study of the performance envelope of a variable sweep wing.pdf

8/18/2019 Study of the Performance Envelope of a Variable Sweep Wing.pdf

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Om Gam Ganapataye Namah STUDY OF THE PERFORMANCE ENVELOPE OF A VARIABLE

SWEEP WING

Minor Project Report

B.Tech. (ASE) Semester VI

By

Vinayak VadlamaniR340308040

Shikhar PurohitR180208035

Abhishree BaniR180208047

Under the guidance of

Prof. Dr. Ugur GUVENProfessor of Aerospace Engineering

Department of Aerospace Engineering,

University of Petroleum & Energy Studies, Dehradun


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FOREWORD

The students undertaking this minor project are sincerely grateful to Prof. Dr. Ugur GUVEN,

Professor of Aerospace Engineering for consenting to take this minor project under his purview

and would like to acknowledge his expert guidance and advice as the mentor/advisor for this

minor project, without whom our ventures into the vast and diverse field of Computational Fluid

Dynamics would have been unfruitful.

April 2011 Vinayak Vadlamani, B.Tech(ASE), VI Semester

Shikhar Purohit B.Tech(ASE),VI Semester

Abhishree Bani B.Tech(ASE),VI Semester


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Certificate

It is hereby certified that this minor project report titled “Study of the Performance Envelope of a

Variable Sweep Wing” by Vinayak Vadlamani, Shikhar Purohit and AbhishreeBani is the

original work of the authors and is thus approved for final submission.

Date: 27thApril 2011 Prof. Dr. Ugur GUVEN


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Table of Contents

Table of Contents............................................................................................................................4Nomenclature...................................................................................................................................6

Abbreviations...................................................................................................................................9List of Tables...................................................................................................................................8

List of Figures...............................................................................................................................10

Abstract..........................................................................................................................................13

1. Introduction & Literature Review..............................................................................................14

1.1 Introduction to Variable Sweep.........................................................................................14

1.2 Purpose and Objectives......................................................................................................15

1.3 Literature Review...............................................................................................................15

2. Sweep Theory............................................................................................................................182.1 Introduction ......................................................................................................................18

3. Fundamentals of Performance Envelopes.................................................................................19

3.1 Introduction........................................................................................................................193.2 Performance Curves...........................................................................................................22

4. Compressibility & its effects on aerodynamic coefficients.......................................................314.1 Introduction........................................................................................................................31

4.2 Compressibility Corrections..............................................................................................32

4.3 Lift Slope...........................................................................................................................35

4.4 Variation of wave drag and lift slope with sweep angle...................................................395. Airfoil Selection & Wing Design............................................................................................42

5.1 Introduction......................................................................................................................42

5.2 Initial Sizing & Weight Estimation..................................................................................42

5.3 Wing Design....................................................................................................................47

5.4 Airfoil Selection Criteria..................................................................................................505.5 Final Airfoil Selection......................................................................................................61

6. Performance curves..................................................................................................................65

6.1 Introduction......................................................................................................................656.2 2D lift Curve: Sectional Lift Coefficient, Cl vs. angle of attack, α..................................66

6.3 3D Lift Curve : Wing Lift Coefficient, CL vs. Angle of attack, α....................................66

6.4 Drag Polar: Total Drag coefficient, CD vs Lift Coefficient, CL........................................67

6.5 Subsonic and Supersonic Lift Curve Slope, α vs Mach number, M............................706.6 Taper Ratio, λ vs. Sweep Angle, Λ.................................................................................726.7 Aspect Ratio, AR versus Sweep Angle, Λ .....................................................................73

7. Effects of Variable Sweep........................................................................................................74

7.1 Mission objectives demanding variable sweep................................................................747.2 Efficient subsonic cruise and loiter..................................................................................74

7.3 Cruise efficiency..............................................................................................................75

7.4 Supersonic Efficiency......................................................................................................76

7.5 Take-off and landing performance...................................................................................787.6 Excessive Longitudinal Stability......................................................................................78

8. Computational Fluid Dynamics Analysis.................................................................................81 8.1 Introduction to CFD..........................................................................................................83

8.2 Purpose..............................................................................................................................838.3 Approach............................................................................................................................84


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3.3.1 Modeling.................................................................................................................84

3.3.2 Meshing...................................................................................................................853.3.3 Solving.....................................................................................................................90

8.4 Postprocessing/Results.............................................................................................................968.4.1 Lift Coefficient, Cl...................................................................................................97

8.4.2. Drag coefficient, Cd................................................................................................98

8.4.3 Static pressure contours.........................................................................................100

8.4.4 Dynamic pressure contours...................................................................................101

8.4.5 Total Pressure Contours......................................................................................102

8.4.6 Wall shear stress contours.....................................................................................103

8.4.7 Turbulent Kinetic Energy(k).................................................................................104

8.4.8 Symmetry Plane Mach number contours..............................................................105

8.5 Result Comparison & Conclusions..................................................................................106

8.5.1. Lift Coefficient, Cl................................................................................................106

8.5.2. Drag Coefficient, Cd.............................................................................................1068.5.3 Static Pressure Contours........................................................................................106

8.5.4 Dynamic Pressure Contours..................................................................................1078.5.5 Total Pressure Contours........................................................................................107

8.5.6 Wall shear stress contours.....................................................................................107

8.5.7 Turbulent kinetic energy (k) contours...................................................................108

8.5.8 Mach number contours..........................................................................................1089 Wind Tunnel Tests....................................................................................................................109

9.1 UPES Wind Tunnel Setup...............................................................................................109

9.2 Wind tunnel model..........................................................................................................109

9.3 Wind Tunnel Test Parameters & Results........................................................................110

9.3.1 Test Readings & Observations..............................................................................1109.4 Results & Conclusions.....................................................................................................112

10 Recommendations...................................................................................................................113

11 Working Mechanism...............................................................................................................11411.1 Introduction....................................................................................................................114

11.2 Working.........................................................................................................................115

11.3 Initial Design & Construction.......................................................................................11511.4 SolidWorks Simulation.................................................................................................116

11.5 Total Cost Estimate........................................................................................................117

Bibliography...............................................................................................................................118

Appendix.....................................................................................................................................119


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Nomenclature

NACA National Advisory Committee on Aeronautics

USAF United States Air Force

USN United States navy

TFX Tactical Fighter Experimental

a0Lift slope for an infinite wing

a0,comp Lift slope for an infinite wing in a subsonic compressible flow

a Lift slope for a finite wing

acompLift slope for a finite wing in a subsonic compressible flow

a∞Freestream speed of sound

A Statistical constant for Eqn. 4.3

AR (=b2 /S) Aspect Ratio

C statistical constants for Eqn. 4.3

Cd,i =kCL2 induced drag coefficient

CD,e total parasite drag coefficient

CD,w wave drag coefficient

CD,0 zero-lift (parasite) drag coefficient

ClSection Lift Coefficient

CLWing Lift Coefficient

Cl,des Design Lift Coefficient

CL,maxMaximum value of CL

CL,min Minimum value of CL

Cp Pressure Coefficient (Compressible case)

Cp,o Pressure Coefficient (Incompressible case)


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Cl,o Lift Coefficient (Incompressible case)

e Ostwald’s Span Efficiency factor

KvsVariable sweep constant for Eqn. 4.3

L/D Lift-to Drag Ratio

McrCritical Mach number

MDrag DivergenceDrag Divergence Mach number

M∞,nComponent of M∞ perpendicular to the half-chord line of the swept wing

M∞Freestream Mach number

SReference Area (of wing)

(t/c)max Maximum Thickness-to-Chord ratio

V∞Freestream/Flight Velocity

Vstall Stalling Speed

Vmax Maximum Speed

W Weight

W0Design take-off weight

WcrewWeight of the crew

WpayloadMaximum payload weight

WfuelWeight of the fuel

WemptyEmpty or structural weight

W/S Wing Loading

αi Induced angle of attack

α Angle of Attack

αstallStall Angle

αL=0 Zero-lift Angle of Attack

µ∞Freestream Absolute Viscosity


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ρ∞Free stream density

Taper Ratio

ΛSweep angle


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LIST OF TABLES

Page

Table 5.1:Maximum Critical Mach numbers for NACA 64-XXX series .................. 53Table 5.1:Stalling angle for various NACA 641-XXX airfoils. .................................. 56

