Improved Unsteady Aerodynamic Influence Coefficients for Dynamic Aeroelastic Response

Transcription

1 Improved Unsteady Aerodynamic Influence Coefficients for Dynamic Aeroelastic Response Quinn Murphy Department of Mechanical Engineering McGill University Montréal, Canada December 212 A thesis submitted to McGill University in partial fulfillment of the requirements for the degree of Master of Engineering c Quinn Murphy 212

2 Abstract Flutter, or the dynamic instability of an aircraft wing due to aerodynamic loads, must be considered when designing an aircraft. For this reason work has been done to improve a method developed at Bombardier Aerospace to analyze the dynamic aeroelastic response of aircraft. This method replaces original Doublet Lattice Method (DLM) aerodynamic data with that from high-fidelity Computational Fluid Dynamics (CFD) codes. The new aerodynamic loads are transmitted to the NASTRAN aeroelastic module through improved Aerodynamic Influence Coefficients (AIC). Previously this high-fidelity data was solely steady, and weighting factors were needed to obtain unsteady data. This gave good results for flutter calculations in the subsonic and transonic regime, however, for improved results in the transonic and supersonic regime, unsteady aerodynamic data was needed. This research incorporates unsteady high-fidelity CFD data into this analysis method. The unsteady CFD data was obtained by means of the Transpiration Method. This allowed for the unsteady movement of the model to be accounted for, while saving computational time needed to deform and remesh the aerodynamic mesh at each time step. The transpiration method was validated with two standard test cases, for both static deflections and unsteady cyclic movement. Once this method was validated, high-fidelity CFD results could then be used in the AIC method. The AIC method begins with a set of baseline modes being obtained for the wing model. From these modes an aerodynamic base is calculated. Using the AIC method this aerodynamic base is transferred to NASTRAN. The natural mode shapes of a new configuration, along with the modal-based AIC method are used to approximate aerodynamic loads for the new configuration. These loads are used in NASTRAN to compute the flutter analysis of the new configuration.

3 Résumé Le flottement, ou instabilité dynamique des ailes d un avion dû à sa charge aérodynamique, doit être considéré lors de la conception d un avion. Pour cette raison, la méthode d analyse de l aéroélasticité d un avion développée par Bombardier Aéronautique, doit être améliorée. La nouvelle méthode, utilisant un code de calcul dynamique des fluides, ou Computational Fluid Dynamics (CFD), remplace la méthode originale du Doublet Lattice Method. La nouvelle charge aérodynamique est transmise au module d aéroélasticité de NASTRAN à l aide d un Coefficient d Influence Aérodynamique (AIC) amélioré. Précédemment, seules les données stationnaires étaient de haute fidélité et un facteur de correction était nécessaire pour compléter les données instationnaires. Ces données fournissaient de bons résultats pour les calculs du flottement en régimes subsonique et transsonique. Cependant, pour les cas de régimes transsonique et supersonique, des données aérodynamiques instationnaires sont nécessaires pour améliorer les résultats. Cette étude introduit donc les données instationnaires de haute fidélité provenant du calcul dynamique des fluides. Les données instationnaires générées par le calcul dynamique des fluides sont obtenues au moyen de la Méthode par Transpiration. Cette méthode permet de prendre en compte la composante instationnaire tout en minimisant le temps de calcul nécessaire pour déformer et r ler le maillage aérodynamique à chaque pas de temps. La méthode par Transpiration a été validé par deux tests standards, pour des cas de déflections statiques et de mouvements cycliques instationnaires. Une fois cette méthode validée, les résultats de haute fidélité du calcul dynamique des fluides peuvent être utilisés pour la méthode AIC. La méthode AIC est initiée avec des modes de bases obtenus pour des ailes, ce qui permet de calculer une base aérodynamique. Cette base aérodynamique est ensuite transférée NASTRAN en utilisant la méthode AIC. La forme des modes naturels de la nouvelle configuration, avec les bases modales de la méthode AIC, permet d approximer la charge aérodynamique de

4 la nouvelle configuration. Cette charge est ensuite utilisée par NASTRAN pour résoudre l analyse du flottement pour la nouvelle configuration.

5 Acknowledgements First and foremost, I would like to thank my family, who have supported me throughout this entire process and my entire career. They have been there for me and are the sole reason I was able to complete this challenge. I would also like to thank both of my supervisors, Eddy Zuppel and Prof. Mathias Legrand, for the support and guidance that they have given to me. This project would not have been possible without their help. Finally, I would like to extend my gratitude to everyone in the Controlled Facilities Engineering Services and Dynamics Groups at Bombardier Aerospace for their continuing support with this project.

