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**MAE 1202: AEROSPACE PRACTICUM**

Lecture 7: Compressible Flow Review and Overview of Airfoils March 11, 2013 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

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**UDPATES Mid-term grades**

Team project: Introduced in Laboratory this week Mid-Term Exam: Monday, March 18, 2013 in class Covers Chapter 4 and Chapter 5.1 – 5.7 Open book / open notes… but no time to study during the exam No computers, cell phones, etc. Sample Mid-Term with Solution on line Review Session: Thursday, March 14, 2013, Crawford Science Tower, Room 112, 8 – 10 pm AIAA Meeting/Fund Raiser Friday, March 15, 2013, 7:00 pm – 11:00 pm, Buffalo Wild Wings (Palm Bay Road)

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**READING AND HOMEWORK ASSIGNMENTS**

Reading: Introduction to Flight, by John D. Anderson, Jr. For March 25, 2013 lecture: Chapter 5, Sections Read these sections carefully, most interesting portions of Ch. 5 Lecture-Based Homework Assignment: Problems: 5.7, 5.11, 5.13, 5.15, 5.17, 5.19 DUE: Friday, March 29, 2013 by 5pm Turn in hard copy of homework Also be sure to review and be familiar with textbook examples in Chapter 5

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**ANSWERS TO LECTURE HOMEWORK**

5.7: Cp = -3.91 5.11: Cp = Be careful here, if you check the Mach number it is around 0.71, so the flow is compressible and the formula for Cp based on Bernoulli’s equation is not valid. To calculate the pressure coefficient, first calculate r∞ from the equation of state and find the temperature from the energy equation. Finally make use of the isentropic relations and the definition of Cp given in Equation 5.27 5.13: cl = 0.97 Make use of Prandtl-Glauert rule 5.15: Mcr = 0.62 Use graphical technique of Section 5.9 Verify using Excel or Matlab 5.17: m = 30° 5.19: D = 366 lb Remember that in steady, level flight the airplane’s lift must balance its weight You may also assume that all lift is derived from the wings (this is not really true because the fuselage and horizontal tail also contribute to the airplane lift). Also assume that the wings can be approximated by a thin flat plate Remember that Equation 5.50 gives a in radians

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HOMEWORK EXAMPLES Can you read this? Who would you hire?

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MODERN ART?

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**Compressible Flow Applications**

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**SUMMARY OF GOVERNING EQUATIONS (4.8) STEADY AND INVISCID FLOW**

Incompressible flow of fluid along a streamline or in a stream tube of varying area Most important variables: p and V T and r are constants throughout flow continuity Bernoulli continuity Compressible, isentropic (adiabatic and frictionless) flow along a streamline or in a stream tube of varying area T, p, r, and V are all variables isentropic energy equation of state at any point

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**MEASUREMENT OF AIRSPEED: SUBSONIC COMRESSIBLE FLOW**

If M > 0.3, flow is compressible (density changes are important) Need to introduce energy equation and isentropic relations cp: specific heat at constant pressure M1=V1/a1 gair=1.4

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**EXAMPLE: TOTAL TEMPERATURE**

Static temperature Vehicle flight Mach number A rocket is flying at Mach 6 through a portion of the atmosphere where the static temperature is 200 K What temperature does the nose of the rocket ‘feel’? T0 = 200(1+ 0.2(36)) = 1,640 K!

