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Shape Optimization of Low Speed Airfoils using MATLAB and - KTH PDF

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Shape Optimization of Low Speed Airfoils using MATLAB and Automatic Differentiation Christian Wauquiez Stockholm 2000 Licentiate’s Thesis Royal Institute of Technology Department of Numerical Analysis and Computing Science Abstract Thegoaloftheprojectistodevelopan innovative tooltoperform shapeoptimizationoflow speed airfoils. This tool is written in Matlab, and is constructed by coupling the Matlab Optimization Toolbox with a parametrised numerical aerodynamic solver. The airfoil shape is expressed analytically as a function of some design parameters. The NACA 4 digits library is used with design parameters that control the camber and the thickness of the airfoil. The solver has to provide fast and robust computation of the lift, pitching moment and drag of an airfoil placed in a low-speed viscous flow. A one-way coupled inviscid - boundary layer model is used. The inviscid flow is computed with a linear vortex panel method, which provides the lift and moment coefficients. The boundary layer is computed using an integral formulation : the laminar part of the flow is computed with a two-equation formulation, and the turbulent part is solved with Head’s model. An e9-type amplification formulation is used to locate the transition area. Finally, the drag coefficient is computed using the Squire-Young formula. In order to be used in optimization, the solver must provide derivatives of the objective function and limiting constraints with the solution for each set of parameters. These derivatives are computed by automatic differentiation, a technique for augmenting computer programs with the computation of derivatives based on the chain rule of differ- ential calculus. The recent Matlab automatic differentiation toolbox ADMAT is used. Finallyasanapplication,sampleoptimizationproblemsaresolvedusingtheMatlabOptimi- zation Toolbox, and the resulting optimal airfoils are analysed. ISBN 91-7170-520-1l TRITA-NA-0004l ISSN 0348-2952l ISRN KTH/NA/R--00/04--SE 1 Contents Introduction 1-TheAerodynamicsSolver..................................................7 1.1- Introduction - Overview of the Model 1.2- Airfoil and Flow Parameters 1.3- Inviscid Flow Model 1.4- Boundary Layer Model 2-AutomaticDifferentiation.................................................39 2.1- Introduction 2.2- Method Fundamentals 2.3- Computer Implementation 2.4- ADMAT, Automatic Differentiation Toolbox for Matlab 2.5- Application of ADMAT to the Aerodynamic Solver 3-AirfoilShapeOptimization................................................51 3.1- Definition of the Optimization Problems 3.2- Solving the Optimization Problems Conclusion................................................................61 References Acknowledgements I would like to thank my supervisor, Associate Professor Jesper Oppelstrup, for his interest and support throughout this work, and Professor Arthur Rizzi for providing information and for many helpful discussions. I would also like to thank, for advice and technical support, Associate Professor Ilan Kroo from Stanford University and Desktop Aeronautics, and Doctor Arun Verma from the Cornell University Theory Center. Financial support from NUTEK through the Parallel Scientific Computing Institute, KTH, and TFR through the Center for Computational Mathematics and Mechanics, KTH, is gratefully acknowledged. 3 Introduction The performance of an airfoil can be characterized by three quantities : the lift, moment and dragcoefficients:Cl,CmandCdrespectively.Theyrepresenttheaerodynamicloadsapplied totheairfoil.Theactualloadsareproportionaltothecoefficientstimesthesquareoftheflow velocity. • Cl corresponds to the force acting on the airfoil in the direction orthogonal to the flow, which allows an aircraft to fly by compensating its weight. • Cmcorrespondstothemomentoftheaerodynamicforcewithrespecttothequarterofthe airfoil chord length. For the equilibrium of an aircraft, the pitching moment of the main winghastobecompensatedbythemomentofanegative-lifttail.Cmshouldthereforenot be too large. • Finally,Cdcorrespondstothecomponentoftheforceintheflowdirection,whichhinders aerodynamic performances and causes fuel consumption. Thepresentworkfocusesonshapeoptimizationofairfoilsinlowspeedviscousflows,based on the analysis ofCl,Cm and Cd. The airfoils are chosen from the NACA 4 digits library [1], in which the shape is expressed analytically as a function of three parameters. The library is presented in section 