Continuous adjoint methods and optimal control theory are applied to a pressure-matching inverse design problem of quasi 1-D nozzle flows. Pontryagin’s Minimum Principle is used to derive the adjoint system and the reduced gradient of the cost functional. The properties of adjoint variables at the sonic throat and the shock location are studied, revealing a logarithmic singularity at the sonic throat and continuity at the shock location. A numerical method, based on the Steger-Warming flux-vector-splitting scheme, is proposed to solve the adjoint equations. This scheme can finely resolve the singularity at the sonic throat. A non-uniform grid, with points clustered near the throat region, can resolve it even better. The analytical solutions to the adjoint equations are also constructed via Green’s function approach for the purpose of comparing the numerical results. The pressure-matching inverse design is then conducted for a nozzle parameterized by a single geometric parameter. In the second part, the adjoint methods are applied to the problem of minimizing drag coefficient, at fixed lift coefficient, for 2-D transonic airfoil flows. Reduced gradients of several functionals are derived through application of a Lagrange Multiplier Theorem. The adjoint system is carefully studied including the adjoint characteristic boundary conditions at the far-field boundary. A super-reduced design formulation is also explored by treating the angle of attack as an additional state; super-reduced gradients can be constructed either by solving adjoint equations with non-local boundary conditions or by a direct Lagrange multiplier method. In this way, the constrained optimization reduces to an unconstrained design problem. Numerical methods based on Jameson’s finite volume scheme are employed to solve the adjoint equations. The same grid system generated from an efficient hyperbolic grid generator are adopted in both the Euler flow solver and the adjoint solver. Several computational tests on transonic airfoil design are presented to show the reliability and efficiency of adjoint methods in calculating the reduced (super-reduced) gradients. / Ph. D.

Nowadays, the competition among airlines seriously depend upon the saving operating costs, with the premise that not to degrade its services quality. Especially in the face of increasingly scarce oil resources, reducing fleets operational fuel consumption, is an important means to improve profits. Aircraft fuel economy is determined by operational management strategies and application technologies. The application of technologies mainly refers to airplane’s engine performance, Weight efficiency and aerodynamic characteristics. A market competitive aircraft should thoroughly consider to all of these aspects. Transport aircraft aerodynamic performance mainly is determined by wing’s properties. Wings that are optimized for efficient flight in cruise conditions need to be fitted with powerful high-lift devices to meet lift requirements for safe takeoff and landing. These high-lift devices have a significant impact on the total airplane performance. The aerodynamic characteristics of the wing airfoil will have a direct impact on the aerodynamic characteristics of the wing, and the wing’s effective cruise hand high-lift configuration design has a significant impact on the performance of transport aircraft. Therefore, optimizing the design is a necessary airfoil design process. Nowadays engineering analysis relies heavily on computer-based solution algorithms to investigate the performance of an engineering system. Computational fluid dynamics (CFD) is one of the computer-based solution methods which are more widely employed in aerospace engineering. The computational power and time required to carry out the analysis increases as the fidelity of the analysis increases. Aerodynamic shape optimization has become a vital part of aircraft design in the recent years. Since the aerodynamic shape optimization (ASO) process with CFD solution algorithms requires a huge amount of computational power, there is always some reluctance among the aircraft researchers in employing the ASO approach at the initial stages of the aircraft design. In order to alleviate this problem, statistical approximation models are constructed for actual CFD algorithms. The fidelity of these approximation models are merely based on the fidelity of data used to construct these models. Hence it becomes indispensable to spend more computational power in order to convene more data which are further used for constructing the approximation models. The goal of this thesis is to present a design approach for assumed wing airfoils; it includes the design process, multi-objective design optimization based on surrogate modelling. The optimization design stared from a transonic single-element single-objective optimization design, and then high-lift configurations were two low-speed conditions of multi-objective optimization design, on this basis, further completed a variable camber airfoil at low speed to high-lift configuration to improve aerodynamic performance. Through this study, prove a surrogate based model could be used in the wing airfoil optimization design.

