AerodynamicShapeDesign-FinalReport-Fall2010.pdf

EML 4905 Senior Design Project

A SENIOR DESIGN PROJECT

PREPARED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF

BACHELOR OF SCIENCE IN MECHANICAL ENGINEERING

Aerodynamic Shape Design Optimization of Winglets

Gianluca Minnella 1737720 Yuniesky Rodriguez 2086370

Jose Ugas 1603084

Advisor: Dr. George S. Dulikravich

Professor: Dr. Sabri Tosunoglu

October 25, 2010

This report is written in partial fulfillment of the requirements in EML 4806. The contents represent the opinion of the authors and not the Department of

Mechanical and Materials Engineering.

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Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering

Ethics Statement

The work submitted in this project is solely prepared by a team consisting of Gianluca Minnella,

Jose Ugas and Yuniesky Rodriguez it is original. Excerpts from others’ work have been clearly identified,

their work acknowledged within the text and listed in the list of references. All of the engineering

drawings, computer programs, formulations, design work, prototype development and testing reported

in this document are also original and prepared by the same team of students.

Gianluca Minnella

Team Member

Jose Ugas

Team Member

Yuniesky Rodriguez

Team Member

Dr. Sabri Tosunoglu

Faculty Professor

Dr. George S. Dulikravich

Thesis Advisor

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Table of Contents

Ethics Statement .................................................................................................................................... 2

Table of Contents ................................................................................................................................... 3

List of Figures .......................................................................................................................................... 6

List of Tables ......................................................................................................................................... 10

Abstract ................................................................................................................................................ 11

Chapter 1 Introduction .................................................................................................................... 12

1.1 Problem Statement ................................................................................................................. 12

1.2 Motivation ............................................................................................................................... 12

1.3 Literature Review .................................................................................................................... 12

1.3.1 The Kutta-Zhukowsky Condition ..................................................................................... 13

1.3.2 Aerodynamic Characteristics of Airfoils .......................................................................... 15

1.3.3 The Finite Wing ............................................................................................................... 17

1.3.4 Flow Fields around Finite Wings ..................................................................................... 17

1.3.5 Downwash an Induced Drag ............................................................................................. 1

1.3.6 The Fundamental Equations of Finite-Wing Theory ......................................................... 3

1.3.7 The Elliptical Lift Distribution ............................................................................................ 6

1.3.8 Winglets ............................................................................................................................ 8

1.3.9 Boeing 757-200 Background and Winglet Benefits ........................................................ 10

1.3.9.1 Technical features: ................................................................................................... 10

1.3.9.2 Range Capability ....................................................................................................... 12

1.3.9.3 Addition of Winglets ................................................................................................. 12

1.3.10 **********KUBRINSKI******* ................................................................................. 15

1.3.11 Optimization ................................................................................................................ 15

1.3.11.1 Optimization Overview........................................................................................... 16

1.3.11.2 Optimization Algorithm .......................................................................................... 17

1.3.11.3 Particle-Swarm ....................................................................................................... 18

1.3.11.4 Pareto Front Overview ........................................................................................... 19

1.3.12 OpenFOAM Software .................................................................................................. 21

1.3.12.1 Case Setup: Mesher ................................................................................................ 24

1.3.12.2 Case Setup: Solver .................................................................................................. 27

1.3.12.3 Case Setup: Parallel Computing ............................................................................. 28

1.3.13 Experimental Aerodynamics and Wind Tunnel Testing .............................................. 32

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1.3.13.1 Test Parameters ..................................................................................................... 33

1.3.13.2 Types of Wind Tunnels ........................................................................................... 36

1.3.13.3 Measurement of Airflow Pressure ......................................................................... 39

1.3.13.4 Static pressure ........................................................................................................ 40

1.3.13.5 Total pressure ......................................................................................................... 41

1.3.13.6 Proximity to walls or model surfaces ..................................................................... 41

1.3.13.7 Pressure rakes ........................................................................................................ 41

1.3.13.8 Pressure Measurement Devices ............................................................................. 41

1.3.13.9 Flow Visualization ................................................................................................... 42

1.3.13.10 Tufts ...................................................................................................................... 42

1.3.13.11 Smoke Flow .......................................................................................................... 43

1.3.13.12 Airfoil Testing ....................................................................................................... 44

1.3.13.13 General Testing Considerations ........................................................................... 44

1.3.13.14 Finite Span Wings ................................................................................................. 44

1.3.13.15 Force Measurements Using a Balance System..................................................... 45

1.3.13.16 Profile Drag by Momentum Loss Measurement .................................................. 45

Chapter 2 Project Formulation and Management ........................................................................... 47

2.1 Overview ................................................................................................................................. 47

2.2 Project Objectives ................................................................................................................... 47

2.3 Design Specifications............................................................................................................... 48

2.4 Constraints and Other Specifications ...................................................................................... 49

Chapter 3 Design Parameters .......................................................................................................... 50

3.1 Overview of Conceptual Designs Developed .......................................................................... 50

3.2 Design Parameter 1 ................................................................................................................. 50

3.3 Design Parameter 2 ................................................................................................................. 51

3.3.1.1.1 Analysis of a Simple Swept-Back Wing ..................................................................... 52

3.3.1.1.2 Analysis of a Wing with Winglets that are Vertically Downwards ............................ 55

3.3.1.1.3 Analysis of a Wing with Winglets that are vertically Upwards ................................. 58

3.4 Design Alternate 3 ................................................................................................................... 60

3.4.1.1.1 Analysis of a Simple Rectangular Wing ..................................................................... 61

3.4.1.1.2 Analysis of Winglets with Camber Pointing Inwards ................................................ 64

3.4.1.1.3 Analysis of a Winglet with Camber Pointing Outwards ............................................ 67

3.5 Proposed Design ..................................................................................................................... 69

Chapter 4 Optimization.................................................................................................................... 69

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4.1 Design Variables ...................................................................................................................... 69

4.2 Objectives ................................................................................................................................ 69

4.3 Optimization Algorithm .......................................................................................................... 69

4.4 Parameters and Ranges for Optimization ............................................................................... 70

4.5 Optimization of 100 Winglets Configurations ......................................................................... 74

4.6 Discontinuous Pareto Front Graphs ........................................................................................ 76

4.7 Optimal Winglet Configurations ............................................................................................. 88

4.7.1.1.1 Simple NACA 2412 .................................................................................................... 89

4.7.1.1.2 Optimal Winglet Configuration #1 ............................................................................ 91

4.7.1.1.3 5.5.2 Optimal Winglet Configuration # 2 .................................................................. 95

Chapter 5 Aerodynamic Analysis ................................................................................................... 100

5.1 6.1 Boeing 757 Simple Wing ................................................................................................. 100

5.2 Original Boeing 757 Winglets ................................................................................................ 103

5.3 6.3 Optimal Boeing 757 Winglets ......................................................................................... 108

Chapter 6 Testing and Evaluation .................................................................................................. 112

6.1 Testing ................................................................................................................................... 112

6.2 Manufacturing....................................................................................................................... 115

Chapter 7 Environmental Impact ................................................................................................... 119

7.1 Environmental Impact of Winglets ....................................................................................... 119

Chapter 8 Conclusion ..................................................................................................................... 121

Chapter 9 Appendix ....................................................................................................................... 122

9.1 Appendix A: Boeing 757-200 technical data ......................................................................... 122

9.2 Appendix B: Engineering Drawings of Parts .......................................................................... 125

9.3 Appendix C: Sample of the three winglets elliptical curves .................................................. 131

Chapter 10 References................................................................................................................. 137

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List of Figures

Figure 1: Kutta-Zhukowski Condition, No Viscosity .............................................................................. 13 Figure 2: Kutta-Zhukowsi Condition, Viscosity ..................................................................................... 13 Figure 3: Starting Vortices .................................................................................................................... 14 Figure 4: Airfoil Pressure Distribution .................................................................................................. 15 Figure 5: Airfoil characteristics ............................................................................................................. 15 Figure 6: Flow around an airfoil............................................................................................................ 16 Figure 7: Vortex Configuration ............................................................................................................. 17 Figure 8: Superposition of elliptical vortices in steady flow ................................................................. 18 Figure 9: Formation of trailing vortices at wing tips ............................................................................ 18 Figure 10: Wing tips flow vortices ........................................................................................................ 19 Figure 11: Formation of trailing vortices at wing tips ............................................................................ 1 Figure 12: Downwash velocity w induced by trailing vortices. .............................................................. 1 Figure 13: Downwash contribution from trailing vortex filament ......................................................... 3 Figure 14: Finite wing Theory parameters ............................................................................................. 4 Figure 15: Finite wing Theory representation ........................................................................................ 4 Figure 16 - Winglet parameters .............................................................................................................. 9 Figure 17: Retrofitting mechanism of blended winglets, created by the Aviation Partners Boeing

Company ..................................................................................................................................................... 13 Figure 18: Retrofitting mechanism of blended winglets, created by the Aviation Partners Boeing

Company for the 737-200 ........................................................................................................................... 13 Figure 19: Visualization of Dominance ................................................................................................. 20 Figure 20: Pareto Fronts for 2 Objectives............................................................................................. 21 Figure 21: OpenFOAM logo .................................................................................................................. 22 Figure 22: Sample Dict file for OpenFOAM .......................................................................................... 23 Figure 23: OpenFOAM Case Directory Chart ........................................................................................ 24 Figure 24: blockMesh ........................................................................................................................... 25 Figure 25: Mesh and Refinement Box for snappyHexMesh ................................................................. 26 Figure 26: Final Stage of snappyHexMesh............................................................................................ 26 Figure 27: MAIDROC Station ................................................................................................................ 29 Figure 28: Tesla-128 Parallel Computing Lab ....................................................................................... 29 Figure 29: Tesla-128 Cluster Diagram................................................................................................... 30 Figure 30: Subdomains Visualization .................................................................................................... 31 Figure 31: Shell Script ........................................................................................................................... 31 Figure 32: vs. air pressure and temperature ................................................................................. 36 Figure 33 - Open Section Wind Tunnel ................................................................................................. 37 Figure 34 - Closed Circuit Wind Tunnel ................................................................................................ 37 Figure 35 - Smoke Wind Tunnel ........................................................................................................... 38 Figure 36 - Pitot Static Probe ................................................................................................................ 40 Figure 37: Tufts Visualization ............................................................................................................... 43 Figure 38: Smoke Flow ......................................................................................................................... 43 Figure 39: Force Balance support ......................................................................................................... 45 Figure 40: Drag by Momentum Loss .................................................................................................... 46 Figure 41: Elliptic winglet Design parameters. ..................................................................................... 48 Figure 42: Blended, Elliptical and Wing-Tip Fence Winglets ................................................................ 50 Figure 43: Isometric View of Domain of Simple Sweptback Wing ....................................................... 53 Figure 44: Residuals vs. Time for Plain Swept-Back Wing .................................................................... 54

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Figure 45: Isometric View of Trefftz Plane Behind the Wing ............................................................... 54 Figure 46: Streamlines for Plain Swept-Back Wing .............................................................................. 55 Figure 47: Isometric View of Trefftz Plane with Winglets Down .......................................................... 56 Figure 48: Residuals vs. Time for Winglets Down ................................................................................ 57 Figure 49: Streamlines at Wing-Tip with Winglets Down ..................................................................... 57 Figure 50: Isometric View of Trefftz Plane for Winglets Up ................................................................. 58 Figure 51: Residuals vs. Time for Winglets Up ..................................................................................... 59 Figure 52: Streamlines for Winglets Up................................................................................................ 60 Figure 53: Plain Wing NACA 2412 Trefftz Plane ................................................................................... 63 Figure 54: Residuals vs. Time for Plain Rectangular Wing .................................................................... 64 Figure 55: Top View of a Cambered In Winglet .................................................................................... 65 Figure 56: Front View of Trefftz Plane for Camber In .......................................................................... 65 Figure 57: Residuals vs. Time for Winglet Camber In ........................................................................... 66 Figure 58: Top View of a Winglet with Camber Out ............................................................................. 67 Figure 59: Trefftz Plane of Winglets with Camber Out ........................................................................ 68 Figure 60: Residuals vs. Time for Winglet Camber Out ........................................................................ 68 Figure 61: Graphical representation of optimization limits ................................................................. 71 Figure 62: Graphic definition of optimization parameters .................................................................. 71 Figure 63: Front View (b vs a) of Elliptic Profile ................................................................................... 72 Figure 64: Side View (b vs c) of Elliptic Profile ...................................................................................... 73 Figure 65: Discontinuous Pareto Front for Objective 1, minimum Cd and maximum Cl ..................... 76 Figure 66: Discontinuous Pareto Front for Objective 2, minimum Cd minimum Cm........................... 77 Figure 67: Discontinuous Pareto Front for Objective 3, maximum Cl/Cd minimum Cd ...................... 77 Figure 68: Discontinuous Pareto Front for Objective 4, Maximum Cl minimum Cm ........................... 78 Figure 69: Discontinuous Pareto Front for Objective 5. Maximum Cl/Cd maximum Cl ....................... 78 Figure 70: Discontinuous Pareto Front for Objective 6, maximum Cl/Cd minimum Cm ..................... 79 Figure 71: Cl, Cd, Cm ............................................................................................................................. 79 Figure 72: Cl vs. Cd ............................................................................................................................... 80 Figure 73: Cm vs Cd .............................................................................................................................. 81 Figure 74: Cm vs Cl ............................................................................................................................... 81 Figure 75: Cl, Cm, Cl/Cd ........................................................................................................................ 82 Figure 76: Cm vs. Cl .............................................................................................................................. 82 Figure 77: Cl/Cd vs .Cm ......................................................................................................................... 83 Figure 78: Cl/Cd vs. Cl ........................................................................................................................... 83 Figure 79: Isometric View of Cl/Cd, Cl, Cd ............................................................................................ 84 Figure 80: Cl vs. Cl/Cd ........................................................................................................................... 84 Figure 81: Cd vs. Cl ............................................................................................................................... 85 Figure 82: Cd vs. Cl/Cd .......................................................................................................................... 85 Figure 83: Isometric View of Cl/Cd, Cd, Cm .......................................................................................... 86 Figure 84: Cd vs. Cm ............................................................................................................................. 86 Figure 85: Cl/Cd vs. Cm ......................................................................................................................... 87 Figure 86: Cl/Cd vs. Cd .......................................................................................................................... 87 Figure 87: Domain of Simple NACA 2412 with a Symmetry Plane ....................................................... 89 Figure 88: Front View of Trefftz Plane for Simple NACA 2412 Wing .................................................... 90 Figure 89: Streamlines at Wing-Tip for Simple NACA 2412 Wing ........................................................ 90 Figure 90: Pressure Field of Simple NACA 2412 Wing .......................................................................... 91 Figure 91: Plot of Residuals vs. Time for Simple NACA 2412 Wing ...................................................... 91 Figure 92: Front View of Optimal Winglet #1 ....................................................................................... 92

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Figure 93: Side View of Optimal Winglet #1 ......................................................................................... 92 Figure 94: Top View of Optimal Winglet #1 ......................................................................................... 93 Figure 95: Front View of Trefftz Plane for Optimal Winglet #1 ............................................................ 93 Figure 96: Side View of Pressure Field of Optimal Winglet #1 ............................................................. 94 Figure 97: Streamlines at Wing-Tip for Optimal Winglet #1 ................................................................ 94 Figure 98: Plot of Residuals vs Time for Optimal Winglet #1 ............................................................... 95 Figure 99: Top View of Optimal Winglet #2 ......................................................................................... 96 Figure 100: Front View of Optimal Winglet # 2 .................................................................................... 96 Figure 101: Side View of Optimal Winglet #2 ....................................................................................... 97 Figure 102: Front View of Trefftz Plane for Optimal Winglet #2 .......................................................... 97 Figure 103: Pressure Field for Optimal Winglet #2 .............................................................................. 98 Figure 104: Streamlines Around Wing-Tip For Optimal Winglet #2 ..................................................... 98 Figure 105: Plot of Residuals vs. T ........................................................................................................ 99 Figure 106: Top View of Domain of Simple, Half 757 Wing with a Symmetry Plane ......................... 101 Figure 107: Front View of Trefftz Plane for Simple 757 Wing ............................................................ 101 Figure 108: Pressure Field around 757 Simple Wing .......................................................................... 102 Figure 109: Streamlines at Wing-Tip of 757 Simple Wing .................................................................. 102 Figure 110: Plot of Residuals vs. Time for Simple 757 Wing .............................................................. 103 Figure 111: Side View of Original Boeing 757 Winglet ....................................................................... 104 Figure 112: Front View of Original Boeing 757 Winglet ..................................................................... 104 Figure 113: Top View of Original Boeing 757 Winglet........................................................................ 105 Figure 114: Front View of Trefftz Plane of Original Boeing 757 Winglet ........................................... 105 Figure 115: Side View of Pressure Field for Original Boeing 757 Winglets ........................................ 106 Figure 116: Streamlines at Wing-Tip for Original Boeing 757 Winglets ............................................. 106 Figure 117: Plot of Residuals vs. Time for Original Boeing 757 Winglets ........................................... 107 Figure 118: Front View of Optimal Boeing 757 Winglets ................................................................... 108 Figure 119: Top View of Optimal Boeing 757 Winglets ...................................................................... 109 Figure 120: Side View of Optimal Boeing 757 Winglets ..................................................................... 109 Figure 121: Front View of Trefftz Plane for Optimal Boeing 757 Winglets ........................................ 110 Figure 122: Side View of Pressure Field for Optimal Boeing 757 Winglets ........................................ 110 Figure 123: Streamlines at Wing-Tip of Optimal Boeing 757 Winglet ............................................... 111 Figure 124: Plot of Residuals vs. Time for Optimal 757 Winglets ...................................................... 111 Figure 125: Clearance for Wind Tunnel .............................................................................................. 113 Figure 126: Smoke Tunnel Test Section at Embry-Riddle................................................................... 113 Figure 127: Test Section of Embry-Riddle Wind Tunnel ..................................................................... 114 Figure 128: 1/8’’ Steel Pin for Retrofitting Winglets .......................................................................... 114 Figure 129: Removing Parts from 3D Printer ..................................................................................... 116 Figure 130: De-powdering Excess of the Parts ................................................................................... 116 Figure 131: Parts Heating in Oven ...................................................................................................... 116 Figure 132: Applying Epoxy to the Wing and Winglets ...................................................................... 117 Figure 133: Parts After Curing Process ............................................................................................... 117 Figure 134: Sanding Parts ................................................................................................................... 118 Figure 135: Drilling Holes in the Wing and Winglets .......................................................................... 118 Figure 136: The structure of the atmosphere below 50 km [6] ......................................................... 120 Figure 137: Boeing 757 aircraft SolidWorks model. These are our target aircraft wings .................. 125 Figure 138: General parts of a commercial airplane. ......................................................................... 126 Figure 139: Comparison of dimension between a 757-200 and a 757-300 aircrafts. ......................... 127 Figure 140: Technical drawings of the wing for a 757 Boeing aircraft. .............................................. 128

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Figure 141: Technical drawings for a rectangular camber wing NACA 2412 with the same average

chord length as the wings of the 757 Boeing aircraft................................................................................ 129 Figure 142: Technical drawing of a random elliptic blended winglet configuration. The optimization

parameters a, b, n, cw and β are shown also in this figure. ....................................................................... 130

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List of Tables

Table 1.1: Technical specifications of Boeing 757-200 aircraft serie ................................................... 11 Table 1.2: .............................................................................................................................................. 33 Table 2.1: Project cost analysis............................................................................................................. 49 Table 3.1: Winglet Comparison ............................................................................................................ 51 Table 3.2: Parameters for Winglets Up/Down ..................................................................................... 52 Table 3.3: Domain for Winglets Up/Down ........................................................................................... 53 Table 3.4: Forces for Simple Swept-Back Wing .................................................................................... 55 Table 3.5: Forces for Winglets Down ................................................................................................... 56 Table 3.6: Forces for Winglets Up ........................................................................................................ 59 Table 3.7: Parameters for Winglets Camber In/Out............................................................................. 61 Table 3.8: Domain Box for Winglets Camber In/Out ............................................................................ 62 Table 3.9: Forces of Simple Rectangular Wing ..................................................................................... 63 Table 3.10: Forces for Winglet Camber In ............................................................................................ 66 Table 3.11: Forces for Winglet Camber Out ......................................................................................... 68 Table 4.1: Variables that define elliptical winglets............................................................................... 70 Table 4.2: Range of Optimization Parameters ..................................................................................... 73 Table 4.3: Aerodynamic coefficientes for the NACA2412 without winlgets ........................................ 74 Table 4.4: Corresponding winglets configurations for maximum Cl and Cl/Cd and minimum Cd and

Cm ............................................................................................................................................................... 74 Table 4.5: Parameters for Optimal Winglets CFD Analysis ................................................................... 88 Table 4.6: Values of Forces for Simple NACA 2412 Wing ..................................................................... 89 Table 4.7: Values of Forces for Optimal Winglet #1 ............................................................................. 91 Table 4.8: Values of Forces for Optimal Winglet #2 ............................................................................. 95 Table 9: Parameters for Boeing 757 CFD Analysis .............................................................................. 100 Table 10: Values of Forces for 757 Simple Wing ................................................................................ 100 Table 11: Values of Forces for Original Boeing 757 Winglets ............................................................ 103 Table 12: Values of Forces for Optimal Boeing 757 Winglets ............................................................ 108 Table 5.13: Comparison of aerodynamic efficiency of 757 with and without winglets ..................... 112 Table 9.1: Types of Boeing 757-200 aircraft [] ................................................................................... 122 Table 9.2: Engines types used by Boeing 757-200 aircrafts [11] ........................................................ 122 Table 9.3: Workload of the Boeing 757-200 aircraft [11] .................................................................. 123 Table 9.4: External dimensions of Boeing 757-200 aircraft [11] ........................................................ 123 Table 9.6: Operational external weights of the Boeing 757-200 aircraft [11] ................................... 123 Table 9.7: Flight performance parameters of the Boeing 757-200 [11] ............................................ 124 Table 9.8: Raw data for the 100 winglet configurations .................................................................... 131 Table 9.9: Data for winglet 1 LMT ...................................................................................................... 133

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Abstract

The problem being addressed is the design, optimization, construction and testing of a wing tip-

winglet configuration. The main objective of this report is to formulate the inverse design procedure so

that the mathematical principles are well-posed both theoretically and numerically, and to design and

optimize a winglet that best matches the obtained results. The inverse method originates from the

attainment of a target pressure distribution for a functional winglet, whereas the optimization method

will be accomplished by implementing non-gradient-based methodology algorithms, in an attempt to

maximize lift while maintaining (or decreasing) the resultant drag unvaried. The significance of the

obtained result parameters will be considered by manufacturing and testing the optimized winglet

configuration in Embry-Riddles’s subsonic wind-tunnel under different angles of attack, while comparing

test results obtained by similar methods for currently manufactured and commercialized winglet

configurations. The intent being, designing and optimizing a wing- tip winglet configuration capable of

reducing induced drag by 2% with respect to currently implemented wing-tip designs.

