The Verification and Validation of Preliminary CFD Results for the Construction of an A Priori Aerodynamic Model.

Transcription

1 Master of Science Thesis The Verification and Validation of Preliminary CFD Results for the Construction of an A Priori Aerodynamic Model. W.G.M. (Wim) Vos February 2006

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3 Verification and Validation of Preliminary CFD Results for the Construction of an A Priori Aerodynamic Model. Master of Science Thesis In order to obtain the degree of Master of Science in Aerospace Engineering at Delft University of Technology W.G.M. (Wim) Vos February 2006 Faculty of Aerospace Engineering Delft University of Technology

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5 Abstract In ancient aviation aircraft were designed and built without much scientific insight and pilots were brave men, willing to fly these machines. Nowadays airplanes are created over years by groups of engineers. This causes the airplanes to be bigger, faster, more complex, etc. And with this also the role of the pilot changed. The brave man from before got more and more people aboard his aircraft. He changed from pioneer to "bus driver". Since these days pilots are responsible for the lives of many people and since they fly with complex machines they have to be trained carefully. Since complex modern aircraft do not allow pilots to fly them without practice flight simulators play an important role in the training process. Flight simulation models describe the properties and responses of an aircraft in order to regenerate them again for on-ground training sessions. Commonly these models are based on data measured during dynamic flight tests manoeuvres where aircraft responses are measured. At the Delft University of Technology another method is developed where flight simulation models consist of an a priori aerodynamic model based on Computational Fluid Dynamics (CFD) computations that is updated by means of an aerodynamic residual model. This report is the second in a row produced at the aerodynamics department. The first one was written as a master s thesis by ir. E. Willem [12] in This second report focusses on the CFD computations that form the basis for the aerodynamic model, using the Cessna Citation II laboratory aircraft as a test bed. To confidently use the CFD data for the construction of an a priori aerodynamic model, the CFD results have to be trustworthy. For this the CFD model has to be designed carefully and a thorough verification and validation has to be performed. It is shown for steady symmetric flight conditions that Euler CFD results can be generated such that they can be used for the generation of an a priori aerodynamic model. Techniques have been developed to ensure accurate enough results and a foundation is laid to continue with more complex flight situations. i

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7 Acknowledgements Six years, or a quarter of my life. That is what many people would call: "the time Wim needed to become an engineer". I myself do not see it like that. The last six years I spend in Delft were not only needed to become an engineer. The last six years were also needed to change from a kid to an adult. The last six years were also needed to find out which direction I want to give to my life. With this master s thesis report this period comes to an end. I performed my master s thesis project in the chair of Aerodynamics at the faculty of Aerospace Engineering of the Delft University of Technology, under the auspices of dr.ir. Hester Bijl. I would like to thank her for the freedom and the support I got during the work and especially for the opportunity she gave me to write a paper about my work. This paper was presented at the EUCASS 2005 conference in Moscow and can be found in appendix B. I also want to express my gratitude to ir. Jelmer Cnossen for the time he spent to guide me throughout this last year, to dr.ir. Leo Veldhuis for the advice on the practical use of CFD modelling and to ir. João Oliveira for the cooperation, the long discussions about our project and the moral support. A special thank you also goes to my parents. They gave me the opportunity to study and live in Delft. And last but not least I want to thank my friends for the great six years we spent together. Wim Geraard Maria Vos Delft, November 2005 iii

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9 Nomenclature Abbreviations 2D 3D CAD CFD Two Dimensional Three Dimensional Computer Aided Design Computational Fluid Dynamics EEPP Estimated Engine Performance Program FSM GPS IMU Flight Simulation Model Global Positioning System Inertial Measurement Unit Variables ṁ Mass Flow [ kg s ] Q Energy Flow [ J s ] A Area [m 2 ] b Span [m] C D Three Dimensional Drag Coefficient [-] C L Three Dimensional Lift Coefficient [-] c l Two Dimensional Lift Coeffient [-] c p Specific Heat Capacity at Constant Pressure [ kj kgk ] D Drag [N] D Experimental Results [-] E Comparison Error [-] F Force [N] F T Trust [N] h Average Cell Size [m 3 ] L Lift [N] M Mach Number [-] v

10 vi O() Order of Accuracy [-] p Pressure [P a] R Gas Constant [ J kgk ] S Simulated Results [-] S Corrected Simulated Results [-] T Temperature [K] T True Results [-] U Uncertainty [-] V Velocity [ m s ] Greek symbols α Angle of Attack [deg] δ Error Estimation [-] Difference [-] δ Error [-] ɛ Error of an Estimation [-] Γ Vortex Distribution [ Nms kg ] γ Specific Heat Ratio [-] µ Mean Value [-] π The Number Pi [-] ρ Air Density [ kg m 3 ] σ Standard Deviation [-] Subscripts D G I i M MA N Free Stream Values Experimental Grid Size Iteration Number Single Point from Series Modelling Modelling Error Numerical

11 vii P PS S T t V X Z Other Parameters Previous Data Simulated Time Step Total Values Validation In X-direction In Z-direction

