3

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Calculation using momentum.for

We tried to use the program momentum.for to calculate the force coefficients. Our idea was to test the precision of the calculations made with the other program, but we had a surprise. We had expected that a kind of error could be found changing the dimensions of the parallelepipedon on whose surfaces the momentum variation is evaluated, so we performed different tests, varying in 4 steps those dimensions from a minimum parallelepipedon that included almost only the rudder to a maximum with doubled dimension, thus with a volume 8 times greater. What we found is shown in figure 6.18. The errors due to the change of the parallelepipedon dimensions are evident: they can be seen better in the drag coefficient figure because the involved forces are in that case smaller. In the same figure, where the four curves are discernible, the bigger the parallelepipedon, the higher the curve. This alone would be a good reason for not using this method to calculate the forces: the position of the cuts influences the result. This error could be caused to the fact that the program has an approximation: on every triangle found with the cuts, as explained in section 5.3.2, the values of the quantities V~ and p are considered constant and equal to the average of the values at the vertexes. This brings to small errors if the triangles are small and thus there is not a big difference between the unknown values at the 3 vertexes; evidently, as the errors are present, the spacing in the volume where the cuts are made is not enough small. If this reasoning is correct, then the most precise calculation is made with a parallelepipedon close to the rudder, where the spacing due to the sources is still small. But there is another important fact: with no incidence the drag is not zero. This cannot be due to errors while cutting, because all the curves shown have about the same value for α = 0◦ . After many reasonings, we found that it could depend on the artificial viscosity. Acting, as its name says, like a viscosity, it creates forces that are not felt by the rudder with the integration of the pressures —program lift.for— but they appear like a loss in momentum. It would be the same with two kinds of experiments, one performed with static pressure measurement made on the rudder surface, the other made with velocity and pressure measurement made on a big parallelepipedon including also the rudder. Figure 6.19 confirms what we have said. Varying the parameter diss1, and proportionally the parameter diss1 too, the drag coefficient with no Chapter 6. Preliminary study

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0.9

0.8

using momentum.for using lift.for

0.7

Lift coefficient

0.6

0.5

0.4

0.3

0.2

0.1

0 0

0.16

5

10 angle of attack [degrees]

15

20

10 angle of attack [degrees]

15

20

using momentum.for using lift.for

Drag coefficient

0.12

0.08

0.04

0

5

Figure 6.18: Lift and drag coefficients calculated with the two programs Chapter 6. Preliminary study

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0.03

0.025

CD at alpha=0

0.02

0.015

0.01

0.005

0 0

0.1

0.2

0.3 0.4 0.5 ’diss2’ coefficient of artificial viscosity

0.6

0.7

0.8

Figure 6.19: Influence of the artificial viscosity

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incidence varies, almost linearly. It must not be forgotten that the artificial dissipation is of high order, thus its influence is not easily predictable as it is different from a real viscosity. Besides the boundary conditions are of impermeability and not of no-slippery, like it is in the reality. With a smaller viscosity coefficient, at α 6= 0◦ , the lift calculated with the integration of the pressure changes only slightly; the drag changes a bit more: in particular reducing the coefficient the drag is reduced. Being unreal, it would be better to keep the artificial viscosity as small as possible. Unfortunately, reducing such coefficients brings the calculation not to converge in some cases. For example, the reduction of the coefficient to 1/4 of the starting value does not create problems to the case with α = 0◦ , while with α = 10◦ lets the computation not converge. Besides, the convergence becomes terribly slow, if the coefficient is too small. Finally, the program momentum.for will not be used any more in this thesis. But it can represent at a future time an useful tool for evaluating how much the artificial viscosity interfere with the inviscid calculation; in particular it could be very useful to test different codes in the resolution of the same problem: the higher the drag found with our program, the worse the code.

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165

Chapter 7 Results In this chapter we want to show the results of the computations that we have performed on many different keels, with the help of many diagrams. We have started from the study, made at different heel angles, on the configuration with the hull and the conventional keel; then we have looked for the influence of the addition of many kinds of winglets on that same shape. The tested winglets differ for aspect ratio, sweep angle and dihedral angle. In the pages in which there are two figures, we will refer to the upper with the letter A and to the lower with the letter B.