Table 5.2:Stalling angle for various NACA 64AXXX airfoils ................................... 57

Table 8.1:Boundary Conditions ............................................................................... 96

Table 9.1:Test Run1 Data ...................................................................................... 106

Table 9.2:Test Run2 Data ...................................................................................... 107

Table 5.1:Test Run 3 Data ..................................................................................... 107

Table 5.1:Test Run 4 Data ..................................................................................... 107

Table 11.1:Final Cost Estimate ............................................................................... 113

Table A.1 :Wo Iteration Counter. ........................................................................... 115


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LIST OF FIGURES

Page

Fig. 1.1: The USAF F-14 - a variable sweep aircraft - cruising at a high sweep angle.................10

Fig. 2.1: Straight wing of airfoil section with Mcr=0.7..................................................................14

Fig. 2.2:Swept-wing of same airfoil section with Mcr=0.7...........................................................15Fig. 3.1: Schematic of CL and CD versus angle of attack...............................................................19

Fig. 3.2: Schematic of variation of lift coefficient with flight velocity ( in level flight)...............20

Fig. 3.3: Schematic of variation of drag coefficient with flight velocity for level flight...............21Fig. 3.4: Schematic of variation of L/D ratio with flight velocity for level flight.........................22

Fig. 3.5: Schematic of variation of L/D ratio with angle of attack................................................22

Fig. 3.6: Schematic of the components of a drag polar.................................................................24

Fig. 3.7: Slope of the drag polar at various points........................................................................25

Fig. 3.8: Illustration of minimum drag and drag at zero-lift.........................................................26

Fig. 4.1: Variation of profile drag with Mach number, illustrating drag divergence....................27

Fig. 4.2: Flat plate in supersonic flow inclined at an angle α, illustrating wave drag...................31

Fig 4.3: Lift slope for infinite and finite wing...............................................................................32Fig. 4.4: Variation of supersonic wave drag with AR...................................................................35

Fig. 4.5: Variation of minimum total drag coefficient with sweep angle......................................36

Fig. 5.1: Wing Design Flowchart...................................................................................................38

Fig. 5.2: Airfoil Nomenclature......................................................................................................47Fig. 5.3: Critical Mach number, Mcr vs. section lift coefficient, cl for NACA 0006, 0009 and

0012(left) and for NACA 1408, 1410 and 1412(right).................................................50

Fig 5.4: Critical Mach number, Mcr vs. section lift coefficient, cl for NACA 2412, 2415, 2418,

2421 and 2224(left) and for NACA 4412, 4415, 4418, 4421 and 4424(right)......................50

Fig 5.5:Critical Mach number, Mcr vs. section lift coefficient, cl for NACA 23012, 23015, 23018,23021 and 23024(left) and for NACA 63-XXX series(right)........................................................51

Fig 5.6:Critical Mach number, Mcr vs. section lift coefficient, cl for NACA 64-006, 64-009, 64l-

012, 642-015, 643-018 and 644-021 and for NACA 64-108, 64-110 and 641-112(right)..............52

Fig 5.7: Critical Mach number, Mcr vs. section lift coefficient, cl for NACA 64-XXX series......53

Fig 5.8: Lift and moment characteristics of NACA 0006(left) and NACA 0009(right)..............54

Fig 5.9: Lift and moment characteristics of NACA 1410 and NACA 1412.................................55

Fig 5.10: Lift and moment characteristics of NACA 64A210 and NACA 64A410....................56Fig. 5.11: NACA 64A210 airfoil profile.......................................................................................58

Fig. 5.12: NACA 64A410 airfoil profile.......................................................................................58

Fig. 5.13: Interpolated airfoils modeled on SolidWorks CAD software.......................................59Fig. 5.14: Side view of wing (from tip).........................................................................................60

Fig. 5.15: Lofted wing model rendered on SolidWorks CAD software (unswept configuration).60

Fig 6.1: Plot of sectional lift coefficient versus angle of attack....................................................62

Fig 6.2: Plot of wing lift coefficient versus angle of attack for different sweep angles..............63

Fig. 6.3: Variation of Oswald Efficiency factor with sweep angle...............................................65

Fig. 6.4: Drag Polar : Plot of CD vs. CL for different angles of attack...........................................66Fig. 6.5: Subsonic and Supersonic Lift Curve Slope Clα versus Mach number............................67

Fig. 6.6: Variation of Taper Ratio with Sweep Angle...................................................................68

Fig. 6.7: Variation of Aspect Ratio with Sweep Angle................................................................69


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Fig. 7.1: Variation of (L/D)max with Mach number ; variation of span loading(W/b2) with leading

edge sweep for F-14........................................................................................................70Fig 7.2: Variation of zero-lift parasite drag with Mach number for different sweep angles for the

F-14..................................................................................................................................71Fig. 7.3: Variation of range parameter (ML/D) versus Mach number for different sweep

configurations ................................................................................................................71

Fig. 7.4 : Wing thickness comparison for various aircraft.............................................................72

Fig. 7.5 : Plot of drag/weight(D/W), thrust/weight(T/W) ratio against Mach number for the F-14

and F-15.........................................................................................................................73

Fig. 7.6: Influence of sweep and speed on ride quality.................................................................74

Fig. 7.7: Effect of pivot position on aerodynamic loading about the longitudinal axis (X)........75

Fig. 7.8: Pivot and apex influence on longitudinal stability (Source: Design for Air Combat,Whitford).........................................................................................................................76

Fig. 7.9: Influence of pivot position on wing span and area (Source: Design for Air Combat,

Whitford).........................................................................................................................76Fig 8.1: The complete Navier-Stokes equations for a 3-D unsteady incompressible flow (Source:

NASA Glenn Research Center)......................................................................................77Fig. 8.2: CFD simulation showing surface pressure coefficient distribution over the lower surface

of the space shuttle at Mach 15 (Source: NASA Ames Research Center)....................78

Fig. 8.3(a) Minimum sweep configuration model.........................................................................80

Fig. 8.3(b) Cruise sweep configuration model...............................................................................80Fig. 8.4 : A 3-view drawing of the geometric model along with dimensions................................81

Fig. 8.5: Tri-Pave face meshing scheme –example mesh..............................................................83

Fig. 8.6: Aircraft Surface Mesh.....................................................................................................83

Fig. 8.7: Symmetry plane mesh.....................................................................................................84

Fig. 8.8: Complete Mesh including flow volume..........................................................................85Fig. 8.9: Boundary conditions applied to mesh/geometry.............................................................85

Fig. 8.10: Residuals showing divergence in the solver.................................................................87

Fig. 8.11: Scaled residuals for 200+ iterations..............................................................................89Fig. 8.12: Scaled residuals for 1000 iterations..............................................................................89

Fig. 8.13: Scaled residuals for 2000 iterations (minimum sweep configuration)........................90

Fig. 8.14: Scaled residuals for 1000 iterations (cruise sweep configuration)..............................91Fig. 8.15: Scaled residuals for 2000 iterations (cruise sweep configuration)..............................92

Fig. 8.16(a): Cl convergence history of minimum sweep configuration model..........................93

Fig. 8.16(b): Cl convergence history of cruise sweep configuration model................................94

Fig. 8.17: Cd convergence history of minimum (top) and cruise (bottom) sweep configuration..95

Fig. 8.18: Static pressure contours of minimum (top) and cruise (bottom) sweep configuration.96Fig. 8.19: Dynamic pressure contours of minimum (top) and cruise (bottom) sweep

configuration.................................................................................................................97

Fig. 8.20: Total pressure contours of minimum (top) and cruise (bottom) sweep configuration..98

