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SailShapeOptimizationwithCFD

THESIS·AUGUST2013

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StigKnudsen

TechnicalUniversityofDenmark

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Sail Shape Optimization with CFD

Stig Staghøj KnudsenAugust 2013

Preface

This Master Thesis is done as part of a Master in Mechanical engineering at DTU MEK.The thesis is done at the section of Fluid Mechanics, Coastal and Maritime Engineering(FVM) in collaboration with OSK-Shiptech, X-Yacht and North Sails. The aim of thethesis is to set up a computational network for optimization of sail design.

The project has received help and input from many people. A thanks to Søren Thys-trup from X-Yacht for providing detailed information and drawing of the X-40. Thanksto Heine Sørensen from North Sails for preparing the initial design of the sails includingdeformation analysis. Thanks to Michael Richelsen for providing valuable inside knowl-edge and great advices. And a special thanks to Kasper Wedersøe for helping with theprocess and contacts. Thanks to OSK-Shiptech for providing the necessary computationalpower at times of low working load on the company cluster. Thanks to Milovan Pericand Albert Gascon from CD-Adapco for advice and support for the simulation setup.Thanks to Mattia Brenner, Claus Abt and Konrad Lorentz for support with FriendshipFramework. Thanks to Frank Pedersen, Casper Schytte, and Brian Kerr for feedback onthe report. And finally thanks to my supervisor Professor Jens Honore Walther for greatcooperation, constructive conversations and valuable inputs to the project.

Keywords: CFD, VOF, STAR-CCM+, Sail design, optimization, Sail Boat, FriendshipFramework

Abstract

The design of sailboat sails are of great interest in the world of yacht racing. The appear-ance of modern simulation technology makes it possible to predict the aerodynamic forceson the sail and the hydrodynamic forces of the water on the sailboat. The purpose of thisthesis is to find a way to optimize design of sailboat sails through the use of simulationbased design. An X-40 sailboat is used as test boat for the optimization with a set of sailsdesign by North Sails DK as baseline.

A CFD model is presented for simulating both aero- and hydrodynamic forces andthe movements of the boat. Traditionally the performance evaluation of sail boats aredone separately from the simulation of forces, but in this case the velocity, leeway angleand heel angle prediction is built into the simulation to reduce the number of simulationsneeded. The simulations are performed in the commercial CFD software STAR-CCM+with the VOF model to handle the water/air interface and a 6-DOF rigid body motionsolver to handle the movements of the boat.

The goal is to optimize the shape of the head sail in a wind range of 4 to 8 m/s. Firstlythe trim and true wind angle is optimized for optimum VMG. Secondly the shape of thesail is optimized by reduced parametric modelling. In both cases the SOBOL algorithmwas used to investigate the design space followed by a pattern search method to refinethe variables further to the optimum VMG.

The trim optimisation showed improvements in VMG from 0.9 to 1.5 % over the windrange and the main trend was a reduction in true wind angle and trim angle of the sails.The shape optimization showed an overall improvement of 0.7 % on VMG. The sail shapecamper was increased, and the camper was moved forward in the lower and middle partof the sail, and backward in the upper part of the sail.

The computational power required by this method and the associated expenses arelarge compared to the budgets of most racing teams. But with the decreasing prices andincreasing performance of simulations the perspective of this sort of optimization couldgrow. The increasing commercial interest in using sails and kites to assist propulsionof commercial ships could lead to an increased interest in optimization of sail and kiteperformance.

Synopsis

Design af effektive sejl til sejlbade har stor interesse indenfor sejlsport. Moderne teknologihar gjort det muligt at forudsige de kræfter, der virker pa baden i form af aerodynamiskekræfter fra vinden og hydrodynamiske kræfter fra vandet. Formalet med dette kandidat-speciale er at finde en metode til optimering af sejl baseret pa simuleringsdrevet design.Sejlbaden X-40 fra danske X-yacht er anvendt som testbad til optimeringen, med et sætsejl designet af North Sails DK som udgangspunkt for optimeringen.

En CFD model til at forudsige aero- og hydrodynamiske kræfter samt bevægelsen afsejlbaden bliver præsenteret. Normalt sker hastigheds forudsigelsen af sejlbade separatfra simulering af kræfterne, men i dette tilfælde bliver forudsigelsen af hastighed, af-driftsvinkel og krængningsvinkel inkluderet direkte i simuleringen for at reducere antalletaf simuleringer. Simuleringerne udføres i det kommercielle CFD program STAR-CCM+,hvor en VOF model anvendes til at beskrive interaktionen mellem luft og vand og en6-DOF løser handterer bevægelserne af baden.

Malet er at optimere formen pa forsejlet til et vindomrade mellem 4 og 8 m/s. Førstoptimeres trimmet af sejlet og den sande vindvinkel for at opna optimalt VMG for degivne designs. Dernæst optimeres formen af sejlet ved reduceret parameter modellering.I begge tilfælde anvendes SOBOL algoritmen til at undersøge forskellige kombinationer afparametre efterfulgt af en patern search optimering for at justere variablerne til optimalVMG.

Trim optimeringen gav en forbedring af VMG pa 0,9 til 1,5 % afhængig af vindstyrken,og tendensen var at det optimale var en reduktion i sand vindvinkel og trim vinkel af se-jlet. Optimeringen af sejlfaconen gav en forbedring pa 0,7 % pa VMG. Dybden af sejletblev større, og dybden blev flyttet længere frem i bunden og midten af sejlet, mens denblev flyttet længere tilbage i toppen.

Metodens krav til beregningskraft og de dertilhørende udgifter er store sammenlignetmed budgetterne for de fleste kapsejladshold. Men de faldende priser og stigende effek-tivitet af numeriske simuleringer kan fa denne type optimering til at blive mere interessanti fremtiden. Den stigende kommercielle interesse i at bruge sejl eller drager til at hjælpefremdriften af kommercielle skibe kan ligeledes føre til en øget interesse om optimering afsejl og drager.

Contents

1 Introduction 1

2 Boat 32.1 Sail and rigging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Full Scale Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Theory 63.1 Sailboat Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.1.1 Apparent Wind and VMG . . . . . . . . . . . . . . . . . . . . . . . 63.1.2 Aerodynamic forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1.3 Hydrodynamic forces . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1.4 Heeling moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1.5 Righting moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Balance of forces and moments . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Sail shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Trim variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.4.1 True wind speed and angle . . . . . . . . . . . . . . . . . . . . . . . 123.4.2 Sheet tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4.3 Sheeting position . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4.4 Forestay tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4.5 Halyard tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.5 CFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.5.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . 133.5.2 RANS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.5.3 Turbulence modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 133.5.4 Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . . . 143.5.5 Volume of Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.5.6 Fluid Structure Interaction . . . . . . . . . . . . . . . . . . . . . . . 153.5.7 Wind profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5.8 Verification and Validation . . . . . . . . . . . . . . . . . . . . . . . 16

3.6 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.6.1 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . 173.6.2 Optimization Methods . . . . . . . . . . . . . . . . . . . . . . . . . 183.6.3 CFD-based Optimization . . . . . . . . . . . . . . . . . . . . . . . . 193.6.4 Pattern Search Algorithms . . . . . . . . . . . . . . . . . . . . . . . 193.6.5 SOBOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Method 204.1 CFD model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.1.1 Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.1.2 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.1.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 244.1.4 Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 Friendship Framework Model . . . . . . . . . . . . . . . . . . . . . . . . . 264.2.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2.2 Trim parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2.3 Shape parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2.4 Optimization algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 294.3 Computation Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Verification and Validation 315.1 Validation of aerodynamic forces in CFD model . . . . . . . . . . . . . . . 31

5.1.1 Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.1.2 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.1.4 Validation Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2 Verification of full model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2.1 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2.2 Temporal Discretization . . . . . . . . . . . . . . . . . . . . . . . . 415.2.3 Domain size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2.4 Moment of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2.5 Simulation Scatter . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.3 Full scale data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6 Sail Trim Optimization 476.1 4 m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.1.1 Design space evaluation . . . . . . . . . . . . . . . . . . . . . . . . 476.1.2 Pattern search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.2 6 m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.2.1 Design space evaluation . . . . . . . . . . . . . . . . . . . . . . . . 546.2.2 Pattern Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.3 8 m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.3.1 Design space evaluation . . . . . . . . . . . . . . . . . . . . . . . . 616.3.2 Pattern Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

7 Sail Shape Optimization 687.1 Design Space Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.2 Pattern Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

8 Conclusion 84

9 Perspective 85

Appendix 87

1 Introduction

Sailboats are powered by the wind, and the physics behind the boats are a complex in-teraction between aero and hydrodynamic forces. The sails act as thin aerofoils providingaerodynamic lift and drag as seen in figure 1. The shape of the sail has a great influenceon the production of aerodynamic forces.

Wind

Boat Speed (V)λ

Lift

Drag

Figure 1: Sail forces

Sail shape has been optimized over many years by experience and trial and error. Butno perfect shape has been found because the shape depends on the rest of the boat andon the conditions which the boat is sailing in. Detailed automated optimization has beenconducted only for high budget racing campaigns.

The common approach to optimization is to reduce the complexity and optimize merelyfor driving force with a penalty of side force, [1]. It is however difficult to determine thepenalty of the side force. A more accurate approach is to do full velocity prediction of theboat by a Velocity Prediction Program (VPP). The program seeks to find a steady statesolution where all forces are in equilibrium. The hydrodynamic and aerodynamic forcescome from empirical models, experiments or simulations. This requires data availablefor variation of different parameters such as boat speed, wind speed, leeway angle, windangle, trim parameters and heel angle. This gives a large matrix of parametric variationto consider.

A new approach is to move the velocity prediction inside of the simulation softwareand predict velocity, heel angle and leeway angle as part of the CFD solution. The aim ofthis thesis is to automate the optimization process and find optimization approach to saildesign, with velocity prediction included in the simulation. The performance of the sailis evaluated by evaluating the performance of the boat. To get the full boat performanceboth aero and hydrodynamic forces are modelled. The interaction between these forcesare modelled by a 6-DOF solver in the CFD software.

The trim of the sails and the steered angle has a large impact on the performance,and thus these are first optimized to find the optimal sailing condition for the present

1

design. The shape of the sail is varied by use of reduced-parametric modelling. Instead ofmodelling points or control point directly connected to the surface, a secondary controlsurface is used to transform the shape. This reduces the number of parameters and thusthe number of evaluations needed. The commercial software Friendship Framework areused for this transformation, for the trim variations and for the optimization. Based onthe performance of the sail in different conditions the shape is optimized to get the bestoverall performance.

As a test case for this project a 40 foot performance cruiser, X-40 from X-yacht, isused. The boat is simulated at 3 different wind strengths for the upwind sailing condition.The goal is to optimize the head sail over the range of these wind strengths for optimalupwind performance. The optimization is limited to the head sail to reduce the numberof variable. For reference of special term, symbols and abbreviations please see appendixA.

2

2 Boat

The example boat used in the current project is an X-40, designed and produced byX-Yacht. The main particulars of the boat are listed in table 1. X-yacht delivered thegeometry of the hull and superstructure in 3D IGES format along with 2D drawings ofrigging and appendages. From these a full 3D model of the X-40 with rigging was created.The model of the hull and appendages is presented in figure 2.

Figure 2: 3D model of hull and apendages

LWL 10.63 mLOA 12.20 mB 3.80 mT 2.10 mTc 0.41 m∇ 7.26 m3

∇c 7.06 m3

Wcrew 885 kg

Table 1: Main particulars

2.1 Sail and rigging

North Sails Denmark provided the baseline design of the sails for the study. The baselinecases are sail shapes of the deformed sails. The deformation is found by a coupling betweena panel CFD method and a membrane FEM model. The sails are subjected to the truewind speed and direction of the baseline case, and the boat speed and heel angle aremodified to fit the sail forces calculated by a panel code. The trim of the sail is adjustedby the sail designer by adjusting the sheeting position. The deformation of the mast isalso included in the model and implemented in the baseline cases. Figure 3 shows anoutlay of the sail and rigging and table 2 shows some of the main rig and sail dimensions.

3

Figure 3: Rig and sails

E 5.60 mP 15.70 mLP 4.56 mTmax 16.57 m

Table 2: Rig and sail dimensions

2.2 Full Scale Comparison

The sail design from North Sails is a copy of the sails made for the X-40 Sirena, shownin figure 4. Full scale data from the boat is available used for rough comparison. Thedata was recorded from racing during the last couple of years fitted by a VVP evaluationsoftware. The data was delivered by crew member Kasper Wedersøe.

4

Figure 4: X-40 Sirena

5

3 Theory

3.1 Sailboat Dynamics

The dynamics of sailboats are a complex interaction between aerodynamic, hydrodynamic,gravity and inertia forces. In the following paragraphs the different forces affecting thesailboat are described. The forces result in motions, and because the boat is free to movein any direction and rotate around any axis a system of six principal motions are used.These principal motions are shown in figure 5 taken from [2]. For the sailboat to reachequilibrium, forces must balance in all motion directions.

Figure 5: Principal motions. Taken from [2]

3.1.1 Apparent Wind and VMG

When the boat moves it experiences a wind different from the wind experienced whenstationary. This experienced wind is called apparent wind. The apparent wind is a vectorsum of the true wind vector and the boat speed vector as shown in figure 6. The apparentwind speed (AWS) is for upwind sailing larger than the true wind speed (TWS). Thismakes fast boats capable of sailing faster than the wind and in popular term the boat issaid to generate wind for itself.

