Numerical Analysis and Design of Upwind Sails
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NUMERICAL ANALYSIS AND DESIGN OF UPWIND SAILS a dissertation submitted to the department of aeronautics and astronautics and the committee on graduate studies of stanford university in partial fulfillment of the requirements for the degree of doctor of philosophy Sriram Shankaran April 2005 °c Copyright 2005 by Sriram Shankaran All Rights Reserved ii I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Antony Jameson (Principal Adviser) 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 equa- tions are increasingly being used by competing syndicates in races like the Americas Cup. For sail configurations, this desire stems from a need to understand the influ- ence of the mast on the boundary layer and pressure distribution on the main sail, the effect of camber and planform variations of the sails on the driving and heeling force produced by them and the interaction of the boundary layer profile of the air over the surface of the water and the gap between the boom and the deck on the performance of the sail. Traditionally, experimental methods along with potential flow solvers have been widely used to quantify these effects. While these approaches are invaluable either for validation purposes or during the early stages of design, the potential advantages of high fidelity computational methods makes them attractive candidates during the later stages of the design process. The aim of this study is to develop and validate numerical methods that solve the inviscid field equations (Euler) to simulate and design upwind sails. The three dimensional compressible Euler equations are modified using the idea of artificial com- pressibility and discretized on unstructured tetrahedral grids to provide estimates of lift and drag for upwind sail configurations. Convergence acceleration techniques like multigrid and residual averaging are used along with parallel computing platforms to enable these simulations to be performed in a few minutes. To account for the elastic nature of the sail cloth, this flow solver was coupled to NASTRAN to provide v estimates of the deflections caused by the pressure loading. The results of this aeroe- lastic simulation, showed that the major effect of the sail elasticity, was in altering the pressure distribution around the leading edge of the head and the main sail. Adjoint based design methods were developed next and were used to induce changes to the camber distribution of the main sail. The goal of the design process was to reduce the leading edge suction peaks that were considered to be detrimental to the growth of the boundary layer. The deflected shape of the sails obtained from the aeroelastic simulation were used by the design process. The design process re- sulted in an camber distribution that allowed smooth entry of the flow through the leading edge of the main sail thereby, reducing the leading edge suction peaks. vi Acknowledgments vii Contents iv Abstract v Acknowledgments vii 1 Introduction 1 1.1 Design Requirements of Racing Yachts . 1 1.2 Models of Fluid Flow . 3 1.3 Analysis with CFD . 5 1.4 Aeroelastic Analysis . 8 1.5 Optimum Design . 10 1.6 Aerodynamic Shape Optimization . 12 1.7 Outline of this study . 16 2 Discretization of Governing Equations 19 2.1 Overview of the Numerical Scheme . 19 2.2 Finite Volume Discretization of the Flow equations . 21 2.3 Spatial Discretization . 23 2.4 Staggered Meshes . 24 2.5 Implementation of the Cell-Vertex Scheme . 29 2.6 Artificial Diffusion . 29 2.7 Analysis of Artificial Compressibility . 30 2.8 Time Integration . 31 viii 2.9 Multigrid Acceleration . 32 2.10 Parallel Implementation . 35 2.10.1 Domain Decomposition, Load Balancing . 35 2.10.2 Parallel implementation of the multigrid algorithm . 38 2.10.3 Speedup of the Parallel Implementation . 39 2.11 Governing equations and analysis of the structural model . 39 2.11.1 Structural Model of the Sail . 41 2.12 Aeroelastic Coupling and Mesh Deformation . 42 3 Analysis of Sail Configurations 46 3.1 Low Mach number, high angle of attack simulations with a compress- ible flow solver . 47 3.1.1 Multi-Element airfoils . 47 3.1.2 Sail simulations . 47 3.2 Effect of Numerical Discretization and diffusion on artificial compress- ibility methods . 48 3.3 Validation of the parallel implementation . 49 3.4 Single and multi-element sail computations with artificial compress- ibility methods . 50 3.4.1 Characteristics of the main sail . 50 3.4.2 Characteristics of the Head and Main sail combination . 52 3.5 Aeroelastic simulations for single and multi-element foils . 54 4 Aerodynamic Shape optimization 76 4.1 The general formulation of the Adjoint Approach to Optimal Design . 77 4.2 Adjoint and Gradient formulations . 79 4.2.1 Adjoint Equations for the Euler equations modified by artificial compressibility method . 83 4.2.2 The need for a Sobolev inner product in the definition of the gradient . 84 4.3 Analysis of the Optimization Procedure . 86 4.4 Mesh movement . 88 ix 4.5 Parallel Implementation . 88 5 Validation of the Optimization Procedure and Results 89 5.1 Shape optimization for airfoils in compressible flow . 90 5.2 Shape optimization of airfoils in incompressible flow . 90 5.3 Three dimensional shape optimization of wings in compressible flow . 95 5.4 Inverse design of wings in incompressible flow . 96 5.5 Inverse design for sail geometries . 96 6 Conclusions 107 6.0.1 Aerodynamic and Aeroelastic analysis . 107 6.0.2 Aerodynamic design . 108 Bibliography 110 x List of Tables xi List of Figures 2.1 Dual mesh representation of the control volume . 25 2.2 Nodal formulation of the finite volume scheme . 25 2.3 Evaluation of fluxes in three dimensions . 26 2.4 Control volume for cell-vertex schemes in three dimensions . 26 2.5 Staggered arrangement of flow variables . 28 2.6 Half-staggered arrangement of flow variables . 28 2.7 Interpolation coefficients for use in the multigrid cycle . 34 2.8 Transfer of solution, residuals and corrections between the fine and coarse mesh . 35 2.9 Domain decomposition of a rectangular region using a bisection method 36 2.10 Halo nodes and the distribution of edges along processor boundaries 37 2.11 Speedup from the parallel implementation . 37 2.12 Boundary conditions for the main sail . 42 2.13 Boundary conditions for the head sail . 43 3.1 Grid and Pressure distribution over a multi-element airfoil geometry at a M = 0.2 and α = 8.2 degrees . 56 3.2 Cp distribution at two sections and convergence history of the com- pressible flow solver . 57 3.3 Potential flow solution from FLO1 at 0,1 and 3 degrees . 58 3.4 Flow over a NACA 0012 airfoil at 0,1 and 3 degrees using a cell-centered scheme . 59 3.5 Flow over a NACA 0012 airfoil at 0,1 and 3 degrees using a nodal scheme 60 xii 3.6 Flow over a NACA 0012 airfoil at 0,1 and 3 degrees using a half- staggered scheme . 61 3.7 Total Pressure losses on the airfoil surface at 0o . 62 3.8 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 .