DESIGN OPTIMIZATION OF HIGH-LIFT CONFIGURATIONS USING A VISCOUS ADJOINT-BASED METHOD

Transcription

1 DESIGN OPTIMIZATION OF HIGH-LIFT CONFIGURATIONS USING A VISCOUS ADJOINT-BASED METHOD 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 By Sangho Kim December 2001

2 c Copyright 2002 by Sangho Kim All Rights Reserved ii

3 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 (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. Antony Jameson 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. Ilan M. Kroo 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. Robert W. MacCormack Approved for the University Committee on Graduate Studies: iii

4 To my Lord, Jesus Christ iv

5 Abstract Aerodynamic Shape Optimization (ASO) has long been a challenging problem in the study of fluid dynamics. A continuous adjoint method for ASO using the compressible Reynolds-Averaged Navier-Stokes (RANS) equations was implemented and tested. Using a viscous continuous adjoint formulation, the necessary aerodynamic gradient information was obtained with large computational savings over traditional finite-difference methods. The resulting implementation was used to determine the accuracy in the calculation of aerodynamic gradient information for use in ASO problems. The accuracy of the derivative information was assessed by direct comparison with finite-difference gradients. Design examples, including inverse problems, drag minimization, and lift maximization were performed for a single-element airfoil. The method was also used to demonstrate the feasibility of aerodynamic design of twodimensional multi-element airfoils, although more complete studies are needed to generate realistic configurations. The viscous design method used a RANS multi-block solver, FLO103-MB, a point-to-point matched multi-block grid system and the Message Passing Interface (MPI) communication standard for both the flow and adjoint calculations. The Spalart-Allmaras turbulence model was implemented to account for high Reynolds number effects. Airfoil shape, element positioning, and angle of attack were used as design variables. Design results that verify the mathematical correctness of the gradient calculation method for high-lift system design and optimization were also shown. v

6 Acknowledgments This research has been made possible by the generous support of the David and Lucille Packard Foundation in the form of a Stanford University School of Engineering Terman Fellowship. I would like to express my utmost gratitude to my advisor, Professor Juan J. Alonso, both for giving me the opportunity to work in this interesting field of research and for his continual guidance, encouragement and support throughout this research. My special thanks go to Professor Antony Jameson for his invaluable advice and for his pioneering work on the adjoint-based design method which served as a starting point for my work. I would also like to thank Professor Robert W. MacCormack for his generosity and help during my early years as a graduate student. I am also grateful to Professor Ilan M. Kroo for showing great interest in my work and for providing valuable suggestions. I would also like to thank all my friends at Stanford, especially my colleagues in the Aerospace Computing Laboratory, my friends in the Korean Aeronautics and Astronautics Students Association and my brothers in the Stanford cell of the New Community Baptist Church. This work would not have been possible without their friendship and their helpful discussions and suggestions on both the technical and non-technical topics. My sincere thanks go to Professors of Astronomy and Space Science at Kyunghee University for their constant encouragement throughout my studies. vi

7 Most importantly, I cannot thank enough my loving wife, Hyun Kim, for her endless support and patience. To my son, Joshua, and my daughter, Hannah, thank you for giving me so much happiness. I would like to express my appreciation to my loving parents, Jungirl Kim and Ockyoung Cho, my two sisters and my brother for all their love and support over the years. vii

8 Contents iv v Acknowledgments vi 1 Introduction CFD-Based Design Adjoint Methods Goals of this Research Gradient Accuracy Study for Viscous Adjoint Methods High-Lift System Design Parallel Implementation Multi-Block Solver Summary Viscous Aerodynamic Sensitivity Accuracy Study Implementation, SYN Results Euler Inverse Design Problem Navier-Stokes Inverse Design Problem Navier-Stokes Drag Minimization Problem Summary of Results viii

9 3 High-Lift System Design Implementation Results FLO103-MB with SA Model Validation Single-Element Airfoil Design Multi-Element Airfoil Design Maximum Lift Maximization Summary of Results Conclusions and Future Work Conclusions of Viscous Sensitivity Accuracy Study Conclusions of High-Lift System Design Conclusions of Design Optimization Future Work A Governing Equations and Discretization 114 A.1 The Navier-Stokes Equations A.1.1 Integral form of the Navier-Stokes Equations A.1.2 Boundary Conditions A.2 Numerical Discretization A.2.1 Finite-Volume Technique A.2.2 Artificial Dissipation A.2.3 Time-Stepping A.2.4 Convergence Acceleration A.2.5 Discrete Boundary Conditions B Adjoint Equations and Discretization 132 B.1 Differential form of the Navier-Stokes Equations B.2 General Formulation of the Optimal Design Problem for the Navier- Stokes Equations B.3 Derivation of the Inviscid Adjoint Terms B.4 Inviscid Adjoint Boundary Condition ix

