Chapter 4: Thin airfoils and finite wings Nomenclature for airfoils and wings
Aerodynamics 2 Chapter 4: Thin airfoils and finite wings Airfoils
NACA airfoils: developed by the National Advisory Committee for Aeronautics (NACA) starting from the late 1920s.
Four-digit series: First digit: maximum camber as % of chord. Second digit: distance of maximum camber from the airfoil leading edge in tenths of the chord. Last two digits: maximum thickness of the airfoil as % of chord. NACA0012 NACA2412
(careful about the “American convention” on airfoil thickness)
Aerodynamics 3 Chapter 4: Thin airfoils and finite wings Airfoils
NACA airfoils: after the four-digit series, NACA developed the five-, six-, seven- and eight-digit series, typically aiming to maximize the extent of laminar flow above and below the wing. DLR, ONERA, TsAGI … and many more! An extensive database (almost 1600 entries!) of airfoil designs is available on the dedicated website of the Aerospace Engineering Department of the University of Illinois at Urbana-Champaign: https://m-selig.ae.illinois.edu/ads.html
Aerodynamics 4 Chapter 4: Thin airfoils and finite wings Wing configuration: monoplane
low wing mid wing shoulder wing
high wing parasol wing (by the use of struts or pylon)
Aerodynamics 5 Chapter 4: Thin airfoils and finite wings Wing configuration: biplane
two wing planes of similar unequal span biplane size (ex. Wright Flyer I) (ex. Curtiss JN-4 Jenny)
sesquiplane inverted sesquiplane (ex. Nieuport 17) (ex. Fiat C.R.1)
Aerodynamics 6 Chapter 4: Thin airfoils and finite wings Wing configuration: biplane
On May 5th, 1925, Mario de Bernardi, piloting a FIAT C.R.1, achieved the world record speed over a 500 km distance, flying at an average speed of 254 km/h.
Aerodynamics 7 Chapter 4: Thin airfoils and finite wings Wing configuration: multiplane
triplane quadriplane (ex. Fokker DR.I) (ex. Armstrong Whitworth F.K.10)
multiplane (ex. Caproni CA.60 Transaereo)
Aerodynamics 8 Chapter 4: Thin airfoils and finite wings Wing configuration: closed wing
Merging or joining structurally the two wing planes at or near the tips stiffens the structure and can reduce induced drag
box wing annular box wing (ex. Santos-Dumont’s 14-bis) (ex. Blériot III)
Aerodynamics 9 Chapter 4: Thin airfoils and finite wings Wing planform: aspect ratio 퐴푅
c(y) y x
b
푏/2 푐 푦 d푦 mean chord: 푐ҧ = −푏/2 planform area: 푆 = 푏 푐ҧ 푏 푏2 푏 aspect ratio: 퐴푅 = = 푆 푐ҧ
Aerodynamics 10 Chapter 4: Thin airfoils and finite wings Wing planform: aspect ratio 퐴푅
low 푨푹 moderate 푨푹 high 푨푹 (structurally efficient (general purpose, (aerodynamically and high roll rate, ex. Lockheed P-80 efficient, ex. ex. Lockheed F-104 Shooting Star) Bombardier Dash 8) Star Fighter)
Aerodynamics 11 Chapter 4: Thin airfoils and finite wings Wing planform: chord variation along y
y x
constant chord tapered wing, compound tapered or rectangular wing trapezoidal (ex. Westland (low cost but not (c decreases Lysander army efficient, ex. with y, ex. cooperation Piper J-3 Cub) Grumman F4F aircraft) Wildcat)
Aerodynamics 12 Chapter 4: Thin airfoils and finite wings Wing planform: chord variation along y
constant chord elliptical wing semi-elliptical wing w/ tapered outer (ex. Supermarine (only LE or TE have section (ex. Spitfire) elliptical shape, ex. many Cessna) Seversky P-35)
Aerodynamics 13 Chapter 4: Thin airfoils and finite wings Supermarine Spitfire
Probably the most famous British fighter aircraft of WWII, it continued in service for many years after the war. Designed by R.J. Mitchell, the aircraft’s elliptical wing had a thin cross-section to permit achieving high speeds..
According to John D. Anderson, Jr., the choice of the elliptic shape had “nothing to do with aerodynamics” …
Aerodynamics 14 Chapter 4: Thin airfoils and finite wings Wing planform: wing sweep
straight swept back forward swept structurally efficient high subsonic low drag at transonic wing for low speed and early super- speeds, but aero- designs, ex. sonic designs, elastic problems, ex. Lockheed P-80 ex. Hawker Sukhoi Su-47 Shooting Star Hunter
Aerodynamics 15 Chapter 4: Thin airfoils and finite wings Wing planform: wing sweep
swing-wing crescent cranked arrow ex. a few military different sweep on prototypes aircrafts, such as outer and inner sections, by General the General compromise between Dynamics Dynamics F-111 shock delay and spanwise F16-XL Aardvark flow control, ex. Handley Page Victor
Aerodynamics 16 Chapter 4: Thin airfoils and finite wings Wing twist
Aerodynamic feature to adjust the lift distribution along wing span.
geometric twist
aerodynamic twist
Aerodynamics 17 Chapter 4: Thin airfoils and finite wings Dihedral and anhedral
Angling the wings up or down spanwise from root to tip can help to resolve various design issues, such as stability and control in flight.
dihedral anhedral improved lateral stability, improved maneuverability ex. Boeing 737 Ilyushin ex. Boeing B-52 Strato- fortress
Aerodynamics 18 Chapter 4: Thin airfoils and finite wings Minor independent surfaces
Winglet: a small vertical fin at the wingtip, usually turned upwards. Reduces the size of vortices shed by the wingtip, and hence also tip drag.