6 Contents Introduction 1 1 Literature Review Current Aeroelastic Analysis Aeroelastic Analysis at Bombardier Notations for Matrix Equations Aerodynamic and Structural Data Mapping Aerodynamic Loads in NASTRAN Flutter Solution Unsteady Aerodynamic Analysis Transonic Aerodynamic Analysis Modal Based AIC Method Transpiration Method Evaluation of CFD Methods DLM Methodology Wind Tunnel Methodology High-Fidelity CFD Methodology i

7 2 Transpiration Boundary Condition Undeflected Aerodynamics Deflection of the Model Natural Mode Shapes Defining the Boundary Conditions Computational Fluid Dynamic Simulations Unsteady Considerations Processing Simulation Results Implementation of High-Fidelity CFD Data into NASTRAN Validation of the Transpiration Method BACT Wing AGARD Wing Modal Based AIC Method in NASTRAN Background Lifts and Moments Unsteady Lifts and Moments Validation of High-Fidelity CFD Results Defining the Deflected Model Results and Comparison to DLM Steady Simulations Unsteady Simulations Conclusions 69 Bibliography 72 ii

8 A Program Scripts 74 A.1 Deflecting Model A.2 Transpiration Boundary Conditions A.3 Processing Output iii

9 List of Figures 1.1 Transpiration Method Concept Transpiration Method Model Linear Regression of Wind Tunnel Data Natural Mode Shapes of AGARD Wing BACT Wing Model Dimensions Deflected Trailing Edge Flap Convergence of Solution Residuals - BACT Wing Pressure Contour Comparison of Pressure Distribution for Actual and Simulated Flap Deflection of AGARD Wing Model Dimensions AGARD Wing Convergence of Solution Residuals - AGARD Wing Pressure Contour for Deflection of AGARD Wing Comparison of Pressure Distribution for Actual and Simulated Deflection of AGARD Wing Real C l for First Natural Mode, M = iv

10 4.2 Real C m for First Natural Mode, M = Real C l for Second Natural Mode, M = Real C m for Second Natural Mode, M = Real C l for Third Natural Mode, M = Real C m for Third Natural Mode, M = Real C l for Fourth Natural Mode, M = Real C m for Fourth Natural Mode, M = Imaginary C l for First Natural Mode, M = Imaginary C m for First Natural Mode, M = Imaginary C l for Second Natural Mode, M = Imaginary C m for Second Natural Mode, M = Imaginary C l for Third Natural Mode, M = Imaginary C m for Third Natural Mode, M = Imaginary C l for Fourth Natural Mode, M = Imaginary C m for Fourth Natural Mode, M = Spanwise Comparison of high-fidelity CFD and DLM results for First Natural Mode, M = Panel Comparison of high-fidelity CFD and DLM results for First Natural Mode, M = Spanwise Comparison of high-fidelity CFD and DLM results for Second Natural Mode, M = Panel Comparison of high-fidelity CFD and DLM results for Second Natural Mode, M = Spanwise Comparison of high-fidelity CFD and DLM results for Third Natural Mode, M = v

11 5.6 Panel Comparison of high-fidelity CFD and DLM results for Third Natural Mode, M = Spanwise Comparison of high-fidelity CFD and DLM results for Fourth Natural Mode, M = Panel Comparison of high-fidelity CFD and DLM results for Fourth Natural Mode, M = Comparison of Real and Imaginary C l for First Natural Mode, M = Comparison of Real and Imaginary C m for First Natural Mode, M = Comparison of Real and Imaginary C l for Second Natural Mode, M = Comparison of Real and Imaginary C m for Second Natural Mode, M = Comparison of Real and Imaginary C l for Third Natural Mode, M = Comparison of Real and Imaginary C m for Third Natural Mode, M = Comparison of Real and Imaginary C l for Fourth Natural Mode, M = Comparison of Real and Imaginary C m for Fourth Natural Mode, M = vi

12 Introduction The Controlled Facilities Engineering Services and Dynamics Groups at Bombardier Aerospace have developed a methodology to analyze the dynamic aeroelastic response of aircraft, or more specifically, the flutter response. Flutter being the dynamic instability of an elastic body, such as an aircraft wing, when subjected to aerodynamic loads [17]. The method was initially investigated as it was determined that a more cost and labor effective tool needed to be developed In-House at Bombardier for such aeroelastic analysis. This method was developed to be used with aerodynamic data obtained from the Doublet Lattice Method (DLM), which is included in the structural solver in use by Bombardier. This method has proven to be very useful in preliminary design stages of an aircraft design program at Bombardier. The limitations introduced by the Doublet Lattice Method have resulted in less reliable aerodynamic calculations in the transonic regime. High-fidelity Computation Fluid Dynamic (CFD) solvers must be used to obtain the aerodynamic data. This need for more accurate aerodynamic analysis is even greater during the preliminary design stage of a program when other forms of data, such as wind tunnel tests, are not available. In order to overcome these deficiencies, work has been done with this new method to incorporate high-fidelity CFD data into the flutter calculations, however, only steady aerodynamic data was able to be obtained and used. The addition of steady high-fidelity 1

13 CFD data proved to give improved results for flutter calculations in the subsonic regime. For the transonic and supersonic regimes the results were not as reliable. This is where the need for unsteady high-fidelity CFD data comes. The incorporation of unsteady aerodynamic data to the methodology will provide more accurate results in the transonic and supersonic regimes. A review of current aeroelastic analysis techniques is presented, with focus on the analysis process of MSC/NASTRAN [14], followed by a review of unsteady and transonic aerodynamic analysis. The modal based Aerodynamic Influence Coefficient (AIC) method is explained briefly as well as how it is incorporated in this project. Finally, the transpiration boundary condition is reviewed and explained. Validation of the transpiration method is presented for both steady and unsteady test cases. These validations are performed with Euler calculations, since the transpiration method, as used in this project, is restricted to such calculations. The mesh used to perform the high fidelity CFD calculations needed to be deformed based on the mode shapes of the structure. This is where the transpiration method was utilized, as surface velocities were modified to simulate these deflections. The high-fidelity CFD data is compared to DLM results to ensure accuracy. Finally, the unsteady data is used in the modal based AIC method to obtain flutter solutions for subsonic and transonic flows. 2

