Aerodynamic Drag " Configuration Aerodynamics - 2
 Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2012 ! !  Drag" !  Induced drag" !  Compressibility effects" !  P-51 example" !  Newtonian Flow" !  Moments" !  Effects of Sideslip Angle" 2 ρV S ≈ C D0 + ε C L ρV S 2 + ε C Lo + C Lα α ( ρV S *2 ) ( Drag = C D ≈ %C D0 ' & ( ) ) Copyright 2012 by Robert Stengel All rights reserved For educational use only ! http://www.princeton.edu/~stengel/MAE331.html ! http://www.princeton.edu/~stengel/FlightDynamics.html ! Induced Drag of a Wing " Induced Drag •  Lift produces downwash (angle proportional to lift)" –  Downwash rotates local velocity vector CW in figure" –  Lift is perpendicular to velocity vector" –  Axial component of rotated lift induces drag" Spitfire! Induced Drag of a Wing" C Di = C Li sin α i ≈ (C L0 + C Lα α ) sin α i ≈ (C L0 + C Lα α ) α i ≡ ε C L ≡ C (1+ δ ) CL = L π eAR π AR where e = Oswald efficiency factor = for elliptical distribution δ = departure from ideal elliptical lift distribution Wing Design Parameters" Straight, Swept, and Tapered Wings " •  Straight at the quarter chord" •  Swept at the quarter chord" •  Progression of separated flow from trailing edge with increasing angle of attack" Taper Ratio Effects " •  Planform" –  –  –  –  –  –  Aspect ratio" Sweep" Taper" Complex geometries" Shape at root" Shape at tip" •  Chord section" •  –  Airfoils" –  Twist" •  Movable surfaces" –  Leading- and trailing-edge devices" –  Ailerons" –  Spoilers" •  Interfaces" –  Fuselage" –  Powerplants" –  Dihedral angle" Taper makes lift distribution more elliptical" –  λ ~ 0.45 is best" –  L/D effect (phugoid)" •  •  •  Tip stall (pitch up)" Bending stress" Roll Damping" Secondary Wing Structures " Airfoil Effects " •  Vortex generators, fences, vortilons, notched or dog-toothed wing leading edges" •  Camber increases zero-α lift coefficient" •  Thickness" –  Boundary layer control" –  Maintain attached flow with increasing α! –  Avoid tip stall" –  increases α for stall and softens the stall break" –  reduces subsonic drag " –  increases transonic drag " –  causes abrupt pitching moment variation (more to follow)" McDonnell-Douglas F-4! LTV F-8! •  Profile design " –  can reduce c.p (static margin) variation with α! –  affects leading-edge and trailing-edge flow separation" Sukhoi Su-22! Leading-Edge Extensions " •  Strakes or leading edge extensions" –  Maintain lift at high α! –  Reduce c.p shift at high Mach number" McDonnell Douglas F-18! General Dynamics F-16! Wingtip Design " •  •  •  Winglets, rake, and Hoerner tip reduce induced drag by controlling the tip vortices" End plate, wingtip fence straightens flow, increasing apparent aspect ratio (L/D)" Chamfer produces favorable roll w/ sideslip" Boeing 747-400! Airbus A319! Boeing P-8A! Yankee AA-1! Design for Satisfactory Stalls " •  •  •  •  •  •  Marked by noticeable, uncommanded changes in pitch, yaw, or roll and/or by a marked increase in buffet" Stall must be detectable" Aircraft must pitch down when it occurs" Up to the stall break, ailerons and rudder should operate properly" Inboard stall strips to prevent tip stall and loss of roll control before