Strake: a small surface, typically longer than it is wide and mounted on the fuselage. Strakes may be located at various positions in order to improve aerodynamic behavior. Leading edge root extensions (LERX) are also sometimes referred to as wing strakes.
Aerodynamics 19 Chapter 4: Thin airfoils and finite wings Control surfaces
Chine: long, narrow sideways extension to the fuselage, blending into the main wing. It improves low speed (high angle of attack) handling, provides extra lift at supersonic speeds for minimal increase in drag. Ex. Lockheed SR-71 Blackbird191.
Moustache: small high-aspect- ratio canard surface having no movable control surface. Typically is retractable for high speed flight. Deflects air downward onto the wing root, to delay the stall. Ex. Dassault Milan192 and Tupolev Tu-144193.
Aerodynamics 20 Chapter 4: Thin airfoils and finite wings Control surfaces: high lift devices
Slat and slot: a leading edge slat is a small airfoil extending in front of the main leading edge. The spanwise gap behind it forms a leading-edge slot. Air flowing up through the slot is deflected backwards by the slat to flow over the wing, allowing the aircraft to fly at lower air speeds without flow separation or stalling. A slat may be fixed or retractable. Flap: a hinged aerodynamic surface, usually on the trailing edge, which is rotated downwards to generate extra lift and drag. Cuff: fixed aerodynamic device which introduces a sharp discontinuity at the LE, typically to improve low-speed characteristics.
Aerodynamics 21 Chapter 4: Thin airfoils and finite wings Control surfaces: devices to delay stall
Vortex generator: small protrusion on the upper leading wing surface; usually, several are spaced along the span of the wing. They increase drag at all speeds. Vortilon: one or more flat plates attached to the underside of the wing near its outer leading edge. At low speeds, it creates a vortex which energizes the boundary layer over the wing. Leading-edge root extension (LERX): placed forward of the leading edge. The primary reason for adding a LERX is to improve the airflow at high angles of attack and low airspeeds, to improve handling and delay stall.
Aerodynamics 22 Chapter 4: Thin airfoils and finite wings Control surfaces: drag reduction devices
Anti-shock body: a streamlined pod shape added to the leading or trailing edge of an aerodynamic surface, to delay the onset of shock stall and reduce transonic wave drag. Examples include the Küchemann carrots on the wing trailing edge of the Handley Page Victor B.2. Fillet: a small curved infill at the junction of two surfaces, such as a wing and fuselage, blending them smoothly together to reduce drag. Fairings of various kinds, such as blisters, pylons and wingtip pods, whose only aerodynamic purpose is to produce a smooth outline and reduce drag.
Aerodynamics 23 Chapter 4: Thin airfoils and finite wings Axis of control of an airplane
The three axes go through the center of gravity (balance point) of the airplane. Ailerons control rotation about the roll axis Elevators control the pitch of an airplane, and thus the angle of attack of the wing. The rudder controls yaw.
Aerodynamics 24 Chapter 4: Thin airfoils and finite wings Parts definition
Aerodynamics 25 Chapter 4: Thin airfoils and finite wings Ailerons
Ailerons
Aerodynamics 26 Chapter 4: Thin airfoils and finite wings Elevators
Ailerons
Aerodynamics 27 Chapter 4: Thin airfoils and finite wings Rudder
Ailerons
Aerodynamics 28 Chapter 4: Thin airfoils and finite wings One spoiler
Ailerons
Aerodynamics 29 Chapter 4: Thin airfoils and finite wings Both spoilers
Ailerons
Aerodynamics 30 Chapter 4: Thin airfoils and finite wings Flaps/slats
Ailerons
Aerodynamics 31 Chapter 4: Thin airfoils and finite wings Flaps/slats
Ailerons
Aerodynamics 32 Chapter 4: Thin airfoils and finite wings The vortex filament
Aerodynamic convention: positive circulation is clockwise
(this is different from what we had in ch. 3)
Aerodynamics 33 Chapter 4: Thin airfoils and finite wings The vortex sheet
훾 = 훾(푠) strength of the vortex sheet per unit length
Aerodynamics 34 Chapter 4: Thin airfoils and finite wings The vortex sheet
The infinitesimal portion 푑푠 of the vortex sheet induces on point P in the flow a velocity of magnitude dΓ d푉 = 2휋푟
dΓ = 훾 d푠 circulation over infinitesimal segment d푠
Aerodynamics 35 Chapter 4: Thin airfoils and finite wings The vortex sheet
For a vortex sheet there is a discontinuous change in the tangential component of velocity across the sheet, while the normal component is preserved across the sheet.
Γ = 푢1 d푠 − 푣2 d푛 − 푢2 d푠 + 푣1 d푛 = 푢1 − 푢2 d푠 + 푣1 − 푣2 d푛 = 훾 d푠
Aerodynamics 36 Chapter 4: Thin airfoils and finite wings The vortex sheet
For d푛 → 0 we have
훾 = 푢1 − 푢2
The local jump in tangential velocity across the vortex sheet is equal to the local sheet strength
Aerodynamics 37 Chapter 4: Thin airfoils and finite wings How to use a vortex sheet?
