A Numerical optimization technique for the design of airfoils in viscous flows

Transcription

1 Rohester Institute of Tehnology RIT Sholar Works Theses Thesis/Dissertation Colletions A Numerial optimization tehnique for the design of airfoils in visous flows Robert MaNeill Follow this and additional works at: Reommended Citation MaNeill, Robert, "A Numerial optimization tehnique for the design of airfoils in visous flows" (1995). Thesis. Rohester Institute of Tehnology. Aessed from This Thesis is brought to you for free and open aess by the Thesis/Dissertation Colletions at RIT Sholar Works. It has been aepted for inlusion in Theses by an authorized administrator of RIT Sholar Works. For more information, please ontat ritsholarworks@rit.edu.

2 "A Numerial Optimization Tehnique for the Design of Airfoils in Visous Flows" by Robert MaNeill A Thesis Submitted In Partial Fulfillment ofthe Requirements for the MASTER OF SCIENCE in Mehanial Engineering Approved by: Dr. P. VnketarunaD Thesis AAIvisor Dr. Kevin Kohnba'ga' Dr. Aliogut Dr. Charles Haines Departrnm Head DEPARTMENT OF MECHANICAL ENGINEERING COLLEGE OF ENGINEERING ROCHESTER INSTITUTE OF TECHNOLOGY JULY 1995

3 PERMISSION TO REPRODUCE: Thesis Title: "A Numerial Optimization Tehnique for the Design of Airfoils in Visous Flows" I, Robert MaNeill, hereby grant permission to the Wallae Memorial Library, of the Rohester Institute of Tehnology to reprodue my thesis in whole or part. Any reprodution an not be used for ommerial use or profit. July 3, 1995

4 ACKNOWLEDGMENTS: I would like to express my appreiation and thanks to the following people: Dr. Venkataraman, the Dean's Offie, and the Graduate Studies Offie for helping me get support for this projet, Dr Ogut for letting me use his lab, Ali for her patiene over the past ouple years, my dog, Max, for being my best friend, and my mother for being my inspiration to persevere through the past five years, espeially the last few months. This work is dediated to the memory of my late step-father, Henry Ernst Kertgens, Jr

5 ABSTRACT The present work involves the implementation of an effiient optimization proedure for the design of airfoils in visous flows The sope of the work is limited to low Reynolds number, inompressible, and unstalled fluid flow. Cubi Bezier urves with orresponding polygons are employed to define the airfoil, the verties of whih are used as design variables in the optimization proess. Invisid onditions about the airfoil are determined using a traditional Hess-Smith-Douglas panel method. Boundary layer alulations are subsequently made based on the invisid results and the solution is updated, thereby aounting for visous effets. A hybrid Generalized Redued Gradient/Sequential Quadrati Programming method is used in onjuntion with the aerodynami model, to optimize the airfoils. Results were obtained for maximum lift and minimum drag problems with and without onstraints. The results of the optimization were validated using CFD

6 Airfoil TABLE OF CONTENTS List of Figures iii Nomenlature v 1 Introdution 1 2. Design Optimization Problem Statement Optimization Proedure Existene and Uniqueness of an Optimum Solution Unonstrained Problems Constrained Problems and the Kuhn- Tuker Conditions Diret Optimization Methods Design Design Variables Optimizer: OptdesX Mathematial Model Invisid Flow Model: The Hess-Smith-Douglas Panel Method Visous Flow Model Program Development Disussion of Program Code Verifiation CFD Modeling Of Optimized Solutions 72

7 3 5 1 Geometry and Grid Generation The Fluent Solver Fluent Model Validation With the NACA Airfoil Optimization Results Problems Considered Problem Results Conlusions and Reommendations 145 List of Referenes 148 Appendies 150 I FORTRAN Files of Analysis Model 1 50 II. Sample Calulation of Aerodynami Coeffiients 179 m. Start Point and Variable Limit Data 1 8 1

8 1 2a 4 LIST OF FIGURES: 2 1 Desription of Usable-Feasible Setor 2 2 Desription of a Convex Curve 2.3 Update Algorithm for the GRG Method 2 4 Overall Algorithm for the GRG Method 3. Typial Airfoil Using Bezier Curves 3. Example of Bezier Curves 3 2b Example of Bezier Curves 3 3 Typial Airfoil Defined by Sheme 2 3. Illustration of OptdesX/Analysis Model Interation 3 5 Nomenlature for Panel Method 3 6 An Airfoil Desribed by Panels 3 7 The i* Panel 3.8 Loal Coordinate System for Panel Method 3 9 Geometrial Interpretation ofeq. (3.20) 4. 1 Subroutine Flow Diagram 4.2 Sloped Line Definition 4.3 Algorithm for Treatment of Laminar Separation 4 4 Coeffiient Comparison Plots for NACA Coeffiient Comparison Plots for NACA Coeffiient Comparison Plots for NACA in

9 7 5.1 Geometry Strategy 5 2 Grid Mapping Strategy 5.3a Sample Geometry 5.3b Close-up of Sample Geometry 5 4a Weighted Grid for Sample Geometry 5 4b Close-up of Weighted Grid for Sample Geometry 5 5 Grid in Viinity of Leading Edge 5.6a Geometry for Cambered Airfoil 5 6b Close-up of Geometry for Cambered Airfoil 5. Grid Mapping Coordinates for Cambered Airfoil Optimized Airfoil Plots IV

10 NOMENCLATURE 0 the null vetor A (m+1) x (n-1) matrix of equality onstraint gradients with respet to Z panel method oeffiient matrix Aj the (i, j)* omponent ofthe oeffiients matrix ai quadrati funtion oeffiients, i = 1, 2, 3 B (m+1) x (m+1) matrix of equality onstraint gradients with respet to Y b; b(y) i* element of b vetor binomial distribution as a funtion of y Cd drag oeffiient Cl lift oeffiient Cm moment oeffiient panel method freestream veloity vetor Cf skin frition oeffiient Cp pressure oeffiient E entrainment veloity F(X) objetive funtion f; arbitrary i* funtion Gr the generalized redued gradient gj(x) j* inequality onstraint H Hessian matrix

11 H shape fator 6*/9 Hi shape funtion parameter hk(x) k* equality onstraint I the identity matrix J3,i 3rd order Bernstein basis m slope n number of design variables A n unit normal vetor L Lagrangian / number of equality onstraints, parameter for Cf independent of Reynolds number /; length of impanel m number of inequality onstraints N number of nodes desribing the airfoil panels n order of binomial distribution P Pi (x,y) pair for the oordinate ofthe Bezier urve ^produt of 8*iVe,i Q Q q (m+1) x (n+m) matrix of all equality onstraints volume flow rate soure strength per unit length R,. x Re, 9 Reynolds number with respet to x Reynolds number with respet to G VI

12 X* r polar length oordinate S searh diretion s urvilinear distane A t unit tangent vetor t dummy variable for the distane along a panel u x veloity omponent V veloity v y veloity omponent X design variable vetor optimal solution X; i* design variable x x-oordinate Xj j* variable of funtion f x, i* panel midpoint x-oordinate Y dependent variable vetor of size m + 1 Y derivative matrix of matrix Y with respet to a Yi i* dependent design variable y index for binomial distribution, 0, 1,2,..., n, y-oordinate yt i* panel midpoint y-oordinate Z independent variable vetor of size n - / VII

13 5* Z; i* independent design variable a step size parameter, angle of attak optimal step size parameter 5j salar parameter for j"1 inequality onstraint displaement thikness s gradient perturbation parameter < > flow potential y vortex strength per unit length X Lagrange multiplier, dimensionless pressure gradient parameter Xj Lagrange multiplier for a orresponding f* inequality onstraint.vk+m Lagrange multiplier for a orresponding (m+k)* equality onstraint u visosity v parameter for determining airfoil oordinates, P(x,y) 0 angular polar oordinate, momentum thikness 0; inlination of i* panel x shear stress V gradient VIII

14 Subsripts i index for design variables, 1,2,..., n, Bezier vertex index, arbitrary funtion index, panel index; 1, 2,...,N j index for inequality onstraints, 1, 2,..., m, arbitrary variable index, panel index, 1, 2,..., N k index for equality onstraints, 1,2,...,/ n normal diretion S indiates soure singularity t tangential diretion V indiates vortex singularity w wall (y = 0) Y indiates gradient with respet to Y Z indiates gradient with respet to Z oo freestream onditions Supersripts -1 inverse of matrix I Bezier vertex index / lower limit IX

15 n order of binomial distribution o 0th iteration q iteration number T vetor transpose u upper limit y binomial distribution index * optimum loal oordinates

16 1. INTRODUCTION Sine the beginning of powered flight there has been intense interest in the area of airfoil design. In the early stages of aerodynamis, mostly ad ho methods were employed to determine suitable wing setions. Over time, however, the design proess has beome more sophistiated. In the past, diret analysis methods were widely used where a designer would define a geometry and then analyze the flow about the airfoil. Generally, panel methods were employed for these analyses. The original design would have to be modified based on the output and the proess would be iterated until satisfatory aerodynami performane was ahieved. As one would expet, this an be a time onsuming proess that does not lend itself well to systemati updating. Analyzing the flow onditions for a speified airfoil shape is the most diret approah but is by no means the most effiient method for optimization. For this reason, there has been onsiderable study into the inverse design problem. The inverse design problem may be defined as a tehnique that yields an airfoil shape as output while the pressure/veloity around the airfoil is speified. The inverse problem an be a very useful approah to airfoil design. Conformal mapping tehniques are mostly used in typial appliations of inverse design problems. The famous Eppler method (Referene 1) employs a onformal mapping tehnique whih requires a user speified veloity distribution. Based on this, an airfoil geometry is alulated. Inidentally, the Eppler ode may also be used for diret design

17 sine it also uses a panel method with an integral boundary layer solver These solution shemes an be muh quiker than diret methods beause no guessing has to be done to ahieve desired onditions. However, being restrited to a speified distribution an limit a solution from being potentially optimum. It would be ideal to use a method with optimization apabilities whih ould searh the design spae, updating based on previous solutions, to find optimum shapes. For this reason, the present work proposes an optimization method involving an objetive funtion of maximum lift oeffiient (l) or minimum drag oeffiient (d) whih takes into aount pressure and veloity distributions. Unlike traditional inverse design problems, the present method is not restrited by initial flow onditions and lends itselfto finding unonventional shapes. Design optimization in general is extremely attrative and has been reeiving a lot of attention lately. With the advanements in omputers and optimization algorithms, one almost unsolvable problems are now very modest in run times. By virtue of their iterative nature, optimization proedures usually take longer than straightforward design tehniques, but the end result is almost always superior. In response to the reent interest for design optimization, there are urrently many easy to use software pakages that make the optimization proess muh less burdensome. Some finite element pakages even have optimization modules and it is foreseeable that CFD will soon inorporate similar ones. Until then, however, it is still desirable to have a method for the optimization of airfoils. Over the past years, many methods for optimizing aerodynami shapes have been developed. For example, Vanderplaats, Hiks, and Murman (Referene 2) advaned a method that optimizes airfoil shapes based on onjugate diretions for loally

18 unonstrained problems and feasible diretions for loally onstrained problems The airfoil geometry is defined by a polynomial approximation for different segments of the shape For this geometry desription method, the design variables beome the segment endpoint oordinates and the polynomial oeffiients. Problems for minimum drag, minimum pithing moment, and maximum lift were addressed with some suess Most of the solutions were, however, limited by loal optima. Another method developed by Vanderplaats and Hiks (Referene 3) allows the designer's intuition to play a role in the optimization. This is atypial of most optimization tehniques. The method starts with many different shapes, eah with desirable harateristis to the user. The approah is to define the airfoil as a linear ombination of the shapes. The optimization problem beomes one of determining the influene of eah shapes on the optimal solution. This method seemed more suessful than the first but a major disadvantage is that solutions are limited by the initially defined shapes. Liebek and Ormsbee proposed yet another method that was very suessful (Referene 4). For the tehnique, boundary layer theory and the alulus of variations were employed to determine the pressure distribution whih provides the maximum lift without separation. This solution sheme produed the very well known Liebek high lift profiles whih were reported to have lift oeffiients as high as 2.8 for Reynolds numbers between five and ten million. Drag oeffiients for the orresponding airfoils were kept as low as A single element airfoil definition was used to define the shape. The analysis is split into two subproblems whih onsists of first determining the pressure distribution for maximum lift without separation, and then determining an airfoil shape to determine

19 the speified pressure distribution (inverse problem) The first step employs a Stratford distribution to model separation Besides the no-separation requirement, onstraints are plaed on the leading edge and trailing edge to assure design feasibility. Also, the satisfation of the Kutta ondition is required. The seond step is aomplished by utilizing onformal mapping methods. Low drag airfoils and fuselages are of speial interest in the field of gliders. For this reason, Coiro and Niolosi (Referene 5) developed a tehnique to determine low drag airfoils, high lift and low drag multi-omponent airfoils, and extended laminar region fuselages. A onstrained-rmnimization method is used as the design optimizer An integral boundary layer method is also used for the aerodynami analysis. Chang, et. al. (Referene 6) proposed a method for the optimization of wing-body onfigurations through the solution of the Euler equations for the flow surrounding the airfoil. A feasible diretion method for onstrained optimization is used as the optimizer. Unlike the other methods desribed, the geometry is ontrolled and analyzed through the use of a flow field grid generator and a set of shape funtions for the definition of the shapes Although these optimization methods vary widely in their sope and solution strategy, they share some ommon harateristis. In the optimization proess, there are three distint omponents: (1) the design variables, (2) the optimizer, and (3) the mathematial model Eah omponent should be effiient individually to effetively streamline the optimization proedure.