6

Apparent wind

True

win

d

Boat Speed (V)

TWA AWA

Figure 6: Apparent and true wind

Sailboats are not capable of sailing directly against the wind. They typically sail at atrue wind angle between 30 and 40°. Due to side force generation the boat moves slightlysideways through the water as shown in figure 7. The difference between the steeredcourse and the course the boat actually moves is the leeway angle λ. When measuringthe efficiency of the boat sailing in upwind condition, the true wind angle (TWA), leewayangle and boat speed must be considered. This efficiency is called Velocity Made Good(VMG) and measures how fast the boat moves in the wind direction. VMG is calculatedby:

VMG = V cos(TWA+ λ) (1)

VMG

True wind

Boat Speed (V)λ

TWA

Figure 7: Velocity made good

3.1.2 Aerodynamic forces

The aerodynamic forces are the ”engine” of the boat. The sails work as aerofoils, resultingin lift and drag forces. For upwind sailing, the lift force is by far the biggest and provides

7

a driving force in the sailing direction, affecting the surge motion, and a side force perpen-dicular to the sailing direction, affecting the sway motion. The driving force is obviouslyimportant since it drives the boat forward, but the side force is of equal importance sinceit needs to be balanced by a counteracting side force from the hydrodynamic forces. It isnot only the sails that introduce aerodynamic forces. The hull, superstructure, crew andrigging introduce a windage that provides drag and thus slow the boat. The aerodynamicforces are acting above the waterline and often their point of action is called the Centreof Effort (CE), shown in figure 9. The centre of effort is interesting because it influencesthe heeling moment, ie balance in roll motion. The center of effort is influenced by thesail trim.

3.1.3 Hydrodynamic forces

The driving force is counteracted by a resistance force of the hull and appendages. Like-wise the side force from the aerodynamic forces is counteracted by the hydrodynamic liftfrom the hull and appendages. In order to provide this lift the boat has to travel slightlysideways through the water with a leeway angle λ as shown in figure 9. The leeway angleis the angle of attack for the keel and hull to generate the side force as a hydrofoil. Likethe aerodynamic forces the hydrodynamic forces can be gathered in a single point. Oftenthis point is used only for the side forces and is thus called Centre of Lateral Resistance(CLR).

Figure 8: Forces and leewy angle. Taken from [2]

3.1.4 Heeling moment

The vertical distance h between CE of the aerodynamic side force and CLR of the hy-drodynamic side force introduce a heeling moment. The heeling moment, as the namesuggests, make the boat heel over. The heeling moment is counteracted by the rightingmoment.

8

Figure 9: Forces and moments in roll. Taken from [2]

3.1.5 Righting moment

The righting moment comes from the horizontal distance between the centre of gravity(CG) of the boat and the centre of buoyancy (CB). The centre of buoyancy moves to theleeward side as the boat heels over. The center of gravity can be changed by moving thecrew to the windward side to further increase the righting moment.

3.2 Balance of forces and moments

To achieve a steady sailing condition the forces and moments must balance in principalmotions, shown in figure [?]. In the following the balance in all motions are described.

Surge: The balance in surge motion is obviously the most important since it ac-counts for the boat speed. The aerodynamic driving force from the sails are balancedby the hydrodynamic resistance of the hull and appendages and the aerodynamicdrag of the hull, rigging and crew.

9

Sway: The sway motion is of importance when the boat is going upwind becausethe large production of side forces from the sails cause a leeway angle. The leewayangle is important because it affects how fast the boat moves against the wind and italso influences the resistance by adding induced drag from the side force production.

Heave: The heave balance is of little importance. Dynamic effects from the hydro-dynamic forces on the hull tend to create a downward suction forcing the boat to aslightly larger draught. For light boats at high speed the effect shifts and tends tomove the boat out of water. This effect is called planing.

Roll: The roll motion has no direct effect on performance but is of importance sinceit controls the heel angle and thus has a large effect on both aero and hydrodynamicforces. The roll motion is a balance between the heeling moment and rightingmoment as described above.

Pitch: The pitch motion is of little importance for most steady sailing conditions.As for the roll motion there is a forcing moment from the aero and hydrodynamicforces and a righting moment from the hydrostatic forces. The stability of the boatis much greater in the longitudinal direction and thus only small changes are seen inthe pitch angle. The pitch balance is of significant importance for boats travellingin waves and for light boats in the down wind sailing condition.

Yaw: The yaw motion has no direct effect on the performance, but to make bal-ance the rudder angle must be changed. The changes of rudder angle change theresistance of the rudder and the distribution of side forces between the rudder andthe keel. The change in distribution of side forces affects the resistance of the keeland even the centre of lateral resistance affecting the heeling moment. The changesare often small and of moderate importance.

If all the above mentioned balances are achieved a steady state is obtained. For non-steady motion the inertia forces become of large importance.

3.3 Sail shape

Sail shape is a complicated matter because the shape is influenced by many factors. Firstlythe design shape and structural properties of the sail is important. Secondly the followingparameters affect the shape while sailing. Numbers refer to numbers in figure 10.

Apparent wind speed and angle

Sheet tension (1)

Sheeting position (2)

Forestay tension (3)

Halyard tension (4)

10

(1)

(2)

(3)

(4)

Figure 10: Trim changes that affects the sail shape

In section 3.4 the effect of the different parameters are described. The shape of thesail when sailing is called the flying shape, and a typical way of describing the shape isby maximum camber, position of maximum camber and the leading and trailing edgeangle as shown in figure 11. These variables are thus not independent since the maximumcamper, and position of maximum camper, influence the leading and trailing edge anglesand vice versa.

Maximum camper

Maximum camper position

Leading edge angle Trailing edge angle

Figure 11: Section geometry

11

3.4 Trim variation

3.4.1 True wind speed and angle

The wind speed and angle determines the aerodynamic load on the sails. The complexthree dimensional flow is influenced by the heel angle, wind profile, and presence of hulland crew. The presence of separation further complicates the flow pattern and thus theresulting forces. The aerodynamic force distribution changes the sail shape and deformsthe sail.

3.4.2 Sheet tension

The tension of the head sail sheet tension affects the shape of the sail dramatically. Firstly,it changes the trim angle of the sail and to some extend the twist. Secondly, it flattensthe sail by stretching the boundaries of the sail with increased tension.

3.4.3 Sheeting position

On most modern sailboats the sheeting position can be varied by moving the head saillead block (C)(and blue circle in figure 10) along the deck trail (G) as shown in figure12. The deck trail usually has an inward facing angle, which means that by moving thelead block forward the sheeting position will move both forward and inward. The inwardmovement changes the overall trim angle, whereas the forward movement changes thetwist. Along with the sheet tension the sheeting position is the most important trimmingparameter of the head sail.

Figure 12: Sheeting system of head sail (from www.harken.com)

3.4.4 Forestay tension

The pressure of the head sail makes the forestay sag as seen in figure 10. The sagging ofthe stay depends on the tension in the stay. The sagging increases the camper of the sailespecially in the middle part. By increasing the forestay tension the sail thus gets flatterand the lift and drag is reduced. In practice the forestay tension is controlled by changingthe tension of the backstay. It does however also depend on the main sail sheet tensionand the pretension of the mast.

3.4.5 Halyard tension

By changing the tension of the halyard the position of maximum camper can be moved.Increase in tension moves it forward and decrease moves i backwards. The effect fromthis variation primarily affects the minimum apparent wind angle that can be obtainedwithout luffing of the sails.

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3.5 CFD

3.5.1 Governing equations

The governing equations are the momentum conservation equations known as Navier-Stokes equations, here written in differential form for incompressible fluids with Einstein’ssummation notation:

∂Ui∂t

+ Uj∂Ui∂xj

= −1

ρ

∂p

∂xi+ ν

∂2Ui∂x2

j

(2)

Equation 2 assume Newtonian fluids, meaning a linear relation between viscous strain andstress. The proportionality factor is the viscosity. For Cartesian coordinates this givestree equations, one for conservation in each direction. Since there are only three equationsand four unknowns, three velocity components and the pressure, an additional equationis needed to close the system. This is known as the closure problem. The additionalequation is the continuity equation:

∂Uj∂xj

= 0 (3)

Again, the incompressible version. The assumption of incompressibility assumes that thedensity is constant, and this is valid for low Mach numbers as expected in this project.

3.5.2 RANS

For turbulent flow the solution of the full Navier-Stokes equations through direct numericalsimulation is difficult and for most engineering purposes impossible. Instead the equationsare solved on a time averaged basis. All quantities are split in a time averaged and afluctuating part. By insertion in equation 2 a time averaged version of Navier-Stokesequations, namely the RANS equations, are obtained:

∂Ui∂t

+ Uj∂Ui∂xj

= −1

ρ

∂p

∂xi+ ν

∂2Ui∂x2

j

− ∂uiuj∂xj

(4)

The new term u′iu′j is a tensor that represents the effect of small velocity fluctuation on the

momentum. This tensor is always of importance in turbulent flow and must be modelledby a turbulence model.

3.5.3 Turbulence modelling

The purpose of the turbulence model is to model the effect of small turbulence on theflow. These are typically modelled by adding one or two equations to model transportof some turbulence properties. In the following two sections, two of the commonly usedtwo-equation models are briefly described.

k− ε turbulence model This turbulence model uses a transport equation to modelthe transport of turbulent kinetic energy k and an additional equation to model thedissipation of turbulent kinetic energy. The two equations are solved alongside the RANSequations and continuity equation. The standard k− ε model is known to perform wellfor small pressure gradients, but to give bad results for strong adverse pressure gradients.Due to this a newer version of the model, called realizable k − ε model, is used. The

13

new model has a different equation for the turbulent dissipation based on more recentscientific experiments. See [3] for more details. This realizable k− ε model is used in thisproject due to its superior performance for free surface flows as documented in [4].

k− ω SST turbulence model As with the k− ε model the transport of turbulentkinetic energy is modelled by a transport equation for k but instead of modelling thedissipation by the dissipation rate ε the specific dissipation rate ω is used. This approachhas proved to give better results close to the wall and in strong pressure gradients thanthe standard k− ε model. The SST extension by Mentor [5] combines the k− ω modelclose to the wall with the k− ε in the free stream.

Wall Treatment Close to solid walls the resolution of flow quantities are difficult dueto large gradients. To resolve the full boundary layer of high Reynolds number flows alarge number of very thin cells are required. This makes the computation too expensive.Instead a wall law can be applied to model the near wall behaviour of quantities such asvelocity and turbulence quantities. The wall laws are typically specific to the turbulencemodel and often consist of a mix of algebraic formulations. In the region closest tothe wall the viscous sublayer, the flow is completely dominated by the viscosity, and alittle further from the wall the flow is dominated by turbulent shear. In between theseregions an overlapping zone exist where both turbulent and viscous forces are important.The velocity in the viscous sublayer is typically modelled by a linear relation while thevelocity in the outer layer is modelled by a logarithmic relation. In the overlapping layera fit between the two are usually used.

3.5.4 Finite Volume Method

To solve the complex non-linear differential equations described above the equations haveto be discretized in small control volumes and the derivatives replaced by difference ap-proximations. For the finite volume method the equations are solved in integral form.The general transport equation on integral form with vector notation:∫

S

ρφV · ndS =

∫S

Γ∇φ · ndS +

∫Ω

qφdΩ (5)

The transport equations are solved in each control volume by calculating the surface inte-grals on the cell faces. Quantities are assumed to be constant at each cell face and thus theintegral can be calculated as a summation of the surface centre quantities. Since all quan-tities are only known in cell centres the surface quantities are estimated by interpolationschemes from the centre values.

3.5.5 Volume of Fluid

To model both air and water flow with a variable interface the Volume of Fluid method isused. This method introduces an additional transport equation describing the amount ofwater and air in each cell. The flow and pressure field solver is not modified but the fluidproperties are modified to match the cell content of the different fluids. The interfacebetween air and water can stretch over many cells, and thus a compression scheme isintroduced to compress the interface to a few cells.

14

3.5.6 Fluid Structure Interaction

To model the changes in speed, heel angle and leeway angle a FSI solver is applied. Thesolver models the movements of the structure based on the hydrodynamic and aerody-namic forces from the flow. The movements depend on the weight and mass momentsof inertia of the structure, and the accelerations are found from Newton’s second law ofmechanics. For surge and sway the equations read:

R + FM = Wax (6)

FLAT + PLAT = Way (7)

The final balance interesting at this point is for the roll motion. Here the mass momentof inertia plays a roll to find the angular acceleration αxx:

FLAT · CE + PLAT · CLR +RM = Ixx · αxx (8)

The equations are stepped in time along with the RANS equations and values for theaccelerations are found at each time step.

3.5.7 Wind profile

As the wind blow over a surface like the sea a velocity gradient is build up. The velocitygradually increases with the height above the surface to develop a thick boundary layer.Often the atmospheric boundary layers are described by a logarithmic function:

U(z) =u∗kloge

(z

z0

), (9)

where u∗ is the friction velocity and z0 is the roughness length depending on theroughness of the underlying surface. k is Von Karmans constant 0.41. On the ocean thesurface roughness depends on the waves, which again depend on the wind velocity. Therelationship between the friction length and the Velocity at 10 m height U10 is accordingto Charnock [6] described by:

z0 =a

g

kU10

loge

(10z0

)2

, (10)

where a is an empirical constant between 0.01 and 0.02 according to [7]. This implicitrelation must be solved iteratively. An example of the velocity profile for 6 m/s is shownin figure 13.

15

0 1 2 3 4 5 6 70

2

4

6

8

10

12

14

16

U [m/s]

z [m

]-

Figure 13: Wind profile 6 m/s

3.5.8 Verification and Validation

To ensure quality and estimate accuracy of CFD simulations, verification and validationmust be performed. Verification is performed to verify that the equations are solved right,whereas validation is a comparison with experimental data. The theory presented in thissection is based on the formulation from [8].

Verification In general there are two types of verification: code verification and solu-tion verification. The code verification is often done by the software developer to makesure their code is solving the equations correctly. A constructed problem, with a knownsolution can be used to verify the implementation. For each new problem a solution veri-fication should be performed. The solution verifications are often performed on problemswhere the solution is unknown, to estimate the uncertainty for the problem. The estima-tion is made by monitoring the important output parameters as the mesh is systematicallyrefined.

If the mesh is fine enough and thus inside the asymptotic range the parameters willfollow a asymptotic convergence as the mesh is refined. This is however rarely the casesince the practical meshes for most RANS solutions are well outside the asymptotic range.Thus care must be taken when the uncertainties are estimated.

