Shape Optimization of Low Speed Airfoils using MATLAB and Automatic Differentiation. Christian Wauquiez

Transcription

1 Shape Optimization of Low Speed Airfoils using MATLAB and Automatic Differentiation Christian Wauquiez Stockholm 2000 Licentiate s Thesis Royal Institute of Technology Department of Numerical Analysis and Computing Science

2 Abstract The goal of the project is to develop an innovative tool to perform shape optimization of low speed airfoils. This tool is written in Matlab, and is constructed by coupling the Matlab Optimization Toolbox with a parametrised numerical aerodynamic solver. The airfoil shape is expressed analytically as a function of some design parameters. The NACA 4 digits library is used with design parameters that control the camber and the thickness of the airfoil. The solver has to provide fast and robust computation of the lift, pitching moment and drag of an airfoil placed in a low-speed viscous flow. A one-way coupled inviscid - boundary layer model is used. The inviscid flow is computed with a linear vortex panel method, which provides the lift and moment coefficients. The boundary layer is computed using an integral formulation : the laminar part of the flow is computed with a two-equation formulation, and the turbulent part is solved with Head s model. An e9-type amplification formulation is used to locate the transition area. Finally, the drag coefficient is computed using the Squire-Young formula. In order to be used in optimization, the solver must provide derivatives of the objective function and limiting constraints with the solution for each set of parameters. These derivatives are computed by automatic differentiation, a technique for augmenting computer programs with the computation of derivatives based on the chain rule of differential calculus. The recent Matlab automatic differentiation toolbox ADMAT is used. Finally as an application, sample optimization problems are solved using the Matlab Optimization Toolbox, and the resulting optimal airfoils are analysed. ISBN TRITA-NA-0004 ISSN ISRN KTH/NA/R--00/04--SE

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4 Contents Introduction - The Aerodynamics Solver Introduction - Overview of the Model.2- Airfoil and Flow Parameters.3- Inviscid Flow Model.4- Boundary Layer Model 2- Automatic Differentiation Introduction 2.2- Method Fundamentals 2.3- Computer Implementation 2.4- ADMAT, Automatic Differentiation Toolbox for Matlab 2.5- Application of ADMAT to the Aerodynamic Solver 3- Airfoil Shape Optimization Definition of the Optimization Problems 3.2- Solving the Optimization Problems Conclusion...6 References Acknowledgements I would like to thank my supervisor, Associate Professor Jesper Oppelstrup, for his interest and support throughout this work, and Professor Arthur Rizzi for providing information and for many helpful discussions. I would also like to thank, for advice and technical support, Associate Professor Ilan Kroo from Stanford University and Desktop Aeronautics, and Doctor Arun Verma from the Cornell University Theory Center. Financial support from NUTEK through the Parallel Scientific Computing Institute, KTH, and TFR through the Center for Computational Mathematics and Mechanics, KTH, is gratefully acknowledged. 3

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6 Introduction The performance of an airfoil can be characterized by three quantities : the lift, moment and drag coefficients : Cl, Cm and Cd respectively. They represent the aerodynamic loads applied to the airfoil. The actual loads are proportional to the coefficients times the square of the flow velocity. Cl corresponds to the force acting on the airfoil in the direction orthogonal to the flow, which allows an aircraft to fly by compensating its weight. Cm corresponds to the moment of the aerodynamic force with respect to the quarter of the airfoil chord length. For the equilibrium of an aircraft, the pitching moment of the main wing has to be compensated by the moment of a negative-lift tail. Cm should therefore not be too large. Finally, Cd corresponds to the component of the force in the flow direction, which hinders aerodynamic performances and causes fuel consumption. The present work focuses on shape optimization of airfoils in low speed viscous flows, based on the analysis of Cl, Cm and Cd. The airfoils are chosen from the NACA 4 digits library [], in which the shape is expressed analytically as a function of three parameters. The library is presented in section.2 of this report. The formulation used to compute the aerodynamic coefficients is an inviscid - boundary layer model. The advantage of this kind of approach is that it provides a fast computation of the flow solution, the disadvantage being that cases with massive flow separation are impossible to handle. Famous codes based on this formulation include for 2D cases Desktop Aeronautics Panda [9], and Mark Drela s ISES [0] and Xfoil [2], and for 3D cases Brian Maskew s VSAERO [3]. The solver is presented in section. In order to perform optimization, the solver has to provide the derivatives of the defined objective function and limiting constraints with the solution for a given set of parameters. The most common method used to compute these derivatives is finite differences. In the present work, a new approach, called automatic differentiation (AD), is used. Is is a chainrule-based technique to compute the derivative of functions defined by computer programs with respect to their input variables, and has been investigated since 960. In 998, a general AD toolbox for programs written in Matlab, called ADMAT [9], has been developped at Cornell University by Arun Verma and Thomas F. Coleman. The present work uses this toolbox. Automatic differentiation is presented in section 2. Finally, airfoil shape optimization is performed using the Matlab Optimization Toolbox [20], which uses a Sequential Quadratic Programming algorithm for non linear constrained problems. The formulation of sample optimization problems and their resolution are presented in section 3. 5

