Sail Shape Optimization with CFD - DANSIS

Master Thesis

Sail Shape Optimization with CFD

Stig Staghøj Knudsen August 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 the thesis 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 Thystrup from X-Yacht for providing detailed information and drawing of the X-40. Thanks to Heine Sørensen from North Sails for preparing the initial design of the sails including deformation analysis. Thanks to Michael Richelsen for providing valuable inside knowledge and great advices. And a special thanks to Kasper Wedersøe for helping with the process and contacts. Thanks to OSK-Shiptech for providing the necessary computational power at times of low working load on the company cluster. Thanks to Milovan Peric and Albert Gasc´on from CD-Adapco for advice and support for the simulation setup. Thanks to Mattia Brenner, Claus Abt and Konrad Lorentz for support with Friendship Framework. Thanks to Frank Pedersen, Casper Schytte, and Brian Kerr for feedback on the report. And finally thanks to my supervisor Professor Jens Honore Walther for great cooperation, constructive conversations and valuable inputs to the project.

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

Abstract The design of sailboat sails are of great interest in the world of yacht racing. The appearance of modern simulation technology makes it possible to predict the aerodynamic forces on the sail and the hydrodynamic forces of the water on the sailboat. The purpose of this thesis is to find a way to optimize design of sailboat sails through the use of simulation based design. An X-40 sailboat is used as test boat for the optimization with a set of sails design by North Sails DK as baseline. A CFD model is presented for simulating both aero- and hydrodynamic forces and the movements of the boat. Traditionally the performance evaluation of sail boats are done separately from the simulation of forces, but in this case the velocity, leeway angle and heel angle prediction is built into the simulation to reduce the number of simulations needed. 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 motion solver 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. Firstly the trim and true wind angle is optimized for optimum VMG. Secondly the shape of the sail is optimized by reduced parametric modelling. In both cases the SOBOL algorithm was used to investigate the design space followed by a pattern search method to refine the variables further to the optimum VMG. The trim optimisation showed improvements in VMG from 0.9 to 1.5 % over the wind range 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 shape camper was increased, and the camper was moved forward in the lower and middle part of the sail, and backward in the upper part of the sail. The computational power required by this method and the associated expenses are large compared to the budgets of most racing teams. But with the decreasing prices and increasing performance of simulations the perspective of this sort of optimization could grow. The increasing commercial interest in using sails and kites to assist propulsion of commercial ships could lead to an increased interest in optimization of sail and kite performance.

Synopsis Design af effektive sejl til sejlb˚ ade har stor interesse indenfor sejlsport. Moderne teknologi har gjort det muligt at forudsige de kræfter, der virker p˚ a b˚ aden i form af aerodynamiske kræfter fra vinden og hydrodynamiske kræfter fra vandet. Form˚ alet med dette kandidatspeciale er at finde en metode til optimering af sejl baseret p˚ a simuleringsdrevet design. Sejlb˚ aden X-40 fra danske X-yacht er anvendt som testb˚ ad til optimeringen, med et sæt sejl designet af North Sails DK som udgangspunkt for optimeringen. En CFD model til at forudsige aero- og hydrodynamiske kræfter samt bevægelsen af sejlb˚ aden bliver præsenteret. Normalt sker hastigheds forudsigelsen af sejlb˚ ade separat fra simulering af kræfterne, men i dette tilfælde bliver forudsigelsen af hastighed, afdriftsvinkel og krængningsvinkel inkluderet direkte i simuleringen for at reducere antallet af simuleringer. Simuleringerne udføres i det kommercielle CFD program STAR-CCM+, hvor en VOF model anvendes til at beskrive interaktionen mellem luft og vand og en 6-DOF løser h˚ andterer bevægelserne af b˚ aden. M˚ alet er at optimere formen p˚ a forsejlet til et vindomr˚ ade mellem 4 og 8 m/s. Først optimeres trimmet af sejlet og den sande vindvinkel for at opn˚ a optimalt VMG for de givne designs. Dernæst optimeres formen af sejlet ved reduceret parameter modellering. I begge tilfælde anvendes SOBOL algoritmen til at undersøge forskellige kombinationer af parametre efterfulgt af en patern search optimering for at justere variablerne til optimal VMG. Trim optimeringen gav en forbedring af VMG p˚ a 0,9 til 1,5 % afhængig af vindstyrken, og tendensen var at det optimale var en reduktion i sand vindvinkel og trim vinkel af sejlet. Optimeringen af sejlfaconen gav en forbedring p˚ a 0,7 % p˚ a VMG. Dybden af sejlet blev større, og dybden blev flyttet længere frem i bunden og midten af sejlet, mens den blev flyttet længere tilbage i toppen. Metodens krav til beregningskraft og de dertilhørende udgifter er store sammenlignet med budgetterne for de fleste kapsejladshold. Men de faldende priser og stigende effektivitet af numeriske simuleringer kan f˚ a denne type optimering til at blive mere interessant i fremtiden. Den stigende kommercielle interesse i at bruge sejl eller drager til at hjælpe fremdriften af kommercielle skibe kan ligeledes føre til en øget interesse om optimering af sejl og drager.

Contents 1 Introduction

1

2 Boat 2.1 Sail and rigging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Full Scale Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4

3 Theory 3.1 Sailboat Dynamics . . . . . . . . . 3.1.1 Apparent Wind and VMG . 3.1.2 Aerodynamic forces . . . . . 3.1.3 Hydrodynamic forces . . . . 3.1.4 Heeling moment . . . . . . . 3.1.5 Righting moment . . . . . . 3.2 Balance of forces and moments . . 3.3 Sail shape . . . . . . . . . . . . . . 3.4 Trim variation . . . . . . . . . . . . 3.4.1 True wind speed and angle . 3.4.2 Sheet tension . . . . . . . . 3.4.3 Sheeting position . . . . . . 3.4.4 Forestay tension . . . . . . . 3.4.5 Halyard tension . . . . . . . 3.5 CFD . . . . . . . . . . . . . . . . . 3.5.1 Governing equations . . . . 3.5.2 RANS . . . . . . . . . . . . 3.5.3 Turbulence modelling . . . . 3.5.4 Finite Volume Method . . . 3.5.5 Volume of Fluid . . . . . . . 3.5.6 Fluid Structure Interaction . 3.5.7 Wind profile . . . . . . . . . 3.5.8 Verification and Validation . 3.6 Optimization . . . . . . . . . . . . 3.6.1 Optimization Problems . . . 3.6.2 Optimization Methods . . . 3.6.3 CFD-based Optimization . . 3.6.4 Pattern Search Algorithms . 3.6.5 SOBOL . . . . . . . . . . . 4 Method 4.1 CFD model . . . . . . . . . . 4.1.1 Domain . . . . . . . . 4.1.2 Mesh . . . . . . . . . . 4.1.3 Boundary conditions . 4.1.4 Solvers . . . . . . . . . 4.2 Friendship Framework Model 4.2.1 Geometry . . . . . . . 4.2.2 Trim parameters . . . 4.2.3 Shape parameters . . .

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4.3

4.2.4 Optimization algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 29 Computation Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Verification and Validation 5.1 Validation of aerodynamic forces in CFD model 5.1.1 Domain . . . . . . . . . . . . . . . . . . 5.1.2 Mesh . . . . . . . . . . . . . . . . . . . . 5.1.3 Results . . . . . . . . . . . . . . . . . . . 5.1.4 Validation Remarks . . . . . . . . . . . . 5.2 Verification of full model . . . . . . . . . . . . . 5.2.1 Spatial Discretization . . . . . . . . . . . 5.2.2 Temporal Discretization . . . . . . . . . 5.2.3 Domain size . . . . . . . . . . . . . . . . 5.2.4 Moment of inertia . . . . . . . . . . . . . 5.2.5 Simulation Scatter . . . . . . . . . . . . 5.3 Full scale data . . . . . . . . . . . . . . . . . . . 6 Sail Trim Optimization 6.1 4 m/s . . . . . . . . . . . . . . 6.1.1 Design space evaluation 6.1.2 Pattern search . . . . . . 6.2 6 m/s . . . . . . . . . . . . . . 6.2.1 Design space evaluation 6.2.2 Pattern Search . . . . . 6.3 8 m/s . . . . . . . . . . . . . . 6.3.1 Design space evaluation 6.3.2 Pattern Search . . . . .

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47 47 47 48 54 54 55 61 61 62

7 Sail Shape Optimization 68 7.1 Design Space Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 7.2 Pattern Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 8 Conclusion

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9 Perspective

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Appendix

87

1

Introduction

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

t

Lif

Boat S λ

peed (V

a Dr

g

)

ind

W

Figure 1: Sail forces

Sail shape has been optimized over many years by experience and trial and error. But no perfect shape has been found because the shape depends on the rest of the boat and on the conditions which the boat is sailing in. Detailed automated optimization has been conducted only for high budget racing campaigns. The common approach to optimization is to reduce the complexity and optimize merely for driving force with a penalty of side force, [1]. It is however difficult to determine the penalty of the side force. A more accurate approach is to do full velocity prediction of the boat by a Velocity Prediction Program (VPP). The program seeks to find a steady state solution where all forces are in equilibrium. The hydrodynamic and aerodynamic forces come from empirical models, experiments or simulations. This requires data available for variation of different parameters such as boat speed, wind speed, leeway angle, wind angle, trim parameters and heel angle. This gives a large matrix of parametric variation to consider. A new approach is to move the velocity prediction inside of the simulation software and predict velocity, heel angle and leeway angle as part of the CFD solution. The aim of this thesis is to automate the optimization process and find optimization approach to sail design, with velocity prediction included in the simulation. The performance of the sail is evaluated by evaluating the performance of the boat. To get the full boat performance both aero and hydrodynamic forces are modelled. The interaction between these forces are 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 of modelling points or control point directly connected to the surface, a secondary control surface is used to transform the shape. This reduces the number of parameters and thus the number of evaluations needed. The commercial software Friendship Framework are used for this transformation, for the trim variations and for the optimization. Based on the performance of the sail in different conditions the shape is optimized to get the best overall performance. As a test case for this project a 40 foot performance cruiser, X-40 from X-yacht, is used. 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 optimal upwind performance. The optimization is limited to the head sail to reduce the number of variable. For reference of special term, symbols and abbreviations please see appendix A.

2

2

Boat

The example boat used in the current project is an X-40, designed and produced by X-Yacht. The main particulars of the boat are listed in table 1. X-yacht delivered the geometry of the hull and superstructure in 3D IGES format along with 2D drawings of rigging 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

LW L LOA B T Tc ∇ ∇c Wcrew

10.63 12.20 3.80 2.10 0.41 7.26 7.06 885

m m m m m m3 m3 kg

Table 1: Main particulars

2.1

Sail and rigging

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

Figure 3: Rig and sails

E P LP Tmax

5.60 15.70 4.56 16.57

m m m 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, shown in figure 4. Full scale data from the boat is available used for rough comparison. The data was recorded from racing during the last couple of years fitted by a VVP evaluation software. The data was delivered by crew member Kasper Wedersøe.

