Supersonic Aircraft Optimization for Minimizing Drag and Sonic Boom

Supersonic Aircraft Optimization for Minimizing Drag and Sonic Boom

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF AERONAUTICS AND ASTRONAUTICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Martin K. Chan August 2003

” Copyright by Martin K. Chan 2003 All Rights Reserved

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I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the Doctor of Philosophy.

___________________________ Ilan M. Kroo (Principal Advisor)

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the Doctor of Philosophy.

___________________________ Antony Jameson

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the Doctor of Philosophy.

___________________________ Juan Alonso

Approved for the University Committee on Graduate Studies.

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Abstract A design tool incorporating classical sonic boom theory, computational fluid dynamics and a multi-objective genetic algorithm was developed for low-boom supersonic aircraft optimization. Both sonic boom and drag were optimized simultaneously and a Pareto optimal set of designs ranging from minimum boom to minimum drag was obtained for each optimization. Since sonic boom was optimized directly, the method had broader applicability than the traditional inverse method. A high-order three-dimensional panel method was used for sonic boom prediction. The traditional linear source model was fast but did not account for wing-body aerodynamic interaction. Euler solutions were expensive for computing sonic booms because a large number of grid points were needed to accurately predict the pressure signature away from the aircraft. For the Mach number and configurations of interest, the panel code showed good agreement with Euler but at a fraction of the cost. A loudness metric was shown to have advantages over initial overpressure and peak overpressure for measuring shaped sonic booms. However, optimization results obtained using calculated loudness raised concerns about the robustness of the solution to atmospheric disturbance. Peak overpressure minimization also produced reasonable sonic boom signatures and appeared more robust to atmospheric uncertainties, but the resulting loudness was not as good. Better convergence was also observed with peak overpressure. Two supersonic business jets were optimized. One was a conventional configuration; the other was a natural laminar flow configuration. Optimization results obtained using loudness and peak overpressure were compared. A non-axisymmetric fuselage was optimized and found to reduce the inviscid drag by 9 to 30 percent at the same sonic boom loudness.

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Acknowledgements First and foremost, I would like to acknowledge God’s hand in leading me into pursuing a doctoral degree, and for His faithfulness in seeing it through to completion over these last six years. Truly all that I have and have accomplished in my life, I owe to Him. Funding for my graduate studies was from the Ministry of Defense of the Republic of Singapore, under the Defense Technology Training Award; and also from DSO National Laboratories (Singapore) under the DSO Scholarship program. Many thanks to my advisor, Ilan Kroo, for his enthusiasm and support, and for the optimization codes that helped jump-start this work. His insight on aircraft design and optimization was invaluable, often influencing the direction of my research. I am also grateful to all who contributed to this work: Professors Antony Jameson and Juan Alonso, for the use of their software, and for their helpful comments pertaining to this thesis; Hyoung Seog Chung, for software support and for providing some of the Euler solutions for comparison; the Aircraft Design research group members, for the stimulating discussions, technical as well as non-technical help, and the enjoyable working environment; Peter Sturdza and Joaquim Martins, for taking care of the Linux cluster. I am thankful to my parents and wife’s family for their support and for taking care of business at home in Singapore while I am away. I am also thankful for my daughter, Emily, who has been such a blessing from God. Last but not least, no words can express the gratitude I feel for my wife, Wendy, for her love, support and sacrifice, especially over the last couple of years.

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Contents Chapter 1 Introduction.....................................................................................................1 1.1 Motivation .............................................................................................................1 1.2 Supersonic Air Transportation ...............................................................................1 1.3 Sonic Boom ...........................................................................................................2 1.3.1 Environmental Impact of Sonic Boom.............................................................3 1.3.2 Factors Influencing Sonic Boom Strength .......................................................4 1.3.3 Low Sonic Boom Design ................................................................................6 1.4 Sonic Boom and Aerodynamic Optimization .........................................................7 1.5 Organization of Thesis...........................................................................................7 Chapter 2 Sonic Boom Prediction....................................................................................9 2.1 Historical Review of Sonic Boom Prediction .........................................................9 2.2 Sonic Boom Analysis...........................................................................................10 2.3 Whitham Sonic Boom Theory..............................................................................11 2.4 Computing Equivalent Area Due to Non-axisymmetric Volume and Lift .............14 2.5 Computing the F-function from CFD ...................................................................15 2.6 Boom Propagation Through Non-Uniform Atmosphere .......................................15 2.7 Ray Tracing and Ray Tube Area Calculation .......................................................17 2.8 Non-linear Steepening and Geometric Acoustics..................................................21 2.9 Near-Field Distance and Validity of Whitham Theory .........................................21 2.10 Shock Wave Rise Time......................................................................................24 Chapter 3 Aerodynamics Analysis.................................................................................26 3.1 Linear Source Method .........................................................................................26 3.2 A502 (Panair) 3-D Panel Method.........................................................................27 3.3 FLO87 3-D Euler Flow Solver.............................................................................29 3.4 Comparison Between A502 and Linear Source Method .......................................29 3.5 Comparison Between A502 and FLO87...............................................................31 vi

3.5.1 Sears-Haack Axisymmetric Test Case...........................................................32 3.5.2 Wing-Body Test Cases..................................................................................34 3.5.3 A502 and Euler CFD Near Field Pressure Comparison .................................41 3.6 A502 and Wind Tunnel Comparison....................................................................45 Chapter 4 Low-Boom Aerodynamic Design ..................................................................48 4.1 Introduction .........................................................................................................48 4.2 Seebass-George Sonic Boom Minimization Method ............................................50 4.3 Limitations of Seebass-George Method................................................................57 4.4 Direct Sonic Boom Optimization .........................................................................59 4.5 Advantages of Direct Sonic Boom Optimization..................................................62 Chapter 5 Sonic Boom Optimization .............................................................................64 5.1 Selection of Sonic Boom Metric ..........................................................................64 5.1.1 Initial Shock and Maximum Overpressure.....................................................64 5.1.2 Loudness.......................................................................................................65 5.1.3 Impulse .........................................................................................................71 5.2 Selection of Sonic Boom Objective Function .......................................................72 5.2.1 Minimize Sonic Boom Metric .......................................................................72 5.2.2 Minimize Difference with Target Sonic Boom Signature ..............................73 5.2.3 Sonic Boom as a Constraint ..........................................................................74 5.3 Handling of Constraints .......................................................................................74 5.4 Selection of Optimization Search Method ............................................................75 5.4.1 The Simplex Method.....................................................................................79 5.4.2 Genetic Algorithm ........................................................................................80 5.4.3 Comparison of Simplex and GA Solutions ....................................................80 5.5 Multi-Objective Optimization ..............................................................................81 5.5.1 Multi-Objective GA ......................................................................................82 5.6 GA Population Size .............................................................................................85

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Chapter 6 Integration of Sonic Boom and Aerodynamics with Optimization..................87 6.1 Integrated A502 Sonic Boom Analysis ................................................................87 6.2 MAKEPAN Automated A502 Model Generator ..................................................89 6.2.1 Fuselage Parameters......................................................................................90 6.2.2 Wing Parameters...........................................................................................90 6.2.3 Canard and Tail Parameters...........................................................................91 6.2.4 Nacelle Parameters........................................................................................92 6.2.5 Near Field Signature Orientation...................................................................94 6.3 GA Optimization Using Multiple Processors .......................................................95 6.4 Simplex Optimization Using Multiple Processors ................................................96 Chapter 7 Aircraft Optimization Examples ....................................................................97 7.1 Conventional Configuration Using dBA Loudness...............................................97 7.2 Conventional Design Using Peak Overpressure ................................................. 108 7.3 Natural Laminar Flow Wing-Canard Configuration ........................................... 113 7.4 Optimization of Asymmetric Fuselage ............................................................... 121 7.5 Effect of Shock Rise Time on Results ................................................................ 127 Chapter 8 Conclusions and Future Work ..................................................................... 129 8.1 Conclusions ....................................................................................................... 129 8.2 Future Work ...................................................................................................... 131 References................................................................................................................... 133

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List of Figures Figure 1.1: Production of a N-wave sonic boom. .............................................................3 Figure 1.2: Sonic boom focusing resulting from acceleration...........................................5 Figure 1.3: Sonic boom carpet. ........................................................................................6 Figure 1.4: Examples of shaped booms............................................................................6 Figure 2.1: Linear versus actual characteristic. ................................................................9 Figure 2.2: Nomenclature for Whitham’s Theory...........................................................12 Figure 2.3: Sonic boom prediction of a Mach 1.5 parabolic body of revolution..............13 Figure 2.4: Rear view of aircraft showing azimuth angle f. ...........................................14 Figure 2.5: Rays, ray tube and ray tube area. .................................................................16 Figure 2.6: Predicted sonic boom in uniform and standard atmosphere. .........................17 Figure 2.7: R-z plane containing the ray. .......................................................................19 Figure 2.8: Ray tracing results for from 40,000 ft for Mach 1.5 steady level flight.........21 Figure 2.9: Wing-Body configuration showing pressure distribution..............................22 Figure 2.10: F-function derived from near field pressure distributions. ..........................23 Figure 2.11: Sonic boom on ground from near field pressure distributions.....................23 Figure 2.12: Sonic boom shock rise times......................................................................24 Figure 2.13: Sonic boom modified to include rise time. .................................................25 Figure 3.1: Nomenclature for linear supersonic theory...................................................27 Figure 3.2: Aerodynamic analysis using A502 panel method. ........................................28 Figure 3.3: Location of nacelle for results shown in Figure 3.4. .....................................30 Figure 3.4: Example of wing shielding of nacelle. .........................................................31 Figure 3.5: Sears-Haack body drag comparison (M=1.5). ..............................................33 Figure 3.6: Computed near field pressures for Sears-Haack body...................................34 Figure 3.7: Comparison of the surface mesh/panels. ......................................................36 Figure 3.8: Comparison of computed drag polar. ...........................................................36 Figure 3.9: Comparison of the upper surface Cp. ...........................................................37 Figure 3.10: Comparison of lower surface Cp................................................................37

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Figure 3.11: Comparison of section Cp. Solid line is Euler, symbols are A502. .............38 Figure 3.12: Comparison of the surface mesh/panels for subsonic leading edge wing. .....................................................................................................................39 Figure 3.13: Comparison of computed drag polar (subsonic leading edge wing). ...........39 Figure 3.14: Comparison of the upper surface Cp for subsonic leading edge wing. ........40 Figure 3.15: Comparison of the lower surface Cp for subsonic leading edge wing. ........40 Figure 3.16: Comparison of section Cp for subsonic leading edge wing.........................41 Figure 3.17: Geometry of near field signature test case..................................................42 Figure 3.18: A502 and Euler near field pressure at 0.4 body-lengths below. ..................43 Figure 3.19: A502 and Euler near field pressure at 0.8 body-lengths below. ..................43 Figure 3.20: A502 and Euler near field pressure at 1.2 body-lengths below. ..................44 Figure 3.21: Sonic boom on ground obtained using near field signatures at 0.4 body-lengths below aircraft at 50,000 ft altitude.....................................................44 Figure 3.22: Sonic boom on ground obtained using near field signatures at 1.6 body-lengths below aircraft at 50,000 ft altitude.....................................................45 Figure 3.23: Cp from A502 analysis of Model 1 (NASA TN-D7160). ...........................46 Figure 3.24: Computed near field signature compared with NASA TN-D7160 for Model 1. ................................................................................................................46 Figure 3.25: Comparison of the sonic boom predicted using A502 and wind tunnel data from NASA TN-D7160. ......................................................................47 Figure 4.1: Framework for low-boom design using Seebass-George method. ................49 Figure 4.2: Boom pressure signatures proposed by Seebass and George. .......................50 Figure 4.3: Darden’s F-function modified from Seebass-George....................................51 Figure 4.4: Effective length of aircraft. ..........................................................................52 Figure 4.5: Front and rear shock area balance. ...............................................................55 Figure 4.6: Example of Seebass-George Method. ..........................................................57 Figure 4.7: Supersonic business jet design using a natural laminar flow wing. ...............58 Figure 4.8: Schematic of direct sonic boom optimization...............................................60 Figure 4.9: Comparison between direct optimization and Seebass-George method. .......61 Figure 4.10: Optimized boom differing from the Seebass-George shape booms.............62

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Figure 5.1: Sonic booms with same initial Dp. ...............................................................65 Figure 5.2: Power spectrum of sonic boom. ...................................................................67 Figure 5.3: Unweighted and A-weighted band pressure level (BPL). .............................68 Figure 5.4: Effect of initial Dp and rise time on dBA level for symmetric N-waves (170 ms duration)...................................................................................................69 Figure 5.5: Effect of initial Dp on dBA level for asymmetric N-waves (170 ms duration). ...............................................................................................................69 Figure 5.6: Effect of secondary rise time on dBA level for symmetric shaped booms with initial Dp of 0.3 psf (180 ms duration).................................................70 Figure 5.7: dBA levels for the sonic booms shown in Figure 5.1....................................71 Figure 5.8: Modifications to pressure signature for studying front and rear shock shaping using loudness as an objective...................................................................73 Figure 5.9: Sonic boom matching objective. ..................................................................73 Figure 5.10: Sonic boom as a constraint function...........................................................74 Figure 5.11: Drag versus change in radius at one fuselage station. .................................75 Figure 5.12: Calculated sonic boom loudness versus change in radius at one fuselage station (x).................................................................................................76 Figure 5.13: Change in near field signature due to small perturbation in geometry.........77 Figure 5.14: Shock coalescent at far field signature due to small perturbation in geometry................................................................................................................78 Figure 5.15: Simplex method.........................................................................................79 Figure 5.16: Two-objective optimization problem. ........................................................82 Figure 5.17: Convergence history for one point on the Pareto front using weighted sum method on a single objective GA. ...................................................................83 Figure 5.18: Example showing children should be ranked globally................................84 Figure 5.19: Pareto front after 40 generations for a 12-variable problem with a population of 120...................................................................................................86 Figure 5.20: Convergence history of a single objective GA for a 12-variable problem with a population of 120...........................................................................86 Figure 6.1: Design of an integrated sonic boom analysis for optimization......................88 Figure 6.2: MAKEPAN aircraft geometry components..................................................89 xi

Figure 6.3: Fuselage parameters. ...................................................................................90 Figure 6.4: Wing parameters..........................................................................................91 Figure 6.5: Canard and tail parameters...........................................................................91 Figure 6.6: Nacelle modeled by MAKEPAN. ................................................................92 Figure 6.7: Nacelle modeling options. ...........................................................................93 Figure 6.8: Nacelle parameters. .....................................................................................94 Figure 6.9: Orientation of near field signature................................................................94 Figure 6.10: Parallel genetic algorithm. .........................................................................95 Figure 6.11: Parallel simplex optimization.....................................................................96 Figure 7.1: Conventional supersonic configuration. .......................................................98 Figure 7.2: Layout and design variables for conventional configuration.........................99 Figure 7.3: Evolution of population from start to 60th generation. ................................ 101 Figure 7.4: Pareto front for conventional configuration with loudness objective. ......... 102 Figure 7.5: Solution histories for conventional configuration with loudness objective. ............................................................................................................. 103 Figure 7.6: Normalized lift of population for conventional configuration with loudness objective................................................................................................ 104 Figure 7.7: Pareto optimal fuselage geometry for conventional configuration with loudness objective................................................................................................ 105 Figure 7.8: Pareto optimal angle of attack for conventional configuration with loudness objective................................................................................................ 105 Figure 7.9: Pareto optimal wing for conventional configuration with loudness objective. ............................................................................................................. 106 Figure 7.10: A502 near field pressure signatures for conventional configuration with dBA objective. ............................................................................................. 106 Figure 7.11: Sonic boom signatures on ground for conventional configuration with loudness objective........................................................................................ 107 Figure 7.12: Pareto front for conventional configuration with peak overpressure objective. ............................................................................................................. 109 Figure 7.13: A502 near field signature for conventional configuration with peak overpressure objective ......................................................................................... 110 xii

Figure 7.14: Sonic boom pressure signatures for conventional configuration with peak overpressure objective. ................................................................................ 110 Figure 7.15: Pareto optimal fuselage geometry for conventional configuration with peak overpressure objective. ........................................................................ 111 Figure 7.16: Pareto optimal wing twist for conventional configuration with peak overpressure objective. ........................................................................................ 111 Figure 7.17: Loudness of Pareto optimal solutions in order of increasing overpressure. ....................................................................................................... 112 Figure 7.18: Population resulting from minimizing loudness and peak overpressure. ....................................................................................................... 112 Figure 7.19: NLF wing-canard supersonic configuration. ............................................ 114 Figure 7.20: Design variables for NLF wing-canard configuration............................... 114 Figure 7.21: Pareto front without viscous drag for NLF configuration. ........................ 116 Figure 7.22: Pareto front with viscous drag for conventional and NLF configurations...................................................................................................... 116 Figure 7.23: A502 near field pressures for NLF configuration. .................................... 117 Figure 7.24: Sonic boom signatures for NLF configuration.......................................... 118 Figure 7.25: Comparison between minimum drag and boom designs for NLF configuration. ...................................................................................................... 118 Figure 7.26: Pareto optimal fuselage for NLF wing-canard configuration. ................... 119 Figure 7.27: Pareto optimal wing tip twist and canard root incidence for NLF configuration. ...................................................................................................... 119 Figure 7.28: Pareto optimal canard area for NLF configuration. .................................. 120 Figure 7.29: Pareto optimal angle of attack for NLF configuration. ............................. 120 Figure 7.30: Normalized lift of population for NLF configuration. .............................. 121 Figure 7.31: Design variables for conventional configuration with nonaxisymmetric fuselage. ........................................................................................ 122 Figure 7.32: Non-axisymmetric fuselage definition. .................................................... 122 Figure 7.33: Pareto front for non-circular fuselage case. .............................................. 124 Figure 7.34: Pareto optimal non-axisymmetric fuselage geometry. .............................. 125 Figure 7.35: Pareto optimal axisymmetric fuselage geometry. ..................................... 125 xiii

Figure 7.36: A502 near field pressure signatures for non-axisymmetric and axisymmetric fuselage. ........................................................................................ 126 Figure 7.37: Sonic boom signatures for non-axisymmetric and axisymmetric fuselage. .............................................................................................................. 127

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List of Tables Table 1.1: Examples of sonic boom intensity...................................................................3 Table 5.1: 1/3 Octave Band Center Frequencies for 13 Hz to 22kHz..............................67 Table 7.1: Weight and dimensions of conventional configuration. .................................98 Table 7.2: Weight and dimensions of NLF configuration............................................. 113 Table 7.3: Low-boom dBA loudness adjusted for shock rise time. ............................... 128

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Chapter 1 Introduction

Introduction 1.1 Motivation The goal of this research was to minimize the sonic boom and aerodynamic drag of a supersonic aircraft via aerodynamic shape optimization. The research was motivated by the desire to reduce the sonic boom to a publicly acceptable level so that unrestricted supersonic flights over land will be permitted.