Validation To verify that the proper assumptions have been made a validation is per-formed. Validation requires a known solution in the form of analytical or experimentaldata. The difference between the calculated values and the experimental data is com-pared with the combined uncertainty of the numerical model, experimental data andinput parameters. This uncertainty is called validation uncertainty, and is calculated by:

Uval =√U2num + U2

exp + U2inp (11)

The numerical uncertainty is estimated by the solution verification and the experi-mental uncertainty is estimated from the experimental scatter, and / or combination ofknown experimental uncertainties. If the error is smaller than the validation uncertaintythe validation is successful to the uncertainty specified by the validation error. This meansthat a large uncertainty of numerical and/or experimental data may lead to a successful

16

validation, but with an uncertainty not satisfying the need for the purpose of the simula-tion.

All uncertainties must be multiplied by a safety factor prior to use in validation pur-poses. According to [8] the safety factor can be taken as 1.2 if the uncertainty fittingerror is negligible and 3 otherwise. For most practical RANS calculations the safetyfactor should be taken as 3 according to [8].

3.6 Optimization

3.6.1 Optimization Problems

In general, optimization is formulated as a minimization of an objective function bychanging a set of variables. Often the variables are restricted within a specified range,called design space. Furthermore constraints functions may exist that limits the solutionand must be taken into account during the optimization. The nature of the optimiza-tion problem depends both on the type of variables and the form of the objective andconstraint function. The variables can be either continues, meaning they can attain anyreal number, or discrete, meaning they are restricted to for example integers. Dependingon the properties of the variables as well as the objective and constraint functions, theoptimization problem may fall into one of a large number of categories, for instance:

Linear programming: Linear objective and constraints functions

Quadratic programming: Quadratic objective function and linear constraint func-tions

Non-linear programming: General objective and constraints functions

The linear programming optimisation is the simplest and often give a fast and accuratesolution. The success of the quadratic programming optimization depends on the natureof the objective function. If the function is convex the optimum is in general easy to find,where as a concave objective function may pose difficulties. General objective functionmeans that the function can have any shape. This means that there is a potential of morelocal minima, see figure 14. Non-linear programming problems often have more than onelocal minimum and thus the optimization algorithm risk getting stuck at a local minimumand not finding the global minimum.

17

Variable

Obj

ectiv

e

Local minima

Global minimum

Figure 14: Global and local minima

One way to try to overcome this issue is by searching the design space with differentcombinations of variables.

3.6.2 Optimization Methods

Numerical optimization methods solves an optimization problem by iteratively improvinga solution estimate, until convergence is achieved according to given tolerance criteria.The methods for updating the variables are numerous, but usually one of the followingstrategies are used for solving non-linear programming problems:

Line search methods combines a method for estimating a descent direction, witha method for finding a minimum in that direction. Numerous methods exist forestimating the descent direction and performing line search.

Trust region methods solves a simplified optimization problem, that is an ap-proximation to the main optimization problem. The simplified problem includeconstraints that limit the length of the step suggested by the algorithm. The limitis often referred to as a trust region radius. The trust region radius is continuouslyupdated, and depends on the success of the previous iterations.

Penalty and damping methods involve augmenting the objective function witha function that increase in undesired parts of the design space, for instance inregions where one or more constraints are violated. These methods are referred toas penalty methods. The same technique can be used for controlling the size of thestep suggested by the algorithm, by adding a term to the objective function thatincreases for large steps. These methods are referred to as damping methods.

18

Many optimization methods use information about the gradient. But the gradient isnot always available. When the gradient is unknown it can be estimated by use of finitedifference methods, but according to [9] this requires a large number of sample pointsand can be inaccurate when noise is present in the objective function. Thus gradient freeoptimization algorithms such as pattern search algorithms are an attractive alternative.

3.6.3 CFD-based Optimization

CFD based optimization as in this project in general have continuous variables but non-linear objective functions. Information about the gradient are usually unknown. Con-straints may exist for CFD optimization problems, but in this project no constraint func-tions are used. The complex flow interactions leave room for more than one local minimasince the appearance and disappearance of separation zones makes the objective variationsoscillate.

3.6.4 Pattern Search Algorithms

The idea of the pattern search is to search in a number of search directions from thebase point. The new sample points form a stencil around the base point. If a significantdecrease of the objective function is found in one of the directions the base point is movedin that direction. The step length and search directions may be changed depending onthe output from the sample points. The optimization is stopped when the change in thevariables have reached a specified tolerance.

The accuracy and stability may be affected by noise in the objective function andnon-continuous behaviour, but often pattern search algorithms are found to give goodresults even in these conditions[9].

3.6.5 SOBOL

One way to explore the design space is to generate random combination of the variables.These combinations can be produced by a random number sequence, but this has thedisadvantage of clustering points, giving a non-uniform coverage of the design space.Instead a quasi-random algorithm like SOBOL sequence [10] is recommended to give amore equal distribution of design points. This ensures that more of the design space iscovered and thus fewer points are needed.

19

4 Method

4.1 CFD model

The commercial CFD code Star-CCM+ ver. 8.02 is used for the project. Both waterand air are simulated to get both aero and hydrodynamic forces affecting the boat. Thespeed, heel angle and leeway angle are adjusted in the model by the 6 DOF solver. In thefollowing sections the CFD setup will be described in further details. Much of the setupis based on the guidelines from Milovan Peric’s contribution to the Gothenburg Workshopon Numerical Ship Hydrodynamics [4] and the guidelines in [11].

4.1.1 Domain

The domain shown in figure 15 extends 2 times Lenght og Water Line (LWL) behind theboat and to the leeward side and 1 times LWL in front and to the windward side. Thedomain is larger in the wake of the flow to capture the disturbed flow behind the sails andnot affect the pressure distribution on the sails. Below the hull 1 times LWL is included,and above 2 times LWL is included. The variations of the dimensions of the domain aretested in the verification study presented in section 5.

Figure 15: Simulation domain

20

(a) Original triangulation (b) Remeshed triangulation

Figure 16: Surface mesh

4.1.2 Mesh

In the following paragraphs the mesh generation is described. For further details on themethods reference is made to the Star-CCM+ user manual [12].

Surface Remeshing To ensure a good quality surface mesh and an even distributionof sizes before the volume mesh is executed the initial geometry tessellation is replaced bya new surface mesh. The size of the surface elements are determined by global and localsettings on the minimum and target size, and by curvature constraints. The curvatureconstraint is set to secure 72 point per circle, which is double the default setting. Thisextra resolution is used to ensure that the curvature of the sails are captured accuratelysince this can have a significant influence on the flow. Figure 16 shows the original andremeshed surface mesh triangulation. As seen the distribution is more smooth for theremeshed surface and finer cells are included where curvature is high. The original modelof the boat and main sail was imported through IGES format and the shown originaltriangulation of these is only the initial attempt to make a triangulation and not usedfor further processing. The new tessellation is based on the CAD geometry and not theshown original triangulation. The head sail on the other hand is imported as STL, whichis by nature triangulated geometry. To get a proper quality for this kind of import a finetriangulation is used, as seen in the figure.

Volume Meshing The trimmer method is used for the volume mesh because it allowsfor anisotropic refinement needed to accurately resolve the free surface interface betweenwater and air. The trimmer works by introducing a background mesh filling the entiredomain with hexahedral cells of the maximum size. Where the mesh intersects the surface

21

mesh the mesh is gradually refined by cutting the cells in half until the surface mesh sizeis match sufficiently. Figure 17 shows a section view of the volume mesh. As seen on thefigure, the far field mesh is very coarse and the cells are gradually refined closer to theboat. The refinement near the water surface is also clear and an important property ofthis is that the water surface should stay within the refinement for the small changes ofheel expected.

Figure 17: Vertical section showing the volume mesh

Figure 18 shows the volume mesh in a horizontal section in the waterline. An arrowshaped refinement in the horizontal dimensions is used to capture the wake field of theyacht. The angle of the arrow edges is 19 °which corresponds to the angle of a Kelvinwave system resulting from boats travelling at reasonable speeds according to [13].

22

Figure 18: Horizontal section showing the volume mesh in the waterline

Prismatic Layer To accurately resolve the boundary layer, layers of prismatic cells arecreated close to the wall boundaries. The layers are controlled by the total height andthe number of layers. Figure 19 shows a vertical section of the mesh with a close up closeto the boat. Here the prismatic layers can be seen along the keel and hull.

Figure 19: Vertical section showing the volume mesh with a closeup close to the boat

23

4.1.3 Boundary conditions

Figure 20 shows the domain with colours for the relevant Boundary types. The light greyon the outer edges is inlet, the orange area is outlet and the solid grey colour is solidwalls.

Figure 20: Boundary conditions

Inlet At the inlet the apparent wind and water velocity is specified. These valuesare based on the estimated boat speed, leeway angle, and true wind velocity and angle.The wind varies with the distance from the water surface to represent the atmosphericboundary layer as described in section 3.5.7. The volume fractions of water and air arespecified to have the water surface in the right position. Because the domain followthe motion of the boat the velocity and volume fraction is updated as the boat speed,heel angle and leeway angle are changed during the simulation. Turbulence intensityand Viscosity ratio are also specified at the inlet. The default values are used for bothquantities.

Outlet At the outlet pressure and volume fractions are specified. The volume fractionis, as for the inlet, specified to keep the water surface in place, and in addition the

24

hydrostatic pressure is specified. As for the inlet the default values for turbulence intensityand viscosity ratio are used.

Boat On all solid parts of the boat the no-slip boundary condition is applied. The boatis part of the solid body used for the motion solver and thus fluid forces are captured onthe surface to be used in the motion equations at each time step.

Sails The sails are of zero thickness and must thus be treated specially. They aremodelled as baffle interfaces meaning that a copy of the sails are made to represent theother side of the sail. Both sides are treated as solid walls with no-slip boundary condition.As for the boat the sails are part of the solid body used for motion calculation and fluidforces are evaluated from these.

Mesh motion The mesh is moving during simulation due to the motions required bythe 6-DOF solver. The motion is a rigid motion of the entire mesh and thus no re-meshingor morphing is needed. Since the velocity, volume fraction and pressure is specified in theglobal coordinate system the values on the boundaries automatically update.

4.1.4 Solvers

Due to the complexity of the problem a wide range of solvers are needed for the simu-lations. The segregated flow solver is used with implicit unsteady time stepping. Thesegregated flow solver is an implementation of the SIMPLE type algorithm, where themomentum and continuity equations are solved uncoupled, and a predictor corrector ap-proach is used to link the equations. The transient term is modelled with first orderdiscretization to ensure stability and all other terms are modelled with second order dis-cretization in space. The volume of fraction transport equation is solved along with theflow equation by the segregated solver. The equations of solid body motion are solved atevery time step to march forward the solution.

25

4.2 Friendship Framework Model

The commercial software Friendship Framework ver. 3.0.8 is used for the present projectunder a non-commercial student license.

4.2.1 Geometry

The baseline geometry of the head sail is imported into Friendship framework by thehorizontal control curves of the NURBS surface received from North Sail. The curves areused in order to facilitate the trim and shape variation to be variable over the length ofthe sail. After the trim and shape modifications the spline curves are transformed to asurface by a lofting operation.

4.2.2 Trim parameters

The trim parameters are applied as rotations around a line from the top of the sail to thetack corresponding to the forestay. An example of a trim angle change is shown in figure21

Trim Angle The trim angle is a rotation around the aforementioned line with no vari-ation from top to bottom of the sail. The angle is free to change ±2. An example of atwist angle change is shown in figure 22

Twist The twist is the difference between the trim angle at the top and the bottom. Itis also free to vary with ±2.

TWA The final parameter is the true wind angle, which is changed by changing a valuein the Star-CCM+ script displayed in appendix C. The true wind angle is not an actualtrim parameter but related to the steering of the boat. The parameter is included in theoptimization due to its important influence on the VMG. The optimal twist and trimangle is largely dependent on the steered TWA.

26

Figure 21: Trim angle change. Blue line is original and red line is with change in trim angle

Figure 22: Twist angle change. Blue line is original and red line is with change in twist angle

27

4.2.3 Shape parameters

For the shape optimization a shift function is used to control the camper and positionof maximum camper of the sail sections. The shift function uses a control surface tocontrol the deformation. For details on shift functions see [14]. The surface, shown infigure 23 along with the sail surface, is a B-Spline surface created by a loft operation on4 control lines. The lines, shown in figure 24, are third degree splines controlled by 3points and the gradients in the end points. The end points correspond to the sides of thesail and thus no deformation is wanted here, which is why these are kept at zero. Thegradients at the end points are not given and are thus found from the movement of themiddle point. The middle point position is thus the only influencing parameter availablefor design modifications. The point can be moved in both horizontal directions, as shownin figure 24, but not in the vertical direction. Thus 2 design variables are present for eachline, the camper C and the camper position CP. The top line is not modified because thisis difficult to accommodate in practice, thus a total of 6 design variables are available forthe shape optimization.

Head Sail

Shift surface

Control points

Figure 23: Shift surface

28

C

CP

Figure 24: Deformation parameters

4.2.4 Optimization algorithm

The optimization is split into two parts. The first is design space investigation and thesecond is a tangent search. The objective used for the trim optimization is the VelocityMade Good (VMG) and for the shape optimization a combination of the VMG for thedifferent wind velocities is used. The medium wind speed accounts for 50 % of theobjective, while the high and low wind speed share the remaining 50 %.

Design space investigation The purpose of the design space investigation is to findthe global trends in the design space. As mentioned in section 3.6 objectives may havemany local minima across the design space, and to make sure that the optimization resultsin a global minimum the design space investigation is needed. For the current project theSOBOL algorithm is used to find the semi-random design variable variations. The outputof the investigation is used as a basis for the pattern search.

Pattern search The TSearch algorithm of Friendship Framework is used. This is apattern search method with constraints. The only constraints for this project are thelimits for the variables. If a constraint is breached the TSearch method steps along thetangent of the constraint. The step size of the method is set to an initial value andreduced along the way if no improvements are found. The method can reuse points fromthe design space investigation if appropriate to save time.

29

4.3 Computation Network

The CFD simulations require extensive computational power and the optimization processrequires user control and input through a graphical user interface. A remote simulationserver delivers the computational power and a local Linux machine runs the optimizationprocess with a graphical user interface. The communication between the two machines isperformed through a SSH tunnel connection controlled by terminal scripts. The commu-nication network is shown in figure 25.