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Chapter 1 Introduction

1.1 Problem Statement

The approach and landing phases of a commercial B757-200 passenger aircraft will be

considered. The intent being, design and test a wing tip-winglet configuration capable of increasing the

lift-to-drag ratio by 2% with respect to currently implemented applications.

The aircraft will be travelling at Mach 0.3 with 8° angle of attack with respect to the free stream,

at an altitude of 6000ft; parameters which reflect actual flight conditions. Aircraft specifications are

given below.

1.2 Motivation

Within the past 15 years, great attention has been devoted to the study drag-inducing flow

structures in an attempt to strive for better aerodynamic efficiency of an aircraft, while attempting to

optimize the volatile consumption of fuel and to increase system life. It was soon understood that these

objectives are strictly correlated to one another and, that in order to achieve one, all must be

accomplished.

Recent avionics have shown that wing-tip disturbances are particularly effective in developing

adverse conditions during takeoff and landing procedures, during which lift to drag ratio of a flying

aircraft is maximized in order to slow down. Vortices are generated at wing’s extremities, which impact

the overall flight safety by inducing high-speed longitudinal currents and considerable rolling effects on

neighboring aircraft; conditions especially unfavorable during low altitude scenarios like take-off and

landing. A number of critical failures involving medium to small aircrafts have been recorded within

recent years during which, the most plausible cause for malfunction points toward the effects of trailing

vortices. It is to compensate for this lack of aerodynamic efficiency that a number of wing-tip devices

have steadily appeared in both the private and commercial sector.

While understanding complex aerodynamics has always been a needed priority, our intent lies in

producing a design project capable of delivering a well thought-out winglet configuration.

1.3 Literature Review

Although it is more efficient and accurate to have finite-wing computations carried out by

computers using readily available computational engineering languages such as FORTRAN, it is incredibly

important to have a firm understanding of the theories involved in aerodynamic shape design. It is for

this purpose that our emphasis begins with the foundations of aerodynamics.

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1.3.1 The Kutta-Zhukowsky Condition

The Kutta-Zhukowski Theorem predicts with remarkable accuracy the magnitude and

distribution of the lift of airfoils up to angles of attack of 15 degrees. This theorem states that the force

(L’) experienced by a body in a uniform stream is equal to the product of the fluid density (ρ), stream

velocity ( ), and circulation (ᴦ ) and acts in a direction perpendicular to the stream velocity.

Experiments have shown that when a body with a sharp trailing edge is set in motion , the action of the

fluid viscosity causes the flow over the upper and lower surfaces to merge smoothly at the trailing edge;

this circumstance, which fixes the magnitude of the circulation around the body, is termed the Kutta-

Zhukowski Condition which may be summarized as follows: A body with a sharp trailing edge in motion

through a fluid creates about itself a circulation of sufficient strength to hold the rear stagnation point at

the trailing edge of finite angle to make the flow along the trailing edge bisector angle smooth. For a

body with a cusped trailing edge where the upper and lower surfaces meet tangentially, a smooth flow

at the trailing edge requires equal velocities on both sides on the edge in the tangential direction.

The Flow around an airfoil at an angle of attack in an inviscid flow develops no circulation and

the rear stagnation point occurs on the upper surface as can be seen by Fig.1. Fig.2 is a sketch of the

streamlines around an airfoil in viscous flow , indicating the smooth flow past the trailing edge, termed

the Kytta-Zhukowsi Condition. This Condition has served as the basis for the calculation of forces around

an airfoil.

Figure 1: Kutta-Zhukowski Condition, No Viscosity

Figure 2: Kutta-Zhukowsi Condition, Viscosity

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Based on The Helmholtz laws however, the circulation around an airfoil and its ‘wake’, being

zero before the motion began, must remain zero. The establishment of the Kutta Condition, therefore,

requires the formation of the so-called starting vortices (see Fig.3) with a combined circulation equal

and opposite to that around the airfoil. The induced flow caused by the vorticity of the airfoil, added to

that caused by the starting vortices in the wake, will be just enough to accomplish the smooth flow at

the trailing edge.

Figure 3: Starting Vortices

The starting vortices are left behind as the airfoil moves farther and farther from its starting

point, but during the early stages of the motion, Figure 3 indicates that their induced velocities assist

those induced by the surface vortices, to satisfy the Condition. It follows that the surface vortex and as a

result, the forces acting on the airfoil, will not be as strong in the early stages, when they are being

influenced by the starting vortices, as they are after the flow is fully established when the surface

vortices must be strong enough by itself to move the rear stagnation point to the trailing edge.

Simultaneously, notice the increase in airspeed around the leading edge, as indicated in Figure 3. The

resulting pressure decrease manifests a ‘leading edge suction’ phenomena by which to opposing

pressure vectors are located adjacent to each other.

A typical pressure distribution of an airfoil is shown in Figure 4, the arrows representing

pressure vectors. In a perfect fluid, the total force on the airfoil is the lift , acting normal to . It’s

magnitude can be represented as the resultant of two components, one normal to the chord line of

magnitude , given by the integral over the chord of the pressure difference between points

and on the upper and lowers surfaces, and the other parallel to the chord line of magnitude

, representing the leading edge suction. In a real fluid, viscous effects alter the pressure

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distribution and friction drag is generated, though at low angles of attack the theoretical pressure

distribution can be taken as a valid approximation.

Figure 4: Airfoil Pressure Distribution

1.3.2 Aerodynamic Characteristics of Airfoils

The history of the development of airfoil shapes is long and involves numerous contributions by

scientists from all over the world. By the beginning of the twentieth century the methods of classical

hydrodynamics had been successfully applied to airfoils, and it became possible to predict the lifting

characteristics of certain airfoils shapes mathematically. In 1929, the National Advisory Committee for

Aeronautics (NACA) began studying the characteristics of systematic series of airfoil in an effort to

determine exact characteristics. The airfoils were composed of a thickness envelope wrapped around a

mean chamber line as shown by Fig.5. The mean chamber line lies halfway between the upper and

lower surfaces of the airfoil and intersects the chord line at the leading and trailing edges.

Figure 5: Airfoil characteristics

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The various families of airfoils are designed to show the effects of varying the geometrical

variables on their aerodynamic characteristics such as lift, drag and moment, as functions of the

geometric angle of attack. The geometric angle of attack is defined as the angle between the flight

path and the chord line of the airfoil. The geometrical variables include the maximum chamber of the

mean chamber line and its distance behind the leading edge, the maximum thickness and its

distance behind the leading edge, the radius of curvature of the surface at the leading edge, and

the trailing edge angle between the upper and lower surfaces at the trailing edge. Theoretical studies

and wind tunnel experiments show the effects of these variables in a way to facilitate the choice of

shapes for specific applications.

The lifting characteristics of an airfoil below stall conditions are negligibly influenced by viscosity

and the resultant of the pressure forces on the airfoil is only slightly altered by the thickness envelope

provided that the ratio of maximum thickness to chord

and the maximum mean chamber

remain small, and the airfoil is operating at a small angle of attack. These conditions are usually met

during standard operations of airfoils. In a real fluid, lift is within 10% of theory for inviscid fluids up to

an angle of attack of of 12 to 15° depending on the geometric factors of Figure 5. Figure 6

shows that at these low angles the streamlines follow the surface smoothly, although particularly on the

upper surface the boundary layer causes some deviation. At angles of attack greater than , called

the stalling angle, the flow separates on the upper surface and the Kutta-Zhukowski Condition no longer

holds and large vortices are formed. At these angles, the flow becomes unsteady and there is a dramatic

decrease in lift, accompanied by an increase in drag and large changes in the moment exerted on the

airfoil by the altered pressure distribution

Figure 6: Flow around an airfoil

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1.3.3 The Finite Wing

It has been shown that, from momentum considerations, a vortex which is stationary with

respect to a uniform flow experiences a force of magnitude in a direction perpendicular to , also

known as the Kutta-Zhukowski Condition. It follows that a stationary line vortex normal to a moving

stream is the equivalent of an infinite span wing, an airfoil, from resultant force calculations. The airfoil-

vortex analogy forms the basis for calculating the properties of the finite wing however, since the lift

and therefore the circulation, is zero at the tips of a wing of finite span and varies throughout the wing

span, additional flow components must be considered. This section is devoted to this addressing these

concepts.

1.3.4 Flow Fields around Finite Wings

Considering a wing of span b in a uniform flow velocity represented by a bound vortex AB of

circulation (see Figure 7). According to the Kutta-Zhukowski Condition a force having magnitude

will be exerted onto the vortex in a direction perpendicular to . Helmotz Laws however, require that

the bound vortex cannot end at the wingtips as it must form a complete circuit, or it must extend to

infinity or a boundary of the flow. Adjunctively, it has been shown that these laws further require that at

the beginning of the motion a starting vortex (CD, Figure 7) of strength equal to and opposite to that of

the bound vortex, be formed. The Vortex Laws are satisfied by including the trailing vortices BD and AC

of strength .

Figure 7: Vortex Configuration

The resulting velocity field is comprised of the uniform flow with a superimposed downward

flow within the rectangle ABCD and an upward flow outside it. This flow, however, is unsteady as the

starting vortex moves downstream with the flow, and the trailing vortices AC and BD are therefore

increasing in length at the rate .

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Note first, that the velocity induced by a given vortex varies with the reciprocal of the distance

from the vortex. Therefore, as time goes on, the starting vortex recedes from the wing position and,

soon after the start the velocities it induces at the wing are negligible compared with those induced by

portions of the trailing vortices near the wing. In practice, b << t for steady flight and the

configuration becomes essentially an elliptical vortex fixed to the wing and extending to infinity.

Figure 8: Superposition of elliptical vortices in steady flow

Actual finite wings are made up of a superposition of elliptical vortex elements of various

strengths (see Figure 8). An infinite number of these elements lead to a continuous distribution of

circulation and therefore of the lift as a function of y extending over –b/2 < y < b/2. In steady flight, the

vortices will in general be symmetrically placed. The trailing vortex lines lying on the xy plane form a

vortex sheet of width b extending from the trailing edge of the wing to infinity.

Figure 9: Formation of trailing vortices at wing tips

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From a physical standpoint, Figure 9 can help visualize the formation of trailing vortices. The

flow field that develops as the consequence of the circulation around the wing is initiated by an under

pressure ( - ) over the upper surface and an overpressure ( + ) over the lower surface.

Figure 10: Wing tips flow vortices

The indicated flow from high to low pressure at the wing tips signifies the formation of the

trailing vortices. In terms of the Vortex Laws and the Kutta-Zhukowski Condition, the formation of the

trailing vortices can be expressed as follows: The circulation about the wing is generated as the

consequence of the action of viscosity in establishing the Kutta Condition at the trailing edge. The

boundary layer that forms adjacent to the surface is a rotational flow resulting from the viscous shearing

action; the rotating fluid elements spill over the wing tips at the rate required to for trailing vortices with

circulation equal to that around the wing. After leaving the wing tips, the trailing vortices follow the

streamlines of the flow and, in conformity with the Vortex Laws, the circulation around them remains

constant.

Trailing vortices may become visible in the presence of dust and moisture. Figure 11is a

photograph of an airplane emitting insecticide dust from its trailing edge. It shows that, because of the

influence of the vortex line, the trailing vortex sheet will roll up along the edge to form a concentrated

vortex which can be clearly seen in Figure 11.

Figure 11: Formation of trailing vortices at wing tips

1.3.5 Downwash an Induced Drag

The main problem of finite-wing theory is the determination of the distribution of airloads on a

wing of given geometry flying at a given speed and orientation in space. The analysis is based on the

assumption that the trailing vortex sheet (see Figure 11) remains undeformed and that at every point

along the span, the flow is essentially two dimensional.

Figure 12: Downwash velocity w induced by trailing vortices.

Notice that the bound vortex with circulation varying along the span represents a wing for which

the center of pressure at each spanwise point lies on the y axis. The lift distribution is continuous and

the trailing vortices therefore form a vortex sheet of total circulation zero, since the flow field is that of

an infinite number of infinitesimally weak elliptic vortices, with the cross section of each being a vortex

pair of zero total circulation. The trailing line vortices are assumed to lie in the z = 0 plane and to be

parallel to the x axis therefore, the effect on the flow at a given point on the bound vortex is therefore a

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downwash w, whose magnitude at each point is given by the integrated effect of the circulation

distribution on the semi-infinite vortex sheet over the range –b/2 < y < b/2 (see Figure 12). The resultant

velocity at the wing has two components and at each point. These define the induced angle

of attack:

By the Kutta-Zhukowski Condition, the force on the bound vortex per unit span has the

magnitude and is normal to V, that is is inclined to the z axis at an angle of . This force has a lift

component normal to given by

and a drag component, termed the induced drag

In most practical applications the downwash is small, that is . It follows that is a

small angle and the above formulas become

Notice that the induced drag is a component of the Kutta-Zhukowski force in the direction of

, that is the plane of flight.

Although the trailing vortex sheet induces a downwash along the span of a lifting wing, it also

induces an up wash velocity field in the regions beyond the wing tips. When another wing flies in such a

region, the incoming flow is effectively skewed up by the up wash so that the resultant aerodynamic

force will cause a forward thrust instead if a backward drag on the second wing. This phenomenon can

be noticed in our daily lives for flying birds. Flock of birds flying in V-shaped formations take advantage

of this effect and studies have shown that in proper configurations, savings higher than 50% in the total

power required for flight can be achieved as compared to that when birds fly far apart at the same

speed.

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In order to calculate the downwash and induced angle of attack at a wing section, we will be

referring to Figure 13 which represents the essential features of Figure 11, shown from the top view of

the z = 0 plane. Notice that the downwash is assumed to be positive outward.

Figure 13: Downwash contribution from trailing vortex filament

By means of the Biot-Savart Law we can express the increment of downwash at the point

induced by the element of the vortex filament of strength extending from to infinity ∞ in the

direction. The entire contribution of the vortex filament at to the downwash is

The total downwash at is the sum of the contributions of

from all parts of the

vortex sheet. Thus after integrating and diving by we obtain the induced angle of attack for the wing

section at the spanwise location :

This equation gives the amount by which the downwash alters the angle of attack of the wing as

a function of the coordinate along the span.

1.3.6 The Fundamental Equations of Finite-Wing Theory

The fundamental equations needed to find the circulation distribution for a finite wing are

expressed as the equations connecting three angles: , the absolute angle of attack (see Fig. 14) that is

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the angle between the direction of the flow for zero lift (Z.L.L) at a given and the flight velocity vector

, the induced angle of attack , and the effective angle of attack .

Figure 14: Finite wing Theory parameters

These equations are

The effective angle of attack is a section property and thus must satisfy the equation for

sectional lift coefficient

Where according to thin wing theory. The meaning of for a finite wing is shown

in the figure below.

Figure 15: Finite wing Theory representation

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If the airfoil section were on a wing of infinite span, the sectional lift coefficient there would

have a higher value of . Therefore, since the absolute angle of attack is determined by wing

geometry, the sectional lift coefficient of a finite wing can be expressed as

where is a function of . The relation between and is given by

Notice that for . The absolute angle of attack can therefore be derived by

first writing

from which,

Where c is the chord length of the airfoil (see Figure 5).

This equation indicates that the sectional circulation on a finite wing, which is proportional to

, is smaller than that of a wing of infinite span, which is proportional to , because of the induced

angle of attack caused by the downwash (see Fig.13). Then the fundamental equation in its final form

is

The only unknown in the above equation is the circulation, and its solution for all span wise

locations solves the airload distribution problem for a given wing. Unfortunately, its solution can only

be obtained for only a few special cases, the most important of these, the elliptical lift distribution.

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1.3.7 The Elliptical Lift Distribution

Equation (above) is readily solved if the distribution is assumed to be known and the chord

distribution is taken as the unknown. This problem of finding a chord distribution that corresponds

to a given circulation distribution simply involves the solution of an algebraic equation. A very important

case is the elliptical circulation distribution, for this distribution represents the wing of minimum

induced drag. Fortunately the properties of wings of arbitrary planforms that do not differ radically from

the most common shapes are close to those of the elliptical wing. It is therefore customary to write the

properties of wings of arbitrary planforms in terms of the properties of the elliptical wing and a

correction factor.

If represents the circulation in the plane of symmetry, the elliptical variation of circulation with

span is written

Then the induced angle of attack then becomes

Which indicates that at any point along the lifting line is constant if the distribution is

elliptical. Therefore if the absolute angle of attack at every spanwise location is the same then the

effective angle of attack is also constant. Thus,

Where is the sectional induced drag coefficient and is the dynamic pressure

.

To summarize for wings with an elliptical distribution and constant lift curve slope and absolute

angle of attack, the nondimensional sectional properties will not vary along the span. Using these

conditions, the product must vary elliptically for

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Notice that in an elliptical planform only the product is independent of . On the other

hand, for a noon elliptical planform, since is nearly constant, must be a specific function of ,

that is, the wing must be twisted is the equation is to satisfied. This condition could occur only at a

specific attitude of the wing.

The wing properties are found by integrating the section properties across the span. The

wing lift-coefficient is defined as the total wing lift divided by the product of the dynamic pressure

and the wing planform area

Notice that the wing lift coefficient and sectional lift coefficient are equal when the sectional lift

coefficients are constant along the span. Under this condition, the induced angle of attack for an

elliptical distribution becomes,

Where is the aspect ratio of the wing and is defined as

The wing induced drag coefficient is given by

Experiments have shown that the extra power needed to compensate the induced drag is

quite significant even at low flight speed. Since for a given lift coefficient the induced drag is inversely

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proportional to the aspect ratio, the extra power can be made smaller by increasing the aspect ratio of

the wing. For this reason, slender wings of larger aspect ratio are often observed on gliders, low power

light planes, long duration reconnaissance military planes, as well as birds migrating over long distances.

For high-lift, high-payload conditions, induced drag is accountable for up to 40% of the total

aerodynamic drag coefficient and, as a result, any attempt to improve such flight characteristic is highly

sought-after and desirable. For design purposes, it essential to understand that properties of wing-tip

vortices change based on the speed, weight and shape of the lift-producing surface. Weight is the main

contributor as the vortices’ strength is virtually proportional to the operating weight of an aircraft and,

as a result, to its lift. Great detail needs to be given to the effects of the generation of great lift forces. At

the same time it is also inversely proportional to the wing-span over the velocity squared therefore

correct dimensioning of the wing plays a major role in the designing wing shapes.

So, in general, being that lift induces a large amount of drag which is strongly correlated to the

strength of the trailing vortices that have origin at the wing tips of an aircraft, particular attention needs

to be devoted to the development of optimized wing tip configurations.

1.3.8 Winglets

We have seen in Figure 9, Figure 10Figure 11 that the vortices trailing behind a finite wing are

formed by the communication of the high and low pressure regions across the lifting surface through

the wing tips. It has been shown that the trailing vortices induce a downwash velocity field at the wing,

which in turn causes an induced drag on the wing.

Mounting end plates would not prevent the pressure communication through the wing tips

because, as sketched in Figure 7 the circulation of the trailing vortices is the same as that about the

wing. Thus during a steady, level flight the strength of the trailing vortices is proportional to the weight

of the airplane and it will remain the same with or without the end plates. Experiments (Minnella, Jugas,

Rodriguez, 2010. See below) with vertical plates mounted on the upper surface of a wing tips, indicate

that the plates could reduce the maximum circumferential velocity of a rolled-up trailing vortex, but

with a corresponding increase in the diameter of the core. The total circulation of the vortex appeared

to be the same as that of the vortex trailing behind wing tips without the plates.

Although the total strength of the trailing vortices behind an airplane cannot be changed, it is

possible to decrease the induced drag of a given airplane by using properly designed end plates, called

winglets, to redistribute the strength of the trailing vortex sheet. Flat end plates are not efficient in that

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they cause viscous drag that is large enough to offset the reduction in induced drag. To be fully effective,

the vertical surface at the tip must efficiently produce significant side forces that are required to reduce

the lift-induced inflow above the wing tip or the outflow below the tip.

Figure 16 - Winglet parameters

A typical winglet is shown above. It is a carefully designed lifting surface mounted at the wing

tip, which can produce a gain in induced efficiency at a small cost in weight, viscous drag, and

compressibility drag. The geometry of a winglet is primarily by the toe-in (or out) angle, cant angle,

leading edge sweep angle, and the chord and aspect ratio of the winglet. Flow surveys behind the tip of

a wing with and without winglets by Fletcher (1976), indicate that the basic effect of the winglets is a

vertical diffusion of the tip vortex flow just downstream of the tip, which leads to drag reduction.