20 To determine appropriate design variables for the problem, an effiient way to define an airfoil whih would lend itself to design perturbation is neessary There are many ways to desribe airfoil geometries but the present work uses a Bezier polygon definition sheme whih is outlined in Referene 7. This method provides an extremely effiient desription tehnique that handles perturbation very niely. Again, it is very important that the optimizer be very effiient to solve the problem. Ideally, the optimization tehnique should determine searh diretions solely from gradient information. There are many effiient methods available. The lass of optimization tehniques referred to as diret methods are by far superior to others in dealing with problems with many design variables and onstraints. The optimization proedure used for the present work is a hybrid method whih ombines the best harateristis of the Generalized Redued Gradient method and Sequential Quadrati Programming. This algorithm is implemented through a software pakage alled OptdesX (Referene 8) Finally, an effiient method for alulating the onditions about the airfoil should be employed to solve the problem. An invisid flow panel method is used with an integral method for the boundary layer effets beause ofthe eonomy of its solution, This method may be somewhat primitive and, hene, less aurate ompared to other more time onsuming flow analyzing tehniques but it is very important that the mathematial model be as effiient as possible in an already time onsuming iterative proess. Also, it ould be advantageous to sarifie some auray in the solution beause if the objetive is satisfatorily improved while the onstraints are satisfied, then the optimization proess may be onsidered suessful, regardless of the preision of the results. Furthermore,

21 sophistiated CFD odes an be used to more aurately determine the flow onditions about the airfoil that is determined to be optimal. The present work first reviews optimization theory followed by an in depth disussion of the Generalized Redued Gradient method. Then an airfoil design setion is dediated to the disussion of the three main omponents of optimization: the design variables, the optimizer, and the mathematial model. Program development is then disussed followed by CFD modeling of the solutions. For the urrent work, the NACA 0012 and the NACA 4412 airfoils were both used as starting points for the optimization. Also, two different Bezier definition shemes were used to desribe the airfoils. Finally, six optimization ases were onsidered and are listed as follows: A. Max Cl, unonstrained D and Cm B. Max Cl, Cd < 0.007, unonstrained m C. Max L, Cd < 0.007, M < D. Min Cd, unonstrained Cl and m E. Min Cd, l ^ 0.7, unonstrained M F. M_nD, L>07, M<-0.15 Considering the two starting points, the two definition shemes, and the six ases, a total of twenty four optimization problems were onsidered. The presentation of the results is followed by a disussion and onlusions. 6

22 2. DESIGN OPTIMIZATION 2.1 Problem Statement Design optimization is a systemati approah meant to improve upon existing designs. Many optimization algorithms have been developed but almost all of them share some ommon features For example, most methods are meant to optimize a design by improving upon an objetive funtion while satisfying some design onstraints, where the objetive and onstraints are funtions of the design variables relevant to the problem The general statement of an optimization problem is as follows: Minimize: F(X) X = [X_, X2>...,XJ (2.1a) Subjet to: gj(x) < 0 j * 1, m (2. lb) hk(x) = 0 k=l,/ (2.1) X1'<X1<X1U i=l,n (2. Id) Where F(X) is the objetive funtion of the design variables, X, gj(x) are the inequality onstraints, and ht(x) are the equality onstraints. Side onstraints used to limit the design to the feasible, or realisti, domain are shown in Eq. (2. Id). The funtions F(X), gj(x), and hk(x) may be linear or nonlinear in X, and may be evaluated by any analytial or numerial tehnique. Also, for most optimization tehniques, the objetive funtion and onstraints have to be ontinuous and have ontinuous first derivatives (for gradient alulations).

23 2.2 Optimization Proedure A fundamental similarity between different optimization tehniques is the basi update and searh method Most methods require an initial set of design variables, X. It is advantageous if the initial guess was in reasonable proximity to a realisti design This is not always neessary for onvergene, but it ould expedite the design proess. From the start point the design is updated iteratively until an optimum is found The ommon form ofthe update is Xq - Xq_1 + Sq (2.2) where q is the iteration number, S is the vetor searh diretion, and is the optimal step size in diretion S As an be seen in Eq. (2.2), previous iterations are updated with the term <x*sq. This update is a one-dimensional searh whih onverts a multiple variable problem (n) into a one variable problem in a. The two main stages of the update onsists of finding a suitable searh diretion, S, and determining the salar The searh diretion would be found suh that it improves the objetive while maintaining the onstraints. The salar is found to be the optimal step size in diretion S. For a method to be suessful, the update should be determined effiiently and reliably. 8

24 2.3 Existene and Uniqueness of an Optimal Solution Beause of the inherent nonlinearity of most optimization problems, it is diffiult to ensure that an optimal solution is global For nonlinear problems, there is a strong tendeny to settle into loal optima (also alled relative optima). This means that there will assuredly be multiple solutions and it will be neessary to attak the problem from different starting points to try to find a global optimum If the same solution is onverged upon from different points, it is reasonable to say that the solution is a global optimum Of ourse, this leads to an inrease in design time to an already time onsuming iterative proess Unonstrained Problems For unonstrained problems, there is a set of well defined riteria for determining whether a solution is a relative optimum. Also, though it is diffiult to do, these onditions may be used to determine whether a solution is a global optimum. The first ondition is that the gradient with respet to all variables (i = 1, n) be the null vetor: VF(X) = '<_F(X)/<2, = 0 (2.3) &(X)/Xn Furthermore, for a rninimum, the seond derivative matrix (Hessian) must be positive definite. Likewise, for a maximum, the Hessian must be negative definite. These are neessary, but not suffiient, onditions to prove that the solution is a relative minimum. To ensure that a solution is definitely a relative rninimum, the Hessian of the objetive

25 ax2 X* funtion must be positive definite for the solution A Hessian matrix of a funtion is defined as the matrix of the seond partial derivatives with respet to all variables The Hessian of the objetive funtion is H <?F(X) 2F(X)?F(X) dx\ ax,exl dxxxn?f(x) o2f(x) ff{x) 3XxdXx 6X2dXn (2.4)?F(X) o2f(x) <?F(X) Kn8Kx KndX2 ax2n For a matrix to be positive definite, all its eigenvalues must be positive in sign. If both of the previously defined onditions are satisfied, then the solution X is at least a relative minimum. The only way to ensure that a design is a global minimum is if the ondition defined by Eq. (2.3) is satisfied and the Hessian is found to be positive definite for all possible values of X. Obviously, this an be diffiult, if not unrealisti, espeially for a design with many design variables. The best way to ahieve a level of onfidene that an optimum is global is to solve the same problem from many different starting points Constrained Problems and the Kuhn-Tuker Conditions Unfortunately, determining whether a solution is a global optimum or even a relative optimum beomes more diffiult with the addition of onstraints to the optimization problem. Unlike the unonstrained problem, the gradient of the objetive does not have to equal the null vetor at the optimum. This an be the ase if at least one onstraint is ative. An ative onstraint may be defined as one whose value is lose to zero (from, g(x) < 0). 10

26 F(X) = onstant Optimum Figure 2.1: Desription of Usable-Feasible Setor Figure 2. 1 (from Referene 9) may be used to intuitively establish the neessary onditions for a onstrained problem. For the optimum, point B, both the gradients of the onstraints and the objetive are perfetly opposite in diretion. This means that, at the optimum, the searh diretion would have to be perpendiular to both VF(X) and Vg(X). This is an important onlusion in that it establishes a neessary ondition for a solution to be optimal, but, unfortunately, this is not a suffiient ondition. This and two other 11

27 X" onditions make up what is known as the Kuhn- Tuker neessary onditions for onstrained optimality. If X is an optimal solution, it must satisfy the following: 1. is feasible (2.5) 2. A,gj(X) = 0 j = l,m, X,Z0 (2.6) 3. VF(X*) + fjavgi(x') + fdak+mvhk(x*) = *=i 0 (2.7) >4>0 (2.8).vm+k unrestrited in sign (2.9) The first ondition is obvious in that a design must be feasible (satisfy all onstraints) to be optimal. Equation (2.6) states that if gj(x) < 0, and, hene, not ative, then the Lagrange multiplier Xj must be zero. The last set of equations is the mathematial statement that, at the optimum, the only possible diretion vetor S would be tangent to both the ative onstraint boundary and the line of onstant objetive funtion and would be perpendiular to both gradients. Also, Eq. (2.7) is alled the Lagrangian L(x,X). At off optimal designs, the Lagrangian is defined as L(X,A)^ F(X) + fdajgj(x) + fdam+thk(^) (210) Where the Lagrange multipliers Xj and A^+t at as salar multipliers to gj(x) and hk(x). While neessary and suffiient for the determination of loal optima, satisfation of the Kuhn-Tuker onditions is not neessarily suffiient to ensure a global optimum. A solution X* an, however, be said to be a global optimum if it satisfies the Kuhn-Tuker onditions and the surfaes of the problem's objetive and onstraints are onvex A onvex surfae an be defined as one where, if any point on the surfae was onneted to 12

28 the optimum, then the entire line onneting them would lie within the feasible region and the surfae defined by g(x) = 0 is onvex. This implies that if the Kuhn-Tuker onditions were satisfied, then the solution would be a global optimum, as is the ase with point B in Fig 2.1. An example of a non-onvex surfae may be seen by referring to Fig 2.2 (from Referene 9). While the line onneting points B and D lies within the feasible region, the lines onneting points A and C, and points A and D would be outside of the onstraint bounds for some portion of the line. Therefore, the surfae defined by g(x) = 0 in Fig. 2.2 is non-onvex and the Kuhn-Tuker onditions ould not be used to determine a global optimum. x2!.(x) = 0 F(X) = onstant Figure 2.2: Desription of a Convex Curve 13

29 As previously mentioned, it is very diffiult to determine whether the suffiieny requirements have been met for a partiular problem. Therefore, it is again suggested that the problem be started from multiple points to ensure a global optimum. 14

30 2.4 Diret Optimization Methods For the present work, a hybrid Generalized Redued Gradient (GRG)/Sequential Quadrati Programming (SQP) method was used as the optimizing proedure The algorithm was made available through the software pakage OptdesX Sine the proedure was not programmed by the author, its desription is not inluded (see Referene 10). However, as an example of a typial optimization method, the GRG method and its implementation is desribed in detail. This method belongs to the family of tehniques known as diret methods. They are named as suh sine they all deal with onstraints in a diret manner. This is opposed to former, less sophistiated methods that rely on onstraint penalty funtions to onvert onstrained into unonstrained optimization problems. The diret methods are typially very effiient and are in almost exlusive use with respet to optimization problems with multiple variables and onstraints The Generalized Redued Gradient method determines optimal solutions by relying heavily on ative onstraints. An ative onstraint is one whih is at its bounds, hene, potentially limiting the solution. For the method, a searh diretion is found to keep any ative onstraints exatly ative for small movements in that diretion. If, for any reason suh as nonlinearity, the urrently ative onstraints beome inative upon movement in the searh diretion, then the onstraint would be fored to its bounds by using Newton's method. The present method solves equality onstrained problems only by adding a slak variable to eah inequality onstraint. Subsequently, the general form of a GRG problem is 15

31 Minimize: F(X) (2.11) Subjet to: gj(x) + XJ+n = 0 j=l,m (2.12) hk(x) = 0 k=l,/ (2.13) Xi/<Xi<Xiu i=l,n (2.14) X^>0 j=l,m (2.15) Due to the requirement of slak variables in Eq (2.12), the number of design variables inreases to n + m. The extra design variables ould make the problem diffiult to solve beause of extra storage requirements, however, this problem ould be alleviated through areful storage of gradient information. The basi idea behind the GRG method is that the total number of design variables may be redued by defining one dependent design variable for eah equality onstraint. Considering the problem stated in Eqs. ( ), the X vetor ontains the original n design variables as well as the m slak variables. For larity and onveniene, X an be split into its omponents and written as follows: X={ZY}T (2.16) Where Z represents n - / independent variables, and Y represents m + / dependent variables. For this definition, no restritions are assigned regarding whih variables are to be ontained in Z and Y. As will be disussed later, the dependent variables will be hosen from the original n variables as well as the m slak variables suh that the problem is balaned. Beause all onstraints are now equal, it is onvenient to ombine Eqs. (2.12) and (2 13) into a single onstraint definition: hj(x) = 0 j = l,m + / (2.17) 16

32 ' _Vfhm+/(X)..V[hm+/(X) and ombine Eqs. (2. 14) and (2 1 5), yielding X/<Xi<X;u i=l,n + m (2.18) where the upper bounds on the slak variables are allowed to be very large Now the optimization problem has been redued to Minimize: F(X) = F(Z, Y) (2 19) Subjet to: hj(x) = 0 j = l,m + / (2.20) X/s-Xi-.X," i=l,n + m (2.21) Differentiating the objetive and the onstraint funtion yields df(x) = VZF(X) dz + V YF(X) dy (2. 22) dhj(x) = Vzhj(X) dz + VYhj(X) dy (3.23) where j = 1, m + / The subsripts Z and Y indiate gradients with respet to the independent and dependent variables, respetfully. For feasibility, the equality onstraint must remain satisfied (assuming they are satisfied initially) for any hange in the independent variables. This means that dhj(x) = 0, j = 1, m + / in Eq. (2.23). Sine Y ontains m + / dependent variables and sine Eq. (2.23) is atually a system of m + / equations, they an be written as vjmx) vjh,(x) dh(x) = Vjh2(X) dz + Vjh2(X) tfy (2.24) or dh(x) = A dz + B dy (2.25) 17

33 where the dimensions of A and B are (m + /) x (n - /) and (m + /) x (m + I), respetively Sine it was established that dh(x) = 0, for any hange of the independent variables dz, Eq. (2.25) an be solved for the orresponding hanges dy in the dependent variables to maintain feasibility: dy = -B-'AdZ (2.26) Now substituting Eq. (2.26) into Eq. (2.22) yields df = VZF(X) dz - VYF(X)T [B"1 A]dZ = {VZTF(X) - VYF(X)T [B"1 A]} dz (2.27) So Gr = df(x)/dz = VZF(X) - [B"1 A]T VYF(X) (2.28) Equation (2.28) defines the generalized redued gradient Gr, and an be viewed as an unonstrained funtion. A searh diretion S an be found using Gr for use in the following equation: Xq = Xq-1 + a*sq (2.29) Also, the dependent variables Y are updated using Eq. (2.26) for every proposed step a. It must be remembered, however, that Eq. (2.26) is a linear approximation to a nonlinear problem. This means that when the onstraints are evaluated for an a, they may not be exatly zero. In other words, the vetor dh(x) from Eq. (2.25) may not be the null vetor. In order to orret this, holding Z fixed, a new dy must be found to drive h(x) to zero So hj(x) + dhj(x) = 0 j=l,m + / (2.30) and dy must be found so that 18

34 ' ' A dhj(x) = -hj(x) j = l,m + / (231) Substituting Eq. (2 31) into Eq. (2.25) yields a new estimate for dy: dy = B1 (-h(x) - dz} (2.32) Next, dy is added to the most reent vetor Y of dependent variables and the onstraints are evaluated again. This is repeated until hj(x) = 0, j = 1, m + /, within a speified tolerane Start i Given: X, S, J, a0,. a-a0 *> X-X + as Yes Update the dependent variables using Eq. (6-95) Interpolate a* for Figure 2.3: Update Algorithm for the GRG Method In summary, a redued gradient is reated from previously hosen dependent variables. The redued gradient is then used to determine a searh diretion in the independent variables. While marhing in this diretion for eah proposed a, the 19