An airfoil surface or the camber line can be replaced by a vortex sheet of variable strength (L. Prandtl, 1912-1922)
(note the similarity with the concept of a boundary layer!)
Aerodynamics 38 Chapter 4: Thin airfoils and finite wings Kutta (yet again!)
At the trailing edge velocity and pressure (!) are unique, hence:
훾 푇퐸 = 0
Aerodynamics 39 Chapter 4: Thin airfoils and finite wings Thin airfoil theory
A thin airfoil can be modeled as a vortex sheet on the camber line. Goal: find the distribution of 훾(푠) that renders the camber line a streamline of the flow and such that the Kutta condition is satisfied.
훾 = 훾(푠)
Aerodynamics 40 Chapter 4: Thin airfoils and finite wings Thin airfoil theory
훾 = 훾(푥)
Aerodynamics 41 Chapter 4: Thin airfoils and finite wings Thin airfoil theory
Aerodynamics 42 Chapter 4: Thin airfoils and finite wings Thin airfoil theory
(− sign because in direction opposite to that of 푧-axis)
Fundamental equation of thin airfoil theory
(훼 in radians!)
Aerodynamics 43 Chapter 4: Thin airfoils and finite wings Thin airfoil theory
Once 훾 = 훾 푥 is found, we can easily find the total
푐 circulation, Γ = 0 훾 푥 d푥, and then from the KJ theorem ′ the lift on the airfoil: 퐿 = 휌 푉∞ Γ (notice that now clockwise circulation is positive)
Aerodynamics 44 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the symmetric airfoil
d푧 If the thin airfoil is symmetric 푧 푥 = 0 and = 0 d푥
Transformation: 휃 = 0 0((LE, 휉 = 0) 휃 = 휋 0 (TE, 휉 = 푐)
x is a some point on the chord line
Aerodynamics 45 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the symmetric airfoil
휃 휃0
The transformation variable
Aerodynamics 46 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the symmetric airfoil
From the mathematical theory of integral equations a solution can be found:
(verify the solution by yourselves, using Glauert’s first integral, and verify also that Kutta condition is satisfied)
Aerodynamics 47 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: Glauert’s integrals
휋 cos(푛휃) 휋 sin(푛휃 ) න d휃 = 0 cos 휃 − cos 휃0 sin 휃0 0
휋 sin 푛휃 sin 휃 න d휃 = −휋 cos(푛휃0) cos 휃 − cos 휃0 0
Aerodynamics 48 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the symmetric airfoil
2휉 cos 휃 = 1 − 푐 휉 휉 sin 휃 = 2 1 − Tapez une équation ici. 푐 푐 휉 훾 1 − 푐 푐 훾 휉 = 2α 푉∞ = 2α 푉∞ − 1 휉 휉 휉 1 − 푐 푐
푐 Γ = න 훾 휉 d휉 푐 휉 0
Aerodynamics 49 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the symmetric airfoil
푐 휋 1 + cos 휃 푐 Γ = න 훾 휉 d휉 = න 2훼 푉∞ sin 휃 d휃 0 0 sin 휃 2
휋 (clockwise = 훼 푐 푉∞ න (1 + cos 휃) d휃 = 휋 훼 푐 푉∞ circulation) 0
′ 2 KJ theorem: 퐿 = 휌 푉∞ Γ = 휋 훼 푐 휌 푉∞ → 푐푙= 2 휋 훼 (same result found already for the Joukowski airfoil!)
d푐 lift slope: 푎 = 푙 = 2 휋 0 d훼 2 휋 rad -1 = 0.10966227 degree -1
Aerodynamics 50 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the symmetric airfoil
d퐿′
푐 ′ ′ 푀LE = න − 휉 d퐿 0
푐 1 = −휌 푉 න 휉 γ ξ d휉 = ⋯ = − 휋 훼 휌 푉2 푐2 ∞ 4 ∞ 0
휋 훼 푐 푐 = − = − 푙 푚 LE 2 4
Aerodynamics 51 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the symmetric airfoil
푐 푐 = 푐 + 푙 푚 푐/4 푚 LE 4 (see slide 20 of ch. 1!)
