MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS

Final Exam Review and Closing Comments

Mechanical and Aerospace Engineering DepartmentFlorida Institute of Technology

D. R. Kirk

OVERVIEW OF ACCOMPLISHMENTThis book is designed for a 1-year course in aerodynamics. Chapters 1 to 6 constitute a solid semester [bold, italics added for emphasis] emphasizing inviscid, incompressible flow. Chapters 7 to 14 occupy a second semester dealing with inviscid, compressible flow. John D. Anderson, Jr.

What we did:Chapters 1-5Why not Chapter 6? 3-D incompressible flow (sources, doublet, etc.)Chapters 7-9, 11 and 12Why not Chapter 10? Fluids II material (nozzles, diffusers, etc.)Multiple examples of applications to flight and projectile mechanics

What would we do if we had more time:Viscous flowLaminar and turbulent boundary layer models for drag predictionExact solutions, Faulkner-Skan equations and Thwaites method

OUTLINEBasic IdeasCan you convey basic ideas in aerodynamics in simple terms: lift, stall, streamline, Kutta-condition, camber, lifting line, separation, etc.Explain in words or pictures what complicated equations are trying to sayStream and Potential Functions: Inviscid, Incompressible FlowWhat is the point? What is the utility? What is weakness?How do you set-up and use these simple models?Flow Over AirfoilsIncompressible flow: Theory vs. experimentCompressible flow (why so complicated?): Theory vs. experimentSupersonic flow: Why does shape of airfoil want to be so different?Flow Over WingsImpact of wing tips? How do you model, how do you proceed?What are implications for design?Flight MechanicsWhat do (1)-(4) imply about aerodynamic design and performance impacts?

KEY CONCEPTS: CHAPTERS 1 and 2Aerodynamic forces and moments (center of pressure)Where do they come from, why do we care?Mach and Reynolds number matching guarantee flow similarityTypes of flowsInviscid vs. ViscousIncompressible vs. CompressibleMach number regimes

Fundamentals PrinciplesConservation of Mass (integral and control volume form)Conservation of Momentum (integral form)Conservation of Energy (algebraic form)Angular velocity, vorticity and circulation (why do we care about these concepts?)

Stream Function and Velocity Potential (how are these related?)

KET CONCEPTS: CHAPTER 3Elementary Flows (Building Blocks, why such a name?)Uniform FlowSource / Sink FlowDoublet FlowVortex FlowWhat is the purpose? Simulate real shapes in a simple mannerCombine (1) + (2) flow over half-body or ovalCombine (1) + (3) flow over a cylinderCombine with (4) flow over a lifting cylinderKutta-Joukowski TheoremCombinations of sources, vortex, uniform flow, tornados, ground effect, etc.Why can we combine so easily (simply add)?Know how to set up y and f for all cases and combined flows (no time to solve)Know how to get velocity components u and v How would you model some basic shapes using these tools?

Homework #4 has many practice problems (nothing more difficult than these)

KEY CONCEPTS: CHAPTER 4, 11 and 12Model an airfoil as a vortex sheetWhat does this mean, why can we do this, why would we want to do this?

Thin airfoil theory: Mean camber line is a streamline of the flow

Symmetric vs. Cambered AirfoilsS+C: Lift coefficient: 2paS+C: Lift slope: 2pS: Moment Coefficient, c/4 = 0C: Moment Coefficient, c/4 = p/4(A2 - A1)

Role of airfoil thickness (incompressible, subsonic, supersonic)What are added complexities (physics and math) associated with compressibility?How can we correct for compressibility (what are strengths and weaknesses)?Also see key concepts/comments for Chapters 7, 8, and 9Chapter 12: 12.1- 12.3

KEY CONCEPTS: CHAPTER 5Airfoils vs. WingsWhat is different about these situationsWhy should we care? When is it important to care?How do we model a wing? Is it accurate?What is lifting line theoryKey resultsElliptical WingsOther WingsWhy do we taper a wing?Why do we sweep wing?Why do we vary AR (or span) as designersWhy do modern commercial airplane wings (A320, B757, etc.) look way they do?Why do modern fighter wings not look like this?