14 Chapter 1 Literature Review 1.1 Current Aeroelastic Analysis The AIC method being developed at Bombardier is the focus of this research, however, in order to introduce the general context of this research area, other existing strategies currently being developed are also briefly reviewed. The pros and cons of each of them are underlined. For the purpose of this research only methods pertaining to fixed wing aircraft wings were examined, and hence other aeroelastic problems are not discussed. The first of the other methods employs the commercial aerodynamic and aeroelastic solver ZAERO, produced by Zona Technologies. This solver is able to perform both aerodynamic and aeroelastic analysis, however it is unable to perform structural finite element solutions and require the import of structural free vibration solutions. Further information on the details of ZAERO can be found in reference [1]. The second method under investigation incorporates the commercial code AERO Suite, produced by CMSoft. This method and AERO Suite are able to perform the full aerodynamic, structural and aeroelastic analysis, unlike ZAERO which requires the addition of the 3

15 structural analysis to be performed by an external source. It has been found to be a very powerful tool in aeroelastic analysis, however, it is also very labor intensive. Again, more details on this method may be found in reference [6]. Although both methods mentioned here are suitable for the aeroelastic analysis required by Bombardier, it is of interest to develop a comparable In-House solution that is less time and labor intensive. This interest is the motivation for the research of the AIC method presented here. A third method for aeroelastic analysis is with the use of the commercial codes produced by ANSYS. This includes their aerodynamics solver ANSYS CFX coupled with their structural solver ANSYS Mechanical. The method is similar to the AIC method, as it requires the structural and aerodynamic analysis to be completed separately and the information to be passed from one solver to the other. Reference [2] describes in greater detail this process and the ANSYS commercial codes. Further to the previously mentioned methods, the Computational Aero Servo Elasticity (CASE) Laboratory in the Department of Mechanical and Aerospace Engineering at Oklahoma State University, in conjunction with the NASA Dryden Flight Research Centre, have developed an aeroelastic analysis tool, Structural Analysis Routines (STARS). This is an multidisciplinary, finite element based solver, which is capable of structural, heat transfer, aerodynamic and aeroelastic analysis. This tool is very powerful but also very labor and time intensive. For information about STARS see reference [12]. As previously mentioned, all of the other methods investigated meet the requirements of Bombardier for aeroelastic analysis, however, it was determined that a more cost and labor effective tool needed to be developed In-House. For this reason the AIC method was further investigated, as it incorporates current analysis techniques used at Bombardier. This will be expanded upon in the following sections. 4

16 1.2 Aeroelastic Analysis at Bombardier The current analysis of aeroelastic problems at Bombardier Aerospace is done with the use of NASTRAN, a commercial structural Finite Element Method (FEM) solver. NASTRAN is able to perform different aeroelastic cases, Static Aeroelasticity, Dynamic Stability and Dynamic Response, by means of a combination of an aerodynamic and structural model. The Static Aeroelastic case will provide information of deflection of an aircraft configuration. This will provide information on the needed structural resistance of a configuration which must then be made aerodynamically efficient. The process to find a balance between structural and aerodynamic efficiency can prove to be very difficult. The Dynamic Stability case, more specifically Flutter, is the case of most interest for the project at hand. Flutter is when the combination of aerodynamic loads and the natural modes of the aircraft structure interact with one another, causing potentially catastrophic results. It is therefore of the utmost importance to be able to predict these interactions, and hence design to stay well outside of the dangerous limits. The current flutter analysis, performed in NASTRAN, utilizes the Doublet Lattice Method with dynamic weighting provided from wind tunnel data. This method has proved to be very reliable for the subsonic regime [15]. The method used by the NASTRAN solver is described below, and is based on Reference [15] and the NASTRAN Aeroelastic User Guide [14]. The models used for the aerodynamic and structural simulations are described in later sections of this thesis. 5

17 1.2.1 Notations for Matrix Equations Throughout this report, the following notation will be used when describing the matrix calculations used. [Q kk ] = [S kj ] [A jj ] 1 [ D jk 1 + ikd jk 2 ] Here the subscript kk refers to the dimensions of the matrix [Q]. The imaginary index is denoted by i. The superscripts 1 and 2 denote the real and imaginary components of the term in the equation Aerodynamic and Structural Data Mapping As the aerodynamic model and structural model of the aircraft are not necessarily the same, a mapping of the data from the aerodynamic grid points to the structural grid points, or interpolation, must be completed. This interpolation is called splining. Splining the data is what enables the transfer of displacements and velocities from the structural model to the aerodynamic model. There are several methods used for spline data, including one-dimensional splines and surface splines. Regardless of splining method chosen, an interpolation matrix will be produced. This matrix, usually denoted as [G kg ], relates the deflection of the structural grid points, {u g }, to the aerodynamic grid points, {u k }, as is written below: {u k } = [G kg ] {u g } (1.1) Aerodynamic Loads in NASTRAN In order to calculate the aerodynamic loads on the model, which are necessary for analyzing the aeroelastic phenomenon, NASTRAN utilizes the Doublet Lattice Method. This method uses lifting surface theory. The aerodynamic data calculated using the DLM is 6