푐푚 푐/4 = 0
푥 = 푐/4 is both CP and AC for the thin symmetric airfoil
Aerodynamics 52 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the symmetric airfoil
We can now refine a bit the qualitative argument presented in ch. 3, slide 29. 훾 푥 = 푢1 − 푢2 푉∞ 푢1 푝1 ′ 푢2 푝2 d퐿 = 휌 푉∞ 훾 푥 d푥 ≈ d퐹푝
= 푝2 − 푝1 d푥 1 Bernoulli: 푝 − 푝 = 휌(푢2 − 푢2) ≈ 휌 푉 훾 푥 2 1 2 1 2 ∞ 훾 훾 푢 ≈ 푉 + , 푢 ≈ 푉 − 1 ∞ 2 2 ∞ 2 푐 푐 푢 ≈ 푉 + α 푉 − 1, 푢 ≈ 푉 − α 푉 − 1 1 ∞ ∞ 푥 2 ∞ ∞ 푥
Aerodynamics 53 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the symmetric airfoil
푐 푢 ≈ 푉 1 + α − 1 ≠ 0 ∀푥 for 훼 ≥ 0 1 ∞ 푥
푐 푥 훼2 푢 ≈ 푉 1 − α − 1 = 0 when = 2 ∞ 푥 푐 훼2+1
When α is sufficiently small, the stagnation point on the 2 pressure side of the airfoil has coordinate: 푥푠푡푎푔푛 ≈ 훼 푐 푧
푉∞ 푥푠푡푎푔푛 푥
Aerodynamics 54 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the cambered airfoil
Let us go back to the fundamental equation of thin airfoil theory (slide 43) which, in Glauert’s coordinates, read:
together with Kutta condition:
To express the solution we must rely on a Fourier cosine 푑푧 series expansion of the even function 푑푥
Aerodynamics 55 Chapter 4: Thin airfoils and finite wings Fourier cosine series expansion
The Fourier cosine series expansion of a generic even function 푓(푥) is: ∞ 1 푓 푥 = 푎 + 푎 cos(푛푥) 2 0 푛 푛=1 with the coefficients given by 1 휋 2 휋 푎0 = න 푓 푥 푑푥 = න 푓 푥 푑푥 휋 −휋 휋 0
1 휋 2 휋 푎푛 = න 푓 푥 cos 푛푥 푑푥 = න 푓 푥 cos 푛푥 푑푥 휋 −휋 휋 0
Aerodynamics 56 Chapter 4: Thin airfoils and finite wings 푑푧 Fourier cosine series expansion of 푑푥
The function z = 푧(휃0) is odd (practical camber lines go 푑푧 to zero at LE and TE), so Τ푑푥 is an even function. The 푑푧 Fourier cosine series expansion of is written as: 푑푥 ∞ 푑푧 = (훼 − 퐴 ) + 퐴 cos(푛휃 ) 푑푥 0 푛 0 푛=1 1 휋 푑푧 with the coefficients given by (훼 − 퐴 ) = 푑휃 0 휋 0 푑푥 0
2 휋 푑푧 and 퐴 = cos 푛휃 푑휃 푛 휋 0 푑푥 0 0
Aerodynamics 57 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the cambered airfoil
푑푧 Inserting 훼 − = 퐴 − σ∞ 퐴 cos 푛휃 into the 푑푥 0 푛=1 푛 0 fundamental equation we have:
∞ 1 휋 훾 휃 sin 휃 푑휃 න = 퐴0 − 퐴푛 cos 푛휃0 2휋푉∞ cos 휃 − cos 휃0 0 푛=1
The result of this equation, satisfying Kutta condition, is:
Aerodynamics 58 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the cambered airfoil
Assignement: Verify that the given expression of 훾(휃) satisfies the fundamental equation, using Glauert’s second integral (slide 48). Verify also that Kutta condition is satisfied.
푑푧 Note: for the limit case of symmetric airfoil = 0 we 푑푥
have 퐴0= 훼 and 퐴푛 = 0, for 푛 = 1, 2, 3 … in this case the solution in slide 58 coincides with that of slide 47.
Aerodynamics 59 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the cambered airfoil
훾 휃 =
, 휉
Generic form of the vortex sheet strength distribution, and schematic description of the first four terms in the series describing the circulation.
Aerodynamics 60 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the cambered airfoil
Note: from slide 58 it is clear that at the LE, for 휃 → 0, it is sin 휃 → 0 and the strength of the vortex sheet is locally
singular (lim |훾| → ∞). The only way for this not to happen is 휃→0
that 퐴0 = 0. When this occurs the angle of attack is called ideal (because the local load at the LE vanishes). Thus
1 휋 푑푧 훼푖푑푒푎푙 = න 푑휃0 휋 0 푑푥
In this case the ideal lift coefficient is 푐푙 = 휋 퐴1 (check slide 63). This is also called the design lift coefficient.
Aerodynamics 61 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the cambered airfoil
Total circulation due to the vortex sheet:
with
Aerodynamics 62 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the cambered airfoil
휋 KJ: 퐿′= 휌 푉 Γ = 휌 푉2 푐 휋 퐴 + 퐴 ∞ ∞ 0 2 1 퐿′ 퐴 푐 = = 2휋 퐴 + 1 푙 1 0 휌 푉2푐 2 2 ∞ 휋 1 푑푧 = 2휋 훼 + න (cos 휃 − 1) 푑휃 = 2휋 훼 − 훼 휋 푑푥 0 0 푙=0 0
Aerodynamics 63 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the cambered airfoil
푑푐 Lift slope: 푎 = 푙 = 2휋 0 푑훼 The lift slope is the same as for a symmetric airfoil!
The zero lift angle is: 휋 1 푑푧 훼 = − න (cos 휃 − 1) 푑휃 푙=0 휋 푑푥 0 0 0
For symmetric airfoil 훼푙=0= 0
Aerodynamics 64 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the cambered airfoil
d퐿′
푐 ′ ′ 푀LE = න − 휉 d퐿 0
푐 1 퐴 = −휌 푉 න 휉 γ ξ d휉 = ⋯ = − 휋 휌 푉2 푐2 퐴 + 퐴 − 2 ∞ 4 ∞ 0 1 2 0
휋 퐴 푐 휋 푐 = − 퐴 + 퐴 − 2 = − 푙 + 퐴 − 퐴 푚 LE 2 0 1 2 4 4 2 1
Aerodynamics 65 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the cambered airfoil
The moment coefficient for a cambered airfoil depends on only three Fourier series coefficients (function of angle of attack and camber).
푐푙 Using the relation in slide 52 푐 = 푐 + 푚 푐/4 푚 LE 4 we have: 휋 푐 = 퐴 − 퐴 푚 푐/4 4 2 1
Thus, the moment coefficient about quarter chord is independent of the angle of attack, which means that 푥 = 푐/4 is the aerodynamic center.