1.2 Supersonic Air Transportation For the last 30 years, the only operational supersonic transport aircraft was the Concorde. However, the crash of Air France Concorde flight 4590 in Paris on 25th July 2000, followed by a series of high profile technical problems and the drop in air travel due to the threat of terrorism eventually led to the decision to retire the aircraft by November 2003. Although hailed a technological marvel, it was also deemed an economic disaster never to be repeated. It carried less than one-fourth the passengers of a 747 for the same amount of fuel; the sonic boom noise eventually ruled out flight over land; it was largely restricted to transatlantic use, since it could not carry enough fuel for longer oceanic missions. The projected global reach of the Concorde dwindled to just New York, London and Paris. Efforts to build a supersonic successor have consistently failed to find a way to overcome huge bills for fuel, maintenance and design. More recently, noise and emissions have also been added to the list of technological hurdles for supersonic transport.

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Since development cost, noise, emissions and sonic boom are related to weight and speed, there is general agreement that the obstacles to developing a viable supersonic transport are easier to overcome for a smaller and slower (Mach 1.5~1.6) aircraft, such as a supersonic business jet (SSBJ). Based on the current trend in the business aviation market, there is a belief that the market could support a supersonic business jet costing up to $80 million (Flight International, 6 May 2003). This belief is based on the success of Cessna’s Citation X, which at Mach 0.92 is the second fastest civil aircraft after the Concorde, despite its relatively small cabin and limited range. Even with sonic boom restrictions, halving transatlantic or transpacific flight times can be highly attractive to some customers. Furthermore the retirement of Concorde may create a niche for supersonic transportation. According to Dassault, who plan to deliver a SSBJ in 10 years, passengers who are accustomed to traveling in the Concorde are now potential customers of a SSBJ (Flight International, 6 May 2003). If unrestricted overland supersonic flight was possible for a low-boom SSBJ, Hartwich1 et al. cited market surveys suggesting that the projected market would double.

1.3 Sonic Boom It is widely known that a sonic boom is heard when an object travels faster than the speed of sound. The classic sonic boom is the N-wave pressure signature illustrated in Figure 1.1. It consists of an initial shock followed by expansion until the rear shock. It is the result of the pressure disturbance caused by an object, in this case an aircraft, moving at supersonic speeds. As the pressure disturbance propagates to the ground, they eventually coalesce into one shock at the front, and another at the rear, producing the N-wave. The N-wave is characterized by its maximum overpressure, rise time and duration. Overpressure is the pressure disturbance relative to the ambient atmospheric pressure. Rise time is the time it takes for the shock to attain its peak overpressure, and duration is defined as the time interval between the positive and negative peak overpressures, as shown in Figure 1.1.

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Near Field

Overpressure, DP Pressure

Mid Field

Ambient Pressure

Rise Time Duration

Far Field Time

Figure 1.1: Production of a N-wave sonic boom. Currently all supersonic aircraft produce N-waves when flying supersonically at cruise altitudes. Examples of sonic booms are given in Table 1.12.

Table 1.1: Examples of sonic boom intensity. Concorde @ Mach 2, 52,000 ft

1.9 psf

SR-71@Mach 3, 80,000 ft

0.9 psf

F-104 @ Mach 1.93, 48,000 ft

0.8 psf

Space Shuttle @ Mach 1.5 60,000 ft

1.3 psf

(landing approach)

1.3.1 Environmental Impact of Sonic Boom According to Ref. 2, no structural damage is expected for a sonic boom overpressure of 1 psf. Significant public reaction can be expected for 1.5 to 2 psf. Rare structural damage may occur for 2-5 psf, though structures in good condition will remain undamaged up to 11 psf. A person’s eardrums will only be harmed at 720 psf. For example, sonic booms of 3

144 psf, resulting from supersonic fly-by from less than 100 ft, have been experienced without injury. Therefore for supersonic flights at typical cruise altitudes, the sonic booms are an annoyance rather than harmful, though some have claimed that the startle effect of sonic boom may result in indirect harm3. For example a surgeon might make a mistake as a result of being startled by the sonic boom. Citing previous studies, Shepherd4 reported that, depending on the number of flights per day, N-wave overpressures ranging from 0.3 to 1 psf were acceptable to more than 90% of those tested. It is not yet clear how wildlife and marine life are affected by sonic booms. Due to considerable variation in hearing ability and how noise influences behavior, the effects of sonic booms on animals vary widely5. Individual animal response also varied widely, due to a number of factors, such as time of day and year, physical condition of the animal, physical environment (such as whether the animal is restrained or unrestrained) and whether or not other physical stressors (e.g. drought) are present. Sonic boom disturbance of the animal's behavior during the reproductive cycle was also suggested to cause lowered reproduction in a variety of animals. Studies have also shown that wildlife quickly adapted to sonic booms in their habitat6.

1.3.2 Factors Influencing Sonic Boom Strength The following are some factors affecting sonic boom strength: •

Aircraft weight

•

Aircraft altitude

•

Aircraft shape and length

•

Aircraft maneuver

•

Location in sonic boom carpet

Aircraft weight has long been known to have significant impact on sonic boom strength. The heavier the aircraft, the greater will be the lift, and greater lift means greater 4

disturbance to the air. Flying at higher altitude will reduce sonic boom since the pressure perturbation from the aircraft varies as square root of the vertical distance from the aircraft. Aircraft shape influences the pressure disturbances around the aircraft, which can have significant affect on sonic boom. A longer aircraft helps reduce sonic boom because spreading the lift along its length helps reduce the pressure disturbances that cause the sonic boom. A maneuvering aircraft creates a louder sonic boom at certain locations. When the aircraft accelerates or turns, the Mach angle changes relative to the ground. Figure 1.2 illustrates how this causes the sonic boom rays to converge, resulting in super booms7. Accelerating aircraft m t2

m t1

Mach cone rays converging

Figure 1.2: Sonic boom focusing resulting from acceleration.

A boom carpet is the area on the ground where the aircraft’s sonic boom can be heard (Figure 1.3). It is approximately 1 mile wide per 1000 ft altitude. Since the disturbance to the air is greatest in the lift direction, the sonic boom is loudest directly under the flight path. Moving sideways across the boom carpet, the sonic boom reduces because of the greater distance traveled and lesser air disturbance in that direction. The boom carpet has finite width because at some point the sonic boom rays do not reach the ground due to refraction. The current research focuses on shaping the aircraft for optimal sonic boom and drag. Hence shape and length is addressed directly. In addition, minimizing the drag will result in weight saving since less fuel will have to be carried. Though not considered here, the effect of aircraft maneuver can be incorporated in future work.

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Boom Carpet

Figure 1.3: Sonic boom carpet.

1.3.3 Low Sonic Boom Design For a given aircraft weight, little can be done to reduce the N-wave by much. In the sixties, it was observed that a sufficiently long aircraft could produce mid-field pressure signatures (see Figure 1.1) on the ground because the pressure disturbances coalesce more slowly in the real atmosphere than in a uniform atmosphere. Mid-field pressure signatures depend on aircraft shape, so it is possible for aircraft designers to shape the sonic booms on the ground - hence the term ‘shaped booms’. Low boom design is all about designing the aircraft to produce shaped booms of desirable characteristics. Figure 1.4 shows two shaped booms proposed by Seebass and George33. The left sonic boom has low overpressures while the other has small shocks.

Minimum Overpressure

Minimum Shock

Pressure

Time

Figure 1.4: Examples of shaped booms.

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1.4 Sonic Boom and Aerodynamic Optimization Current supersonic aircraft were designed only for aerodynamic efficiency. As a result, they all have unacceptable sonic booms on the ground. On the other hand, aircraft designed solely for low sonic boom may show poor aerodynamic performance because the optimal aircraft shape for sonic boom consists of a blunt nose. Meeting the requirements of one without the other is difficult enough. Yet it is necessary to address both because sonic boom is critical for environmental acceptability, just as aerodynamic efficiency is critical for range and operating cost. Perhaps the only way to meet the demanding challenges of both is to employ numerical shape optimization techniques8. Even with optimization, one still has to determine how the two objectives are handled together.

1.5 Organization of Thesis The current research on low-boom design optimization began with the development of a sonic boom prediction computer program based on classical sonic boom theory. Chapter 2 describes the theory and how it was incorporated with CFD in a manner consistent with the assumptions made in the theory. In addition to the aerodynamic properties of the aircraft, the aerodynamics analysis must also provide the flow field data near the aircraft as input to the sonic boom prediction. Chapter 3 discusses aerodynamic analyses of varying computational cost and fidelity that were used in this research, namely linear source method, a 3-D panel method, and a 3-D Euler solver. Results are presented to demonstrate the validity of the codes and to justify the use of the panel method for the current work. Chapter 4 presents the framework for the direct sonic boom optimization method developed in this research. A description of the Seebass-George low-boom design method is first given, since it is commonly found in sonic boom literature. The limitations

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of the Seebass-George method are discussed and a comparison made with the present method. Chapter 5 discusses the details of sonic boom optimization within the framework established in Chapter 4. First issue addressed is that of the sonic boom metric. An example will demonstrate that the traditionally used initial shock overpressure is inadequate for shaped booms. Results are presented to support the use of the calculated loudness instead. Following that is the selection of the objective function for sonic boom optimization. The search method, whether gradient or non-gradient-based, is discussed. The handling of constraints is described. Finally, an approach for multi-objective optimization is discussed for dealing with both the sonic boom and performance objectives. Chapter 6 describes how the sonic boom and aerodynamic analyses were integrated with optimization. Two important areas that have not been addressed in previous chapters are discussed. These are firstly the automated A502 model generator, which enables the A502 input file to be created automatically during the optimization; and secondly, the parallelization of the optimization. In Chapter 7, four examples of the multi-objective optimization of sonic boom and drag are presented and discussed in order to demonstrate the capability of the method developed.

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Chapter 2 Sonic Boom Prediction

Sonic Boom Prediction This chapter describes the sonic boom prediction method used in this research. A historical review of theoretical developments related to the method used in this research is given, followed by a description of the method used in this research.

2.1 Historical Review of Sonic Boom Prediction The foundation of classical sonic boom theory is the paper by Whitham9 in 1952 on the flow pattern around supersonic projectiles. In that paper Whitham set out to remedy the failure of linear theory as a description of the flow. His fundamental hypothesis was that linear theory gives valid approximations to the flow everywhere provided the approximate characteristics (x+ b r = constant) are replaced by the exact ones (y(x, r) = constant), as illustrated in Figure 2.1. In doing so, he formulated the ‘F-function’ which related the flow field, and hence sonic boom, to the lengthwise area distribution of an axisymmetric body.

Figure 2.1: Linear versus actual characteristic. Several years later, Walkden10 extended Whitham’s work to lifting wing-body configurations by treating them as equivalent bodies of revolution. Wind tunnel tests in 9

the sixties and seventies, such as by Carlson11 and Hunton,12validated the method of Whitham and Walkden. Early sonic boom analysis assumed a uniform atmosphere. In 1969, Hayes13 et al. showed that Whitham’s theory could be generalized for non-uniform atmosphere using geometric acoustics. In addition to variation in atmospheric properties, like speed of sound and wind, Hayes also included aircraft maneuvering in the calculation. This was a significant development because atmospheric gradients and aircraft maneuver have a large impact on the sonic boom. The developments highlighted so far embody classical or standard sonic boom theory, where the F-function is calculated using linearized flow area rule or derived from wind-tunnel measurement. In the early 90’s, Cheung14 and Siclari15 obtained the Ffunction from near field pressure results that were computed using high fidelity CFD. The use of CFD was motivated by hypersonic flight research, where linear aerodynamics theory is not valid. Although CFD was used to obtain the F-function, Whitham’s theory was still used to determine the sonic boom on the ground. Consequently, the near field pressure signature had to be computed several body-lengths away so that both the axisymmetric and linear flow assumptions of the theory are not violated. This is very costly since the mesh resolution needs to be high all the way out to where the near field pressure signature is needed. To improve the computational efficiency, Page and Plotkin16 developed the multipole method to compute the F-function from distances no further than the wingspan.

2.2 Sonic Boom Analysis For the current research, sonic boom is predicted as follows: 1. Compute Whitham F-function using near field pressure signature. The near field signature was obtained either from linear source method, 3-D panel method or 3D Euler method.

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2. Perform ray tracing to compute age and amplitude parameters. 3. Knowing the age and amplitude parameters, apply Whitham’s theory on the Ffunction to obtain the far field pressure signature on the ground. 4. To account for wave reflection off the ground, the pressure signature on the ground is multiplied by a reflection factor of 1.9.

2.3 Whitham Sonic Boom Theory Whitham’s method is based on replacing the approximate characteristic of x+br=constant by the exact characteristic curve of y(x, r)=constant. Referring to Figure 2.2, the modified characteristic in a uniform medium is given by: x = y + br -

†

where g

(g + 1)M 4 2b 3

rF(y)

(2.1)

= ratio of specific heats

M

= Mach Number

r

= vertical distance from body

b

= M 2 -1

F(y)

= Whitham F-Function

† Equation (2.1) is similar to linear theory except for the additional term involving the F-function F(y), which is the age parameter. The characteristic is still an approximation since higher terms in the perturbation velocity are ignored. Furthermore, it has been simplified by assuming br/y is large. This is an important point because it means Whitham’s results are valid only at large distances from the body.

11

M

y, t m x+br=constant line

r

x x-br

br

Figure 2.2: Nomenclature for Whitham’s Theory. For an axisymmetric body, the F-function is defined as:

F(y) =

†

1 2p

Ú

y 0

S"(t) dt y-t

(2.2)

where S” is the second derivative of the cross-sectional area of the body as measured by the normal projections of cuts aligned with the Mach angle m . y is measured from the nose along the body axis. The modified characteristic will invariably intersect each other resulting in multiple values at some locations. Figure 2.3(ii) illustrates this as part of an example of applying Whitham’s method for the sonic boom prediction of a parabolic body of revolution. This does not happen in practice because of the presence of shocks. The shock condition proposed by Whitham is that if two regions of supersonic flow are separated by a shock, then to the first order in strength, the direction of the shock bisects the Mach directions of the two regions of the flow. This leads to Whitham’s Area-Balance rule which states that the “lobes” cut off on each side of the shock (shaded region in Figure 2.3(ii)) must be equal in area. The numerical approach for determining the area balance is given by Middleton17.

12

Figure 2.3: Sonic boom prediction of a Mach 1.5 parabolic body of revolution. In a uniform medium, Hayes13 showed that the pressure signature is related to the Ffunction by:

DP gM 2 = F(y) P 2br

†

where DP P

(2.3)

= overpressure = ambient pressure

The equation above is sometimes generalized as follows:

13

DP 1 = F(y) P B

† where

†

1 B

(2.4)

1 gM 2 = B 2br

(2.5)

is called the amplification parameter.

2.4 Computing Equivalent Area Due to Non-axisymmetric Volume and Lift

†

Walken10 applied Lomax’s (Lomax, 1955) idea of equivalent areas to sonic boom. The equivalent area is the area of an equivalent non-lifting axisymmetric body which produces the same sonic boom. The equivalent area AE(x, f) is a function of azimuthal angle f shown in Figure 2.1 and consists of two components: a body component AB and a lift component AL. AE (x, f ) = AB (x, f ) + AL (x, f )

(2.6)

†

f

Figure 2.4: Rear view of aircraft showing azimuth angle f.