KEY CONCEPTS: CHAPTER 7, 8, and 9What are isentropic relations?When can we use them?Why would we use them? (replace energy equation, simple, algebraic)When do they break down?

If flow speeds are greater than Mach 1, shock waves are present in the flow (why?)How do flow properties across normal and oblique shock waves change?Is it important to capture these effects?

Expansion processes

Make use of Appendix A, B, and C as well as q-b-M diagram Dont waste time calculating, but know where these appendicies and figures come from (what are equations that generate them)

BASIC CONCEPTSCHAPTERS 1-2

KEY CONCEPTSAerodynamic forces and moments (center of pressure)Where do they come from, why do we care?Mach and Reynolds number matching guarantee flow similarityTypes of flowsInviscid vs. ViscousIncompressible vs. CompressibleMach number regimes

Fundamentals PrinciplesConservation of Mass (integral and control volume form)Conservation of Momentum (integral form)Conservation of Energy (algebraic form)Angular velocity, vorticity and circulation (why do we care about these concepts?)

Stream Function and Velocity Potential (how are these related?)

WHAT DOES EULERS EQUATION TELL US?Eulers Equation (Differential Equation)Relates changes in momentum to changes in force (momentum equation)Relates a change in pressure (dp) to a chance in velocity (dV)Assumptions:Steady flow and no friction (inviscid flow), body forces, and external forces

dp and dV are of opposite signIF dp increases dV goes down flow slows downIF dp decreases dV goes up flow speeds up

Incompressible and Compressible flows, Irrotational and Rotational flows

BERNOULLIS EQUATIONIf flow is irrotational p+1/2rV2 = constant everywhereRemember:Bernoullis equation holds only for inviscid (frictionless) and incompressible (r=constant) flowsRelates properties between different points along a streamline or entire flow field if irrotationalFor a compressible flow Eulers equation must be used (r is a variable)Both Eulers and Bernoullis equations are expressions of F=ma expressed in a useful form for fluid flows and aerodynamicsConstant along a streamline

WHAT CREATES AERODYNAMIC FORCES?Aerodynamic forces exerted by airflow comes from only two sourcesPressure, p, distribution on surfaceActs normal to surface

Shear stress, tw, (friction) on surfaceActs 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

SOME DEFINITIONSRelative Wind: Direction of VWe used subscript to indicate far upstream conditionsAngle of Attack, a: Angle between relative wind (V) and chord line

Total aerodynamic force, R, can be resolved into two force componentsLift, L: Component of aerodynamic force perpendicular to relative windDrag, D: Component of aerodynamic force parallel to relative wind

Center of Pressure: It is that point on an airfoil (or body) about which the aerodynamic moment is zero

Aerodynamic Center: It is that point on an airfoil (or body) about which the aerodynamically generated moment is independent of angle of attack

SAMPLE DATA TRENDSLift coefficient (or lift) linear variation with angle of attack, aCambered airfoils have positive lift when a=0Symmetric airfoils have zero lift when a=0At high enough angle of attack, the performance of the airfoil rapidly degrades stall

clCambered airfoil haslift at a=0At negative a airfoilwill have zero lift

AIRFOIL DATA (APPENDIX D)NACA 23012 WING SECTIONclcm,c/4Re dependenceat high aaclcdcm,a.c.cl vs. aIndependent of Recd vs. clDependent on Recm,a.c. vs. cl very flat

HOW DOES AN AIRFOIL GENERATE LIFT?Flow velocity over the top of airfoil is faster than over bottom surfaceStreamtube A senses upper portion of airfoil as an obstructionStreamtube A is squashed to smaller cross-sectional areaMass continuity rAV=constant, velocity must increaseStreamtube A is squashedmost in nose region(ahead of maximum thickness) AB

HOW DOES AN AIRFOIL GENERATE LIFT?As velocity increases pressure decreasesIncompressible: Bernoullis EquationCompressible: Eulers EquationCalled Bernoulli EffectWith lower pressure over upper surface and higher pressure over bottom surface, airfoil feels a net force in upward direction LiftMost of lift is producedin first 20-30% of wing(just downstream of leading edge)