the stall" Strakes for improved high-α flight" Spanwise Lift Distribution of 3-D (Trapezoidal) Wings " Straight Wings (@ 1/4 chord) " (McCormick) " TR = taper ratio, λ ! •  Spanwise Lift Distribution of 3-D Wings " For some taper ratio between 0.35 and 1, lift distribution is nearly elliptical" Wing Twist Effects " •  Washout twist" C L2− D (y)c(y) C L3− D c Straight and Swept Wings " (NASA SP-367) " •  Wing does not have to have a geometrically elliptical planform to have a nearly elliptical lift distribution" •  Sweep moves lift distribution toward tips" –  reduces tip angle of attack" –  typical value: 2° - 4°" –  changes lift distribution (interplay with taper ratio) " –  reduces likelihood of tip stall; allows stall to begin at the wing root" •  separation burble produces buffet at tail surface, warning of stall" –  improves aileron effectiveness at high α" Clδ A Induced Drag Factor, δ ! C Di = C L (1 + δ ) π AR Oswald Efficiency Factor, e ! C Di = CL π eAR •  Approximation for e (Pamadi, p 390)" •  Graph for δ (McCormick, p 172)" e≈ 1.1C Lα RC Lα + (1 − R)π AR where R = 0.0004κ − 0.008κ + 0.05κ + 0.86 Lower AR! κ= P-51 Mustang " AR λ cos Λ LE P-51 Mustang Example " Wing Span = 37 ft (9.83 m) Wing Area = 235 ft (21.83 m ) Loaded Weight = 9, 200 lb (3, 465 kg) Maximum Power = 1, 720 hp (1, 282 kW ) C Do = 0.0163 AR = 5.83 λ = 0.5 http://en.wikipedia.org/wiki/P-51_Mustang! C Lα = π AR ) # AR & , +1 + + % ( $ ' + * e = 0.947 δ = 0.0557 ε = 0.0576 = 4.49 per rad (wing only) C Di = ε C L = C (1 + δ ) CL = L π eAR π AR http://www.youtube.com/watch?v=WE0sr4vmZtU! Drag Due to Pressure Differential" C Dbase = C pressurebase Sbase Mach Number Effects < # Sbase & % ( γM $ S ' C Dwave ≈ Prandtl factor " ≈ 1− M C Dcompressible M −1 C DM ≈ M −1 Air Compressibility Effect " Lockheed P-38! •  •  Drag rises due to pressure increase across a shock wave" Subsonic flow" Transonic flow" –  Airspeed is less than sonic at some points, greater than sonic elsewhere" •  Supersonic flow" –  Local airspeed is greater than sonic virtually everywhere" 0.029 Sbase S S C friction wet Sbase ( M < 1) ( M < 1) [ Hoerner ] Blunt base pressure drag " γ = specific heat ratio) “The Sonic Barrier”! ( M > 1) ( M > 1) Effect of Chord Thickness on Wing Pressure Drag " Lockheed F-104! •  Thinner chord sections lead to higher Mcrit, or drag-divergence Mach number" –  Local airspeed is less than sonic (i.e., speed of sound) everywhere" •  ≈ ( M > 2, C Dincompressible ≈ Shock Waves in! Supersonic Flow! S •  Critical Mach number" –  Mach number at which local flow first becomes sonic" –  Onset of drag-divergence" –  Mcrit ~ 0.7 to 0.85" Air Compressibility Effect on Wing Drag" Pressure Drag on Wing Depends on Sweep Angle Sonic Booms" http://www.youtube.com/watch?v=gWGLAAYdbbc" Transonic! " Sweep Angle! Effect on Wing Drag! Supersonic! Subsonic! Incompressible! Talay, NASA SP-367! M critswept = Transonic Drag Rise and the Area Rule " •  Richard Whitcomb (NASA Langley) and Wallace Hayes (Princeton) " •  YF-102A (left) could not break the speed of sound in level flight; F-102A (right) could" M critunswept cos Λ Transonic Drag Rise and the Area Rule " •  Cross-sectional