Representative Post-Stall Modeling of T-tailed Regional Jets and Turboprops for Upset Recovery Training

by

Tianhang Teng

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Aerospace Science and Engineering University of Toronto

c Copyright 2016 by Tianhang Teng Abstract

Representative Post-Stall Modeling of T-tailed Regional Jets and Turboprops for Upset Recovery Training

Tianhang Teng Master of Applied Science Graduate Department of Aerospace Science and Engineering University of Toronto 2016

Loss-of-control resulting from airplane upset is a leading cause of worldwide commer- cial aircraft accidents. In order to provide pilots with sufficient stall recovery training using ground simulators, flight models will need to be improved. In this thesis, a methodology for generating a generic representative post-stall aircraft model is developed. Current type specific post-stall models are too expensive to generate, and are not practical for routine training. The representative model offers a much cheaper way to train crew for upset conditions, at the same time providing sufficiently realistic cues seen in an upset condition. The methodology provides a foundation for generating full scale representative models using wind tunnel data and physics-based approaches. An example of applying the methodology to predict stall regime aircraft behavior is provided for one aerodynamic coefficient.

ii Acknowledgements

First, I would like to express my sincere gratitude to my thesis supervisor Professor Peter Grant for the continuous support of my MASc study and related research, for his patience, motivation, and immense knowledge. His guidance helped me throughout the entire research and writing portions of this thesis. I could not have imagined having a better advisor and mentor for my MASc study.

I would like to thank Stacey Liu for her tremendous support in the lab, for help on technical knowledge, as well as moral support at difficult times.

Thanks also go out to the Plane of Sky (POS) group, Tony Zhang, Zane Luo, and Jacek Khan for their feedback on some of the issues related to this study, for the sleepless nights we were working together before deadlines, and for all the fun we have had in the last few years.

Finally, I would like to thank my parents and my girlfriend for their constant support during the course of my studies.

iii Contents

1 Introduction1 1.1 Background...... 1 1.2 Upset Recovery Training...... 2 1.3 Post Stall Modeling...... 3 1.4 Project Definition...... 4 1.5 Organization...... 5

2 Literature Review6 2.1 Existing Post-Stall Model...... 6 2.2 Semi-Analytical Methods...... 9 2.2.1 Linear Methods...... 9 2.2.2 Non-Linear Methods...... 11 2.3 Stall Aerodynamics...... 13 2.3.1 G Break/Pitch Non-Linearity...... 14 2.3.2 Roll-Off...... 15 2.3.3 Reduced Lateral/Directional Stability...... 15 2.3.4 Reduced Control Effectiveness...... 15 2.4 Configuration Effects...... 16 2.4.1 Tail Effects...... 16 2.4.2 Nacelle Effects...... 19 2.4.3 Wing Effects...... 19 2.4.4 Fuselage Effects...... 21 2.4.5 Engine/Power Effects...... 21 2.5 Published Wind Tunnel Data...... 22

3 Basic Methodology 26 3.1 Model Structure...... 26 3.2 Modeling Approach...... 28

iv 3.3 Configuration Deltas (∆)...... 30

4 Generation of Configuration ∆’s 32 4.1 Overview...... 32 4.2 Semi-Analytical Approach...... 33

4.2.1 Coefficient of Lift CL ...... 39

4.2.2 Coefficient of Pitching Moment Cm ...... 42

4.2.3 Coefficient of Rolling Moment Cφ due to β ...... 43 4.2.4 Flap Effects...... 46

4.2.5 Engine Power Effects CT ...... 48

4.2.6 Dynamic Effects Clp and Clr ...... 51 4.3 Empirical Approach...... 53 4.3.1 Effects of Nacelle-Tail Interaction on Pitching Moment...... 54 4.3.2 Effects of Wing-Tail Interaction on Pitching Moment...... 58 4.3.3 Effects of Tail Location on Pitching Moment...... 59 4.3.4 Effects of Tail Size and Incidence on Pitching Moment...... 59 4.3.5 Effects of Fuselage on Pitching Moment...... 60

5 Full Scale Example 62

6 Conclusions 67 6.1 Summary of Work...... 67 6.2 Future Research Needs...... 69

References 70

v List of Figures

2.1 Enhanced post stall model developed by NASA LaRC [1]...... 7 2.2 Blending method used on merging post stall model onto existing pre-stall model [2]...... 8 2.3 Classical vortex-lattice method...... 11 2.4 Pitching moment for conventional tailed aircraft versus T-tailed aircraft. Reprinted from [3]...... 14

2.5 Schematic of Tail Effect on Cm for T-Tail (Data adapted from NASA technical report [4])...... 17

2.6 Effect of horizontal tail size on Cm. Adapted from Ray [4]...... 18

2.7 Effect of horizontal tail incidence angle on Cm. Adapted from Ray [4].. 18

2.8 Nacelle Location effect on Cm(α), dashed lines are nacelle off condition (Reprinted from Ray [4])...... 19

2.9 Sweep effect on Cm(α) (Reprinted from NASA technical report [4])... 20

2.10 Effect of fuselage shape and size on Cm. Adapted from Ray [4]...... 21

2.11 Effect of fuselage length on Cm. Adapted from Ray [4]...... 21

2.12 Wind tunnel data of CT effect on a straight wing turboprop aircraft... 22

3.1 Example Pitch Moment Coefficients for Target pre-stall (T P ), Target es- timated pre-stall/stall/post-stall (T E), and Target Stretched/offset pre- stall/stall/post-stall ( T M )...... 29 3.2 Example Pitch Moment Coefficients for Target pre-stall (T P ), Target Stretched/offset stall (T M ) and Target stall model (T S). Blending occurs between two dashed vertical lines...... 30

4.1 Spanwise lift distribution for angle of attack ranging from 0◦ to 25◦ at 5◦ interval...... 34 4.2 Viscous remapping procedure for a generic swept wing using a generic cambered airfoil...... 35 4.3 Schematic of the de-α process...... 37

vi 4.4 Prediction of CL after every iteration of de-α ...... 38 4.5 Comparison between α-correction method and crude direct remap method 38 4.6 3-D effect on the spanwise lift distribution. Reprinted from [5]...... 38 4.7 Example of spanwise lift after correcting for 3-D effect...... 38 4.8 Comparison of de-α method vs. wind tunnel data using wing modeled from Ostawari [6]...... 41 4.9 2-D lift curve of NACA 4415 airfoil...... 41 4.10 Comparison between the potential flow and corrected span-wise lift distri- bution...... 41 4.11 2-D lift curve of the artificially generated NACA 0024 airfoil...... 42 4.12 Comparison of wing configuration from Shortal [7] with wind tunnel data 42 4.13 Comparison of wing configuration from Shortal [7] with wind tunnel data 43 4.14 2-D pitching moment curve of the artificially generated NACA 0024 airfoil 43 4.15 Spanwise lift distribution for straight wing at α = 15◦ β = −10◦ ..... 44 4.16 Comparison of the spanwise lift of wings under sideslip for higher α ... 45

4.17 Comparison of Cr vs. β for wing configuration from Ray [8]...... 45

4.18 Comparison of Crβ for wing configuration from Ray [8]...... 45 4.19 Definition of the three parameters. Plot shows two-dimensional trailing- edge flap effects...... 46 4.20 2-D lift curve of clean airfoil versus predicted flap 35 configuration.... 47

4.21 Comparison of predicted CL and Cm against wind tunnel data for flap 35 48 4.22 Augment spanwise lift distribution to capture accelerated flow...... 49

4.23 Comparing predicted CT effects against wind tunnel data for high and low

CT ...... 50 4.24 A schematic showing the two side of the wing at non-zero rolling rate.. 51 4.25 Spanwise lift distribution under non-zero p or r ...... 52 4.26 Spanwise lift distribution of an already stalled wing under non-zero p .. 52

4.27 Predicted Clp for a straight wing...... 52

4.28 Predicted Clr for a straight wing...... 52 4.29 Isolation of nacelle-tail interaction N ⊗ VH ...... 56 4.30 N ⊗ VH effect for various tail shadow angles...... 56 4.31 Spline points on the interpolation curve...... 56 4.32 Extrapolation of nacelle-tail interaction from tail shadowing angle.... 56 4.33 ∆ term subtracted from the two interoplated curves...... 57 4.34 Comparison of wind tunnel data between target configuration and baseline + ∆...... 57

vii 4.35 Isolation of wing-tail interaction W ⊗ VH ...... 58 4.36 W ⊗ VH effect for various tail shadow angles...... 58 4.37 Look-up graph for horizontal tail size...... 60 4.38 Look-up graph for horizontal tail incidence angle...... 60 4.39 Look-up graph for fuselage cross-section size...... 60 4.40 Look-up graph for fuselage length...... 60

4.41 Adding the ∆Cm of fuselage onto total aircraft configuration...... 61

5.1 Specifications for the baseline aircraft. Reprinted from Ray [4]...... 62

5.2 Cm plot of the baseline aircraft and the target aircraft...... 63

5.3 Cm plot of the baseline aircraft and the target aircraft after correcting for stall point...... 63 5.4 Interpolating/Extrapolating for fuselage length ∆...... 64 5.5 Interpolating/Extrapolating for fuselage cross-sectional size ∆...... 64 5.6 Interpolating/Extrapolating for tail shadowing angle ∆...... 65 5.7 Obtaining ∆ term for tail moment arm change...... 65 5.8 Four ∆ effects that transforms the baseline aircraft to the target aircraft 65

5.9 Progression of the Cm curve after adding the delta terms...... 66 5.10 Final predicted stall model for the target aircraft...... 66

viii List of Tables

2.1 References to public post-stall wind tunnel data with configuration vari- ations (longitudinal). Red cells indicate high importance for post-stall modelling; yellow indicates modest importance...... 24 2.2 References to public post-stall wind tunnel data with configuration varia- tions (lateral). Red cells indicate high importance for post-stall modelling; yellow indicates modest importance...... 25

3.1 Overview of geometric changes and which ones are covered in this thesis. 30

4.1 Details of the wing modeled from Ostawari [6]...... 40 4.2 Details of the wing modeled from Shortal [7]...... 41 4.3 Details of the wing modeled from Ray [8]...... 45

ix Nomenclature

α angle of attack, degrees β sideslip angle, degrees φ Euler roll angle, degrees θ Euler pitch angle, degrees ψ Euler yaw angle, degrees p roll rate, deg/s q pitch rate, deg/s r yaw rate, deg/s X force along the body x-axis, N Y force along the body y-axis, N Z force along the body z-axis, N L rolling moment about the body x-axis, N · m M pitching moment about the body y-axis, N · m N yawing moment about the body z-axis, N · m

ωss steady-state rate (wind-axis roll rate) V true airspeed, knots or m/s b wing span, m c¯ mean aerodynamic chord, m S wing area

Ci non-dimensional aerodynamic coefficient: i = X,Y,Z,L,D,l,m,n

δf flap deflection, degrees

δr rudder defection, degrees: positive trailing edge left

δs stabilizer deflection, degrees: positive trailing edge down ∆ configuration delta term Γ vortex strength CFIT controlled flight into terrain C.G. center of gravity EUR enhanced upset recovery (NASA’s full-scale enhanced flight model)

x FAA U.S. Federal Aviation Administration IFS in-flight simulator LaRC Langley Research Center LOC loss-of-control NASA U.S. National Aeronautics and Space Administration NTSB U.S. National Transportation Safety Board URT upset recovery training UTIAS University of Toronto Institute for Aerospace Studies NPRM Notice of Proposed Rule Making ICATEE International Committee for Aviation Training in Extended Envelopes SUPRA Simulation of Upset Recovery in Aviation LLT Lifting Line Theory CFD Computational Fluid Dynamics VLM Vortex Lattice Method Subscripts X force component along the body x-axis Y force component along the body y-axis Z force component along the body z-axis L lift component D drag component l, r rolling moment component m pitching moment component n yawing moment component l local 2-D lift component T thrust f.o. forced oscillatory component of total angular rate Other Notations ˆ non-dimensional value ˙ time derivative

xi Chapter 1

Introduction

1.1 Background

Loss Of Control-Inflight (LOC-I) is one of the leading causes of fatal aircraft accidents in recent history, and has surpassed the previous leading cause of world-wide commer- cial aircraft accidents: Controlled Flight Into Terrain (CFIT). In a recent report by the Boeing Company titled “Statistical Summary of Commercial Jet Airplane Accidents”, between 2003 to 2012 LOC-I accounted for 18 accidents and 1,648 fatalities out of a total of 75 accidents and 4,269 total fatalities from all causes [9], making LOC-I the largest category of fatal aviation accidents during this period.

While the number of CFIT accidents has largely been reduced through the introduction of the Enhanced Ground Proximity Warning System, the number of LOC-I accidents has remained relatively constant. According to the International Civil Aviation Orga- nization (ICAO), “LOC-I is an extreme manifestation of a deviation from the intended flightpath” [10]. Aircraft enter such upset conditions when an aircraft operates beyond its designated flying envelope. This is typically defined in the Upset Recovery Aid as the aircraft having a pitch angle of greater than 45◦ nose-up, less than 10◦ nose-down, a bank angle of greater than 45◦, or flying at an inappropriate speed [11]. Of all the LOC-I acci- dents that occurred worldwide over the past decade, the leading cause was aerodynamic stall [12]. Studies from the FAA showed that between 1993 and 2007, airplane upsets are primarily caused by aerodynamic stall (36%), followed by flight control system mal- function (21%), pilot spatial disorientation (11%), contaminated airfoils (11% excluding stall), atmospheric disturbance (8%), and other/undetermined causes (13%) [12]. LOC-I resulting from a stall goes into a flight condition where the aircraft motion often cannot be predictably controlled by pilot inputs, where small state and control changes may lead

1 Chapter 1. Introduction 2 to large responses, or where the aircraft experiences high angular rates, divergent, and/or oscillatory behaviour. This results in the pilot having difficulty maintaining stable flight, which can potentially lead to fatal accidents. High profile recent examples of fatal air accidents include Air France 447 [13] and Colgan Air 3407 [14], both of which led to the death of all passengers and crew onboard. In both cases, the pilots were unable to recover the aircraft from an aerodynamic stall that developed into LOC-I before subsequently crashing. It is therefore proposed that better pilot training in stall recovery would be crucial in reducing stall related accidents. Stall recovery training is just one aspect of the more general Upset Prevention and Recovery Training (UPRT).

