Thiết kế cánh máy bay wing design

In chapter 4, aircraft preliminary design – the second step in design process – was introduced. Three parameters were determined during preliminary design, namely: aircraft maximum takeoff weight (WTO); engine power (P), or engine thrust (T); and wing reference area (Sref). The third step in the design process is the detail design. During detail design, major aircraft component such as wing, fuselage, horizontal tail, vertical tail, propulsion system, landing gear and control surfaces are designed onebyone. Each aircraft component is designed as an individual entity at this step, but in later design steps, they were integrated as one system – aircraft and their interactions are considered. This chapter focuses on the detail design of the wing. The wing may be considered as the most important component of an aircraft, since a fixedwing aircraft is not able to fly without it. Since the wing geometry and its features are influencing all other aircraft components, we begin the detail design process by wing design. The primary function of the wing is to generate sufficient lift force or simply lift (L). However, the wing has two other productions, namely drag force or drag (D) and nosedown pitching moment (M). While a wing designer is looking to maximize the lift, the other two (drag and pitching moment) must be minimized. In fact, wing is assumed ad a lifting surface that lift is produced due to the pressure difference between lower and upper surfaces. Aerodynamics textbooks may be studied to refresh your memory about mathematical techniques to calculate the pressure distribution over the wing and how to determine the flow variables. Basically, the principles and methodologies of “systems engineering” are followed in the wing design process. Limiting factors in the wing design approach, originate from design requirements such as performance requirements, stability and control requirements, producibility requirements, operational requirements, cost, and flight safety. Major performance requirements include stall speed, maximum speed, takeoff run, range and endurance. Primary stability and control requirements include lateraldirectional static stability, lateraldirectional dynamic stability, and aircraft controllability during probable wing stall.

Introduction

In Chapter 4, the preliminary design phase of aircraft development is explored, focusing on three key parameters: maximum takeoff weight (WTO), engine power (P) or thrust (T), and wing reference area (Sref) Following this, the detail design phase commences, where critical components such as the wing, fuselage, horizontal tail, vertical tail, propulsion system, landing gear, and control surfaces are designed individually Although each component is treated as a separate entity during this stage, they will be integrated into a cohesive aircraft system in subsequent design phases, taking into account their interactions.

This chapter delves into the detailed design of the aircraft wing, a crucial component essential for flight in fixed-wing aircraft The design process begins with the wing, as its geometry significantly impacts all other aircraft elements The primary role of the wing is to generate lift (L), while also producing drag (D) and a nose-down pitching moment (M) Wing designers aim to maximize lift while minimizing drag and pitching moment Lift is created through the pressure difference between the wing's upper and lower surfaces For a deeper understanding of the pressure distribution and flow variables, consulting aerodynamics textbooks is recommended.

The wing design process adheres to the principles and methodologies of systems engineering, influenced by various limiting factors stemming from design requirements These include performance, stability and control, producibility, operational needs, cost, and flight safety Key performance metrics encompass stall speed, maximum speed, takeoff distance, range, and endurance Additionally, essential stability and control criteria focus on lateral-directional static and dynamic stability, as well as aircraft controllability during potential wing stalls.

During the wing design process, eighteen parameters must be determined They are as follows:

1 Wing reference (or planform) area (SW or Sref or S)

3 Vertical position relative to the fuselage (high, mid, or low wing)

4 Horizontal position relative to the fuselage

10 Mean Aerodynamic Chord (MAC or C)

15 Incidence (iw) (or setting angle,  set )

16 High lifting devices such as flap

This chapter focuses on the methodology for calculating or selecting the remaining 17 wing parameters, following the preliminary design step where only the planform area has been determined Notably, the design of ailerons, which is a complex aspect of wing design with diverse requirements, will not be addressed here; instead, it will be explored in detail in Chapter 12, which is dedicated to control surfaces Additionally, the positioning of the horizontal wing relative to the fuselage will be discussed in Chapter 7, after the fuselage and tail designs have been completed.

The wing design process initiates with a known variable, S, from which fifteen additional parameters are derived, ensuring the wing generates adequate lift while minimizing drag and pitching moment These design objectives must be met across all flight operations and missions Additional parameters, including wing tips, winglets, engine installation, fairings, vortex generators, and structural considerations, will be explored in subsequent chapters The flowchart in Figure 5.1 outlines the wing design process, beginning with the known variable S and culminating in optimization, with detailed explanations of each design step provided later in the chapter.

(Performance, stability, producibility, operational requirements, cost, flight safety)

Select or design wing airfoil section

Determine other wing parameters (AR, i w ,  t )

Calculate Lift, Drag, and Pitching moment

Select/Design high lift device

Select/Determine sweep and dihedral angles ( )

In the wing design process, calculating wing lift, drag, and pitching moment is essential, and advancements in aerodynamics have led to the development of various tools and techniques for this task Over the past few decades, numerous aerodynamic software options, including CFD software based on Navier-Stokes equations, vortex lattice methods, and thin airfoil theory, have become available, though they can be costly and time-consuming At the initial stages of wing design, employing such complex software may be unnecessary; instead, Lifting Line Theory offers a simpler and effective approach to accurately determine wing lift, drag, and pitching moment.

This chapter concludes with practical steps for wing design and airfoil selection, featuring two fully solved examples—one focused on airfoil selection and the other on complete wing design As highlighted in Chapter 3, wing design is a crucial element in the iterative aircraft design process, necessitating multiple revisions until all aircraft components reach their optimal configuration Consequently, wing parameters will undergo several adjustments to satisfy the various design requirements effectively.

Number of Wings

One of the decisions a designer must make is to select the number of wings The options are:

The number of wings higher than three is not practical Figure 5.2 illustrates front view of three aircraft with various configurations

Figure 5.2 Three options in number of wings (front view)

Modern aircraft predominantly utilize a monoplane design, with only a handful of biplanes still in operation today Notably, contemporary aircraft do not feature a three-wing configuration, marking a significant evolution in aviation design.

The choice to use multiple wings in aircraft design was historically driven by manufacturing technology limitations, as a single long wing could not maintain structural integrity and rigidity However, advancements in manufacturing techniques and the introduction of strong, lightweight materials like advanced aluminum and composites have rendered this concern obsolete Additionally, to comply with restrictions on wing span, increasing the number of wings has been a practical solution.

In conventional modern aircraft design, a single wing configuration is typically the most practical choice; however, certain design considerations, particularly aircraft controllability, may necessitate multiple wings A shorter wingspan enhances roll control due to a reduced mass moment of inertia around the x-axis, making multi-wing designs advantageous for faster rolling capabilities Historical examples from the 1940s and 1950s include maneuverable biplanes and triplanes Nevertheless, alternatives to monoplane designs come with drawbacks such as increased weight, decreased lift, and limited pilot visibility Therefore, it is advisable to start with a monoplane design and only consider additional wings if specific performance requirements are unmet.

Wing Vertical Location

The vertical location of the wing relative to the fuselage centerline is a critical parameter determined early in the wing design process This choice significantly impacts the design of various aircraft components, including the tail, landing gear, and center of gravity There are four primary options for positioning the wing vertically.

4 Parasol wing a High wing b Mid wing c Low wing b Parasol wing

Figure 5.3 Options in vertical wing positions

Wing Design 6 a Cargo aircraft C-130 (high wing)

(Photo courtesy of Tech Sgt Howard Blair, U.S Air Force) b Passenger aircraft Boeing 747 (low wing)

(Photo courtesy of Philippe Noret – AirTeamimages) c Military aircraft Scorpions (mid wing)

(Photo courtesy of Photographer’s Mate 3rd Class Joshua Karsten, U.S Navy) d Home-built Pietenpol Air Camper (parasol wing)

(Photo courtesy of Adrian Pingstone)

Figure 5.4 Four aircraft with different wing vertical positions

Figure 5.3 illustrates the schematics of four wing configurations, highlighting the front views of aircraft fuselage and wings Typically, cargo and some general aviation (GA) aircraft feature high wings, while most passenger aircraft are designed with low wings Conversely, mid-wing designs are common in fighter planes and certain GA aircraft, whereas hang gliders and many amphibious aircraft utilize parasol wings The primary factor in selecting wing location is based on operational requirements, although stability and producibility also play significant roles in certain design scenarios.

Figure 5.4 presents four aircraft models, each showcasing different wing locations This section analyzes the pros and cons of each configuration, with the final decision hinging on a comprehensive evaluation of these factors in relation to design requirements Each option carries a specific weight based on its alignment with the design criteria, and the overall summation of these weights will determine the ultimate choice.

The high wing configuration offers distinct advantages and disadvantages that influence its suitability for various flight operations This design can enhance stability and visibility, making it ideal for certain missions, while also presenting limitations that may render it less effective for others.

1 Eases and facilitates the loading and unloading of loads and cargo into and out of cargo aircraft For instance, truck and other load lifter vehicles can easily move around aircraft and under the wing without anxiety of the hitting and breaking the wing

2 Facilitates the installation of engine on the wing, since the engine (and propeller) clearance is higher (and safer), compared with low wing configuration

3 Saves the wing from high temperature exit gasses in a VTOL 2 aircraft The reason is that the hot gasses are bouncing back when they hit the ground, so they wash the wing afterward Even with a high wing, this will severely reduce the lift of the wing structure Thus, the higher the wing is the farther the wing from hot gasses

4 Facilitates the installation of strut This is based on the fact that a strut (rod or tube) can handle higher tensile stress compared with the compression stress In a high wing, struts have to withstand tensile stress, while struts in a low wing must bear the compression stress Figure 3.5d shows a parasol wing with strut

5 Item 4 implies that the aircraft structure is heavier when struts are employed

6 Facilitates the taking off and landing from sea In a sea-based or an amphibian aircraft, during a take-off operation, water will splash around and will high the aircraft An engine installed on a high wing will receive less water compared with a low wing Thus, the possibility of engine shut-off is much less

7 Facilitates the aircraft control for a hang glider pilot, since the aircraft center of gravity is lower than the wing

8 High wing will increase the dihedral effect (C l  ) It makes the aircraft laterally more stable The reason lies in the higher contribution of the fuselage to the wing dihedral effect ( l W

2 Vertical Take Off and Landing

9 The wing will produce more lift compared with mid and low wing, since two parts of the wing are attached 9at least on the top part)

10 For the same reason as in item 8, the aircraft will have lower stall speed, since CLmax will be higher

11 The pilot has better view in lower-than-horizon A fighter pilot has a full view under the aircraft

12 For an engine that is installed under the wing, there is less possibility of sand and debris to enter engine and damage the blades and propellers

