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.
P a g e | 2
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
P a g e | 3
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 4
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 5
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 6
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 7
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 8
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 9
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 10
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 11
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 12
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 13
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 14
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 15
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 16
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 17
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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 .
P a g e | 18
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 19
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 2
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 3
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 4
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 5
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 6
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 7
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 8
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 9
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 10
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 11
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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)
P a g e | 12
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 13
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 14
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 15
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 16
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 17
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 18
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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)
P a g e | 19
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 20
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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:
P a g e | 21
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 22
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 23
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 24
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 25
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 26
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 27
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 28
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 29
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 30
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 31
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 32
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 33
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 34
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 35
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
.
P a g e | 36
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 37
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 38
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 40
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 41
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 42
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 43
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 44
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 45
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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,
P a g e | 46
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 47
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 48
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 49
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
9 Optimal Winglet #2 2 NACA 2412 Wing 18''/12'' 23.3
10 Optimal Winglet #5 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
P a g e | 50
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 51
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 52
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 53
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 54
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 55
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 56
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 47: Isometric View of Trefftz Plane with Winglets Down
Table 3.5: Forces for Winglets Down
Force Functions Value
Coefficient of Lift (CL) 0.30548
Coefficient of Drag (CD) 0.0429454
Lift/Drag ratio (CL/CD) 7.11
Coefficient of Moment (CM) -0.020751
P a g e | 57
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 58
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 59
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
Force Functions Value
Coefficient of Drag (CD) 0.0633643
Coefficient of Lift (CL) 0.461229
Lift/Drag ratio (CL/CD) 7.28
Coefficient of Moment (CM) -0.0156387
Figure 51: Residuals vs. Time for Winglets Up
P a g e | 60
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 61
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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˚
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) 181.38 m2
Mean Aerodynamic Chord (Lref) 5.34 m
Wing Span 34.2 m
Center of Rotation (Center of Pressure) (2.23, -0.23, 0)
Lift Direction (Vector Value) (0, 1, 0)
P a g e | 62
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Drag Direction (Vector Value) (1, 0, 0)
Pitch Axis (Vector Value) (0, 0, 1)
# of Nodes on Cluster 10
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.
P a g e | 63
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
Force Functions Value
Coefficient of Drag (CD) 0.114784
Coefficient of Lift (CL) 0.622182
Lift/Drag ratio (CL/CD) 5.42
Coefficient of Moment (CM) -0.0329155
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.
P a g e | 64
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 65
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 55: Top View of a Cambered In Winglet
Figure 56: Front View of Trefftz Plane for Camber In
P a g e | 66
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 57: Residuals vs. Time for Winglet Camber In
Table 3.10: Forces for Winglet Camber In
Force Functions Value
Coefficient of Drag (CD) 0.106857
Coefficient of Lift (CL) 0.69013
Lift/Drag ratio (CL/CD) 6.46
Coefficient of Moment (CM) -0.0610928
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
P a g e | 67
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 68
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
Force Functions Value
Coefficient of Drag (CD) 0.107395
Coefficient of Lift (CL) 0.676833
Lift/Drag ratio (CL/CD) 6.30
Coefficient of Moment (CM) -0.0641388
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.
P a g e | 69
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 70
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 71
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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:
P a g e | 72
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 73
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 74
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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%
P a g e | 75
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 76
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 77
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 78
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
A B C D Pareto front Objective 5 Raw data Wing 1
P a g e | 79
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
A B C D Pareto front Objective 6 Raw data Wing 1
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
P a g e | 80
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 72: Cl vs. Cd
High CL
High CD
Low CL
Low CD
P a g e | 81
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 73: Cm vs Cd
Figure 74: Cm vs Cl
Desirable Area 1>CL>0.95
-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
P a g e | 82
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 75: Cl, Cm, Cl/Cd
Figure 76: Cm vs. Cl
Low CL
Low Cm
Desirable Area 1>CL>0.95
-0.1<CM<-0.092
High CL
High Cm
Desirable Area 1>CL>0.95
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
P a g e | 83