1.2 Upset Recovery Training

Unlike previous decades when commercial pilots were often recruited from the military, far fewer modern pilots have military backgrounds. A common training among air force pilots involves aerobatics and stunts, where practicing spin recovery and stall recovery are required. As the sophistication of aircraft evolves, the perceived need for pilots to receive training beyond the normal operational maneuvering envelope diminishes. The develop- ment of autopilots and various stall protection systems has effectively masked the need for civil pilots to train outside of their aircraft’s normal operating envelope [15]. There- fore it is important that pilots receive proper stall training as part of their flight readiness.

Pilots can train through a few methods: in-flight aircraft training, classroom instruction, in-flight simulation and ground-based simulation. Ideally, pilots should receive upset training in actual aircraft, however such training is costly and dangerous, particularly for large transport aircraft. While it is possible to perform upset recovery training in smaller, more agile aircraft, the difference between handling qualities of the aircraft may limit the transfer of training. In-flight simulators are capable of offering the handling qualities of large transport aircraft, however there are very few in-flight simulators in the world, and the high cost and low availability make them impractical for repeated training on a large scale. Ground-based simulators offer a cost-efficient, safe option for pilot training. Pilots can obtain experiences that are close to actual flight, without endangering themselves or the aircraft. Studies have been done in the past examining the effectiveness of each method in performing upset recovery manoeuvres [16]. Although the results failed to identify the sole contributor to improve upset recovery capability, they did conclude that the use of simulators, both ground based and in-flight, were effective in improving pilots’ upset recovery capabilities. Chapter 1. Introduction 3

According to a recent Notice of Proposed Rule Making (NPRM) from the United States Federal Aviation Administration (FAA) [17], ground-based flight simulator stall recovery training will become a requirement for civil aviation pilots. The use of ground-based sim- ulators is being proposed as they are safe, inexpensive, accessible, and have reached a high level of technological maturity. The FAA NPRM states: “The simulator must include aerodynamic modeling for high angle of attack maneuvers to at least ten degrees beyond the stall angle of attack or as required to execute a recovery from a fully stalled flight condition”. Unfortunately the flight models in most current flight training simulators are unlikely to meet this modeling requirement. A lack of sufficient aerodynamic mod- els that cover the post-stall region limits the current ground-based simulators to only perform upset prevention training instead of upset recovery training, where pilots are trained to avoid impending stall rather to recover. To date, simulators were not required to cover the stall regime, and thus real-time models for post-stall were not required. Typically wind tunnel testing is conducted for the normal flight envelope at angles of attack up to and just beyond stall. Characteristics in sideslip are usually measured up to the angle of attack for stall warning activation and out to sideslip angles representative of crosswind landings [1]. Limited data are acquired at angles of attack significantly beyond stall primarily because the focus of the testing is configuration development for the purpose of predicting performance and certification characteristics. Typically when a simulation database is derived from the wind tunnel data, it is common practice to implement a database that is a rectangular function of angle of attack and sideslip near the α and β axis [1], leaving regions of both large α and β unfilled, resulting in regions of extrapolated or estimated data where direct wind tunnel data are not available. Loss of control accidents have been known to achieve flight conditions far beyond the normal flight envelope and well beyond stall conditions where aerodynamic characteristics are available. Therefore to reach this goal of enhanced simulator training into the post-stall regime, better modeling of the aircraft aerodynamics in the stall and post-stall regime is required.

1.3 Post Stall Modeling

Due to the complex flow around an aircraft, the full airflow cannot be determined analyt- ically and solved in real time. Instead, aerodynamic models take data from pre-generated lookup tables to determine forces and moments at the given aircraft state. These values, stored in their dimensionless form, are collected from experimental wind tunnel data, flight tests, or CFD analysis. The size of the database collected is generally tolerable Chapter 1. Introduction 4 for nominal flight conditions. However, the database size grows significantly with the increase of angle of attack and sideslip angle due to non-linearity and coupling of param- eters. The vast database required to simulate stall using the conventional method, along with the associated high cost and time consumption, makes aircraft manufacturers reluc- tant to include simulations into the stall region. Also, the unsteady and highly non-linear behavior results in uncertainty about the accuracy of the model, making manufacturers concerned regarding a possible negative transfer of training. There have been a number of stall and post-stall flight models created for commercial aircraft in the past, but most of these were specific to certain aircraft models [2][1]. These models were built using a large set of expensive wind-tunnel tests and/or flight tests to generate aerodynamic databases covering the full range of flight conditions for a single geometric configuration. It would be prohibitively expensive to enforce this level of modeling for all post-stall simulators, and in addition the typical methodology of enforcing the matching of time histories between flight test and flight models to within a certified level of tolerance be- comes questionable for post-stall modeling when aircraft may become unstable, as the aircraft can exhibit chaotic behavior in stall.

1.4 Project Definition

As part of a collaborative project between the University of Toronto Institute for Aerospace Studies (UTIAS), the United States Federal Aviation Agency (FAA), and Bombardier Aerospace (BA), this thesis aims to develop a methodology for creating a generic repre- sentative aerodynamic model for two class types of aircraft: twin engine T-tailed turbo- prop transports, and rear mounted engine, T-tailed regional jet transports. The models should be tunable based on aircraft configuration details such as wing geometry (size, aspect ratio, and sweep) or tail position, without depending on a large amount of ad- ditional wind tunnel data. The resultant models must be able to represent most if not all of the aircraft’s stall characteristics, at the same time being significantly cheaper to implement both in cost and time. Such representative models would provide pilots with some hands-on experience in stall recovery, at the same time being more accessible than the current expensive extended models. The FAA NPRM explicitly states: “It is recog- nized, however, that strict time-history based evaluation against flight test data may not adequately validate the aerodynamic model in an unstable flight regime, such as stalled flight, particularly in cases where significant deviations are seen in the aircraft’s stability and control. As a result, the objective testing requirements defined in Table A2A do not prescribe strict tolerances on any parameter at angles of attack beyond the stall angle of Chapter 1. Introduction 5 attack. In lieu of mandating objective tolerances to flight test data at angles of attack at and beyond the stall, a Statement of Compliance (SOC) will be required to define the source data and methods used to develop the stall aerodynamic model which incorporates defined stall characteristics as applicable for the simulated aircraft type.” [17] The feasi- bility of using representative models was previously examined by Schroeder et al. [18], and showed insignificant differences in the training benefits of the high fidelity model over the representative model. This approach of allowing for a lower fidelity representa- tive model saves both time and money compared to type specific models, although it is understood that they are not as accurate as specific models.

1.5 Organization

The remainder of this document is organized as follows:

• Chapter 2: A literature review of the stall characteristics, past attempts at post- stall modeling, as well as varying aircraft configuration effects on stall.

• Chapter 3: The basic approach behind post-stall modeling, which includes an overview of the model structure, basic application of the method, as well as intro- duction to configuration ∆ terms.

• Chapter 4: The details in generating the configuration ∆ terms. Two approaches will be discussed: semi-analytical and empirical.

• Chapter 5: A full scale example of how the method is applied.

• Chapter 6: Conclusions and future work Chapter 2

Literature Review

2.1 Existing Post-Stall Model

Despite the prohibitive cost and time associated with generating post-stall aerodynamic models using the conventional method of wind tunnel tests, aircraft specific models have been developed for research use. Wind tunnel studies of aircraft configurations past stall were carried out as far back as World War II. Previous efforts in developing post-stall simulator models had been limited to military use, and it is not until recent years that extensive sets of wind tunnel tests were conducted to model transport aircraft in the post-stall region for flight simulators.

A wind tunnel study of the post stall behaviors of a twin engine transport jet was con- ducted by Shah et al. at the NASA Langley Research Center (LaRC) in 2002 [19]. The data coverage was extremely broad in many aspects, with large angle of attack and sideslip ranges, and control surface deflections both within and beyond the normal range of motion. Conventional static tests were conducted in the 14x22-foot tunnel at a dy- namic pressure of 10lb/ft2, corresponding to a Reynolds Number of 0.54 ∗ 106 based on mean aerodynamic chord. Data were acquired at angles of attack from −30◦ to +90◦, and at angles of sideslip from 45◦ to +45◦. Forced oscillation tests were also conducted to capture dynamic effects. The model was oscillated in a sinusoidal motion over a range of frequencies and amplitudes that corresponded to typical full-scale motions. Oscillation tests were conducted individually in the pitch, roll, and yaw body axes to acquire aerody- namic damping characteristics. Data was also taken in the spin facility where rotations occurred around the velocity vector, such that the data could be used to better replicate aerodynamic forces and moments during spin type motions.

6 Chapter 2. Literature Review 7

Given the extensive dataset compiled from the wind tunnel study, a high fidelity model was generated for the specific type of aircraft studied. The same team later published the post-stall model in 2004 [20], naming it the Enhanced Upset Recovery (EUR) model. The model successfully captures many of the significant stall characteristics such as a loss of lift, the highly non-linear pitching moment, along with many other stall characteristics as seen in Figure 2.1.

(a) CL (b) Cm

Figure 2.1: Enhanced post stall model developed by NASA LaRC [1]

In addition to the work done by NASA LaRC, work has also been done at UTIAS by previous MASc students. Liu made use of the set of data from LaRC along with a public domain post-stall NASA T2 model and developed a post-stall version for the Boeing 747 aircraft [2]. The Boeing 747 is different from the twin engine aircraft examined by LaRC in many aspects. However, due to the scarcity of post stall data for the Boeing 747 air- craft, the twin engine aircraft data had to be used, and numerous alterations had to be carried out to the data before fitting it to the existing pre-stall Boeing 747 model. Scaling and blending were carried out to transition between the Boeing 747 pre-stall model and the post-stall portion of the new model. An example is shown in Figure 2.2. Using these blending methods, the existing Boeing 747 model was successfully extended to angle of attack of 85◦, and sideslip angle of +/ − 45◦. Chapter 2. Literature Review 8

Figure 2.2: Blending method used on merging post stall model onto existing pre-stall model [2]

The importance of post stall modeling had also been recognized by organizations in Eu- rope. The SUPRA research project - Simulation of Upset Recovery in Aviation - was funded by the European Union 7th Framework Program to enhance the flight simula- tion envelope for upset recovery simulation. Multiple aspects of stall modelling were tackled. CFD tools were used to complement the wind tunnel data. In addition to the wind tunnel and CFD data based approach, the aerodynamic model included differen- tial equations to capture hysteresis and other unsteady effects that occur during stall [21].

However, despite the accomplishments by NASA, SUPRA, and UTIAS in developing high fidelity post-stall models, this methodology is not readily applicable for commercial use. These types of models depend on extensive wind tunnel data or CFD data, which is extremely expensive and time consuming to collect. Thus this method is impractical for commercial aircraft manufactures to generate for every aircraft available. An alternative solution to this problem lies in the feasibility of representative models. Representative models do not have the rigid requirement of exactly matching the time history of the aircraft to within predefined tolerances. Instead, these types of models would capture the basic trend and behaviors in the post-stall region. By relaxing the fidelity requirements, this type of model can be generated more easily and hence at a lower cost.

The feasibility of representative post-stall modeling has been examined in the past. Don- aldson et al. [22] built a representative model of a P-8A aircraft and found that some Chapter 2. Literature Review 9 useful training could be accomplished in full-flight simulators using representative mod- els. More recently, Gingras et al. [23] developed a representative model of a B737-800. This model and another representative model were subsequently tested by Schroeder et al. [18] in a simulator and were also compared to a type specific model built by Boe- ing [18]. Results from these studies showed that the test pilots’ evaluations of the models varied widely, but that for many measures the evaluations of the models were not statis- tically different. In addition, a quasi-transfer of training study using airline pilots found a wide range in the performance of the pilots trained using the two different models. Many of the results however, showed insignificant difference in the training benefit of the high fidelity model over the representative model [18]. Similar representative models have also been developed at TUDelft [24] and Lockheed Aeronautical Systems Company [25]. However, although both of these tools claims to have predicted stall behavior, the exact method of development was not fully disclosed.

2.2 Semi-Analytical Methods

Due to the relaxation in the accuracy of the representative model, many traditional and relatively simple computational methods can be considered for modeling. In contrast to generating the aerodynamic database with wind tunnel data, potential flow solvers hold promise for generating aerodynamic data. However, these solvers are inviscid in nature; they can only predict aerodynamics in the pre-stall region with good accuracy. To make use of these tools for the post stall regime, modifications must be made. Nevertheless, the relatively simple and direct way that these methods can be applied make them an attractive candidate for representative modeling.

2.2.1 Linear Methods

One of the most widely used potential flow computational methods is the Vortex Lattice Method (VLM). The VLM models lifting surfaces, such as wings of an aircraft as an infinitely thin sheet of discrete vortices. By enforcing the no flow condition through the lifting surface and the Kutta condition they can be used to compute lift and induced drag. The influence of thickness and viscosity is neglected. The vortex lattice methods are built on the theory of potential flow. Potential flow is a simplification of the real flow experienced in nature, where all viscous effects are neglected, and turbulence, dissipation and boundary layers are not considered. Chapter 2. Literature Review 10

The method makes a few assumptions:

• The flow field is incompressible, inviscid, and irrotational.

• The lifting surfaces is thin. The influence of thickness on aerodynamic forces are neglected.

For any given vortex filament, a velocity will be induced at an arbitrary point P, which is governed through the Biot-Savart Law, −→ −→ Γ Z d l × −→r V = (2.1) p 4π |−→r |

The vortex lattice method approach superimposes a finite number of horseshoe vortices of different strength Γ on the plane of the wing surface, as shown in Figure 2.3. Each vortex filament will induce a normal velocity at any arbitrary point on the plane, and the total velocity induced at the point from the horseshoe can be calculated from the superposition of the filaments. By covering the entire wing with a series of horseshoe vortices, the total Γ at point P on the plane can be found through superposition. The central problem of VLM is to solve this relationship between the induced velocity and the circulation such that the sum of the induced velocity and normal component of the freestream velocity is zero, that is, the flow is tangent to the planform surface of the wing.