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8 - The Aerodynamic Solver The solver used to provide the objective function as well as the limiting constraints of the optimization is presented. The solution is computed using an inviscid-boundary layer model, and consists in the lift, moment, and drag coefficients : Cl, Cm and Cd respectively..- Introduction - Overview of the Model The general Navier-Stokes equations are very powerful since they give a complete description of all possible flow situations. However it is very time-consuming to obtain a numerical solution using them. In our particular case, the incompressible turbulent flow past an airfoil, the viscous effects are important only in a small region near the profile. In this region, the Navier-Stokes equations can be approximated by the so-called boundary layer equations. Outside, viscous effects can be neglected, and one can use an inviscid flow model. Navier-Stokes Euler V V Boundary Layer The inviscid flow model The inviscid part of the flow can be solved in several ways. A finite difference discretization of the steady Euler equation on a grid around the airfoil can be used, as in ISES [0], or a panel method, as in Xfoil [2] and Panda [9]. One advantage of the first option is that by applying a periodic boundary condition to the outer boundary, cascade flows can be simulated. This is done in MISES [3], an extension of ISES. In the present work, a panel method is used. A variety of such methods exists, they differ in the choice of the singularity used to represent the velocity potential on the airfoil (sources, doublets or vortices), and by the choice of the Kutta condition, an extra condition that one must add to the final system of equations in order to obtain a unique solution. Details about different panel methods can be found in [2]. The present work uses a linear vortex distribution, which gives a good solution accuracy, even with only a few panels. The Kutta condition chosen is especially well suited for linearly varying singularity distributions. The inviscid flow solver provides the tangential velocity distribution on the airfoil s surface (Ue). The pressure distribution is then computed from the velocity field using the Bernoulli equation. The lift and moment coefficients, as well as the pressure drag, are calculated by integrating the pressure over the body surface. 7

9 - The Aerodynamic Solver The boundary layer model The boundary layer formulation consists of a model for the laminar part of the flow, a transition criterion, and a model for the turbulent part of the flow. The flow models rely on one or two differential equations derived from the integration of the Falker-Skan equations accross the boundary layer thickness, and on additional semi-empirical equations which close the sytem. Xfoil and ISES use the same viscous formulation : a set of two differential equations (integral momentum and kinetic energy shape parameter equations) for both laminar and turbulent flows, and different closure relations depending on the flow regime. Panda on the other hand uses simpler formulations : Thwaites one equation method [4] for the laminar part, and Head s two equations method [5] for the turbulent part. The present work initially used simple models, thus Thwaites and Head s. But since Thwaites model cannot represent separated flow, Xfoil s laminar boundary layer model, which can describe thin separated regions, has been implemented instead. Concerning the transition criterion, Panda uses Michel s criterion, and Xfoil uses a more advanced e9-type formulation. The second method has been chosen after testing. Details about the reasons of these choices are found in sections.4. and.4.2. The results provided by the boundary layer solver include the displacement thickness δ, the momentum thickness θ, the shape factor H, and the skin friction coefficient Cf. These quantities are used to compute the drag coefficient. Different methods can be used depending on the coupling between the inviscid flow and the boundary layer. The coupling The coupling between the two models is as follows : The effect of the boundary layer is that it modifies the shape of the airfoil as seen by the external flow. This gives the inviscid flow a zero normal velocity condition on a boundary obtained by adding the boundary layer displacement thickness δ to the airfoil s surface. The boundary layer equations depend on the external tangential velocity distribution. Inviscid Flow with zero normal velocity condition on Airfoil +δ Ue δ Boundary Layer Two different approaches are possible for the coupling of inviscid - boundary layer flows. Two-way coupled computation In this case, the modification of the boundary where the zero normal velocity condition has to be met due to the boundary layer thickness is taken into account. The solution begins with the inviscid flow problem, which produces the velocity field. This data is then fed into the boundary layer model which results the local wall friction coefficient and the displacement thickness. Then a second iteration is performed, now with modified 8

10 .2- Airfoil Shape Parameters surface geometry. This modification can be obtained by displacing the body panels according to the local displacement thickness, and the procedure is reiterated until a converged solution is obtained. Another way to account for the displacement effects is to modify the boundary condition instead of the geometry. In this case the normal flow is made non-zero to account for the effect of δ. This formulation, known as the transpiration velocity concept, states that : V n d ( U on the airfoil s surface. dx e δ ) Using a two-ways coupled model, the total drag is found by adding the friction drag, obtained by integrating the skin friction coefficient Cf, to the pressure drag, obtained by integrating the inviscid pressure distribution. A study of such iterative methods can be found in [4], where it appears that convergence is not easy to obtain. A more recent approach is the one used in ISES and Xfoil. The transpiration velocity concept is used, and the entire nonlinear equation set is solved simultaneously as a fully coupled system by a Newton-Raphson method. This method provides more robust results than the iterative approach. One-way coupled computation In this case the effect of the boundary layer thickness is neglected. One single iteration is performed : the external tangential velocity is computed by the inviscid model with the condition V.n0 on the airfoil s surface, and then fed into the boundary layer model. The drag coefficient is obtained using the Squire-Young formula [8], which in effect computes the momentum deficit. It is a function of some of the boundary layer results (momentum thickness and shape factor) at the trailing edge. Compared to coupled computations, the accuracy of the lift is very good, but not as good for the drag. However the sensitivity of the results with respect to airfoil shape or flow parameters is well reproduced. This method is used in Panda, as well as in the present work..2- Airfoil Shape Parameters The shape of the airfoil can be chosen in the famous Naca 4 digits library []. This simple library is interesting because the shape is expressed analytically as a function of three parameters, which control the maximum camber, maximum camber location, and maximum thickness of the airfoil. y Maximum Thickness Maximum Camber x Maximum Camber Location