Within each azimuthal plane, the volume component is calculated in the same way as for an axisymmetric body while the lift component is given by:

AL (x, f ) =

†

b rU•2

Ú

x 0

L(x, f )dx

(2.7)

where U• is the flight velocity, r is the ambient density and L(x, f ) is the lift component in the f plane at axial station x.

† 14

Once the equivalent area has been calculated, the F-function and hence sonic boom can be calculated using Whitham’s method.

2.5 Computing the F-function from CFD Although the F-function for any arbitrary aircraft configuration can be calculated from the equivalent area, it can also be derived from the near-field pressure signature using equation (2.3):

F(y) =

†

2br DP gM 2 P

If nonlinear CFD is used, equation (2.1) can be used to adjust for curvature in the characteristics:

y = x - br +

†

(2.8)

(g + 1)M 4 2b 3

rF(y)

(2.9)

Hence any method that gives the near field pressure signature, such as wind tunnel or CFD, could also be used to obtain the F-function. The selection of the methods used in this research will be discussed in the following chapter under aerodynamic analysis. One important condition is that the near field pressure signature must be far away enough such that the axisymmetric and far field assumptions for Whitham’s theory are not violated.

2.6 Boom Propagation Through Non-Uniform Atmosphere Whitham’s theory was originally formulated for sonic boom propagation through a uniform medium, where the properties are constant. By using geometrical acoustics, Hayes13 showed that Whitham’s theory could be extended to standard atmosphere, where

15

pressure, density and speed of sound vary with altitude. In this case the amplitude parameter is:

1 gM 2 = B 2bro

†

where ro

r o ao3 Ao ra 3 A

(2.10)

= reference radius near the aircraft

a

= ambient speed of sound

A

= normal ray tube area (see Figure 2.5)

Subscript o denote values at reference radius

Flight Path

Mach Cone t

m

t + Dt

x

m

f

f + Df

f y

A

Ray

z

Ray 2 Ray 1

Ray 4 Ray 3

Figure 2.5: Rays, ray tube and ray tube area. Rays, ray tubes and the ray tube area (Figure 2.5) are basic concepts from geometric acoustics. Acoustic signals are propagated along rays. A bunch of neighboring rays make a ray tube and the normal ray tube area is the area intercepted by the tube and the plane normal to the ray tube. The calculation of the ray tube area is described in a later section.

16

Figure 2.6: Predicted sonic boom in uniform and standard atmosphere. In a non-uniform atmosphere the age parameter L is given by: L(s) = u

where s †

u

g + 1 s ds Ú 2g 0 a B

(2.11)

= distance along acoustic ray = aircraft velocity

The above expression for L differs from Hayes13 by u because here it is in terms of length rather than time. Note that the Equation (2.11) reverts to the uniform medium case of equations (2.1) and (2.3) when r and a are constant. Figure 2.6 compares the sonic boom predicted using uniform and standard atmosphere formulation. The difference illustrates the need to account for the atmospheric gradients in the analysis.

2.7 Ray Tracing and Ray Tube Area Calculation In geometric acoustics the signal is propagated along rays. Each ray is the trajectory of a wave front and ray tracing determines the shape of the ray. For any given ray, one can visualize a ray tube as a tube consisting of neighboring rays, and the ray tube area in a given direction is the area intercepted by the ray tube and the plane normal to that 17

direction. In a uniform medium, rays are straight. In a non-uniform medium, particularly where the speed of sound varies, the rays will be refracted as they pass through the medium (just as light rays are refracted as they pass from pass from air into water). A general approach for ray tracing and ray tube area calculation for arbitrary atmospheric conditions (including wind) and flight maneuvers is presented in Hayes13. For the current purpose, wind was not included explicitly, so a simpler approach by Plotkin18 was implemented. In addition, the equations were further simplified for steady and level cruise condition. Consider a ray tube defined by four corner rays as shown in Figure 2.5. The area of a horizontal cut through the tube is given by: (2.12)

A H = Dt Df J

† where J = Jacobian =

†

∂ (x r , y r ) ∂x r ∂y r ∂y r ∂x r = ∂ (t, f ) ∂t ∂f ∂t ∂f

Let d be the angle of the ray tube with respect to the vertical. The normal ray tube area A is: A = Dt Df J cos d

†

(2.13)

(2.14)

Consider a small radius ro from the aircraft. Since ro is small, the rays have not undergone much refraction and are therefore straight. Hence d=m at ro. From geometry, the normal ray tube area at ro is: Ao = VDt Df ro cos m

†

(2.15)

where V is the flight velocity and m the Mach angle. The ratio of ray tube areas A/Ao is:

18

A J cos d = Ao M ao ro cos m

†

(2.16)

In the absence of wind, a ray will lie in a plane at angle b to the x-z plane, as shown in Figure 2.7. x b do

y

r Ray d z

Vertical Plane Containing Ray

Figure 2.7: R-z plane containing the ray. The ray shape in the r-z plane is given by: -1/ 2

È a2 ˘ R = Ú z Í 2 o 2 -1˙ o a sin d Î ˚ o z

†

dz

From geometry cosdo = cos m cos f - sin m sin g

†

†

(2.17)

tan b =

sin f cos g tan m + cos f sin g

(2.18)

(2.19)

For steady level flight, the final expression for the Jacobian is: J = Q M ao sin b cosdo ∂d∂fo + R M ao cos b ∂b ∂f

(2.20) 19

†

where Q =

z

ao 2 a

È ÎÍ

ao 2 a

Ú ( ) ( ) zo

˘-3 / 2 - sin 2 do ˙ dz ˚

(2.21)

†

∂do (cos m sin f ) = ∂f sin do

(2.22)

†

∂b tan m cosf = 2 ∂f sin f + tan 2 m

(2.23)

†

Figure 2.8 shows the results of ray tracing for Mach 1.5 steady level flight at 40,000 ft altitude. From equation (2.4) the overpressure D p is obtained by multiplying the Ffunction by P / B . Figure 2.8(iii) is a plot of P / B verses altitude. Note that the value on the ground calculated using uniform atmosphere is only half of that calculated in standard atmosphere. In addition Figure 2.8(iv) also shows a significant difference in the † † age parameter, with the same amount of aging occurring in just half the vertical distance in the uniform case. This means that in the standard atmosphere, the sonic boom will take longer to develop into its far field shape, i.e. N-wave. The differences observed here explain the difference in the sonic booms shown in Figure 2.6. For these reasons, it was deemed necessary to use age and amplitude parameters obtained for standard atmosphere in the current low-boom design research.

20

Figure 2.8: Ray tracing results for from 40,000 ft for Mach 1.5 steady level flight.

2.8 Non-linear Steepening and Geometric Acoustics The nonlinear effect of signature aging (i.e. nonlinear steepening) is assumed to have negligible effect on the ray and ray tube area. Although no comprehensive theory is available to justify this assumption, Hayes19 showed that the neglected effects correspond to higher order terms. Hence the ray parameters are independent of the F-function or the aircraft configuration. In other words, for the same flight conditions, the same ray parameters are applicable to any aircraft.

2.9 Near-Field Distance and Validity of Whitham Theory When using wind tunnel or CFD to predict sonic boom, there is the temptation to use near field pressure data close to the aircraft. Since the height of the wind tunnel test section is limited, the closer the near field measurement, the larger the test model can be; 21

the larger the test model, the better the near field measurements. In the case of CFD, the closer the near field pressure needed, the smaller the grid size and hence computation cost. Unfortunately, the closer the near field pressure signature, the more it violates the conditions necessary for Whitham theory to hold. Recall firstly that Whitham’s theory is an axisymmetric solution, which is approximately true at large distances from the aircraft, but not at closer distances where significant amounts of cross flow exist. Secondly, the modified characteristic is derived by assuming br>>y. Hence the modified characteristic does not apply when r is too small.

Figure 2.9: Wing-Body configuration showing pressure distribution. Figure 2.10 shows the F-function obtained from the near field pressure signatures computed by a 3-D panel method at several distances for a supersonic business jet (Figure 2.9) cruising at Mach 1.5 and 2O angle of attack. Note that the F-function is different when calculated from near field pressures less than two body lengths from the aircraft. Since the pressures were computed using linear theory, the variation in the Ffunction can only be attributed to the cross flow effect. This is further supported by the observation that the variation does not exist in front of the wing, where the body is axisymmetric. Figure 2.11 shows the resulting sonic boom calculated for the various near field locations, illustrating the need for the near field pressure to be sufficiently far away

22

from the aircraft. The degree of cross flow is configuration dependent, so a parametric study not unlike the one shown here may be necessary to ensure sufficient distance.

Figure 2.10: F-function derived from near field pressure distributions.

Figure 2.11: Sonic boom on ground from near field pressure distributions.

23

On a final note, Page and Plotkin16 developed a method to account for the cross flow in the F-function formulation. It requires the near field pressure to be known on a cylinder around the aircraft, and is particularly well suited for CFD. The radius of the cylinder only needs to be large enough to contain the entire aircraft, which results in a considerably smaller computational grid. This method was not pursued in this research because, for reasons discussed in the next chapter, a 3-D linear panel method was used for the design.

2.10 Shock Wave Rise Time Sonic boom theory as described above assumes shock waves to be zero-thickness pressure jumps. Real sonic booms however display a finite rise time. Rise time is important because it affects the high frequency content of the sonic boom energy, which in turn affects the human auditory response.

Figure 2.12: Sonic boom shock rise times (from Ref. 20). Sonic boom rise time is believed to be caused by atmospheric absorption of sound, which molecular absorption theory shows to be several orders of magnitude larger than classical viscous absorption7. Figure 2.12 shows the shock rise times as a function of shock overpressure predicted by molecular absorption theory and classical viscous absorption theory from Darden20. Also shown is a set of experimental data, which appeared to support the molecular absorption theory. Additional rise time in the experimental data was attributed to atmospheric turbulence. 24

The empirical model, which is a straight-line fit through the experimental data, is (Ref. 39):

T=

0.003 DP

where T is the rise time in seconds and D P the shock strength in pounds per square foot. The above empirical model or the line representing molecular absorption theory is used † to modify the solutions obtained by sonic boom theory. Needleman39 et al. did this by simply adding the rise time to the signature, resulting in the lengthening of the signature, as illustrated in Signature output by sonic boom theory

Signature modified with rise time

Figure 2.13: Sonic boom modified to include rise time. The development and validation of a sonic boom propagation method that accounts for rise time are ongoing areas of research20. Such a method will probably provide a statistical variation of the sonic boom for a given aircraft configuration.

25

Chapter 3 Aerodynamics Analysis

Aerodynamic Analysis This chapter discusses three methods used in the aerodynamic analysis of supersonic aircraft: a linear source method, the A502 3-D panel method and FLO87, a 3-D Euler method. In the current research, the near field pressure signature was needed for sonic boom prediction, while lift and drag were needed to determine the aircraft performance. The chapter also explains why A502 was favored over the other two methods for lowboom aircraft optimization.

3.1 Linear Source Method Classical sonic boom theory was derived from linearized aerodynamic theory, where the flow disturbance is created by a linear distribution of sources and sinks. Using the idea of equivalent area, the linear source distribution method was extended to aircraft of arbitrary shape. Under the assumptions of linearized aerodynamic theory, a perturbation potential j can be defined such that the perturbation velocity is given by the partial derivative of j in the direction of interest. For an axisymmetric body with area distribution S(x), the perturbation potential j at radial distance r is given by21:

j (x,r) = -

†

1 2p

Ú

x- br 0

S ¢(x1 )dx1 (x - x1 ) 2 - b 2 r 2

(3.1)

For non-axisymmetric lifting cases, S’(x) is replaced by the equivalent area. The pressure can be calculated from the perturbation potential using the following:

26

CP =

†

P - P• = -2j x - j y2 - j z2 2 1 r U • 2

The wave drag for the lineal source distribution is22:

Dw = -

†

r•U•2 4p

l

l

0

0

Ú Ú

S ¢¢(x1 ) S ¢¢(x 2 )ln x1 - x 2 dx1dx 2

(3.3)

From the Cp distribution at r, the F-function is:

F(y) =

†

(3.2)

2br DP br = CP 2 gM P 2

(3.4)

where y = x - br (refer to Figure 3.1) M

(3.5)

y, t m x+br=constant line

r

x br

x-br

Figure 3.1: Nomenclature for linear supersonic theory. Note that because the near field was computed using the unmodified linear theory, the linear characteristics are used to compute the F-function.

3.2 A502 (Panair) 3-D Panel Method A502, also known as Panair23, is a computer program developed at Boeing to solve the aerodynamic properties for arbitrary aircraft configurations at subsonic and supersonic speeds. The program uses a higher-order (quadratic doublet, linear source) panel method, 27

based on the solution of the linearized potential flow boundary-value problem. Results are generally valid for cases that satisfy the assumptions of linearized potential flow theory – small disturbance, not transonic, irrotational flow and negligible viscous effects. Once the solution is found for the aerodynamic properties on the surface of the aircraft, A502 can then easily calculate the flow properties at any location in the flow field, hence obtaining the near field pressure signature needed for sonic boom prediction. In keeping with the axisymmetric assumption of sonic boom theory, the near field pressure is obtained at 15 body-lengths below the aircraft. The F-function is computed using Equations (3.4) and (3.5) as before, since the panel method is based on linear theory.

Figure 3.2: Aerodynamic analysis using A502 panel method. One of the difficulties of using A502 is the creation of the input file, where every panel corner is defined. Moreover, the points at the interface of a network of panels, such as wing-body junctions, must coincide exactly. Another problem is that A502 is unable to give reliable gradients. This is firstly because the geometry data is restricted to 10character width in the input file, and secondly, data is written to scratch files with limited precision. Being a legacy code comprising 500 subroutines, attempts to modify the

28

program to overcome these limitations were unsuccessful, whether by increasing the data width or complexifying24 the program.

3.3 FLO87 3-D Euler Flow Solver FLO87 is a three-dimensional Euler solver developed by Jameson25. It solves the steady three-dimensional Euler equations using a modified explicit multistage Runge-Kutta time stepping scheme. Multigrid26 and implicit residual smoothing are used to achieve fast convergence. With FLO87, the F-function is computed from the near field signature using Equation (3.4) as in the previous two methods. However, since the Euler equations are nonlinear, the characteristics would not be the same as the linear ones. Instead Cheung14 suggested that the initial location of each characteristic be determined using the modified linear characteristics:

y = x - br +

(g + 1)M 4 2b 3

rF(y)

(3.6)

3.4 Comparison Between A502 and Linear Source Method †

The linear source model produces the fastest results of the three methods used. A typical computation takes less than a second, while A502 takes about 30 seconds. However this advantage is lost for lifting aircraft configurations, because the lift has to be first determined by some other method, say CFD, in order to obtain the equivalent area. Although both are based on linearized aerodynamics, A502 models the aircraft geometry accurately whereas the linear source method lumps the aircraft into a line source. With the line source, it is difficult to account for non-axisymmetric effects, such as the shielding effect of the wing. Figure 3.4 is the near field pressure signature computed by A502 for a nacelle, modeled as a parabolic body, at three locations shown by Figure 3.3. Location A is over the fuselage, B is at the wing tip and C is under the 29

fuselage. The nacelle on symmetry plane is twice the size of nacelle at outboard since it is modeled as a single body. To help in the comparison, all three were aligned along the same Mach plane. From the pressure signatures, it was apparent that the nacelle at A did not affect the sonic boom under the aircraft. In comparison, the effect of the nacelle at B and C is clearly felt. This result has significant impact on low-boom design as it suggested that the upper fuselage could possibly be optimized to improve drag without affecting the shaped boom below. Although not used for aircraft low-boom optimization, the linear source method was used to compute the flow around axisymmetric bodies for the validation of the panel and Euler codes, as well as for testing the sonic boom optimization.

B

A

C

TOP VIEW B

A

C SIDE VIEW

Mach Line

Figure 3.3: Location of the nacelle for results shown in Figure 3.4.

30

Figure 3.4: Example of wing shielding of nacelle.

3.5 Comparison Between A502 and FLO87 FLO87 is more accurate than A502 since it solves the nonlinear Euler equations while A502 relies on linearized theory. However, since the Mach number and configurations of interest in this research fall within the regime of linear aerodynamics, the results produced by the two methods were expected to be similar.

31

The main reason for choosing A502 over FLO87 is computational cost, which was particularly important in the present work because a large number of analyses are performed during optimization. A typical run time for aircraft aerodynamic loads analysis using A502 is 30 seconds on a 1 GHz Intel Pentium III CPU, while FLO87 takes about 30 minutes on an SGI Origin 300 with a 600 MHz R14000 processor. For sonic boom prediction, FLO87 will be even costlier because the computational grid resolution between the aircraft and near field signature has to be very fine. Compounding the problem, because the multipole method was not available, the near field has to be far away for sonic boom theory to hold. In contrast, A502, being a boundary element method, is able to compute the pressure at any distance from the aircraft at almost no additional cost. Four test cases are presented below to demonstrate the validity of A502 for drag and sonic boom prediction. The first test case - the Sears-Haack body – showed that A502 agreed with theoretical and Euler results for wave drag. Near field pressures were also compared to verify that A502’s off-body results were consistent with linear theory. The second and third test cases were studied to show that A502 indeed produced the same surface pressures as FLO87 for lifting supersonic aircraft configurations. In the final test case, a comparison was made with Euler near field signatures for a lifting aircraft to validate A502 for near field pressures prediction of lifting configurations.