35 dependent variable vetor Y is updated as previously disussed This updating method is quite similar to Newton's method for solving simultaneous linear equations for dy However, the present method assumes the gradient information in A and B to be onstant. After finding the minimum in the searh diretion, and, hene, a*, the proess is repeated until the onvergene riteria is satisfied. Figure 2.3 (from Referene 9) shows the algorithm for this updating sheme. The last requirement is to hoose the dependent variables, Y, (1) suh that the B matrix is not singular and (2) so that the solution an move some distane in a searh diretion without violating the side onstraints. The seond requirement is satisfied by piking dependent variables whih are a suffiient distane from the limits of their respetive side onstraints. The first requirement, however, is a little less obvious, it is met by starting with the matrix of gradients: Vrh,(X) Vrh2(X) (2.33) Lvrhm+/(X)i (m+l)x(n+m) To this point, the independent and dependent variables have not been divided. Sine there are more olumns than rows in matrix Q, it is desirable to find a nonsingular submatrix that would orrespond to B, hene, defining the dependent variables. This an be done by employing Gaussian elimination with pivoting on matrix Q. To find the dependent variables, start with the following matrix equation: QX = I (2.34) 20

36 Where I is a (m + I) x (m + I) identity matrix The first step is to searh row one of Q for the element of largest magnitude, omitting those that are at the side onstraint limits Next, the matrix is then pivoted on that element and regular Gaussian elimination operations are performed. This proess is repeated for the remaining rows. After this proedure is omplete, the right hand side of the equation ontains B'\ The remaining olumns of Q will ontain the produt B"1 A from Eq. (2 28). Start Given: X, F, (g,j=l, m),(h.. /. =!,/) j * Add slak variables to inequality onstraints Calulate the gradients of the objetive and all onstraints Calulate the redued gradient Determine the searh diretion S Perform the one-dimensional searh with respet to the independent variables XVx'-'+a'S' / / / Update the dependent variables using Newton's method x-x +.xd Figure 2.4: Overall Algorithm for the GRG Method 21

37 The method just desribed for determining the dependent variables will satisfy the requirements of a nonsingular B matrix orresponding to variables that are not at their onstraints. However, several potentially hazardous situations are not aounted for and need to be addressed separately For instane, the method does not guarantee that a side onstraint will not be violated during a one-dimensional searh. If this was to our, a smaller step size a would need to be hosen suh that the onstraint was not violated Another situation that may arise is when a row or a olumn of Q ontains all zeros. This ould either mean that a redundant onstraint exists whih may be temporarily deleted from the set (temporarily beause the onstraint may beome independent later in the optimization proess) and the number of dependent variables redued by one, or it ould mean that no onstraint is a funtion of the design variable. The latter is a degenerate ase where no dependent variable should be used for the design variable in question. A final situation may our where the only nonzero pivot element is at one of its side onstraint limits. In this ase, there is no hoie but to inlude this variable in the dependent variable set. After alulating dy it may be found that the variable moves away from its limit. However, if the variable moves in a manner suh that it violates its onstraint, that variable is set to its limiting value. This ould prevent Newton's method from onverging whih would indiate that the step size a should be redued. Figure 2.4 shows the overall algorithm for the GRG method. The simplest way of finding the searh diretion S is to take the negative of the generalized redued gradient: S = -Gr (2.35) 22

38 For subsequent iterations, other more sophistiated methods suh as a onjugate diretion or a variable metri method may be used See Referene (9) for a more in depth disussion of those methods. Also, a first estimate for a may be found as the distane to the nearest side onstraint. For independent variables: dz,/da = S, (2.36) and for the dependent variables (using Eq. (2.32)): dy da ~, = Y = -B_1AS (2.37) To find the a whih will drive the i* independent variable to its bound: Z,- +a dzj -r-*- i = Z, + asj = Zj or Zf (238) yielding Zl ' -Z ' a = &i if S; < 0 (2.39a) a= Zu ' - Z J/ ' if Si > 0 (2.39b) Based on the above linear approximation, the a whih drives an independent variable to either its upper or lower bound is the minimum a for all independent variables Z;, i = 1, n - /. Similarly, for the dependent variables (using Eq. (2.37)): Yi+aYi=Y/orY/' (2.40) yielding Yl -Y a=-jrl y, if Y; < 0 (2.41a) 23

39 Yu - Y a=^-z-l y, if Yi > 0 (2.41b) Again, the a whih drives a dependent variable to its bound is the minimum a for all dependent variables Y, i = 1, m + / The minimum of the values of a determined by Eqs (2 39) and (2 41) is then taken as the step size for the one-dimensional searh This one-dimensional searh method is atually very effiient with quadrati polynomial interpolation usually being suffiient. Also, to hasten onvergene, quadrati approximations an be used on the omponents of dy while using Newton's method on Eq. (2.32). Up to this point, it has been assumed that the initial design was feasible. Sometimes, however, it is unavoidable to start from an infeasible design If the initial design is not feasible, the first step is to obtain a feasible point from whih feasibility an be maintained. This is given top priority beause the GRG method is based on the requirement that the onstraints are satisfied exatly and remain so throughout the optimization proess. There are a few different methods for finding a feasible start point suh as employing Newton's method, minimizing the sum of onstraint violations, or minimizing the objetive while heavily penalizing the sum of the onstraint violations. See Referene (9) for a more detailed disussion of these methods as they apply to initially infeasible start points. In general, the GRG method is a very effiient diret method. Its appliation is espeially worthwhile if funtion and gradient evaluations are ostly. However, the method does have diffiulties in some areas. For instane, if the problem is highly 24

40 nonlinear, the use of Newton's method to maintain feasibility during the one dimensional searh may beome ill-onditioned and may not onverge, requiring very small steps for onvergene Also, if there are many inequality onstraints, storage requirements may beome large as does the solution of the dependent variable subproblem. Furthermore, the method by whih the GRG algorithm returns infeasible solutions to the onstraint boundary (Newton's method) is not as effiient as some other algorithms but, on the other hand, a feasible solution is guaranteed at the end of every iteration, whih has its virtues Despite the mentioned problems, the Generalized Redued Gradient method is an effiient diret method for solving onstrained optimization problems. 25

41 3. AIRFOIL DESIGN There are three main parts involved in the optimization proess: (1) the design variables, (2) the optimizer, and (3) the mathematial model. It is very important that all of these omponents be robust to failitate an effiient solution to an optimization problem In the following setions, the individual faets to the proess are disussed in detail 3.1 Design Variables In design optimization problems, it is highly important that design variables be hosen judiiously to failitate a streamlined analysis. For the airfoil optimization problem, an effiient method for desribing the airfoil shape is required. Trying to ontrol individual points along the airfoil surfae would beome too umbersome in the analysis beause too many points would need to be used to ahieve any degree of surfae resolution. To minimize the number of design variables, a Bezier parameterization sheme, as outlined in Referene 7, was used to define the airfoil. This method of definition an be easily used to desribe airfoils, traditional or ontemporary. In general, airfoil definition using Bezier urves lends itself extremely well to the optimization proess. Airfoil onstrution involves a total of four Bezier urves, two for the top surfae and two for the bottom surfae Eah parametri urve is ompletely defined by a Bezier polygon onsisting of four verties The order of a Bezier urve is defined as the number of verties minus one, hene, the urves are ubi. Adjaent urves share endpoints or polygon verties. A typial airfoil as defined by Bezier parameterization may be seen in Fig

42 Figure 3.1: Typial Airfoil Using Bezier Curves As previously mentioned, Bezier parameterization lends itself very well to airfoil definition and several features of the desription method enhane its usefulness. The Bernstein basis funtions are used for the Bezier parameterization. Referring to Figs. 3.2a and 3.2b (from Referene 7), the following useful properties have been used to aid in defining design variables for the airfoil geometry: (1) the basis funtions are real, (2) as previously mentioned, the order of the Bezier urve is defined as one less than the number of verties of its orresponding polygon, (3) the first and last points of a Bezier urve share ommon points with the first and last points of its polygon, (4) at the endpoints, the 27

43 ..1(3-01 slopes of the polygon and the urve are equal, and (5) the urve is onfined within the onvex hull of the polygon (a) (b) Figures 3.2a and 3.2b: Examples of Bezier Curves A ubi urve has four orresponding polygon verties: B0, Bi, B2, and B3, where eah B; represents a point in two-dimensional, x-y spae. A parameter v, lying between one and zero, is used to haraterize any point P(x,y) on the urve whih may be found by the equation P(v) = (*,>) = 5, J3,(v) 1=0 (3.1) where J3,i is the Bernstein basis, or blending funtion, and is defined as JiM)= fi\ v'(l-v)j j 11 3-i... (3.2) with f-i\ \U 3! (3-3) 28

44 Notie that this is nothing more than the probability density funtion of the binomial distribution where parameters n and p equal 3 and v, respetfully The binomial distribution is defined as g(y)= f{\-pty y = 0,l,2,...,n (3.3) A more in depth disussion of the usefulness and appliability of Bezier parameterization of airfoils, as developed by Venkataraman, is onsidered in Referene (7). Also see Referene (1 1) for a more in depth disussion of the theory behind Bezier urves. To onstrut an airfoil from four Bezier urves, some onstraints need to be defined. First, the leading edge of the airfoil is shared by the leftmost top and bottom urves (refer to Fig. 3.1). To ensure a ontinuous and properly defined leading edge, the three verties losest to it should be required to make a vertial line. This will make the leading edge the leftmost point. Similarly, the rear two segments share the trailing edge but no onstraint is plaed on the adjaent verties. Next, a ontinuity restrition must be imposed on the shared verties between the left and right segments for both the top and bottom surfaes. The three verties in the viinity of the shared vertex should all lie on a straight line to guarantee slope ontinuity For the optimization problem at hand, two different definition shemes were used. Both used a horizontal line onstraint for the upper surfae but one used a horizontal line onstraint for the lower surfae while the other used an unrestrited sloped line. For onveniene, the horizontal line definition is alled sheme 1 and the sloped line definition is alled sheme 2. The different shemes were used to investigate the effet of different definition methods on final solutions. Sheme 1 would normally lend itself to traditional airfoil definition while sheme 2 would normally be 29

45 better for more ontemporary airfoils with high amber Figure 3.1 shows a typial airfoil shape as defined by sheme 1 and Fig. 3.3 shows a typial airfoil shape as defined by sheme Bl.l m B14(B21) B / B B32 B23 B43 -B11(B 44) B33 *7m 4(B41) J Generated Airfoil Bezier Polygons B24(B B I/C Figure 3.3: Typial Airfoil Defined by Sheme 2 The atual design variables are desribed by some of the polygon verties Beause of the definition onstraints just outlined, it is unneessary to use all of the verties as design variables. Therefore, the onstraints also help to redue the number of design variables. Also, the leading edge oordinates as well as the trailing edge oordinates need not be design variables sine their loations are fixed at (0,0) and (1,0), respetfully. Sheme 1, referring to Fig. 3.1, requires the determination of both the absissa and 30

46 ordinate values for Bu, BZ3, B3._, and B3,3, the absissa values for BI4, B2.2, B34, and B42, and the ordinate values for Bu, and B4,3 (the first number in the subsript refers to the segment number and the seond number refers to the vertex number). Sheme 2, referring to Fig 3 3 requires the determination of both the absissa and ordinate values for Bu, Bz3, B3,2, B3,3 and B3,4, the absissa values for Bi,4, Bz2, and B42, and the ordinate values for Bi>2, and B43. This makes fourteen total design variables for sheme 1 and fifteen total design variables for sheme 2. In addition to slope ontinuity onstraints, some limitations are plaed on verties suh that only realisti airfoils are allowed. This is done by onstraining the magnitude of the absissas and by onstraining the distane between adjaent ordinates. For example, the differene between the loations of Bi,3 and Bi2 may be onstrained to be less than 30% of the hord. Likewise, the absissa value of B23 may be onstrained to be less than 10% ofthe hord. These are simple linear onstraints whih are easy to analyze. There are 18 of these onstraints for sheme 1 and 20 for sheme 2. Also, maximum and minimum values were set for all of the design variables (side onstraints) suh that the design was onstrained to be realisti. 31

47 3.2 Optimizer: OptdesX To run the optimization of the design variables, an interative software pakage alled OptdesX was used. This software allows a user reated analysis model to be linked to it via program subroutines whih are used to define the design variables, the onstraints, and the mathematial model in general. OptdesX is extremely flexible in the way it allows adjustment of the design variables and design funtions. This flexibility allows for quik redefinition of problems, making it easier to investigate the effet of different parameters on the optimal solution. Another major advantage of the software is that it allows the user to effiiently save intermediate results whih ould be quikly realled for further study. OptdesX works by updating design variables based on a user speified optimization algorithm and gradient alulation method. These updated variable values are then sent to the user supplied analysis model where the mathematial model for the problem is exeuted. After the alulations are ompleted, funtion values are relayed bak to OptdesX. This proess, illustrated in Fig 3.4, is repeated until a searh diretion is found. Then the hosen optimization algorithm determines a suitable step length, iterates the solution, and repeats the proess until onvergene is satisfied. OptdesX has two optimization methods for ontinuous variables available within the software. The first is a hybrid GRG-SQP method (whih is alled GRG in OptdesX) and the seond is a traditional SQP method. The first is said to be a more robust method than the plain SQP in that it is said to be able to solve a wider variety of problems This may be due to the fat that, although solutions may go slightly infeasible during an iteration, feasible solutions are guaranteed by the GRG method at the end of eah 32

48 iteration. For these and other reasons, the GRG method is suggested as the first method to try On the other hand, the SQP method inluded in OptdesX is potentially the fastest and most effiient algorithm of the two However, during iterations, the solution may go highly infeasible whih may ause trouble in many analysis models This ill onditioning an be expeted in highly nonlinear mathematial models whih is the ase for the airfoil problem at hand. For this reason, the GRG method was used exlusively for the airfoil optimization problem. As previously mentioned, the GRG method used in OptdesX is atually a GRG- SQP hybrid. This method was developed by the reators of OptdesX. For a more omplete desription of the method, see Referene 10. Some major points of the algorithm are addressed here. The algorithm is said to be robust and effiient as would be expeted from a hybrid of two effiient and ommonly used diret optimization methods. An iteration of the GRG algorithm has two parts: (1) a searh diretion is determined, and (2) a step size is then determined for the searh diretion. Gradients are alulated one for eah iteration and the analysis model may be alled and evaluated many times during an iteration. The first step in the algorithm (searh diretion determination) is borrowed from the ommon SQP method (see Referene 9). The step size determination is borrowed from the traditional GRG method whih is desribed in setion 2.5. The hybrid algorithm takes the best features from the respetive methods and ombines them into an effiient, robust optimization tehnique There are two methods of gradient alulation available in OptdesX: (1) the entral differene method, and (2) the forward differene method. Gradient alulations, 33