area of the total configuration should gradually increase and decrease to minimize transonic drag" Talay, NASA SP-367! Sears-Haack Body! http://en.wikipedia.org/wiki/Sears-Haack_body! Supercritical Wing " NASA Supercritical ! Wing F-8! Airbus A320! Supersonic Biplane " •  Concept of Adolf Busemann (1935)" •  Richard Whitcomb s supercritical airfoil " •  –  Wing upper surface flattened to increase Mcrit" –  Wing thickness can be restored" Shock wave cancellation at one specific Mach number" 2-D wing" •  •  Important for structural efficiency, fuel storage, etc." http://en.wikipedia.org/wiki/Adolf_Busemann! •  Kazuhiro Kusunose et al , Tohoku U (PAS, 47, 2011, 53-87)" (–)" •  •  (+)" •  Adjustable flaps" Tapered, variably spaced 3-D wings" Fuselage added" Pressure Distribution on Supercritical Airfoil ~ Section Lift! Supersonic Transport Concept " Large Angle Variations in Subsonic Drag Coefficient (0° < α < 90°) " •  Rui Hu, Qiqi Wang, Antony Jameson, Stanford, MIT, AIAA-2011-1248" •  •  Optimization of biplane aerodynamics" Sketch of possible configuration" •  •  All wing drag coefficients converge to Newtonian-like values at high angle of attack" Low-AR wing has less drag than high-AR wing at given α! Lift-to-Drag Ratio vs Angle of Attack " Lift vs Drag for Large Variation in Angle-of-Attack (0° < α < 90°)" •  L/D is an important performance metric for aircraft" •  High-AR wing has best overall L/D" •  Low-AR wing has best L/D at intermediate angle of attack" Subsonic Lift-Drag Polar" L CL q S CL = = D CD q S CD •  •  Low-AR wing has less drag than high-AR wing, but less lift as well" High-AR wing has the best overall L/D" € Lift-Drag Polar for a Typical Bizjet " •  Lift-Drag Polar: Cross-plot of CL(α) vs CD(α)" Note different scaling for lift and drag! •  L/D equals slope of line drawn from the origin" –  Single maximum for a given polar" –  Two solutions for lower L/D (high and low airspeed)" Newtonian Flow and High-Angle-of-Attack Lift and Drag Newtonian Flow " Newtonian Flow" •  No circulation" •  Cookie-cutter flow" •  Equal pressure across bottom of the flat plate" Normal Force = ! Mass flow rate $ # & (Change in velocity ) ( Projected Area ) ( Angle between plate and velocity ) " Unit area % " N = ( ρV ) (V ) ( S sin α ) (sin α ) Lift and drag coefficients = ( ρV ) ( S sin α ) Normal Force = ! Mass flow rate $ # & ( Change in velocity ) ( Projected Area ) ( Angle between plate and velocity ) " Unit area % Newtonian Lift and Drag Coefficients " Lift = N cos α #1 & = ( 2sin α ) % ρV ( S $2 ' #1 & ≡ C N % ρV ( S = C N qS $2 ' C L = ( 2sin α ) cos α Drag = N sin α C D = 2sin α Application of Newtonian Flow " •  Hypersonic flow (M ~> 5)" Space Shuttle in! Supersonic Flow! –  Shock wave close to surface (thin shock layer), merging with the boundary layer" –  Flow is ~ parallel to the surface" –  Separated upper surface flow" C L = ( 2sin α ) cos α •  All Mach numbers at high angle of attack" C D = 2sin α –  Separated flow on upper (leeward) surfaces" High-Angle-ofAttack Research Vehicle (F-18)! Airplane Forces and Moments Resolved into Body Axes " Force Vector" Moments of the Airplane ! X # B f B = # YB # " ZB $ & & & % Moment