Conventionally, a wing is divided into panels, each containing a horseshoe vortex. The span-direction side of the lattice (bound vortex) is placed at the 1/4 point of the panel, and a control point (where zero normal velocity will be enforced) is placed at the 3/4 point, as shown in Figure 2.3. At any control point P, the normal velocity induced by all the horseshoe vortices can be obtained from the Biot-Savart law. When flow-tangency condition is applied at all the control points, a system of simultaneous algebraic equation results which can be solved for the unknown Γ’s. Once the distribution of Γ is computed for each lattice, the lift can be obtained through Kutta-Joukowski theorem, which states that,

L = ρV∞Γ (2.2) where L is the aerodynamic force per unit width, and V∞ is the freestream velocity.

To apply VLM to an aircraft, the following procedure is used:

1. Divide the planform up into a lattice of quadrilateral panels, and put a horseshoe vortex on each panel. Chapter 2. Literature Review 11

Figure 2.3: Classical vortex-lattice method

2. Place the bound vortex on the 1/4 chord element line of each panel, as seen on Figure 2.3.

3. Place a control point on the 3/4 chord point of each panel (at the midpoint in the span-wise direction).

4. Determine the strengths of each Γn required to satisfy the boundary conditions by solving a system of linear equations.

5. Calculate the lift at each panel by using the Kutta-Joukowski theorem.

The total lift of the surface is obtained from summing up the lift from each panel. The above description acts as an overview of a classical VLM adaptation. The details of the method are beyond the scope of this thesis, and can be found in [26].

2.2.2 Non-Linear Methods

In order to predict stall effects correctly, non-linear corrections must be applied to the simple potential flow calculations. These extensions to classical potential flow methods are done by applying changes to capture viscous effects during their computation pro- cesses. A number of these corrections have been examined in the past. Chapter 2. Literature Review 12

One of the earliest non-linear methods, by Tani [27] in 1934, is the Iterative Γ-Distribution approach for handling non-linear section lift curves using lifting line theory (LLT). In his technique, a span-wise bound vorticity distribution is first assumed; this distribution is then used to compute the distribution of induced velocities and hence induced angles of attack αi and effective angles of attack αeff along the lifting line. The distribution of αeff is then used to look up the operating CL of the local section using the known nonlinear CL vs. α data for the airfoil. A new distribution is then calculated from the span-wise CL distribution. The iteration is carried out until the distribution converges. This method was applied for analysis of wings up to the onset of stall, that is, until a wing angle of attack at which some section on the wing has CL equal to the local section CLmax . At higher angles of attack where some sections on the wing might have a negative lift-curve slope, this successive-approximation approach appears to have failed.

An entirely different approach to the use of non-linear section data called α-correction approach was developed by Tseng [28] in 1988. Although their primary focus was on vortex-dominated flows on low-aspect-ratio fighter type wings at high α, they incorpo- rated the effect of boundary-layer separation by iteratively reducing the angle of attack at each section of the wing. In their method, the α reduction at any given wing sec- tion is determined by the difference between the potential-flow CL and the viscous CL, with the latter being obtained from the non-linear section CL vs. α curve. In contrast to the iterative Γ-distribution approach, which can be used only in LLT-based methods because of the explicit use of the induced and effective angles of attack in the itera- tion, the α-reduction approach can be incorporated in LLT and VLM. As a result, it can be used on swept, low-aspect-ratio wings, and multiple-wing configurations. More recently, the α-correction approach was used by van Dam et al. [29] for rapid estimation of CLmax and other high-lift characteristics for airplane configurations. In their work, the authors show the effectiveness of using the α-correction approach for wing CLmax prediction with two-dimensional aerodynamic characteristics obtained using CFD com- putations. Additionally, the results of Ref. [29] also demonstrate how the α-correction approach is successful in estimating the CLmax of wings with part-span flaps, particularly when the start of the trailing-vortex wake over the flapped portion is displaced vertically to account for the downward location of the flap trailing edge.

In 2006, Mukherjee [30] developed a decambering approach for predicting post-stall aero- dynamic characteristics of wings using known section data. As angle of attack increases to the stall point, the boundary layer on the upper surface of an airfoil thickens and Chapter 2. Literature Review 13

finally separates. It is this flow separation that causes the viscous CL and Cm to deviate from the potential-flow theory predictions. Mukherjee argues that the effect of separa- tion can be viewed as an effective change in the chordwise camber distribution because of the boundary-layer displacement thickness and separation. In this approach, the ef- fective chordwise camber distribution at each section of the wing is modified to account for the viscous effects at high angles of attack. This approach is similar in concept to the α-correction approach [28] and can be incorporated in LLT and VLM. It differs from the

α-correction approach in its capability to use both the CL and Cm data for the section and in the use of a two-variable function for the decambering. Also, unlike all of the earlier methods, the current approach uses a multidimensional Newton iteration that accounts for the cross-coupling effects between the sections in predicting the decamber- ing for each step in the iteration. In addition, by changing the effective camber on the airfoil, both the modified lift distribution as well as the airfoil pitching moment about the aerodynamic center are captured using a single camber modification.

Despite capturing both lift and moment changes on a stalled airfoil, the computation of the multidimensional Newton iteration in the decambering approach can be demand- ing for large number of spanwise nodes. In addition, the results from the decambering method in Reference [30] seem to deviate past the onset of stall, and validations were only shown for straight wing configurations. Therefore, a modified approach based on the simpler α-correction method is used in this study, and is detailed in Section 4.2. Al-

though the Cm of the airfoil is treated less rigoriously, it is assumed that for the fidelity of the representative model, Cm of the airfoil is small when compared to Cm contribution from the tail or effects due to sweep.

2.3 Stall Aerodynamics

Aircraft aerodynamic stall can be very complex. At stall, air flow over part of the aircraft begins to separate. The turbulent wake caused by the separation greatly reduces the lift and significantly increases drag. Aircraft can exhibit many characteristic behaviors that signify a stall. Examples include the loss in lift (g-break) past stall, a sudden roll-off due to one wing stalling before the other, or a pitch instability due to the horizontal tail blanketed by the wake of the wing. In order to successfully train pilots in stall prevention and/or recovery, models developed for this purpose should capture these stall characteristics in order to properly cue the pilots. ICATEE recommends the following characteristics to be represented in the post-stall flight model [31]: Chapter 2. Literature Review 14

2.3.1 G Break/Pitch Non-Linearity

The G break is a loss of lift observed when an aircraft begins to enter stall. As the flow over the lifting surfaces begins to separate, negative pressure over the top of the surfaces is no longer maintained, and thus the surfaces produce significantly lower lift than they would in attached flow. When the angle of attack is further increased, the separation

increases. This causes a maxima in the lift versus α curve, where CLα becomes negative past the stall point.

Typical static pitch stability in the pre-stall region is given by a linear negative slope of the Cm vs. α curve. Past stall, the pitching moment becomes highly nonlinear. There are many causes that can affect the pitching behavior. For swept wings, the tips of the wing tend to stall before the root, leading to an unstable nose-up moment; the down- wash produced by the wing on the tail could change as a result of separation on the wing, which can lead to a change in effectiveness of the tail; the tail of the aircraft could become blanketed by the wake of the separation, thus thus further perpetuating the loss of pitch stability. The Cm vs. α curve could increase or decrease in slope around stall depending on the exact situation. The tail blanketing issue is especially prominent in T-tail aircraft. Figure 2.4 shows a graphical depiction of the tail blanketing effect.

Figure 2.4: Pitching moment for conventional tailed aircraft versus T-tailed aircraft. Reprinted from [3]

For the high tail configuration at very high angles of attack, the wake of the separated wing will make contact with the horizontal tail. This means at high α, the wake will Chapter 2. Literature Review 15 more strongly disrupt the airflow over the tail, significantly reducing the effectiveness of the tail. Not only will this cause an instability in the aircraft, this also reduces the elevator control authority, making recovery difficult. A more detailed description can be found in Section 2.4.1

2.3.2 Roll-Off

A roll-off can occur when one of the wing stalls before the other, leading to a sudden rolling moment on the aircraft. Since stall is typically chaotic in nature, it is very likely that the stall will occur on the two sides of the wing at different times. The strength of the roll-off is determined by the span-wise location of the separation point on the wing. Separations occurring outboard will cause a much greater rolling moment than separations occurring inboard. In addition, for many turbo-prop aircraft, the engines on either side of the wing spin in the same direction. This will cause an asymmetry in the rolling moment and asymmetrical flow over the wing, which will likely initiate the onset of roll.

2.3.3 Reduced Lateral/Directional Stability

Reduced lateral and directional stability can occur if the vertical tail becomes affected by the separated flow. The vertical tail, fuselage, wing sweep and wing dihedral are often the main contributors to directional stability. At stall conditions, the stabilizing effects of wing sweep and dihedral are diminished or even reversed due to the separated flow over the lifting surfaces. In many cases, roll damping is also reduced or reversed when the wing becomes stalled. Similarly to the pitch break discussion, if the vertical tail becomes blanketed by the wake of the fuselage, it can lose effectiveness, greatly reducing the directional stability. In addition, the asymmetric stalling on the two sides of the wing will cause a differential drag, which further aggravates the problem.

2.3.4 Reduced Control Effectiveness

Similar to the above discussions, the separation and blanketing of control surfaces will greatly reduce their effectiveness. Flight control surfaces on an aircraft require clean flow over them to provide adequate control authority. In some cases, a reversal of the control effects can be observed. In the case of an aileron at the onset of stall, the reduction in angle of attack (or alternatively the change in effective camber) due to up aileron up may un-stall the wing section, which leads to an increase in lift rather than a decrease. Chapter 2. Literature Review 16

2.4 Configuration Effects

The contribution of aircraft components on the corresponding aerodynamic forces and moments are an integral part of this thesis. In order to develop a representative model that can be applied to a variety of aircraft, the stall properties resulting from each air- craft configuration must be understood. These trends will help to decouple the overall stall behavior of any aircraft into the contribution from individual components, aiding in both developing and validating the desired stall model.

Extensive research have been done in predicting aircraft behavior with a given con- figuration for pre-stall conditions in the past. One of such research resulted in the US Air Force development of the USAF Stability and Control Data Compendium (USAF DATCOM), which calculates the static, dynamic and control derivative characteristics of fixed-wing aircraft. For any given geometric configuration and flight condition, a com- plete set of stability and control derivatives can be determined. Despite being a powerful tool to understand and predict aircraft configuration effects, it is limited to pre-stall flight conditions. It would be ultimately desirable to generate a similar compendium for the post-stall region, but this would be an enormous challenge, and is beyond the scope of this thesis. Nevertheless, a simplified study on the effects of configuration changes is discussed in the following sections. Note that the full impact of configuration effects can be very complex, and depends on many variables. For this discussion, only the effects that are modeled in this thesis will be described.

2.4.1 Tail Effects

T-tailed aircraft possess a notorious possibility for pitch instability in the stall region. Due to the position of the high horizontal tail, the aircraft has the potential to enter a phenomenon called “deep stall”, where the wake of the wing, fuselage, and nacelles completely shadows the tail, disabling any restoring pitching moment the tail might provide [32]. The pitching behavior of a swept wing and a straight wing configuration can be different, and the exact shape of the Cm curve can depend on many factors. Generally, the pitching effect of a T-tailed aircraft can be divided into three regions, as shown in Figure 2.5.

• In the early portion of region A, stall has not developed, and Cm remains relatively linear. Towards the end portion of region A, local stall begins to occur on the wing. Chapter 2. Literature Review 17

Figure 2.5: Schematic of Tail Effect on Cm for T-Tail (Data adapted from NASA technical report [4])

In some swept wing aircraft, a steepening of the Cm curve, also known as a pitch break, is artificially designed into the wing by either adding a twist near the root of the wing, or only deploying slats outboard and not inboard. When the root is forced to stall first, the center of lift will be shifted backward, causing a nose down moment, as shown in region A of Figure 2.5. In some straight wing aircraft, this pitch break can be caused by a reduction of lift due to stall initiating from inboard of the wing. This will shift the lift center slightly forward which leads to a pitch up moment, but more importantly the inboard separation will reduce the downwash on the tail and even introduce upwash, which can increase the efficiency of the tail, leading to a nose down moment. However, in many other cases, such as a swept wing configuration without artificial twists, the pitch break is not observed or is even reversed to be nose up. For this configuration, the tips of the wing usually stall first, shifting the lift center forward and causing a nose up moment, which will transition from the linear portion directly into region B.

• In region B, the wake begins to interact with the tail, causing it to lose effectiveness, leading to a “pitch bucket”. For swept wings without artificial twist, the forward shift of the lift center also contributes to the loss of pitch stability. In addition, when the tips of the wing are stalled, the tip vortices shift further inboard and becomes stronger, which increases the downwash on the tail, further contributing to a nose up moment. Towards the later part of this region, the wing will become fully Chapter 2. Literature Review 18

stalled, and no downwash will encounter the tail. This will naturally restore pitch stability, however the tail will also be partially shadowed by the wing wake around this angle of attack, and the effect of diminishing downwash will be overpowered by the wake shadowing, resulting in loss of pitch stability.

• Finally, at even higher angles of attack, the tail is completely engulfed in the wake, and thus pitch stability is completely lost. At extremely high angles of attack when the wake clears the top of the tail, pitch stability is then restored. As the tail is shifted forward or upward, the angle in which the wake hits the tail is increased. This leads to a deeper pitch bucket in region B and a worst instability in region C. On the contrary, if the tail is shifted backward or downward, the angle in which the wake hits the tail is decreased. This leads to a shallower pitch bucket, and a milder instability at full wake immersion.