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13 Contents Abstract Acknowledgements Nomenclature Contents List of Figures List of Tables i iii v x xii xiii 1 Project Outline Flight Simulation Models Three Dimensional CFD Calculations Verification and Validation Outline of this Report Verification and Validation Methods Terminology Verification Method Iteration Number Grid Size Correction Factor and Confidence Interval Validation Method Model Choices D CAD Model Size of the Computational Domain Circumstances Determination of the Width of the Domain Determination of the Length and Height of the Domain Dimensions of the Computational Domain Grid Generation Simplified Engine Model The Pressure at the Numerical Outflow The Velocity at the Numerical Inflow The Temperature at the Numerical Inflow Consequences of the Simplified Engine Model Numerical Approximation of Reality ix

14 x CONTENTS 4 Results From Output to Result Flow Through Nacelles Engines On Verification and Validation Verification Iterative Convergence Grid Convergence Corrected Results with Confidence Intervals Final Remarks on Verification Validation Uncertainties in Flight Test Results Confidence Intervals for Flight Test Results Validation of the Results Conclusions and Recommendations Conclusions Recommendations A Shifting Airfoil Position vs CFD Accuracy A-1 B EUCASS 2005 Moscow B-1

15 List of Figures 2.1 Iterative behavior for a random aerodynamic force, i.e. the lift Solution for an aerodynamic force, i.e. the lift, on different grids The wiggles at the vertical tailplane Example of a computational domain used in this work Flight test data points within the flight envelope of the Cessna Citation II Lift coefficient versus the size of the computational domain for an angle of attack of 6.8 degrees Error in lift coefficient versus the size of the computational domain for an angle of attack of 6.8 degrees An example of the grid used throughout this work The engine of the Cessna Citation, a black box approach The force in X-direction versus angle of attack ( engines on ) The lift coefficient versus angle of attack ( flow through nacelles ) The drag coefficient versus angle of attack ( flow through nacelles ) The lift coefficient versus angle of attack ( flow through nacelles ), CFD vs flight test The drag coefficient versus angle of attack ( flow through nacelles ), CFD vs flight test Relative two dimensional error in lift coefficient versus the angle of attack Lift coefficient versus angle of attack for corrected CFD data ( flow through nacelles ) The lift coefficient versus angle of attack ( engines on ), CFD vs flight test The drag coefficient versus angle of attack ( engines on ), CFD vs flight test The lift coefficient versus angle of attack The drag coefficient versus angle of attack An iterative convergence for the lift coefficient, with indication of the estimated final result Grid convergence for the lift coefficient for different points in the flight envelope Grid convergence for the drag coefficient for different points in the flight envelop.e Corrected values for C L with confidence intervals ( engines on ) Corrected values for C D with confidence intervals ( engines on ) CFD and flight test results for C L with confidence intervals ( engines on ) CFD and flight test results for C D with confidence intervals ( engines on ). 39 A.1 Pressure distribution inside the computational domain for different box sizes A-1 A.2 Pressure distribution inside the computational domain for different box sizes A-2 B.1 Computational domain B-2 xi

16 xii LIST OF FIGURES B.2 Error in c l in two dimensions as function of the size of the computational domain B-3 B.3 Flight envelope of the Cessna Citation II (X-axis: velocity [kt], Y-axis: height [flight level]) B-4 B.4 Comparison of the lift and drag curves (flow through nacelle) B-5 B.5 Lift curve for corrected CFD data (flow through nacelle) B-5 B.6 Lift and drag curves (engines on) B-5 B.7 Relative difference between FTI and CFD results, engines on B-6

17 List of Tables 3.1 Dimensions of the computational domain Target cell size on different aircraft segments Different grids used for the grid convergence study Grid convergence data for the lift coefficient (CI = confidence interval) Grid convergence data for the drag coefficient (CI = confidence interval).. 33 xiii

18 xiv LIST OF TABLES

19 1 Project Outline In this first chapter, the importance of verification and validation of Computational Fluid Dynamics results within this project will be emphasized. In addition to this, thorough explanation of the research project, Flight Simulation Models based on Computational Fluid Dynamics and Flight Test Identification will be given. This project, funded by Technologiestichting STW, was started in 2003 as a joint project at the Control and Simulation and Aerodynamics division of the faculty of Aerospace Engineering at Delft University of Technology, the Netherlands. The function of the Aerodynamics section within this project is to provide the required Computational Fluid Dynamics (CFD) data. 1 Once the importance of providing verified and validated CFD data within the research project is made clear, the methods used for verification and validation of the CFD results are discussed. These methods are the main theme of this work. 1.1 Flight Simulation Models The generation of Flight Simulation Models (FSM) is an important branch in the field of Control and Simulation. FSM are used to train pilots in an environment that approximates real flight conditions. For this it is important that the FSM is able to simulate the flight conditions as accurately as possible. FSM s are commonly based on data measured during dynamic flight tests manoeuvres where aircraft responses are measured. This is immediately the main advantage of flight tests, real conditions for a real aircraft. However, this technique requires an aircraft equipped with a system that is able to measure all the aircraft responses. Such a system, existing of GPS sensors (global positioning system), IMU s (Inertial Measurement Unit), gyroscopes, accelerometers, magnetometers, air data measurement system, angle of attack vanes, et cetera, is very expensive and needs to be thoroughly calibrated before it can be used. Another drawback of this method is the lack of flight test data when a FSM of an aircraft needs to be constructed in the development phase. A last drawback of flight tests is the difficulty of predicting aircraft behavior at the borders of the flight envelope since this is a very dangerous region to fly in. Another option is to base the FSM on wind tunnel tests. Wind tunnel tests measure the 1 The first steps in this area were taken by ir. E. Willem during his master s thesis project in More information on his work can be found in his master s thesis report [12]. 1