7.1

Conventional keel

The computations have been made at angles of heel of 0◦ , 10◦ , 20◦ and 30◦ , at leeway angles λ of 0◦ , 2.5◦ , 5◦ , 7.5◦ and 10◦ . The first results, made integrating the pressures only on the surfaces of the fin keel, are shown in figure 7.1. The lift curve slope decreases as the heel angle increases; all the sets of results are fitted with a straight line, with a least square interpolation. The approximation, as we had already seen while making the tests on the code, is very good. The drag coefficient, which should represent the induced drag coefficient, has a peculiarity: when the leeway angle is zero, it has a considerable value of about 0.014 for all the heel angles. During the tests this had not happened, as we expected: with an inviscid code and without any lift the resistance should be zero. The reason of the presence of that resistance can be found with the help of the pressure coefficient plot shown in the figures A and Abis of appendix D and figure 7.6B. If we integrate the pressures on all the boat 166

Chapter 7 (pages 167 to 190 are omitted) Please contact: Marco Trucchi - [email protected]

Marco Trucchi - [email protected]

7.3

”Fluid Mechanics of Yacht Keels”

Some graphical representations and interesting thoughts

In appendix D we have put some images that represent the conventional keel we have studied and a T-shaped keel derived from the first with winglets of aspect ratio 1, no sweep angle and 20◦ of dihedral angle. The angle of heel is zero and the leeway angle is 5◦ for both the keels. In figure B and C the surface streamlines are shown. The windward side, in depressure, is shown on the left, while the leeward side, in overpressure, is on the right. It is evident what has already been explained in this thesis: the flow tends to go from one side to the other passing on the tip. In this way two winglets put as in figure C are invested by a flow very different from the free-stream, so they can produce particular forces: see section 2.7. Figures D and E can be confronted to see where the pressure is lower or higher on the keel surface. The vertical part seems to have the same pressure distribution in the two cases. It is very interesting to observe the winglet pressure distribution. On the left winglet the zone at low pressure is evident, and the winglet has an upward lift. The corresponding force on the right winglet should be downwards, and so it is in effect if we integrate the pressure, but it is not evident as on the right; in fact it is smaller. Everything is in perfect accord with what we have said in all this chapter. In the figures F and G we have made vorticity plots. The tip vortex can be seen in the two cases: in reality in the inviscid case there should be only a vortex layer, with no thickness, behind the keel, but due to the omnipresence of the viscosity in every computer code the flow behaves in some way like if it was viscous. We can understand qualitatively what are the main differences. The conventional keel has one core only; the T-keel has two cores, one for each winglet tip, but the left one is decisely stronger than the right one. This corresponds to the fact that the vertical forces produced on the winglets are of different intensity. The vorticity is stronger on the conventional keel than on the T-shaped one; on the latter the vortex seems to be dissipated earlier. Note that these two graphics are only indicative. Now let us look better at figure B, remembering that we have found many times that the most efficient winglet is the one attached on the depressure side (the windward side, left side in our calculations). It seems that on the overpressure side of the keel (figure B on the right) the flow near the keel tip is more deflected near the leading edge than near the trailing edge . The opposite happens on the depressure side of the keel (figure B on the left), Chapter 7. Results

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Figure 7.19: Winglets on an airplane where the flow is more deflected near the trailing edge. Moreover on this side the flow is more deflected than on the other, and this happens on a greater part of the tip chord. We could think that the streamlines that are most deflected are the same ones on both sides: they are first near the leading edge on the overpressure side, then they pass on the other side going around the tip and are near the trailing edge on the depressure side. Figure C compared with figure B shows that the deflection on the depressure side is prevented (and, we already know, exploited to produce upward lift and driving force) by attaching our kind of winglets; but on the overpressure side the deflected flow near the leading edge is almost the same in the two figures, because the winglet is not attached in that zone but behind. Considering Marchaj’s idea of figure 2.29 as true, as we have confirmed with our calculations, we could infer that to obtain the best performing winglets we should put them where the flow is most deflected: with short chord near the leading edge on the overpressure side and with longer chord near the trailing edge on the depressure side. We have to say, to be honest, that we have had this idea looking at the winglets often attached to the wing

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tips of the aeroplanes, like the ones of figure 7.19. In the figure it is clear that the most important winglet is the one on the depressure side; in fact it often happens on aeroplanes that there is no winglet attached on the overpressure side. Clearly the case of the yachts is different, as we have already explained: both sides of their keels must be able to be alternatively the depressure and the overpressure one, depending on the tack. Thus it is not possible to have one winglet different from the other, unless we want to privilege one sailing direction. Having to choose, it is clear that it is better to attach the winglets in the back, to exploit each of them at its maximum when it is on the depressure side.