18 then used to produce an Aerodynamic Influence Coefficient matrix. This matrix defines the aerodynamic properties of the model. This matrix is created from the relationship between the lifting pressure on a panel and the change in angle of attack of that panel with regard to the flow reproduced by applying a downwash to the panel. The basic equation of this applied downwash is displayed below: {w j } = [A jj ] { } fj q (1.2) where {w j } : Downwash [A jj ] : Aerodynamic Influence Coefficient Matrix; a function of Mach Number, M and Reduced Frequency, k {f j } : Pressure on lifting element q : Dynamic pressure of airflow The displacement of the structural grid points is accounted for in the downwash through the substantial differentiation matrix, [D]. The contribution to the downwash is then represented by the following equation. {w j } = [ D jk 1 + ikd jk 2 ] {u k } + {w j g } (1.3) where [ Djk 1 ] [ D jk 2 ] : Real and Imaginary parts of Substantial Differentiation Matrix {w j g } : Initial static aerodynamic downwash 7

19 From the pressure data of each panel of the aerodynamic model, the forces and moments can be obtained. The force is found by multiplying the pressure on the panel by the area of the aerodynamic panel. The total force theoretically acts through the quarter chord point of the airfoil, or aerodynamic panel. The moment is then calculated by multiplying this force by the distance between the quarter chord line and the mid-chord line of the panel. This is all done by means of the following equation: {P k } = [S kj ] {f j } (1.4) where {P k } : Lifts and Moments of each aerodynamic box [S kj ] : Integration Matrix Finally, the Aerodynamic Influence Coefficient (AIC) Matrix can be formed by combining Equations (1.2), (1.3) and (1.4) to produce [Q kk ] = [S kj ] [A jj ] 1 [ D jk 1 + ikd jk 2 ] (1.5) This Aerodynamic Influence Coefficient Matrix, [Q kk ], is calculated by using only the aerodynamic model, and hence must be applied to the structural model. Applying the aerodynamic matrix to the structural model is done by using the splining method described earlier. To reduce this problem from a continuous, and hence infinite size problem, a modal approach is used to form an eigenvalue problem. This is also done for the mass and stiffness matrices of the aeroelastic model. Once this modal approach is taken, Equation (1.5) can 8

20 be transformed into the following AIC Matrix. [Q ii ] = [φ ai ] T [G ka ] T [W T F ACT ] [Q kk ] [G ka ] [φ ai ] (1.6) where [Q ii ] : Generalized Aerodynamic Matrix [φ ai ] : i Normal mode vectors for physical a-set [G ka ] : Interpolation Matrix reduced to a degrees of freedom [W T F ACT ] : Correction Factor Matrix, based on wind tunnel data When the extra points from the aerodynamic model are not needed for the aeroelastic calculations, the generalized aerodynamic matrix, [Q hh ] is the [Q ii ] matrix of Equation (1.6). Also, the Correction Factor Matrix, which is based on wind tunnel data was used in previous research as the steady high-fidelity CFD data was considered. This lead to the introduction of a factor, [W T F ACT ] being needed to modify the unsteady portion of the [Q kk ] matrix. In this research, both steady and unsteady aerodynamic analyses were considered and hence this weighting factor is not needed Flutter Solution In order to complete the aeroelastic flutter analysis, the p k Method developed in Reference [8] is employed by the structural finite element solver MSC/NASTRAN. This is done by including the Generalized Aerodynamic Matrices, defined in the previous section, into the fundamental equation for modal flutter analysis using the p k Method, Equation (1.7) 9

21 written below: [ ( M hh p 2 + B hh 1 4 ρ cv Q I ) ( hh p + K hh 1 k 2 ρv 2 Q R ) ] hh p {u h } = (1.7) k where Q I hh : Imaginary part of Generalized Aerodynamic Matrix, a function of Mach Number, M and Reduced Frequency, k Q R hh : Real part of Generalized Aerodynamic Matrix, a function of Mach Number, M and Reduced Frequency, k p : Eigenvalue k : Reduced Frequency, k = ( ω c ) ( c ) = Im (p) 2V 2v This method solves for a flutter solution by means of an iterative process aiming to match the eigenvalues with the frequency in Equation (1.7). This is done for each mode by first selecting an airspeed. An initial value of frequency, and hence reduced frequency is then chosen. The eigenvalues, p are then calculated from Equation (1.7). These eigenvalues are then compared with the reduced frequency, k. The values must match, if this is not the case, the value of frequency, ω is modified to match the imaginary part of the selected mode s frequency. This is repeated until the process converges, and is then completed for all modes. To obtain the full flutter solution, the entire process is performed again for all necessary airspeeds. 1