20 2 CHAPTER 1. PROJECT OUTLINE responses of a scaled aircraft to real, but scaled conditions. This scaling causes errors in certain parameters, e.g. the Reynolds number. In addition, the construction of a scale model of the aircraft is again quite expensive, as is the use of the wind tunnel. A third option is to use Computational Fluid Dynamics (CFD) data for the generation of an FSM. CFD data is relatively cheap to obtain and no scale factors are used. The main disadvantage of using CFD is that it computes data for a simulated aircraft in simulated conditions. This means that models of the physical phenomena are used which bring along errors in the results as well as simulation errors (e.g. discretization errors). The research project proposed by the Control and Simulation division aims to develop a method to generate FSM as fast and cheap as possible. For this reason, the third option is chosen: generating the FSM based on CFD results. Since CFD results give the responses of a simulated aircraft in simulated conditions the results are only used to generate an a priori aerodynamic model that will provide the most important aerodynamic characteristics of the aircraft. Afterwards, this a priori model will be updated with a residual model based on flight test data to define the exact responses to real conditions. This new method can be used for both existing aircraft and aircraft that are still in the development phase. The a priori model can be used for the theoretical shape to give some insight in the aerodynamic properties that can be expected. A complete aerodynamic model can be constructed contiguous when a first prototype is available. To test the method, an FSM of an existing aircraft is developed. The laboratory aircraft available at the faculty of Aerospace Engineering at the Delft University of Technology is the Cessna Citation II business jet, equipped with a complete data acquisition system. 1.2 Three Dimensional CFD Calculations At the Aerodynamics division of the faculty of Aerospace Engineering at the Delft University of Technology the work within the field of CFD is mainly focussed on the improvement of the techniques currently used to reduce computational time and to increase the accuracy of results. However, Willem [12] was the first to use full scale three dimensional (3D) CFD calculations. For this reason it is important to investigate more these 3D results before using them in the development of a FSM. In this first part of the joint research project, the main interest of the Aerodynamics division lies in the verification and validation of the CFD results. The correctness of the results has to be understood fully for elementary cases before more elaborate cases can be studied. This is the reason this work only concentrates on steady symmetric flight conditions. 1.3 Verification and Validation To be able to discuss the reliability of CFD results two steps are required. The first step is a verification of the results. verification is defined as a process for assessing simulation numerical uncertainty and, when conditions permit, estimating the sign and magnitude of the numerical error itself and the uncertainty in that error estimator [10]. The second step consists of the validation of these results. validation is defined as a process for assessing simulation model uncertainty by using benchmark experimental data and, when conditions permit, estimating the sign and magnitude of the modelling error itself [10].

21 1.4. OUTLINE OF THIS REPORT 3 Both steps are performed according to the work of Stern [10]. The uncertainties defined by Stern are used to generate confidence intervals for both the CFD and flight test results. The confidence intervals for the CFD results are generated with the verification techniques, while the confidence intervals for the flight test results are computed by the Control and Simulation division using the uncertainties in measured data. Plotting both results with their confidence intervals provides an ultimate validation of the numerical results. The level of reliability of the CFD data achieved after verification and validation will determine to which extent the a priori aerodynamic model will be able to describe the responses of an aircraft to flight conditions. In this work the main issues concerning verification and validation are the models that are used and the computations that are performed. 1.4 Outline of this Report In the following chapter (chapter 2) the terminology used by Stern in his work on verification and validation is depicted. It also includes a more in-depth description of the general techniques used for verification and validation Chapter 3 focusses on the preprocessing that is required before actual CFD calculations can be performed. The generation of a three dimensional CAD (Computer Aided Design) model will be addressed briefly. More attention is given to the generation of a grid in general, and in particular to the development of a method that gives an estimate for the size of the computational domain needed to obtain sufficiently accurate results. This chapter gives details on the simplified engine models that are used throughout this work. One where the engines are hollow tubes where the air can flow through and a second where the engines are modelled with an outflow boundary at the engine inlet and an inflow boundary at the engine outlet. Finally this chapter explains why Euler CFD computations are performed and the consequences this has on the results. In chapter 4 the results for chosen points of the steady symmetric are presented. Euler computations are done for two different engine conditions: flow through nacelles and engines on. The results are presented through C L -α and C D -α curves, compared with flight test results and examined. Chapter 5 emphasizes the actual verification and validation of the results. The methods described in chapter 2 are adjusted for proper use within this work. Afterwards results are verified and confidence intervals are generated so that a final validation can be performed. Finally, the conclusions from this work are presented, together with recommendations for future work.