Fig. 8.21: Wall shear stress contours for minimum (top) and cruise (bottom) sweep

configuration.................................................................................................................99

Fig. 8.22: Turbulent kinetic energy contours for minimum (top) and cruise (bottom) sweep

configuration..............................................................................................................100

Fig. 8.23: Mach number contours at the symmetry plane for minimum (top) and cruise (bottom)sweep configuration....................................................................................................101


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Fig. 9.1: Front view of model in wind tunnel .............................................................................106

Fig. 9.2: Top view of model setup in wind tunnel.......................................................................106 Fig. 11.1: Schematic diagram of initially proposed mechanism..................................................110

Fig. 11.2: Construction drawing of curve profile for working mechanism.................................112Fig. 11.3: Screenshot of the animation showing the simulation of the working mechanism....113


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ABSTRACT

Aircraft designers are faced with a daunting challenge while drafting the dimensions of a multi-

role aircraft with regard to wing design particularly and the phenomenon of drag divergence does

not make things easier for designers. Variable sweep offers a compromised solution that presents

variable-geometry wing as a means for incorporating a dynamic wing area and wing sweep

characteristics – which permit the wing geometry to be optimally customized for each mission

segment. This minor project has sought to understand and study the characteristics of a variable-

sweep wing, particularly the performance envelope. Computational Fluid Dynamics (CFD) has

been used as a tool to investigate the effects of variable sweep and to determine aerodynamic

coefficients mainly. A wind tunnel investigation was also carried out for basic aerodynamic

investigations to corroborate the CFD study. Also, an original working mechanism for changing

the sweep angle of wings was devised and constructed to aid in better understanding of

mechanical aspects of variable sweep. This report presents the findings of the three dimensional

CFD analysis carried out for two sweep configurations (takeoff and cruise sweep

configurations), a comparison between the theoretical and experimental studies and also

discusses a few aspects related to the physical sweep changing mechanism.

Keywords : variable sweep, computational fluid dynamics


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1. Introduction& Literature Review

1.1 Introduction to Variable sweep& its background

The concept of variable sweep wings has its roots in the development & emergence of multi-role

aircraft, in the late 1950s and early 60s. With huge advancements in propulsion and airframe

technology in the late 1950s, aircraft capable of performing multiple mission types were being

conceived. When aircraft designers were faced with challenges in the design of multi-role

aircraft, particularly wing design, variable sweep emerged as a possible solution. Variable sweep

offers an alternative in a variable-geometry wing as opposed to a common fixed-geometry wing

that can be varied in flight or on ground for optimum performance for a given mission segment.

But, there is a condition that the aerodynamic gains of variable sweep must offset its weight and

volume penalties. Figure 1 shows the most successful fighter aircraft in terms of production that

incorporates variable sweep, the Grumman F-14 Tomcat.

Fig. 1.1: The USAF F-14 - a variable sweep aircraft - cruising at a high sweep angle


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1.2 Purpose and Objectives

The following thesis/report is the result of the culmination of the minor project performed in theacademic year 2010-11, semesters V and VI, as part of the curriculum for a four year B.Tech

degree in Aerospace Engineering. The primary purpose of this minor project thesis is to

investigate and summarize the effects of variable sweep, particularly the performance envelope.

The minor project focused on both theoretical and practical aspects of aerospace engineering

through division of work on three main fronts:

• 3-D computational fluid dynamics analysis,

• Wind tunnel investigation of a scale model

• Fabrication of an original working mechanism for changing the sweep angle of

wings

A 3-D CFD analysis was chosen over a 2-D CFD analysis for an obvious reason that the sweep

theory is after all a completely three dimensional phenomenon and a 2D study would have lead

to limited results and incomplete understanding of the effects of variable sweep. The overall

objective of this minor project was to establish conclusions on the effects of variable sweep and

to demonstrate the capabilities of a working mechanism and its future potential.

1.3 Literature Review

In the course of investigation of various literature and text on the subject of variable sweep and

the sweep theory, many resources and text were found, both online and as physical text or books.

Adolf Busemann (1935) at the 5th Volta Conference first put forward the idea of the swept wing

concept. Later, Robert Jones (1945) from NACA gave a simple yet elegant mathematical

formulation for the explanation of the sweep theory. General text series in aerospace engineering

like titles on aerodynamics by Anderson and Clancy stress on the fact that the drag divergence

can be delayed by utilizing swept wings for high-speed aircraft.


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Ray Whitford’s Design for Air Combat discusses the origins of the development of variable

sweep experimental aircraft in the late 1950s and early 60s and the requirement for that

prompted their development. Further, J.D. Anderson’s title Aircraft Performance & Design was

consulted for understanding the fundamentals of performance curves related to aerodynamic

coefficients. Compressibility and its effects have been well explained in the standard text on

aerodynamics by J.D.Anderson – Fundamentals of Aerodynamics.

For matters pertaining to airfoil selection and wing design, technical papers from the NASA

Technical Reports Server (NTRS) were extensively used. NACA Report No. 824 by Abbott, vonDoenhoff and Stivers (1945) served as the primary reference for airfoil data required in pursuit

for the correct airfoil. This report is an excellent and comprehensive presentation of NACA four-

digit, five-digit, 6- and 7-series airfoils in development up to 1945 and also supplements with

additional data for predicted critical Mach number and aerodynamic characteristics of various

airfoil sections that have been extremely helpful during airfoil selection.

In the matters of wing design, Daniel P. Raymer’s Aircraft Design: A Conceptual Approach

which is regarded as the standard text on aircraft design by many was referred extensively and

the complete wing design for this project is based on empirical data and formulae from this book.

J.D. Anderson’s Aircraft Performance & Design served as a supplementary text here. For

understanding the effects of variable sweep, Ray Whitford’s Design for Air Combat was heavily

relied upon and extensively employed.

Further, technical reports describing studies performed by Loftin(1947) and Harper and

Maki(1964) also shed some light on aerodynamic characteristics of NACA 6A-series airfoils and

the stall characteristics of swept wings respectively. The former text discusses the aerodynamic

characteristics of the modified NACA 6A-series as compared to the original NACA 6-series. The

latter report is a fair guide that can aid in determining the actions necessary empirically to


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achieve a given set of wing characteristics with reference to stalling nature of swept wings that

was fairly unknown at the time of publication of this text.

During the CFD analysis of the two configuration models, the GAMBIT 2.2 Tutorial and

GAMBIT 2.2 Modeling Guide were extremely handy to address issues related with meshing.

And while working with FLUENT, the FLUENT User Guide and Documentation were overly

resourceful during the case setup and monitor setup. Also, Cornell University’s FLUENT tutorial

and resources like Introduction to CFD Basics by Rajesh Bhaskaran and Lance Collins were a

good quick read to understand the underlying principles of Computational Fluid Dynamics as asupplement to Anderson’s text on CFD – Computational Fluid Dynamics:The Basic Approach

with Application.


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2. Introduction to Sweep Theory & Variable Sweep

2.1 Introduction to Sweep Theory

The introduction of swept-wings into aerodynamics was prompted due to the ever increasing

speed of aircraft. Faster planes meant more drag but supersonic flight was the real player in

bringing swept wings into the fray. In the 1930s, when supersonic flight was considered

impossible by some aeronautical engineers and the speed of sound was looked upon as some sort

of natural barrier which could not be broken, leading aerodynamicists from all over the world

including the likes of Theodore Von Karman, Ludwig Prandtl and Adolf Busemann gathered at

the 5th Volta Conference to discuss flight at Mach numbers greater than unity. Busemann who

invented the swept-wing concept, explained why they were to play a major role in aircraft going

supersonic, most of his efforts were associated with the implications of compressibility.

Although Busemann and others tried to establish a mathematical framework for the sweep theory

but it was the mathematical genius of Robert T. Jones from NACA in 1945 who gave a simple

and comprehensive analysis of swept wing performance. This project will intend to look into

those same performance parameters which these aerodynamicists established so well.