3.5.1 Sears-Haack Axisymmetric Test Case The Sears-Haack body gives the minimum wave drag for given volume. Its radius distribution is given by27: Ê r ˆ2 1+ 1- x 2 2 2 = 1x x ln Á ˜ x Ë ro ¯

†

(-1 £ x £ 1)

(3.7)

From linear theory, the drag coefficient is:†

32

CD =

†

4 p 2 ro2 l2

(3.8)

Figure 3.5 is a comparison of the drag for various ratio of maximum radius over length calculated by A502, FLO87 and linear aerodynamics theory. The results show good agreement with each other, thereby validating A502 for drag prediction.

Figure 3.5: Sears-Haack body drag comparison (M=1.5). Figure 3.6 shows near field pressure signatures computed at radial distances R/L of 0.25 and 1.0 using the linear source method, A502 and FLO87. Based on SSBJ fuselages observed in literature, a maximum radius/length ratio of 0.03 was chosen. A502 and the linear source solutions were nearly identical, which was expected since both were based on linearized theory. A502 showed good agreement with FLO87 at R/L=0.25, except at the front. At greater distances from the body the agreement generally became poorer. Since the current problem was expected to be within the realm of linear theory, the discrepancy was attributed to insufficient grid resolution for computing the near field signature reliably. The final test case presented later will verify that with proper grid refinement, Euler and A502 do give similar results.

33

Figure 3.6: Computed near field pressures for Sears-Haack body.

3.5.2 Wing-Body Test Cases The purpose of these test cases was to verify that A502 would perform just as well as FLO87 for predicting the aerodynamics properties of supersonic aircraft. To address concerns that A502 may be unreliable for cases with subsonic leading edges, two similar configurations were analyzed, differing only by the wing sweep angle. The first had a 34

leading edge sweep of 19o, which had a supersonic leading edge at a free stream Mach number of 1.5. The second configuration, with a sweep of 55o had a subsonic leading edge (normal Mach number component of 0.87). Fuselage length and wing span were 128 and 72.4 ft respectively. Utilizing symmetry, only half the aircraft was modeled. Figure 3.7 and Figure 3.12 compare the surface mesh of FLO87 with the panels of the A502 model for both configurations. The C-H grid for the FLO87 model consisted of 193 nodes in the Cdirection, 73 in the direction normal to the surface, and 49 in the semi-span direction. In contrast, the A502 model was made up of 28 around the chord, 11 span-wise, 40 along the fuselage, and 6 around half of the fuselage cross-section. FLO87 took 28 minutes on an SGI Origin 300, while A502 took only 21 seconds on a 1 GHz Pentium III Linux box. Despite the large difference in computational cost, A502 showed good agreement with most of the FLO87 results for both configurations. Figure 3.9, Figure 3.10, Figure 3.14, and Figure 3.15 are comparisons of FLO87 and A502 surface pressures at 2 degree angle of attack. In each figure, the upper half is FLO87 and the lower half, A502. The results are nearly identical. Figure 3.8 shows good agreement in lift and drag for the low-swept case, while the agreement was poorer for the highly swept one (Figure 3.13). The difference can be explained from the wing section pressure distributions shown in Figure 3.11 and Figure 3.16. For the highly swept case, A502 predicted larger pressure peaks, probably because the Mach number normal to the leading edge was in the transonic regime. In addition the panel resolution was too coarse to capture the leading edge suction. The combined effect would account for the higher drag predicted by A502. Better agreement was found by increasing the A502 panel density at the leading edge, but that also increased the computation time. For the current purpose, the slight discrepancy was considered acceptable.

35

Figure 3.7: Comparison of the surface mesh/panels.

Figure 3.8: Comparison of computed drag polar.

36

Figure 3.9: Comparison of the upper surface Cp.

Figure 3.10: Comparison of lower surface Cp. 37

Figure 3.11: Comparison of section Cp. Solid line is Euler, symbols are A502.

38

Figure 3.12: Comparison of the surface mesh/panels for subsonic leading edge wing.

Figure 3.13: Comparison of computed drag polar (subsonic leading edge wing).

39

Figure 3.14: Comparison of the upper surface Cp for subsonic leading edge wing.

Figure 3.15: Comparison of the lower surface Cp for subsonic leading edge wing. 40

Figure 3.16: Comparison of section Cp for subsonic leading edge wing. Solid line is Euler, symbols are A502

3.5.3 A502 and Euler CFD Near Field Pressure Comparison It was difficult to make any conclusions from the near field pressure signatures for the previous wing-body examples because insufficient grid resolution resulted in poorly defined shocks there. Chung28 computed the near field signature for a wing-canard configuration using QSP10729, an integrated sonic boom prediction tool employing fully nonlinear CFD with a H-type mesh that had been optimized for near field signature prediction. Figure 3.17 is the A502 model showing the top view of the configuration. The free stream Mach number was 1.6; the angle of attack was 2.23 degrees; the lift 41

coefficient was 0.1 and the drag coefficient 0.00955. Leading edge sweep angles for the wing and canard were 60 and 45 degrees respectively. Both wing and canard had uncambered biconvex airfoils with 4 and 2 percent thicknesses respectively.

Figure 3.17: Geometry of near field signature test case. In order to compare the A502 near field pressures with QSP107, Whitham’s method was applied to the A502 off-body pressures to account for non-linear steepening. This was done by first calculating the F-function from the A502 near field results, and then recomputing the pressure signature at that location using Whitham’s method. In theory, Whitham’s method assumes the calculated signature is at a large distance from the aircraft, which was clearly not the case here. However, since the signatures were so close that the degree of nonlinear steepening was still small, any error from the theory would also be small. Figure 3.18 to Figure 3.20 are near field pressure signatures computed by A502 and QSP107 at 0.4, 0.8 and 1.2 body-lengths below the aircraft. A502 results with and without Whitham’s method are shown to illustrate the effect of nonlinear steepening. The results showed good agreement, except at the aft end of the signatures, where the QSP107 grid resolution was poorer28.

42

Figure 3.18: A502 and Euler near field pressure at 0.4 body-lengths below.

Figure 3.19: A502 and Euler near field pressure at 0.8 body-lengths below.

43

Figure 3.20: A502 and Euler near field pressure at 1.2 body-lengths below.

Figure 3.21: Sonic boom on ground obtained using near field signatures at 0.4 body-lengths below aircraft at 50,000 ft altitude.

44

Figure 3.22: Sonic boom on ground obtained using near field signatures at 1.6 body-lengths below aircraft at 50,000 ft altitude. Sonic booms on the ground were calculated using the near field signatures to assess the impact of the discrepancies in the signatures. Figure 3.21 and Figure 3.22 are sonic booms from the signatures at 0.4 and 1.2 body-lengths below the aircraft. Except for the location of the intermediate shock, the A502 and Euler boom signatures were identical.

3.6 A502 and Wind Tunnel Comparison To further evaluate the near field results computed by A502, two wind tunnel test cases were obtained from NASA TN-D7160. Since the wind tunnel near field data was measured at 3.6 body-lengths below the model, Whitham’s method was applied to the A502 off-body results to account for nonlinear steepening (as in the previous section). Figure 3.24 shows that this approach achieved better agreement with the measured results. Though two test cases were analyzed, for conciseness, only one is presented since the second case did not offer additional insight. Figure 3.23 shows the geometry of Model 1 overlaid with the Cp on the lower surface. Figure 3.24 is the comparison of calculated and measured results. The agreement is reasonably good considering the uncertainties present. Firstly, with the wind tunnel model being only 17.52 cm in length, it is unlikely 45

that it was manufactured to the exact specifications. Secondly, the measurements for the pressure signature would have significant uncertainty at such scale. Thirdly, Whitham’s theory assumes the pressure signature is calculated far from the aircraft, which was not the case here. Figure 3.25 is a comparison of the sonic boom predicted on the ground using the measured and calculated near field signatures.

Figure 3.23: Cp from A502 analysis of Model 1 (NASA TN-D7160).

Figure 3.24: Computed near field signature compared with NASA TN-D7160 for Model 1.

46

Figure 3.25: Comparison of the sonic boom predicted using A502 and wind tunnel data from NASA TN-D7160.

47

Chapter 4 Low-Boom Aerodynamic Design

Low-Boom Aerodynamic Design This chapter presents the low-boom aerodynamic design methodology used in this research. A description and comparison with the Seebass-George method is also given, since it is commonly found in sonic boom literature.

4.1 Introduction The object of low-boom aerodynamic design is to minimize the sonic boom via aerodynamic shaping of the aircraft. Applying Whitham and Walkden’s theories, the earliest works focused only on reducing the peak overpressure in N-waves. This was motivated by the assumption that all pressure signatures reaching the ground would be Nwaves. Darden30 cited Jones31 where he showed that the lower bound for the N-wave shocks occurred for extremely blunt area distribution, corresponding to having a delta function at x=0 in the F-function. Not surprisingly, aircraft shapes derived from these areas suffered huge drag penalties. In the 1960’s, it was observed that a sufficiently long aircraft could produce midfield pressure signatures on the ground. Hayes32, as cited by Darden30, showed that the characteristics coalesce more slowly in the real atmosphere than in a uniform atmosphere, thus increasing the possibility that mid-field signatures could reach the ground. Unlike the N-wave, mid-field pressure signatures depend on airplane shape, thereby giving aircraft designers a means for shaping the sonic booms on the ground. Following this idea, Seebass and George33 formulated a generic F-function that would produce mid-field signatures (a.k.a. shaped booms) on the ground with certain desirable characteristics. Similar to Jones, the Seebass-George F-function had a delta function at the nose, resulting in blunt nose shapes. Concerned with the drag penalty, Darden30 modified 48

Seebass-George’s F-function for lesser degrees of nose bluntness. The Seebass-George Ffunction has a simple analytical form that can be inverted to produce the equivalent area distribution, which was used to shape the aircraft. Figure 4.1, taken from Mack and Needleman34, illustrates how the Seebass-George method may be incorporated to the aircraft design cycle.

Mission Requirements Range, Payload, Dp, Altitude, Weight, Length etc

Ideal Boom Signature & Equivalent Area Ae Numerical Model Dp Lift

Volume

Model Equivalent Area & Boom Signature

Compare Equivalent Area & Boom Signature

Ae

Ae

Dp

Dp

NO

Performance and Mission Analysis

Agreement YES

Figure 4.1: Framework for low-boom design using Seebass-George method. 49

The Seebass-George low-boom design method is an inverse approach because the aircraft shape is derived from a given sonic boom pressure signature. The current proposed method takes a more direct approach, using recent techniques in numerical aerodynamic shape optimization to shape the aircraft in relation to the predicted boom, rather than the area distribution. P max

P max Pf

(a) Overpressure Minimized

(b) Shock Minimized

Figure 4.2: Boom pressure signatures proposed by Seebass and George35.

4.2 Seebass-George Sonic Boom Minimization Method This section describes the Seebass-George sonic boom minimization method with the nose bluntness modifications made by Darden30. The method solves for the required equivalent area Ae distribution to produce the sonic boom signatures shown in Figure 4.2. Both signatures were thought to be less disturbing to people. The first signature minimizes the overpressure, while the other minimizes the shock intensities. The F-function illustrated in Figure 4.3 produces the pressure signatures given in Figure 4.2. It is defined by:

†

F(y) = B(x - y f ) + C

†

(0 £ x £ y /2) ( y /2 £ x £ y ) ( y £ x £ l)

†

F(y) = B(x - y f ) - D

†

( l £ x £ l)

F(y) = 2xH / y f

f

F(y) = C(2x / y f -1) - H(2x / y f - 2)

†

†

†

†

f

f

f

(4.1a)

(4.1a)

(4.1b) (4.1a)

50

slope B

H slope S

C yf

l

l y

r slope S

D slope B

Pf Pr

Figure 4.3: Darden’s F-function modified from Seebass-George. Slope s is the slope of the area balance line and is given by:

s=

†

1 L

(4.2)

L is the age parameter. In the course of the nonlinear steepening, the area balance line rotates towards the vertical position, eventually coinciding with the front shock at the design point. yr is the intersection of the rear area balance line with F(y) in the wake which corresponds to the location of the rear shock. yf determines the nose bluntness, which is usually specified by the designer. l is the effective length of the aircraft. l in general is not the same as the length of the aircraft because of angle of attack and non-planar effects (Figure 4.4).

51

Fuselage Length Aircraft Length Aircraft Eff. Length Fuselage Eff. Length

m

a

Figure 4.4: Effective length of aircraft. When B=0 the signature will be overpressure minimized (Figure 4.2(a)), while 0
Pf =

†

1 B

PC

(4.3)

In addition, H can be determined from the front shock area balance if yf is known. Figure 4.5(a) shows the front area balance which is used to calculate H. The area balance is given by:

A1 = A2 A more convenient form is obtained by adding the common region A3 to both A1 and A3:

52

A1+A3 = A2+A3

leading to:

1 2

†

yf 0

yf 0

F(y)dy

(4.4)

F(y)dy = y f ( H2 + C4 )

(4.5)

Hence H is determined from Equations (4.4) and (4.5):

H=

†

Ú

Using F(y) from Equations (4.1a) and (4.1b):

Ú †

LC 2 =

LC 2 C yf 2

(4.6)

Having specified B, yf, P f and finding C and H, the remaining four unknowns in the F-function (i.e. D,l, l, yr) can be found from the following four conditions: •

Base area assumption (Equation 4.7)

•

Rear shock area balance (Equation 4.11)

•

Ratio of front to rear shocks (Equation 4.14)

•

Rear area balance line connecting F(yr) to F(l) (Equation 4.15)

Assuming the base area of the equivalent body consists of the lift component only, from Equation (2.7):

Ae (l) =

bw rU 2

(4.7)

53 †

where w is the weight of the aircraft. The expression for the area distribution Ae(x) is obtained by first inverting the definition of the F-function: x

Ae (x) = 4 Ú F(y) x - y dy 0

†

(4.8)

Substituting Equation (4.1) into (4.8): Ae (x) =

3/2 32 H 5 / 2 8 x + 1( x - y f /2) ( x - y f /2) 15 y f 15

{( 3y f /2 + 2x )(1/ y f )(2C - 4H ) + 5(2H - C )} + 3/2 Ê 2ˆ 1( x - y f ) 4 ( x - y f ) {(2C / y f )Á - ˜( 3y f + 2x ) + Ë 15 ¯ 2 4 4 2 C + ( H / y f )( 3y f + 2x ) - H + B( 3y f + 2x ) 3 15 3 15 2 2 8 3/2 - By f + C} -1( x - l) ( x - l) (C + D) 3 3 3

(4.9)

†where 1(x-a) is the Heaviside unit step function.

Hence Equation (4.7) becomes: 3/2 bw 32 H 5 / 2 8 = l + ( l - y f /2) 2 rU 15 y f 15

{( 3y f /2 + 2l)(1/ y f )(2C - 4H ) + 5(2H - C )} + Ê 2ˆ 2 {(2C / y f )Á - ˜( 3y f + 2l) + C Ë 15 ¯ 3 4 4 2 + ( H / y f )( 3y f + 2l) - H + B( 3y f + 2l) 15 3 15 2 2 8 3/2 - By f + C} - ( l - l ) (C + D) 3 3 3 4( l - y f )

3/2

(4.10)

†

54

y r

l A2 slope s

A6

H

A1

slope s C

A3

A5 A4

LC

(a) Front

(b) Rear

Figure 4.5: Front and rear shock area balance. From Figure 4.5(b) the rear shock area balance is: A4 = A5 A4+A6 = A5+A6 Therefore

Ú †

yr l

F(y)dy =

1 2

[F(l) + F(y r )](y r - l)

To evaluate the above, for a given aircraft’s F-function in the range 0 £ y £ l and assuming a cylindrical wake, the analytical expression for the F-function at y>l and hence at yr is given by Darden30:

F(y r ) =

† and

†

(4.11)

Ú

yr l

1 p yr - l

2 F(y)dy = p

†

Ú

l 0

1- x F(x)dx yr - x

Ê y r - l ˆ1/ 2 Ú 0 F(x)tan ÁË l - x ˜¯ dx l

-1

(yr > l)

(4.12)

(yr > l)

(4.13)

From Figure 4.3, the ratio of the front to rear shock strengths has the following relation: 55

Pf C = Pr F(y r ) - F(l)

†

(4.14)

A ratio of 1.0 is typically chosen so that both shocks are equally minimized. Since yr is the location of the intersection of the area balance line with the F-function, the following must be true: F(yr) = F(l) + s(yr – l)

(4.15)

Combining Equations (4.14) and (4.15) gives an expression for yr:

yr =

†

LC +l Pf /Pr

(4.16)

The nonlinear set of equations for determining D , l , l a n d yr were solved by nonlinear optimization using the Nelder Mead36 method. An initial value for l and l is first chosen. D is then determined from Equation (4.10). y r calculated from Equation (4.16) is used to obtain F(yr) in Equation (4.12). Both yr and F(yr) are then used along with Equation (4.1d) to compute the area in the right-hand-side of the rear area balance Equation (4.11), while the left-hand-side is computed using Equation (4.13). Since the two areas must be the same, the optimizer will repeat the above procedure with a new value of l and l until Equation (4.11) is satisfied. With l , l, C, D, H and yf known for the F-function, the required equivalent area is found using Equation (4.9). Figure 4.6 is an example of a low-boom design obtained using the Seebass-George method for a 70,000 lb Mach 1.5 business jet with 0.3 psf front and rear shocks on the ground. For the purpose of illustrating the method, the wing planform was selected and the lengthwise lift distribution was assumed to be proportional

56

to the wing area. The fuselage volume was then determined by taking the difference between the required and lift equivalent areas.