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193

Conclusions We want to summarize here the work we have done, and the most important results we have found. First of all we fully examined which characteristics had to be considered important in a study like ours. We determined that some of them can be taken as representative of the performance of a yacht fin keel: the lift curve slope, the effective aspect ratio and the L/D curve. Other important parameters we considered are the CL of maximum L/D and the z-quote of the centre of effort. We then successfully wrote some flexible programs in standard Fortran 77. Two of them create, on the basis of user specified shape parameters, the files containing the geometry and the mesh characteristics that are needed by the FAN system, the set of codes we used to solve the 3D steady incompressible and inviscid flows around the submerged part of a yacht. The created shape is made up of a hull and a fin keel; two winglets can be attached to the keel tip. The yacht can be also heeled, up to a maximum angle of 30◦ ; the shapes attainable with the programs are satisfactorily similar to the actual. Other tools make calculations and analyze the results of the computations. In particular two programs evaluate the hydrodynamic forces and moments, the first integrating the pressure on the surfaces and the other applying the momentum theorem to a closed domain containing the same surfaces. The friction drag on the keel is evaluated in first approximation with an empirical method, based on the well known experimental results of [1]. We have tested on three low aspect ratio rudders our codes and methods by means of the comparison of the results with experimental data taken from an external source. The comparison showed that the inviscid results are very good for the usual (small) angles of incidence that can be found in normal sailing conditions on a keel. The results are more precise, as we expected, as the aspect ratio of the rudder rises. On the other hand we noted that the 194

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friction drag estimate was not very reliable; the results obtained with the use of such estimate were generally pessimistic if compared to the reality, because they showed too much drag force. We found a very interesting fact by using on the test geometry the program that makes the force calculations using the momentum theorem: we noted that the drag results depend on the value of the artificial viscosity coefficients. For this reason the program is unuseful to study the forces, but it can be used to evaluate the influence of the artificial viscosity of different flow solvers on the same problem, to understand which of them can be considered ‘the least viscous’. The main computations have been performed on an inverse tapered keel of aspect ratio 1.4, attached to a classic shape hull, with heel angles from 0◦ to 30◦ and leeway angles from 0◦ to 10◦ . On the same keel many winglets of different aspect ratio, sweep angle and dihedral angle have been attached to compare their performance. Our conclusions are that, despite the approximations, the methods we have used can represent a good tool for the hydrodynamic design of a fin keel, while they are inadequate for the hull: we neglected too important phenomena present near the water surface to obtain good results on it. We also found that studying a conventional keel without winglets it is possible to obtain quite precise results without making calculations at all different heel angles, but evaluating the forces just for the unheeled case and then applying simple formulas to find the forces at the other heel angles. This does not work for a T-shaped keel, in which the situation is very different when an angle of heel is present. For what concerns the positive effects of the winglets we found that adding such devices to a conventional keel the lift curve slope and the effective aspect ratio rise, while the centre of lateral resistance lowers. A higher improvement can be found in the effective aspect ratio if a angle of heel is present. In all cases with our estimate on the friction drag we found a worsening in the L/D curve. This characteristic could be considered even more important than the first ones, so it could let us think that there is no hydrodynamic convenience in attaching the winglets to a conventional fin keel; the reasons of the diffusion of T-shaped keels should be searched in the other positive features that they have. Anyway we do not have to forget that with a more correct calculation of the friction we could find that winglets improve also the L/D curve; our results about this should be checked again in other ways. The parametric study that we made attaching different winglets to the Conclusions

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same initial conventional keel showed that the variation of sweep angle is almost uninfluential; instead increasing the aspect ratio and the dihedral angle the winglet positive effects become more evident. Interesting thoughts have been made about the different behavior of each of the two winglets: we found that the most important is the one attached on the depressure side of the keel. But as a keel must be symmetric we do not have the possibility of eliminate the other one, unless the yacht can sail always on the same tack. With our calculations and flow and pressure visualizations we have understood why winglets attached to aeroplane wing tips usually have their peculiar shape. Possible future developments on this subject could consist in finding a better estimate of the friction drag and in evaluating the wave drag and the unsteady behavior; the results could then be tested in the towing tank. New calculations could be made to find the influence of the addition of different tip winglets to different kinds of conventional keel, especially with different aspect ratio.