22 1.3 Unsteady Aerodynamic Analysis Unsteady aerodynamic analysis is a computationally intensive process. In order to achieve a time accurate solution for a situation such as flutter, the solver must simulate the conditions of the flow accurately at each time step. This implies that for a flutter solution the deformation of the wing must be prescribed at each time step. This deformation is prescribed by the natural modes of the structure of the wing. In this study, the natural modes, and their accompanying natural frequencies were calculated using NASTRAN and are described in more detail in Chapter 2. Then, using the chosen method, the mesh is deformed for each time step. A simulation is then run to convergence at each time step with each deformed mesh. The results of this time accurate simulation are then post processed to determine real and imaginary components of the aerodynamic forces and loads. This post processing is achieved by analyzing the unsteady data using the Fast Fourier Transform. The FFT analyzes the lift generated by the unsteady simulations as a signal input and is able to produce information on the frequency and real and imaginary components of this signal. This process was done using MATLAB and the results of which can be found in Chapter Transonic Aerodynamic Analysis When studying the aerodynamics in the transonic regime, Mach numbers between.8 and 1.2, there are unique difficulties which arise. These difficulties manifest themselves due to the governing equations which are used to solve the problem, namely the linearized velocity potential equation: ( 1 M 2 ) 2 ˆφ x ˆφ y 2 = (1.8) 11

23 These methods and equations will be discussed further in Chapter 1.7 with regard to the Doublet Lattice Method. The linearized velocity potential equation, Equation (1.8), is only valid for small perturbations, i.e. thin bodies at small angles of attack, and is only valid for Subsonic and Supersonic Mach numbers. Because of this, the results for the DLM in the transonic regime do not show the same accuracy as the subsonic results. For this reason high-fidelity Computational Fluid Dynamics, such as Euler or Navier-Stokes, must be used to solve these problems. This will also be discussed in Chapter Modal Based AIC Method The Aerodynamic Influence Coefficient Matrix was described above. It was shown that the AIC Matrix used in the Doublet Lattice Method (DLM) is formed from a purely aerodynamic source. This is due to the fact that the AIC matrix is produced as a result of downwash applied to the aerodynamic panels. However, Chen et al. [16] developed a procedure to produce a modal based AIC matrix, which is created from the modal displacements of the model, and thus contains both aerodynamic and structural information. This method begins by examining the pressure difference created by a deformation of the model by the natural modes of the structure. This is expressed as C pij = N (φ ij ) (1.9) where N : Nonlinear operator φ ij : Natural modes of the structure 12

24 The nonlinear operator represents the nonlinear equations which high-fidelity CFD codes utilize in solving the aerodynamics of the problem. This can be replaced by a linear operator, L as the model displacements are assumed to be small, as in linear aerodynamics. Equation (1.9) then becomes C pij = L (φ ij ) (1.1) Now, as in the Rayleigh-Ritz method [7], the mode shapes of k number of modes can be determined by i number of baseline modes and coefficients q ki. h kj = q ki φ ij (1.11) The coefficients, q ki, are determined through the following least-square procedure: [ (φij q ki = T ) ] 1 T φ ij φij h kj (1.12) This same method and least-square procedure is applied to the linear operator in Equation (1.1): L = C pij [ (φij T φ ij ) 1 φij T ] h kj (1.13) This linear operator is now defined as the modal-based AIC matrix, [A m ], and is used to connect any set of mode shapes to a related pressure difference. This is shown below: C p = [A m ] h (1.14) Now the modal-based AIC matrix contains both aerodynamic information, from the baseline modes, as well as structural information from the deformed mode shapes. Therefore, it is possible to use this modal-based AIC matrix to determine the pressure difference for a 13

25 new structure with slightly different modes. This is different from the original AIC matrix developed by NASTRAN which only uses the downwash on the aerodynamic panels; this new pressure difference is related to the modes shapes. This method is also well suited for use with high-fidelity CFD codes. 1.6 Transpiration Method In order to perform the unsteady time dependent aerodynamic solutions, the aerodynamic model needed to deform at each time step. In the past this was accomplished by actually deforming the aerodynamic model and remeshing the entire domain. This is very time consuming, both in terms of man hours and computational hours. In order to circumvent this, a method known as the Transpiration Method was employed. This method was first introduced by Lighthill [13], which discuses the method of accounting for viscous effects in the boundary layer on the inviscid flow outside by modifying the surface conditions and implementing the Method of Equivalent Sources. The method introduces a velocity to the surface of the model which in turn simulates a deflected surface. This is described in Figure 1.1 below: The original surface normal has x, y and z components which can ˆn New V New ˆn Original V Transpiration V Original Figure 1.1: Transpiration Method Concept be modified by changing the velocity boundary condition on this surface. This boundary condition modification is achieved through the addition of fluid velocity applied to the 14

26 surface. This can be seen in Figure 1.1, where V Original is the original surface tangent fluid velocity, which has the surface normal, ˆn Original, and the deformed surface has the surface normal, ˆn New and resulting surface fluid velocity, V New, due to the application of transpiration velocity, V T ranspiration. The fluid velocity boundary condition is governed by the equation of flow tangency and the fact that there must be no flow through the surface. This is stated in the equations below for both the steady and unsteady cases. V ˆn = (1.15) V ˆn = V Body ˆn (1.16) Equation (1.15) simply states the fluid velocity normal to the surface must be zero, whereas Equation (1.16) states that the fluid velocity must remain normal to the surface of the body, and hence assume the velocity of the moving body to ensure that there is no flow through the surface of the body. It is important to note that, the velocity in Equation (1.16), V Body, is not the transpiration velocity, V T ranspiration, shown in Figure 1.1. This procedure is then implemented over the entire deformed surface. The surface deformation is calculated by means of a structural solver (for this project NASTRAN was used) and the resulting deformed normals then define the boundary conditions for each surface element. This procedure can also be applied to sudden deformations, or discontinuities, such as flaps or ailerons. This was the first validation case that was studied in this thesis when investigating the accuracy of this method. The model used is shown in Figure 1.2. The results of this can be found in Chapter 3. Using the transpiration method for sudden discontinuities also eliminates the issues of grid stretching and ill-proportioned volumes around the discontinuities. The transpiration method was studied and verified 15