10 B.5 Derivation of the Viscous Adjoint Terms B.6 The Viscous Adjoint Field Operator B.7 Viscous Adjoint Boundary Conditions B.8 Adjoint Discretization and Solution Procedures B.8.1 Discretization of the Viscous Adjoint Equation B.8.2 Solution Methodology and Convergence Acceleration for the Viscous Adjoint Equation B.8.3 Discrete Adjoint Boundary Conditions C Turbulence Models 161 C.1 Baldwin-Lomax Model C.2 Spalart-Allmaras Model D Parallelization 166 Bibliography 168 x

11 List of Tables 2.1 Summary of Gradient Study for Korn to NACA64A410 Inverse Problem Summary of Design Results for Inverse Problem Summary of Gradient Study for RAE2822 C d Minimization Summary of Design Results for RAE2822 Airfoil Drag Minimization Summary of Design Results for RAE2822 Muti-Element Airfoil Summary of Design Results for 30P 30N Muti-Element Airfoil Summary of Maximum Lift Maximization Design Results For RAE Summary of Maximum Lift Maximization Design Results For 30P30N 105 xi

12 List of Figures 1.1 Flowchart of the Design Process Definitions of Gap, Overlap, and Deflection Angles Illustration of the Effect of Flaps and Slats on the Lift Curve Convergence History for the Average Density Residual of a Typical FLO103 Calculation Convergence History of the Adjoint Solver for the Average Residual in the First Costate Variable Viscous Mesh for RAE2822 Airfoil Perturbed Viscous Mesh for RAE2822 Airfoil Close-Up of Nose Region for Perturbed Mesh Close-Up of Tail Region for Perturbed Mesh A Typical Sine Bump on RAE Inviscid Inverse Design: Finite-Difference Gradients for Varying Flow Solver Convergence Inviscid Inverse Design: Adjoint Gradients for Varying Flow Solver Convergence Inviscid Inverse Design: Finite-Difference Gradients for Varying Step Sizes Inviscid Inverse Design: Adjoint Gradients for Varying Step Sizes Inviscid Inverse Design: Adjoint Gradients for Varying Adjoint Solver Convergence Levels Inviscid Inverse Design: Comparison of Finite-Difference and Adjoint Gradients for Medium Mesh xii

13 2.14 Inviscid Inverse Design: Comparison of Finite-Difference Gradients for Varying Mesh Sizes Inviscid Inverse Design: Comparison of Adjoint Gradients for Varying Mesh Sizes Navier-Stokes Inverse Design: Finite-Difference Gradients for Varying Flow Solver Convergence Navier-Stokes Inverse Design: Adjoint Gradients for Varying Flow Solver Convergence Navier-Stokes Inverse Design: Finite-Difference Gradients for Varying Step Sizes Navier-Stokes Inverse Design: Adjoint Gradients for Varying Step Sizes Navier-Stokes Inverse Design: Adjoint Gradients for Varying Adjoint Solver Convergence Levels Navier-Stokes Inverse Design: Comparison of Finite-Difference and Adjoint Gradients for Medium Mesh Navier-Stokes Inverse Design: Comparison of Finite-Difference Gradients for Varying Mesh Sizes Navier-Stokes Inverse Design: Comparison of Adjoint Gradients for Varying Mesh Sizes Navier-Stokes Inverse Design: Comparison of Finite-Difference and Adjoint Gradients on Coarse Mesh Navier-Stokes Inverse Design: Comparison of Finite-Difference and Adjoint Gradients on Fine Mesh Typical Navier-Stokes Inverse Design Calculation, Korn airfoil to NACA 64A Typical Navier-Stokes Inverse Design Calculation, RAE 2822 airfoil to NACA 64A Navier-Stokes Drag Minimization: Finite-Difference Gradients for Varying Flow Solver Convergence Navier-Stokes Drag Minimization: Adjoint Gradients for Varying Flow Solver Convergence xiii

