numerical analysis and design of upwind sailsaero-comlab.stanford.edu/papers/ssriram.thesis.pdf ·...

NUMERICAL ANALYSIS AND DESIGN OF UPWIND

SAILS

a dissertation

submitted to the department of aeronautics and astronautics

and the committee on graduate studies

of stanford university

in partial fulfillment of the requirements

for the degree of

doctor of philosophy

Sriram Shankaran

April 2005

c© Copyright 2005 by Sriram Shankaran

All Rights Reserved

ii

I certify that I have read this dissertation and that in

my opinion it is fully adequate, in scope and quality, as

a dissertation for the degree of Doctor of Philosophy.

Antony Jameson(Principal Adviser)




Juan J. Alonso




Margot Gerritsen

Approved for the University Committee on Graduate

Studies:

iii

To all things alive

iv

Abstract

The use of computational techniques that solve the Euler or the Navier-Stokes equa-

tions are increasingly being used by competing syndicates in races like the Americas

Cup. For sail configurations, this desire stems from a need to understand the influ-

ence of the mast on the boundary layer and pressure distribution on the main sail,

the effect of camber and planform variations of the sails on the driving and heeling

force produced by them and the interaction of the boundary layer profile of the air

over the surface of the water and the gap between the boom and the deck on the

performance of the sail. Traditionally, experimental methods along with potential

flow solvers have been widely used to quantify these effects. While these approaches

are invaluable either for validation purposes or during the early stages of design, the

potential advantages of high fidelity computational methods makes them attractive

candidates during the later stages of the design process.

The aim of this study is to develop and validate numerical methods that solve

the inviscid field equations (Euler) to simulate and design upwind sails. The three

dimensional compressible Euler equations are modified using the idea of artificial com-

pressibility and discretized on unstructured tetrahedral grids to provide estimates of

lift and drag for upwind sail configurations. Convergence acceleration techniques like

multigrid and residual averaging are used along with parallel computing platforms

to enable these simulations to be performed in a few minutes. To account for the

elastic nature of the sail cloth, this flow solver was coupled to NASTRAN to provide

v

estimates of the deflections caused by the pressure loading. The results of this aeroe-

lastic simulation, showed that the major effect of the sail elasticity, was in altering

the pressure distribution around the leading edge of the head and the main sail.

Adjoint based design methods were developed next and were used to induce

changes to the camber distribution of the main sail. The goal of the design process

was to reduce the leading edge suction peaks that were considered to be detrimental

to the growth of the boundary layer. The deflected shape of the sails obtained from

the aeroelastic simulation were used by the design process. The design process re-

sulted in an camber distribution that allowed smooth entry of the flow through the

leading edge of the main sail thereby, reducing the leading edge suction peaks.

vi

Acknowledgments

vii

Contents

iv

Abstract v

Acknowledgments vii

1 Introduction 1

1.1 Design Requirements of Racing Yachts . . . . . . . . . . . . . . . . . 1

1.2 Models of Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Analysis with CFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Aeroelastic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Optimum Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6 Aerodynamic Shape Optimization . . . . . . . . . . . . . . . . . . . . 12

1.7 Outline of this study . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Discretization of Governing Equations 19

2.1 Overview of the Numerical Scheme . . . . . . . . . . . . . . . . . . . 19

2.2 Finite Volume Discretization of the Flow equations . . . . . . . . . . 21

2.3 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Staggered Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Implementation of the Cell-Vertex Scheme . . . . . . . . . . . . . . . 29

2.6 Artificial Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.7 Analysis of Artificial Compressibility . . . . . . . . . . . . . . . . . . 30

2.8 Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

viii

2.9 Multigrid Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.10 Parallel Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.10.1 Domain Decomposition, Load Balancing . . . . . . . . . . . . 35

2.10.2 Parallel implementation of the multigrid algorithm . . . . . . 38

2.10.3 Speedup of the Parallel Implementation . . . . . . . . . . . . . 39

2.11 Governing equations and analysis of the structural model . . . . . . . 39

2.11.1 Structural Model of the Sail . . . . . . . . . . . . . . . . . . . 41

2.12 Aeroelastic Coupling and Mesh Deformation . . . . . . . . . . . . . . 42

3 Analysis of Sail Configurations 46

3.1 Low Mach number, high angle of attack simulations with a compress-

ible flow solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1.1 Multi-Element airfoils . . . . . . . . . . . . . . . . . . . . . . 47

3.1.2 Sail simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2 Effect of Numerical Discretization and diffusion on artificial compress-

ibility methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3 Validation of the parallel implementation . . . . . . . . . . . . . . . . 49

3.4 Single and multi-element sail computations with artificial compress-

ibility methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4.1 Characteristics of the main sail . . . . . . . . . . . . . . . . . 50

3.4.2 Characteristics of the Head and Main sail combination . . . . 52

3.5 Aeroelastic simulations for single and multi-element foils . . . . . . . 54

4 Aerodynamic Shape optimization 76

4.1 The general formulation of the Adjoint Approach to Optimal Design . 77

4.2 Adjoint and Gradient formulations . . . . . . . . . . . . . . . . . . . 79

4.2.1 Adjoint Equations for the Euler equations modified by artificial

compressibility method . . . . . . . . . . . . . . . . . . . . . . 83

4.2.2 The need for a Sobolev inner product in the definition of the

gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3 Analysis of the Optimization Procedure . . . . . . . . . . . . . . . . . 86

4.4 Mesh movement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

ix

4.5 Parallel Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 88

5 Validation of the Optimization Procedure and Results 89

5.1 Shape optimization for airfoils in compressible flow . . . . . . . . . . 90

5.2 Shape optimization of airfoils in incompressible flow . . . . . . . . . . 90

5.3 Three dimensional shape optimization of wings in compressible flow . 95

5.4 Inverse design of wings in incompressible flow . . . . . . . . . . . . . 96

5.5 Inverse design for sail geometries . . . . . . . . . . . . . . . . . . . . 96

6 Conclusions 107

6.0.1 Aerodynamic and Aeroelastic analysis . . . . . . . . . . . . . 107

6.0.2 Aerodynamic design . . . . . . . . . . . . . . . . . . . . . . . 108

Bibliography 110

x

List of Tables

xi

List of Figures

2.1 Dual mesh representation of the control volume . . . . . . . . . . . . 25

2.2 Nodal formulation of the finite volume scheme . . . . . . . . . . . . . 25

2.3 Evaluation of fluxes in three dimensions . . . . . . . . . . . . . . . . 26

2.4 Control volume for cell-vertex schemes in three dimensions . . . . . . 26

2.5 Staggered arrangement of flow variables . . . . . . . . . . . . . . . . . 28

2.6 Half-staggered arrangement of flow variables . . . . . . . . . . . . . . 28

2.7 Interpolation coefficients for use in the multigrid cycle . . . . . . . . . 34

2.8 Transfer of solution, residuals and corrections between the fine and

coarse mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.9 Domain decomposition of a rectangular region using a bisection method 36

2.10 Halo nodes and the distribution of edges along processor boundaries 37

2.11 Speedup from the parallel implementation . . . . . . . . . . . . . . . 37

2.12 Boundary conditions for the main sail . . . . . . . . . . . . . . . . . . 42

2.13 Boundary conditions for the head sail . . . . . . . . . . . . . . . . . . 43

3.1 Grid and Pressure distribution over a multi-element airfoil geometry

at a M = 0.2 and α = 8.2 degrees . . . . . . . . . . . . . . . . . . . . 56

3.2 Cp distribution at two sections and convergence history of the com-

pressible flow solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3 Potential flow solution from FLO1 at 0,1 and 3 degrees . . . . . . . . 58

3.4 Flow over a NACA 0012 airfoil at 0,1 and 3 degrees using a cell-centered

scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5 Flow over a NACA 0012 airfoil at 0,1 and 3 degrees using a nodal scheme 60

xii

3.6 Flow over a NACA 0012 airfoil at 0,1 and 3 degrees using a half-

staggered scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.7 Total Pressure losses on the airfoil surface at 0o . . . . . . . . . . . . 62




3.11 Sail geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.12 Pressure distributions along sections at 1, 25 and 85 percent of the

height of main sail . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.13 Spanwise force distributions . . . . . . . . . . . . . . . . . . . . . . . 66

3.14 Variation of Lift and Drag with wind incidence . . . . . . . . . . . . . 66

3.15 Effect of mast on variation of Lift and Drag with wind incidence . . . 67

3.16 Effect of heeling angle on variation of Lift and Drag . . . . . . . . . . 67

3.17 Twist,camber and chord distribution of the head sail . . . . . . . . . 68

3.18 Twist,camber and chord distribution of the main sail . . . . . . . . . 68


height of head sail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69


height of the main sail . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.21 Spanwise force distributions on the head sail . . . . . . . . . . . . . . 71

3.22 Spanwise force distributions on the main sail . . . . . . . . . . . . . . 71

3.23 Pressure distribution over the pressure and suction side of the head

and sail combination . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.24 Original and deformed sail sections for the head sail . . . . . . . . . . 73

3.25 Original and deformed sail sections for the main sail . . . . . . . . . . 73


height of head sail after aeroelastic analysis . . . . . . . . . . . . . . . 74


height of main sail after aeroelastic analysis . . . . . . . . . . . . . . 75

5.1 Comparison of the gradients from SYN75 and SYN82 . . . . . . . . . 91

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5.2 Comparison of the first co-state variable from SYN75 and SYN82 . . 91

5.3 Comparison of the second co-state variable from SYN75 and SYN82 . 92

5.4 Comparison of the third co-state variable from SYN75 and SYN82 . . 92

5.5 Comparison of the fourth co-state variable from SYN75 and SYN82 . 93

5.6 Initial pressure distribution for the RAE-2822 airfoil . . . . . . . . . . 93

5.7 Drag minimization for the RAE-2822 airfoil . . . . . . . . . . . . . . 94

5.8 Final and target pressure distribution for the RAE-2822 airfoil . . . . 94

5.9 Initial and final pressure distribution, o is the target pressure distribu-

tion, x is the computed pressure distribution for the redesigned airfoil 95

5.10 Initial and final pressure and section geometries . . . . . . . . . . . . 97

5.11 Initial and final pressure distributions at 5 %, 50 % and 95 % of the

wing span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.12 Initial pressure distribution over a NACA 0012 wing . . . . . . . . . . 99

5.13 Final pressure distribution and modified section geometries along the

wing span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.14 Final computed and target pressure distributions at 0 % and 20 %of

the wing span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.15 Final computed and target pressure distributions at 40 % and 60 % of

the wing span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.16 Final computed and target pressure distributions at 80 % and 100 %

of the wing span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.17 Final computed and target pressure distributions at 0, 25, 75 and 100 %

of the wing span at 3 degrees angle of attack . . . . . . . . . . . . . . 103

5.18 Initial (o) and final(+,x) pressure distribution at 15, 32, 75 and 85%

height on the main sail . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.19 Initial and redesigned camber line at 15,32,75 and 85% of height . . . 106

6.1 Components of the overall design process for upwind sails . . . . . . . 109

xiv

Chapter 1

Introduction

1.1 Design Requirements of Racing Yachts

Races like the Americas Cup have seen significant improvements in the designs of both

the hull and the sails over the last two decades. Competing syndicates are constantly

pushing the aerodynamic and structural limits of the designs as improvements of less

than 0.5 % in the speed of the boat result in savings of around 25-35 seconds, which

is near the margin of victory for these races. For the windward leg of the race, a good

measure of the performance of a design is the distance that the boat travels directly

to windward in a given time. This performance index (called the speed-made-good by

the boat) is dependent on both the speed of the boat and the true sailing course, which

in turn are dependent on the aerodynamic and hydrodynamic forces produced by the

sails and the hull. In general, the windward performance of the boat can be improved

by reducing the resistance of the hull and the drag of the sails. However changes in

the aerodynamic and hydrodynamic forces alter the equilibrium of the sail boat which

then has to adjust its speed. Hence, the design of sailing yachts has to be carried out

in an environment where the analysis and design procedures for the sails and the hull

are integrated to realize a meaningful overall design. Traditionally, designers have

used Velocity Prediction Programs (VPP) which essentially solve for the equations

of equilibrium of the boat. These programs typically estimate the aerodynamic and

hydrodynamic forces using simple potential flow solvers. These provide quick answers,

1

CHAPTER 1. INTRODUCTION 2

enabling the designer to evaluate a wide array of designs. However, there exists large

regions of rotational flow and significant viscous interaction, where the assumptions

of potential flow are not valid. Hence there exists the need to develop and validate

alternate design techniques using more realistic models of the flow.

There exist a variety of tools that a designer can exploit to make improvements

to an existing design. Experimental techniques have been a favorite choice with most

designers and have been used successfully for downwind designs. This method intro-

duces no approximations to the physical properties of the fluid, and hence carefully

performed wind-tunnel tests can provide good estimates of the aerodynamic and hy-

drodynamic forces developed by the sail boat. However, experimental facilities are

usually expensive to build and maintain and have slower turn-around times than

computational models. For sail geometries, experimental testing usually does not

provide the designer with detailed descriptions of the pressure and velocity over the

sail geometry, as it is difficult to mount sensors that do not interfere with the flow

physics. Hence, experimental methods can usually only provide macroscopic esti-

mates of quantities of interest. Further, the twist in the onset profile of the wind

and the interaction of the free-surface with the hull are difficult to accurately model

through experimental methods. Over the last decade, experimental facilities in New

Zealand, California and Italy have been built that allow for twisted onset flow. Es-

timates of the flying shape of the sail and the variation of sail shape and trim for

varying weather conditions are typically observed using cameras mounted along the

sail and colored ribbons at various positions along the height of the sail.

Alternatively, computational models are finding increasing acceptance within the

sailing community [1]. Computational models have the capability of providing de-

tailed estimates of the aerodynamic and hydrodynamic forces along with deflections

developed by the sail. The steadily decreasing cost of computational simulations

is making this option more attractive than experimental methods during the initial

stages of design. Computational Fluid Dynamics (CFD) uses numerical solution pro-

cedures for mathematical models that describe the evolution of the flow-field. Hence,

it is possible to obtain solutions to a hierarchy of mathematical models that can be

used to continuously refine an initial design. Linear potential flow models are used by


many sail designers to estimate the forces produced by the sails and the hull. These

models are easy to implement and are computationally inexpensive while providing

the designer with valuable insights during the early stages of the design process. How-

ever, these models do not account for rotational flow fields and neglect viscous effects.

Rotational effects can be modeled by the full Euler equations of inviscid flow. For

upwind sail configurations an inviscid fluid model is valid for angles of attack up to

20 degrees provided there is no significant separation along the trailing edge, and the

sails have been trimmed so that leading edge separation is not too large either. These

models have the potential to accurately predict the induced drag which typically ac-

counts for 15 % of the total drag. The development of computational tools that solve

the Euler equations could also lay the foundation for the introduction of viscous ef-

fects either by solving the Navier-Stokes equations or by coupling a boundary layer

solution to the inviscid solution.

The key requirements for effective computational tools are

• sufficient accuracy

• acceptable computational cost

• rapid turn around

• reliability

• procedures to optimize the design.

The issues which need to be addressed in order to satisfy these requirements are

examined in more detail in the next sections leading to an outline of the thesis in

Section 1.7.

1.2 Models of Fluid Flow

There exist a variety of mathematical models for the flow-field that have been used

by the aerodynamic community. The most general description of the behavior of the

fluid particles involves descriptions of the time-evolution of fluid properties and are


described by the Boltzmann’s equations. It is easy to see that such particle-based

formulations quickly become intractable for all but the simplest flows due to the large

number of particles that need to simulated, and also the lack of universal physical

models to account for the interaction of various fluid particles.

Simplifying assumptions to the Boltzmann equations lead to the Navier-Stokes

equations which been found to consistently provide accurate descriptions of the flow

features for a variety of flow regimes. The flow around upwind and downwind sails and

around the hull/keel/appendages occur at high Reynolds numbers and hence, they

are turbulent in nature. The range of length and time scales present in turbulent flows

pose considerable difficulty both in the mathematical formulation and the numerical

resolution of the observed phenomena. Various models to predict the evolution of

the turbulent structures and the turbulent eddy viscosity have been developed for a

range of fluid flows. Direct Numerical Simulations that rely on the solution of the

Navier-Stokes equations in their original form, try to resolve all the scales associated

with turbulence. Due to the large range of scales involved with turbulence, these

methods have only been used for simple geometries to reduce the computational cost

of the simulation, mostly with the aim of gaining better insights into the physics

of turbulence. Alternatively, Large Eddy Simulations resolve some of the turbulent

scales while modeling the others. These have allowed more complex geometries to be

analyzed, but they are still prohibitively expensive for use in an industrial setting.

Consequently the Reynolds Averaged Navier-Stokes equations are generally used for

industrial simulations. The major stumbling block with this approach is the need for

models to predict the evolution of the turbulent quantities. Turbulence models for

the particular problem at hand have to be tested and tuned to arrive at meaningful

estimates of the quantities of interest. The quest for a universal turbulence model

is ongoing and until significant inroads are made in this area, engineers are left with

models that exhibit drastically different behavior for different flow regimes.

A further approximation that eliminates the viscous terms from the governing

equations, leading to the Euler equations, allows for the fluid to be compressible or

incompressible and the flow to be rotational or irrotational. Accordingly this set of

equations is capable of providing better estimates than the potential flow solvers.


Further, the twist in the onset flow can be also be easily included in the bound-

ary conditions for the Euler equations. The routine use of inviscid calculations in

the aeronautical world has resulted in well established numerical and computational

techniques. Analysis of shock structures in transonic and supersonic flight has led to

a wide array of computational schemes embedded in Finite Volume, Finite Element

and Finite Difference techniques with various flavors to identify the shock structures

and pressure distributions around airfoils, wings and complete aircraft configurations.

Significant developments in the numerical analysis of the Euler equations has resulted

in fast solvers that use multigrid and residual averaging techniques along with effi-

cient time-stepping algorithms to advance the solution very rapidly to a steady state.