13 There is a lower possibility of human accident to hit the propeller and be pulled to the engine inlet In few rare accidents, several careless people has died (hit the rotating propeller or pulled into the jet engine inlet)

14 The aerodynamic shape of the fuselage lower section can be smoother

15 There is more space inside fuselage for cargo, luggage or passenger

16 The wing drag is producing a nose-down pitching moment, so it is longitudinally stabilizing This is due to the higher location of wing drag line relative to the aircraft center of gravity (M Dcg < 0) b Disadvantages

1 The aircraft frontal area is more (compared with mid wing) This will increase aircraft drag

2 The ground effect is lower, compared with low wing During takeoff and landing operations, the ground will influence the wing pressure distribution The wing lift will be slightly lower than low wing configuration This will increase the takeoff run slightly Thus, high wing configuration is not a right option for STOL 3 aircraft

3 Landing gear is longer if connected to the wing This makes the landing gear heavier and requires more space inside the wing for retraction system This will further make the wing structure heavier

4 The pilot has less higher-than-horizon view The wing above the pilot will obscure part of the sky for a fighter pilot

5 If landing gear is connected to fuselage and there is not sufficient space for retraction system, an extra space must be provided to house landing gear after retraction This will increase fuselage frontal area and thus will increase aircraft drag

6 The wing is producing more induced drag (Di), due to higher lift coefficient

7 The horizontal tail area of an aircraft with a high wing is about 20% larger than the horizontal tail area with a low wing This is due to more downwash of a high wing on the tail

8 A high wing is structurally about 20% heavier than low wing

9 The retraction of the landing gear inside the wing is not usually an option, due to the required high length of landing gear

10 The aircraft lateral control is weaker compared with mid wing and low wing, since the aircraft has more laterally dynamic stability

Although, the high wing has more advantages than disadvantages, but all items do not have the same weighing factor It depends on what design objective is more significant than

3 Short Take Off and Landing

Wing Design 9 other objectives in the eyes of the customer The systems engineering approach delivers an approach to determine the best option for a specific aircraft, using a comparison table

This section outlines the advantages and disadvantages of a low wing configuration, highlighting that many reasons overlap with those of a high wing design and thus will not be reiterated Typically, low wing specifications are assessed in comparison to high wing configurations.

1 The aircraft take off performance is better; compared with a high wing configuration; due to the ground effect

2 The pilot has a better higher-than-horizon view, since he/she is above the wing

3 The retraction system inside the wing is an option along with inside the fuselage

Airfoil

This section focuses on the process of determining the optimal airfoil section for a wing, which is the second most critical parameter after wing planform area The airfoil section plays a vital role in achieving the ideal pressure distribution on the wing's surfaces, ensuring that the required lift is generated while minimizing aerodynamic drag and pitching moment While aircraft designers possess basic knowledge of aerodynamics and airfoils, this article will provide a uniform overview of airfoil concepts and governing equations It will cover airfoil selection and design, key airfoil parameters, and the most significant factors influencing airfoil performance A review of NACA 4 airfoils, the precursor to NASA 5, will be included, with a focus on selection criteria The section will conclude with a detailed example demonstrating the airfoil selection process for a candidate wing.

5.4.1 Airfoil Design or Airfoil Selection

The primary role of a wing is to generate lift, achieved through a specialized cross-section known as an airfoil While the wing is a three-dimensional structure, the airfoil itself is a two-dimensional section In addition to lift, the airfoil and wing also produce drag and a pitching moment Wings can feature either a constant or a varying cross-section, which will be further explored in section 5.9.

There are two ways to determine the wing airfoil section:

Designing an airfoil is a complex and time-consuming process that requires advanced knowledge of aerodynamics, typically at a graduate level This design must be validated through costly wind tunnel testing Major aircraft manufacturers like Boeing and Airbus possess the necessary expertise and financial resources to create custom airfoils for each aircraft model In contrast, smaller aircraft companies, experimental aircraft producers, and homebuilt aircraft enthusiasts often lack the same level of expertise and budget, making the airfoil design process more challenging for them.

4 National Advisory Committee for Aeronautics

5 National Administration for Aeronautics and Astronautics

Wing Design 12 manufacturers do not afford to design their airfoils Instead they select the best airfoils among the current available airfoils that are found in several books or websites

The advancement of high-speed and powerful computers has significantly simplified airfoil design compared to thirty years ago Today, several computational fluid dynamics (CFD) software packages are available for designing airfoils to meet various needs Airfoil design is not limited to aircraft; it is also crucial for numerous applications, including jet engine axial compressor and turbine blades, steam power plant turbines, wind turbine propellers, and impeller blades for both centrifugal and axial pumps The efficiency of these mechanical and aerospace devices is heavily dependent on the design of their blades, commonly referred to as "airfoils."

For aircraft designers with ample time, budget, and manpower, designing a custom airfoil is advisable, with references available in the textbook for guidance However, it's crucial to integrate the airfoil design effectively into the overall aircraft design process Conversely, junior aircraft designers with limited resources should consider selecting an airfoil from an existing database to streamline their project.

Aerodynamics textbooks present various theories for analyzing airflow around airfoils, with potential-flow and boundary-layer theories being foundational for airfoil design Over the years, these theories have evolved significantly, particularly with advancements in computing technology Nowadays, the cost-effectiveness of conducting potential-flow and boundary-layer analyses is striking, often amounting to less than one percent of traditional wind-tunnel tests Consequently, there is a growing reliance on versatile computer codes that minimize the need for wind-tunnel testing, enabling the customization of airfoils for specific applications.

Richard Eppler, a renowned airfoil designer from Germany, has developed a reliable airfoil design code over the past 45 years, utilizing conformal mapping techniques This code integrates a conformal-mapping approach for airfoil design with specific velocity-distribution characteristics, a panel method for analyzing potential flow around airfoils, and an integral boundary-layer method It features an option for computing aircraft-oriented boundary-layer developments, accommodating variations in Reynolds number and Mach number based on the aircraft's lift coefficient and local wing chord Additionally, users can input a local twist angle and calculate the aircraft's drag polar, which includes both induced drag and parasite drag.

The code is compatible with most personal computers, workstations, and servers, with execution times varying based on the system The analysis method, which is the most computationally intensive part, runs in just a few seconds on a PC Written in standard FORTRAN 77, the code requires a FORTRAN compiler to convert the source code into executable form A sample input and output case is provided, along with a separate plot-post-processing code that handles graphics routines This post-processing code generates an output file suitable for direct printing and can be customized for use with other plotting devices, including screens.

The code is highly efficient and has proven effective for Reynolds numbers ranging from 3×10⁴ to 5×10⁷ It includes a compressibility correction for velocity distributions, applicable as long as the local flow remains subsonic This code is available for purchase in North America exclusively through Mark D Maughmer.

If you're not ready to design your own airfoil, it's advisable to choose from established airfoil designs, with NACA and Eppler being two reliable sources Eppler airfoils are detailed in Reference 1, while NACA airfoils can be found in a book by Abbott and Von Donehoff, originally published in the 1950s and still widely available in aerospace libraries Both references provide essential airfoil coordinates, pressure distribution data, and graphs such as lift coefficient (Cl), drag coefficient (Cd), and moment coefficient (Cm) across various angles of attack Notably, Eppler airfoil names start with the letter "E" followed by three digits Further information on NACA airfoils will be discussed in Section 5.3.4.

A typical flight operation includes several phases: takeoff, climb, cruise, turn, maneuver, descent, approach, and landing The cruise phase is where the airfoil performs optimally, as the aircraft spends the majority of its flight time in this stage During cruising, the lift (L) equals the weight (W) of the aircraft, and the drag (D) matches the engine thrust (T) Therefore, the wing must generate an adequate lift coefficient while minimizing the drag coefficient, both of which are influenced by the airfoil section The governing equations for cruising flight are essential for understanding these dynamics.

Equation 5.2 pertains to aircraft equipped with jet engines, while equation 5.3 applies to those with propeller-driven engines, with the variable “n” ranging from 0.6 to 0.9 This indicates that only partial engine throttle is utilized during cruising flight, rather than maximum engine power or thrust The precise value of “n” will be established in subsequent design phases, although an initial recommendation of 0.75 is suggested for airfoil design Maximum engine thrust is primarily reserved for take-off or achieving maximum cruising speed Since meeting cruising flight requirements is a key criterion for airfoil design, equations 1 through 3 will be employed in the airfoil design process, as detailed in this section The following section outlines the procedure for selecting the appropriate wing airfoil.

6 RR 1, Box 965 Petersburg, PA 16669 USA

5.4.2 General Features of an Airfoil

An airfoil is defined as any section of an aircraft wing cut parallel to the xz plane, typically featuring a positive camber with the thicker part positioned at the front As an airfoil-shaped body moves through the air, it creates variations in static pressure on its upper and lower surfaces A typical airfoil section illustrates several geometric parameters, where a straight mean camber line designates a symmetric airfoil, while a curved line indicates a cambered airfoil, which usually has positive camber In positive cambered airfoils, the static pressure on the upper surface is lower than the ambient pressure, whereas the lower surface experiences higher static pressure This pressure difference arises from the faster airflow over the upper surface compared to the lower surface Additionally, as the angle of attack of the airfoil increases, the pressure difference between the upper and lower surfaces also becomes more pronounced.

Figure 5.5 Airfoil geometric parameters a Small angle of attack b Large angle of attack

Figure 5.6 Flow around an airfoil

Leading edge radius x-location of

Wing Design 15 a Small angle of attack b Large angle of attack Figure 5.7 Pressure distribution around an airfoil

Pressure is defined as the force applied over an area, and in the context of an airfoil in a flow field, the aerodynamic force can be calculated by multiplying the total pressure by the surface area Total pressure is obtained by integrating the pressure across the entire surface of the airfoil The characteristics of the aerodynamic force, including its magnitude, location, and direction, depend on various factors such as the geometry of the airfoil, the angle of attack, the properties of the flow, and the airspeed relative to the airfoil.

Figure 5.8 The pressure center movement as a function of angle of attack

The center of pressure (cp) is the point where the resultant force acts on an airfoil, and its location is influenced by the aircraft's speed and angle of attack As speed increases, the cp shifts toward the aft, moving from a position near the leading edge at lower speeds to closer to the trailing edge at higher speeds Additionally, the aerodynamic center plays a crucial role in aircraft stability and control, making it an important concept in aerodynamics.

