Numerical Analysis and Design of Upwind Sails

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

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)

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.

Juan J. Alonso

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.

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 equations 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 influence 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 compressibility 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 aeroelastic 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 resulted 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

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 compressible 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 compressibility methods ...... 48 3.3 Validation of the parallel implementation ...... 49 3.4 Single and multi-element sail computations with artificial compressibility 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 compressible 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 3.8 Total Pressure losses on the airfoil surface at 1o ...... 62 3.9 Total Pressure losses on the airfoil surface at 3o ...... 63 3.10 Total Pressure losses on the airfoil surface at 5o ...... 63 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 3.19 Pressure distributions along sections at 1, 25 and 85 percent of the height of head sail ...... 69 3.20 Pressure distributions along sections at 1, 25 and 85 percent of the 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 3.26 Pressure distributions along sections at 1, 25 and 85 percent of the height of head sail after aeroelastic analysis ...... 74 3.27 Pressure distributions along sections at 1, 25 and 85 percent of the height of main sail after aeroelastic analysis ...... 75

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

xiii 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 distribution, 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 introduces no approximations to the physical properties of the fluid, and hence carefully performed wind-tunnel tests can provide good estimates of the aerodynamic and hydrodynamic 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 estimates 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 detailed 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 procedures 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 CHAPTER 1. INTRODUCTION 3

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 effects 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 CHAPTER 1. INTRODUCTION 4

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. CHAPTER 1. INTRODUCTION 5

Further, the twist in the onset flow can be also be easily included in the boundary 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 efficient 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 CHAPTER 1. INTRODUCTION 6

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 unstructured 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 quality 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 CHAPTER 1. INTRODUCTION 7

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 collapsing 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 corrections 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 generated 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 CHAPTER 1. INTRODUCTION 8

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 CHAPTER 1. INTRODUCTION 9

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 geometries 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. CHAPTER 1. INTRODUCTION 10

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 CHAPTER 1. INTRODUCTION 11

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. CHAPTER 1. INTRODUCTION 12

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 gradients. 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 CHAPTER 1. INTRODUCTION 13

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 control 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 effectively 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 successfully 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]. CHAPTER 1. INTRODUCTION 14

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 CHAPTER 1. INTRODUCTION 15

variation δy(x), the variation of the cost is δI.Now suppose that δI can be expressed to ﬁrst order as Z 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 Z 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

∂G A = , (1.4) ∂y 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 CHAPTER 1. INTRODUCTION 16

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 n+1 n n yj = yj − λGj . (1.7)

This may be regarded as a forward Euler discretization of a time dependent process with λ = ∆t. Hence, ∂y = −G. (1.8) ∂t 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 discretization 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 analysis. 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. CHAPTER 1. INTRODUCTION 17

Finite Volume techniques in conjunction with unstructured grids are used to discretize 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 approach. 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 depend 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 investigated where the target pressure distribution is prescribed through a combination of experience and engineering intuition. CHAPTER 1. INTRODUCTION 18

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 engineering 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 common 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 compressible 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 typically 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. CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 21

2.2 Finite Volume Discretization of the Flow equations

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 continuity 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 incompressible, inviscid fluid can be cast in the following form: CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 22

½ ¾ ∂w ∂F ∂G ∂H + P + + = 0. (2.1) ∂t ∂x ∂y dz Here, the dependent variables w, the inviscid ﬂux vectors f, g and h and the preconditioning matrix P are described by

         p   u   v   w           u   u2 + p   vu   wu  w = ,F = ,G = ,H = ,      2     v   uv   v + p   wv          w uw vw w2 + p

  Γ2 0 0 0      0 1 0 0  P =   . (2.2)  0 0 1 0  0 0 0 1

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 Z Z d pdV + Γ2 (u · n) dS = 0. (2.3) dt V S

Conservation of Momentum Z Z Z d udV + u(u · n)dS = − pndS, (2.4) dt V S S

Spatial discretization of equation (2.3) and (2.4) leads to a separate equation for each CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 23

sub-domain in the computational mesh.

d X V w + F .n S = 0, (2.5) dt i i k k k k 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 possible 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 traditionally 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 vertices 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 CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 24

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 affects 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 CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 25

Median Dual

Centroid dual

Dirichlet region

Figure 2.1: Dual mesh representation of the control volume

4 5 3 S P 6 2 10 1 11 9 Q R 12 8 7

Figure 2.2: Nodal formulation of the ﬁnite volume scheme CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 26

¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

4 ¡

¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

¡ 2 5

¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

Figure 2.3: Evaluation of ﬂuxes in three dimensions

Figure 2.4: Control volume for cell-vertex schemes in three dimensions CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 27

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 pressure 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 equations 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. CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 28

u u u u

v p v p v p v p v

u u u u

v p v p v p v p v

u u u u

v p v p v p v p v

u u u u

v p v p v p v p v

u u u u

Figure 2.5: Staggered arrangement of ﬂow variables

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

p p p p

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

p p p

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

p p p

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

p p p

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

Figure 2.6: Half-staggered arrangement of ﬂow variables CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 29

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 during 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 computational point. The JST scheme [32] has been widely proven to be a robust frame-work for numerical diffusion. This scheme can be represented as

³ ´ 2 4 dj+ 1 = ² 1 ∆wj+ 1 − ² 1 ∆wj+ 3 − 2∆wj+ 1 + ∆wj− 1 . 2 j+ 2 2 j+ 2 2 2 2 When used for problems with embedded supersonic regions, the above scheme switches to a locally ﬁrst order scheme to prevent oscillations. For incompressible CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 30

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 analogy 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 = U 2 + Γ2(ψ2 + η2 + ξ2), and U = uψ + vη + wξ. CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 31

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 (V w) + PQ = 0, (2.6) dt i i 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 (V w) + P [Q − D ] = 0, (2.7) dt i i i CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 32

or d (V w) + R = 0, (2.8) dt i i 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 equations 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) CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 33

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

h L2hδv2h + I2hRh = 0, (2.10)

h where I2h represents the aggregation or restriction operator. The correction to the present ﬁne grid solution can be represented as

new 2h vh = vh + Ih δvh,

2h where Ih represents an interpolation operator. We can add and subtract the following from equation (2.10)

L2hv2h − F2h = R2h, to get

h L2h(v2h + δv2h) − F2h + I2hRh = 0.

This leads to the full approximation scheme (FAS)

+ h L2h(v2h) − F2h + I2hRh − R2h = 0.

Then + h + vh = vh + I2h(v2h − 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 CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 34

a1 p a2 n3

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.

CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 35

¡ ¡ ¡

fine grid nodes

¢ ¢ ¢

£ £ £

¢ ¢ ¢

£ £ £ coarse grid nodes

Figure 2.8: Transfer of solution, residuals and corrections between the ﬁne and coarse mesh 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 implemented in a computer program that used the Message Passing Interface (MPI) standard 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 CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 36

1 2 3

4 5 7

9 10 y 11 6 14 15 13 8 12 16

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 algorithms 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. CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 37

inter-processor boundary

edges that are shared between processors

Figure 2.10: Halo nodes and the distribution of edges along processor boundaries

16 Actual SpeedUp Ideal SpeedUp

10 SpeedUp 8

2 2 4 6 8 10 12 14 16 Number of Processors

Figure 2.11: Speedup from the parallel implementation CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 38

2.10.2 Parallel implementation of the multigrid algorithm

The use of multigridding techniques for flow analysis necessitates the need to exchange 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 implementation 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 interpolation 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 CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 39

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 structural 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 CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 40

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.