11 - The Aerodynamic Solver A wide variety of airfoils can be obtained by varying the three parameters, as shown on the previous figure. A numbering system is used to define NACA 4-digits wing sections. The first digit indicates the maximum value of the mean-line ordinate in percent of the chord. The second integer indicates the distance from the leading edge to the location of the maximum camber in tens of the chord. The last two integers indicate the section thickness in percent of the chord. Thus, the NACA 452 has 4% camber located at 50 % of the chord from the leading edge, and is 2% thick. We note that these digits do not really have to be integers. By extension of the notation, a NACA would have 4.23% camber located at 22.% of the chord from the leading edge, and be 7.2% thick The wing section is obtained by combining the camber line and the thickness distribution as shown on the following figure y y c y th θ y th x x U x y th cosθ y U y c + y th sinθ And : x L x+ y th cosθ y L y c y th sinθ where ( x U, y U ) and ( x L, y L ) are points on the upper and lower surface respectively. The thickness distribution and the camber line are given by : y th 5τc x -- c 0.26x x c x c x c -- 4 c And : y c ---- m for p 2 2p x -- x c -- c 2 x -- p c m for ( p) 2 2p + 2 p x -- x -- c c 2 x -- p c In these expressions, c is the airfoil chord length, m is the maximum camber, p is the maximum camber location, and τ is the maximum thickness. 0

12 .3- Inviscid Flow Model.3- Inviscid Flow Model The incompressible potential flow governed by the Laplace equation is solved numerically with a panel method, which provides the tangential external velocity. The pressure is then obtained using the Bernoulli equation, and Cl and Cm are derived by integrating the pressure over the airfoil. Theoretical equation of the flow Inviscid Incompressible Irrotational α U airfoil A n For an irrotational flow, the velocity is the gradient of a quantity called the velocity potential. V ( u, v) φ Substituting this into the continuity equation for an inviscid incompressible flow leads to : 2 φ x 2 2 φ + or : (Laplace equation) y 2 0 φ 0 On the airfoil s surface A, the external Neumann boundary condition must be satisfied : V n φ V, n n where can be related to integral boundary layer quantities through the transpiration velocity model. In our case, the effect of the boundary layer is neglected and V n is simply set to zero, which produces the classical zero normal velocity condition. General solution of the incompressible potential flow A general solution to the Laplace equation is obtained by adding a distribution of vortices γ on the airfoil s surface to the potential of the free stream. The solution at any field point P is thus given by : φ v φ P u x+ v y + γφ v ds, where is the potential of a unit strength vortex : ( r, θ) φ v θ 2π being the polar coordinate of P relative to (ds). A

13 - The Aerodynamic Solver This equation has to satisfy the boundary condition for every point on A, which gives : φ P n 0 ( u, v ) n γ φ v ds 0 n A This is the basis equation for the panel method. To construct a numerical solution the body is divided into N flat panels and the boundary condition equation is specified on each of them at a "collocation point" defined as the panel mid-point. 3 Panel corner point Collocation point 2 N- N Moreover, the integration is performed on each panel and the boundary condition becomes : Singularity element ( u, v ) n γ φ v ds 0 n i panel Now, the integral on each panel must be computed : N panel γ φ v ds n panel γ φ v s γ φ v d, ds x y panel n ( u, v) n We will first compute u and v in the panel coordinate system, and then transform them back in the global coordinate system. We now consider the coordinates (x,y) in the panel coordinate system. They are obtained using the following transformation : x y p cosα i sinα i sinα i cosα i x x 0 y y 0 G, where ( x 0, y 0 ) are the coordinates of the panel origin in the global coordinate system. y Global CS y p Panel CS Panel i α i x p x 2

14 .3- Inviscid Flow Model On each panel, we choose a linear vortex distribution γ ( x) γ 0 + γ x. y p γ 0 γ 0 + γ x x p L This is simply the superposition of a constant-strength element and a linearly varying element. For simplicity, we consider the two elements separately to compute the potential and the velocity induced by one panel. γ 0 φ γ π L 0 y atan dx x x 0 0 So : And : φ γ u 0 γ x φ γ v 0 γ y L γ π 0 γ π y ( x x 0 ) 2 + y 2 d x 0 L x x 0 ( x x 0 ) 2 + y 2 d x 0 Details of the computation of the integrals appear in [2]. The result is : γ 0 γ u γ ( θ and : 2π 2 θ ) v γ ln--- 2π We consider the linear term now : r 2 r φ γ x L γ y x 2π 0 atan dx x x So : And : u γ x v γ x φ γ x x φ γ x y L γ π 0 γ π L 0 x 0 y ( x x 0 ) 2 + y 2 d x 0 x 0 ( x x 0 ) ( x x 0 ) 2 + y 2 d x 0 3

15 - The Aerodynamic Solver Solving the integrals gives : And : u γ x v γ x γ r zln--- 2x( θ 4π r 2 θ ) 2 γ r xln--- + L+ z( θ 2π 2 θ ) r 2 Now, what we really want is a piecewise linear continuous vortex distribution on the whole airfoil surface. So we have to set the strength of γ at the beginning of each panel equal to the strength of the vortex at the end point of the previous panel as shown on the following figure. γ j γ j y p P(x,y) r j j- j θ j γ j+ r j+ j+ θ j + x p j+2 γ j+ 2 The relation between the vortex strengths of the elements shown above and the panel end values γ 0 and γ is : γ j γ 0 and γ j+ γ 0 + γ L Thus, rearranging the expressions for u and v in terms of γ j and γ j+ gives : u v z γ j + γ j r j + γ ( j x j+ x ) j + ( γ j + γ ) j ( x x ) ln + j 2π x j + x j r ( j 2π( x j+ x j ) θ j + θ j ) γ ( j x j+ x ) γ j + ( j + γ ) x x j ( ) r j j z γ j+ γ j x j+ x ln j + ( θ 2π( x j + x j ) r j + 2π x j+ x j z j+ θ j ) These two equations can be divided into velocity induced by γ j and γ j+, such that : ( uv, ) ( u a, v a ) + ( u b, v b ), where the subscripts a and b represent the contribution due to the leading and trailing singularity respectively. By rearranging the equations, we obtain the a part of the velocity : u a v a γ j 2π( x j+ x j ) z r ln j + + x r ( j+ x ) ( θ j + θ j ) j γ j 2π( x j+ x j ) x r ( j + x) ln j + r ( x j+ x ) j + z ( θ j + θ j ) j+ 4