WHY DOES AN AIRFOIL STALL?Key to understandingFriction causes flow separation within boundary layerSeparation then creates another form of drag called pressure drag due to separation

STALL CHARACTER: NACA 4412 VERSUS NACA 4421Both NACA 4412 and NACA 4421 have same shape of mean camber lineThin airfoil theory predict that linear lift slope and aL=0 should be the same for both

Leading edge stall shows rapid drop of lift curve near maximum liftTrailing edge stall shows gradual bending-over of lift curve at maximum lift, soft stallHigh cl,max for airfoils with leading edge stall

Flat plate stall exhibits poorest behavior, early stalling

Thickness has major effect on cl,max

INVISCID, INCOMPRESSIBLE FLOWCHAPTER 3

KET CONCEPTSElementary Flows (Building Blocks, why such a name?)Uniform FlowSource / Sink FlowDoublet FlowVortex FlowWhat is the purpose? Simulate real shapes in a simple mannerCombine (1) + (2) flow over half-body or ovalCombine (1) + (3) flow over a cylinderCombine with (4) flow over a lifting cylinderKutta-Joukowski TheoremCombinations of sources, vortex, uniform flow, tornados, ground effect, etc.Why can we combine so easily (simply add)?Know how to set up y and f for all cases and combined flows (no time to solve)Know how to get velocity components u and v How would you model some basic shapes using these tools?

Homework #4 has many practice problems (nothing more difficult than these)

SUMMARY OF STREAM AND POTENTIAL FUNCTIONSTABLE 3.1

LIFTING FLOW OVER A CYLINDERKutta-Joukowski Theorem

FLOW OVER AIRFOILSINCOMPRESSIBLE: CHAPTER 4COMPRESSIBLE: CHAPTER 11

KEY CONCEPTSModel an airfoil as a vortex sheetWhat does this mean, why can we do this, why would we want to do this?

Thin airfoil theory: Mean camber line is a streamline of the flow

Symmetric vs. Cambered AirfoilsS+C: Lift coefficient: 2paS+C: Lift slope: 2pS: Moment Coefficient, c/4 = 0C: Moment Coefficient, c/4 = p/4(A2 - A1)

Role of thickness

CENTER OF PRESSURE AND AERODYNAMIC CENTERCenter of Pressure: It is that point on an airfoil (or body) about which the aerodynamic moment is zeroThin Airfoil Theory:Symmetric Airfoil:Cambered Airfoil:

Aerodynamic Center: It is that point on an airfoil (or body) about which the aerodynamically generated moment is independent of angle of attackThin Airfoil Theory:Symmetric Airfoil:Cambered Airfoil:

PREVIEW: COMPRESSIBILITY CORRECTIONEFFECT OF M ON CPFor M < 0.3, r ~ constCp = Cp,0 = 0.5 = constEffect of compressibility(M > 0.3) is to increaseabsolute magnitude of Cp and M increasesCalled: Prandtl-Glauert RulePrandtl-Glauert rule applies for 0.3 < M < 0.7 (Why not M = 0.99?)SoundBarrier ?

RESULTVelocity Potential Equation: Nonlinear EquationCompressible, Steady, Inviscid and Irrotational FlowsNote: This is one equation, with one unknown, fa0 (as well as T0, P0, r0, h0) are known constants of the flowVelocity Potential Equation: Linear EquationIncompressible, Steady, Inviscid and Irrotational Flows

RESULTAfter order of magnitude analysis, we have following results

May also be written in terms of perturbation velocity potential

Equation is a linear PDE and is rather easy to solve (see slides 19-22 for technique)

Recall:Equation is no longer exactValid for small perturbationsSlender bodiesSmall angles of attackSubsonic and Supersonic Mach numbersKeeping in mind these assumptions equation is good approximation

CRITICAL MACH NUMBER, MCRAs air expands around top surface near leading edge, velocity and M will increaseLocal M > M