27 Figure 1.2: Transpiration Method Model extensively in Reference [5]. This method has also demonstrated its effectiveness in aeroelastic analysis, as is shown in the works of Fisher and Arena [3], [4]. The methodology used to implement the Transpiration method can be found in Chapter Evaluation of CFD Methods There are several methods available to produce the aerodynamic data required for use with flutter analysis. The first method, currently used at Bombardier, is the Doublet Lattice Method, DLM. This along with Wind tunnel data, are the main sources of aerodynamic data for flutter calculations. These methods will be described in the following sections. The advancement of Computation Fluid Dynamic (CFD) solvers has offered the opportunity to utilize this method in addition to wind tunnel data which can be time consuming and difficult to obtain. These CFD methods will also be discussed. 16

28 1.7.1 DLM Methodology The Doublet Lattice Method, DLM, is based on potential flow theory and utilizes the main simplification of neglected viscosity. It is based on the linearized velocity potential equation stated below [9]: ( 1 M 2 ) 2 ˆφ x ˆφ y 2 = (1.17) Values of lift and moments can be extracted from this and used to form the aerodynamic matrices needed for the aeroelastic solver Wind Tunnel Methodology In a similar manner to that of the Doublet Lattice Method, the lifts and moments needed to construct the aerodynamic matrices can be extracted from pressures and velocity data on the surface for the model. The main difference being this data is obtained from pressure taps positioned at strategic locations on the wing of the model. There are usually between five and ten span-wise locations along the wing at which this data is measured. The data is measured at set flow conditions, but at varying angles of attack. This data is then correlated to the same locations and divisions as used in the DLM. A linear regression is performed on the data at the different angles of attack to produce the change in lift and moment, or more specifically the change in lift and moment coefficients, C lα and C mα respectively. The linear regression line is defined by the equation: y = ax + b. The slope of this line, a, determines the variation of the lift and moment coefficients with regard to the angle of attack. An example of this linear regression for the coefficient of lift is displayed in Figure

29 Figure 1.3: Linear Regression of Wind Tunnel Data [15] High-Fidelity CFD Methodology The Computational Fluid Dynamics methodology to obtain the variation of the coefficients of lift and moment is very similar to the methodology of experimental or wind tunnel testing. A finite volume model is generated to reproduce the testing environment and aircraft geometry that is tested in the wind tunnel. To increase efficiency and reduce the computational time required to obtain convergence of the CFD simulation, a half-model of the aircraft is used. Once the model has been created, steady aerodynamic analysis is performed by means of a Euler CFD code. This type of code, neglecting viscosity, does not require a model accuracy capable of capturing the boundary layer, thus producing the mesh is much easier. Again, similar to the wind tunnel method, the model is divided into portions that coincide with the DLM. This way the data obtained is easily comparable with DLM results. Additionally, the aerodynamic data obtained can be easily transferred to the aeroelastic solver that is configured to accept DLM data. As with the wind tunnel data reduction, the CFD analysis must be performed within the linear regime of aerodynamic loads; this is usually confined to angles of attack between 5 and 5. 18

30 Chapter 2 Transpiration Boundary Condition The Transpiration Method was described theoretically in the previous Chapter. The following will describe the methodology utilized to implement the Transpiration Boundary Conditions for both steady and unsteady simulations. This chapter also describes, briefly, the approach used to process the outputs of the simulations. It references the scripts and files used in the appropriate appendices. 2.1 Undeflected Aerodynamics The first step in performing the Transpiration Method is to determine a base set of aerodynamic data for the model to be studied. This entails running a simulation on the same model as the desired one to be studied, however, at an undeflected or unmodified state. This simulation must be run at the same conditions as the desired simulation. From this base simulation, the velocities on the surface are used in Equation (1.15), along with the deformed normals of the desired deflected model to produce the Transpiration Velocities. These velocities are then applied to the boundary conditions of the model to simulate a deflected state. This will be described in the following sections. 19

31 2.2 Deflection of the Model In order to determine the deflected state of the model, the natural modes of the model are calculated using a structural solver. For this thesis MSC/NASTRAN was used. A structural model was created with the same families as the aerodynamic model, that is, both models were divided into the same planar divisions. Once the natural modes of the model had been calculated, the eigenvectors were then used to define the new deflected model. This was done using a MATLAB script (found in Appendix A.1) which would output a file of the deflected points of the model that could then be read by another script. 2.3 Natural Mode Shapes As previously stated, the natural modes of the structure needed to be calculated using the structural solver in NASTRAN. These modes then needed to be verified. For the model used in this research there are several sources of validation, as the model used was the benchmark case of the AGARD standard aeroelastic configuration wing model. The mode shapes of the model can be seen in Figure 2.1 as taken from the work of Yates [1]. It can been seen in Figure 2.1 that the first mode is almost purely bending. This pure bending is defined by the increase in displacement away from the wing root progressing along to the tip. There is little torsion in this first mode, however, this can be influenced by changing the stiffness of the wing structure. For modes 2 and 3, there is also the influence of twisting. All of the extracted modes were quantified by the associated eigenvectors and natural frequencies. These eigenvectors provided the information needed to define the boundary conditions for the transpiration method, which is described in the next section. 2