Friendship framework provides Java macros, Terminal scripts and STL geometry files.The terminal scripts are executed in the local and remote terminal to control the processes.The geometry file and Java macros are copied to the remote server and used for simulationsby STAR-CCM+. STAR-CCM+ is run through a queuing system in batch mode, and theoutput from the simulations is CSV files containing time history of output parameters.The CSV files are copied back to the local machine and read by Friendship Frameworkfor further processing. Examples of terminal scripts are listed in appendix B and JAVAmacros are listed in appendix C. The scripts are run internally at OSK-shiptech and thusprotected by a firewall. The sshpass is used to avoid password prompt. This is an unsafemethod on open networks. Instead sshagent can be used. All company specific detailsare removed from the scripts for safety reasons.

Local linux machine Remote simulation server

Terminal scrips

STAR-CCM+

Local terminal

Java macros

SSH

.csv

Java macros

Remote terminal

.stl

.csv.stl

Figure 25: Flow chart of simulation network

30

5 Verification and Validation

The validation of the CFD model consists of two parts. The first part focuses on theaerodynamic part of the simulation by comparison with experimental wind tunnel data.The second part investigates the full aero and hydrodynamic model including solid bodymotion.

5.1 Validation of aerodynamic forces in CFD model

Since the focus of this project is on sail design the aerodynamic forces on the sails are ofgreat importance. To validate the aerodynamic forces a series of experimental data fromwind tunnel tests are used. The data presented in [15] are measurements of forces andpressures on a set of America’s Cup class (AC33) sails performed in the wind tunnel ofYacht Research Unit (YRU), University of Auckland. The test setup shown in figure 26consists of a test plate with the sails suspended above and with pressure transducers builtinto the sails.

Figure 26: Wind tunnel with test sails

31

5.1.1 Domain

To validate the model against the wind tunnel results, a domain similar to the windtunnel geometry, is constructed. Only the part of the wind tunnel above the test plateis modelled as seen in figure 27. The bottom of the domain is then modelled as a freeslip wall producing a flat velocity profile. The sides and top of the tunnel section and thetest plate are all modelled as no slip walls, and the sails are modelled as baffle interfaceswith no slip conditions. The pressure is reported in the points displayed in figure 28corresponding to the measurement point of the experiment.

Inlet

Free slip wall

Tunnel walls

Test plate

Tunnel roof

Figure 27: Domain for validation study

Figure 28: Measuring points for validation study

32

5.1.2 Mesh

The mesh consists of hexahedral cells for most of the domain created by the trimmermethod. Close to no-slip boundaries prismatic layers are included to resolve boundarylayer gradients. An illustration of the finest mesh used is shown in figures 29 and 30. Fromthis mesh a series of meshes were constructed with varying cell size scaled by a factor 3

√2.

The size and average y+ values of the meshes are listed in table 3. The y+ average is takenas the average of y+ values on the sail surfaces, and a plot of the y+ distribution is shownin figure 31. In principle the number of cells should scale with one over the base size tothe power of 3 and thus half between each step, but because the surface size depends onthe curvature of the surface and the geometric limitation, the number of cells are reducedless than that. This also means that the meshes are not completely geometrically similaraffecting the mesh conversion according to [8].

Figure 29: Finest mesh for validation study seen from above

33

Figure 30: Finest mesh for validation study seen from side

# Base size Number of cells y+ avg1 1.00 5.484.973 1.92 1.26 3.384.183 2.43 1.59 2.140.462 3.14 2.00 1.421.954 3.95 2.52 984.854 5.06 3.17 675.508 6.27 4.00 492.513 7.58 5.04 371.064 9.19 6.35 267.782 11.210 8.00 208.398 13.511 10.08 155.318 15.712 12.70 117.742 18.7

Table 3: Validation meshes

34

Figure 31: Wall Y + on the sail surfaces

5.1.3 Results

In the following sections the results from the validation study are presented. Validationis done for lift, drag and pressure coefficients on the sail surface. Validations are donein accordance to the recommendations presented in [8]. The validation is done on meshnumber 8 corresponding to the mesh resolution used for the full simulation model.

Lift The lift shows an oscillatory convergence with grid refinement, as seen in figure32, and a fit for estimating discretization error is thus difficult. Instead the fluctuationamplitude of the 8th finest grids is used multiplied with a safety factor of 3. The resultingdiscretization error is 5.6 %. Both turbulence models give similar results, but the k − ωSST show some large oscillations in some of the meshes, as seen in figure 33. These oscilla-tion may represent actual physical oscillations due to oscillating separation, but since theyare only present in a few of the meshes the cause is more likely to be numerical instability.

The round off error is neglected because of its relatively small size. The convergenceerror is neglected because the simulation is run until the solution is stable. No informationis available on the experimental uncertainty of the lift and drag, but an uncertaintysimilar to the measurement accuracy of the dynamic pressure, i.e. 3% is used. Thevalidation result is presented in table 4. The error E is an order of magnitude smallerthan the validation uncertainty Uval. According to [8] this can mean two things. Eitherthe validation is successful or the uncertainty is too big. The validation uncertainty is inthis case relatively large, but this is also related to the large safety factor of 3. The errorof 0.6 % is very small but this is more a coincident. The validation is also successful witha much lower safety factor. This means that the expected uncertainty is in the order of3.7 to 6.3 % for the lift on mesh number 8, depending on the choice of safety factor. Theuncertainty is independent of the mesh size for the 8th finest meshes due to the oscillatoryconvergence.

35

0 2 4 6 8 10 12 141.22

1.24

1.26

1.28

1.3

1.32

1.34

hi/h

1

CL

k−εk−ω SST

Figure 32: Grid convergence for lift

0 5 10 15−1.27

−1.265

−1.26

−1.255

−1.25

−1.245

time [s]

CL

k−ε

CFDAVG

0 5 10 150.145

0.146

0.147

0.148

0.149

0.15

0.151

time [s]

CD

k−ε

0 5 10 15−1.26

−1.255

−1.25

−1.245

−1.24

−1.235

time [s]

CL

k−ω SST

0 5 10 150.146

0.147

0.148

0.149

0.15

0.151

0.152

0.153

time [s]

CD

k−ω SST

Figure 33: Lift and drag time series

Drag The drag coefficient is plotted as a function of relative cell size in figure 34 andshows a better convergence than the lift. It does however show some oscillatory behaviour.The error is fitted with a polynomial and with a safety factor of 3 this gives a discretization

36

uncertainty of 15.3 % for the mesh number 8. The oscillations in the drag appear to becoupled with the oscillations in lift. The validation result is presented in table 4. As forthe lift, the error of drag is almost an order of magnitude smaller than the validationuncertainty, and the validation is successful for an uncertainty of 15.6 %. The largediscretization uncertainty can be reduced by using one of the finer meshes, and due tothe size of the experimental error the validation is also valid for these meshes with anuncertainty around of 3 %.

0 2 4 6 8 10 12 140.142

0.144

0.146

0.148

0.15

0.152

0.154

0.156

0.158

0.16

0.162

hi/h

1

CD

k−εk−ω SSTk−ε fit

Figure 34: Grid convergence for drag

CL CD

Udiscr 0.0701 0.02320Uexp 0.0391 0.00468Uval 0.0803 0.02367E 0.0072 0.00160E(%) 0.6% 1.1%

Table 4: Validation table Lift and Drag

37

Pressure Figures 35 and 36 show the numerical and experimental pressure coefficients.In general they show good agreement. There is however a difficulty in predicting theleading edge separation of the main sail. The experimental results show larger separationon the 3 lower sections whereas the numerical results show higher degree of separationin the uppermost section of the mainsail. The difference for the lower sections may beattributed to the fact that the experimental sail has non-zero thickness where as thenumerical model has zero thickness. The upper section is situated in the wake of thehead sail tip vortex and thus experiences a complex separation which is difficult to pre-dict, as described in [16]. As the lift, the pressure shows oscillating mesh convergenceand thus the discretization uncertainty is estimated in the same way. The validation re-sults are presented in table 5. The simulation is validated although the error is up to 12 %.

The descretization error is large compared to the result of [16], however a directcomparison is difficult due to difference in mesh type. In the reference a fully structuredmesh providing a better mesh similarity between the meshes is used. This is expectedto provide a better convergence. The experimental uncertainty is taken from [16] and isin the order of 13 to 30 %. This large experimental uncertainty makes the validation ofpressure less accurate. The pressure simulation is thus validated to an accuracy of 17 -30 %.

0 0.2 0.4 0.6 0.8 1

−2

−1.5

−1

−0.5

0

0.5

1

x/c

Cp

Head section 1 (HS1)

ExperimetalNumerical

0 0.2 0.4 0.6 0.8 1

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

x/c

Cp

Head section 2 (HS2)

0 0.2 0.4 0.6 0.8 1

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

x/c

Cp

Head section 3 (HS3)

0 0.2 0.4 0.6 0.8 1

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

x/c

Cp

Head section 4 (HS4)

Figure 35: Experimental and numerical pressure coefficients on head sections

38

0 0.2 0.4 0.6 0.8 1

−1.5

−1

−0.5

0

0.5

1

x/c

Cp

Main section 1 (MS1)

ExperimetalNumerical

0 0.2 0.4 0.6 0.8 1

−1.5

−1

−0.5

0

0.5

1

x/c

Cp

Main section 2 (MS2)

0 0.2 0.4 0.6 0.8 1

−2

−1.5

−1

−0.5

0

0.5

1

x/c

Cp

Main section 3 (MS3)

0 0.2 0.4 0.6 0.8

−1.5

−1

−0.5

0

0.5

1

x/c

Cp

Main section 4 (MS4)

Figure 36: Experimental and numerical pressure coefficients on main sections

HS1 HS2 HS3 HS4 MS1 MS2 MS3 MS4Unum 0.593 0.554 0.435 0.349 0.165 0.176 0.243 0.405Uexp 0.681 0.703 0.687 0.66 0.811 0.811 0.764 0.687Uval 0.903 0.895 0.813 0.747 0.828 0.830 0.802 0.798E 0.442 0.419 0.368 0.104 0.329 0.283 0.268 0.248E(%) 8% 9% 8% 2% 12% 9% 8% 9%

Table 5: Validation table pressure

5.1.4 Validation Remarks

The validation presented above show relatively large uncertainties. These uncertaintiesare most likely related to the complex nature of the flow around the sails. For detailedstudies of sails where the absolute values of the forces are of importance a detailed study ofthe flow is recommended to decide on the final mesh resolution. The validation presentedabove are for a scale model and the change in Reynolds number to a full scale model posea potential source of error. The validation above is used as guideline to the resolution ofthe full scale mode.

39

5.2 Verification of full model

5.2.1 Spatial Discretization

To test the mesh quality and to estimate the discretization uncertainty for the full sim-ulation model, 7 systematically refined meshes are run. Table 6 shows the properties ofthe meshes. The y+ values are larger than for the validation study above because theresolution is the same but the overall size of the sail is larger. Figure 37 show the meshconvergence for 4 of the relevant output parameters as a function of the relative basesize. In general the convergence is good and a fit is used for each parameter to estimatethe discretization uncertainty, which is shown in table 7. Only heel angle show a largerdiscretization uncertainty. The most important parameter is the Velocity Made Good,which shows just above 1 % uncertainty with a safety factor of 3 for the reference meshused for further calculations. Since the discretization uncertainty is in the order of theimprovements of the optimization it is recommended that the final results are rerun witha finer mesh for verification.

# Base size Number of cells y+ avg1 1.00 2,102,549 452 1.12 1,800,627 513 1.26 1,508,921 574 1.41 1,298,445 645 1.59 1,130,284 716 1.78 957,527 807 2.00 834,330 89

Table 6: Validation meshes for full model

40

1 1.2 1.4 1.6 1.8 23.44

3.46

3.48

3.5

3.52

3.54

3.56

3.58

hi/h

1

[m/s

]

Boat speed

1 1.2 1.4 1.6 1.8 218.6

18.7

18.8

18.9

19

19.1

19.2

hi/h

1

[°]

Heel Angel

1 1.2 1.4 1.6 1.8 2

5

5.2

5.4

5.6

5.8

hi/h

1

[°]

Leeway Angle

1 1.2 1.4 1.6 1.8 22.5

2.52

2.54

2.56

2.58

2.6

2.62

hi/h

1

[m/s

]

Velocity Made Good

Figure 37: Discretization study for full model

Safety factor Un

Boat Speed 3 1.3%Heel Angle 3 16.8%Leeway Angle 3 0.5%VMG 3 1.2%

Table 7: Safety factor and discretization uncertainty for full model

5.2.2 Temporal Discretization

To investigate transient convergence error a series of different time steps are evaluated.The time series of the output parameters are displayed in figure 38. The highest timestep of 0.1 s gives an unstable solution, while the lower time steps give a stable solution.The value used for the preceding simulations is 0.05 s. For the heel and leeway angle the3 small time steps give very similar results, but for the boat speed there is a differencewhich is reflected again in VMG. Figure 39 show the time averaged VMG as a function ofthe relative time step along with a fit. This fit gives an uncertainty of 6 % for the valueused in the baseline model with a safety factor of 3.

41

0 10 20 30 40 506.5

6.6

6.7

6.8

6.9

7

Time [s]

[kts

]

Boat speed

∆ t = 0.1∆ t = 0.05∆ t = 0.01∆ t = 0.005

0 10 20 30 40 5012

14

16

18

20

22

24

Time [s]

[°]

Heel angle

0 10 20 30 40 503

4

5

6

7

8

Time [s]

[°]

Leeway angle

0 10 20 30 40 50

4.9

5

5.1

5.2

5.3

5.4

5.5

5.6

Time [s]

[kts

]

VMG

Figure 38: Time series of output parameters for different time steps

42

0 2 4 6 8 10 12 14 16 18 202.64

2.66

2.68

2.7

2.72

2.74

2.76

∆ ti / ∆ t

1

VM

G [m

/s]

CFDfit

Figure 39: Influence of time step on VMG

5.2.3 Domain size

To test if the domain is large enough to avoid boundary effects a series of different domainsizes are evaluated. A 50 % and a 100 % increase in domain size are used and the timeseries of the output parameters are displayed in figure 40. Surprisingly the solution showsincreasing destabilization for increasing domain size. An explanation for this could bethat the far ends of the domain experience large changes in volume fraction as the boatchanges heel angle. The water surface moves past more cells in a single time step andcauses large changes in the density and thus may lead to destabilization of the pressuresolver. This problem may be solved by reducing the time step or changing the surfacecompression. It is difficult to judge the domain size convergence based on these resultsbecause they are influenced by large oscillations in the heel angle. Figure 41 shows thewave field as a scalar plot for the 6 m/s case. Some boundary effects appear to be present;wave build up outside the Kelvin wedge, and a slightly larger domain might have removedthis. The influence of this is probably insignificant since the elevations are in the order ofa few centimetres.

