Geometric Parameterisation and Aerodynamic Shape Optimisation Name: Feng Zhu Supervisor: Prof Ning Qin Email: [email protected] Address: Department of Mechanical Engineering University of Sheffield S1 3JD PhD Thesis i Abstract Aerodynamic optimisation plays an increasingly important role in the aircraft industry. In aerodynamic optimisation, shape parameterisation is the key technique, since it determines the design space. The ideal parameterisation method should be able to provide a high level of flexibility with a low number of design variables to reduce the complexity of the design space. In this work, the Class/Shape Function Transformation (CST) method is investigated for geometric representation of an entire transport aircraft for the purpose of aerodynamic optimisation. It is then further developed for an entire passenger transport aircraft, including such components as the wing, horizontal tail plane, vertical tail plane, fuselage, belly fairing, wingtip device, nacelle, flap tracking fairing and pylon. This work presents the parameterisation of these components in detail using the CST methods for the reference of future aerodynamic optimisation work. The intersection line calculation method between CST components is presented for future entire aircraft optimisation. The performance of the CST has been tested as well, and it found a few drawbacks of the CST methods; for example, it cannot provide some key intuitive design parameters and can lose the accuracy in the wing leading edge area. Therefore, two derivatives of the CST method are proposed: one is called the intuitive CST method (iCST), which is to transform the CST parameters to intuitive design parameters; the other is called the RCST method, which is able to increase the fitting accuracy of the original CST method with fewer design variables. Their performances are studied by comparing them regarding their accuracy in inversely fitting a wide range of aerofoils. Finally, the CST method is also developed to represent the shock control bump, which has better curvature continuity than cubic polynomials. The aerodynamic optimisation study based on adjoint approaches is carried out using the above parameterisation methods. Optimisation was performed on two-dimensional cases to make a preliminary investigation of the performances of the above parameterisation methods. The results showed that all of CST, iCST and RCST parameterisation methods are able to successfully reduce the drag. The results of the CST methods showed the i lower order CST is able to provide fast convergence, and the high order CST is able to provide more flexibility and more local control of the shape to reach better optimal solution. The iCST providing intuitive parameters is improving the process of setup constraints, which is useful for aerofoil optimisation. The RCST showed good performance in aerodynamic optimisation in terms of convergence rate, number of design variables, low order of polynomials and smoothness of the shape. This work provides a reference to designer for choosing suitable parameterisation method in these three methods regarding specific requirement. The shock control bump optimisation on 2D aerofoil is performed to compare three shock control bump parameterisation methods. The results showed the CST parameterisation method is promising for shock control bump optimisation. Three-dimensional optimisation tests, including wing and winglet drag minimisation, were performed using the above parameterisation methods. The results showed that the CST methods are able to handle three-dimensional wing optimisation. It also investigated the effect of the order of CST method in optimisation. The results showed the lower order CST already performed well in optimisation in terms of optimal results and convergence rate. The optimisation also discussed the importance of using Cmx constraint in aerodynamic optimisation. In the winglet test cases, it showed the CST methods and adjoint approach are able to perform winglet optimisation. The drag of four winglets are successfully reduced. The downward winglet showed the potential benefits in terms of lower wing root bending momentum. At the end, the shock control bump optimisation using CST method on 3D wing has been performed. The results showed the mesh adjoint methods is able to identify the sensitive area for deploying shock control bumps and the CST shock control bump successfully reduced the wave drag. ii Acknowledgements I sincerely appreciate my supervisor Prof. Ning Qin for his guidance. He offered me this opportunity and led me into a very interesting aerodynamic design and optimisation area, and has provided me with endless helpful advice to overcome all difficulties throughout the years. I also would like to thank present and former colleagues in the Aerodynamics Group at the University of Sheffield for their fruitful discussions. This work was funded by a Scholarship from Airbus within the CFMS programme. I would like to thank Stefano Tursi, Murray Cross, Francois Gallard and Kasidit Leoviriyakit from Airbus. This work would have been impossible without their support and help. I would also like to thank DLR(German Aerospace Research Center) for providing the experimental version of the TAU solver with flow and mesh adjoint capability. I am indebted to Dr Caslav Illic for his availability and all the help and explanations he gave me on the TAU solver. Finally, my gratitude goes to my family for always supporting me through difficult times over the past few years. iii List of Contents Abstract… ............................................................................................................................ i Acknowledgements ............................................................................................................ iii List of Contents .................................................................................................................. iv List of Figures .................................................................................................................. viii List of Tables .................................................................................................................. xvii Nomenclature ................................................................................................................. xviii Chapter 1 Introduction ........................................................................................................ 1 1.1 Background ......................................................................................................................... 1 1.2 Outline of thesis .................................................................................................................. 4 PART I Chapter 2 Literature Review of Geometric Parameterisation ............................................. 