∂ui ∂ui ²ii = , ²ji = ²ij = . ∂xi ∂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 represent the body. While triangular elements allow for ease of representation of arbitrary geometries, they usually turn out to be stiffer than quadrilateral elements. Higher CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 41

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 elements 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 CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 42

Translational degrees of freedom

suppressed at the mast

Leech is allowed to move freely 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 CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 43

mast

head

stay

Translational degrees of freedom supressed along the stay leech

luff

clew foot

All degrees of freedom suppressed

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 CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 44

∂∆x XN i + K (∆x − ∆x ) = 0, ∂t ij i j j=1 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 overcome 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 µ ¶ 1 ∂ui ∂uj ²ij = + , i, j = 1, 2, 3. 2 ∂xj ∂xi For an isotropically elastic solid the stress tensor is defined as

σij = λ²kkδij + 2µ²ij,

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 ∂σ ij = 0. ∂xij Dividing by the shear modulus µ leads to an equation that depends only on the λ parameter λ/µ. Alternatively, one can introduce Poisson’s ratio ν 2(λ+µ) and consider this to be the user defined parameter. It is again possible to increase the rigidity CHAPTER 2. DISCRETIZATION OF GOVERNING EQUATIONS 45

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 aeroelastic 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 performance 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 simulations 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 validate 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 numerical 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 CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 48

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 diffusion on artificial compressibility methods

As discussed in the previous chapter, it is possible to implement the numerical procedure 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 analyzed. All three schemes used the same numerical scheme, namely the finite volume scheme, artificial compressibility corrections, second order construction of the convective 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 distributions 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 CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 49

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 velocity components thereby yielding more accurate solutions. The cell-vertex scheme has advantages over the cell-centered scheme when used with three dimensional unstructured 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 computational 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 CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 50

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 compressibility 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 variety 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 coordinate 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 conditions 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 CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 51

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. CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 52

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 interaction 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 CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 53

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. CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 54

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 CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 55

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. CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 56

+ ++ ++ ++ ++ ++ ++ ++ + + + + + + + + + + + + + + + ++ + ++++ + +++ + + + + + + + + + + + ++++ + + + +++ + + ++ + + + ++ + + + + ++ + + + + ++ + + ++ + + + + + +++ + ++ + +++ + + ++ +++ + + ++ + +++ + + +++ ++++ + + + ++++++++ +++++ + ++ ++++++ + ++ Cp + + +++++ + + + +++++++++ + + + +++++++++++++++++++++++ + ++ + + + ++++++++++++++++++++++++ + + + + ++ ++ + + ++ ++ + + + ++ ++ + + + + ++ + + + ++ + + + + + + + + + ++ + + ++ + + ++ + + ++ + ++ ++ + + ++ + + ++ + + + ++ ++ + + + + + + ++ + + + +++ +++ + + + + + ++ + + ++ ++ + ++ + + + +++ + ++ + + ++ + +++++ + ++ ++++ + +++++++ ++ + + ++++ + + ++ ++ ++ ++ + ++ ++ ++ ++++ + + + +++++++++++++++++++++++ ++ ++ ++ ++++++ + +++ + +++ + + +++++++++++++ ++ +++ +++++++++++ + ++++ + +++++ ++++++ +++ ++++ + ++++++++++++++++++ +++ ++ +++++ +++ ++++++++++ +++++++++++++ + +++ ++++++++++++++++++ + ++++ +++ + + +++ + + + + + +

+ 3.00 2.00 1.00 0.00 -1.00 -2.00 -3.00 -4.00 -5.00 -6.00 -7.00

AGARD-AR-303 MACH 0.200 ALPHA 8.230 Figure 3.1: Grid and PressureCL 3.5407 distribution CD 0.0483 CM -0.6802 over a multi-element airfoil geometry at a NODES 11158 NCYC 2500 RES0.530E-04 M = 0.2 and α = 8.2 degrees CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 57

+ 8.00 4.00 0.00 -4.00 -8.00 -12.00 -16.00 8.00 4.00 0.00 -4.00 -8.00 -12.00 -16.00

+ SAIL M6 SAIL M6 MACH 0.100 ALPHA 8.000 Z 1.077 MACH 0.100 ALPHA 8.000 Z 2.154 CL 1.3734 CD 0.1474 CM -0.5968 CL 1.4962 CD 0.1595 CM -0.6785 + NCYC 500 RES0.615E-03 NCYC 500 RES0.615E-03 Nsup Log(Error)

0.00 100.00 200.00 300.00 400.00 500.00 600.00 -0.20 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 -12.00 -10.00 -8.00 -6.00 -4.00Work -2.00 0.00 2.00 4.00

SAIL M6 MACH 0.100 ALPHA 8.000 Figure 3.2: Cp distribution at twoRESID1 0.278E+01 sections RESID2 0.615E-03 and convergence history of the compressible ﬂow solver WORK 499.00 CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 58

+++++++++++ +++ +++++++ + ++++++ + +++++ + +++++ ++ Cp + ++++++++++++++++++++ + +++ Cp ++++ ++++ +++ ++++ +++++ +++ +++ ++++ + +++++++++++++++++++ +++ ++ ++++ +++ ++++++ +++ ++ ++++ +++ +++++ ++++ ++ ++++ + ++ ++++ +++ + ++++ + +++++ ++++ + +++++ + +++++ +++ ++ +++++ + ++++ +++ +++++ + + +++++ ++++ ++ +++++ + ++++ ++++ ++++ + +++++ +++ ++++ +++++++ ++ ++++ + +++++++ ++++ + +++++ + ++++ + +++ + ++++ +++ +++ + +++ ++++ +++ ++ ++++ ++ +++ ++++ +++ +++ + +++ ++ ++ ++ ++ ++ ++ + ++ ++ + ++ ++ ++ + ++ + + ++ ++ ++ ++

+ ++ + + ++ ++ ++ ++ ++ ++ ++ ++ ++ 1.20 0.80 0.40 0.00 -0.40 -0.80 -1.20 -1.60 -2.00 1.20 0.80 0.40 0.00 -0.40 -0.80 -1.20 -1.60 -2.00

NACA 0012 NACA 0012 ALPHA 0.000 ALPHA 1.000 CL 0.0000 CD 0.0000 CM 0.0000 CL 0.1206 CD 0.0000 CM -0.0013 GRID 256 GRID 256

+++ ++++ + ++ ++ + ++ ++ + +++ +++ +++ + +++ +++ +++ +++ +++ + +++ +++ +++ ++++ Cp ++++ ++++ + ++++ ++++ ++++ ++++ +++++++++++++++++++ +++ +++++ +++++++++ +++ +++ +++++++ +++ ++ +++++++ +++ + ++ ++++++ +++ ++ ++++++ +++ + +++++ +++ ++ +++++ ++ + +++++++ + +++++ + ++++ + +++ + +++ + + ++ + ++ ++ + ++ + ++ + ++ + + + + + + + ++ ++ ++ ++ 1.20 0.80 0.40 0.00 -0.40 -0.80 -1.20 -1.60 -2.00

NACA 0012 ALPHA 3.000 Figure 3.3: Potential ﬂowCL 0.3617solution CD 0.0000 CM -0.0040 from FLO1 at 0,1 and 3 degrees GRID 256 CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 59

+++++++++++ ++ +++++ + ++++ + +++ + +++ Cp + ++++++++++++++ + + Cp ++ +++ + ++++ ++++++ ++ +++ ++++++ +++++++++ + +++ ++++ + ++++ ++++ ++ ++ +++++ ++ + + + + ++ ++ +++++ + ++ + + + ++ ++++ + + + ++ ++ ++++ + + + + ++++ + + + + ++ ++ ++++ + + + + ++ + ++++ + + + ++ + ++++ + + + + ++++ + + +++ ++ ++++ + + + +++ ++++ + + ++ ++ ++++ ++++ +++ + + ++ + ++ ++ ++ + ++ + ++ + ++ ++ + ++ ++ + + + ++ ++ ++ ++ + ++ + ++ +

++ ++ + ++ + + ++ ++ ++ + +++ ++ 0.1E+01 0.8E+00 0.4E+00 -.2E-15 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01 0.1E+01 0.8E+00 0.4E+00 -.2E-15 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01

NACA 0012 - SCHEME : QUAD. CELL-CENT. RHCUSP-SLIP NACA 0012 - SCHEME : QUAD. CELL-CENT. RHCUSP-SLIP MACH 0.000 ALPHA 0.000 MACH 0.000 ALPHA 1.000 CL 0.0000 CD -0.0001 CM 0.0000 CL 0.1199 CD 0.0000 CM -0.0013 GRID 161X33 NCYC 80 RES0.834E-03 GRID 161X33 NCYC 80 RES0.262E-02