14 2.30 Navier-Stokes Drag Minimization: Finite-Difference Gradients for Varying Step Sizes Navier-Stokes Drag Minimization: Adjoint Gradients for Varying Step Sizes Navier-Stokes Drag Minimization: Adjoint Gradients for Varying Adjoint Solver Convergence Levels Navier-Stokes Drag Minimization: Comparison of Finite-Difference and Adjoint Gradients for Medium Mesh Typical Navier-Stokes Drag Minimization Calculation, RAE 2822 Airfoil Typical Navier-Stokes Drag Minimization Calculation, RAE 2822 Airfoil Grid around the 30P30N Multi-Element Configuration Grid around the 30P30N Multi-Element Configuration : Close-Up between Slat and Main Elements Grid around the 30P30N Multi-Element Configuration : Close-Up around Flap Element Perturbed Grid for the 30P30N : 0.02 chord Bump on Main, 25 Slat Deflection Change and 15 Flap Deflection Change Close-Up of Perturbed Mesh between Slat and Main Elements Close-Up of Perturbed Mesh around Flap Element Convergence History of C l and Density Residual for the 30P30N Multi- Element Airfoil Using the Spalart-Allmaras Turbulence Model Comparison of Experimental and Computational Pressure Coeffient Distributions for the 30P30N Multi-Element Airfoil Comparison of Experimental and Computational Lift Coeffient vs. Angle of Attack for the 30P30N Multi-Element Airfoil Comparison of Computational and Experimental Velocity Profiles for the 30P30N Multi-Element Airfoil Block Grid around the RAE2822 Airfoil Typical Navier-Stokes Drag Minimization Calculation at Fixed α = 2.79, RAE 2822 Airfoil xiv

15 3.13 Typical Navier-Stokes Drag Minimization Calculation at Fixed C l = 0.83, RAE 2822 Airfoil Typical Navier-Stokes Lift Maximization Calculation at Fixed α = 2.79, RAE 2822 Airfoil Typical Navier-Stokes Lift Maximization Calculation at Fixed C d = , RAE 2822 Airfoil Example of the 30P30N Multi-Element Euler Inverse Design Multi-Element Airfoil Drag Minimization Calculation at Fixed α = 16.02, 30P30N Multi-Element Airfoil Drag Minimization Calculation at Fixed C l =4.04, 30P30N Multi-Element Airfoil Lift Maximization Calculation at Fixed α = 16.02, 30P30N Multi-Element Airfoil Lift Maximization Using Settings Only at Fixed α = 16.02, 30P30N Multi-Element Airfoil Lift Maximization Calculation at Fixed C d = , 30P30N Design Curve of Approach I for the RAE2822 C lmax Maximization RAE2822 C lmax Maximization Calculation at a Fixed α = Accuracy of Adjoint Gradient Using α near C lmax of the RAE Design Curve of Approach II for the RAE2822 C lmax Maximization RAE2822 C lmax Maximization Calculation Including α as a Design Variable Design Curve of the 30P30N C lmax Maximization Multi-Element Airfoil C lmax Maximization Calculation for the 30P30N Convergence History for Lift Maximization of RAE2822 Using Angle of Attack as the Only Design Variable Convergence History for Drag Minimization of RAE2822 at Fixed Angle of Attack A.1 Finite-Volume Mesh Cell for Evaluation of the Flux Balances xv

16 A.2 Auxiliary Control Volume for the Discretization of the Viscous Terms. 122 A.3 Multigrid Cycle Descriptions A.4 Solid Boundary for the Finite-Volume Scheme B.1 Solid Boundary for the Finite-Volume Scheme D.1 Parallel Speedup of SYN103-P Viscous Design Code xvi

17 Chapter 1 Introduction Typical problems in applied aerodynamics are usually tackled using one or a combination of three different fundamental approaches: experimental fluid dynamics, theoretical fluid dynamics and computational fluid dynamics (CFD). Experimental and theoretical approaches were the first used in the development of the field. Initially, the focus centered on hydrodynamics - the study of incompressible fluids. During the late 1960s and early 1970s, CFD was introduced as an important new third approach, opening the aerodynamic community to the three-approach world of theory, experiment and computation. These three approaches have been individually and cooperatively playing leading roles in the analysis and design of aerodynamic configurations. Historically, the most successful contribution of experimental methods to aerodynamic design is to be found in the well-known success story of the Wright brothers. The aerodynamic performance of the Wrights first glider in 1900 was approximately equal to that of others prior to that time. But the Wrights started making steady progress in 1901 and their 1902 glider was the first truly effective heavier-than-air craft. Behind this progress and success, one must acknowledge their development of a state-of-the-art wind tunnel and their hundreds of laborious tests. Unlike previous experiments using wind tunnels, the Wrights created instruments that quantified the lift and drag of wing segments and they were able to identify a highly-efficient wing 1