16 .3- Inviscid Flow Model And the b part of the velocity : u b γ j+ 2π( x j+ x j ) z r ln j+ + ( x x r j )( θ j + θ j ) j v b γ j+ 2π( x j+ x j ) x x r ( ) ln j + x j r ( j+ x ) z θ j + ( j + θ j ) j+ To transform these velocity components back to the global coordinate system, a rotation by the panel orientation angle is performed as given by : α i u v G cosα i sinα i sinα i cosα i u w p () The expressions above can be included in an induced velocity function F, which will compute the velocity (u,v) at an arbitrary point (x,y) in the global coordinate system due to the j-th panel. u a, v a u b, v b F( γ j, γ j+, xzx,, j, y j, x j+, y j+ ) Discretization of geometry In most cases involving thick airfoils, a denser paneling is used near the leading and trailing edges. A frequently used method for dividing the chord into panels with larger density near the edges is the full cosine method. It is shown on the figure below. y β β β Corner point Collocation point x If nine chordwise panels are needed, then the semicircle is divided by this number, and β π 9. The corresponding x stations are found by using the following formula : x c -- ( cosβ) 2 Then the airfoil points coordinates are computed using the airfoil shape function. 5

17 - The Aerodynamic Solver Influence coefficients In this phase, the zero normal flow boundary condition is implemented. For example, the velocity induced by the jth element with a unit strength at the first collocation point is obtained by : u a, v a u b, v b j F( γ j, γ j+, x, z, x j, y j, x j+, y j+ ) This shows that the velocity at each collocation point is influenced by the two edges of the jth panel. When adding the influence of the j+ panel and on, the local induced velocity will have the form : ( uv, ) ( u a, v a ) γ + [( u b, v b ) + ( u a, v a ) 2 ]γ [( u b, v b ) N + ( u a, v a ) N ]γ N + ( u b, v b ) N γ N+ This equation can be reduced to the form : Such that for the first and last terms : And for all other terms : ( uv, ) ( uv, ) γ + ( uv, ) 2 γ ( uv, ) N γ N + ( uv, ) N ( uv, ) ( u a, v a ) and : ( u, v) N + ( u b, v b ) N ( uv, ) j [( u b, v b ) j + ( u a, v a ) j ] a ij + γ N+ The influence coefficient is then defined as the velocity component normal to the surface. The contribution of a unit strength singularity element j at collocation point is therefore : a j ( uv, ) j n, where : n i ( sinα i, cosα i ) y n i ( sinα i, cosα i ) t i ( cosα i, sinα i ) α i x 6

18 .3- Inviscid Flow Model Establishing boundary conditions The free stream normal velocity component is found as : RHS i ( u, v ) ( sinα i, cosα i ) Specifying the boundary condition equation for each (i to N) of the collocation points results in N linear equations with the unknowns (j to N+). γ j a a 2 a N + a 2 a 22 a 2N + a N a N2 a NN + γ RHS γ 2 RHS 2 γ N+ RHS N An additional condition must be established in order to obtain a unique solution. Physical considerations lead to the choice of the Kutta condition, which specifies that the circulation at the trailing edge must be zero : γ TE 0 With our model, the circulation is given by γ TE γ + γ N +, and the Kutta condition is : γ + γ N+ 0 This extra condition is added to the system of equations to give : a a 2 a N + a 2 a 22 a 2N + a N a N2 a NN + 0 γ γ 2 γ N+ RHS RHS 2 RHS N 0 (2) The above set of equations has well defined diagonal, and can be solved for standard methods of linear algebra. Modification of the Kutta condition γ i by using With the implementation of the Kutta condition above, strange results can be obtained. Indeed the condition γ + γ N + 0 makes possible results such as : γ e6 and γ N + e6 A better Kutta condition would require in addition that both vorticities must be small. We apply the following : γ 0 and γ N + 0 7

19 - The Aerodynamic Solver It is thus necessary to introduce an additional unknown. A wake panel with a constantstrength vortex γ w is added to the model as shown on the following figure. The panel leaves the trailing edge at its median angle and extends to infinity, so that in practice the far portion (starting vortex) will have no influence. 3 2 N- N W In the panel coordinate system, the velocity induced by the constant-strength vortex the wake panel at point Px ( i, y i ) is : u iw β 2π iw γ w r i v iw π ln γ w r i γ w of where β iw is the angle at which P sees the wake panel, and r i and r i are the distances from P to the wake panel end points, as shown on the following figure. Px (, y ) i i Y r i x β iw r i X Wake panel x These velocity components are computed at each collocation point, and transfered in the global coordinate system using equation (). The influence coefficient is then obtained by : a iw ( uv, ) iw n i, where : n i ( sinα i, cosα i ) This is then included in the influence matrix, as well as the new Kutta condition. We obtain : a a 2 a N + a w a 2 a 22 a 2N + a 2w a N a N2 a NN + a Nw γ RHS γ RHS 2 2 RHS γ N+ N 0 γ W 0 (3) Which can be solved for. 8 γ i