++ +++ + ++ ++ + ++ ++ + ++ ++ ++ ++ + ++ ++ ++ ++ + ++ ++ ++ ++ Cp ++ + ++ ++ ++ ++ ++ ++++++ + + + + + + ++++ + + + + + ++ ++ + + + + ++ ++ + + + + + ++ + + + + + + + + ++ ++ + + + + + ++ + + + + + + +++ + ++ + + + ++ + ++ + ++ + + + + + + + + ++ + ++ ++ ++ 0.1E+01 0.8E+00 0.4E+00 -.2E-15 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01

NACA 0012 - SCHEME : QUAD. CELL-CENT. RHCUSP-SLIP MACH 0.000 ALPHA 3.000 Figure 3.4: Flow over a NACA 0012CL 0.3598 CD airfoil 0.0001 CM -0.0039 at 0,1 and 3 degrees using a cell-centered scheme GRID 161X33 NCYC 80 RES0.734E-02 CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 60

++++++++++ +++ +++++ + ++++ + +++ + +++ ++ Cp + + +++++++++++++++ ++ Cp ++++ ++++++ + +++ ++++++ + ++ +++ +++++ ++++++++++++++ ++ +++ ++++ + ++ + + + + + +++++ ++ + + + ++ + ++++ ++ + + + ++ ++ ++++ + + + ++ ++ ++++ + + + + + ++++ + + + + ++ ++ ++++ + + + ++ ++++ + + + ++ ++ ++++ + + + + + ++++ + + +++ ++++ + + +++ ++ ++++ + ++ ++++ + + ++++ ++ +++ + ++ + + ++ + ++ ++ + + ++ + + ++ + + ++ + ++++ +++ + ++ + ++ + + +

++ ++

+ ++ + + ++ ++ ++ + + + ++ ++ 0.1E+01 0.8E+00 0.4E+00 -.2E-15 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01 0.1E+01 0.8E+00 0.4E+00 -.2E-15 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01

NACA 0012 NACA 0012 MACH 0.000 ALPHA 0.000 MACH 0.000 ALPHA 1.000 CL 0.0006 CD 0.0003 CM -0.0001 CL 0.1200 CD 0.0003 CM -0.0014 GRID 161X33 NCYC 200 RES0.730E-07 GRID 161X33 NCYC 200 RES0.829E-07

+ +++ + ++ + ++ ++ + ++ ++ ++ + ++ ++ ++ ++ + ++ ++ ++ ++ + ++ ++ Cp ++ ++ ++ ++ + ++ ++ +++ + + +++++ + + + + + + + ++ +++ + + + + + ++ + + + + ++ ++ + + + + + + + + ++ + + + + + + ++ + + + + ++ + + +++ + ++ + + + + + + + + ++++ + ++ + + + + + ++ + ++ + ++ ++ 0.1E+01 0.8E+00 0.4E+00 -.2E-15 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01

NACA 0012 MACH 0.000 ALPHA 3.000 Figure 3.5: Flow over a NACA 0012CL 0.3587 CDairfoil 0.0005 CM -0.0040 at 0,1 and 3 degrees using a nodal scheme GRID 161X33 NCYC 200 RES0.187E-06 CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 61

+++++++++++ ++ +++++ + ++++ + +++ + +++ Cp + ++++++++++++++ + + Cp ++++ +++++++ ++ +++ ++++++ + +++ ++ +++ +++++ ++++++ ++++++ ++ ++ ++++ ++ + + + + + + +++++ + ++ + + ++ ++ ++++ + + + + ++ ++ ++++ ++ + + + ++++ + + + + ++ ++ ++++ + + + + ++ ++++ + + ++ ++ ++++ + + + + ++++ + + + +++ ++ +++++ + + + +++ ++++ + +++ ++ ++++ + +++ +++ + +++ ++ + ++ ++ ++ + ++ + ++ + ++ ++ + + + + ++ + + + ++ ++ + ++ + ++ + ++ +

++ ++ + ++ + + ++ ++ ++ ++ ++ ++ 0.1E+01 0.8E+00 0.4E+00 -.2E-15 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01 0.1E+01 0.8E+00 0.4E+00 -.2E-15 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01

NACA 0012 - SCHEME : HALF-STAGGERED NACA 0012 - SCHEME : HALF-STAGGERED MACH 0.000 ALPHA 0.000 MACH 0.000 ALPHA 1.000 CL 0.0006 CD -0.0001 CM -0.0001 CL 0.1211 CD -0.0001 CM -0.0014 GRID 161X33 NCYC 150 RES0.169E-06 GRID 161X33 NCYC 150 RES0.212E-06

++ +++ + ++ + ++ ++ ++ + ++ ++ ++ + ++ ++ ++ ++ + ++ ++ ++ ++ ++ Cp + ++ ++ ++ ++ ++ ++ + ++++++ + + + + + ++++ + + + + + ++ ++ + + + + ++ ++ + + + + + ++ + + + + + + + + ++ ++ + + + + + ++ ++ + +++ + + ++ + ++ + + + + + ++ + ++ + + ++ + + + + + + ++ ++ + ++ 0.1E+01 0.8E+00 0.4E+00 -.2E-15 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01

NACA 0012 - SCHEME : HALF-STAGGERED MACH 0.000 ALPHA 3.000 Figure 3.6: Flow over a NACA 0012CL 0.3620 CD airfoil -0.0001 CM -0.0040 at 0,1 and 3 degrees using a half-staggered scheme GRID 161X33 NCYC 150 RES0.366E-06 CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 62

Total Pressure Error at 0 deg angle of attack 3 cell−center cell−vertex 2 half−staggerred

−1

−2

Percentage Error −3

−4

−5

−6

−7 0 20 40 60 80 100 120 140 160 180 Points along the airfoil surface,Lower surface trailing edge to upper surface trailing edge

Figure 3.7: Total Pressure losses on the airfoil surface at 0o

Total Pressure Error at 1 deg angle of attack 4 cell−center cell−vertex 3 half−staggerred

−1

Percentage Error −2

−3

−4

−5

−6 0 20 40 60 80 100 120 140 160 180 Points along the airfoil surface,Lower surface trailing edge to upper surface trailing edge

Figure 3.8: Total Pressure losses on the airfoil surface at 1o CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 63

Total Pressure Error at 3 deg angle of attack 6 cell−center cell−vertex half−staggerred

0 Percentage Error

−2

−4

−6 0 20 40 60 80 100 120 140 160 180 Points along the airfoil surface,Lower surface trailing edge to upper surface trailing edge

Figure 3.9: Total Pressure losses on the airfoil surface at 3o

Total Pressure Error at 5 deg angle of attack 4 cell−center cell−vertex half−staggerred 2

−2

−4 Percentage Error

−6

−8

−10 0 20 40 60 80 100 120 140 160 180 Points along the airfoil surface,Lower surface trailing edge to upper surface trailing edge

Figure 3.10: Total Pressure losses on the airfoil surface at 5o CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 64

Twisted inflow with boundary layer profile 10 m

Main Sail

24 m

Jib

2.3 m

10 m

Figure 3.11: Sail geometry CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 65

Cp + ++ +++ Cp ++ +++ +++ ++++ ++++ +++++++ +++++ +++++++ ++++++++ ++++++ +++++++++++ ++++++ +++++++++++++++++++++++++++++ +++++++++ ++++++++++++++++++++++ + ++++++++++++ ++++ +++++++++++++++++++++++++++++++ ++ +++ + + +

+++ 1.00 0.50 0.00 -0.50 -1.00 -1.50 -2.00 1.00 0.50 0.00 -0.50 -1.00 -1.50 -2.00

M2F TNZ M2F TNZ ALPHA 18.000 Z 0.023 ALPHA 18.000 Z 0.336 CL 0.7986 CD 0.2178 CM -0.3186 CL 0.7677 CD 0.0882 CM -0.3240 NCYC 200 RES0.914E-01 NCYC 200 RES0.914E-01