49 _ bi whihever method is used, are required and are very important for a suessful exeution of the optimization algorithms. For a generi funtion of n variables, ffx), the gradient vetor is expressed as Vf(x) 5/5/ dj dx2 dxnx _ IT (3.4) Central differene derivatives of a funtion are alulated by the formula djj i(*b *2> > xj + E>, xn)~ji[x\> x2> > xj ~8> > xn) dxj ~ 2e (3.5) and forward differene derivatives are alulated using the formula dj Ji\x\i x2> > xj + 8> > xn)~)i\x\> x2> > xj> > xn) dx} 8 (3.6) Where e is the derivative perturbation. This perturbation parameter may be set by the user in OptdesX. Choosing an appropriate gradient method and perturbation an signifiantly influene the suess of an optimization. When onsidering numerial derivatives, there are trade-offs between round-off error and trunation error. If the funtions being evaluated are fairly smooth, the round-off error is usually negligible and a forward differene method may be used with a small perturbation to minimize trunation error But for highly nonlinear funtions, round-off error may beome signifiant. To redue this, a larger perturbation may be used but this tends to inrease trunation error. To keep both round-off and trunation error to a rninimum, a entral differene method with an inreased perturbation may be used sine it has a smaller order of trunation error than the 34

50 It forward differene method However, the entral differene method requires twie the number of analysis alls for the alulation of funtion gradients Sine, for the airfoil optimization problem, was by far the most time onsuming part of iterating, the forward differene gradient alulator was hosen with a modest perturbation (e = 0.001). was also believed that sine double position arithmeti was used in the analysis model, the gradient alulation error would be minimized. It has been the author's experiene that OptdesX is an extremely effiient optimizer In fat, when running other, nonrelated analysis model that were less ompliated, optimization times were as low as five seonds. In general, OptdesX served as a robust tool whih aided in the suess ofthe urrent work. Updated Variables OptdesX Analysis Model Updated Funtions Figure 3.4: Illustration of OptdesX/Analysis Model Interation 35

51 3.3 Mathematial Model For the optimization problem, an aerodynami analysis model needed to be developed. The model was required to perform two main tasks: first, the model needed to alulate airfoil oordinates based on the Bezier verties speified as design variables, and, seond, the model needed to alulate the flow onditions and aerodynami oeffiients for the defined airfoil. The airfoil oordinates are generated by employing Eqs (3.1) and (3 2) for the four Bezier segments neessary to define an airfoil From this operation the, verties are then onverted into (x,y) oordinates for use in the aerodynami setion. To analyze the flow about an airfoil, the basi but adequate Hess-Smith-Douglas panel method is employed to determine invisid onditions whih are then used in an integral boundary layer method to approximate visous effets. The development of both of these models are desribed in detail here Invisid Flow Model: The Hess-Smith-Douglas Panel Method Panel methods have been suessfully utilized for many years for use in many different problems and are still used today despite the advanes in CFD. These methods are appropriately named sine the body surfae is approximated as a olletion of panels. There are many different panel methods where different types of singularities are used. These range from the very sophistiated with ombinations of soures and doublets oriented normal to the surfae, to the basi whih makes use of a vortex ating at the trailing edge with soures distributed along the body surfae. For the present model, the latter is used. It is a basi but effiient panel method whih helps in reduing optimization 36

52 time Realling potential flow theory, the law of superposition allows the ontribution of the individual singularities to be broken up into their respetive omponents: $ = 4>. + 4>s + <k (3.8a) Where < >ao represents the potential of the uniform flow, <J>S represents the potential of the soure distribution, and <j>v represents the potential of the vortex distribution. These potentials are expressed as < >a = Vao(x os a + y sin a) (3.8b).s = 271 Inrds (38)?v =-J 2t Qds (3.8d) where for <J>S and <Jv. the integration is performed over the body surfae Figure 3 5 (from Referene 12) shows that s is the distane along the surfae and (r,0) are polar oordinates for a point (x, y) in the flow field. The soure potential < )s has a strength per unit length of q(s). Likewise, the vortex potential has a strength per unit length ofy(s). Figure 3.5: Nomenlature for Panel Method 37

53 The By virtue of the superposition priniple, Eqs. (3 8) automatially satisfies the Laplae equation and the boundary onditions far from the body (infinity) However, for Eq (3.8) to model the flow in the viinity of the body, the boundary onditions of flow tangeny and the Kutta ondition must be satisfied. For invisid flow, it is expeted that flow be perfetly tangent to a body sine there is no boundary layer showing the flow Also, the Kutta ondition must be satisfied whih is stated as: the flow from a sharp-tailed airfoil must leave the trailing edge smoothly; that is, the veloity at the trailing edge must be finite. Some orollaries may be drawn form the Kutta ondition and are as follows: 1. trailing edge serves as a stagnation point from whih a streamline emanates. 2. The stagnation streamline from a trailing edge follows the bisetor ofthe trailing edge angle. 3. The flow veloities on the top and bottom surfaes are equal at equal distanes near the trailing edge, hene, ensuring total pressure reovery. Therefore, the problem beomes one to determine the soure and vortex strengths suh that these onditions are satisfied. The soure strength may be thought of as governed by the flow tangeny ondition and, similarly, the vortex strength by the Kutta ondition. This simplifies the problem in that a single, onstant vortex strength may be applied over the whole airfoil while soure strengths are allowed to vary to satisfy flow tangeny. A single vortex strength may be used sine the Kutta ondition only applies the trailing edge Solution of Eq. (3.8) to determine the strengths, even with the established simplifiation, an be very diffiult beause of the omplexity of the airfoil surfae. For this reason, the surfae is simplified by breaking it up into straight line segments alled 38

54 - panels, onneted by the endpoints whih are alled nodes. These lines make evaluation of the integrals in Eqs (3.8) and (3 8d) muh easier Also, the airfoil oordinates generated by the Bezier definition an be used as the node loations. The desribed surfae simplifiation is illustrated in Fig. 3 6 (from Referene 12). Distributing the soures and vorties along the panels makes Eq. (3 8) beome iv r a( \ < > = Kx, (x os a +.ysin.) + Z J I ^^ In r J -i j,_i._,l 27t panel 2t J y "j ~9 ds (3.9) Equation (3.9) an be used to losely approximate the exat solution with the auray of the solution inreasing with the number of panels used (to a ertain limit). Panel Nodes Figure 3.6: An Airfoil Desribed by Panels To further simplify the problem, the soure distribution is approximated with onstant strengths at eah panel while, from panel to panel, the strengths are allowed to vary: q(s) = qi on panel i, i = 1, 2,..., N. Again, the auray of the approximation inreases with the number of panel used to model the airfoil surfae. Now, the parameters to determine are the N soure strengths q; and the vortex strength y. These parameters must still be found suh that they satisfy the flow tangeny 39

55 Ith ondition and the Kutta ondition. This may be done by imposing the Kutta ondition and flow tangeny at N ontrol points. The midpoints of the panels are seleted as the ontrol points, sine, as may be seen later, the veloity at the panel endpoints beomes infinite. Flow tangeny will be imposed by requiring that the normal veloity at eah i* panel be zero. Also, the Kutta ondition may be satisfied by equating the veloity omponents tangential to the first and last panels, as defined by Fig i Figure 3.7: The i* Panel For this method, the panel is defined as the panel between the i* and (i+l)* nodes and its inlination with the x-axis is 9;, as shown in Fig. 3.7 (from Referene 12). So, sin 0i = (yi+, - y,)ai (3.11a) os 9. = (Xj+i - Xi)/li (3.11b) where li is the length ofthe i* panel. It follows that the normal to the i4 panel is A A A n, =-sing, i + osg, j (3.12) 40

56 Following the numbering sheme of Fig 3 6, the normal vetor should point outward from the body Similarly, a unit vetor tangent to the i* panel may be defined suh that it is direted from node i to node i+1: AAA t, = os 0, i + sin0, j (3.13) The ontrol point oordinates may be defined as >>^^- (3.14b) Where the veloity omponents at these points are defined as wr-=au-. yt\ (3.15a) vi = \^i, i. (3.15b) Now the flow tangeny ondition may be written as 0 = -m, sin0,+ v, os0, fori= 1,2,...,N (3.16) and the Kutta ondition may be expressed as u\ os 0i + vi sin 0i = -wn os 0n - vm sin 0n (3 17) The veloity omponents u, and v; are made up of ontributions from the onset flow, the soures on eah panel, and the vorties on eah panel. The veloities indued on a panel from all the soures and vorties are proportional to the strengths of the soure and vortex on that panel. It follows that w, A' N = V os a + Z?y uso +rll "Vij (3 1 8a) 41

57 ~ Ith v,,= Fw N N sin a +?. v^+^zv Vjf (3.18b) Where, for example, vvy is the y-omponent of veloity at the midpoint of the panel due to the vortex distribution from the j* panel. Figure 3.8: Loal Coordinate System Now it beomes more onvenient to work in loal oordinate (x,y ) for the evaluation of sij, vsij, Wvij, and Wij This oordinate system is defined in Fig 3.8 (from Referene 12). Subsequently, after determining (u,v), the global veloity omponents an be evaluated from u = u os 6j - v sin 0, (3.19a) v = u sin 0, - os 6] (3.19b) From this, it an be shown that MSg 1 (7 7 x -t. * *2 o _ Jn..2 2*-J(/-,)2+/ 2/r *2 1/2 2, In (x -/)'+>< " /=/, f=0 (3.20a) 42

58 -.=0 Ith % 1 a 2*1 J'' > (x -0+3' 1-1 y tan 2/r x -. <* (3.20b) Where t is a dummy variable for the distane along the panel. Also, (x*,y) are the loal oordinates orresponding to (xj,y;). Figure 3 9 (from Referene 12) may be seen for a geometrial interpretation of these results: Figure 3.9: Geometrial Interpretation of Eq. (3.20) Now the veloity omponents an be written as -1 rij+\ (3.21a) vs = Vl 2n VQ fiij 2k (3.21b) Where, as shown in Fig. 3.9, ry is the distane from the midpoint of the panel to the j* node, rij+i is the distane from the midpoint ofthe i* Q+lf1 panel to the node, and frj is the 43

59 ~ ~ - ~ -J angle defined by these lines. It an be shown, using a onvenient FORTRAN expression, that (5,j is Pij = 7. ifi=j (3 22a) = ATAN 2 (yt (*, " *, +1 X*,: - >J+l- )(x, *7 ) 0-i x} ) (x, ~ ^ +1 Xtt x +1 )(>>, >y ), >. ) if i * j (3.22b) Similarly, the loal veloity omponents due to the vortex distribution may be found as U^ = dt = -pl 2Joix*-tf+}*ldt-2x (3.23a) If/. x 1-. -t Vvv=2x3o(x*-tf+/ldt=2xln 1, rij+\ ri} (3.23b) Now the flow tangeny ondition may be written as N E4.?, +AiN+\r=bi /=i (3.24a) Where A,= -usij sin Oj+vsjj os 0t -u^ (os 6, sin 9, - sin 6, os Bt) (sin 6 sin v*sij 0t- os, 6, os 0y) (3.24b) So, 2;r Ay = sin(0, -6>.) In ir+i + os (0, -0;)#j (3.25) 44

60 .v Similarly, 2^A,A.+1 = N r os(0,-0,)ln,=1 'Vi smtfi-oppij (3 26) and b, = V* sin(0, a) (3.27) Also, the Kutta ondition from Eq (3.17) an be rewritten as 2L,AN+lj q. + AN+XN+Xy -bn+l (328) and, in similar fashion to Eqs (3.25) through (3 27), we find 2*Ajv+1</ Zsin(^ = -9. )fy os (0b -$) ln -^ ;=1 rk (3.29) 2ttAn+i,n+i = ZZsin(^ N N -0j)]n -^+ os (0to -0,)^ (3.30) j.=ij=i 'V Z>at+; = - F, os(0i - a) F* os(^j - a) (3 31) Now, the soure strengths q*, i = 1, 2,..., N, and the vortex strength y an be found by solving the N+l system of equations defined in Eqs. (3 24a) and (3.28). After this, the tangential veloity at eah ontrol point an be alulated. Manipulating Eqs. (3. 13), (3. 18), (3.19), (3.21), and (3.23) yields Vti=VMos(9i-a) sin(0, 0j )f3ij ij+i os {0t -9j)\n rij (3.32) 2k N sin(0, ) ln + os (0, 0, )/#,> and sine Vn,i = 0, the pressure oeffiient an be alulated by 45

61 V,2 P(xi, >,) = (333) Finally, the invisid aerodynami oeffiients may be alulated from N _=H ZsA rs osa + T.Pldy, LM = l y w=i sina (334a) N \ f,v A osa+ [2-V&, M=l / M = llp\dxixpdyiyi 1=1 v sina (3 34b) (3.34) Where dx, =x,+] -x, and dy, = >,+; -y{. This onludes the steps required to analyze the invisid flow ondition about an airfoil. The Hess-Smith-Douglas panel method presented is very effiient and heap to alulate. The next step is to determine a suitable model for the visous flow effets. 46

62 3 3 2 Visous Flow Model After the results are obtained form the previously desribed panel method, they an be used to help alulate a visous orretion to the invisid solution. The orretion for the present method has two forms: (1) skin frition is alulated as a orretion to the updated d, and (2) the normal veloity at eah panel is alulated and used to realulate the pressure distribution, hene, updating Cd for form drag effets. In order to update the invisid solution, an integral boundary layer alulator was employed. The visous flow model utilized separate methods to alulate both the laminar and the turbulent flow regions. Also, two different transition riteria models were applied to the boundary layer approximator. Disussion of the speifis of the boundary layer model is preeded by a short review of the relevant theory. As implied above, skin frition is an important parameter in the solution of the boundary layer The alulation of the skin frition is based on shear stresses within the boundary layer: du x =u dy (3.35) At the surfae, or wall, of a body, the shear stress is du Tw = U-r- (336) This wall shear stress, xw, may be expressed in dimensionless terms: C'-T& <337) 47

63 Cf is alled the skin frition oeffiient and, as will be seen later, it beomes important in orreting the drag oeffiient, D, for skin frition effets The boundary layer displaement thikness is an important parameter for orreting the pressure distribution for the presene of a boundary layer To do this, the normal omponent of the veloity needs to be alulated V\ = Jimv(x,>0 (3.38) To alulate the right hand side of Eq (3.38), the ontinuity equation (div V = 0) may be used to get fvdv v(xp)=)o\ip(x>y'w (339) This an be written as v(*,>) = ^r\}0 [Ve{x) - -y dip dx (340) For large ompared to 8, u(x,y*) Ve(x) (u -> Ve as y/8-oo). Therefore, (3.41) Where i'w-f (x,y) 1- v> (3.42) Ve{x) J 48