Figure 4.6: Example of Seebass-George Method.

4.3 Limitations of Seebass-George Method The limitations of the Seebass-George low boom design method can be summarized as follows: •

For flat-top and ramp shaped booms only

•

Drag not considered

•

Reduced design space due to equivalent area constraints

•

Lift distribution including wing-body interferences must be specified

•

Error in equivalent area matching has unpredictable effect on boom signature

By design, the Seebass-George method works only for flat top and ramp shaped sonic booms. In addition, the method was formulated with no regard to drag minimization, other than Darden’s modification for nose bluntness. It is likely that a more 57

aerodynamically efficient design could be achieved with a different mid-field signature of similar loudness. Smooth volume and lift variation is required by the Seebass-George method. This significantly constrains the design degrees of freedom for drag minimization. For the natural laminar flow (NLF) aircraft shown in Figure 4.7, where low wing sweep is necessary to prevent transition, the abrupt change in volume and lift at the wing leading edge results in an impractical fuselage shape when conforming to the equivalent area.

Figure 4.7: Supersonic business jet design using a natural laminar flow wing. (Picture courtesy of ASSET Research Corp.) Although the Seebass-George method offers a simple approach to designing aircraft with shaped sonic boom, the resulting design usually does not reproduce the specified shaped booms because of wing-body aerodynamic interaction. One has to iterate between volume and lift in order to match the equivalent area. A suggested future work is to apply recent CFD based aerodynamic shape optimization techniques37 to solve the areamatching problem. Even for a conventional supersonic aircraft with highly swept wing, it is difficult to match the area profile exactly because of complex aerodynamic interactions of the wing, tail and body and nacelles. Any subsequent error in the area will, in general, have an 58

unpredictable effect on the sonic boom. Depending on the configuration, the area deviation at certain locations may have greater impact on the boom than at other locations. The use of higher fidelity aerodynamic analysis and the need to minimize the sonic boom for the natural laminar flow configuration led to the approach described in the next section.

4.4 Direct Sonic Boom Optimization The low-boom design approach employed here utilizes numerical optimization techniques to minimize the sonic boom. Mathematically, optimization involves maximizing or minimizing a scalar objective J(x) whose value is a function of the design variables x, subject to any constraints gi(x) present. For example

Minimize

J(x) = 100(x 2 - x12 ) 2 + (1- x1 ) 2

Subject to

g(x) = x12 + x 22 £ 1

† where x1 and x 2 are the design variables. In this example, the objective function J is † minimized with the constraint that x1 and x2 are within a circle of radius 1.

The fundamental difference between this method and the Seebass-George method is not in the use of optimization, but in the use of the sonic boom signature instead of the equivalent area in the optimization. The objective function in the optimization is a value computed directly from the sonic boom signature, whereas if numerical optimization is applied to the Seebass-George method, the objective would be to minimize the difference in the equivalent areas. Figure 4.8 is an example of how the direct method works. In this case, the objective is to minimize the maximum overpressure of the sonic boom subject to the given lift and drag constraints. The optimization begins with an initial set of design variables x. A 59

numerical model of the aircraft is created and the aerodynamic loads and near field pressure signature are computed. Using Whitham’s method, the sonic boom on the ground is calculated and the objective J(x) is determined from the pressure signature on the ground. A new set of design variables is computed and the process is repeated until the constraints and optimality conditions are satisfied.

Minimize:

f(x) = maximum overpressure

subject to:

Lift = Weight CD < Drag Limit x’s Aero Analysis

Near Field Pressure Signature

Lift and Drag

Sonic Boom Propagation

Sonic Boom on Ground

Compute Constraints

Compute f(x)

Check Optimality Conditions

No

Search for New x’s

Yes Stop

Figure 4.8: Schematic of direct sonic boom optimization. Although the examples above used the maximum overpressure as the objective, there were a few other sonic boom objectives considered. These will be discussed in a later chapter. In addition, the sonic boom could be used as a constraint while drag is minimized as the objective. 60

To demonstrate that the direct method works, a simple test was performed where the shape of an axisymmetric body was optimized for sonic boom. The sonic boom was constrained in that the overpressure had to fall between P max and P min . The design variables were the radii at 8 lengthwise locations. In order to prevent the optimizer from eliminating the body altogether, the length and the base area of the body were also constrained. In order to compare with known results, the length, base area, Pmax and P min were obtained from a Seebass-George example. The aerodynamics was analyzed using a discretized linear source method. Figure 4.9 shows the optimized sonic boom satisfying its constraint and the optimized shape having good agreement with Seebass-George. Since the radii agreed very well, the small differences observed in the near field pressure signature was attributed to discretization error.

Figure 4.9: Comparison between direct optimization and Seebass-George method.

61

4.5 Advantages of Direct Sonic Boom Optimization In the direct optimization of the sonic boom, various types of sonic boom objectives can be used, and the optimized shaped booms could have many forms. For example, the shaped boom shown in Figure 4.10 was obtained by minimizing the calculated dB(A) loudness of the sonic boom.

Figure 4.10: Optimized boom differing from the Seebass-George shape booms. In the example shown in Figure 4.9, where the optimized solution was compared with the Seebass-George method, the sonic boom still satisfied the constraints, despite the discrepancy in the F-functions. This is because the optimization was directly driven by the sonic boom, thereby ensuring that the sonic boom (not F-function, or equivalent area) would be optimal.

62

Since the shaped boom and area distribution is no longer under the limitations imposed by the Seebass-George method, the direct optimization approach can be used to optimize the sonic boom for the NLF and any aircraft configuration. An example is presented in Chapter 7.

63

Chapter 5 Sonic Boom Optimization

Sonic Boom Optimization In the last chapter, the case was made for pursuing a low-boom design approach using optimization. This chapter discusses the details of the sonic boom optimization, in particular, the objective function, search method and constraints. The handling of multiple objectives, i.e. sonic boom and drag, is also addressed.

5.1 Selection of Sonic Boom Metric Before embarking on the optimization, one must first decide on the metric by which the designs are measured. For conventional aircraft design, the lift to drag ratio or maximum take-off weight is often used as the metric of performance. For sonic boom, it is unclear what the appropriate metric should be that would translate to better public acceptability. The following are some possible sonic boom metrics: •

Initial shock overpressure (Dp)

•

Maximum overpressure

•

Loudness metric (example dBA)

•

Impulse

5.1.1 Initial Shock and Maximum Overpressure Initial shock overpressure (or initial D p) is the most common metric used because of historical work with N-wave sonic booms, in which case, it is the same as the maximum overpressure. A recent example of its use is DARPA’s Quiet Supersonic Platform program (initiated Fall 2000), which specifies an initial Dp of 0.3 psf as the sonic boom goal. Maximum overpressure was considered in Seebass and George’s sonic boom minimization method, resulting in the flat-top shaped boom of the method. 64

The problem with using the initial overpressure as a metric is that, firstly, it does not account for rise time, which has a major impact on the sonic boom noise. Secondly, it does not account for what happens after the initial shock. This does not matter for Nwaves, but is a problem for shaped booms. Figure 5.1 are four sonic boom examples. The first is an N-wave, the second is based on results published by Farhat38 et al., the third is a flat top shaped boom and the fourth is a ramp-type shaped boom. Based on initial overpressure, all the sonic booms shown in Figure 5.1 are equally good (or bad).

1.0

(a)

1.0

0.5

0.5

0.0

0.0

1.0

(c)

1.0

0.5

0.5

0.0

0.0

(b)

(d)

Figure 5.1: Sonic booms with same initial Dp.

5.1.2 Loudness Concern over whether the initial shock is an adequate measure of public annoyance led to the consideration of loudness as a metric for sonic boom39. The loudness level is a numerical value calculated using the entire pressure signature. However, there is no general agreement on which unit for loudness is best. Brown and Haglund40 studied various loudness metrics for sonic boom, i.e. dB, dBA, dBC, Stevens’s Mark VI and VII phons, PLdB etc. and concluded that, with the exception of dBC, their sensitivities were all similar. For the rest of their study, they chose dBA because of the simplicity of its calculation. Citing previous studies done in human response testing, they proposed sonic boom goals of 72 and 65 dBA for limited and unlimited overland supersonic flights respectively. Laboratory studies conducted later by Leatherwood and Sullivan41 on 65

human test subjects confirmed the correlation between dB loudness of sonic booms and how people rated loudness. In keeping with Brown and Haglund’s proposal, dBA was also used in the present work. dBA is the abbreviation for A-weighted decibel. Weightings are generally applied because the human ear responds differently to each frequency band, and different weightings are used (for example A and C) for different types of sound (impulse, white noise, etc.). The procedure for calculating dBA was obtained from Brown and Haglund40, Johnson and Robinson42, and Shepherd and Sullivan43. This involved calculating the onethird-octave band pressure levels from the sonic boom’s power spectrum, which is described below. For any waveform defined by p(t), the total energy is proportional to 2

• { p(t)} dt Ú-•

(5.1)

which according to Parseval’s Theorem is equal to †

•

Ú-• F(w ) 2dw

1 2p

or

1 p

•

Ú0

2

F(w ) dw

(5.2)

where F(w ) is the Fourier transform of p(t). Figure 5.2 shows the power spectrum of a † sample sonic boom calculated from the pressure-time signature using a Fast Fourier

Transform. The energy within any frequency band w1 to w2 is therefore 1 p

w2

Úw

1

2

F(w ) dw

(5.3)

The one-third-octave band energy level was computed using the above. The frequency † for each band is given in Table 5.1. Since the human auditory range is between 20 Hz and

20 kHz, not all frequencies from zero to infinity are needed. For loudness calculations, the above energy values (dimensions pressure2 x time) are converted to pressure levels by dividing the 1/3-octave band energy levels by 70 ms42, and normalized with the standard reference pressure of 2x10-5 N/m2. In addition, because the calculation is for a single 66

acoustic event, the band pressure levels (BPL) are reduced by 3 dB to account for two separate bangs heard from each sonic boom signature. Finally, A-weighting is applied to the BPLs and summed logarithmically to yield the dBA level. Figure 5.3 illustrates the difference between the A-weighted and unweighted band pressure levels, with band frequency increasing from left to right.

Figure 5.2: Power spectrum of sonic boom.

Table 5.1: 1/3 Octave Band Center Frequencies for 13 Hz to 22kHz. Band No. Center Freq. (Hz) Band No. Center Freq. (Hz) Band No. Center Freq. (Hz)

1

2

3

4

5

6

7

8

9

10

11

13.9

17.5

22.1

27.8

35.1

44.2

55.7

70.2

88.4

111

140

12

13

14

15

16

17

18

19

20

21

22

177

223

281

354

445

561

707

891

1123

1414

1782

23

24

25

26

27

28

29

30

31

32

33

2245

2828

3564

4490

5657

7127

8980

11.3k

14.3k

18.0k

22.7k

67

Figure 5.3: Unweighted and A-weighted band pressure level (BPL). Band frequencies are given in Table 5.1. Figure 5.4 is a plot of initial Dp (from 0.3 to 2.4 psf) versus the calculated dBA for symmetric N-waves of rise times of 1, 3, 6, 12 and 24 ms. For every doubling of Dp, the sonic boom loudness increased by 6 dBA. For an N-wave of given rise time, minimizing dBA would have the same effect as minimizing initial Dp. Figure 5.5 are results not previously found in sonic boom literature, showing the effect of the asymmetry on the loudness of N-waves. The rise time is fixed in this case. Each solid curve has fixed Dp at the rear while initial Dp varies from 0.3 to 2.4 psf. The dotted line is the loudness for the symmetric case. It is important to note that the effect of reducing initial Dp depends on rear Dp value. The results provide some important insights into sonic boom loudness. Firstly, the loudness is dominated by the strongest shock in the signature, which is evident when comparing the loudness of the 2.4 initial Dp points with the loudness of the 2.4 rear D p curve. Consequently, initial D p by itself is not a good metric for evaluating shaped booms.

68

DP

DP Rise Time

Figure 5.4: Effect of initial Dp and rise time on dBA level for symmetric Nwaves (170 ms duration).

DP

Rise Time

DPrear

Figure 5.5: Effect of initial Dp on dBA level for asymmetric N-waves (170 ms duration). 69

Figure 5.6 shows the change in dBA levels for symmetric shaped booms with initial Dp of 0.3 psf as the secondary rise time t increases from 3 to 24 ms. The maximum overpressure Pmax, which is fixed for each curve, ranges from 0.3 to 2.4 psf. The curve with P max of 0.3 psf is a flat top boom while the rest are ‘peaky’ shaped booms. The results illustrate the widely known observation that if the secondary rise time is greater than 20 ms, the maximum overpressure will not add to the loudness of the sonic boom. This has tremendous benefit to sonic boom mitigation. It also means that maximum overpressure is not a suitable metric for shaped sonic boom.

Pmax DP

t

Figure 5.6: Effect of secondary rise time on dBA level for symmetric shaped booms with initial Dp of 0.3 psf (180 ms duration). Figure 5.7 shows the calculated dBA levels for the sonic booms given in Figure 5.1. All the signatures have the same initial Dp, but the dBA levels for sonic booms (a), (b) and (c) are not the same. Signature (b) has a strong shock located behind the nose shock, while the tail shock in (c) is greater than all the shocks in (a). On the other hand, the dBA levels for signatures (a) and (d) are similar despite the difference in maximum overpressure because the secondary rise time is large. 70

From the sonic boom examples of Figure 5.1, it is evident that initial Dp or maximum overpressure is an inadequate metric for shaped sonic booms. Instead a loudness metric, in this case dBA level, was preferred.

1.0 0.5

(a)

0.0

1.0 0.5 0.0

1.0 0.5 0.0

1.0 0.5

(b)

(c)

(d)

0.0

Figure 5.7: dBA levels for the sonic booms shown in Figure 5.1.

5.1.3 Impulse The sonic boom impulse is the area under its pressure-time signature. Seebass and George suggested that the sonic boom impulse was one of the parameters to be minimized. Darden published parametric studies with sonic boom impulse as one of the metrics. It has been suggested that the impulse might be important when considering the effect of sonic booms on structures, but no study has been made to establish the link. For the present work, impulse was not considered, though it could easily be incorporated if desired.

71

5.2 Selection of Sonic Boom Objective Function The objective function is the parameter that is minimized (or maximized) in the optimization. The following describes some objective functions used in low-boom design.

5.2.1 Minimize Sonic Boom Metric With reference to the discussion on sonic boom metric, the natural choice for the sonic boom objective function was to minimize the calculated sonic boom loudness. However, optimized results revealed a problem not previously anticipated, which cast doubt as to whether loudness is a good objective function for optimization. On the other hand, minimizing peak overpressure was found to be a viable alternative to minimizing loudness. This will be discussed in the next chapter. When using the loudness as the objective, it is important to remember that the whole boom signature is affecting the loudness. Sufficient degrees of freedom must therefore be present. For example, if the space spanned by the set of design variables is able to shape the shock at the front but not at the rear, the optimizer would terminate before the optimal front shock is found because the rear shock would eventually dominate the loudness. This may mislead one into thinking that the optimal shape applied to the front shock as well. Although it makes sense to optimize the entire signature, it was useful to be able to study the front and rear shocks separately. By doing so, one could reduce the dimension of the problem when performing parametric studies to, say, determine the parameters that affect the front shock. The approach used here for shaping only the front shocks was to reset all the negative overpressures to zero in the signature (Figure 5.8). This approach was found to work reasonably well and was also applied to rear shock shaping by resetting all positive overpressures to zero.

72

Actual

Modified for shaping front shock

Modified for shaping rear shock

Figure 5.8: Modifications to pressure signature for studying front and rear shock shaping using loudness as an objective. Calculating the loudness for the positive and negative overpressures may also have advantages for optimizing the entire pressure signature. As mentioned previously, the optimizer will stall when the worst part of the sonic boom signature can no longer be improved. By having an objective function composed of the sum of the loudness of all three signatures (full, positive and negative), designs favoring either the front or rear shock would continue to be searched.

5.2.2 Minimize Difference with Target Sonic Boom Signature An approach commonly seen is to treat the optimization as an inverse design problem. In this case, a sonic boom signature is first specified, say from the Seebass-George method, and the objective is then to minimize the difference with the specified signature. Makino44 et al. followed such an approach to correct the geometry derived using linear theory in order to reproduce the desired shaped boom with CFD (Figure 5.9).