Vector" ! L # B mB = # M B # " NB Incremental Moment Produced By Force Distribution " i k x y z fx r×f = j fy Aerodynamic Force and Moment Vectors of the Airplane " ! f $ ! X # x & # B # fy & dx dy dz = # YB fB = ∫ Surface # & # # & " ZB " fz % $ & & & % fz # % ( yfz − zfy ) m = % ( zfx − xfz ) % % ( xf − yf ) y x $ = ( yfz − zfy ) i + ( zfx − xfz ) j + ( xfy − yfx ) k & # −z y & # fx & ( % ( ( = rf = % z −x ( % f (  % ( y ( % −y x ( % f ( ( '$ z ' $ % ( ' $ & & & % " $ ( yfz − zfy ) $ zf − xf mB = ∫ ( x z) Surface $ $ ( xf − yf ) y x # % " L ' $ B ' dx dy dz =$ M B ' $ NB ' # & % ' ' ' & Tail Design Effects" •  •  Horizontal Tail Location and Size ! •  •  •  •  •  Aerodynamics analogous to those of the wing" Longitudinal stability" –  Horizontal stabilizer" –  Short period natural frequency and damping" •  15-30% of wing area" ~ wing semi-span behind the c.m." Must trim neutrally stable airplane at maximum lift in ground effect" Effect on short period mode" Horizontal Tail Volume: Typical value = 0.48" Lockheed Martin F-35! North American F-86! Directional stability" –  Vertical stabilizer (fin)" •  •  •  •  •  Ventral fins" Strakes" Leading-edge extensions" Multiple surfaces" Butterfly (V) tail" –  Dutch roll natural frequency and damping" •  •  VH = Stall or spin prevention/ recovery" Avoid rudder lock (TBD)! Cmα ,Cmq ,Cmα ,Cnβ ,Cnr ,Cnβ Sht lht S c Vertical Tail Location and Size ! •  Analogous to horizontal tail volume" •  Effect on Dutch roll mode" •  Powerful rudder for spin recovery" –  Full-length rudder located behind the elevator" –  High horizontal tail so as not to block the flow over the rudder" •  Vertical Tail Volume: Typical value = 0.18" Curtiss SB2C! Piper Tomahawk! VV = Svt lvt S b Pitching Moment of the Airplane Pitching Moment " •  Pressure and shear stress differentials times moment arms integrate over the airplane surface to produce a net pitching moment" Center of mass establishes the moment arm center" •  Pitching Moment " •  Distributed effects can be aggregated to local centers of pressure" Body - Axis Pitching Moment = M B =− ∫∫ #Δpz ( x, y ) + Δsz ( x, y )%( x − xcm ) dx dy $ & surface + ∫∫ #Δpx ( y, z ) + Δsx ( y, z )% Δpx ( z − zcm ) dy dz $ & surface I M B ≈ −∑ Z i ( xi − x cm ) i=1 I + ∑ Xi ( zi − zcm ) + Interference Effects + Pure Couples i=1 Pure Couple " •  Net force = 0" •  Net moment ≠ 0" Rockets! Cambered Lifting Surface! Net Center of Pressure" •  Local centers of pressure can be aggregated at a net center of pressure (or neutral point) along the body x axis" xcpnet •  •  •  Cross-sectional area, A! x positive to the right" At small α! –  Positive lift with dA/dx > 0" –  Negative lift with dA/dx < 0" Fuselage! !x C # "( cp n )wing + ( xcpCn ) fuselage + ( xcpCn )tail + $ = C Ntotal Static Margin " Static Margin " •  Static margin reflects the distance between the center of mass and the net center of pressure" •  Body axes" •  Normalized by mean aerodynamic chord" •  Does not reflect z position of c.p.! Static Margin = SM = 100 ( xcm − xcpnet ) B c ,% ≡ 100 ( − hcpnet ) % Static Margin = SM = ( ( 100 xcm − xcpnet c ), % ) ≡ 100 − hcpnet % Pitch-Moment Coefficient Sensitivity to