43

0 10 20 30 40 505

5.5

6

6.5

7

7.5

Time [s]

[kts

]

Boat speed

Baseline50 % increase100 % increase

0 10 20 30 40 5010

12

14

16

18

20

22

24

Time [s]

[°]

Heel angle

0 10 20 30 40 500

5

10

15

20

25

Time [s]

[°]

Leeway angle

0 10 20 30 40 503

3.5

4

4.5

5

5.5

6

Time [s]

[kts

]

VMG

Figure 40: Time series of output parameters for different Domain sizes

Figure 41: Wave field

44

5.2.4 Moment of inertia

The actual moment of inertia of the boat is unknown, but as a reference an estimatebased on a radius of gyration of 0.35 times the width are used as a starting point. Theboat probably has a larger moment of inertia due to the mast and sail and thus a rangeof moments of inertia are tested. The results are presented in figure 42 as time seriesof output parameters. The highest and lowest moments of inertia give similar resultsfor both average value and fluctuation amplitude. The average moment of inertia givesa higher fluctuation amplitude and a different average value for the velocity. A furtherinvestigation of the effect of moment of inertia is recommended for future studies alongwith a possible coupling with time step. For the current project the high value of themoment of inertia is used. Tests with higher values have been tried with uncontrollableoscillations as a result.

0 10 20 30 40 506.7

6.75

6.8

6.85

6.9

6.95

7

Time [s]

[kts

]

Boat speed

I = 1e5I = 5e4I = 1.32e4

0 10 20 30 40 5018

18.5

19

19.5

20

20.5

21

Time [s]

[°]

Heel angle

0 10 20 30 40 504

4.5

5

5.5

6

Time [s]

[°]

Leeway angle

0 10 20 30 40 505.15

5.2

5.25

5.3

5.35

5.4

Time [s]

[kts

]

VMG

Figure 42: Time series of output parameters for different mass moments of inertia

5.2.5 Simulation Scatter

The tessellation of the sail in Friendship Framework and the following mesh generation andsimulation in STAR-CCM+ was found to create some scatter in the result. Consecutiverun of the same setting gave a scatter in the order of 0.1 %.

45

5.3 Full scale data

Full scale data from racing are available from the racing yacht Sirena. The data hasbeen fitted by a VPP data processing program to get values for the right wind speeds. 5series of data are available from different races, and they are plotted along with the CFDcalculations in figure 43. As seen in the figure the data show a very large scatter andthe uncertainty of these data are considered too large for validation purposes. It doeshowever show that the CFD calculation lies in between the different data series and thusare in the right range.

20 25 30 35 40 45 502

3

4

5

6

7

84 m/s

TWA[°]

Boa

t spe

ed [k

ts]

20 25 30 35 40 45 505

5.5

6

6.5

7

7.5

86 m/s

TWA[°]

Boa

t spe

ed [k

ts]

20 25 30 35 40 45 504.5

5

5.5

6

6.5

7

7.5

88 m/s

TWA[°]

Boa

t spe

ed [k

ts]

Series 1Series 2Series 3Series 4Series 5CFD

Figure 43: Full scale measurements and CFD results

46

6 Sail Trim Optimization

6.1 4 m/s

6.1.1 Design space evaluation

From the baseline case a series of parameter variations are run based on the SOBOLalgorithm. The varied parameters are trim angle, twist angle and true wind angle. Atotal of 20 variations are used and the result of the successful runs are displayed in table8. Some runs failed due to instability or failure of the mesh generation. Most of theruns actually proved to give a smaller VMG, but a few runs gave improvements in theperformance. The best run, nr. 7 gave a performance improvement of 0.69 %.

# Trim[°] Twist[°] TWA[°] V[kts] Leeway[°] Heel[°] VMG[kts] Dev.[%]7 -1.250 -0.750 33.125 6.029 4.198 9.273 4.794 0.6919 -0.125 -1.625 34.688 6.141 4.176 9.808 4.782 0.443 -0.5 -0.5 36.25 6.273 4.173 10.449 4.775 0.295 0.5 -1.5 33.75 6.044 4.149 9.141 4.769 0.170 0 0 38 6.413 4.064 10.972 4.761 0.001 1 -1 37.5 6.347 4.114 10.636 4.745 -0.3418 -0.625 0.875 35.938 6.204 4.209 10.093 4.742 -0.402 -1 1 32.5 5.911 4.223 8.576 4.738 -0.489 1.75 -1.75 35.625 6.159 4.155 9.642 4.733 -0.598 0.75 1.25 38.125 6.359 4.131 10.584 4.707 -1.1312 1.25 0.75 34.375 6.005 4.152 8.814 4.698 -1.3216 0.375 1.875 33.438 5.917 4.209 8.490 4.685 -1.606 -1.5 0.5 38.75 6.396 4.21 11.299 4.681 -1.6813 0.25 -0.25 31.875 5.784 4.275 8.048 4.671 -1.8910 -0.25 0.25 30.625 5.66 4.371 7.373 4.636 -2.6311 -0.75 -1.25 39.375 6.408 4.288 11.513 4.635 -2.6517 1.375 -1.125 30.938 5.613 4.42 6.900 4.578 -3.844 1.5 1.5 31.25 5.519 4.51 6.421 4.479 -5.92

Table 8: Result of the design space evaluation 4 m/s

Figure 44 displays the relation between the input parameters, and the objective func-tion and a regression curve. There is a clear dominance by the true wind angle whichoverrules the variations in the other parameters. It is difficult to distinguish the effect oftwist and trim angle based on this.

47

−2 −1 0 1 24.45

4.5

4.55

4.6

4.65

4.7

4.75

4.8

Trim angle [°]

VM

G [k

ts]

VMG − Trim angle

−2 −1 0 1 24.45

4.5

4.55

4.6

4.65

4.7

4.75

4.8

Twist angle [°]

VM

G [k

ts]

VMG − Twist angle

30 31 32 33 34 35 36 37 38 39 404.45

4.5

4.55

4.6

4.65

4.7

4.75

4.8

TWA [°]

VM

G [k

ts]

VMG − TWA

Figure 44: Velocity made good as a function of trim variables - red line shows cubic regression

6.1.2 Pattern search

The pattern search uses the SOBOL design space investigation as a basis for furtheroptimization. It starts in the best point nr. 7 and systematically steps in different di-rections to find a new optimum position. Table 9 shows the parameter variations andcorresponding output parameters and Figure 45 shows a graphic representation of theiteration process. The stencil size is gradually reduced when no improvements are found.After 15 evaluations the optimizer is stopped and the best trim found is at a true windangle of 33.175°, trim angle of -1.45°and twist angle of -0.55°. Running the optimizerfurther might have given further improvement, but the variable changes are now smallerthan what is practically changeable and the variations in the objective function are closeto the expected scatter.

48

# Trim[°] Twist[°] TWA[°] V[kts] Leeway[°] Heel[°] VMG[kts] Dev.[%]0 -1.250 -0.750 33.125 6.029 4.198 9.273 4.794 0.701 -1.050 -0.750 33.125 6.026 4.185 9.210 4.793 0.682 -1.450 -0.750 33.125 6.034 4.230 9.361 4.796 0.733 -1.450 -0.550 33.125 6.039 4.191 9.288 4.803 0.894 -1.450 -0.550 33.625 6.075 4.236 9.512 4.796 0.745 -1.450 -0.550 32.625 5.992 4.180 9.021 4.797 0.776 -1.725 -0.275 33.125 6.024 4.209 9.274 4.790 0.617 -1.375 -0.550 33.125 6.036 4.187 9.260 4.800 0.838 -1.525 -0.550 33.125 6.024 4.223 9.281 4.789 0.599 -1.450 -0.475 33.125 6.029 4.225 9.297 4.793 0.6610 -1.450 -0.625 33.125 6.024 4.206 9.248 4.790 0.6011 -1.450 -0.550 33.313 6.055 4.205 9.385 4.802 0.8712 -1.450 -0.550 32.938 6.024 4.202 9.206 4.802 0.8613 -1.422 -0.550 33.125 6.027 4.217 9.254 4.791 0.6414 -1.478 -0.550 33.125 6.026 4.189 9.275 4.793 0.6615 -1.450 -0.522 33.125 6.028 4.215 9.294 4.792 0.66

Table 9: Result of the pattern search 4 m/s

0 5 10 154.785

4.79

4.795

4.8

4.805VMG

[kts

]

0 5 10 15−2

−1.5

−1

−0.5

0Trimming parameters

[°]

Twist angleTrim angle

0 5 10 1532.6

32.8

33

33.2

33.4

33.6

33.8

34TWA

[°]

Figure 45: Parameter variation from pattern search and corresponding objective evaluations

The result of the trim optimization shows a modest improvement compared to thebaseline case of 0.89 % as seen in table 10. The change is a reduction in the trim angle,

49

meaning that the sail is sheeted at a tighter angle, and a reduction in twist means thatsheeting angle is reduced in the top compared to in the bottom. Figure 46 shows thesections of the sail for the original and trimmed case. In an actual sailing situation thiscan be achieved by increasing the sheet tension and moving the sheeting point slightlyforward. In addition the boat is steered closer to the wind. As seen in table 10 theboat speed is actually slightly reduced and the leeway angle is increased, but the VMG isincreased anyway due to the reduction in true wind angle. The boat moves faster againstthe wind because it steers a tighter course with only a slight speed reduction in boatspeed.

Baseline Trimmed Change Change [%]Trim[°] 0.000 -1.450 -1.450 -Twist[°] 0.000 -0.550 -0.550 -TWA[°] 38.000 33.125 -4.875 -12.83V[kts] 6.413 6.039 -0.374 -5.83Leeway[°] 4.064 4.191 0.127 3.12Heel[°] 10.972 9.288 -1.684 -15.35AWA[°] 20.545 18.475 -2.070 -10.07AWS [m/s) 6.874 6.799 -0.075 -1.09VMG[kts] 4.761 4.803 0.042 0.89

Table 10: Comparison of baseline and trimmed 4 m/s

50

Figure 46: Section geometry of original (blue) and trimmed (red) sail for 4 m/s

Figure 47 to 50 show constrained stream lines and pressure coefficients on the sailsurfaces for the baseline and trimmed sails. The flow is very similar, but there is a reducedupward flow in the lower part of the leeward side of the head sail. This is probably dueto the the decrease in apparent wind angle.

51

Figure 47: Constrained streamlines baseline sail 4 m/s

Figure 48: Constrained streamlines trimmed sail 4 m/s

52

Figure 49: Pressure coefficient baseline sail 4 m/s

Figure 50: Pressure coefficient trimmed sail 4 m/s

53

6.2 6 m/s

6.2.1 Design space evaluation

As for the 4 m/s case 20 variations are performed in the SOBOL search. The results ofthe successful runs are shown in table 11. As for the 4 m/s case most of the variationsshow no improvement and only 4 points show improvement, compared to the baselinecase. The best point nr. 2 show a 1.08 % improvement.

# Trim[°] Twist[°] TWA[°] V[kts] Leeway[°] Heel[°] VMG[kts] Dev.[%]2 -1.000 1.000 32.500 3.519 5.063 18.701 5.423 1.08

7 -1.250 -0.750 33.125 6.890 5.136 19.242 5.410 0.845 0.500 -1.500 33.750 6.925 5.074 19.354 5.395 0.5616 0.375 1.875 33.438 6.879 5.034 18.905 5.386 0.390 0.000 0.000 35.000 7.016 5.122 20.120 5.365 0.009 1.750 -1.750 35.625 7.052 5.060 20.250 5.348 -0.3210 -0.250 0.250 30.625 6.566 5.219 16.777 5.322 -0.803 -0.500 -0.500 36.250 7.093 5.207 20.896 5.316 -0.9217 1.375 -1.125 30.938 6.535 5.253 16.424 5.274 -1.6914 -1.750 1.750 36.875 7.109 5.252 21.059 5.272 -1.731 1.000 -1.000 37.500 7.121 5.285 21.226 5.226 -2.594 1.500 1.500 31.250 6.505 5.360 16.386 5.222 -2.668 0.750 1.250 38.125 7.131 5.329 21.354 5.177 -3.5115 -1.625 -0.125 38.438 7.097 5.441 21.438 5.115 -4.656 -1.500 0.500 38.750 7.120 5.444 21.585 5.105 -4.8511 -0.750 -1.250 39.375 7.137 5.377 21.813 5.068 -5.53

Table 11: Result of the design space evaluation 6 m/s

Figure 51 displays the relation between the input parameters, and the objective func-tion and a regression curve. Again there is a clear dominance by the true wind anglewhich overrules the variations in the other parameters. It is difficult to distinguish theeffect of twist and trim angle based on this.

54

−2 −1 0 1 2−5.5

−5.4

−5.3

−5.2

−5.1

−5

Trim angle [°]

VM

G [k

ts]

VMG − Trim angle

−2 −1 0 1 2−5.5

−5.4

−5.3

−5.2

−5.1

−5

Twist angle [°]

VM

G [k

ts]

VMG − Twist angle

30 31 32 33 34 35 36 37 38 39 40−5.5

−5.4

−5.3

−5.2

−5.1

−5

−4.9

TWA [°]

VM

G [k

ts]

VMG − TWA

Figure 51: Velocity made good as a function of trim variables - red line shows cubic regression

6.2.2 Pattern Search

Again the pattern search is started from the best result of the SOBOL design space in-vestigation. In this case it starts in point nr. 2 and systematically steps in differentdirections to find a new optimum position. Table 12 shows the parameter variations andcorresponding output parameter, and Figure 52 shows a graphic representation of theiteration process. The stencil size is gradually reduced when no improvements are found.After 16 evaluations the optimizer is stopped and the best trim found is at a true windangle of 32.688°, trim angle of -1.4° and twist angle of 1°. Running the optimizer furthermight have given further improvement, but the variable changes are now smaller thanwhat is practically changeable and the variations in the objective function are close tothe expected scatter.