Title: Solution of inverse problem for a flow around an airfoil Author: Mgr. Jan Šimák Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Miloslav Feistauer, DrSc., dr. h. c., Department of Numerical Mathematics Abstract: The method described in this thesis deals with a solution of an inverse problem for a flow around an airfoil. It can be used to design an airfoil shape according to a specified velocity or pressure distribution along the chord line. The method is based on searching for a fixed point of an operator, which combines an approximate inverse and direct operator. The approximate inverse operator, derived on the basis of the thin airfoil theory, assigns a corresponding shape to the specified distribution. The resulting shape is then constructed using the mean camber line and thickness function. The direct operator determines the pressure or velocity distribution on the airfoil surface. We can apply a fast, simplified model of potential flow solved using the Fredholm integral equation, or a slower but more accurate model of RANS equations with a k-omega turbulence model. The method is intended for a subsonic flow.

The research presented in this dissertation investigates the use of all-at-once methods applied to aerodynamic design. All-at-once schemes are usually based on the assumption of sufficient continuity in the constraints and objectives, and this assumption can be troublesome in the presence of shock discontinuities. Special treatment has to be considered for such problems and we study several approaches. Our all-at-once methods are based on the Sequential Quadratic Programming method, and are designed to exploit the structure inherent in a given problem. The first method is a Reduced Hessian formulation which projects the optimization problem to a lower dimension design space. The second method exploits the sparse structure in a given problem which can yield significant savings in terms of computational effort as well as storage requirements. An underlying theme in all our applications is that careful analysis of the given problem can often lead to an efficient implementation of these all-at-once methods. Chapter 2 describes a nozzle design problem involving one-dimensional transonic flow. An initial formulation as an optimal control problem allows us to solve the problem as as two-point boundary problem which provides useful insight into the nature of the problem. Using the Reduced Hessian formulation for this problem, we find that a conventional CFD method based on shock capturing produces poor performance. The numerical difficulties caused by the presence of the shock can be alleviated by reformulating the constraints so that the shock can be treated explicitly. This amounts to using a shock fitting technique. In Chapter 3, we study variants of a simplified temperature control problem. The control problem is solved using a sparse SQP scheme. We show that for problems where the underlying infinite-dimensional problem is well-posed, the optimizer performs well, whereas it fails to produce good results for problems where the underlying infinite-dimensional problem is ill-posed. A transonic airfoil design problem is studied in Chapter 4, using the Reduced SQP formulation. We propose a scheme for performing the optimization subtasks that is based on an Euler Implicit time integration scheme. The motivation is to preserve the solution-finding structure used in the analysis algorithm. Preliminary results obtained using this method are promising. Numerical results have been presented for all the problems described. / Ph. D.