+++++++++++++++ ++++++ +++++++ ++++ ++++ ++++ ++++ +++ +++++ ++++++ + +++++ +++++ ++ +++++ ++++++ +++++++ ++++++ ++ ++++

+ ++ +++ Cp ++++ ++++++ +++++++ +++++++ ++++++++ + +++++++++++ ++++++++++ ++++++++++++++++++++++++++++++++++++++++ 1.00 0.50 0.00 -0.50 -1.00 -1.50 -2.00

M2F TNZ ALPHA 18.000 Z 0.526 Figure 3.12: Pressure distributionsCL 0.7794 along CD 0.0495 CM sections -0.3384 at 1, 25 and 85 percent of the height of main sail NCYC 200 RES0.914E-01 CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 66

Lift and Drag distribution along the height of the sail 0.9 Cl Cd 0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Height along the span

Figure 3.13: Spanwise force distributions

Variation of Lift and Drag with wind incidence 1.3

1.2 22.5 deg

20.0 deg 1.1 17.5 deg 1

15.0 deg 0.9

Cl 0.8 12.5 deg

0.7 10 deg 0.6

7.5 deg 0.5

0.4 5 deg

0.3 0.05 0.1 0.15 0.2 0.25 0.3 Cd

Figure 3.14: Variation of Lift and Drag with wind incidence CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 67

Effect of mast on variation of Lift and Drag with wind incidence 1.6 without mast with mast

1.4

1.2 22.5 deg 20.0 deg

17.5 deg 1 15.0 deg Cl

0.8 12.5 deg

10 deg 0.6

7.5 deg

0.4 5 deg

0.2 0.05 0.1 0.15 0.2 0.25 0.3 Cd

Figure 3.15: Eﬀect of mast on variation of Lift and Drag with wind incidence

Effect of heel on variation of Lift and Drag with wind incidence 1.3

1.2 22.5 deg

20.0 deg 1.1 17.5 deg 1

15.0 deg 0.9

Cl 0.8 12.5 deg

0.7 10 deg 0.6

7.5 deg 0.5 no heel 25 degrees heel

0.4 5 deg

0.3 0.05 0.1 0.15 0.2 0.25 0.3 Cd

Figure 3.16: Eﬀect of heeling angle on variation of Lift and Drag CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 68

twist, camber and chord distribution of the head sail

25 twist distribution in deg chord distribution in m camber distribution as % of chord

1 2 3 4 5 6 7 8 9 10 11 12 Height along the span

Figure 3.17: Twist,camber and chord distribution of the head sail

twist, camber and chord distribution of the main sail

twist distribution in deg chord distribution in m camber distribution as % of chord 20

0 10 15 20 25 30 35 40 45 50 Height along the span

Figure 3.18: Twist,camber and chord distribution of the main sail CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 69

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + Cp + + + Cp + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 1.00 0.50 0.00 -0.50 -1.00 -1.50 -2.00 -2.50 -3.00 1.00 0.50 0.00 -0.50 -1.00 -1.50 -2.00 -2.50 -3.00

M2F TNZ M2F TNZ ALPHA 19.000 Z 2.587 ALPHA 19.000 Z 9.509 CL 0.8872 CD 0.2378 CM -0.5607 CL 0.8018 CD 0.1745 CM -0.5067 NCYC 500 RES0.180E-07 NCYC 500 RES0.180E-07

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + Cp + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 1.00 0.50 0.00 -0.50 -1.00 -1.50 -2.00 -2.50 -3.00

M2F TNZ ALPHA 19.000 Z 16.409 Figure 3.19: Pressure distributionsCL 0.5918 along CD 0.1136 CM sections -0.4305 at 1, 25 and 85 percent of the height of head sail NCYC 500 RES0.180E-07 CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 70

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Cp

Cp + + + + + + + + + + ++ + + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 1.00 0.50 0.00 -0.50 -1.00 -1.50 -2.00 -2.50 -3.00 1.00 0.50 0.00 -0.50 -1.00 -1.50 -2.00 -2.50 -3.00

M2F TNZ M2F TNZ ALPHA 19.000 Z 3.885 ALPHA 19.000 Z 10.532 CL 0.8119 CD 0.3222 CM -0.5526 CL 0.7847 CD 0.2737 CM -0.5064 NCYC 500 RES0.180E-07 NCYC 500 RES0.180E-07

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Cp + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 1.00 0.50 0.00 -0.50 -1.00 -1.50 -2.00 -2.50 -3.00

M2F TNZ ALPHA 19.000 Z 18.929 Figure 3.20: Pressure distributionsCL 0.9173 along CD 0.2608 CM sections -0.5521 at 1, 25 and 85 percent of the height of the main sail NCYC 500 RES0.180E-07 CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 71

Lift and Drag distribution along the height of the head sail 1.2 Cl Cd

0.8

0.6

0.4

0.2

−0.2 0 5 10 15 20 25 Height along the span

Figure 3.21: Spanwise force distributions on the head sail

Lift and Drag distribution along the height of the head sail 1.4

1.2

Cl Cd 0.8

0.6

0.4

0.2

0 0 5 10 15 20 25 30 35 Height along the span

Figure 3.22: Spanwise force distributions on the main sail CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 72

AIRPLANE AIRPLANE

CP from -1.0000 to -0.5000 CP from -0.6000 to -0.1000

AIRPLANE AIRPLANE

CP from -1.0000 to -0.5000 CP from -0.6000 to -0.1000

Figure 3.23: Pressure distribution over the pressure and suction side of the head and sail combination CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 73

Deformed and original section geometry along the height of the head sail 2 Original Deformed

1.5

0.5

0 Z = 3.6 Z = 8 Z = 14 Z = 18 Z = 25

−0.5 2 4 6 8 10 12 14

Figure 3.24: Original and deformed sail sections for the head sail

Deformed and original section geometry along the height of the main sail 1.2 Original Deformed

0.8 Z = 19

Z = 24 Z = 14 0.6

0.4 Z = 8.5

0.2 Z = 31

Z = 5 0

−0.2 10 11 12 13 14 15 16 17 18 19 20

Figure 3.25: Original and deformed sail sections for the main sail CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 74

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + + ++ Cp + Cp + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 1.00 0.50 0.00 -0.50 -1.00 -1.50 -2.00 -2.50 -3.00 1.00 0.50 0.00 -0.50 -1.00 -1.50 -2.00 -2.50 -3.00

M2F TNZ M2F TNZ ALPHA 19.000 Z 2.587 ALPHA 19.000 Z 9.509 CL 0.8897 CD 0.1600 CM -0.4765 CL 1.0856 CD 0.0757 CM -0.5412 NCYC 500 RES0.538E-02 NCYC 500 RES0.538E-02

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ + + + + + + + + Cp + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 1.00 0.50 0.00 -0.50 -1.00 -1.50 -2.00 -2.50 -3.00

M2F TNZ ALPHA 19.000 Z 16.409 Figure 3.26: Pressure distributionsCL 1.0096 along CD -0.0163 CM sections -0.5429 at 1, 25 and 85 percent of the height of head sail after aeroelastic analysisNCYC 500 RES0.538E-02 CHAPTER 3. ANALYSIS OF SAIL CONFIGURATIONS 75

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Cp Cp + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 1.00 0.50 0.00 -0.50 -1.00 -1.50 -2.00 -2.50 -3.00 1.00 0.50 0.00 -0.50 -1.00 -1.50 -2.00 -2.50 -3.00

M2F TNZ M2F TNZ ALPHA 19.000 Z 3.885 ALPHA 19.000 Z 10.532 CL 0.7112 CD 0.2590 CM -0.4177 CL 0.9060 CD 0.2505 CM -0.4739 NCYC 500 RES0.538E-02 NCYC 500 RES0.538E-02

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Cp

++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 1.00 0.50 0.00 -0.50 -1.00 -1.50 -2.00 -2.50 -3.00