20 .3- Inviscid Flow Model Results are now presented using the regular and modified Kutta conditions. The test case is a NACA 992 at zero angle of attack, with 00 panels. This example clearly shows the improvement of the solution. Calculation of the velocity The velocity is obtained by adding the tangential components of (u,v) of each panel to the tangential component of the external flow velocity. So we have to build the N N + matrix b of coefficients b ij such that : b ij ( uv, ) ij t i, where : t i ( cos, sin ) α i α i U e And the vector of terms : ( U e ) i u, v ( ) ( cosα i, sinα i ) Then, we have : U e b b 2 b N + γ U e U e2 b 2 b 22 b 2N + γ 2 U + e 2 U en b N b N2 b NN + γ N+ U e N Which gives the tangential velocity at each airfoil collocation point. 9

21 - The Aerodynamic Solver Computation of the pressure The Bernoulli equation applied to a streamline between the upstream infinity and a point on the airfoil s surface gives : 2 2 p + --ρu 2 e p + --ρ 2 U So : Then : Gives : 2 2 p p + --ρ 2 U --ρu 2 e Cp p p ρ U U e 2 Cp U 2 We can thus compute the pressure coefficient at each airfoil collocation point. Computation of the aerodynamic coefficients We are looking for the lift coefficient Cl, and the moment coefficient Cm, which represents the moment of the aerodynamic force with respect to the point of coordinates ( c 4, 0). Dimensionless coefficients are obtained by integrating Cp. y Cp j ( x j+, y j+ ) panel j ( x j, y j ) x f xj f yj The elementary forces and acting on panel j is obtained as : f xj Cp j ( y j+ y j ) and : f yj Cp j ( x j+ x j ) m j And the elementary moment is given by : m j f y y j + + j xj x f j+ + x j c + 2 yj By doing this for each panel and by adding the elementary forces and moments, the total load applied to the airfoil is obtained : (Fx, Fy) and Cm. 20

22 .3- Inviscid Flow Model The lift coefficient is then simply the component of F normal to the flow direction : C l sinα F x + cosα F y And the pressure drag coefficient is the component of F in the flow direction : C dp F x cosα + F y sinα Validation of the inviscid flow model First, the accuracy of the solver is tested. The lift, moment and drag coefficients of a NACA 442 airfoil at 5 degrees angle of attack are computed with 50, 00, 200, 400 and 800 panels. The results is a clear convergence for all coefficients, as can be seen on the following figures. Note that the pressure drag coefficient tends to zero. This result is known as the d Alembert paradox : if the effect of the displacement thickness is not taken into account, the pressure drag of an airfoil is theoretically equal to zero. Fortunately, the method we use to compute the drag (the Squire-Young formula, see.4.4) provides directly the total drag, so we do not have to estimate the pressure and friction parts of the drag separately. The results are then compared to the solution provided by Xfoil, which uses exactly the same formulation and has been validated with respect to experimental data [2]. On the following graphs, our program is referred to as PMBL (Panel Method / Boundary Layer). Graphical results obtained with the two programs are seen in the following figure. The test case is the inviscid flow past a Naca 452 at zero angle of attack. PMBL Xfoil 2

23 - The Aerodynamic Solver The lift coefficients Cl computed by the two programs are compared in several test cases. Each case shows the sensitivity, i.e. the evolution, of the results with respect to the angle of attack and to shape parameters. The moment coefficients Cm computed by the two programs are now compared in two test cases. Cm does not vary much with respect to the angle of attack and the thickness, so we test the sensitivity with respect to the camber and camber location parameters only. 22

24 .4- Boundary Layer Model.4- Boundary Layer Model An integral model is used : the laminar part of the flow is computed with a two-equation formulation previously used in ISES [0], and the turbulent part is solved with Head s model [5]. An e9-type amplification formulation also used in ISES is used to locate the transition area. Finally, the drag coefficient is computed using the Squire-Young formula [8], and the results are compared to the solution provided by Xfoil [2]..4.- Preliminary Work 2 x 4 3 Upper surface boundary layer x Lower surface boundary layer Before using them in the boundary layer model, the external velocity distribution and the coordinates of the airfoil must be made dimensionless. The external velocity is thus divided by the far field flow velocity and the airfoil point coordinates are divided by the airfoil chord length. The boundary layer should start at the stagnation point, and follow the flow along the upper or lower surface toward the trailing edge. This requires identifying the stagnation point in the inviscid solution and using the panel discretization to compute the arc length along the surface from the stagnation point, which is the length x in the boundary layer equations. The discretization of each side can be given directly by the panels used for the external flow computation. But in order to be able to solve the boundary layer accurately without having to use many panels for the inviscid flow solution, it is more convenient to build a uniform arc length grid on each side. This grid is based on a spline interpolation of the airfoil panels corner points. The boundary layer model uses the external tangential velocity, as well as its first spatial derivative. The value of the velocity at each mesh point is obtained using a spline interpolation of the velocity field known at the panel nodes. The derivative of the velocity is obtained by differentiating the spline interpolation. 23