Space discretizations techniques are also well understood, and can provide level of

accuracy needed for engineering estimates. Growth in computing power has added

fuel to the development of this technology, and the routine use of parallel computing

techniques has reduced the computational time of inviscid analysis to the order of a

few minutes. Hence, inviscid flow models can been incorporated into the design cycle

to replace potential flow solvers in the design process.

1.3 Analysis with CFD

Discretization of the flow equations requires the subdivision of the computational

domain into a grid of sufficiently small cells. The first choice to be made is the type

of grid.

Numerical solutions to the governing partial differential equations of fluid flow

were first obtained using computational grids that were structured in nature. It is

easy to obtain higher order accurate flow solutions and boundary conditions with

these grids and hence a wide variety of numerical techniques have been developed

for structured grids. However, for complex geometries, the generation of body-fitted

structured grids is not straight-forward. Structured grid generation techniques have

reached a state of considerable maturity, but typical turn-around times of the grid

generation process are still of the order of weeks, or even months. Unstructured grid

generation techniques that divide the computational domain into arbitrary polyhedral


elements can handle complex geometries with greater ease than structured grids.

They also have the potential to be automated, and hence might be incorporated in

a design environment where repeated changes to the geometry have to be performed

while the design evolves.

Over the last two decades well established grid generation techniques for unstruc-

tured grids have been developed. The Delaunay criterion and the advancing front

technique are two of the most widely used techniques by these researchers. While

the use of the Delaunay criterion results in meshes of the highest possible mesh qual-

ity for a given distribution of mesh points, the advancing front algorithm allows the

user more control over point-placement within the computational domain. However,

unstructured grid generation techniques for viscous flows have not yet reached the

maturity that allows them to be completely automated. Nevertheless, a variety of

grid generation methods for both structured and unstructured grids has alleviated

the problem of grid generation, and it is now possible to produce structured and

unstructured grids for very complex geometries.

The nature of unstructured grids lends itself naturally to Finite Volume and Finite

Element methods. Depending on the nature of the underlying discretization, either

the elements of the unstructured grid or the dual of the mesh can be used to construct

non-overlapping control volumes around each computational node. A reconstruction

of the fluxes along the edges of the control volume along with artificial dissipation

terms to prevent odd-even coupling can be shown to be identical to a Finite Element

approximations with linear basis functions and a compact stencil spanning the control

volume. Some of the first calculations over a complete aircraft configuration were

performed with a finite volume technique of this type which was mathematically

presented as a Finite Element method [28].

Spatial discretization of the convective terms of the governing equations along

with the numerical diffusion terms results in a set of ordinary differential equations

that can be integrated in time to obtain time accurate and steady state flow solutions.

These ODEs can be integrated explicitly using a multistage Runge-Kutta scheme with

an appropriate choice of coefficients that maximize the stability of the time evolution.

To accelerate convergence to steady state, multigrid and residual averaging techniques


can be used.

The best choice of coarser grids for unstructured multigrid schemes is still an open

problem. Generating a series of meshes repeatedly from a grid generator, edge col-

lapsing techniques which ‘contract’ a given mesh, and agglomeration methods which

fuse cells from a given mesh are three approaches that have been tried. Generating

the meshes repeatedly from a grid generator places a considerable burden on the

grid generator, and is not easy to incorporate in an automated procedure. Further,

the need to transfer the solution and the residuals to coarser meshes and the correc-

tions to a finer mesh necessitates fast algorithms to locate points within cells. Edge

collapsing algorithms use heuristic ideas to repeatedly collapse edges from a given

mesh. It is possible to automate this procedure to generate coarser meshes from a

given fine mesh. Further, this method has the advantage of being able to compute

the interpolation coefficients for the multigrid while the coarser meshes are gener-

ated thereby avoiding the need to use additional search algorithms. Agglomeration

multigrid techniques ‘fuse’ cells from a given mesh to generate the coarser meshes.

This gives rise to cells on the coarser meshes which have arbitrary shape and hence

require efficient data structures to implement the underlying numerical schemes on

them. While the coarser meshes are obtained using a set of heuristic ideas, it is also

possible to automate this procedure. Further the coefficients for the multigrid cycle

are automatically obtained during the agglomeration cycle and the flux balances on

the coarser meshes can be free of any interpolation errors. Further research is needed

to determine whether agglomeration is superior to edge collapsing, or whether some

other technique can yield a faster rate of convergence.

Spatial discretization of the convective terms of the governing equations along

with the numerical diffusion terms results in a set of ordinary differential equations

that can be integrated in time to obtain time accurate and steady state flow solutions.

These ODEs can be integrated explicitly using a multistage Runge-Kutta scheme with

an appropriate choice of coefficients that maximize the stability of the time evolution.

To accelerate convergence to steady state, multigrid and residual averaging techniques

can be used.

The above mentioned numerical algorithms are widely used in the aerodynamic


design of aircraft by most commercial civil transport manufacturers across the world.

They were designed to treat compressible flows with embedded shock structures.

However, in the incompressible limit they require some modifications. Chorin [30]

proposed the idea of artificial compressibility to enable the re-use of the well developed

numerical techniques for compressible flows for incompressible flows. The key idea

of this method is to augment the continuity equation by a time dependent pseudo-

pressure term. While the value of this pressure is not physically meaningful during

the evolution of the system of equations, in the steady state it provides the pressure

which satisfies the momentum equations, and it drops out of the continuity equation,

thereby satisfying both the continuity and momentum equations simultaneously. This

idea has been used by a number of researchers [3], [4], [31] to convert computational

programs developed for compressible flows to handle incompressible flows and has

proven to be robust and accurate for steady state problems.

1.4 Aeroelastic Analysis

The pressure distribution acting on a lifting surface is determined by its shape and

in most aerodynamic applications this shape is fixed under the assumption that the

geometry does not change appreciably under the action of the aerodynamic loads.

However, this is not the case when the lifting surface is an elastic membrane like a sail,

since the twist and camber of such wings under the load may be quite different from

their unloaded values. The flexible behavior of sails necessitates the need to perform

aeroelastic simulations with analysis methods which can treat large displacements.

Classical models to study the behavior of membranes under static and unsteady loads

have been extensively studied. In applications to sails, the structural analysis needs to

take into account geometric nonlinearities. However, since the strains remain small,

constitutive laws for the material can be considered to be linear, with the result that

the tension in the structure is a linear function of the local deformation.

Charvet [5] presented a scheme to estimate the steady equilibrium configuration of

a sail. This analysis decomposed the large displacements into two steps. The first step

computed the large displacements of an inextensible sail and the second considered the


elastic displacements of an elastic sail. This approach is satisfactory for structures

whose Young’s modulus is large thereby limiting the elastic deformations to small

displacements. Another approach, pursued in the works of Jackson et. al. [6] and

Fukusawa et. al. [7], considered small displacements to an arbitrary elastic structure.

Large-displacement analysis of an elastic membrane is an ongoing quest [24], [25] and

no satisfactory analysis has been performed to date.

Another important feature of the structural deformations induced in sail geome-

tries by the aerodynamic loading is the possibility of wrinkles which are usually local

in their presence. Due to the highly non-linear nature of the formation of these

wrinkles, most sail designers have neglected the effect of these wrinkles. Studies by

Miller et. al. [8], [9], argued that the most important effect of wrinkles is to locally

increase the average strain in the normal direction due to a strain or displacement in

the longitudinal direction. Under these assumptions, they accounted for wrinkles by

locally increasing the Poisson’s ratio in regions where wrinkles are formed and using

the Hookean material properties which now become dependent on the local state of

strain. This approximate theory attempts only to estimate the average wrinkle strain,

and does not identify the shape of the wrinkle. Further, the emergence of wrinkles

is based on generalized assumptions originating from the magnitude of the principal

strain in each element of the finite element model.

Another important feature of modern day rigs for races like the Americas Cup is

the flexibility of the mast. The masts tend to be flexible to exploit the advantages

of automatic shape changes under heavy wind conditions. Studies which account for

the flexibility of the mast usually employ an incremental procedure, where the sail

and the mast are deflected in turn with appropriate boundary conditions along the

point of attachment [21], [23], [22]. An integrated structural simulation of a complete

sail rig has still not been reported in the open literature, and could be invaluable in

the quest to achieve improved designs.


1.5 Optimum Design

The search for optimal designs that maximize/minimize a performance index has been

the aim of designers in all engineering disciplines. Identification of a possible set of

design variables that have the maximum influence on the performance of a design

along with suitable representations of the state of the system are needed to cast the

optimization problem in a mathematical frame-work. Typical design problems in most

engineering fields are multi-disciplinary in nature. The design of sails boats is multi-

disciplinary due to the tightly coupled interaction between the sails and hull. Within

this multi-disciplinary environment, it is possible to identify optimization problems

that are confined to a single discipline provided the constraints from other disciplines

are satisfied. Aerodynamic shape optimization is one such area that involves the

identification of an optimum shape to improve the aerodynamic characteristics of the

design. It is possible to cast the problem of identifying an optimum sail, hull or keel

geometry under this umbrella. If the drag of a given sail shape has to be reduced,

the span loading has to be altered. However, altering the span loading changes the

heeling moment and the equilibrium of the boat and hence suitable constraints have

to be provided to achieve a meaningful design. On the other hand, it is often desirable

to determine a sail shape that provides a favorable pressure distribution that inhibits

separation of the boundary layer. While this class of optimization problems (herein

referred to as shape optimization problems) can be studied within the single discipline

of aerodynamics, a major difficulty is the large dimensionality of the design space.

To overcome this problem, sail designers typically optimized a given design using a

combination of parametric studies [2] and experience.

The task of locating minima in the design space requires some knowledge of the

topology (typically slope/gradient and curvature information) of the design space and

a ‘search’ algorithm that navigates through the design space to a minima. Due to the

complexity of the problem many attempts have been based on techniques which do

not explicitly compute the gradients in the design space. Some of these approaches

use evolution or genetic algorithms to evolve the design towards an optimum. These

algorithms use a collection of candidate designs that are then modified using heuristic


rules based on some knowledge of the design space to identify a new set of designs.

An advantage of these approaches is that they are relatively easy to implement and

do not usually require gradient evaluations. However, their computational complexity

can become prohibitive with a large number of design variables. Hence, successful

optimization has required an experienced user to judiciously select a minimal set of

design functions which adequately defines the design space in which the search for

the optimum navigates.

Alternately, first order gradient based methods estimate the first derivative of

the change of the cost function with respect to the choice of design variables. This

estimate of the gradients is then used to predict a new design configuration that leads

to an improvement in the performance index. The main challenge for this approach is

to estimate the gradients accurately and cheaply. Initially, finite-difference methods

were used to estimate the gradient of the cost function with respect to the design

variables. Hence, these methods require one or two flow solutions to obtain the

gradient with respect to each design variable. The formulation of the optimization

problem in a control theory context leads to the idea of adjoint systems which allow

evaluation of the gradients with respect to a large number of design variables with

minimal computational effort. While the complexity of this approach scales with the

number of performance indices, this is not a difficulty for aerodynamic design as the

primary performance measures are lift and drag.

Once the gradients have been evaluated a variety of algorithms can be used to

evolve the design. The simplest method, called the steepest descent method, takes

a step in the direction of the negative gradient. Hence, this approach requires an

estimate of the step-size which is usually obtained by trial and error. Alternatively,

Newton methods make use of the curvature of the topology and the slope to navigate

through the design space have also been used. These methods require estimates of

the second derivative of the cost function with respect to the design variables, and

again can quickly become expensive or intractable. Under these circumstances the

use of adjoint based design methods combined with the steepest descent technique

has proven to be a good compromise which provides a robust and efficient tool for

aerodynamic shape optimization, as described in the next section.


1.6 Aerodynamic Shape Optimization

Aerodynamic shape design has long been a challenging objective in the study of fluid

dynamics. CFD has played an important role in the aerodynamic design process since

its introduction for the study of fluid flow. However, CFD has mostly been used in

the analysis of aerodynamic configurations in order to aid in the design process rather

than to serve as a direct design tool in aerodynamic shape optimization. Although

several attempts have been made in the past to use CFD as a direct design tool, it has

not been until recently that the focus of CFD applications has shifted to aerodynamic

design [42, 43, 44, 45, 46, 47]. This shift has been mainly motivated by the availability

of high performance computing platforms and by the development of new and efficient

analysis and design algorithms. In particular, automatic design procedures which use

CFD combined with gradient-based optimization techniques, have made it possible

to remove difficulties in the decision making process faced by the aerodynamicist.

Gradient-based optimization techniques typically identify a control function to be

optimized and a suitable cost function whose optimum location in the design space

is the quest of the algorithm. The control function can either be parameterized to

reduce the number of design variables or represented in forms which account for all

possible variations subject to applicable constraints. To determine future designs

within a design space, estimates of the slope in the design space are evaluated and an

algorithm to determine possible movement within the design space is used to move

towards a better/optimum design. Finding a fast and efficient way to determine the

gradients is critical to this method as is the need for an intelligent search algorithm.

Gradient information can be computed using a variety of approaches. The finite-

difference method is probably the most straight-forward way of computing these gra-

dients. In the finite-difference method, small steps are taken in each and every one of

the design variables independently, in order to find the sensitivity of the cost function

with respect to those design variables. Since each of these steps requires a complete

flow solution, the computational cost of this method is proportional to the number

of design variables, and, consequently, it cannot be afforded for problems with design

spaces of large dimensionality. Further, the accuracy of the gradients is sensitive to


the choice of the step used to perturb each design variable which can be alleviated by

alternative methods whose accuracy is independent of the choice of step size, such as

the complex step method [59] and automatic differentiation [60].

As an alternative choice, the control theory approach has dramatic computational

cost advantages when compared to any of these methods. The foundation of con-

trol theory for systems governed by partial differential equations was laid by J.L.

Lions [48]. The control theory approach is often called the adjoint method, since

the necessary gradients are obtained via the solution of the adjoint equations of the

governing equations of interest. The adjoint method is extremely efficient since the

computational expense incurred in the calculation of the complete gradient is effec-

tively independent of the number of design variables. The only cost involved is the

calculation of one flow solution and one adjoint solution whose complexity is similar

to that of the flow solution. Control theory was applied in this way to shape design

for elliptic equations by Pironneau [50] and it was first used in transonic flow by

Jameson [42, 43, 51]. Since then this method has become a popular choice for design

problems involving fluid flow [45, 52, 53]. In fact, the method has even been success-

fully used for the aerodynamic design of complete aircraft configurations [44, 54].

Gradient formulations which require the solution of an adjoint system have fallen

into the categories of the discrete and continuous approaches. In the former, the

adjoint system to the discretized flow equations are assembled to obtain the gradient,

thereby necessitating the need to the formulate different adjoint systems for different

discretizations. In the latter, the adjoint system to the original flow equations in

partial difference form is used to estimate the gradients eliminating the need to refor-

mulate the adjoint equations. Studies by Siva Nadarajah and Jameson [49] concluded

that there is no particular benefit in using either one of these methods due to the

trade-offs between the complexity of the discretization of the adjoint equations for

the continuous and discrete approaches, the accuracy of the resulting estimates of the

gradient, and the computational costs required by each method to reach an optimum.

Jameson and Vassberg also compared discrete adjoint versus continuous gradients for

the Brachistochrone problem in which an exact optimal solution is known and showed

that in this case the continuous gradient is slightly more accurate [55].


Jameson’s initial work and Jameson and Reuther’s later works are based on the

continuous adjoint method. Anderson and Venkatakrishnan explored the continuous

adjoint on unstructured grids [53]. Anderson and Nielsen have also implemented the

discrete adjoint on unstructured grids [57]. In their work, Anderson et al. presented

the accuracy of the adjoint sensitivity derivatives in aerodynamic design using the

Navier-Stokes equations, and also presented some design examples including a wing

drag minimization and an inviscid multi-element airfoil shape design.

Using the control theory approach it is possible to obtain Frechet derivatives

of the cost function for a set of design variables which allow for the all possible

shapes of the control surface in question, usually a wing or a sail geometry. This

approach mandates the use of all the computational points in the mesh that represent

the control surface and hence could be in the order of a few thousand. Estimating

gradients for these design variables can quickly become expensive if intelligent choices

on the mesh perturbation and gradient calculations are not made. A recent study

by Jameson and Sangho Kim [73] has enabled gradient calculations to eliminate this

need by rewriting the formulations in terms that depend only on the flow and adjoint

solution on the control surface. This finding has far reaching implications to the

world of design using unstructured grids. Earlier formulations of the gradients under

the umbrella of adjoint methods required mesh displacement strategies and residual

evaluations for perturbations in each design variable. While it is possible to arrive

at efficient choices to perform this on structured grids, the lack of structure with

unstructured grids requires intelligent mesh perturbation techniques. Hence earlier

researchers with unstructured grids used a parametric representation of the control

surface. This reduced set of design variables might be incapable of recovering all

possible shapes. The use of the reduced gradient formulation eliminates this difficulty

and allows the designer to view the shape as a free surface.

For the class of aerodynamic shape optimization problems which are investigated

in this study, the design space is essentially infinitely dimensional. The problem is

one of choosing an optimum curve or curved surface, as in classical problems in the

calculus of variations and trajectory optimization. Suppose that the performance

is measured by a cost function I which depends on a function y(x), where under a


variation δy(x), the variation of the cost is δI.Now suppose that δI can be expressed

to first order as

δI =

∫G(x)δy(x)dx, (1.1)

where G(x) is the gradient. Then by setting

δy(x) = −λG(x), (1.2)

one obtains an improvement

δI = −λ∫G2(x)dx, (1.3)

unless the gradient is zero. Thus the vanishing of the gradient is a necessary condition

for a local minimum.

In order to accelerate the search, one may resort to using the Newton method.

Here, the search direction is based on the equation represented by the vanishing of

the gradient and is solved by the standard Newton iteration for nonlinear equations.

Suppose that the Hessian is denoted by

A =∂G∂y, (1.4)

then the result of a step δy may be linearized as

G(y + δy) = G(y) + Aδy. (1.5)

This is set to zero for a Newton step; therefore

δy = −A−1G. (1.6)

The Newton method is generally effective if the Hessian can be evaluated accurately

and cheaply.