55

# Trim[°] Twist[°] TWA[°] V[kts] Leeway[°] Heel[°] VMG[kts] Dev.[%]0 -1.200 1.000 33.000 6.905 5.071 19.184 5.436 1.321 -1.000 1.000 33.000 6.899 5.029 19.119 5.434 1.292 -1.400 1.000 33.000 6.906 5.078 19.220 5.437 1.333 -1.400 1.200 33.000 6.887 5.056 19.174 5.423 1.084 -1.400 0.800 33.000 6.897 5.060 19.211 5.431 1.225 -1.400 1.000 33.500 6.900 5.162 19.385 5.388 0.436 -1.400 1.000 32.500 6.866 5.089 18.938 5.441 1.427 -1.675 1.000 31.813 6.782 5.054 18.554 5.426 1.138 -1.325 1.000 32.500 6.846 5.067 18.889 5.427 1.159 -1.475 1.000 32.500 6.862 5.096 18.936 5.437 1.3510 -1.400 1.075 32.500 6.835 5.064 18.882 5.418 0.9911 -1.400 0.925 32.500 6.841 5.082 18.891 5.422 1.0612 -1.400 1.000 32.688 6.887 5.084 19.087 5.444 1.4713 -1.400 1.000 32.945 6.901 5.060 19.194 5.438 1.3514 -1.372 1.000 32.688 6.871 5.067 18.998 5.432 1.2615 -1.428 1.000 32.688 6.880 5.051 19.038 5.441 1.4216 -1.400 1.028 32.688 6.865 5.073 19.005 5.428 1.17

Table 12: Result of the pattern search 6 m/s

0 2 4 6 8 10 12 14 165.38

5.4

5.42

5.44

5.46VMG

[kts

]

0 5 10 15 20−2

−1.5

−1

−0.5

0

0.5

1

1.5Trimming parameters

[°]

Twist angleTrim angle

0 5 10 15 2031.5

32

32.5

33

33.5TWA

[°]

Figure 52: Parameter variation from pattern search and corresponding objective evaluations

The results of the trim optimization shows a modest improvement compared to the

56

baseline case of 1.47 %. The changes of the different parameters are outlined in table13. The change is a reduction in the trim angle meaning that the sail is sheeted at atighter angle, and an increase in twist means that sheeting angle is increased in the topcompared to in the bottom. Figure 53 show the sections of the sail for the original andtrimmed case. In an actual sailing situation this can be achieved by increasing the sheettension and moving the sheeting point slightly backwards. In addition the boat is steeredcloser to the wind. As seen in table 13 the boat speed is actually slightly reduced butthe VMG is increased anyway due to the reduction in true wind angle and reduction inleeway angle. The boat moves faster against the wind because it steers a tighter coursewith only a slight speed reduction in boat speed and on top of that the boat slides lesssideways through the water.

Baseline Trimmed Change Change [%]Trim[°] 0.000 -1.400 -1.400 -Twist[°] 0.000 1.000 1.000 -TWA[°] 35.000 32.688 -2.313 -6.61V[kts] 7.016 6.887 -0.129 -1.84Leeway[°] 5.122 5.084 -0.038 -0.74Heel[°] 20.120 19.087 -1.033 -5.13AWA[°] 20.762 19.615 -1.146 -5.52AWS [m/s) 9.048 9.067 0.019 0.21VMG[kts] 5.365 5.444 0.079 1.47

Table 13: Comparison of baseline and trimmed 6 m/s

57

Figure 53: Section geometry of original and trimmed sail for 6 m/s

Figure 54 and 55 show constrained streamlines on the sail surfaces. The flow is almostidentical in the two cases. Figure 54 and 55 show the pressure coefficient on the sailsurfaces. As for the streamlines the two cases are almost identical. The pressure isslightly lower on the windward side of the main sail on the trimmed than on the baseline.This is due to the small change in apparent wind angle. The suction in the leeward sideof the head sail is slightly bigger on the trimmed sail than on the baseline sail. This canprobably be accounted to the tighter trim angle.

58

Figure 54: Constrained streamlines baseline model 6 m/s

Figure 55: Constrained streamlines final model 6 m/s

59

Figure 56: Pressure coefficient baseline model 6 m/s

Figure 57: Pressure coefficient final model 6 m/s

60

6.3 8 m/s

6.3.1 Design space evaluation

As for the 4 and 6 m/s case, 20 variations are performed in the SOBOL search. Theresults of the successful runs are shown in table 14. As for the 4 and 6 m/s cases most ofthe variations show no improvement, and only 3 points show improvements compared tothe baseline case. The best point nr. 7 shows a 0.75 % improvement.

# Trim[°] Twist[°] TWA[°] V[kts] Leeway[°] Heel[°] VMG[kts] Dev.[%]7 -1.250 -0.750 31.750 7.052 5.519 21.28 5.612 0.75

19 -0.125 -1.625 33.625 7.229 5.579 22.69 5.602 0.570 0.000 0.000 34.000 7.230 5.605 22.71 5.570 0.005 0.500 -1.500 32.500 7.060 5.511 21.33 5.562 -0.152 -1.000 1.000 31.000 6.908 5.556 20.18 5.549 -0.3912 1.250 0.750 33.250 7.049 5.520 21.29 5.496 -1.349 1.750 -1.750 34.750 7.208 5.633 22.81 5.491 -1.4316 0.375 1.875 32.125 6.930 5.569 20.36 5.484 -1.5513 0.250 -0.250 30.250 6.745 5.635 18.88 5.465 -1.9018 -0.625 0.875 35.125 7.216 5.799 23.68 5.453 -2.113 -0.500 -0.500 35.500 7.267 5.885 24.20 5.452 -2.121 1.000 -1.000 37.000 7.342 6.003 24.88 5.370 -3.608 0.750 1.250 37.750 7.368 6.072 25.18 5.316 -4.5710 -0.250 0.250 28.750 6.427 6.105 17.26 5.274 -5.3215 -1.625 -0.125 38.125 7.387 6.418 26.15 5.265 -5.4817 1.375 -1.125 29.125 6.385 6.136 17.15 5.214 -6.4011 -0.750 -1.250 39.250 7.393 6.613 26.52 5.148 -7.584 1.500 1.500 29.500 6.338 6.302 16.73 5.140 -7.72

Table 14: Result of the design space evaluation 8 m/s

Figure 58 displays the relation between the input parameters, and the objective func-tion and a regression curve. Again there is a clear dominance by the true wind anglewhich overrules the variations in the other parameters. It is difficult to distinguish theeffect of twist and trim angle based on this.

61

−2 −1 0 1 25.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

Trim angle [°]

VM

G [k

ts]

VMG − Trim angle

−2 −1 0 1 25.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

Twist angle [°]

VM

G [k

ts]

VMG − Twist angle

28 30 32 34 36 38 40

5

5.2

5.4

5.6

5.8

TWA [°]

VM

G [k

ts]

VMG − TWA

Figure 58: Velocity made good as a function of trim variables - red line show cubic regression

6.3.2 Pattern Search

Again the pattern search is started from the best result of the SOBOL design space in-vestigation. In this case it starts in point nr. 7 and systematically steps in differentdirections to find a new optimum position. Table 15 shows the parameter variations andcorresponding output parameter and Figure 59 shows a graphic representation of the it-eration process. The stencil size is gradually reduced when no improvements are found.After 12 evaluations the optimizer is stopped and the best trim found is at a true windangle of 31.15°, trim angle of −1.45° and twist angle of −0.675°. Running the optimizerfurther might have given further improvement, but the variable changes are now smallerthan what is practically changeable and the variations in the objective function are closeto the expected scatter.

62

# Trim[°] Twist[°] TWA[°] V[kts] Leeway[°] Heel[°] VMG[kts] Dev.[%]0 -1.250 -0.750 31.750 7.052 5.519 21.280 5.612 0.751 -1.050 -0.750 31.750 7.030 5.567 21.120 5.591 0.372 -1.450 -0.750 31.750 7.063 5.534 21.375 5.620 0.883 -1.450 -0.550 31.750 - - - - -4 -1.450 -0.950 31.750 7.060 5.568 21.412 5.615 0.795 -1.450 -0.750 32.350 7.120 5.572 21.969 5.617 0.846 -1.450 -0.750 31.150 7.008 5.481 20.778 5.624 0.967 -1.725 -0.750 30.325 6.895 5.579 20.123 5.585 0.268 -1.375 -0.750 31.150 6.995 5.493 20.699 5.612 0.759 -1.525 -0.750 31.150 7.005 5.494 20.776 5.621 0.9110 -1.450 -0.675 31.150 7.012 5.490 20.805 5.626 1.01

11 -1.450 -0.675 31.375 7.016 5.561 21.010 5.608 0.6712 -1.450 -0.675 30.925 6.983 5.518 20.566 5.617 0.84

Table 15: Result of the pattern search 8 m/s

0 2 4 6 8 10 125.58

5.59

5.6

5.61

5.62

5.63VMG

[kts

]

0 5 10 15−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4Trimming parameters

[°]

Twist angleTrim angle

0 5 10 1530

30.5

31

31.5

32

32.5TWA

[°]

Figure 59: Parameter variation from pattern search and corresponding objective evaluations

The results of the trim optimization shows a modest improvement compared to thebaseline case of 1.01 %. The changes of the different parameters are outlined in table16. The change is a reduction in the trim angle meaning that the sail is sheeted at atighter angle, and a reduction in twist meaning that sheeting angle is decreased in the topcompared to in the bottom. Figure 60 show the sections of the sail for the original and

63

trimmed case. In an actual sailing situation this can be achieved by increasing the sheettension and moving the sheeting point slightly forward. In addition the boat is steeredcloser to the wind. As seen in table 16 the boat speed is actually slightly reduced butthe VMG is increased anyway due to the reduction in true wind angle and reduction inleeway angle. The boat moves faster against the wind because it steers a tighter coursewith only a small reduction in boat speed, and on top of that the boat slides less sidewaysthrough the water.

Baseline Trimmed Change Change [%]Trim[°] 0.000 -1.450 -1.450 -Twist[°] 0.000 -0.675 -0.675 -TWA[°] 34.000 31.150 -2.850 -8.38V[kts] 7.230 7.012 -0.218 -3.02Leeway[°] 5.605 5.490 -0.115 -2.05Heel[°] 22.707 20.805 -1.902 -8.38AWA[°] 21.736 20.308 -1.428 -6.57AWS [m/s) 11.029 11.061 0.032 0.29VMG[kts] 5.570 5.626 0.056 1.01

Table 16: Comparison of baseline and trimmed 8 m/s

64

Figure 60: Section geometry of original and trimmed sail for 8 m/s

Figure 61 and 62 show constrained streamlines on the sail surfaces. The flow is almostidentical in the two cases. There is a decrease in the upward going flow in the top of theleeward side of the head sail on the trimmed sail compared to the baseline. Figure 61and 62 show the pressure coefficient on the sail surfaces. The pressure on the windwardside is in general smaller on the trimmed model, and the suction on the leeward side isalso slightly smaller. This is somewhat a surprise for the head sail since the apparentwind angle change is similar to the trim change, which means they should cancel eachother. The apparent wind speed is also almost the same and thus should not influencethe pressure. The explanation may be found in the sail interaction. The pressure on themain sail is naturally decreased since it is not trimmed and thus experiences a decreasein angle of attack. The main sail twists the air to a higher angle of attack for the headsail. This is called upwash and is a well documented phenomena, see [2]. The increasedpressure on the main sail reduces the upwash and thus decreases the apparent wind anglefor the head sail resulting in a reduced pressure on the head sail.

65

Figure 61: Constrained streamlines baseline model 8 m/s

Figure 62: Constrained streamlines trimmed model 8 m/s

66

Figure 63: Pressure coefficient baseline model 8 m/s

Figure 64: Pressure coefficient trimmed model 8 m/s

67

7 Sail Shape Optimization

The sail shape optimization is performed on the baseline design with the trim modifica-tions from the above trim optimization. Except for the 4 m/s case, where the settingshad a mistake. Due to this the values of 4 m/s in this section cannot be compared tothose of the previous section.

7.1 Design Space Investigation

As for the trim investigations a SOBOL design space investigation of 20 points has beenconducted. The results of the investigation shows a small improvement in two of the pointas seen in table 17. The best design point nr. 8 has an improvement of 0.49 %.

# C1 C2 C3 CP1 CP2 CP3 Obj[kts] Dev[%]8 0.188 0.313 0.313 0.313 0.238 0.613 5.314 0.49

1 0.250 -0.250 0.250 0.350 0.650 0.350 5.293 0.100 0.000 0.000 0.000 0.500 0.500 0.500 5.288 0.003 -0.125 -0.125 0.125 0.275 0.725 0.725 5.287 -0.0212 0.313 0.188 -0.063 0.238 0.763 0.388 5.286 -0.0314 -0.438 0.438 0.188 0.388 0.313 0.238 5.279 -0.166 -0.375 0.125 0.375 0.725 0.575 0.275 5.266 -0.4115 -0.406 -0.031 0.344 0.444 0.369 0.406 5.246 -0.7916 0.094 0.469 -0.156 0.744 0.669 0.706 5.235 -1.0119 -0.031 -0.406 -0.031 0.369 0.594 0.631 5.218 -1.325 0.125 -0.375 -0.125 0.425 0.275 0.575 5.207 -1.5213 0.063 -0.063 -0.313 0.688 0.613 0.538 5.185 -1.942 -0.250 0.250 -0.250 0.650 0.350 0.650 5.173 -2.177 -0.313 -0.188 -0.188 0.613 0.538 0.313 5.166 -2.3017 0.344 -0.281 -0.406 0.294 0.519 0.256 5.084 -3.8510 -0.063 0.063 -0.438 0.463 0.688 0.763 5.010 -5.26

Table 17: Result of SOBOL variations for shape optimization

In figure 65 the objective function is plotted as a function of the individual designvariables along with a quadratic regression line. All of the graphs show a large scatterand only C3 shows some pattern. The trend is that C3 should have a value around 0.2.This means that the upper section of the sail should have more camper. The numberof design evaluations in the investigation is too small for the large number of designparameters. Thus there is a large possibility that not all of the local minima have beencovered. More points could thus improve the result, but time and resource limitationsprevented more design evaluations in the project.