64 h+8* Equation 3.42 defines the displaement thikness whih represents the distane the external flow streamlines are displaed by the boundary layer That is, for a height h in the free stream, a height is required to allow the same volume rate of flow By evaluating Eq. (3.41) at y = 0, we get 4.--o=^/> <343> This formula may be used as a orretion to the right hand side of the system of equations desribed in Eq. (3 24a). Reevaluation of the system with the update would yield a pressure distribution whih aounted for boundary layer effets Also, before proeeding with the disussion of the boundary layer model, a few more things should be introdued. The momentum thikness is another method for quantifying the boundary layer thikness. It plays a role in some empirial formulas for drag alulations. The momentum thikness 0 is stated as 0 = 1, d 1-77 f* (3.44) '0 Ve\ Ve The momentum thikness and the displaement thikness may be related through a shape fator H: H = S/0 (3.45) All of these major outputs of a boundary layer analysis (8*, 0, Cf) are all onneted by the Karmen momentum integral equation: do Q dve 1 49

65 p02 - A derivation of Eq 3 46 may be seen in Referene 12 Unfortunately, there are too many unknowns in the equation to make it useful by itself However, other equations may be used in onjuntion with it to solve the boundary layer problem This is the fous of many integral methods To solve the laminar region of the boundary layer, Thwaites method is employed This method makes use of the momentum integral equation without having to make any assumptions about the form of the veloity profile. This is one of the virtues of the method sine veloity profile fitting is usually an ineffiient and often unreliable approah to boundary layer alulation The basi idea behind the method is to supplement Eq. (3.46) with equations involving the unknowns 0, H, and Cf. First, to make things easier, 0 and x are made dimensionless by forming,9=m (3.47a) *_,= <347b> H and Cf are already dimensionless but Cf is a strong funtion of Reynolds number To alleviate this, a parameter whih is independent ofreynolds number is introdued. l = (R,,0,)i2 (3.48) Next, the momentum integral equation is multiplied by Re,e, yielding pvjdm )L-S- + -^-(2 + [i dx [i dx dve, = / (3.49) Also, a dimensionless pressure gradient parameter X is defined as: 50

66 - - p02 dv. X = -^ \i dx (3.50) The two previous equations an be ombined and written as pve de2 r, ^-JL- = 2[l-(2 + H)X] ' (351) H dx From Thwaites, the right hand side an be aurately approximated by a linear equation: 2[l-(2+H)A]* /1 (3.52) This and the definition of A, may be substituted into Eq (3 51), yielding ove dq2 6p02 ^-L-~ = (i dx [i dx dve rr (353) After multiplying by Ve5 and some rearrangement, the equation beomes K._+9.s,._*J._(^)_a4^ (3,54) This equation, with any given Ve(x) and an initial value of 0, an be solved for 0(x) All that is required is the integration of a first order ordinary differential equation (ODE). One 0 is known, X an be alulated from Eq. (3.50), and using the empirial Thwaites orrelation formulas suggested in Referene (12), ](X) and H(x) an be alulated: /(?.) = X-1.8A2 for0<>.<0.1 (3.55a) X = X + X for -0.1 <X<0 (3.55b) /_"(/.) = A2 for 0 < X < 0 1 (3.56a) = for < X < 0 (3.56b) The initial ondition for the solution ofthe ODE is 0 at the stagnation point: 51

67 6(0)= M2 (3,57) Whih is determined by solving Eq (3.54) at the initial onditions (Ve(0) * V0x). Then Eq. (3.54) may be solved for 0(x) analytially For the present method, two different transition riteria are used. While evaluating the upper surfae, Mihel's riteria is used but while evaluating the lower surfae, the point of minimum pressure is taken as the transition point. Transition starts at a ritial value of Reynolds number For example, for a smooth flat plate, ReCr i 2.8 x IO6 (Referene 12). Transition Reynolds number is a funtion of many different parameters. Most important are the imposed pressure gradients from the ideal solution and surfae roughness. The ritial Reynolds number is lowered for inreased surfae roughness and a positive value of dp/dx. Mihel's method predits that transition should be expeted when ( 22,400^ 046 i?e)e>1.174il + -^J*Jf (3.58) This method aounts for the effet of pressure gradients sine 0 grows more rapidly in positive pressure gradients. It does not, however, aount for surfae roughness but this method should still be good for airfoil analysis sine published data doesn't show a strong dependeny on surfae roughness (see Referene (13)). Mihel's method is used for the top surfae alulations but, due to instabilities in the method, transition is fixed to a point based on the point of minimum pressure for the lower surfae. 52

68 dq E = -f- o'=d-q- These methods are used to fix transition to a speifi point whih isn't what atually ours. As a matter of fat, transitional flow ours over a segment of the airfoil not at a speifi point. Within this region of transition, the flow osillates between laminar and turbulent Skin frition is typially high in this region and sine the present method approximates transition as ourring at a point, the inreased drag is not aounted for The transition method for the boundary layer boundary layer program is basially used to toggle flow alulations from laminar to turbulent. Despite the defiienies of the transition model, it is used sine there are no other integral methods for the predition of transition. Finally, Head's method is used to predit the turbulent flow region of the boundary layer. The method is based on the onept of an entrainment veloity. The volume flowrate within the boundary layer at x is Q(x) = ^X)Udy (3.59) Where 8(x) is the boundary layer thikness. The rate at whih Q inreases with x is known as the entrainment veloity, E: ax (3 60) The displaement thikness may be written as e (3.61) Whih leads to =JVe(8-8*) (3.62) 53

69 Whih an be written as E=~(VeQHp (3 63) Where Ht.^f (3 64) It was assumed that a dimensionless entrainment veloity E/Ve was only dependent on Hi whih, in turn, is dependent on H Curves have been fitted to many sets of experimental data, the equations of whih were reported in Referene (12), and are as follows: tyj$)hx ) = (7/! (3.65) ' e "X _71= (//-l.ir~1287 for H < 1 6 (3.66a) = (_ )~3064 forh>1.6 (3.66b) Head's method employs another empirial equation known as the Ludwieg-Tillman skin frition law: Cj = x 10"678//tf;jp68 (3.67) These three equations along with the momentum integral equation omprise the four equations required to solve for the four unknowns 0, H, Hi, and Cf. The boundary layer method also handles flow separation, albeit primitive. The disussion of separation riteria and the model's reation to separated flow is addressed in Setion

70 Both Thwaites' and Head's method produe the required data to perform the aforementioned invisid solution update. The implementation of the analysis methods desribed here is disussed in depth in hapter 4. 55

71 4. PROGRAM DEVELOPMENT The Bezier definition theory desribed in Setion 3.1 and the mathematial model for the aerodynami analysis needed to be enoded to interfae with OptdesX This hapter desribes the main aspets of the developed program. Also, the ode was validated by omparing the obtained results against published data for some well known airfoil setions. 4.1 Disussion of Program As mentioned in Chapter 3, a program needed to be developed to exeute an aerodynami analysis model. FORTRAN was the programming language used to aomplish this and a opy of the soure ode may be seen in Appendix I. As seen in Fig (end of hapter), there were many subroutines used to exeute the analysis. Following the numbering shown in Fig. 4.1, the main steps in the program may be desribed as follows: (1) Updated variables are sent from OptdesX to subroutine anafun. Anafun serves as the main program for the model even though it is just a subroutine of OptdesX OptdesX ommuniates with the analysis model through variable and funtion subroutine alls (avdsa and afdsa, respetively). Then from anafun, most of the other subroutines are alled. (2) Subroutine oord is then alled from anafun. This subroutine serves one of the main funtions of the analysis model - airfoil definition. The funtion of this subroutine is 56

72 to determine the airfoil oordinates based on the Bezier verties whih are speified as the design variables reeived from OptdesX. Based on the parameter 'npoints', 'nat' nodes and 'npanels' panels are defined where npanels = nat - 1. (3) Also, subroutine amult is alled from oord to perform repetitive multipliations involved in alulating the airfoil oordinates. (4) Next, subroutine ambient is alled from anafun and is basially used to alulate sin a and os a. The subroutine used to have the funtion of setting ambient onditions, but this has sine been diretly inorporated into the user interfae of OptdesX (5) Psslope is the next subroutine alled by anafun. the funtion of this routine is to alulate the slopes of the individual panels. Also, the atual panel lengths are alulated and stored. (6) One of the other main analysis subroutines, oeffht, is alled next. It is in oeffiit that the invisid panel method problem is set up. Coeffht sets up the system of equations as desribed by Eqs. (3.24a) and (3.28). The oeffiient matrix is stored in a 'nat x nat' matrix a(i, j), and the freestream veloity vetor of length 'nat' is stored in (i). Also, for reasons explained later, a opy of (i) is stored in the vetor tempo(i). (7) Next, for the solution of the invisid problem, [A] x =, subroutine solsys is alled. The method of solution is Gaussian elimination with partial pivoting. In order to redue spae requirements, the solution is stored and returned in the vetor (i). Unfortunately the original vetor is needed for the visous update, hene, the vetor is stored in tempo before it is sent to solsys. It should be mentioned that the present work is 57

73 based from an existing program that alulated optimum airfoils subjeted to ideal flow (see Referenes (14) and (12)), so the storage of was not required (8) After returning from solsys, subroutine veldist is alled. What started originally as a subroutine to alulate the invisid pressure and veloity distributions turned into a gateway and setup subroutine for the boundary layer alulator As an be seen in Fig 4 1 veldist is very ative in alling other subroutines. The major funtions of veldist are: it still alulates veloity and pressure distribution for both the invisid and visous solutions; (9) it alls ldm for the invisid flow oeffiients whih are alulated aording to Eqs (3 34a) to (3.34), it determines the stagnation point for the airfoil and subsequently splits up the airfoil panels to define upper and lower surfaes; it finds the transition point for the minimum pressure transition model whih is used for the lower surfae, (10) it alls intgrl twie for the visous solutions of both the upper and lower surfaes; it updates the vetor (or more appropriately, the tempo vetor) with a normal veloity orretion obtained from information alulated in intgrl, and (17) it alls solsys for the reevaluation of the [AJ x = problem with the inorporated visous effets. Some of these features are addressed in more detail later. In the desription of veldist, steps 11 through 16 were skipped. Intgrl, too, has many routines whih it alls. Subroutine thwats is used to alulate Eqs. (3.55) and (3.56) for the laminar region predited by Thwaites' method. The rest of the funtions and subroutines (12 through 16) are used to analyze the turbulent boundary layer region predited by Head's method. The subroutine runge2 is used to perform a seond-order 58

74 Runge-Kutta method on the system of two first-order ODE's defined by Eqs (3.49) and (3.65). Lastly, ldm is alled again from anafun for the alulation of the aerodynami oeffiients whih are orreted for the presene of a boundary layer Sine muh of the program was done by others, disussion of the speifi aspets are limited to that whih the author ontributed. The disussion of these features are presented in the order that they are exeuted in the program. x., y. xio, yio Figure 4.2: Sloped Line Definition First, a slight modifiation in the optimization onstraints was made to aommodate a sloped line definition for the three Bezier verties B8, B9> and Bi0. This definition method was referred to as sheme 2 in Setion 3.1. Before modifiation, onstraints were plaed on y9, and y_0 suh that they were fored to be equal to y8, thus ensuring a horizontal line definition sheme (sheme 1) for the bottom surfae To allow a free slope of the line defined by the points B8, B9, and Bio, y. was made to be a design 59

75 variable. Then y10 was onstrained to lie on the line defined by y8 and y9. Referring to Fig 4.2, the slope of the line may be found as and knowing x8 x9 the equation of a line to be m= >! > (4 i) y-yx =m{x-xx) (4.2) Where (xi, yo is a referene point. If B8 is taken as the referene point, then the onstraint on yio may be expressed as >,o = "*(*.. -*.) + >. (4.3) This onstraint allows the line to be sloped (positive or negative) while maintaining the three points on the same line. The next feature developed was the method for determination of the stagnation point. Beause ofthe onvention by whih the tangent vetor is defined (1th to (i+1)"1 node, starting from the trailing edge in a lokwise diretion), the veloity on the bottom surfae is negative. This makes determination of the stagnation point simple The program looks through the veloity data starting at the trailing edge until it finds a non-negative number. After finding the first non-negative veloity, say at the i* panel, the panels may be split into an upper and a lower surfae To alulate the veloity gradient for the first panel after the stagnation point (this is required for the boundary layer alulation), the adjaent panel on the other side of the stagnation point is inluded in that surfae and is made to be the first panel. After alulations are made, however, the data from that first panel is disarded. This is done beause the data would be redundant with the data found by analyzing the same panel but on the other surfae. One the panels are split into sides, they 60

76 x,2 are renumbered suh that the first panel is at the stagnation point, not the trailing edge This is done beause the boundary layer model alulates starting from the stagnation point Another feature developed for the program was the minimum pressure riteria to fix transition. The point of minimum pressure is found in similar fashion to the stagnation point. The pressure data is sanned starting at the trailing edge in a lokwise diretion The program looks for a value of pressure that is greater than the previous, and in satisfying this riteria, the point of minimum pressure is found. Reall that this method for fixing transition is only used for the lower surfae. To alulate the normal veloity orretion as stated in equation (3 43), the displaement thikness needs to be alulated. This is straightforward beause the momentum thikness 0 and the shape fator H are already alulated in the solution of both Thwaites and Head's method. Reall these parameters are related by the equation H = 6*19. So, to alulate the orretion for eah i* panel, 8*; needs to be multiplied with its respetive Ve,;. After doing this for every panel, the gradient for eah produt 8*iVe,i needs to be omputed. Before doing this, the data should be rearranged suh that it starts at the trailing edge and is numbered in a lokwise fashion. The gradients for eah panel are then alulated by a forward differene method whih employs a three point quadrati approximation of the funtion. If the produts are denoted as P;, i = 1, npanels, then the funtion is P, (x) = a0 + ax x, + a2 i = 1, 2, 3 Given (x,, P, ) (4 4) 61

77 ~IL Where x, is the surfae length for the panel So the derivative of Pi is dx = ax+2a2xx (4.5) and it an be shown for a quadrati equation with three known points that the oeffiients ai and &2 are - P a, =-2 P l--a2(xx+x2) (4.6) fp-p^ x, x,; * *-! / \ *1 (4.7) Substituting Eqs. (4.6) and (4.7) into Eq. (4.5) and after some manipulation it may be shown that (P2-P1Xx3-x1) (P2-PxXx3-xx) (^-/^Xx.-x,) (^-/^(x.-x,) p._ X2 X1 X3 X, fr x3 x2 (48) Then, after finding the gradients, every i* omponent an be added to the respetive omponent of the vetor tempo(i), with the sum being renamed as (i): (i) = tempo(i) + ^-(VeiS:) (4.9) dx This updated vetor is all that is needed to reevaluate the [A] x = problem beause the oeffiient matrix A will not hange with visous effets. Skin fntion is also alulated for both the upper and lower surfaes by the program and is done so that it ould be added to the form drag found from the visous update just outlined. It was established from Eq. (3 37) that 62