Aerodynamics 66 Chapter 4: Thin airfoils and finite wings Thin airfoil theory
For thin symmetric arfoil 푥 = 푐/4 is both AC and CP.
For thin cambered airfoil 푥 = 푐/4 is AC (but not CP since 푐푚 푐/4 ≠ 0 because of the airfoil camber) To find the center of pressure consider that the total moment about the CP vanishes, i.e. ′ 푀푐/4 ′ ′ ′ 푀CP = 퐿 Δ푥 + 푀푐/4 = 0 AC CP
Δ푥 푐푚 푐/4 휋 퐴1−퐴2 푐 = − = → 푥CP = + Δ푥 푐 푐푙 4 푐푙 4 (notice that 푥퐶푃 changes with lift!)
Aerodynamics 67 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: comparison w/ exp
푧/푐
푧/푐
Abbott & van Doenhoff
Aerodynamics 68 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: comparison w/ exp
푧/푐
푧/푐
Abbott & van Doenhoff
Aerodynamics 69 Chapter 4: Thin airfoils and finite wings Thin (or thick?) airfoil: the AC
Thin airfoil theory should, in principle, not be applied if the thickness of the airfoil is large. For a thick airfoil the AC is not at c/4 (but close …)
′ ′ 푐 푀 = 푀 + 푥 − 퐿′+ AC 푐/4 AC 4 AC 푥 1 푐 = 푐 + AC − 푐 푚 AC 푚 푐/4 푐 4 푙
푑푐 푑푐푚 푐/4 푥 1 푑푐 푥 1 푚 AC = 0 = + AC − 푙 = 푚 + AC − 푎 푑훼 푑훼 푐 4 푑훼 0 푐 4 0
푥AC 1 푚0 = − (either larger or smaller than 1/4) 푐 4 푎0
Aerodynamics 70 Chapter 4: Thin airfoils and finite wings Thin (or thick?) airfoil: the AC
푥AC 푐
푥AC 푐
Aerodynamics 71 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the effect of flaps
z
z z
z
Aerodynamics 72 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the effect of flaps
Flaps are the main mechanism to generate control forces on airplanes (cf. slides 68 and 69) since they change the effective camber of the airfoil/wing.
Deploying flaps has a minor effect on lift slope 푎0, but:
• 훼푙=0 is substantially decreased (+) • max lift coefficient is increased (+) • the angle of stall is reduced (−) (but this reduction is not large enough to detract from the advantages of shifting the lift curve to the left)
Aerodynamics 73 Chapter 4: Thin airfoils and finite wings Thin airfoil theory: the effect of flaps
+ h
h h
h
(for h ≠ 0 drag increases …)
Aerodynamics 74 Chapter 4: Thin airfoils and finite wings Other HLD: the leading edge slat
leading edge slat activated
Aerodynamics 75 Chapter 4: Thin airfoils and finite wings Other HLD
Airfoil with LE slat and TE multi-element flap
Aerodynamics 76 Chapter 4: Thin airfoils and finite wings Other HLD
훼 = 25°
Aerodynamics 77 Chapter 4: Thin airfoils and finite wings Thin airfoil theory
Exercises
1. Consider a thin airfoil, with mean camber line in the form of a circular arc, 푥 푥 2 with mean camber given by 푧 = 4푓 − , where 푓 is the maximum 푐 푐 camber. Find 훼푙=0, 푐푙, 푐푚 AC, 푐푚 LE, 푥CP, 훼푖푑푒푎푙, and the design lift coefficient.
2 2. For the thin airfoil of mean camber line given by 푧 = 푓 sin 휃0 (휃0 defined as in slide 45), with 푓 the maximum camber, answer the same question of the previous exercise. z 3. Consider the airfoil with flap shown in the figure, with the chord c equal to the segment AC and h, F < 1. Answer the same question of ex. 1.
Aerodynamics 78 Chapter 4: Thin airfoils and finite wings The numerical vortex panel method
The panel method, based on a distribution of potential sources (or doublets or vortices, and sometimes a combination of different singularities), is a common numerical technique to treat the flow over non-lifting or lifting bodies. XFOIL (https://web.mit.edu/drela/Public/web/xfoil/) – and similar codes – are based on the vortex panel method. This differs from the lumped vortex element method. Panels, i.e. straight-line segments, have a surface distribution 훾(푠) of vorticity. The strength of the vortex sheet on each panel of the airfoil must be determined.
Aerodynamics 79 Chapter 4: Thin airfoils and finite wings The numerical vortex panel method
Simple implementation: each of the N straight panels carries a vortex distribution of linearly varying strength between the end points of each panel. End point values are the unknowns of the problem: 훾푗 푗 = 1, 2, … N + 1 , and are found by imposing that the body surface is a streamline of the flow, plus Kutta condition: 훾1 = −훾N+1. z
x
Aerodynamics 80 Chapter 4: Thin airfoils and finite wings The numerical vortex panel method
훾푗+1
훾푗 훾푗−1
훾푗−2
Example of linear distribution of 훾 over neighboring panels. The strength of the vortex sheet at the end point of panel 푗 − 1 is equal to the strength at the first point of panel 푗.