Aside from the tail position, the tail incidence angle and tail size can also play a major role in the pitching behavior of the aircraft. Altering both tail size and incidence angle changes the effectiveness of the tail. Changing the tail size will both offset the Cm curve as well as alter its slope, while changing the tail incidence angle will only add an offset

to the Cm curve, and is often used to help trim of the aircraft. These effects can be seen from wind tunnel data in Figure 2.6 and 2.7.

Figure 2.6: Effect of horizontal tail size on Figure 2.7: Effect of horizontal tail inci- Cm. Adapted from Ray [4] dence angle on Cm. Adapted from Ray [4] Chapter 2. Literature Review 19

2.4.2 Nacelle Effects

In the cases of engine nacelles mounted at the rear of the aircraft, the shadowing effect of the wake coming off the nacelle can be significant. Similar to the tail shadowing discussion, the wake coming off the nacelles can influence the flow over the tail at high angles of attack, reducing their effectiveness. As shown in Figure 2.8, the nacelle wake can result in a strong instability in the pitching behavior of the aircraft, depending on the position of the nacelle relative to the tail. It is argued in a study by Ray et al. that the tail shadowing effect due to nacelles are stronger than that of the wing and fuselage [4]. A more forward mounted nacelle will produce a wake that shadows the tail at earlier angles of attack, but the effect will be milder, as indicated in the left most graph; a more aft mounted nacelle will produce a wake that shadows the tail at higher angles of attack, however the effect will be much stronger, as indicated in the center graph. However, if the nacelle is moved considerably backward, then the wake will almost never make contact with the tail, thus the instability is not observed.

Figure 2.8: Nacelle Location effect on Cm(α), dashed lines are nacelle off condition (Reprinted from Ray [4])

2.4.3 Wing Effects

A wing can vary in multiple ways, including sweep, taper ratio, aspect ratio, dihedral, and it can have other added surfaces such as flaps, slats, or other wing dressings. Varying each of these can lead to significant changes in aircraft stall behavior, as can be found in any aircraft design reference book. In this section, discussions will be limited to what is directly used in this thesis. As shown in the wind tunnel data in Figure 2.9, increasing Chapter 2. Literature Review 20 the wing sweep has a marked effect on the stall angle of attack and also tends to shift the Cm curve up as the angle of attack approaches stall. When a swept wing is stalled, the tips of the wing tend to stall first, thus losing lift at a more aft location, leading to a pitch up moment. The higher the sweep, the more dominant this effect will become.

Figure 2.9: Sweep effect on Cm(α) (Reprinted from NASA technical report [4])

Extending the flaps and slats will also change the way stall occurs on a wing. By changing the effective camber on the wing, the stall point angle of attack can change. When flaps are extended, the CL vs. α curve is shifted up and left, resulting in a higher CLmax but shifts the stall point to a lower angle of attack. Extending the slats will delay stall by shifting the CLmax to a higher angle of attack. This will be examined in more detail in Section 4.2.4. Not only is the variation in the wing itself important, the attachment of the wing onto a fuselage is also important to roll stability. A sideslip leads to airflow wrapping around the fuselage, causing a local increase of α on one side of the fuselage and a decrease on the other. When the wing is at the top of the body, the angle of attack distribution is such as to produce a negative rolling moment, and thus introduces a virtual dihedral. Conversely, when the airplane has a low wing, a virtual anhedral is introduced by the fuselage interference. These needs to be considered when modeling the wing under non-zero sideslip. Chapter 2. Literature Review 21

2.4.4 Fuselage Effects

The fuselage can have a major impact on the pitch behavior of the aircraft. Only lon- gitudinal coefficients will be discussed in this section. Three fuselage parameters are considered in this document: length, cross-sectional size and shape. From wind tunnel data in Figure 2.10[4], it can be seen that a larger forebody creates a destabilizing effect on the pitch behavior of the aircraft. Increasing the length of the forebody also creates a destabilizing effect, as shown in wind tunnel data in Figure 2.11[4]. Since the fuselage itself contribute to pitch instability, having a larger cross-section or longer fuselage will exaggerate the pitch instability. As noted in Reference [4], the change in forebody shape did not seem to alter the pitching behavior significantly.

Figure 2.10: Effect of fuselage shape and Figure 2.11: Effect of fuselage length on size on Cm. Adapted from Ray [4] Cm. Adapted from Ray [4]

2.4.5 Engine/Power Effects

The engine and propellers can affect the behavior of aircraft in a number of ways. First, the thrust setting directly effects the pitching moment of the aircraft through the thrust acting about an axis that is offset from the center of mass. Second, the effect of engine power is reflected through the accelerated air flow in the slipstream directly aft of the spinning propellers, increasing the dynamic pressure in the slipstream region. When the wing is directly behind the propellers, this results in a much higher lift with higher power settings, while keeping flow attached out to a higher angle of attack locally within the slipstream. This can significantly increase the lift provided by the wing for large diameter propellers, which can be seen in Figure 2.12. This slipstream can also eventually reach the tail, affecting the dynamic pressure at the tail. Chapter 2. Literature Review 22

Figure 2.12: Wind tunnel data of CT effect on a straight wing turboprop aircraft

In addition, twin engine turbo-prop aircraft that have both propellers spinning in the same direction can lead to an asymmetric slipstream roll-up that affects the aft fuselage and tail, which can be hard to predict. Lastly, at non-zero angles of attack, the rising propeller blade will see a different blade pitch angle than the descending blade, resulting in a differential lift and drag on the two sides of a propeller [33]. This asymmetric blade pitch effect will lead to a net yawing moment at non-zero α’s, and a net pitching moment at non-zero β’s from the differential lift, and a normal force from the differential drag. The calculations for the blade asymmetric effects can be found in Section 4.2.5.

2.5 Published Wind Tunnel Data

There exists a substantial amount of post-stall wind tunnel studies for various aircraft configurations in the published literature. The majority of these studies are targeted at examining a particular stall behavior, such as the effect of sweep on CL in stall, and by no means constitutes a comprehensive set of data that can be implemented into a flight model. It should be noted that in many stall conditions, the aerodynamics in wind tunnels are known to behave differently than the aerodynamics in flight tests due to Reynolds number and/or Mach number effects. Nevertheless, wind tunnel data provides an excellent starting point for determining the behavior of an aircraft in stall. In this work the data will either be used for trend interpolation, and/or for methodology vali- dation purposes. Focus is placed on finding available data that can be used to determine Chapter 2. Literature Review 23 how specific aircraft parameters, e.g. wing sweep, tail height etc., effect the aerodynamic force and moment coefficients near and past stall for T-tailed jets and turboprop aircraft. Tables 2.1 and 2.2 summarize the available wind tunnel data found in the literature. It should be noted that red cells represent parameters that are thought to have a strong effect on the aerodynamic coefficient(s), yellow cells represent modest effects and unhigh- lighted cells represent relatively weak effects. From these tables it can be seen that not all important contributions are covered by available wind tunnel data, and also that not all contributions deemed important are covered by the available wind tunnel data. Chapter 2. Literature Review 24 δe ˙ α q D C α, β 5 , 7 7 4 46 47 δe ˙ α q 46 L C α, β 3 , 5 7 35 5 , 35 7 , 36 37 34 , 38 4 , 39 – 42 4 , 34 38 43 46 4 , 38 43 3 , 47 δe 44 44 ˙ α q 46 m C α, β 3 – 5 , 7 34 7 , 36 37 7 , 36 37 4 , 39 40 4 , 32 34 38 43 44 4 , 38 45 46 4 4 , 38 43 44 4 , 45 3 , 47 Sweep Taper AR Position Size Flaps/Slats Position Size Position Size Forebody Aftbody Section Length Position Size Power Position Blade Longitudinal Coefficients Refs Wing Horizontal Tail Vertical Tail Fuselage Shape Nacelle Propeller Table 2.1: References toimportance public for post-stall post-stall wind tunnel modelling; data yellow with indicates configuration modest variations importance. (longitudinal). Red cells indicate high Chapter 2. Literature Review 25 δr δa r 46 p 49 49 46 n C α, β 34 , 50 34 40 34 , 43 52 46 52 43 47 , 53 δr δa r 46 p 49 49 51 46 l C α, β 34 , 48 8 , 34 34 , 43 52 46 48 52 43 47 , 53 54 δr δa r 46 p 34 34 46 Y C α, β 34 8 , 34 40 34 , 43 52 46 52 43 47 Sweep Taper AR Position Size Flaps/Slats Position Size Position Size Forebody Aftbody Section Length Position Size Power Position Blade Lateral Coefficients Refs Wing Horizontal Tail Vertical Tail Fuselage Shape Nacelle Propeller Table 2.2: Referencesimportance to for public post-stall modelling; post-stall yellow wind indicates modest tunnel importance. data with configuration variations (lateral). Red cells indicate high Chapter 3

Basic Methodology

The development of a generic representative stall model is rather different from the development of an aircraft specific model. Given the past examples of generating aircraft specific post-stall models, it would be beneficial to generate a class of representative post- stall models by using an existing aircraft specific post-stall model as a baseline. One way to capture the stall behavior of a target aircraft is to employ the use of configuration add-on delta (∆) terms that are added onto the existing post-stall model of the baseline aircraft. These ∆’s capture changes in the aerodynamic coefficients due to geometry differences between the baseline and target aircraft of interest. For this method to work well the baseline and target aircraft must be similar and therefore the ∆ terms should be relatively small. Since different classes of aircraft differ significantly in their post-stall behaviors, configuration ∆ terms will be constrained only to aircraft within the same class, which puts less demand on the calculation of these ∆s. It should be noted that this method requires a similar basic stall behavior between the baseline and target aircraft, such as similar stall onset locations on the wing, and no differences in special treatments such as artificial vortex generators. As part of the overall collaborative project, an aircraft specific model for a twin engine regional jet as well as a twin engine turboprop transport aircraft are to be developed, and will be assumed to be available. The methodology described in this thesis will focus on finding a way to generate the configuration delta (∆) terms, as well as incorporating them into the certified pre-stall models.

3.1 Model Structure

The aerodynamic model or aerodynamic database of a flight model typically consists of

look-up tables and/or equations for each of the six aerodynamic coefficients: CL, CD,

CY , Cm, Cl, and Cn, representing the forces in 3 directions and the moments in the

26 Chapter 3. Basic Methodology 27

3 directions. While the exact structure for each aerodynamic coefficient may vary, the model structure for certified pre-stall models are often relatively simple. This reduces the amount of wind tunnel test down to a practical level. In the most simplified case, such as small perturbations while flying near cruise, stability derivatives can often be used. This means significantly smaller number of tests are required to generate the stability deriva- tives and the aerodynamic coefficients can be obtained by multiplying the derivatives by the aerodynamic states. However, even in standard flight simulator models this level of simplicity is not sufficient, and the flight models will require many more additional terms. Nevertheless, in pre-stall flight models, the dynamic effects are often represented as stability derivatives that are a function of α and β. In the stalled region, dynamic effects are no longer linear with the angular rates, and thus full-dimensional look-up tables will be required to capture the coefficients as a function of all of the variables of interest. Non-linear coupling of effects such as these greatly increases the amount of experimental tests required to fill the look-up tables for a post-stall flight model, thus making generation of post-stall models difficult.

In a conventional aerodynamic model, each coefficient can generally be considered as a sum of four effects: the basic static aerodynamic effects (ST), the effects due to changes in control inputs (CTL), the dynamic effects (DYN), and unsteady effects (UNS), as shown in equation below.

˙ Ci = Ci,ST (α, β, δf , M, ...)+∆Ci,CT L(α, β, δ)+∆Ci,DY N (α, p, q, r, ω, ...)+∆Ci,UNS(α, ˙ β, ...) (3.1) Given a baseline model with the above model structure, each configuration change found to have an impact will have a ∆ term appended to the model to capture the effect of the varying geometry on the corresponding aerodynamic coefficient. The predicted representative model for the desired aircraft will result from summing up the ∆ effects and adding them to the baseline model. After adding the ∆ terms, the model takes the form of, config config X X Ci =Ci,ST + ∆Ci,ST j + Ci,CT L + ∆Ci,CT Lk j k (3.2) config config X X + Ci,DY N + ∆Ci,DY N m + Ci,UNS + ∆Ci,UNSn m n Thus, the key component in arriving at a generic model with the above structure lies in finding the corresponding configuration ∆ terms. The unsteady ∆ terms will be more difficult to predict due to its time dependency, and will not be included in this thesis. Chapter 3. Basic Methodology 28

3.2 Modeling Approach

A representative post-stall model of a Target (T) aircraft will be developed by extending an existing pre-stall model. Extension of an existing pre-stall flight model to a Target aircraft is accomplished by adjusting a Baseline (B) post-stall aerodynamic model of a similar aircraft (within the same class) based on the differences in aircraft parameters. Note that this method requires a post-stall model of a Baseline aircraft. By only extend- ing models within a given class of aircraft, the parameter range for the extension should be relatively small. In addition, as the model will be used for stall recovery training, control should be relatively good and thus the extension needs only go into the near post-stall region (the FAA NPRM only dictates 10◦ past-stall). These two facts com- bined with the requirement that the models be only “representative of the aircraft type” suggest that the methods used for extending the Baseline model to the Target aircraft can be relatively crude. In most of the discussions in the later sections, the post-stall model is assumed to be accurate up to α of 25◦ and β’s up to +/- 10◦.

To better illustrate the process of morphing a baseline model into a different aircraft, the following nomenclature is defined:,

jk Ci where i = L, D, Y, l, m, n and j = B - Baseline aircraft T - Target aircraft and k = P - existing Pre-stall model E - Estimated stall model M - Transformed E to match P S - Stall model, blended P and M

The basic methodology for extending a pre-stall model of a Target (T) aircraft is as follows:

1. Create ∆’s in aerodynamic coefficients relative to Baseline coefficients as a function of ∆’s in aircraft parameters. These ∆’s will be created based on a combination of wind-tunnel data, semi-analytic methods, and parameter estimation from flight test data. The calculation of these ∆’s are discussed in more detail in Chapter 4.

2. Extend pre-stall model of Target aircraft to post-stall regime. The aerodynamic Chapter 3. Basic Methodology 29

T S coefficients are designated as Ci .