18 CHAPTER 1. INTRODUCTION 2 shape from the data which was systematically obtained through their tunnel testing. One year later, the world s first power-driven, heavier-than-air machine built by Wilbur and Orville was flown at Kitty Hawk, North Carolina, on December 17, The development of the governing equations of fluid flow is one of the most fundamental contributions of theoretical fluid dynamics. Benefitting from earlier works of Newton, D Alembert and Bernoulli, Leonhard Euler ( ) derived the governing equations for an inviscid flow in These equations have come to be known as the Euler equations. The Navier-Stokes equations that describe the behavior of a viscous fluid were derived independently by Louis Navier and George Stokes during the nineteenth century. Adding the conservation of energy to their original versions which included only the equations of conservation of mass and momentum, the complete set of Euler and Navier-Stokes equations remain today the fundamental governing laws used to analyze fluid flows. No general solutions for these systems of nonlinear partial differential equations are presently known. However, for geometrically simple configurations, and with the use of insightful approximations and breakthrough ideas, some solutions of the governing equations have been obtained. In this way, theoretical aerodynamics has complemented experiments both as a qualitative tool and also as a quantitative tool. Examples of this type of solution to the governing equations include the first calculation of lift using Kutta and Joukouski s circulation theory, and Ludwig Prandtl s prediction of the skin friction drag of a body using his Boundary-Layer Theory. In an even more practically important example, following the ideas of Lanchester, Prandtl and his colleagues developed their Finite Wing and Airfoil Theories and discovered new concepts such as induced drag and optimal elliptic planforms. The development of numerical methods began with Newton and Gauss, and gained momentum around the turn of the twentieth century. Some of the early fundamental contributions include Richardson s relaxation technique for the numerical solution of Laplace s equation and Courant, Friedrich, and Lewy s numerical work on the solution of hyperbolic partial differential equations. Although these methods were intellectually stimulating, they were impractical and rather purely academic, since the actual computations had to be carried out by hand in the early twentieth century.

19 CHAPTER 1. INTRODUCTION 3 However, in the 1960 s, the advent of high speed digital computers led to the current ongoing revolution of numerical methods by allowing a large number of calculations to be carried out using computer power. Since then, with rapid improvements in computer performance and progressive development of accurate and stable numerical algorithms, computational methods have gained acceptance in the aerodynamic community and are now used extensively in both the aerodynamic analysis and design processes. Thus we see that advances in fluid dynamics have mainly relied on three different approaches: theoretical, experimental, and computational. This three-approach paradigm will continue to mature with further improvements of all three methods. Indeed, the importance of cooperation between these methods should not be overlooked. For example, CFD-based aerodynamic predictions often use theory and experimental data in order to interpret, understand, and validate the computational results; the converse is also true. In comparison with theoretical techniques, the main advantage of CFD techniques is that they do not require linearization of the governing equations and can treat both complex physics in a variety of flow regimes and complex geometries. In comparison with experimental methods, they usually require lower costs and time to obtain the solution to a given problem. On the other hand, experimental methods can provide both credible and accurate data, while theoretical techniques provide understanding of the flow physics, since they yield not only solutions, but also trends and dependencies on the variation of relevant parameters in the flow. Further advances in aerodynamics will rely on a balance of these three approaches; the true balance may be realized by an effort to maximize the advantages of each method while complementing any disadvantages of the others. The objective of this research work is to develop CFD-based design methods while maintaining a balanced link to both the theoretical and experimental approaches. 1.1 CFD-Based Design 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

20 CHAPTER 1. INTRODUCTION 4 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 [1, 2, 3, 4, 5], it has not been until recently that the focus of CFD applications has shifted to aerodynamic design [6, 7, 8, 9, 10, 11]. 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. Typically, in gradient-based optimization techniques, a control function to be optimized (an airfoil shape, for example) is parameterized with a set of design variables, and a suitable cost function to be minimized or maximized is defined (drag coefficient, lift/drag ratio, difference from a specified pressure distribution, etc). Then, a constraint, the governing equations in the present study, can be introduced in order to express the dependence between the cost function and the control function. The sensitivity derivatives 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 to a minimum or maximum 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 algorithm. This is particularly true in problems which involve a very large number of design variables, as is the case in a typical three-dimensional wing shape design. Gradient information can be computed using a variety of approaches. The finitedifference 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

21 CHAPTER 1. INTRODUCTION 5 of design variables, and, consequently, it cannot be afforded for problems with design spaces of large dimensionality. The finite-difference method is potentially the worst possible way of computing gradient information, since the cost is proportional to the number of design variables in the problem, and the accuracy of the gradient information depends strongly on the choice of step size (which is not known a priori). It must be mentioned, however, that alternative methods exist, whose accuracy is independent of the choice of step size, such as the complex step method [12] and automatic differentiation [13]. Unfortunately, their computational cost is still proportional to the number of design variables in the problem. 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 [14]. 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 [15] and it was first used in transonic flow by Jameson [6, 7, 16]. Since then this method has become a popular choice for design problems involving fluid flow [9, 17, 18, 19]. In fact, the method has even been successfully used for the aerodynamic design of complete aircraft configurations [8, 20]. It is interesting to point out that the different existing approaches to the adjoint method can be classified into two categories: the continuous adjoint and discrete adjoint methods. If the adjoint equations are directly derived from the governing equations and then discretized, they are termed continuous, and if instead they are directly derived from the discretized form of the governing equations then they are referred to as discrete. In theory, the discrete adjoint method gives gradients which