Conclusions

196

Appendix A Details of the geometry definition In this appendix we want to go deeper inside the methods we have inserted in the keel.for and tkeel.for programs to create the geometry of the domain.

A.1

Hull and conventional keel

The definition of the support points for the curve and the surface components has to be carefully made if the interpolated curve or surface is to be an appropriate representation of the actual geometry. This is best explained via a practical example that shows some of the difficulties associated with the definition of the splines. In figure A.1a a bad choice of support points for a curve spline interpolation is shown: the interpolated curve differs a lot from the exact curve we wanted to represent. Besides this error in the shape, the splined curve has kinks that create problems while generating the mesh for the computation. In our case if a yacht surface has kinks due to the interpolation, the module ST cannot triangulate it. To avoid this kind of problems the distribution has to be done with care, putting for example more points where the curve —or surface— has a faster variation in direction: this is clear in figure A.1b. Another thing that must be born in mind is that the triangular mesh generated from the surface triangulation module tends to be distorted if the quadrangles of the surface component grids are stretched. The warnings for this problem appear if the ratio between the dimensions of the quadrangles 197

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Figure A.1: Selection of support points for curve interpolation is more than 20. With our program the warnings usually do not appear, or if they appear they show a stretching never greater than 23, so they can be ignored with sufficient safeness. Curve components. The first thing that the program keel.for does is to create the curve components, to which the curve segments correspond directly. To do this, it makes some geometric calculations with the parameters inputted by the user to find the coordinates of the junctions of the curves. These, besides the two fixed points of the keel root section placed in (0,0,0) and (1,0,0), are: the vertexes of the parallelepipedon, the bow, the stern, the four points shown with circles in figure 5.3. Our advice is to follow the following reasonings while looking at the figures at the beginning of this chapter. Parallelepipedon edges. The 8 vertexes of the parallelepipedon, plus the two points at the junction of the segments 9, 26 and 28 and the segments 11, 27 and 24, are found with simple geometric relations rotating an initial cuboid of the heel angle, anticlockwise if seen from behind. The dimensions of this cuboid can be inserted directly into the program, in the variables xm (maximum x value), ym (maximum y value) and zm (maximum depth). Otherwise the program finds their values for them, depending on the yacht dimensions and on the variable farfield, so it is possible varying the latter to create different domains quickly but with reasonable ratios between

Appendix A. Details of the geometry definition

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the dimensions. The relations are: xm = ym = (xstern − xbow ) · farfield zm = 1.7 · (yacht draft) · farfield .

(A.1) (A.2)

Following the FELISA system [18], that is similar to the FAN system in the definition of the geometry but it solves steady compressible flows, we would put a value for farfield of 4.0. We will see that this value is exaggerated for our study. For the parallelepipedon edges, like for all the other straight segments, it is sufficient to give the two limit points. For the other curves it is necessary to give a certain number of points, as we have described before. It is convenient that the points of a curve segment coincide with the points of the surface that has that segment as boundary, otherwise the different splining may let the ST module generate errors. This is the easiest way to avoid the problem, and it is the one used in all this program. When adding the winglets in the program tkeel.for we will use another method for some of the segments. Keel root section (segments 1 and 7). For the definition of the shape of the profile we have used equation A.3, taken from [1] and valid for the thickness of the four-digit NACA profiles: y=±

√ t (0.2969 x − 0.1260x − 0.3516x2 + 0.2843x3 − 0.1015x4 ) , (A.3) 0.2

where t is the maximum thickness (e.g. for the NACA 0012 profile t = 0.12). The value for the z-coordinate is 0 for all the points. We have used a distribution of np=41 points suggested by Prof. J. Peir´o, made to minimize the difference between the splined profile and the real profile. The user can change the number np of the points, and modify the file naca0012.dat from which the abovementioned point distribution is taken to insert any wanted profile. Hull section at water level (segments 20 and 21). To let the connection between the surfaces easier, the hull sections are defined by a set of np points, just like the keel. Here the shape depends on the parameters of figure 5.5. The profile is found with two cubic curves: the first goes from the bow to the maximum width point, the second from there to the stern. The cubic has the classic form: y(x) = ax3 + bx2 + cx + d , Appendix A. Details of the geometry definition