4

Figure 4: X-40 Sirena

5

3 3.1

Theory 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 the sailboat are described. The forces result in motions, and because the boat is free to move in 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 reach equilibrium, 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 when stationary. This experienced wind is called apparent wind. The apparent wind is a vector sum of the true wind vector and the boat speed vector as shown in figure 6. The apparent wind speed (AWS) is for upwind sailing larger than the true wind speed (TWS). This makes fast boats capable of sailing faster than the wind and in popular term the boat is said to generate wind for itself. 6

Boat Speed (V) AWA

Tru

ew ind

TWA

p Ap

ar

e

w nt

ind

Figure 6: Apparent and true wind

Sailboats are not capable of sailing directly against the wind. They typically sail at a true wind angle between 30 and 40°. Due to side force generation the boat moves slightly sideways through the water as shown in figure 7. The difference between the steered course and the course the boat actually moves is the leeway angle λ. When measuring the efficiency of the boat sailing in upwind condition, the true wind angle (TWA), leeway angle 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 calculated by: V M G = V cos(T W A + λ) (1) Boat Spe λ

ed (V) TWA

VM

G

ind

Tr

w ue

Figure 7: Velocity made good

3.1.2

Aerodynamic forces

The aerodynamic forces are the ”engine” of the boat. The sails work as aerofoils, resulting in 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 perpendicular to the sailing direction, affecting the sway motion. The driving force is obviously important since it drives the boat forward, but the side force is of equal importance since it needs to be balanced by a counteracting side force from the hydrodynamic forces. It is not only the sails that introduce aerodynamic forces. The hull, superstructure, crew and rigging introduce a windage that provides drag and thus slow the boat. The aerodynamic forces are acting above the waterline and often their point of action is called the Centre of Effort (CE), shown in figure 9. The centre of effort is interesting because it influences the heeling moment, ie balance in roll motion. The center of effort is influenced by the sail trim. 3.1.3

Hydrodynamic forces

The driving force is counteracted by a resistance force of the hull and appendages. Likewise the side force from the aerodynamic forces is counteracted by the hydrodynamic lift from the hull and appendages. In order to provide this lift the boat has to travel slightly sideways through the water with a leeway angle λ as shown in figure 9. The leeway angle is the angle of attack for the keel and hull to generate the side force as a hydrofoil. Like the aerodynamic forces the hydrodynamic forces can be gathered in a single point. Often this 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 hydrodynamic side force introduce a heeling moment. The heeling moment, as the name suggests, make the boat heel over. The heeling moment is counteracted by the righting moment.

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 the leeward side as the boat heels over. The center of gravity can be changed by moving the crew 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 principal motions, shown in figure [?]. In the following the balance in all motions are described. ˆ Surge: The balance in surge motion is obviously the most important since it accounts for the boat speed. The aerodynamic driving force from the sails are balanced by the hydrodynamic resistance of the hull and appendages and the aerodynamic drag of the hull, rigging and crew.

9

ˆ Sway: The sway motion is of importance when the boat is going upwind because the large production of side forces from the sails cause a leeway angle. The leeway angle is important because it affects how fast the boat moves against the wind and it also influences the resistance by adding induced drag from the side force production. ˆ Heave: The heave balance is of little importance. Dynamic effects from the hydrodynamic forces on the hull tend to create a downward suction forcing the boat to a slightly larger draught. For light boats at high speed the effect shifts and tends to move the boat out of water. This effect is called planing. ˆ Roll: The roll motion has no direct effect on performance but is of importance since it controls the heel angle and thus has a large effect on both aero and hydrodynamic forces. The roll motion is a balance between the heeling moment and righting moment as described above. ˆ 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 hydrodynamic forces and a righting moment from the hydrostatic forces. The stability of the boat is much greater in the longitudinal direction and thus only small changes are seen in the pitch angle. The pitch balance is of significant importance for boats travelling in waves and for light boats in the down wind sailing condition. ˆ Yaw: The yaw motion has no direct effect on the performance, but to make balance the rudder angle must be changed. The changes of rudder angle change the resistance of the rudder and the distribution of side forces between the rudder and the keel. The change in distribution of side forces affects the resistance of the keel and even the centre of lateral resistance affecting the heeling moment. The changes are often small and of moderate importance.

If all the above mentioned balances are achieved a steady state is obtained. For nonsteady 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. Firstly the design shape and structural properties of the sail is important. Secondly the following parameters affect the shape while sailing. Numbers refer to numbers in figure 10. ˆ Apparent wind speed and angle ˆ Sheet tension (1) ˆ Sheeting position (2) ˆ Forestay tension (3) ˆ Halyard tension (4)

10

(3) (4)

(2) (1) Figure 10: Trim changes that affects the sail shape

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

Trailing edge angle Maximum camper

Maximum camper position

Figure 11: Section geometry

11

3.4 3.4.1

Trim variation True wind speed and angle

The wind speed and angle determines the aerodynamic load on the sails. The complex three dimensional flow is influenced by the heel angle, wind profile, and presence of hull and crew. The presence of separation further complicates the flow pattern and thus the resulting forces. The aerodynamic force distribution changes the sail shape and deforms the 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 flattens the 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 sail lead block (C)(and blue circle in figure 10) along the deck trail (G) as shown in figure 12. The deck trail usually has an inward facing angle, which means that by moving the lead block forward the sheeting position will move both forward and inward. The inward movement changes the overall trim angle, whereas the forward movement changes the twist. Along with the sheet tension the sheeting position is the most important trimming parameter 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 of the stay depends on the tension in the stay. The sagging increases the camper of the sail especially in the middle part. By increasing the forestay tension the sail thus gets flatter and the lift and drag is reduced. In practice the forestay tension is controlled by changing the tension of the backstay. It does however also depend on the main sail sheet tension and 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 from this variation primarily affects the minimum apparent wind angle that can be obtained without luffing of the sails. 12

3.5 3.5.1

CFD Governing equations

The governing equations are the momentum conservation equations known as NavierStokes equations, here written in differential form for incompressible fluids with Einstein’s summation notation: ∂Ui 1 ∂p ∂ 2 Ui ∂Ui + Uj =− +ν 2 (2) ∂t ∂xj ρ ∂xi ∂xj Equation 2 assume Newtonian fluids, meaning a linear relation between viscous strain and stress. The proportionality factor is the viscosity. For Cartesian coordinates this gives tree equations, one for conservation in each direction. Since there are only three equations and four unknowns, three velocity components and the pressure, an additional equation is needed to close the system. This is known as the closure problem. The additional equation is the continuity equation: ∂Uj =0 ∂xj

(3)

Again, the incompressible version. The assumption of incompressibility assumes that the density 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 numerical simulation is difficult and for most engineering purposes impossible. Instead the equations are solved on a time averaged basis. All quantities are split in a time averaged and a fluctuating part. By insertion in equation 2 a time averaged version of Navier-Stokes equations, namely the RANS equations, are obtained: ∂Ui 1 ∂p ∂ 2 Ui ∂ui uj ∂Ui + Uj =− +ν 2 − ∂t ∂xj ρ ∂xi ∂xj ∂xj

(4)

The new term u0i u0j 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 modelled by a turbulence model. 3.5.3

Turbulence modelling

The purpose of the turbulence model is to model the effect of small turbulence on the flow. These are typically modelled by adding one or two equations to model transport of some turbulence properties. In the following two sections, two of the commonly used two-equation models are briefly described. k −  turbulence model This turbulence model uses a transport equation to model the transport of turbulent kinetic energy k and an additional equation to model the dissipation of turbulent kinetic energy. The two equations are solved alongside the RANS equations and continuity equation. The standard k −  model is known to perform well for 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 −  model, is used. The 13

new model has a different equation for the turbulent dissipation based on more recent scientific experiments. See [3] for more details. This realizable k −  model is used in this project due to its superior performance for free surface flows as documented in [4]. k − ω SST turbulence model As with the k −  model the transport of turbulent kinetic energy is modelled by a transport equation for k but instead of modelling the dissipation by the dissipation rate  the specific dissipation rate ω is used. This approach has proved to give better results close to the wall and in strong pressure gradients than the standard k −  model. The SST extension by Mentor [5] combines the k − ω model close to the wall with the k −  in the free stream. Wall Treatment Close to solid walls the resolution of flow quantities are difficult due to large gradients. To resolve the full boundary layer of high Reynolds number flows a large 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 as velocity and turbulence quantities. The wall laws are typically specific to the turbulence model and often consist of a mix of algebraic formulations. In the region closest to the wall the viscous sublayer, the flow is completely dominated by the viscosity, and a little further from the wall the flow is dominated by turbulent shear. In between these regions 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 the velocity in the outer layer is modelled by a logarithmic relation. In the overlapping layer a 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 have to be discretized in small control volumes and the derivatives replaced by difference approximations. For the finite volume method the equations are solved in integral form. The general transport equation on integral form with vector notation: Z Z Z ρφV · ndS = Γ∇φ · ndS + qφ dΩ (5) S

Ω

S

The transport equations are solved in each control volume by calculating the surface integrals on the cell faces. Quantities are assumed to be constant at each cell face and thus the integral can be calculated as a summation of the surface centre quantities. Since all quantities are only known in cell centres the surface quantities are estimated by interpolation schemes 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 is used. This method introduces an additional transport equation describing the amount of water and air in each cell. The flow and pressure field solver is not modified but the fluid properties are modified to match the cell content of the different fluids. The interface between air and water can stretch over many cells, and thus a compression scheme is introduced 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. The solver models the movements of the structure based on the hydrodynamic and aerodynamic forces from the flow. The movements depend on the weight and mass moments of inertia of the structure, and the accelerations are found from Newton’s second law of mechanics. For surge and sway the equations read: R + FM = W ax

(6)

FLAT + PLAT = W ay

(7)

The final balance interesting at this point is for the roll motion. Here the mass moment of 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 the accelerations 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 velocity gradually 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 (z) = loge , (9) k z0 where u∗ is the friction velocity and z0 is the roughness length depending on the roughness of the underlying surface. k is Von Karmans constant 0.41. On the ocean the surface roughness depends on the waves, which again depend on the wind velocity. The relationship between the friction length and the Velocity at 10 m height U 10 is according to Charnock [6] described by:  2 a kU 10    , z0 =  (10) g log 10 e

z0

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

15

16 14 12

z [m]-

10 8 6 4 2 0

0

1

2

3

4

5

6

7

U [m/s]

Figure 13: Wind profile 6 m/s

3.5.8

Verification and Validation

To ensure quality and estimate accuracy of CFD simulations, verification and validation must 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 this section is based on the formulation from [8]. Verification In general there are two types of verification: code verification and solution verification. The code verification is often done by the software developer to make sure their code is solving the equations correctly. A constructed problem, with a known solution can be used to verify the implementation. For each new problem a solution verification should be performed. The solution verifications are often performed on problems where the solution is unknown, to estimate the uncertainty for the problem. The estimation is made by monitoring the important output parameters as the mesh is systematically refined. If the mesh is fine enough and thus inside the asymptotic range the parameters will follow a asymptotic convergence as the mesh is refined. This is however rarely the case since 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 performed. Validation requires a known solution in the form of analytical or experimental data. The difference between the calculated values and the experimental data is compared with the combined uncertainty of the numerical model, experimental data and input parameters. This uncertainty is called validation uncertainty, and is calculated by: q 2 2 + U2 + Uexp (11) Uval = Unum inp The numerical uncertainty is estimated by the solution verification and the experimental uncertainty is estimated from the experimental scatter, and / or combination of known experimental uncertainties. If the error is smaller than the validation uncertainty the validation is successful to the uncertainty specified by the validation error. This means that 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 simulation. All uncertainties must be multiplied by a safety factor prior to use in validation purposes. According to [8] the safety factor can be taken as 1.2 if the uncertainty fitting error is negligible and 3 otherwise. For most practical RANS calculations the safety factor should be taken as 3 according to [8].