The automation of airfoil design process is an ongoing effort within the field of turbo-machinery design, with significant focus on developing new reliable and consistent methods that can meet the needs of the engineers. A wide variety of approaches has been in use for inverse airfoil design process which benefit from theoretical inverse design, statistical methods, empirical discoveries and many other ways to solve the design problem. This thesis work also develops a tool in Python to be used in airfoil aerodynamic design process that is simple, fast and accurate enough for initial design of turbo-machinery blades with focus on turbine airfoils used for operation in aircraft engines. To convey the decision-making process during development a simplified case is presented. The underlying considerations are discussed. Other available methods in the literature used for similar problems, are also evaluated and compared to demonstrate the advantages and limitations of the methods used within the tool. The inverse design problem is formulated as a multi-objective optimization problem to handle various different objectives that are relevant for aerodynamic design of turbo-machinery airfoils. Test runs are made and the results are discussed to assess how robust the tool is and how the current capabilities can be modified or extended. After the development process, the tool is verified to be a suitable option for real-life design optimization tasks and can be used as a building block for a much more comprehensive tool that may be developed in the future. / Automatisering av processen för design av vingprofiler kräver fortlöpande insatser inom området turbomaskindesign, med stort fokus på att utveckla nya tillförlitliga och konsekventa metoder som kan tillgodose ingenjörernas behov. Ett stort antal olika tillvägagångssätt har provats för omvänd design av vingprofiler såsom teoretisk invers design, statistiska metoder, empiriska upptäckter och många andra sätt att lösa designproblemet. Detta avhandlingsarbete är också ett lyckat försök att utveckla ett verktyg i Python som ska användas i den aerodynamiska designprocessen; det är enkelt, snabbt och noggrant för den initiala designen av turbomaskinblad med fokus på turbinblad som för användning i flygmotorer. För att förmedla beslutsprocessen under utvecklingen presenteras ett förenklat fall. De underliggande övervägandena diskuteras. Andra tillgängliga metoder i litteraturen som används för liknande problem utvärderas och jämförs för att visa fördelarna och begränsningarna med de metoder som används i verktyget. Det omvända designproblemet formuleras som ett multi-objektivt optimeringsproblem för att hantera olika mål som är relevanta för aerodynamisk design av turbomaskiner. Testkörningar görs och resultaten diskuteras för att bedöma hur robust verktyget är och hur de nuvarande funktionerna kan modifieras eller utökas. Efter utvecklingsprocessen verifieras verktyget som ett lämpligt alternativ för verkliga designoptimeringsuppgifter och kan användas som en byggsten för ett mycket mer omfattande verktyg som kan utvecklas i framtiden.

Thesis (MScEng (Department of Mechanical and Mechatronic Engineering))--Stellenbosch University, 2006. / ENGLISH ABSTRACT: Wind conditions in South Africa determine the need for a small-scale wind turbine to produce useable power at windspeeds below 7m/s. In this project, a range of windspeeds, within which optimal performance o the wind turbine is expected, was selected. The optimal performance was assessed in terms of the Coefficient of Power(Cp), which rates the turbines blade's ability to extract energy form the avalible wind stream. The optimisation methods employed allowed a means of tackling the multi-variable problem such that the aerodynamic characteristics of the blade were ideal throughout the wind speed range. The design problem was broken down into a two-dimensional optimisaion of the airfoils used at the radial stations, and a three-dimensional optimisation of the geometric features of the wind rotor. by means of blending various standard airfoil profiles, a new profile was created at each radial station. XFOIL was used for the two-dimensional analysis of these airfoils. Three-dimensional optimisn involved representation of the rotor as a simplified model and use of the Blade Element Momentum(BEM) method for analysis. an existimg turbine blade, on which the design specifications were modelled, was further used for comparative purposes throughout the project. The resulting blade design offers substantial improvements on the reference design. The application of optimisation methods has successfully aided the creation of a wind turbine blade with consistent peak performance over a range of design prints. / Sponsored by the Centre for Renewable and Sustainable Energy Studies, Stellenbosch University

This thesis introduces a design optimization methodology that is based on minimizing the Cost of Energy (COE) of a Horizontal Axis Wind Turbine (HAWT) that is to be operated at a specific wind site. In the design methodology for the calculation of the Cost of Energy, the Annual Energy Production (AEP) model to calculate the total energy generated by a unit wind turbine throughout a year and the total cost of that turbine are used. The AEP is calculated using the Blade Element Momentum (BEM) theory for wind turbine power and the Weibull distribution for the wind speed characteristics of selected wind sites. For the blade profile sections, either the S809 airfoil profile for all spanwise locations is used or NREL S-series airfoil families, which have different airfoil profiles for different spanwise sections, are used,. Lift and drag coefficients of these airfoils are obtained by performing computational fluid dynamics analyses. In sample design optimization studies, three different wind sites that have different wind speed characteristics are selected. Three scenarios are generated to present the effect of the airfoil shape as well as the turbine power. For each scenario, design optimizations of the reference wind turbines for the selected wind sites are performed the Cost of Energy and Annual Energy Production values are compared.