(A.4) 199

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Figure A.2: Support cubic for distribution of points so it needs four conditions to define the variables a, b, c and d. Two conditions are found by imposing the passage or the cubic in two points (for example, for the first cubic, the bow and the maximum width point), other two are found by imposing the value of the derivative y 0 (x) in the same two points (e.g. tan β1 in the bow and 0 at the maximum width point). The value for the z-coordinate is constant and it is given by the user (hull draft). The distribution of points is in this case regular, that is the ∆x interval is constant. Segments 29 and 25. These are obtained by rotating and scaling the curves 20 and 21, in a way such that their points of maximum width pass in the points found while defining the midsection, in the paragraph ‘Heel angle and radius factor’ of section 5.1.1. Note that it is important that the curve 29 has the last ID number, because it disappears in case of maximum heel (Θ = 30◦ ), and the FAN system does not allow to skip any ID number, as it would happen if it was not the last. Hull longitudinal curves (segments 22 and 23). Each of these curves is found with an interpolation equal to the one used for segments 20 and 21, just explained. The only difference is that the cubic is in this case in the form z = z(x), and both the derivatives depend on the user’s parameters. For all the points it is y = 0. The distribution is not regular, for this reason: trying to keep the quadrangle of the grids of the surfaces 9 and 11 not stretched, but being in them the longitudinal dimension reduced from the big value of the segments 29 and 25 to the small of the segments 1 and 7, we have to reduce the spacing near the keel connection. To achieve this, the user can insert the values angle3 and angle4 in the program; they correspond respectively to γ1 and γ2 of figure A.2. The cubic in the figure, passing for (0,0) and (1,1), depends only upon the value of the Appendix A. Details of the geometry definition

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Figure A.3: Cubic for the segments 4 two angles. Varying these, it is possible to transform a constant distribution of points (on the ξ-axis of the figure), to a distribution with more or less points distributed near 0 or 1 (on the ψ-axis). A low value of γ1 brings more points near 0, and the same happens with γ2 near 1. In this case the irregular distribution is made on the z-coordinate; the z of the water surface corresponds to ψ = 1, while z = 0 corresponds to ψ = 0. The default values are angle3= 2◦ and angle4= 70◦ . The values of the z-coordinate are the same for both the segments, in order to have an ordinate grid on the surfaces 9 and 11. As we have a nonreversible function x = x(z), for every given value of z the program uses an iterative method of bisection to find the corresponding x. Keel tip rounding (segments 4 and 5). Both the segments are defined by a cubic curve. The problem was that it was not possible to express it like with z = z(x) or x = x(z) like for the curves 20 and 22, being the orientation unfavorable in many possible cases, like the one shown in figure A.3 for the segment 4. The cubic splining is made using the auxiliary axes (x0 , y 0 ) with origin O0 , rotated in that particular way to obtain always the possibility of finding a function y 0 = y 0 (x0 ) that describes the segment 4. The coordinates of the points of the curve found are then retranslated into the normal coordinates (x,z). For all the points y = 0. The distribution is found from a regular spacing made on the x0 -axis. For the segment 5 the method is nearly the same. The main difference consists in the fact that for all the points we want the same z-coordinates of the curve 4, in order to have an ordinate grid on the surfaces 1 and 2. So an iterative method, similar to bisection, is used to find the points belonging Appendix A. Details of the geometry definition