78 ~ - _ == J \pvl and drag an be defined as \rvds D.J \^s (410) Rearranging yields ^-Wtp^-W^ > This an be approximated as nx I.C, S M < D.J "^ nx (4.12) JBC_... _> =1 Where nx is the number of points on a surfae and Sj is the length of the j* panel. The skin frition drag oeffiients from both surfaes are added together and then the sum is added to the form drag, yielding a total drag oeffiient: CD CD, form + CD, skinfrktion 13) Lastly, subroutine intgrl has the ability to handle flow separation in a very limited apaity. For Thwaites method, laminar separation is predited to our when \{X) from Eq. (3.55) vanishes sine the parameter is proportional to the wall shear stress (wall shear stress equals zero at separation). By the orrelation formulas, \{X) is predited to vanish when X = One a laminar boundary layer separates, it usually reattahes in the form of a turbulent boundary layer. This is known as the phenomenon of laminar separation bubbles. The ode simulates this to a degree in that if the ritial X is reahed 63

79 then it skips over the transition riteria and immediately starts to alulate the boundary layer aording to Head's method. The algorithm may be seen in Figure 4.3. Thwaites' Method yes yr X < no transition point reahed? no yes i t Head's Method Figure 4.3: Laminar Separation Algorithm The treatment of turbulent separation, on the other hand, has no basis. In general, turbulent separation is predited when H = 2.4 (Referene 12). If this riteria is met, then the onditions from the last, non-separated panel are imposed on the remainder and skin frition alulations are stopped for those separated panels. This, of ourse, is a very rude method for handling separation. However, while exeuting the program it was very rarely found that turbulent separation was ahieved. 64

80 4.2 Code Verifiation Regardless of the sophistiation, or lak thereof, of any numerial model, there must be some way to test the output before any onfidene in the method's validity is warranted In fat, it would be ideal if there were multiple approahes to verify results For this reason, some onsiderable time was spent refining the ode. At one point during the programming, it was thought that the ode was at a suffiient level of auray beause the aerodynami oeffiients were mathing fairly losely with published data. However, after lose srutiny, it was found that some bogus assumptions were made with respet to flow onditions. One indiator that the solution was inorret was the fat that the alulated ideal (invisid) and real (visous) lift oeffiients were extremely lose in magnitude. Contrary to what might be expeted, as illustrated in Referene (12), the ideal and real lift oeffiients should be substantially different. This is beause the visous update to the invisid solution has a tendeny to orrupt the lift alulations. This is a inherent disadvantage to the method being used. However, the ideal lift, if the method is being exeuted orretly, should give a very good approximation to the real lift. As it turned out, the drag oeffiients were fairly lose the published data beause the skin frition drag dominated over the form drag sine the ases being studied were of fairly low thikness. This example just proves that it was neessary to metiulously examine initial results before aepting the model as valid. After resolving the mentioned problems and after refining other aspets of the ode, the program was exeuted for three different NACA series airfoils: (1) the NACA 0012 symmetri airfoil, (2) the NACA 4412 ambered airfoil, and (3) the NACA

81 ambered airfoil Both the 0012 and the 4412 were analyzed beause they were used as start points in the optimization analysis and the was hosen arbitrarily Aerodynami oeffiients were obtained for all the test ases for angles of a attak between -10 and 10 degrees, where the analyses were exeuted in 2 degree inrements Also the Reynolds number for all the runs was six million. The results of the analyses are shown on oeffiient plots in Figures 4.4 through 4.6 for the respetive airfoils (data plots from Referene 13). Ideal lift oeffiients were used for reasons desribed above, and the orresponding total real drag oeffiients were also plotted. Moment oeffiients are not shown sine they are alulated about the leading edge, not the aerodynami enter. For the most part, the 0012 data looks deent. The lift urve mathes very niely with the published data. A trend for low drag values in the negative a region is exhibited on the drag oeffiient urve. This was not a big onern sine the present work only investigates an angle of attak of four degrees. However, this is a definite area that ould use improvement. Besides the negative side of the urve, the data is in very lose proximity to the published values. The NACA 4412 analysis was not as suessful as the 0012 but the data follows the general trend of the urves. Also, unlike the 0012 analysis, the program seemed to handle the negative side of the urves better than the positive for the Again deent results were obtained in the viinity ofa = 4. The last analysis that was performed showed more tendenies toward poor results at the extremes of the 's investigated. Despite this, the results are believed to show an adequate level of auray for the problem at hand. The poor results at the high and low 66

82 angle of attaks ould be due to the lak of sophistiation of the separation model, thus, not apturing the derease in lift and inrease in drag expeted from poor pressure reovery. Nevertheless, It is believed that the model has a suffiient level of sophistiation for the projet sope. Improvement of the separation model ould be the fous of later studies. It should also be added that, By virtue of the solution method, the results shouldn't be expeted to have a high level of auray. The fous of the projet was to reate an effiient solution method for the optimization proess and this was satisfied sine the alulation of panel methods with an integral boundary layer model is very heap, omputationally speaking. If the oeffiients are being improved during the optimization proess while satisfying the onstraints, then the optimization may be thought to be suessful. Indeed, the key onsideration is that the analysis model has to be valid but not neessarily preise beause other more aurate modeling tehniques may be used afterward to better determine flow harateristis. This leads into the next hapter whih disusses the CFD modeling ofthe optimized solutions. 67

83 ' ' _= oo S 1 o o On 00 > G\ VO e 3 O o 1 VI o VO H m t CO -* > <U T3 h u. oi U.ST O i 8J "to o CN 3 (-0 CO O CO rs m 68

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85 _^ -., _l. JP <i CM > <o <M m ^ -^ <5 CJ Q P7 'fud!jjdoo 6o~/p UCIJD3$ '? liu^id'jj3o0 lu^luow < < z 2 ". o "* S 3 J..a 1 j w f V" <#f j$l/ Jn p Jr 5 ^JjT f! 3_C r 1 k Vv >l % * -f^^yi _? C^&.!_ V j j.i L_ J -i,wvn i SJ ~^ Jl^k 1 ' f < 1 ta. J. \ f J L i "*~.hf "*Wi }. j_ > j _i P ' i., _. i, i i ^^. ~v 2 L ^ i ' * _ ' 1 r t to Q. E o O *-* u '5 E o U ^g^ ^- j r h 'juatoi})oo-> f/'i uo//>as '. i

86 t~ _. m / \ / \ / <H 1 \ / \ / *> fuoiotjjaoo tuaujory <o CN o m CN < < z 2 vo 2 3 O M o E '5 a. E o U u '5 U o U <o m d > <s 'j '{UJ/JIJJ30J ll U0IJ.03S <5> "3 'i.u3p!jjooo I.U81UOH 71

87 5. CFD MODELING OF OPTIMIZED SOLUTIONS It was addressed last hapter that a more sophistiated solver ould be used to further verify the optimization results and to better determine the aerodynami oeffiients for the airfoils. The purpose of this hapter is to desribe the proess involved in modeling an airfoil using Computational Fluid Dynamis (CFD) for a speifi software pakage - Fluent. The results of the atual runs are presented with the optimization results in Chapter 6. The software used to model the airfoils was Fluent version 4 23 and was run off of a server with an Ultrix version 4.3 operating system. Modeling an airfoil in Fluent was not a trivial task. In fat, it took numerous modeling iterations to determine the appropriate onditions for the problem. Most of the modeling time was spent modifying the geometry and grid to aquire reasonable results. Unfortunately, CFD is not yet at a level of sophistiation where a user an simply enter a geometry, set the fluid onditions, and run the model. Instead, areful onsideration must be given to the eah speifi step ofthe problem. Similar to finite elements, the main steps in CFD modeling are geometry generation, grid generation, and boundary ondition setting. One these steps are performed, a model may be analyzed by the CFD solver. The proess by whih these steps were exeuted is presented in the following setions. Also, a NACA 0012 was analyzed to determine the validity of the modeling assumptions. The results of this analysis are also inluded. 72

88 5.1 Geometry and Grid Generation Before beginning the modeling proess, a geometry and a orresponding grid need to be reated. This is by far the most diffiult part of CFD modeling and should be given areful onsideration before starting. Fluent has a preproessor alled PreBFC where both the geometry and grid are reated. It is a CAD-like program that allows a geometry to be reated through the use of points, lines, urves, and so on. The grid generator has the option of reating a Cartesian, an axisymmetri, or a body fitted grid. Being most appropriate, a body fitted grid was used for the airfoil problem. It is a good idea to first sketh out a geometry before it is atually generated. Careful onsideration should be given to how the grid will be mapped to this geometry. The strategy determined for the airfoil problem may be seen in Fig The drawing shows an airfoil entered in a irle (the irle is not to sale) with a yli boundary onneting the two entities. Atually, there are two yli boundaries, CYC and CYCP, overlapping eah other. The geometry was reated as shown to aommodate an O-type grid whih an be thought of as a retangular grid that is wrapped around the airfoil suh that the two opposing sides beome overlapped. These opposing sides of the grid are the yli boundaries whih impose the flow onditions from one fae onto the other That is, flow property ontinuity is imposed aross the faes. The geometry's orresponding retangular omputational grid mapping sketh may be seen in Fig 5 2. It is important to make these two skethes at the same time beause the reation of one depends on the 73

89 other. In both figures, the dashed lines indiate that the orresponding endponits are to be mapped at the same Ith grid index. With that being said, the geometry may be reated as shown in Fig The airfoil surfae was reated by onneting data points generated from the optimization program with a spline (atually, a ubi Bezier spline similar to those used in the original geometry definition). After reation, the surfae was broken into eight subsetions with eah subsetion being assigned a different zone number. Zone numbers are used by Fluent to keep trak of data for the respetive setions of interest. As will be shown later, the segments were given different zone numbers so that wall fore data ould be obtained for eah subsetion. The wall fore data an then be used to alulate the aerodynami oeffiients. Just one zone number ould have been used for the airfoil surfae but this would have resulted in wall fores being alulated for the whole skin at one. This is undesirable beause the auray of the wall fore alulation would suffer; that is, the wall fores would be more aurate if many smaller setions were used. After varying the number of subsetions, it was determined that having eight segments yielded good results Also, the outer bound was broken into subsetions that orrespond to the individual airfoil segments. These boundary segments, in turn, served to define the flow inlets and outlets. Sine only an angle of attak offour degrees was investigated, the inlets may be defined as suburves L3 through L7, and the outlets defined as L8, LI, and L2. One aspet of the geometry that affeted the auray of the solution was the outer boundary radius. If the boundary radius is too small, then it ould have an adverse affet on the flow onditions in the viinity of the airfoil, hene, rendering a solution to be 74

90 invalid One indiator that a solution had been influened by the boundary was that, when looking at pressure ontours for the solved model, many errati isobars ould be seen emanating from the airfoil to the boundary. To prevent this from happening, it was neessary to make many initial runs, inreasing the boundary radius until it was found that the pressure ontours looked normal. It was deided that a radius of fifteen times the hord length (whih was equal to one) was suffiient to prevent the boundary from affeting the solution. The final geometry for a sample airfoil (NACA 0012) may be seen in Figs. 5 3a and 5.3b where Fig. 5.3a shows the whole geometry and Fig. 5 3b shows a lose-up view of the airfoil The next step after the geometry had been reated was to map the grid. A 220 x 70 ell grid was used for the airfoil problem. These dimensions were arrived at by, again, making multiple iterations to test the affet of different grid sizes on the solution. The mapping was performed aording to the drawing shown in Fig 5.2. Sine the flow field is so large ompared to the size of the airfoil, it is desirable that the grid be weighted so that there would be a higher grid density towards the airfoil surfae Otherwise, for the grid size hosen, there would not be enough resolution in the viinity of the airfoil surfae to provide any degree of auray Therefore, grid weighting for this problem serves two purposes: (1) it provides grid resolution near the airfoil surfae, and (2) it lessens the grid size requirements. The grid weighting parameters were adjusted through multiple trials and it was found that speifying a weighting fator of 70 on the endpoints PI and PIP of urves CYC and CYCP, respetfully, yielded the best 75

91 results. This weighting fator is higher than what would typially be used but it seemed reasonable in light of the large boundary size. The weighted grid may be seen in Figs 5.4a and 5 4b Weighting was also performed for urves Bl and Tl at the endpoint P5 This turned out to be very important for the solution of the problem If the weighting was not performed then the grid aspet ratios were too high near the stagnation point Aspet ratios are reommended to be as lose to one as possible in regions of high gradients and should not exeed five in any ase. As is the ase for the present model, high aspet ratios an not always be avoided but they should be minimized. So, sine the gradients in the viinity of the leading edge an be very high, the grid needed to be weighted to minimize the aspet ratios of the surrounding ells. A lose-up view of the grid around the leading edge may be seen in Fig Cell aspet ratios were definitely lessened to a large degree as a result of the weighting proedure. However, it would have been better if the aspet ratio ould have been redued for all the ells around the airfoil. This ould have potentially been done by inreasing the grid density but the results obtained from this setup were fairly aurate. One other problem that needed to be resolved was that for ambered airfoils with downward sloping trailing edges, there were problems with the grid validity. The problem was that the grid would map inorretly at the intersetion of the trailing edge and the yli boundaries. To orret the problem, the yli boundary needed to be angled suh that it extended from the trailing edge at a slope that biseted the angle defined by the top and bottom surfaes near the trailing edge. This is illustrated by Figs. 5.6a and 5.6b. 76

92 Subsequently, the grid mapping oordinates needed to be modified and are shown in Fig 5.7 Inidentally, having the yli boundary as shown helps the solver satisfy the Kutta ondition at the trailing edge sine it has diffiulties with grids that are oriented at awkward angles with respet to the flow streamlines. 77

93 5.2 The Fluent Solver After the grid has been ompleted it an be brought into the Fluent solver where boundary onditions are speified Flow parameters suh as density and visosity are speified as well. Also, if speial models need to be utilized, they are turned on in the solver module of Fluent Sine the analysis model for the optimization used dimensionless variables, it beame neessary to quantify a suitable flow speed. This an be done easily sine the Reynolds number was held onstant at six million throughout the optimization runs. Reynolds number is defined as,=^ M (51) Choosing ambient onditions for density (p = kg/m3) and visosity (n= 1 79 x IO'5 Ns/m2) and, hoosing a unit hord ( = 1 m) yields A,(1.79_l<r'AWX6_l<.) p (1225kg /m3)(\m) A renormalized group theory (RNG) turbulene model was used in all the problem solutions. It was reommended in the Fluent manual (Referene 15) that the RNG turbulene model be used for airfoil problems where separation may be enountered. Although the RNG model was used for all models, a trial run was made with the default k- s turbulene model and there was negligible differene. This was probably due to the fat that the problem being onsidered did not have any flow separation Default underrelaxation parameters were used after trying to solve the same problem from many different ombinations. Either the solutions tended toward divergene for inreased values or iterations progressed too slowly. Therefore it was deided to use 78