25 - The Aerodynamic Solver.4.2- The Laminar Boundary Layer Theoretical equation of the flow By analyzing and comparing the order of magnitude of the terms of the steady incompressible Navier-Stokes equations, we can derive the following Prandtl boundary layer equations (details of the computations and approximations can be found in [6]) : u v + 0 x y ρ u u u + v p u + µ x y x y 2 p 0 y 2 (4) (5) (6) Note : x and y are not Cartesian coordinates. x is measured along the airfoil s surface with x0 locating the stagnation point, and y is measured normally to the surface. The most important assumptions used in the derivation are : The boundary layer thickness is very small compared to L for large Reynolds numbers, The tangential velocity u is much larger than the normal component v, The pressure is essentially constant across the boundary layer (in the y direction). These results naturally lead to the concept of an integral boundary layer formulation, where the properties are assumed to depend only on x. Integral momentum equation By combining the boundary layer equations, and integrating the resulting expression from zero to infinity with respect to y, the well known Von Karman integral momentum equation is obtained. dθ dx θ 2 δ du e + θ dx U e --C 2 f See [6] for more details about the derivation. The displacement and momentum thicknesses δ and θ, and the skin friction coefficient C f are defined as follows. u δ dy θ C f 0 U e u u dy U e 0 τ w U e u , where : τ 2 ρu2 w µ e y y 0 24

26 .4- Boundary Layer Model Note : The displacement thickness can be seen as the displacement of the airfoil s surface needed to construct a constant velocity profile through the boundary layer having the same flow volume as in the real case. y Ue y Ue real BL thickness δ( x) x Airfoil s surface Airfoil s surface Displacement thickness x δ ( x) This flow volume equivalence can be analytically expressed as : 0 ( u) dy U e δ U e By introducing the shape factor H δ θ, the momentum equation can be written as : dθ dx θ du e + ( 2 + H) dx U e --C 2 f (7) Kinetic energy integral equation If the momentum equation is multiplied by u and then integrated, the kinetic energy integral equation results : dθ θ du e C, (8) dx dx D U e where the kinetic energy thickness θ, and the dissipation coefficient are defined by : θ u u dy and : C µ u D dy y 2 0 U e U e ρu3 e 0 Then, by introducing the second shape parameter H θ θ and substracting equation (7) from equation (8), the kinetic energy equation can be written as : 2 C D θ dh [ H ( H ) ]----- θ du e C dx U e dx D H C f (9) 25

27 - The Aerodynamic Solver Thwaites model This simple model relies on the integral momentum equation, and is derived as follows. Multiplying equation (7) by the Reynolds number based on the momentum thickness Re θ ReθU e, we obtain : dθ ReU e L [ ( 2+ H)λ) ], dx C f where : L ReθU e and : λ Reθ, 2 2dU e dx Then, Thwaites found that the right hand side can be approximated by the linear formula : 2l [ ( 2+ H)λ) ] λ When this and the definition of λ are substituted in the equation, we obtain : dθ ReU 2 e Reθ dx 2dU e dx Which leads to the differential equation : d Re ( θ 2 U6 dx e ) 0.45U5 e The value of θ at the stagnation point is known : θ( x 0) Re du e ( 0) dx Starting from that, the integration is performed as follows : Re( θ 2 U6 e ) i i x i 0.45 U5 e dx x i The integral is evaluated using a 5th-order Gauss quadrature, which gives a better accuracy than the st-order trapezoidal rule recommended in [6] or [7] : x i x i U 5 dx e ( x) dx U e x m -- dx U e ( x m ) 5U e x m + -- dx , where dx x i x i and ( x + x. ) 2 26

28 .4- Boundary Layer Model Once θ is known, λ can be calculated. The shape factor H and the skin friction coefficient C f are then computed from semi-empirical formulas given by Cebeci and Bradshaw [4] : H( λ) λ λ 2 for 0 < λ < for λ < λ < L( λ) And : C f , ReVeθ with : L( λ) λ.8λ 2 for 0 < λ < λ λ + for λ < λ < As all the one-equation methods, Thwaites method can not represent separated flows since it uniquely ties the shape parameter to the local pressure gradient which is, in fact, a nonunique relationship in separating flows. Therefore, for some cases, laminar separation (detected by the vanishing of C f ) is obtained before transition. After that, following what is done in Panda, laminar separation is considered as a trigger for transition and the computation is carried on with the turbulent flow model. There are thus two different effective transition criteria, which may impair the differentiability of the solution with respect to the design parameters. A slight modification of the airfoil geometry can make a change in the active criterion, and since this implies a jump in the start of the turbulent flow, a resulting jump is obtained in the drag results. As an example, we consider a 2 % thick airfoil, with 4% maximum camber. The location of the maximum camber p varies between 20 and 30% of the chord length by increment of %. As p increases, we have first laminar separation on both sides up to p22%, then transition happens on the lower side from p23%, and finally transition takes place on the upper side as well from p27%. The discontinuities in the solution which are seen on the above graph result from the move of the start of the turbulent flow : from 9.% of the chord for p22% to 5,5% for p23%, and from 26.5% of the chord for p26% to 43% for p27%. 27