32 Figure 2.1: Natural Mode Shapes of AGARD Wing [1] 21

33 2.4 Defining the Boundary Conditions In order to define the boundary conditions for the Transpiration Method another MATLAB script was written. This can be found in Appendix A.2. This script reads the undeflected points of the model, as well as the deflected points of the model, to produce the new deflected surface normals. Using these normals, along with the undeflected aerodynamic data in Equation (1.15), the transpiration velocities were defined. This could be done for any of the natural modes of the model simply by choosing the desired eigenvectors. This enabled the ability to perform several simulations at any number of deflected states with minimal effort. These transpiration velocities were then output in a manner which could be read by the computation fluid dynamic solver to define the boundary conditions of the simulation. 2.5 Computational Fluid Dynamic Simulations Once the transpiration velocities, and hence new boundary conditions, were defined, they were transferred to the CFD solver. For this research Metacomp s CFD++ was the solver that was used. The simulation was run at the same conditions as the undeflected case. The results of the steady simulation validation can be found in the following Chapter. Once the undeflected simulation is complete, and initial setup of the deflected simulation is complete, several simulations, all with different simulated deflections, can be performed by simply providing CFD++ with modified transpiration boundary conditions. No further setup is required, hence several simulations can be set up and completed with very little setup time. 22

34 2.6 Unsteady Considerations In order to perform an unsteady simulation, one that would be needed for such calculations as a flutter analysis, the boundary conditions needed to be defined at every time step. This was done in a similar manner to the steady boundary conditions, where the eigenvectors were used to deflect the model. To account for the model deflecting with time, two methods were explored. The first method involved the magnitude of the transpiration velocities which were applied to the surface of the model being scaled from zero magnitude to maximum value at a specific frequency. This frequency corresponds to the natural frequency of the natural mode of which the model was deflected. Hence, for Mode 1, or First Bending, the model was deflected in this mode and the velocities varied at the first natural frequency of the model. The second method involved scaling the eigenvectors from zero to maximum magnitude at the natural frequency of the natural mode of which the model was deflected. Then, using the scaled eigenvectors at each time step, the transpiration velocities were calculated and were applied as the boundary conditions for the corresponding time step. This second method was determined to be a more accurate representation of the actual deflection of the model. This is due to the fact that for the first method, it is assumed that the transpiration velocities vary linearly with the deflection of the model. This is not the case, and as such, calculating the transpiration velocities based on modified eigenvectors instead of scaled maximum transpiration velocities, was chosen as the method to perform the unsteady simulations. 23

35 2.7 Processing Simulation Results User defined probes at locations along the surface of the wing were created. Within CFD++ it is possible to define a probe location which records all the primitive variables in the simulation at each time step and writes them to a single file. This file was read by a MATLAB script and the data was processed using the Fast Fourier Transform, as described in section 1.3. This script can be found in Appendix A.3. Once this was complete, the results were used to replace the aerodynamic data obtained by the DLM used in the structural solver to perform aeroelastic flutter analysis. 2.8 Implementation of High-Fidelity CFD Data into NAS- TRAN As mentioned in the previous section, once the simulation results were processed they were then used to replace the aerodynamic data produced by the DLM. Replacing the DLM data was done by manipulating the aerodynamic matrices NASTRAN uses to calculate its flutter results. This method is explained in detail in Chapter 4. 24

36 Chapter 3 Validation of the Transpiration Method The validation of the Transpiration Method was done by means of two test cases. The two cases were run at steady condition in order to first validate the ability of the Transpiration Method in simulating deflections. The first was the BACT Wing with a deflected trailing edge flap, and the second was a deflected AGARD Wing. The two cases are described below. 3.1 BACT Wing The first validation test case was the Benchmark Active Controls Technology (BACT) Wing, developed by The Structural Division of NASA Langley Research Centre s Benchmark Models Program. The dimensions of the BACT wing, along with the placement of the control surfaces can be seen in Figure 3.1 and have been taken from Reference [5]. There are three control surfaces on the wing, a trailing edge flap and an upper and lower surface spoiler. The control surfaces are centred along the 6% span, and are 3% 25

37 Figure 3.1: BACT Wing Model Dimensions [5] the wing span. The trailing edge flap has a width of 25% the wing chord. The airfoil of the BACT wing is the NACA 12 symmetric airfoil. In order to compare the transpiration method, an actually deflected trailing edge flap was modelled. The two meshes are compared in Figures 3.2a and 3.2b. Both meshes contained on the order of 1, 6, tetrahedral cells, with concentration of the mesh at the trailing edge flap. The increased number of cells is to improve the precision of the simulations and to capture the flow conditions imposed by the trailing edge flap. This number is much larger than other studies done using this method, as in reference [5], however with the computational resources available the time required to run the simulations was not an issue. It can be seen in Figures 3.2a and 3.2b that the deflection of the flap is significant, at 1. This is to demonstrate the ability of the transpiration method to simulate large deflections. The simulation was executed at Mach.77 with the wing at an angle of attack of and the trailing edge flap at 1 (downward) deflection, to match with the simulations 26