M2F TNZ ALPHA 19.000 Z 18.929 Figure 3.27: Pressure distributionsCL 1.1717 along CD 0.2199 CM sections -0.5570 at 1, 25 and 85 percent of the height of main sail after aeroelastic analysisNCYC 500 RES0.538E-02 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 optimized (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 effective 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

∂IT ∂IT δI = δw + δF. (4.1) ∂w ∂F CHAPTER 4. AERODYNAMIC SHAPE OPTIMIZATION 78

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 dependence 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 · ¸ · ¸ ∂R ∂R δR = δw + δF = 0. (4.3) ∂w ∂F

Next, introducing a Lagrange Multiplier ψ, we have µ· ¸ · ¸ ¶ ∂IT ∂IT ∂R ∂R δI = δw + δF − ψT δw + δF , ∂w ∂F ∂w ∂F which can be rearranged as µ · ¸¶ µ · ¸¶ ∂IT ∂R ∂IT ∂R δI = − ψT δw + − ψT δF, ∂w ∂w ∂F ∂F

Choosing ψ to satisfy the adjoint equation · ¸ ∂R T ∂I ψ = , (4.4) ∂w ∂w the ﬁrst term is eliminated and hence

δI = GδF, (4.5) where · ¸ ∂IT ∂R G = − ψT . (4.6) ∂F ∂F CHAPTER 4. AERODYNAMIC SHAPE OPTIMIZATION 79

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 ∂xi Kij = ,J = det(K), ∂ξj Then the transformed equations are

∂F ∂(S f ) i = ij j = 0. ∂ξi ∂ξi

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 CHAPTER 4. AERODYNAMIC SHAPE OPTIMIZATION 80

cost function has the form Z 1 2 I = (p − pt) dS 2 B 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 Z Z 1 2 δI = (p − pt)δpdS + (p − pt) dδS. B 2 B

The variation in the ﬂow solution can be expressed as

∂ (δFi(w)) = 0. ∂ψi

Here the ﬂux changes are

δFi = δSijfj + Ciδw, where ∂f C = S j . i ij ∂w Consequently one can augment the cost variation by Z ∂δF ψT i dξ, D ∂ξi which can be integrated by parts to obtain

Z Z Z T T ∂δFi T ∂ψ ψ dξ = niψ δFidξB − δFidξ. D ∂ξi B D ∂ξ

Now choose ψ to satisfy the adjoint equation

T ∂ψ Ci = 0, ∂ξi CHAPTER 4. AERODYNAMIC SHAPE OPTIMIZATION 81

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

Z ZZ Z T 1 2 ∂ψ (p − pt) dδS − (δS21ψ2 + δS22ψ3 + δS23ψ4) pdξ1dξ3 − (δSijfj)dξ, 2 B B D ∂ξ 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

∂f δF = δS f + S j δw. i ij j ij ∂w

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

∗ ∂w δw = δw = δxj, ∂xj and since ∂δF i = 0, ∂ξ it follows that Z Z T ∂(δSijfj) T ∂fj ∗ ψ dξ = − ψ Sij δw dξ, D ∂ξi D ∂w CHAPTER 4. AERODYNAMIC SHAPE OPTIMIZATION 82

or Z Z T ∂(δSijfj) ∂w ψ dξ = Ci δxjdξ. D ∂ξi D ∂xj A similar relationship can be derived in the general case with boundary movement [73]. Now, Z Z ∂ ψT δRdξ = C (δw − δw∗)dξ ∂ξ i D ZD i ∂ψT = C (δw − δw∗)dξ ∂ξ i ZD i T ∗ = ψ Ci(δw − δw )dξB. (4.7) B

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:

ZZ T ∗ δI = ψ (δS2jfj + C2δw )dξ1dξ3 − ZZ B

(δS21ψ2 + δS22ψ3 + δS23ψ4)pdξ1dξ3 (4.8) B

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 savings in the computational cost. The discrete adjoint does not provide for such a transformation. CHAPTER 4. AERODYNAMIC SHAPE OPTIMIZATION 83

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 ∂ψ − [Ai] = 0, (4.9) ∂t ∂xi where    p     φ  ψ = 1 . (4.10)    φ2    φ3 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 ∂φ − Γ2 i = 0. (4.11) ∂t ∂xi 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. CHAPTER 4. AERODYNAMIC SHAPE OPTIMIZATION 84

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 Zb I = F (y, y0 )dx,

a with ﬁxed end points y(a) and y(b). Under a variation δy(x),

Zb µ ¶ ∂F ∂F 0 δI = δy + δy dx ∂y ∂y0 a Zb µ ¶ ∂F d ∂F = − δydx. ∂y dx ∂y0 a

Thus deﬁning the gradient as

∂F d ∂F g = − , ∂y dx ∂y0 and the inner product as Zb (u, v) = uvdx,

a we ﬁnd that δI = (g, δy),

Then if we set δy = −λg, λ > 0, CHAPTER 4. AERODYNAMIC SHAPE OPTIMIZATION 85

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, y0 , y00 , g = g(y, y0 , y00 ),

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] Z hu, vi = (uv + ²u0 v0 )dx, where ² is a parameter that controls the weight of the derivatives. If we deﬁne a gradient g such that δI = hg, δyi,

Then we have Z δI = (gδy + ²g0 δy0 )dx Z ∂ ∂g = (g − ² )δydx ∂x ∂x = (g, δy) , where ∂ ∂g g − ² = g, ∂x ∂x and g = 0 at the end points. Thus g is obtained from g by a smoothing equation. Now the step yn+1 = yn − λngn CHAPTER 4. AERODYNAMIC SHAPE OPTIMIZATION 86

gives an improvement δI = −λnhgn, gni, 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]

dF = −G, dt where the λ is the time step ∆t. Let A be the Hessian matrix with element

2 ∂Gi ∂ I Aij = = . ∂Fi ∂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

1 I(F) = I∗ + (F − F ∗)A(F − F ∗) + ... 2 CHAPTER 4. AERODYNAMIC SHAPE OPTIMIZATION 87

Correspondingly, G(F) = A(F − F ∗) + ...

As F approaches F ∗, the leading terms become dominant. Then, setting F , the search process approximates dF = −A∗F dt 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 = RT R = I.

Setting v = RT F, the search process can be represented as

dv = −Mv. dt

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 = λP G, where P is a preconditioner for the search. An ideal choice is P = A−1, so that the CHAPTER 4. AERODYNAMIC SHAPE OPTIMIZATION 88

corresponding time dependent process reduces to

dF = −F, dt 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 computationally 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 reduced 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 transonic 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 design 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 inverse 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 incompressible 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 CHAPTER 5. VALIDATION OF THE OPTIMIZATION PROCEDURE AND RESULTS91

Gradients

0.4 syn75 syn82

0.3

0.2

0.1

−0.1

−0.2

−0.3

20 40 60 80 100 120 140 160 points on the airfoil surface, lower trailing edge to upper trailing edge

Figure 5.1: Comparison of the gradients from SYN75 and SYN82

First co−state variable 0.8 syn75 syn82

0.6

0.4

0.2

−0.2

−0.4

−0.6 0 20 40 60 80 100 120 140 160 180 points on the airfoil surface, lower trailing edge to upper trailing edge

Figure 5.2: Comparison of the ﬁrst co-state variable from SYN75 and SYN82 CHAPTER 5. VALIDATION OF THE OPTIMIZATION PROCEDURE AND RESULTS92

Second co−state variable 1 syn75 syn82

0.5

−0.5

−1

−1.5

−2 0 20 40 60 80 100 120 140 160 180 points on the airfoil surface, lower trailing edge to upper trailing edge

Figure 5.3: Comparison of the second co-state variable from SYN75 and SYN82

Third co−state variable 1 syn75 syn82

0.5

−0.5

−1

−1.5

−2 0 20 40 60 80 100 120 140 160 180 points on the airfoil surface, lower trailing edge to upper trailing edge

Figure 5.4: Comparison of the third co-state variable from SYN75 and SYN82 CHAPTER 5. VALIDATION OF THE OPTIMIZATION PROCEDURE AND RESULTS93