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23 2 Verification and Validation Methods In the work of Willem [12] the verification and validation of CFD results were only briefly discussed. The lack of computational power made it hard to perform many different CFD calculations, which lead to few results that could be verified and validated. However, verification and validation are two very important topics in CFD computations. Therefor within this report verification and validation are discussed more thoroughly. Since the development of CFD techniques, engineers have been trying to determine adequate techniques for verification and validation. However, no consensus between different parties working on this subject was reached. In response to this, Stern published a set of two papers on Comprehensive Approach to Verification and Validation of CFD Simulations in 2001 [10][11] with the main intention to reach consensus on this concept, in terms of definitions, and useful methodology and procedures for the verification and validation process of CFD calculations. This work provides a pragmatic approach for estimating errors and uncertainties in CFD simulations and is an extension on his previous work on validation and verification. In this work the same methods are used for two reasons: 1) Stern presented them in an easily implementable way, intended for practical use. 1 2) Stern developed them for CFD simulations on an already developed CFD code, without requiring availability of the source code for specified objectives, geometry, conditions and available benchmark information. In this chapter, first the key concept, definitions and derivations of equations for simulation errors and uncertainties are given (section 2.1). This is followed by a section on verification (section 2.2) and one on validation (section 2.3) of CFD simulations. 2.1 Terminology The most important terms in this chapter are the error δ, the uncertainty U and the error estimate δ. The error δ is defined as the difference between a simulated or experimental value and the truth. Since errors are rarely known exactly they need to be estimated. The uncertainty U is an estimate of the error δ such that the interval ±U contains the true value of δ in 95 % of the cases. This interval gives information about the range of likely magnitudes of δ but no information about the sign is given. For some situations, simulation errors can be estimated including both magnitude and sign. Such errors are 1 The first paper [10] explains the methods and the second paper [11] gives an application of the method for a RANS simulation of a cargo/container ship 5

24 6 CHAPTER 2. VERIFICATION AND VALIDATION METHODS referred to as an error estimate δ. The error in a simulation δ S is defined as the difference between simulated results S and true values T. This simulation error can be divided into a modelling error and a numerical error: δ S = S T = δ SM + δ SN. (2.1) Here the modelling error δ SM = M T and δ SN = S M. The simulation S and model M values are obtained by numerical and exact solutions of the continuous equations used to model the true values, respectively. Also, an uncertainty equation corresponding to the error equation (2.1) can be given: U 2 S = U 2 SM + U 2 SN. (2.2) Similar to equation (2.1), U S is the uncertainty in the simulation and U SM and U SN are the modelling and numerical uncertainties, respectively. For certain conditions, some additional information on the numerical error δ SN is present and it can be written as: δ SN = δ SN + ɛ SN, (2.3) where δ SN is an estimate of the magnitude and sign of δ SN, and ɛ SN is the error in that estimate (and is estimated as an uncertainty since only a range bounding its magnitude and not its sign, can be defined). Following this, the corrected simulation value S c can also be defined: S c = S δ SN. This leads to the following simulation error equation: (2.4) δ S = S c T = δ SM + ɛ SN. (2.5) After introduction of the terminology used in this chapter, verification and validation can be discussed. Stern uses the numerical errors δ SN to verify the CFD simulations. Model errors δ SM are used in the validation phase. 2.2 Verification Method Verification was defined by Stern (see section 1.3) as a process for assessing simulation numerical uncertainty U SN and, when conditions permit, estimating the sign and magnitude δ SN of the simulation numerical error itself and the uncertainty ɛ SN in that error estimate. The error δ SN can be decomposed in different error contributions [10]: δ SN = δ I + δ G + δ T + δ P, (2.6) where the subscript I,G,T and P stand for the iteration number, the grid size, the time step and other parameters, respectively. Since the errors are assumed to be independent, the simulation numerical uncertainty is then given by: U 2 SN = U 2 I + U 2 G + U 2 T + U 2 P. (2.7) The value of U SN will be used in this work to generate confidence intervals of the CFD results later in this work. Since only steady flight conditions are considered for now the uncertainty related to the time step (UT 2) is zero. For simplicity, the influence of other parameters (U P 2 ) is assumed to be small and also neglected. Equation (2.7) then reduces to: U 2 SN = U 2 I + U 2 G. (2.8) The simulation numerical accuracy is thus dependant on only two factors, i.e. the iteration number and the grid size.