Busemann originated the concept on the basis of the theory that swept-wings would have less

drag at high speeds than conventional straight wings. In supersonic flight, the main spoilsport is

the abrupt increase in drag due to shock waves for the freestream Mach number M∞greater than

the critical Mach number Mcr1. Hence it is desirable to increase Mcr as muchas possible in high-

speed airplane design. The following explanation simply compares a straight wing and a swept

wing.

V∞ M∞ Airfoil section with Mcr=0.7

Fig. 2.1 : Straight wing of airfoil section with Mcr=0.7

1Mcr: Mach number at which flow over some part of the airfoil first becomes sonic


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For a straight wing (refer Fig. 1) with an airfoil section with Mcr= 0.7, the airfoil experiences the

incident freestream velocity and Mach number at the leading edge which is same everywhere as

there is a zero angle of sweep. Hence, Mcr for the whole wing is itself 0.7. Now, if the wing is

swept by 30o(refer Fig. 2), then the same airfoil section on the new swept-wing will experience

only the component of flow normal to the leading edge of the wing. Hence all aerodynamic

properties including Mach number at this locality will be governed by the normal component of

flow.

Hence, the effective Mach number will be M∞cos 30o and the critical Mach number for the

swept-wing 0.7/cos 30o = 0.808. This means that the freestream Mach number can be further

increased. Hence, by sweeping the wings of subsonic aircraft, drag divergence is delayed to

higher mach numbers.

30o

Mcr for swept wing=0.7/cos 30o

Component parallel to section

Airfoil section with Mcr=0.7

Fig. 2.2 : Swept-wing of same airfoil section with Mcr=0.7

But in real scenario of 3-D flow over a wing the actual Mcr for swept wing, if Λ is the sweep

angle, then the following relation persists.

Mcr for airfoil < Actual Mcr for swept-wing <

Another explanation of how Mcr is increased by sweeping the wing is that the thickness-to-chord

ration (t/c) for a swept-wing wing is less than a straight wing or that the airfoil section is


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effectively thinner which in effect increases the critical mach number or Mcr. But a downside for

increasing the wing sweep is that the lift is essentially reduced for the same velocity and angle of

attack.

Also for supersonic flight, swept wings make the leading edge of the wings fall inside the Mach

cone rather than outside it, thereby avoiding the component of M∞ normal to the leading edge

which would have been supersonic.


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3. Fundamentals of Performance Envelopes

3.1. Introduction

Pushing the envelope has become such a popular remark in commonplace usage that it has taken

on the tag of a cliché. It is generally used to denote an act of exploring unchartered territories or

a new frontier by venturing beyond the limits. It is phrase that originated in the corridors of the

aerospace research industry, used by aerospace design engineers whose prototypes while being

tested were often pushed to the edge of their operational capabilities. The “ performance

envelope” of any flight vehicle in aerospace terminology is the set of curves defining the

maximum permissible values of crucial parameters such as velocity, lift etc. under which theaircraft is expected to perform safely. The basic custom is as follows: based upon the design

specifications and their knowledge and experience, the engineers express their probable

expectations within which safe behavior and control is anticipated, in form of a projected

performance envelope. Then, it is up to the test pilots to get behind the controls of the prototype

to judge its actual performance and to fly it to the absolute limit of its operational capability as a

calculated risk, in order to sketch out the actual performance envelope.

Performance Envelopes in aerodynamics deal with the most fundamental parameters in the

subject: the non-dimensional aerodynamic coefficients. These quantities are of greater

importance than the aerodynamic forces and moments themselves. The reason for this being that

the aerodynamic coefficients are dependent on less factors than the aerodynamic force itself.

The three aerodynamic coefficients are defined as follows:

1 Lift coefficient,

(3.1)

2 Drag coefficient,

(3.2)

3 Moment Coefficient,

(3.3)


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Hence, they are also much easier and accurately determinable than their dimensional

counterparts. In the words of Dr. J.D Anderson, author of Fundamentals of Aerodynamics,

“they are fundamental quantities, which make the difference between intelligent engineering and

simply groping in the dark ”. The evidence is in the form of the following equations:

Dimensional Analysis leads us to the following conclusions in form of Eqns. (3.4) and (3.5).

, , , ,, (3.4) , , (3.5)

, , (3.6)

Looking at Eqn. 3.4, we see that lift is a function of freestream density, freestream velocity,

reference area, angle of attack, viscosity of fluid and speed of sound in the fluid respectively, that

is a total of 6 parameters whereas from eqns. (3.5) and (3.6), CL and CD depend only upon three

namely angle of attack, Reynolds Number and the freestream Mach number respectively, the last

two being similarity parameters which help us in scaling the flow.2

Moreover, it can be inferred that since aerodynamic coefficients are independent of the reference

area S, which means that CL and CD allow for comparison between planes with different

reference area or simply different aircraft.

3.2. Performance Curves

The various performance envelopes are as follows:

3.2.1. CL, Lift Coefficient versus α, Angle of Attack and CD, Drag Coefficient versus α, Angle

of Attack

The generic variations of CL and CD, versus angle of attack are shown in figure 1.

2 To be noted, these relations are for an airplane of a given shape only.


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Fig. 3.1 : Schematic of CL and CD versus angle of attack

CLincreases linearly with the angle of attack until a maximum angle is reached at which the

aircraft wing stalls and CL peaks and then drops when angle of attack is increased further. From

this we arrive at a relation, the lowest possible velocity at which the aircraft can maintain steady,

level flight is dictated by the value of CL,max

.

, (3.7)

Hence, from this performance curve, without any aid of extra data, CL,max is determinable from

the physical laws of aerodynamics of flow over wings.

3.2.2. CL, Lift Coefficient versus V∞, Flight Velocity

Another performance curve, is used to find the maximum possible velocity in flight, Vmax. From

eqn. (3.1), it can be seen that for each value of V∞ there is a specific value of CL. The curve in

fig. 3.2 shows the variation of CL whole range of velocity from Vmax to Vstall.


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Fig. 3.2: Schematic of variation of lift coefficient with flight velocity ( in level flight)

In other words, the values of CL in the curve are the ones needed to maintain level flight over the

whole range of velocity. Thus designers must design the airplane to enable the aircraft to achieve

such values of CL.

3.2.3. CD, Drag Coefficient versus V∞, Flight Velocity

Performance Curves 1 and 2 give designers an estimate of the lift which the aircraft needs to

achieve. But this has to be done keeping in mind the degree of drag produced. For an efficient

design we need necessary lift with low drag. For this we have a curve of the drag coefficient

versus the flight velocity.


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Fig. 3.3: Schematic of variation of drag coefficient with flight velocity for level flight

A poor aerodynamic design with necessary values of CL but high values of CD will generate a

plot denoted by the dashed curve. On the other hand, a design with lower values of drag would

lead to a curve like the one denoted by the solid curve. An observation will show that the latterplot leads to higher value of Vmax as compared to the former.

3.2.4. (L/D), Lift-to Drag Ratio versus V∞, flight velocity

A correct predictor of aerodynamic efficiency is the lift-to-drag ratio which is nothing but the

ratio of the lift coefficient to the drag coefficient. For a good aerodynamic design, the L/D ratio

should be high enough for the aircraft to climb smoothly, hence the maximum value of the ratiogives the best climb rate for the aircraft.


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3.2.6CD, Drag Coefficient versus CL, Lift Coefficient (Drag Polar)

By far, the most important plot in applied aerodynamics is the Drag Polar. It is a performancecurve which covers all aerodynamic aspects in one single plot of the drag coefficient C D versus

the lift coefficient CL. The drag polar is a complete and concise plot of the overall aerodynamics

of an aircraft. Basically, the drag polar is a relation between CD and CL in which CD is expressed

as a function of CL. Both the equation and the plot are designated as “Drag Polar”.