Angle of attack Leading edge

Wing Incidence

The wing incidence (i w), also known as the wing setting angle (αset), is the angle between the fuselage center line and the wing chord line at the root The fuselage center line, which is typically parallel to the cabin floor, lies in the plane of symmetry This angle can either remain constant throughout all flight operations or be variable during flight If a variable wing incidence is chosen, it eliminates the need to determine the wing setting angle during aircraft manufacturing; however, a mechanism must be designed to adjust the wing incidence for different flight phases It is important to note that variable wing incidence is generally not recommended.

Wing design poses significant safety and operational challenges, particularly when incorporating a variable angle setting This requires a single shaft for wing rotation, controlled by the pilot, but such mechanisms are not entirely reliable due to issues like fatigue, weight, and stress concentration Historically, only one aircraft, the Vought F-8U Crusader, featured a variable incidence wing In contrast, flying wings like the Northrop Grumman B-2 Spirit lack a fuselage and therefore do not have a variable wing incidence; however, determining the wing's angle of attack remains essential for operational effectiveness.

A highly convenient option for wing design is to maintain a constant wing setting angle, which can be securely attached to the fuselage through welding, screws, or other manufacturing methods This approach enhances safety compared to variable setting angles Designers must carefully determine the optimal angle for wing attachment to the fuselage, ensuring that the wing incidence meets essential design requirements.

1 The wing must be able to generate the desired lift coefficient during cruising flight

2 The wing must produce minimum drag during cruising flight

3 The wing setting angle must be such that the wing angle of attack could be safely varied (in fact increased) during take-off operation

4 The wing setting angle must be such that the fuselage generates minimum drag during cruising flight (i.e the fuselage angle of attack must be zero in cruise)

Figure 5.26 Wing setting angle corresponds with ideal lift coefficient i w

The design requirements align with the wing airfoil's angle of attack, which corresponds to the ideal lift coefficient of the airfoil Once the ideal lift coefficient is established, the C l -α graph can be referenced to determine the appropriate wing setting angle Table 5.7 provides an overview of the wing incidence for various aircraft.

The typical number for wing incidence for majority of aircraft is between 0 to 4 degrees

As a general guidance, the wing setting angle in supersonic fighters, is between 0 to 1 degrees; in

GA aircraft, between 2 to 4 degrees; and in jet transport aircraft is between 3 to 5 degrees

No Aircraft Type Wing incidence

3 Sukhoi Su-27 Jet fighter 0 o Mach 2.35

4 Embraer FMB-120 Brasilia Prop-driven transport 2 o 272

6 Antonov An-26 Turbo-prop Transport 3 o 235

7 BAe Jetstream 31 Turbo-prop Business 3 o 282

8 BAe Harrier V/STOL close support 1 o 45' 570

9 Lockheed P-3C Orion Prop-driven transport 3 o 328

10 Rockwell/DASA X-31A Jet combat research 0 o 1485

13 Beech Super King Air B200 Turbo-prop Transport 3 o 48' 289

16 McDonnell MD-11 Jet Transport 5 o 51‟ Mach 0.87

Table 5.7 Wing setting angle for several aircraft (Ref 4)

The wing setting angle can be adjusted during the design process to optimize performance For instance, a fuselage with significant unsweep at the rear to accommodate aft cargo doors may achieve minimum drag at a small positive angle of attack, necessitating a reduction in wing incidence Additionally, enhancing landing performance by maximizing weight on the braked wheels may require a slight reduction in wing incidence, provided it does not noticeably affect cabin comfort While shortening the nose gear can achieve similar results, this approach is limited in passenger aircraft due to the need for a level cabin floor on the ground, whereas it is less of a concern for fighter aircraft.

Aspect Ratio

Aspect ratio (AR) 10 is defined as the ratio between the wing span; b (see figure 5.31) to the wing Mean Aerodynamic Chord (MAC or C)

10 Some textbooks are using symbol A instead of AR

The wing planform area with a rectangular or straight tapered shape is defined as the span times the mean aerodynamic chord:

Thus, the aspect ratio shall be redefined as:

 (5.19) a AR = 26.7 b AR = 15 c AR = 6.67 d AR = 3.75 e AR = 1

Figure 5.27 Several rectangular wings with the same planform area but different aspect ratio

This equation is applicable only to rectangular wings, and should not be used for other geometries like triangles, trapezoids, or ellipses unless the span is redefined As illustrated in Example 5.4, when only the wing planform area is known, the designer has numerous options for selecting the wing geometry For example, if an aircraft's wing reference area is established at 30 m², various design options can be explored.

1 A rectangular wing with a 30 m span and a 1 m chord (AR 0)

2 A rectangular wing with a 20 m span and a 1.5 m chord (AR 333)

3 A rectangular wing with a 15 m span and a 2 m chord (AR = 7.5)

4 A rectangular wing with a 10 m span and a 3 m chord (AR = 3.333)

5 A rectangular wing with a 7.5 m span and a 4 m chord (AR = 1.875)

6 A rectangular wing with a 6 m span and a 5 m chord (AR = 1.2)

7 A rectangular wing with a 3 m span and a 10 m chord (AR = 0.3)

8 A triangular (Delta) wing with a 20 m span and a 3 m root chord (AR = 13.33; please note that the wing has two sections (left and right))

9 A triangular (Delta) wing with a 10 m span and a 6 m root chord (AR = 3.33)

While there are additional options available, we will focus on the taper ratio parameter later Figure 5.27 illustrates various rectangular wings with differing aspect ratios, all sharing the same planform area but varying in span and chord dimensions.

The lift equation indicates that identical lift can be achieved with the same lift coefficient, regardless of wing area However, the wing lift coefficient depends on non-dimensional aerodynamic factors, including airfoil shape and aspect ratio Notably, the aspect ratio of the 1903 Wright Flyer was 6.

When selecting the optimal aspect ratio for wing design, it is crucial to evaluate its impact on key flight characteristics, including aircraft performance, stability, control, cost, and manufacturability Understanding these effects will guide designers in determining the most effective wing geometry for their specific needs.

1 From aerodynamic points of view, as the AR is increased, the aerodynamic features of a three-dimensional wing (such as C L ,  o ,  s , C Lmax , C Dmin ) are getting closer to its two- dimensional airfoil section (such as C l , o, s, C lmax , C dmin ) This is due to reduction of the influence of wing tip vortex The flow near the wing tips tends to curl around the tip, being forced from the high-pressure region just underneath the tips to the low-pressure region on top (Ref 3) As a result, on the top surface of the wing, there is generally a spanwise component of flow from the tip toward the wing root, causing the streamlines over the top surface to bend toward the root Similarly, on the bottom surface of the wing, there is generally a spanwise component of flow from the root toward the wing tip, causing the streamlines over the bottom surface to bend toward the tip

2 Due to the first item, as the AR is increased, the wing lift curve slope (C L ) is increased toward the maximum theoretical limit of 2 1/rad (see figure 5.28) The relationship (Ref

3) between 3d wing lift curve slope (C L ) and 2d airfoil lift curve slope (C l ) is as follows:

For this reason, a high AR (longer) wing is desired

Figure 5.28 The effect of AR on C L versus angle of attack graph

3 As the AR is increased, the wing stall angle (s) is decreased toward the airfoil stall angle Since the wing effective angle of attack is increased (see figure 5.28) For this reason, the horizontal tail is required to have an aspect ratio lower than wing aspect ratio to allow for a higher tail stall angle This will result in the tail to stall after wing has stalled, and allow for a safe recovery For the same reason, a canard is desired to have an aspect ratio to be more than the wing aspect ratio For this reason, a high AR (longer) wing is desired

4 Due to the third item, as the AR is increased, the wing maximum lift coefficient (CLmax) is increased toward the airfoil maximum lift coefficient (C lmax ) This is due to the fact that the wing effective angle of attack is increased (see figure 5.28) For this reason, a high

AR (longer) wing is desired

5 As the AR is increased, the wing will be heavier The reason is the requirement for structural stiffness As the wing gets longer, the wing weight (W w ) moment arm (T) gets larger (since

The wing root experiences higher stress due to the weight of the longer wing, necessitating a stronger structure to support it This increased strength leads to a heavier wing, which in turn raises costs Therefore, a low aspect ratio (AR) wing, which is shorter, is preferred to mitigate these issues.

6 As the AR is increased, the aircraft maximum lift-to-drag ratio is increased Since

3d wing (low AR) increasing AR

The wing induced drag factor (K), Oswald span efficiency factor (e), and aircraft zero-lift drag coefficient (C Do) are critical parameters in aerodynamics, as detailed in References 7 and 8 A high aspect ratio (AR) wing is preferred because it minimizes drag, which is why gliders typically feature long wings with large aspect ratios.

7 As the AR is increased, the wing induced drag is decreased, since the induced drag (C Di ) is inversely proportional to aspect ratio For this reason, a low AR (shorter) wing is desired

8 As the AR is increased, the effect of wing tip vortex on the horizontal tail is decreased

The flow leakage around the wing tips generates trailing vortices, creating downwash that affects the tail's effective angle of attack This downwash reduces the tail's angle, impacting the aircraft's longitudinal stability and control.

9 As the AR increases, the aileron arm will be increased, since the aileron are installed outboard of the wing This means that the aircraft has more lateral control

10 As the AR increases, the aircraft mass moment of inertia around x-axis (Ref 9) will be increased This means that it takes longer to roll In another word, this will reduces the maneuverability of aircraft in roll (Ref 10) For instance, the Bomber aircraft B-52; that has a very long span; takes several seconds to roll at low speed, whilst the fighter aircraft F-16 takes a fraction of a second to roll For this reason, a low AR (shorter) wing is desired for a maneuverable aircraft The tactical supersonic missiles have a low AR of around 1 to enable them to roll and maneuver as fast as possible

11 If the fuel tank is supposed to be inside wing, it is desirable to have a low aspect ratio wing This helps to have a more concentrated fuel system For this reason, a low AR (shorter) wing is desired

12 As the aspect ratio is increased, the wing stiffness around y-axis is decreased This means that the tendency of the wing tips to drop during a take-off is increased, while the tendency to rise during high speed flight is increased In practice, the manufacture of a very high aspect ratio wing with sufficient structural strength is difficult

Taper Ratio

The taper ratio (λ) is the proportion of the tip chord (Ct) to the root chord (Cr), applicable to wings and both horizontal and vertical tails The definitions of root chord and tip chord are illustrated in Figure 5.31.

The geometric result of taper is a smaller tip chord In general, the taper ratio varies between zero and one

0  where three major planform geometries relating to taper ratio are rectangular, trapezoidal and delta shape (see Figure 5.29)

A rectangular wing planform is generally considered aerodynamically inefficient, despite its benefits in performance, cost, and ease of manufacture This design results in a larger downwash angle at the wing tip compared to the root, leading to a reduced effective angle of attack at the tip Consequently, this can cause the wing tip to stall more readily.