3.6 3.6.1

Optimization Optimization Problems

In general, optimization is formulated as a minimization of an objective function by changing a set of variables. Often the variables are restricted within a specified range, called design space. Furthermore constraints functions may exist that limits the solution and must be taken into account during the optimization. The nature of the optimization problem depends both on the type of variables and the form of the objective and constraint function. The variables can be either continues, meaning they can attain any real number, or discrete, meaning they are restricted to for example integers. Depending on the properties of the variables as well as the objective and constraint functions, the optimization problem may fall into one of a large number of categories, for instance: ˆ Linear programming: Linear objective and constraints functions ˆ Quadratic programming: Quadratic objective function and linear constraint functions ˆ Non-linear programming: General objective and constraints functions

The linear programming optimisation is the simplest and often give a fast and accurate solution. The success of the quadratic programming optimization depends on the nature of 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 function means that the function can have any shape. This means that there is a potential of more local minima, see figure 14. Non-linear programming problems often have more than one local minimum and thus the optimization algorithm risk getting stuck at a local minimum and not finding the global minimum.

17

Objective

Local minima Global minimum Variable Figure 14: Global and local minima

One way to try to overcome this issue is by searching the design space with different combinations of variables. 3.6.2

Optimization Methods

Numerical optimization methods solves an optimization problem by iteratively improving a solution estimate, until convergence is achieved according to given tolerance criteria. The methods for updating the variables are numerous, but usually one of the following strategies are used for solving non-linear programming problems: ˆ Line search methods combines a method for estimating a descent direction, with a method for finding a minimum in that direction. Numerous methods exist for estimating the descent direction and performing line search. ˆ Trust region methods solves a simplified optimization problem, that is an approximation to the main optimization problem. The simplified problem include constraints that limit the length of the step suggested by the algorithm. The limit is often referred to as a trust region radius. The trust region radius is continuously updated, and depends on the success of the previous iterations. ˆ Penalty and damping methods involve augmenting the objective function with a function that increase in undesired parts of the design space, for instance in regions where one or more constraints are violated. These methods are referred to as penalty methods. The same technique can be used for controlling the size of the step suggested by the algorithm, by adding a term to the objective function that increases for large steps. These methods are referred to as damping methods.

18

Many optimization methods use information about the gradient. But the gradient is not always available. When the gradient is unknown it can be estimated by use of finite difference methods, but according to [9] this requires a large number of sample points and can be inaccurate when noise is present in the objective function. Thus gradient free optimization 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 nonlinear objective functions. Information about the gradient are usually unknown. Constraints may exist for CFD optimization problems, but in this project no constraint functions are used. The complex flow interactions leave room for more than one local minima since the appearance and disappearance of separation zones makes the objective variations oscillate. 3.6.4

Pattern Search Algorithms

The idea of the pattern search is to search in a number of search directions from the base point. The new sample points form a stencil around the base point. If a significant decrease of the objective function is found in one of the directions the base point is moved in that direction. The step length and search directions may be changed depending on the output from the sample points. The optimization is stopped when the change in the variables have reached a specified tolerance. The accuracy and stability may be affected by noise in the objective function and non-continuous behaviour, but often pattern search algorithms are found to give good results 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 the disadvantage 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 a more equal distribution of design points. This ensures that more of the design space is covered and thus fewer points are needed.

19

4 4.1

Method CFD model

The commercial CFD code Star-CCM+ ver. 8.02 is used for the project. Both water and air are simulated to get both aero and hydrodynamic forces affecting the boat. The speed, heel angle and leeway angle are adjusted in the model by the 6 DOF solver. In the following sections the CFD setup will be described in further details. Much of the setup is based on the guidelines from Milovan Peric’s contribution to the Gothenburg Workshop on 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 the boat and to the leeward side and 1 times LWL in front and to the windward side. The domain is larger in the wake of the flow to capture the disturbed flow behind the sails and not 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 are tested 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 the methods reference is made to the Star-CCM+ user manual [12]. Surface Remeshing To ensure a good quality surface mesh and an even distribution of sizes before the volume mesh is executed the initial geometry tessellation is replaced by a new surface mesh. The size of the surface elements are determined by global and local settings on the minimum and target size, and by curvature constraints. The curvature constraint is set to secure 72 point per circle, which is double the default setting. This extra resolution is used to ensure that the curvature of the sails are captured accurately since this can have a significant influence on the flow. Figure 16 shows the original and remeshed surface mesh triangulation. As seen the distribution is more smooth for the remeshed surface and finer cells are included where curvature is high. The original model of the boat and main sail was imported through IGES format and the shown original triangulation of these is only the initial attempt to make a triangulation and not used for further processing. The new tessellation is based on the CAD geometry and not the shown original triangulation. The head sail on the other hand is imported as STL, which is by nature triangulated geometry. To get a proper quality for this kind of import a fine triangulation is used, as seen in the figure. Volume Meshing The trimmer method is used for the volume mesh because it allows for anisotropic refinement needed to accurately resolve the free surface interface between water and air. The trimmer works by introducing a background mesh filling the entire domain 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 size is match sufficiently. Figure 17 shows a section view of the volume mesh. As seen on the figure, the far field mesh is very coarse and the cells are gradually refined closer to the boat. The refinement near the water surface is also clear and an important property of this is that the water surface should stay within the refinement for the small changes of heel expected.

Figure 17: Vertical section showing the volume mesh

Figure 18 shows the volume mesh in a horizontal section in the waterline. An arrow shaped refinement in the horizontal dimensions is used to capture the wake field of the yacht. The angle of the arrow edges is 19 °which corresponds to the angle of a Kelvin wave 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 are created close to the wall boundaries. The layers are controlled by the total height and the number of layers. Figure 19 shows a vertical section of the mesh with a close up close to 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 grey on the outer edges is inlet, the orange area is outlet and the solid grey colour is solid walls.

Figure 20: Boundary conditions

Inlet At the inlet the apparent wind and water velocity is specified. These values are 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 atmospheric boundary layer as described in section 3.5.7. The volume fractions of water and air are specified to have the water surface in the right position. Because the domain follow the 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 intensity and Viscosity ratio are also specified at the inlet. The default values are used for both quantities. Outlet At the outlet pressure and volume fractions are specified. The volume fraction is, 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 intensity and viscosity ratio are used. Boat On all solid parts of the boat the no-slip boundary condition is applied. The boat is part of the solid body used for the motion solver and thus fluid forces are captured on the 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 are modelled as baffle interfaces meaning that a copy of the sails are made to represent the other 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 fluid forces are evaluated from these. Mesh motion The mesh is moving during simulation due to the motions required by the 6-DOF solver. The motion is a rigid motion of the entire mesh and thus no re-meshing or morphing is needed. Since the velocity, volume fraction and pressure is specified in the global 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 simulations. The segregated flow solver is used with implicit unsteady time stepping. The segregated flow solver is an implementation of the SIMPLE type algorithm, where the momentum and continuity equations are solved uncoupled, and a predictor corrector approach is used to link the equations. The transient term is modelled with first order discretization to ensure stability and all other terms are modelled with second order discretization in space. The volume of fraction transport equation is solved along with the flow equation by the segregated solver. The equations of solid body motion are solved at every 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 project under a non-commercial student license. 4.2.1

Geometry

The baseline geometry of the head sail is imported into Friendship framework by the horizontal control curves of the NURBS surface received from North Sail. The curves are used in order to facilitate the trim and shape variation to be variable over the length of the sail. After the trim and shape modifications the spline curves are transformed to a surface 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 the tack corresponding to the forestay. An example of a trim angle change is shown in figure 21 Trim Angle The trim angle is a rotation around the aforementioned line with no variation from top to bottom of the sail. The angle is free to change ±2. An example of a twist angle change is shown in figure 22 Twist The twist is the difference between the trim angle at the top and the bottom. It is also free to vary with ±2. TWA The final parameter is the true wind angle, which is changed by changing a value in the Star-CCM+ script displayed in appendix C. The true wind angle is not an actual trim parameter but related to the steering of the boat. The parameter is included in the optimization due to its important influence on the VMG. The optimal twist and trim angle 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 position of maximum camper of the sail sections. The shift function uses a control surface to control the deformation. For details on shift functions see [14]. The surface, shown in figure 23 along with the sail surface, is a B-Spline surface created by a loft operation on 4 control lines. The lines, shown in figure 24, are third degree splines controlled by 3 points and the gradients in the end points. The end points correspond to the sides of the sail and thus no deformation is wanted here, which is why these are kept at zero. The gradients at the end points are not given and are thus found from the movement of the middle point. The middle point position is thus the only influencing parameter available for design modifications. The point can be moved in both horizontal directions, as shown in figure 24, but not in the vertical direction. Thus 2 design variables are present for each line, the camper C and the camper position CP. The top line is not modified because this is difficult to accommodate in practice, thus a total of 6 design variables are available for the shape optimization.

Head Sail

Control points

Shift surface Figure 23: Shift surface

28

CP C

Figure 24: Deformation parameters

4.2.4

Optimization algorithm

The optimization is split into two parts. The first is design space investigation and the second is a tangent search. The objective used for the trim optimization is the Velocity Made Good (VMG) and for the shape optimization a combination of the VMG for the different wind velocities is used. The medium wind speed accounts for 50 % of the objective, while the high and low wind speed share the remaining 50 %. Design space investigation The purpose of the design space investigation is to find the global trends in the design space. As mentioned in section 3.6 objectives may have many local minima across the design space, and to make sure that the optimization results in a global minimum the design space investigation is needed. For the current project the SOBOL algorithm is used to find the semi-random design variable variations. The output of the investigation is used as a basis for the pattern search. Pattern search The TSearch algorithm of Friendship Framework is used. This is a pattern search method with constraints. The only constraints for this project are the limits for the variables. If a constraint is breached the TSearch method steps along the tangent of the constraint. The step size of the method is set to an initial value and reduced along the way if no improvements are found. The method can reuse points from the design space investigation if appropriate to save time.