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to the cubic curves in x0 and y 0 coordinates, that, when retranslated in the normal coordinates, give exactly the z-coordinate wanted. Surface components. To understand how the surfaces are defined, it is advisable to follow the reasonings of this section on the figures put at the beginning of this chapter. It must remembered, even if it is no more mentioned, that the points of each grid must be ordinated as explained in section 4.2 so that the normal to the surface points inside the domain. Faces of the parallelepipedon. These surfaces are easily defined. Being flat with no curvature, it is sufficient to give the coordinates of the four vertexes corresponding to each face. No more efforts are therefore needed. Surfaces 10 and 13. To define these surfaces, the defining grid is formed by np points in the longitudinal direction. In the vertical direction the number of points depends on the angle that the surface has to represent, that on its turn depends on the heel angle. The grid is formed by successive rows of np points, found rotating and scaling the curve 20 so that the maximum width of each row corresponds to a point of the midsection curve already defined. As it can be seen from the figures, the first and the last point of each row are always in the bow and the stern. The angle between two different rows is equal to the deth degrees, being deth a parameter; the default is 7.5 degrees, found with some attempts in order to create not stretched quadrangles. Like for curve 29, the surface 13 must have the last ID number, as it disappears when the heel angle is 30 degrees. Surfaces 9 and 11. First of all the program defines the shape of the midsection curves, with an interpolation method very similar to the one used for the curves 4 and 5. The midsection shape is visible on any of the views from behind the yacht. On such curve the same number of points of the curves 22 and 23 is distributed, with the same distribution function; this however does not mean that the points are at the same height, as the difference of z-coordinate to cover with the distribution of points is smaller in the midsection curve than in the other two curves. Like for the surface 10 and 13, the grid is formed by rows of np points. The shape of each row is a weighted average between the shapes of the curve 29 or 25 and the keel root profile (curve 1 or 7). The weight of each curve decreases while moving away from it. It is possible to regulate how much the influence of each curve spreads towards the other by changing the two parameters angle1 and angle2, like in figure A.2. The parameter angle1 Appendix A. Details of the geometry definition

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regulates the influence of the keel root section: the smaller the parameter is, the more the influence spreads. The same happens for the other parameter with the curve 29 or 25. Each shape is then rotated and scaled so that it passes on three corresponding points of the curves 22, 23 and the midsection. The result is excellent. It can be noted that between the upper and the lower surfaces that define each hull side, thus between the surfaces 13 and 9 or the surfaces 10 and 11, there is smoothness only in corrispondence of the midsection, where in fact all the splining is made analitically. In the rest of the junction there is an edge, more emphasized near the bow and the stern; this is a consequence of the particular shape of the upper surfaces, made thinking only about the heel rotations and their related problems. Anyway this should not affect much our calculations, both for the fact that the edge is not so prominent and, again, the waves are not represented in our study; if at a future time a code is found that computes the wave drag, then it will be necessary to redifine in some way the upper surfaces of the hull side. Keel surfaces (surfaces 1 and 2). Here again the surfaces are defined by rows of np points, each representing a NACA 00XY profile. The number of rows of the non-rounded part is regulated automatically by the parameter nw1; as the chord length of the keel root section is 1, and it is defined by np points, then we should expect to find a correct grid spacing putting nw1 = np · (keel draft); and so it is. Like for the other yacht lateral surfaces, it is first necessary to define the shape of the curve that passes through the points of maximum thickness of all the sections. This curve is easy to define in its straight part, between the root section and the last section before the tip rounding, because the user enters the XY digits of the NACA 00XY profiles of both the sections. In the tip rounding the points are distributed on the z-coordinate like on the curves 4 and 5; this part of the curve is found with a cubic spline, with the same method employed for the curve 5. Once that the curve of the maximum thicknesses is defined, it is trivial to define the surface. Each row is in fact horizontal, its first and last points are respectively on the trailing and on the leading edge, and its maximum thickness point must be on the defined curve. The result is a perfectly smooth keel, with a realistic representation of the tip. Curve segments and surface regions. We have already said, but we repeat here for sake of clarity, that in our study the curve segments correspond Appendix A. Details of the geometry definition

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Region 1 2 3 4 5 6 7 8 9 10 11 12 13

”Fluid Mechanics of Yacht Keels”

Curve segments 1, 2, 4, 6, -5, -3 -7, 3, 5, -6, -4, -2 9, 26, -20, 27, -11, -8 13, 17, -28, -9, -16 -12, 16, 8, -19 -15, 19, 11, 24, -18 14, 18, -10, -17 12, 15, -14, -13 29, -22, -1, 23 25, -21 -23, 7, 22, -25 -27, 21, -26, 28, 10, -24 20, -29

Table A.1: Connectivity of surface regions directly to the curve components, and no rigid motion is employed. On the other hand, the definition of the surface regions require, besides indicating the component on which each of them is placed, the ordered set of curve segments that form the closed loop around the region, following the right-hand rule (see figure 4.5 on page 95) and remembering that all the normals to the surface components are pointed inside the domain. Each region is placed on a dedicated surface component, except the regions 3 and 12, that are placed on the same plane representing the top face of the parallelepipedon. In table A.1 it is shown the connectivity of the regions. The table is valid for heel angles smaller than 30◦ ; for Θ = 30◦ , in fact, the surface 13 disappears; besides, disappearing also the curve 29, the loop defining the surface 9 changes and becomes: 20, -22, -1, 23.