Figure 5.9: Sonic boom matching objective44.

73

5.2.3 Sonic Boom as a Constraint Instead of casting the sonic boom as an objective, it could be considered a constraint while the aerodynamic performance is maximized. Figure 5.10 is an example where drag is minimized subject to the sonic boom overpressure limits shown. The result was obtained from a non-lifting axisymmetric test case

Figure 5.10: Sonic boom as a constraint function.

5.3 Handling of Constraints Depending on the approach, lift, drag or sonic boom may be constraints in the optimization. Lift is always constrained to ensure level flight. When minimizing sonic boom, drag may also be a constraint and vice versa. In addition, the design variables need to be bounded to avoid unreasonable geometries from arising. Variables that exceed their bounds could simply be replaced by their limit value. However it was found that the addition of a small penalty helped to distinguish several out-of-bound points and steer the optimizer in the right direction to satisfy the bounds. The following is an example of how the upper bound xb of variable x was implemented when x>xb: J(x) = J(x b ) + wgb

(5.4) 74

†

where gb = max(0, x - x b ) w = weighting † Hence gb is active only when x>xb. Note because J is computed at xb instead of x, it does

not matter how small the weighting on gb is. Lift, drag and sonic boom constraints are handled similarly, except that now care must be given to ensure that the weighting applied to the penalty g is large enough.

5.4 Selection of Optimization Search Method The search methods are the process by which the optimizer finds a better solution. They can be broadly classified into two types: •

Gradient methods

•

Non-gradient methods

Figure 5.11: Drag versus change in radius at one fuselage station.

75

X

Figure 5.12: Calculated sonic boom loudness versus change in radius at one fuselage station (x). Gradient methods, as the name implies, utilize gradient information to determine the search direction, while non-gradient methods are based on function comparisons only. Where the objective and constraint functions are smooth, gradient methods are most efficient, i.e. they find the optimum point with the least number of function evaluations. Although non-gradient methods generally require more function evaluations to find the optimum, they can, however, better handle non-smooth functions. Figure 5.11 is an example of a smooth function while Figure 5.12 shows a non-smooth one. The first is a plot of the variation in inviscid drag computed by A502 for a supersonic aircraft for changes in radius from -0.3 to +0.3 ft at the location indicated by X in Figure 5.12. The nominal radius at X is 3 ft. Figure 5.12 is a plot of the calculated sonic boom loudness from the same A502 runs, showing that the calculated sonic boom loudness is nonsmooth. This non-smoothness is due to shock coalescence and its effect on dBA loudness. Figure 5.13 and Figure 5.14 compare the near and far field signatures at both sides of the discontinuity at location C in Figure 5.12. The small change to the near field 76

signature was enough to cause the shock coalescing at D. This sudden increase in shock strength results in the sudden change in calculated loudness. Although the shock in question is not dominant, the effect on the calculated loudness is enough to cause the sudden change.

Figure 5.13: Change in near field signature due to small perturbations in geometry. Bearing in mind that the non-smoothness is worse in higher dimensions, these results suggest that gradient-based methods are not suitable unless something is done to address the non-smoothness, such as using an approximate model45,46,47 for the loudness. Even so, 77

past experience has shown that gradients cannot be computed reliably with A502 (see Section 3.2). The drag variation shown in Figure 5.11 appears smooth because the step size shown is larger that what is usually required for gradient calculations. For this reason, gradient methods were not used in the present research.

Figure 5.14: Shock coalescent at far field signature due to small perturbation in geometry. Two search methods were used in this work: •

The Simplex or Polytope method48

•

A Genetic Algorithm (GA)

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5.4.1 The Simplex Method For an n-dimensional problem, the simplex method (not to be confused with the betterknown simplex method for linear programming) uses n+1 points at each stage. At the beginning, the simplex consists of the initial point and one point in each coordinate direction. At each iteration, the simplex is updated by replacing its worst point with a better point. The method assumes that a better point can be found in the direction along the line from the worst point to the centroid of the best n-points. Figure 5.15 illustrates this for a 2-dimensional case, where the goal is to minimize J(x) and J(x1)<J(x2)<J(x3). There are three cases to consider:

•

If the new point xr is neither the best or worst point, xr replaces the worst point.

•

If the new point xr is the new best point, the search direction is assumed to be good and the simplex is expanded to point xe.

•

If the new point xr is worse than the worst point, the simplex is contracted along the search direction, i.e. point xC1 and xC2 in Figure 5.15.

If none of the above produces an improvement, the simplex is shrunk about the best point x1 in all coordinates by some specified amount. The solution is found when all the points converge within some specified limit.

X1

X C1

Xe X centroid X C2

Xr

X3 X2 Figure 5.15: Simplex method.

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5.4.2 Genetic Algorithm The GA was real-coded (as opposed to binary coded). The mating pool consisted of everyone in the population. Each member in turn randomly picked a mate, other than itself. Two offspring were produced by linear crossover49: one was the average of both parents; the other was extrapolated on the side of the better parent. The better offspring replaced the worse parent only if the offspring was better, hence preserving the best solutions. Both offspring were mutated uniformly with a 10% probability rate. Mutation range was set at either 10% or 20% of the variable range.

5.4.3 Comparison of Simplex and GA Solutions The simplex method was very effective for minimizing the drag. Both simplex and the GA gave similar minimum drag designs, although the GA took nearly 10 times more functional evaluations. However, when applying the simplex method to sonic boom loudness, it was found that its solution was always worse than that of the GA. Going back to the non-smooth objective function shown in Figure 5.12, the difference is due to the presence of many local minima in the sonic boom objective function. Multiple local minima present a major obstacle to most optimizers, including simplex and gradientbased methods, since they are formulated to find a local minimum point only, with no guarantee of global minimum. If many local minima exist the only way for the optimizer to find the global optimum point is to use a starting point that leads to it. This is only feasible if the design space is already well understood, which is usually not the case. In practice, the optimization is run from a few starting points and the global solution is assumed to be found if all the runs converge to the same solution. The simplex method is a little more robust than gradient methods in this area because it uses information from more than one location, and therefore has a slightly better chance of finding a better minimum. Genetic algorithms are much better because they start with an initial population that spans the entire design space. This is why the GA often gives better sonic boom solutions than the simplex method.

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5.5 Multi-Objective Optimization So far the discussion on optimization has been either for drag or sonic boom. However the goal here is to optimize both. The classical approach to multi-objective optimization is to optimize the weighted sum of all the objectives:

Minimize J(x) = Â

M m=1

w m J m (x)

(5.5)

where M is the number of objectives, and wm is the weight given to objective m. In this † and sonic boom are being minimized so M=2. case, drag If the objectives are conflicting, i.e. the improvement of one objective can only be achieved at the expense of the other, the optimal solution would comprise a set of nondominated solutions. A solution x1 is said to dominate the other solution x2 if both of the following conditions are true:

•

x1 is no worse that x2 in all objectives.

•

x1 is strictly better than x2 in at least one objective.

The resulting non-dominant set is called the Pareto-optimal set. It is often called the Pareto front because it is located on a front in the objective set. An example of a twoobjective minimization is shown in Figure 5.16. J1 and J2 are the two objectives defined by: J1 (x) = x12 + x 22 J 2 (x) = (x1 - 2) 2 + x 22

The Pareto front, indicated by the solid line, was obtained by minimizing the following † weighted sum with the Simplex method: J(x) = aJ1 (x) + (1- a )J 2 (x)

†

(5.6)

81

for incremental values of a ranging from 0 to 1. Note that 0 corresponded to the single objective minimization of J2 and 1 corresponds to the single objective minimization of J1.

Figure 5.16: Two-objective optimization problem. The solid line is the Pareto front.

5.5.1 Multi-Objective GA With GAs, one can obtain the Pareto front from a single simulation. The entire front can be found in about the same number of generations as a single objective GA run. Figure 5.17 is a plot of the GA’s convergence history for a single point on the Pareto front using the weighted sum method. From the convergence history, the GA took about 7 generations to find the Pareto solution. If, say, 10 points were needed to define the entire Pareto front, it will then take a total of 70 generations to obtain the front. In comparison, Figure 5.16 shows the objective values for the initial population and the tenth generation. By the tenth generation, the Pareto front was already clearly defined by the population.

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Figure 5.17: Convergence history for one point on the Pareto front using weighted sum method on a single objective GA. The multi-objective GA used here was modified from the existing single objective GA (described above) by incorporating the following features: •

Dominance-based selection

•

Niche count to determine the better of same rank members and to improve spread of solutions on the Pareto front

Dominance-based selection means that instead of the objective value, selection was based on dominance. Every member of the population was assigned a rank number according to the number of members it was dominated by. The rank number was computed by first setting it to one for everyone. As every member was checked for dominance, its rank number was increased by 1 whenever it was dominated. Nondominated solutions will have a rank number of 1 at the end of the check. During the tournament selection, the person with the lowest rank number won. Ideally the GA is run until everyone’s rank number is one, i.e. every member lies somewhere on the Pareto front.

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With the current GA, the children must be ranked globally. They were initially compared with their parents only. However, this was incorrect because it overlooked the fact that others might still dominate the children. For example, Figure 5.18 shows three solutions on the Pareto front. Solutions 1 and 3 are used to produce the child. Even though the child was not dominated by the parents, and could potentially replace one of them, the child was still dominated by solution 2.

Parent 1 1

Child

2

Parent 2 3

Figure 5.18: Example showing children should be ranked globally. When using only the rank number for selection, the GA did nothing when the rank numbers were the same. To overcome this problem, the niche count was added as a tiebreaker. The niche count is traditionally used for promoting diversity in GAs to capture other optimal solutions in multimodal problems. It is also used to promote an even distribution of points along the Pareto front. For a two-objective problem with a population of N, the niche count nc for solution i was computed as follows49:

†

†

Ê J1(i) - J1( j ) ˆ 2 Ê J 2(i) - J 2( j ) ˆ 2 dij = Á max + Á max min ˜ min ˜ Ë J1 - J1 ¯ Ë J 2 - J 2 ¯

(5.7)

ÏÔ Sh(d) = Ì1ÔÓ 0

(5.8)

a

( ) d s sh

, if d £ s sh otherwise

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N

nc i = Â Sh(dij )

(5.9)

j=1

†

dij is the normalized distance between any two solutions i and j. Sh(d) is the sharing function. ssh is the distance beyond which the sharing function is zero. a determines the variation in the sharing function with distance. a=1 was chosen, which corresponded to a linear decrease in Sh(d) with distance. The niche count was normalized as follows so that its value did not exceed one:

nc inorm =

†

nc i

Â

N j=1

(5.10)

nc j

By doing so, it could be added to the rank number to distinguish between members of same rank number without demoting the rank of the member.

5.6 GA Population Size It is not clear how the required population size is related to the number of design variables. Some estimates exist for binary-coded GAs, but none for real-coded ones. In general, it has been observed that the population size is related to the complexity of the problem49. The more multi-modal, the larger the population should be. A larger population is required if no mutation is used, in order to ensure that the necessary building blocks are present in the crossover. However, the number of generations required for crossover methods is smaller. To illustrate the effect of population size on the Pareto optimal solution, a simple 12variable two-objective minimization problem is presented: 12

J1 = Â (x i - y *i ) 2 i=1 12

J 2 = Â (x i - z*i ) 2 i=1

† where y* and z* are two different target solutions. †

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Figure 5.19: Pareto front after 40 generations for a 12-variable problem with a population of 120. Figure 5.20 is the convergence history for a 12-variable single objective GA with population of 30, 60 and120. The objective function is J1 as defined above.

Figure 5.20: Convergence history of a single objective GA for a 12-variable problem with a population of 120.

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Chapter 6 Integration of Sonic Boom and Aerodynamics with Optimization

Integration of Sonic Boom and Aerodynamics With Optimization This chapter describes how the sonic boom and aerodynamic analyses were integrated with optimization in accordance with the direct sonic boom optimization framework introduced in Section 4.4. Two important areas that have not been addressed in previous chapters will be discussed here. These are firstly the automated A502 model generator, which enables the A502 input file to be created automatically during the optimization; and secondly, the parallelization of the optimization.

6.1 Integrated A502 Sonic Boom Analysis To facilitate integration with optimization software, the sonic boom prediction code should output the value of the objective function J(x) for a given set of input design variables x (Figure 6.1(a). With this in mind, an integrated sonic boom analysis package was developed, which consisted of four stand-alone programs,:

•

A502

•

MAKEPAN

•

BOOM

•

LOUD

A502 is the aerodynamic analysis program described in Chapter 3. MAKEPAN is a program written to automate the creation of A502’s input file for an aircraft model consisting of wing, fuselage, canard, horizontal tail and nacelle. BOOM is the sonic boom propagation code written based on the theory presented in Chapter 2. Finally, 87

LOUD is the program for calculating sonic boom loudness from its pressure signature. The current version was extracted from PCBOOM50.

X

Optimizer: Min J(x) subject to g(x)

MAKEPAN A502 model generator

x Input File for Full Aircraft Model Evaluate Functions J(x) and g(x)

A502 Aero Analysis Near Field Signature

J(x), g(x) (a) Basic structure of optimization software

Lift, Drag

Boom Propagation Code Far Field Signature Loudness or Peak DP Code

Integrated sonic boom analysis package

dBA or Peak DP

(b) Evaluation of objective function J(x) and constraints g(x)

Compute J(x) and g(x)

Figure 6.1: Design of an integrated sonic boom analysis for optimization. To compute the sonic boom for an aircraft, one only has to specify the parameters in the MAKEPAN input file. Running MAKEPAN creates the A502 input file for the aircraft model. After running A502, BOOM reads the A502 output file containing the offbody results and computes the far field sonic boom on the ground. Finally program LOUD takes the output boom signature from BOOM and calculates the sonic boom

88

loudness. If desired, the loudness calculation can be replaced by some other sonic boom objective, such as peak overpressure. For the optimization, the above four programs were executed in sequence by the optimizer (Figure 6.1). Instead of rewriting the MAKEPAN input file each time, a second input file containing only the design variables x was created. This new input file is read after the baseline parameters are loaded, thereby overwriting the baseline values. During optimization, only this second input file is updated, while the main input file that is read first is untouched.

6.2 MAKEPAN Automated A502 Model Generator This section describes the capabilities of MAKEPAN, the program written to automate the creation of the A502 aircraft model. The discussion is intended to give the reader an idea of the parameterization of the aircraft geometry and the level of detail that can be expected in the aircraft shape produced by the current sonic boom optimization. Details of the modeling procedure will not be presented, since it is nothing more than simple but tedious geometric calculations. The most basic aircraft model generated by MAKEPAN consists of a fuselage and a trapezoidal wing. One can add an outer wing, canard, horizontal tail and nacelle, as illustrated in Figure 6.1.

Outer Wing Canard

Hor. Tail Wing

Nacelle

Fuselage

Plane of Symmetry

Figure 6.2: MAKEPAN aircraft geometry components.

89

6.2.1 Fuselage Parameters The fuselage shape is defined by an Akima spline51 through any number of specified radii. Figure 6.3 is an example where it is specified by 4 internal sections. For section i, rin(i) is the nominal fuselage radius at location xin(i). Frup and frlo are factors by which the upper and lower fuselage is scaled about the wing plane. This feature was added in order to study the benefits of non-axisymmetric fuselage optimization on the drag of lowboom designs. Wing-fuselage intersection is computed based on an axisymmetric fuselage of nominal radius rin.

TOP VIEW

Wing

rin(2)

rin(1)

rin(3)

rin(4)

Symmetry Plane

xin(2) flen

SIDE VIEW frup(1) x rin(1)

frup(2) x rin(2)

frup(3) x rin(3)

frup(4) x rin(4)

Wing Plane frlo(1) x rin(1)

frlo(3) x rin(3)

frlo(2) x rin(2)

frlo(4) x rin(4)

SECTION VIEW

rin frup x rin Wing Plane

frlo x rin rin

Figure 6.3: Fuselage parameters.

6.2.2 Wing Parameters The wing is specified by parameters described in Figure 6.4. A biconvex airfoil thickness is used throughout and twist varies linearly. 90

WING PARAMETERS xrtle : root location rtcord : root chord tcrt : root thickness camber : root camber wins : root incidence sweep : leading edge sweep dihed : dihedral taper : taper ratio ytip : tip/break pt semispan twist : tip/break pt twist tctip : tip/break pt thickness cambert : tip/break pt camber

OUTER WING PARAMETERS sweep2 : leading edge sweep dihed2 : dihedral taper2 : taper ratio ytip2 : tip semispan twis2 : tip twist tctip2 : tip thickness camber2 : tip camber

sweep2 ytip2

sweep ytip xrtle

rtcord

Figure 6.4: Wing parameters.

6.2.3 Canard and Tail Parameters Figure 6.5 describes the parameters defining the canard and tail geometry. Biconvex airfoil thicknesses are also used and twist varies linearly.