6 2.1 Discrete methods................................................................................................................ 7 2.2 Analytical methods ............................................................................................................. 8 2.3 Polynomial, spline methods, CAD-based and free-form deformation ............................. 10 2.4 PARSEC parameterisation methods ................................................................................. 23 2.5 Class/shape function transformation (CST) methods ...................................................... 34 2.6 Comparison of parameterisation methods ...................................................................... 41 Chapter 3 Development of CST and PARSEC Methods in Two-Dimensional Aerofoils 48 3.1 Combination of CST and PARSEC: the intuitive CST method ............................................ 48 3.2 Geometric inverse fitting test and results of iCST methods ............................................. 52 3.2.1. Geometric inverse fitting .......................................................................................... 53 3.2.2. Inverse fitting test results ......................................................................................... 53 3.3 CST method with rational function (RCST) ....................................................................... 58 3.4 Geometric fitting results of RCST ..................................................................................... 60 iv 3.5 Conclusion ........................................................................................................................ 64 Chapter 4 CST Parameterisation Method for the Entire Aircraft ..................................... 66 4.1 Parameterisation for wing type geometries ..................................................................... 66 4.1.1. Standard CST for wing type geometries ................................................................... 66 4.1.2. Fitting accuracy of the standard CST method for a wing .......................................... 70 4.1.3. RCST method for wing type geometries ................................................................... 86 4.1.4. Fitting accuracy of the RCST for a wing .................................................................... 87 4.2 CST parameterisation method for wing tip device ........................................................... 93 4.3 CST Parameterisation for fuselage (simplified forward, mid and tail cone parts) ......... 111 4.3.1 Cylindrical fuselage ................................................................................................. 112 4.3.2 Nose fuselage .......................................................................................................... 113 4.3.3 Rear fuselage........................................................................................................... 116 4.4 CST parameterisation for belly-fairing ............................................................................ 117 4.5 CST parameterisation for the nacelle ............................................................................. 123 4.6 CST parameterisation for flap tracking fairing (FTF) and pylon ...................................... 126 4.7 CST parameterisation for three-dimensional shock bump local modification ............... 130 4.8 Calculation of intersection line ....................................................................................... 136 PART II Chapter 5 Governing Equation and Numerical Solver ................................................... 147 5.1 Governing equation ........................................................................................................ 148 5.2 Reynolds-averaged Navier-Stokes (RANS) simulation and turbulence model ............... 151 5.2.1 Spalart-Allmaras turbulence model ........................................................................ 153 5.3 Finite volume method .................................................................................................... 154 5.4 Central convective fluxes ................................................................................................ 157 5.5 Construction of gradient................................................................................................. 160 5.6 Temporal discretisation .................................................................................................. 162 v Chapter 6 Discrete Adjoint Approach and Numerical Optimisation .............................. 165 6.1 Common methods to calculate sensitivities ................................................................... 167 6.2 Discrete adjoint methods ............................................................................................... 173 6.2.1 Discrete adjoint equation ....................................................................................... 173 6.2.2 Discrete adjoint solver with mesh deformation ..................................................... 177 6.3 Numerical optimisation .................................................................................................. 182 6.4 Mesh deformation .......................................................................................................... 185 6.5 Optimisation framework ................................................................................................ 195 Chapter 7 Optimisation in Two-Dimensions .................................................................. 198 7.1 Two-dimensional aerofoil optimisation ......................................................................... 198 7.2 Shock bump optimisation in the two-dimensional aerofoil ........................................... 210 Chapter 8 Optimisation in Three-Dimensions ................................................................ 217 8.1 Wing optimisation using CST methods ........................................................................... 218 8.1.1 Influence of different order of the CST methods on wing optimisation ................ 218 8.1.2 Wing optimisation with rolling momentum constraint .......................................... 231 8.2 Wing optimisation using RCST methods ......................................................................... 239 8.3 Winglet optimisation ...................................................................................................... 247 8.4 Shock bump optimisation on the wing ........................................................................... 262 Chapter 9 Conclusion and Future Work ......................................................................... 270 9.1 Summary ......................................................................................................................... 