Quasi-Newton methods estimate A or A−1 from the changes of the gradient

recorded during successive steps. For a discrete problem, it requires N steps to


obtain a complete estimate of the Hessian. Therefore, as the dimensionality of the

design space increases, this method requires in more memory to compute the Hessian

and more steps to reach an optimum. This motivates the search for an alternative.

Steepest descent methods provide an alternative search scheme. Here a step is

taken in the negative gradient direction. Denoting the iterations with the superscript

n, we have

yn+1j = yn

j − λGnj . (1.7)

This may be regarded as a forward Euler discretization of a time dependent process

with λ = ∆t. Hence,∂y

∂t= −G. (1.8)

The simplicity of steepest descent methods is off-set by the need to identify step

sizes and the potentially large number of steps that might be required to reach an

optimum. However, for typical cost functions of interest in aerodynamic problems,

the design space seems to be rather benign, with the result that steepest descent

methods provide accurate answers.

1.7 Outline of this study

The first part of the thesis describes the development of the analysis tool aimed at

providing a viable alternative to linear potential flow models. Towards this objective,

the flow solver has to be both robust and have fast turn-around times. In this study,

the numerical solution procedure that simulates the flow around the sails uses a dis-

cretization of the computational domain into unstructured tetrahedra and hence, it

can easily be extended to include the geometry of the deck and the hull in the anal-

ysis. Simulations of the hull-appendages can also be performed using unstructured

grids and eventually it would be possible to couple the sail simulations with the hull

calculations to provide an integrated analysis tool.


Finite Volume techniques in conjunction with unstructured grids are used to dis-

cretize the governing equations of motion of an incompressible flow equations. Spa-

tially second order accurate schemes with numerical diffusion and multistage Runge-

Kutta time integration schemes are used to advance the solution to a steady state.

Non-nested multigrid methods, where the meshes are regenerated, along with implicit

residual averaging techniques are used to obtain converged solutions in about 75-100

multigrid cycles. The algorithm is parallelized to reduce turn-around time of the

simulations to the order of a few minutes. To predict the flying shape of sails, this

flow solver is coupled to the commercial structural analysis package, NASTRAN.

Using this tool, the variations of the lift and drag for different wind and sail setting

is studied. Multiple sail geometries are also analyzed to study the interaction of the

main sail with the genoa.

The second part of the thesis addresses the quest for optimum sail design. Here

the aerodynamic shape optimization problem is cast under the control theory ap-

proach. Accordingly the shape of the sail is identified as the control mechanism that

is modified to meet the required performance criteria. Hence, an adjoint system to

the governing flow equations is introduced and solved using the same techniques as

those used for the flow solver. Gradient formulations which use the solution to an

adjoint system are used together with steepest descent search methods to identify an

optimum in the design space. Each point in the computational mesh that describes

the sail geometry is used to alter the sail shape, thereby allowing all possible shapes

to be recovered during the optimization procedure. Gradient formulations which de-

pend only the surface geometry information have largely made possible the use of

unstructured grids in this design methodology as they eliminate the contribution of a

field integral to the gradient formulation. This field integral is typically computed by

perturbing each design variable and computing a new residual at each mesh point in

the computational mesh proving to be quite expensive for design variables which run

in the order of a few thousand. Shape modifications to an existing design are made

to improve the performance of the design. In this study, inverse problems are inves-

tigated where the target pressure distribution is prescribed through a combination of

experience and engineering intuition.


In the next two chapters, the numerical implementation of the analysis tool is

discussed with emphasis on the discretization of the fluid-flow equations. To make

more realistic predictions an aeroelastic package is used to predict the flying shape of

the sail. This aeroelastic package is used to analyze a head and main sail geometry

that is representative of those used in the Americas Cup. In the last two chapters

the design philosophy is laid out with particular reference to the gradient calculations

on unstructured grids. Results for an inverse design exercise are provided to validate

the design process, and then it is used to alter the shape of the sails to eliminate

undesirable flow features.

Chapter 2

Discretization of Governing Fluid

and Structural Equations

2.1 Overview of the Numerical Scheme

Traditionally, panel methods with corrections to account for the boundary layer and

wake have been used to model the fluid flow around sails [10], [14]. For most en-

gineering purposes, these simplified linear potential flow models provide reasonably

accurate estimates of the forces and moments on upwind sails, and they have been

exclusively used by sail designers over the last couple of decades [11], [15], [16], [17].

However, the flow around sails possess substantial regions of rotation, the most com-

mon feature being the shedding of vorticity from sharp edges. With a potential flow

model, the user is required to set up vortex-sheet discontinuities in the flow field and

then ‘adjust’ and ‘fit’ them to the surrounding flow. This requires prior knowledge

of where the sheets begin, and becomes complicated for all but the simplest situa-

tions. However, potential flow codes have been successfully used in Americas Cup

campaigns and continue to the mainstay of most designers.

Further, the desire to incorporate the effect of twist in the onset flow and viscous

phenomena necessitate the use of more advanced numerical models that solve the

complete field equations. However, these non-linear models require the solution of

the coupled partial differential equations governing the evolution of the fluid which

19

CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 20

are much more computationally expensive than panel methods. Advances in both

numerical techniques and the growth in computing power over the last two decades

have alleviated these problems. The use of parallel computing techniques along with

the use of multigrid and residual averaging techniques have enabled flow solutions to

be performed in the order of minutes [27]. Thus, it is possible to obtain numerical

solutions to the field equations with turnaround times that are acceptable to the

overall design process while allowing for non-linear models to be incorporated in the

design process. The solution to the structural equations, static and time-dependent,

has been extensively studied and have reached a stage of considerable maturity such

that routine analysis can be performed without much intervention by the end-user.

Commercial finite element packages provide a range of options to handle linear and

non-linear deformations under a variety of operational conditions. These methods

have the robustness and flexibility needed for analysis and design.

This chapter discusses the numerical scheme used to solve the governing equations

for both the fluid and the structure. The governing equations of motion of a com-

pressible inviscid fluid are modeled using the Euler equations, modified using the idea

of artificial compressibility to handle incompressible flows. In the following sections

the finite volume approach to discretize the governing equations on an unstructured

grid are presented, along with Runge-Kutta time integration techniques and residual

averaging and multigrid methods. The combined use of these techniques enables a

flow solution to be obtained in about 75-100 multigrid cycles. Further, the use of

parallel computing methods reduce the cost of these computations to the order of a

few minutes. The pressure loads obtained from the fluid solver are fed to a structural

analysis program to estimate the deflections. Because of the large deflections typi-

cally observed in sail geometries, a non-linear model provided by NASTRAN is used

within the structural solver. This non-linear model breaks the loading into a series of

small steps, which are applied sequentially. The deflected shape is used to modify the

computational mesh for the flow solver, using standard mesh deformation techniques

in order to obtain a new pressure loading. Finally an iterative process that couples

the flow and structural solver is used to arrive at the steady flying shape of the sail.


2.2 Finite Volume Discretization of the Flow equa-

tions

A vast repertoire of computational codes have been developed by Antony Jameson to

analyze aerodynamic configurations in transonic flight [28], [29]. These codes model

the fluid as a compressible fluid, and a variety of numerical techniques have been

developed to efficiently solve the governing equations of a compressible fluid with

embedded supersonic regions. In the limit of truly incompressible flow, or zero Mach

number, alternative methods are needed to preserve the accuracy, robustness and

convergence properties of the flow solution procedure. The fundamental difference

between a compressible fluid model and an incompressible one is the loss of of the

evolution equation for the density. Since the density is constant, a constraint must

be imposed on the continuity equations to ensure a divergence-free velocity field. In

addition, the eigenvalues resulting from the system of conventional hyperbolic Euler

equations for compressible flows become infinite in the limit of incompressible flow.

This is due to the fact that the sound speed becomes unbounded. Hence, the use of

compressible flow solvers in the incompressible flow limit, introduces widely varying

eigen speeds, resulting in extremely stiff equations. To overcome this difficulty, the

present work uses the artificial compressibility method, an approach first proposed

by Chorin in 1967 [30] as a method to solve viscous flows. Artificial compressibility

methods introduce a psuedotemporal equation for the pressure through the continu-

ity equation. This approach removes the troublesome sound waves associated with

compressible flow formulations as the Mach number approaches zero. The eigenvalues

of the original system are now replaced with an artificial set that renders the new

set of equations well-conditioned for numerical computation. When combined with

multigrid acceleration procedures, artificial compressibility proves to be particularly

effective [31]. Converged solutions of incompressible flows over a main sail can be

obtained in about 75-100 multigrid cycles.

Using the idea of artificial compressibility, the equations of motion of an incom-

pressible, inviscid fluid can be cast in the following form:


∂w

∂t+ P

{∂F

∂x+∂G

∂y+∂H

dz

}= 0. (2.1)

Here, the dependent variables w, the inviscid flux vectors f , g and h and the precon-

ditioning matrix P are described by

w =

p

u

v

w

, F =

u

u2 + p

uv

uw

, G =

v

vu

v2 + p

vw

, H =

w

wu

wv

w2 + p

,

P =

Γ2 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

. (2.2)

This set of equations has no physical meaning until the steady state is reached. At

steady state, the time dependent pressure term drops from the continuity equation

resulting in the true steady state equations for an incompressible flow. Further, Γ

can be selected to accelerate the time decay to steady state.

Using the finite volume approach, the governing equations can be cast in the

integral form for each computational volume in the domain as follows,

Conservation of Mass

d

dt

∫

V

pdV +

∫

S

Γ2 (u · n) dS = 0. (2.3)

Conservation of Momentum

d

dt

∫

V

udV +

∫

S

u(u · n)dS = −∫

S

pndS, (2.4)

Spatial discretization of equation (2.3) and (2.4) leads to a separate equation for each


sub-domain in the computational mesh.

d

dtViwi +

∑

k

Fk.nkSk = 0, (2.5)

where p is the pressure, u is the velocity vector, n is the unit normal at the surface of

the control volume, V and S are the volume and surface area of the control volume

respectively, F is the flux through the control volume and the summation of the fluxes

is over the control volume that surrounds each node of the mesh.

2.3 Spatial Discretization

A variety of approaches for the spatial discretization of the governing equations for

unstructured meshes have been studied. These ideas involve identification of a possi-

ble set of locations at which the flow variables are stored, the construction of control

volumes around each computational point and the details of the integration of the

fluxes in each control volume. Cell-centered or cell-vertex schemes have been tra-

ditionally used within the aerodynamic community for compressible flow equations.

The first trade-off between these two approaches is between a better representation of

the flow field versus the increased cost of memory due to the fact that on triangular

and tetrahedral meshes, the number of cells is usually larger than the number of ver-

tices by a factor of approximately six. The use of cell-vertex schemes requires special

treatment along boundary edges/faces to compute the fluxes which is circumvented

in cell-centered schemes through the use of ghost/halo cells behind the boundary.

The best choice of a control volume for unstructured meshes is not entirely clear.

Typical choices for cell-vertex schemes include the median dual, the centroid dual

and the Dirichlet tessellation of domain (figure 2.1). Most numerical algorithms on

unstructured meshes use either the medial dual or the centroid dual mesh for the

construction of the spatial discretization operator [33], [34]. The use of cell-centered

schemes leads to the natural choice of control volumes which are the triangles around

each control point.

Another important consideration while choosing the control volumes is the ability


of the spatial discretization operator to integrate a linear variation of the flow and/or

flux variables exactly. This property of the spatial operator, usually called linearity

preserving, guarantees that the order of accuracy of the scheme is preserved on an

irregular mesh, a highly stretched mesh or an adapted mesh. Using a median or a

centroid dual mesh and a Green-Gauss integration around the control volume can be

shown to be equivalent to a Galerkin discretization of the gradient on linear elements

which is known to be linearity preserving.

The computer programs that implement cell-vertex schemes on unstructured meshes

can utilize some of the geometric properties of the triangular and tetrahedral tessel-

lations. Figure 2.2 shows a two dimensional triangular grid and the control volumes

surrounding nodes P and Q, which are formed as the union of the triangles meeting

at P and Q. These control volumes share a common edge SR which is an internal

edge for the control volumes surrounding S and R. Thus the flux across SR only af-

fects the vertices P and Q. Similarly every internal edge only influences two nodes,

and hence the accumulation of the flux balances of all the nodes can be performed

by looping through the edges of the mesh and distributing the flux across each edge

to the two nodes it influences. A similar method can be used in three dimensions

(figure 2.3) where two vertices (4 and 5) share a common face 123, and hence the

flux balances can be accumulated by looping over the faces and transferring the flux

across each face between the two vertices it influences. By grouping the umbrella of

faces around each edge as illustrated in figure 2.4, the accumulation of the fluxes in

three dimensions can also be reformulated as a loop over the edges in which the flux

is transferred between the two vertices joined by each edge. This is equivalent to the

use of the median dual as the control volume for each edge.

2.4 Staggered Meshes

Researchers working in the area of incompressible flow have traditionally approached

the numerical solution of the governing Navier-Stokes equations in a different way.

To satisfy the constraint of a divergence-free flow field for incompressible flow, they

interpret the role of the pressure in the momentum equations as a projection operator


Dirichlet region

Median Dual

Centroid dual

2

4

3

5

6

O

1

Figure 2.1: Dual mesh representation of the control volume

7

6

54

3

21

8R

S

Q

P

910

11

12

Figure 2.2: Nodal formulation of the finite volume scheme


1

2

3

4

5

� � � � � � � � � � � ��

� � � � � � � � � � � ��

Figure 2.3: Evaluation of fluxes in three dimensions

Figure 2.4: Control volume for cell-vertex schemes in three dimensions


that projects a given velocity field onto a divergence free field. Fractional step or time-

split method are the most popular among these methods. The pressure field that

leads to a divergence free velocity field is typically obtained through the solution of

the Poisson’s equation. This numerical scheme is typically implemented on staggered

meshes (figure 2.5) that store the velocity at the cell faces (u on the j faces and v on

the i faces) and the pressure at the cell center. One of the prime motivation for this

approach is to reduce the decoupling between the velocity and the pressure terms [69]

and hence a decrease in the amount of numerical diffusion required to stabilize the

scheme. While these methods involve no over-head for cartesian meshes, the use

of curvilinear meshes requires storage of both velocity components at each edge to

implement finite volume schemes. Other disadvantages of this approach are that some

of the velocity components are not defined at the boundaries and extension to higher

order is difficult.

Another approach, the use of a half-staggered mesh (figure 2.6) offsets some of

these disadvantages while permitting better coupling between the velocity and pres-

sure fields. In this scheme, the velocity components are stored at the vertices of the

cell and pressure is stored at the cell-center. This allows for the momentum equa-

tions to obtain a pressure distribution around each node and the Poisson equation

for pressure in each cell to be influenced by the velocity at the corners of the cell.

When used in the context of finite volume schemes for hyperbolic equations, the half-

staggered arrangement retains its advantages for curvilinear grids but it is still hard

to implement it on an unstructured grid.

In the next chapter, some results obtained by using a half-staggered arrangement

are presented for two-dimensional flow around airfoils and compared with results from

cell-centered and cell-vertex schemes. Although the estimates of lift and drag from the

different schemes were within acceptable engineering limits, the half-staggered scheme

resulted in pressure distributions that exhibited large errors, especially around the

leading edge. The cause of this discrepancy is not clear and warrants further study.


v

u

p v

vv

u

pv

u

pv

u

pv

u

p

vv

u

pv

u

pv

u

pv

u

p

vv

u

pv

u

pv

u

pv

u

p

u

u

vp

u

u

vp

u

u

p

u

v

Figure 2.5: Staggered arrangement of flow variables

u,v u,v

p

u,v

p

u,v

p

u,v

u,v u,v

p

u,v

p

u,v

p

u,v

u,v u,v

p

u,v

p

u,v

p

u,v

u,v u,v

p

u,v

p

u,v

p

u,v

u,v u,v u,v u,v u,v

p

Figure 2.6: Half-staggered arrangement of flow variables


2.5 Implementation of the Cell-Vertex Scheme

As no immediate benefit was observed from using the staggered arrangement, a cell-

vertex scheme was used in this study for the implementation of the finite volume

scheme on an unstructured tetrahedral mesh. Median-dual mesh cells constructed

from planes bisecting each edge of the mesh are used to accumulate the fluxes at each

node. Boundary conditions are then enforced along the triangular faces that lie on the

boundary to account for the one-sided control volumes for the nodes on the boundary.

The rest of the discussion in this section outlines the details of the implementation of

the spatial discretization operators when used with artificial compressibility methods,

the evaluation of the numerical diffusion terms, and the multigrid algorithm.

2.6 Artificial Diffusion

Numerical diffusion schemes for the solution of transonic and supersonic flows received

wide attention in the late seventies and early eighties. Numerous research efforts dur-

ing this period led to the development of a mathematical framework to add numerical

diffusion to the discretized governing equations with an emphasis to produce shock

profiles that were free of oscillations. This mathematical frame-work can be inher-

ited for incompressible flows that use the artificial compressibility method with some

modifications that limit the amount of numerical diffusion. Local Extremum Dimin-

ishing (LED) schemes that guarantee that new extrema are not generated during the

evolution of the solution have proven to be robust and efficient. These schemes limit

the reconstructed solution and fluxes at cell interfaces by using limiters that can be

constructed from gradient information from a stencil of points around each computa-

tional point. The JST scheme [32] has been widely proven to be a robust frame-work

for numerical diffusion. This scheme can be represented as

dj+ 12

= ε2j+ 1

2∆wj+ 1

2− ε4

j+ 12

(∆wj+ 3

2− 2∆wj+ 1

2+ ∆wj− 1

2

).