(a) Given ∆’s in parameters between Baseline and Target aircraft, sum up aero- dynamic coefficient ∆’s and add to Baseline model to obtain Estimated Target T E stall model, Ci . See Figure 3.1 for a hypothetical example.

M T E X BS Ci = (∆Ci)j + Ci (3.3) j=1

T E (b) Stretch and offset estimated Target stall model, Ci , to match Target pre- T P T M stall model, Ci , in pre-stall regime resulting model is designated Ci , see Figure 3.1 for an example. (c) Blend the Target pre-stall model with the Stretched and offset estimated stall model to create the total Target model, see Figure 3.2 for an example.

T S T P Region 1 α < α1 Ci = Ci (3.4) T S T P T M α − α1 Region 2 α1 < α < α2 Ci = (1 − K)Ci + KCi where K = (3.5) α2 − α1 ¯ T S T E Region 3 α > α2 Ci = Ci (3.6)

(a) Cm (b) Cmq

Figure 3.1: Example Pitch Moment Coefficients for Target pre-stall (T P ), Target es- timated pre-stall/stall/post-stall (T E), and Target Stretched/offset pre-stall/stall/post- stall ( T M ) Chapter 3. Basic Methodology 30

(a) Cm (b) Cmq

Figure 3.2: Example Pitch Moment Coefficients for Target pre-stall (T P ), Target Stretched/offset stall (T M ) and Target stall model (T S). Blending occurs between two dashed vertical lines 3.3 Configuration Deltas (∆)

The key to constructing a valid representative model lies in finding accurate configu- ration ∆’s to capture the changes between the baseline and target aircraft. Once the configuration ∆’s that link the baseline aircraft to the target aircraft are found, then the baseline model can be readily transformed into the model of the target aircraft. An intuitive approach is to include a ∆ term for every configuration change systematically (tail height, tail longitudinal position, nacelle location, etc.). However, the number of

Longidutinal Lateral Static Dynamic Static Dynamic Wing X NA X X Horizontal Tail X O NA NA Vertical Tail NA NA O O Nacelle X NA O NA Fuselage X NA O NA Engine/Power X NA X NA X = Covered in this thesis O = Discussed in a separate thesis within the same project NA = Not examined

Table 3.1: Overview of geometric changes and which ones are covered in this thesis Chapter 3. Basic Methodology 31 geometric features on an aircraft is vast, thus to capture every single change would re- quire a huge number of ∆ terms, and additionally not all configuration changes affect the aircraft behavior significantly. Some geometric effects are orders of magnitude smaller than others, and thus for representative modeling these should be ignored. In addition, it is found that the aerodynamic effects from many geometric changes in an aircraft are not independent. The full set of configuration effects examined are listed in Section 2.4. It should be noted that the configuration ∆’s are intended to capture the trend and patterns seen in the aerodynamic coefficients when varying aircraft geometry. The exact magnitude of the ∆’s are still subject to scaling with real data. The categories of geometric changes captured using configuration δ’s in this thesis are shown in Table 3.1. Chapter 4

Generation of Configuration ∆’s

4.1 Overview

An integral part of generating the representative stall model is to predict the desired ∆ terms that capture the changes between the baseline and target aircraft. Ideally it would be desirable to find a way to directly calculate the effect of each and every geometry change on the aerodynamic coefficients. However, this cannot be done in simple ways due to the complex flow conditions in a stall that lead to strong interactions between geometric changes. To overcome this, two simpler approaches are considered. Some components of an aircraft have relatively small influences on each other. For these cases, the variation in geometry on these components can be examined independently from the rest of the aircraft. For example, the lift of an aircraft is mostly produced by the main wing. Thus when estimating total lift of the aircraft, only the effect of varying the main wing is studied, without much consideration to the rest of the aircraft. For estimation of these effects, simpler computational methods can be used to estimate the geometry change. These methods are referred to as the semi-analytical approach. However, in many other cases the aerodynamic interaction is very significant. For example, the signature pitching behavior of a T-tailed aircraft is caused mainly by the downwash and wake of the wing and nacelle interacting with the tail, and not the effect of the wing and tail alone. In addition, some components of the aircraft such as the fuselage can be hard to model directly. In these cases, an empirical approach is used, where the trend and behaviors of varying geometry is extracted from empirical wind tunnel and flight test data. The following sections will discuss the details of generating a number of ∆ terms using both approaches. It should be noted that ideally in order to have a full collection of ∆’s needed to generate a stall model, extensive wind tunnel data examining configuration parametric changes are required. However for this thesis, performing wind

32 Chapter 4. Generation of Configuration ∆’s 33 tunnel tests was not possible, and application of the methods for generating the ∆’s was limited to coefficients where sufficient public data was available. This lack of data also means assumptions had to be made, such that configuration changes which had minor effects on the overall behavior of the aircraft were neglected.

4.2 Semi-Analytical Approach

The semi-analytical approach attempts to model the post-stall aerodynamic coefficients using modifications to simple Vortex Lattice Methods (VLM) to predict post stall be- havior. Due to the nature of these simple computational methods, this approach only approximates the effects of varying the wing configurations where cross-component inter- ferences such as wake shadowing are not present. This approach relies on the main stall characteristics of the wing to be the same between the two aircraft. If, for example, the target aircraft wing stalls inboard and the baseline aircraft stalls outboard (at the tips) this method will likely be inaccurate for some of the aerodynamic coefficients. However, the method is able to provide estimates without any empirical wind tunnel data for the target aircraft. It should be noted that this approach is very suitable for predicting the wing effects, as the wing can have multiple variations such as sweep, taper, aspect ratio and dihedral. Through the semi-analytical approach, all of the geometric changes on the wing can be estimated in unison, and thus one ∆ term can capture the entire wing change. This approach allows for a cheap and fast prediction for post-stall region that is based on physics and does not rely on empirical data.

The current approach first uses the AVL [55] software package to implement conven- tional VLM computations and provide linear inviscid estimates of the lift profile of a desired wing. In a typical VLM, the lifting surface is divided into several span-wise and chord-wise lattices. Through linear VLM calculations, the span-wise lift distribution for a given wing geometry (sweep, taper, AR) can be obtained for a range of α. For a general wing configuration, the span-wise lift distribution is shown in Figure 4.1 for a range of angles of attack. Note that in the conventional VLM solution, stall behaviors are not captured, and thus the magnitude of lift can go unrealistically high at high α’s.

For post-stall conditions, the inviscid flow approximation cannot be used. To account for viscosity, a correction is made to the raw VLM predictions. This approach makes use of the viscous Cl vs. α curve of the 2-D airfoil, which is either assumed to be known from wind tunnel studies, or can be estimated using XFOIL [56]. In the inviscid VLM compu- Chapter 4. Generation of Configuration ∆’s 34

Figure 4.1: Spanwise lift distribution for angle of attack ranging from 0◦ to 25◦ at 5◦ interval

tations, all airfoils are assumed to have linear lift curves, with a constant Clα (often 2π). With increasing angle of attack, the boundary layer on the upper surface of an airfoil separates, which causes the viscous Cl to deviate from the inviscid potential flow theory predictions. It is desired to first map the local Cl of each span station to the 2-D lift distribution, then eliminate the deviation due to viscosity in the 2-D case, before finally updating the change back in the 3-D span-wise lift. One crude approach is to directly reduce the lift from the inviscid 2-D lift curve to the viscous 2-D lift curve. Figure 4.2 shows a graphical schematic overview of the correction process. A typical span-wise lift distribution on a wing is shown as the blue curve on the right. To correct for this, each local span node Cl on the computed span-wise lift is mapped onto the corresponding linear 2-D airfoil Cl vs. α curve, shown as the blue curve on the left plot, which the VLM solver originally used to obtain the span-wise lift distribution. Next, each point on the linear inviscid 2-D curve is adjusted to the realistic viscous airfoil lift curve. For pre-stall regions, no changes will be made. However, for regions at and past stall, local lift is significantly reduced. This will then be reflected in the corrected span-wise lift.

Although this crude approach accounts for the viscous effects locally, each span section is treated independently. When one portion of the wing loses lift, the resultant vor- tices decrease, thus affecting the surrounding nodes. This effect is not captured in the crude approach. Rather than directly reducing the lift at every cross-section, a modified α-correction method is used in this thesis to compute the span-wise lift distribution, similar to the decambering approach mentioned in Section 2.2.2[30]. The deviations be- Chapter 4. Generation of Configuration ∆’s 35

Figure 4.2: Viscous remapping procedure for a generic swept wing using a generic cam- bered airfoil tween the inviscid and viscous lift curves can be approximated as an effective change in the angle of attack on the airfoil (de-α), which can be carried out by adding a local twist to the wing. If the effective α is taken into account, then the potential flow prediction will more closely match the viscous CL for high angles of attack. In the 3-D case, chang- ing the effective α on one section is likely to have a significant effect on the neighboring sections. To account for these effects, the entire wing must be solved together, and a N-dimensional Newton iteration is used to predict the de-α parameter at each of the N sections. The problem can be set up as follows:

J · δx = −F (4.1)

where x is a vector containing the de-α parameter αde,i to be applied onto the wing. F is the residual vector with,

Fi = (Cl)viscous,i − (Cl)potential,i (4.2)

The Jacobian J is a matrix of the effect of perturbing each span section on the effect of the entire wing, which can be calculated as,

∂Fi Ji,j = (4.3) ∂αde,j

Equation 4.1 is repeated iteratively to reduce the residual to zero. For each iteration step, F and J are determined, and δx is computed using this equation. The iteration procedure can be summarized using the following steps. An example will be given following the description of the methods. Chapter 4. Generation of Configuration ∆’s 36

1. Assume starting values of the de-α variable. Initially the αde,i are chosen to be zeros, indicating a clean/unmodified wing.

2. Using VLM compute for the span-wise lift distribution for the given wing with the current set of de-α parameters.

3. Compute the effective angle of attack of each span section, denoted by αeff corre-

sponding to the section CL. This is obtained by,

αeff = (Cl)/a0 − αde (4.4)

where the Cl is the local Cl of the current wing predicted by VLM, and a0 is the

Clα of the 2-D airfoil.

4. Locate the Cl on the viscous lift curve at the αeff computed in step 3. The residual

Fi is then computed from the difference between Cl of the viscous curve at αeff

and Cl directly from VLM predictions.

5. Compute the Jacobian using finite differences.

6. Compute δx using Equation 4.1.

7. Update the de-α parameter by adding δx to the total αde’s. Repeat the entire process until F has converged to within a specified tolerance.

To show the de-α process more clearly, consider Figure 4.3. Starting with an unmod- ified wing, use the VLM to compute the span-wise lift along the wing. Each local Cl can be plotted on the 2-D inviscid lift curve which the VLM tool used for the original calculation. This is denoted as point V LM1 in red. Next, the effective α is computed.

In this diagram, the (Cl)/a0 is denoted as αtot,1. For the first iteration, no de-α is done on the wing, and thus αeff,1 = αtot,1. This results in the residual being the difference between the V LM1 prediction on the inviscid curve and the viscous curve at αeff . This is denoted with F1 in red. Once the residual is found, compute the Jacobian and solve for the new de-α parameter. The computed de-α is aimed at reducing the residual. For the next iteration, the wing is twisted by the de-α parameter calculated in the previous iteration. However, due to the coupling of each span section, the Cl from the second execution of VLM will not be exactly equal to the target Cl from the de-α, shown in the diagram as a slight difference between the target V LM1,T and the Cl calculated from the second iteration of VLM, shown as V LM2 in blue. Once again, compute the αeff,2.

This time, since de-α is no longer zero, it will not coincide with αtot,2. Next, locate the Chapter 4. Generation of Configuration ∆’s 37

corresponding Cl on the viscous curve at αeff,2 and the residual will be the difference between this Cl and V LM2. This process will be repeated until the residual converges.

Figure 4.3: Schematic of the de-α process

Once the span-wise lift is corrected through iteration, the overall 3-D aerodynamic coef-

ficients of the wing can be obtained by integrating the local Cl along the wing. Figure

4.4 shows the CL prediction of a straight wing after subsequent iterations, and Figure 4.5 shows a comparison between the predicted CL between the crude direct remap approach and the α-correction approach. It is seen that the results from the latter predicted the stall onset and post-stall behaviors more accurately.

In addition to the 2-D viscous effects, it is known that 3-D viscous effects greatly change how a wing behaves post stall. One of the major effects on a 3-D wing is inclusion of span-wise flow as well as the increased downwash towards the root of the wing. Harper

et. al [5] found that for swept and tapered wings, there is a gradient of Clmax where the root of the wing stalls significantly later than the tip, as shown in Figure 4.6. The root

of the wing can often reach a significantly higher Clmax than the Clmax of the 2-D airfoil,

thus making the root less prone to stalling. In some cases, Clmax on the tip can drop

below the Clmax of a 2-D airfoil. This effect is implemented into the method by applying Chapter 4. Generation of Configuration ∆’s 38

Figure 4.5: Comparison between α- Figure 4.4: Prediction of CL after every iteration of de-α correction method and crude direct remap method

Figure 4.6: 3-D effect on the spanwise lift Figure 4.7: Example of spanwise lift after distribution. Reprinted from [5] correcting for 3-D effect a linear gradient to the 2-D lift curve during remapping, where

∗ CL3D (α , y) = G(y)CL2D (α) (4.5) where α∗ = G(y)α (4.6) y where G(y) = (G − G ) + G (4.7) L 0 L L where y is the distance from the root of the wing and L is the semi-span. The 2-D Cl(α) curve is increased towards the root and will be GLCl(α) at the root and G0Cl(α) at the tip, resulting in Figure 4.7. This modification is added onto the basic α-correction method, and this method will be used in the remainder of this thesis. Chapter 4. Generation of Configuration ∆’s 39

For simple wing geometries (uniform airfoil selection, no wing dressing, no flaps/slats), results obtained from this method can be reasonably accurate. The entire estimation process for each wing configuration is straightforward, except for the determination of the 2-D lift-curve slope and the selection of GL and G0. These two parameters are de- termined by fitting the pre-stall lift and pitching moment to match existing wind tunnel data. This therefore requires an estimate to the 3-D CL and Cm for the wing-body of the target aircraft in the pre-stall regime.