The gain in induced efficiency for a winglet is greater for a wing that has larger loads near the

tip. If the winglet was set vertically on the wing tip, it would behave like an endplate, that is, its own

normal force would contribute nothing to lift. On the other hand, if the winglet lay in the plane of the

wing, its effect would be that of an irregular extension of the span, causing a large increase in the

bending moment at the wing root and therefore a weight penalty for the wing structure. In practice, the

winglet generally has an outward cant angle so that its influence is a mixture of both effects. The best

cant angle will be a compromise between induced efficiency and drag caused by mutual interference at

the junction of the wing tip and the winglet. Winglet toe-in angle provides design freedom to trade small

reductions in induced efficiency increment for larger reductions in the weight penalties caused by the

increased bending moment at the wing root.

For high effectiveness of the winglet for cruise conditions, the leading edge of the winglet is

placed near the crest of the wing-tip section with its trailing edge near the trailing edge of the wing (see

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Figure 16). In front of the upper winglet mounted above the wing tip, a shorter lower winglet may also

be mounted below the wing tip. A lower winglet in combination with a larger upper winglet produces

relatively small additional reductions in induced drag at cruising speeds, but may improve overall

winglet effectiveness at both high-lift and supercritical conditions.

The combined upper and lower winglets mounted on a jet transport wing were investigated in a

wind tunnel by Withcomb (1976). It was shown that at a Mach number of 0.78 and wing lift coefficient

of 0.44, the addition of winglets reduced the induced drag by about 20% and an increase in the wing lift-

drag ratio of approximately 9%. These results clearly showed the effectiveness of winglets.

1.3.9 Boeing 757-200 Background and Winglet Benefits

On January 13, 1982 the first Boeing 757-200 was assembled and on February 19 it did its first

successful flight. On December 21, same year, after 1380 hours of flight testing for more than 10 month,

the U.S. FAA certified the 757-200. The first delivery of this aircraft occurred next day and was made to

Eastern Airlines. Subsequently in January 1983 the British Civil Aviation Authority certified the above

aircraft for flight over the United Kingdome.

Entering into the Boeing 757-200 specifications we can say it is a midsize airplane with two

engines that allows it to operate in a short or medium range flights. It was designed on the final of the

70’s by the Boeing Company. Top technology was used in order to bring down noise pollution, increase

passenger comfort and operating performance. Although its sales toke about a year to reach high levels

this plane has had a versatile adaptation throughout the world. It has been created in several

configurations such as freighter or jetliner.

It was originally designed to carry 200 passengers in a regular configuration but it can

accommodate up to 228 passengers which brings it capacity into the range of the 757-300 and 737-900.

Its takeoff weights varies from 220 000 pounds to 255 000 pounds increasing its payload range.

1.3.9.1 Technical features:

One of the most important technical parameters that highlight the 757-200 design are the high

bypass ratio engines from the Rolls Royce or from the Pratt & Whitney companies that combined with

the sweptback-twisted wing design makes it one of the quietest more fuel efficient airplanes in the

world. The thrust of the above engines varies from 36 600 to 43 500 pounds. Its fuel consumption

oscillates around 43 % less per seat than other older trijets aircrafts. The most important technical

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aspects for our objective are being shown in Table 1.1. More in deep technical features are also shown

in the Appendix 9.1.

The 757 wing has some important changes with respect to Boeings previous designs. For

example it is not so swept back but it is thicker at the base allowing a longer wing span. The upper

surface is slightly curve than the lower originating a camber wing airfoil and the leading edge is also

slightly sharper. These last changes contribute to the lift force and drag reduction producing better

aerodynamic performance and burning less fuel. Another important contribution of the wing design is

that it allows for the engines to use less power during takeoff and landing procedures. For example with

respect to the 737-200, a much smaller plane, the 757-200 can flight for about 1740 nautical miles or 5

500 feet more. Therefore it can reach a cruiser speed of Mach 0.82 much faster than others.

Other improvement had been included in the design of the Boeing 757 airplane class, such as

the usage of lightweight materials. Aluminum alloys for the wing skins produced a saving of 610 pounds.

Graphite or epoxy composites were used in the control surfaces such as elevators, rudder and ailerons.

Aerodynamics fairings, engine cowlings and landing gear doors introduced a total weight saving of 1 100

pounds. Another impressive inclusion in the design of this plane is the use of carbon brakes which add

time to the service life with respect to the steel brakes and also reduces about 650 pounds in weight

too.

Table 1.1: Technical specifications of Boeing 757-200 aircraft serie

757-200

Passengers Typical 2-class configuration Typical 1-class configuration

200 228

Cargo 1,670 cu ft (43.3 cu m) Engines

maximum thrust Rolls-Royce RB211-

535E4 40,200 lb (179 kN)

Rolls-Royce RB211-535E4B

43,500 lb (193.5 kN)

Pratt & Whitney PW2037 36,600 lb (162.8 kN)

Pratt & Whitney PW2040

40,100 lb (178.4 kN) Maximum Fuel

Capacity 11,489 gal (43,490 l)

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Maximum Takeoff Weight

255,000 lb (115,680 kg)

Maximum Range 3,900 nautical miles (7,222 km)

Cruise Speed Mach 0.80 Basic Dimensions

Wing span Overall Length

Tail Height Interior Cabin Width Body Exterior Width

124 ft 10 in (38.05 m) 155 ft 3 in (47.32 m)

44 ft 6 in (13.6 m) 11 ft 7 in (3.5 m) 12 ft 4 in (3.7 m)

1.3.9.2 Range Capability

In 1990 the Federal Aviation Administration granted 180- minutes certification for the 757-200

of extended-range twin (engine) operation or ETOPS. This certification was given for both type engines

this plane has, the Rolls Royce Rb211-535E4, RB211-535C and the Pratt & Whitney PW2000 series. This

certification was given as prove of the 757-200s series flight reliability. As an example of this is the fact

that the 757-200 can fly 4 500 statute miles with full payload.

1.3.9.3 Addition of Winglets

Few years later with the increments of the oil and gas prices a new way of increasing flight

efficiency was the introduction of winglets. The Boeing Aviation Partners Inc. created blended winglets

that reduced about 5 % fuel consumption. They were available for the 757-200 as an addition to the

already available 1 030 airplanes. The mechanism of incorporating the winglets was called retrofitting.

The next Figure 17 shows some of the basic parts to perform the assembly of winglets into this plane

wings and Figure 18 shows a real life winglet assembly components for the 737-200 airplane.

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Figure 17: Retrofitting mechanism of blended winglets, created by the Aviation Partners Boeing Company

Figure 18: Retrofitting mechanism of blended winglets, created by the Aviation Partners Boeing Company for the 737-200

Winglets were made with the purpose of reducing not only the fuel consumption, but more

importantly they were meant to reduce the induced drag. The blended winglets were registered by the

Boeing Aviation Partners Company at the U. S. patent office on September 20 1994 with the patent

number 53482531. These winglets brought more benefits to airplanes than the ones already mentioned.

A list of them is in the patent document and we already summarize them below.

Benefits of blended winglets for 757

Up to 5% drag reduction

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Built-in fuel hedge

Improved takeoff performance

Reduced engine maintenance costs

Increased payload-range

Lower airport noise and emissions

Improved operational flexibility

Dramatically enhanced appearance

Higher airplane residual value

2Some of the most important facts winglets do in favor of the Boeing series of 757 planes are

also being named below2.

The 757 has carried more than 1.3 billion passengers, more than four times the population

of the United States and Canada combined.

In 18 years of operation, the 757 fleet has flown the equivalent of nearly 25,000 roundtrips

between the Earth and the Moon.

The 757 fleet has produced over 24 million hours of service for its operators, equivalent to

about 2,750 years of continuous service.

The 757 Freighter can hold over 6 million golf balls.

At 255,000 pounds (115,660 kilograms), the 757 weighs as much as a diesel train

locomotive.

The surface area of a pair of 757 wings is 1,951 square feet (181 square meters), about the

same as the floor space of a three-bedroom house in the U.S.

There are about 626,000 parts in a 757. About 600,000 bolts and rivets fasten those parts

together. The length of all wires in the twinjet is about 60 miles (100 kilometers).

Airlines fly the versatile 757 on a wide variety of routes. The twinjet is used to serve city

pairs as far as 4,281 statute miles (6,890 kilometers) and as close as 65 statute miles (105 kilometers).

The common 757/767 cockpit type-rating permits flight crews trained on the 757 to also

fly the 767.

Of the company's (year-end 2000) unfilled announced orders for 1,612 commercial jets , 4.9

percent (79) are for 757 twinjets.

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Thanks to all of these benefits and the retrofitting mechanism of winglets it is possible that

airplanes that are not longer in production can increase their performance, such as the Boeing 757-200

series. And for those new airplanes some of them already have winglets embedded to their wing as part

of their original design.

For a wing having an infinitely large wing span, trailing vortices would not be of particular

significance as the wing would successfully redistribute and de-strengthen its wing tip characteristics. In

modern avionics, however, due to span-wise restrictions which involve maneuverability at airports,

aerodynamicists are forced to take into account this flight interference and develop optimized wings

that administer this behavior. Recently, it has been found that rather than drastically altering the wing

foil in an attempt to improve aerodynamic efficiency, wing tip modifications such as winglets, capably

diminish the unwanted trailing disturbances.

Winglets are wing-tip devices designed to:

Reduce the induced drag component of lift by redistributing and de-strengthening the

trailing vortices.

Increase the payload capabilities of an aircraft by providing and additional lift

component.

Improve the strain distribution of a wing by applying lift components at the tip-sections.

Contribute a positive-traction component to the aircraft thereby reducing loading on the

propulsion system.

1.3.10 **********KUBRINSKI*******

Surface pressure distribution along the wing is only one of the parameters that go into the

design and construction of an optimal wing, and in turn a winglet. For correct design, several factors

have to be taken into account; these are span wise load distribution, the local chord multiplied by the

sectional lift coefficient because induced drag depends on this. Also the optimal pressure distribution

must be enforced through a range of angle of attack, including at conditions of forward slip and side slip.

The design of winglets must also include influence of the wing, fuselage, empennage and the location of

the center of gravity. Good design must take all these factors into account, with these methods and a

fully developed boundary layer around the whole wing surface, a better wing will be designed.

1.3.11 Optimization

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1.3.11.1 Optimization Overview

As explained in the literature review, there are various ways to achieve an optimal result for a

problem or experiment. A given problem may have a great number of variables and outputs, and it is an

engineers’ duty to retrieve all this information and construct a design that will best perform the given

task. Due to the need for higher efficiency of systems in a world that is rapidly modernizing, the time has

come for systems and processes to reach 100% efficiency. To reach this ultimate efficiency the

complexity and size of models has increased dramatically. Now with the advent of computer aided

design, the scenario has changed; now all the variables and constants in a model can be accounted for.

During the last few decades in engineering, optimization was mainly performed with a single

objective function. These problems mainly used gradient based methods that looked for global

minimums and maximums; they base its results by using a step size or a change in the variable to be

optimized that determines how to obtain the best results in the least amount of time. These classical

methods of optimization follow a point by point approach seeking of the best solution. With time

passing, and the complexity of systems growing, multi-objective functions have now become the norm.

It can be said that optimization algorithms have evolved with time, and have been termed, Evolutionary

Optimization Algorithms. These relatively new algorithms use a set of multiple candidate solutions,

population, and follow an iterative procedure that produce a set of the best compromised results. A plot

of these best comprised results is termed a Pareto front.

In the case of the optimization of winglets, there will be two objective functions and five

variables. These two objective functions, output parameters, are the greatest CL and the least possible

CD. Since these two objective functions will produce extreme values of these two objective functions. So

a trade-off or compromise has to be reached within the design so as to satisfy these two functions,

Pareto front. A compromised solution is needed because the optimization algorithm may choose one

design as its best fit to provide the greatest possible CL; however, a completely different design is

determined to be the best solution to minimize the most drag and produce the least CD. Since this will

most likely happen, a compromise between the two designs has to be met that will satisfy both

objective functions.

In order to reach our goal of improving aerodynamic efficiency by 2% with respect to current

winglet design of winglet, a multi-objective optimizer that is accurate, efficient and conceptually simple

is desired. The algorithm that best suites our needs is a response surface method-based hybrid

optimizer that was designed by Marcelo J. Colaco, George S. Dulikravich and Debasis Sahoo.

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1.3.11.2 Optimization Algorithm

The optimizer designed by Colaço, Dulikravich and Sahoo, is a hybrid optimizer based on a highly

accurate response surface method. Response surface method (RSM) seeks the relationship between

explanatory variables and response variables. RSM was first developed in the field of statistics and has

branched off into several disciplines, namely multi-objective optimization. RSM is mainly used to obtain

an optimal response by using a second degree polynomial to interpolate. Even though this method is

only an approximation, a sample model is easy to estimate and apply, even if little is known about the

process; and it is a close approximation to reality.

The response surface the hybrid optimizer uses several radial basis functions and polynomials as

interpolants. The RSM is able to interpolate linear as well as highly non-linear functions in multi-

dimensional spaces. Radial basis functions (RBF) are real-valued functions whose values depends on

their distance from the origin, so that,

Any function that satisfies the above condition is termed a radial function.

Due to the uniqueness of this hybrid optimizer code that tethers different algorithms for

separate functions, its accuracy and robustness is close to the best commercial optimizers available.

Utilizing the RBF for interpolation has the advantage of reducing computational time while still

maintaining a high level of accuracy. A detailed analysis of the hybrid optimizer is provided in Appendix

The bullets listed below are the main tasks the hybrid algorithm runs repeatedly over several

levels of grid refinement. Initially, the optimization procedure starts with a very coarse grid and over the

course of multiple iterations, the mesh is refined.

1. Generate an initial population, using the real function (not the interpolated one) f(x). Call this

population Preal.

2. Determine the individidual that has the minimum value of the objective function over the entire

population Preal and call this individual xbest.

3. Determine the individual that is more distant from the xbest, over the entire population Preal. Call

this individual xfar.

4. Generate a response surface, with the methodology in Section 2, using the entire population

Preal as training points. Call this function g(x).

5. Optimize the interpolated function g(x) using the hybrid optimizer H1, defined above, and call

the optimum variable of the interpolated function as xint. During the generation of the internal

population to be used in the H1 optimizer, consider the upper and lower bounds limits as the

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minimum and maximum values of the population Preal in order to not extrapolate the response

surface.

6. If the real objective function f(xint) is better than all objective function of the population Preal,

replace xfar by xint. Else, generate a new individual, using the sobol pseudo-random generator

within the upper and lower bounds of the variables, and replace xfar by this new individual.

7. If the optimimum is achieved, stop the procedure. Else, return to step 2.

The driven component for this methodology is the particle swarm method. Particle Swarm method

is based on the social behavior of various species and tries to equilibrate the individuality and sociality of

individuals to seek the optimal interest.

1.3.11.3 Particle-Swarm

The Particle Swarm Optimization method (PSO) was introduced by Kennedy and Eberhart3 and it

is based in the intelligent unite behavior of the organisms in a swarm to reach a collective goal; when at

the same time the behavior of a single organism in the swarm seems completely inefficient but it is

intended to find it’s particular best solution to the global problem. This method is very useful when

optimizing unconstrained functions but, if a number of constrains are added the problem turns to be

more complicated. Therefore several approaches had been introduced such as penalty, repair, and

constraint-preserving methods4.

How is this algorithm implemented? Well, let say that a particle xi has memory of which one is

the best solution yi that has being found and it travels through the search dominium with a velocity vi. If

this velocity is continuously adjusted with respect to its particular best and the global best solution

found by the rest of the swarm them we define the swarm as:

(1)

With i = 1, 2, 3…

After each iteration of the PSO algorithm, the best particular solution yi of each element is

compared to its actual performance and set to a better performance. So if the function to be optimized

is defined as we can find yi as the next equation shows.

(2)

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The general best solution is updated to the position with the best performance within the

swarm with the next formula:

(3)

Now the particles velocity and position are updated separately for each dimension j, by the next

formula:

(4)

Where

and

are two random number between 0 and 1scaled by the acceleration

coefficients and to determine the stochastic nature of the algorithm. , .

The standard PSO algorithm is summarized below:

1. Set the iteration number t to zero, and randomly initialize swarm S within the

search space.

2. Evaluate the performance

of each particle.

3. Compare the personal best of each particle to its current performance, and set

to the better performance, according to equation (7.2).

4. Set the global best

to the position of the particle with the best performance

within the swarm, according to equation (7.3).

5. Change the velocity vector for each particle, according to equation (7.4).

6. Move each particle to its new position, according to equation (7.5).

7. Let: .

8. Go to step 2, and repeat until convergence.

1.3.11.4 Pareto Front Overview

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In most multi-objective problems, including this one, seeking the optimal solution and the range

of solutions is driven by dominance. Comparing two different solutions, this dominance elects the better

solution, taking into account both objectives. This can be stated as the following, Solution 1 is said to

dominate solution 2 if:

1. The solution 1 is no worse than 2 in all the objectives

2. The solution 1 is strictly better than 2 in at least one objective.

If either solution 1 or 2 is not better than the other on the basis of the above statements, they

are referred to as non-dominated with respect to one another. Figure 19 illustrates the above

statements.

Figure 19: Visualization of Dominance

Figure 19 shows that alternative 1 is non-dominating with respect to the other solutions.

Solution 1 is better than the other three in objective ‘f2’ but worse in objective ‘f1’. Solution 3 is equal to

solution 4 for ‘f2’ and worse than solution 4 in objective ‘f1’; therefore solution 3 is dominated by

solution 4. Solution 2 is dominated by both 3 and 4 because it is the least desirable in both objective ‘f1’

and ‘f2’. When two solutions are independently non-dominated by a third solution, it is not necessary

that the former two solutions be non-dominated with respect to each other. The compatibility of trade-

off solutions in multi-objective problems cases shows is demonstrated by non-dominance.

Most multi-objective optimization algorithms use a population of decision variable sets that

search for optimal sets. The two main sets this population can be divided into during any generation is:

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1. The non-dominated set, which is composed of solutions that are not dominated

by any other solution in the whole population

2. The dominated set, which is composed of all the solutions excluded from the

dominated set.

The set of solutions belonging to the non-dominated set during a certain generation for a

surface called the Pareto front. The Pareto front can be visualized as a curve in a 2-D objective problem

and as a 3D object in case of a 3 objective problem. Solutions that are not dominated by any other

solution in the whole feasible space are known as globally optimal solutions. The Pareto front comprised

of these solutions is termed the global Pareto front.

Figure 20: Pareto Fronts for 2 Objectives

Figure 20 illustrates four different kinds of Pareto fronts for a problem with 2 objectives. To

have flexibility in an optimal design it is required to compute a set of solutions that are biased towards

one or more objectives. Achieving a uniform distribution of solutions over the whole range of the global

Pareto front is important, because it demands the presence among the members of the non-dominated

population set.

1.3.12 OpenFOAM Software

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All CFD studies performed throughout were done using a Linux based, open source software

called OpenFOAM (Open Field Operation and Manipulation).

OpenFOAM is a software package with applications in the engineering and science fields. It’s

diverse uses across many disciplines and it being open source makes OpenFOAM a very important tool

for an engineer. It is capable of solving for chemical reactions, turbulence, solid dynamics,

electromagnetics and finance.

Figure 21: OpenFOAM logo

Open source, Linux software gives the user an advantage that is not present in standard

programs like ANSYS and SolidWorks; this being that the user can freely modify any aspect of the

software that he or she deems necessary. To someone who is well versed in programming and CFD this

is critical, this means that he or she can customize the software to suit their needs. Another advantage

of OpenFOAM is that it is free. While student versions of modeling software that perform CFD analysis

are available, the meshers for these have a limit on the number of vertices that can be applied on the

domain. A mesh is a collection of vertices, edges and faces that is unstructured and in the shape of a

grid; this grid defines the shape of an object.

The shape of the faces of the grid varies for each mesher, OpenFOAM uses hexahedral polygons

for its faces. When performing CFD analysis in the field of aerodynamics, a large domain box is desirable.

For this reason, to obtain coherent results, a large amount of vertices of this grid are needed. Using

student versions of ANSYS or SolidWorks, the number of nodes available is set at 1 million nodes or

vertices. For small domains, this will generate a mesh fine enough to give an accurate analysis. However

since the limit is set at 1 million nodes, when these elements are spread out over a large domain box,

the mesh becomes too coarse and results will not coincide with experimental values. In contrast to

these other software, OpenFOAM has no limit on the number of nodes when meshing. When setting up

the case experiment, the user specifies how many nodes he or she wants the mesher to use for the

object and domain box. For meshes greater than 1 million nodes the mesher should be run on a

computer cluster. OpenFOAM has a limit when meshing, no more than a million nodes per processor.

So, if 3 million nodes were desired to achieve a fine mesh, OpenFOAM would be run on 3 processors

that are connected with a parallel computer network. In theory, if one has a computer cluster at their

disposal, there are no limits to the number of nodes used to define a surface and domain.

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While OpenFOAM is very advantageous in terms of the quality of CFD you can perform, it does

have its disadvantages, one of these being that it is not user friendly. Most CFD software has a GUI

based program that is user-friendly and is designed for ease of use. OpenFOAM lacks this; it is comprised

of C++ files that are created and modified by the user to suit his or her needs. OpenFOAM has stored

functions that are called up by the user writing different files written in C++ programming language;

these files are called dicts. Each specific function for OpenFOAM has a different dict, which can be

entirely modified. So depending on what analysis the user wants to run, he or she writes and modifies

those specific dicts. Figure 22 shows a sample dict for snappyHexMesh, OpenFOAM’s mesher, as is seen

the file is written entirely in C++ so a strong background in programming is a prerequisite. Within this

dict file is where the options for snappyHexMesh are modified: refinement region, level of desired

refinement and mesh layering.