When used for problems with embedded supersonic regions, the above scheme

switches to a locally first order scheme to prevent oscillations. For incompressible


flows, the first order term is dropped and the higher order diffusion term is retained to

provide background smoothing. Another choice of numerical diffusion operators based

on the SLIP constructions [36] introduces flux limiters to provide a high resolution

scheme without oscillations. In these schemes, a limited average of the flow variables

is used to construct a flux limiter which is then introduced as an anti-diffusion term

along with a first order diffusive term. A variety of choices exist for the form of the

limited average and the JST scheme can also be rewritten under the class of SLIP

scheme for a particular choice of the limited average.

2.7 Analysis of Artificial Compressibility

In Equation (2.2), Γ is called the artificial compressibility parameter due to the anal-

ogy that may be drawn between the above equations and the equations of motion for a

compressible fluid whose equation of state is given by p = Γ2ρ. Thus, ρ is an artificial

density and Γ may be referred to as an artificial speed of sound. When the tempo-

ral derivatives tend to zero, the set of equations satisfy precisely the incompressible

Euler equations, with the consequence that the correct pressure may be established

using the artificial compressibility formulation. The preconditioning matrix, P , may

be viewed as a device to create a well posed system of hyperbolic equations that are

to be integrated to steady state along lines similar to well established compressible

flow Finite Volume formulations. In addition, the artificial compressibility parameter

may be viewed as a relaxation parameter for the pressure iteration.

The eigen values of the system of equations in equation (2.1) are given by

λ1 = U, λ2 = U, λ3 = U + a, λ4 = U − a,

where,

a2 = U2 + Γ2(ψ2 + η2 + ξ2),

and

U = uψ + vη + wξ.


The terms ψ, η, ξ represent the slopes of the characteristic system of waves, and are

arbitrary and defined.

The choice of Γ is crucial in determining the convergence and stability properties of

the numerical scheme. Typically, the convergence rate and stability of the scheme are

dictated by the slowest system waves and the stability of the scheme by the fastest.

In the limit of large Γ, the difference in wave speeds can be large. Although this

situation would presumably lead to a more accurate solution through the penalty

effect in the pressure equation, very small time steps would be required to ensure

stability. Conversely, for small Γ, the difference in the maximum and minimum

wave speeds may be significantly reduced, but at the expense of accuracy. Thus a

compromise between the extremes is achieved by choosing Γ to be

Γ2 = C(u2 + v2 + w2),

where C is a constant of the order of unity. In regions of high velocity and low

pressure where suction occurs, Γ is large to improve accuracy, and in regions of low

velocity, Γ is correspondingly reduced.

2.8 Time Integration

Under these assumptions on the choice of the preconditioner, P , the application of the

Finite Volume method for a cell-vertex scheme results in a set of ordinary differential

equation for each node of the computational mesh,

d

dt(Viw) + PQi = 0, (2.6)

where Vi is the volume around each node and Qi represents the flux through the faces

of the control volume. To prevent odd-even decoupling at adjacent nodes which may

lead to oscillatory solutions, a dissipation term is added to the flux calculation to

modify the above equation to

d

dt(Viw) + P [Qi −Di] = 0, (2.7)


ord

dt(Viw) +Ri = 0, (2.8)

where Ri is the residual at each node in the computational mesh.

The resulting system is integrated in time using an explicit multistage scheme with

coefficients that maximize the stability region of the time-stepping scheme. To further

accelerate convergence to steady state, local time-stepping and residual averaging

techniques are used. Detailed numerical analysis of the spatial discretization and the

time stepping scheme can be obtained from the following references [32], [36], [67].

2.9 Multigrid Acceleration

Multigrid techniques are widely used to accelerate the convergence of a system of equa-

tions to steady state. A general framework for the development of full-approximation

multigrid methods for non-linear equations can be outlined as follows.

Consider,

Lu = F ,

discretized on a mesh with spacing h as

Lhvh = Fh.

This can be rewritten as

Lh(vh + δvh) = Fh,

where δv represents a correction to the present estimate vh or

Lhδvh +Rh = 0, (2.9)


where Rh is the residual. On the coarse grid, the above equation can be replaced by

L2hδv2h + Ih2hRh = 0, (2.10)

where Ih2h represents the aggregation or restriction operator. The correction to the

present fine grid solution can be represented as

vnewh = vh + I2h

h δvh,

where I2hh represents an interpolation operator. We can add and subtract the following

from equation (2.10)

L2hv2h −F2h = R2h,

to get

L2h(v2h + δv2h)−F2h + Ih2hRh = 0.

This leads to the full approximation scheme (FAS)

L2h(v+2h)−F2h + Ih

2hRh −R2h = 0.

Then

v+h = vh + Ih

2h(v+2h − v2h).

For unstructured grids, the nature of the grids to be used in the multigrid cycle is a

question of ongoing debate. In the present work a series of non-nested meshes are used

for the multigrid cycle and the solution and residual from each mesh are aggregated to

the coarser mesh while interpolating the correction from the coarser to the fine mesh.

Detailed descriptions of the multigrid scheme can be obtained from [37]. Each mesh

of the multigrid cycle was separately generated by a grid generator (MESHPLANE).

A detailed description of the multigrid scheme can be obtained from [37]. The

initial solution from a particular mesh is advanced in pseudo-time to obtain new


n1

n2

n3

a1

a2

a3

p

Figure 2.7: Interpolation coefficients for use in the multigrid cycle

estimates of the flow variables. On transfer to the next level of the multigrid, the

solution for the coarse grid mesh points are interpolated from the four nodes of the

fine mesh tetrahedron that contains this node. Further, the accumulated residual

at each fine mesh point is distributed to each node of the tetrahedron in the coarse

mesh that that encloses the fine mesh node. The interpolating factors for each node

are computed from weights which are based on the volume included by a given node

and opposite face of the tetrahedron (figure 2.7). This reduces to a second order

interpolation scheme on equilateral tetrahedra and has been found to be sufficient

for the present calculations. The solution that is transferred to the coarse mesh and

the estimate of the residuals from the fine mesh are used by the coarse mesh to

remove/convect error terms in the residuals that can be most efficiently tackled by

the coarse mesh. Further levels in the multigrid cycle involve the same operations

are before, thereby using grids that are coarser and coarser to convect the error

terms out of the computational domain faster. The ascend of the multigrid cycle

estimates a correction from each grid which is then interpolated to the fine mesh

points (figure 2.8). The corrections from the coarser mesh are transferred using

similar interpolating factors as for the aggregation operations. Multigrid cycles which

progress in the shape of a W have been known to provide faster convergence to steady

state than the V cycle.


fine grid nodes

coarse grid nodes

� � ��

� � ��

Figure 2.8: Transfer of solution, residuals and corrections between the fine and coarsemesh

2.10 Parallel Implementation

To make use of the availability of parallel computing facilities, the numerical scheme

and the computational methodology described in the previous sections were imple-

mented in a computer program that used the Message Passing Interface (MPI) stan-

dard to enable parallel computations. The rest of the this section describes the parallel

implementation of the flow solver on an unstructured tetrahedral grid.

2.10.1 Domain Decomposition, Load Balancing

Computational tests performed by Jameson [38] showed that the use of a domain

decomposition algorithm reduced the stride in the numbering of the vertices at each

edge of the mesh resulting in a reduction of computational times by a factor of three.

The domain decomposition algorithms used in this study was a modified form of

the coordinate bisection method that led to sub-domains with approximately equal

number of computational nodes (figure 2.9).

To construct a partition of the computational mesh for parallel implementation,

the above mentioned domain decomposition method was reused and the resulting

sub-domains were distributed among the available processors while balancing the


9

8

7

6

54

3

10y

x

16

15

11

12

1314

21

Figure 2.9: Domain decomposition of a rectangular region using a bisection method

load among them. This methodology worked well for the problems in this study.

Extensions to viscous flows will require more sophisticated graph partitioning algo-

rithms that distribute the computational nodes equally to all the processors while

minimizing the number of edges that are shared between the processors.

A set of sub-domains which is produced by the above partitioning method is

distributed among the set of available processors by considering a combination of

computational complexity for each domain along with the cost of communication

across processor boundaries. No further attempt was made to redistribute the points

within each sub-domain to minimize the cost of communication arising from edges

that are shared across processor boundaries. Once the partitions are distributed

among the processors, data structures that allow for the exchange of information

along processor boundaries are constructed. As the flow solver uses an edge-based

data structure to accumulate the fluxes at each vertex, the edges surrounding the

nodes that lie within a partition are accumulated. If an edge connects a point across

processor boundaries, this edge is duplicated in the two processors and ‘halo’ nodes

are constructed for both processors. This idea is illustrated further in figure 2.10.


inter-processor boundary

edges that are shared between processors

Figure 2.10: Halo nodes and the distribution of edges along processor boundaries

2 4 6 8 10 12 14 162

4

6

8

10

12

14

16

Number of Processors

Spe

edU

p

Actual SpeedUpIdeal SpeedUp

Figure 2.11: Speedup from the parallel implementation


2.10.2 Parallel implementation of the multigrid algorithm

The use of multigridding techniques for flow analysis necessitates the need to ex-

change information between a fine and coarse grid point and vice-versa. While the

residuals from the fine grid points need to be accumulated at the coarse grid points,

the corrections from the coarse grid points need to be transferred back to the fine grid

points. In this study a non-nested approach to multigridding has been used and the

sequence of meshes are repeatedly generated from a mesh generator. Hence, efficient

methods have to be used to generate the interpolation coefficients by identifying the

cell in the fine grid or coarse grid that contains a given point.

A ‘naive’ implementation of the point search algorithm results in an algorithm

with complexity kO(n2), where n is the number of nodes in the computational mesh.

This is clearly not acceptable for computational meshes where the number of nodes

are typically of the order of a few hundred thousand. Octree-based search algorithms

are known to be efficient for such problems. The computational complexity of an

octree based search algorithm is O(log(n)). Due to the superior performance of the

octree-based search routine, this method was implemented to perform point searches.

To implement the octree searches, a tree data structure to hold the octants and

its extents is first determined for each mesh. Each octant is allowed to hold a certain

number of points. Once an octant contains more than an user specified set of points,

then the octant is sub-divided to create 8 new octants. Using this data structure, a

given point is identified within an octant and the node closest to a point in this octant

is determined. The cells that meet this node are checked to see if they contain the

search point. The efficiency of the octree method lies in its ability to quickly localize

the search process to a small region of the computational mesh.

The octree based search routine has been found to be very useful for the imple-

mentation of the non-nested multigrid methods as the major cost during the pre-

processing step is associated with the point search routines to compute the inter-

polation coefficients between successive meshes. Using the octree data structure,

interpolation coefficients between a fine grid node and the next coarser mesh in the

multigrid cycle is constructed in a pre-processing step. Further, to reduce the cost

communication across processors during each multigrid cycle, sub-domains on the


coarser grids are constructed and distributed to the processors so as to conform to

the division and distribution of the fine mesh. Typical computational times for the

building of the octree and the subsequent point search algorithms are in the range of

a few minutes for a mesh sequence containing a million nodes.

An alternate implementation of the octree based point search algorithm was also

implemented and found to very efficient. This method uses the sub-domains created

by the domain decomposition algorithm to identify the domain that contains the given

search point. Then the cells in that sub-domain are queried to see if they contain the

given point. As the sub-domains are already created for load balancing, this approach

eliminates the need to construct the octree data structure. Hence the computational

time for this method was comparable to that of the octree based searches and it is

exteremely trivial to implement.

2.10.3 Speedup of the Parallel Implementation

Several test cases were used to test the implementation of the parallel flow solver.

Figure 2.11 shows the typical speed-up observed for these cases. The meshes in

these studies typically contained about half a million nodes in the fine mesh. Due

to the reduction in the stride of the node numbering for the edges in each sub-

domain, more than linear speed-up was observed for 4 and 8 processor runs. However,

as more processors we employed the domain decomposition algorithms resulted in

partitions that were sub-optimal leading to increased communication cost among the

processors thereby resulting in less than linear speed-up. Hence, there is a definite

need to improve the domain decomposition algorithm for larger problems. Most of

the calculations for this study were performed using 8 processors.

2.11 Governing equations and analysis of the struc-

tural model

In order to determine the static or dynamic displacements of a structure under the

action of external and/or body forces the elasticity equations need to solved. For a


general three dimensional structure these equations involve fifteen unknowns (three

displacements, six stresses and six strains) and hence a set of fifteen equations are used

to describe the state of the structure. Under varying assumptions of the properties of

the model, these equations can be simplified and the principle of virtual work along

with the ‘Unit Displacement Theorem’ can be used to lay the foundation for the finite

element method which has become the choice for most structural analysis problems.

For a general three dimensional structure the components of strain can be related

to the displacement field through he following equations.

εii =∂ui

∂xi

, εji = εij =∂ui

∂xj

.

Now, the strains can be related to the stresses and if the material is assumed

to be isotropic, the stress-strain relations are related through the Young’s modulus

and the Poisson’s ratio. Using the equilibrium equations, the elasticity equations in

three dimensions can be formulated. To arrive at a set of discrete set of equations,

a Galerkin formulation can be used to reduce the strong form of the problem into

a weak formulation. This involves multiplying the equilibrium equations by a shape

function and integrating over the domain of interest and reducing the integral using

integration-by-parts. To numerically solve this set of equations, the domain of interest

is broken into finite elements which are typically triangles or quadrilaterals. Assuming

a particular form of the shape function (linear for Galerkin formulations) along with

a basis to represent the displacement field leads to a set of linear equations relating

the unknown displacement field and the external forces. This set of linear equations

usually assumes the form

[k]x = F,

where k is the global stiffness matrix, x is the displacement vector under investigation

and F is the external force at the nodes at which the displacement is sought. The form

of the stiffness matrix is dependent on the nature and type of elements used to repre-

sent the body. While triangular elements allow for ease of representation of arbitrary

geometries, they usually turn out to be stiffer than quadrilateral elements. Higher


order approximations for the variation of the displacements and strains within in each

element can also be used to capture sharply varying changes in the displacement field.

Most commercial finite element packages provide means to choose elements and the

basis functions to approximate the displacement field. Triangular or quadrilateral ele-

ments with a linear basis are typically chosen for most problems. Further, procedures

to assemble the stiffness matrix and solve the resulting system of equations are also

provided, thereby allowing the structural analysis package to be used as black-box

that dumps out the displacements for a given loading.

2.11.1 Structural Model of the Sail

The sail cloth was discretized into finite elements (after neglecting the presence of

batten pockets) with a set of quadrilateral membrane elements with four nodes. These

elements withstand all external forces through tension and are essentially incapable

of resisting bending moments. The translational and rotational degrees of freedom

along the foot of the main sail was suppressed. Along the mast, the translational

degrees of freedom was inhibited while allowing for rotational motion. For the head

sail, the point of attachment of the foot to the rig was constrained. The leech of

the main and head sail were allowed to move freely to induce a geometric twist

due to the aerodynamic loading. The mast was assumed to be rigid during the

structural and aeroelastic calculations. The presence of battens and tension cables

and other structural elements of the sail rig was neglected from this analysis. The

linear system of equations relating the displacements to the force field was advanced to

a steady state by a iterative process that incrementally added the load while obtaining

a converged displacement field for each step. This non-linear model to predict the

deflected shape of the sail was included in anticipation of large deflections of the sail

geometry. Wrinkling of the structure, which is an important consideration especially

around the leading edge (luff) and at the sail tip, is not anticipated by this model

but the use of methodology to large deformations allows for wrinkling models to be

included at a later stage.

For Americas Cup sails it is important to account for the yarn layout as it gives rise


Leech is allowed

to move freely

Translational degrees of freedom

suppressed at the mast

Mast is assumed to be rigid

Translational and

rotational degrees of freedom suppressed at the boom

Figure 2.12: Boundary conditions for the main sail

to anisotropy in the material properties of the sail. Hence the assumption of isotropy

that is used in the present study is not completely realistic. Also the absence of

battens would cause the aeroelastic procedure to over-estimate the deflections.

2.12 Aeroelastic Coupling and Mesh Deformation

The pressure loading from the flow solver is fed to the structural analysis to estimate

the deflected shape of the sail. To enable the transfer of loads and displacements to

be conservative, the fluid mesh on the surface and the structural mesh were made

identical, eliminating the need for interpolation. The deflected shape of the sail is used

to deform the computational mesh. The popular ‘spring-analogy’ method was used to

track the mesh deformations. While this method was restrictive in terms of the nature

of the deflections and quality of the deformed mesh, it provides a simple tool to track

mesh deformations. Another method which provided increased robustness relied on

solving the elasticity equations within the computational domain of the fluid to predict

the mesh deformations. A number of authors have reported successful implementation


the stayfreedom supressed alongTranslational degrees of

All degrees of freedom suppressed

mast

clew foot

leech

luff

head

stay

Figure 2.13: Boundary conditions for the head sail

of this method for aeroelastic problems, adaptive refinement techniques and in surface

propagation problems. The deformed mesh is then used to compute a new pressure

loading for the sail. This iterative process offers no guarantee of convergence but

typically predicts the deflected shape to reasonable accuracy in a few steps (typically

4-7 for sail geometries).

Mesh deformation techniques in aeroelastic computations are typically posed as

problems in structural mechanics. The most popular ‘spring-analogy’ method de-

termines the new position of the nodes of the computational mesh by imposing the

boundary displacements as initial conditions and solves for the equations of static

equilibrium for each node. To model the computational mesh as a structural mem-

ber, a stiffness is associated with each edge of the mesh. This stiffness is typically

inversely proportional to the length of the edge. This allows for control of points

which are bunched close to each other.

The spring method can be mathematically conceptualized as solving the following

equation


∂∆xi

∂t+

N∑j=1

Kij(∆xi −∆xj) = 0,

where the Kij is the stiffness of the edge connecting node i to node j and its value

is inversely proportional to the length of this edge, ∆xi is the displacement of node

i and ∆xj is the displacement of node j, the opposite end of the edge. The position

of static equilibrium of the mesh is computed using a Jacobi iteration with known

initial values for the surface displacements.

The spring method has been known to either degrade the quality of the mesh or

produce ‘inverted’ meshes when the boundary deformations are not small. To over-

come some of these failings, some researchers have used the spring-analogy method

in conjunction with edge-swapping routines to ensure that the quality of the mesh

does not degrade during the computations.