29

4.3

Computation Network

The CFD simulations require extensive computational power and the optimization process requires user control and input through a graphical user interface. A remote simulation server delivers the computational power and a local Linux machine runs the optimization process with a graphical user interface. The communication between the two machines is performed through a SSH tunnel connection controlled by terminal scripts. The communication 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 simulations by STAR-CCM+. STAR-CCM+ is run through a queuing system in batch mode, and the output 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 Framework for further processing. Examples of terminal scripts are listed in appendix B and JAVA macros are listed in appendix C. The scripts are run internally at OSK-shiptech and thus protected by a firewall. The sshpass is used to avoid password prompt. This is an unsafe method on open networks. Instead sshagent can be used. All company specific details are removed from the scripts for safety reasons. Local linux machine

Remote simulation server STAR-CCM+ .csv .stl

Java Terminal macros scrips

.stl

Local terminal

Remote terminal SSH

Figure 25: Flow chart of simulation network

30

Java macros

.csv

5

Verification and Validation

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

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 of great importance. To validate the aerodynamic forces a series of experimental data from wind tunnel tests are used. The data presented in [15] are measurements of forces and pressures on a set of America’s Cup class (AC33) sails performed in the wind tunnel of Yacht Research Unit (YRU), University of Auckland. The test setup shown in figure 26 consists of a test plate with the sails suspended above and with pressure transducers built into 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 wind tunnel geometry, is constructed. Only the part of the wind tunnel above the test plate is modelled as seen in figure 27. The bottom of the domain is then modelled as a free slip wall producing a flat velocity profile. The sides and top of the tunnel section and the test plate are all modelled as no slip walls, and the sails are modelled as baffle interfaces with no slip conditions. The pressure is reported in the points displayed in figure 28 corresponding to the measurement point of the experiment.

Tunnel walls

Tunnel roof

Test plate Inlet Free slip wall 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 trimmer method. Close to no-slip boundaries prismatic layers are included to resolve boundary layer gradients. An illustration of the finest mesh used is shown in figures 29 and 30. From √ this 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 taken as the average of y + values on the sail surfaces, and a plot of the y + distribution is shown in figure 31. In principle the number of cells should scale with one over the base size to the power of 3 and thus half between each step, but because the surface size depends on the curvature of the surface and the geometric limitation, the number of cells are reduced less than that. This also means that the meshes are not completely geometrically similar affecting 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 1 1.00 2 1.26 3 1.59 4 2.00 5 2.52 6 3.17 7 4.00 8 5.04 9 6.35 10 8.00 11 10.08 12 12.70

Number of cells y+ avg 5.484.973 1.9 3.384.183 2.4 2.140.462 3.1 1.421.954 3.9 984.854 5.0 675.508 6.2 492.513 7.5 371.064 9.1 267.782 11.2 208.398 13.5 155.318 15.7 117.742 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. Validation is done for lift, drag and pressure coefficients on the sail surface. Validations are done in accordance to the recommendations presented in [8]. The validation is done on mesh number 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 figure 32, and a fit for estimating discretization error is thus difficult. Instead the fluctuation amplitude of the 8th finest grids is used multiplied with a safety factor of 3. The resulting discretization error is 5.6 %. Both turbulence models give similar results, but the k − ω SST show some large oscillations in some of the meshes, as seen in figure 33. These oscillation may represent actual physical oscillations due to oscillating separation, but since they are 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 convergence error is neglected because the simulation is run until the solution is stable. No information is available on the experimental uncertainty of the lift and drag, but an uncertainty similar to the measurement accuracy of the dynamic pressure, i.e. 3% is used. The validation result is presented in table 4. The error E is an order of magnitude smaller than the validation uncertainty Uval . According to [8] this can mean two things. Either the validation is successful or the uncertainty is too big. The validation uncertainty is in this case relatively large, but this is also related to the large safety factor of 3. The error of 0.6 % is very small but this is more a coincident. The validation is also successful with a much lower safety factor. This means that the expected uncertainty is in the order of 3.7 to 6.3 % for the lift on mesh number 8, depending on the choice of safety factor. The uncertainty is independent of the mesh size for the 8th finest meshes due to the oscillatory convergence.

35

1.34

1.32

CL

1.3

1.28

1.26

1.24 k−ε k−ω SST 1.22

0

2

4

6

8

10

12

14

hi/h1

Figure 32: Grid convergence for lift

k−ε

k−ε

−1.245

0.151

−1.25

0.15 0.149

CL

CD

−1.255 0.148

−1.26 0.147 −1.265 −1.27

CFD AVG 0

5

10

0.146 0.145

15

0

5

time [s]

10

15

10

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time [s]

k−ω SST

k−ω SST

−1.235

0.153 0.152

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0.151 CL

CD

−1.245

0.15 0.149

−1.25

0.148 −1.255 −1.26

0.147 0

5

10

0.146

15

0

5

time [s]

time [s]

Figure 33: Lift and drag time series

Drag The drag coefficient is plotted as a function of relative cell size in figure 34 and shows 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 be coupled with the oscillations in lift. The validation result is presented in table 4. As for the lift, the error of drag is almost an order of magnitude smaller than the validation uncertainty, and the validation is successful for an uncertainty of 15.6 %. The large discretization uncertainty can be reduced by using one of the finer meshes, and due to the size of the experimental error the validation is also valid for these meshes with an uncertainty around of 3 %.

0.162

0.16

0.158

0.156

CD

0.154 k−ε k−ω SST k−ε fit

0.152

0.15

0.148

0.146

0.144

0.142

0

2

4

6

8

10

12

hi/h1

Figure 34: Grid convergence for drag

Udiscr Uexp Uval E E(%)

CL 0.0701 0.0391 0.0803 0.0072 0.6%

CD 0.02320 0.00468 0.02367 0.00160 1.1%

Table 4: Validation table Lift and Drag

37

14

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 the leading edge separation of the main sail. The experimental results show larger separation on the 3 lower sections whereas the numerical results show higher degree of separation in the uppermost section of the mainsail. The difference for the lower sections may be attributed to the fact that the experimental sail has non-zero thickness where as the numerical model has zero thickness. The upper section is situated in the wake of the head sail tip vortex and thus experiences a complex separation which is difficult to predict, as described in [16]. As the lift, the pressure shows oscillating mesh convergence and thus the discretization uncertainty is estimated in the same way. The validation results 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 direct comparison is difficult due to difference in mesh type. In the reference a fully structured mesh providing a better mesh similarity between the meshes is used. This is expected to provide a better convergence. The experimental uncertainty is taken from [16] and is in the order of 13 to 30 %. This large experimental uncertainty makes the validation of pressure less accurate. The pressure simulation is thus validated to an accuracy of 17 30 %. Head section 1 (HS1)

Head section 2 (HS2)

−2

−2.5 Experimetal Numerical

−1.5

−2 −1.5 Cp

Cp

−1 −0.5

−1 −0.5

0

0

0.5 1

0.5 0

0.2

0.4

0.6

0.8

1

1

0

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x/c

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0.5 0

0.2

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Head section 4 (HS4) −2.5

Cp

Cp

Head section 3 (HS3) −2.5

1

0.6 x/c

0.6

0.8

1

1

x/c

0

0.2

0.4

0.6 x/c

Figure 35: Experimental and numerical pressure coefficients on head sections

38

Main section 1 (MS1)

Main section 2 (MS2)

−1.5

−1.5 Experimetal Numerical

−1

−1

Cp

−0.5

Cp

−0.5 0

0

0.5

0.5

1

0

0.2

0.4

0.6

0.8

1

1

0

0.2

0.4

x/c

0.6

0.8

1

x/c

Main section 3 (MS3)

Main section 4 (MS4)

−2

−1.5

−1.5

−1

−1 Cp

Cp

−0.5 −0.5

0 0 0.5

0.5 1

0

0.2

0.4

0.6

0.8

1

1

0

0.2

x/c

0.4 x/c

0.6

0.8

Figure 36: Experimental and numerical pressure coefficients on main sections

HS1 Unum 0.593 Uexp 0.681 Uval 0.903 E 0.442 E(%) 8%

HS2 0.554 0.703 0.895 0.419 9%

HS3 0.435 0.687 0.813 0.368 8%

HS4 MS1 0.349 0.165 0.66 0.811 0.747 0.828 0.104 0.329 2% 12%

MS2 MS3 MS4 0.176 0.243 0.405 0.811 0.764 0.687 0.830 0.802 0.798 0.283 0.268 0.248 9% 8% 9%

Table 5: Validation table pressure

5.1.4

Validation Remarks

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

39

5.2 5.2.1

Verification of full model Spatial Discretization

To test the mesh quality and to estimate the discretization uncertainty for the full simulation model, 7 systematically refined meshes are run. Table 6 shows the properties of the meshes. The y + values are larger than for the validation study above because the resolution is the same but the overall size of the sail is larger. Figure 37 show the mesh convergence for 4 of the relevant output parameters as a function of the relative base size. In general the convergence is good and a fit is used for each parameter to estimate the discretization uncertainty, which is shown in table 7. Only heel angle show a larger discretization 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 mesh used for further calculations. Since the discretization uncertainty is in the order of the improvements of the optimization it is recommended that the final results are rerun with a finer mesh for verification. # 1 2 3 4 5 6 7

Base size 1.00 1.12 1.26 1.41 1.59 1.78 2.00

Number of cells y+ avg 2,102,549 45 1,800,627 51 1,508,921 57 1,298,445 64 1,130,284 71 957,527 80 834,330 89

Table 6: Validation meshes for full model

40

Boat speed

Heel Angel

3.58

19.2

3.56

19.1 19

3.52 [°]

[m/s]

3.54

18.9

3.5 18.8

3.48

18.7

3.46 3.44

1

1.2

1.4

1.6

1.8

18.6

2

1

1.2

1.4

hi/h1

1.6

1.8

2

1.8

2

hi/h1

Leeway Angle

Velocity Made Good

5.8

2.62

5.6

2.6 2.58

[°]

[m/s]

5.4

2.56

5.2 2.54 5

2.52 1

1.2

1.4

1.6

1.8

2.5

2

1

1.2

1.4

hi/h1

1.6 hi/h1

Figure 37: Discretization study for full model

Boat Speed Heel Angle Leeway Angle VMG

Safety factor Un 3 1.3% 3 16.8% 3 0.5% 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 time step 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 the 3 small time steps give very similar results, but for the boat speed there is a difference which is reflected again in VMG. Figure 39 show the time averaged VMG as a function of the relative time step along with a fit. This fit gives an uncertainty of 6 % for the value used in the baseline model with a safety factor of 3.