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**MEASUREMENT OF AIRSPEED: SUBSONIC COMRESSIBLE FLOW**

So, how do we use these results to measure airspeed p0 and p1 give Flight Mach number Mach meter M1=V1/a1 Actual Flight Speed Actual Flight Speed using pressure difference What is T1 and a1? Again use sea-level conditions Ts, as, ps (a1=340.3 m/s)

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**MEASUREMENT OF AIRSPEED: SUPERSONIC FLOW**

What can happen in supersonic flows? Supersonic flows (M > 1) are qualitatively and quantitatively different from subsonic flows (M < 1)

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**HOW AND WHY DOES A SHOCK WAVE FORM?**

Think of a as ‘information speed’ and M=V/a as ratio of flow speed to information speed If M < 1 information available throughout flow field If M > 1 information confined to some region of flow field

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**MEASUREMENT OF AIRSPEED: SUPERSONIC FLOW**

Notice how different this expression is from previous expressions You will learn a lot more about shock wave in compressible flow course

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**SUMMARY OF AIR SPEED MEASUREMENT**

Subsonic, incompressible Subsonic, compressible Supersonic

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**HOW ARE ROCKET NOZZLES SHAPPED?**

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**MORE ON SUPERSONIC FLOWS (4.13)**

Isentropic flow in a streamtube Differentiate Euler’s Equation Since flow is isentropic a2=dp/dr Area-Velocity Relation

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**CONSEQUENCES OF AREA-VELOCITY RELATION**

IF Flow is Subsonic (M < 1) For V to increase (dV positive) area must decrease (dA negative) Note that this is consistent with Euler’s equation for dV and dp IF Flow is Supersonic (M > 1) For V to increase (dV positive) area must increase (dA positive) IF Flow is Sonic (M = 1) M = 1 occurs at a minimum area of cross-section Minimum area is called a throat (dA/A = 0)

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**2: OUTLET 1: INLET TRENDS: CONTRACTION M1 < 1 M1 > 1 V2 > V1**

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**2: OUTLET 1: INLET TRENDS: EXPANSION M1 < 1 M1 > 1 V2 < V1**

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**PUT IT TOGETHER: C-D NOZZLE**

1: INLET 2: OUTLET

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**MORE ON SUPERSONIC FLOWS (4.13)**

A converging-diverging, with a minimum area throat, is necessary to produce a supersonic flow from rest Supersonic wind tunnel section Rocket nozzle

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Chapter 5 Overview

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**HOW DOES AN AIRFOIL GENERATE LIFT?**

Lift due to imbalance of pressure distribution over top and bottom surfaces of airfoil (or wing) If pressure on top is lower than pressure on bottom surface, lift is generated Why is pressure lower on top surface? We can understand answer from basic physics: Continuity (Mass Conservation) Newton’s 2nd law (Euler or Bernoulli Equation) Lift = PA

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**HOW DOES AN AIRFOIL GENERATE LIFT?**

Flow velocity over top of airfoil is faster than over bottom surface Streamtube A senses upper portion of airfoil as an obstruction Streamtube A is squashed to smaller cross-sectional area Mass continuity rAV=constant: IF A↓ THEN V↑ Streamtube A is squashed most in nose region (ahead of maximum thickness) A B

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**HOW DOES AN AIRFOIL GENERATE LIFT?**

As V ↑ p↓ Incompressible: Bernoulli’s Equation Compressible: Euler’s Equation Called Bernoulli Effect With lower pressure over upper surface and higher pressure over bottom surface, airfoil feels a net force in upward direction → Lift Most of lift is produced in first 20-30% of wing (just downstream of leading edge) Can you express these ideas in your own words?

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**AIRFOILS VERSUS WINGS Why do airfoils have such a shape?**

How are lift and drag produced? NACA airfoil performance data How do we design? What is limit of behavior?