A more robust mesh movement scheme that overcomes this limitation can be

constructed by modeling the domain as an elastic solid and solving the equilibrium

equations for the stress field. In terms of the displacement vector u the strain tensor

can be written as

εij =1

2

(∂ui

∂xj

+∂uj

∂xi

), i, j = 1, 2, 3.

For an isotropically elastic solid the stress tensor is defined as

σij = λεkkδij + 2µεij,

where λ and µ are the Lame constants, δij is the Kroneckar delta and the summation

convention has been invoked. If there is no distributed body force the stress field

satisfies the equation∂σij

∂xij

= 0.

Dividing by the shear modulus µ leads to an equation that depends only on the

parameter λ/µ. Alternatively, one can introduce Poisson’s ratio ν λ2(λ+µ)

and consider

this to be the user defined parameter. It is again possible to increase the rigidity


of the mesh in regions of small element size and/or bad element aspect ratio, by

modifying the coefficients λ and µ.

Further research needs to be performed to identify the optimal mesh deformation

technique for aeroelastic calculations of sail geometries.

Chapter 3

Analysis of Sail Configurations

This chapter presents the results obtained from using the flow solver and the aeroe-

lastic package on sail geometries. The incompressible flow solver is used to study

the effect of the mast, apparent wind angle and heel on the aerodynamic perfor-

mance of the sail configurations. Aeroelastic simulations were performed to study the

importance of sail elasticity on the pressure distribution over the sails.

The behavior of the finite volume scheme for the compressible flow equations

in the low Mach number regime and at high angles of attack is analyzed first. As

representative examples, multi-element airfoil configurations and three dimensional

sail computations are performed to test the robustness and accuracy of the numerical

scheme. After verifying that these simulations provide good engineering estimates, the

artificial compressibility correction is next introduced, and it is then used to study

airfoils in two dimensional flow and single and multi-element sail configurations in

three dimensional flow. Finally the flow solver is coupled to a structural model to

obtain steady deflected shapes of the sail configurations.

46

CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 47

3.1 Low Mach number, high angle of attack simu-

lations with a compressible flow solver

3.1.1 Multi-Element airfoils

The operational conditions of a high lift system provide an aerodynamic analogy to

the thrust producing mechanism of sails in close-hauled conditions. In order to vali-

date the numerical scheme, a three element airfoil configuration was analyzed using a

two dimensional version of the finite volume scheme on unstructured grids. The grids

were generated using a mesh generator which uses Delaunay criterion to triangulate a

set of points. The grid shown in figure 3.1 contains 20,000 triangles in the fine mesh.

The finite volume scheme is implemented on a dual of the underlying Delaunay trian-

gulated mesh with modified Runge-Kutta time stepping schemes, residual averaging,

the JST scheme for the numerical diffusion and multigrid techniques to get to steady

state. The pressure distribution over the three elements is shown in figure 3.1. The

distribution of entropy on the surface is shown using normals in figure 3.1. As ex-

pected, the trailing edge of the slat, main and the flap show larger entropy than

the remainder of the section. The overall pressure distribution follows engineering

intuition and compares well with a structured grid flow solver, FLO103 [39]. The

suction peaks around the leading edge of the main, flap and the slat are critical to

the production of high lift and are recovered well by the numerical scheme. Quantities

of engineering interest (lift and drag) typically converge in under hundred iterations.

The results of the tests with this configuration confirmed the robustness of the nu-

merical scheme for low Mach number, high angle of attack problems on unstructured

grids.

3.1.2 Sail simulations

Low mach number compressible flow simulations over a sail geometry were performed

next. Simulations were performed for Mach numbers up to 0.1 and angles of attack

ranging from 8 to 20 degrees. The numerical scheme was robust within this range of

flow conditions. However, further reduction in the Mach number required increased


levels of numerical diffusion to arrive at converged solutions. Figure 3.2 shows the Cp

distribution at two sections along the sail and also shows typical convergence rates for

this geometry. The presence of the mast induces a sharp leading edge suction peak

and a strong pressure gradient which could have adverse effects on the boundary

layer development on the upper surface of the sail. The oscillations near the trailing

edge are due to a combination of problems. The Kutta condition is not strongly

enforced in these flow simulations but is usually recovered due to the concentration

of computational points around the trailing edge. Further, as the thickness of the

sail geometry is rather small, most mesh generation programs result in a mesh with

poor quality along the trailing edge. However, it must be pointed out that the use of

artificial compressibility methods greatly reduces the magnitude of these oscillations

(see results in the subsequent sections). While the wind speeds for these calculations

are beyond the reach of most racing boats, this test re-enforces belief in the underlying

numerical scheme for sail geometries at high angles of attack.

3.2 Effect of Numerical Discretization and diffu-

sion on artificial compressibility methods

As discussed in the previous chapter, it is possible to implement the numerical proce-

dure that integrates the governing equations using different arrangements of the flow

variables. In order to study the behavior of three different arrangements (cell-vertex,

cell-centered and the half-staggered scheme) an airfoil in incompressible flow was an-

alyzed. All three schemes used the same numerical scheme, namely the finite volume

scheme, artificial compressibility corrections, second order construction of the convec-

tive fluxes, SLIP construction for the numerical diffusion and Runge-Kutta multistage

time integration schemes. For the half-staggered arrangement, at the solid wall, flow

tangency and pressure boundary conditions were applied. All three approaches used

a ‘vortex-corrected’ far-field boundary condition. To compare the pressure distribu-

tions obtained from these simulations, potential flow solutions were obtained using

FLO1 (figure 3.3). The deviation of the computed solution on the airfoil surface from


the Bernoulli’s equation was recorded for each test and used as a basis for comparing

the three flow solutions. The pressure distribution for the cell-centered, cell-vertex

and the half-staggered arrangement are shown in figures 3.4, 3.5, 3.6 respectively. The

error in satisfying the Bernoulli’s equation is shown in figures 3.7, 3.8, 3.9, 3.10. It can

be seen from these plots that the cell-centered scheme has the least error, followed

by the cell-vertex scheme. The half-staggered arrangement seems to display large

errors near the leading edge stagnation point, a feature that is less prominent for the

cell-vertex and the cell-centered schemes. Further research needs to be performed

to study the cause of this behavior as it seems to contradict the popular belief that

half-staggered arrangements lead to less decoupling between the pressure and veloc-

ity components thereby yielding more accurate solutions. The cell-vertex scheme has

advantages over the cell-centered scheme when used with three dimensional unstruc-

tured grids and as the difference in the accuracy of the solution was not appreciable,

cell-vertex schemes were used for the subsequent problems in this study.

3.3 Validation of the parallel implementation

Parallel implementation of the flow solver was tested against solutions from the single

processor version of the solver and previously existing solutions from other computa-

tional programs (FLO87).

A variety of geometries ranging from wing to complete aircraft configurations were

analyzed with the compressible version of this parallel program. The meshes were

generated using MESHPLANE, an unstructured grid generator which uses the Delau-

nay criterion to connect points in the field to form tetrahedra. Single grid calculations

typically take 300 to 400 flow cycles to reach a converged solution on a mesh with

350000 nodes. Coarser levels of the multigrid cycle were regenerated with MESH-

PLANE by typically halving the number of nodes in the mesh. The triangulation of

the surface was retained in the coarser meshes to recover the geometry but it was

found to be not so critical for some of the problems. Typically, 3 levels of multigrid

were used for the aircraft configurations and converged solutions can be obtained in

about 60-70 multigrid cycles. The results for these simulations compare well with


the results from a single processor version of the program thereby establishing the

robustness of the parallel implementation of the flow solver.

3.4 Single and multi-element sail computations with

artificial compressibility methods

Having identified a suitable numerical discretization scheme for the artificial com-

pressibility method and validated the parallel implementation of the flow solver, the

flow around sail configurations was analyzed to obtain estimates of lift, drag and

heeling moment along with detailed pressure distributions at various heights of the

sail. The aim of this study was to characterize the performance of the sail for a va-

riety of conditions, thereby allowing the designer to judge the quality of the design.

Simulations were performed with the main sail alone and for a head and main sail

combination.

The computational domain typically extended 10 body lengths in all three co-

ordinate directions. For computations with the main sail alone, the foot of the sail

coincided with the symmetry plane and for computations with the head and the main

sail, the symmetry plane was off-set from the sail geometries. Twisted inflow condi-

tions were prescribed at the inlet to simulate the twisted boundary layer profile of

the incoming air-stream. The meshes were generated with MESHPLANE. Typically,

around 2 million cells were used for the single sail computations and around 4 million

cells were used for the head and main sail combination. Meshes for the coarser levels

in the multigrid cycle were regenerated using MESHPLANE and typically the number

of cells/nodes in the mesh was halved for each coarser mesh. For all the simulations a

W cycle was used for the multigrid calculations. 75-100 multigrid cycles were needed

to obtain converged estimates of the lift and the drag.

3.4.1 Characteristics of the main sail

This section discusses the performance of the main sail. Figure 3.12 shows the pressure

distribution at 3 sections along the height of the sail. The pressure distribution near


the foot exhibits a sharp peak around the leading edge of the upper surface. However,

due to the absence of a gap between the foot of the sail and the free-surface, the results

along the lower sections will at best be qualitative. Typically, these sections do not

produce much lift due to the tendency of the flow to equalize the pressure on the

upper and lower surface by using the gap between the foot of the sail and the deck.

Along the mid and upper sections of the sail, the flow enters the sail at the optimum

angle suggesting that the twist of these sections have been aligned with the incoming

flow. The flow is smooth through the remainder of the sail planform with a mild

deviation near the trailing edge of each section. As the current numerical scheme

does not explicitly impose the Kutta-condition but hopes to recover it through the

distribution of points in the field and the geometry, the flow does not pass smoothly

over the trailing edge. Further, the presence of a blunt trailing edge attenuates the

problem while also clouding the physics of an inviscid flow around a sharp trailing

edge. Figure 3.13 shows the distribution of the span-wise force coefficients. The sail

sections operate at roughly the same lift coefficient from the foot to the tip. While this

is desirable in light wind conditions when the heeling moment produced by the sail

can be stabilized by the allowable ballast weight, it is not so desirable in heavy wind

conditions. As the sails sections have been twisted to account for the upwash created

by the bound vortex, the occurrence of tip induced stall has been reduced allowing the

sail sections to stall at the same time. On the flip side, the uniform loading produced

by the main sail results in a large tip vortex and hence an associated increase in drag.

The forces generated by the main sail under a range of close-hauled incident wind

angles is shown in figure 3.14. This figure exhibits the typical behavior of sails sailing

towards the wind. Maximum lift coefficients of around 1.6 at 22.5 degrees of wind

incidence with an associated L/D of 8.83 are typical of the sails used in Americas

Cup.

To study the effect of the mast on these simulations, the above experiments were

repeated with a mast. The mast was assumed to have an elliptic cross-section and

was oriented with the tangent to luff at each section to minimize the influence of

the mast. The major axis of the mast was 8 inches long and the minor axis was 1

inch wide. Figure 3.15 shows the effect of the mast with increasing angle of attack.


Until about 10 degrees the effect of the mast is not very pronounced. At higher

angles of attack, the presence of the mast induces higher lift and drag coefficients.

Experimental studies by Milgram [12] shows that the presence of the mast could

increase the form and pressure drag of the sail to match the induced drag depending

on its shape and orientation to the incoming air-stream. The interaction of the mast

with the pressure distribution on the sail is strongly influenced by viscous effects

originating from the development of the boundary layer in adverse pressure gradients

and hence the current study is only able to a qualitative picture of the flow physics.

The performance of the sail usually degrades as it heels mainly due to the inter-

action of the sail with the free surface of the sea and the change in the sail trim to

the incoming air-stream. Due to restrictions on the weight of the ballast, most high

performance yachts sail upwind at a heel angle while paying the associated loss in lift

and drag. Numerical experiments were performed to study the effect of the heel with

an aim to identify the maximum allowable heel angle. Figure 3.16 shows the lift and

drag at various apparent wind angles for two different heel angles. This figure shows

that the reduction in lift and increase in drag is more pronounced at higher incidence

angles and can be as high as 15 %.

3.4.2 Characteristics of the Head and Main sail combination

A head and main sail combination [13] was analyzed next. The planform and section

characteristics of the main and head sail are shown in figures 3.17 and 3.18. It can

be seen that the head sail has a triangular planform while the main has an elliptic

chord distribution. The twist distribution of the head and the main sail increases

from the foot to offset the upwash created by the other sail. Further, the maximum

camber and it position also increases from the foot to the head of the sail. Sails are

usually designed to have increasing camber towards the head to provide for favorable

pressure gradients that would delay the onset of separation or reduce the increase in

drag from a turbulent/separated boundary layer. This configuration was tested at a

heel angle of 25 degrees with the apparent wind angle of 19 degrees at a height of 10

meters along the sail. The onset flow was twisted to result in a parabolic distribution


of the velocity magnitudes.

The span-wise loading of the head and the main sail are shown in figure 3.21

and 3.22. The head sail displays a desirable pressure loading where the lift gradually

tapers towards the tip and has an elliptic distribution. This results in a weaker

trailing vortex at the tip and hence reduced drag. Both the lift and the drag have

oscillations near the foot possibly due to the gap between the foot and the symmetry

plane. The loading of the main sail displays a gradual increase towards the tip which

is an undesirable feature. This feature become more prominent above the head sail

showing that favorable pressure gradients induced by the head sail have significant

influence on the pressure profiles of the main sail sections. The presence of this region

on the main sail with large suction peaks could be detrimental to the development of

the boundary layer and possible separation. Due to the absence of a viscous model

in the present analysis it is difficult to make quantitative estimates of the loss in lift

by the main sail. One of the potential applications of a shape optimization design

procedure would be to determine the optimal shape of the main sail that either

eliminates separation or delays its onset.

The pressure distribution around the leading edge of the main sail shows the

influence of the jib. The leading edge peak has been suppressed providing a more

favorable pressure gradient leading to a reduced probability of separation and stall.

The favorable influence of the head sail allows the main sail to be set at a higher

angle of attack without flow separation and stall. The pressure distribution on the

head sail shows some undesirable features. Along the leading edge of the mid and

upper sections, the flow enters at an angle different from the optimum. This leads to

a small region of ‘inverted’ pressure profiles which could be offset by altering the twist

or camber distribution of these sections. However, due to the upwash created by the

bound vortex around the main sail, the task of identifying the optimum head sail twist

and camber is not straight-forward. Further, these simulations were performed on an

undeformed head and main sail combination and as discussed in the next section,

the aeroelastic effects cause the twist and camber distribution to be altered thereby

eliminating some of the undesirable pressure profiles on the head sail.


3.5 Aeroelastic simulations for single and multi-

element foils

The flexibility of the sail cloth and its inherent desire to wrinkle pose major obstacles

to the development of accurate computational tools to perform structural analysis.

Finite Element methods that allow for large deformations have been used to study the

structural behavior of sail shapes. As the flying shape of the sail is also determined

by the rig used to hoist the sail, a thorough analysis needs to include tension cables,

stays, kicking-straps, the flexibility of the mast and the presence of battens along wit

suitable physical description for the finite elements. To date, an analysis of this nature

has not been performed. In the present study, a simple structural model that neglects

the structural members of the rig (stays, tensional cables etc.) and the presence of

the battens. The mast was assumed to be rigid and quadrilateral membrane elements

were used to model the sail cloth. Isotropic material properties were used for the

sail cloth. Suitable boundary conditions along the mast and the foot of the sail were

prescribed and a non-linear model capable of predicting large deflections was used for

the structural analysis. Quantitative experimental data of the flying shape was not

available. Hence, the aim of the aeroelastic analysis was to estimate the nature of

the deflections and the effect of mesh deformation on both the flow solution and the

aeroelastic computation.

The aeroelastic analysis typically takes about 5-7 iterations between the flow solver

and the structural analysis program. The coupling between the two programs is weak

and convergence is assumed to be attained when the maximum deflections fall below

a particular threshold. For the computations in this section, the aeroelastic analysis

was assumed to have converged when the deflections where below the thickness of

the sail (1 mm for these calculations). The flow solution was typically converged 4

orders of magnitude for the initial solution. The flow solutions were performed using

8 processors of an SGI Origin 2000 and takes about 15 minutes for the first solution.

Subsequent flow simulations were obtained in under 2 minutes. The pressure loading

obtained from the flow solver was imposed on the structure and a non-linear analysis

methods (SOL129 in NASTRAN) was used to obtain the deflections. Typically, the


pressure loading was broken into 25 incremental steps. The structural simulations

were performed on a single processor and typically take about 2-3 minutes. The

spring method was used to deform the CFD mesh. While this method is not entirely

desirable for large deformations that are typically encountered with sail geometries,

it is extremely trivial to implement and usually takes about 10-20 seconds when

computed with multiple processors (typically 8). During the aeroelastic iterations it

was sometimes observed that the deformed mesh degrades in quality which adversely

affects the flow solution and hence the overall aeroelastic procedure. This difficulty

was reduced to some extent by breaking the deflections into incremental steps and

deforming the mesh after each step. Alternately, the mesh generation program could

be used to regenerate the mesh but this would entail transferring the flow solution

from the previous iterations using interpolation coefficients. This approach was not

explored for the results presented here.

The deflected shape of the head and the main sail are shown in figures 3.24

and 3.25. It can be seen from these plots that the lower sections of the head and

the main sail do not undergo appreciable deformation. The largest deflections occur

in the mid-sections of the main sail. Due to the absence of battens in the structural

model, it is believed that the deflections predicted by the aeroelastic procedure would

be greater than those observed on the true flying shape.

As the point of attachment of the main sail to the mast and the leading edge of

the head sail was not allowed to move, the pressure loading altered the twist of the

sail geometry. This had a favorable influence on the pressure distribution, especially

on the head sail (figures 3.26 and 3.27). The pressure distribution over the head

and sail after the aeroelastic simulation highlights the need to perform aeroelastic

analysis for sail geometries. While the lift and the drag of the deformed shape is not

significantly different from the undeformed shape, the pressure distribution over the

sail sections shows that the twist and the camber distribution can be altered by the

pressure loading that can potentially alter the flow around the head and the main

sail.