Figure 22: Sample Dict file for OpenFOAM

Figure 23 shows the structure for a sample CFD case, this structure will vary slightly for a

different discipline. Within the case directory there will be two main folders, one folder for the mesher

and one for the solver. The mesher folder contains the dicts for blockMesh and for snappyHexMesh,

also the surface being analyzed. The solver directory is split up into three subparts, the 0, constant and

system folders. The 0 directory contains all the initial conditions for the case; initial conditions include

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the pressure, temperature and velocity fields. Depending on what conditions want to be studied, these

are changed accordingly. The constant directory holds the dict file for the solver and within the System

branch goes the controlDict. The control dict is the file that dictates the time step, max courant number,

start time and end time for the analysis.

Figure 23: OpenFOAM Case Directory Chart

To help users become acquainted with the software, there are very instructive tutorials and

online guides that go in depth and help users set up sample cases. There are sample cases are for

different disciplines that guide the user, and give a general idea as to which of OpenFOAM’s functions

will have to be used so that those dicts can be written and modified.

1.3.12.1 Case Setup: Mesher

OpenFOAM utilizes two different meshers, blockMesh and snappyHexMesh. To run CFD on any

object a domain box has to be created to house this object. On OpenFOAM this domain is created using

blockMesh, blockMesh is simply a mesher that splits the prescribed domain into blocks of eight vertices;

this is illustrated in Figure 24.

Case Directory

Mesher

Constant

blockMesh

System

snappyHexMesh

Solver

0

Initial Conditions

Constant

rhoCentralFoam

System

controlDIct

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Figure 24: blockMesh

The size of this domain is determined by what needs to be studied. For research on winglets, to

study the effects of wing-tip vortices and curtain effect, the domain box should extend at least seven

chord lengths behind the wing. The domain box should start at least two chord lengths in front of the

wing, for the flow to fully develop and to properly analyze the stagnation point. The reason for this

being that the effects of induced drag are only noticed very far behind the wing and any reduction from

the curtain effect will only be noticed from this distance. A good rule of thumb for choosing the height

of the domain box would be forty times the height of the wing. A domain box of these parameters is

sufficient to perform a correct analysis of the wake left by the wing. The values of the domain, min (x, y,

z) to max (x, y, z), are input into the blockMeshDict. This file is simply a dictionary written in C++ that

controls blockMesh, and this file is provided in appendix.

After constructing the domain box, the next step is to mesh the object and the domain box.

OpenFOAM reads the object being analyzed as a STL file and refers to it as a triSurface. STL is a format

from stereolithography CAD software. This extension is commonly used for computer-aided

manufacturing and rapid prototyping. SnappyHexMesh is a mesher that uses hexahedra and split-

hexahedra elements to mesh iteratively around a given surface. For snappyHexMesh to start, a point

inside the domain, a location (x, y, z) anywhere inside the domain but outside the triSurface has to be

specified. With this location, snappyHexMesh can find the triSurface, mesh around it and run successive

iterations around the surface until the level of specified refinement has been fulfilled. Figure 25 shows

how the mesh conforms around the triSurface; the darker grey area is the refinement box. If a certain

area inside the domain requires more extensive analysis a refinement box can be specified. Everywhere

inside this box the mesh will be finer allowing for a more detailed examination. As stated in the

literature review, the effect of winglets are noticed very far downstream, so it is desirable to specify a

refinement box around this region, for more accurate results.

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Figure 25: Mesh and Refinement Box for snappyHexMesh

Observing Figure 25, it is noticeable that the inside of the triSurface is actually meshed on the

inside. The way snappyHexMesh operates is that the surface is meshed iteratively with hexahedral

elements that penetrate the surface, and in the final stage of meshing these elements are snapped to

the surface of the object. This final stage, Figure 26, allows for a more detailed and finer mesh that isn’t

equipped in other CFD software. Another advantage of this software that isn’t present elsewhere is that

there is no limit to the number of nodes that OpenFOAM uses to mesh. Other software programs such

as ANSYS and SolidWorks have a limit to the number of nodes the mesher uses. For the CFD analysis

performed in this research, a large domain box is needed to study the resulting vortices far downstream

of the plane, for this reason if only a small amount of nodes are used, the results obtained using

SolidWorks or ANSYS will not be accurate.

Figure 26: Final Stage of snappyHexMesh

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1.3.12.2 Case Setup: Solver

OpenFOAM utilizes various solvers; cases can be set up for inviscid, compressible,

incompressible flows and Newtonian and non-Newtonian fluids to name a few. The solver used

throughout is rhoCentralFOAM, it is a compressible flow solver that is built in with the Navier-Stokes

Equations and takes viscosity into account. Running a case until convergence with this solver takes

between five – six days using ten nodes of a parallel computer. To cut down on excess run time the

viscosity of air was neglected and set equal to zero, therefore significantly reducing computational time

from five days to two days. Another advantage of neglecting viscosity is that by the solver will now only

focus on induced drag and no other undesirable factors which may skew results.

A subroutine was written for OpenFOAM to output CL, CD and CM (coefficient of lift, drag and

moment). Using these values one can assign a score to the performance of the winglet. Ideally, the best

winglet configuration is the one that outputs the highest CL/CD ratio, this means that the winglet is

decreasing the most drag while maximizing lift, this is obviously the objective of a winglet. A reduction in

the coefficient of moment is also an objective, since these winglets are being optimized for commercial

aircraft which fly solely in high lift, low speed conditions. Winglets that reduce the coefficient of

moment act like a dihedral, the upward angle from horizontal of the wings that reduces the rolling

moment. For the flight regime of commercial aircraft this dihedral effect is very desirable, because the

more stable the plane is the better. However the opposite is true for military aircraft, these planes are

designed to be unstable so as to be optimal for dogfighting and sudden maneuvers.

OpenFOAM has built in equations that output the coefficients of lift, drag and moment.

For coefficient of lift:

(5)

Where L is the lift force, ρ is the density of air, v is true airspeed and A is the planform area.

Planform area can be calculated as half of the surface area of the wing. In the case of fixed-wing aircraft

the wing is the only lifting surface and lift is perpendicular to the flight direction, so the wing is the only

structure to consider when calculating for planform area, the fuselage, horizontal stabilizers and

winglets are not taken into account since they produce little to no lift. To better visualize the planform

area it can be thought of as the area of the wing as viewed from above the plane.

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For coefficient of drag:

(6)

Where FD is drag force, ρ is the density of air, v is the true airspeed and l is the mean average

chord length. For a rectangular wing, the mean average chord length is simply the chord length because

there will not be a variation in the length of the chord. However, for a wing where the chord length

varies this value can be calculated as the planform area divided by the wingspan.

For coefficient of moment:

(7)

Where M is the pitching moment force, q is the dynamic pressure and l is the mean average

chord length. To calculate the coefficient of moment, OpenFOAM also requires that the center of

rotation, or center of pressure, be inputted. For a rectangular wing the center of pressure can be

estimated to be the center of gravity of the wing; since there is no change of the chord length along the

span of the wing, pressure will be acting at this point. For a wing with a varying chord length, swept

forward or back wings, the center of pressure is calculated to be 25%-30% of the chord length from the

leading edge. Another value that OpenFOAM requires is the vector value of the pitch axis. Pitch axis is

the axis about which the wing will pitch and can also be visualized as the axis that goes through the

center of rotation and is perpendicular to the long axis of the plane.

1.3.12.3 Case Setup: Parallel Computing

Once the case is completely set up, it is now ready to run on a computer cluster. Running a case

on a cluster has several advantages; many cases can be run simultaneously and cases converge much

quicker because computing power has significantly increased. For a detailed, extensive study in

aerodynamics numerous case studies have to be conducted. A personal computer has the computing

power to run one case and reach convergence in a desirable time; but running over a 100 cases on a

single computer is unfathomable. To save on computing time and power, cases are run on parallel

computers to yield a lower computing time.

The Tesla-128 Parallel Computer Lab has two different workstations; one is the MAIDROC, this is

the workstation that grants access to users. This workstation is the security station that can be can

accessed remotely by secure shell ('ssh') and secure FTP ('sftp'). The advantage of this system is that, to

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access the MAIDROC one only needs to be connected to the internet, so one does not have to be on-

site. Once access is granted to the MAIDROC the user can then patch into the Tesla Computing Lab.

Using a secure shell, the user can upload their zipped case file directly into the Telsa, of course to do

this, a user account at the MAIDROC is needed, the command to upload a sample case into the Tesla, i.e.

‘scp solver_Optimizer_13.tgz [email protected]:/home/winglets/case_studies’

‘scp’, is a command for secure copy and is a Linux function. Solver_Optimizer_13.tgz is the

zipped case directory that is being uploaded into the case_studies directory for the user winglets.

Figure 27: MAIDROC Station

Figure 28: Tesla-128 Parallel Computing Lab

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Figure 29 shows a diagram of the Tesla Computing Lab at FIU. The front-end workstation, Figure

27 acts as the security checkpoint for the Tesla that queries the user name and password for anyone

trying to access the lab. The queue master is the head node that distributes the cases between the

nodes, it acts as the supervisor of the cluster telling individual nodes which cases to run. Each node

pictured below has 2 processors, there are 64 nodes, and therefore the Tesla has 128 processors.

Figure 29: Tesla-128 Cluster Diagram

OpenFOAM has a function that decomposes one domain into several subdomains, this way it

can be run in parallel. Figure 30 gives a good visualization as to how a domain is decomposed. It helps to

visualize the domain of the case as a cube. Now depending on how many nodes are desired to be used,

the domain is split evenly. In Figure 30, the domain is split into 27 cubes, it is split 3 times in the x, y and

z direction, so for this example 27 nodes will be used to analyze each subdomain. Since each node now

has a very small domain to analyze, convergence of the case will be reached very rapidly. The number of

subdomains to split the job into depends on how refined the mesh of the body is and the computing

time desired. For a faster computing time the more subdomains the domain is divided into the faster

the case reaches convergence. Along the faces where each domain touches another, processors will

communicate with each other to analyze sections of the mesh that are at these intersections. This type

of parallel computing, where nodes of the cluster communicate with each other many times each

second is referred to as fine grain parallelism and is the hardest to program for.

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There are three main types of parallel computing, these are: fine grain, coarse grain and

embarrassing parallelism. A computer task is fine grain parallelism if subtasks, communicate between

each other many times per second. If the subdomains communicate very few times between each other

each second, it is referred to as coarse grained parallelism. Embarrassingly parallel computing are the

easiest to parallelize, because subtasks never communicate with each other.

Figure 30: Subdomains Visualization

After the job has been decomposed the final stage is to submit the job to the task manager. This

is accomplished by using the ‘qsub’ command. The most common form to do this is by writing a shell

script that is called up by the ‘qsub’. Figure 31 shows a sample shell script written in C++. The task

manager is a feature of the Tesla-128 that comes with its operating system of Rocks 5.1. The task

manager keeps track of jobs submitted and keeps job in queue when all the nodes are already running

previous cases.

Figure 31: Shell Script

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1.3.13 Experimental Aerodynamics and Wind Tunnel Testing

Discussions of aerodynamics principles almost always revolve around an understanding of the

term relative velocity and relative wind. Recall that relative wind refers to a body of some sort immersed

in a fluid (air) which is in motion relative to the body. The key point is that the pressure on the surface of

the body and the forces which result from those pressures are the same regardless of whether the body

is stationary and the air is moving or the air is stationary and the body is moving through it. As long as

the relative motion is identical, the aerodynamic forces will be the same. This physical fact explains why

the testing done in the development of a flight vehicle can be, and almost always is, a complementary

mixture of wind tunnel testing and flight testing.

The case of a stationary model exposed to a moving air stream is, of course, the relative wind

condition which exists in a wind tunnel. The model moving through air that is stationary, presuming no

surface wind in the atmosphere, is the relative wind case provided by flight testing. Either is equally

valid. Which method of testing is preferable depends upon the importance of time, cost, safety and data

accuracy to the project being tested. Wind tunnel testing is frequently the quickest and cheapest way to

evaluate the performance of a new design. This is due to the fact that a wind tunnel model can be built

far more quickly and less expensively than a flyable prototype aircraft. Picture the savings possible if

several different configurations generated by the preliminary design group must be evaluated before

selecting a final design. Another aspect of possible cost and time savings is the instrumentation

necessary to measure and record the aerodynamic data being sought. Wind tunnel instrumentation is

stationary outside the tunnel so size, weight, and power needed rarely pose an issue. Just the opposite

is true for flight testing. The safety consideration is that of danger to a fight crew and also damage to

people and buildings on the ground in the event of a crash. The wind tunnel clearly avoids this concern.

The final factor, which usually falls in favor of flight testing, is the quality of the data gathered. There is

always an element of uncertainty in the accuracy of the data recorded for a subscale model. Final

commitment to production of a certain design is usually dependent upon the prototype full scale

vehicle’s demonstration that it can in fact perform as required. The same sort of uncertainty exists

concerning calculated performance, the huge computer programs sometimes referred to as the

“numerical wind tunnel” or more generally as CFD, Computational Fluid Dynamics. Flight test data

involves its own difficulties and inaccuracies and is not infallible, but it will probably always serve as the

final proof.

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Table 1.2:

Source of relative velocity

Type of testing

Time span required

Project cost ($)

Safety Instrumentation Data accuracy

Moving model, stationary air

Flight test Years

Dangerous

Packaging difficult, telemetry required

Best available

Stationary model, moving air

Wind tunnel

test

Months Safety Stationary, fairly easy Good

1.3.13.1 Test Parameters

We will concern ourselves almost entirely with low speed aerodynamics. Most of the testing

equipment is low speed and compressibility factors are not to be accounted for.

Low speed means velocities at which the compressible nature of the moving air is not noticeable

in the aerodynamics phenomena observed. Thus, low speed flow is synonymous with incompressible

flow. Air is, of course, actually compressible and compressibility effects always exist. The traditional

upper limit for incompressible flow is the velocity at which the compressibility effects produce results

1% different from data calculations made assuming incompressibility. This generally occurs at a velocity

of about 300 mph or Mach number of about 0.4.

If an algebraic expression expresses a relation among physical quantities, it can have meaning

only if the terms involved are alike dimensionally. For example, two numbers may be equal, but if they

represent unlike physical quantities they may not be compared. This requirement of dimensional

homogeneity in physical equations is useful in determining the combinations in which the variables

occur and to establish direct meaningfulness when scaled testing is required. The theorem states that

any physical equation can be expressed in terms of dimensionless combinations of the variables.

Therefore, and function of N variables

May be expressed in terms of (N – K) products

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Where each product is a combination of an arbitrary selected set of K independent variables

and one other, whereas K is equal to the number of fundamental dimensions required to describe the

variables P. If the problem is one in mechanics, all quantities P may be expressed in terms of mass,

length, time, and K = 3. In thermodynamics, all quantities may be expressed terms of mass, length, time,

and temperature, and K = 4. The arbitrarily selected set of K variables may contain any of the quantities

of , with the restriction that the K set itself may not form a dimensionless combination.

When considering the force experienced by a body that is in motion through an idealized fluid,

assume that the force will depend on the following parameters:

Where the parameters are given below

Symb

ol Name

Dimensi

on

F Force

ρ Density

V Velocity

l Size of the body (chord length)

μ Coefficient of viscosity

a Speed of Sound

Therefore following the analogy, the system can be represented by

There are six variables and three fundamental dimensions therefore by choosing , and as

the K set, the product become

The theorem guarantees that the products above can be made dimensionless and, upon

applying this condition we obtain

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The above equations represent similarity parameters used in aerodynamic testing.

Aerodynamic forces are inertia forces, meaning that they result from the model changing the

momentum of the moving air. The model changes the velocity or direction of the airflow, or both, if

aerodynamic forces are present.

Reynolds number is the key factor for wind tunnel testing. To reasonably expect two tests to

produce comparable data, we must insure that the tests were run at or near the same Reynolds

number.

Mach number is not significant in low speed testing because, by definition, compressibility does

not have a noticeable effect on the obtained data. Froude number involves gravity and is important in

testing free flight models, for which we do not currently have equipment.

In summary, in low speed testing we desire to accomplish tests at

An expedient for calculating Reynolds number, Figure 32 gives values for

as a function of air

pressure and temperature. The value at sea level standard atmosphere conditions is

.

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Figure 32:

vs. air pressure and temperature

Models of different sizes tested at different conditions of velocity, temperature, and pressure

should produce the same force coefficients if the flow patterns around the models are geometrically

similar. This will occur if the test conditions are such that the similarity parameters are the same for all

model tests being compared.

1.3.13.2 Types of Wind Tunnels

There are a variety of sizes and types of wind tunnels, but they are generally classified as

belonging to two major categories, opened circuit or closed circuit.

Open circuit refers to a tunnel in which the air passes through a basically straight duct and does

not recirculate. The air is simply exhausted into the atmosphere. This type of tunnel is known as Eiffel

tunnel, named after Gustav Eiffel, the builder of the famed Eiffel tower in Paris. He was interested in

experimenting with aerodynamics phenomena and generated his relative velocity by dropping models

from the tower. This proved to be inconvenient, at best, as you can easily imagine. Still, he built a

simple wind-tunnel as an expedient to conducting an experiment. A schematic diagram of an open

circuit tunnel is shown in Figure 33. In this case and in the case of tunnels specifically built for engine

testing, contaminants are put into the airflow during the test which we do not want recirculating

through the experiment.

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Figure 33 - Open Section Wind Tunnel

A closed circuit tunnel employs a duct which guides the air around a closed path, resulting in air

continually recirculating through the test section. It is also called a Prandtl tunnel, after pioneer

aerodynamicist Ludwig Prandtl. A diagram of a closed circuit tunnel is shown in Figure 34 below.

Figure 34 - Closed Circuit Wind Tunnel

A few parameters must be defined:

Test section or jet – area in which the model is normally mounted for testing.

Diffuser – any diverging passage, but specifically the one immediately

downstream of the test section.

Bellmouth, entrance cone or extraction cone – the converging passage

immediately upstream of the test section.

Fan – the propeller which moves the air through the tunnel.

There are some other classifications of for wind tunnel types which are used frequently. The test

section may be open or closed, meaning the test section is enclosed by walls. The presence of walls,

however, prevents realistic deflection of the moving air for tests of models which deflect the air by a

large amount. Tilt engines, very high lift wings, and helicopters fall into this category. For this class of

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testing, test section walls are not used. The result is an open jet or open test section tunnel. Both open

circuit and closed circuit tunnels may be open test section. Tunnels are sometimes classified by the

cross-sectional shape of the test section. Common shapes are round, square, elliptical, rectangular, and

octagonal. Rectangular is probably the most common, though in many cases a “rectangular” test section

has fillets in the corners which render the section actually octagonal. A height to width ratio of 7x10 is

very common for rectangular sections.

There are several types of special purpose tunnels which deserve to be mentioned. Probably the

most common special purpose tunnel is a smoke tunnel. A typical one is shown in Figure 35. The

frequently have very narrow test section, used only for visualization of flow around a short span

segment of airfoils. The tunnel used was built with a 18x24 inch test section allowing visualization of

three-dimensional flow fields, particularly tip vortex patterns. Smoke is injected into the airstream from

a row of parallel tubes. The smoke is generally not smoke at all, as it is usually oil vapor created by

electrically heating thin oil until it boils. Burning of materials which produce real smoke is messy and

hazardous. In order to keep the smoke streams clearly defined, it is necessary to have steady laminar

flow. As a result, smoke tunnels are frequently very low speed with a large entrance cone and many

turbulence- damping screens and/or honeycombs. The tunnel used seemed to operate best at an air-

speed of about 5 feet per second and its maximum speed is about 30 feet per second.

Figure 35 - Smoke Wind Tunnel

Variable density tunnels are constructed so that the entire circuit is a pressure vessel. They can

be pressurized and/or cooled to produce higher air density, facilitating high Reynolds number testing.

Or, some can be evacuated to lower pressure, which allows high Mach number testing with less power.

These are obviously very complex and expensive facilities. NASA Langley complete construction in 1982

P a g e | 39


of one which they call the National Transonic Facility (NTF). It can be pressurized to nine atmospheres

and uses liquid nitrogen to cool the air to . It has an 8.2 ft. square test section. Industry and

government tunnel are generally much larger than privately owned ones, but sill they are not large

enough for full scale models. There are two very large tunnels called full scale tunnels in the United

States. NASA Langley has one with a 30 x 60 foot section, and NASA Ames has a 40 x 80 foot test section.

The Ames tunnel is the world’s largest and has completed construction, also in 1982, to add a new 80 x

120 test section. There are a few tunnels with vertical test sections called spin tunnels. They are used for

testing spin characteristics of free flight models, and sometimes for parachute testing. Tunnels

specifically built for testing airfoil models which span the entire test section from wall to wall are called

two-dimensional tunnels. Most airfoil data in standard texts and reference books is acquired in this type

tunnel.

With the emphasis on reduction of fuel consumption in the automotive industry, there is

currently much research being done on aerodynamic reduction. Many automakers are presently doing

wind tunnel work in Lockheed’s large wind tunnel in Marietta, Georgia, though a couple have their own

tunnels. Closed circuit tunnels are more prevalent than open circuit. Open circuit tunnels are simpler,

and thus cheaper to build. The problem with them is that due to large size, the inlet must usually be

outside the building which encloses the test section. This means that the air being drawn into the tunnel

is subject to weather: wind gusts, rain, insects, dust, temperature, humidity. It is also easy to picture

that open circuit tunnels require more power than closed circuit. Closed circuit tunnels get to capture

and recycle some of the kinetic energy of the air, while open circuit tunnels just dump the moving air

into the surrounding atmosphere. The additional power required, typically 10-15%, for an open circuit

usually does not justify the cost and space required to build the rest of a closed circuit. It is control of

the quality of the air going through the test section that throws the decision in favor of a closed circuit.