38 (a) Actual (b) Simulated Figure 3.2: Deflected Trailing Edge Flap performed in reference [5]. The results of the two simulations can be seen in Figures 3.4a and 3.4b. In order to ensure the convergence of the simulations, the residuals were monitored throughout the simulations. The simulation was considered to have converged to an acceptable level once the residuals had decreased by five orders of magnitude. The solution residuals for the momentum are shown in Figure 3.3, for both the actually deflected flap simulation and the transpiration method simulation, to illustrate this point. As a note, the residuals shown are relative values, showing the five order of magnitude drop, not the absolute values of the residuals. Qualitatively the two pressure contours show very good agreement. This is further illustrated by means of a comparison of the pressure distribution at a cut located at the 6% span of the wing. This is shown in Figure 3.5 below. Figure 3.5 shows excellent agreement between the actual and simulated flap deflection, with the only discrepancy being noted at the interface of the control surface and the wing. This discrepancy was also noted in reference [5] and was thought to be caused by the sudden change in geometry at the flap interface. Reference [5] stated that the fact that a Euler solver is used for the simulation it would not be expected that any separation and boundary 27

39 1 6 Actual Deflection x momentum y momentum z momentum 1 6 Transpiration x momentum y momentum z momentum Residual Residual Iteration Iteration Figure 3.3: Convergence of Solution Residuals - BACT Wing layer-shock interactions caused by the large surface deflection, would be detected. That is to say that the transpiration method is only as accurate as the limitations of the inviscid assumptions applied to the method. One other explanation for the discrepancy at the control surface interface, is that the transpiration method, as applied here, is applied to a set of panels on the control surface, whereas in other research [5], [3], [4] it was applied to every node. This was not possible with the tools presently in use, and was not considered to be necessary as the point of the BACT test case was to determine if the transpiration method showed the potential to be 28

40 (a) Actual Flap Deflection (b) Simulated Flap Deflection Figure 3.4: Pressure Contour Figure 3.5: Comparison of Pressure Distribution for Actual and Simulated Flap Deflection of 1 used in conjunction with the AIC method. Also, as shown in the following test case, when only a deflection is modelled using the transpiration method the results are very accurate. This simulation demonstrated the ability of the transpiration method to simulate large surface deflections and discontinuities. To further test the accuracy of the method, the AGARD wing was then simulated with an arbitrary deflection. The results of the AGARD 29

41 wing simulations are shown in the following section. 3.2 AGARD Wing The second test case was the AGARD Wing. This is a standard aeroelastic test case and serves as a good reference for the current application of the Transpiration Method. The AGARD Wing in this study is defined by a half span of 2.5ft, a root chord of 1.833ft and a tip chord of 1.28 ft. The quarter-chord sweep angle is 45 with the elastic axis sweep angle at These dimensions can be seen in Figure 3.6 and have all been taken from Reference [1]. The airfoil of the AGARD wing was a constant NACA 65A4 profile. As with the BACT wing, two simulations were completed to test the validity of the transpiration method for steady flows. The first being an actually deflected wing, whereas the second had the transpiration boundary condition applied. These two models, with their respective meshes are shown in Figures 3.7a and 3.7b. The simulation for the AGARD wing was executed at Mach.687 with the wing at an angle of attack of. The results of the pressure contours of the two simulations can be seen in Figures 3.9a and 3.9b. Again, the residuals of the simulation were monitored to ensure the convergence of the solution. As with the BACT simulations, the AGARD solution was accepted once the residuals had decreased by five orders of magnitude. The residuals of the AGARD simulations - both the actually deflected and transpiration - can be seen in Figure 3.8. Again, the residuals shown are the relative values showing the five order of magnitude drop. Again, as with the BACT wing simulations, the two results show acceptable agreement. This is better illustrated with a comparison of the pressure distribution at a cut located at the 6% span of the wing. This is shown in Figure

42 Quarter-chord line Elastic axis Figure 3.6: AGARD Wing Model Dimensions Once again, with the simulation of an arbitrarily deflected AGARD wing, the transpiration method has shown to provide well matching results when compared to an actually deflected model. The final validation case was to apply this method to the Global Express wing. The results from this test were not as good as the previous cases, however, the trends suggested that the method would perform well if more time was spent refining the model 31

43 (a) Actual (b) Simulated Figure 3.7: AGARD Wing 1 6 Actual Deflection x momentum y momentum z momentum 1 5 Transpiration x momentum y momentum z momentum Residual 1 3 Residual Iteration Iteration Figure 3.8: Convergence of Solution Residuals - AGRAD Wing 32

44 (a) Actual (b) Simulated Figure 3.9: Pressure Contour for Deflection of AGARD Wing Cp Actually Deflected + + Transpiration X Figure 3.1: Comparison of Pressure Distribution for Actual and Simulated Deflection of AGARD Wing and technique for this case. This was forgone in respect to time constraints of this research. The results obtained from the steady validation cases provided sufficient information that this method is suitable for the use of deforming the aerodynamic model at each time step in an unsteady simulation. The results of unsteady simulations of an AGARD wing experiencing flutter are presented in section