Fourth co−state variable

syn75 syn82

0.3

0.2

0.1

−0.1

−0.2

20 40 60 80 100 120 140 160 points on the airfoil surface, lower trailing edge to upper trailing edge

Figure 5.5: Comparison of the fourth co-state variable from SYN75 and SYN82

++ ++ ++ ++ ++ +++ +++ ++++ ++++ ++++ + ++++ ++++ ++++ ++++ ++ + + + + + + + + + + Cp + + + + + + + + ++ + + ++ ++ + ++ + ++ + ++ + + + + ++ + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++++++++++ + ++ + + + + + + + ++ ++ ++ ++ 0.1E+01 0.8E+00 0.4E+00 -.2E-15 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01

RAE 2822 MACH 0.750 ALPHA 0.703 Figure 5.6: Initial pressureCL 0.5999 CD 0.0062distribution CM -0.1334 for the RAE-2822 airfoil GRID 161X33 NDES 0 RES0.785E-05 GMAX 0.000E+00 CHAPTER 5. VALIDATION OF THE OPTIMIZATION PROCEDURE AND RESULTS94

oooo+o+o+o+o+o+o+o+o+o+ooo ++o+o+o+o+o+o+o++++ +++o+o+o+o+o+o++ ++o+oo oo+o+o+++ +ooo oo ++ +oo oo + +o oo++ +o o + +o o + +o o + +o o + o o + + + + + + + + o Cp o + + o + o+ + ooo o o o + + +oo o + o+ o +o o +o o + o+ + +o o + o+ o o+o o + o+ +o o o+ + o+ o+ o o+ +o o+ +o o+ o o+ o+ + o+ o+ +o o+ +o o+ o + o+ + o o o+ + o o+ o+ + o o+ + o o+ o+ + o +o o+ + o o+ + o o+ + o o+o + oo + o+ + +oo oo +o o ++o+o+o+o+o+++ + o +o o+ + o +o + o+ o+ +oo+ o+ +oo+ +o+ o+ +o+ 0.1E+01 0.8E+00 0.4E+00 0.3E-07 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01

RAE 2822 MACH 0.750 ALPHA 0.866 Figure 5.7: DragCL minimization 0.6000 CD 0.0026 CM -0.1243 for the RAE-2822 airfoil GRID 161X33 NDES 40 RES0.191E-04 GMAX 0.569E-02

o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o ooo+o+ +o+o+o+o+o+oo oo++ ++o+o+o+o+ oo+ o+o+ o+ o+ o+ o+ o+ o+ + o+ o+ o o+ +o o+ o+o+o+o +o + o+ +o+ 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+oo+++ o + + +o + o+ oo+ +o+ o+ o +o+ o+ o +o+ o +o+ +o+ 0.1E+01 0.8E+00 0.4E+00 -.2E-15 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01

RAE 2822 : INVERSE TO SHOCK FREE SOLUTION MACH 0.750 ALPHA 0.763 Figure 5.8: Final and targetCL 0.6000 pressure CD 0.0025 CM -0.1242 distribution for the RAE-2822 airfoil GRID 161X33 NDES 40 RES0.466E-05 GMAX 0.255E-02 CHAPTER 5. VALIDATION OF THE OPTIMIZATION PROCEDURE AND RESULTS95

oo ++ oo ++ o + o + +o ++++++ + +++ +o + +++ ++ o + ++ + ++ +o ++ o +ooo ++ + +++o+o+ooo + + +++o+ooo + + +++o+o+ooo o + + ++++o +o +o +o Cp + + Cp +o +o +++ + + + +o +o +++ + + + + + + +o + ++ + + + + + +o + + + + + o o o o o +o +o + + + + oo+o+o+o+ + + + + + o+ o+ o +o + + + + + oo+o+o+o+ + + o+ o +o + + + + oo+ o+o+o++ + o+ o +o + + o+ o+o+ + o+ o +o + + o+ +o+ + o +o + + + o+o+o+o + o+ o + + + + + + o + + oo + +o + + + + + + + +o + + o+ +

++ o + + o ++ +

+ o + ++ ++ o 0.1E+01 0.8E+00 0.4E+00 0.0E+00 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01 0.1E+01 0.8E+00 0.4E+00 0.0E+00 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01

NACA 0012 TO ONERA NACA 0012 TO ONERA MACH 0.000 ALPHA 1.796 MACH 0.000 ALPHA 2.015 Figure 5.9: InitialCL 0.2117 CD and 0.0041 CM ﬁnal -0.0029 pressure distribution, oCL is 0.2116 the CD 0.0059 target CM -0.0053 pressure distribution, x is the computedGRID 160X32 pressureNDES 0 RES0.469E-03 GMAX distribution 0.100E-05 for the redesignedGRID 160X32 NDES 90 RES0.195E-04 airfoil GMAX 0.161E-04 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 distribution 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 recovered 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 CHAPTER 5. VALIDATION OF THE OPTIMIZATION PROCEDURE AND RESULTS96

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 corresponded 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 coefficients 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 CHAPTER 5. VALIDATION OF THE OPTIMIZATION PROCEDURE AND RESULTS97

Cp = -2.0 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

Tip Section: 91.8% Semi-Span Cl: 0.280 Cd:-0.01369 Cm:-0.1042

Cp = -2.0 Cp = -2.0

Root Section: 6.6% Semi-Span Mid Section: 49.2% Semi-Span Cl: 0.241 Cd: 0.02383 Cm:-0.1179 Cl: 0.406 Cd: 0.00203 Cm:-0.1871

Figure 5.10: Initial and ﬁnal pressure and section geometries CHAPTER 5. VALIDATION OF THE OPTIMIZATION PROCEDURE AND RESULTS98

o+o+o+o+o+o+o+o+o+o+o+oo +o+o+o+o+o+ ++o+o o+o+o +o +o +o+o+ +o +o+o+o+o+o+o+o +o+o+o +o +o+o+o+o+o +o+o o +o+o +o +o+o+o ++o o+o o +o+o +o o +o+ + +o+o + +o +o +o +o+o +o +o o +o+o oo o + +o oo+o+o+ + +o+o+o+ + +o o+o ooo++ o+ o oo+o+o+o+o+o+o+o+o +o +o+ o ++ o+ + +o o+o+o++ +o+ o + Cp o o ++o o + o+ oo+o++ +o+ +o Cp +o o + o+ o o + ++o oo + + + o++ o+ +o o ++ o+ o o oo+ o+ o +o oo+ o+ + o++ o+ + +o o + o+ + oo+ o+ o o++ +o o+ o+ +o + oo+ o+ o o+ o o++ o+ o o+ o+ +o oo+ + + + o+ + o++ o+ o o o o+ o+ +o o o+ o+ + o++ o o+ o+ o o+ + +o o+ o + +oo+ o+ + + + +o o+ o+ o o o+ o o + o + o+ + + +o o+ o+ o o o+ + o+ o +o+ o+ o o o+ ++ o o +o+ o o+ + o+ ++ + + oo o + +o o o+ oo+ o + o+ o +o o+ + ++ ++ 0.1E+01 0.8E+00 0.4E+00 0.3E-07 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01 0.1E+01 0.8E+00 0.4E+00 0.3E-07 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01

SHARKX6 (JCV: 16 DEC 99) SHARKX6 (JCV: 16 DEC 99) MACH 0.780 ALPHA 1.400 Z 16.548 MACH 0.780 ALPHA 1.400 Z 66.191 CL 0.2787 CD 0.0120 CM -0.1352 CL 0.4341 CD 0.0018 CM -0.2010 GRID 192X32 NCYC 80 RES0.683E-03 GRID 192X32 NCYC 80 RES0.683E-03

ooo+o+o+o+o+o oo+o+++ +o+o+o+o+o+o+o+ oo++ o+o+o+ o++ o+o+o o+ +o+o o+ +o+ o+ o+o + +o +o o +o +o + oooooo +o Cp oooo oo +o oooo +++ + + + + + + + o o +o +o oo ++++ + + +o +o ooo +++ +o o +o oo+++ + +o + oo++ +o o +o o++ +o +o oo++ + o +o o++ + o +o + o+ + o +o + o+ + o o + o+ + o+ o+ o + o o+ o + + o+ o o+ o+ o o o+ o o ++ o+ + + ++ oo o+ o +o o o+ + ++ 0.1E+01 0.8E+00 0.4E+00 0.3E-07 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01