1.3.13.3 Measurement of Airflow Pressure

The term pressure without a descriptive adjective is nearly useless in aerodynamic work.

Consider the equation expressing conservation of energy of energy in incompressible flow without

transfer of heat or external work in or out of the flowing air. We refer to it as Bernoulli’s equation:

Total pressure = Static Pressure + dynamic pressure

It is clear that for this expression to make sense, we must make a habit of distinguishing

between various types of pressures. We also need to distinguish between these free stream properties

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and surface pressure on any model we might be testing. Different instruments are used these types of

pressures, and particular care must be given when choosing the proper one.

1.3.13.4 Static pressure

Static pressure is analogous to potential energy when we look at Bernoulli’s equation as an

energy conservation equation. Dynamic pressure is analogous to kinetic energy.

So, to measure static pressure accurately, we must be careful to prevent any accidental

inclusion of the part of the dynamic pressure. We do this by making sure that the static pressure hole is

in a surface parallel to the velocity direction. This can be easily accomplished in wind tunnel by simply

drilling a hole in the wall of the tunnel. The air has no choice but to flow parallel to the wall. The second

consideration is that the local velocity at the pressure measurement location must be the same as the

free stream velocity. Once again, this is fairly easy to determine in a wind tunnel. As long as no rapid

changes in wall shape occur near the static pressure hole, the local velocity will be the same as the free

stream (outside the wall boundary layer, of course). And, the static pressure at the wall is the same as at

any point in that cross-sectional plane.

Use of a probe for measuring static pressure is not quite so easy. A simple static pressure probe

could be made by drilling a hole in the side of a tube with the leading edge plugged. The same two

problems mentioned above must be dealt with, though. If the probe is not perfectly aligned with the

flow, the static pressure reading will be in error. It will be high if the hole is on the upwind side and low

on the downwind side. The problem is corrected fairly well by using eight holes equally spaced around

the circumference of the probe body. These all feed into a common manifold, which almost cancels out

the misalignment errors on the opposite sides. A typical pitot-static probe is shown below.

Figure 36 - Pitot Static Probe

The second problem is that of insuring that the local velocity is equal to the free stream velocity.

The fore-and-aft position of the static holes takes care of this requirement. The local velocity is high, and

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static pressure low, near the tip of the probe due to crowding together of the streamlines as they flow

past the tip. The static holes are a safe distance downstream of the tip to avoid locally high velocity.

However, the holes cannot be too close to the stem because the local stagnation of air against the stem

reduces local velocity and raises static pressure. Probes are commercially available in many diameters,

with inch being the most common.

1.3.13.5 Total pressure

Total pressure is more easily measured. Almost any opening located at a stagnation point will

work, as long as the probe is aligned correctly. The most common Pitot tube is simply a tube cut off flat

on the end and pointed into the oncoming flow. This simple tube is accurate for misalignments up to

about 10°.

1.3.13.6 Proximity to walls or model surfaces

Care must be taken in not allowing a probe to get close enough to a wall to change the

streamlines around the probe. Crowding of streamlines raises the local velocity between the probe and

the wall, affecting both the total and static pressure readings. Total pressure readings are less sensitive

than static. The probe can be nearly touching a surface ( ) before 1% error occurs in the total

pressure. Static pressure is much more sensitive and 5 diameters seems like a minimum safe distance

producing about 1% error.

1.3.13.7 Pressure rakes

Measurement of a distribution of pressures across a section of flowing air can be accomplished

by moving a single probe in steps across the area, called a probe traverse. It can be done manually, or

traversing mechanics are available commercially which are driven by electric motors or hand cranks.

However, in order to make measurements simultaneously at all locations a multiple probe, called a rake,

is frequently used. Experiments will be performed using a 12-tube rake to measure velocity distribution

in the tunnel test section, and a 20-tube rake to measure momentum loss in the wake of an airfoil. Most

rakes are total pressure tubes. Length is not critical, and the tubes simply need to be cut off square on

the end. Also, spacing between tubes is not a problem either. Pitot-static, or static, rakes are seldom

used. As, previously addressed, the squeezing phenomena of the streamlines between individual tubes

causes significant static pressure errors. If a static rake is used, its accuracy must be checked accurately.

1.3.13.8 Pressure Measurement Devices

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We use manometers for most of our pressure measurements because of the fact that

manometers are primary standards, and because most of the performed experiments require

measurement of relatively few pressures. It is more typical of industry testing to use electrical sensing

devices called transducers to measure pressure. These devices can be read and are frequently wired

directly into analog-to-digital signal converters which in turn send the digital signals directly into the

data reduction computer. Calibration and maintenance of this equipment is a demanding task.

1.3.13.9 Flow Visualization

Flow visualization techniques represent the response of the experimental aerodynamics world

to the proverbial statement that a picture is worth a thousand words. Few, if any, different methods

exist which are as helpful in clarifying the nature of the particular airflow pattern. The importance of

being able to see the moving air cannot be overestimated whether the problem is as simple as

understanding flow around an airfoil or as complex as finding and eliminating an aircraft vibration

problem. First-hand observation of flow visualization tests is very instructive, and photographs are an

incredibly important tool for further studies. A variety of flow visualization techniques exist; Tufts and

Smoke Flow being the most widely used.

1.3.13.10 Tufts

Tufts are the easiest to use and are probably the most common. They are particularly

useful in finding regions of separated flow. Tufts can be almost any light and flexible threads or yarns

that are usually taped to the surface of the model. In very low speed flow, spanwise strips of tape are

suitable and the tufts must be very flexible so that the drag of the tufts will pull them with enough force

to make them trail in the local flow direction. Thus, tufts clearly show local flow direction when the

boundary layer is not separated. For high speed flow, the strips of tape are less likely to blow loose if

they are aligned chordwise and must be made out of stronger material in order to avoid breakage.

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Figure 37: Tufts Visualization

Tufts are usually a contrasting color with the surface for improved visibility. Still photos at a

relatively slow shutter speed (about 1/50 second) will show clearly defined tufts where attached flow

has blown the tufts tightly back along the model surface. Separated flow will cause the tufts to wave

around, and they appear blurred in the photos. Tufts can be further implemented with the use of

whisker pole which is comprised of one or two tufts on the end of a long pole allowing the tuft to be

held briefly in whatever position is desired.

1.3.13.11 Smoke Flow

Most smoke generators for low flow speed heat up oil until it vaporizes, then inject the vapor

into the air flow in several parallel streams. If the smoke density is adjusted so that it has no tendency to

either sink or drift upward, the smoke streams will closely approximate streamlines in the flow pattern.

Figure 38: Smoke Flow

Photography is again a beneficial addition to this flow visualization method. Thus, the inside of

the test section must be painted flat black for contrast.

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1.3.13.12 Airfoil Testing

The term airfoil data refers to the results of tests on a constant chord model which spans the

test section from wall to wall. It is synonymous with two-dimensional data and also with the airfoil

section data. Aerodynamic force coefficients measured in this manner are denoted by lower case

subscripts.

1.3.13.13 General Testing Considerations

Test sections made especially for airfoil testing are rectangular in cross section, with

height about 2.5 – 4 times their width. For high angle of attack tests, the airfoil should have a chord less

than 40% of the test section height. For low angle of attack data, a chord less than 70% of the height is

sufficient to avoid unduly large wall interference effects on the measured data. When measuring

velocity profiles in the wake of an airfoil to determine momentum loss, the most accurate and

convenient measurement device is a total pressure rake with a static pressure tap in the wall at the

same cross-sectional plane as the tips of the total pressure probes. Because the static pressure in the

airfoil wake may not be the same as free stream static pressure right at the trailing edge of the airfoil,

proper positioning of the total pressure rake of at least 0.7c downstream of the trailing edge to give the

wake static pressure a chance to equalize to free stream static pressure.

1.3.13.14 Finite Span Wings

Upon specializing an airfoil section data to use on a particular finite span wing shape,

several key differences must be accounted for:

The spanwise lift distribution must fall to zero at the tips of the wing. Therefore,

even though they use exactly the same airfoil shape, a finite span of wing will have a maximum

lift coefficient only about 90% of that of a twp-dimensional wing.

A finite wing span, or finite aspect ratio wing has tip vortices and an

accompanying increase in downwash. The velocity distribution in the tip vortex also causes an

angle of attack variation across the span. The result is a sharply reduced lift curve slope,

. It can be calculated by

The shorter chord at the tip of a tapered planform produces a lower Reynolds

number at the tip compared to at the root, making the tip likely to stall at a lower angle of

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attack. This results in roll control problems unless the wing is somehow adjusted to correct it –

usually twisted about 2 degrees.

Finite span or 3-dimensional wing data is denoted by using upper case

subscripts on aerodynamic coefficients .

1.3.13.15 Force Measurements Using a Balance System

There are three ways to measure forces on a model in a wind tunnel. Forces on the model can

be retrieved by using an instrumented model mounting system called a balance as shown in the picture

below.

Figure 39: Force Balance support

1.3.13.16 Profile Drag by Momentum Loss Measurement

Aerodynamic forces on an airfoil are accompanied by a change in momentum of the air flowing

over the model. It is possible to measure the change in momentum of the moving air and calculate the

force which must have existed to produce the change in momentum. This method works particularly

well in the case of drag on streamlined bodies like airfoils, as long as no large separation regions exist.

Separation includes recirculating flow, which brings exchange of angular momentum into the problem.

Considering linear momentum only, Newton’s second law states that force equals the tie derivative of

momentum. In the case of streamwise component of the momentum,

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Where ; V is volume and

.

Substituting, , yields .

Dividing by

, allows to think in terms of drag coefficient:

Figure 40: Drag by Momentum Loss

Substituting the definitions of dynamic pressure, q,

,

allows for further

specialization of the obtained drag equation. Notice that is different from only within the wake of

the model. Outside the wake, is unaffected by the model and is the same as . Therefore adopting

terminology by which , and results in a modified version of the coefficient of drag

equation:

; where y is the vertical position and c is chord length.

Thus, if dynamic pressure is measure as a function of vertical position within the airfoil wake, we

can calculate fairly easily.

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Chapter 2 Project Formulation and Management

This chapter covers the division of responsibilities by team member along with the time

management guidelines set up in order to successfully reach the project main and partial objectives. A

more detailed explanation of each objective will be carried out in the upcoming sections of this project.

2.1 Overview

Within recent times, significant improvements have been brought about by the worldwide

scientific community in the field of aerodynamics. Particularly, aerodynamic efficiency has been a hot

topic for quite some time, and the first commercial applications of efficiency-inducing aerodynamic

components have been steadily appearing on the market. Soaring fuel prices combined with

technological advances in the field of computational fluid dynamics have obligated aerospace and

aeronautical engineers to develop and test innovative design methodologies capable of delivering

aerodynamically-efficient and cost-effective designs. This project has been formulated with the intent of

exploring the field of aerodynamics, and to develop knowledge and skills necessary to design an

aerodynamic component capable of improving aerodynamic efficiency.

Different fields such as the aerospace, aeronautical, and automotive would gladly welcome

initiatives such as our own, however, in an attempt to link the pursued mechanical engineering

bachelor’s degree to a specialization in aerospace engineering, this project concentrated on airplane

applications. The intent being, improve the aerodynamic efficiency of a Boeing 757-200 commercial

aircraft by the implementation of wing tip devices called Winglets.

2.2 Project Objectives

The design and implementation of elliptic winglets to an existing Boeing 757-200 airplane, in an

attempt to increase the aerodynamic efficiency

of the aircraft by 8% with respect to the

standard configuration without Winglets, and by 2% with respect to the standard configuration with

Winglets.

In order to fulfill the above main objective the elliptical winglet optimization procedure will be

implemented first into a NACA 2412 airfoil with the same planform area as the 757-200 wings and with

the same average chord length. This partial objective is necessary to be carried out first to save

computational time in the validation of the optimization algorithm we decided to use.

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2.3 Design Specifications

Elliptical Winglets are parameterized according to the Lame equation shown next and the

parameters shown on next figure.

Where a, is the winglet total addition to the wing span and b is the total height of the winglet. The

exponent n will define de radio of curvature of the winglet.

A total of 5 variables or parameters are necessary to define and design an elliptical winglet. The

other 2 parameters are the tilt back angle (β) of the winglet and the chord length of the winglet tip airfoil

(cw). All of these parameters are being shown in the next figure for better understanding.

Figure 41: Elliptic winglet Design parameters.

Multi-objective optimization is performed on 100 randomized-parameter configurations within

prescribed limits (see optimization results). CFD is performed and compared to wind tunnel testing on

the 3 optimal Winglet configurations.

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2.4 Constraints and Other Specifications

Aircraft is undergoing approach/takeoff phases of flight. Free stream velocity Mach 0.3 at 8

degrees angle of attack at an altitude of 6000ft. (air properties are based on altitude). Payload is *****

Table 2.1: Project cost analysis

Part No.

Part Quantity (Ea.)

Application Totat Price $

1 Winglet Camber In 2 NACA 2412 Wing 18''/12'' 23.3

2 Winglet Camber Out 2 NACA 2412 Wing 18''/12'' 23.3

3 Winglet Symmetrical Airfoil 2 NACA 2412 Wing 18''/12'' 23.3

4 NACA 2412 Wing 18'' 1 Wind Tunnel 140

5 NACA 2412 Wing 12'' 1 Smoke Tunnel 70

6 Original Boeing 757 Winglet 1 Boeing 757 Wing 19'' 23.3

7 Optimal Boeing 757 Winglet 1 Boeing 757 Wing 19'' 23.3

8 Optimal Winglet #1 2 NACA 2412 Wing 18''/12'' 23.3



11 Boeing 757 Wing 1 Wind Tunnel 218.36

12 West System Marine Epoxy 2 qt. Manufacturing 81.29

13 Slow Hardener 1 pt. Manufacturing 42.77

14 Embry-Riddle Wind Tunnel Testing 5hrs. Testing 875

Total= 1613.82

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Chapter 3 Design Parameters

The elliptical winglet approach we will take in order to create the optimal winglet configuration

for the Boeing 757-200 aircraft will be based in the Lame equation presented in chapter

3.1 Overview of Conceptual Designs Developed

As mentioned earlier, the purpose of winglets is to minimize wingtip vortices that cause induced

drag. The best method for reducing these vortices is attaching endplates at the wingtip, so as to

dissipate the flow of high to low pressure. After knowing this, a preliminary study must be done to

determine which basic configuration of endplates will provide the best CL/CD ratio. Based on an

extensive literature review, three main parameters for the design of a winglet were determined. These

three parameters were: the transition from the wing-tip to the winglet, whether the winglet goes

straight up or down and finally the camber of the airfoil of the winglet.

Each of these key parameters has several alternatives and to determine the best configuration

of all three parameters, CFD cases have to be setup and analyzed to determine the best combination.

The alternatives for the transition of the winglet were whether it will be a blended, ellipitical or wing-tip

fence. The second design parameter is to determine whether the best performing winglet is one that

points up or down. Finally, the last parameter also has two alternatives. The alternatives are whether

the camber in the airfoil of the winglet point towards the fuselage, camber is in, or away from the

fuselage, camber is out.

3.2 Design Parameter 1

For the design of the optimal winglet, three different existing winglets were analyzed and rated

to see which one is the best all-around design so that it could be improved upon. The winglets that to be

considered were, wing-tip fence, blended and blended elliptical winglets.

Figure 42: Blended, Elliptical and Wing-Tip Fence Winglets

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Table 3.1 is a chart comparing all the winglets to the standards we chose as the most vital for a

desired winglet, they are payload contribution, de-strengthening of vortices, positive traction

component, wing flutter, retrofitting capabilities, manufacturing, housing capabilities. These were the

characteristics to look for in a desirable winglet and the scale for the comparison ranges from -5 to 5.

Table 3.1: Winglet Comparison

In this comparison, the most efficient and desirable winglet is a blended elliptical winglet. While

it causes some wing flutter, it is the winglet that most contributes to adding a positive traction

component, and de-strengthens wingtip vortices in turn reducing induced drag.

3.3 Design Parameter 2

The second design parameter is whether t the winglets will be vertically up and vertically down.

Modern planes all have winglets that are vertically pointed up, but it is advantageous to know how

winglets that are vertically down will perform, this test is performed to better understand the concept of

induced drag and how winglets destroy these vortices.

The setup for analysis consists of three cases: a simple sweptback wing, no winglets, the same

wing with vertical upward winglets and lastly the same wing with vertical downwards winglets. To avoid

bias of any kind, both winglets constructed for these cases have the same airfoil and no transition from

wing-tip to winglets. So they can be thought of as end-plates attached to each wing-tip. This is done to

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just focus on the fact of which end-plate is more effective; a vertical plate that goes above or beneath

the wing and if so which is the most effective.

These three cases all have the same initial conditions. Since wingtip vortices are more prevalent

at take-off conditions for low speed and high lift conditions, the wings will be run at a Mach number of

0.3 and at 6,000 ft. The pressure and temperature of air at this height is 81.22 kpa and 276.26 K

respectively. Speed was calculated to be 99.96 m/s for a commercial airplane at this height.

3.3.1.1.1 Analysis of a Simple Swept-Back Wing

A simple swept-back wing refers to the fact that this wing has no attachments like winglets,

leading or trailing edge devise and no flaps. It is just a plain wing with a varying chord length along the

span.

Table 3.2: Parameters for Winglets Up/Down

Parameter Value

Mach Number (M) 0.3

Angle of Attack (α) 5.0

Pressure (p) 81.22 KPa

Height (h) 1830 m (6000 ft)

Temperature (T) 276.26 K

Density of Air (ρ) 1.2798 kg/m3

Kinematic Viscosity (ν) 0 m2/s

Free Stream Velocity (U) 99.96 m/s

Planform Area (Aref) 151.9 m2

Mean Aerodynamic Chord (Lref) 5.367 m

Wing Span 28.3 m

Center of Rotation (Center of Pressure) (5.14, 2.856, 14.336)

Lift Direction (Vector Value) (0, 1, 0)

Drag Direction (Vector Value) (1, 0, 0)

Pitch Axis (Vector Value) (0, 0, 1)

# of Nodes on Cluster 10

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Table 2 shows all the parameters that were input into OpenFOAM so it can solve for coefficients

of lift, drag and moment. These parameters were kept constant for all three cases of simple wing,

winglets up and winglets down.

Table 3.3: Domain for Winglets Up/Down

Axis Minimum Maximum

X -20 60

Y -24 30

Z -26 55

Table 3 contains the values of the domain box that was constructed for the simple swept-back

wing. Since winglets up and down were constructed at the wing-tip of this same wing, the domain box

was kept constant for all three cases.

Figure 9 shows the domain box for the case of a simple sweptback wing. This will be the control

case to compare with the experimental cases, winglets up and down. If the lift over drag ratio improves

with the winglets attached at the wing-tip then it is proves that these do indeed serve a purpose. Once it

is determined that winglets improve the flight characteristics of a plane the best configuration of a

winglet will be determined.

Figure 43: Isometric View of Domain of Simple Sweptback Wing

Using rhoCentralFoam, the compressible flow solver for OpenFOAM, this case ran until

convergence, this can be proved by plotting the residuals from the solver with respect to time. Residuals

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are simply the error from a result and when the residuals become higher in order, this means that the

solution is no longer in a transient state and has reached steady-state. As shown in figure 10, the

residuals have reached an order greater than 10 and the plots of both Uz and Uy have plateaued.

Figure 44: Residuals vs. Time for Plain Swept-Back Wing

Figure 45 shows the Trefftz plane of the pressure field ten meters behind the wing. A Trefftz

plane is a plane that is downstream of an aircraft and is perpendicular to the wake. Observing figure 11,

the wing-tip vortices are clearly shown in the Trefftz plane.

Figure 45: Isometric View of Trefftz Plane Behind the Wing

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Table 3.4: Forces for Simple Swept-Back Wing

Force Functions Value

Coefficient of Lift (CL) 0.382751

Coefficient of Drag (CD) 0.0597286

Lift/Drag ratio (CL/CD) 6.41

Coefficient of Moment (CM) -0.0329124

Figure 46: Streamlines for Plain Swept-Back Wing

Streamlines were applied to each wingtip to study the vortices at these locations. Observing

Figure 46 the streamlines are shown as round concentric circles. This figure is important because when

winglets are attached to each wing-tip the change in the streamlines will quickly become apparent.

3.3.1.1.2 Analysis of a Wing with Winglets that are Vertically Downwards

Now that the control case has been solved those results can be used to compare with the

experimental cases, winglets up and down. If the lift/drag ratio increases, which it should, then we have

proven the fact that winglets decrease wing-tip vortices, makes an aircraft more aerodynamic which can

be translated directly to fuel savings. When designing optimal winglets, every possibility has to be

explored to determine the best winglet configuration, which is why a wing-tip down assembly was

constructed.

Figure 47, shows an isometric view of the Trefftz plane of the pressure field that is 10 meters

behind the wing. Comparing Figure 45 and Figure 47, already a reduction in wing-tip vortices is

apparent; this will obviously translate to a stronger lift/drag ratio for this wing.

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Figure 47: Isometric View of Trefftz Plane with Winglets Down

Table 3.5: Forces for Winglets Down






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Figure 48: Residuals vs. Time for Winglets Down

Figure 48, illustrates the fact that the solution for winglets down has converged due to the fact

that the residuals have plateaued and have reached a high order.