The total drag on an airplane can be written as

, , (3.8)

where CD,e is the total parasite drag, CD,w is the wave drag and the term kCL2 is the

induced drag. Eqn. (3.8) can also be rewritten as

, (3.9)where CD,0 = CD,e + CD,w is called the zero-lift (parasite) drag coefficient

Eqn. (3.9) is called the drag polar of the airplane. It is valid for both subsonic and

supersonic flight. The plot can also be viewed as that of the resultant aerodynamic force

in polar coordinates, hence the label drag polar. Figure 3.6 is the plot of eqn. (3.9) and

hence the curve is also called the drag polar .


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F

Fig. 3.6 : Schematic of the components of a drag polar

Since eqn. (3.9) contains a squared-term of CL, hence the profile of the drag polar curve is

parabolic. The tangent of the curve would give a ration of CL /CD which is nothing but the L/D

ratio. The intercept of the curve on the x-axis is the zero-lift drag coefficient CD,0. As we go up

the curve, the slope increases at first, reaching a maximum value and then decreases again.

Figure 3.7 illustrates this observation.

Fig. 3.7: Slope of the drag polar at various points


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Interestingly, the tangent from the origin locates the point (2) on the curve of maximum L/D

ratio for the airplane. This point is called the design point for the airplane and the corresponding

value of CL is called the design-lift coefficient, C L,des.

For symmetric airfoils and in the case of zero incidence between the wing chord and axis of

symmetry of the fuselage, the zero-lift drag is equal to the minimum drag. But in the case of real

airplanes, when the plane is pitched at zero-lift angle of attack, parasite drag may be slightly

higher than the minimum drag. In this case the curve gets vertically shifted upwards as shown in

figure 3.8. The equation for the drag polar in that case would be

, , (3.10)

Fig. 3.8: Illustration of minimum drag and drag at zero-lift.


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4. Compressibility and its effect on aerodynamic coefficients

4.1 Introduction

When Adolf Busemann first conceived about swept wings, it was primarily an outcome of an

endeavor to reduce a new kind of drag encountered at supersonic speeds. Frank Whittle, another

mastermind, inventor of the jet engine, came tantalizingly close to designing the first plane to

break the so-called “Sound barrier” when a prototype of his design, broke up when it came close

to Mach 1. This problem of ever increasing drag near Mach 1 started myths of an “unbreakable”

sound barrier, a wall which no plane could ever cross. But aerospace engineers soon found a

reason to this in the form of the explanation of the phenomena of “drag divergence”.

Fig. 4.1: Variation of profile drag with Mach number, illustrating drag divergence

We will describe the concept of drag divergence which was introduced in the first chapter, here

in full detail. Figure 1 is a plot of drag coefficient, Cd versus the freestream Mach number M∞,

which vividly describes the concept of drag divergence. To follow drag divergence, it is

necessary to know what critical Mach number is.


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It is known that the flow over an airfoil expands around the top surface near the leading edge,

hence the velocity and thus the Mach number increase. This fact directly suggests that on the

airfoil surface a particular velocity is reached before the free stream flow reaches that particular

value.

Therefore, it is very realistic for a flow to be locally sonic on some point on the upper surface of

the airfoil even though the freestream flow is subsonic. Hence, by definition the freestream Mach

number at which sonic flow is first achieved locally somewhere on the airfoil is called the

Critical Mach number , Mcr of the airfoil.

Now, refer figure 1, we can clearly observe that cd remains fairly constant till the critical Machnumber is encountered. If the freestream Mach number is increased the local point of minimum

pressure at which the flow first achieved sonic speed is surrounded by a small “bubble” of

supersonic flow. Even now cd remains rationally low.

However, after this point if M∞ is further increased, the plot reveals a dramatic and abrupt rise in

cd. This corresponds to the first instance of shock waves appearing in the flow which in turn

cause an adverse pressure gradient leading to flow separation which explains the massive

increase in drag. This phenomenon is called Drag Divergence. And the freestream Mach number

at which cd begins to increase rapidly is called the Drag Divergence Mach number .

1.0 (4.1) The discovery of drag divergence gave aerospace designers an upper hand in their battle against

breaking the voodoo of the sound barrier. Designers soon realized that they could not reduce or

limit drag divergence but could possibly delay it. Experiments showed that it was possible to

increase the critical Mach number for a particular wing/airfoil section by sweeping the wings

either backwards or forwards. The mathematical genius of Robert T. Jones of NACA gave a

simple sweep theory which has already been discussed in chapter 1, which we restate here in

brief.

The reason for the increase in critical Mach number by the effect of sweep can be explained by

any one or both of the following reasons:


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i. By sweeping the wing, the airfoil effectively sees only the normal component of the Mach

number to the leading edge

ii. By sweeping the wings, the thickness-to-chord ratio is effectively lower i.e. the airfoil is

thinner

Recalling that this topic of drag divergence is nothing but a consequence of shock waves and

their effects which are in turn an effect of compressibility, we state that drag divergence is an end

result of variable density encountered at M∞> 0.3, if we may use this reference.

4.2 Compressibility Corrections

4.2.1. Subsonic

The thin airfoil theory was an initial aerodynamic theory evolved during the early days of flight

when aircraft speeds were limited to subsonic maxima from 1900s to 1940s. But with the advent

of the high-powered reciprocating engines and eventually with the introduction of the jet engine,

speeds of fighter aircraft began to increase to 500 mph and faster. Since at high subsonic speeds

of the order M∞=0.3 itself, compressibility effects come into picture, the incompressible flow

theory in which density was assumed practically as constant, failed outright in such scenarios.

But aerodynamicists who painstakingly collected data in low-speed aerodynamics did not want

to totally discard such data, called for relatively simple corrections rather than resorting to ab-

initio methods. Such methods called compressibility corrections.

The first and most popular of these corrections is the Prandtl-Glauert compressibility correction,

which is based on the linearized perturbation velocity potential function. This theory is limited to

thin airfoils at small angles of attack. Also to be noted is that it is a purely subsonic theory and

gives consistent results only upto M∞=0.7.

, (4.2)


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Eqn. (3.2) is called the Prandtl-Glauert rule and states that if the pressure distribution over an

airfoil for incompressible case is known then the compressible pressure distribution over the

same airfoil can be obtained from the above equation. Since this equation relates compressible

and incompressible pressure coefficients, the same must be true for the lift coefficient. Hence,

the corrected relation for Cl is

, (4.3)

Since the Prandtl-Glauert corrections are based on potential flow theory, D’Alembert’s paradox

prevails here too in that drag is zero for inviscid, subsonic, compressible flow. However if the

Mach number is high enough to produce local supersonic flow then with the presence of shock

waves, a positive wave drag is produced an d’Alembert’s paradox no longer prevails. (Anderson,

2010)

3.2.2.Supersonic

Wave Drag

For supersonic flows i.e. for M∞ greater than unity, the aerodynamics experience a complete shift

in paradigm, courtesy of shock waves. Shock waves due to their presence in supersonic flow,

create a new type of drag called wave drag. Now, consider a thin supersonic airfoil by a flat

plate which is inclined at an angle α to the supersonic free stream as shown in figure 4.2.


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Fig. 4.2 : Flat plate in supersonic flow inclined at an angle α, illustrating wave drag

On the top surface, due to the presence of an expansion wave at the leading edge, the flow field

is turned away from the free stream. At the trailing edge, the flow is turned back towards the free

stream.

The expansion and shock waves at the leading edge result in a surface pressure distribution in

which pressure at the top surface is less than the freestream pressure, while the pressure at the

bottom surface is greater than the freestream pressure. This results in an aerodynamic force

normal to the plate whose components parallel and perpendicular to the relative wind, the lift and

drag coefficients respectively.

Approximate relations for cl and cd are

(4.4)

(4.5)

Eqns. (4.4) and (4.5) are approximate expressions useful for thin airfoils at small to moderate

angles of attack. It is interesting to note that cl and cd both decrease as M∞ increases. This can be

seen in figure 1, where after Mach 1, cd begins to decrease. But surprisingly in any flight regime


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including supersonic flight, lift and drag increase with velocity contrary to eqns. (4.4) and (4.5),

as the dynamic pressure increases.