SHARKX6 (JCV: 16 DEC 99) MACH 0.780 ALPHA 1.400 Z 115.834 Figure 5.11: Initial and ﬁnal pressureCL 0.3122 CD -0.0139 distributions CM -0.1244 at 5 %, 50 % and 95 % of the GRID 192X32 NCYC 80 RES0.683E-03 wing span CHAPTER 5. VALIDATION OF THE OPTIMIZATION PROCEDURE AND RESULTS99

Cp = -2.0 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

Tip Section: 87.8% Semi-Span Cl: 0.262 Cd:-0.00437 Cm:-0.0473

Cp = -2.0 Cp = -2.0

Root Section: 9.8% Semi-Span Mid Section: 48.8% Semi-Span Cl: 0.308 Cd: 0.04594 Cm:-0.1176 Cl: 0.348 Cd: 0.01749 Cm:-0.0971

Figure 5.12: Initial pressure distribution over a NACA 0012 wing CHAPTER 5. VALIDATION OF THE OPTIMIZATION PROCEDURE AND RESULTS100

Cp = -2.0 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

Tip Section: 87.8% Semi-Span Cl: 0.291 Cd:-0.00239 Cm:-0.0489

Cp = -2.0 Cp = -2.0

Root Section: 9.8% Semi-Span Mid Section: 48.8% Semi-Span Cl: 0.294 Cd: 0.03309 Cm:-0.1026 Cl: 0.333 Cd: 0.01115 Cm:-0.0806

Figure 5.13: Final pressure distribution and modiﬁed section geometries along the wing span CHAPTER 5. VALIDATION OF THE OPTIMIZATION PROCEDURE AND RESULTS101

+o +o +o +o +o +o +o+o+o +o +o +o+o+o +o +o +o+o +o+o+o +o+o +o+o+o +o+o+o +o+o+o +o+o +o+o+o +o+o +o+o+o ++o+o ++o+o +o+o+oo +o +o+o+oo +o Cp ++o+o+o Cp ++o+o+o o +o ++o+oo o +o ++o+oo + ++o+oo + ++o+oo +o+oo ooo o +o+oo ooo o ++o oo+o+ + + + +o+o+ o ++o oo+o+ + + + +o+o+ o +o +o oo+ + o+o+ + +o +o oo+ + o+o+ + +o oo+ + o+o+ +o oo+ + o+o+ o ++o o+ + o+ o o ++o o+ + o+ o + oo o+o+ o++o++o + oo o+o+ o++o++o o+o+ o+ +o o+o+ o+ +o o+o+ o+ o+o+ o+ o+o+ +o+ o+o+ +o+ o+o+ o+o+ o+o+ +o+ o+o+ +o+ oo+o+ oo+o+ +ooo+ oo++ +o+ +ooo+ oo++ +o+ +o+ oo++ o +o+ oo++ o o+o+ oo++ o+o+ oo++ oo+o+o+++ +o+ oo+o+o+++ +o+ o o +o+ +o+ +o+ +o+ o o ++ ++ o o +o+ +o+ o+ o+ +o+ +o+ +o +o +o+ +o+ 0.1E+01 0.8E+00 0.4E+00 0.3E-07 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01 0.1E+01 0.8E+00 0.4E+00 0.3E-07 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01

NACA 0012 WING TO ONERA M6 TARGET NACA 0012 WING TO ONERA M6 TARGET MACH 0.840 ALPHA 3.060 Z 0.00 MACH 0.840 ALPHA 3.060 Z 0.00 Figure 5.14: FinalCL 0.2814 CD computed 0.0482 CM -0.1113 and target pressure distributionsCL 0.2814 CD 0.0482 CM -0.1113 at 0 % and 20 %of the wing span GRID 192X32 NDES 50 RES0.162E-02 GRID 192X32 NDES 50 RES0.162E-02

+o+o ++o+o+ +o+o+ oo+o+ + o+ +o o o o+ +o o+o+ + + o+o+o+o+o +o o o +o + +o + o + + + o+o +o o+ o o +o +o +o o +o ++o ++o o +o+o+oo +o ++oo + +o+o +o +o +o+oo + +o +o ++o+o+o +o ++o+o o +o +o+o+o+oo +o+o+oo +o+o+o+o+o+o+o +o +o +o Cp Cp +o oo+o+o+o+o+o+o+ oo+o+o+o+o+o+o oo+o+ + o+o+ o oo+o+ + +o+o oo++ o+o + +o oo+o++ +o+ +o oo+o++ +o+ oo+o++ o+o ooo+o++ o+ ooo+o++ ++o+ ooo+++ o+o+ ooo+++ o +o+ oo ooo+++ +o +o++ oo ooo+++ o +o +o++o+ o+ oo+++ o o o++o o+ oo++ +o o +o+ o+o++ooo++ ++ o++o+ o+o+o+ooo+++ + +o++ + oo++++ +o+ + +++ o++o+ +o +o+ + +o+ o +o+ oo +o+ + +o+ + +o+ o +o+ o +o+ + ++ o+ ++ +o+ o o oo ++ oo ++ + + +o +o + o+ o o ++ ++ o o o o +o+ +o+ +o+ +o+ 0.1E+01 0.8E+00 0.4E+00 0.3E-07 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01 0.1E+01 0.8E+00 0.4E+00 0.3E-07 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01

NACA 0012 WING TO ONERA M6 TARGET NACA 0012 WING TO ONERA M6 TARGET MACH 0.840 ALPHA 3.060 Z 0.40 MACH 0.840 ALPHA 3.060 Z 0.60 Figure 5.15: FinalCL 0.3269 CD computed 0.0145 CM -0.0865 and target pressureCL distributions 0.3356 CD 0.0081 CM -0.0735 at 40 % and 60 % of the wing spanGRID 192X32 NDES 50 RES0.162E-02 GRID 192X32 NDES 50 RES0.162E-02 CHAPTER 5. VALIDATION OF THE OPTIMIZATION PROCEDURE AND RESULTS102

+o+o+o+o+ ++o o+o++ +o+o o oo+o+ +o+o o+o+o+ + +o+ + o+o+ o o+o o o+o+ +o+o + o+o + +o+o+o+ +o o o+o+o+o+o+o+o+o+ o o+ +o o+ + +o o +o+o+o+o o +o +o +o +o + o +o o +o + +o + Cp Cp o +++ +o+oooo+o+o+o oooo+oo + +o+o +o +o ooo+o+ + + + +o+ +o+o o +o +o +o ooo+++ +o+o +o +o + ooo+++ +++o++ o +o o +o ooo+++ o o++o + + +o +o ooo+++ +o +o o++o oo+o+o+o +o ooo+++ ++o++ o o + o +o+o+o + oo oo++ +o+o o++o oo + o+ +o+o+o o +o+o+oo++ ++o++ ooooooo+o+++ + +o+o + o o+o++++ o++o ooo++++++ +o+o o + + o+ +o+ ooo+++ +o+o+o +o oo +o+ ooo ooo+++ +o+o+o +o+ o++o+oo+++ + o+ o+ + + + +++ o+ o+ +o o+ o+ o+ o + o+ o+ +o o +o+ ++ +o+ o + o + o +o+ + + o o+ +o + o+ o + o +o ++ +o o + +o+ o+ +o+ ++ o 0.1E+01 0.8E+00 0.4E+00 0.3E-07 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01 0.1E+01 0.8E+00 0.4E+00 0.3E-07 -.4E+00 -.8E+00 -.1E+01 -.2E+01 -.2E+01