22 CHAPTER 1. INTRODUCTION 6 are closer in value to exact finite-difference gradients. On the other hand, the continuous adjoint method has the advantage that the adjoint system has a unique form independent of the scheme used to solve the flow field system. Shubin and Frank have presented a comparison between the continuous and discrete adjoint methods for quasi-one-dimensional flows [21]. Recently Nadarajah and Jameson performed a detailed gradient comparison study of the continuous and discrete adjoint approaches using the Euler equations [22]. This work 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. In their work, the discrete adjoint gradients have slightly better agreement with step-optimized finite-difference gradients as expected by the theory but the difference vanishes as the number of cells in the mesh increases. However, the human and computational costs of derivation and calculation of the discrete adjoint are greater. 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 the continuous gradient is slightly more accurate [23]. Burgreen and Baysal used the discrete adjoint for shape optimization of wings on structured grids [24]. On unstructured grids, Elliot and Peraire computed the discrete adjoint of the Euler equations which were solved by a multistage Runge-Kutta scheme [25]. Jameson s initial work and Jameson and Reuther s later works are based on the continuous adjoint method. Ta saan, Kuruvila, and Salas also conducted shape optimization on structured grids using the continuous adjoint approach [26]. Instead of decoupling the design problem into separate parts (flow solver, adjoint solver, optimization algorithm) as in Jameson and Reuther s work, they have implemented a strong coupling of the systems through the use of multigrid algorithms. Anderson and Venkatakrishnan explored the continuous adjoint on unstructured grids [19]. Anderson and Nielsen have also implemented the discrete adjoint on unstructured grids [27]. In their work, Anderson et al. presented the accuracy of the adjoint sensitivity derivatives in aerodynamic design using the Navier-Stokes equations, and also

23 CHAPTER 1. INTRODUCTION 7 presented some design examples including a wing drag minimization and an inviscid multi-element airfoil shape design. For the present research, the continuous adjoint method on multi-block structured grids has been used. 1.2 Adjoint Methods The progress of a design procedure is measured in terms of a cost function I, which could be, for example the drag coefficient or the lift to drag ratio. For the flow about an airfoil or wing, 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, say. Then I = I (w, F), and a change in F results in a change [ ] I T δi = w I [ ] I T δw F II δf, (1.1) in the cost function. Here, the subscripts I and II are used to distinguish the contributions due to the variation δw in the flow solution from the change associated directly with the modification δf in the shape. This notation is introduced to assist in grouping the numerous terms that arise during the derivation of the full Navier Stokes adjoint operator in Appendix B, so that it remains easy to recognize the basic structure of the approach as it is sketched in the present section. 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 multiple flow solutions. This corresponds to eliminating δw from (1.1). 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. (1.2)

24 CHAPTER 1. INTRODUCTION 8 Then δw is determined from the equation δr = [ ] R δw w I Next, introducing a Lagrange Multiplier ψ, we have δi = = [ ] I T w { I T I w ψt [ ] I T δw F [ ]} R w Choosing ψ to satisfy the adjoint equation I II [ ] R δf = 0. (1.3) F II ([ ] R δf ψ T w { I T δw F ψt I [ ] ) R δw δf F II ]} (1.4) δf. (1.5) [ R F II [ ] T R ψ = I w w (1.6) the first term is eliminated, and we find that δi = GδF, (1.7) where G = IT F ψt [ ] R. (1.8) F The advantage is that (1.7) is independent of δw, with the result that the gradient of I with respect to an arbitrary number of design variables can be determined without the need for additional flow-field evaluations. In the case that (1.2) is a partial differential equation, the adjoint equation (1.6) is also a partial differential equation and determination of the appropriate boundary conditions requires careful mathematical treatment. The computational cost of a single design cycle is roughly equivalent to the cost of two flow solutions since the adjoint problem has similar complexity. When the number of design variables becomes large, the computational efficiency of the control theory approach over the traditional finite-differences, which require direct evaluation