68

-0.5 0 0.54.95

5

5.05

5.1

5.15

5.2

5.25

5.3

C1

VM

G [k

ts]

VMG - C1

-0.5 0 0.54.95

5

5.05

5.1

5.15

5.2

5.25

5.3

C2 []

VM

G [k

ts]

VMG - C2

-0.5 0 0.54.95

5

5.05

5.1

5.15

5.2

5.25

5.3

C3

VM

G [k

ts]

VMG - C3

0.2 0.4 0.6 0.84.95

5

5.05

5.1

5.15

5.2

5.25

5.3

CP1

VM

G [k

ts]

VMG - CP1

0.2 0.4 0.6 0.84.95

5

5.05

5.1

5.15

5.2

5.25

5.3

CP2

VM

G [k

ts]

VMG - CP2

0.2 0.4 0.6 0.84.95

5

5.05

5.1

5.15

5.2

5.25

5.3

CP3

VM

G [k

ts]

VMG - CP3

Figure 65: Velocity made good as a function design variables - red line show cubic regression

7.2 Pattern Search

The pattern search is started from point nr 8, but was interrupted after 6 iterations. Thesearch was then started again from point 4 of the first search and continued for another 13iterations. However a couple of the points are reused from the first pattern search. Thesecond pattern search did not succeed in improving the result further and the best point isthus point 4 of the first TSearch. This point shows an overall performance improvementof 0.72 %. The variations in input parameters and objective function are displayed in

69

figure 66.

# C1 C2 C3 CP1 CP2 CP3 Obj[kts] Dev [%]1-0 0.1875 0.3125 0.3125 0.3125 0.2375 0.6125 5.314 0.491-1 0.2375 0.3125 0.3125 0.3125 0.2375 0.6125 5.316 0.541-2 0.2375 0.3625 0.3125 0.3125 0.2375 0.6125 5.319 0.591-3 0.2375 0.3625 0.3625 0.3125 0.2375 0.6125 5.307 0.371-4 0.2375 0.3625 0.2625 0.3125 0.2375 0.6125 5.326 0.721-5 0.2375 0.3625 0.2625 0.3425 0.2375 0.6125 - -1-6 0.2375 0.3625 0.2625 0.2825 0.2375 0.6125 5.309 0.402-0 0.2375 0.3625 0.2625 0.3125 0.2375 0.6125 5.326 0.722-1 0.2875 0.3625 0.2625 0.3125 0.2375 0.6125 - -2-2 0.1875 0.3625 0.2625 0.3125 0.2375 0.6125 5.319 0.592-3 0.2375 0.4125 0.2625 0.3125 0.2375 0.6125 5.314 0.492-4 0.2375 0.3125 0.2625 0.3125 0.2375 0.6125 5.310 0.412-5 0.2375 0.3625 0.3125 0.3125 0.2375 0.6125 5.319 0.592-6 0.2375 0.3625 0.2125 0.3125 0.2375 0.6125 - -2-7 0.2375 0.3625 0.2625 0.3425 0.2375 0.6125 - -2-8 0.2375 0.3625 0.2625 0.2825 0.2375 0.6125 5.309 0.402-9 0.2375 0.3625 0.2625 0.3125 0.2675 0.6125 5.322 0.642-10 0.2375 0.3625 0.2625 0.3125 0.2075 0.6125 5.310 0.432-11 0.2375 0.3625 0.2625 0.3125 0.2375 0.6425 5.321 0.622-12 0.2375 0.3625 0.2625 0.3125 0.2375 0.5825 5.313 0.482-13 0.25625 0.3625 0.2625 0.3125 0.2375 0.6125 5.313 0.49

Table 18: Result of pattern search for shape optimization

70

0 2 4 6 8 10 12 14 16 18 205.305

5.31

5.315

5.32

5.325

5.33

5.335VMG

[kts

]

0 5 10 15 20

0.2

0.25

0.3

0.35

0.4

0.45

0.5C

[°]

C1C2C3

0 5 10 15 200.2

0.3

0.4

0.5

0.6

0.7

[°]

CP

CP1CP2CP3

Figure 66: Parameter variation from pattern search and corresponding objective evaluationsfor shape optimization

Table 19s show a comparison of the original design and the optimized design. From thetable it is seen that the strongest improvement is for the 8 m/s case where the improvementis 1.79 %. Speed, leeway and heel angles are increased for all wind speeds. This fits wellwith what could be expected for a sail with a greater camper. The actual deformationof the sail is visualized by three sections in figures 67 to 69. As indicated by C1-3 thecamper is increased in all sections. The two lower sections gained more camper furtherforward, while the top section moved camper more aft.

71

Baseline Optimized Change Change [%]Variables C1 0.000 0.2375 0.2375 -

C2 0.000 0.3625 0.3625 -C3 0.000 0.2625 0.2625 -CP1 0.500 0.3125 -0.1875 -CP2 0.500 0.2375 -0.2625 -CP3 0.500 0.6125 0.1125 -

4 m/s V[kts] 5.883 5.928 0.045 0.77Leeway[°] 4.277 4.423 0.146 3.42Heel[°] 8.744 9.545 0.802 9.17AWA[°] 18.718 18.615 -0.102 -0.55AWS [m/s) 6.724 6.743 0.019 0.28VMG[kts] 4.674 4.700 0.027 0.57

6 m/s V[kts] 6.864 6.895 0.031 0.45Leeway[°] 5.073 5.244 0.171 3.37Heel[°] 18.875 19.606 0.731 3.88AWA[°] 19.663 19.548 -0.114 -0.58AWS [m/s) 9.058 9.065 0.007 0.08VMG[kts] 5.427 5.439 0.012 0.22

8 m/s V[kts] 7.008 7.136 0.129 1.84Leeway[°] 5.481 5.515 0.034 0.61Heel[°] 20.778 22.351 1.573 7.57AWA[°] 20.315 19.997 -0.318 -1.57AWS [m/s) 11.060 11.097 0.038 0.34VMG[kts] 5.624 5.724 0.101 1.79Objective 5.288 5.326 0.038 0.72

Table 19: Comparison of baseline and optimized design

72

Figure 67: Section geometry of original (blue) and optimized (red) sail for 4 m/s

73

Figure 68: Section geometry of original (blue) and optimized (red) sail for 6 m/s

74

Figure 69: Section geometry of original (blue) and optimized (red) sail for 8 m/s

Figures 70 and 71 show constrained streamlines of the baseline and optimized sail for4 m/s condition. On the windward side of the head sail the flow reflects further upwardsdue to the increased camper of the sail. A separation-like rotation is observed near theleading edge. On the leeward side of the head sail the increased camper creates a trailingedge separation. The leading edge separation on the main sail is increased maybe dueto the increased effect of the head sail increasing the downwash effect. This effect canalso be seen on the pressure distribution on the windward side of main sail, as seen infigures 72 and 73. The pressure on the windward side of the head sail is reduced and thepressure centre is moved backwards. This can be attributed to the increase and forwardmovement of the camper. The increased camper also increases the suction on the leewardside of the head sail, while the suction on the main sail is unchanged.

75

Figure 70: Constrained streamlines baseline sail 4 m/s

Figure 71: Constrained streamlines optimized sail 4 m/s

76

Figure 72: Pressure coefficient baseline sail 4 m/s

Figure 73: Pressure coefficient optimized sail 4 m/s

77

Figures 74 to 77 display the constrained streamlines and pressure coefficient for thebaseline and optimized design for the 6 m/s condition. The trend of upward moving flowon the windward side of the head sail is also present in the 6 m/s case but it is not as strongas for 4 m/s, and the swirling separation is not seen in the streamline plot. The upwardfacing flow on the leeward side of the head sail has been reduced. As for 4 m/s trailingedge separation occurs at the leeward side of the head sail for the optimized design. Theleading edge separation on the windward side of main sail has also increased as for the4 m/s condition. In contrast to the 4 m/s condition the pressure on the windward sideof the head sail is not reduced, but it is moved backward. The windward pressure of themain sail is slightly increased while the leeward suction remains the same. The suctionon the leeward side of the head sail is only increased a little, which is reflected in therelatively small improvement of boat speed found in the 6 m/s condition.

78

Figure 74: Constrained streamlines baseline sail 6 m/s

Figure 75: Constrained streamlines optimized sail 6 m/s

79

Figure 76: Pressure coefficient baseline sail 6 m/s

Figure 77: Pressure coefficient optimized sail 6 m/s

80

Figures 78 to 81 display the constrained streamlines and pressure coefficient for thebaseline and optimized design for the 8 m/s condition. The upward facing flow on thewindward side of the head sail has increased as for the other conditions and the swirlingseparation near the leading edge is strong. On the leeward side the trailing edge separationalso becomes significant for the optimized sail. In contrast to the other conditions, theleading edge separation on the windward side of the main sail has not been increased.The main sail flow is similar in baseline and optimized case. The windward pressure isreduced on both sails, while the leeward suction has been increased on the head sail andreduced on the main sail. The increase in suction on the head sail can be accounted tothe increase in camper, and the decrease on the main sail is probably due to the trailingedge separation of the head sail disturbing the flow.

81

Figure 78: Constrained streamlines baseline sail 8 m/s

Figure 79: Constrained streamlines optimized sail 8 m/s

82

Figure 80: Pressure coefficient baseline sail 8 m/s

Figure 81: Pressure coefficient optimized sail 8 m/s

83

8 Conclusion

A simulation setup for simulating the physics of sailboats is made in the commercial CFDsoftware STAR-CCM+. The simulation includes both aero and hydrodynamics of theboat and calculates their effect on boat speed, leeway angle and heel angle. The simula-tion is used as a basis for an optimization of sail trim and sail shape. The X-40 is usedas a test boat for the optimization, and the sail design used as a baseline is a design fromNorth Sails Denmark. Only the head sail trim and shape is changed during the project.

To verify and validate the CFD model two different studies are conducted. Firstlya study of the aerodynamic forces on sails are performed on a set of AC33 class sails.The result is validated against experimental wind tunnel data for pressure, lift and dragforces. The verification shows an oscillatory behaviour of the discretization uncertainty.The lift is validated with a validation uncertainty of 6.3 % while uncertainty of the drag isaround 15.6 % for the chosen mesh density. The pressure validation in general shows goodagreement with the experimental results, but again the oscillatory behaviour of the dis-cretization gives a large uncertainty. Combined with the large experimental uncertaintythis gives a validation uncertainty for the pressure of 17 - 30 % for the different sections.

A verification of the full simulation model is conducted with 7 different meshes and thediscretization uncertainty is evaluated for the important output parameters. The studyagain shows some oscillations but in general the mesh convergence is much better. Onlythe heel angle shows a large discretization uncertainty of 16.8 %, while the most importantoutput parameter VMG shows 1.2 % uncertainty. Finally full scale measurements fromracing results of an X-40 are presented for a rough comparison. The data in general showtoo much scatter for validation purposes, but they do however illustrate that the resultsof the simulation are well inside the scatter of the data.

To modify and optimize the sail the commercial software Friendship Framework isused. The framework control the parametric variations, optimization of the variables,and execution of the simulations. The software is run on a local machine and the simu-lations are run via SSH at a remote simulation cluster.

A trim optimization was run to optimize trim angle, twist angle and true wind angleto get the best VMG. The trim optimization was run with a design space investigationby the SOBOL algorithm and a tangent search method from the best point in the designspace investigation. In general the trend was to decrease the true wind angle and thetrim angle. While the twist angle was decreased for 4 and 8 m/s and increased for 6 m/s.The outcome was an improvement of 0.89, 1.47 and 1.01 % on VMG for 4, 6 and 8 m/srespectively.

The shape optimization was run on the basis of 6 design variables deforming the sailshape within the borders of the sail with a shift surface. The 6 variables change thecamper and the position of the maximum camper of the sail. As with the trim optimiza-tion, the shape optimization contained a design space investigation and a pattern search.The design space investigation was, due to time and resource limitations, limited to 20designs, which is somewhat insufficient for the number of variables. It anyhow succeededin finding some improvements. With the following pattern search the total improvement

84

was 0.72 %. This may seem like a small difference, but it corresponds to passing anothersimilar boat with an unoptimized sail within 4 to 30 min depending on the wind speed.

The sail has an increased camper for all sections. The camper is moved forward in thelower and middle part of the sail and backwards in top part. Surprisingly the increase incamper gives more improvement in the 8 m/s condition than in the two others. However,it shows the purpose of simulation which is to enlighten our view on physics phenomena.

9 Perspective

In this project the trim and shape optimization was separated. A straight forward im-provement of this approach is to combine these optimizations. Either by including thetrim optimization as a sub-optimization for each design point in the shape optimizationor by including all variables in one optimization. In both cases the computational timerequired will be much larger than for the separated case used in this project. On theother hand it may bring improvements not foreseen by the present method.

The trim optimization shows an interesting potential. A future inclusion of more vari-ables could make this a good tool for predicting performance with variation of the differentvariables. The output of an extended analysis could be used for improvement of trimmingduring racing. Inclusion of a structural code to evaluate the actual deformation of the sailand rigging could make the tool more accurate and relevant for high-end racing programs.

The present cost of computational power makes this sort of optimization too expen-sive to be used in most of the sailing world. However with the ever decreasing cost andincreasing computational power the future may bring opportunities for this sort of opti-mization to be spread wider.

Simulation and optimization of sails are not only relevant for pleasure crafts. An in-creasing pressure on the world society to reduce consumption of non-renewable energyincreases the interest on fitting sails ore kites on commercial ships. This brings anothercommercial perspective into play and increases demand for efficient sail or kite design.

85

References

[1] Rousselon N., 2008, Optimization for Sail Design, paper presented at the ModeFron-tier Conference, June 2008, Trieste, Italy

[2] Fossati, F., Aero-Hydrodynamics And the Performance of Sailing Yachts, AdlardColes Nautical, 2009

[3] Shih, T.-H., Liou, W.W., Shabbir, A., Yang, Z. and Zhu, J. 1994. A New k− ε EddyViscosity Model for High Reynolds Number Turbulent Flows – Model Developmentand Validation, NASA TM 106721.