1.2 of this report. The formulation used to compute the aerodynamic coefficients is an inviscid - boundary layer model. The advantage of this kind of approach is that it provides a fast computation of the flow solution, the disadvantage being that cases with massive flow separation are impossible to handle. Famous codes based on this formulation include for 2D cases Desktop Aeronautics’ Panda [9], and Mark Drela’s ISES [10] and Xfoil [12], and for 3D cases Brian Maskew’s VSAERO [3]. The solver is presented in section 1. In order to perform optimization, the solver has to provide the derivatives of the defined objective function and limiting constraints with the solution for a given set of parameters. The most common method used to compute these derivatives is finite differences. In the present work, a new approach, called automatic differentiation (AD), is used. Is is a chain- rule-based technique to compute the derivative of functions defined by computer programs withrespecttotheirinputvariables,andhasbeeninvestigatedsince1960.In1998,ageneral AD toolbox for programs written in Matlab, called ADMAT [19], has been developped at Cornell University by Arun Verma and Thomas F. Coleman. The present work uses this toolbox. Automatic differentiation is presented in section 2. Finally,airfoilshapeoptimizationisperformedusingtheMatlabOptimizationToolbox[20], which uses a Sequential Quadratic Programming algorithm for non linear constrained problems. The formulation of sample optimization problems and their resolution are presented in section 3. 5 1- The Aerodynamic Solver The solver used to provide the objective function as well as the limiting constraints of the optimization is presented. The solution is computed using an inviscid-boundary layer model, and consists in the lift, moment, and drag coefficients : Cl, Cm and Cd respectively. 1.1- Introduction - Overview of the Model The general Navier-Stokes equations are very powerful since they give a complete description of all possible flow situations. However it is very time-consuming to obtain a numerical solution using them. In our particular case, the incompressible turbulent flow past an airfoil, the viscous effects are important only in a small region near the profile. In this region, the Navier-Stokes equations can be approximated by the so-called boundary layer equations.Outside,viscouseffectscanbeneglected,andonecanuseaninviscidflowmodel. Navier-Stokes Euler V V Boundary Layer The inviscid flow model Theinviscidpartoftheflowcanbesolvedinseveralways.Afinitedifferencediscretization ofthesteadyEulerequationonagridaroundtheairfoilcanbeused,asinISES[10],orapanel method, as in Xfoil [12] and Panda [9]. One advantage of the first option is that by applying aperiodicboundaryconditiontotheouterboundary,cascadeflowscanbesimulated.Thisis done in MISES [13], an extension of ISES. In the present work, a panel method is used. A variety of such methods exists, they differ in the choice of the singularity used to represent the velocity potential on the airfoil (sources, doublets or vortices), and by the choice of the Kutta condition, an extra condition that one must add to the final system of equations in order to obtain a unique solution. Details about different panel methods can be found in [2]. The present work uses a linear vortex distri- bution, which gives a good solution accuracy, even with only a few panels. The Kutta condition chosen is especially well suited for linearly varying singularity distributions. The inviscid flow solver provides the tangential velocity distribution on the airfoil’s surface (Ue). The pressure distribution is then computed from the velocity field using the Bernoulli equation. The lift and moment coefficients, as well as the pressure drag, are calculated by integrating the pressure over the body surface. 7 1- The Aerodynamic Solver The boundary layer model The boundary layer formulation consists of a model for the laminar part of the flow, a transition criterion, and a model for the turbulent part of the flow. The flow models rely on one or two differential equations derived from the integration of the Falker-Skan equations accross the boundary layer thickness, and on additional semi-empirical equations which close the sytem. Xfoil and ISES use the same viscous formulation : a set of two differential equations (integral momentum and kinetic energy shape parameter equations) for both laminar and turbulent flows, and different closure relations depending on the flow regime. Panda on the other hand uses simpler formulations : Thwaites’ one equation method [4] for the