41

Boat speed

Heel angle

7

24

6.9

22

[°]

[kts]

20 6.8 6.7

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

6.6 6.5

0

10

20 30 Time [s]

18 16 14

40

12

50

0

10

Leeway angle

20 30 Time [s]

40

50

40

50

VMG

8

5.6 5.5

7

5.4 5.3

[°]

[kts]

6 5

5.2 5.1 5

4

4.9 3

0

10

20 30 Time [s]

40

50

0

10

20 30 Time [s]

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

42

2.76 CFD fit 2.74

VMG [m/s]

2.72

2.7

2.68

2.66

2.64

0

2

4

6

8

10 ∆ ti / ∆ t1

12

14

16

18

20

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 domain sizes are evaluated. A 50 % and a 100 % increase in domain size are used and the time series of the output parameters are displayed in figure 40. Surprisingly the solution shows increasing destabilization for increasing domain size. An explanation for this could be that the far ends of the domain experience large changes in volume fraction as the boat changes heel angle. The water surface moves past more cells in a single time step and causes large changes in the density and thus may lead to destabilization of the pressure solver. This problem may be solved by reducing the time step or changing the surface compression. It is difficult to judge the domain size convergence based on these results because they are influenced by large oscillations in the heel angle. Figure 41 shows the wave 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 removed this. The influence of this is probably insignificant since the elevations are in the order of a few centimetres.

43

Boat speed

Heel angle

7.5

24 22

7 6.5

18 [°]

[kts]

20

16

6

5

14

Baseline 50 % increase 100 % increase

5.5

0

10

20 30 Time [s]

12 40

10

50

0

10

Leeway angle

20 30 Time [s]

40

50

40

50

VMG

25

6

20

5.5 5

[°]

[kts]

15

4.5

10 4 5 0

3.5 0

10

20 30 Time [s]

40

3

50

0

10

20 30 Time [s]

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 estimate based on a radius of gyration of 0.35 times the width are used as a starting point. The boat probably has a larger moment of inertia due to the mast and sail and thus a range of moments of inertia are tested. The results are presented in figure 42 as time series of output parameters. The highest and lowest moments of inertia give similar results for both average value and fluctuation amplitude. The average moment of inertia gives a higher fluctuation amplitude and a different average value for the velocity. A further investigation of the effect of moment of inertia is recommended for future studies along with a possible coupling with time step. For the current project the high value of the moment of inertia is used. Tests with higher values have been tried with uncontrollable oscillations as a result. Heel angle 21

6.95

20.5

6.9

20 [°]

[kts]

Boat speed 7

6.85 6.8

19 I = 1e5 I = 5e4 I = 1.32e4

6.75 6.7

19.5

0

10

20 30 Time [s]

40

18.5 18

50

0

10

Leeway angle

20 30 Time [s]

40

50

40

50

VMG

6

5.4 5.35

5.5 [kts]

[°]

5.3 5

5.25 4.5

4

5.2

0

10

20 30 Time [s]

40

5.15

50

0

10

20 30 Time [s]

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 and simulation in STAR-CCM+ was found to create some scatter in the result. Consecutive run 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 has been fitted by a VPP data processing program to get values for the right wind speeds. 5 series of data are available from different races, and they are plotted along with the CFD calculations in figure 43. As seen in the figure the data show a very large scatter and the uncertainty of these data are considered too large for validation purposes. It does however show that the CFD calculation lies in between the different data series and thus are in the right range. 6 m/s 8

7

7.5

6

7

Boat speed [kts]

Boat speed [kts]

4 m/s 8

5 4 3 2 20

6.5 6 5.5

25

30

35 TWA[°]

40

45

5 20

50

25

30

35 TWA[°]

8 m/s 8

Series 1 Series 2 Series 3 Series 4 Series 5 CFD

7.5

Boat speed [kts]

7 6.5 6 5.5 5 4.5 20

25

30

35 TWA[°]

40

45

50

Figure 43: Full scale measurements and CFD results

46

40

45

50

6 6.1 6.1.1

Sail Trim Optimization 4 m/s Design space evaluation

From the baseline case a series of parameter variations are run based on the SOBOL algorithm. The varied parameters are trim angle, twist angle and true wind angle. A total of 20 variations are used and the result of the successful runs are displayed in table 8. Some runs failed due to instability or failure of the mesh generation. Most of the runs actually proved to give a smaller VMG, but a few runs gave improvements in the performance. The best run, nr. 7 gave a performance improvement of 0.69 %. # Trim[°] Twist[°] TWA[°] V[kts] 7 -1.250 -0.750 33.125 6.029 19 -0.125 -1.625 34.688 6.141 3 -0.5 -0.5 36.25 6.273 5 0.5 -1.5 33.75 6.044 0 0 0 38 6.413 1 1 -1 37.5 6.347 18 -0.625 0.875 35.938 6.204 2 -1 1 32.5 5.911 9 1.75 -1.75 35.625 6.159 8 0.75 1.25 38.125 6.359 12 1.25 0.75 34.375 6.005 16 0.375 1.875 33.438 5.917 6 -1.5 0.5 38.75 6.396 13 0.25 -0.25 31.875 5.784 10 -0.25 0.25 30.625 5.66 11 -0.75 -1.25 39.375 6.408 17 1.375 -1.125 30.938 5.613 4 1.5 1.5 31.25 5.519

Leeway[°] Heel[°] VMG[kts] Dev.[%] 4.198 9.273 4.794 0.69 4.176 9.808 4.782 0.44 4.173 10.449 4.775 0.29 4.149 9.141 4.769 0.17 4.064 10.972 4.761 0.00 4.114 10.636 4.745 -0.34 4.209 10.093 4.742 -0.40 4.223 8.576 4.738 -0.48 4.155 9.642 4.733 -0.59 4.131 10.584 4.707 -1.13 4.152 8.814 4.698 -1.32 4.209 8.490 4.685 -1.60 4.21 11.299 4.681 -1.68 4.275 8.048 4.671 -1.89 4.371 7.373 4.636 -2.63 4.288 11.513 4.635 -2.65 4.42 6.900 4.578 -3.84 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 function and a regression curve. There is a clear dominance by the true wind angle which overrules the variations in the other parameters. It is difficult to distinguish the effect of twist and trim angle based on this.

47

VMG − Twist angle 4.8

4.75

4.75

4.7

4.7 VMG [kts]

VMG [kts]

VMG − Trim angle 4.8

4.65 4.6

4.65 4.6

4.55

4.55

4.5

4.5

4.45 −2

−1

0 Trim angle [°]

1

4.45 −2

2

−1

0 Twist angle [°]

1

2

39

40

VMG − TWA 4.8 4.75

VMG [kts]

4.7 4.65 4.6 4.55 4.5 4.45 30

31

32

33

34

35 TWA [°]

36

37

38

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 further optimization. It starts in the best point nr. 7 and systematically steps in different directions to find a new optimum position. Table 9 shows the parameter variations and corresponding output parameters and Figure 45 shows a graphic representation of the iteration 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 wind angle of 33.175°, trim angle of -1.45°and twist angle of -0.55°. Running the optimizer further might have given further improvement, but the variable changes are now smaller than what is practically changeable and the variations in the objective function are close to the expected scatter.

48

# Trim[°] Twist[°] TWA[°] V[kts] 0 -1.250 -0.750 33.125 6.029 1 -1.050 -0.750 33.125 6.026 2 -1.450 -0.750 33.125 6.034 3 -1.450 -0.550 33.125 6.039 4 -1.450 -0.550 33.625 6.075 5 -1.450 -0.550 32.625 5.992 6 -1.725 -0.275 33.125 6.024 7 -1.375 -0.550 33.125 6.036 8 -1.525 -0.550 33.125 6.024 9 -1.450 -0.475 33.125 6.029 10 -1.450 -0.625 33.125 6.024 11 -1.450 -0.550 33.313 6.055 12 -1.450 -0.550 32.938 6.024 13 -1.422 -0.550 33.125 6.027 14 -1.478 -0.550 33.125 6.026 15 -1.450 -0.522 33.125 6.028

Leeway[°] Heel[°] VMG[kts] Dev.[%] 4.198 9.273 4.794 0.70 4.185 9.210 4.793 0.68 4.230 9.361 4.796 0.73 4.191 9.288 4.803 0.89 4.236 9.512 4.796 0.74 4.180 9.021 4.797 0.77 4.209 9.274 4.790 0.61 4.187 9.260 4.800 0.83 4.223 9.281 4.789 0.59 4.225 9.297 4.793 0.66 4.206 9.248 4.790 0.60 4.205 9.385 4.802 0.87 4.202 9.206 4.802 0.86 4.217 9.254 4.791 0.64 4.189 9.275 4.793 0.66 4.215 9.294 4.792 0.66

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

TWA

Trimming parameters 34

0

33.8 33.6 33.4

Twist angle Trim angle

−1

[°]

[°]

−0.5

33.2 33

−1.5

32.8 −2

0

5

10

32.6

15

0

5

10

15

VMG 4.805

[kts]

4.8

4.795

4.79

4.785

0

5

10

15

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

The result of the trim optimization shows a modest improvement compared to the baseline 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 that sheeting angle is reduced in the top compared to in the bottom. Figure 46 shows the sections of the sail for the original and trimmed case. In an actual sailing situation this can be achieved by increasing the sheet tension and moving the sheeting point slightly forward. In addition the boat is steered closer to the wind. As seen in table 10 the boat speed is actually slightly reduced and the leeway angle is increased, but the VMG is increased anyway due to the reduction in true wind angle. The boat moves faster against the wind because it steers a tighter course with only a slight speed reduction in boat speed. 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 -12.83 V[kts] 6.413 6.039 -0.374 -5.83 Leeway[°] 4.064 4.191 0.127 3.12 Heel[°] 10.972 9.288 -1.684 -15.35 AWA[°] 20.545 18.475 -2.070 -10.07 AWS [m/s) 6.874 6.799 -0.075 -1.09 VMG[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 sail surfaces for the baseline and trimmed sails. The flow is very similar, but there is a reduced upward flow in the lower part of the leeward side of the head sail. This is probably due to 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 6.2.1

6 m/s Design space evaluation

As for the 4 m/s case 20 variations are performed in the SOBOL search. The results of the successful runs are shown in table 11. As for the 4 m/s case most of the variations show no improvement and only 4 points show improvement, compared to the baseline case. The best point nr. 2 show a 1.08 % improvement. # Trim[°] Twist[°] TWA[°] V[kts] 2 -1.000 1.000 32.500 3.519 7 -1.250 -0.750 33.125 6.890 5 0.500 -1.500 33.750 6.925 16 0.375 1.875 33.438 6.879 0 0.000 0.000 35.000 7.016 9 1.750 -1.750 35.625 7.052 10 -0.250 0.250 30.625 6.566 3 -0.500 -0.500 36.250 7.093 17 1.375 -1.125 30.938 6.535 14 -1.750 1.750 36.875 7.109 1 1.000 -1.000 37.500 7.121 4 1.500 1.500 31.250 6.505 8 0.750 1.250 38.125 7.131 15 -1.625 -0.125 38.438 7.097 6 -1.500 0.500 38.750 7.120 11 -0.750 -1.250 39.375 7.137

Leeway[°] 5.063 5.136 5.074 5.034 5.122 5.060 5.219 5.207 5.253 5.252 5.285 5.360 5.329 5.441 5.444 5.377

Heel[°] VMG[kts] Dev.[%] 18.701 5.423 1.08 19.242 5.410 0.84 19.354 5.395 0.56 18.905 5.386 0.39 20.120 5.365 0.00 20.250 5.348 -0.32 16.777 5.322 -0.80 20.896 5.316 -0.92 16.424 5.274 -1.69 21.059 5.272 -1.73 21.226 5.226 -2.59 16.386 5.222 -2.66 21.354 5.177 -3.51 21.438 5.115 -4.65 21.585 5.105 -4.85 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 function and a regression curve. Again there is a clear dominance by the true wind angle which overrules the variations in the other parameters. It is difficult to distinguish the effect of twist and trim angle based on this.