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**AIRFOIL THICKNESS: WWI AIRPLANES**

English Sopwith Camel Thin wing, lower maximum CL Bracing wires required – high drag German Fokker Dr-1 Higher maximum CL Internal wing structure Higher rates of climb Improved maneuverability

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AIRFOIL NOMENCLATURE Mean Chamber Line: Set of points halfway between upper and lower surfaces Measured perpendicular to mean chamber line itself Leading Edge: Most forward point of mean chamber line Trailing Edge: Most reward point of mean chamber line Chord Line: Straight line connecting the leading and trailing edges Chord, c: Distance along the chord line from leading to trailing edge Chamber: Maximum distance between mean chamber line and chord line Measured perpendicular to chord line

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**NACA FOUR-DIGIT SERIES**

First digit specifies maximum camber in percentage of chord Second digit indicates position of maximum camber in tenths of chord Last two digits provide maximum thickness of airfoil in percentage of chord Example: NACA 2415 Airfoil has maximum thickness of 15% of chord (0.15c) Camber of 2% (0.02c) located 40% back from airfoil leading edge (0.4c) NACA 2415

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**WHAT CREATES AERODYNAMIC FORCES? (2.2)**

Aerodynamic forces exerted by airflow comes from only two sources: Pressure, p, distribution on surface Acts normal to surface Shear stress, tw, (friction) on surface Acts tangentially to surface Pressure and shear are in units of force per unit area (N/m2) Net unbalance creates an aerodynamic force “No matter how complex the flow field, and no matter how complex the shape of the body, the only way nature has of communicating an aerodynamic force to a solid object or surface is through the pressure and shear stress distributions that exist on the surface.” “The pressure and shear stress distributions are the two hands of nature that reach out and grab the body, exerting a force on the body – the aerodynamic force”

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**RESOLVING THE AERODYNAMIC FORCE**

Relative Wind: Direction of V∞ We use subscript ∞ to indicate far upstream conditions Angle of Attack, a: Angle between relative wind (V∞) and chord line Total aerodynamic force, R, can be resolved into two force components Lift, L: Component of aerodynamic force perpendicular to relative wind Drag, D: Component of aerodynamic force parallel to relative wind

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MORE DEFINITIONS Total aerodynamic force on airfoil is summation of F1 and F2 Lift is obtained when F2 > F1 Misalignment of F1 and F2 creates Moments, M, which tend to rotate airfoil/wing A moment (torque) is a force times a distance Value of induced moment depends on point about which moments are taken Moments about leading edge, MLE, or quarter-chord point, c/4, Mc/4 In general MLE ≠ Mc/4 F1 F2

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**VARIATION OF L, D, AND M WITH a**

Lift, Drag, and Moments on a airfoil or wing will change as a changes Variations of these quantities are some of most important information that an airplane designer needs to know Aerodynamic Center Point about which moments essentially do not vary with a Mac=constant (independent of a) For low speed airfoils aerodynamic center is near quarter-chord point, c/4

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AOA = 2°

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AOA = 3°

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AOA = 6°

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AOA = 9°

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AOA = 12°

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AOA = 20°

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AOA = 60°

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AOA = 90°

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**SAMPLE DATA: SYMMETRIC AIRFOIL**

Lift (for now) Angle of Attack, a A symmetric airfoil generates zero lift at zero a

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**SAMPLE DATA: CAMBERED AIRFOIL**

Lift (for now) Angle of Attack, a A cambered airfoil generates positive lift at zero a

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**SAMPLE DATA Lift (for now) Cambered airfoil has lift at a=0**

Lift coefficient (or lift) linear variation with angle of attack, a Cambered airfoils have positive lift when a = 0 Symmetric airfoils have zero lift when a = 0 At high enough angle of attack, the performance of the airfoil rapidly degrades → stall Lift (for now) Cambered airfoil has lift at a=0 At negative a airfoil will have zero lift

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**SAMPLE DATA: STALL BEHAVIOR**

Lift (for now) What is really going on here What is stall? Can we predict it? Can we design for it?