AGARD-AR-303 MACH 0.200 ALPHA 8.230

CL 3.5407 CD 0.0483 CM -0.6802

NODES 11158 NCYC 2500 RES0.530E-04

3.

00

2.00

1.

00

0.00

-1.

00 -

2.00

-3.

00 -

4.00

-5.

00 -

6.00

-7.

00

Cp +

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++

++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++

+

++

+

3.

00

2.00

1.

00

0.00

-1.

00 -

2.00

-3.

00 -

4.00

-5.

00 -

6.00

-7.

00

Cp

+

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++

+

+

3.

00

2.00

1.

00

0.00

-1.

00 -

2.00

-3.

00 -

4.00

-5.

00 -

6.00

-7.

00

Cp

+

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

++++++

Figure 3.1: Grid and Pressure distribution over a multi-element airfoil geometry at aM = 0.2 and α = 8.2 degrees


SAIL M6 MACH 0.100 ALPHA 8.000 Z 1.077

CL 1.3734 CD 0.1474 CM -0.5968

NCYC 500 RES0.615E-03

8.

00

4.00

0.

00 -

4.00

-8.

00 -

12.0

0 -

16.0

0

Cp

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

++

+

+

+

+

++

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++

+

+

SAIL M6 MACH 0.100 ALPHA 8.000 Z 2.154

CL 1.4962 CD 0.1595 CM -0.6785

NCYC 500 RES0.615E-03

8.

00

4.00

0.

00 -

4.00

-8.

00 -

12.0

0 -

16.0

0

Cp

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++

+++

+

+

+

+

+

+

+++++++++

+++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

++

SAIL M6

MACH 0.100 ALPHA 8.000

RESID1 0.278E+01 RESID2 0.615E-03

WORK 499.00

0.00 100.00 200.00 300.00 400.00 500.00 600.00

Work -12

.00

-10

.00

-8.

00 -

6.00

-4.

00 -

2.00

0.

00

2.00

4.

00

Log

(Err

or)

-0.

20

0.00

0.

20

0.40

0.

60

0.80

1.

00

1.20

1.

40

Nsu

p

Figure 3.2: Cp distribution at two sections and convergence history of the compress-ible flow solver


NACA 0012 ALPHA 0.000

CL 0.0000 CD 0.0000 CM 0.0000

GRID 256

1.

20

0.80

0.

40

0.00

-0.

40 -

0.80

-1.

20 -

1.60

-2.

00

Cp

+

+++++++++

++++++++

++++++++++

++++++++++++

++++++++++++++

++++++++++++

+++++++++++++

+++++++++++++++++++++++++++++++++++++

+++++

+

+

+

+

+

+++++

+

+

+

+

+

+

+++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+


CL 0.1206 CD 0.0000 CM -0.0013

GRID 256

1.

20

0.80

0.

40

0.00

-0.

40 -

0.80

-1.

20 -

1.60

-2.

00

Cp

+

+++++++++

++++++++

+++++++++++

++++++++++++++

+++++++++++++++

+++++++++++++++

+++++++++++++++++++++++++++++++++++++++++

+++++++

+

+

+

+++++

+

+

+

+

+

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++

+


CL 0.3617 CD 0.0000 CM -0.0040

GRID 256

1.

20

0.80

0.

40

0.00

-0.

40 -

0.80

-1.

20 -

1.60

-2.

00

Cp

+

+++++++++

++++++++

+++++++++++++

+++++++++++++++++++

+++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++

++

+

+

+

+

+

+

+

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++

+

Figure 3.3: Potential flow solution from FLO1 at 0,1 and 3 degrees


NACA 0012 - SCHEME : QUAD. CELL-CENT. RHCUSP-SLIP MACH 0.000 ALPHA 0.000

CL 0.0000 CD -0.0001 CM 0.0000

GRID 161X33 NCYC 80 RES0.834E-03

0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++

+

+

+

+

+

+++++

++

+

+

+

+

+

++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++


CL 0.1199 CD 0.0000 CM -0.0013


0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++

+

+

++++++

+

+

+

+

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++


CL 0.3598 CD 0.0001 CM -0.0039


0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++

+

+

+

+

+

+

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++

Figure 3.4: Flow over a NACA 0012 airfoil at 0,1 and 3 degrees using a cell-centeredscheme


NACA 0012 MACH 0.000 ALPHA 0.000

CL 0.0006 CD 0.0003 CM -0.0001

GRID 161X33 NCYC 200 RES0.730E-07

0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++

+

+

+

+

+

+++++

+

+

+

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++


CL 0.1200 CD 0.0003 CM -0.0014

GRID 161X33 NCYC 200 RES0.829E-07

0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++

+

+

+

+++++

++

+

+

+

+

+

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++


CL 0.3587 CD 0.0005 CM -0.0040

GRID 161X33 NCYC 200 RES0.187E-06

0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++

++

+

+

+

+

+

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++

Figure 3.5: Flow over a NACA 0012 airfoil at 0,1 and 3 degrees using a nodal scheme


NACA 0012 - SCHEME : HALF-STAGGERED MACH 0.000 ALPHA 0.000

CL 0.0006 CD -0.0001 CM -0.0001

GRID 161X33 NCYC 150 RES0.169E-06

0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++

+

+

+

+

+

+++++

++

+

+

+

+

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++


CL 0.1211 CD -0.0001 CM -0.0014

GRID 161X33 NCYC 150 RES0.212E-06

0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++

+

+

+

++++++

+

+

+

+

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++


CL 0.3620 CD -0.0001 CM -0.0040

GRID 161X33 NCYC 150 RES0.366E-06

0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++

+

+

+

+

+

+

+

+

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++

Figure 3.6: Flow over a NACA 0012 airfoil at 0,1 and 3 degrees using a half-staggeredscheme


0 20 40 60 80 100 120 140 160 180−7

−6

−5

−4

−3

−2

−1

0

1

2

3Total Pressure Error at 0 deg angle of attack

Points along the airfoil surface,Lower surface trailing edge to upper surface trailing edge

Per

cent

age

Err

or

cell−centercell−vertexhalf−staggerred

Figure 3.7: Total Pressure losses on the airfoil surface at 0o

0 20 40 60 80 100 120 140 160 180−6

−5

−4

−3

−2

−1

0

1

2

3



Per

cent

age

Err

or




0 20 40 60 80 100 120 140 160 180−6

−4

−2

0

2

4



Per

cent

age

Err

or



0 20 40 60 80 100 120 140 160 180−10

−8

−6

−4

−2

0

2



Per

cent

age

Err

or




24 m

Twisted inflowwith boundary layerprofile

Main Sail

10 m

10 m

2.3 m

Jib

Figure 3.11: Sail geometry


M2F TNZ ALPHA 18.000 Z 0.023

CL 0.7986 CD 0.2178 CM -0.3186

NCYC 200 RES0.914E-01

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00

Cp

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+++

+

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

M2F TNZ ALPHA 18.000 Z 0.336

CL 0.7677 CD 0.0882 CM -0.3240

NCYC 200 RES0.914E-01

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00

Cp

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

M2F TNZ ALPHA 18.000 Z 0.526

CL 0.7794 CD 0.0495 CM -0.3384

NCYC 200 RES0.914E-01

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00

Cp

++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+++++++

+++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Figure 3.12: Pressure distributions along sections at 1, 25 and 85 percent of the heightof main sail


0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Lift and Drag distribution along the height of the sail

Height along the span

ClCd

Figure 3.13: Spanwise force distributions

0.05 0.1 0.15 0.2 0.25 0.30.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3Variation of Lift and Drag with wind incidence

Cd

Cl

5 deg

7.5 deg

10 deg

12.5 deg

15.0 deg

17.5 deg

20.0 deg

22.5 deg

Figure 3.14: Variation of Lift and Drag with wind incidence


0.05 0.1 0.15 0.2 0.25 0.30.2

0.4

0.6

0.8

1

1.2

1.4

1.6Effect of mast on variation of Lift and Drag with wind incidence

Cd

Cl

5 deg

7.5 deg

10 deg

12.5 deg

15.0 deg

17.5 deg

20.0 deg

22.5 deg

without mastwith mast

Figure 3.15: Effect of mast on variation of Lift and Drag with wind incidence

0.05 0.1 0.15 0.2 0.25 0.30.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3Effect of heel on variation of Lift and Drag with wind incidence

Cd

Cl

5 deg

7.5 deg

10 deg

12.5 deg

15.0 deg

17.5 deg

20.0 deg

22.5 deg

no heel25 degrees heel

Figure 3.16: Effect of heeling angle on variation of Lift and Drag


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

5

10

15

20

25


twist, camber and chord distribution of the head sail

twist distribution in degchord distribution in mcamber distribution as % of chord

Figure 3.17: Twist,camber and chord distribution of the head sail

10 15 20 25 30 35 40 45 50

0

5

10

15

20


twist, camber and chord distribution of the main sail

twist distribution in degchord distribution in mcamber distribution as % of chord

Figure 3.18: Twist,camber and chord distribution of the main sail


M2F TNZ ALPHA 19.000 Z 2.587

CL 0.8872 CD 0.2378 CM -0.5607

NCYC 500 RES0.180E-07

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp

++

+++++++++++++++++++++++++++++++

++

++

+++++ + + ++

++

++

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + ++

+

M2F TNZ ALPHA 19.000 Z 9.509

CL 0.8018 CD 0.1745 CM -0.5067

NCYC 500 RES0.180E-07

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp ++

++

+++++++++++++++++++++++++++++

++

++

+++++

+ + ++

++

++

++

++ + + + + + + + + + + + + + + + + + + + + + + + + +

+

+

M2F TNZ ALPHA 19.000 Z 16.409

CL 0.5918 CD 0.1136 CM -0.4305

NCYC 500 RES0.180E-07

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp ++

++

++++++++++++++++++++++++++++

++

++++

+++

++

+ + + ++

++

++

++ + + + + + + + + + + + + + + + + + + + + + + + + +

++

Figure 3.19: Pressure distributions along sections at 1, 25 and 85 percent of the heightof head sail


M2F TNZ ALPHA 19.000 Z 3.885

CL 0.8119 CD 0.3222 CM -0.5526

NCYC 500 RES0.180E-07

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp

++

++++++++++++++++++++++++++++++++++++++++ + + + + + + +

++

++

++

++ + + + + + + + + + + + + + + + + + + + + + +

+

+

M2F TNZ ALPHA 19.000 Z 10.532

CL 0.7847 CD 0.2737 CM -0.5064

NCYC 500 RES0.180E-07

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp

++

++++++++++++++++++++++++++++++++++++++++ + + + +

++

++

++

++

+ + + + + + + + + + + + + + + + + + + + + + + + ++

+

M2F TNZ ALPHA 19.000 Z 18.929

CL 0.9173 CD 0.2608 CM -0.5521

NCYC 500 RES0.180E-07

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp

++

+++++++++++++++++++++++++++++++++++++++

+ + + ++

++

++

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + ++

+

Figure 3.20: Pressure distributions along sections at 1, 25 and 85 percent of the heightof the main sail


0 5 10 15 20 25−0.2

0

0.2

0.4

0.6

0.8

1

1.2Lift and Drag distribution along the height of the head sail


ClCd

Figure 3.21: Spanwise force distributions on the head sail

0 5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

1

1.2

1.4Lift and Drag distribution along the height of the head sail


ClCd

Figure 3.22: Spanwise force distributions on the main sail


AIRPLANE CP from -1.0000 to -0.5000




Figure 3.23: Pressure distribution over the pressure and suction side of the head andsail combination


2 4 6 8 10 12 14−0.5

0

0.5

1

1.5

2

Z = 25

Deformed and original section geometry along the height of the head sail

Z = 18Z = 14Z = 8Z = 3.6

OriginalDeformed

Figure 3.24: Original and deformed sail sections for the head sail

10 11 12 13 14 15 16 17 18 19 20−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Z = 5

Z = 8.5

Z = 14

Z = 19

Z = 24

Z = 31

Deformed and original section geometry along the height of the main sail

OriginalDeformed

Figure 3.25: Original and deformed sail sections for the main sail


M2F TNZ ALPHA 19.000 Z 2.587

CL 0.8897 CD 0.1600 CM -0.4765

NCYC 500 RES0.538E-02

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp ++

++++++++++++++++++++++++++++++++++++++

+

++ + +

+ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

M2F TNZ ALPHA 19.000 Z 9.509

CL 1.0856 CD 0.0757 CM -0.5412

NCYC 500 RES0.538E-02

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp

++

+++++++++++++++++++++++++++++++++++

++

++

++

++

++

++

++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

M2F TNZ ALPHA 19.000 Z 16.409

CL 1.0096 CD -0.0163 CM -0.5429

NCYC 500 RES0.538E-02

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp

+++++

++

++++++++++++++++++++++++++++++

+++++ + +

++

++

++

++

+ + + + + + + + + + + + + + + + + + + + + + + + + + + ++

Figure 3.26: Pressure distributions along sections at 1, 25 and 85 percent of the heightof head sail after aeroelastic analysis


M2F TNZ ALPHA 19.000 Z 3.885

CL 0.7112 CD 0.2590 CM -0.4177

NCYC 500 RES0.538E-02

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp

++++++++++++++++++++++++++++++++++++++

+++

++ + + + + + + +

++

+ + + + + + + + + + + + + + + + + + + + + + + + + + ++

+

M2F TNZ ALPHA 19.000 Z 10.532

CL 0.9060 CD 0.2505 CM -0.4739

NCYC 500 RES0.538E-02

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp

+++

++++++++++++++++++++++++++++++++++++

+

+

+

+ + + + + + + ++

+ + + + + + + + + + + + + + + + + + + + + + + + ++

+ ++

+

M2F TNZ ALPHA 19.000 Z 18.929

CL 1.1717 CD 0.2199 CM -0.5570

NCYC 500 RES0.538E-02

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp

+++++++++++++++++++++++++++++++++++++++

+

+

+

++

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++

Figure 3.27: Pressure distributions along sections at 1, 25 and 85 percent of the heightof main sail after aeroelastic analysis

Chapter 4

Aerodynamic Shape optimization

With the availability of high performance computing platforms and robust numerical

methods to simulate fluid flows, it is possible to shift attention to automated design

procedures which use CFD combined with gradient-based optimization techniques.

Typically, in gradient-based optimization techniques, a control function to be opti-

mized (the sail shape, for example) is parameterized with a set of design variables

and a suitable cost function to be minimized is defined. For aerodynamic problems,

the cost function may be the lift or drag or a specified target pressure distribution.

Then, a constraint, the governing equations, can be introduced in order to express

the dependence between the cost function and the control function. The sensitivity

of the cost function with respect to the design variables are calculated in order to

get a direction of improvement. Finally, a step is taken in this direction and the

procedure is repeated until convergence is achieved. Finding a fast and accurate way

of calculating the necessary gradient information is essential to developing an effec-

tive design method since this can be the most time consuming portion of the design

process. This is particularly true in problems which involve a very large number of

design variables as is the case in a typical three dimensional sail shape design.

The control theory approach has dramatic computational cost advantages over

the finite-difference method of calculating gradients. The control theory approach

is also called the adjoint method as the necessary gradients are obtained through

the solution of an adjoint system of equations of the governing equations of interest.

76

CHAPTER 4. AERODYNAMIC SHAPE OPTIMIZATION 77

The adjoint method is extremely efficient since the computational expense incurred

in the calculation of the complete gradient is effectively independent of the number of

design variables. Control theory was applied in this way to shape design for elliptic

equations by Pironneau [50] and to transonic flow by Jameson [42].

In this study, a continuous adjoint formulation has been used to derive the adjoint

system of equations. Hence, the adjoint equations are derived directly from the

governing equations and then discretized. This approach has the advantage over the

discrete adjoint formulation in that the resulting adjoint equations are independent

of the form of discretized flow equations. The adjoint system of equations has a

similar form to the governing equations of the flow and hence the numerical methods

discussed in the previous chapters can be reused for the adjoint equations. The

gradient formulation is derived to be independent of the mesh modification which is

critical for this design methodology to work on unstructured meshes. If the gradient

depends on the form of the mesh modification, then the field integral in the gradient

calculation has to be recomputed for mesh modifications corresponding to each design

variable. Using the gradients computed with this new formulation, a steepest descent

method is used to improve an existing design.

4.1 The general formulation of the Adjoint Ap-

proach to Optimal Design

The aerodynamic properties which define the cost function are functions of the flow-

field variables, w, and the physical location of the boundary, which may be represented

by the function, F . Then

I = I(w,F),

and the variation of the cost function can be expressed as

δI =∂IT

∂wδw +

∂IT

∂F δF . (4.1)


Using control theory, the governing equations of the flow field are introduced as a

constraint in such a way that the final expression for the gradient does not require

re-evaluation of the flow-field. In order to achieve this, δw must be eliminated from

equation (4.1). Suppose that the governing equation, R, which expresses the depen-

dence of w and F within the flow field domain D can be written as

R(w,F) = 0. (4.2)

Then δw is determined from the expression for the variation in R

δR =

[∂R

∂w

]δw +

[∂R

∂F]δF = 0. (4.3)

Next, introducing a Lagrange Multiplier ψ, we have

δI =∂IT

∂wδw +

∂IT

∂F δF − ψT

([∂R

∂w

]δw +

[∂R

∂F]δF

),

which can be rearranged as

δI =

(∂IT

∂w− ψT

[∂R

∂w

])δw +

(∂IT

∂F − ψT

[∂R

∂F])

δF ,

Choosing ψ to satisfy the adjoint equation

[∂R

∂w

]T

ψ =∂I

∂w, (4.4)

the first term is eliminated and hence

δI = GδF , (4.5)

where

G =∂IT

∂F − ψT

[∂R

∂F]. (4.6)


In this way the gradient with respect to the shape is obtained at the cost of one flow

and one adjoint solution.