11 In some older textbooks, taper ratio was defined as the ratio between root chord and the tip chord

Wing design plays a crucial role in optimizing aerodynamic efficiency, particularly in minimizing induced drag The spanwise lift distribution often deviates from the ideal elliptical shape, making it essential to consider planform tapering There are three primary wing shapes: rectangular wings (λ = 1), straight tapered trapezoidal wings (0 < λ < 1), and delta wings (λ = 0), each offering unique advantages in drag reduction and overall performance.

Figure 5.29 Wings with various taper ratio

The smaller tip chord compared to the root chord results in a lower tip Reynolds number and a reduced tip induced downwash angle, leading to a lower angle of attack at which stall occurs Consequently, the wing tip may stall before the root, which negatively impacts lateral stability and control Additionally, a rectangular wing planform is structurally inefficient due to excess outboard area that contributes little lift Implementing wing taper can effectively address these issues.

1 The wing taper will change the wing lift distribution This is assumed as an advantage of the taper, since it is a technical tool to improve the lift distribution One of the wing design objective is to generate the lift such that the spanwise lift distribution be elliptical The significance of elliptical lift distribution will be examined in the next section Based on this item, the exact value for taper ratio will be determined by lift distribution requirement

2 The wing taper will increase the cost of the wing manufacture, since the wing ribs will have different shapes Unlike a rectangular planform that all ribs are similar; each rib will have different size If the cost is of major issue (such as for homebuilt aircraft), do not taper the wing

3 The taper will reduce the wing weight, since the center of gravity of each wing section

(left and right) will move toward fuselage center line This results in a lower bending

Wing Design 47 moment at the wing root This is an advantage of the taper Thus, to reduce the weight of the wing, more taper (toward 0) is desired

4 Due to item 3, the wing mass moment of inertia about x-axis (longitudinal axis) will be decreased Consequently, this will improve the aircraft lateral control In this regard, the best taper is to have a delta wing ( = 0)

5 The taper will influence the aircraft static lateral stability (C l ), since the taper usually generates a sweep angle (either on the leading edge or on quarter chord line) The effect of the weep angle on the aircraft stability will be discussed in section 5.8

The taper ratio significantly impacts various aircraft features, serving as a compromise among aerodynamic, structural, performance, stability, cost, and manufacturability requirements While a non-tapered wing may be advantageous for aspects like cost and manufacturability, a tapered wing is preferred for stability, performance, and safety Initial taper ratio estimates stem from lift distribution calculations, with the final value determined through comprehensive investigations into aircraft performance and other factors Employing a systems engineering approach, particularly a weighted parametric table, is essential for accurately establishing the taper ratio Additionally, Table 5.9 provides examples of taper ratios for different aircraft, while Figure 5.30 illustrates the typical effects of taper ratio on lift distribution.

Figure 5.30 The typical effect of taper ratio on the lift distribution

In normal flight conditions, the aerodynamic forces on lifting surfaces, such as lift and drag, are represented at the Aerodynamic Center (AC) along with a pitching moment that remains constant regardless of the angle of attack Techniques for locating the planform aerodynamic center are detailed in most aerodynamic textbooks.

In wing design, compressibility effects influence the aerodynamic center (ac), which typically ranges from 25% to 30% of the Mean Aerodynamic Chord (MAC) As speeds approach transonic and supersonic ranges, the ac shifts rearward, nearing the 50% chord point at transonic speeds The aerodynamic center is located in the wing's plane of symmetry, and for practical purposes, the MAC is often assessed using a half wing The length of the MAC for a general planform can be calculated using a specific integral.

In a trapezoidal planform with a constant taper and sweep angle, the Mean Aerodynamic Chord (MAC) is calculated using the local chord (C) and the aircraft's lateral axis (y), as illustrated in figure 5.31.

Figure 5.31 Mean Aerodynamic Chord and Aerodynamic Center in a straight wing

Table 5.9 illustrates the aspect ratio for several jet and prop-driven aircraft.

The Significance of Lift and Load Distributions

Lift distribution, defined as the non-dimensional lift coefficient (CL) per unit span along the wing, indicates that each segment of the wing generates a specific amount of lift The total lift produced by the wing is the sum of these individual contributions Notably, lift distribution approaches zero at the wingtips due to pressure equalization occurring at y = -b/2 and +b/2, resulting in no lift generation at these points.

The variation of the product of lift coefficient and sectional chord (C C L) along the span is known as "load distribution," which, along with lift distribution, plays a crucial role in wing design Lift distribution is primarily used for aerodynamic calculations, whereas load distribution is essential for wing structural design and controllability analysis.

In the 1930s, it was believed that an elliptical lift distribution required the chord to vary elliptically along the wing span, leading to the conclusion that the wing planform must be elliptical This belief resulted in the design of several aircraft, including the iconic Supermarine Spitfire, which featured an elliptical wing shape However, modern understanding reveals that multiple parameters can create an elliptical lift distribution, indicating that an elliptical wing planform is not a necessity.

In wing design, both lift and load distribution are crucial factors that significantly impact aircraft performance, airworthiness, stability, control, and cost Ideally, both distributions should be elliptical, making them the primary design objectives in the wing design process Figure 5.32 illustrates an elliptical lift distribution from a front view of the wing, where the horizontal axis represents y/s, with y as the location on the y-axis and s as the semispan (s = b/2) In this illustration, no high lift devices, such as flaps, are deflected, and the fuselage's effect is disregarded The elliptical lift and load distributions offer several desirable properties that enhance overall wing performance.

1 If the wing tends to stall (CLmax), the wing root is stalled before the wing tip (CLroot C Lmax while C Ltip < C Lmax ) In a conventional aircraft, the flaps are located inboard, while the ailerons are installed outboard of the wing In such a situation, ailerons are active, since the flow over the wing outboard section is healthy This is of greater importance for spin recovery (which often happens after stall); since the aileron (in addition to rudder) application are very critical to stop the autorotation Thus, the elliptical lift distribution provision guarantees the flight safety in the event of stall (see figure 5.33)

Figure 5.32 Elliptical lift distribution over the wing

2 The bending moment at the wing root is a function of load distribution If the load distribution is concentrated near to the root, the bending moment is considerably less that when it is concentrated near the tip The center of an elliptical load distribution is closer to the wing root, thus it leads to a lower bending moment, which results in a less bending stress and a less stress concentration at wing root (see figure 5.34) This -b/2 +b/2

Wing Design 50 means a lighter wing spar and lighter wing structure that is always one of the design requirements The load distribution is a function of the lift distribution

3 The center of gravity of each wing section (left or right) for an elliptical load distribution is closer to the fuselage center line This means a lower wing mass moment of inertia about x-axis which is an advantage in the lateral control Basically, an aircraft rolls faster when the aircraft mass moment of inertia is smaller

4 The downwash is constant over the span for an elliptical lift distribution (Ref 3) This will influence the horizontal tail effective angle of attack

5 For an elliptical lift distribution, the induced angle of attack is also constant along the span

6 The variation of lift over the span for an elliptical lift distribution is steady (gradually increasing from tip (zero) to the root (maximum)) This will simplify the wing spar(s) design root tip root tip a Non-elliptical (tip stalls before the root) b Elliptical (root stalls before the tip)

Figure 5.33 Lift distribution over the half wing root tip root tip a Non-elliptical (load is farther from root) b Elliptical (load is closer to root)

Figure 5.34 Load distribution over a half wing

When analyzing lift distribution, it's important to note that adding the fuselage's contribution to the wing lift may result in a non-elliptical distribution, primarily due to the minimal lift generated by the fuselage.

Total lift generated by a half wing Total lift generated by a half wing

In conventional aircraft, the wing is connected to the fuselage, which influences lift distribution This section examines an ideal scenario, allowing readers to adjust lift distribution by factoring in the fuselage's role, as shown in Figure 5.35 for a low wing configuration Additionally, Figure 5.36 illustrates how deflected flaps affect lift distribution Ultimately, the objective in wing design is to achieve an elliptical lift distribution, independent of contributions from the fuselage, flaps, or other components.

Figure 5.35 The fuselage contribution to the lift distribution of a low wing configuration

Figure 5.36 The flap contribution to the lift distribution

In Section 5.15, a mathematical technique will be introduced to determine the lift and load distribution along the wing

Sweep Angle

The leading edge sweep (ΛLE) is defined as the angle between a constant percentage chord line along the wing's semispan and the lateral axis perpendicular to the aircraft's centerline (y-axis) Similarly, the angle between the wing's leading edge and the y-axis is also referred to as leading edge sweep Trailing edge sweep (ΛTE) describes the angle between the wing's trailing edge and the longitudinal axis (y-axis) of the aircraft Additionally, the quarter chord sweep (ΛC/4) is the angle between the wing's quarter chord line and the y-axis, while the 50 percent chord sweep (ΛC/2) refers to the angle between the wing's 50 percent chord line and the y-axis.

Figure 5.37 Five wings with different sweep angles a b c d

Wings can be classified based on their angle of sweep: if the angle is greater than zero, it is termed aft sweep, while an angle less than or equal to zero is referred to as forward sweep Various wing designs are illustrated, with figure 5.37 showcasing different sweep angles Specifically, figure 5.37a depicts a wing without sweep, while figures 5.37b to 5.37d present wings with leading edge, trailing edge, quarter chord, and 50 percent chord sweeps, respectively Since the mid-1940s, many high-speed aircraft, such as the North American F-86 Saber, have adopted swept wings, typically featuring a design where the leading edge has a greater sweep than the trailing edge.

Sweep angle refers to the angle at which a wing is angled back or forward, with specific distinctions such as aft leading edge sweep and forward trailing edge sweep Among the four types of sweep angles, quarter chord sweep and leading edge sweep are the most significant In subsonic conditions, lift due to angle of attack predominantly occurs at the quarter chord, which is also where the crest is typically located This section primarily focuses on the characteristics, advantages, and disadvantages of leading edge sweep angle, unless otherwise noted.

Figure 5.38 The effective of the sweep angle of the normal Mach number

Basically, a wing is being swept for the following four design goals:

1 Improving the wing aerodynamic features (lift, drag, pitching moment) at transonic, supersonic and hypersonic speeds by delaying the compressibility effects

2 Adjusting the aircraft center of gravity

3 Improving longitudinal and directional stability

4 Increasing pilot view (especially for fighter pilots)

Stagnation streamline ( lateral curvature exaggerated)

This section will elaborate on the items mentioned, and readers are encouraged to consult the technical textbooks listed at the end of the chapter for additional information The sweep angle significantly impacts various flight characteristics.