25 2.2. VERIFICATION METHOD Iteration Number To visualize the influence of the iteration number on the accuracy of the simulation numerical results, figure 2.1 gives an example of how the results for an aerodynamic force (i.e. the lift) iterate to the final result. After a certain number of iterations the iterative behavior is oscillatory convergent. Stern gives as an estimate for the uncertainty in the case of an oscillatory convergence: U = 1 2 (S up S low ), (2.9) where S up and S low are the upper and lower bounds of the solution oscillation near the final iteration, respectively. 2.5 x L [ N ] iteration number [ ] Figure 2.1: Iterative behavior for a random aerodynamic force, i.e. the lift The same approach is followed to achieve an error correction term, δi, however it is formulated in a slightly different way. For the oscillatory convergent section near the final iteration in figure 2.1 the mean and standard deviation can easily be computed. The mean is given by: µ = N i=500 y i N 500, and the standard deviation by: (2.10) σ = N i=500 (y i µ) 2. (2.11) N 500 In equation (2.10) and equation (2.11), y i is the quantitate of interest at iterate i from the series and N is the number of solutions in the series. Throughout this work the mean is used as an estimate of the numerical solution when N goes to infinity, while the standard deviation gives the uncertainty for this solution: δ I = µ S (2.12) and ɛ I = σ. (2.13) In equation (2.12) S is the numerical solution of the CFD computation after a certain number of iterations within the region of oscillatory convergence. δi is an estimate for the magnitude and sign of the error (equation (2.3)) caused by iterative effects, and ɛ I is the uncertainty in this estimate.

26 8 CHAPTER 2. VERIFICATION AND VALIDATION METHODS Grid Size The second part of the simulation numerical error is an error introduced by a discrete domain (equation (2.6)). To determine this error, solutions on different grids are generated, from coarse grids to dense grids. Figure 2.2 shows the behavior of the solution as a function of the grid size. The X-axis represents the reference length, given by: reference length = 3 volume of the box number of cells. (2.14) Contrary to oscillatory convergence for the iterative error a monotonic convergence is noticeable as the solution is given as a function of the reference length x Lift [ N ] reference length [ m ] Figure 2.2: Solution for an aerodynamic force, i.e. the lift, on different grids For the study of the convergence of the solution as a function of the reference length different grids are used. These grids are labeled 1, 2, 3, 4, 5 and 6 from a coarse to a dense grid, respectively. Since a second order discretization scheme is used for the CFD calculations the solution on a grid with reference length h can be written as: S h = S + O(h 2 ) = S + c 1 h , (2.15) where S is the solution on an infinitely fine grid. This function describes the convergence behavior in figure 2.2. A small remark has to be made here. Since the discretization of the boundary conditions has an order smaller than 2, equation (2.15) is not exactly correct. It is better to write equation (2.15) as: S h = S + O(h x ), (2.16) where x < 2 but x is close to 2. Since it is very hard to determine the exact value of x, x is chosen to be equal to 2 in the remaining of this report. Looking back to equations (2.3) and (2.6) the error introduced by a discrete domain, δ G is discussed. An estimate for the magnitude and sign of the error caused by a discretization of the domain, δ G, and the uncertainty in this estimate ɛ G can be expressed as follows: δ G = c 1 h 2 (2.17) and ɛ G = U(c 1 h 2 ) (2.18)

27 2.3. VALIDATION METHOD 9 where U(x) stands for the uncertainty of x. In this subsection about the influence of the grid size on the error in the CFD computations only the influence of the number of cells (or the reference length) in the computational domain is included. In section 3.2 the influence of the length, width and height of this computational domain is discussed. This influence is not included in the final verification and the generation of a confidence interval since it is analyzed by means of a one or two dimensional method. The fact that one or more dimensions are neglected makes it impossible to use the error estimates for the verification of the three dimensional CFD simulations. The error estimates in section section 3.2 are only used to obtain an idea of how large the computational domain should be to capture most of the aerodynamic phenomena around the Cessna Citation II in steady symmetric flight conditions Correction Factor and Confidence Interval The solution S for the variable of interest of the CFD computations is corrected using the methods described previously. This means that the final value S c is equal to: S c = S + δ I + δ G. (2.19) And the uncertainty for this corrected value S c is given by: U 2 S c = ɛ 2 I + ɛ 2 G. 2.3 Validation Method (2.20) Validation was defined by Stern (see section 1.3) as a process for assessing simulation model uncertainty U SM by using benchmark experimental data and, when conditions permit, estimating the sign and magnitude δsm of the modelling error itself. Stern [10] starts his validation procedure by defining the comparison error, E: E = D S, (2.21) where D is the value for a variable of interest given by experiments (flight tests in the case of this work) and S the same value given by the CFD simulations. The uncertainty for E is given by: U 2 E = U 2 D + U 2 SMA + U 2 SP D + U 2 SN, (2.22) where U D, U SMA and U SP D are the uncertainty in the experimental results, the uncertainty due to simulation modelling errors and the uncertainty due to the use of previous data (such as fluid properties), respectively. Ideally, it can be postulated that if the absolute value of E is less than its uncertainty U E, validation is achieved (i.e. E is zero considering the resolution imposed by the noise U E ). However, in reality, there is no known approach that gives an estimate of U SMA. This means that U E cannot be estimated, which leaves a more stringent validation test as the practical alternative. The validation uncertainty is defined as: U 2 V = U 2 E U 2 SMA = U 2 D + U 2 SP D + U 2 SN. (2.23) In this work the uncertainty due to previous data is considered small and left out of consideration since no starting information on the airflow is provided. This means that the verification uncertainty becomes: U 2 V = U 2 D + U 2 SN. (2.24) Equation (2.24) shows that validation is achieved if the confidence intervals for the experimental (flight test) data and the CFD computations have an overlap.