Figure 49: Streamlines at Wing-Tip with Winglets Down

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Figure 49 shows the streamlines at each wing-tip. Comparing Figure 46 and Figure 47 it is

apparent to see how winglets affect wing-tip vortices. Instead of the nice concentric circles that were

prevalent in the simple wing, now with winglets attached a more elliptical streamline is noticeable. The

streamlines are now stretched toward the tip of the winglets; since the flow from high-low pressure now

has to travel a further distance, this means that vortex will now be somewhat more dissipated.

This visual representation is backed by the values of the coefficients of drag, lift and moment.

Now that these vortices have been weakened by the endplates, the ratio of lift/drag is considerably

stronger and even the coefficient of moment is more desirable. With a coefficient of moment closer to

zero, the aircraft will be more stable, giving the commercial aircraft more stability.

3.3.1.1.3 Analysis of a Wing with Winglets that are vertically Upwards

Now a winglets up configuration will be compared to previous results to determine whether a

winglet that is vertically up is more aerodynamic and more stable than winglets that are down. It was

previously shown that winglets that are pointed down decrease drag and increase lift on a wing. So one

can hypothesize that winglets up winglets up will provide the best ratio of the three; since winglets up is

currently the configuration used on all modern planes.

Figure 50: Isometric View of Trefftz Plane for Winglets Up

Figure 50 shows a Trefftz plane of the pressure field 10 meters behind the wing. Like Figure 47, a

reduction in vortices at the wing-tip is clear to see. However, now it is also easy to see that an area of

high pressure goes up and around the winglet. This effect of the high pressure going up all the way to

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the wing-tip of the winglet creates even more stability for the commercial aircraft. In the previous case

of winglets down, the area of high pressure wasn’t able to get around the winglet, thus not providing

the stability as shown here. Since the winglet is pointing down, the high pressure generated by lift is

unable to go around the winglet and push against the outside of the winglet. Before regarding any

numbers or values, one can hypothesize that winglets up will yield the best lift/drag ratio and provide

the most stability.

Table 3.6: Forces for Winglets Up






Figure 51: Residuals vs. Time for Winglets Up

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Figure 52: Streamlines for Winglets Up

Figure 52 proves that the solution has converged; therefore the forces in Table 3.6 can be used

for comparison against the other cases. As predicted, winglets up was a clear winner, it provided the

best lift to drag ratio and the most stability. The coefficient of moment is very close to zero, because of

the analysis that was discussed previously. Also winglets up increase the lift and decrease drag more

efficiently than both winglets down and the plain wing. Comparing Figure 49 and Figure 52 various

similarities are apparent. For one, both streamlines are elliptical in nature and in both the streamlines

tend to go towards the wing-tip of the winglet, instead of concentric circles that is evident from Figure

46. Figure 52, also better illustrates the area of high pressure that covers the outside of the winglet,

leading to a more stable winglet. This fact is proven by the coefficient of moment which is close to 0.

Due to all these factors, the curtain effect prevalent very downstream of the aircraft will be significantly

reduced.

In conclusion, between the alternatives of no winglets, winglets up or winglets down; the

obvious winner is a winglet vertically up.

3.4 Design Alternate 3

The next key parameter in the design of a winglet is whether the camber in the airfoil will be

pointed in, towards the fuselage, or the camber will point outwards, away from the fuselage. As

discussed in the literature review, a symmetrical airfoil provides no lift at a zero degree angle of attack,

therefore commercial aircraft have cambered airfoils that will provide extra lift and therefore create an

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area of higher pressure at the bottom of the wing. Since lift creates wing-tip vortices and drag is a

function of lift, it makes sense to say that winglets destroy these vortices by creating negative lift. This

negative lift is obviously going to be a function of where the camber in the airfoil is. So now a study will

be conducted on various winglets to determine the placement of the camber in an airfoil of a winglet.

The coefficients of moment, lift and drag will be analyzed to determine the best winglet. Now that it is

known that winglets up are the best performing, all the analyses from now on will be done on winglets

that are vertically up.

3.4.1.1.1 Analysis of a Simple Rectangular Wing

This case will be used as the control to compare these results with those of the experimental

cases, camber in/out. In all experiments a datum is needed to give the results obtained a point of

reference. This rectangular wing has no change in chord length along the span.

Table 3.7: Parameters for Winglets Camber In/Out

Parameter Value

Mach Number (M) 0.3

Angle of Attack (α) 8.0˚


Height (h) 1830 m (6000 ft)







Wing Span 34.2 m

Center of Rotation (Center of Pressure) (2.23, -0.23, 0)


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Table 3.7 gives the values that were input into OpenFOAM to solve for the coefficients of lift,

drag and moment. The conditions here are the same as in the previous cases except that now the wing

is given an 8˚ angle of attack. The airfoil used for the wing and winglet is a NACA 2412, and using a

higher angle of attack will generate more lift thus producing more induced drag. So if more wing-tip

vortices are present it will be clearer when looking at the solution which winglet is more effective and is

dissipating vortices at the wing-tips.

Table 3.8: Domain Box for Winglets Camber In/Out

Axis Minimum Maximum

X -20 60

Y -24 30

Z -26 55

Table 3.8 gives the dimensions that were used to construct the domain box for the next three

cases. The same domain box was used because the wing was used in all three cases, only the winglet

varied for each case.

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Figure 53: Plain Wing NACA 2412 Trefftz Plane

Figure 53 is a Trefftz plane of the pressure field 10 meters behind the wing. Even though the

wing here varies from the winglets up/down study, Figure 45, there are various similarities in the

solution. To begin with, when there is no winglet attached at the end of the wing, wing-tip vortices are

very much present and visible and is contributing to the drag. Also, when applying streamlines at the

wing-tip, the vortices can be seen as round concentric circles, there is nothing to dissipate these

vortices; therefore drag will be high, especially at a high angle of attack.

Table 3.9: Forces of Simple Rectangular Wing






Table 3.9 gives the coefficients of lift, drag and moment. These values will be used to compare

to the experimental cases of winglets camber in/out. Figure 54 shows a plot of the residuals of Uy and Uz

vs. time, the fact that the solution has converged gives validity to the results obtained.

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Figure 54: Residuals vs. Time for Plain Rectangular Wing

3.4.1.1.2 Analysis of Winglets with Camber Pointing Inwards

The best way to visualize a winglet with the camber pointing in is to imagine that the ends of a

cambered wing are simply lifted up vertically to form a winglet. Camber out is the opposite of this, from

the wing-tip to the top of the winglet, the airfoil of the wing is reversed. The transition for both of these

winglets is elliptically blended. Figure 55 is magnified and is a top view of the winglet to better visualize

this. As can be seen, the winglet is basically a continuation of the wing, just turned up vertically.

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Figure 55: Top View of a Cambered In Winglet

Figure 56: Front View of Trefftz Plane for Camber In

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Figure 57: Residuals vs. Time for Winglet Camber In

Table 3.10: Forces for Winglet Camber In






Figure 56 shows the Trefftz plane of the pressure field 10 meters behind the wing. As shown in the figure the wing-tip

vortices have been significantly reduced. Comparing

Table 3.9 and

Table 3.10 it shows that indeed this observation is true because there is a decrease in drag. The

increase in lift can be attributed to the positive traction component of the winglet, the elliptical

translation of this winglet is what provides this effect. Surrounding the winglet, an area of high pressure

is clearly seen. This area of high pressure extends all the way to the tip of the winglet and provides

stability to the wing; this is exactly what is needed for a commercial airplane. One can tell that the

coefficient of drag has decreased in this case study just by looking at the streamlines in Figure 56. This

figure shows how the streamlines have transformed and have become more elliptical in nature. This

elliptical shape extending towards the tip of winglet is what dissipates the energy from the wing-tip

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vortex, thus leading to a decrease in drag. As illustrated in Figure 53, without any end-plates flow is free

to travel from high to low pressure and creating tremendous drag. With end plates flow is forced to

travel up and around the winglet, thus weakening the flow considerably and lowering drag.

3.4.1.1.3 Analysis of a Winglet with Camber Pointing Outwards

Figure 58 helps visualize what camber out what really means. As stated before it is the opposite

of camber in and because the cambered part of the airfoil points away from the fuselage.

Figure 58: Top View of a Winglet with Camber Out

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Figure 59: Trefftz Plane of Winglets with Camber Out

Figure 60: Residuals vs. Time for Winglet Camber Out

Table 3.11: Forces for Winglet Camber Out






Evaluating the values for the force functions, it is clear that a winglet with a camber in the airfoil pointing outwards does not

provide optimal results. Comparing

Table 3.10 and

Table 3.11 shows that the winglets with the camber pointing towards the fuselage reduced

more drag and provided extra lift. Also the coefficient of moment was closer to zero with camber in

winglets than with winglets cambered out. This last part can be explained by referring to Figure 59, in it

is clear that the area outside the winglet does not have the high pressure as shown in the previous case.

This fact leads to the wing having less stability. Even though this case provided a better lift/drag ratio

than the control case, the best performing winglet without a doubt is a winglet with camber pointing

inwards, toward the fuselage.

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3.5 Proposed Design

After extensive CFD analysis to determine the best configuration of 3 key parameters it is safe to

say that the best winglet to optimize is the one has that has an elliptically blended transition, points

completely upwards and the camber in the airfoil points inwards, towards the fuselage. All possible

alternatives for the three parameters have been proposed and tested, and this proposed configuration

will be the best performing winglet.

Chapter 4 Optimization

4.1 Design Variables

The first step in optimization is to determine the design variables. The design variable is a

parameter that the engineer or designer is able to modify that changes a system. These design variables

need to have a range.

4.2 Objectives

An objective is a numerical value that is to be maximized or minimized. For the optimization of

winglets, two objectives are deemed essential. These are that the winglets maximize lift and minimize

drag. When determining objectives for a problem, it can be multidisciplinary. For example, a designer or

engineer may want to reduce weight of a fuselage, increase its structural integrity while keeping the

design under a budget.

4.3 Optimization Algorithm

Once the objective functions, parameters and ranges for these parameters are chosen,

optimization of the system can begin. The ideal setup for optimization would be to tether a shape

generator to an optimization algorithm. The logic behind this is that the shape generator performs CFD

analysis on an initial geometry. The values of lift, drag and moment would be input to the optimizer, the

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optimizer would then command the shape generator how to modify the geometry, to better suit the

objective functions. This process would repeat in a loop until the optimum winglet whose criterion

meets those of the objective functions.

However, the TESLA parallel-computing lab at FIU has a Linux server, and there is no shape

generator available that runs in a LINUX based environment. So a more robust optimization will be

performed. This involves creating 100 different winglets in SolidWorks and performing CFD analysis on

them. These 100 cases will be created by using a random number generator to pick values for each of

the five design variables; this is done to avoid bias when picking the values. Each winglet will be

completely different from the one before, since the parameters that define it will be completely

random. Once these 100 winglets have reached convergence on OpenFOAM and their respective

coefficient of lift, moment and drag are obtained; these values will be input to the optimizer. Based on

these values that have been inputted, a population of winglets will be created and a non-gradient based

algorithm will determine the most suitable winglet.

4.4 Parameters and Ranges for Optimization

In order to be able to create several models of winglets to analyze in the Tesla it was necessary to

parameterize the winglet configuration. A total of 6 variables were defined to fully characterize the

elliptic profile of the winglets, their names and meanings can be found in

Table 4.1: Variables that define elliptical winglets

Lower limit Variables Upper limit

0 a (addition to span) [m] 4.005

1.335 b (height) [m] 5.34

1.5 n (curvature ratio, lame equation exponent) 19

1.602 Cw (winglet chord length) [m] 5.34

0 β (winglet tilt back angle) [°] 45

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Figure 61: Graphical representation of optimization limits

Figure 61 shows the side view of the 4 extreme of the winglet configurations we chose in order

to optimize the best elliptical winglet configuration that outputs the higher CL/CD ratio for the given

rectangular wing. The coordinate reference system was chosen to match the one from the design

software SolidWorks. Some of the limit of these parameters will vary with respect to the parameters of

table 1. Therefore their relations are expressed in the next figure considering an arbitrary elliptical

configuration.

Figure 62: Graphic definition of optimization parameters

Tilt back angle of the winglet trailing edge curve (α), tilt back angle of the winglet leading edge

curve (β), winglet chord length (Cw) and winglet high or ellipse vertical length (b). From the above figure

we can define all the variables we need to design a winglet in SolidWorks by the following trigonometric

relations:

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Using the above relations we created an excel sheet that outputs random elliptical profiles.

These profiles are then converted into a leading curve and a trailing curve and loaded to the SolidWorks

model to create the winglet. The initial point for the leading curve is the initial point of the leading edge

and for the trailing curve this point coincides with the initial point of the trailing edge. The ellipse profile

is being generated randomly between the limits we chose for our optimization parameters which are

presented in the table 1.

Figure 63: Front View (b vs a) of Elliptic Profile

For Figure 63 and Figure 64, the blue and green curves represent the leading edge of two

different winglet configurations. The red and yellow curves are the trailing edge of the respective

winglet configuration. This definition makes the blue and red curve one elliptical winglet profile and the

0

1

2

3

4

5

6

0 1 2 3 4 5 6

b

a

Δb1L Δb1T Δb2L Δb2T

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green and yellow curve another winglet profile. Figure 63 is the front view of these profiles and Figure

64 is the side view of the same winglet profiles.

Figure 64: Side View (b vs c) of Elliptic Profile

From Figure 64 the changes between the tilt back angle β (for the leading edge) and α (for the

trailing edge) that occurs for the different elliptical profiles are possible to see.

Table 4.2: Range of Optimization Parameters

Lower limit Parameter Upper limit

0 a (ellipse horizontal length) 0.75 C

0.25 b (ellipse vertical length) C (wing chord length)

1.5 n (ellipse equation exponent) 19

0.3 Cw (winglet chord length) C

0 β (tilt back angle of winglet leading curve) 45°

The parameters from Table 4.2 will define the parameters represented in Figure 62 depending

on the wing thickness and chord length ratio. This will assure to optimize a customized elliptic winglet

for a specific wing airfoil. Considering the range of the above parameters a total of 100 elliptical profiles

were created to be optimized. The optimal shape will be the one that gives the higher ratio between the

lift coefficient and the drag coefficient (CL/CD). The result of this optimization will be tested in a wind

tunnel with a scaled model to validate computational work.

0

1

2

3

4

5

6

0 1 2 3 4 5 6

b

c

Δc1L Δc1T Δc2L Δc2T

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4.5 Optimization of 100 Winglets Configurations

In order to determine the best winglet configuration a response surface method-based hybrid

optimizer was used to create a Pareto front for the optimal results. This optimizer was developed by

Marcelo Colaҫo .This optimizer only outputs results for a combination of two parameters for example,

Maximum Cl and Minimum Cd. Therefore we created 6 graphs in order to better analyze these results.

From the Figure 65 to the Figure 70 the results for all the possible combinations of our project target can

be seen.

Also to determine which of these results will be the best for our target wing, NACA2412, we divided

each of them by the coefficients of lift, drag, moment, and lift/drag of the naked wing shown on table 1.

These results help us to determine the winglet configurations that accomplished these parameters

which are shown on Table 4.4. Then we place these points (A to D) also in the Figure 65 through Figure

70 to see their position with respect to the Pareto front. However in order to determine the best winglet

configuration is necessary to consider the simultaneous effect of all the coefficients and not chose a

configuration base on only the limit value of one coefficient. Therefore configuration 1 from the same

table was determined. Table 4.3 contains the coefficient values for the NACA2412 without any winglets.

Table 4.3: Aerodynamic coefficientes for the NACA2412 without winlgets

Naked wing

Cd Cl Cm Cl/Cd

0.0547969 0.606192 -0.0664281 11.06

Table 4.4: Corresponding winglets configurations for maximum Cl and Cl/Cd and minimum Cd and Cm

Nam

e

Ob

ject

ive

Re

sult

Optimized winglets variables Winglet Optimized

Coefficients Change with respect to

wing

β a b cw n Cd Cl Cm Cl/Cd Cd Cl Cm Cl/Cd

Min Cd

A

2

45

.029

8

0.3

870

9

5.3

853

2

1.6

334

9

1.5

226

2

0.0

583

3

0.7

043

9

-0.0

76

99

12

.018

7

6.4

5%

16

.20

%

15

.91

%

8.6

7%

1

45

.029

8

0.3

870

9

5.3

853

2

1.6

349

2

1.5

226

2

0.0

583

3

0.7

044

-0.0

77

12

.018

9

6.4

5%

16

.20

%

15

.91

%

8.6

7%

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Max Cl

B

1

28

.55

75

4.0

41

82

5.3

85

32

2.5

65

22

8.1

63

36

0.0

86

43

1.0

64

78

-0.1

19

88

12

.31

27

57

.72

%

75

.65

%

80

.47

%

11

.33

%

4

28

.55

75

4.0

41

82

5.3

85

32

2.5

65

22

8.1

63

36

0.0

86

43

1.0

64

78

-0.1

19

88

12

.31

27

57

.72

%

75

.65

%

80

.47

%

11

.33

%

Min Cm

C 4

44

.34

16

0.3

87

09

2.4

09

16

1.3

37

89

1.5

22

62

0.0

61

58

0.6

56

68

-0.0

71

1

10

.79

76

12

.37

%

8.3

3%

7.0

4%

-2.3

7%

Max Cl/Cd

D 3

45

.02

98

3.6

81

73

2.9

47

43

3.5

39

66

4.8

74

28

0.0

78

67

0.9

95

82

-0.1

13

4

12

.61

19

43

.56

%

64

.27

%

70

.71

%

14

.03

%

1

44

.60

92

7

3

.38

95

67

4.6

46

77

3

1.9

32

98

1

1.5

22

62

0.0

64

62

9

0.8

02

75

8

-0.0

79

31

12

.42

09

6

17

.9%

32

.4%

19

.4%

12

.3%

From the above table we can see that for configurations A and B we obtained duplicated winglets

configurations from different optimization objectives with the same minimum Cd coefficient.

Configurations C and D are the best fit options for minimum Cm and maximum aerodynamic efficiency.

Configuration 1 is the optimal winglet configuration resulting from analyzing the combined effect of all

the four coefficients we target in this project. The process of choosing this configuration is completely

up to the authors’ decision and analysis of the optimization results. This process will be described below

with the help of 3D graphs represented from Figure 71 to Figure 86.

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4.6 Discontinuous Pareto Front Graphs

Figure 65: Discontinuous Pareto Front for Objective 1, minimum Cd and maximum Cl

Figure 65 shows the results for maximum Cl and minimum Cd, objective 1. Where A represents the

winglet configuration that correspond to Minimum Cd, B is for maximum Cl, C is for minimum Cm and D

is for maximum Cl/Cd. As we can see from Figure 65 through Figure 66 the Pareto front for all the

objectives is discontinued. The values given for the naked wing are shown in these graphs to see the

improvement of the winglet configurations.