1 The sweep angle, in practice, tends to increase the distance between leading edge and trailing edge Accordingly, the pressure distribution will vary

2 The effective chord length of a swept wing is longer (see Figure 5.38) by a factor of 1/cos () This makes the effective thickness-to-chord ratio thinner, since the thickness remains constant

3 Item 2 can be also translated into the reduction of Mach number (Mn) normal to the wing leading edge to M cos () Hence, by sweeping the wing, the flow behaves as if the airfoil section is thinner, with a consequent increase in the critical Mach number of the wing For this reason, a classic design feature used to increase M cr is to sweep the wing (Ref 11)

4 The effect of the swept wing is to curve the streamline flow over the wing as shown in figure 5.38 The curvature is due to the deceleration and acceleration of flow in the plane perpendicular to the quarter chord line Near the wing tip the flow around the tip from the lower surface to the upper surface obviously alters the effect of sweep The effect is to unsweep the spanwise constant-pressure lines; isobar To compensate, the wing tip may be given additional structural sweep

5 The wing aerodynamic center (ac) is moved aft by the wing aft sweep at about few percent The aft movement of the ac with increase in sweptback angle occurs because the effect of the downwash pattern associated with a swept wing is to raise the lift coefficient on the outer wing panel relative to the inboard lift coefficient Since sweep movers the outer panel aft relative to the inner portion of the wing, the effect on the center of lift is an aft ward movement The effect of wing sweep on ac position is shown in figure 5.39 for aspect ratios of 7 and 10 and for taper ratios of 0.25 and 0.5

Figure 5.39 Effect of wing sweepback on ac position for several combinations of AR and 

6 The effective dynamic pressure is reduced, although not by as much as in cruise

7 The sweep angle tends to change the lift distribution as sketched in figure 5.40 The reason becomes clear by looking at the explanations in item 5 As the sweep angle is increased, the Oswald efficiency factor (e) will decrease (Equation 5.25).

Figure 5.40 Typical effect of sweep angle on lift distribution

The Oswald span efficiency for a straight wing and swept wing are given respectively by equation 5.27a and 5.27b (Ref 14)

Equation 5.27a is for a straight wing and Equation 5.27b is for a swept wing where sweep angle is more than 30 degrees When the Oswald span efficiency is equal to 1, it indicates that the lift distribution is elliptical, otherwise it is non-elliptic

8 The wing maximum lift coefficient can actually increase with increasing sweep angle However, the maximum useful lift coefficient actually decreases with increasing sweep angle, due to loss of control in pitch up situation Whether or not pitch up occurs depends not only on the combination of sweep angle and aspect ratio, but also an airfoil type, twist angle, and taper ratio Thus, the sweep angle tends to increase stall speed (V s )

The maximum lift coefficient of the basic wing without high lift device is governed by the following semi-empirical relationship (Ref 6):

Wing Design 56 where sweep angle () is in degrees and C lmax denotes the maximum lift coefficient for the outer panel airfoil section

9 Wing sweep tends to reduce the wing lift curve slope (C L ) A modified equation based on Prandtl-Glauert approximation is introduced by Ref 6 as follows:

10 The aircraft pitching moment will be increased, provided the aircraft cg is forward of aircraft ac The reason is that wing aerodynamic center is moving aft with increase in sweep angle

11 An aft swept wing tends to have tip stall because of the tendency toward outboard, spanwise flow This causes the boundary layer to thicken as it approaches the tips For the similar reason, a swept forward wing would tend toward root stall This tends to have an influence opposite to that of wing twist

12 On most aft swept wing aircraft, the wing tips are located behind the aircraft center of gravity Therefore, any loss of lift at the wing tips causes the wing center of pressure to move forward This in turn will cause the aircraft nose to pitch up This pitch up tendency can cause the aircraft angle of attack to increase even further This may result in a loss of aircraft longitudinal control For the similar reason, a forward swept wing aircraft would exhibit a pitch down tendency in a similar situation

13 Tip stall on a swept wing is very serious If the outboard section of a swept wing stalls, the lift loss is behind the wing aerodynamic center The inboard portion of the wing ahead of the aerodynamic center maintains its lift and produces a strong pitch-up moment, tending to throw the aircraft deeper into the stall Combined with the effect of tip stall on the pitching moment produced by the tail, this effect is very dangerous and must be avoided by options such as wing twist

14 A swept wing produce a negative rolling moment because of a difference in velocity components normal to the leading edge between the left and right wing sections (Ref

13) The rolling moment due to aft sweep is proportional to the sine of twice the leading edge sweep angle

The dihedral effect (Cl β) becomes more negative with a swept wing, indicating that such wings possess an inherent dihedral effect, potentially eliminating the need for additional dihedral or anhedral for lateral-directional stability This sweep angle enhances the dihedral effect, contributing to greater spirally stability in the aircraft However, this increase in negative dihedral effect also leads to a decrease in the Dutch roll damping ratio, creating a design conflict that necessitates a careful compromise.

Twist Angle

Wings can exhibit different twists, categorized as negative twist (washout) when the wing tip has a lower incidence than the wing root, and positive twist (wash-in) when the wing tip has a higher incidence Typically, wings are designed with negative twist, meaning the angle of attack at the wing tip is lower than at the root, which reduces the angle of attack along the span Many modern aircraft feature varying airfoil sections along the span, leading to different zero lift angles of attack, known as aerodynamic twist Often, the airfoil section at the wing tip is thinner than that at the root In some cases, both sections may share the same thickness-to-chord ratio, but the root airfoil section will have a higher (more negative) zero-lift angle of attack compared to the tip section.

Wing Design 66 a geometric twist b Aerodynamic twist Figure 5.47 Wing twist

Geometric twist occurs when the tip and root incidence angles differ, while aerodynamic twist arises when the tip and root airfoil sections are dissimilar Each type of twist presents unique advantages and disadvantages, requiring designers to make informed selections to meet specific design requirements The decision to apply twist is a critical aspect of the design process, with the precise amount of twist being determined through calculations This section will explore both geometric and aerodynamic twist in detail.

Aerodynamic twist is often preferred over geometric twist in wing design due to its manufacturing convenience In aerodynamic twist, different sections of the wing can have varying rib configurations while maintaining a consistent incidence across the entire wing In contrast, geometric twist requires each wing section to have a unique incidence, necessitating a linear decrease in the angle of attack from the root to the tip This is typically achieved by twisting the main wing spar, which automatically applies the necessary rib twist Alternatively, the wing can be divided into inboard and outboard sections, where the inboard portion maintains the wing setting angle while the outboard portion is adjusted to create the desired twist In some cases, both aerodynamic and geometric twists can be utilized together.

There are two major goals for the employing the twist in wing design process: tip root

1 Avoiding tip stall before root stall

2 Modification of lift distribution to elliptical one

In addition to two above-mentioned desired goals, there is another one unwanted output in twist:

When the wing root stalls before the wing tip, pilots can still control the aircraft using the ailerons, as the outboard section remains unstalled, enhancing safety during wing stalls The importance of elliptical lift distribution is highlighted in section 5.7 However, a significant drawback of wing twist is the associated loss of lift, typically due to negative twist As the angle of attack decreases, the lift coefficient also declines Therefore, the twist angle must be limited to prevent negative lift in the outer wing sections, adhering to the zero-lift angle of attack criterion.

To optimize wing performance, it is crucial to avoid negative lift generated by the outboard portion of the wing, as this decreases overall lift The ideal geometric twist angle typically ranges from -1 to -4 degrees, and it must be precisely determined to ensure that the wing tip stalls after the root and that the lift distribution remains elliptical Figure 5.48 demonstrates the impact of negative twist angles on lift distribution, while Table 5.11 provides twist angle data for various aircraft, including the Cessna 208, Beech 1900D, Beechjet 400A, AVRO RJ100, and Lockheed C-130 Hercules, which all feature both geometric and aerodynamic twists.

Figure 5.48 The typical effect of a (negative) twist angle on the lift distribution b/2 y/s root

Wing incidence at root (i w ) (deg)

Wing angle at tip (deg)

19 Piper PA-28-161 Warrior 2,440 2 -1 -3 a Geometric twist (Ref 4 and 12)

10 Kawasaki T-4 12,544 t/C = 10.3% t/C = 7.3% 3 b Aerodynamic twist (Ref 4) Table 5.11 Twist angles for several aircraft

Dihedral Angle

The dihedral angle (Γ) of an aircraft wing is defined as the angle between the wing's chord-line plane and the "xy" plane, observed from the front view This imaginary chord line plane is formed by connecting all chord lines along the wing's span A positive dihedral occurs when the wing tip is elevated above the xy plane, while a negative dihedral, or anhedral, is present when the wing tip is lower For balanced aircraft performance, both the left and right wing sections must maintain identical dihedral angles Understanding the advantages and disadvantages of different dihedral angles is crucial, and this section will outline these characteristics along with design recommendations for determining the optimal dihedral angle.

The main purpose of wing dihedral is to enhance an aircraft's lateral stability, which refers to its ability to return to its original flight position after being disturbed by external factors like gusts This phenomenon, often referred to as dihedral stability, occurs because the wing dihedral angle generates a restoring rolling moment Lateral static stability is primarily characterized by a stability derivative known as the aircraft dihedral effect.

C l  dC l ) that is the change in aircraft rolling moment coefficient due to a change in aircraft sideslip angle ()

When a level-wing aircraft experiences a disturbance, such as a gust affecting one side of the wing, it results in an undesired positive roll, causing one wing (e.g., the left wing) to rise while the other (the right wing) drops This drop leads to a temporary loss of lift on the right wing, causing the aircraft to accelerate and slip toward the right, creating a positive sideslip angle (β) To regain stability, a laterally statically stable aircraft must generate a negative rolling moment, which is associated with a negative dihedral effect (C l β < 0) The wing's dihedral angle plays a crucial role by increasing the angle of attack (Δα) and producing a normal velocity (Vn = VΓ), thereby helping the aircraft return to its original wing-level position.

Wing design involves understanding the relationship between airspeed components along the x-axis (U) and y-axis (V), where an increase in angle of attack leads to a corresponding increase in lift, resulting in a negative rolling moment contribution Interestingly, the left wing section experiences an opposite effect, also contributing to a negative rolling moment The rolling moment due to sideslip is proportional to the geometric wing dihedral angle; a positive dihedral angle results in a negative sideslip derivative (C l β) Aircraft require a minimum negative rolling moment from the dihedral effect to avoid excessive spiral instability, as too much dihedral can reduce dutch roll damping While a more negative C l β enhances spiral stability, it simultaneously decreases dutch roll stability.