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29 3 Model Choices This chapter gives a short overview of the preprocessing involved in the numerical CFD simulations performed throughout this work. Preprocessing can be seen as all the actions that have to be taken before the CFD calculations can be started. Specifically for this work, the preprocessing consists of the generation of a three dimensional CAD model, the determination of the size of the computational domain with grid generation and the development of a simplified engine model. These themes are discussed chronologically in this chapter D CAD Model The three dimensional CAD model is generated from a data series of surface points of the Cessna Citation II. These points form the skeleton of the 3D model. From these, several surfaces are generated. This makes it easy to remove a certain part from the model, adapt it and add it again. The trailing edge of the Cessna Citation II has a finite thickness that is smaller than one percent of the aerodynamic chord of the wing. These small surfaces might lead to problems in the grid generation. For this reason, the trailing edge is sharpened in the 3D CAD model. The 3D CAD model is generated in Rhinoceros NURBS 3D modeling [8]. Willem [12] encountered some problems during the generation of the CAD model from the given set of surface points of the Cessna Citation II at the position of the vertical tailplane where wiggles in the surface arose. Figure 3.1.a shows this problem. The wiggles were removed by a manual correction of the points in this region. Figure 3.1.b shows the corrected section, as used in this work. Also the pylons and nacelles are an addition to Willem s model as can be seen in Figure Size of the Computational Domain The computational domain is the domain where the flow equations are solved. The size of the domain strongly influences the accuracy of the results as well as the computational time. This section gives an insight into the computational domain used in this work and how to determine a certain domain. 11

30 12 CHAPTER 3. MODEL CHOICES (a) Willem s model showing wiggles at the base of the vertical tailplane (b) The new model with pylons and nacelles Figure 3.1: The wiggles at the vertical tailplane Circumstances The choice for a computational domain is strongly influenced by the circumstances (e.g. which aircraft, points of the flight envelope to be computed, et cetera). In this work, only steady symmetric flight conditions are studied. This means that only half a model is needed for the computations. Figure 3.2 gives an idea of how the domain used in this work appears. It is clear that one side of the box coincides with the symmetry plane of the Citation II. The airplane is also in a horizontal position, i.e. the X-axis of the aircraft is parallel with the X-axis of the domain (or XY-plane), see figure 3.2. This means that to get a certain angle of attack the flow in the domain is not horizontal. This option is preferred since horizontal flow with an aircraft under a certain inclination with the domains XY-plane requires a different grid for every angle of attack which requires a phenomenal increase in preprocessing time. Figure 3.2: Example of a computational domain used in this work The dimensions of the computational domain are determined for a well known series of computations that has to be done. Since the results of CFD computations are compared with flight test results, the dimensions of the domain are optimized for the points of the flight envelope flown in the flight tests. In figure 3.3 these points of the flight envelope are shown. The flight test data points consist of two series of points, measured at two different altitudes. The color of the dots changes as the angle of attack changes (the bigger

31 M max = SIZE OF THE COMPUTATIONAL DOMAIN 13 the angle of attack the lighter the dots become). The following subsections give a general overview of the determination of the size of the domain for this very special case, namely the Cessna Citation II in a well defined region of the flight envelope Alt max = 43000ft Tropopause = 11km Altitude [ FL ] V stall = 85 kt (empty W) V stall = 98 kt (MaxTOW) V max = 262 kt) TAS [ kt ] Figure 3.3: Flight test data points within the flight envelope of the Cessna Citation II Determination of the Width of the Domain To determine the lateral dimension of the box (Y-direction in figure 3.2) the influence of the side plane (gray plane in figure 3.2) on the angle of attack is estimated. The Cessna Citation II (or its wing, the most important source of disturbance in the domain) is replaced by a horseshoe vortex according Prandtl s lifting line theory [1]. Since the velocity on the boundary is equal to the free stream velocity, the presence of the side plane can be seen as if there was a mirrored horseshoe vortex with respect to this surface. The mirrored vortex generates a change in angle of attack on the real wing. By limiting the maximum allowed change in angle of attack at the wing, the position of the mirrored vortex and the side of the computational domain can be computed. The Kutta-Jouwkowsky theorem gives the following relation between the lift and the vortex strength of a horseshoe vortex: L (y 0 ) = ρ V Γ(y 0 ), (3.1) where ρ and V are the density and velocity in the undisturbed flow, respectively. The total lift is found by integrating this equation: L = ρ V b 2 b 2 Γ(y)dy, (3.2) where b is the span of the aircraft. If a constant vortex distribution is assumed (Γ(y) = Γ) equation (3.2) can be rewritten as: Γ = L ρ V b. (3.3)