4.3 Lift Slope

The lift slope is the slope of the linear portion of the lift curve i.e. the plot of lift coefficient CL

versus the angle of attack α. For an airfoil and finite-wing the lift slope differs. We know that

finite wings generate less lift as compared to infinite wings due to induced drag and starting

vortex. But, at zero lift, there are no induced effects i.e. αi=Cd,i=0 which means αL=0 is the same

for both cases as shown from the graph.

Fig 4.3 : Lift slope for infinite and finite wing


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The relation between a0, the lift slope for an infinite wing and a, the lift slope for a finite wing

can be related as follows:

(4.6) Integrating, we get

const (4.7)We know that the induced angle of attack αiis given by

(4.8)Substituting this in eqn. (4.8), we get

const (4.9)Differentiating eqn. (4.9) with respect to α, and solving we get,

(4.10)

Now, from eqn. (4.10) to find the lift slope for a swept wing in a compressible flow, we have the

following derivation.

Let a0 be the lift slope for an infinite wing for incompressible flow and a0,comp be the lift slope

for an infinite wing in a subsonic compressible flow. Hence from the Prandtl-Glauert rule, we

have

, (4.11)


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Assuming that eqn. (4.11) holds good for subsonic compressible flow as well and supposing the

lift slope for a finite wing for compressible flow as acomp, the compressible counterpart of eqn.

(4.10) is

.

,, (4.12)Substituting eqn. (3.11) in eqn. (3.12) and simplifying, we get,

(4.13)

Eqn. (3.13) is for estimating the lift slope for high-AR straight wing in compressible flow.

Helmhold’s eqn. for low-aspect ratio straight wings for incompressible flow modified by the

Prandtl-Glauert rule is,

(4.14)

Eqn.(3.14) is for estimating the lift slope for a low aspect ratio straight wing in a compressible

flow with subsonic M∞.

Finally, for a swept wing, applying the Prandtl-Glauert’s rule where M∞ is replaced by M∞,n

which is the component of M∞ perpendicular to the half-chord line of the swept wing. If the half-

chord line is swept by the angle Λ, then M∞,n=M∞cosΛ. The resulting equation is

(4.15)


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3.4 Variation of wave drag and lift slope with sweep angle

Now consider the variation of wave drag coefficient CD,w with aspect ratio AR. Consider a low-aspect ratio straight wing at supersonic speeds, the wave drag coefficient for a flat plate is given

by

, (4.16)

For a finite aspect ratio plate, the wave drag coefficient will be

, 1

(4.17)

Where R is given by

Fig. 4.4: Variation of supersonic wave drag with AR


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Figure 4 shows the graph of equation (4.17). It is clearly visible that for low-aspect ratios thewave drag coefficient drops significantly. This shows the advantage of low-aspect ratio wings for

supersonic flight.

Now for swept wings, we have already discussed that supersonic wave drag can be reduced by

sweeping the wings inside the Mach cone i.e. to have a subsonic leading edge. From the

pioneering supersonic wind tunnel work performed by Walter Vicenti of NACA in 1947, in

figure 5, the minimum total drag coefficient is plotted versus wing sweep angle for M∞=1.53.

This data includes both positive sweep angles representing swept back wings and negative sweepangles representing swept-forward wings.

Fig. 4.5 : Variation of minimum total drag coefficient with sweep angle

It is interesting to note that the curve is near symmetrical with regard to the positive and negative

sweep angles. It is also noted that the wave drag is same in magnitude for the same degree of

sweep irrespective of the sweep direction.


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Also, just after sweep angle of 49o on both side, CD,min begins to drop considerably. This is due to

the fact that at M∞=1.53 the mach angle is given by µ=sin-1

(1/M) = sin-1

(1/1.53) = 41o. Hence,

the wings are totally inside the Mach cone after 49o which indicates the obvious decrease in total

drag.

Refer figure 4.5, which shows the variation of the lift slope as a function of aspect ratio for

tapered swings at M∞=1.53. The dashed lines represent the Mach cones. These were obtained

from the experimental data obtained by Walter Vicenti in a ground-breaking experiment at

NACA Ames labs in 1947 on the effect of aspect ratio on the lift curve for straight wings at

supersonic speeds.


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5. Airfoil Selection & W

5.1 Introduction

The design of an airplane wi

precision and decision-makin

exhaustive process which inv

below:

5.2 Initial Sizing & Weight

Before preliminary wing desi

the design take-off weight,

complete wing design & ana

basic fighter jet comparable t

reference for initial approxim

AirfoilSelection

Max. thickness-to-

chord ratio

ing Design

g is a unique design process for each aircra

g skills on the account of the designer. The

lves selection of various parameters, as illu

Fig. 5.1: Wing Design Flowchart

stimation

n can be undertaken, it is necessary to have

0, so that wing loading and related aspects

lysis. The aircraft whose wing design is u

the specifications of the F-14 Tomcat whos

tions.

WingDesign

Wing sweep

WingLoading

WingDihedral

WingLo

Initial Sizing &Weight

Estimation

t, that requires crafted

design of a wing is an

trated in the flowchart

a rough estimation for

can be ascertained for

der consideration is a

data will be used as a

Verticalation

AspectRatio

TaperRatio


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From reference 1, we have a basic formula for the design take-off weight, which is:

(5.1)

where W0 is the design take-off weight, Wcrew is the weight of the crew, Wpayload is the maximum

payload weight that the aircraft can carry, Wfuelis the weight of the fuel the aircraft carries and

Wempty is the empty or structural weight of the aircraft.

Wcrewand Wpayload are based on the mission/design requirements, whereas Wfueland Wempty are

expressed as fractions of W0, which are in turn functions of W0.The equation 5.1 after

rearranging becomes,

(5.2)

4.2.1 , Empty Weight Fraction Estimation

From reference 1, table 3.1 shows the data for a statistical curve-fit equation, given as follows,

for the general trend observed for empty weight fractions seen in aircraft.

0

(5.3)where A and C are statistical constants available for different types of aircraft

Kvs is the variable sweep constant (1.04, if variable sweep3 / 1.00 if fixed sweep)

Hence from the table, the constants for jet fighter aircraft were A=2.34, C=-0.13 and since we

are incorporating variable sweep, Kvs=1.04, therefore we have,

3 A variable sweep wing is heavier than a fixed sweep wing due to the extra weight of the mechanism involved in

changing the sweep angle.


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2.4336 00.13 (5.4)

4.2.2 , Fuel Weight Fraction Estimation

To estimate fuel fraction, we have to take into account the mission profile of the aircraft and

hence the various mission segment weight fractions. From references 1 and 2, we have different

standard values and formulae for calculation of mission segment weight fractions as follows:

1 Warmup& takeoff (From ref. 1, table 3.2, typical historical values for initial sizing)

0.97 (5.5)

2 Climb& Accelerate to Cruise (From ref. 1, table 3.2, typical historical values for initial

sizing)

0.985 (5.6)

3 Cruise to Destination

Assuming data for initial sizing based upon F-14 specifications,

Cruise Range, R=500 Nautical miles=926 km=926000 m

Cruise velocity, Vcruise= Mach 0.72= 218.33 m/s

Specific Fuel Consumption, C=0.5 hr-1= 0.0001389 s-1

L/D ratio=0.866(L/D)max=0.866 X 15=12.99


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.

0.955 (5.7)

4 Acceleration to High speed/ Interception (From ref.2 , fig. 2.3, value plotted from graph for

Mcruise=0.72)

0.98 (5.8)

5 Cruise back (From ref. 1, table 3.2, typical historical values for initial sizing)

0.955 (5.9)

6 Loiter (from ref.1,eqn. 3.8 for loiter weight fraction)

Assuming the data for loiter before landing for a fighter jet operating on and off an aircraft

carrier,

Endurance, E=0.5 hr= 3600 s

Specific Fuel Consumption, C=0.4 hr-1

=.000111 s-1

L/D ratio = (L/D)max=15

/

.