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**REAL EFFECTS: VISCOSITY (m)**

To understand drag and actual airfoil/wing behavior we need an understanding of viscous flows (all real flows have friction) Inviscid (frictionless) flow around a body will result in zero drag! This is called d’Alembert’s paradox Must include friction (viscosity, m) in theory Flow adheres to surface because of friction between gas and solid boundary At surface flow velocity is zero, called ‘No-Slip Condition’ Thin region of retarded flow in vicinity of surface, called a ‘Boundary Layer’ At outer edge of B.L., V∞ At solid boundary, V=0 “The presence of friction in the flow causes a shear stress at the surface of a body, which, in turn contributes to the aerodynamic drag of the body: skin friction drag” p.219, Section 4.20

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**TYPES OF FLOWS: FRICTION VS. NO-FRICTION**

Flow very close to surface of airfoil is Influenced by friction and is viscous (boundary layer flow) Stall (separation) is a viscous phenomena Flow away from airfoil is not influenced by friction and is wholly inviscid

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**COMMENTS ON VISCOUS FLOWS (4.15)**

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**THE REYNOLDS NUMBER, Re Within B.L. flow Outside B.L. flow**

One of most important dimensionless numbers in fluid mechanics/ aerodynamics Reynolds number is ratio of two forces: Inertial Forces Viscous Forces c is length scale (chord) Reynolds number tells you when viscous forces are important and when viscosity may be neglected Outside B.L. flow Inviscid (high Re) Within B.L. flow highly viscous (low Re)

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**LAMINAR VS. TURBULENT FLOW**

Two types of viscous flows Laminar: streamlines are smooth and regular and a fluid element moves smoothly along a streamline Turbulent: streamlines break up and fluid elements move in a random, irregular, and chaotic fashion

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**LAMINAR VS. TURBULENT FLOW**

All B.L.’s transition from laminar to turbulent Turbulent velocity profiles are ‘fuller’ cf,turb > cf,lam

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FLOW SEPARATION Key to understanding: Friction causes flow separation within boundary layer Separation then creates another form of drag called pressure drag due to separation

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**REVIEW: AIRFOIL STALL (4.20, 5.4)**

Key to understanding: Friction causes flow separation within boundary layer B.L. either laminar or turbulent All laminar B.L. → turbulent B.L. Turbulent B.L. ‘fuller’ than laminar B.L., more resistant to separation Separation creates another form of drag called pressure drag due to separation Dramatic loss of lift and increase in drag

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**SUMMARY OF VISCOUS EFFECTS ON DRAG (4.21)**

Friction has two effects: Skin friction due to shear stress at wall Pressure drag due to flow separation Total drag due to viscous effects Called Profile Drag Drag due to skin friction Drag due to separation = + Less for laminar More for turbulent More for laminar Less for turbulent So how do you design? Depends on case by case basis, no definitive answer!

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**COMPARISON OF DRAG FORCES**

Same total drag as airfoil

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**TRUCK SPOILER EXAMPLE Note ‘messy’ or turbulent flow pattern High drag**

Lower fuel efficiency Spoiler angle increased by + 5° Flow behavior more closely resembles a laminar flow Tremendous savings (< $10,000/yr) on Miami-NYC route

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**LIFT, DRAG, AND MOMENT COEFFICIENTS (5.3)**

Behavior of L, D, and M depend on a, but also on velocity and altitude V∞, r ∞, Wing Area (S), Wing Shape, m ∞, compressibility Characterize behavior of L, D, M with coefficients (cl, cd, cm) Matching Mach and Reynolds (called similarity parameters) M∞, Re M∞, Re cl, cd, cm identical

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**LIFT, DRAG, AND MOMENT COEFFICIENTS (5.3)**

Behavior of L, D, and M depend on a, but also on velocity and altitude V∞, r ∞, Wing Area (S), Wing Shape, m ∞, compressibility Characterize behavior of L, D, M with coefficients (cl, cd, cm) Note on Notation: We use lower case, cl, cd, and cm for infinite wings (airfoils) We use upper case, CL, CD, and CM for finite wings

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**SAMPLE DATA: NACA 23012 AIRFOIL**