0.50

0.60

0.70

0.80

0.90

1.00

1.10

0.05 0.06 0.06 0.07 0.07 0.08 0.08 0.09

C l

CdPareto front Objective 1 A B C D Raw data Wing 1

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Figure 66: Discontinuous Pareto Front for Objective 2, minimum Cd minimum Cm

Figure 67: Discontinuous Pareto Front for Objective 3, maximum Cl/Cd minimum Cd

-0.14

-0.13

-0.12

-0.11

-0.10

-0.09

-0.08

-0.07

-0.06

-0.05

0.05 0.06 0.06 0.07 0.07 0.08 0.08 0.09

Cm

CdPareto front Objective 2 A B C D Raw data Wing 1

10.50

11.00

11.50

12.00

12.50

13.00

0.05 0.06 0.06 0.07 0.07 0.08 0.08 0.09

Cl/

Cd

Cd

A B C D Pareto front Objective 3 Raw data Wing 1

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Figure 68: Discontinuous Pareto Front for Objective 4, Maximum Cl minimum Cm

Figure 69: Discontinuous Pareto Front for Objective 5. Maximum Cl/Cd maximum Cl

-0.13

-0.12

-0.11

-0.10

-0.09

-0.08

-0.07

-0.06

0.56 0.66 0.76 0.86 0.96 1.06

Cm

Cl

Pareto front Objective 4 A B C D Raw data Wing 1

10.50

11.00

11.50

12.00

12.50

13.00

0.56 0.66 0.76 0.86 0.96 1.06 1.16

Cl/

Cd

Cl


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Figure 70: Discontinuous Pareto Front for Objective 6, maximum Cl/Cd minimum Cm

Figure 71: Cl, Cd, Cm

10.00

10.50

11.00

11.50

12.00

12.50

13.00

-0.13 -0.12 -0.11 -0.10 -0.09 -0.08 -0.07 -0.06

Cl/

Cd

Cm


Desirable Area 1>CL>0.95

0.078<CD<0.08 -0.01<CM<-0.092 High CL

High CD

High Cm

Low CD

Low CM

Low CL

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Figure 72: Cl vs. Cd

High CL

High CD

Low CL

Low CD

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Figure 73: Cm vs Cd

Figure 74: Cm vs Cl


-0.1<CM<-0.092

High CL

High Cm

Low CL

Low Cm

Desirable Area 0.078<CD<0.08 -0.1<CM<-0.092

High CD

High Cm

Low CD

Low Cm

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Figure 75: Cl, Cm, Cl/Cd

Figure 76: Cm vs. Cl

Low CL

Low Cm


-0.1<CM<-0.092

High CL

High Cm


12.4<CL/CD<12.5 -0.1<CM<-0.092

Low CL/CD

Low CL

Low Cm

High CL/CD

High CL

High Cm

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Figure 77: Cl/Cd vs .Cm

Figure 78: Cl/Cd vs. Cl

Low CL/CD

Low CL

High CL/CD

High CL


12.4<CL/CD<12.5

High CL/CD

High Cm

Low CL/CD

Low Cm

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Figure 79: Isometric View of Cl/Cd, Cl, Cd

Figure 80: Cl vs. Cl/Cd

Low CL

Low CL/CD

Low CL

High CL/CD


12.4<CL/CD<12.5


12.4<CL/CD<12.5 0.078<CD<0.08

High CL/CD

Low CL

Low CD

Low CL/CD

Low CL

Low CD

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Figure 81: Cd vs. Cl

Figure 82: Cd vs. Cl/Cd

High CL/CD

Low CD

Low CD

Low CL/CD

Desirable Area 0.078<CD<0.08

12.4<CL/CD<12.5

High CD

High CL

Low CL

Low CD

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Figure 83: Isometric View of Cl/Cd, Cd, Cm

Figure 84: Cd vs. Cm

High CD

High CM

Low CD

Low CM

Desirable Area -0.1<CM<0.95

0.078<CD<0.08


12.4<CL/CD<12.5

0.078<CD<0.08

Low CL/CD

Low CD

Low CM

High CL/CD

Low CD

Low CM

High CL/CD

High CD

High CM

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Figure 85: Cl/Cd vs. Cm

Figure 86: Cl/Cd vs. Cd

Low CM

High CL/CD

Low CM

Low CL/CD


12.4<CL/CD<12.5

High CM

High CL/CD

Desirable Area 0.078<CD<0.08

12.4<CL/CD<12.5

High CD

High CL/CD

Low CD

High CL/CD

Low CD

Low CL/CD

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4.7 Optimal Winglet Configurations

Table 4.5: Parameters for Optimal Winglets CFD Analysis

Parameter Value

Mach Number (M) 0.3



Height (h) 1830 m (6000 ft)







Wing Span 17.1 m

Center of Rotation (Center of Pressure) (2.285, -0.1377, -6.86)





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4.7.1.1.1 Simple NACA 2412

Table 4.6: Values of Forces for Simple NACA 2412 Wing






Figure 87: Domain of Simple NACA 2412 with a Symmetry Plane

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Figure 88: Front View of Trefftz Plane for Simple NACA 2412 Wing

Figure 89: Streamlines at Wing-Tip for Simple NACA 2412 Wing

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Figure 90: Pressure Field of Simple NACA 2412 Wing

Figure 91: Plot of Residuals vs. Time for Simple NACA 2412 Wing

4.7.1.1.2 Optimal Winglet Configuration #1

Table 4.7: Values of Forces for Optimal Winglet #1






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Figure 92: Front View of Optimal Winglet #1

Figure 93: Side View of Optimal Winglet #1

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Figure 94: Top View of Optimal Winglet #1

Figure 95: Front View of Trefftz Plane for Optimal Winglet #1

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Figure 96: Side View of Pressure Field of Optimal Winglet #1

Figure 97: Streamlines at Wing-Tip for Optimal Winglet #1

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Figure 98: Plot of Residuals vs Time for Optimal Winglet #1

4.7.1.1.3 5.5.2 Optimal Winglet Configuration # 2

Table 4.8: Values of Forces for Optimal Winglet #2






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Figure 99: Top View of Optimal Winglet #2

Figure 100: Front View of Optimal Winglet # 2

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Figure 101: Side View of Optimal Winglet #2

Figure 102: Front View of Trefftz Plane for Optimal Winglet #2

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Figure 103: Pressure Field for Optimal Winglet #2

Figure 104: Streamlines Around Wing-Tip For Optimal Winglet #2

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Figure 105: Plot of Residuals vs. T

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Chapter 5 Aerodynamic Analysis

Table 9: Parameters for Boeing 757 CFD Analysis

Parameter Value

Mach Number (M) 0.3



Height (h) 1830 m (6000 ft)







Wing Span 17.1 m

Center of Rotation (Center of Pressure) (5.97, -0.34, 4.62)





5.1 6.1 Boeing 757 Simple Wing

Table 10: Values of Forces for 757 Simple Wing




Coefficient of Moment (CM) 0.0488408


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Figure 106: Top View of Domain of Simple, Half 757 Wing with a Symmetry Plane

Figure 107: Front View of Trefftz Plane for Simple 757 Wing

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Figure 108: Pressure Field around 757 Simple Wing

Figure 109: Streamlines at Wing-Tip of 757 Simple Wing

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Figure 110: Plot of Residuals vs. Time for Simple 757 Wing

5.2 Original Boeing 757 Winglets

Table 11: Values of Forces for Original Boeing 757 Winglets






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Figure 111: Side View of Original Boeing 757 Winglet

Figure 112: Front View of Original Boeing 757 Winglet

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Figure 113: Top View of Original Boeing 757 Winglet

Figure 114: Front View of Trefftz Plane of Original Boeing 757 Winglet

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Figure 115: Side View of Pressure Field for Original Boeing 757 Winglets

Figure 116: Streamlines at Wing-Tip for Original Boeing 757 Winglets

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Figure 117: Plot of Residuals vs. Time for Original Boeing 757 Winglets

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5.3 6.3 Optimal Boeing 757 Winglets

Table 12: Values of Forces for Optimal Boeing 757 Winglets






Figure 118: Front View of Optimal Boeing 757 Winglets

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Figure 119: Top View of Optimal Boeing 757 Winglets

Figure 120: Side View of Optimal Boeing 757 Winglets

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Figure 121: Front View of Trefftz Plane for Optimal Boeing 757 Winglets

Figure 122: Side View of Pressure Field for Optimal Boeing 757 Winglets

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Figure 123: Streamlines at Wing-Tip of Optimal Boeing 757 Winglet

Figure 124: Plot of Residuals vs. Time for Optimal 757 Winglets

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Table 5.13: Comparison of aerodynamic efficiency of 757 with and without winglets

Simple 757 Wing Original 757 Winglet

Optimal 757 Winglet

Coefficient of Lift 0.496619 0.514469 0.529444

Coefficient of Drag 0.095009 0.0956025 0.0960115

Coefficient of Moment

0.0488408 0.0623464 0.0647884

CL/CD 5.227 5.38 5.514

Desired Improvement of CL/CD

8% 2% ----

Actual Improvement of CL/CD

5.2% 2.43% ----

Chapter 6 Testing and Evaluation

6.1 Testing

With the rapid modernization of computers and their increased role in industry, computers are

now playing new roles in the field of engineering and the sciences. Prior to CFD analysis and computer

clusters, wind tunnels were the norm in the fields of aerodynamic design and testing. Throughout the

years, CFD programs have been evolving and the size and power of computers have been growing; as a

result, wind tunnel testing has now played a less crucial role in design. However, the scientific method is

based on gathering experimental, empirical evidence through observation and experimentation to test a

hypothesis. In order to prove the validity of our computational analysis, experiments have to be

conducted in a wind tunnel. The empirical values of Cd, CL, CM and ultimately CL/CD will be compared to

their analytical counterparts attained through OpenFOAM simulation.

To test prototypes on a wind tunnel, the cross sectional area of the test section of the wind

tunnel is a crucial value to know. The size of the prototype has to adhere to certain restrictions so as to

avoid blockage of the wind tunnel. At least 7 inches of clearance between the wall of the wind tunnel

and each side of the wing have to be given. This is also done to allow the vortices at the wing-tip to

properly develop. 7 inches is a good general rule of thumb to remember, if this rule is not adhered to,

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flow will not properly develop and experimental results will not reflect those of CFD simulations. Figure

125 illustrates this rule of thumb well.

Figure 125: Clearance for Wind Tunnel

Embry-Riddle Aeronautical University currently has two excellent wind tunnels in which testing

can be performed. One is a wind tunnel with a 40’’x30’’ test cross section, the second is a smoke tunnel

with a 24’’x18’’ test cross section. Smoke tunnels have the added advantage of flow visualization, this

way flow separation point and boundary layer can be seen with ease. Acknowledgements have to be

given to Mr. Snorri Gudmundsson and Mr. Michael Scheppa for giving us full access to all their testing

facilities and for their help in guidance in testing.

Figure 126: Smoke Tunnel Test Section at Embry-Riddle

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Figure 127: Test Section of Embry-Riddle Wind Tunnel

To verify the computational work performed throughout, several configurations of winglets will

be tested. To corroborate our initial testing on the effect of the camber on the airfoil in the winglet;

winglets with airfoils that are symmetrical, cambered in, out will be tested on a NACA 2412 wing. To

take advantage of both the wind and smoke tunnels, two sets of NACA 2412 wings will be

manufactured, one for each tunnel. Calculating for the least amount of blockage, the wing for the wind

tunnel will have a span of 18’’ while the wing for the smoke tunnel will be 12’’ spanwise. These

dimensions were calculated as desirable for wind to develop over the wingtips. The chord length of both

wings will remain constant, so as to be able to retrofit all winglets on both wings and test them in both

tunnels. The only factor between both wings that varied was the span. Maintaining the same chord

length between both wings has the advantage of significantly saving money, creating separate winglets

for each wing would be very costly. Since the objective is to find CL,CD,Cm , the span of the wing will not

skew results since these are dimensionless parameters. To retrofit the wing to the winglets, two holes

will be drilled in the wing and winglet; the holes will be 1/8’’ diameter to allow a slender steel pin to

connect both.

Figure 128: 1/8’’ Steel Pin for Retrofitting Winglets

Once the reasoning for the preliminary analysis is complete, the next logical step is to test all the

optimal winglets. Winglet configurations #’s 1,2 and 5, chosen as optimal from the Pareto front graphs,

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will be tested for its aerodynamic efficiency on the smoke and wind tunnels using both NACA 2412

wings.

To test the optimal Boeing 757 winglet, the 757 wing will have to be constructed. This wing will

only be tested in the wind tunnel. Flow visualizations on the NACA 2412 winglets will provide sufficient

data as to flow separation point and boundary layer, such that more is not necessary. Two winglets will

be tested on this wing, the optimal Boeing 757 winglet and the original 757 winglet. The best of these

will be found experimentally, and if these results coincide with those of OpenFOAM, more validity will

be given to CFD analysis as a reliable tool for aerodynamic design and analysis.

6.2 Manufacturing

These several configurations of winglets will be manufactured using a Z-510 3D Printer, rapid

prototype machine. The 3D printer was chosen as the best option for constructing the wing + winglets, it

is one of the most efficient rapid protyping machine in industry. It is capable of creating physical models

from CAD files in hours instead of days. Parts however are very fragile and epoxy resin has to be applied

to insure complete infiltration. As illustrated in Figure 129, parts are extremely fragile and great care has

to be taken when removing them from the bin. The procedure for the manufacturing of these winglets

was as follows:

1. Excavate and remove parts from bin of 3D printer with extreme care, parts are very

fragile, and break easily.

2. De-powder excess dust and powder with a small vacuum cleaner.

3. To strengthen parts apply a marine grade epoxy, before applying epoxy, heat parts in a

200˚ F oven for 30 minutes.

4. Apply epoxy resin until the part is completely infiltrated with the epoxy.

5. To speed up curing process of epoxy, place parts in a 100˚ oven for 45 minutes and let

the parts completely de-gas.

6. Sand down parts to remove any imperfections, until it is perfectly smooth. This part is

important because if part is not smooth interferences will exist with the wind and

results will be skewed.

7. Drill 1/8’’ holes into winglets and wing to allow space for steel pins to be inserted.

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Figure 129: Removing Parts from 3D Printer

Figure 130: De-powdering Excess of the Parts

Figure 131: Parts Heating in Oven

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Figure 132: Applying Epoxy to the Wing and Winglets

Figure 133: Parts After Curing Process

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Figure 134: Sanding Parts

Figure 135: Drilling Holes in the Wing and Winglets

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Chapter 7 Environmental Impact

7.1 Environmental Impact of Winglets

The introduction of winglets into the modern aviation industry can be very beneficial in the

economical sense. But will it be also beneficial to the environment? To address this question we look

into several articles to create a global and or local impact description of the issue in question. When

keeping track of the aviation history we found that the three more important aspects that define the

environmental impact of the aviation industry are the noise pollution, the local air quality and the

climate changes5. If we look into the future clearly the last one is the most important of them.

According to the Royal Commission on Environmental Pollution6 There are other points to consider

when analyzing the effect of aircraft in flight into the environment7, for example the stratospheric ozone

reduction that leads to an increment of the UV radiation surface, or the regional or global pollution

associated to chemical changes in the troposphere that can reach several kilometers downwind of

airports, and also there is the local pollution based on the noise level and the decrement of the air

quality cause by COx emissions from the aircrafts.

Now entering into what winglets do or change in aircrafts according to the Aircraft Research

Association in Bedford, UK we can clearly see that they affect the noise pollution factor in a positive

way. Their blended or elliptical transition from the wing tip helps reduce the stress concentration points

and their shape helps to reduce vortices formation or induced drag making the engines work less while

the takeoff or landing procedures8,9.

With respect to the climate or global pollution the winglets did not act directly, but indirectly they

help with the reduction of NOx and CO2 emissions while reducing the fuel engines burn during takeoff

and landing up to 6% for some aircrafts and about 3.2% for the 757-2002. During flight airplanes engines

have several types of emissions such as CO2, NOx, water vapor, hydrocarbons and sulfate or sulfur

oxides in form of particles. All of them can affect the chemical balance and composition of the

atmosphere in a long term or short term range. The way winglets influence on this factor is indirectly

and minimal because they only reduce the engines fuel usage during takeoff and landing procedures. As

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a matter of fact during flight winglets increase the friction drag because they constitute an addition of

surface area into the airplane structure10.

When the airplane is flying the engines emit CO2 as result of the kerosene combustion and it mixes

very well into the atmosphere. But the NOx emissions product of the elevated temperature the engines

work are quickly reacting leading to concentration changes of ozone and methane. However this effect

only occurs in the lower stratosphere. In the troposphere ozone is created by the NOx emissions.

However since ozone has a short life period its concentration increments or decrements are limited by

to a short distribution horizontal and vertically in the atmosphere. On the other hand methane life is

long enough to allow the reduction of it to be spread throughout the entire atmosphere. This change

will contribute to the greenhouse effect, because methane is one of the gases that permit through the

short wave solar radiation and absorb and emit again the long wave thermal radiation crating the

heating of the region near the tropopause. This is also clearly shown on Figure 136.

Figure 136: The structure of the atmosphere below 50 km [6]

Summarizing the overall impact of winglets into the environment we can say that they majorly

influence indirectly into the noise pollution, the local climate change, and the global pollution by

increasing the engines performance. This way engines burn less fuel and reduce the greenhouse gases

emission. Since engines have to work less they will produce less noise.

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Chapter 8 Conclusion

Large commercial transport planes are not likely to be critically affected by trailing vortices from

a preceding aircraft, however, due respect has to be given to these type of flow structures. Wing tip

vortices have been recorded to have lasted up to 3 minutes after the passing of a large aircraft, and

having moved a total of 200 ft. vertically while producing substantial downwind up to 300 m/s

downwind. Aerodynamic efficiency, L/D Ratio, leads to a series of factor which impact the economics of

flying with the extent of fuel consumption and fatigue on the aircraft, noise and many more. These are

all motives why rotational flow patterns are so highly regarded and studied. Thanks to winglets,

aircrafts will be able to more efficiently consume fuel, extend their range capabilities, reduce takeoff

and landing thrust settings, and cut down on noise and emissions.

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Chapter 9 Appendix

9.1 Appendix A: Boeing 757-200 technical data

Table 9.1: Types of Boeing 757-200 aircraft [11]

Specifications Boeing 757-200

Versions

B757-200 initial production passenger aircraft

B757-200 Freighter

developed by Pemco Aeroplex as conversion to existing 757 aircraft

B757-200M combi; mixed cargo/passenger version; only one delivered to Royal Nepal Airlines

B757-200PF package freighter; developed for United Parcel Service

B757-200SF modified 200 coverted by Boeing to Special Freighter

B757-200X extended range version; more than 1.000 km increase in range

VC-32A military version; 4 built for U.S. Air Force

Table 9.2: Engines types used by Boeing 757-200 aircrafts [11]

Engines

Type Thrust

2 Pratt & Whitney PW2037 36,600 lb st (162,8 kN)

2 Pratt & Whitney PW2040 40,100 lb st (178,4 kN)

2 Rolls Royce RB211-535E4 40,200 lb st (178,8 kN)

2 Rolls Royce RB211-535E4-B 43,500 lb st (193,5 kN)

Fuel Capacity 42.684 liters (11,276 US gallons)

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Table 9.3: Workload of the Boeing 757-200 aircraft [11]

Accomodation

flightdeck 2 provision for an observer

cabin attendents 5-7

standard interior arrangements

9 from 178 in 2-class to 239 passengers in all tourist

galley 2 1 front starboardside; 1 rear

3 in 239 passenger version (midships)

toilets 4 1 front portside; 3 rear

3 1 front portside; 2 rear of mid-ships (239 passenger version)

Table 9.4: External dimensions of Boeing 757-200 aircraft [11]

Dimensions; external

wingspan 38,05 mtr

wing chord; at root 8,20 mtr

wing chord; at tip 1,73 mtr

wing aspect ratio 7,8

length; overall 47,33 mtr

length; fuselage 46,97 mtr

tailplane span 15,22 mtr

tail height 13,49 mtr min

13,74 mtr max

wheel track 7,32 mtr

wheelbase 18,29 mtr

Table 9.5: Operational external weights of the Boeing 757-200 aircraft [11]

Weight and loadings

operating weight empty 57.840 - 57.975 kg

operating weight empty (freighter) 50.475 - 50.605 kg

freighter revenue load 32.755 kg

freighter payload 757-200 SF 27.215 kg

max. take-off weight (PW2037 & RB211-535E4) 99.790 kg

max. take-off weight (PW2040 & RB211-535E4-B) 115.665 kg

max. landing weight (PW2037 & RB211-535E4) 89.815 kg

max. landing weight (PW2040 & RB211-535E4-B) 95.255 kg

max. ramp weight (PW2037 & RB211-535E4) 100.245 kg

max. ramp weight (PW2040 & RB211-535E4-B) 116.120 kg

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Table 9.6: Flight performance parameters of the Boeing 757-200 [11]

Performance

max. operating Mach 0.86

cruising speed M 0.80

approach speed (PW2037 & RB211-535E4) 132 knots (245 km/h)

approach speed (PW2040 & RB211-535E4-B) 137 knots (254 km/h)

cruising height: PW2037 / PW2040 11.675 mtr (38,300 ft) / 11.795 mtr (38,700 ft)

cruising height: RB211-535E4 / RB211-535E4-B 10.790 mtr (35,400 ft) / 10.880 mtr (35,700 ft)

take-off field length: PW2037 / PW2040 1.814 mtr (5,950 ft) / 1.677 mtr (5,500 ft)

take-off field length: RB211-535E4 / RB211-535E4-B 2.378 mtr (7,800 ft) / 2.104 mtr (6,900 ft)

landing field length: PW2037 / PW2040 1.463 mtr (4,800 ft) / 1.418 mtr (4,650 ft)

landing field length: RB211-535E4 / RB211-535E4-B 1.555 mtr (5,100 ft) / 1.494 mtr (4,900 ft)

range: PW2037 / PW2040 4.769 km (2,570 Nm) / 4.398 km (2,376 Nm)

range: RB211-535E4 / RB211-535E4-B 7.278 km (3,930 Nm) / 6.843 km (3,695 Nm)

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9.2 Appendix B: Engineering Drawings of Parts

This appendix contains the detailed solidworks technical drawings of the boeing 757-200. The first

figure is a 3-D picture of the model we used in solidworks to optimize winglets for.

Figure 137: Boeing 757 aircraft SolidWorks model. These are our target aircraft wings

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Figure 138: General parts of a commercial airplane.

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Figure 139: Comparison of dimension between a 757-200 and a 757-300 aircrafts.

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Figure 140: Technical drawings of the wing for a 757 Boeing aircraft.

HA

LF

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Figure 141: Technical drawings for a rectangular camber wing NACA 2412 with the same average chord length as

the wings of the 757 Boeing aircraft.

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Figure 142: Technical drawing of a random elliptic blended winglet configuration. The optimization parameters a, b,

n, cw and β are shown also in this figure.