laminar part, and Head’s two equations method [5] for the turbulent part. The present work initially usedsimplemodels,thusThwaites’andHead’s.ButsinceThwaites’modelcannotrepresent separated flow, Xfoil’s laminar boundary layer model, which can describe thin separated regions, has been implemented instead. Concerning the transition criterion, Panda uses Michel’s criterion, and Xfoil uses a more advanced e9-type formulation. The second method has been chosen after testing. Details about the reasons of these choices are found in sections 1.4.1 and 1.4.2. Theresultsprovidedbytheboundarylayersolverincludethedisplacementthickness d * ,the momentum thickness q , the shape factor H, and the skin friction coefficient Cf. These quantitiesareusedtocomputethedragcoefficient.Differentmethodscanbeuseddepending on the coupling between the inviscid flow and the boundary layer. The coupling The coupling between the two models is as follows : • The effect of the boundary layer is that it modifies the shape of the airfoil as seen by the externalflow.Thisgivestheinviscidflowazeronormalvelocityconditiononaboundary obtained by adding the boundary layer displacement thickness d * to the airfoil’s surface. • The boundary layer equations depend on the external tangential velocity distribution. Ue Inviscid Flow with zero normal velocity condition Boundary Layer on Airfoil +d * d * Two different approaches are possible for the coupling of inviscid - boundary layer flows. Two-way coupled computation In this case, the modification of the boundary where the zero normal velocity condition has to be met due to the boundary layer thickness is taken into account. The solution begins with the inviscid flow problem, which produces the velocity field. This dataisthenfedintotheboundarylayermodelwhichresultsthelocalwallfrictioncoefficient and the displacement thickness. Then a second iteration is performed, now with modified 8 1.2- Airfoil Shape Parameters surface geometry. This modification can be obtained by displacing the body panels according to the local displacement thickness, and the procedure is reiterated until a converged solution is obtained. Another way to account for the displacement effects is to modify the boundary condition instead of the geometry. In this case the normal flow is made non-zero to account for the effect of d * . This formulation, known as the transpiration velocity concept, states that : V(cid:215) n = --d----(U d * ) on the airfoil’s surface. dx e Using a two-ways coupled model, the total drag is found by adding the friction drag, obtained by integrating the skin friction coefficient Cf, to the pressure drag, obtained by integrating the inviscid pressure distribution. A study of such iterative methods can be found in [14], where it appears that convergence is not easy to obtain. A more recent approach is the one used in ISES and Xfoil. The transpiration velocity concept is used, and the entire nonlinear equation set is solved simultaneously as a fully coupled system by a Newton-Raphson method. This method provides more robust results than the iterative approach. One-way coupled computation In this case the effect of the boundary layer thickness is neglected. One single iteration is performed : the external tangential velocity is computed by the inviscid model with the conditionV.n=0 on the airfoil’s surface, and then fed into the boundary layer model. The drag coefficient is obtained using the Squire-Young formula [8], which in effect computes the momentum deficit. It is a function of some of the boundary layer results (momentum thickness and shape factor) at the trailing edge. Compared to coupled computations, the accuracy of the lift is very good, but not as good for the drag. However the sensitivity of the results with respect to airfoil shape or flow parameters is well reproduced. This method is used in Panda, as well as in the present work. 1.2- Airfoil Shape Parameters The shape of the airfoil can be chosen in the famous Naca 4 digits library [1]. This simple library is interesting because the shape is expressed analytically as a function of three parameters, which control the maximum camber, maximum camber location, and maximum thickness of the airfoil. y 2412 Maximum Thickness 8412 Maximum Camber 4212 x 4812 4406 Maximum Camber Location 4420 9

Description:
The airfoils are chosen from the NACA 4 digits library [1], in which the shape is expressed . Maximum Thickness. Maximum Camber Location. 2412. 8412. 4212. 4812 . are the coordinates of the panel origin in the global coordinate system.
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