CANARD PARAMETERS

TAIL PARAMETERS

canxrtle canrtcord cantcrt cancamber canins cansweep cantaper canytip cantwis cantctip

tailxrtle tailrtcord tailtcrt tailcamber tailins tailsweep tailtaper tailytip tailtwis tailtctip

: root location : root chord : root thickness : root camber : root incidence : leading edge sweep : taper ratio : tip leading edge : tip twist : tip thickness

Canard

canrtlex

canrtcord

: root location : root chord : root thickness : root camber : root incidence : leading edge sweep : taper ratio : tip leading edge : tip twist : tip thickness Hor. Tail

Plane of Symmetry tailxrtle

tailxrtcord

Figure 6.5: Canard and tail parameters. 91

6.2.4 Nacelle Parameters In order to avoid the complexity of calculating component intersections, nacelles in MAKEPAN are limited to externally mounted types (Figure 6.6). For simplicity, the nacelle pylon is omitted.

Figure 6.6: Nacelle modeled by MAKEPAN. Figure 6.7 illustrates three ways of modeling the nacelle with A502. The first and most sophisticated method was to specify the normal flow through the panels at the inlet and base of the nacelle, as shown in Figure 6.7(a). Unfortunately this feature worked only for subsonic flow. The next method (b) was to make the nacelle hollow. This was a reasonable approximation for the engine operating at design point, with no intake spillage and overexpansion of the exhaust plume. It did not matter that the flow going through the nacelle was nothing like in a real engine, since only the outer flow was important for sonic boom. However, reflection of shocks within the nacelle appeared to cause the divergent solutions at times, so it too was not adopted. The third method was to model the nacelle as an equivalent parabolic body. As shown in Figure 6.7(c), the diameter of the equivalent body gives a frontal area that is equal to the difference between the nacelle’s frontal area Anc (Figure 6.7b) and the stream tube area AO entering the engine. Though geometrically not as accurate as the other two, it is a good enough approximation for sonic boom and wave drag for the current purpose. Figure 6.8 below describes the parameters for the nacelle model.

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Specify Inlet Flow

Specify Exit Flow shock

(a)

Flow Through Nacelle

(b)

Ao

A nc

shock

A nc - Ao

(c)

shock

Figure 6.7: Nacelle modeling options.

93

NACELLE PARAMETERS elen xnc rnc engx0, engy0, engz0

: nacelle length : x for nacelle radii distribution : nacelle ext radii distribution : location of engine origin rnc

xnc elen

Wing

Nacelle

Fuselage engx0

Figure 6.8: Nacelle parameters.

6.2.5 Near Field Signature Orientation Since the A502 model is aligned with the axes while the onset flow is not, the near field signature for boom prediction is not parallel with the x-axis (Figure 6.9). MAKEPAN computes the near field orientation, which is a function of angle of attack.

Free Stream Mach Line R

Mach Angle

Near Field Line Angle of Attack

Figure 6.9: Orientation of near field signature.

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Create SubDirectories

Create New Population

x MPI J(x) MPI

Select Mate l cd case directory l MAKEPAN

Create Offspring 1 Population

l A502 l BOOM l LOUD / Peak DP

Repeat for N Generations

- CPU 1 -

Create Offspring 2 Population

Evaluate Offspring 1 Population

Evaluate Offspring 2 Population

x MPI

n CPU’s

J(x) x MPI J(x)

Elimination

New Population

Figure 6.10: Parallel genetic algorithm.

6.3 GA Optimization Using Multiple Processors Running the GA on multiple processors saves time because the evaluation of the population’s objective function can be carried out in parallel. The parallel GA is essentially the same as the single processor GA. In fact, the main program still runs on one processor. Figure 6.10 illustrates how the parallel GA works. A subdirectory must first be created for each population member so that the A502 scratch files do not conflict 95

when running A502 in parallel. Two new populations comprising the interpolated and extrapolated offspring are generated. MPI is used to execute the integrated sonic boom analysis in parallel, and is run only when evaluating the initial population and offspring populations. Once the entire population has been evaluated, the sonic boom and aerodynamic results are consolidated from all the subdirectories and the GA continues on one processor until time to evaluate another population.

6.4 Simplex Optimization Using Multiple Processors The simplex algorithm is essentially the same for single and multiple processor runs. With multiple processors, the evaluation of the objective functional is carried out in parallel using MPI (Figure 6.11). The first time this happens is with the initial simplex. Instead of computing the expansion and contraction sequentially, this is all done at once in parallel. The evaluation of the simplex after a general contraction along all the vertices is also performed in parallel.

Create SubDirectories

Create Initial Simplex

Expansion / Contraction

x MPI

n+1 Cases

J(x)

x MPI

4 Cases

MPI

n Cases

J(x)

General Contraction

x J(x)

Repeat until converged

Figure 6.11: Parallel simplex optimization.

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Chapter 7 Aircraft Optimization Examples

Aircraft Optimization Examples This chapter discusses four examples of aircraft sonic boom optimization using the multiobjective genetic algorithm. Two candidate low-boom configurations were optimized. The first is a conventional supersonic configuration, featuring a highly swept wing with aft-mounted nacelles (Figure 7.1); the second is a natural laminar flow (NLF) wingcanard configuration (Figure 7.19). Using an integrated MDO program based on area-rule aerodynamics, a baseline design for the conventional configuration that satisfied mission requirements was obtained. The NLF configuration was put together solely to demonstrate that the current method could be applied to configurations excluded by the Seebass method. No previous MDO was therefore carried out for its baseline. In the first example, the conventional configuration was optimized using dBA loudness, which was the natural choice for sonic boom objective function because of its good performance as a sonic boom metric. However, some undesirable elements observed in the optimized boom signature led to the second example, which minimized the peak (or maximum) overpressure instead of loudness. The third example is the optimization of the wing-canard NLF configuration. The fourth example investigates the non-axisymmetric shaping of the fuselage to take advantage of wing shielding for improving drag without affecting the sonic boom below the aircraft.

7.1 Conventional Configuration Using dBA Loudness The weight and dimensions of the conventional configuration are given in Table 7.1. It consisted of an axisymmetric fuselage, a cranked delta wing, a trapezoidal horizontal tail and nacelles mounted on the aft body.

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Table 7.1: Weight and dimensions of conventional configuration. Weight: 70,000 lbs

Wing Span: 64.8 ft

Mach: 1.5

Wing Area: 1650 sq ft

Altitude: 50,000 ft

Wing Leading Edge Sweep: 55 deg

Fuselage Length: 108 ft

Leading Edge Extension Sweep: 74.6 deg

Cabin Diameter: 6.2 ft

Horizontal Tail Area: 180 sq ft

Crew Station Diameter: 5 ft

Horizontal Tail Aspect Ratio: 2.9

Nacelle Length: 37 ft

Horizontal Tail Leading Edge Sweep: 55 deg

Nacelle Diameter: 4.16 ft

Horizontal Tail Taper Ratio: 0.28

Stream Tube Area: 13.3 sq ft Figure 7.1 is a plan view of the A502 model showing the geometry and panel density for this configuration. The wing consisted of 28 panels around the chord and 11 along the semi-span; the fuselage had 12 panels around the cross-section and 40 lengthwise; the tail had 16 panels around the chord and 4 in the semi-span; the nacelle had 8 panels in the cross-section direction and 8 panels lengthwise.

Figure 7.1: Conventional supersonic configuration. 11 design variables were used for this optimization. They are described in Figure 7.2. The values in parenthesis are the variable bounds.

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x1-6

: fuselage diameter

x7

: horizontal tail x-location (100 to 110 ft)

x8

: horizontal tail root incidence (-3 to 3 deg.)

x9

: wing twist at break point (0 to –2 deg.)

x10

: wing twist at tip (0 to –3 deg.)

x11

: angle of attack (1.5 to 3.5 deg.) X10

X9

X1

X2

X3

X4

X5

X8

X6

X7

Figure 7.2: Layout and design variables for conventional configuration. The fuselage was defined by an Akima spline through the six fuselage variables, x1 to x6, located at 0.05, 0.1, 0.3, 0.5, 0.6 and 0.8 fuselage length respectively. Initially 11 interior points were used, but the optimized results were found to be reproducible with the six defined here. The number of variables on the fuselage was limited for reasons of computational cost. The reduced design space necessitates using the horizontal tail for reducing the aft shock of the sonic boom. Longitudinal stability was not constrained because, for the fixed tail size and allowable movement, stability was expected to be achievable by shifting the center of gravity. Two fuselage points were specified near the nose to allow for blunt nose shapes commonly featured in low-boom designs in the past. Wing plan-form was fixed because preliminary MDO had determined that it was constrained by low speed requirements. Wing twist, which varied linearly from root to break point and to tip, was included to improve the induced drag. A parabolic wing camber was defined such that the leading edge slope was aligned with the free stream: yc = tan(a+q) x (1-x) where 0 £ x £ 1 where a = angle of attack in radians q = wing twist in radians 99

Though not the same as the local angle of attack, this was a simple way of eliminating the leading edge suction peak, which was believed to be detrimental to the sonic boom shaping. The objective was to minimize sonic boom dBA loudness and drag: Minimize

J1 = Sonic boom dBA loudness J2 = Drag

Subject to:

Lift ≥ Weight xmin < x < xmax

Lower bounds for x3 and x4 were set by cabin diameter constraints, while the bounds for the other fuselage radii were estimated to avoid severely varying profiles that would cause A502 to fail. Although a population of 120 was earlier shown to be sufficient for a 12 variable problem, a population size of 204 was selected here to account for the more complex sonic boom function. Using a Linux cluster consisting of 12 1-GHz Intel Pentium III processors to evaluate the population in parallel, it took a little under 22 minutes to evaluate each generation. No termination criteria were defined. Instead the optimization was run for 20 generations at a time until the Pareto front showed little change and the average rank of the population was reasonably small.

100

Figure 7.3: Evolution of population from start to 60th generation. Figure 7.3 shows the population of the 1st, 20th, 40th and 60th generation. Lift-to-drag ratio (L/D) is plotted instead of drag because it is a more meaningful parameter for aircraft design. Since minimizing drag is the same as maximizing L/D, the Pareto front moves in the direction of increasing L/D and decreasing loudness. The objective values plotted include the penalty added for constraint violations. However, at the end of the optimization, none of the constraints were violated (Figure 7.6), so the Pareto front shown reflects the actual values.

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Figure 7.4: Pareto front for conventional configuration with loudness objective. The simulation was stopped after the 80th generation as the Pareto front began showing little change after 60 generations. The average rank, maximum L/D and minimum loudness histories, plotted in Figure 7.5, also showed little change after 60 generations. The population at the end of the optimization is shown in Figure 7.4. Loudness ranged from 84.5 to 88.5 dBA, while inviscid L/D ranged from 14.3 to 20.6.

102

Figure 7.5: Solution histories for conventional configuration with loudness objective.

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Also shown in Figure 7.4 is the L/D based on the total drag, which ranged from 7.2 to 8.5 along the Pareto front. The total drag is the sum of the inviscid drag output by A502, and the viscous drag calculated using PASS, an aircraft design program developed by Kroo52. In addition to skin friction and nacelle base drag, PASS also accounts for drag associated with miscellaneous items like air conditioning system, cooling systems, and the many necessary protuberances that exist on an airplane. The computed viscous drag (Cdviscous=0.011, with reference area 1200 ft2) was assumed constant for the whole simulation because the changes to the geometry were not significant enough to affect it.

Figure 7.6: Normalized lift of population for conventional configuration with loudness objective. Figure 7.7 shows the variation in the fuselage of the Pareto optimal solution set. Cabin diameter constraint at x3 was active in all cases. The fuselage was widest in the optimal boom solution. As the fuselage became narrower, the boom loudness increased while the drag decreased until the minimum drag solution was reached. Angle of attack increased by no more than 0.5 degrees (Figure 7.8) going from low-boom to low-drag, while wing downward twist increased by 1 degree at the tip (Figure 7.9). From these results two main features distinguished between low-boom and low-drag designs. Firstly the low-boom design had a blunt nose while the low-drag one did not. Secondly, the aft fuselage diameter was clearly smaller for the low-drag design because of wave drag area104

ruling. These differences are already well known and therefore validate the optimization results.

Figure 7.7: Pareto optimal fuselage geometry for conventional configuration with loudness objective.

Figure 7.8: Pareto optimal angle of attack for conventional configuration with loudness objective.

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Figure 7.9: Pareto optimal wing for conventional configuration with loudness objective. Figure 7.10 is the A502 near field pressure signatures (without nonlinear aging) for the minimum drag and minimum boom solutions. Significant differences existed in the front half of the signature, resulting in the sonic boom signatures shown in Figure 7.11.

Figure 7.10: A502 near field pressure signatures for conventional configuration with dBA objective. For the minimum drag signature, the initial shock was 0.6 psf, followed 30 ms later by a second shock, also 0.6 psf in magnitude. Positive peak overpressure was 0.8 psf, 106

occurring at the top of the second shock. From the area balancing of the F-function, the first shock was traced back to the nose, while the second was from the wing. Peak negative overpressure of –0.7 psf was located at one of the two shocks at the rear. Both rear shocks were of similar magnitude of 0.3 psf, and separated by about 5 ms.

Figure 7.11: Sonic boom signatures on ground for conventional configuration with loudness objective. For the minimum boom signature, the initial shock was 0.4 psf. The peak positive overpressure, slightly below 0.6 psf, was located 1 ms after the initial shock. Three other shocks, including the wing shock, were seen behind the peak overpressure. The rear of the signature consisted of one shock of magnitude 0.3 psf. The peak negative overpressure was about –0.6 psf. From the boom signatures, it was deduced that the 4 dBA difference between the two was the result of two things. Firstly, the initial shock was reduced from 0.6 to 0.4 psf, and secondly, the wing shock was reduced from 0.6 to 0.3 psf. In the minimum drag case, both the initial shock and wing shock dominated, hence the reduction in the two would lower the loudness.

107

An obvious concern in the minimum boom signature was that the initial shock and the peak overpressure were separated by only 1 ms. This is the result of the optimizer exploiting the loudness calculation since a sensitivity study confirmed that a separation of just under 1 ms was optimal in this case. Since shock rise times are typically in the order of 3 ms, it appears likely that the shocks would have merged in practice. Therefore, even though loudness is a good metric for assessing sonic boom acceptability, it is unclear that it makes a good sonic boom objective function. Further study is therefore needed to determine the robustness of loudness-minimized solution.

7.2 Conventional Design Using Peak Overpressure The optimization of the conventional configuration was repeated using peak overpressure and the results compared with the results obtained by minimizing loudness. Both positive and negative overpressures were considered. The sonic boom objective was thus: Minimize

J1 = max (DPmax , -DPmin) J2 = Drag

All variables and constraints were the same as before. This time, however, the Pareto front converged after 40 generations, which suggested that peak overpressure was a better behaved function than loudness. Figure 7.12 is a plot of peak overpressure and both inviscid and viscous L/D for the population. Peak overpressure of the Pareto front ranged from 0.48 to 0.69 psf. Inviscid L/D ranged from 14.6 to 19.5, while viscous L/D was between 7.2 and 8.2. At maximum L/D, peak overpressure was lower than before (Figure 7.3). However the minimum loudness was 2 dBA higher.

108

Figure 7.12: Pareto front for conventional configuration with peak overpressure objective. Near field and sonic boom signatures for the minimum drag and minimum sonic boom solutions are shown in Figure 7.13 and Figure 7.14. Again the near field signature of the low boom design was characterized by the strong shock at the nose. The sonic boom signatures were similar to the previous ones (Figure 7.11), except now the initial overpressures have increased a little to match the peak overpressures, since a larger initial overpressure would reduce the drag. The peak overpressures were lower (but not to the level of the previous initial shocks) because they were forced to by the objective function. In the previous optimization the peak overpressures were allowed to be higher because 109

loudness is primarily determined by shock strength, i.e. the change in overpressure, rather than the absolute overpressure. One good thing about overpressure-minimized sonic booms is they appear more robust to atmospheric disturbance, since the shocks are well separated and the overpressures are lower.

Figure 7.13: A502 near field signature for conventional configuration with peak overpressure objective

Figure 7.14: Sonic boom pressure signatures for conventional configuration with peak overpressure objective.

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Figure 7.16 shows similar variation in the Pareto optimal fuselage shapes as before (Figure 7.7), but wing twist showed no discernible trend.

Figure 7.15: Pareto optimal fuselage geometry for conventional configuration with peak overpressure objective.

Figure 7.16: Pareto optimal wing twist for conventional configuration with peak overpressure objective.

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Figure 7.17: Loudness of Pareto optimal solutions in order of increasing overpressure.

Figure 7.18: Population resulting from minimizing loudness and peak overpressure. Figure 7.17 is a plot of loudness for the Pareto optimal designs in order of increasing overpressure. Loudness generally increased as overpressure increased. This seemed to suggest that peak overpressure might also be a suitable sonic boom objective function, despite being an inadequate sonic boom metric. This is because the optimization would match the initial shock with the maximum overpressure elsewhere in the boom 112

signature (Figure 7.14), since having a lower initial shock penalized drag without benefiting the sonic boom objective. With nothing to reduce the initial shock below the peak overpressure, it was not surprising that the minimum loudness was worse than before. Figure 7.18 is a comparison of loudness versus inviscid L/D for the population optimized using peak overpressure and loudness. Although the loudness-minimized solutions were quieter, the two would be very similar if shocks were to merge in the loudness-minimized signatures.