Aerodynamics 81 Chapter 4: Thin airfoils and finite wings The numerical vortex panel method
A condition of no through-flow (푉푛,푖 = 0) must be applied at the center of the 푖푡ℎ panel, on the control point. This yields N equations for N + 1 unknowns. The extra equation needed is Kutta condition. z
x
Aerodynamics 82 Chapter 4: Thin airfoils and finite wings The numerical vortex panel method
z
dVn dV (induced by panel N-2 on control point of panel i)
x
panel N-2
Aerodynamics 83 Chapter 4: Thin airfoils and finite wings The numerical vortex panel method
For the generic panel 푖 the equation to be enforced is: N+1
푉푛,푖 = 퐴푖,푗 훾푗 + 퐵푖 푉∞ = 0 푗=1 where 퐴푖,푗 represents the influence of the vorticity of panel 푗 on the control point of panel 푖, and 퐵푖 represents the influence of the free-stream on panel 푖. Both 퐴푖,푗 and 퐵푖 are function of the geometry of the section, because of orientation and spacing of the panels.
Aerodynamics 84 Chapter 4: Thin airfoils and finite wings The numerical vortex panel method
The system of linear equations eventually reads:
훾 ⋮ 퐴1,1 퐴1,2 … … … 퐴1,N 퐴1,N+1 1 퐵1 훾 퐴 퐴 2 퐵2 2,1 2,2 … … ⋮ …
⋮ ⋮ ⋮ = −푉∞ ⋮ … … … 퐴N,1 퐴N,2 … … 퐴N,N 퐴N,N+1 훾N 퐵 … N 1 0 0 0 0 1 훾N+1 0
Aerodynamics 85 Chapter 4: Thin airfoils and finite wings The numerical vortex panel method
Once the distribution strengths 훾푖 have been calculated, we can have surface tangential velocities at the center of each 2 푉푖 panel (푉푖), and surface pressure coefficients, 푐푝,푖 = 1 − 2 푉∞
z chord (c)
x
Aerodynamics 86 Chapter 4: Thin airfoils and finite wings The numerical vortex panel method
The lift coefficient can be calculated, e.g. assuming a small angle of attack 훼 (so that cos 훼 is close to one), as the integration of surface pressure coefficient acting in the 푧- direction, i.e. N 푥푖 − 푥푖+1 푐푙 = 푐푝,푖 1 푐 푖=1 4 Pitching moment coefficient will similarly be the sum of the panel moments about the 1/4 chord point. N 푥 − 푥 푥 + 푥 1 푐 = 푐 푖 푖+1 푖 푖+1 − 푚 푐/4 푝,푖 푐 2푐 4 푖=1
Aerodynamics 87 Chapter 4: Thin airfoils and finite wings The numerical vortex panel method
Aerodynamics 88 Chapter 4: Thin airfoils and finite wings The inverse problem
For design purposes it is desirable to specify the surface pressure (or velocity) distribution, for example to achieve enhanced airfoil performances such as: • preserve a laminar boundary layer over a considerable chordwise extent • delay boundary-layer separation and hence reduce the form drag of the airfoil • a relatively low suction peak on the upper surface of the airfoil would result in a higher critical Mach number for the airfoil, meaning that it could operate efficiently at a higher subsonic cruise speed than an airfoil with a conventional pressure distribution • … and calculate the shape of the airfoil that will produce the specified pressure (or velocity) distribution.
Aerodynamics 89 Chapter 4: Thin airfoils and finite wings The inverse problem
푐푝 z/푐
푥/푐 푥/푐
Adjoint optimization of an airfoil to match a specified pressure distribution (free-stream Mach number of 0.4, angle of attack of 2 deg., Reynolds number of 5 × 106)
Aerodynamics 90 Chapter 4: Thin airfoils and finite wings The inverse problem
Further shape optimization objectives are:
• optimize lift coefficient • minimize skin friction or form drag (or other types of drag) • maximize the lift-to-drag ratio (aerodynamic efficiency) • limit system vibrations (aeroelasticity problem) • reduce aerodynamic noise • stabilize (or mitigate) a shock • …
Several optimization strategies are covered in the course by Prof. J. Pralits
Aerodynamics 91 Chapter 4: Thin airfoils and finite wings 3D wings
Aerodynamics 92 Chapter 4: Thin airfoils and finite wings Lift for 3D wings
Aerodynamics 93 Chapter 4: Thin airfoils and finite wings Wing-tip vortices
Aerodynamics 94 Chapter 4: Thin airfoils and finite wings ′ Downwash and induced drag, 퐷푖
′ 퐷푖
퐿′
푉∞
Effect of downwash on a local airfoil section of a 3D wing
Aerodynamics 95 Chapter 4: Thin airfoils and finite wings Vortex line and Biot-Savart law
l dl
Γ
푽
Aerodynamics 96 Chapter 4: Thin airfoils and finite wings Vortex line and Biot-Savart law
+∞ ΓΓ cos 휎 If 1 → −∞ and 2 → +∞ then 푉 = න 푑푙 2휋ℎ 2 4휋 −∞ 푟 Γ +∞ cos3 휎 Γ +휋/2 Γ = න 2 푑푙 = න cos 휎 푑휎 = 4휋 −∞ ℎ 4휋ℎ −휋/2 2휋ℎ
l dl This result agree with the case of the 2D potential vortex!