32 14 CHAPTER 3. MODEL CHOICES Now that the vortex strength is known it is possible to compute change in vertical velocity induced by the semi-infinite tip vortex with constant strength Γ at the position of the wing. This change is given by: δv Z = Γ 4π(2 width b), (3.4) where (2 width b) is the distance between the mirrored tip vortex and the wing tip. This change in vertical velocity will change the angle of attack α at the wing: δα = arctan V Z + δv Z V X α. (3.5) Using equations (3.3), (3.4) and (3.5) an expression for the width of the box as a function of the change in angle of attack at the wing can be derived: width = 1 8π(tan (α δα)v X V Z ) L ρ V b + b 2. (3.6) To determine the actual width of the domain for this work, the error in α due to the presence of boundary is compared with how accurately α can be determined in flight tests. Flight test data provides values for α with an accuracy of 0.1. Due to the fact that in future work this accuracy might increase, the error in α caused by the side of the domain is chosen to be half of the actual accuracy of α from the flight test data. For the points of the flight envelope given in figure 3.3 this requires a width of 40m. This value is used throughout the work Determination of the Length and Height of the Domain To determine the length and height of the computational domain (X- and Z-direction in figure 3.2, respectively) a different strategy is followed. The most important wing airfoil of the Cessna Citation II is placed in a two dimensional computational domain. The influence of the distance between the airfoil and the borders of this 2D domain on the lift and drag coefficients is then studied. The Cessna Citation has a NACA23014 airfoil at the root of the wing and a NACA23012 airfoil at its tip. Both airfoils are similar in shape but differ in thickness. The thickest profile (NACA23014) will have a bigger influence on the airflow around the airfoil since it generates more lift. The average airfoil of the wing of the Cessna Citation II lies somewhere between the NACA23014 and NACA Choosing the NACA23014 airfoil for a two dimensional analysis is too strict, while choosing the NACA23012 airfoil is too narrow. Errors in lift and drag coefficient caused by the influence of the boundaries of the computational domain are overestimated and underestimated respectively. The real maximum error in lift and drag coefficient will be smaller than the error from a two dimensional analysis with the NACA23014 airfoil. Using the NACA23014 airfoil to determine the influence of boundaries will provide a domain size large enough for the three dimensional CFD computations. The normalized NACA23014 airfoil is placed into a 2D computational domain and the lift and drag coefficients are computed. This is done with an Euler CFD computation. Figure 3.4 shows how the lift coefficient changes with the size of the computational domain. A computational domain with a size of 20 chords for example means that the airfoil is 20 chords away from every boundary of the domain. Now assuming that an airfoil inside a 100 chord computational domain is not influenced at all by the boundaries (this can be checked by using a potential code (updated with a boundary layer model) for two dimensional airfoils 1 ) the error in lift coefficient can be plotted against the size of the domain, 1 For this work this was done with the X-foil software. X-foil is an interactive program for the design and analysis of subsonic isolated airfoils [2].

33 3.2. SIZE OF THE COMPUTATIONAL DOMAIN 15 see figure c L [ ] height/length [ chords ] Figure 3.4: Lift coefficient versus the size of the computational domain for an angle of attack of 6.8 degrees c L [ ] height/length [ chords ] Figure 3.5: Error in lift coefficient versus the size of the computational domain for an angle of attack of 6.8 degrees To determine the length and height of the computational domain, the error in lift and drag coefficient from the two dimensional CFD computations is compared with the accuracy of these coefficients if they are computed from flight test results. Flight test data provides values for lift and drag coefficients with an accuracy of 0.01 and 0.05, respectively. To obtain errors within this accuracy when using CFD computations a width and height of approximately 20 chords is required. With an aerodynamic chord of approximately 2 meters, the box would have a length and height of 80 meters. Since this two dimensional analysis is too strict (all 3D effects to diminish the effect of the boundaries are not present) and 80 meters is quite large, the length and height of the box is reduced to 60 meters. This value is used throughout the work. A final remark has to be made here. The airfoil is placed in the center of the computational domain for this analysis. Another option is to vary the position of the airfoil inside the domain. This option has been tried, however results turned out to be better for an airfoil in the center of the domain. Appendix A gives the results for an airfoil placed at different positions inside the computational domain.