After taking a step in the negative gradient direction, the gradient is recalculated

and the process is repeated to follow the path of steepest descent until a minimum

is reached. In order to avoid violating constraints the gradient can be projected into

an allowable subspace within which the constraints are satisfied. In this way one can

devise procedures which must necessarily converge at least to a local minimum and

which can be accelerated by the use of more sophisticated descent methods such as

conjugate gradient or quasi-Newton algorithms.

4.2 Adjoint and Gradient formulations

In applying the adjoint method one may apply the above procedure directly to the

partial differential equations to derive a continuous adjoint equation, which must then

be discretized to obtain a numerical solution. Alternatively one may derive a discrete

adjoint equation directly after first discretizing the flow equations. In this work the

first procedure has been adopted because it allows more flexibility in the formulation

of the gradient.

The procedure is illustrated here for the Euler equations. These are represented

in transformed coordinates ξi on a fixed computational domain.

Let

S = JK−1,

where

Kij =∂xi

∂ξj, J = det(K),

Then the transformed equations are

∂Fi

∂ξi=∂(Sijfj)

∂ξi= 0.

Consider the case of an inverse problem where one wishes to find the shape which

brings the pressure as close as possible to the specified target pressure, pt. Hence, the


cost function has the form

I =1

2

∫

B(p− pt)

2dS

over the design surface B, which for convenience is assumed to be the surface ξ2 = 0.

Now a shape modification induces a change δp in the pressure and consequently

δI =

∫

B(p− pt)δpdS +

1

2

∫

B(p− pt)

2dδS.

The variation in the flow solution can be expressed as

∂

∂ψi

(δFi(w)) = 0.

Here the flux changes are

δFi = δSijfj + Ciδw,

where

Ci = Sij∂fj

∂w.

Consequently one can augment the cost variation by

∫

DψT ∂δFi

∂ξidξ,

which can be integrated by parts to obtain

∫

DψT ∂δFi

∂ξidξ =

∫

Bniψ

T δFidξB −∫

D

∂ψT

∂ξδFidξ.

Now choose ψ to satisfy the adjoint equation

CTi

∂ψ

∂ξi= 0,


with the boundary condition

ψ2ηx + ψ3ηy + ψ4ηz = p− pt,

where ηx, ηy, ηz are the components of the surface normal. Then the boundary inte-

grals involving δp and the field integral involving δw are eliminated and the gradient

is reduced to

1

2

∫

B(p− pt)

2dδS −∫ ∫

B(δS21ψ2 + δS22ψ3 + δS23ψ4) pdξ1dξ3 −

∫

D

∂ψT

∂ξ(δSijfj)dξ,

where typically the first term is negligible and can be dropped.

The evaluation of the field integral requires the evaluation of the metric variations

δSij throughout the domain. The true gradient should not depend on the way the

mesh is modified. Consider the case of a mesh variation with a fixed boundary. Then

δI = 0,

but there is a variation in the transformed flux

δFi = δSijfj + Sij∂fj

∂wδw.

Here the true solution is unchanged, so the variation δw is actually the variation δw∗

due to the mesh movement δx at fixed ξ. Therefore

δw = δw∗ =∂w

∂xj

δxj,

and since∂δFi

∂ξ= 0,

it follows that ∫

DψT ∂(δSijfj)

∂ξidξ = −

∫

DψTSij

∂fj

∂wδw∗dξ,


or ∫

DψT ∂(δSijfj)

∂ξidξ =

∫

DCi∂w

∂xj

δxjdξ.

A similar relationship can be derived in the general case with boundary movement [73].

Now,

∫

DψT δRdξ =

∫

D

∂

∂ξiCi(δw − δw∗)dξ

=

∫

D

∂ψT

∂ξiCi(δw − δw∗)dξ

=

∫

BψTCi(δw − δw∗)dξB. (4.7)

Hence on the wall boundary

C2δw = δF2 − δS2jfj.

Thus by choosing ψ to satisfy the adjoint equation and the adjoint boundary

condition, we have the following expression for the reduced gradient:

δI =

∫ ∫

BψT (δS2jfj + C2δw

∗)dξ1dξ3 −∫ ∫

B(δS21ψ2 + δS22ψ3 + δS23ψ4)pdξ1dξ3 (4.8)

It has been confirmed in numerical experiments performed by Jameson and Kim [73]

that these alternate formulations yield computed values of the gradient which are in

close agreement, and that the optimization procedure converges to essentially the

same result whichever is used. On a structured mesh one can explicitly define mesh

deformations which allow the field terms to be evaluated easily. On an unstructured

mesh this is not the case and the reduction to a boundary integral yields large sav-

ings in the computational cost. The discrete adjoint does not provide for such a

transformation.


4.2.1 Adjoint Equations for the Euler equations modified by

the artificial compressibility method

Although the adjoint equation represents a linear set of partial differential equations

for the adjoint variables, they are of the same form of the flow equations. The

numerical solution procedures developed for the flow equations are applied to the

adjoint system with the appropriate boundary conditions. The adjoint co-state flux

terms are modified to account for the introduction of the artificial compressibility

terms in the governing flow equations. The methodology followed here is derived

from the work of Cowles and Martinelli [74]. The adjoint field equations can be

expressed as a time dependent system of the form

∂ψ

∂t− [Ai]

T ∂ψ

∂xi

= 0, (4.9)

where

ψ =

p

φ1

φ2

φ3

. (4.10)

Hence, this system can be integrated to steady state using a preconditioner similar

to that used in the method of artificial compressibility. The adjoint ‘continuity’

equation is augmented by a time derivative of the adjoint pressure p to

∂p

∂t− Γ2∂φi

∂xi

= 0. (4.11)

The form of Γ is identical to that used for the flow equations since the magnitude

of the eigenvalues of the flux Jacobians for the two systems are identical. Together

with equation (4.11), the adjoint system is discretized and solved in a manner that is

consistent with that used for the flow equation.


4.2.2 The need for a Sobolev inner product in the definition

of the gradient

Another key issue for successful implementation of the continuous adjoint method is

the choice of an appropriate inner product for the definition of the gradient. It turns

out that there is an enormous benefit from the use of a modified Sobolev gradient,

which enables the generation of a sequence of smooth shapes. This can be illustrated

by considering the simplest case of a problem in calculus of variations [55].

Choose y(x) to minimize

I =

b∫

a

F (y, y′)dx,

with fixed end points y(a) and y(b). Under a variation δy(x),

δI =

b∫

a

(∂F

∂yδy +

∂F

∂y′δy

′)dx

=

b∫

a

(∂F

∂y− d

dx

∂F

∂y′

)δydx.

Thus defining the gradient as

g =∂F

∂y− d

dx

∂F

∂y′,

and the inner product as

(u, v) =

b∫

a

uvdx,

we find that

δI = (g, δy),

Then if we set

δy = −λg, λ > 0,


we obtain an improvement

δI = −λ(g, g) ≤ 0,

unless g = 0, the necessary condition for a minimum. Note that g is a function of

y, y′, y

′′,

g = g(y, y′, y

′′),

Now each step

yn+1 = yn − λngn

reduces the smoothness of y by two classes. Thus the computed trajectory becomes

less and less smooth, leading to instability.

In order to prevent this we can introduce a modified Sobolev inner product [72]

〈u, v〉 =

∫(uv + εu

′v′)dx,

where ε is a parameter that controls the weight of the derivatives. If we define a

gradient g such that

δI = 〈g, δy〉,

Then we have

δI =

∫(gδy + εg

′δy

′)dx

=

∫(g − ∂

∂xε∂g

∂x)δydx

= (g, δy) ,

where

g − ∂

∂xε∂g

∂x= g,

and g = 0 at the end points. Thus g is obtained from g by a smoothing equation.

Now the step

yn+1 = yn − λngn


gives an improvement

δI = −λn〈gn, gn〉,

but yn+1 has the same smoothness as yn, resulting in a stable process.

In applying control theory for aerodynamic shape optimization, the use of a

Sobolev gradient is equally important for the preservation of the smoothness class

of the redesigned surface and it has been employed to obtain all the results in the

next chapter.

4.3 Analysis of the Optimization Procedure

Once the gradient has been determined, any number of optimization algorithms can be

utilized to determine the desired shape modification. In this work, a steepest descent

method is used in which small steps are taken in the negative gradient direction

δF = −λG

This can be thought of as a simulation of the following time dependent process [75]

dFdt

= −G,

where the λ is the time step ∆t. Let A be the Hessian matrix with element

Aij =∂Gi

∂Fi

=∂2I

∂Fi∂Fj

.

Suppose that a locally minimum value of the cost function I∗ = I(F) is attained

when F = F∗. Then the gradient G∗ = G(F) must be zero, while the Hessian matrix

A∗ = A(F) must be positive definite. Since G∗ is zero, the cost function can be

expanded as a Taylor series in the neighborhood of F∗ with the form

I(F) = I∗ +1

2(F − F∗)A(F − F∗) + ...


Correspondingly,

G(F) = A(F − F∗) + ...

As F approaches F∗, the leading terms become dominant. Then, setting F , the

search process approximatesdF

dt= −A∗F

Also, since A∗ is positive definite it can be expanded as

A∗ = RMRT ,

where M is a diagonal matrix containing the eigenvalues of A∗, and

RRT = RTR = I.

Setting

v = RTF ,

the search process can be represented as

dv

dt= −Mv.

The stability region for the forward Euler time stepping scheme is a unit circle centered

at -1 on the negative real axis. Thus for stability we must choose

µmax∆t = µmaxλ < 2,

while the asymptotic decay rate, given by the smallest eigenvalue, is proportional to

e−µmint. In order to improve the rate of convergence, one can set

δF = λPG,

where P is a preconditioner for the search. An ideal choice is P = A−1, so that the


corresponding time dependent process reduces to

dF

dt= −F,

for which all eigenvalues are equal to unity, and F is reduced to zero in one time step

by the choice δt = 1.

With problems of the present complexity the calculation of the Hessian is com-

putationally infeasible. However, the smoothing operator which maps the gradient

to a Sobolev space proves to be a very effective preconditioner, and it is used in the

present work.

4.4 Mesh movement

The same tools used for mesh deformation during the aeroelastic analysis are reused

to induce mesh modifications during the design cycle. Due to the nature and the

magnitude of the geometry modifications, the spring method provides reasonable

answers. As in the aeroelastic analysis, the solution of the elasticity equations provides

a more robust tool to perform mesh modifications. However, the solution of the

elasticity equations is computationally more expensive. Hence, the spring method

has been used to obtain the results presented in the next chapter.

4.5 Parallel Implementation

The modules and data structures developed to solve the flow equations are reused to

solve the adjoint system in parallel. The gradient calculation and the mesh movement

are also executed in parallel though these are relatively inexpensive steps in the

optimization strategy. The parallel implementation of the design methodology enables

inverse design problems for incompressible flows to be performed in about 30 minutes

(for a mesh with 300000 nodes) using 8 processors of an SGI Origin 300.

Chapter 5

Validation of the Optimization

Procedure and Results

The aerodynamic shape optimization procedure for unstructured grids and the re-

duced gradient formulation described in the previous chapter were used to obtain the

optimal shape of sail geometries. To validate the design procedure, the method was

initially applied to airfoils and wings in compressible flows where comparative data

is available from previously developed structured grid codes [58].

In the following sections, two dimensional shape optimization of airfoils in tran-

sonic flows are presented first. Then, using the idea of artificial compressibility, the

flow and the adjoint equations are modified to perform shape optimization of airfoils

in incompressible flow. Three dimensional flows around wing geometries were then

investigated. Wings in transonic flows were initially investigated to validate the de-

sign process and then the flow and adjoint equations were modified to redesign wings

in incompressible flow. After proving the feasibility of the design methodology, sail

geometries were redesigned to remove the sharp suction peaks that were observed in

the flying shapes obtained by the aeroelastic simulations.

89

CHAPTER 5. VALIDATION OF THE OPTIMIZATION PROCEDURE AND RESULTS90

5.1 Shape optimization for airfoils in compressible

flow

The unstructured adjoint technology was initially validated for two-dimensional in-

verse design and drag minimization problems. Figures 5.6 and 5.7, show the result

of drag minimization for the RAE 2822 airfoil in transonic flow (M∞ = 0.75). The

lift was constrained to be 0.6 and the angle of attack was perturbed to maintain the

lift. The final geometry is shock-free and the drag was reduced by 36 drag counts.

Figures 5.6 and 5.8 show the result of an inverse design for the RAE 2822 airfoil.

Here the target pressure distribution was a shock-free profile obtained from the drag

minimization exercise. As can be seen from these pictures, the final pressure profile

almost exactly matches the target pressure distribution.

A comparison of the gradients from a well documented structured grid adjoint

solver (SYN82) and a version which uses the same numerical schemes and gradient

formulations but using unstructured grids (SYN75) is shown in figure 5.1. These

gradients are for an inverse problem and as can be seen from the plot, they match

well except neat the leading edge of the airfoil where the unstructured solver predicts a

smaller gradient. However, the overall design process was not affected. The difference

between the gradients is attributed to the difference in the flow and adjoint solution

near the leading edge. The differences in the adjoint solution are highlighted in

figures 5.2, 5.3, 5.4 and 5.5.

5.2 Shape optimization of airfoils in incompress-

ible flow

To redesign airfoils in incompressible flow, the flow solver and adjoint solvers were

modified using the idea of artificial compressibility. An inverse design problem was

identified to validate the design procedure. The pressure distribution over an Onera

M6 wing section was prescribed as target to the design process. The initial airfoil

shape corresponded to the NACA 0012 airfoil section. It can be seem from figure 5.9


20 40 60 80 100 120 140 160

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Gradients

points on the airfoil surface, lower trailing edge to upper trailing edge

syn75syn82

Figure 5.1: Comparison of the gradients from SYN75 and SYN82

0 20 40 60 80 100 120 140 160 180−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8First co−state variable


syn75syn82

Figure 5.2: Comparison of the first co-state variable from SYN75 and SYN82


0 20 40 60 80 100 120 140 160 180−2

−1.5

−1

−0.5

0

0.5

1Second co−state variable


syn75syn82

Figure 5.3: Comparison of the second co-state variable from SYN75 and SYN82

0 20 40 60 80 100 120 140 160 180−2

−1.5

−1

−0.5

0

0.5

1Third co−state variable


syn75syn82

Figure 5.4: Comparison of the third co-state variable from SYN75 and SYN82


20 40 60 80 100 120 140 160

−0.2

−0.1

0

0.1

0.2

0.3

Fourth co−state variable


syn75syn82

Figure 5.5: Comparison of the fourth co-state variable from SYN75 and SYN82

RAE 2822 MACH 0.750 ALPHA 0.703

CL 0.5999 CD 0.0062 CM -0.1334

GRID 161X33 NDES 0 RES0.785E-05 GMAX 0.000E+00

0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++++++++++++++++++

++

++

++

++

++

+++++++++++++++++++++++++++++++++++++++++++

+

+

+

+

+

+

+

+

++++++++++++++

+++++++++

++++++++

+++++++++++++++++++++++++

+

++++++++++++++++++++++

+

Figure 5.6: Initial pressure distribution for the RAE-2822 airfoil


RAE 2822 MACH 0.750 ALPHA 0.866

CL 0.6000 CD 0.0026 CM -0.1243

GRID 161X33 NDES 40 RES0.191E-04 GMAX 0.569E-02

0.1E

+01

0.8E

+00

0.4E

+00

0.3E

-07

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++++++++++++++++

++

++

++

++

++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

+

+

+

+

+

+

++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++

++

++

++

++++++++++++++

+oooooooooooooooooooo

oo

oo

oo

ooooooooooooooooooooooooooooooooooooooooooooooo

o

o

o

o

o

o

o

o

ooooooooooooooooooo

oooooooooooooooooooooooooooooooooooooooo

oo

oo

oo

oooooooooooooo

o

Figure 5.7: Drag minimization for the RAE-2822 airfoil

RAE 2822 : INVERSE TO SHOCK FREE SOLUTION MACH 0.750 ALPHA 0.763

CL 0.6000 CD 0.0025 CM -0.1242


0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++++++++++++++++++++

++

++

++

++

+++++++++++++++++++++++++++++

++++++++++++++++

++

+

+

+

+

+

+

+++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++

++

++

++

++++++++++++++

+oooooooooooooooooo

oo

oo

oo

oo

ooooooooooooooooooooooooooooooooooooooooooooooo

o

o

o

o

o

o

o

o

oooooooooooooooooooooooo

oooooooooooooooooooooooooooooooooo

oo

oo

oo

oooooooooooooo

Figure 5.8: Final and target pressure distribution for the RAE-2822 airfoil


NACA 0012 TO ONERA MACH 0.000 ALPHA 1.796

CL 0.2117 CD 0.0041 CM -0.0029


0.1E

+01

0.8E

+00

0.4E

+00

0.0E

+00

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++++++++++++++++++++++++++++++++++++

++++

+

+

+

+

+++

+

+

+

+

+

+

+++++++++++++++++++++++++++ + + + + + + + + + + + + + + + +

++

NACA 0012 TO ONERA MACH 0.000 ALPHA 2.015

CL 0.2116 CD 0.0059 CM -0.0053


0.1E

+01

0.8E

+00

0.4E

+00

0.0E

+00

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++

++++++++++++++++++++++++++++++++++++++++++

+

+

+

+++

+

+

+

+

++++

+

+

+++++++++++++++++++++++ + + + + + + + + + + + + + + +

++

+oo

oooooooooooooooooooooooooooooooooooooooooo

o

o

o

o

o

o

o

o

o

oooo

o

o

oooooooooooooooooooo o o o o o o o o o o o o o o o o o o

oo

o

Figure 5.9: Initial and final pressure distribution, o is the target pressure distribution,x is the computed pressure distribution for the redesigned airfoil

that the target pressure distribution is almost fully recovered by the design process.