29 - The Aerodynamic Solver Two-equation model This more advanced model is based on the integral momentum and kinetic energy equations. It adequately describes thin separated regions. Thus the computation of the laminar flow is always carried out until transition, and there are no longer any discontinuities in the solution. Closure relations Since the system of equations (7) and (9) contains too many unknowns, it must be supplemented by other equations. Semi-empirical relations derived from the Falkner-Skan one-parameter velocity profile family are used to close the system. The following functional dependencies are assumed : H H ( H) ( 4 H)2 for H < 4 H ( 4 H)2 for H 4 H C Re θ f f for H < ( H) ( H)2 H for H 7.4 H 6 for H < 4 ( 4 H) for ( H 4) 2 H 4 Note : The correlation between H* and H indicates that H* can never be lower than.55 which occurs for H 4. Equation (9), however, does not guarantee that dh*/dx 0 at this point. Now H 4 is obtained very seldom, but still there are cases where the solution suddenly ceases to exist because (9) indicates dh*/dx * 0 close to H 4. In this case we force transition to be able to continue the calculation downstream. Practical equations 2C D Re θ f H 2 H ( ) ( 4 H) 5.5 The equations are written in terms of H and ω Reθ 2. Multiplying equation (7) by Re θ and arranging the derivative to make dθ 2 dx appear gives : Vx ( ) dω ( 2 + H)ωWx ( ) f 2 dx ( H) And multiplying equation (9) by appear results in : Re θ H, and expanding the derivative to make dh dx 28 ωv( x)gh ( ) dh ( H)ωWx ( ) f, dx 3 ( H) where V, W du e dx, g( H) d( lnh ) dh, and f 3 ( H) f 2 ( H) f ( H). U e

30 .4- Boundary Layer Model Initial values at the stagnation point They are chosen such that dω dx and dh dx are equal to zero. This choice avoids any initial transient and allows large steps to be taken from x0. H(0) and ω( 0) are computed from : And : H f ( H) , with the root H 2 + H f ( H) f ω ( H 0 ) W0 ( )( 2 + H 0 ) Implicit scheme We apply a backward Euler with slight modification to avoid implicit iterations. In the step from x n to x n +, f and f 3 are linearized around H n and g is taken as g n. This results in a system of two bilinear equations in H H n + and ω ω n + which can be solved exactly. V n+ V n W + 2 x n + ω + Wn + Hω f' n H ω 2 x n f n + f' n H n 0 This procedure is used until transition is predicted or until the trailing edge point is reached Locating Transition Transition from laminar to turbulent flow is a result of the growth of disturbances which make the laminar boundary layer unstable. Accurate prediction of the transition region is a crucial point in obtaining a good drag estimate since it separates the laminar flow region, where the skin-friction drag is low, from the turbulent flow region, where the skin friction drag increases dramatically. Michel s criterion gh W n V n + n gv ω n + W x x n + Hω f'3n H f 3n + f' 3n H n 0 This criterion is based on the idea that transition starts at a specific Reynolds number based on the distance x from the start of the boundary layer. The value of the transition Reynolds number depends on many factors, the most important being the pressure gradient imposed on the boundary layer by the inviscid flow and the surface roughness. For incompressible flows without heat transfer, Michel [5] examined a variety of data and concluded that, for airfoil-type applications, transition should be expected when : 22.4 Re θ > Re θmax ( Rex ), Re x 0.46 where : Re θ ReU e θ and : Re x ReU e x. 29

31 - The Aerodynamic Solver This formula does account for the effect of the pressure gradient, because the momentum thickness grows more rapidly in a positive pressure gradient. However, it does not include the effect of surface roughness, but being based on data taken on airfoils, it should be good for wing analysis. Implementation of this criterion reveals that for cases where the external flow velocity is not monotone past the suction peak, the function Re θ Re θmax is not monotone either, and can therefore have several zeros. Since transition is predicted as soon as the function vanishes, this can result in a discontinuity in the solution with respect to the design parameters, as illustrated on the following example. We consider a 2% thick airfoil with 4% maximum camber. For a slight modification of the maximum camber location from 7 to 72%, the active root of the criterion changes. Transition therefore moves suddenly forward by approximately 7%, and a jump in the drag results is obtained. Transition then keeps moving forward, which explains the increase of the drag. This behavior is suspect with respect to reality. It is moreover problematic for the present work, because it makes the drag solution discontinuous and non differentiable. Another transition criterion is thus considered. 30

32 .4- Boundary Layer Model ISES e9-type method ISES uses a spatial amplification theory based on the Orr-Sommerfeld equation, also known in the literature as the e 9 method. This equation governs the growth and decay of infinitesimal wave-like disturbances in two or three dimensional shear layers. Since the unstable growth of disturbances is known to be the precursor of free transition in boundary layers, the calculation of the growth of such disturbances is a good basis for the prediction of transition. The procedure is to compute the maximum amplification ratio downstream and to assume that transition occurs when the amplitude has grown by more than a factor of e This is a wholly empirical assumption. The exponent "9" actually can vary between 7 and depending on quantities such as free stream turbulence, surface roughness, and background noise level as discussed in Cebeci and Bradshaw [4]. Using the Falker-Skan profile family, the spatial amplification curve envelopes can be related to the local boundary layer parameters. The procedure is described in []. The envelopes are then approximated by straight lines, giving : ñ dñ ( H) [ Re, (0) dre θ Re θ0 ] θ where ñ is the logarithm of the maximum amplification ratio. The slope dñ dre θ and the critical Reynolds number are expressed by the following empirical formulas : Re θ0 dñ { [ 2.4H tanh(.5H 4.65) ] dre } 2 θ And : log 0 ( Re θ0 ) H tanh + H H 0.44 For simple cases such as symmetric airfoils at zero angle of attack and flat plates, Re θ is uniquely related to the streamwise coordinate x, and equation (0) immediately gives the amplitude ratio ñ as a unique function of x. Transition is assumed to occur where ñ( x) 9. For more general cases, where the external flow velocity may not be monotone past the suction peak, it is more physically realistic to use x as the spatial amplification coordinate rather than Re θ. This is where the present transition criterion becomes better than Michel s formulation. Using some basic properties of the Falkner-Skan profile family, the conversion from x is accomplished as follows by Drela [0] : Re θ to dñ dx dñ dre θ dre θ dx dñ dre θ 2 -- x du e ρu e θ2 dx µx θ -- U e 3