34 16 CHAPTER 3. MODEL CHOICES Dimensions of the Computational Domain As mentioned in the previous sections the computational domain has the shape of a box. The symmetry plane of the aircraft coincides with one of the planes of this box, and the aircraft is placed in the center of this plane. The dimensions of the computational domain are given in table 3.1 length: height: width: 60m 60m 40m Table 3.1: Dimensions of the computational domain 3.3 Grid Generation Inside the computational domain an unstructured all hexahedral grid is generated automatically. All hexahedral grids might cause problems during grid generation since some shapes are very hard to approximate using only hexahedral shapes. However, with a good knowledge of the model (with some small adaptations, eg. the sharpening of the trailing edge) and some experience with the grid generator, these problems do not occur often. The grid is generated with the Hexpress grid generator from Numeca International [6] in a similar way as done by Willem [12]. The grid that is used to generate the results in chapter 4 has cells in its domain and this grid is visualized in figure 3.6. Figure 3.6: An example of the grid used throughout this work The properties that have to be defined to generate the grid are the size of the starting grid and the target cell size on each part of the aircraft. The starting grid consists of uniform, beam shaped cells and the size of the beam shaped cells can be assigned. The target cell size after refinement is defined on each segment of the aircraft. The target cell sizes used in this work can be found in table 3.2. The grids that were generated for the purpose of the gird convergence study (subsection 5.1.2) have all the same target cell sizes. The only difference in their generation is the size of the initial beam shaped cells. 3.4 Simplified Engine Model In chapter 4 results will be presented for two engine conditions. The first one is the situation where the nacelles are modelled as hollow pipes, called flow through nacelles. This

35 3.4. SIMPLIFIED ENGINE MODEL 17 aircraft segment target cell size (reference length) fuselage: 0.1m wing: 0.1m nacelle (inside): 0.08m nacelle (outside) 0.08m vertical tailplane 0.05m pylon 0.01m Table 3.2: Target cell size on different aircraft segments can be seen as a first step from an engineless aircraft to an aircraft that has working engines. For the second situation a simplified engine model is generated. CFD techniques for turbomachinery make it possible to perform CFD computations throughout the engine of an aircraft. These computations are very time consuming, especially when the interaction between the flow within the engine and around the aircraft is taken into account fully but the presence of working engines in the CFD model, however, has a major influence on the accuracy of the results. Therefore a simplified engine model is developed [3]. Figure 3.7: The engine of the Cessna Citation, a black box approach The turbofan engines are replaced by an outflow and an inflow condition in the CFD model. The numerical outflow boundary is the inlet of the engine, chosen at the position of the fan of the engine. The numerical inflow boundary is the outlet of the engine, chosen in the mixing plane of the hot and the cold exhaust. Figure 3.7 gives a 2D graphical representation of how the engine is considered in the CFD model. At the numerical outflow, the pressure need to be defined, while at the numerical inflow values for velocity and temperature are required. These values are not known from flight test data and need to be computed The Pressure at the Numerical Outflow The pressure at the numerical outflow boundary, p outflow, is computed by assuming isentropic flow from the undisturbed air towards the engine inlet: T outflow = T t k, (3.7) p outflow = p t, k γ γ 1 (3.8) ρ outflow = p outflow, RT outflow (3.9)

36 18 CHAPTER 3. MODEL CHOICES where k = 1 + γ 1 2 M 2 outflow. In equations (3.7), (3.8) and (3.9) flight test data gives values for T t and p t, γ is considered to be equal to 1.4. The only unknown is k or the engine inlet Mach number M outflow. To compute this Mach number flight test data will be used, as well as the Estimated Engine Performance Program (EEPP) by Pratt and Whitney. Estimated Engine Performance Program (EEPP) The EEPP designed by Pratt and Whitney is a special software code that predicts engine properties for the Cessna Citation II s JT15D-4 series turbofan engines. The software is based on tables with engine data measured on the ground that are extrapolated to flight conditions. Inputs for the EEPP are measured during flight tests (altitude, total air temperature, flight Mach number and input fan speed). Outputs used in this work are the mass flow of air throughout the engine and the net-thrust generated by the engine. The main problem with the EEPP is this use of tables. Both input and output values are rounded of to the nearest value found in the table (no interpolation). This method non-interpolation gives in certain conditions output values that are reasonably different from the true values. This mass flow through the engine is given by: ṁ = ρ outflow V outflow A outflow. (3.10) Rewriting this formula for V 2 and replacing V = M γrt gives: M 2 outflow γrt outflow = ṁ 2 ρ 2 outflow A2 outflow. (3.11) Using equations (3.7), (3.8) and (3.9) this equation becomes: ck γ+1 γ 1 k + 1 = 0. (3.12) Here c is equal to: c = (γ + 1) RT t 2γ ( ṁoutflow p t A outflow ) 2. (3.13) Equation (3.12) can now be solved for k. This is done with Matlab. Since this is, for γ = 1.4, a sixth degree linear equation, six different values for k are found. Since only one value for k gives and acceptable value for M outflow it has to be selected with care. Once the value of k is known equation (3.8) gives the value for the pressure on the numerical outflow The Velocity at the Numerical Inflow The velocity at the numerical inflow, V inflow, can be found by the assumption that there is a straight exhaust stream. For simplicity this exhaust is believed to be directed entirely in the X-direction in figure 3.2. Straight outflow from an engine means that all the thrust is generated by the acceleration of the mass flow inside the engine: F T = ṁ(v inflow V ). (3.14) In this equation V is known from the flight test data. F T and ṁ are calculated with the EEPP program again.