94 the default values Typially, the underrelaxation parameters an be inreased to expedite the onvergene of a solution For the solution of the first airfoil problem, a run time of about 12 hours was required to reah onvergene. However, for subsequent models, solution from previous models ould be used as analysis starting points This drastially redued the run time to about 1.5 hours. A validation ase was run using the desribed modeling and is presented in the following setion 79

95 E E E E E E E E E E Fluent Model Validation With the NACA 0012 Airfoil As was the ase with the analysis model, the CFD model was run for a well known airfoil for whih there is published data. This should be done for all types of CFD models sine the solutions an be thrown off dramatially for small hanges in modeling setup So, the NACA 0012 was hosen as a test ase and was modeled as desribed above After solution, wall fore data was aquired through postproessing and is as follows: WALL FORCES BY ZONE : UNITS = NEWTONS WALL NORMAL FORCES ZONE X-DIR. Y-DIR. SHEAR X-DIR. FORCES Y-DIR. Wl E+00 0.OOOE+00 W E+01 1 W E E E+00 W E E+02 W E+02 1 W E+02 W E+00 W E+01 WA 1.625E E E E+02.OOOE E E E E E E E E E-01 Table 5.1 Wall Fore Data The total fores in the respetive diretions may be added and then the sums may be resolved onto the L and D axis defined by the four degree angle of attak After finding the total drag and lift fores, their respetive oeffiients an be alulated (see Appendix II for sample alulations of Cd and l) The following table ompares the Fluent results with published data for the NACA 0012 (Referene (13)): 80

96 Fluent Published % Differene Ct Cp Table 5.2 Comparison of Validation Results Both values of the aerodynami oeffiients are in good agreement to the published data. It an, therefore, be onluded that the model developed is valid for alulating the flow about airfoils. 81

97 / ;> fh U. & u +i CO u y_ / Sl u 3 en <u to E o <u 0 82

98 '-_ -f & w # t Ua fn 1/3 V. (50 II C 3 o. J* an to s T3 4. L_n LT\ 4. Z: 83

99 ,.-r v. f / ;? o 0" o 'O. O O Q 0 h u d Tj _ V^1 Figure 5.3a: Sample Geometry 84

100 4> E o v. & 1 a Oti+H o 3 i 6 o o l^ * 85

101 , PP-P^P ^:pp;pp\\\ Figure 5.4a: Weighted Grid for Sample Geometry 86

102

103 " V ' > 7 ' A r 1 I I An;. -"I I 'VVfi V^'AlA P.HH'rt1.ii;N l-l..?-; N N '. v;mv< ; / 1 1! :. '/.' i \ \ \ v".\. \ ^UY.V ^ V V\/ v 7 ' \ < V.1 V A A i \- \M*va4 V\.(VVV i \ V,V\AV I > A / X V>A A<V '/V'AaAv, ^ttpii./j / hii'iulii- MMlkii}... ihili'ii'ln \, Pip. i / / / /ff ' /* 7 T3 < A/./ v / a - -ff a ''Vr v < \ v 7 v. ''/./',',V(..(..(v v :< X / ' / \ / / / e ' *" >VVVVVy\\'V7'/ v\/ ^ /.,\ "S./

104 Q~ P'' r^l O 'f.j O V "^ ^v-s;^ t+h++ph Figure 5.6a: Geometry for Cambered Airfoil 89

105 S _a u ve u. V3 =2 2! CT 3 E o <5 <*- O O. 3 U B1 O 90

106 -o u. U -O U r w. en <u "" hi rrs 3 Ml a to u O O U Oft ex a. eo r\i 91

107 6. OPTIMIZATION RESULTS As previously mentioned, OptdesX optimization software was used to run the optimization problems. The software was run from a DEC server loaded with an OSF/1 version 3.2 operating system. Eah problem ase was analyzed by stopping after every iteration to save results Consequently, atual run times are not available but every iteration was on the order of 3-4 minutes with almost all of the time spent in the analysis model. Approximately twentyfive analysis alls and 1 gradient all were made per iteration. Also, on average, optimization runs took about five or six iterations to onverge. In the following setions, the problem ases that were investigated are defined and the orresponding results are presented. Also, Fluent omparisons to seleted optimization solutions are presented. 6.1 Problems Considered As in most airfoil optimization methods, the aerodynami oeffiients are of speial interest. For this reason, they were inorporated into the analysis model and through using OptdesX, their defining funtions an be speified as either an objetive (funtion to be optimized) or a onstraint (funtion that limits feasible design spae). The problems to be onsidered are defined as follows: Problem A: Max L, unonstrained D and M Problem B: Max l, d < 0 007, unonstrained M 92

108 Problem C: Max L D < 0.007, M < Problem D: Min D, unonstrained L and M Problem E: Min D, L > 0 7, unonstrained M Problem F: Min D, L > 0.7, M< where all oeffiients are the visous, or real, versions. These were optimized and onstrained instead of the ideal oeffiients beause the real oeffiients apture the influene ofthe boundary layer, whereas the ideal oeffiients do not. Also, it has been presented that two distint airfoil definition shemes were to be investigated. The first, sheme 1, requires the three Bezier verties Bg, B9, and Bio to define a horizontal line. The seond definition approah, sheme 2, relaxes the requirement on the same three points in that they have to lie on a line without any onstraint imposed on the slope ofthe line. The last point of investigation are the different starting points to be onsidered. It has been mentioned that the NACA 0012 and the NACA 4412 airfoils would be used as different starting points to the optimization problems. To be onsistent in solution proedure, the side onstraints on all design variables and the limits of all onstraints were held fixed throughout the optimization trials. These limits for both definition shemes and starting points may be seen in Appendix III Also, the same hybrid GRG-SQP method was used to optimize all the airfoil shapes Furthermore, the same gradient perturbation of0.001 was also held onstant. This allowed for a basis of omparison between the solutions 93

109 This investigation outline totals to twenty-four separate problems for onsideration The following setion presents the results for eah ase Also, if warranted, omparisons between ases are made 94

110 6.2 Problem Results The results from the various optimization solutions are presented here. They are addressed in the order defined in the last setion. Plots of the progression of the airfoil shapes at eah iteration are presented. Some iterations may be omitted for larity Tabular data of the aerodynami oeffiients for eah trial is presented along with the plots. Also, Fluent modeling results from seleted problems are presented and ompared at the end of the setion. Problem A: Max l, unonstrained D and Cm NACA 0012 Start Point: Sheme 1: From Fig. 6.1 the potential of the method is apparent. The solution airfoil looks nothing like the original. The flat portion on the bottom surfae is harateristi of the horizontal line definition sheme. Also, many harateristis of a high lift airfoil an be seen. For example, the surfae distane on the top surfae inreased greatly while the opposite happened for the bottom surfae. This would result in a greater pressure differential between the two surfaes, hene, ausing greater lift. Another high lift harateristi that the airfoil possesses is amber. As seen in Table 6. 1 the optimization inreased the visous lift oeffiient by 260 %. But, as would be expeted for an unonstrained D problem, the drag oeffiient inreased by 101 %. 95

111 Sheme 2: Again, similar harateristis were exhibited by the problem solution shown in Fig Although, the bottom front surfae is niely rounded opposed to the flat surfae seen in Fig 6.1 Again a very high inrease in lift may be seen in Table 6 2 (306 %). A omparison of the two solutions from Figs 6. 1 and 6.2 may be seen in Fig. 6 3 Although, the two airfoils were started from the same point they turned out fairly different, but some similar tendenies may also be seen. NACA 4412 Start Point: Sheme 1: As may be seen in Fig. 6 4, some problems were enountered in that the solution had a kink on the lower surfae. This is an unaeptable airfoil beause slope ontinuity is usually required in an airfoil. The kink ourred beause the x-oordinates of Bg, B., and Bio were all very lose together The only way to fix this was to modify the limits of the onstraints x9-xl0, and x8-x9. They were set to be greater than The modified shape may be seen in Fig. 6.5 It tended towards the same shape as the previous but the kink disappeared. For this airfoil solution, the lift was inreased by 75% (see Table 6.3) Sheme 2: This solution, seen in Fig. 6.6, attained the highest lift out of all the maximum Cl problems. One an notie from Table 6.4 that the lift 96

112 inreased by 84 % So, from an aerodynami standpoint, this was the best setion ahieved, but from an optimization standpoint, the NACA 0012 sheme 2 solution was the best with a 306 % inrease in the objetive Also, A omparison plot between this and the previous solution may be seen in Fig The two solutions were almost idential on the top surfae but very different on the bottom, most likely beause of the different definition shemes. Another observation that may be made is that the solutions from the different start points were very dissimilar Problem B: Max Cl, Cd < 0.007, unonstrained m NACA 0012 Start Point: Sheme 1: Figure 6.8 shows a solution with similar harateristis to the first problem onsidered in that it has a flat bottom-front surfae. But notie that the point of maximum thikness on the top surfae has dereased. This is due to the ative D onstraint. Table 6.5 shows that the lift was inreased by 168 % whih is onsiderably less than the first problem but is still quite good. Sheme 2: Again, a derease in thikness is shown when omparing this solution (Fig. 6.9) to its unonstrained ounterpart. The Cd onstraint was again ative and, as shown in Table 6.6, the lift was inreased by 219 %. This is still a very good solution onsidering its start point. The two solutions from Figs 97

113 6 8 and 6 9 are ompared in Fig 6 10 They are very lose in shape exept for the bottom- front Regardless of this differene, the aerodynami oeffiients were almost idential. Both of these solutions were run through Fluent and were verified to be lose in aerodynami harateristis (see the end of the setion for Fluent omparison). NACA 4412 Start Point: Sheme 1: Similar to what was seen before, there is a kink in the solution of Fig Again, the limits were adjusted to be greater than 0.05 yielding the shape in Fig For this setion, the onstraint was slightly inative and the resulting lift was only inreased by 5.1 % (see Table 6.7). Still, the lift was omparable to the previous two solutions. Sheme 2: Again, this ombination produed the highest lift even though the inrease was low (20.8 %) aording to Table 6.8. The solution may be seen in Fig and the omparison between this and the last solution may be seen in Fig Problem C: Max L, D < 0.007, M < NACA 0012 Start Point: Sheme 1 : Unfortunately, for all the onstrained m problems, the moment oeffiient had little to no affet on the solution sine the onstraints were only ative, 98

114 for if at all, for the first one or two iterations. For this reason the figures for these airfoils will be presented but numerial results will not be disussed See Figure 6.15 the results of this airfoil. There atually were some signifiant differenes between this airfoil and its orresponding onstrained d problem Sheme 2: The solution to this problem, seen in Fig. 6.16, was very similar to the its onstrained Cd ounterpart NACA 4412 Start Point: Sheme 1: As would be expeted, the kink was again present in the solution shown in Fig However, no attempt to modify the limits was made beause it would almost ertainly be the same as the onstrained Cd solution. Sheme 2: Another idential solution is shown in Fig Both solutions to the NACA 4412 start point problem never had the m onstraint ative, but the NACA 0012 trials had Cm ative for the first iteration, hene, sending the solutions in slightly different diretions. Problem D: Min d, unonstrained d, and Cm NACA 0012 Start Point Sheme 1: 99

115 The solution shown in Fig 6.19 is quite different from those seen in the Max L problem solutions. The NACA 0012 start point airfoils for this set ofmn D problems had diffiulties initially moving from the 0012 airfoil In fat, the solutions often needed to be physially perturbed to get them to move. This was done by moving one vertex at a time until the solution moved It usually only took a single perturbation to move the solution As shown in Table 6.9, the drag was redued by 33 1 % whih is very good Sheme 2: A similar shape to Fig is shown in Fig The drag for this shape was redued by 28.5 % (see Table 6.10). It is worth noting that this and most ofthe other shapes for the Min Cd problems have very low form drag. This an be seen by notiing that the total drag to the skin frition drag are almost equal. A omparison of Figs and 6.20 is shown in Fig where it an be seen that the two shapes are very similar NACA 4412 Start Point: Sheme 1: It an be seen from the solution in Fig that there were still tendenies for kinks on the bottom surfae but the problem resolved itself by the time a solution was found. The final shape is thinner than the original whih ould be expeted from a minimum drag problem. The drag for this setion was redued by an impressive 45.7 % (see Table 6.11). Not only did this 100

116 airfoil have the largest redution in drag but it also had the lowest value for D. Sheme 2: The shape determined for this solution was similar to the previous and is shown in Fig A omparison of the two is shown in Fig 6 24 Also, as indiated in Table 6 12, the drag for this airfoil was redued by 27.4 % Problem E: Min D, L > 0.7, unonstrained M NACA 0012 Start Point: Sheme 1: With the addition of the Cl onstraint, the solution hanged quite a bit. The solution may be seen in Figure For this shape, the drag was redued by 19.6% (see Table 6.13). Sheme 2: The shape hanged signifiantly for this solution as well. Refer to Fig for a plot of the results. As indiated in Table 6 14, the drag was redued by 13.2 %. A omparison plot for this and the last solution is shown in Fig Again, the shapes were fairly lose. NACA 4412 Start Point: Sheme 1: One again, the solution for this ombination tended to kink as shown in Fig The overall drag redution was 26.3 % as indiated in Table This solution had the lowest drag as it did in the previous problem. 101

117 Sheme 2: The solution for this ase was fairly similar to the orresponding unonstrained Cd problem and may be seen in Figure 6 29 The drag was redued by 23.9 % (see Table 6 16). a omparison of this and the last solution is shown in Fig Problem F: Min D, L> 0.7, M< NACA 0012 Start Point; Sheme 1: Like the previous onstrained M problems, this and the other forthoming solutions did not deviate muh, if at all, from the onstrained Cd problems Again, only the plots are presented. The present solution may be seen in Fig Sheme 2, The solution for this airfoil is presented in Fig NACA 4412 Start Point: Sheme 1; Figure 6.33 shows the solution for this airfoil. Sheme 2: Refer to Fig for the solution to this airfoil Now that the results have been presented, some general observations may be made. As was the ase, it ould be expeted that sheme 2 would produe the highest lift 102

118 airfoils sine it allows for higher amber Furthermore, it seems logial that the 4412 start point would produe the highest lift sine it starts with more lift than the 0012 A similar ase ould be made for the minimum drag airfoils. The horizontal definition sheme lends itself better to low drag shapes. Sine the airfoils started at approximately the same D, it makes sense that either start point ould yield the lowest drag. It was hoped that the present method would allow the airfoils to onverge upon some global optimum, this, however, was not the ase The nonlinearity of the analysis model was too great whih made the solutions settle into loal optima. Many pairs of the ases that were started from the same point onverged upon very similar solutions whih indiates that the optima are a strong funtion ofthe starting airfoil shape. As previously mentioned, Fluent runs were made to verify the results of seleted optimized airfoils. The results are as follows (% differene alulated based on Fluent results): 1) Unonstrained Max Cl, 4412 Start Point, Sheme 2: Fluent Analysis Model % Differene Cn CL ) Max L with D < 0.007, 0012 Start Point, Sheme 2: Fluent Analysis Model % Differene Cn CL ) Max L with D < 0.007, 0012 Start Point, Sheme 1: Fluent Analysis Model % Differene Cn l.,