1 푙 = ℎ tan 휎 → 푑푙 = ℎ 2 푑휎 Γ cos 휎
푽
Aerodynamics 97 Chapter 4: Thin airfoils and finite wings Vortex line and Biot-Savart law
If 1 and 2 are two values characterized by the two angles 휎1 and 휎2, the velocity in P has magnitude
Γ 휎2 Γ 푉 = න cos 휎 푑휎 = (sin 휎 − sin 휎 ) 4휋ℎ 4휋ℎ 2 1 휎1
l If the end point 1 (ordl eventually both end points) of the segment 1-2 lies to the left of the point of interest P, then 휎1 is a negative angle.Γ If the filament is semi- infinite, then Γ 푉 = 4휋ℎ (half the value found for an infinite vortex filament!)
Aerodynamics 98 Chapter 4: Thin airfoils and finite wings The horseshoe vortex
푥 → ∞
Simple horseshoe vortex model representation of a wing
Aerodynamics 99 Chapter 4: Thin airfoils and finite wings The horseshoe vortex
Downwash distribution along y for a single horseshoe vortex (unphysical …)
Aerodynamics 100 Chapter 4: Thin airfoils and finite wings Prandtl’s lifting line theory
Superposition of many horseshoe vortices, each with a different length of the bound vortex, but with all bound vortices lying on the lifting line
Aerodynamics 101 Chapter 4: Thin airfoils and finite wings Prandtl’s lifting line theory
Infinite horseshoe vortices superimposed along the lifting line
lifting 푑푤 line
continuous vortex sheet behind the lifting line
Goal of the analysis: find 횪 = 횪(풚)
Aerodynamics 102 Chapter 4: Thin airfoils and finite wings Prandtl’s lifting line theory
On 푦0 the vertical velocity 푑푤 induced by the 푑푤 infinitesimal segment 푑푥 of a vortex filament positioned in a generic 푦 is:
푑Γ 푑푦 푑Γ 푑Γ 푑푦 (notice that < 0 at 푦 in figure) 푑푤 = = 푑푦 0 4휋 푦 − 푦0 4휋(푦 − 푦0)
푏/2 (푑Γ/푑푦) → 푤(푦0) = න 푑푦 4휋(푦 − 푦0) −푏/2
Aerodynamics 103 Chapter 4: Thin airfoils and finite wings Prandtl’s lifting line theory
From slide 95 the induced angle of attack at the generic section 푦0 is
−1 −푤(푦0) 푤(푦0) 훼푖 푦0 = tan ≈ − Tapez푉∞une équation ici.푉∞ 푏/2 1 (푑Γ/푑푦) 훼푖(푦0) = න 푑푦 4 휋 푉∞ (푦0 − 푦) −푏/2
Velocity components at the lifting line
Aerodynamics 104 Chapter 4: Thin airfoils and finite wings Prandtl’s lifting line theory
if thin profile Section lift coefficient:
푐푙 푦0 = 푎0 훼eff 푦0 − 훼푙=0 = 2휋 훼 − 훼푖 푦0 − 훼푙=0
Tapez une équation ici.
geometric twist: α = α(푦0) wash-out
aerodynamic twist: 훼푙=0 = 훼푙=0(푦0)
Aerodynamics 105 Chapter 4: Thin airfoils and finite wings Fundamental equation of Prandtl’s LLT
1 퐿′ 푦 = 휌 푉2 푐(푦 ) 푐 푦 ≈ 휌 푉 Γ 푦 0 2 ∞ 0 푙 0 ∞ 0
2 Γ 푦0 푐푙 푦0 = = 푎0 훼(푦0) − 훼푖(푦0) − 훼푙=0(푦0) 푉∞ 푐 푦0
푏/2 2 Γ 푦0 1 (푑Γ/푑푦) 훼 푦0 = + 훼푙=0(푦0) + න 푑푦 푎0 푉∞ 푐 푦0 4 휋 푉∞ (푦0 − 푦) −푏/2
This is the monoplane equation
Aerodynamics 106 Chapter 4: Thin airfoils and finite wings Prandtl’s lifting line theory
The fundamental equation of Prandtl’s lifting line theory simply states that the geometric angle of attack is the sum of the effective angle plus the induced angle of attack. It is an integro-differential equation for Γ = Γ 푦0 , with 푦0 ranging along the span from −푏/2 to +푏/2. Not easy to solve …
Consequences: 푏/2 1. ′ 퐿 푦0 = 휌 푉∞ Γ 푦0 → 퐿 = 휌 푉∞ −푏/2 Γ 푦0 푑푦0 .1 푏 ′ ′ ′ 2 퐷푖 = 퐿 sin 훼푖 ≈ 퐿 훼푖 → 퐷푖 ≈ 휌 푉∞ 푏 Γ 푦0 훼푖 푦0 푑푦0 .2 − 2 3. 퐶퐿 and 퐶퐷,푖 readily available
Aerodynamics 107 Chapter 4: Thin airfoils and finite wings Drag
The total drag of a subsonic finite wing includes the induced drag, the skin friction drag and the pressure drag
퐷 = 퐷푖+ 퐷푓푟푖푐푡푖표푛 + 퐷푝푟푒푠푠푢푟푒 → 퐶퐷
also called profile drag, and related to viscous effects At moderate angle of attack, the profile drag coefficient for a finite wing is essentially the same as for its airfoil sections.