The anhedral wing design serves a destabilizing function, contrasting with dihedral wings Its use aims to balance various wing parameters, such as sweep angle and vertical position, which influence lateral stability While increased lateral stability can reduce rolling control, it is essential to carefully select wing parameters to meet both stability and controllability needs Although the primary purpose of the dihedral angle is to enhance lateral stability, considerations for wing sweep and vertical positioning also involve performance and operational requirements.

Cargo aircraft typically feature a high wing design to facilitate efficient loading and unloading operations, which enhances lateral stability To reduce this inherent stability, designers may incorporate an anhedral angle to the wings, improving the aircraft's rolling controllability without affecting its operational capabilities High wing aircraft naturally exhibit a dihedral effect, while low wing aircraft often lack this characteristic, necessitating a greater dihedral angle for stability Conversely, swept wing designs also play a role in the overall aerodynamic performance of the aircraft.

Restoring moment xy plane airstream

Aircraft with a Wing Design 71 often exhibit excessive dihedral effect (C l β) due to their sweep angle To counteract this in high-wing aircraft, implementing negative dihedral, or anhedral, can be effective Achieving a balance between lateral stability and roll control is crucial when determining the appropriate dihedral angle.

Boeing 737-100 (Transport) Figure 5.51 Four aircraft with different dihedral angles

The dihedral effect of aircraft wings significantly influences ground and water clearance, as wings, nacelles, and propellers require a minimum distance from the ground and water surfaces An increased dihedral angle enhances this clearance, while an anhedral angle reduces it Notably, aircraft with high aspect ratios and flexible wings, like the record-breaking Voyager, experience additional dihedral angle due to wing deformation during flight, which must be factored into their wing design.

Applying a dihedral angle to a wing decreases its effective planform area (Seff), leading to a reduction in lift compared to a wing without dihedral, which is not ideal To minimize this lift reduction, it's advisable to use the lowest possible dihedral angle when designing the wing The relationship between effective wing planform area and dihedral angle can be quantified for better understanding and application.

No Aircraft Type Wing position

1 Pilatus PC-9 Turboprop Trainer Low-wing 7

2 MD-11 Jet Transport Low-wing 6

3 Cessna 750 Citation X Business Jet Low-wing 3

4 Kawasaki T-4 Jet Trainer High-wing -7

5 Boeing 767 Jet Transport Low-wing 4 o 15'

6 Falcon 900 B Business Jet Transport Low-wing 0 o 30'

7 C-130 Hercules Turboprop Cargo High-wing 2 o 30'

8 Antonov An-74 Jet STOL Transport Parasol-wing -10

9 Cessna 208 Piston Engine GA High-wing 3

10 Boeing 747 Jet Transport Low-wing 7

11 Airbus 310 Jet Transport Low-wing 11 o 8'

12 F-16 Fighting Falcon Fighter Mid-wing 0

13 BAE Sea Harrier V/STOL Fighter High-wing -12

14 MD/BAe Harrier II V/STOL Close Support High-wing -14.6

16 Fairchild SA227 Turboprop Commuter Low-wing 4.7

17 Fokker 50 Turboprop Transport High-wing 3.5

18 AVRO RJ Jet Transport High-wing -3

Table 5.12 Dihedral (or Anhedral) angles for several aircraft

Table 5.12 presents the dihedral and anhedral angles for various aircraft, highlighting their wing vertical positions Typically, dihedral angles range from -15 to +10 degrees, as illustrated in Figure 5.51, which features four aircraft with differing dihedral angles Table 5.13 provides standard dihedral angle values for both swept and unswept wings across various vertical positions, serving as a useful reference for initial angle selection However, the precise dihedral angle is ultimately established during the stability and control analysis of the entire aircraft, taking into account the design of other components such as the fuselage and tail to assess overall lateral stability.

For optimal lateral controllability and stability in aircraft design, the recommended dihedral effect (C l β) should range from -0.1 to +0.4 1/rad To meet all design specifications, adjustments to the dihedral angle may be necessary If a single dihedral angle for the entire wing fails to meet these requirements, consider dividing the wing into inboard and outboard sections for more precise control.

Wing Design 73 each with different dihedral angle For instance, you may apply dihedral angle to the outboard plane, in order to keep the wing level in the inboard plane

No Wing Low wing Mid-wing High wing Parasol wing

1 Unswept 5 to 10 3 to 6 -4 to -10 -5 to -12

2 Low subsonic swept 2 to 5 -3 to +3 -3 to -6 -4 to -8

3 High subsonic swept 3 to 8 -4 to +2 -5 to -10 -6 to -12

4 Supersonic swept 0 to -3 1 to -4 0 to -5 NA

5 Hypersonic swept 1 to 0 0 to -1 -1 to -2 NA

Table 5.13 Typical values of dihedral angle for various wing configurations

High Lift Device

5.12.1 The Functions of High Lift Device

A key objective in wing design is to enhance the wing's ability to generate lift, represented by the maximum lift coefficient (CLmax) During trimmed cruising flight, the lift produced equals the aircraft's weight The airspeed at which the aircraft achieves its maximum lift coefficient is known as stall speed.

Two key design objectives in aircraft development are maximizing payload weight and minimizing stall speed (Vs) According to equation 5.36, increasing the maximum lift coefficient (CLmax) enhances payload capacity while reducing stall speed, which is crucial for safe take-offs and landings A lower stall speed contributes to safer operations, while a higher payload weight improves aircraft efficiency and lowers flight costs Additionally, a greater CLmax enables a smaller wing area, resulting in a lighter wing structure Therefore, wing designers must focus on maximizing CLmax, which can be achieved in-flight by temporarily increasing wing camber when high lift devices are deflected downward.

High lift devices are essential for aircraft during take-off and landing, as they enable the wings to generate a higher lift coefficient at low speeds During these critical phases, the aircraft operates at speeds just above stall speed, necessitating the use of these devices to ensure safe operations Airworthiness standards outline the correlation between take-off speed, landing speed, and stall speed, emphasizing the importance of high lift devices in maintaining aircraft performance and safety.

V   (5.38) where k is about 1.1 for fighter aircraft, and about 1.2 for jet transports and GA aircraft

The use of high lift devices alters the camber of the airfoil section and wing, resulting in a positive increase in camber This modification affects the pressure distribution along the wing chord, as illustrated by the pressure coefficient (CP) in figure 5.52.

Leading edge high lift devices enhance the boundary layer energy of aircraft wings Since the early 1930s, various types of high lift devices have been incorporated into nearly all aircraft designs These devices are essential for achieving significant increases in lift during flight.

At the airfoil level, a high lift device deflection tends to cause the following six changes in the airfoil features:

2 Maximum lift coefficient (C lmax ) is increased,

3 Zero-lift angle of attack (o) in changed,

5 Pitching moment coefficient is changed, and

7 Lift curve slope is increased

Figure 5.52 Example of pressure distribution with the application of a high lift device

Figure 5.53 illustrates the effects of high lift devices, highlighting three key advantages and several negative side effects While a plain flap decreases the stall angle, both slotted flaps and leading edge slats increase it Notably, the Fowler flap and leading edge slat enhance the lift curve slope (CLα), whereas the leading edge flap shifts the zero-lift angle of attack (αo) to the right.

A reduction in stall angle is unfavorable because it can cause the wing to stall at a lower angle of attack During take-off and landing, a high angle of attack is essential for successful operations, as it helps minimize both the take-off and landing distance, which is crucial for airports with limited runway lengths Additionally, an increase in pitching moment coefficient necessitates a larger horizontal tail area to maintain balance in the aircraft.

 f x/C Pressure distribution of original wing

Pressure distribution of the wing when HLD deflected

The Wing Design 75 effectively reduces drag coefficient, enhancing acceleration during take-off and landing While the use of high lift devices can introduce three undesirable side effects, the benefits they provide significantly outweigh these drawbacks.

To ensure safe take-off and landing, aircraft can temporarily enhance their natural maximum lift coefficient (CLmax) using mechanical high lift devices, allowing for increased lift without altering the aircraft's pitch This adjustment is crucial during critical flight operations such as take-off and landing Table 5.14 illustrates the maximum lift coefficients for various aircraft, comparing values with and without flap deflection.

Figure 5.53 Typical effects of high lift device on wing airfoil section features

During cruising flight, maximum lift coefficient is unnecessary due to high speeds High Lift Devices (HLD) are wing components that enhance lift when deflected downward Typically found in the inboard section of the wing, these devices are primarily used during take-off and landing.

Table 5.14 Maximum lift coefficient for several aircraft

Two main groups of high lift devices are:

1 leading edge high lift device (LEHLD or flap), and

2 trailing edge high lift devices (TEHLD)

Wing trailing edge flaps come in various types, with the most common being split flaps, plain flaps, single-slotted flaps, double-slotted flaps, triple-slotted flaps, and fowler flaps.

Wing Design 76, depicted in figure 5.54a, features downward-deflected elements that enhance the wing's camber, resulting in an increased CLmax The most prevalent leading edge devices include the leading edge flap, leading edge slat, and Kruger flap, as illustrated in figure 5.54b.

A significant challenge in utilizing high lift devices is managing the gap between these devices and the main wing, which can either be sealed or left open, each presenting its own issues An open gap allows airflow from the lower surface to escape to the upper surface, negatively affecting pressure distribution Conversely, sealing the gap with a diaphragm can lead to blockages from ice in colder, humid conditions, creating operational concerns This article discusses the technical features of various high lift devices, including plain flaps, split flaps, single slotted flaps, double slotted flaps, triple slotted flaps, Fowler flaps, trailing edge high lift devices, leading edge flaps, leading edge slats, and Kruger flaps.

1 The plain flap (figure 5.54-a) is the simplest and earliest type of high lift device It is an airfoil shape that is hinged at the wing trailing edge such that it can be rotated downward and upward However, the downward deflection is considered only A plain flap increases the lift simply by mechanically increasing the effective camber of the wing section In terms of cost, a plain flap is the cheapest high lift device In terms of manufacturing, the plain flap is the easiest one to build Most home build aircraft and many General Aviation aircraft are employing the plain flap The increment in lift coefficient for a plain flap at 60 degrees of deflection (full extension) is about 0.9 If it is deflected at a lower rate, the CL increment will be lower Some old GA aircraft such as Piper 23 Aztec D has a plain flap It is interesting to know that the modern fighters such aircraft F-15E Eagle and MIG-29 also employ plain flaps

2 In the split flap (figure 5.54-b), only the bottom surface of the flap is hinged so that it can be rotated downward The split flap performs almost the same function as a plain flap However, the split flap produces more drag and less change in the pitching moment compared to a plain flap The split flap was invented by Orville Wright in 1920, and it was employed, because of

Wing Design 77 its simplicity, on many of the 1930s and 1950s aircraft However, because of the higher drag associated with split flap, they are rarely used on modern aircraft

Aileron

Ailerons, located at the outboard sections of an aircraft's wings, are similar to trailing edge plain flaps but can be deflected both upward and downward Unlike flaps, ailerons operate differentially, with one aileron moving up while the other moves down, allowing for effective lateral control during flight Due to their critical role and the extensive materials involved in their design, ailerons will be explored in detail in a separate chapter (Chapter 12).