270 9.2 Future work .................................................................................................................... 275 References ....................................................................................................................... 277 Publication List ............................................................................................................... 301 Appendices ...................................................................................................................... 302 Appendix A: Derivatives of Bezier curve ................................................................................. 302 Appendix B: Value of rational shape function of RCST method at the trailing edge .............. 303 vi Appendix C Value at boundary of CST function with class parameters N1=1.0 and N2=1.0 .. 305 Appendix D Partial differentiation of geometry of fuselage and belly-fairing ........................ 305 Appendix E Objective function using target lift iteration ........................................................ 308 vii List of Figures Figure 1.1 Transport aircraft fuel efficiency (from Penner 1999) ...................................... 2 Figure 2.1 Bezier curve with control points...................................................................... 12 Figure 2.2 The control box of FFD for ONERA M6 wing (Widhalm et al. 2007) .......... 22 Figure 2.3 PARSEC method parameters definition .......................................................... 24 Figure 2.4 PARSEC parameters for DTE (Sobieczky 1998) ............................................ 27 Figure 2.5 PARSEC parameters for local control bump (Sobieczky 1998) ..................... 28 Figure 3.1 The intuitive CST parameterisation method.................................................... 49 Figure 3.2 Geometric fitting for RAE 2822 using iCST, CST 12 and PARSEC 12 ........ 54 Figure 3.3 Geometric fitting for RAE 5214 using iCST, CST 12 and PARSEC 12 ........ 54 Figure 3.4 Geometric fitting for SC-20714 using iCST, CST 12 and PARSEC 12 ......... 54 Figure 3.5 Geometric fitting for NLF 414F using iCST, CST 12 and PARSEC 12 ......... 55 Figure 3.6 Geometric fitting for NLF 416 using iCST, CST 12 and PARSEC 12 ........... 55 Figure 3.7 Geometric fitting for HSNLF 213 using iCST, CST 12 and PARSEC 12 ...... 56 Figure 3.8 Geometric fitting for S805A using iCST, CST 12 and PARSEC 12 .............. 56 Figure 3.9 Geometric fitting for S809 using iCST, CST 12 and PARSEC 12 ................. 57 Figure 3.10 Geometric fitting for S825 using iCST, CST 12 and PARSEC 12 ............... 57 Figure 3.11 Geometric fitting for RAE 2822 using 6th order RCST................................ 60 Figure 3.12 Geometric fitting for RAE 5214 using 6th order RCST................................ 61 Figure 3.13 Geometric fitting for NASA SC-20714 using 6th order RCST .................... 61 Figure 3.14 Geometric fitting for NLF 414F using 6th order RCST ................................ 61 Figure 3.15 Geometric fitting for NLF 416 using 6th order RCST .................................. 62 Figure 3.16 Geometric fitting for HSNLF 213 using 6th order RCST ............................. 62 Figure 3.17 Geometric fitting for S805A using 6th order RCST ..................................... 62 Figure 3.18 Geometric fitting for S809 using 6th order RCST ........................................ 63 Figure 3.19 Geometric fitting for S825 using 6th order RCST ........................................ 63 Figure 4.1 Wing aerofoil section definition in the CST method ....................................... 67 Figure 4.2 Leading edge x coordinates distribution and error .......................................... 71 Figure 4.3 Leading edge height distribution and error ..................................................... 72 viii Figure 4.4 Non-dimensional trailing edge thickness distributions leading edge and error ........................................................................................................................................... 72 Figure 4.5 Tangential value of twist angle distributions and error ................................... 72 Figure 4.6 Local chord distributions and error ................................................................. 73 Figure 4.7 The error contour of wing inverse fitting with BPOX 6-BPOY 6 (left figure in metre) and BPOX 6-BPOY 10 (right figure in metre) ....................................................... 74 Figure 4.8 The error contour of wing inverse fitting with BPOX 6-BPOY 12 (left figure in metre) and BPOX 10-BPOY 6 (right figure in metre) ....................................................... 75 Figure 4.9 The error contour of wing inverse fitting with BPOX 10-BPOY 10 (left figure in metre) and BPOX 12-BPOY 6 (right figure in metre) ................................................... 75 Figure 4.10 The error contour of wing inverse fitting with BPOX 12-BPOY 10 (left figure in metre) and BPOX 12-BPOY 12 (right figure in metre) ................................................. 75 Figure 4.11 The hybrid mesh of F6 wing for CFD study ................................................. 77 Figure 4.12 The sections index and position on the wing................................................. 78 Figure 4.13 Comparisons of pressure distribution and wing shape on section 1 .............. 79 Figure 4.14 Comparisons of pressure distribution and wing shape on section 2 .............. 80 Figure 4.15 Comparisons of pressure distribution and wing shape on section 3 .............. 80 Figure 4.16 Comparisons of pressure distribution and wing shape on section 4 .............. 81 Figure 4.17 Comparisons of pressure distribution and wing shape on section 5 .............. 82 Figure 4.18 Comparisons of pressure distribution and wing shape on section 6 .............. 83 Figure 4.19 Comparisons of pressure distribution and wing shape on section 7 .............. 83 Figure 4.20 Comparisons of pressure distribution and wing shape on section 8 .............. 84 Figure 4.21 The HTP model using the CST methods ....................................................... 85 Figure 4.22 The VTP model using the CST methods ....................................................... 86 Figure 4.23 The error contour of wing inverse fitting with RCST BPOX 6-BPOY 6 (in metre) ................................................................................................................................ 87 Figure 4.24 Comparisons of pressure distribution and wing shape at 10% span ............. 89 Figure 4.25 Comparisons of pressure distribution and wing shape at 20% span ............. 89 Figure 4.26 Comparisons of pressure distribution and wing shape at 30% span ............. 90 Figure 4.27 Comparisons of pressure distribution and wing shape at 40% span ............. 90 Figure 4.28 Comparisons of pressure distribution and wing shape at 50% span ............. 90 ix