54

VMG − Twist angle −5

−5.1

−5.1 VMG [kts]

VMG [kts]

VMG − Trim angle −5

−5.2 −5.3 −5.4 −5.5 −2

−5.2 −5.3 −5.4

−1

0 Trim angle [°]

1

−5.5 −2

2

−1

0 Twist angle [°]

1

2

39

40

VMG − TWA −4.9

VMG [kts]

−5 −5.1 −5.2 −5.3 −5.4 −5.5 30

31

32

33

34

35 TWA [°]

36

37

38

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 investigation. In this case it starts in point nr. 2 and systematically steps in different directions to find a new optimum position. Table 12 shows the parameter variations and corresponding output parameter, and Figure 52 shows a graphic representation of the iteration 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 wind angle of 32.688°, trim angle of -1.4° and twist angle of 1°. Running the optimizer further might have given further improvement, but the variable changes are now smaller than what is practically changeable and the variations in the objective function are close to the expected scatter.

55

# Trim[°] Twist[°] 0 -1.200 1.000 1 -1.000 1.000 2 -1.400 1.000 3 -1.400 1.200 4 -1.400 0.800 5 -1.400 1.000 6 -1.400 1.000 7 -1.675 1.000 8 -1.325 1.000 9 -1.475 1.000 10 -1.400 1.075 11 -1.400 0.925 12 -1.400 1.000 13 -1.400 1.000 14 -1.372 1.000 15 -1.428 1.000 16 -1.400 1.028

TWA[°] V[kts] Leeway[°] 33.000 6.905 5.071 33.000 6.899 5.029 33.000 6.906 5.078 33.000 6.887 5.056 33.000 6.897 5.060 33.500 6.900 5.162 32.500 6.866 5.089 31.813 6.782 5.054 32.500 6.846 5.067 32.500 6.862 5.096 32.500 6.835 5.064 32.500 6.841 5.082 32.688 6.887 5.084 32.945 6.901 5.060 32.688 6.871 5.067 32.688 6.880 5.051 32.688 6.865 5.073

Heel[°] VMG[kts] Dev.[%] 19.184 5.436 1.32 19.119 5.434 1.29 19.220 5.437 1.33 19.174 5.423 1.08 19.211 5.431 1.22 19.385 5.388 0.43 18.938 5.441 1.42 18.554 5.426 1.13 18.889 5.427 1.15 18.936 5.437 1.35 18.882 5.418 0.99 18.891 5.422 1.06 19.087 5.444 1.47 19.194 5.438 1.35 18.998 5.432 1.26 19.038 5.441 1.42 19.005 5.428 1.17

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

Trimming parameters

TWA

1.5

33.5

1 33

0.5 0 −0.5

[°]

[°]

Twist angle Trim angle

32.5

−1

32

−1.5 −2

0

5

10

15

31.5

20

0

5

10

15

20

VMG 5.46

[kts]

5.44

5.42

5.4

5.38

0

2

4

6

8

10

12

14

16

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 table 13. The change is a reduction in the trim angle meaning that the sail is sheeted at a tighter angle, and an increase in twist means that sheeting angle is increased in the top compared to in the bottom. Figure 53 show the sections of the sail for the original and trimmed case. In an actual sailing situation this can be achieved by increasing the sheet tension and moving the sheeting point slightly backwards. In addition the boat is steered closer to the wind. As seen in table 13 the boat speed is actually slightly reduced but the VMG is increased anyway due to the reduction in true wind angle and reduction in leeway angle. The boat moves faster against the wind because it steers a tighter course with only a slight speed reduction in boat speed and on top of that the boat slides less sideways 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 -6.61 V[kts] 7.016 6.887 -0.129 -1.84 Leeway[°] 5.122 5.084 -0.038 -0.74 Heel[°] 20.120 19.087 -1.033 -5.13 AWA[°] 20.762 19.615 -1.146 -5.52 AWS [m/s) 9.048 9.067 0.019 0.21 VMG[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 almost identical in the two cases. Figure 54 and 55 show the pressure coefficient on the sail surfaces. As for the streamlines the two cases are almost identical. The pressure is slightly 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 side of the head sail is slightly bigger on the trimmed sail than on the baseline sail. This can probably 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 6.3.1

8 m/s Design space evaluation

As for the 4 and 6 m/s case, 20 variations are performed in the SOBOL search. The results of the successful runs are shown in table 14. As for the 4 and 6 m/s cases most of the variations show no improvement, and only 3 points show improvements compared to the baseline case. The best point nr. 7 shows a 0.75 % improvement. # Trim[°] Twist[°] TWA[°] V[kts] 7 -1.250 -0.750 31.750 7.052 19 -0.125 -1.625 33.625 7.229 0 0.000 0.000 34.000 7.230 5 0.500 -1.500 32.500 7.060 2 -1.000 1.000 31.000 6.908 12 1.250 0.750 33.250 7.049 9 1.750 -1.750 34.750 7.208 16 0.375 1.875 32.125 6.930 13 0.250 -0.250 30.250 6.745 18 -0.625 0.875 35.125 7.216 3 -0.500 -0.500 35.500 7.267 1 1.000 -1.000 37.000 7.342 8 0.750 1.250 37.750 7.368 10 -0.250 0.250 28.750 6.427 15 -1.625 -0.125 38.125 7.387 17 1.375 -1.125 29.125 6.385 11 -0.750 -1.250 39.250 7.393 4 1.500 1.500 29.500 6.338

Leeway[°] Heel[°] VMG[kts] Dev.[%] 5.519 21.28 5.612 0.75 5.579 22.69 5.602 0.57 5.605 22.71 5.570 0.00 5.511 21.33 5.562 -0.15 5.556 20.18 5.549 -0.39 5.520 21.29 5.496 -1.34 5.633 22.81 5.491 -1.43 5.569 20.36 5.484 -1.55 5.635 18.88 5.465 -1.90 5.799 23.68 5.453 -2.11 5.885 24.20 5.452 -2.12 6.003 24.88 5.370 -3.60 6.072 25.18 5.316 -4.57 6.105 17.26 5.274 -5.32 6.418 26.15 5.265 -5.48 6.136 17.15 5.214 -6.40 6.613 26.52 5.148 -7.58 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 function and a regression curve. Again there is a clear dominance by the true wind angle which overrules the variations in the other parameters. It is difficult to distinguish the effect of twist and trim angle based on this.

61

VMG − Twist angle 5.8

5.7

5.7

5.6

5.6 VMG [kts]

VMG [kts]

VMG − Trim angle 5.8

5.5 5.4

5.5 5.4

5.3

5.3

5.2

5.2

5.1 −2

−1

0 Trim angle [°]

1

5.1 −2

2

−1

0 Twist angle [°]

36

38

1

2

VMG − TWA 5.8

VMG [kts]

5.6 5.4 5.2 5

28

30

32

34 TWA [°]

40

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 investigation. In this case it starts in point nr. 7 and systematically steps in different directions to find a new optimum position. Table 15 shows the parameter variations and corresponding output parameter and Figure 59 shows a graphic representation of the iteration 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 wind angle of 31.15°, trim angle of −1.45° and twist angle of −0.675°. Running the optimizer further might have given further improvement, but the variable changes are now smaller than what is practically changeable and the variations in the objective function are close to the expected scatter.

62

# 0 1 2 3 4 5 6 7 8 9 10 11 12

Trim[°] Twist[°] -1.250 -0.750 -1.050 -0.750 -1.450 -0.750 -1.450 -0.550 -1.450 -0.950 -1.450 -0.750 -1.450 -0.750 -1.725 -0.750 -1.375 -0.750 -1.525 -0.750 -1.450 -0.675 -1.450 -0.675 -1.450 -0.675

TWA[°] V[kts] 31.750 7.052 31.750 7.030 31.750 7.063 31.750 31.750 7.060 32.350 7.120 31.150 7.008 30.325 6.895 31.150 6.995 31.150 7.005 31.150 7.012 31.375 7.016 30.925 6.983

Leeway[°] 5.519 5.567 5.534 5.568 5.572 5.481 5.579 5.493 5.494 5.490 5.561 5.518

Heel[°] VMG[kts] Dev.[%] 21.280 5.612 0.75 21.120 5.591 0.37 21.375 5.620 0.88 21.412 5.615 0.79 21.969 5.617 0.84 20.778 5.624 0.96 20.123 5.585 0.26 20.699 5.612 0.75 20.776 5.621 0.91 20.805 5.626 1.01 21.010 5.608 0.67 20.566 5.617 0.84

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

Trimming parameters

TWA

−0.4

32.5

−0.6

32

−0.8 −1

31.5

−1.2

[°]

[°]

Twist angle Trim angle

31

−1.4 30.5

−1.6 −1.8

0

5

10

30

15

0

5

10

15

VMG 5.63

[kts]

5.62 5.61 5.6 5.59 5.58

0

2

4

6

8

10

12

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

The results of the trim optimization shows a modest improvement compared to the baseline case of 1.01 %. The changes of the different parameters are outlined in table 16. The change is a reduction in the trim angle meaning that the sail is sheeted at a tighter angle, and a reduction in twist meaning that sheeting angle is decreased in the top compared 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 sheet tension and moving the sheeting point slightly forward. In addition the boat is steered closer to the wind. As seen in table 16 the boat speed is actually slightly reduced but the VMG is increased anyway due to the reduction in true wind angle and reduction in leeway angle. The boat moves faster against the wind because it steers a tighter course with only a small reduction in boat speed, and on top of that the boat slides less sideways through the water. Trim[°] Twist[°] TWA[°] V[kts] Leeway[°] Heel[°] AWA[°] AWS [m/s) VMG[kts]

Baseline Trimmed Change Change [%] 0.000 -1.450 -1.450 0.000 -0.675 -0.675 34.000 31.150 -2.850 -8.38 7.230 7.012 -0.218 -3.02 5.605 5.490 -0.115 -2.05 22.707 20.805 -1.902 -8.38 21.736 20.308 -1.428 -6.57 11.029 11.061 0.032 0.29 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 almost identical in the two cases. There is a decrease in the upward going flow in the top of the leeward side of the head sail on the trimmed sail compared to the baseline. Figure 61 and 62 show the pressure coefficient on the sail surfaces. The pressure on the windward side is in general smaller on the trimmed model, and the suction on the leeward side is also slightly smaller. This is somewhat a surprise for the head sail since the apparent wind angle change is similar to the trim change, which means they should cancel each other. The apparent wind speed is also almost the same and thus should not influence the pressure. The explanation may be found in the sail interaction. The pressure on the main sail is naturally decreased since it is not trimmed and thus experiences a decrease in angle of attack. The main sail twists the air to a higher angle of attack for the head sail. This is called upwash and is a well documented phenomena, see [2]. The increased pressure on the main sail reduces the upwash and thus decreases the apparent wind angle for 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 modifications from the above trim optimization. Except for the 4 m/s case, where the settings had a mistake. Due to this the values of 4 m/s in this section cannot be compared to those of the previous section.