Angle of Attack " Effect of Static Margin on Pitching Moment" •  For small angle of attack and no control deflection" M B = Cm q Sc ≈ $Cmo − C Nα ( − hcpnet ) α & q Sc % ' ≈ $Cmo − C Lα ( − hcpnet ) α & q Sc % ' •  For small angle of attack and no control deflection" ( ) M B = Cm q Sc ≈ Cmo + Cmα α q Sc # ∂C & = %Cmo + m α ( q Sc = (Cmo + Cmα α ) q Sc $ ∂α ' = in trimmed (equilibrium) flight •  Typically, static margin is positive and ∂Cm/∂α is negative for static pitch stability" Pitch-Moment Coefficient Sensitivity to Angle of Attack " •  Cmα For small angle of attack and no control deflection" $x −x ' ≈ −C Nαnet ( − hcpnet ) ≈ −C Lαnet ( − hcpnet ) = −C Lαnet & cm cpnet ) c % ( $ xcm − xcpwing ' $ xcm − xcpht ' $ lwing ' $ lht ' ≈ −C Lαwing & ) − C Lαht & ) = −C Lαwing & ) − C Lαht & ) %c ( c c % c ( % ( % ( referenced to wing area, S! = Cmαwing + Cmαht Horizontal Tail Lift Sensitivity to Angle of Attack " ) , +# ∂C L & = C Lαht % ( +$ ∂α 'horizontail * -aircraft tail ( ) aircraft ( = C Lαht reference Vht : ε: ∂ε ∂α : ηelas : Airspeed at the horizontal tail [Flow over body (±), Scrubbing (–), Propeller slipstream (+)] Downwash angle due to wing lift at the horizontal tail Sensitivity of downwash angle to angle of attack Correction for aeroelastic effect # Static Margin (%) & = −C Lαtotal % ( $ ' 100 •  •  Tail Moment Sensitivity to Angle of Attack " ( Cmαht = − C Lαht ) # Vht & # ∂ε & # S &# l & % ( %1− (ηelas % ht (% ht ( ht V $ S '$ c ' $ N ' $ ∂α ' ( = − C Lαht VHT ) # Vht & # ∂ε & % ( %1− (ηelasVHT ht V $ N ' $ ∂α ' S l = ht ht = Horizontal Tail Volume Ratio Sc ) # ∂ε & # S &# V & %1− (ηelas % ht (% ht ( ht $ $ S '$ VN ' ∂α ' Downwash effect on aft horizontal tail" Upwash effect on a canard (i.e., forward) surface" Effects of Static Margin and Elevator Deflection on Pitching Coefficient" •  Zero crossing determines trim angle of attack, i.e., sum of moments = 0" •  Negative slope required for static stability" •  Slope, ∂Cm/∂α, varies with static margin" •  Control deflection shift curve up and down, affecting trim angle of attack" ∂Cm ∂α ( ) M B = Cmo + Cmα α + Cmδ E δ E q Sc αTrim = − (Cmo + CmδEδ E ) Cmα Subsonic Pitching Coefficient vs Angle of Attack (0° < α < 90°) " Lateral-Directional Effects of Sideslip Angle Rolling and Yawing Moments of the Airplane " Sideslip Angle Produces Side Force, Yawing Moment, and Rolling Moment! Distributed effects can be aggregated to local centers of pressure " I LB ≈ ∑ Z i ( yi − y cm ) Rolling Moment! i=1 I − ∑Yi ( zi − zcm ) + Interference Effects + Pure Couples i=1 I N B ≈ ∑Yi ( xi − x cm ) Yawing Moment! i=1 I − ∑ Xi ( yi − ycm ) + Interference Effects + Pure Couples i=1 !  Sideslip usually a small angle ( ±5 deg)" !  Side force generally not a significant effect" !  Yawing and rolling moments are principal effects" Side Force due to Sideslip Angle! Y≈ ∂CY qS • β = CYβ qS • β ∂β Yawing Moment due to Sideslip Angle ! N≈ % ρV ( ∂ Cn % ρV ( ' * Sb • β = Cnβ ' * Sb • β ∂β & ) & ) •  Fuselage, vertical tail, and wing are main contributors" ( ) CYβ ≈ CYβ (C ) Yβ Vertical Tail (C ) (C ) Yβ Yβ Fuselage Wing Fuselage ( ) + CYβ $ ∂C ' S ≈ & Y ) ηvt Vertical Tail ∂β (vt S % ≈ −2 SBase ; SB = π d Base S ≈ −C DParasite, Wing − kΓ Vertical Tail ( ) + CYβ Wing ηvt = Vertical tail efficiency π AR k= 1+ 1+ AR Γ = Wing dihedral angle, rad Yawing Moment due to Sideslip Angle ! Vertical tail contribution " (C ) nβ Vertical Tail S l ≈ −CYβvt ηvt vt vt  −CYβvt ηvtVVT Sb !  