To be consistent with the model structure, rather than directly use the prediction of the target aircraft, a model prediction for both the baseline aircraft and the target air- craft are made, and a ∆ term is then obtained from the difference between the two predictions. First, this is done such that a ∆ term can be generated and readily added onto the model structure proposed earlier. Second, any systematic errors in modeling will show up in both predictions, and thus the difference between the two will best rep- resent the effect of the configuration change itself. In the following sections that utilize the Semi-Analytical approach, it is assumed that if the direct modeling of each wing configuration is accurate, then the ∆ term will also be accurate.

This approach will be applied to estimate the lift, pitching moment, and rolling moment of the wing, with and without flaps, as well as estimating engine/propeller slipstream effects. The exact application of the basic approach differs for each coefficient, and will be discussed in details in the subsequent sections. For each of the subsections, two types of validations will be considered where possible. One comparison is done with a straight wing as seen on most turboprop transport aircraft. The second comparison is done with a swept wing as seen on most regional jet aircraft.

4.2.1 Coefficient of Lift CL The most apparent feature of a stalled aircraft is the loss of lift. For swept wings, the boundary layer grows toward the tips of the wing due to span-wise flow, which leads to regions with higher likelihood of separation near the tips of the wing. Thus, flow sepa- ration is generally seen to initiate from the tip portion of the wing and spreads inward towards the root. The lift on a wing is affected by the sweep, taper, aspect ratio, dihe- dral, as well as other artificial dressings on the wing such as vortilons, vortex generators or fences. Due to the complexity in modeling a wing with dressings, only variations of the clean wing is discussed here. Chapter 4. Generation of Configuration ∆’s 40

Using the α-correction approach discussed above, the span-wise lift distribution repre- senting a stalled wing can be generated for a range of α’s. To obtain the overall lift of the wing, the corrected span-wise lift distribution is integrated for all angles of attack to obtain the total CL for the wing. It is important to note that the span-wise lift distribu- tion produced by AVL is in the form of local Cl. Therefore, the simple sum of the local

Cl will not result in the total CL of the wing. Instead, the local Cl must be converted to lift force, and summed up, before converting back into CL. It is known that,

L C = (4.8) L 1 2 2 ρv S

And along the wing,

Z b 1 2 L = ρv Cl(y)dy (4.9) 2 0 Z b and S = clocal(y)dy (4.10) 0

where Cl,Span(y) is the local Cl along the wing and clocal is the local chord, then,

R b C (y)c (y)dy C = 0 l,Span local (4.11) L R b 0 clocal(y)dy

To check the accuracy of the α-correction method in predicting CL for different wing configurations, both a straight wing and a swept wing are modeled using the above method. In a study by Ostawari [6], the wind tunnel data of 2-D NACA 4415 airfoil as well as a finite aspect ratio straight wing are presented. The details of the wing being modeled can be found in Table 4.1.

Span AR c¯ cr S Sweep Taper 2.745 9 0.305 0.305 0.837 0 1

Table 4.1: Details of the wing modeled from Ostawari [6]

The predicted CL and the wind tunnel data are presented in Figure 4.8. The 2-D airfoil used to make the viscous correction is shown in Figure 4.9. The prediction matches the wind tunnel data very well. The resultant span-wise lift after the correction can be seen in Figure 4.10. The shape of the lift distribution makes physical sense, as the lift near the tip is reduced the most. Chapter 4. Generation of Configuration ∆’s 41

Figure 4.8: Comparison of de-α method vs. wind tunnel data using wing modeled from Ostawari [6]

Figure 4.10: Comparison between the po- Figure 4.9: 2-D lift curve of NACA 4415 tential flow and corrected span-wise lift airfoil distribution

For the validation of the swept wing, the wing configuration is chosen from wind tunnel data from Shortal et. al. [7]. The details of the wing modeled are shown in Table 4.2.

Span AR c¯ cr S Sweep Taper 30 6 5.18 6.667 150 30 0.5

Table 4.2: Details of the wing modeled from Shortal [7]

In this study, the airfoil used is NACA 0024. Although the airfoil used is listed, the 2-D Chapter 4. Generation of Configuration ∆’s 42

lift curve of the airfoil is not available. Thus, some artificial tuning is done on a similar airfoil NACA 0012, such that the airfoil lift data used resembles closely to a realistic

airfoil lift curve. This artificially generated 2-D Cl curve is shown in Figure 4.11. Using

this airfoil data, the 3-D CL of the wing is predicted and shown in Figure 4.12. Once

again, the prediction of the wing CL matches the wind tunnel results very closely.

Figure 4.11: 2-D lift curve of the artificially Figure 4.12: Comparison of wing configu- generated NACA 0024 airfoil ration from Shortal [7] with wind tunnel data

4.2.2 Coefficient of Pitching Moment Cm For a T-tailed aircraft, the pitching behavior is often considered the most characteristic stall feature. For highly swept wings, the pitching moment from the wing itself can be significant. Since separation on swept wings usually occurs near the tip of the wings which is further towards the rear, any reduction in lift at those locations will contribute to a net pitch up moment of the aircraft. The greater the sweep, the more dominant this effect will be. Straight wings are less affected by this phenomenon, since there is a smaller longitudinal position offset between the tip and the root of the wing. The

method for predicting Cm is similar to the CL prediction described above. Using the same corrections done to the span-wise lift, instead of integrating the lift distribution for the total lift, the pitching moment is calculated using,

R b C (y)c (y)d (y)dy C = 0 l,Span local m.c. (4.12) m R b 0 clocal(y)dy

where dm.c.(y) is the distance in the longitudinal direction from each span station to the moment center of the wing, normalized byc ¯. Chapter 4. Generation of Configuration ∆’s 43

Using the de-α method, the Cm of the swept wing from the above section is predicted and shown in Figure 4.13. The Cm of the straight wing is not significant, and thus not included in this thesis. In addition to the pitching moment contributed by the shifting of lift center on a stalled wing, the Cm of the airfoil itself had to be added into the result.

Since the Cm of the airfoil is not available, the Cm curve of a similar airfoil had to be used, and is shown in Figure 4.14. The total prediction of the pitching moment from the wing matches very well after adding in the Cm of the airfoil. It captures reversal in pitch stability once the wing has stalled, which is a key feature during stalling of swept wings.

Figure 4.13: Comparison of wing configu- Figure 4.14: 2-D pitching moment curve of ration from Shortal [7] with wind tunnel the artificially generated NACA 0024 air- data foil

4.2.3 Coefficient of Rolling Moment Cφ due to β

To distinguish the rolling moment from the local lift, Cφ is used in this section when referring to rolling moment. The rolling moment of an aircraft is mainly from the main wing due to the long moment arm, and is caused by an asymmetrical lift distribution on the two sides of the wing. Although asymmetric flow over the two sides of the wings mostly occurs at non-zero β, at high angles of attack, differences in lift can occur even without sideslip. This Cφ0 can be caused by one side of the wing stalling before the other, or in the case of a turboprop, the propeller slipstream may create an asymmetric flow over the wing. This behavior cannot be predicted using the α-correction approach, and is not discussed here. An aircraft is said to be statically stable in roll if the Cφβ is negative. This means that at positive sideslip angle, the port side wing will need to have a higher lift than the starboard side to have positive stability. Chapter 4. Generation of Configuration ∆’s 44

First the span-wise lift distribution is computed for a range of non-zero β’s. At non-zero sideslip angles, the lift profile on each wing will no longer be symmetrical. An example of the span-wise lift distribution at non-zero β is shown in Figure 4.15.

Figure 4.15: Spanwise lift distribution for straight wing at α = 15◦ β = −10◦

The span-wise lift is then corrected using the de-α. The resultant rolling moment can be calculated simply by summing up the moment contribution from the local lift on each section about the center axis of the wing, as follows:

R b C (y)c (y) y dy C = 0 l,Span local b (4.13) φ R b 0 clocal(y)dy

y where b is the normalized distance along the span at each span station.

From the predicted span-wise lift, at low angles of attack, the port side wing starts with a higher lift than the starboard side. However, as the wing goes higher in angle of attack and approaches stall, the port side stalls first, reducing the lift. Eventually, the port side wing will have completely stalled, while the starboard wing just begins to stall. This will cause the port side wing to have a lower lift than the starboard side, causing a reversal in the rolling moment and a loss of lateral stability. This is demonstrated in Figure 4.16, where the lift distribution obtained from the estimation is presented for low, medium, and high angles of attack.

In a study by Ray [8], the lateral wind tunnel data of a wing-body configuration was examined. Since the rolling moment of a fuselage is negligible, validation of the rolling prediction of the wing is done against this wing-body data. The details of the wing model are listed in Table 4.3. Chapter 4. Generation of Configuration ∆’s 45

Figure 4.16: Comparison of the spanwise lift of wings under sideslip for higher α

Span AR c¯ cr S Sweep Taper 117.86cm 7.8 16.99cm 24.13cm 0.178m2 28 0.253

Table 4.3: Details of the wing modeled from Ray [8]

Figure 4.17: Comparison of Cr vs. β for Figure 4.18: Comparison of Crβ for wing wing configuration from Ray [8] configuration from Ray [8]

The Cφ is computed for a range of β between -4 to 16 degrees. It is assumed that the behaviors will be mirrored over the β = 0 axis. Figure 4.17 shows the Cφ vs. β for a few angles of attack. The Cφβ can be calculated from the slope of the Cφ vs. β plots. Since the wing rolling moment data is compared against the wing-body wind tunnel data, some Chapter 4. Generation of Configuration ∆’s 46 corrections need to be made. As discussed in Section 2.4, near the junction between the wing and body, the airflow is bent, creating a virtual dihedral effect. This will lead to an offset on the Cφβ . This effect was subtracted from the wind tunnel data by matching the Cφβ at zero α, then compared with the predicted Cφβ for post-stall angles of attack, shown in Figure 4.18. The predicted Cφβ captures the loss of static roll stability after the wings have stalled.

4.2.4 Flap Effects

The wing flap extension also has a strong effect on the aerodynamic forces and moments of the wing. The wing cross-section with flaps extended can be viewed as a change in the effective airfoil. In general, when flaps are deployed, the CLmax will be increased, while stall will occur at a lower angle of attack. In a study done by Olson [57], the change in the 2-D lift curve of an airfoil with flaps extended can be calculated from the airfoil and flap geometry. He showed that the CL vs. α curve can be captured using three 0 parameters: ∆f CLmax , ∆f CL0 , and CLα , defined in Figure 4.19.

Figure 4.19: Definition of the three parameters. Plot shows two-dimensional trailing-edge flap effects Chapter 4. Generation of Configuration ∆’s 47

These parameters can be calculated using,

∆f CL0 = ηδαδCLα δf (4.14) (C0 ) Lα δf >0 c0 cf 2 (C ) = c (1 − c0 sin (δf )) (4.15) Lα δf =0 ! (C ) + ∆ C0 0 Lmax δf =0 f L0 cL = min (4.16) max 0.533 ∆y ( Rec )0.08 + 1 (C + ∆ C0 ) c 3∗106 2 L0 f L0

Using these calculated parameters, the effective 2-D airfoil lift curve with flaps extended can be obtained. The method for estimating the CL and Cm of the overall wing with flaps is then modified, where the viscous remapping process would map the clean sections of the wing to the 2-D lift curve of the clean airfoil, and the sections of the wing with flaps to the 2-D lift curve of the effective airfoil. Due to the limitation of available wind tunnel data, only a model of the flap extension of a straight wing is made. Using the geometry of the flap and the base airfoil, the modified lift curve is calculated for the wing of the turboprop aircraft. Figure 4.20 shows the 2-D lift curve of the clean airfoil versus the lift curve of a flap 35 configuration.

Figure 4.20: 2-D lift curve of clean airfoil versus predicted flap 35 configuration

Using these two 2-D airfoil lift curves and applying the viscous correction method, the

3-D wing CL and Cm can be predicted for both flap 0 and flap 35 conditions. A ∆ term can be computed using the difference between the two predictions. This ∆ term is then added onto the wind tunnel curve of the flap 0 configuration. This is shown in Figure 4.21 (a) and (b), where both graphs shows the comparison of adding the computed ∆ term onto the flap 0 configuration. Chapter 4. Generation of Configuration ∆’s 48

(a) CL (b) Cm

Figure 4.21: Comparison of predicted CL and Cm against wind tunnel data for flap 35

Examining the CL plot, it can be seen that the model for the flap effect of a 2-D airfoil works well in predicting the flap effect of the overall wing, as the dotted curve matches the flap 35 configuration wind tunnel data. Note that the comparison is made between the model of the wing against the wind tunnel data of wing-body. Although the fuselage does not provide much lift, wing-body interactions can contribute to the mismatch of the flap 35 comparison around stall.

The same issue of comparing the modeled wing against wing-body wind tunnel data also occurs when comparing the Cm. However, the pitching behavior of the wing and wing-body is significantly different. To allow for comparison, the Cm curves of the wing with no flap and flap 35 are generated. Then, a difference ∆ is obtained between the two predictions. This ∆ term is then added onto the wind tunnel data for the no flap condition and compared against the wind tunnel data of the flap 35 condition. The re- sults shows that the effect of an extended flap on the lift and pitching moment can be estimated correctly. The same method can be applied onto the lateral coefficient using the method discussed in this section along with the method discussed in the Cr section. However, the results are not shown due to a lack of validation data.