33 - The Aerodynamic Solver Using the empirical relations : And : ρu e θ p( H) ( 6.54H 4.07) H µx 2 x du e mh ( ) ( H 4)2 dx H ph ( ) U e The spatial amplification rate is expressed as a function of H and θ : dñ ( H, θ) dx dñ mh ( ) ( H) ph ( ) dre θ 2 θ -- And this amplification rate is then integrated downstream from the instability point x cr, where Re θ Re θ0 : ñ( x) x x cr dñ dx dx Again transition is assumed to occur when ñ( x) 9. Using this criterion, there is no longer a discontinuity in the solution. Implementation Transition to turbulence is not an instantaneous process. Rather, over a certain length of the airfoil, the flow is intermittently laminar and turbulent. Since no method exists for describing this transition process, we have to adopt the fiction of a transition point. After using a laminar boundary layer model up to the transition point, one switches to a different method for the turbulent part. For an accurate and reliable solution, it is essential that no discontinuities are admitted as the transition point moves across a grid point. The transition is first located in an interval (i, i+), such that Re θ ( x i ) < Re θmax ( x i ) and Re θ ( x i+ ) > Re θmax ( x i+ ) for Michel s criterion, or ñ( x i ) < 9 and ñ( x i+ ) > 9 for the e9-type criterion. This interval is then treated as two subintervals as shown in the following figure. Laminar step Turbulent step i x tr i+ x x tr is the exact point where Re θ ( x) Re θmax ( x), or ñ( x) 9. It is found as the root of a linear interpolation of Re θ ( x) Re θmax ( x), or ñ( x) 9, on the interval. 32

34 .4- Boundary Layer Model.4.3- The Turbulent Boundary Layer For a large enough Reynolds number, the boundary layer can become turbulent i.e. experience an unsteadiness due to an unstable response to small disturbances. This unsteadiness is due to weak viscosity which is unsufficient to damp out disturbances that naturally appear and results in a random unstable flow. In such a case, there is little hope of following the fluid motion in detail. It is necessary and, for practical purposes, sufficient to use some form of average description for the flow. For this we define the time-average of any flow quantity by : axy (, ) lim -- T T to + T to a( x, y, t) dt and label the fluctuating part of a ( a a ) with the symbol a, so : axyt (,, ) axy (, ) + a'xyt (,, ) Theoretical equation of the flow By time-averaging and comparing the order of magnitude of the terms of the steady incompressible Navier-Stokes equations, the following turbulent boundary layer equations can be derived : u x v y ρ u u x v u p y x p y y v µ ρv' y 2 y u µ ρu'v' y () (2) (3) Note : x and y are not Cartesian coordinates :x is measured along the airfoil s surface with x0 locating the transition point, and y is measured normally to the surface. Calculation details can be found in [6]. The approximations that lead to the equations above are the same as in the laminar case, the difference here is that the neglected terms are less negligible in this case. Von Karman momentum integral equation The Von Karman integral equation can also be derived from the turbulent equations. The differences with respect to the laminar case are : The inclusion in equation (2) of a term ρu'v', called Reynolds stress. Fortunately this term vanishes at the airfoil s surface because of the no-slip condition. The term p y is non zero in equation (3). However the boundary layer is so thin that we can assume that the pressure is constant in the y direction, The integral quantities δ, θ, H and are now expressed with time-averaged velocities. C f 33

35 - The Aerodynamic Solver Head s model Head s model is described in detail in [2]. It is a typical integral method, wherein some analytical procedures have been carried out before the numerical problem is posed. It is a reasonably accurate and especially fast method. The model uses the Von Karman equation and, as in the laminar case, some semi-empirical relations to close the system. The method has been derived as follows. We consider the volume rate of flow within the boundary layer at x : δ( x) Qx ( ) udy, 0 where δ( x) is the boundary layer thickness. Combining this with the definition of the displacement thickness, we find : δ δ Q U e We introduce the entrainment velocity : E dq dx d Ue ( δ δ ) dx Which we write : E d ( Ue θh dx ) With : δ δ H θ Head assumed that the dimensionless entrainment velocity E U e depends only on H and that H, in turn, is a function of H. Cebeci and Bradshaw [4] fit several sets of experimental data to the following formulas : d ( Ue θh dx ) ( H 3) U e (4) And : H kh ( ) ( H.).287 for H ( H ) for H >.6 The fourth equation used to solve for the unknowns θ, H, H and C f is the Ludwieg-Tillman skin friction law : C f ( H ) Re θ 34

36 .4- Boundary Layer Model Explicit scheme To integrate the Head equations, a 2nd order Runge Kutta method is used. We have : dθ dx θ ( 2 + H) du e C dx 2 f U e dh H dx du e dx U e dθ ( H θ dx θ 3) dy Which we write : f( Y, x) with : Y ds θ H The initial values at the beginning of the turbulent boundary layer are given by the value at the end of the laminar run. The procedure follows : f f( Y i, x ), then Y Yi + ( x i i+ x i )f f2 f( Y, x i+ ) f f2 Yi+ Yi + ( x i+ x i ) The procedure is used until one of these two things happen : The trailing edge point is reached, Turbulent separation takes place. Turbulent separation The turbulent boundary layer equations can not always be solved because of the closure relation H kh ( ), which is not defined for all values of H and H. It is obvious on the following graph of the function. 35