A.2

T-shaped keel

The most difficult part to be generated is the winglet junction. We will explain the methods only for the left side, but clearly on the right side the geometry is perfectly mirrored. The description will be brief; to know all the details it is advisable to read the program list. As the winglet section are defined —to obtain a NACA profile— with Appendix A. Details of the geometry definition

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the same distribution of points of the vertical part, this means that that the junctions (surfaces 13 and 14) must have the same number of points np of the profiles. This is a problem because the winglet connects on a segment with a smaller number of points, being this segment only a part of the keel tip chord (see figure 5.21). The trick we use is to modify the lateral surface of the vertical part: this surface flattens smoothly in the back while approaching the junction, till it arrives to curve 29 which is a straight line. The same happens for the curve 34. The shape of the vertical part is not affected too much by these modifications because the back of the keel profiles is usually already flat enough. We have made some attempts and we have find that it works fairly well even with high percentages of coverage of the keel tip. Being the curve 29 straight it is easy to place on it the np points, regularly distributed. The curves 30 to 33 follow the lines of two columns of the lateral part surface grid; in particular the curves 31 and 33 follow the backest line, thus the trailing edge. The curves 3 and 5 are redefined so to end where the curves 31 and 33 end respectively. The surfaces 15 and 16 are defined with exactly the same method used for the linear part of the surfaces 1 and 2: here, in fact, the leading and trailing edges are two straight lines. The curves 35 and 36 are the intersection between the winglet surfaces and two auxiliary planes, such that both the curves are leaned outwards; they are found with analitical geometry calculations applied to every column of the winglet surfaces. The surface 13 is then an interpolation between the curves 29, 35, 30 and 31. The same happens for the surface 14 with the curves 32, 33, 34 and 36. It must be noted that it is also possible to modify the vertical dimension of the patch by toggling the internal parameters pripia and ulpia, moving also the vertical position of the winglet junction. The winglet tip is not rounded; so the surface 17 is defined by mean of 4 only points, being it part of a plane. That plane is always vertical, even if a dihedral angle is present. Finally, the connectivity of the redefined surfaces 1, 2 and 9 and of the new surface regions is shown in table A.2. For the others see table A.1. If Θ = 30◦ the surface 23 and the curve 51 disappear; the connectivity for the surface 9 becomes 20, -22, -1, 23

Appendix A. Details of the geometry definition

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Region 1 2 9 13 14 15 16 17 18 19 20 21 22 23

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Curve segments 1, 2, 4, 6, -5, 34, -32, -30, -29, -3 3, 41, 42, 43, -44, 5, -6, -4, -2, -7 51, -22, -1, 23 29, 30, -35, -31 36, 32, -34, -33 35, 37, -39, -38 -37, -36, 38, 40 39, -40 -41, 31, 45, -42 44, -43, -46, 33 -47, -45, 48, 49 46, 47, -50, -48 50, -49 20, -51

Table A.2: Connectivity of surface regions — T-shaped keel

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Appendix B Conversion factors

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Symbol Length mil in ft nm Speed kn Mass lb ton Force lbf ton

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Name

SI equivalent

mil inch foot nautical mile

2.54 10−5 m 0.0254 m 0.3048 m 1852 m

knot

0.5144 m/s

pound long ton

0.4535924 kg 1016.046976 kg

pound-force 4.448222 N long ton-force 9964.01728 N

Appendix B. Conversion factors

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Appendix C Computer files and their availability For additional information related to this final project please contact: Marco Trucchi [email protected] +33.(0)6.81.23.37.17 List of the Fortran 77 program files, and files related to the thesis: File name keel.for tkeel.for testa.for lift.for momentum.for profile.for abeta.for *.tex *.ps *.eps thesis.ps

Size 44 Kb 68 Kb 22 Kb 16 Kb 23 Kb 13 Kb 657 b 350 Kb 20.7 Mb 21.4 Mb

Description Hull and keel generator for the FAN system Hull and T-keel generator for the FAN system Test geometry generator for the FAN system Force and moment calculations integrating the pressure Force calculations using the Momentum Theorem Geometry slicing and pressure profiles Free-stream velocity angles LATEX 2.09 files of this final project Postscript figures of this final project Postscript file of this whole final project

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Appendix D Some figures The explanation of the figures shown in this appendix can be found in section 7.3 (page 191).