7.1

Design Space Investigation

As for the trim investigations a SOBOL design space investigation of 20 points has been conducted. The results of the investigation shows a small improvement in two of the point as seen in table 17. The best design point nr. 8 has an improvement of 0.49 %. # 8 1 0 3 12 14 6 15 16 19 5 13 2 7 17 10

C1 0.188 0.250 0.000 -0.125 0.313 -0.438 -0.375 -0.406 0.094 -0.031 0.125 0.063 -0.250 -0.313 0.344 -0.063

C2 0.313 -0.250 0.000 -0.125 0.188 0.438 0.125 -0.031 0.469 -0.406 -0.375 -0.063 0.250 -0.188 -0.281 0.063

C3 0.313 0.250 0.000 0.125 -0.063 0.188 0.375 0.344 -0.156 -0.031 -0.125 -0.313 -0.250 -0.188 -0.406 -0.438

CP1 0.313 0.350 0.500 0.275 0.238 0.388 0.725 0.444 0.744 0.369 0.425 0.688 0.650 0.613 0.294 0.463

CP2 0.238 0.650 0.500 0.725 0.763 0.313 0.575 0.369 0.669 0.594 0.275 0.613 0.350 0.538 0.519 0.688

CP3 Obj[kts] 0.613 5.314 0.350 5.293 0.500 5.288 0.725 5.287 0.388 5.286 0.238 5.279 0.275 5.266 0.406 5.246 0.706 5.235 0.631 5.218 0.575 5.207 0.538 5.185 0.650 5.173 0.313 5.166 0.256 5.084 0.763 5.010

Dev[%] 0.49 0.10 0.00 -0.02 -0.03 -0.16 -0.41 -0.79 -1.01 -1.32 -1.52 -1.94 -2.17 -2.30 -3.85 -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 design variables along with a quadratic regression line. All of the graphs show a large scatter and 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 number of design evaluations in the investigation is too small for the large number of design parameters. Thus there is a large possibility that not all of the local minima have been covered. More points could thus improve the result, but time and resource limitations prevented more design evaluations in the project.

68

VMG - CP1 5.3

5.25

5.25

5.2

5.2 VMG [kts]

VMG [kts]

VMG - C1 5.3

5.15 5.1

5.15 5.1

5.05

5.05

5

5

4.95 -0.5

0 C1

4.95 0.2

0.5

5.25

5.25

5.2

5.2

5.15 5.1

5.15 5.1

5.05

5.05

5

5 4.95 0.2

0.5

0.6

0.8

VMG - CP3 5.3

5.25

5.25

5.2

5.2 VMG [kts]

VMG [kts]

VMG - C3

5.15 5.1

5.15 5.1

5.05

5.05

5

5 0 C3

0.4 CP2

5.3

4.95 -0.5

0.8

VMG - CP2 5.3

VMG [kts]

VMG [kts]

VMG - C2

0 C2 []

0.6 CP1

5.3

4.95 -0.5

0.4

4.95 0.2

0.5

0.4

0.6

0.8

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. The search was then started again from point 4 of the first search and continued for another 13 iterations. However a couple of the points are reused from the first pattern search. The second pattern search did not succeed in improving the result further and the best point is thus point 4 of the first TSearch. This point shows an overall performance improvement of 0.72 %. The variations in input parameters and objective function are displayed in 69

figure 66. # 1-0 1-1 1-2 1-3 1-4 1-5 1-6 2-0 2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 2-9 2-10 2-11 2-12 2-13

C1 0.1875 0.2375 0.2375 0.2375 0.2375 0.2375 0.2375 0.2375 0.2875 0.1875 0.2375 0.2375 0.2375 0.2375 0.2375 0.2375 0.2375 0.2375 0.2375 0.2375 0.25625

C2 0.3125 0.3125 0.3625 0.3625 0.3625 0.3625 0.3625 0.3625 0.3625 0.3625 0.4125 0.3125 0.3625 0.3625 0.3625 0.3625 0.3625 0.3625 0.3625 0.3625 0.3625

C3 0.3125 0.3125 0.3125 0.3625 0.2625 0.2625 0.2625 0.2625 0.2625 0.2625 0.2625 0.2625 0.3125 0.2125 0.2625 0.2625 0.2625 0.2625 0.2625 0.2625 0.2625

CP1 0.3125 0.3125 0.3125 0.3125 0.3125 0.3425 0.2825 0.3125 0.3125 0.3125 0.3125 0.3125 0.3125 0.3125 0.3425 0.2825 0.3125 0.3125 0.3125 0.3125 0.3125

CP2 0.2375 0.2375 0.2375 0.2375 0.2375 0.2375 0.2375 0.2375 0.2375 0.2375 0.2375 0.2375 0.2375 0.2375 0.2375 0.2375 0.2675 0.2075 0.2375 0.2375 0.2375

CP3 Obj[kts] 0.6125 5.314 0.6125 5.316 0.6125 5.319 0.6125 5.307 0.6125 5.326 0.6125 0.6125 5.309 0.6125 5.326 0.6125 0.6125 5.319 0.6125 5.314 0.6125 5.310 0.6125 5.319 0.6125 0.6125 0.6125 5.309 0.6125 5.322 0.6125 5.310 0.6425 5.321 0.5825 5.313 0.6125 5.313

Table 18: Result of pattern search for shape optimization

70

Dev [%] 0.49 0.54 0.59 0.37 0.72 0.40 0.72 0.59 0.49 0.41 0.59 0.40 0.64 0.43 0.62 0.48 0.49

CP

C 0.5

0.7 C1 C2 C3

0.45 0.4

0.6 0.5

CP1 CP2 CP3

[°]

[°]

0.35 0.3

0.4

0.25 0.3

0.2 0

5

10

15

0.2

20

0

5

10

15

20

VMG 5.335 5.33

[kts]

5.325 5.32 5.315 5.31 5.305

0

2

4

6

8

10

12

14

16

18

20

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

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

71

Variables

4 m/s

6 m/s

8 m/s

C1 C2 C3 CP1 CP2 CP3 V[kts] Leeway[°] Heel[°] AWA[°] AWS [m/s) VMG[kts] V[kts] Leeway[°] Heel[°] AWA[°] AWS [m/s) VMG[kts] V[kts] Leeway[°] Heel[°] AWA[°] AWS [m/s) VMG[kts] Objective