7.3 Natural Laminar Flow Wing-Canard Configuration The NLF configuration considered here consisted of an axisymmetric fuselage, a NLF wing and canard. The weight and fuselage were based on the conventional configuration. Table 7.2 is a summary of its weight and dimensions. Figure 7.19 is a plan view of the A502 model for this configuration. The wing was modeled with 28 panels around the chord and 11 along the semi-span; the fuselage was modeled with 6 panels along half the cross-section and 40 lengthwise; the canard was modeled with 16 panels around the chord and 4 in the semi-span; and the nacelle was modeled with 8 panels around the cross-section and 8 lengthwise.

Table 7.2: Weight and dimensions of NLF configuration. Weight: 70,000 lbs

Wing Span: 64.8 ft

Mach: 1.5

Wing Area: 1750 sq ft

Altitude: 50,000 ft

Wing Leading Edge Sweep: 19 deg

Fuselage Length: 108 ft

Canard Aspect Ratio: 2.9

Cabin Diameter: 6.2 ft

Canard Sweep: 19 deg

Crew Station Diameter: 5 ft

Canard Taper Ratio: 0.28

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Figure 7.19: NLF wing-canard supersonic configuration. For the optimization, 11 design variables (and their bounds) were defined as follows: x1-6

: fuselage diameter

x7

: canard area (100 to 300 sq ft)

x8

: canard x-location (0.2 to 0.45 fuselage)

x9

: canard root incidence (-3 to 1 deg)

x10

: wing twist at tip (0 to -3.5 deg)

x11

: angle of attack (1 to 3 deg) X10

X9 X8

X1

X2

X7

X3

X4

X5

X6

Figure 7.20: Design variables for NLF wing-canard configuration. The fuselage was taken from the conventional configuration. Wing twist varied linearly from root to tip. Stability and control were not constrained. The objective was to minimize sonic boom peak overpressure and drag as follows: 114

Minimize

J1 = max (DPmax , -DPmin) J2 = Drag

Subject to: Lift ≥ Weight xmin < x < xmax

The same constraints and bounds were imposed on the fuselage. A population of 204 was run for 40 generations. Figure 7.21 shows the population after 80 generations. The overpressures on the Pareto front ranged from 0.55 to 0.8 psf, while inviscid L/D ranged from 7.5 to 17. As expected, maximum inviscid L/D was (13%) lower than for the conventional case (Figure 7.12) because of the low wing sweep. A sudden drop in the L/D was observed around 0.575 psf as nose bluntness approaches its upper bound. As before, PASS was used to calculate the viscous drag and the results with viscous drag is plotted in Figure 7.22. Two sets of viscous results are shown. One assumed 95% laminar flow over the wing while the other assumed only 5%. The 95% laminar wing has a 40% greater L/D than the 5% laminar wing. Also shown for comparison in Figure 7.22 is the final population for the conventional configuration. The comparison is a good illustration of how a NLF concept may come up on top in terms of total drag reduction even though it is loses out in inviscid drag. Of course, its success is dependent on the amount of laminar flow it can achieve. Another thing brought out by Figure 7.22 is that the sonic boom overpressure for the NLF configuration is worse than the conventional configuration because the lift is less spread out than the conventional configuration.

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Figure 7.21: Pareto front without viscous drag for NLF configuration.

Figure 7.22: Pareto front with viscous drag for conventional and NLF configurations. The near field and sonic boom signatures are shown in Figure 7.23 and Figure 7.24. The presence of the canard is felt not only in the near field pressure signature, but also in the sonic boom on the ground. For the minimum boom case, three shocks of 116

approximately equal overpressure of 0.5 psf could be seen at the front of the boom signature. These shocks originate from the nose, canard and wing. The maximum overpressure, only slightly greater at 0.55 psf, occurred at the rear shock, which was not surprising because of the lack of degrees of freedom at the back. Adding a horizontal tail would likely improve the situation. What was surprising was that the front three shocks had lower overpressures than the peak value, especially with the nose hitting its upper bound (Figure 7.26). In fact it appeared that the optimizer was trying to circumvent the upper bound constraint by increasing the angle of attack (Figure 7.29). To maintain the lift, greater downward twist was applied to the wing (Figure 7.27).

Figure 7.23: A502 near field pressures for NLF configuration. For the minimum drag case, the sonic boom had two shocks in the front and one at the rear. All three shocks had overpressures of 0.8 psf. The initial overpressure was the result of the merging of the nose and canard shocks. These results are consistent with those seen previously. Figure 7.25 shows the lower surface Cp for the minimum drag and boom solutions. It also serves as a comparison between the two designs, especially the canards. The fact that the optimizer did not try to remove the canard meant that the canard was necessary. For the minimum boom case, the canard carried about 10% of the total lift while it carried nearly no load for the minimum drag case. This raised the question as to why the canard 117

was not replaced by adding fuselage volume in the latter case. The answer is that the current fuselage parameterization did not provide the degree of freedom to produce the effect of the canard volume.

Figure 7.24: Sonic boom signatures for NLF configuration.

Figure 7.25: Comparison between minimum drag and boom designs for NLF configuration.

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Figure 7.26: Pareto optimal fuselage for NLF wing-canard configuration.

Figure 7.27: Pareto optimal wing tip twist and canard root incidence for NLF configuration.

119

Figure 7.28: Pareto optimal canard area for NLF configuration.

Figure 7.29: Pareto optimal angle of attack for NLF configuration.

120

Figure 7.30: Normalized lift of population for NLF configuration.

7.4 Optimization of Asymmetric Fuselage The purpose of this optimization was to investigate if wing shielding could be taken advantage of to shape the upper fuselage for drag reduction without penalizing the sonic boom below the aircraft. Up until recently53, the optimization of a non-axisymmetric fuselage section was never incorporated into low-boom design, probably because of traditional stemming from the use of axisymmetric aerodynamics. The conventional configuration was chosen because the wing covered a large section of the fuselage. To better isolate the effect, the wing had no twist and camber. Figure 7.31 illustrates the 12 design variables, which are described below, with bounds given in parenthesis: x1-6

: fuselage diameter

x7

: horizontal tail x-location (100 to 110 ft)

x8

: horizontal tail root incidence (-3 to 3 deg.)

x9-11

: fuselage camber parameter (0.05 to 1.95)

x12

: angle of attack (1.5 to 3.5 deg.)

121

X8 X1

X2

X3

X4

X5

X6

X7 X9

X11

X10 Fuselage Camber

Figure 7.31: Design variables for conventional configuration with nonaxisymmetric fuselage. z

r a q

Wing Plane

y b

r

Figure 7.32: Non-axisymmetric fuselage definition. Figure 7.32 shows the two half ellipses that define the non-axisymmetric fuselage as described in Section 6.2.1. r is the radius of the baseline circular cross-section section. The co-ordinates of the non-circular fuselage section are: y = r cos q z = a sin q

for upper fuselage

z = b sin q

for lower fuselage

122

The non-circular fuselage cross-sectional area is: S = 12 (par + pbr)

†

In order to reduce the number of design variables, the area of the fuselage section was constrained to be equal to the axisymmetric area pr2. Hence a + b = 2r By inspection, the above is satisfied if a = fr and b = (2-f)r, for 0 £ f £ 2. When f = 1, the fuselage cross-section is circular. When f > 1, the fuselage deforms upwards; when f < 1, it deforms downwards. This was similar to adding camber to the fuselage, though not exactly the same. As in the previous examples, r was defined by an Akima spline through x1 to x6. Another spline through x9 to x11 (Figure 7.31) was used to define the variation of f along the fuselage. Loudness was minimized in this investigation. The same cabin diameter and lift constraints were imposed. Population size was increased to 240 because of an additional design variable. For the first time, mutation was used in the optimization to improve robustness. A uniform mutation was set to occur with a 10% probability rate. Maximum allowable mutation range was 20% of the variable range. The above optimization was repeated with an axisymmetric fuselage in order to provide a basis for comparison. The same population size and mutation were used.

123

Figure 7.33: Pareto front for non-circular fuselage case. Figure 7.33 shows the Pareto front obtained after 80 generations. The nonaxisymmetric fuselage was clearly an improvement over the axisymmetric one. Inviscid L/D in the low-boom region improved by nearly 30%, while at the low-drag end it improved by about 9%. The drag penalty for sonic boom reduction was therefore lower for the non-axisymmetric case. In fact, for the same loudness as the low-boom axisymmetric case, the inviscid L/D of the non-axisymmetric case was nearly as high as the maximum inviscid L/D for the axisymmetric case. Was the drag reduction due to the upper fuselage exploiting wing shielding to lower the drag? Figure 7.34 and Figure 7.35 appear to suggest that. Figure 7.34 shows the radius and side profiles of the non-axisymmetric fuselage in the Pareto optimal set. The upper fuselage profile for the minimum boom and drag solutions were very similar. In fact, all the upper fuselage profiles in the Pareto set were quite similar. In contrast, the lower surface exhibits the familiar distinctions between low-boom and low-drag solutions, also seen in the axisymmetric result here (Figure 7.35).

124

Figure 7.34: Pareto optimal non-axisymmetric fuselage geometry.

Figure 7.35: Pareto optimal axisymmetric fuselage geometry. Finally Figure 7.36 and Figure 7.37 show the near and far field signatures for the non-axisymmetric fuselage case. Both signatures share similar features with the axisymmetric fuselage results. Hence we conclude that the upper fuselage differences had little impact on the sonic boom below the aircraft.

125

(a) Non-axisymmetric fuselage

(b) Axisymmetric fuselage Figure 7.36: A502 near field pressure signatures for non-axisymmetric and axisymmetric fuselage.

126

(a) Non-axisymmetric fuselage

(b) Axisymmetric fuselage Figure 7.37: Sonic boom signatures for non-axisymmetric and axisymmetric fuselage.

7.5 Effect of Shock Rise Time on Results No shock rise time had been applied to the results presented thus far. Hence the loudness values reported thus far were unrealistically high because finite rise time is always observed in real sonic booms. Initially, a constant 3 ms rise time was added to all shocks during optimization in the manner suggested by Needleman39 et al. (Section 2.10, Figure 127

2.13), i.e. by simply adding rise time to the signature, resulting in the lengthening of the signature. Two problems were identified which led to the decision to omit rise time in the optimization. Firstly, with rise time, the merging of shocks resulted in larger changes to calculated loudness, thereby magnifying the non-smoothness in the loudness function. Secondly, as seen in the above examples involving loudness minimization, the optimization had a tendency of creating multiple small shocks that were at times separated by a mere 1 ms. With rise time implemented as described above, the shock could be separated by an even smaller amount (say 10-10!), and 3 ms will still be added. It was thought that the latter problem would be resolved by applying rise time after the optimization. Unfortunately, as the results have shown, the shock separation was still too small. On hindsight, it might be better to keep the rise time in the optimization, but with one key difference. Instead of simply adding the rise time behind the shock and extending the duration of the sonic boom, the duration should be fixed. By doing so, closely spaced shocks would merge and then penalized in the subsequent loudness calculation. Table 7.3 presents some of the low-boom loudness adjusted for rise time. Two rise time models are used. The first treats all shocks as having a rise time of 3 ms, the second is an empirical model from Needleman39 et al. In both models, the rise time was added as described above. With the empirical model, the optimized conventional configuration meets the 72 dBA-criteria suggested by Brown40 for limited overland supersonic flight.

Table 7.3: Low-boom dBA loudness adjusted for shock rise time. dBA Minimized

Dp Minimized

No Rise Time

84.3

86.7

Constant 3 ms

73.3

75.5

Needleman & Darden (1991)

66.1

69.5

128

Chapter 8 Conclusions and Future Work

Conclusions and Future Work 8.1 Conclusions A rapid preliminary design optimization method incorporating classical sonic boom theory, a 3-D panel method and a multi-objective GA was successfully developed. Previous supersonic aircraft optimization focused only on drag or sonic boom, but the method developed minimized both sonic boom and drag simultaneously. Since the optimal design for the two objectives are different, the solution of the optimization is a Pareto front where improvement in sonic boom would result in an increase in drag. The Pareto optimal solutions obtained were consistent with previously known trends, thereby validating the method. In contrast to traditional inverse design approach based on area distribution, sonic boom is optimized directly. This fundamental difference enables the method to work for any configuration, unlike inverse design, which is restricted to configurations with gradually varying lift distribution. To demonstrate this capability, the method was successfully applied to a low-swept natural laminar flow configuration as well as a conventional arrow wing configuration. In both cases, the initial and peak overpressures for optimal sonic boom were lower than that of optimal drag by about 30%. Loudness gives a better assessment for public acceptability of shaped booms than initial or peak overpressure. However, loudness has the disadvantage of being a nonsmooth function. A GA was therefore used for its optimization. For multi-objective optimization such as here, the Pareto optimal solution obtained by the GA helped to offset the high cost of the GA. Another problem discovered was that the low-boom 129

signatures obtained with loudness minimization were sometimes questionable because of closely spaced multiple shocks with higher peak overpressures. In a real atmosphere, these shocks may coalesce into stronger shocks because their separation is less than the rise time predicted for these shocks. This problem may be eliminated by including rise time in the optimization in a manner that does not move the location of each shock. Peak overpressure, on the other hand, was found to be a possible alternative for sonic boom objective, despite being an inadequate measure of sonic boom acceptability. Reasonably low loudness was achieved, though not as good as the results obtained by minimizing loudness directly. The main advantage is that solutions may be more robust to atmospheric uncertainties as peak overpressures are lower and shock separations greater. There is therefore little possibility of shocks merging to form stronger shocks. Another advantage is that better convergence was observed, which suggested that the function is better behaved. If so, more efficient optimization methods, perhaps gradientbased, may be used. The A502 panel code was well suited for this work. Its rapid turnaround time was critical to the viable use of a GA. Though a linear source model is faster, it does not account for wing-body aerodynamic interactions, which have significant impact on drag and sonic boom. For the Mach number and configurations of interest, A502 results showed good agreement with an Euler flow solver but at a fraction of the time. For more detailed design, however, an Euler solver is still better because it can model complex flow phenomenon such as nacelle inlet spillage and jet exhaust plume. With the rapid improvements seen in computer processor speed, parallel computing and CFD algorithms, it is envisaged that A502 would eventually be replaced with an Euler solver in the current application. An initial overpressure of 0.4 psf was achieved for a Mach 1.6 70,000 lb conventional configuration. The L/D including viscous drag was around 7.5. If sonic boom was ignored, the maximum L/D was around 8.5. Sonic boom loudness for optimal drag was 4 dBA higher than for optimal sonic boom loudness. Inviscid L/D for optimal 130

boom was 25% lower than the optimal inviscid L/D. Using an empirical model for rise time the predicted sonic boom loudness was below 70 dBA, which is within previously suggested goals for limited supersonic flights over land. However further sonic boom acceptability studies are required before one can conclude that the sonic boom is indeed acceptable. Significant improvement in drag was achieved by shaping the upper fuselage differently from the lower fuselage for the conventional configuration. For the same sonic boom loudness the inviscid L/D increased by nearly 30 percent. L/D for the optimal sonic boom design was nearly as high as the optimal L/D for the axisymmetric fuselage case, and the loss in aerodynamic efficiency due to sonic boom improvements was also less. This suggests that the upper fuselage can be optimized to reduce drag without significant impact to the sonic boom below the aircraft.

8.2 Future Work The present optimization did not include length or wing plan form parameters because they would affect other mission constraints. Since sonic boom is also influenced by length and how lift is distributed, it is likely that further improvement can be achieved by varying wing plan form and aircraft length. Since that would affect weight, performance, stability and control etc, for the results to be viable, the current analysis would have to be integrated with an aircraft MDO program such as PASS. Better models for predicting rise time and the scatter in measured sonic booms are needed. Until they become available, there needs to be some way of dealing with the uncertainty. This concerns not only sonic boom loudness prediction, but also the question of whether peak overpressure is better than loudness for sonic boom optimization. Instead of deciding between loudness and peak overpressure, one might consider using both, since that would be possible with the multi-objective GA. Hence the problem becomes the minimization of loudness, overpressure and drag.

131

Off-track sonic boom should also be investigated to ensure that the low-boom designs do not create louder booms on either side of the flight path. Since current results suggest limited overland flights, one might also consider optimizing the aircraft to reduce the width of sonic boom corridor – if it is to be narrower than the boom carpet. Solution convergence can probably be improved with a better GA. The current selection is random, meaning that bad solutions are just as likely to be chosen as good ones. Other stochastic optimization methods may also be explored. However one must be careful not to place too much emphasis on convergence rate because of the tradeoff with diversity. It would be interesting to apply recent CFD based aerodynamic shape optimization techniques to solve the area-matching problem. Instead of area, one could match near field pressure signature computed by the Seebass-George method. Since the degree of nonlinear steepening is still small in the near field, an inverse method should work very well. Gradient-based optimizers with remote adjoints for the near field would be ideal in this case. Formation flying for drag reduction is a well-known phenomenon. It is, however, not known how formation flying may help or worsen sonic boom. Perhaps there is also a Pareto optimal set of formations for long-range quiet supersonic flight.

132

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