119 These two (2 and 3) were both run to see if they were atually as lose in aerodynami oeffiients as predited. As an be seen, they were fairly lose whih is somewhat odd beause the shapes are dramatially different in the viinity of the bottomfront segment. 4) Max L with D < and M < , 0012 Start Point, Sheme 2: Fluent Analysis Model % Differene Cn CL ) Unonstrained Min D, 0012 Start Point, Sheme 2: Fluent Analysis Model % Differene Cn CL ) Min D with L > 0.7, 4412 Start Point, Sheme 1: Fluent Analysis Model % Differene Cn L ) Min D with L > 0.7 and M < , 0012 Start Point, Sheme 2: Fluent Analysis Model % Differene Cn CL Exept for the Cd value in number (4), the optimization results were within a reasonable amount of error. One must keep in mind that the fous of the program was 104

120 more to have a valid mathematial model that would be effiient for the optimization proess Beause of the lak of sophistiation of the analysis program, the data is atually better than expeted 105

121 0.15 Unonstrained Max Cl Problem (horizontal defintion) 0.05 o Figure 6.1: Problem A: NACA 0012 Start Point, Sheme 1 106

122 r~ 0.15 Max Cl (real): NACA 0012 start pt., unonstrained (trial12) t i i r n r i iteration: qqqq.q.oq o alpha = 4 deg. J L J 1 I L x/ Figure 6.2: Pornlem A: NACA 0012 Start Point, Sheme 2 107

123 0.14 Comparison of Solutions for Unonstrained Max Cl Problem horizontal line definition - sloped line definition x/ Figure 6.3: Problem A: NACA 0012 Start Point Comparison 108

124 0.14 Unonstrained Max Cl Problem (horizontal definition) alpha = 4 deg. NACA 4412 start pt. Figure 6.4: Problem A: NACA 4412 Start Point, Sheme 1 109

125 Unonstrained Max Cl Problem (horizontal definition o alpha = 4 deg. NACA 4412 start pt. 0.5 x/ Figure 6.5: Problem A: NACA 4412 Start Point, Sheme 1 (modified limits) 110

126 .o - ' Unonstrained Max Cl Problem (sloped definition) o o o o o o o o I iteration: o 1 4- O + o + o ^.. + o o ^. + o 4 + +% ^s. N,+.o 5o alpha = 4 deg. NACA 4412 start pt x/ Figure 6.6: Problem A: NACA 4412 Start Point, Sheme 2 111

127 0.14 Comparison of Solutions for Unonstrianed Max Cl Problem i i i sloped line definition - o horizontal line definition NACA 4412 start pt. i ^ x/ Figure 6.7: Problem A: NACA 4412 Start Point Comparison 112

128 0.15 Max Cl with Cd < (horizontal definition) ~i r i i i r 0-1 Wfj/tj **V*.,oO0Oo * alpha = 4 deg J I I L J L x/ Figure 6.8: Problem B: NACA 0012 Start Point, Sheme 1 113

129 Max Cl with onstrained Cd (line definition) alpha = 4 deg " x/ Figure 6.9: Problem B: NACA 0012 Start Point, Sheme 2 114

130 0.14 Comparison of Solutions for Max Cl with Cd < o " x/ Figure 6.10: Problem B: NACA 0012 Strart Point Comparison 115

131 ' _ --.. ' ' ' '". "^ _-:' " " 0W. 121 _- Max Cl Problem with Cd < (horizontal definition) Q9<?9 9 <p 9 iteration: o 0.1. * *" V> 0- o- -.. Q- ->" ---O 1 ^ 9 Px ^O ^ Q.'-X Cl- Cl" ^\ Q."'X ^S> <* 3- \N 'z ' t / x '<* 0.06 / \s'.(. 6o / \\ <t / N^At tf \ ^^^- '. Cr "/ \<<, 1 \^ f \<* r o \> Q V -.3 0? x fo ^ r^^ %_ ^b_ os^-^~~p:~rp - - " ' Ar-^-' alpha = 4 deg. ^^^o f^.'-"'" " NACA 4412 start pt. n r\a i ~-i = ' ~ ^ [ I I I I i i x/ Figure 6.11: Problem B: NACA 4412 Start Point, Sheme 1 116

132 ' *" ^ ' 0.12 Max Cl Problem with Cd < (horizontal definition) o o o o oj-1- r". O C* -' "T,a - " alpha = 4 deg. NACA 4412 start pt Figure 6.12: Problem B: NACA 4412 Start Point, Sheme 1 (modified limits) 117

133 _ - "" --^ Mall " 0.12 Max Cl Problem with Cd < (sloped definition) >> i i i i i i i 0ooooo0n iteration: 0.1 o X''-' 2-- n/x / \ o 0.08 o/ \x.. o 3.. / X q/ o 4o ^\. o/ Xv o 0.06 r?y X y X. o v \\ o / V. o \p \0 A T \ \P ( TS o *-' - ^L X-_ *"*! ^k alpha = 4 deg. ^^^OuCCOOSOOP^^ MAP A _1 19 tart r_- ^. INnUn H'f I pi. n r\a i i i i i i i i i x/ Figure 6.13: Problem B: NACA 4412 Start Point, Sheme 2 118

134 0.12 Comparison of Solutions for Max Cl Problem with Cd < sloped line definition - horizontal line definition NACA 4412 start pt x/ Figure 6.14: Problem B: NACA 4412 Start Point Comparison 119

135 0.12 Max Cl Problem with Cd < and Cm < (horiz. def.) i i i alpha = 4 deg x/ Figure 6.15: Problem C: NACA 0012 Start Point, Sheme 1 120

136 / r Max Cl Problem with Cd < and Cm < (sloped def.) ' 1 1 0o0^uo60 o o o o o --.. o o --.^.. o " o -... o -_ - ^ o. o '<: " - - o -. ^ O ^ O alpha = 4 deg " x/ Figure 6.16: Problem C: NACA 0012 Start Point, Sheme 2 121

137 " o ;._._- o "" ^~~~~Ppt>'^ ' 0.12 Max Cl Problem with Cd < & Cm < (horiz. def.) 1 i ' i i i o o o iteration: 0.1 o p y ^X p o. // \x o / X fin 6 Q/ X ' 0 o/ X Of \n O q/ X o x j \ J 9 VO \ < > o o o o o o. o V ^K, ^%h~> ^AnP ' --"" ^:^SCnrjan^? ''''- - " i n r\a ~i ~-_r T i i i i i i r " alpha = 4 deg. NACA 4412 start pt. o^ o x/ Figure 6.17: Problem C: NACA 4412 Start Point, Sheme 1 122

138 o ***.^ppz~ O O Max Cl Problem with Cd < & Cm <-0.15 (sloped def.) o.- / yx 0 /PP O / / O// O// o'oooo'oq -> ^ "*^^ i -v. "s_ i o o. o i i i iteration: 0- o X O X v \ \ o s. N. O P// X x Xs o o 0.04 y \\ V\o 0.02 \o 0 > o 0 o o o o o o o oo^_^ 0.02 ^pp. alpha = 4 deg. NACA 4412 start pt. n r\a 1 i i i i i i i i x/ Figure 6.18: Problem C: NACA 4412 Start Point, Sheme 2 123

139 0.1 Unonstrained Min Cd Problem (horiz. def.) oo oo iteration: o QQQgQGEHDeoooe-'0-o alpha = 4 deg x/ Figure 6.19: Problem D: NACA 0012 Start Point, Sheme 1 124

140 Unonstrained Min Cd Problem (sloped def.) alpha = 4 deg x/ Figure 6.20: Problem D. NACA 0012 Start Point, Sheme 2 125

141 .o " - / _r ^^y Comparison of Solutions for Unonstrained Min Cd Problem 0.1 I j j 1 j 1 j j 0.08 ""^^ / - sloped line definition ^V \. / horizontal line definition X f X^ 0 \ V p? 0.02 \\ y^y ' x^x. x ^^-^^ A^ y -^An r\a x/ Figure 6.21: Problem D: NACA 0012 Start Point Comparison 126

142 - Unonstrained Min Cd Problem (horizontal definition) Figure 6.22: Problem D. NACA 4412 Start Point, Sheme 1 127

143 0.1 i i i Unonstrained Min Cd Problem (sloped definition) i^^ '"'"'^v>\ i i i i i iteration: #> ff *r /o 4 0d \ ppp b\\ N o.n \ 4o Jr P\ 0.04 \ Jf Os \ J ^ \ J o, \ 7 <* \ 0.02? '**, \ - < A \ X V^v. alpha = 4 deg. ^^aoe^x^f33ss^^ NACA start pt. r\ r\a i i i i i i i i i x/ Figure 6.23: Problem D: NACA 4412 Start Point, Sheme 2 128

144 ,0 0.1 Comparison of Solutions for Unonstrained Min Cd Problem i i i sloped line defintion horzontal line defintion NACA 4412 start pt. J L J L J I I L x/ Figure 6.24: Problem D: NACA 4412 Start Point Comparison 129

145 0.1 Min Cd Problem with Cl > 0.7 (horiz. def.) i 1 r 0.08 ^o> oqpqpoo_0-0x)i)q iteration: alpha = 4 deg " x/ Figure 6.25: Problem E: NACA 0012 Start Point, Sheme 1 130

146 Min Cd Problem with Cl > 0.7 (sloped def.) ^o^o-oa^ alpha = 4 deg x/ Figure 6.26: Problem E: NACA 0012 Start Point, Sheme 2 131

147 0.1 Comparison of Solutions for Min Cd Problem with Constrained Cl i i r J L J I I L J 1_ x/ Figure 6.27: Problem E: NACA 0012 Start Point Comparison 132

148 .<-) " _> - 9' ' /&T- ' -*.>... ' -- -,,, ' - 4 _ i i i Min Cd Problem with Cl > 0.7 (horizontal defintion) l ^ X^^q^o>x. OX\ /r^ ' i iteration: Qfe\ 0- PPP o. X ox 1- ^ JZ5T' PNX -.- o. n\ 2 of/ 0. n X yi o. n x 4 9> P. n Q7' X b \ \ 6 o d' o \ N \ T \ 0/ on x p.n \ d on \ O.N J \ 0.02 f ON \ 0. \ o.\,c1\ '^- V'x 0oOOo_rv-rv-_-^<:^r^^ " aloha v ^^^A-UTji7(VlJ^ft_ ix)c- \J *' ^- - aipi ^ id -. ^.^ XX vn x ^. ^..,. ^ \ \ ^ \ ^ -^'".'A ^ ^ ^ '-- dpn *r u^. NACA 4412 start pt. _ ^ " L 1 L 1, x/ Figure 6.28: Problem E: NACA 4412 Start Point, Sheme 1 133

149 - " x 1 - Min Cd Problem with Cl > 0.7 (sloped defintion) 0.1 i i i y^ ^*^>. i 1 1 i^^_^ iteration: i /Xo Oo>X -- / %xav r, / / o /o o X\ / X 7o jo o X /O o \ P \ f o \ f X * 9 \ 0.02 f A " l b \ " \ 0.02 \D. x^-x XnOoq_^ ^^f^3?^^ " '" alpha = 4 deg. NACA 4412 start pt. i i i i i i i i i n r\a x/ Figure 6.29: Problem E: NACA 4412 Start Point, Sheme 2 134

150 Comparison of Solutions for Min Cd Problem with Constrained Cl i r i r sloped line defintion - horizontal line defintion NACA 4412 start pt J L J I I l_ J L x/ Figure 6.30: Problem E: NACA 4412 Start Point Comparison 135

151 .O 1" 0.1 Min Cd Problem with Cl > 0.7 and Cm < (horiz. def.) 1,<P &0O-0GVG trot iteration: alpha = 4 deg x/ Figure 6.31: Problem F: NACA 0012 Start Point, Sheme 1 136

152 0.1 Min Cd Problem with Cl > 0.7 and Cm < (sloped def.) ^oop-ptfp-b-a^ alpha = 4 deg " x/ Figure 6.32: Problem F: NACA 0012 Start Point, Sheme 2 137

153 0.1 Min Cd Problem with Cl > 0.7 & Cm < (horiz. def.) alpha = 4 deg NACA 4412 start pt. j x/ Figure 6.33: Problem F: NACA 4412 Start Point, Sheme 1 138

154 0.1 Min Cd Problem with Cl > 0.7 & Cm < (sloped defintion) 1 r iteration: alpha = 4 deg. NACA 4412 start pt J L x/ J I I L Figure 6.34: Problem F: NACA 4412 Start Point, Sheme 2 139

155 Table 6.1: Problem A: NACA 0012 Start Point, Sheme 1 Iteration Skin Frition Real Drag Ideal Lift Real Lift Ideal Moment Real Moment Table 6.2: Problem A: NACA Start Point, Sheme 2 Iteration Skin Real Ideal Lift Real Lift Ideal Real Frition Drag Moment Moment Table 6.3: Poblenl A: NACA 4412 Start Joint, Sheme 1 Iteration Skin Real Ideal Lift Real Lift Ideal Real Frition Drag Moment Moment Table 6.4: Problem A: NACA 4412 Start Point, Sheme 2 [Iteration Skin Real Ideal Lift Real Lift Ideal Real Frition Drag Moment Moment

156 '' ' " > Table 6.5: Problem B: NACA 0012 Start Point, Sheme 1 Iteration Skin Frition Real Drag Ideal Lift Real Lift Ideal Moment Real 1 Moment j Table 6.6: Problem B: NACA 0012 Start Point, Sheme 2 Iteration Skin Frition Real Drag Ideal Lift Real Lift Ideal Moment Real Moment J Table 6.7: Problem B: NACA 4412 Start Point, Sheme 1 Iteration Skin Frition Real Drag Ideal Lift Real Lift Ideal Moment Real Moment Table 6.8: Problem B: NACA 4412 Start Point, Sheme 2 Iteration Skin Frition Real Drag Ideal Lift Real Lift Ideal Moment Real Moment i li i i inn:

157 Table 6.9: Problem D: NACA 0012 Start Point, Sheme 1 Iteration Skin Frition Real Drag Ideal Lift Real Lift Ideal Moment Real Moment Table 6.10: Problem D: NACA 0012 Start Point, Sheme 2 Iteration Skin Real Ideal Lift Real Lift Ideal Real 1 Frition Drag Moment Moment Table 6.11I: Problem D: NACA 4412 Start Point, Sheme 1 Iteration Skin Real Ideal Lift Real Lift Ideal Real Frition Drag Moment Moment