5.3 Three dimensional shape optimization of wings

in compressible flow

The design methodology was then applied to wing shapes in transonic flow. Inverse

design computations were performed to validate the design process and the gradient

calculations. Figure 5.13 shows the result of an inverse design calculation, where the

initial geometry was a wing with NACA 0012 sections and the target pressure distri-

bution was the pressure distribution over the Onera M6 wing. Figures 5.14, 5.15, 5.16

show the target and computed pressure distribution at 4 span-wise sections. It can be

seen from these plots that the target pressure distribution is almost perfectly recov-

ered in 50 design cycles. The results from this test case show that the design process

is capable of recovering pressure distributions that are significantly different from the

initial distribution and can also capture shocks and other discontinuities in the target


pressure distribution.

Another test case for the inverse design problem used the wing from an airplane

(SHARK [76]) that was designed for the Reno Air Races. The initial and target

pressure distributions are shown the figure 5.10. As can be seen from these plots,

the initial pressure distribution has a weak shock in the outboard sections of the

wing. The target pressure distribution is shock-free. The computed (after 50 design

cycles) and target pressure distributions along three sections of the wing are shown in

figure 5.11. Again the design process captures the target pressure with good accuracy

in about 50 design cycles.

5.4 Inverse design of wings in incompressible flow

To validate the design process for three dimensional incompressible flows, the test

problem in the previous section was used. The initial wing had the planform of the

Onera M6 but had NACA 0012 airfoil sections. The target pressure distribution cor-

responded to the steady state pressure distribution over the Onera M6 wing. Three

levels of multigrid were used to obtain steady state flow and adjoint solutions. The

meshes were generated using an automated grid generator and interpolation coeffi-

cients were accumulated in a pre-processing step. The parallel implementation of the

flow and adjoint solvers were used to reduce the computational time of the design

process. Modifications to the shape of the wing were transmitted to the interior mesh

using the spring deformation method which worked well for this problem.

Figure 5.17 show that the target pressure distribution has been recovered in about

50 design cycles.

5.5 Inverse design for sail geometries

The results of the flow and aeroelastic simulations show that the interaction of the

head sail with the main reduces the development of sharp pressure gradients around

the leading edge of the main sail. This interaction is crucial to the performance of the

main sail as it allows the main sail to be set at a higher angle to the center-line of the


SHARKX6 (JCV: 16 DEC 99) Mach: 0.780 Alpha: 1.400 CL: 0.280 CD: 0.00624 CM: 0.0000 Design: 60 Residual: 0.1528E+00 Grid: 193X 33X 49

Cl: 0.241 Cd: 0.02383 Cm:-0.1179 Root Section: 6.6% Semi-Span

Cp = -2.0

Cl: 0.406 Cd: 0.00203 Cm:-0.1871 Mid Section: 49.2% Semi-Span

Cp = -2.0

Cl: 0.280 Cd:-0.01369 Cm:-0.1042 Tip Section: 91.8% Semi-Span

Cp = -2.0

Figure 5.10: Initial and final pressure and section geometries


SHARKX6 (JCV: 16 DEC 99) MACH 0.780 ALPHA 1.400 Z 16.548

CL 0.2787 CD 0.0120 CM -0.1352


0.1E

+01

0.8E

+00

0.4E

+00

0.3E

-07

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++++

++

++

++

++

++

++++++++++++++++++++++++++++++++++++++++

++

+

+

+

+++

+

+

+

+

++++++++

+++++

++++

++++++++++++++++++++++++ + + +

++

++

++

++

++

++ooooo

oo

oo

oo

oo

oo

ooooooooooooooooooooooooooooooooooooooooooo

o

oooo

o

o

o

ooooooooooooo

oooo

ooooooooooooooooooo o o o o o o o o

oo

oo

oo

oo

oo

o


CL 0.4341 CD 0.0018 CM -0.2010


0.1E

+01

0.8E

+00

0.4E

+00

0.3E

-07

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++++

++

++

++

++++++++++++++++++++++++++++++++++++++++++++

+

+

+++

+

+

+

+

+++++++++++++

+++++++

+++++++++++++++++++++ + ++

++

++

++

++

++

+

+ooooooo

oo

oo

oo

ooooooooooooooooooooooooooooooooooooooooooo

o

oooo

o

o

o

o

o

oooooooooooo

oooooo

oooooooooooooooooo o o o o o oo

oo

oo

oo

oo

oo

oo


CL 0.3122 CD -0.0139 CM -0.1244


0.1E

+01

0.8E

+00

0.4E

+00

0.3E

-07

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++++++++

++++++++++++++++++++++++++++++++++++++++++++

++++

+

+

+

+++

+

+

+

+

+

+++++++++++

+++++++++++++++++++++++++++++ + + + + + + ++

++ + +

++

+ooooooo

oo

ooooooooooooooooooooooooooooooooooooooooo

oooooo

o

oooo

o

o

o

o

o

o

ooooooooooo

oooooooooooooooooooooooo o o o o o o o o o o oo

oo

o oo

oo

Figure 5.11: Initial and final pressure distributions at 5 %, 50 % and 95 % of thewing span


NACA 0012 WING TO ONERA M6 TARGET Mach: 0.840 Alpha: 3.060 CL: 0.325 CD: 0.02319 CM: 0.0000 Design: 0 Residual: 0.2763E-02 Grid: 193X 33X 33


Cp = -2.0


Cp = -2.0


Cp = -2.0

Figure 5.12: Initial pressure distribution over a NACA 0012 wing


NACA 0012 WING TO ONERA M6 TARGET Mach: 0.840 Alpha: 3.060 CL: 0.314 CD: 0.01592 CM: 0.0000 Design: 50 Residual: 0.1738E+00 Grid: 193X 33X 33


Cp = -2.0


Cp = -2.0


Cp = -2.0

Figure 5.13: Final pressure distribution and modified section geometries along thewing span


NACA 0012 WING TO ONERA M6 TARGET MACH 0.840 ALPHA 3.060 Z 0.00

CL 0.2814 CD 0.0482 CM -0.1113

GRID 192X32 NDES 50 RES0.162E-02

0.1E

+01

0.8E

+00

0.4E

+00

0.3E

-07

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++

++++++++++++++++++++++++++++++++++++++++++++++++++++

++++

+

+

+

++

++++

+

+

+

+

+

+

+

++

+

+

++++

+++++

++++++++++++++++++++++++++++++

+ ++

+

++ + +

++

++

+oo

oo

oooooooooooooooooooooooooooooooooooooooooooooooooooooo

o

o

o

o

oooo

o

o

o

o

o

o

oooo

o

oooo

ooooo

ooooooooooooooooooooooooo o o o o o

o oo

o

oo o o o

oo

oo


CL 0.2814 CD 0.0482 CM -0.1113


0.1E

+01

0.8E

+00

0.4E

+00

0.3E

-07

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++

++++++++++++++++++++++++++++++++++++++++++++++++++++

++++

+

+

+

++

++++

+

+

+

+

+

+

+

++

+

+

++++

+++++

++++++++++++++++++++++++++++++

+ ++

+

++ + +

++

++

+oo

oo

oooooooooooooooooooooooooooooooooooooooooooooooooooooo

o

o

o

o

oooo

o

o

o

o

o

o

oooo

o

oooo

ooooo

ooooooooooooooooooooooooo o o o o o

o oo

o

oo o o o

oo

oo

Figure 5.14: Final computed and target pressure distributions at 0 % and 20 %of thewing span


CL 0.3269 CD 0.0145 CM -0.0865


0.1E

+01

0.8E

+00

0.4E

+00

0.3E

-07

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++

++

++++++++++++++++++++++++++++++++++++++++++++++++++++

++

+

+

+

+

+++

+

+

+

+

+

+

+

+

+

+

+++++++

++

+

+

+++++++++++++++++++++++++

++

+

+

+

+ ++ + + + + + + +

++

++o

oo

oooooooooooooooooooooooooooooooooooooooooooooooooo

ooooo

o

o

o

o

oooo

o

o

o

o

o

o

o

o

ooooooo

ooooooooooooooooooooooooooooo

oo

o

o

o

o o o o o o o o o oo

oo

o


CL 0.3356 CD 0.0081 CM -0.0735


0.1E

+01

0.8E

+00

0.4E

+00

0.3E

-07

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++

++

++++++++++++++++++++++++++++++++++++++++++++++++++++

++

+

+

+

+

+++

+

+

+

+

+

+

+

+

+

+

+++++++++++++++

++++++++++++++++++

+++

+

+

++++ + + + + + + + + +

++

++o

oo

ooooooooooooooooooooooooooooooooooooooooooooooooooo

oooo

o

o

o

o

oooo

o

o

o

o

o

o

o

o

ooooooooooooooo

oooooooooooooooooo

ooo

o

o

o oo o o o o o o o o o o

oo

oo

Figure 5.15: Final computed and target pressure distributions at 40 % and 60 % ofthe wing span



CL 0.3176 CD 0.0011 CM -0.0547


0.1E

+01

0.8E

+00

0.4E

+00

0.3E

-07

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

+

+

+++

+

+

+

+

+

+

+

+

+

++++++++++++++++++++++++

+++++

+

+

+

+

+

+++

++++++ + + + + + + + + +

++

++o

oo

ooooooooooooooooooooooooooooooooooooooooooooooooooo

oooo

o

o

o

o

oooo

o

o

o

o

o

o

o

o

oooooooooooooooooooooo

oooooo

o

o

o

o

o

ooo

o o o o o o o o o o o o o o oo

oo

o


CL 0.4846 CD 0.0178 CM -0.1518


0.1E

+01

0.8E

+00

0.4E

+00

0.3E

-07

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

+

++++

++

+

+

+

+

+

+

+

+++++++++++++++++++++++

+

+

+

+

++

++

++++++++++++ + + + + + + + + + ++ +

+ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

o

o

o

ooo

o

o

o

o

o

o

o

o

o

oooooooooooooooooooooo

o

o

o

o

ooo

oooooooo o o o o o o o o o o o o o o o

oo

o

Figure 5.16: Final computed and target pressure distributions at 80 % and 100 % ofthe wing span

boat. These results also show that the region above the head sail has large suction

peaks which is a cause of concern. The aerodynamic shape optimization procedure

validated in the previous sections was used to redesign the main sail, with an aim of

reducing the pressure gradient around the luff of the main sail.

The cost function was defined as follows

I =

∫

B

(p− pt)2dB,

where p is the pressure distribution at the beginning of each design cycle, pt is the

pressure distribution obtained by smoothing the pressure distribution on the main

sail obtained from the aeroelastic analysis and the integral is taken over the surface

of the main sail. The lift was constrained by perturbing the angle of attack.

Figure 5.18 show that a significant portion of the leading edge of the main sail

has been redesigned to allow for smooth entry of the flow. The associated reduction

in sharp suction peaks should have a favorable affect on the growth of the boundary

layer over the upper surface. The change to the sections is shown in figure 5.19.


NACA 0012 TO ONERA M6 ALPHA 3.060 Z 0.000

CL 0.1890 CD 0.0186 CM -0.0594

NCYC 10 RES0.194E-02 DESIGN CYCLE 50

0.1E

+01

0.5E

+00

0.0E

+00

-.5E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+o+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+

o

+

o

+

o

+

o

+

o

+

o

+o

+o

+o+o

+o

+o

+o

+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o +o +o +o +o +o +o +o +o+o

+o


CL 0.2089 CD 0.0063 CM -0.0543


0.1E

+01

0.5E

+00

0.0E

+00

-.5E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o+o

+

o

+

o

+

o

+

o

+

o

+

o

+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o +o +o +o +o +o +o +o +o +o

+o


CL 0.2299 CD 0.0032 CM -0.0573


0.1E

+01

0.5E

+00

0.0E

+00

-.5E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+o+o+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+o+

o

+

o

+

o

+

o

+

o

+

o

+

o

+o

+o+o+o

+o

+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o +o +o +o +o +o +o +o +o +o

+o


CL 0.3340 CD 0.0219 CM -0.1151


0.1E

+01

0.5E

+00

0.0E

+00

-.5E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+o+o+o+o

+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o +o +o +o +o +o +o +o +o +o

+o

Figure 5.17: Final computed and target pressure distributions at 0, 25, 75 and 100 %of the wing span at 3 degrees angle of attack


The shape changes induced by the design method can be realized on an actual sail

by using battens. As the majority of the shape changes are induced near the leading

edge of the main sail, it is important to account for the presence of the mast to ensure

that the new shape provides favorable pressure gradients to the boundary layer.


TNZ : INVERSE DESIGN ALPHA 19.000 Z 4.975

CL 0.7850 CD 0.2799 CM -0.4489


0.1E

+01

0.5E

+00

0.0E

+00

-.5E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+o+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o

+

o

+o

+

o+

o

+

o

+o+o

+o+o

+o+o

+o

+o

+o

+o

+o

+o

+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+

o+o+

o+o+o

+o


CL 0.9992 CD 0.2965 CM -0.5381


0.1E

+01

0.5E

+00

0.0E

+00

-.5E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+o

+o

+o

+o

+o

+o

+o

+o

+o

+o

+o

+o

+o

+o+o+o+o+o+o+o+o+o+o+o

+o

+o

+o

+o

+o

+o

+o

+o

+o

+o

+o

+o+o+o+

o+

o

+

o

+

o

+

o

+

o

+

o+o+o

+o

+o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+

o+o+

o+o+

o+o+o+o


CL 1.4461 CD 0.2735 CM -0.6545


0.1E

+01

0.5E

+00

0.0E

+00

-.5E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+o+o

+o+o

+o+o+o+o+o+o+o+o+o+o+o

+o

+o

+o

+o

+o

+o

+o

+o

+o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o+o

+o

+o+

o+

o

+

o

+o

+

o

+

o

+

o

+

o

+o+

o

+

o

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+

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o

+

o+

o+o+

o+o+o+o+o+o+o

+o+o+o

+o+o+o+o+o+o


CL 1.5637 CD 0.2928 CM -0.6885


0.1E

+01

0.5E

+00

0.0E

+00

-.5E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+o

+o+o+o

+o+o+o+o+o+o+o+o+

o+o

+o

+o

+o

+o

+o

+o

+o

+o

+o

+o

+

o

+

o

+

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o

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o

+

o

+

o

+

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o+o

+o+

o+

o

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o

+

o

+

o

+

o+

o+o+o+

o+o+o+o+

o+o+o+o+o+o+o+o+o+o+o

Figure 5.18: Initial (o) and final(+,x) pressure distribution at 15, 32, 75 and 85%height on the main sail


11 12 13 14 15 16 17 18 19−3

−2

−1

0

1

2

3

Initial and deformed sections at 15 percent height

x

y

InitialRedesign

11 12 13 14 15 16 17 18

−2

−1

0

1

2

3


x

y

InitialRedesign

11 12 13 14 15 16 17

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3


x

y

InitialRedesign

11.5 12 12.5 13 13.5 14 14.5 15 15.5 16 16.5

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5


x

y

InitialRedesign

Figure 5.19: Initial and redesigned camber line at 15,32,75 and 85% of height

Chapter 6

Conclusions

6.0.1 Aerodynamic and Aeroelastic analysis

The use of robust and accurate numerical methods for the Euler equations along

with the use of parallel computing environments enable designers to characterize the

aerodynamic performance of the design. Due to the fast turn-around times of these

simulations it will now be possible to substitute potential flow solvers with these non-

linear models of the flow field, thereby obtaining improved estimates of the forces on

sail configurations. As the computational tools that were developed in this study were

based on unstructured grids, including the other components of the sail-boat into the

analysis procedure will be straight-forward. Hence it is reasonable to expect that the

analysis tools developed in this study can be used as building blocks to develop an

integrated computational tool to determine the forces and moments generated during

the motion of the boat.

An obvious extension to the analysis tools developed in this study is the need to

include viscous effects in the mathematical models. Numerical solutions to the RANS

equations should be the goal. Computational tools that solve the RANS equations

are quite popular within the CFD community. However, it requires a trained user

to extract the quantities of engineering interest from these simulations. The first

major hurdle is encountered during the grid generation phase. Turbulence models

could potentially pose another hurdle. As an intermediate step, it is conceivable

107

CHAPTER 6. CONCLUSIONS 108

that inviscid flow solvers coupled with boundary layer codes could alleviate some of

the difficulties. The fast turn-around times of these simulations could provide an

attractive alternative to the designer.

The results of the aeroelastic simulations confirm that the elastic nature of the sail

cloth could play an important role in the aerodynamic performance of a design. The

techniques used in this study take first step towards building an integrated aeroelastic

package. To provide accurate estimates of the flying shape it will be necessary to

develop an improved structural analysis code that takes into account the various

elements of the rigs, wrinkling, flexibility of the mast, and presence of the battens.

The appropriate choice of finite element discretization procedures is also an open

question that needs to be addressed in the future.

6.0.2 Aerodynamic design

While analysis tools (aerodynamic and aeroelastic) can provide the designer with

insights into the performance of the design, an automated design tool that arrives at

the optimum design is invaluable. This study has confirmed the feasibility of adjoint

based shape optimization procedures for determining the optimal shape of the sail

sections. As the changes made by the design procedure are small and localized,

alternate design methods would have to perform iterative analysis of a large number

of candidate designs. Hence, the adjoint design method can provide a unique tool for

sail designers.

The multi-disciplinary nature of the design process for sailboats is illustrated in

figure 6.1. This flow chart (from [77]) shows the interaction of the various forces and

moments that result in a particular speed-made good (Vmg) of a design. The develop-

ment of an integrated computational tool to optimize the windward performance of

the boat is clearly feasible. As part of the current research some of the building blocks

for this computational tool have been developed. However they require refinement

and improvement, and this will be the task for the future.

CHAPTER 6. CONCLUSIONS 109

L/DSail Area

Keel

Area

hull

Fs /RΕ

A.R.

E

Sail Polar Diagram

A.R.

Camber

Sail Plan

Hull Side

Fr

Vs

Vmg

Β

Driving ForceHeeling Force

Apparent CourseBoat Speed

Fh

heel angle

Stability

Fs

Force

leeway

R

Hull resistance

ΘΛ

Figure 6.1: Components of the overall design process for upwind sails

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