Flow separation Stall Lift Coefficient cl Moment Coefficient cm, c/4 a

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**AIRFOIL DATA (5.4 AND APPENDIX D) NACA 23012 WING SECTION**

Re dependence at high a Separation and Stall cd vs. a Dependent on Re cl cl vs. a Independent of Re cd R=Re cm,a.c. vs. cl very flat cm,a.c. cm,c/4 a cl

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**EXAMPLE: BOEING 727 FLAPS/SLATS**

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**EXAMPLE: SLATS AND FLAPS**

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**AIRFOIL DATA (5.4 AND APPENDIX D) NACA 1408 WING SECTION**

Flaps shift lift curve Effective increase in camber of airfoil Flap extended Flap retracted

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**PRESSURE DISTRIBUTION AND LIFT**

Lift comes from pressure distribution over top (suction surface) and bottom (pressure surface) Lift coefficient also result of pressure distribution

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**PRESSURE COEFFICIENT, CP (5.6)**

Use non-dimensional description, instead of plotting actual values of pressure Pressure distribution in aerodynamic literature often given as Cp So why do we care? Distribution of Cp leads to value of cl Easy to get pressure data in wind tunnels Shows effect of M∞ on cl

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**EXAMPLE: CP CALCULATION**

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**COMPRESSIBILITY CORRECTION: EFFECT OF M∞ ON CP**

For M∞ < 0.3, r ~ const Cp = Cp,0 = 0.5 = const M∞

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**COMPRESSIBILITY CORRECTION: EFFECT OF M∞ ON CP**

Effect of compressibility (M∞ > 0.3) is to increase absolute magnitude of Cp as M∞ increases Called: Prandtl-Glauert Rule For M∞ < 0.3, r ~ const Cp = Cp,0 = 0.5 = const M∞ Prandtl-Glauert rule applies for 0.3 < M∞ < 0.7

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**OBTAINING LIFT COEFFICIENT FROM CP (5.7)**

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**COMPRESSIBILITY CORRECTION SUMMARY**

If M0 > 0.3, use a compressibility correction for Cp, and cl Compressibility corrections gets poor above M0 ~ 0.7 This is because shock waves may start to form over parts of airfoil Many proposed correction methods, but a very good on is: Prandtl-Glauert Rule Cp,0 and cl,0 are the low-speed (uncorrected) pressure and lift coefficients This is lift coefficient from Appendix D in Anderson Cp and cl are the actual pressure and lift coefficients at M∞

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**CRITICAL MACH NUMBER, MCR (5.9)**

As air expands around top surface near leading edge, velocity and M will increase Local M > M∞ Flow over airfoil may have sonic regions even though freestream M∞ < 1 INCREASED DRAG!

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**CRITICAL FLOW AND SHOCK WAVES**

MCR

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**CRITICAL FLOW AND SHOCK WAVES**

‘bubble’ of supersonic flow

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**AIRFOIL THICKNESS SUMMARY**

Note: thickness is relative to chord in all cases Ex. NACA 0012 → 12 % Which creates most lift? Thicker airfoil Which has higher critical Mach number? Thinner airfoil Which is better? Application dependent!

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**AIRFOIL THICKNESS: WWI AIRPLANES**

English Sopwith Camel Thin wing, lower maximum CL Bracing wires required – high drag German Fokker Dr-1 Higher maximum CL Internal wing structure Higher rates of climb Improved maneuverability

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**THICKNESS-TO-CHORD RATIO TRENDS**

Root: NACA 6716 TIP: NACA 6713 F-15 Root: NACA 64A(.055)5.9 TIP: NACA 64A203

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**MODERN AIRFOIL SHAPES Boeing 737 Root Mid-Span Tip**

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**SUMMARY OF AIRFOIL DRAG (5.12)**

Only at transonic and supersonic speeds Dwave=0 for subsonic speeds below Mdrag-divergence Profile Drag Profile Drag coefficient relatively constant with M∞ at subsonic speeds

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Chapter 12 Compressible Flow

Chapter 12 Compressible Flow

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