4.2.5 Engine Power Effects CT The propulsion system of an aircraft can also greatly affect the aerodynamics, particularly for turbo-prop aircraft where the slipstream from the propellers flows directly over the wings and/or tail. The slipstream of the high velocity flow exiting the engine or propeller Chapter 4. Generation of Configuration ∆’s 49

disk increases the dynamic pressure of the flow impacting the aircraft. Wind-tunnel data

for turbo-prop aircraft with large diameter propellers show large changes in CLmax with

CT where at high values of CT the CLmax can be double or even triple that of CT=0 case. This additional lift due to thrust is modeled by scaling up the local lift at the sections of the wing directly behind the propeller, thus simulating the effect of higher dynamic pressure. The scale factor will depend on many parameters such as air speed, propeller specifications, engine position, and even fuselage proximity. Due to the complexity in modeling the scale factor directly, it will be determined empirically by examining the pre-stall data which is assumed to be known. Firstly, the span-wise lift distribution of the wing is generated and corrected using the same method as the previous sections. To account for the accelerated flow, the local lift is scaled up in the span-stations that the slipstream would encounter. This will produce a modified span-wise lift distribution as shown in Figure 4.22.

Figure 4.22: Augment spanwise lift distribution to capture accelerated flow

The total CL of the wing is then calculated by integrating the lift distribution using

Equation 4.11 as done for CL. The results for comparing high and low CT ’s are shown in Figure 4.23. Reasonable predictions are achieved from this approach. The ∆ term can be obtained by subtracting the two. This validation is only done for the turboprop

configuration, since it is effected so strongly by CT . Rear mounted turbofan engines like the ones on regional jets would not effect the lift as significantly as the flow does not pass over the main wing. Chapter 4. Generation of Configuration ∆’s 50

Figure 4.23: Comparing predicted CT effects against wind tunnel data for high and low CT

In addition to the lift effects caused by the propeller or turbo-fan slipstream, propellers have a blade asymmetry effect, where at non-zero angles of attack, the propeller blade on the upward sweep will see a different blade pitch angle than the downward sweep, resulting in a differential thrust on the two sides of the propeller. This asymmetric blade pitch effect will lead to a net yawing moment at non-zero α’s, and a net pitching moment at non-zero β’s. This effect is often referred to as the P-factor, and is discussed by McCormick in the book Aerodynamics Aeronautics and Flight Mechanics [54]. Also, the differential drag from the two sides of the propeller will cause an in-plane normal force. The normal force and the moment can be captured using the Equations 4.17 and 4.18,

σqA ¯ aJ π 2 π PN = 2 {CL + 2π ln[1 + ( J ) ] + J Cd}α (4.17) −σqAR 2π ¯ α J 2 π 2 π NP = 2 { 3J CL + 2 [1 − ( π ) ln(1 + [ J ] )] − J Cd}α (4.18)

where PN is the lateral induced force, NP is the induced yawing moment, σ is the blade area / disc area, and J is the advance ratio. Similar equations can be derived for the pitch moment and vertical force for a propeller at sideslip angles. Unfortunately there was no data available to validate these equations. Chapter 4. Generation of Configuration ∆’s 51

4.2.6 Dynamic Effects Clp and Clr In this section, the effects of p and r on the rolling moment are presented. As the wing undergoes a non-zero roll rate p, the side of the descending wing sees a higher effective angle of attack than the wing on the ascending side. This effect is proportional to the distance from the pivot center, as the induced vertical airspeed is proportional to the radius of pivot. This can be seen in the schematic in Figure 4.24. At very high rolling rates, the local angle of attack on the tip can reach a negative value. A similar effect is seen with non-zero yaw rates. A yaw rate will swing one side of the wing forward into the incoming flow and swing the other side away. At non-zero α’s, the backward-pivoting side will reach a higher local α than the forward-pivoting side. Likewise, the yaw rate effect is also proportional to the radius from the pivot center.

Figure 4.24: A schematic showing the two side of the wing at non-zero rolling rate

To model the Clp and Clr effects, a linear gradient of angle of attack is imposed on the computed span-wise lift distribution. This will cause the span-wise lift to skew to one side, as shown in Figure 4.25. The gradient is proportional to the roll and yaw rate, and is adjusted by an empirical factor determined by matching the pre-stall data of the

target aircraft. The Cl are then calculated by summing up the span-wise lift multiplied by the moment arm at every given α and angular rate.

For low α’s, the Cl vs. p and Cl vs. r plots are relatively linear. However, as the angle of attack is increased past stall, the side of the wing with the higher effective α will go further into stall, while the side of the wing with lower effective α will be less stalled, possibly unstalled. This phenomenon will cause Clp to become non-linear, and even un- stable. An example of a span-wise lift distribution of a wing past stall under non-zero rolling rate is shown in Figure 4.26. In this case the left side of the wing has a higher local angle of attack than the right side, however since the wing has stalled already, the Chapter 4. Generation of Configuration ∆’s 52

Figure 4.25: Spanwise lift distribution un- Figure 4.26: Spanwise lift distribution of der non-zero p or r an already stalled wing under non-zero p

Figure 4.27: Predicted C for a straight lp Figure 4.28: Predicted Clr for a straight wing wing increase in angle of attack actually decreases lift, which would cause the aircraft to fur- ther roll to the left (i.e. be unstable).

A straight wing is modeled for a range ofp ˆ = -0.1 to 0.1, andr ˆ = -0.1 to 0.1. The plots are shown in Figure 4.27 and 4.28. It can be seen that near stall angles of attack ◦ ◦ (around 13 ), both the Clp and Clr predictions shows non-linearity. At around α = 15 , the Clp begins to become unstable. Unfortunately these results could not be validated against wind tunnel data directly, as none was available at the time of this thesis. How- ever, the predictions shows similar trends to previous post-stall models [2]. Chapter 4. Generation of Configuration ∆’s 53

4.3 Empirical Approach

Another approach to generating the configuration ∆ terms is using an empirical wind tunnel data interpolation approach. The post-stall aerodynamic behavior of an aircraft can be significantly affected by the interactions between components such as the wing and tail. For the majority of post-stall behaviors that are too complicated to capture through analytical methods, wind tunnel data trends are used. Numerous wind tunnel studies have been done in the past exploring the effects of varying different components of the aircraft such as tail position, fuselage size or wing configuration, etc. Wind tunnel data from such publicly available studies was collected, and configuration ∆’s were empir- ically generated from the data. The method could also be applied using CFD generated aerodynamic results. Note that these methods will rely on the stall characteristics of the target aircraft being similar to the baseline aircraft. If stall on the main wing occurs at a different location between the target and baseline aircraft these predictions could be significantly in error. Practically if this method was applied and the resulting model was flown by test pilots it would be noted as deficient resulting in more detailed analysis of the corrections being made.

To generalize from the parametric studies, a look-up table is created for each config- uration change, where the change in aerodynamic coefficients resulting from geometric changes can be extracted. These tables are presented in graphs, which will be referred to as look-up graphs in some parts of this thesis. The look-up graphs contain the effect of varying each configuration, and the ∆ terms can be obtained from the difference between the baseline and target aircraft data interpolated on the graph, rather than modeling them directly. Once these look-up graphs are determined, they can be used for any aircraft within the class. It should be noted that in order for this approach to work, sufficient empirical data which examines the effect of varying aircraft configuration must be available.

The exact method for generating these look-up graphs varies for each configuration. Some can be extracted easily, while others need further analysis. The general scheme of this method can be captured in the following steps:

1. Analyze the stall behaviors and identify the significant independent parameter(s)

2. Isolate the aerodynamic effects caused by the significant independent parameters if necessary Chapter 4. Generation of Configuration ∆’s 54

3. Insert tunable spline points based on the significant independent parameters

4. Interpolate/Extrapolate using the significant independent parameters

These will be discussed in the subsequent section using examples to help explain each step.

4.3.1 Effects of Nacelle-Tail Interaction on Pitching Moment

The first step in the approach involves identifying the significant parameter(s) that effect the aforementioned behavior the most. For a T-Tailed aircraft, rather than considering the effect of moving the tail up/down or forward/backward, Anemaa [32] described the use of the angle between the wing and the horizontal tail as well as the angle between the nacelle and the tail (tail shadowing angle) to capture the effect of the relative positions of the tail, nacelle, and wing. These effects have been previously discussed in the literature review section. In this discussion, the effect of varying the tail moment arm when moving the tail forward/backward is not considered, and therefore the change in wake shadowing angle will be done by shifting the tail up/down. In a study by Ray [4], the nacelle shadow- ing effect on the tail was found to be much more significant than the wing shadow. In this section, the empirical approach will be described using the nacelle-tail interaction as an example. The wing-tail shadowing effect will be discussed in the subsequent Section 4.3.2.

Once the wake shadowing parameter has been identified, the second step involves finding the interaction between the nacelle and the tail. This is isolated from the empirical data using the simple assumption of:

(AB) = A + B + A ⊗ B (4.19)

where (AB) is the overall effect with the two components on, A, B are the effect of the components alone, and A ⊗ B is the component interaction. This applies to all other types of component interaction effects. Applying this to three components yields,

(ABC) = A + B + C + A ⊗ B + A ⊗ C + B ⊗ C + A ⊗ B ⊗ C (4.20)

Given sufficient data these equations can be used to find the effect of individual parts of aircraft as well as their interactions. Unfortunately, data for individual components are not always available, and often isolations have to be done on compound effects. Using the above additive assumption, the isolation of a single contribution can be done through Chapter 4. Generation of Configuration ∆’s 55 a recursive application of the aforementioned process.

In a number of wind tunnel studies [4][38] pitching moment data was found for aircraft with different combinations of fuselages (F), wings (W), vertical tails (V), horizontal tails (H) and nacelles (N). In this discussion it is desired to find the wake shadowing effect on the tail, namely the interaction effect between the tail (VH) and the nacelle (N). The fuselage and the wing can be viewed as a single wing-body (FW) entity, and the vertical and horizontal tail can be combined into a single tail term (VH). Thus, substituting in the aircraft components into Equation 4.19 and 4.20 with A = FW , B = VH and C = N,

(FW )(VH)N = FW + VH + N + FW ⊗ VH + FW ⊗ N + N ⊗ VH + FW ⊗ VH ⊗ N (4.21) (FW )(VH) = FW + VH + FW ⊗ VH (4.22) (FW )N = FW + N + FW ⊗ N (4.23)

To identify the nacelle-tail interaction, the equations above can be combined and the interaction term N ⊗ VH can be isolated,

N ⊗ VH = (FW )(VH)N − (FW )(VH) − (FW )N + FW − FW ⊗ VH ⊗ N (4.24)

According to Ray [4], the nacelle shadowing effect is much greater than the wing and body contribution. The contribution to pitching moment due to FW ⊗ VH ⊗ N is assumed relatively small compared to the nacelle-tail shadowing, and thus is omitted. Given wind tunnel data for the complete aircraft (FWVHN), the nacelle-off (FWVH), the tail-off configuration (FWN) and just the wing-body (FW), the nacelle-tail interaction can be determined. An example of this breakdown is shown in Figure 4.29, the wake shadowing effect of the nacelle onto the tail is isolated and shown by the purple line.

This process is done on multiple tail height conditions, and the isolated tail shadowing effects are plotted in Figure 4.30. The isolated nacelle-tail interaction shows the expected trend: the nacelle and tail do not interact pre-stall and the interference only occurs when the angle of attack is high enough such that the wake from the nacelle shadows the tail, causing unstable pitch behavior.

The next step involves putting in tuning points in the curve to best capture the curve features. These points should have enough degrees of freedom to transition between the plots of different tail shadowing angle configurations. The raw data are then splined Chapter 4. Generation of Configuration ∆’s 56

Figure 4.29: Isolation of nacelle-tail inter- Figure 4.30: N ⊗ VH effect for various tail action N ⊗ VH shadow angles using these points. The tuning points for the N ⊗ VH curve in Figure 4.29 are shown in Figure 4.31. This is done for as many sets of tail shadowing angles as available from the empirical data in order to proceed with interpolation/extrapolation. In this demon- stration, data for the two other tail shadowing angles are also splined, and are shown in Figure 4.32.

Figure 4.31: Spline points on the interpo- Figure 4.32: Extrapolation of nacelle-tail lation curve interaction from tail shadowing angle

Once the three nacelle-tail interaction curves are isolated and splined, they can be used as a basis to interpolate/extrapolate for the baseline and target aircraft. Both the baseline and target configurations are interpolated/extrapolated within this look-up table. For the nacelle-tail shadowing effect, the look-up table will consist of the blue, green, and red curve in Figure 4.32. As a simple demonstration, the baseline aircraft is chosen to be Chapter 4. Generation of Configuration ∆’s 57

the low shadowing angle curve (blue), and the target aircraft is chosen to have a higher tail shadowing angle, which is extrapolated to be the light dotted blue curve. This is achieved by linearly extrapolating each of the spline points using the shadowing angle

parameter in both α and Cm direction. The progression of the spline points of the same color indicates the direction of extrapolation.

The ∆ term is then generated by subtracting the baseline interaction curve from the target interaction curve in Figure 4.32, in this case the low tail configuration and the extrapolated higher tail configuration. The ∆Cm from the change in tail shadowing in this case is shown in Figure 4.33. A set of look-up tables (graphs) will be generated to capture each configuration change, and the effect of all aircraft transformations of this effect will come from these graphs.

Figure 4.33: ∆ term subtracted from the Figure 4.34: Comparison of wind tun- two interoplated curves nel data between target configuration and baseline + ∆

The final prediction will come from adding the ∆ onto the baseline total aircraft. Figure 4.34 shows a comparison between the empirical data of the target aircraft versus the

baseline + ∆ approach. The final predicted Cm for the change in tail shadow angle is compared against wind tunnel data from Ray. It can be seen that the dotted blue and the green match reasonably well. The small mismatch at the ”pitch bucket” is likely due to the assumption of the FW ⊗ VH ⊗ N interaction being relatively small. It is assumed that the current prediction falls within the accuracy of the ∆ terms, and thus the secondary effect is not examined. However, this can certainly be added to improve result if one wishes and the data exists. Chapter 4. Generation of Configuration ∆’s 58

4.3.2 Effects of Wing-Tail Interaction on Pitching Moment

Despite the nacelle-tail wake shadowing being one of the major contributions to pitch stability, the wing-tail interaction is another important factor in determining the pitching behavior of an aircraft. Using the same concept of tail shadowing angle, the wing-tail interaction can be isolated and extracted using a similar method as above. Assuming the same notation for aircraft components, the desired interaction term is the interaction between the tail (VH) and the wing (W), namely W ⊗ VH. Substituting in the aircraft components into equation 4.19 and 4.20,

FW (VH) = F + W + VH + F ⊗ W + F ⊗ VH + W ⊗ VH + F ⊗ W ⊗ VH(4.25) FW = F + W + F ⊗ W (4.26) F (VH) = F + VH + F ⊗ VH (4.27)

To identify the wing-tail interaction, the equations above can be combined and the in- teraction term W ⊗ VH can be isolated,

W ⊗ VH = FW (VH) − FW − F (VH) + F − F ⊗ W ⊗ VH (4.28)

The contribution to pitching moment due to F ⊗ W ⊗ VH is relatively small compared to the tail shadowing, and is thus omitted. Given wind tunnel data for the complete aircraft (FWVH), the wing-body (FW), the wing-off configuration (FVH) and just the fuselage, the wing-tail interaction can be determined. An example of this breakdown is shown in Figure 4.35.