25 CHAPTER 1. INTRODUCTION 9 of the gradients by individually varying each design variable and recomputing the flow field, becomes compelling. Once equation (1.8) is established, an improvement can be made with a shape change δf = λg where λ is positive, and small enough that the first variation is an accurate estimate of δi. Then δi = λg T G < 0. After making such a modification, the gradient can be recalculated and the process repeated to follow a path of steepest descent until a minimum is reached. In order to avoid violating constraints, such as a minimum acceptable airfoil thickness, the gradient may be projected into an allowable subspace within which the constraints are satisfied. In this way, procedures can be devised which must necessarily converge at least to a local minimum. Alternatively, the gradient information provided by the adjoint method can be used in conjunction with more sophisticated constrained, non-linear optimization procedures such as quasi-newton methods or sequential quadratic programming approaches. Although smoothing can serve as a very effective preconditioner to the steepest descent step and therefore reduce the cost of optimization when compared with more sophisticated methods, in specific situations where both linear and nonlinear constraints are necessary, these sophisticated algorithms may play a key role. In practical implementations of the adjoint method, a design code can be modularized into several components such as the flow solver, adjoint solver, geometry and mesh modification algorithms, and the optimization algorithm. After parameterizing the configuration of interest using a set of design variables and defining a suitable cost function, which is typically based on aerodynamic performance, the design procedure can be described as follows. First solve the flow equations for the flow variables, then solve the adjoint equations for the costate variables subject to appropriate boundary

26 CHAPTER 1. INTRODUCTION 10 conditions which will depend on the form of the cost function. Next evaluate the gradients and update the aerodynamic shape based on the direction of steepest descent. Finally repeat the process to attain an optimum configuration. A summary of the design process and a comparison with the finite-difference method are illustrated in Figure Goals of this Research Most of the early work in the formulation of the adjoint-based design framework used the potential and Euler equations as models of the fluid flow. The development of the adjoint system for the Euler equations, as well as the assessment of the accuracy of the gradient information that can be obtained from these adjoint equations have been presented in the past [7, 28]. Aerodynamic design calculations using the Reynolds- Averaged Navier-Stokes equations as the flow model have only recently been tackled. The extension of adjoint methods for optimal aerodynamic design of viscous problems is necessary to provide the increased level of modeling which is crucial for certain types of flows. This cannot only be considered an academic exercise. It is also a very important issue for the design of viscous dominated applications such as the flow in high lift systems. In 1997, a continuous adjoint method for Aerodynamic Shape Optimization (ASO) using the compressible Navier-Stokes equations was formulated and it has been implemented directly in a three dimensional wing problem [16, 29]. In 1998, an implementation of three-dimensional viscous adjoint method was used with some success in the optimization of the Blended-Wing-Body configuration [30]. Since these design calculations were carried out without the benefit of a careful check on the accuracy of the resulting gradient information, it was considered appropriate to conduct a series of numerical experiments in two dimensions that would establish the soundness of the viscous adjoint implementation. This research consists of two main portions: a gradient accuracy study for the adjoint-based Navier-Stokes design method, and the application of the present viscous adjoint method to two-dimensional high-lift aerodynamic optimization.

27 CHAPTER 1. INTRODUCTION 11 Define Design Variables Parameterize Configuration Define Cost Function *note: process in the dashed box are repeated N times where N is the number of design variables. Solve Flow Equations Adjoint Finite Difference Solve Adjoint Equations Perturb Shape & Modify Grid Perturb Shape & Modify Grid Solve Flow Equations Evaluate Gradient Evaluate Gradient c o s t Adjoint Finite Difference Optimization Optimum? No New Shape 0 Adjoint vs. Finite Difference in Cost, N = 100 Yes Stop Figure 1.1: Flowchart of the Design Process

28 CHAPTER 1. INTRODUCTION 12 The majority of the work focuses on the development and assessment of the mathematical formulation of the viscous adjoint design method. To that effect, all necessary formulae were carefully derived and implemented in a computer program. This program was used to establish the validity of the resulting gradient information so that it could later be used in realistic viscous design. As a preliminary demonstration of the validity of this gradient information we have used the problem of two-dimensional high-lift system design. The design examples in this thesis are not meant to represent realistic high-lift configurations but rather, they are meant to show that the viscous adjoint formulation is capable of producing useful gradient information for high-lift design purposes. The application of this new design methodology to real-world design examples is left for future work Gradient Accuracy Study for Viscous Adjoint Methods In the first half of this research, a continuous adjoint formulation for aerodynamic shape optimization using the compressible Reynolds-Averaged Navier-Stokes equations and the Baldwin-Lomax turbulence model is investigated, and the complete viscous adjoint is implemented for two-dimensional airfoil design. In particular, this work focuses on a study of the accuracy of the gradient information resulting from the application of a viscous adjoint to the problems of inverse design and drag minimization. The results of the gradient accuracy study are directly compared with optimum step, finite-difference gradients. The study also focuses on the effects of mesh resolution, flow solver convergence, adjoint solver convergence, and step size, for the calculation of finite-difference and adjoint gradients. The comparisons made here are meant to validate the use of a viscous adjoint formulation in aerodynamic design, and to answer questions regarding the level of convergence in the flow and adjoint solutions necessary for sufficiently accurate gradient information. We also verify the effectiveness, consistency, and robustness of the viscous adjoint method. Finally, the merits of this adjoint implementation are validated on the design of single-element airfoils, including inverse design, drag coefficient(c d ) minimization, lift coefficient(c l ) maximization, and lift-to-drag ratio(l/d) maximization problems.