Flow over airfoil may havesonic regions even thoughfreestream M < 1

DESIGN OPTIONS: SWEEP, AERA RULE, SUPERCRITICAL AIRFOILSSweep:Makes airfoil thinner increases critical Mach numberSweeping wing usually reduces lift for subsonic flight

Area Rule: Drag created related to change in cross-sectional area of vehicle from nose to tail

Supercritical Airfoils: Designed to delay and reduce transonic drag rise, due to both strong normal shock and shock-induced boundary layer separation

FLOW OVER WINGSCHAPTER 5

KEY CONCEPTSAirfoils vs. WingsWhat is different about these situationsWhy should we care? When is it important to care?How do we model a wing? Is it accurate?What is lifting line theoryKey resultsElliptical WingsOther WingsWhy do we taper a wing?Why do we vary AR (or span) as designersWhy do modern commercial airplane wings (A320, B757, etc.) look the way they do?Why do modern fighter wings not look like this?

PHYSICAL INTERPRETATIONFinite Wing Consequences:Tilted lift vector contributes a drag component, called induced drag (drag due to lift) CL < cl and CD > cdLift slope is reduced relative to infinite wing (a < a0)Chord linea: Geometric Angle of Attackai: Induced Angle of Attackaeff: Effective Angle of Attack

PRANDTLS LIFTING LINE EQUATIONFundamental Equation of Prandtls Lifting Line TheoryIn Words: Geometric angle of attack is equal to sum of effective angle of attack plus induced angle of attackMathematically: a = aeff + ai

Only unknown is G(y)V, c, a, aL=0 are known for a finite wing of given design at a given aSolution gives G(y0), where b/2 y0 b/2 along span

KEY RESULTTrue for all finite wings in generalDefine a span efficiency factor, e (also called span efficiency factor)Elliptical planforms, e = 1For all other planforms, e < 10.60 < e < 0.99

Span Efficiency FactorKey Points:Goes with square of CLInversely related to AR

Also called drag due to liftFor Elliptical PlanformsArbitrary Finite Wing

SUMMARY: TOTAL DRAG ON SUBSONIC WINGAlso called drag due to liftProfile DragProfile Drag coefficient relatively constant with M at subsonic speedsLook up(Infinite Wing)May be calculated fromInviscid theory:Lifting line theory

IMPORTANT STATEMENTSFundamental Equation of Thin Airfoil TheoryThe camber line is a streamline of the flowFundamental Equation of Prandtls Lifting-Line TheoryThe geometric angle of attack is equal to the sum of the effectiveangle of attack plus the induced angle of attack

GENERAL LIFT DISTRIBUTION (2/4)Substitute expression for G(q) and dG/dy into fundamental equation of Prandtls lifting line theoryLast term on the right (integral term) is a standard form and may be simplified as:Equation is evaluated at a given spanwise location (q0), just as fundamental equation of Prandtls lifting line theory is evaluated at a given spanwise location (y0)Only unknowns in equation are AnsWritten at q0 equation is 1 algebraic equation with N unknownsWrite equation at N spanwise locations to obtain a system of N independent algebraic equations with N unknowns

SUPERSONIC AIRFOILS AND WINGSREVIEW: CHAPTER 7SHOCK WAVES / EXPANSIONS: CHAPTERS 8 AND 9

KEY CONCEPTSWhat are isentropic relations?When can we use them?Why would we use them? (replace energy equation, simple, algebraic)When do they break down?

If flow speeds are greater than Mach 1, shock waves are present in the flow (why?)How do flow properties across normal and oblique shock waves change?Is it important to capture these effects?