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9.3 Appendix C: Sample of the three winglets elliptical curves

Table 9.7: Raw data for the 100 winglet configurations

Winglet #

Cd Cl Cm Cl/Cd β a b cw n

1 0.068345 0.739014 -0.08103 10.813 20.256 2.072 5.236 2.903 16.037

2 0.062514 0.770204 -0.07896 12.32 41.98 2.815 3.914 1.587 1.615

3 0.065624 0.804262 -0.08222 12.256 37.917 3.515 5.211 2.02 2.763

4 0.070218 0.764275 -0.08376 10.884 11.842 0.931 2.253 1.256 3.072

5 0.067351 0.75718 -0.08103 11.242 25.344 2.111 3.46 1.981 2.336

6 0.060672 0.730628 -0.0815 12.042 10.527 1.7556 2.757 3.046 2.137

7 0.065482 0.812137 -0.08276 12.402 40.327 3.6661 4.094 1.888 2.667

8 0.069146 0.781814 -0.08404 11.307 5.397 2.2063 2.887 3.307 2.232

9 0.063772 0.767287 -0.08277 12.032 16.151 2.4574 4.794 2.554 4.123

10 0.062145 0.766581 -0.08438 12.335 31.91 2.418 3.93 1.049 2.105

11 0.063235 0.667225 -0.07022 10.551 41 1.2233 2.366 1.16 2

12 0.070695 0.808459 -0.08918 11.436 29.377 3.8778 2.227 3.574 2.229

13 0.061459 0.75213 -0.08357 12.238 39.056 1.943 3.764 1.226 3.089

14 0.062271 0.776261 -0.08791 12.466 21.496 2.7937 4.496 0.719 1.88

15 0.060168 0.733081 -0.07816 12.184 41.067 1.344 5.155 1.54 1.928

16 0.059584 0.700233 -0.07525 11.752 29.899 0.6588 4.824 1.265 2.577

17 0.068826 0.8302 -0.08882 12.062 32.157 3.2135 5.239 3.284 4.922

18 0.066075 0.799084 -0.09252 12.094 23.951 3.1134 3.375 0.911 4.938

19 0.062924 0.70472 -0.07618 11.2 27.83 1.3662 2.893 3.069 3.387

20 0.062624 0.726496 -0.08057 11.601 18.925 1.208 4.912 3.561 6.806

21 0.066892 0.810707 -0.09482 12.12 14.872 3.2984 3.541 1.291 5.186

22 0.064379 0.757304 -0.08657 11.763 6.831 1.7084 4.574 1.16 5.215

23 0.064242 0.77116 -0.08835 12.004 25.847 2.03 4.626 2.283 15.844

24 0.066089 0.804213 -0.09182 12.169 30.477 2.833 5.095 1.153 7.784

25 0.065827 0.795359 -0.09081 12.083 30.986 3.98 2.925 0.705 1.544

26 0.059417 0.706493 -0.0777 11.89 43.125 0.743 3.822 3.064 4.387

27 0.06354 0.744834 -0.08159 11.722 29.882 1.9812 2.743 3.273 3.042

28 0.066317 0.787868 -0.08838 11.88 29.312 3.0467 2.506 2.751 3.161

29 0.066876 0.768412 -0.08131 11.49 27.907 2.904 5.175 3.569 2.438

30 0.064514 0.723628 -0.08025 11.217 26.882 2.7982 1.828 3.164 2.083

31 0.065261 0.774121 -0.08562 11.862 41.846 1.939 4.847 1.155 16.207

32 0.071363 0.847006 -0.09821 11.869 1.06 3.7765 5.12 3.391 11.341

33 0.066056 0.711902 -0.07714 10.777 12.515 1.596 4.755 3.408 7.573

34 0.065468 0.766487 -0.08501 11.708 29.542 2.1634 4.634 0.647 5.046

35 0.068243 0.790893 -0.08652 11.589 35.837 3.19 3.939 1.39 4.941

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36 0.063118 0.739882 -0.08225 11.722 6.538 1.695 2.755 3.14 4.211

37 0.063322 0.726681 -0.08138 11.476 25.252 2.3288 2.33 0.763 2.507

38 0.067203 0.798933 -0.09319 11.888 5.741 3.752 1.488 2.929 3.912

39 0.063122 0.748223 -0.07863 11.854 33.559 1.965 4.317 0.88 1.696

40 0.060796 0.729888 -0.08134 12.006 23.249 1.1835 4.156 2.729 2.498

41 0.062762 0.750355 -0.08359 11.956 14.43 1.4421 4.922 1.975 1.976

42 0.066072 0.791038 -0.08906 11.972 23.023 2.2761 4.469 2.629 5.757

43 0.064845 0.788515 -0.09081 12.16 11.311 3.0116 3.828 2.79 1.748

44 0.064222 0.767301 -0.08861 11.948 4.464 2.472 2.706 2.938 2.391

45 0.067417 0.829948 -0.09878 12.311 9.916 3.813 3.597 1.776 3.439

46 0.068852 0.839568 -0.09787 12.194 33.055 4.0018 3.583 1.149 7.023

47 0.066373 0.781072 -0.08644 11.768 29.007 3.156 1.895 3.616 2.265

48 0.065685 0.78912 -0.09349 12.014 1.464 3.5236 1.97 0.814 1.654

49 0.058473 0.706604 -0.07876 12.084 19.077 0.391 4.39 2.539 3.375

50 0.063025 0.763955 -0.08544 12.121 30.319 2.181 3.846 1.246 2.616

51 0.064263 0.773836 -0.08777 12.042 25.394 2.081 4.777 3.029 5.556

52 0.063056 0.765545 -0.08766 12.141 24.596 2.594 3.322 1.85 1.925

53 0.062625 0.751446 -0.08567 11.999 13.927 1.298 5.332 3.632 6.468

54 0.06254 0.741703 -0.08328 11.86 39.294 2.025 2.142 2.666 3.495

55 0.067656 0.845959 -0.09499 12.504 34.655 3.9618 4.867 1.58 3.259

56 0.064607 0.780937 -0.0901 12.087 39.025 2.471 3.662 0.639 7.535

57 0.065082 0.790231 -0.09227 12.142 5.752 2.683 4.58 2.575 2.78

58 0.066747 0.818742 -0.09355 12.266 32.439 3.2601 4.876 1.456 4.249

59 0.06535 0.790547 -0.09246 12.097 1.492 2.3175 4.833 2.821 10.41

60 0.066592 0.816557 -0.09471 12.262 35.163 3.8322 2.411 3.601 3.079

61 0.064056 0.771703 -0.08791 12.047 18.218 2.353 4.296 1.24 2.371

62 0.060086 0.727413 -0.08126 12.106 30.974 0.885 4.618 3.045 9.57

63 0.067359 0.812104 -0.09393 12.056 9.178 2.792 4.755 3.258 4.56

64 0.060754 0.722218 -0.08164 11.888 41.982 1.4058 2.269 1.133 6.789

65 0.067391 0.823019 -0.09566 12.213 26.693 3.619 2.868 1.775 6.247

66 0.066922 0.82277 -0.09703 12.294 10.565 3.643 3.786 2.01 2.862

67 0.067354 0.823617 -0.09503 12.228 29.265 3.166 5.223 2.213 12.226

68 0.068173 0.835251 -0.09659 12.252 35.518 3.65 4.069 1.863 8.408

69 0.065146 0.779695 -0.09019 11.968 19.255 2.1604 5.196 1.248 8.303

70 0.066688 0.814796 -0.09252 12.218 34.239 3.483 3.721 2.228 3.505

71 0.063533 0.768312 -0.08736 12.093 42.123 2.136 3.535 0.776 11.434

72 0.064287 0.772613 -0.08901 12.018 9.88 2.4105 4.587 1.297 1.967

73 0.06717 0.822062 -0.0941 12.239 35.301 3.163 4.146 3.038 11.429

74 0.062079 0.731916 -0.08237 11.79 38.466 1.845 1.647 1.036 16.279

75 0.066765 0.804322 -0.09186 12.047 10.416 2.7521 4.106 3.678 4.329

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76 0.064709 0.791236 -0.08889 12.228 36.699 3.725 3.431 0.621 1.538

77 0.063924 0.769012 -0.08911 12.03 16.983 2.654 3.031 1.265 2.457

78 0.065399 0.778909 -0.08957 11.91 30.393 2.321 4.333 0.933 11.976

79 0.061975 0.746793 -0.08442 12.05 14.886 1.487 4.655 3.094 2.624

80 0.062324 0.75143 -0.08487 12.057 28.114 1.8535 3.487 2.825 3.718

81 0.063528 0.766799 -0.08746 12.07 26.178 1.81 4.793 2.333 9.609

82 0.061137 0.740792 -0.08309 12.117 30.863 1.127 5.028 3.531 10.697

83 0.065376 0.791213 -0.09223 12.103 23.738 3.489 2.182 0.938 2.252

84 0.062364 0.746287 -0.08532 11.967 17.207 1.9317 2.645 2.3 3.721

85 0.062996 0.767336 -0.08747 12.181 11.465 1.942 4.669 3.673 3.698

86 0.05907 0.716277 -0.08117 12.126 9.469 0.782 4.749 2.185 2.064

87 0.062191 0.753393 -0.08513 12.114 12.423 1.86 4.114 3.197 2.176

88 0.059447 0.701039 -0.07962 11.793 1.669 0.5577 3.423 3.525 1.554

89 0.06319 0.770538 -0.08295 12.194 44.584 2.1656 5.203 2.466 2.738

90 0.060548 0.715177 -0.07982 11.812 40.097 1.4208 2.025 0.569 2.616

91 0.064168 0.762937 -0.08814 11.89 2.378 1.752 5.303 1.168 2.779

92 0.062075 0.750418 -0.08386 12.089 36.597 1.916 3.431 2.698 2.772

93 0.06368 0.756973 -0.08532 11.887 21.644 1.6553 5.149 0.587 3.536

94 0.059731 0.726842 -0.08058 12.169 39.352 0.8296 5.211 2.908 9.181

95 0.063824 0.770738 -0.08632 12.076 35.732 2.071 4.445 2.316 5.194

96 0.058911 0.699069 -0.07691 11.867 42.233 0.9 2.501 3.303 1.723

97 0.059066 0.708716 -0.07824 11.999 31.552 0.5712 4.157 3.695 3.273

98 0.066915 0.81989 -0.09328 12.253 33.365 3.1562 4.935 3.584 6.177

99 0.061373 0.737534 -0.08479 12.017 12.92 2.0366 2.633 1.429 1.663

100 0.063422 0.764602 -0.0891 12.056 7.298 2.474 2.974 2.794 2.282

Max 0.847006 12.504 44.584 4.0018 5.332 3.695 16.279

Min 0.058473 -0.09878 1.06 0.391 1.488 0.569 1.538

Table 9.8: Data for winglet 1 LMT

1 LMT

cw ß c a b n

2.576 34.751 5.34 3.4672 1.599 8.761

tx twx x0 s φ α

1.798 0.867 17.145 34.29 83.612 -45.963

Leading edge Trailing edge Median edge

x y z x y z x y z

Δc1L Δb1L Δa1L Δc1T Δb1T Δa1L Δc1M Δb1M Δa1M

0 0 0 5.34 0 0 1.7979 0 0

0 0 0.035 5.34 0 0.035 1.7979 0 0.035

0 0 0.07 5.34 0 0.07 1.7979 0 0.07

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0 0 0.1051 5.34 0 0.1051 1.7979 0 0.1051

0 0 0.1401 5.34 0 0.1401 1.7979 0 0.1401

0 0 0.1751 5.34 0 0.1751 1.7979 0 0.1751

0 0 0.2101 5.34 0 0.2101 1.7979 0 0.2101

0 0 0.2452 5.34 0 0.2452 1.7979 0 0.2452

0 0 0.2802 5.34 0 0.2802 1.7979 0 0.2802

0 0 0.3152 5.34 0 0.3152 1.7979 0 0.3152

0 0 0.3502 5.34 0 0.3502 1.7979 0 0.3502

0 0 0.3852 5.34 0 0.3852 1.7979 0 0.3852

0 0 0.4203 5.34 0 0.4203 1.7979 0 0.4203

0 0 0.4553 5.34 0 0.4553 1.7979 0 0.4553

0 0 0.4903 5.34 0 0.4903 1.7979 0 0.4903

0 0 0.5253 5.34 0 0.5253 1.7979 0 0.5253

0 0 0.5604 5.34 0 0.5604 1.7979 0 0.5604

0 0 0.5954 5.34 0 0.5954 1.7979 0 0.5954

0 0 0.6304 5.34 0 0.6304 1.7979 0 0.6304

0 0 0.6654 5.34 0 0.6654 1.7979 0 0.6654

0 0 0.7004 5.34 0 0.7004 1.7979 0 0.7004

0 0 0.7355 5.34 0 0.7355 1.7979 0 0.7355

0 0 0.7705 5.34 0 0.7705 1.7979 0 0.7705

0 0 0.8055 5.34 0 0.8055 1.7979 0 0.8055

0 0 0.8405 5.34 0 0.8405 1.7979 0 0.8405

0 0 0.8756 5.34 0 0.8756 1.7979 0 0.8756

0 0 0.9106 5.34 0 0.9106 1.7979 0 0.9106

0 0 0.9456 5.34 0 0.9456 1.7979 0 0.9456

0 0 0.9806 5.34 0 0.9806 1.7979 0 0.9806

0 0 1.0156 5.34 0 1.0156 1.7979 0 1.0156

0 1E-05 1.0507 5.34 1E-05 1.0507 1.7979 1E-05 1.0507

0 1E-05 1.0857 5.34 1E-05 1.0857 1.7979 1E-05 1.0857

1E-05 1E-05 1.1207 5.34 1E-05 1.1207 1.7979 1E-05 1.1207

1E-05 1E-05 1.1557 5.34 1E-05 1.1557 1.7979 1E-05 1.1557

1E-05 2E-05 1.1908 5.34 2E-05 1.1908 1.7979 2E-05 1.1908

1E-05 2E-05 1.2258 5.34 2E-05 1.2258 1.7979 2E-05 1.2258

1E-05 3E-05 1.2608 5.34 3E-05 1.2608 1.7979 3E-05 1.2608

2E-05 3E-05 1.2958 5.34 3E-05 1.2958 1.7979 3E-05 1.2958

2E-05 4E-05 1.3308 5.34 4E-05 1.3308 1.7979 4E-05 1.3308

3E-05 5E-05 1.3659 5.34 5E-05 1.3659 1.7979 5E-05 1.3659

4E-05 7E-05 1.4009 5.34 7E-05 1.4009 1.7979 7E-05 1.4009

5E-05 8E-05 1.4359 5.3399 8E-05 1.4359 1.7979 8E-05 1.4359

6E-05 0.0001 1.4709 5.3399 0.0001 1.4709 1.7979 0.0001 1.4709

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7E-05 0.0001 1.506 5.3399 0.0001 1.506 1.798 0.0001 1.506

9E-05 0.0002 1.541 5.3399 0.0002 1.541 1.798 0.0002 1.541

0.0001 0.0002 1.576 5.3399 0.0002 1.576 1.798 0.0002 1.576

0.0001 0.0002 1.611 5.3398 0.0002 1.611 1.798 0.0002 1.611

0.0002 0.0003 1.646 5.3398 0.0003 1.646 1.798 0.0003 1.646

0.0002 0.0003 1.6811 5.3398 0.0003 1.6811 1.798 0.0003 1.6811

0.0002 0.0004 1.7161 5.3397 0.0004 1.7161 1.798 0.0004 1.7161

0.0003 0.0005 1.7511 5.3397 0.0005 1.7511 1.798 0.0005 1.7511

0.0003 0.0006 1.7861 5.3396 0.0006 1.7861 1.798 0.0006 1.7861

0.0004 0.0007 1.8212 5.3395 0.0007 1.8212 1.798 0.0007 1.8212

0.0004 0.0008 1.8562 5.3395 0.0008 1.8562 1.798 0.0008 1.8562

0.0005 0.0009 1.8912 5.3394 0.0009 1.8912 1.798 0.0009 1.8912

0.0006 0.0011 1.9262 5.3392 0.0011 1.9262 1.7981 0.0011 1.9262

0.0007 0.0012 1.9612 5.3391 0.0012 1.9612 1.7981 0.0012 1.9612

0.0008 0.0015 1.9963 5.339 0.0015 1.9963 1.7981 0.0015 1.9963

0.001 0.0017 2.0313 5.3388 0.0017 2.0313 1.7981 0.0017 2.0313

0.0011 0.002 2.0663 5.3386 0.002 2.0663 1.7982 0.002 2.0663

0.0013 0.0023 2.1013 5.3384 0.0023 2.1013 1.7982 0.0023 2.1013

0.0015 0.0026 2.1364 5.3381 0.0026 2.1364 1.7982 0.0026 2.1364

0.0017 0.0031 2.1714 5.3378 0.0031 2.1714 1.7983 0.0031 2.1714

0.002 0.0035 2.2064 5.3375 0.0035 2.2064 1.7983 0.0035 2.2064

0.0023 0.004 2.2414 5.3371 0.004 2.2414 1.7984 0.004 2.2414

0.0026 0.0046 2.2764 5.3367 0.0046 2.2764 1.7985 0.0046 2.2764

0.003 0.0053 2.3115 5.3362 0.0053 2.3115 1.7985 0.0053 2.3115

0.0035 0.0061 2.3465 5.3357 0.0061 2.3465 1.7986 0.0061 2.3465

0.0039 0.0069 2.3815 5.335 0.0069 2.3815 1.7987 0.0069 2.3815

0.0045 0.0079 2.4165 5.3343 0.0079 2.4165 1.7988 0.0079 2.4165

0.0051 0.009 2.4516 5.3336 0.009 2.4516 1.7989 0.009 2.4516

0.0058 0.0102 2.4866 5.3327 0.0102 2.4866 1.7991 0.0102 2.4866

0.0066 0.0115 2.5216 5.3317 0.0115 2.5216 1.7992 0.0115 2.5216

0.0074 0.0131 2.5566 5.3306 0.0131 2.5566 1.7994 0.0131 2.5566

0.0084 0.0148 2.5916 5.3294 0.0148 2.5916 1.7996 0.0148 2.5916

0.0095 0.0167 2.6267 5.328 0.0167 2.6267 1.7998 0.0167 2.6267

0.0107 0.0188 2.6617 5.3265 0.0188 2.6617 1.8 0.0188 2.6617

0.0121 0.0213 2.6967 5.3247 0.0213 2.6967 1.8003 0.0213 2.6967

0.0137 0.024 2.7317 5.3228 0.024 2.7317 1.8006 0.024 2.7317

0.0154 0.027 2.7668 5.3206 0.027 2.7668 1.8009 0.027 2.7668

0.0173 0.0304 2.8018 5.3182 0.0304 2.8018 1.8013 0.0304 2.8018

0.0195 0.0342 2.8368 5.3154 0.0342 2.8368 1.8017 0.0342 2.8368

0.0219 0.0384 2.8718 5.3124 0.0384 2.8718 1.8022 0.0384 2.8718

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0.0246 0.0432 2.9068 5.3089 0.0432 2.9068 1.8027 0.0432 2.9068

0.0277 0.0486 2.9419 5.305 0.0486 2.9419 1.8033 0.0486 2.9419

0.0312 0.0547 2.9769 5.3007 0.0547 2.9769 1.804 0.0547 2.9769

0.0351 0.0616 3.0119 5.2957 0.0616 3.0119 1.8048 0.0616 3.0119

0.0396 0.0695 3.0469 5.2901 0.0695 3.0469 1.8057 0.0695 3.0469

0.0447 0.0784 3.082 5.2836 0.0784 3.082 1.8067 0.0784 3.082

0.0506 0.0887 3.117 5.2762 0.0887 3.117 1.8078 0.0887 3.117

0.0573 0.1006 3.152 5.2677 0.1006 3.152 1.8091 0.1006 3.152

0.0652 0.1144 3.187 5.2578 0.1144 3.187 1.8107 0.1144 3.187

0.0745 0.1306 3.222 5.2461 0.1306 3.222 1.8125 0.1306 3.222

0.0856 0.1501 3.2571 5.2321 0.1501 3.2571 1.8146 0.1501 3.2571

0.0991 0.1738 3.2921 5.2151 0.1738 3.2921 1.8173 0.1738 3.2921

0.1161 0.2036 3.3271 5.1936 0.2036 3.3271 1.8206 0.2036 3.3271

0.1384 0.2428 3.3621 5.1655 0.2428 3.3621 1.825 0.2428 3.3621

0.1701 0.2984 3.3972 5.1255 0.2984 3.3972 1.8311 0.2984 3.3972

0.2235 0.3921 3.4322 5.0582 0.3921 3.4322 1.8416 0.3921 3.4322

0.9116 1.5994 3.4672 4.1902 1.5994 3.4672 1.977 1.5994 3.4672

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Chapter 10 References

1 The Aviation Partners Boeing Co. Unites states Patent. Blended Winglet, September 1994.

2 http://www.boeing.com/commercial/757family/pf/pf_facts.html

3 Kennedy and R.C. Eberhart. “Particle swarm optimization”, in Proceedings of the IEEE International,

Conference on Neural Networks, IV, pages 1942-1948. 4 A New Particle Swarm Optimiser for Linearly Constrained Optimisation, Ulrich Paquet, Andries P. Engelbrecht

5 ALLEN, J.E. Global energy issues affecting aeronautics: a reasoned conjecture, Progress in Aerospace

Sciences, 1999. 6 Royal Commission on Environmental Pollution, The environmental effects of civil aircraft in flight, November

2002. 7 ALLEN, J.E. aviation and the environmental challenge, June 2003. 8 SMITH, H. College of aeronautics blended wing body development programme, Proceedings of ICAS 2000

Congress, Harrogate, September 2000, paper 1.1.4. 9 ICAO, Operational opportunities to minimise fuel use and reduce emissions, proposed ICAO Circular, CAEP/5-

IP/4, January 2001. 10 KUCHEMANN, D. The Aerodynamic Design of Aircraft, Pergamon, 1978. 11

http://www.boeing.com/commercial/757family 11

The Aviation Partners Boeing Co. Unites states Patent. Blended Winglet, September 1994. 11

http://www.boeing.com/commercial/757family/pf/pf_facts.html 11 ALLEN, J.E. Global energy issues affecting aeronautics: a reasoned conjecture, Progress in Aerospace

Sciences, 1999. 11 Royal Commission on Environmental Pollution, The environmental effects of civil aircraft in flight, November

2002. 11 ALLEN, J.E. aviation and the environmental challenge, June 2003. 11 SMITH, H. College of aeronautics blended wing body development programme, Proceedings of ICAS 2000

Congress, Harrogate, September 2000, paper 1.1.4. 11 ICAO, Operational opportunities to minimise fuel use and reduce emissions, proposed ICAO Circular,

CAEP/5-IP/4, January 2001. 11 KUCHEMANN, D. The Aerodynamic Design of Aircraft, Pergamon, 1978.

11

http://www.boeing.com/commercial/757family

http://www.boeing.com/commercial/757family/pf/pf_facts.html

http://www.boeing.com/commercial/757family

http://www.boeing.com/commercial/757family/pf/pf_facts.html

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