29 CHAPTER 1. INTRODUCTION High-Lift System Design The second part of the research addresses the validity of this design methodology for the problem of high-lift design. For an aircraft in the take-off or landing condition, the stall speed is given by 2W V stall =. ρ AC Lmax This speed is the lowest possible speed at which the aircraft is able to generate an amount of lift, L, equal to its weight, W. In the expression, A is the wing area, ρ is the free stream density and C Lmax is able to generate. is the maximum lift coefficient that the aircraft It is clear then that, in order to reduce the stall speed, and therefore minimize the take-off and landing runway length, a higher value of C Lmax is desirable. At the take-off condition, where a higher rate of climb is also desirable, this increase in C Lmax must be realized without sacrificing the L/D of the aircraft. This last consideration is not as important in the landing condition, since a steeper approach angle is often desirable and a higher drag can therefore be tolerated. High-lift system devices such as flaps and slats were developed early on in the history of aviation and, with the help of large amounts of wind-tunnel testing, they have managed to produce respectable values of C Lmax and runway lengths. A flap is a portion of the trailing-edge section of an airfoil that can be deflected downwards in order to increase the effective camber of the airfoil section and, consequently, increase the lift coefficient. Leading-edge devices such as slats are a part of the leading-edge of the airfoil section that allows a secondary flow to pass through the gap between the slat and the main section and prevents the main section from stalling until a higher angle of attack. The relative position of the various elements in a high-lift system is customarily measured using rigging parameters. The rigging parameters that will be used in this work include gaps, overlaps, and deflection angles of both flaps and slats. The slat gap is the distance measured from the slat trailing-edge to the point on the main element whose surface normal passes through the trailing-edge of the slat. The flap gap is measured in a similar manner. The element overlaps are measured

30 CHAPTER 1. INTRODUCTION 14 Figure 1.2: Definitions of Gap, Overlap, and Deflection Angles in a direction which is parallel to the undeflected configuration chord line. Finally, angular deflections are measured between the undeployed chord line and the deflected device chord line. The definition of the rigging parameters for a typical three-element high-lift system configuration is illustrated in Figure 1.2, and the general effect of flaps and slats is illustrated in Figure 1.3. Traditionally, high-lift designs have been realized by careful wind tunnel testing. This approach is both expensive and challenging due to the extremely complex nature of the flow interactions that appear. Computational fluid dynamics (CFD) analyses have recently been incorporated to the high-lift design process [31]. Eyi, Lee, Rogers and Kwak have performed design optimizations of a high-lift system configuration using a chimera overlaid grid system and the incompressible Navier-Stokes equations [32]. Besnard, Schmitz, Boscher, Garcia and Cebeci performed optimizations of high-lift systems using an Interactive Boundary Layer (IBL) approach [33]. Since all of these earlier works on multi-element airfoil design obtained the necessary gradients by finite-difference methods, they were either only able to span a small design space

31 Main Airfoil CHAPTER 1. INTRODUCTION Flap Slat 15 Flap angle 40 deg Slats Extended Increase Flap Angle Lift Coefficient Lift Coefficient Slats Retracted Flap angle 0 deg Angle of Attack Angle of Attack Figure 1.3: Illustration of the Effect of Flaps and Slats on the Lift Curve (usually restricted to relative positioning of the elements in the high-lift system), or they were forced to use simplified approximations to the full Navier-Stokes equations. In the latter half of this work, the present viscous adjoint method is applied to two-dimensional high-lift system designs, removing the limitations on the dimensionality of the design space by making use of the viscous adjoint design methodology. The motivation for our study of high-lift system design is twofold. On the one hand, we would like to improve the take-off and landing performance of existing high-lift systems using an adjoint formulation, while on the other hand, we would like to setup numerical optimization procedures that can be useful to the aerodynamicist in the rapid design and development of high-lift system configurations. In addition to difficulties involved in the prediction of complex flow physics, multi-element airfoils provide an additional challenge to the adjoint method: the effect of the changes in the shape of one element must be felt by the other elements in the system. While preliminary studies of the adjoint method in such a situation have already been carried out [19], this research is designed to validate the adjoint method for complex applications of the type. Emphasis is placed on the validation and not on the creation of realistic designs, which is beyond the scope of this work Parallel Implementation Because of the increased number of mesh points needed to resolve the boundary layer, the additional cost of computing the viscous terms, the need to use a turbulence