Figure 5.56 Typical location of the aileron on the wing

In this section, it is crucial to avoid using the entire wing's trailing edge for flaps, leaving approximately 30 percent of the wing outboard for ailerons, as illustrated in Figure 5.56 The aileron design process involves determining three key parameters: aileron chord, span, and deflection (both up and down) The primary design requirements for ailerons stem from the need for effective roll controllability in aircraft A comprehensive discussion on aileron design and techniques will be provided in Chapter 12.

Lifting Line Theory

In section 5.7, the wing design process involves calculating the lift force generated by the wing, allowing designers to adjust parameters to meet design goals and requirements While this technique is rooted in Aerodynamics, a straightforward yet accurate method is introduced to aid in wing design A solid understanding of Aerodynamics is essential for wing designers, making this section a review of key concepts For more in-depth information, please refer to Reference 16.

This section introduces a technique for calculating the lift generated by a wing without the need for advanced CFD software To utilize this method, essential wing data must be gathered, including wing area, airfoil characteristics, aspect ratio, taper ratio, incidence angle, and details on high lift devices By simultaneously solving several aerodynamic equations, one can accurately determine the lift produced by the wing and analyze the lift distribution along its span, allowing for verification of whether the distribution is elliptical.

The technique is initially introduced by Ludwig Prandtl and is called “lifting-line theory” in

1918 Almost every Aerodynamics textbook has the details of this simple and remarkably

Wing Design 85 utilizes an accurate yet classical linear technique for calculating lift distribution along a wing's span and total lift coefficient However, it is crucial to note that this method does not account for stall, making it unsuitable for use beyond the stall angle of the airfoil section The technique applies to wings with both flaps up and down, and due to the symmetric geometry of the wing, only one half needs to be analyzed initially This method can later be expanded to include both left and right wing halves, with a demonstration of its application provided at the end of the chapter.

Figure 5.57 Dividing a wing into several sections

Figure 5.58 Angles corresponding to each segment in lifting-line theory

Step 1 Divide one half of the wing (semispan) into several (say N) segments The segments along the semispan could have equal span, but it is recommended to have smaller segments in

In wing design, particularly near the wing tip, a higher number of segments (N) is preferred for improved accuracy For instance, a wing can be divided into seven equal segments, each possessing distinct chord lengths and potentially unique spans Additionally, it is possible to assign a specific airfoil section to each segment, taking into account aerodynamic twist It is essential to define the geometry, such as chord and span, alongside the aerodynamic properties, including angles of attack (α) and zero-lift angles (α₀).

C l ) of each segment for future application

Step 2 Calculate the corresponding angle () to each section These angles are functions of lift distribution along the semispan as depicted in figure 5.58 Each angle () is defined as the angle between the horizontal axis and the intersection between lift distribution curve and the segment line In fact, we originally assume that the lift distribution along the semispan is elliptical This assumption will be corrected later

The angle θ ranges from 0 degrees for the last segment to nearly 90 degrees for the first segment The values of angle θ for other segments can be derived from the corresponding triangles, as illustrated in figure 5.58 For example, in figure 5.58, angle θ6 represents the angle associated with segment 6.

Step 3 Solve the following group of equations to find A1 to An:

The lifting-line equation, also known as the monoplane equation, is fundamental to aerodynamic theory and was originally developed by Prandtl In this equation, N represents the number of segments, while α indicates the angle of attack for each segment, and α₀ denotes the zero-lift angle of attack Additionally, the coefficients An serve as intermediate unknowns, and the parameter μ is defined as follows: b.

The equation \( \mu = C_i \cdot \alpha \) (5.41) defines the relationship between the segment's mean geometric chord \( C_i \), the lift curve slope \( C_{l\alpha} \) in radians, and the wing span \( b \) In cases where the wing is twisted (denoted as \( \alpha_t \)), this twist angle must be applied linearly across all segments, resulting in a reduction of the angle of attack for each segment by subtracting the corresponding twist angle from the wing's setting angle During take-off operations with deflected flaps, the inboard segments exhibit a higher zero-lift angle of attack \( \alpha_o \) compared to the outboard segments.

Step 4 Determine each segment‟s lift coefficient using the following equation:

Now you can plot the variation of segment‟s lift coefficient (CL) versus semispan (i.e lift distribution)

Step 5 Determine wing total lift coefficient using the following equation:

C L i   (5.43) where AR is the wing aspect ratio

Please note that the lifting-line theory has other useful features, but they are not covered and used here

Determine and plot the lift distribution for a wing with the following characteristics Divide the half wing into 10 sections

S = 25 m 2 , AR = 8,  = 0.6, iw = 2 deg, t = -1 deg, airfoil section: NACA 63-209

If the aircraft is flying at the altitude of 5,000 m ( = 0.736 kg/m 3 ) with a speed of 180 knot, how much lift is produced?

By using the Reference 2, we can find the airfoil section‟s features A copy of the airfoil graphs are shown in figure 5.22 Based on the C l - graph, we have the following data:

o = - 1.5 degrees, C l = 6.3 1/rad The application of the lifting-line theory is formulated through the following MATLAB m-file clc clear

The wing design features an aspect ratio (AR) of 8 and a taper ratio (alpha_twist) of -1, contributing to its aerodynamic efficiency The twist angle is set at -1 degrees, while the wing setting angle (i_w) is 2 degrees The lift curve slope (a_2d) is calculated at 6.3 (1/rad), with a zero-lift angle of attack (alpha_0) of -1.5 degrees The wingspan (b) is determined using the formula b = sqrt(AR*S), ensuring optimal performance characteristics.

The root chord (Croot) is calculated using the formula Croot = (1.5*(1+lambda)*MAC)/(1+lambda+lambda^2), where lambda represents the taper ratio and MAC is the mean aerodynamic chord The angle theta is defined as a range from pi/(2*N) to pi/2, divided into N segments The angle of attack for each segment, alpha, varies from i_w + alpha_twist to i_w, decreasing by alpha_twist/(N-1) for each segment The position z is determined by z = (b/2)*cos(theta), where b is the wingspan The mean aerodynamic chord (c) for each segment is derived from c = Croot * (1 - (1-lambda)*cos(theta)) Finally, the parameter mu is calculated as mu = c * a_2d / (4 * b), linking the mean aerodynamic chord to the lift coefficient in two-dimensional flow.

LHS = mu * (alpha-alpha_0)/57.3; % Left Hand Side

% Solving N equations to find coefficients A(i): for i=1:N for j=1:N

B(i,j) = sin((2*j-1) * theta(i)) * (1 + (mu(i) * (2*j-1)) / sin(theta(i))); end end

A=B\transpose(LHS); for i = 1:N sum1(i) = 0; sum2(i) = 0; for j = 1 : N sum1(i) = sum1(i) + (2*j-1) * A(j)*sin((2*j-1)*theta(i)); sum2(i) = sum2(i) + A(j)*sin((2*j-1)*theta(i)); end end

CL1=[0 CL(1) CL(2) CL(3) CL(4) CL(5) CL(6) CL(7) CL(8) CL(9)]; y_s=[b/2 z(1) z(2) z(3) z(4) z(5) z(6) z(7) z(8) z(9)]; plot(y_s,CL1,'-o') grid title('Lift distribution’) xlabel(‘Semi-span location (m)’) ylabel (‘Lift coefficient’)

Figure 5.59 shows the lift distribution of the example wing as an output of the m-file

Figure 5.59 The lift distribution of the wing in example 5.5

Lift Distribution semi-span Location (m)

The wing's distribution is not elliptical, making it less than ideal and requiring modifications, such as increasing wing twist, to enhance performance Currently, the total lift coefficient of the wing stands at CL = 0.268, indicating the lift generated by this design.

To evaluate the effectiveness of lifting line theory, a comparison was made between the actual cruise lift coefficients of selected aircraft and those predicted by the theory, as detailed in table 5.17 For example, the Bellance aircraft exhibited a cruising lift coefficient of 0.2193, while the theoretical value calculated using lifting line theory was 0.229, resulting in a discrepancy of just 4.5 percent This minor difference indicates that lifting line theory is a reliable tool for analyzing low-speed aircraft and can be effectively applied in the initial stages of wing design.

Table 5.17 Characteristics of several aircraft to check the accuracy of lifting line theory

Accessories

Aircraft wings may feature various accessories designed to enhance airflow, depending on the type and flight conditions Accessories like wingtips, fences, vortex generators, stall stripes, and strakes are utilized to boost wing efficiency This section will explore several practical considerations regarding these enhancements.

A strake, or leading edge extension, is an aerodynamic surface attached to an aircraft's fuselage that optimizes airflow and manages wing vortexes This design enhances lift, boosts directional stability, and improves maneuverability, particularly at high angles of attack.

In wing design, highly swept strakes along the fuselage forebody are utilized to connect wing sections effectively Aircraft designers strategically select the location, angle, and shape of these strakes to achieve optimal aerodynamic interaction Notably, fighter aircraft such as the F-16 and F-18 have incorporated strakes to enhance wing efficiency at high angles of attack The intricate design of strakes requires advanced computational fluid dynamics (CFD) software, which is beyond the scope of this discussion.

Stall fences are essential components on swept wings, designed to prevent the boundary layer from drifting towards the wing tips due to the spanwise pressure gradient Typically positioned about 35 percent of the span from the fuselage centerline, these fences create a side lift that generates a strong trailing vortex This vortex enhances airflow over the wing's top surface, mixing fresh air into the boundary layer and effectively sweeping it off the wing into the external flow Consequently, this action reduces the amount of boundary layer air flowing outboard at the rear of the wing, ultimately improving the outer panel's maximum lift coefficient.

Figure 5.60 Example of a stall fence

Leading edge snags can create a vortex that functions as a boundary layer fence, with the under-wing fence, known as a vertilon, being the ideal solution In practice, pylons that support engines under the wing serve a similar purpose to leading edge fences Several high subsonic transport aircraft, including the DC-9 and Beech Starship, have successfully implemented fences on their swept lifting surfaces Designing these fences requires advanced computational fluid dynamics (CFD) software, which is beyond the scope of this book.