0.987 (5.10)

7 Landing(From ref. 1, table 3.2, typical historical values for initial sizing)

0.955 (5.11)


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The final formula for the estimation of fuel weight fraction stands as,

1.06 1 (5.12)

Where =

=0.83865

01.06 10.83865 0.1710 (5.13)

Now, assuming the weight of the crew supposing 2 pilots are required to fly the aircraft and that

the aircraft can carry a maximum payload of upto 8000kg, we have

150 (5.14)

8000 (5.15)

Substituting values into eqn. 4.2, we have

– . – ..Kg (5.16)Taking an initial approximation of design take-off weight W0= 35000 kg, we calculate till the

equation converges. After 22 iterations4 the equation successfully converged to a value of,

38311.93761

4 Refer Appendix for iterations/calculations


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5.3 Wing Design

5.3.1 Wing Loading, W/S

Wing Loading, defined as ratio of the weight of the aircraft to the reference area of the wing, is

an important parameter in wing design which affects various flight characteristics like stall

speed, climb rate, takeoff and landing distances and turn performance. But aerodynamically,

wing loading determines the design lift coefficient as seen above due to its effect upon wetted

area and wing span. The wing loading has a strong interdependence with maximum takeoff

weight. As the wing loading is decreased, it implies that the reference area has been increased for

the same weight. This in turn implies that for the extra wing surface, the structural (empty)

weight increases. Moreover, additional drag is generated, to overcome which more fuel will be

required. All these consequences result in an overall increase in maximum takeoff weight itself,

thus the selection of an optimal wing loading is imperative for good aircraft performance.

For the interim selection of the wing loading of our aircraft, wing loading is chosen a fixed value

from table 5.5 (Typical values from historical trends) for wing loading from Ref. 1,

342 /

5.3.2 Maximum Thickness-to-Chord ratio, (t/c)max

The airfoil thickness ratio, t/c, is a measure of the thickness of an airfoil and serves as a

parameter which affects aerodynamic performance through maximum lift and stallcharacteristics. Drag directly increases with increase in thickness as a result of a favorable

tendency for flow separation. Thickness also relates statistically to weight of the wing structure,

approximately inversely.

For initial selection of the t/c ratio, fig. 4.14 from Ref.1 which shows the historical trend for

airfoil thickness by comparison with the design Mach number. For the aircraft to be designed,


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the operational envelope would stretch from subsonic to supersonic speed, hence for subsonic

cruise speed of Mach 0.8, the graph corresponds to a t/c ratio of 14% which proves to be too

thick according to modern standards on comparison. But, according to the NACA Report TN-824,

the critical Mach number decreases with increasing thickness, hence the optimum thickness shall

be chosen as 10% as wing sweep reduces the effective t/c ratio.

10 %

5.3.3 Aspect Ratio, AR (=b2 /S)

Aspect ratio is defined as the ratio of the square of span to the reference area which

characterizies the finiteness of a real wing as supposed to an infinite wing for which the aspect

ratio is infinite. The aspect ratio directly affects the induced drag, the stalling angle and also the

maximum subsonic L/D among other aerodynamic factors. A higher aspect ratio indicates

reduced induced drag. But the decision between high aspect ratio and low aspect ratio lies in the

tradeoff between aerodynamic advantages and increased structural weight.

For selection of aspect ratio, from table 4.1 of ref. 1, we have a table from statistical data for

determining aspect ratio, for the unswept configuration, from which we have (refer Appendix for

calculation):

3.5 5.3.4 Taper Ratio, Taper Ratio is defined as the ratio of the tip chord to the centerline root chord. Taper directly

affects the lift distribution over the span. Taper was defined in order to approximate the ideal

elliptical wing shape which according to Prandtl’s Lifting Line theory is the optimum lift


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distribution. An unswept wing with taper of about 0.45 eliminates unwanted effects of constant

chord rectangular wing. But a swept wing increases the spanwise component of air towards the

tip creating more lift outboard creating an undesired lift distribution. Hence, it becomes

necessary to incorporate taper for swept wings.

From figure 4.23 from ref. 1, we choose the taper ratio for the wing for the unswept

configuration,

λ 0.4 5.3.5 Wing Twist

Wing twist is normally incorporated into wings to prevent tip stall and to alter the lift distribution

to more of an elliptical distribution. Twist may be either in the form of geometric twist i.e. the

actual variation in the incidence of the airfoils with respect to the root airfoil; or aerodynamic

twist which is the angle between the zero-lift angle of attack of an airfoil and the zero-lift angle

of attack of the root airfoil. No geometric twist shall be incorporated but aerodynamic twist will

be included to prevent tip stall.

5.3.6 Wing Vertical Position

The wing’s vertical position with respect to the fuselage is an important parameter with respect

to the functional/operational capabilities of an aircraft. The vertical wing position can be either

low, mid or high. Here, due to the visibility factor, superior aerobatic maneuverability and for the

aircraft to carry bombs under the wings, the mid-wing configuration is chosen.


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5.3.7 Wing Dihedral

The wing dihedral is the angle of the wing with respect to the horizontal when seen from the

front. A finite wing dihedral tends to roll an aircraft during banking due to a rolling moment

produced by sideslip. A swept wing produces an effective dihedral due to the rolling moment

produced. A swept back wing of 10 degrees sweep produces an equivalent dihedral of 1 degree.

From table 4.2 which shows dihedral guidelines for initial estimation, from ref. 1, the dihedral

angle is chosen as 0ofrom a common range for mid-wing, subsonic or supersonic swept wing

aircraft as

0

5.4 Airfoil Selection Criteria

Since Ludwig Prandtl and his colleagues at University of Gottingen took the giant leap in the

analysis of airplane wings by introducing the concept of study of the section of a wing – an

airfoil, aerodynamics has never looked back since then. The study of airfoils has been the

fundamental key in the genesis of general theories of lift. The classical (thin airfoil) theory and

modern (vortex panel method) theory have relied heavily upon aerodynamic characteristics of

airfoils. The birth of powered flight, the successful flight of the Wright Brothers` airplane, was

also the result of the development of early wing sections based upon extensive wind tunnel

testing. From 1884, when Horatio F. Phillips developed the first patented airfoil shapes, to the


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late 1920`s, airfoils were mostly customized. The pioneer in the development of generic airfoils

was NACA, headed by talented aerodynamicists such as Eastman Jacobs, began on the quest of

meticulous testing of airfoils and assimilation of aerodynamic data through extensive wind

tunnel tests. The result was four NACA series of airfoils: the 4-digit, 5-digit, 6-digit and 7-digit

series of airfoils.

Fig. 5.2: Airfoil Nomenclature

The NACA series of airfoils are defined by two basic parameters, in addition to other features as

shown in fig. 5.1:

i.Mean camber line, the locus of the midpoints between the upper and lower surfaces measured

perpendicular to the mean camber line itself

ii.Thickness distribution,the thickness along the chord, which is the distance between the upper

and lower surface measured perpendicular to the chord line

Combining both of which, any airfoil profile, cambered or symmetric, is derived. The systematic

modification of these two parameters to obtain the desired pressure distribution resulted in the

NACA series of airfoils, which exist today.

NACA designated the series of airfoils with a logical numbering system on the basis of the

airfoil geometry. Here, we shall discuss about the numbering system of the NACA 6-digit series

airfoil, as we have chosen this particular series for airfoil selection later. The NACA 6-digit


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series airfoils are designated by six-digit number usually and sometimes followed by a statement

showing the type of mean camber line used. For example, the following airfoil NACA 653-218

denotes that the first digit,"6", stands for the series designation; the 2nd digit,"5", indicates the

point of minimum pressure in tenths of chord from the leading edge; the 3rd

digit as subscript,

"3", gives the extent of the lift coefficient in tenths above and below the design lift coefficient5,

where favorable pressure gradients exist on both surfaces; the 4th

digit trailing the dash, "2",

gives the design lift coefficient in tenths; the 5th

and 6th

digits together, "18", express the

maximum thickness as percentage of chord.