NACA 0012 WING TO ONERA M6 TARGET NACA 0012 WING TO ONERA M6 TARGET MACH 0.840 ALPHA 3.060 Z 0.80 MACH 0.840 ALPHA 3.060 Z 1.00 Figure 5.16: FinalCL 0.3176 CDcomputed 0.0011 CM -0.0547 and target pressure distributionsCL 0.4846 CD 0.0178 CM -0.1518 at 80 % and 100 % of the wing spanGRID 192X32 NDES 50 RES0.162E-02 GRID 192X32 NDES 50 RES0.162E-02 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 Z 2 I = (p − pt) dB, B 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. CHAPTER 5. VALIDATION OF THE OPTIMIZATION PROCEDURE AND RESULTS103

++oo +o+o + + o o o +o ++oo ++ + +o o o +o + o +o + +o o ++oo + +o+o + +o+o+o

Cp o 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 oo+o+o+o+o+o+o+o+o+o+o+o+o+o o o+o+o+o+ +o+o o +o+o+o+ o+o ++o o +o+o o ++o +o +o + +o+o +o +o +o +o o +o +o o o o o +o o +o +o +o +o +o +o o+ o+ o+ o+ o+ +o o o +o +o +o +o + + + + +o +o +o +o +o +o + o +o +o + o+ o+ o+ +o oo+o +o +o + + +o +o +o +o o +o +o + o+ o+ o+ +o oo+o+o+o++ +o +o o +o o oo+o+o + o+ o+ +o o+o+o++ + +o +o + o oo+o++ o+ + ooo+ oo+o+ +o +o + o+o+ oo+o++ +o o +oo++o+o+o++ +o +o oo oo+o++ o+ o+ +o +o +o+ oo++ +o + +o + +o o+oo++ +o oo o+ o ++ o + + ++ +o+ o o + + + +o o + o+ o o o ++ ++ o+ o+ ++ o o +o+ +o+ +o+ 0.1E+01 0.5E+00 0.0E+00 -.5E+00 -.1E+01 -.2E+01 -.2E+01 0.1E+01 0.5E+00 0.0E+00 -.5E+00 -.1E+01 -.2E+01 -.2E+01

NACA 0012 TO ONERA M6 NACA 0012 TO ONERA M6 ALPHA 3.060 Z 0.000 ALPHA 3.060 Z 0.250 CL 0.1890 CD 0.0186 CM -0.0594 CL 0.2089 CD 0.0063 CM -0.0543 NCYC 10 RES0.194E-02 DESIGN CYCLE 50 NCYC 10 RES0.194E-02 DESIGN CYCLE 50

++o +o+o + ++oo o o o + o + + + +o +o +o o +o + +o + +o +o o + +o o +o +o ++o +o o+o+ +o+ +o o+o+ + o+o o+o+ +o+o o+o+o+ +o+o Cp o+o+ 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 oo +o +o +o + + + +o +o +o + + oo+o+o++ +o +o o o +o +o oo+o+o++ + +o o +o o o +o +o o +o ooo++ + +o o+ + +o o +o +o +o +o +o +o + + + +o +o +o +o +o +o oo+++ o o+ +o +o o+o+o+o+o +o + o+ +o +o o o oo+oo++ooo++ +o + o+o+o+o+ + + +o o+ +o o+ ++ o oo+o+ +o o +o + oo+o++ + o +o +o+ o oo++ + + + ooo+o++ o ++o o + + + o o o +o+ + o o +o + + o o + + ++ o o o+ + +o+ +o +

+o+ +o+ 0.1E+01 0.5E+00 0.0E+00 -.5E+00 -.1E+01 -.2E+01 -.2E+01 0.1E+01 0.5E+00 0.0E+00 -.5E+00 -.1E+01 -.2E+01 -.2E+01

NACA 0012 TO ONERA M6 NACA 0012 TO ONERA M6 ALPHA 3.060 Z 0.750 ALPHA 3.060 Z 1.000 Figure 5.17: FinalCL 0.2299 CD computed 0.0032 CM -0.0573 and target pressure distributionsCL 0.3340 CD 0.0219 CM -0.1151 at 0, 25, 75 and 100 % of the wing spanNCYC 10at RES0.194E-02 3 degrees DESIGN CYCLE 50 angle of attack NCYC 10 RES0.194E-02 DESIGN CYCLE 50 CHAPTER 5. VALIDATION OF THE OPTIMIZATION PROCEDURE AND RESULTS104

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. CHAPTER 5. VALIDATION OF THE OPTIMIZATION PROCEDURE AND RESULTS105

+ + + + + + + + + + + + + + o o o o o + + + + + + o o o o + + + + + + + + o o o o + + + + o o o + + o + o o o o o o + o o + + o o o o + + o o + + o o + o + o o + + o o + + o o + o o Cp + 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 0.1E+01 0.5E+00 0.0E+00 -.5E+00 -.1E+01 -.2E+01 -.2E+01 0.1E+01 0.5E+00 0.0E+00 -.5E+00 -.1E+01 -.2E+01 -.2E+01

TNZ : INVERSE DESIGN TNZ : INVERSE DESIGN ALPHA 19.000 Z 4.975 ALPHA 19.000 Z 10.981 CL 0.7850 CD 0.2799 CM -0.4489 CL 0.9992 CD 0.2965 CM -0.5381 NCYC 30 RES0.144E-02 DESIGN CYCLE 20 NCYC 30 RES0.144E-02 DESIGN CYCLE 20

+ + + + + + + + o o + + + + + + + + + + + + + + + + + + + o+ + + + + + o + o + + + o + o o o o o o o o o o o o o o o o o o o o o o o o + o + o+ o o o o + + o o + o o o o o o + o o+ o + o+ o+ o+ o+ + o o+ o + +o o+ +o o+ +o o o+ +o o+ + +o o+ +o o+ +o + o+ Cp +o 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

0.1E+01 0.5E+00 0.0E+00 -.5E+00 -.1E+01 -.2E+01 -.2E+01 0.1E+01 0.5E+00o 0.0E+00 -.5E+00 -.1E+01 -.2E+01 -.2E+01

TNZ : INVERSE DESIGN TNZ : INVERSE DESIGN ALPHA 19.000 Z 23.366 ALPHA 19.000 Z 26.494 Figure 5.18: InitialCL 1.4461 CD 0.2735 (o) CM -0.6545 and ﬁnal(+,x) pressure distributionCL 1.5637 CD 0.2928 CM -0.6885 at 15, 32, 75 and 85% height on theNCYC main 30 RES0.144E-02 sail DESIGN CYCLE 20 NCYC 30 RES0.144E-02 DESIGN CYCLE 20 CHAPTER 5. VALIDATION OF THE OPTIMIZATION PROCEDURE AND RESULTS106

Initial and deformed sections at 15 percent height Initial and deformed sections at 32 percent height

Initial Initial Redesign Redesign 3 3

2 2

1 1 y y

0 0

−1 −1

−2 −2

−3 11 12 13 14 15 16 17 18 19 11 12 13 14 15 16 17 18 x x Initial and deformed sections at 75 percent height Initial and deformed sections at 85 percent height

Initial Initial 3 Redesign 2.5 Redesign

2.5 2

2 1.5 1.5

1 1

y 0.5 y 0.5

0 0

−0.5 −0.5

−1 −1 −1.5

−1.5 −2

11 12 13 14 15 16 17 11.5 12 12.5 13 13.5 14 14.5 15 15.5 16 16.5 x x 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 development 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

Vmg

Boat Speed Apparent Course

Vs Β

R Fh Fr

Heeling Force Driving Force Hull resistance

Λ Θ

leeway heel angle Sail Area E L/D

Hull Side Stability Sail Polar Diagram Force

Ε Fs /R Sail Plan hull

Camber

Keel A.R. Area

A.R.

Figure 6.1: Components of the overall design process for upwind sails Bibliography

[1] S. Collie, M. Gerritsen and P. Jackson, A Review of Turbulence Modeling for Use in Sail Flow Analysis School of Engineering Report, No. 603, University of Auckland, New Zealand.

[2] T. Doyle, M. Gerritsen and G. Iaccarino, Optimization of Yard Sectional Shape and Conﬁguration of a Modern Clipper Ship, Proceedings of the 17th International HISWA Symposium of Yacht Design and Yacht Construction, November, 2002.

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