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 77: Cl/Cd vs .Cm
Figure 78: Cl/Cd vs. Cl
Low CL/CD
Low CL
High CL/CD
High CL
Desirable Area 1>CL>0.95
12.4<CL/CD<12.5
High CL/CD
High Cm
Low CL/CD
Low Cm
P a g e | 84
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
Desirable Area 1>CL>0.95
12.4<CL/CD<12.5
Desirable Area 1>CL>0.95
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
P a g e | 85
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 86
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
Desirable Area -0.1<CM<0.95
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
P a g e | 87
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 85: Cl/Cd vs. Cm
Figure 86: Cl/Cd vs. Cd
Low CM
High CL/CD
Low CM
Low CL/CD
Desirable Area -0.1<CM<0.95
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
P a g e | 88
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
4.7 Optimal Winglet Configurations
Table 4.5: Parameters for Optimal Winglets CFD Analysis
Parameter Value
Mach Number (M) 0.3
Angle of Attack (α) 8.0
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) 95.85 m2
Mean Aerodynamic Chord (Lref) 5.367 m
Wing Span 17.1 m
Center of Rotation (Center of Pressure) (2.285, -0.1377, -6.86)
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
P a g e | 89
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
4.7.1.1.1 Simple NACA 2412
Table 4.6: Values of Forces for Simple NACA 2412 Wing
Force Functions Value
Coefficient of Lift (CL) 0.606192
Coefficient of Drag (CD) 0.0547969
Coefficient of Moment (CM) -0.0664281
Lift/Drag ratio (CL/CD) 11.06
Figure 87: Domain of Simple NACA 2412 with a Symmetry Plane
P a g e | 90
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 88: Front View of Trefftz Plane for Simple NACA 2412 Wing
Figure 89: Streamlines at Wing-Tip for Simple NACA 2412 Wing
P a g e | 91
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
Force Functions Value
Coefficient of Lift (CL) 0.788834
Coefficient of Drag (CD) 0.0638685
Coefficient of Moment (CM) -0.0787166
Lift/Drag ratio (CL/CD) 12.35
P a g e | 92
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 92: Front View of Optimal Winglet #1
Figure 93: Side View of Optimal Winglet #1
P a g e | 93
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 94: Top View of Optimal Winglet #1
Figure 95: Front View of Trefftz Plane for Optimal Winglet #1
P a g e | 94
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 96: Side View of Pressure Field of Optimal Winglet #1
Figure 97: Streamlines at Wing-Tip for Optimal Winglet #1
P a g e | 95
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
Force Functions Value
Coefficient of Lift (CL) 0.802758
Coefficient of Drag (CD) 0.0646293
Coefficient of Moment (CM) -0.0793139
Lift/Drag ratio (CL/CD) 12.42
P a g e | 96
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 99: Top View of Optimal Winglet #2
Figure 100: Front View of Optimal Winglet # 2
P a g e | 97
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 101: Side View of Optimal Winglet #2
Figure 102: Front View of Trefftz Plane for Optimal Winglet #2
P a g e | 98
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 103: Pressure Field for Optimal Winglet #2
Figure 104: Streamlines Around Wing-Tip For Optimal Winglet #2
P a g e | 99
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 105: Plot of Residuals vs. T
P a g e | 100
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Chapter 5 Aerodynamic Analysis
Table 9: Parameters for Boeing 757 CFD Analysis
Parameter Value
Mach Number (M) 0.3
Angle of Attack (α) 8.0
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
Mean Aerodynamic Chord (Lref) 5.367 m
Planform Area (Aref) 79.68 m2
Wing Span 17.1 m
Center of Rotation (Center of Pressure) (5.97, -0.34, 4.62)
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
5.1 6.1 Boeing 757 Simple Wing
Table 10: Values of Forces for 757 Simple Wing
Force Functions Value
Coefficient of Lift (CL) 0.496619
Coefficient of Drag (CD) 0.095009
Coefficient of Moment (CM) 0.0488408
Lift/Drag ratio (CL/CD) 5.227
P a g e | 101
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 102
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 108: Pressure Field around 757 Simple Wing
Figure 109: Streamlines at Wing-Tip of 757 Simple Wing
P a g e | 103
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
Force Functions Value
Coefficient of Lift (CL) 0.514469
Coefficient of Drag (CD) 0.0956025
Coefficient of Moment (CM) 0.0623464
Lift/Drag ratio (CL/CD) 5.38
P a g e | 104
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 111: Side View of Original Boeing 757 Winglet
Figure 112: Front View of Original Boeing 757 Winglet
P a g e | 105
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 113: Top View of Original Boeing 757 Winglet
Figure 114: Front View of Trefftz Plane of Original Boeing 757 Winglet
P a g e | 106
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 115: Side View of Pressure Field for Original Boeing 757 Winglets
Figure 116: Streamlines at Wing-Tip for Original Boeing 757 Winglets
P a g e | 107
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 117: Plot of Residuals vs. Time for Original Boeing 757 Winglets
P a g e | 108
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
5.3 6.3 Optimal Boeing 757 Winglets
Table 12: Values of Forces for Optimal Boeing 757 Winglets
Force Functions Value
Coefficient of Lift (CL) 0.529444
Coefficient of Drag (CD) 0.0960115
Coefficient of Moment (CM) 0.0647884
Lift/Drag ratio (CL/CD) 5.5514
Figure 118: Front View of Optimal Boeing 757 Winglets
P a g e | 109
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 119: Top View of Optimal Boeing 757 Winglets
Figure 120: Side View of Optimal Boeing 757 Winglets
P a g e | 110
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 121: Front View of Trefftz Plane for Optimal Boeing 757 Winglets
Figure 122: Side View of Pressure Field for Optimal Boeing 757 Winglets
P a g e | 111
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 123: Streamlines at Wing-Tip of Optimal Boeing 757 Winglet
Figure 124: Plot of Residuals vs. Time for Optimal 757 Winglets
P a g e | 112
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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,
P a g e | 113
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 114
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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,
P a g e | 115
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 116
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 129: Removing Parts from 3D Printer
Figure 130: De-powdering Excess of the Parts
Figure 131: Parts Heating in Oven
P a g e | 117
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 132: Applying Epoxy to the Wing and Winglets
Figure 133: Parts After Curing Process
P a g e | 118
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 134: Sanding Parts
Figure 135: Drilling Holes in the Wing and Winglets
P a g e | 119
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 120
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 121
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 122
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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)
P a g e | 123
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 124
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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)
P a g e | 125
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 126
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 138: General parts of a commercial airplane.
P a g e | 127
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 139: Comparison of dimension between a 757-200 and a 757-300 aircrafts.
P a g e | 128
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
Figure 140: Technical drawings of the wing for a 757 Boeing aircraft.
HA
LF
P a g e | 129
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 130
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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.
P a g e | 131
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 132
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 133
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 134
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 135
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 136
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
P a g e | 137
Aerodynamic Shape Design Optimization of Winglets ∙ FIU ∙ College of Engineering
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
Top Related