5.4.1 Design Lift Coefficient, Cl,des

For the airfoil selection, the design lift coefficient is fundamental to airfoil selection as it makes

the task of an aircraft designer easier by supplying a figure that decides the optimum lift

coefficient of the airfoil for which the lift-to-drag ratio is maximum. Here, we choose the design

lift coefficient by considering conditions at cruise.

From reference, we assume data as follows

L=Wcruise= 36500 kgf = 358065 N (Refer the Appendix for calculations)

Vcruise= Vcos Λ = 218.33 cos 20=205.16 m/s (Choosing Λ=20o as subsonic cruise sweep angle)

Assuming wing loading , W/S = 342 kg/m2 from ref. 1, we have

Reference Area, S= W/(W/S)= 36500/342 =106.725 m2

We know, that CL is given by the formula,

(5.17)

Now, the 3D design lift coefficient, can be calculated as,

...

= 0.3458

5 Design Lift Coefficient is the lift coefficient at the point in the drag polar of an airfoil where the L/D ratio is

maximum or alternatively it is also described as the theoretical lift coefficient for an airfoil such that the angle of

attack is such that the slope of the camber line at the leading edge is parallel to the freestream velocity.(Ref. 2)


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And the 2D design lift coefficient is calculated by the formula,

= .

, 0.4

Hence, an airfoil with a design lift coefficient of 0.4 from NACA 6-digit airfoil series shall be

chosen. The NACA 64-XXX/64AXXX series of airfoils were specially designed for producing

laminar boundary layer over a larger stretch of the airfoil. From the 2nd

digit, it is evident that

laminar flow is preserved till 40% chord from the leading edge.

5.3.2 Critical Mach Number, Mcr

As discussed thoroughly in the initial chapters, the concept of swept wings was an outcome of

the effort to increase the critical Mach number, Mcr of the wing section. The NACA Report TN-

824 presents a exhaustive compilation of airfoil data, including plots of Critical Mach number,

Mcr versus the low-speed section lift coefficient. The analysis of various plots (refer fig. 4.3)

revealed that the NACA 6-series airfoil sections showed considerably high critical Mach number

than the NACA 24-, 44- and 230-series airfoil sections.


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Fig. 5.3: Critical Mach number, Mcr vs. section lift coefficient, cl for NACA 0006, 0009 and

0012(left) and for NACA 1408, 1410 and 1412(right)

Fig 5.4: Critical Mach number, Mcr vs. section lift coefficient, cl for NACA 2412, 2415, 2418,2421 and 2224(left) and for NACA 4412, 4415, 4418, 4421 and 4424(right)


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Fig 5.5:Critical Mach number,Mcrvs. section lift coefficient, clfor NACA 23012, 23015, 23018, 23021 and

23024(left) and for NACA 63-XXX series(right)

It was also noted that the Critical Mach number peaked near the design lift coefficient. It can also

be noticed that the Mcrdecreases with increasing thickness. Hence, we infer low-thickness airfoils

provide the maximum Mcr. This also reveals that the design lift coefficient needs to be in the

“favorable” range i.e. where Mcr is sufficiently high.


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Fig 5.6:Critical Mach number, Mcr vs. section lift coefficient, cl for NACA 64-006, 64-009, 64l-

012, 642-015, 643-018 and 644-021 and for NACA 64-108, 64-110 and 641-112(right)


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Fig 5.7: Critical Mach number, Mcr vs. section lift coefficient, cl for NACA 64-XXX series

A study of the Critical Mach number characteristics of NACA 64-XXX series airfoils reveals

that this series of airfoils shows a considerably higher Mcr than other NACA airfoils.

Interestingly the NACA 64-2XX shows higher Mcr range as compared to the NACA 64-4XX.

Also to be noted is that with increasing design lift coefficient/camber the Mcr decreases. We also

see that in the case of NACA 64-XXX series airfoils, the maximum critical mach numbers were:

Table 5.1: Maximum Critical Mach numbers for NACA 64-XXX series

Airfoil Mcr,max

64-0XX 0.82

64-2XX 0.79

64-4XX 0.70


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5.4.3 Stalling Angle, αstall Due to excessive sweep, the spanwise component of flow over the wing increases the tip loading.

Moreover, for a wing using the same airfoil throughout, at high angles of attack, the tip stalls

first before the root, the pilot loses control and the wing control surfaces such as ailerons become

ineffective. This problem of “tip stall” can be averted by choosing different airfoils at the root

and tip such that the root stalls first and the stall is much gradual.

In this case, the control surfaces are still effective and the wake from the stalled root induces

vibrations in the horizontal tail indicating the pilot that stall is imminent. Hence, another prime

consideration during airfoil selection is the prevention of tip stall by incorporating aerodynamic

twist. Using different airfoils at root and tip, it can be assured that the root of the wing stalls first

as compared to the tip.Again, we refer to the NACA Report TN-824 which contains aerodynamic

characteristics like lift curve and drag polar.

Fig 5.8: Lift and moment characteristics of NACA 0006(left) and NACA 0009(right)


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Analyzing the graphs for the NACA 0006 and NACA 0009 airfoils, we observe that the stalling

angle increases. The NACA 0006 airfoil stalls at near about 9 degrees while the NACA 0009

airfoils stalls near about 11 degrees.6

Fig 5.9: Lift and moment characteristics of NACA 1410 and NACA 1412

Similarly comparing the NACA 1410 and 1412 airfoils, we see that the former stalls at 14 deg

and the latter at 15 deg. This brings us to the conclusion that the general trend is that stalling

angle increases with thickness.

Comparing the variation of stalling angle with thickness for the NACA 641-XXX series, we have

6 For Reynolds Number, Re= 6 x 10

6


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Table 5.2 : Stalling angle for various NACA 641-XXX airfoils

Airfoil αstall(deg)

641-012 10

641-112 10

641-212 11

641-412 12

Here, we notice that the general trend is that the stalling angle increases with camber or design

lift coefficient .

Fig 4.10: Lift and moment characteristics of NACA 64A210 and NACA 64A410

Now, comparing the stalling angles for the NACA 64A- series, this is the modified version of the

64- series, without a cusp having a flatter trailing edge.

Table 5.3 : Stalling angle for various NACA 64AXXX airfoils


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Airfoil αstall(deg)

64A210 10

64A410 11

5.5 Final Airfoil Selection

5.5.1 NACA 64-XXX series vs NACA 64A-XXX series

The NACA 64-A series airfoils are preferred here over the traditional NACA 64-XXX series

although the latter has less average thickness which translates to a greater Mcr. The 64- series are

very thin near the trailing edge which is a major problem in structural design and fabrication.

With the trailing edge cusp removed, the NACA 64A- series have more straight sides from

roughly 80% chord to the trailing edge and with a mean camber line of 0.8, it has superior

aerodynamics as compared to the other mean lines used. Hence, the NACA 64A-XXX is the

airfoil series chosen.

5.5.2Airfoil (at root)

The 64A series is an extremely popular airfoil used in modern fighter jets since its

development for favorable laminar flow up to 40% chord. The character A indicates a

modification where the back slope of the airfoil is straight and the trailing edge is thick, this

delays separation further backward. We have chosen the NACA 64A210 airfoil at the root which

has a the point of minimum pressure at 40% chord from the leading edge with a design lift

coefficient of 0.4 which was calculated and a thickness of 10% chord. Having a stalling angle of

about 10 degrees, the root will stall first in case the wing stalls. Hence, allowing for greater

control and stability of the aircraft.

NACA 64A210


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5.5.3Airfoil at tip

We have chosen the NAC

increased camber and a hig

prevent tip stall. The same thi

Mach number. As mention

incorporated to prevent tip sta

Fig. 5.11: NACA 64A210 airfoil profile

64A410 airfoil for the tip. The design lift c

er angle of attack at 11 degrees which is

ckness of 10% of chord shall be maintained

ed, airfoils of different thickness at roo

ll.

Fig. 5.12: NACA 64A410 airfoil profile

NACA 64A410

efficient of 0.4 means

our main objective to

o obtain a high critical

t an

study of the performance envelope of a variable sweep wing.pdf

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