Expansion processes

Make use of Appendix A, B, and C as well as q-b-M diagram Dont waste time calculating, but know where these appendicies and figures come from (what are equations that generate them)

SUMMARY OF NORMAL SHOCK RELATIONS

Sheet1

g1.4

M1M2rho2/rho1u1/u2p2/p1T2/T1P02/P01

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0.38776822515.7087661392114.178333333320.00052735550.0031910869

0.3875752735.7142857143116.520.38750.0030447526

Downstream Mach Number, M2

Density Ratio, Rho1/Rho2

Static Pressure Ratio, P2/P1

Static Temperature Ratio T2/T1

Total Pressure Ratio, P02/P01

Upstream Mach Number, M1

M2, P02/P01

r2/r1, p2/p1, T2/T1

Chart1 (2)

11111

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0.3875752735.7142857143116.520.38750.0030447526

Downstream Mach Number, M2

Total Pressure Ratio, P02/P01

Density Ratio, Rho2/Rho1

Static Pressure Ratio, P2/P1

Static Temperature Ratio, T2/T1

Upstream Mach Number, M1

M2, P02/P01

r2/r1, p2/p1, T2/T1

Sheet2

Sheet3

MEASUREMENT OF AIRSPEED:SUPERSONIC FLOW (M > 1)Rayleigh Pitot Tube Formula

SUMMARY OF SHOCK RELATIONSNormal ShocksOblique Shocks

q-b-M RELATIONDeflection Angle, qShock Wave Angle, bWeakStrongM2 > 1M2 < 1Detached, Curved Shock

SWEPT WINGS: SUPERSONIC FLIGHTIf leading edge of swept wing is outside Mach cone, component of Mach number normal to leading edge is supersonic Large Wave DragIf leading edge of swept wing is inside Mach cone, component of Mach number normal to leading edge is subsonic Reduced Wave DragFor supersonic flight, swept wings reduce wave drag

EXAMPLE OF SUPERSONIC AIRFOILShttp://odin.prohosting.com/~evgenik1/wing.htm

FLIGHT MECHANICS

WING LOADING (W/S), SPAN LOADING (W/b) AND ASPECT RATIO (b2/S)Span loading (W/b), wing loading (W/S)and AR (b2/S) are related

Zero-lift drag, D0 is proportional to wing area

Induced drag, Di, is proportional to squareof span loading

Take ratio of these drags, Di/D0

Re-write W2/(b2S) in terms of AR and substitute into drag ratio Di/D0

1: For specified W/S (set by take-off or landing requirements) and CD,0 (airfoil choice), increasing AR will decrease drag due to lift relative to zero-lift drag2: AR predominately controls ratio of induced drag to zero lift drag, whereas span loading controls actual value of induced drag

FURTHER IMPLICATIONS FOR DESIGN: VMAXMaximum velocity at a given altitude is important specification for new airplaneTo design airplane for given Vmax, what are most important design parameters?Steady, level flight: T = D

Steady, level flight: L = W

Substitute into drag equation

Turn this equation into a quadraticequation (by multiplying by q)and rearranging

Solve quadratic equation and setthrust, T, to maximum availablethrust, TA,max

FURTHER IMPLICATIONS FOR DESIGN: VMAXTA,max does not appear alone, but only in ratio (TA/W)maxS does not appear alone, but only in ratio (W/S)Vmax does not depend on thrust alone or weight alone, but rather on ratios(TA/W)max: maximum thrust-to-weight ratioW/S: wing loadingVmax also depends on density (altitude), CD,0, peARWe can increase Vmax byIncrease maximum thrust-to-weight ratio, (TA/W)maxIncreasing wing loading, (W/S)Decreasing zero-lift drag coefficient, CD,0

THRUST REQUIRED VS. FLIGHT VELOCITY Zero-Lift TR(Parasitic Drag)Lift-Induced TR(Induced Drag)Zero-Lift TR ~ V2(Parasitic Drag)Lift-Induced TR ~ 1/V2(Induced Drag)

THRUST REQUIRED VS. FLIGHT VELOCITY At point of minimum TR, dTR/dV=0(or dTR/dq=0)Zero-Lift Drag = Induced DragAt minimum TR and maximum L/D

POWER REQUIREDZero-Lift PRLift-Induced PRZero-Lift PR ~ V3Lift-Induced PR ~ 1/V

POWER REQUIREDAt point of minimum PR, PTR/dV=0