32 CHAPTER 1. INTRODUCTION 16 model, and the slower convergence of the Navier-Stokes equations on highly stretched meshes, the computational costs for viscous design are at least an order of magnitude greater than for inviscid design. Therefore, the reduction of computational costs is crucial to the further development of practically effective viscous design methods, especially the methods for computationally intensive problems such as the high-lift optimization problem. The design method, which is greatly accelerated by the use of control theory, can be further enhanced by the use of parallel computing. In this research a parallel implementation using a domain decomposition approach and the MPI standard for communication is used Multi-Block Solver Although, as will be shown later, the adjoint method is fairly insensitive to the convergence level of either the flow or the adjoint systems, an accurate, robust and fast solver for both the flow and the adjoint equations is required for a usable design method. For the single-element airfoil designs and the corresponding gradient accuracy study, FLO103, a RANS solver developed by Martinelli [34, 35], is used to satisfy the requirements of accuracy, convergence, and robustness. FLO103 solves the steady two-dimensional RANS equations using a modified explicit multistage Runge-Kutta time-stepping scheme. A finite-volume technique and second order central differencing in space are applied to the integral form of the Navier-Stokes equations. The Jameson-Schmidt-Turkel(JST) scheme with adaptive coefficients for artificial dissipation is used to prevent odd-even oscillations, and to allow the clean capture of shock waves and contact discontinuities. In addition, local time-stepping, implicit residual smoothing, and the multigrid method are applied to accelerate convergence to steady-state solutions. The prediction of high-lift flows poses a particularly difficult challenge for both CFD and turbulence modeling. Even in two-dimensions, the physics involved in the flow around a geometrically-complex high-lift device are quite sophisticated. For better prediction of free wakes and shear layers and to capture phenomena such

33 CHAPTER 1. INTRODUCTION 17 as separation and a stall, a more advanced turbulence model than the Baldwin- Lomax algebraic model is required. Here the one-equation Spalart-Allmaras model was implemented to partially satisfy this requirement. The difficulty in developing a flow solver for high-lift systems becomes even greater since the solver should be capable of parallel computing on arbitrary multi-block meshes generated to conform to the surfaces of these multi-element airfoils. In this study FLO103-MB, a multiblock RANS solver based on FLO103 [34] and the three-dimensional RANS solver developed by Reuther, Alonso, Vassberg, Jameson and Martinelli [9], is used for all multi-element flow-field predictions. FLO103-MB satisfies the requirements of accuracy, convergence, and robustness that are necessary in this work. 1.4 Summary The remaining chapters in this thesis describe the numerical studies carried out for the two main portions of this research: the viscous adjoint gradient accuracy study and the application of the adjoint method to high-lift system design. The mathematical underpinnings of this research are provided in a series of appendixes. The sensitivity accuracy study for viscous flows is discussed in Chapter 2. Chapter 2 begins with a discussion of the implementation of the adjoint method for aerodynamic shape optimization problems using the compressible Reynolds-Averaged Navier-Stokes equations. The shortcomings of finite-differencing and the advantages of the adjoint method are then discussed and the results of comparisons of gradient calculations are presented. Design examples for single-element airfoils are also presented in this chapter. Chapter 3 discusses the implementation of the adjoint method for high-lift system design work. After careful validation of the flow predictions around a three-element airfoil configuration, several design results are presented. The results are summarized in tables at the end of each Chapter. Chapter 4 discusses the main results of this research and provides some ideas for future work. In Appendix A the compressible Navier-Stokes equations and their Reynolds- Averaged form are reviewed, together with the appropriate boundary conditions.

34 CHAPTER 1. INTRODUCTION 18 Appendix A also discusses the discretization and solution procedure for our numerical implementations. Appendix B documents the mathematical formulation of the viscous adjoint equations and boundary conditions, including their discretization. Both turbulence models used in this work: the Baldwin-Lomax algebraic model and the Spalart-Allmaras one equation model, are described in Appendix C. Finally, the parallel implementation and its performance are presented in Appendix D.