Side force contributions times respective moment arms" –  Non-dimensional stability derivative" ( ) Cnβ ≈ Cnβ Vertical Tail ηvt = ηelas VVT ( ) S l = vt vt = Vertical Tail Volume Ratio Sb ( ) + Cnβ Wing ( ) + Cnβ Fuselage contribution " (C ) lvt  Vertical tail length (+) = distance from center of mass to tail center of pressure = xcm − xcpvt [x is positive forward; both are negative numbers] % Vvt ( ' 2* ∂β & V ) N Fuselage Propeller Yawing Moment due to Sideslip Angle ! nβ 1+ ∂σ ( ) + Cnβ Fuselage = −2K VolumeFuselage Sb 1.3 " % K = $1− dmax Length fuselage ' # & Wing (differential lift and induced drag) contribution " (C ) nβ Wing = 0.75C LN Γ + fcn ( Λ, AR, λ ) C LN Rolling Moment due to Sideslip Angle ! L ≈ Clβ qSb • β ( ) C lβ ≈ C lβ Wing ( ) + C lβ Wing − Fuselage Rolling Moment due to Sideslip Angle ! •  Crossflow effects depend on vertical location of the wing" ( ) + C lβ Vertical Tail •  Dihedral effect" •  Vertical tail effect" Example of Configuration and Flap Effects ! NACA 641-012 Chord Section Lift, Drag, and Moment (NACA TR-824) ! Rough ~ Turbulent! CL, 60° flap! CD! “Drag Bucket”! CL, w/o flap! Cm, w/o flap! Cm, 60° flap! α! Smooth ~ Laminar! CL! CDo Estimate (Raymer) ! Next Time: Aircraft Performance Reading Flight Dynamics, 107-115, 118-130 Virtual Textbook, Parts 6,7 Downwash and Elasticity Also Effect Elevator Sensitivity " Supplemental Material ) , # Vtail & # ∂ε & #S & +# ∂C L & = (C LδE )aircraft = % % ( ( %1− (ηelas % ht ( (C LδE )ht +$ ∂δ E 'horizontail $S ' $ VN ' $ ∂α ' * -aircraft tail reference Pitch Up and Deep Stall " Anatomy of a Cirrus Stall Accident " •  Possibility of stable equilibrium (trim) points with same control setting" –  Low α" –  High α! •  High-angle trim is called deep stall" –  Low lift" –  High drag" •  Large control moment required to regain low-angle trim" http://www.youtube.com/watch?v=7nm_hoHhbFo! http://www.youtube.com/watch?v=IpZ8YukAwwI&feature=related ! Some Videos " !  XF-92A, 1948" http://www.youtube.com/watch?v=hVjaiMXvCTQ! !  First flight of B-58 Hustler, 1956" http://www.youtube.com/watch?v=saeejPWQTHw! !  Century series fighters, bombers, 1959" http://www.youtube.com/watch?v=WmseXJ7DV4c&feature=related! !  Bird of Prey, 1990s, and X-45, 2000s" http://www.youtube.com/watch?v=BMcuVhzCrX8&feature=related! ... flap! Cm, 60° flap! α! Smooth ~ Laminar! CL! CDo Estimate (Raymer) ! Next Time: Aircraft Performance Reading Flight Dynamics, 107-115, 118-130 Virtual Textbook, Parts 6,7 Downwash and Elasticity... '' # & % '' '' '' & Tail Design Effects" •  •  Horizontal Tail Location and Size ! •  •  •  •  •  Aerodynamics analogous to those of the wing" Longitudinal stability" –  Horizontal stabilizer" – ... Lift Sensitivity to Angle of Attack " ) , +# ∂C L & = C Lαht % ( +$ ∂α ''horizontail * -aircraft tail ( ) aircraft ( = C Lαht reference Vht : ε: ∂ε ∂α : ηelas : Airspeed at the horizontal tail