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FIG.A, λ = 0◦

Appendix D. Some figures

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cp:

-1.52

-1.15

-0.79

-0.42

-0.05

0.32

Y

Z

X

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FIG.B, λ = 5◦

Appendix D. Some figures

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Appendix D. Some figures

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FIG.C, λ = 5◦

Appendix D. Some figures

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Appendix D. Some figures

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FIG.D, λ = 5◦

Appendix D. Some figures

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-1.30 -1.52

-0.86 -1.08

-0.42 -0.64

0.02 -0.20

0.47 0.25

Y

X

Z

cp

Y

Z

X

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FIG.E, λ = 5◦

Appendix D. Some figures

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-1.30 -1.52

-0.86 -1.08

-0.42 -0.64

0.02 -0.20

0.47 0.25

Y

X

Z

cp

Y

Z

X

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FIG.F, λ = 5◦

Appendix D. Some figures

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Appendix D. Some figures

0.48 0.37

0.59

0.70

0.92 0.81

1.03

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1.14

Vorticity

1.25

Z

Y

X

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FIG.G, λ = 5◦

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0.48 0.37

0.59

0.81 0.70

1.03 0.92

1.25 1.14

Vorticity

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X

Z

Y

Z

Y

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Bibliography [1] Abbott I.H., Doenhoff A.E. von, Theory of wing sections: including a summary of airfoil data, new ed., New York, Dover, 1959 [2] Bussi G., La spinta e il suo costo, dispense del corso di Motori per Aeromobili, Politecnico di Torino, Levrotto & Bella, 1981 [3] Chiocchia G., Lezioni di aerodinamica instazionaria, dispense del corso di Principi di Aeroelasticit`a, Politecnico di Torino, 1996 [4] Colella P., Puckett E.G., Modern numerical methods for fluid flow, draft downloaded from Internet, November 1994 [5] Computational Dynamics Research Ltd., FAN System, version 1.0 — User’s Manual, Swansea, 1995 [6] Cone C.D., The theory of induced lift and minimum induced drag of non-planar lifting systems, NACA Technical Reports, R–139, 1962 [7] Farmer J., Martinelli L., Jameson A., A fast multigrid method for solving incompressible hydrodynamic problems with free surfaces, AIAA–93–0767, AIAA 31st Aerospace Sciences Meeting, Reno, January 1993 [8] Garrett R., The symmetry of sailing, London, Adlard Coles Nautical Ltd, 1987 [9] Germano M., Appunti di Gasdinamica, dispense del corso, Politecnico di Torino, 1994 [10] Hammitt A.G., Technical yacht design, St. Albans, Adlard Coles Nautical Ltd, 1975

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[11] Hirsch C., Numerical computation of internal and external flows, Vol. 1 and 2, Chichester, Wiley, 1990 [12] Kramer M., Boundary layer stabilization by distributed damping, Journal of Aerospace Science 24, 1957 [13] Lausetti A., L’aeroplano – Progetto, strutture, installazioni, Torino, Levrotto & Bella [14] Marchaj C.A., Aero-hydrodynamics of sailing, 2nd ed., London, Adlard Coles Nautical Ltd, 1988 [15] Marchaj C.A., Sailing theory and practice, 2nd ed., London, Adlard Coles Nautical Ltd, 1988 [16] Mattioli E., Aerodinamica, 3a ed., Torino, Levrotto & Bella, 1992 ´ J., The simulation of 3D incom[17] Peraire J., Morgan K., Peiro pressible flows using unstructured grids, chapter 16 in Frontiers of Computational Fluid Dynamics 1994 (Caughey D.A., Hafez M.M. Eds.), John Wiley & Sons, 1994 ´ J., FELISA System, version 1.1 — [18] Peraire J., Morgan K., Peiro User’s Manual, London, 1994 [19] Phillips-Birt D., Sailing yacht design, 2nd ed., 1972 [20] Quori F., Aerodinamica, Torino, Levrotto & Bella, 1993 [21] Skene N.L., Skene’s elements of yacht design, 8th ed., 1981

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