Baseline Optimized Change Change [%] 0.000 0.2375 0.2375 0.000 0.3625 0.3625 0.000 0.2625 0.2625 0.500 0.3125 -0.1875 0.500 0.2375 -0.2625 0.500 0.6125 0.1125 5.883 5.928 0.045 0.77 4.277 4.423 0.146 3.42 8.744 9.545 0.802 9.17 18.718 18.615 -0.102 -0.55 6.724 6.743 0.019 0.28 4.674 4.700 0.027 0.57 6.864 6.895 0.031 0.45 5.073 5.244 0.171 3.37 18.875 19.606 0.731 3.88 19.663 19.548 -0.114 -0.58 9.058 9.065 0.007 0.08 5.427 5.439 0.012 0.22 7.008 7.136 0.129 1.84 5.481 5.515 0.034 0.61 20.778 22.351 1.573 7.57 20.315 19.997 -0.318 -1.57 11.060 11.097 0.038 0.34 5.624 5.724 0.101 1.79 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 for 4 m/s condition. On the windward side of the head sail the flow reflects further upwards due to the increased camper of the sail. A separation-like rotation is observed near the leading edge. On the leeward side of the head sail the increased camper creates a trailing edge separation. The leading edge separation on the main sail is increased maybe due to the increased effect of the head sail increasing the downwash effect. This effect can also be seen on the pressure distribution on the windward side of main sail, as seen in figures 72 and 73. The pressure on the windward side of the head sail is reduced and the pressure centre is moved backwards. This can be attributed to the increase and forward movement of the camper. The increased camper also increases the suction on the leeward side 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 the baseline and optimized design for the 6 m/s condition. The trend of upward moving flow on the windward side of the head sail is also present in the 6 m/s case but it is not as strong as for 4 m/s, and the swirling separation is not seen in the streamline plot. The upward facing flow on the leeward side of the head sail has been reduced. As for 4 m/s trailing edge separation occurs at the leeward side of the head sail for the optimized design. The leading edge separation on the windward side of main sail has also increased as for the 4 m/s condition. In contrast to the 4 m/s condition the pressure on the windward side of the head sail is not reduced, but it is moved backward. The windward pressure of the main sail is slightly increased while the leeward suction remains the same. The suction on the leeward side of the head sail is only increased a little, which is reflected in the relatively 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 the baseline and optimized design for the 8 m/s condition. The upward facing flow on the windward side of the head sail has increased as for the other conditions and the swirling separation near the leading edge is strong. On the leeward side the trailing edge separation also becomes significant for the optimized sail. In contrast to the other conditions, the leading 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 is reduced on both sails, while the leeward suction has been increased on the head sail and reduced on the main sail. The increase in suction on the head sail can be accounted to the increase in camper, and the decrease on the main sail is probably due to the trailing edge 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 CFD software STAR-CCM+. The simulation includes both aero and hydrodynamics of the boat and calculates their effect on boat speed, leeway angle and heel angle. The simulation is used as a basis for an optimization of sail trim and sail shape. The X-40 is used as a test boat for the optimization, and the sail design used as a baseline is a design from North 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. Firstly a 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 drag forces. 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 is around 15.6 % for the chosen mesh density. The pressure validation in general shows good agreement with the experimental results, but again the oscillatory behaviour of the discretization gives a large uncertainty. Combined with the large experimental uncertainty this 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 the discretization uncertainty is evaluated for the important output parameters. The study again shows some oscillations but in general the mesh convergence is much better. Only the heel angle shows a large discretization uncertainty of 16.8 %, while the most important output parameter VMG shows 1.2 % uncertainty. Finally full scale measurements from racing results of an X-40 are presented for a rough comparison. The data in general show too much scatter for validation purposes, but they do however illustrate that the results of the simulation are well inside the scatter of the data. To modify and optimize the sail the commercial software Friendship Framework is used. 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 simulations are run via SSH at a remote simulation cluster. A trim optimization was run to optimize trim angle, twist angle and true wind angle to get the best VMG. The trim optimization was run with a design space investigation by the SOBOL algorithm and a tangent search method from the best point in the design space investigation. In general the trend was to decrease the true wind angle and the trim 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/s respectively. The shape optimization was run on the basis of 6 design variables deforming the sail shape within the borders of the sail with a shift surface. The 6 variables change the camper and the position of the maximum camper of the sail. As with the trim optimization, 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 20 designs, which is somewhat insufficient for the number of variables. It anyhow succeeded in 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 another similar 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 the lower and middle part of the sail and backwards in top part. Surprisingly the increase in camper 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 improvement of this approach is to combine these optimizations. Either by including the trim optimization as a sub-optimization for each design point in the shape optimization or by including all variables in one optimization. In both cases the computational time required will be much larger than for the separated case used in this project. On the other hand it may bring improvements not foreseen by the present method. The trim optimization shows an interesting potential. A future inclusion of more variables could make this a good tool for predicting performance with variation of the different variables. The output of an extended analysis could be used for improvement of trimming during racing. Inclusion of a structural code to evaluate the actual deformation of the sail and 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 expensive to be used in most of the sailing world. However with the ever decreasing cost and increasing computational power the future may bring opportunities for this sort of optimization to be spread wider. Simulation and optimization of sails are not only relevant for pleasure crafts. An increasing pressure on the world society to reduce consumption of non-renewable energy increases the interest on fitting sails ore kites on commercial ships. This brings another commercial 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 ModeFrontier Conference, June 2008, Trieste, Italy [2] Fossati, F., Aero-Hydrodynamics And the Performance of Sailing Yachts, Adlard Coles Nautical, 2009 [3] Shih, T.-H., Liou, W.W., Shabbir, A., Yang, Z. and Zhu, J. 1994. A New k −  Eddy Viscosity Model for High Reynolds Number Turbulent Flows – Model Development and Validation, NASA TM 106721. [4] Larsson, L., Stern, F., Visonneau, M., A Workshop on Numerical Ship Hydrodynamics - Proceedings, Volume II, Gothenburg 2010 [5] Menter, F.R. 1994. Two-equation eddy-viscosity turbulence modeling for engineering applications, AIAA Journal 32(8) pp. 1598-1605. [6] CHARNOCK, H, Wind Stress on Water - An Hypothesis - Wind Stress on s Water Surface, 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 Power Generation, Springer Berlin Heidelberg, 2013 [8] E¸ca, L., Vaz, G., Hoekstra, M., 2010. Code verification, solution verification and validation 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 Offshoreteknik,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 pressure distribution database for the validation of numerical codes. International J. Small Craft Technol., Trans. RINA 153(B1), 47-58. [16] Viola, I.M., et al. Upwind sail aerodynamics: A RANS numerical investigation validated with wind tunnel pressure measurements. Int. J. Heat Fluid Flow (2012) [17] Eld´en, L, Wittmeyer-Koch, L, Bruun Nielsen, H.,Introduction to Numerical Computation, Studentlitteratur, Edition 1:4, 2004

86

Appendix A: Nomenclature with illustration

Nomenclature

FA

Total aerodynamic force

α

Angular acceleration

FM

Aerodynamic driving force

αxx

Acceleration in surge

FM

Driving force

β

Apparent wind angle

FLAT Aerodynamic side force

λ

Leeway angle

FLAT Aerodynamic side force

µ

Fluid dynamic viscosity

FV ERT Aerodynamic vertical force

∇

Displacement

F EM Finite Element Method

∇c

Displacement of canoe body

g

Gravitational acceleration

ν

Fluid kinetic viscosity

h

Heeling arm

U 10

Wind Velocity at 10 m

IGES Initial Graphics Exchange Specification - CAD exchange format

ρ

Fluid density

θ

Heel angle

a

Empirical constant

ax

Acceleration in surge

ay

Acceleration in sway

k

Von Karmans constant

kts

Nautical knots

LOA

Length over all

LW L Length of waterline LP

Maximum width of head sail - maximum leech perpendicular

AW S Apparent Wind Speed

P

Luff length of main sail

B

Breadth

PI

Hydrodynamic heel force

CD

Drag coefficient

PLAT Hydrodynamic side force

CL

Lift coefficient

PLAT Hydrodynamic side force

AW A Apparent Wind Angle - β

CAD Computer Aided Design

PV ERT Hydrodynamic vertical force

CB

Center of buoyancy

R

CE

Centre of Effort of the aerodynamic forces R

Hydrodynamic resistance of hull and appendages Hydrodynamic resistance

CF D Computational Fluid Dynamics

RM

CG

SSH Secure Shell - network protocol for secure connection

Centre of gravity

CLR Centre of Lateral Resistance

Righting moment

CSV Comma Separated Values - file for- ST L STereo Lithography - CAD exchange format mat for storing tabular data as text E

Foot length of main sail

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 P P Velocity Prediction Program W

Weight of boat

Wcrew Crew weight z0

Roughness length

Head

Leac

Luff

s Fore

h

Head

tay

Leach

Luff

Main Sail

Bac

Tack

Clew Boom

Mast

ksta

y

Head Sail

Clew

Sheet Lead block

Rudde

r

Hull

Keel

Nomenclature illustration

Tack

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 connection sshpass -p 'password' ssh user@RemoteMachine 'bash -s' < runRemoteDir.sh # All needed files are copied to RemoteMachine through ssh. "Case/" is case name input from Friendship Framework sshpass -p 'password' scp * user@RemoteMachine:/directory/Case/ # The script runRemote is executed in a terminal on RemoteMachine trough ssh connection sshpass -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 directory cd /directory # Make new directory with name input "Case" from Frienship Framework mkdir Case

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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 "Case" specified by Friendship Framework cd /directory/Case # 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 "Dir" specified by Friendship Framework sshpass -p 'password' scp *csv user@LocalMachine:Dir/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 remesh new StarScript(getActiveSimulation(),new java.io.File(resolvePath("changeGeometry.java" ))).play(); // Run a script to change setting of the sim file new StarScript(getActiveSimulation(),new java.io.File(resolvePath("setParameters.java" ))).play(); // Run a script to run simulation new StarScript(getActiveSimulation(),new java.io.File(resolvePath("run.java"))).play(); // Run a script to write results from the simulation to csv files new 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 simulation Simulation simulation_0 = getActiveSimulation(); // Define units for import Units 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 manager PartImportManager partImportManager_0 = simulation_0.get(PartImportManager.class); // Get the part Head LeafMeshPart leafMeshPart_1 = ((LeafMeshPart) simulation_0.get(SimulationPartManager.class).getPart("Head")); // Replace part Head with stl file HeadSail.stl partImportManager_0.reimportStlPart(leafMeshPart_1, resolvePath("HeadSail.stl"), units_0, true, 1.0E-5); // Define units for rotation Units 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 rotation LabCoordinateSystem labCoordinateSystem_0 = simulation_0.getCoordinateSystemManager().getLabCoordinateSystem(); // Rotate head sail to initial heel angle. "HeelAngle" specified -1-

changeGeometry.java

7. august 2013 12:09

heelangle from Friendshi framework simulation_0.get(SimulationPartManager.class).rotateParts(new NeoObjectVector(new Object [] {leafMeshPart_1}), new DoubleVector(new double[] {-1.0, 0.0, 0.0}), new NeoObjectVector(new Object[] {units_0, units_0, units_0}), HeelAngle, labCoordinateSystem_0); // Get surface for defining feature lines PartSurface partSurface_0 = leafMeshPart_1.getPartSurfaceManager().getPartSurface("|6ms|trimmed6ms|trimesh0"); // Create part feature curves PartCurve 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 definition Region region_0 = simulation_0.getRegionManager().getRegion("Domain"); // Get region feature curve FeatureCurve featureCurve_10 = ((FeatureCurve) region_0.getFeatureCurveManager().getObject("HeadOutline")); // Set region feature curve featureCurve_10.getPartCurveGroup().addObjects(partCurve_8); // Get mesh controller MeshPipelineController meshPipelineController_0 = simulation_0.get(MeshPipelineController.class); // Generate mesh meshPipelineController_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 simulation Simulation simulation_0 = getActiveSimulation(); // Get the field function TWA UserFieldFunction userFieldFunction_0 = ((UserFieldFunction) simulation_0.getFieldFunctionManager().getFunction("TWA")); // Set the field function TWA to value from Friendship Framework userFieldFunction_0.setDefinition("TWA/180*$pi"); // Get the solver ImplicitUnsteadySolver implicitUnsteadySolver_0 = ((ImplicitUnsteadySolver) simulation_0.getSolverManager().getSolver( ImplicitUnsteadySolver.class)); // Set solver time step to value from Friendship Framework implicitUnsteadySolver_0.getTimeStep().setValue(TimeStep); // Get stopping criterion PhysicalTimeStoppingCriterion physicalTimeStoppingCriterion_0 = ((PhysicalTimeStoppingCriterion) simulation_0.getSolverStoppingCriterionManager(). getSolverStoppingCriterion("Maximum Physical Time")); // set stopping criterion to value from Friendship Framework physicalTimeStoppingCriterion_0.getMaximumTime().setValue(SimulationTime); } }

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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 simulation Simulation simulation_0 = getActiveSimulation(); // Run the simulation simulation_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 simulation Simulation simulation_0 = getActiveSimulation(); // Get the boat speed monitor MonitorPlot monitorPlot_0 = ((MonitorPlot) simulation_0.getPlotManager().getObject("BoatSpeed Monitor Plot")); // Export boat speed monitor data to csv file monitorPlot_0.export(resolvePath("BoatSpeed.csv"), ","); // Get the heel angle monitor MonitorPlot monitorPlot_1 = ((MonitorPlot) simulation_0.getPlotManager().getObject("HeelAngle Monitor Plot")); // Export heel angle monitor data to csv file monitorPlot_1.export(resolvePath("HeelAngle.csv"), ","); // Get the leeway angle monitor MonitorPlot monitorPlot_2 = ((MonitorPlot) simulation_0.getPlotManager().getObject("LeewayAngle Monitor Plot")); // Export leeway angle monitor data to csv file monitorPlot_2.export(resolvePath("LeewayAngle.csv"), ","); // Save simulation results in sim file simulation_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