[4] Larsson, L., Stern, F., Visonneau, M., A Workshop on Numerical Ship Hydrodynam-ics - Proceedings, Volume II, Gothenburg 2010

[5] Menter, F.R. 1994. Two-equation eddy-viscosity turbulence modeling for engineeringapplications, AIAA Journal 32(8) pp. 1598-1605.

[6] CHARNOCK, H, Wind Stress on Water - An Hypothesis - Wind Stress on s WaterSurface, Quarterly Journal of the Royal Meteorological Society - 1955, Volume 81,Issue 350, pp. 639-640

[7] Emeis, Stefan, Wind Energy Meteorology - Atmospheric Physics for Wind PowerGeneration, Springer Berlin Heidelberg, 2013

[8] Eca, L., Vaz, G., Hoekstra, M., 2010. Code verification, solution verification andvalidation in RANS solvers. In: ASME 29th Int. Conf. OMAE2010, Shanghai, China.

[9] Nocedal, J, Wright, S. J., Numerical Optimization, 2nd edition, Springer 2006

[10] Sobol,I.M., Distribution of points in a cube and approximate evaluation of integrals.Zh. Vych. Mat. Mat. Fiz. 7: 784-802 (in Russian); U.S.S.R Comput. Maths. Math.Phys. 7: 86-112 (in English), 1967

[11] Best Practice Guidelines for Marine Applications of Computational Fluid Dynamics,MARNET-CFD group, European Union, 2003

[12] Star-CCM+ Users guide Version 8.04, CD Adapco, 2013

[13] Terndrup Pedersen, P, Andersen, P and Aage, C., Grundlæggende Skibs- og Off-shoreteknik,version 6, MEK - DTU 2004

[14] Friendship Framework Documentation ver. 3.0.8

[15] Viola, I.M., Pilate, J., Flay, R.G.J., 2011. Upwind sail aerodynamics: a pressuredistribution database for the validation of numerical codes. International J. SmallCraft Technol., Trans. RINA 153(B1), 47-58.

[16] Viola, I.M., et al. Upwind sail aerodynamics: A RANS numerical investigation vali-dated with wind tunnel pressure measurements. Int. J. Heat Fluid Flow (2012)

[17] Elden, L, Wittmeyer-Koch, L, Bruun Nielsen, H.,Introduction to Numerical Compu-tation, Studentlitteratur, Edition 1:4, 2004

86

Appendix A: Nomenclature with illustration

Nomenclature

α Angular acceleration

αxx Acceleration in surge

β Apparent wind angle

λ Leeway angle

µ Fluid dynamic viscosity

∇ Displacement

∇c Displacement of canoe body

ν Fluid kinetic viscosity

U10 Wind Velocity at 10 m

ρ Fluid density

θ Heel angle

a Empirical constant

ax Acceleration in surge

ay Acceleration in sway

AWA Apparent Wind Angle - β

AWS Apparent Wind Speed

B Breadth

CD Drag coefficient

CL Lift coefficient

CAD Computer Aided Design

CB Center of buoyancy

CE Centre of Effort of the aerodynamicforces

CFD Computational Fluid Dynamics

CG Centre of gravity

CLR Centre of Lateral Resistance

CSV Comma Separated Values - file for-mat for storing tabular data as text

E Foot length of main sail

FA Total aerodynamic force

FM Aerodynamic driving force

FM Driving force

FLAT Aerodynamic side force

FLAT Aerodynamic side force

FV ERT Aerodynamic vertical force

FEM Finite Element Method

g Gravitational acceleration

h Heeling arm

IGES Initial Graphics Exchange Specifica-tion - CAD exchange format

k Von Karmans constant

kts Nautical knots

LOA Length over all

LWL Length of waterline

LP Maximum width of head sail - maxi-mum leech perpendicular

P Luff length of main sail

PI Hydrodynamic heel force

PLAT Hydrodynamic side force

PLAT Hydrodynamic side force

PV ERT Hydrodynamic vertical force

R Hydrodynamic resistance of hull andappendages

R Hydrodynamic resistance

RM Righting moment

SSH Secure Shell - network protocol forsecure connection

STL STereo Lithography - CAD exchangeformat

T Total draft

t time

Tc Draft of canoe body

Tmax Luff length of head sail

u∗ Friction velocity

Uexp Experimental uncertainty

Uinp Parameter uncertainty

Unum Numerical uncertainty

Uval Validation uncertainty

V Boat Speed

V PP Velocity Prediction Program

W Weight of boat

Wcrew Crew weight

z0 Roughness length

KeelRud

der

Hull

HeadSail

MainSail

Bac

ksta

y

Forestay

Sheet

Luff

LuffLeac

h

Leac

h

Tack

TackClew

Clew

Head

Head

Boom

Mast

Lead block

Nomenclature illustration

Appendix B: Batch Scripts

runActions.sh 7. august 2013 11:01

#!/bin/sh

############ runActions.sh# 07-08-2013# Batch script for controling communicaltion and execution of simulaton process###########

# The script runRemoteDir is executed in a terminal on RemoteMachine trough ssh connectionsshpass -p 'password' ssh user@RemoteMachine 'bash -s' < runRemoteDir.sh

# All needed files are copied to RemoteMachine through ssh. "<entry>Case</entry>/" is case name input from Friendship Frameworksshpass -p 'password' scp * user@RemoteMachine:/directory/<entry>Case</entry>/

# The script runRemote is executed in a terminal on RemoteMachine trough ssh connectionsshpass -p 'password' ssh user@RemoteMachine 'bash -s' < runRemote.sh

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runRemoteDir.sh 7. august 2013 11:36

#!/bin/sh

############ runRemoteDir.sh# 07-08-2013# Batch script for creating directory on remote machine###########

# Go to directorycd /directory

# Make new directory with name input "<entry>Case</entry>" from Frienship Frameworkmkdir <entry>Case</entry>

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runrRemote.sh 7. august 2013 11:34

#!/bin/sh

############ runRemote.sh# 07-08-2013# Batch script for controlling execution of simulation on remote machine and file tranfer of results###########

# Go to the directory "<entry>Case</entry>" specified by Friendship Frameworkcd /directory/<entry>Case</entry>

# Submit the job to the queuing system vi a the script RunScript~/bin/RunScript filename.sim

# Tranfer csv files back to local machine to the directory "<entry>Dir</entry>" specified by Friendship Frameworksshpass -p 'password' scp *csv user@LocalMachine:<entry>Dir</entry>/computation/

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Appendix C: STAR-CCM+ Java Macros

Master.java 7. august 2013 11:55

///////////// STAR-CCM+ macro: Master.java// 07-08-2013// Java macro for controlling simulation steps. Calls the other macros.///////////

package macro;

import java.util.*;import star.common.*;

public class Master extends StarMacro

public void execute() execute0();

private void execute0()

// Run a script to change geometry and remeshnew StarScript(getActiveSimulation(),new java.io.File(resolvePath("changeGeometry.java"))).play();

// Run a script to change setting of the sim filenew StarScript(getActiveSimulation(),new java.io.File(resolvePath("setParameters.java"))).play();

// Run a script to run simulationnew StarScript(getActiveSimulation(),new java.io.File(resolvePath("run.java"))).play();

// Run a script to write results from the simulation to csv filesnew StarScript(getActiveSimulation(),new java.io.File(resolvePath("getResults.java"))).play();

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changeGeometry.java 7. august 2013 12:09

///////////// STAR-CCM+ macro: changeGeometry.java// 07-08-2013// Java macro for:// * changing head sail geometry// * rotating to initial heel angle// * defining feature lines for head sail// * execute meshing///////////

package macro;

import java.util.*;

import star.common.*;import star.base.neo.*;import star.meshing.*;

public class changeGeometry extends StarMacro

public void execute() execute0();

private void execute0()

// Get the active simulationSimulation simulation_0 =getActiveSimulation();

// Define units for importUnits units_0 =simulation_0.getUnitsManager().getPreferredUnits(new IntVector(new int[] 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));

// Awake import managerPartImportManager partImportManager_0 =simulation_0.get(PartImportManager.class);

// Get the part HeadLeafMeshPart leafMeshPart_1 =

((LeafMeshPart) simulation_0.get(SimulationPartManager.class).getPart("Head"));

// Replace part Head with stl file HeadSail.stlpartImportManager_0.reimportStlPart(leafMeshPart_1, resolvePath("HeadSail.stl"), units_0,

true, 1.0E-5);

// Define units for rotationUnits units_1 =simulation_0.getUnitsManager().getPreferredUnits(new IntVector(new int[] 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));

// Define coordinatesystem for rotationLabCoordinateSystem labCoordinateSystem_0 =simulation_0.getCoordinateSystemManager().getLabCoordinateSystem();

// Rotate head sail to initial heel angle. "<entry>HeelAngle</entry>" specified

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changeGeometry.java 7. august 2013 12:09

heelangle from Friendshi frameworksimulation_0.get(SimulationPartManager.class).rotateParts(new NeoObjectVector(new Object[] leafMeshPart_1), new DoubleVector(new double[] -1.0, 0.0, 0.0), newNeoObjectVector(new Object[] units_0, units_0, units_0), <entry>HeelAngle</entry>,labCoordinateSystem_0);

// Get surface for defining feature linesPartSurface partSurface_0 =leafMeshPart_1.getPartSurfaceManager().getPartSurface("|6ms|trimmed6ms|trimesh0");

// Create part feature curvesPartCurve partCurve_8 =leafMeshPart_1.createPartCurvesOnPartSurfaces(new NeoObjectVector(new Object[] partSurface_0), true, true, true, true, true, true, true, 31.0, false);

// Get region for feature definitionRegion region_0 =simulation_0.getRegionManager().getRegion("Domain");

// Get region feature curveFeatureCurve featureCurve_10 =

((FeatureCurve) region_0.getFeatureCurveManager().getObject("HeadOutline"));

// Set region feature curvefeatureCurve_10.getPartCurveGroup().addObjects(partCurve_8);

// Get mesh controllerMeshPipelineController meshPipelineController_0 =simulation_0.get(MeshPipelineController.class);

// Generate meshmeshPipelineController_0.generateVolumeMesh();

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setParameters.java 7. august 2013 12:45

///////////// STAR-CCM+ macro: setParameters.java// 07-08-2013// Java macro for setting parameters in the simulation file///////////package macro;

import java.util.*;

import star.common.*;import star.base.neo.*;

public class setParameters extends StarMacro

public void execute() execute0();

private void execute0()

// Get the simulationSimulation simulation_0 =getActiveSimulation();

// Get the field function TWAUserFieldFunction userFieldFunction_0 =

((UserFieldFunction) simulation_0.getFieldFunctionManager().getFunction("TWA"));

// Set the field function TWA to value from Friendship FrameworkuserFieldFunction_0.setDefinition("<entry>TWA</entry>/180*$pi");

// Get the solverImplicitUnsteadySolver implicitUnsteadySolver_0 =

((ImplicitUnsteadySolver) simulation_0.getSolverManager().getSolver(ImplicitUnsteadySolver.class));

// Set solver time step to value from Friendship FrameworkimplicitUnsteadySolver_0.getTimeStep().setValue(<entry>TimeStep</entry>);

// Get stopping criterionPhysicalTimeStoppingCriterion physicalTimeStoppingCriterion_0 =

((PhysicalTimeStoppingCriterion) simulation_0.getSolverStoppingCriterionManager().getSolverStoppingCriterion("Maximum Physical Time"));

// set stopping criterion to value from Friendship FrameworkphysicalTimeStoppingCriterion_0.getMaximumTime().setValue(<entry>SimulationTime</entry>);

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run.java 7. august 2013 12:11

///////////// STAR-CCM+ macro: run.java// 07-08-2013// Java macro for running the simulation///////////

package macro;

import java.util.*;

import star.common.*;

public class run extends StarMacro

public void execute() execute0();

private void execute0()

// Get the simulationSimulation simulation_0 =getActiveSimulation();

// Run the simulationsimulation_0.getSimulationIterator().run();

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getResults.java 7. august 2013 12:20

///////////// STAR-CCM+ macro: getResults.java// 07-08-2013// Java macro for exporting results to csv files///////////package macro;

import java.util.*;

import star.common.*;import star.base.neo.*;

public class getResults extends StarMacro

public void execute() execute0();

private void execute0()

// Get active simulationSimulation simulation_0 =getActiveSimulation();

// Get the boat speed monitorMonitorPlot monitorPlot_0 =

((MonitorPlot) simulation_0.getPlotManager().getObject("BoatSpeed Monitor Plot"));

// Export boat speed monitor data to csv filemonitorPlot_0.export(resolvePath("BoatSpeed.csv"), ",");

// Get the heel angle monitorMonitorPlot monitorPlot_1 =

((MonitorPlot) simulation_0.getPlotManager().getObject("HeelAngle Monitor Plot"));

// Export heel angle monitor data to csv filemonitorPlot_1.export(resolvePath("HeelAngle.csv"), ",");

// Get the leeway angle monitorMonitorPlot monitorPlot_2 =

((MonitorPlot) simulation_0.getPlotManager().getObject("LeewayAngle Monitor Plot"));

// Export leeway angle monitor data to csv filemonitorPlot_2.export(resolvePath("LeewayAngle.csv"), ",");

// Save simulation results in sim filesimulation_0.saveState(resolvePath("Baseline_4ms_Results.sim"));

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Appendix D: Streamline plots

3D streamlines 4 m/s baseline model

3D streamlines 4 m/s trimmed model

3D streamlines 4 m/s optimized model

3D streamlines 6 m/s baseline model

3D streamlines 6 m/s trimmed model

3D streamlines 6 m/s optimized model

3D streamlines 8 m/s baseline model

3D streamlines 8 m/s trimmed model

3D streamlines 8 m/s optimized model

DTU Mechanical Engineering

Section of Fluid Mechanics, Coastal and Maritime Engineering

Technical University of Denmark

Nils Koppels Allé, Bld. 403

DK- 2800 Kgs. Lyngby

Denmark

Phone (+45) 4525 1360

Fax (+45) 4588 4325

www.mek.dtu.dk