Vortex generators are small, vertically positioned wings with a low aspect ratio, strategically placed at an angle on aircraft surfaces like wings, fuselage, or tails Their span is designed to extend just beyond the boundary layer's edge, where they generate lift and create tip vortices These vortices mix with higher energy air, increasing the kinetic energy within the boundary layer This process enables the boundary layer to extend further into adverse pressure gradients before separation occurs Vortex generators come in various sizes and shapes, enhancing aerodynamic performance.

Modern high subsonic jet transport aircraft frequently incorporate numerous vortex generators on their wings, tails, and nacelles to delay local wing stall, although this can lead to a significant increase in aircraft drag The optimal quantity and orientation of these vortex generators are typically established through a series of flight tests, earning them the label "aerodynamic afterthoughts." They are often installed after tests reveal specific flow separations For instance, the Northrop Grumman B-2A strategic penetration bomber employs small drop-down spoiler panels in front of its weapon bay doors to create vortices for cleaner weapon releases Similarly, the Hawker Beechcraft Beech King 1900D, a twin turboprop regional airliner, features small horizontal vortex generators on its fuselage near the wing roots.

Figure 5 61 The Hawker Beechcraft Beech King 1900D (note on winglets)

The significant pressure difference between the upper and lower surfaces of a wing generates tip vortices at the wingtips These vortices roll around the edges of the wing, leading to a decrease in lift at the wingtips, effectively reducing the wing's span Experimental studies have demonstrated that wings with this phenomenon exhibit diminished performance.

Wing design with square or sharp edges offers the widest effective span To address the loss in performance, three solutions can be implemented: tip-tanks, increased wing span, and winglets Winglets are small, nearly vertical lifting surfaces positioned rearward and/or downward at the wing tips, enhancing aerodynamic efficiency.

Aerodynamic analysis of winglets, including aspects like lift, drag, and local flow circulation, can be conducted using classical aerodynamic techniques The need for wingtips varies based on the aircraft's mission and configuration, as they contribute to the overall weight Numerous transport aircraft, such as the Pilatus PC-12, Boeing 747-400, McDonnell Douglas C-17A Globemaster III, and Airbus 340-300, feature winglets For instance, the Hawker Beechcraft Beech King 1900D, a twin turboprop regional airliner, is equipped with winglets to enhance its performance in hot and high conditions.

Wing Design Steps

At this stage, we are in a position to summarize the chapter In this section, the practical steps in a wing design process are introduced (see figure 5.1) as follows:

Primary function: Generation of the lift

1 Select number of wings (e.g monoplane, bi-plane) See section 5.2

2 Select wing vertical location (e.g high, mid, low) See section 5.3

3 Select wing configuration (e.g straight, swept, tapered, delta)

4 Calculate average aircraft weight at cruise:

1 (5.44) where W i is the aircraft at the beginning of cruise and W f is the aircraft at the end of cruising flight

5 Calculate required aircraft cruise lift coefficient (with average weight):

6 Calculate the required aircraft take-off lift coefficient:

The coefficient of 0.85 is derived from the aircraft's take-off angle, which is approximately 10 degrees During this phase, around 15 percent of the lift is generated by the vertical component of the engine thrust, calculated as sin(10).

7 Select the high lift device (HLD) type and its location on the wing See section 5.12

8 Determine high lift device geometry (span, chord, and maximum deflection) See section 5.12

9 Select/Design airfoil (you can select different airfoil for tip and root) The procedure was introduced in Section 5.4

10 Determine wing incidence or setting angle (iw) It is corresponding to airfoil ideal lift coefficient; C li (where airfoil drag coefficient is at minimum) See section 5.5

11 Select sweep angle (0.5C) and dihedral angles () See sections 5.9 and 5.11

12 Select other wing parameter such as aspect ratio (AR), taper ratio (and wing twist angle (twist) See sections 5.6, 5.7, and 5.10

13 Calculate lift distribution at cruise (without flap, or flap up) Use tools such as lifting line theory (See section 5.14), and Computational Fluid Dynamics)

14 Check the lift distribution at cruise that must be elliptic Otherwise, return to step 13 and change few parameters

15 Calculate wing lift at cruise (CLw) Do not employ HLD at cruise

16 The wing lift coefficient at cruise (CLw) must be equal to the required cruise lift coefficient (step 5) If not, return to step 10 and change wing setting angle

17 Calculate wing lift coefficient at take-off (CL_w_TO) Employ flap at take-off with the deflection of  f and wing angle of attack of:  w =  sTO – 1 Note that  s at take-off is usually smaller than s at cruise Please note that the minus one (-1) is for safety

18 The wing lift coefficient at take-off (CL_w_TO) must be equal to take-off lift coefficient (step 6) If not, first, play with flap deflection (f), and geometry (Cf, bf); otherwise, return to step 7 and select another HLD You can have more than one for more safety

20 Play with wing parameters to minimize the wing drag

21 Calculate wing pitching moment (Mow) This moment will be used in the tail design process

22 Optimize the wing to minimize wing drag and wing pitching moment

A fully solved example will demonstrate the application of these steps in the next section.

Wing Design Example

This section presents a comprehensive example of wing design, providing a complete solution while intentionally omitting certain details for the reader to explore These omitted aspects closely resemble the solutions discussed in other examples throughout this section.

Design a wing for a normal category General Aviation aircraft with the following features:

S = 18.1 m 2 , m = 1,800 kg, V C = 130 knot (@ sea level), V S = 60 knot

Assume the aircraft has a monoplane high wing and employs the split flap

The number of wings and wing vertical position are stated by the problem statement, so we do not need to investigate these two parameters

The aircraft, characterized as a high wing, low subsonic, mono-wing design, has been assigned a selection of -5 degrees of anhedral based on Table 5.8 This anhedral angle will be subject to revision and optimization as the design of other aircraft components progresses during the lateral stability analysis.

The aircraft is classified as a low subsonic prop-driven normal category model, designed with cost-efficiency in mind To achieve this, we have opted for a zero sweep angle at 50 percent of the wing chord, although tapering the wing may necessitate incorporating sweep angles at both the leading and trailing edges.

To expedite wing design, we choose an airfoil from the NACA series While the detailed design of an airfoil is beyond the scope of this textbook, selecting the appropriate airfoil involves several key calculations.

The aircraft has a split flap, and the split flap generates an CL of 0.55 when deflected 30 degrees Thus:

Thus, we need to look for NACA airfoil sections that yield an ideal lift coefficient of 0.4 and a net maximum lift coefficient of 1.5

By referring to Reference 2 and figure 5.23, we find the following seven airfoil sections whose characteristics match with or is close to our design requirements (all have C li = 0.4, C lmax  1.5):

In comparing the seven airfoil sections, as outlined in Table 5.18, the optimal airfoil is determined by several key characteristics: the lowest C mo, the lowest C dmin, the highest  s, the highest (Cl/Cd)max, and docile stall quality Analyzing the data reveals clear conclusions regarding the performance of each candidate airfoil.

1- The NACA airfoil section 662-415 yields the highest maximum speed, since it has the lowest C dmin (i.e 0.0044)

2- The NACA airfoil section 642-415 yields the lowest stall speed, since it has the highest maximum lift coefficient (i.e 2.1)

3- The NACA airfoil section 66 2 -415 yields the highest endurance, since it has the highest (Cl/Cd)max (i.e 150)

4- The NACA 632-415 and 642-415 yield the safest flight, due to its docile stall quality 5- The NACA airfoil section 64 2 -415 delivers the lowest longitudinal control effort in flight, due to the lowest C mo (i.e -0.056)

No NACA C dmin C mo  s (deg)

Table 5.18 A comparison between seven airfoil candidates for the wing in example 5.6

Due to the non-maneuverable nature of the general aviation (GA) aircraft, a sharp stall quality is not feasible, making the NACA 641-412 airfoil unsuitable For optimal safety, it is essential to select the best airfoil design.

When prioritizing maximum endurance, the NACA airfoil section 662-415 excels due to its high (Cl/Cd)max ratio Conversely, if minimizing costs is the primary concern, the NACA 662-415 is preferred for its low C dmin For designs focusing on stall speed, stall quality, and reduced longitudinal control power, the NACA airfoil section 642-415 stands out as the optimal choice A comparison table that incorporates weighted design requirements can effectively illustrate these preferences.

The NACA airfoil section 642-415 stands out as the most suitable choice for this wing, excelling in three key criteria The characteristics graphs of this airfoil are illustrated in Figure 5.62.

The wing setting angle is initially set to correspond with the ideal lift coefficient of the airfoil, which is 0.416 According to Figure 5.62, this results in an initial angle of 2 degrees However, the value of the wing setting angle (i w = 2 deg) may require adjustments based on further calculations to meet the design specifications.

5 Aspect ratio, Taper ratio, and Twist angle

Wing setting angle Ideal lift coefficient

The lift distribution of an aircraft is significantly influenced by three key parameters: aspect ratio, taper ratio, and twist angle To achieve an optimal elliptical lift distribution, various combinations of these parameters can be explored For this analysis, an aspect ratio of 7 (AR = 7) has been chosen based on Table 5.6 To minimize manufacturing costs and simplify construction, a twist angle of zero (αt = 0) is assumed Additionally, a taper ratio of 0.3 (λ = 3) is tentatively proposed.

To assess the lift distribution of the wing, we need to determine whether it is elliptical and if the lift generated at cruise matches the aircraft's weight The lifting line theory will be utilized to analyze the lift distribution and calculate the wing lift coefficient.

Figure 5.63 The lift distribution of the wing (AR = 7,  = 0.3,  t =0, i w =2 deg)

A MATLAB m-file has been created based on example 5.5 to implement the lifting-line theory The output of this m-file, illustrated in Figure 5.63, displays the lift distribution of the wing Additionally, the m-file calculates the lift coefficient.

The results indicate two key observations: first, the lift coefficient exceeds the required value, measuring at 0.4557 compared to the target of 0.356; second, the lift distribution deviates from the ideal elliptical shape Consequently, modifications to certain wing features are necessary to address both issues.

After several trial and errors, the following wing specifications are found to satisfy the design requirements:

By using the same m-file and these new parameters, the following results are obtained:

- Elliptical lift distribution as shown in figure 5.64

Figure 5.64 The lift distribution of the wing (AR = 7, = 0.8,  t =-1.5, i w =1.86 deg)

The wing design meets the aircraft's cruise requirements, and the next step is to develop the flap system and define its parameters to ensure compliance with take-off requirements.