Figure 4.35: Isolation of wing-tail interac- Figure 4.36: W ⊗VH effect for various tail tion W ⊗ VH shadow angles Chapter 4. Generation of Configuration ∆’s 59

This process is done on multiple tail height conditions (low, medium, high tail height), and the isolated tail shadowing effects are plotted in Figure 4.36. The isolated interaction term shows that the wing does not significantly shadow the tail during stall. Rather, at high angles of attack, the downwash incurred on the tail decreases, which changes the effectiveness of the tail. This process for the wing-tail interaction is the same as the process described in Section 4.3.1.

4.3.3 Effects of Tail Location on Pitching Moment

The previous discussion generated a basis graph for altering the tail shadowing angle when the tail is moved up and down, however it does not taken into consideration the change in tail lever arm when the tail is moved forward/backward. Thus when the tail shadowing angle is changed as a result of forward/backward shift of the tail, an additional correction must be included. Since the shift effectively changes only the lever arm of the horizontal tail, the correction is done by scaling the pitching contribution of the tail based on the change in lever arm. This could be done using analytical approach, however to be consistent with the ∆ approach, the change in lever arm can also be made into a basis graph, where the effect can be looked up. This plot can be generated by taking the lift provided by the tail multiplied by the tail moment arm normalized byc ¯ at a range of angles of attack.

4.3.4 Effects of Tail Size and Incidence on Pitching Moment

The effect of varying the horizontal tail size on the Cm of the overall aircraft is much simpler than the tail shadow angle, assuming the change in tail size does not significantly change the wing-tail interaction. A variation in tail size changes the effectiveness of the tail. A larger horizontal tail will provide more down moment. This can be seen in the look-up graph in Figure 4.37. Rather than combining this effect into a single tail volume metric, the tail size is considered independently due to the potential non-linearity from coupling effects between tail size and tail lever arm in stall. The ∆ term can be obtained through interpolation of the look-up graph.

Similar effects are seen on the tail incidence angle. With a higher incidence angle, the restoring moment on the tail is increased due to higher angles of attack at the tail. The look-up graph for this configuration change is shown in Figure 4.38. The interpolation method for these effects is the same as tail shadowing angle discussion, and the ∆ term can be obtained the same way. Chapter 4. Generation of Configuration ∆’s 60

Figure 4.37: Look-up graph for horizontal Figure 4.38: Look-up graph for horizontal tail size tail incidence angle

4.3.5 Effects of Fuselage on Pitching Moment

The fuselage contains a large area that produces drag and very little lift. It extends far in front of the C.G. and has an unstable contribution to the pitch moment of an aircraft. A longer forebody will increase the pitch up tendency compared to a shorter one. Unlike the wing, the fuselage is harder to compute using semi-analytical means, and thus it is easier to examine using the empirical approach. In the studies by Ray and Kettles [4][38], the pitching behavior of fuselage shape, width, and length are examined, and the Cm plots are shown in Figure 4.39 and 4.40.

Figure 4.39: Look-up graph for fuselage Figure 4.40: Look-up graph for fuselage cross-section size length

In the Section 2.4, it was discussed that the effect of varying fuselage shape did not effect the pitching behavior significantly, and thus no ∆ term is presented. However, it can be seen in Figure 4.39 that varying fuselage cross-sectional size greatly influences the Cm. Chapter 4. Generation of Configuration ∆’s 61

This can be used as a look-up graph between wbody/b = 0.091 and 0.125, where wbody is the average width of the forebody and b is the span. Similarly, the effect of varying fuselage length affects the pitching behavior in the same way. The look-up graph for varying fuselage length between lbody/c¯ of 3.28 and 4.45 is shown in Figure 4.40, where lbody is the length of the forbody. As an example, a ∆ term is obtained by subtracting the two fuselage sizes. Within the same studies by Ray, the complete aircraft pitching moment data are also available for both fuselage sizes. The obtained ∆ term is then added onto the Cm curve of the complete aircraft with the baseline fuselage to compare with the data of the complete aircraft with the target fuselage size. Both nacelle on and nacelle off conditions are examined. The comparison between baseline curve + ∆ term versus the target curve are shown in Figure 4.41. In both cases, it can be seen that majority of the differences between the configuration of large and small fuselage can be captured through the differences between the fuselage themselves.

(a) With Nacelle (b) Without Nacelle

Figure 4.41: Adding the ∆Cm of fuselage onto total aircraft configuration Chapter 5

Full Scale Example

Having obtained the configuration ∆’s that capture the changes between the baseline aircraft and the target aircraft, the final model of the target aircraft can be derived by summing up all the various ∆ terms and adding them to the baseline model. Two aircraft in the class of a twin rear engine, T-tailed regional jet are examined in this section, and the static Cm will be derived for the target aircraft. The baseline aircraft is chosen to be an experimental T-tailed aircraft examined by Ray [4]. A drawing for this configuration is shown in Figure 5.1. The full specification of the aircraft can be found in the original Ray paper. The target aircraft is chosen to be a T-tailed, twin rear engine regional jet from the Bombardier product line. It differs from the baseline aircraft in many geometric features.

Figure 5.1: Specifications for the baseline aircraft. Reprinted from Ray [4]

62 Chapter 5. Full Scale Example 63

The Cm plot for both aircraft is presented in Figure 5.2. The blue curve shows Cm for the baseline aircraft configuration from Ray, and the green curve shows the Cm for the target T-tailed aircraft taken from a flight model that is considered accurate into the post stall regime. Normally this curve would not be accurate beyond stall, but in this case it will be used as the ground truth model that the predicted target will be compared against.

Note that the Cm data for the target aircraft at high angles of attack will be used only as a comparison to the predicted result.

Figure 5.3: C plot of the baseline aircraft Figure 5.2: C plot of the baseline aircraft m m and the target aircraft after correcting for and the target aircraft stall point

The first step in transforming the baseline model into the target model is to match the stall point. To do this, the baseline model is extended in its linear region by an offset obtained from the difference in the location of the CLmax between the two aircraft. This can be seen in Figure 5.3. Next, a list of characteristic configuration differences between the two aircraft are determined. Four major configuration changes were identified:

• Forebody length

• Forebody cross-sectional size

• Tail shadowing angle

• Tail moment arm

Other parameters such as tail size and wing configuration are very similar between the two aircraft and their effects are omitted in this example. Each configuration change is converted to its non-dimensional characteristic parameter and the corresponding look-up tables are used to interpolate through these non-dimensional parameters to find the be- haviors for each of the two aircraft. The details were discussed in Chapter 4. Chapter 5. Full Scale Example 64

For the forebody length, the look-up graph contains data for lbody/c¯ of 3.28 and 4.45,

where lbody is the length of the forebody. From geometry, the ratio is found for the base- line and target aircraft, which is 3.10 and 4.97, respectively. The corresponding curves are extrapolated from the basis graph and are shown in Figure 5.4. Examining the fore-

body cross-sectional size, the basis graph contains data with wbody/b = 0.091 and 0.125,

where wbody is the max width of the forebody. From geometry, the baseline aircraft has

a ratio of wbody/b of 0.125, and the target aircraft has a ratio of 0.103. Both curves with the corresponding fuselage size are interpolated and shown in Figure 5.5.

Figure 5.4: Interpolating/Extrapolating Figure 5.5: Interpolating/Extrapolating for fuselage length ∆ for fuselage cross-sectional size ∆

Next, the effect of tail shadowing angle is examined. In the previous discussion for gen- erating ∆ terms for tail shadowing angle, the basis graph contained data for shadowing angles of 17, 22 and 27 degrees. The baseline aircraft has a shadowing angle of ≈ 17 degrees, and the target aircraft has an angle of ≈ 12 degrees. The basis graph with the extrapolated curves are shown in Figure 5.6. Lastly, due to the change in the tail for-

ward/backward position, the moment arm is different, which leads to a change in Cmα . This is corrected by applying a shift to the moment arm. This effect can also be put into the form of a basis graph, which is shown in Figure 5.7. It is calculated by multiplying the lift of the tail by the moment arm, and is simply a linear plot indicating the equivalent

Cm change due to the offset of the tail.

From these four plots, the ∆ terms resulting from each configuration change can be ob- tained by subtracting the baseline curve from the target curve. The resulting ∆ terms for the four configuration changes mentioned above are gathered and shown in Figure 5.8. To obtain the final prediction, these ∆ terms are added onto the baseline model (after Chapter 5. Full Scale Example 65

Figure 5.6: Interpolating/Extrapolating Figure 5.7: Obtaining ∆ term for tail mo- for tail shadowing angle ∆ ment arm change

Figure 5.8: Four ∆ effects that transforms the baseline aircraft to the target aircraft stretching to match the stall point), and the progression can be seen in Figure 5.9.

Finally, since the pre-stall portion of the target model is usually certified, the final pre- diction is stretched and offset to best match the prestall of the target model. The model resulting from this study will arise from taking the certified pre-stall model and blending with the post-stall of the best prediction, resulting in the graph in Figure 5.10. It can be seen that the prediction in the post stall region matches the target aircraft significantly better than before the transformation. In the post stall region, significantly more pitch up behavior is predicted on the target aircraft just past stall, which matches the behavior of the actual target aircraft. Chapter 5. Full Scale Example 66

(a) Adding ∆ fuselage length (b) Adding ∆ tail lever arm

(c) Adding ∆ fuselage size (d) Adding ∆ tail shadowing

Figure 5.9: Progression of the Cm curve after adding the delta terms

Figure 5.10: Final predicted stall model for the target aircraft Chapter 6

Conclusions

6.1 Summary of Work

Typical flight models for ground-based flight simulators are only designed to be accurate within the aircraft’s normal flight envelope. An aircraft in an upset condition however, can experience states that are far beyond the normal flight envelope. Previous methods for extending flight models into the post-stall regime proved to be costly and impracti- cal. Therefore simpler, less costly models that are representative are being proposed as a means for training pilots in flight simulators.

In this thesis, a methodology for creating a generic representative post-stall flight model the class of both turboprop and regional jet aircraft has been developed. The resultant model will be generic such that it can be applied to a class of aircraft with relatively simple tuning, and yet representative enough to reproduce the important characteristics of a stall. Previous studies suggest that no statistical differences exist between training quality derived from representative models versus aircraft specific models, and thus rep- resentative models can be an attractive candidate for upset recovery training.

The basic method for creating the generic representative model involves using an ex- isting aircraft specific post-stall model as a baseline for each class of aircraft and adding configuration ∆ terms to capture the geometry changes. These configuration ∆ terms will sufficiently represent the transformation from the baseline to the target aircraft. The combined model with all of the ∆ terms added is then stretched and offset to match the pre-stall model of the target aircraft. A method for blending the modified model with the existing pre-stall model has also been discussed.

67 Chapter 6. Conclusions 68

The groundwork of this methodology lies in finding the proper configuration ∆ terms to successfully capture the aircraft changes. Due to the nonlinearities in the post-stall aerodynamics, these aircraft geometric changes cannot be predicted using analytical com- putations. Two approaches are examined in this thesis: a semi-analytical approach, and an empirical approach. The semi-analytical approach uses a modified α-correction ap- proach to examine geometry variation on the wing, wherein a traditional VLM solution is first obtained and then corrected for viscous effects found in the stall condition. This approach can provide a reasonable estimate for wings of varying sweep, taper, aspect ratio, as well as effects of flap extension. For aerodynamic behaviors that are heavily affected by component flow interactions, an empirical approach is used. This approach uses trends in existing wind tunnel studies and interpolates/extrapolates for the behav- ior of the target aircraft. In both approaches the aerodynamic model is predicted for both the baseline aircraft and the target aircraft, and the ∆ terms can be generated by subtracting the two predictions.

The initial derivation and applications of the ∆ terms showed reasonable accuracy in modeling stall behaviors. Prediction of both a straight wing and a swept wing were able to reproduce G-break, pitch nonlinearity, and loss of roll stability; predictions of tail shadowing captured the pitch instability; additional changes such as fuselage size were quantified.

This thesis is one of two theses aimed at developing a methodology for generic rep- resentative post-stall model. The behaviors of the following configuration parameters modeled in this thesis are: • Wing (Sweep, Taper, Aspect Ratio, Flaps, Engine Power)

• Horizontal Tail Position

• Rear Nacelle Position

• Horizontal Tail Size

• Horizontal Tail Incidence Angle

• Fuselage Size

• Fuselage Length The majority of the predictions presented are for longitudinal coefficients. The second of the two theses will cover the lateral effects. Chapter 6. Conclusions 69

6.2 Future Research Needs

In this project, the baseline post-stall model that was assumed as a foundation for the methodology was not complete. As a result the full application of the methodology to construct a complete post-stall model of a target aircraft was not done. In addition, not all of the configuration changes are captured through the limited number of geometry variations examined in this thesis. In order to best capture the behavior of the target aircraft, additional geometry variations need to be examined and similar analysis will need to be done. These additional terms will be required to capture more dynamic effects, both longitudinal and lateral. Furthermore, consideration will have to be devoted in capturing the time-variant unsteady effects associated with stall. Only then will a more complete baseline model be realized. Finally, additional scaling and offsetting will be required to transition the pre-stall of the target model smoothly into the post-stall representative model. References

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