퐷푓푟푖푐푡푖표푛+퐷푝푟푒푠푠푢푟푒 The profile drag coefficient: 푐푑 = 1 for a finite 휌 푉2 푆 2 ∞ wing is available from airfoil data. Thus, the total drag coefficient takes the form: 퐶퐷= 푐푑 +퐶퐷,푖
Aerodynamics 108 Chapter 4: Thin airfoils and finite wings Elliptical lift distribution
Let us consider an elliptic circulation/lift distribution along y
2 2 푦 Γ(푦) = Γ 1 − 0 푏
Aerodynamics 109 Chapter 4: Thin airfoils and finite wings Elliptical lift distribution
Let us consider an elliptic circulation/lift distribution along y
2 2 푦 Γ(푦) = Γ 1 − 0 푏
푑Γ 훾 = ቤ → ∞ 푑푦 푏 푦→±2 vortex sheet of infinite strength at the tips
(slide 103)
Aerodynamics 110 Chapter 4: Thin airfoils and finite wings Elliptical lift distribution
Change of variables: 푏 푏 푦 = cos 휃 → 푑푦 = − sin 휃 푑휃 2 2 with the variable 휃 going from 휋 to 0.
Aerodynamics 111 Chapter 4: Thin airfoils and finite wings Elliptical lift distribution
Change of variables: 푏 푏 푦 = cos 휃 → 푑푦 = − sin 휃 푑휃 2 2
with the variable 휃 going from 휋 to 0. Then Γ 휃 = Γ0 sin 휃 and
(from Glauert’s first integral with n = 1, slide 48)
Aerodynamics 112 Chapter 4: Thin airfoils and finite wings Elliptical lift distribution
Γ0 Constant downwash: 푤(푦) = − 2푏 Induced angle of attack:
For an elliptical lift distribution both the downwash and the induced angle of attack are constant along the span 푦 (and both tend to zero as the span becomes infinite)
휌
푏 휌 휌 푉 Γ 휋 ∞ 0 4
Aerodynamics 113 Chapter 4: Thin airfoils and finite wings Elliptical lift distribution
1 2 4 휌 푉∞ 푆 퐶퐿 4퐿 2 2 푉∞푆 퐶퐿 Γ0 = = = 휌 푉∞푏 휋 휌 푉∞푏 휋 푏 휋
푆 퐶 퐶 2 훼 = 퐿 = 퐿 with 퐴푅 = 푏 /푆 푖 휋 푏2 휋 퐴푅
Since 훼푖 is constant, we have (cf. slide 107): 푏 퐷 퐶2 퐷 ≈ 퐿 훼 = 휌 푉 훼 Γ 휋 → 퐶 = 푖 = 퐿 푖 푖 ∞ 푖 0 퐷,푖 1 4 휌 푉2 푆 휋 퐴푅 2 ∞ “Lift-induced drag” coefficient!
Aerodynamics 114 Chapter 4: Thin airfoils and finite wings Elliptical lift distribution
2 At zero lift 퐶 is zero (and same 퐶퐿 퐷,푖 퐶퐷,푖 = for ) 휋 퐴푅 훼푖 As 퐴푅 increases (to infinity for
퐶퐷,푖 two-dimensional flow) 퐶퐷,푖 decreases (to zero); same for 훼푖 At low speed (take-off or landing) 퐿 is large and 퐷푖 makes up a large part of the total drag (of the order of 60%). Even at relatively high cruising speeds, induced drag is 퐶퐿 typically 25% of the total drag
Aerodynamics 115 Chapter 4: Thin airfoils and finite wings Elliptical lift distribution
퐶2 퐶 = 푐 +퐶 = 푐 + 퐿 퐷 푑 퐷,푖 푑 휋 퐴푅 퐶퐷,푖
Drag polar for an elliptic untwisted wing of aspect ratio 퐴푅 = 5
Aerodynamics 116 Chapter 4: Thin airfoils and finite wings Lift slope for elliptical lift distribution
Ex. Untwisted wing, elliptical lift distribution
훼푖 constant along 푦 훼eff = 훼 − 훼푖 constant along 푦
Also, 훼푖 = 0 when 퐶퐿 = 0, which means that at zero-lift 훼eff = 훼, and thus 훼퐿=0 = 훼푙=0 (wing & airfoil)
Observations show that 푎 < 푎0, i.e. the finite wing has a reduced lift slope compared to an airfoil. By how much?
Aerodynamics 117 Chapter 4: Thin airfoils and finite wings Lift slope for elliptical lift distribution
푑퐶퐿 = 푎0 푑(훼 − 훼푖)
퐶퐿 = 푎0 훼 − 훼푖 + 푐표푛푠푡푎푛푡 퐶 퐶 = 푎 훼 − 퐿 + 푐표푛푠푡푎푛푡 퐿 0 휋 퐴푅 푎 퐶 1 + 0 = 푎 훼 + 푐표푛푠푡푎푛푡 퐿 휋 퐴푅 0 푑퐶 푎 2휋 푎 = 퐿 = 0 = 푎0 2 푑훼 1 + 1 + 휋 퐴푅 퐴푅 for thin profile
Aerodynamics 118 Chapter 4: Thin airfoils and finite wings Elliptical lift distribution: effect of 퐴푅 on 퐶퐿
2휋 2휋 퐴푅 퐶 Thin profiles: 푎 = = = 퐿 2 1 + 2 + 퐴푅 훼 − 훼퐿=0 퐴푅
푎 퐶 = 퐿 2휋 2휋 훼 − 훼퐿=0
퐴푅
Aerodynamics 119 Chapter 4: Thin airfoils and finite wings Elliptical lift distribution: effect of 퐴푅 on 퐶퐷,푖
퐴푅 퐶 = 퐿 2 + 퐴푅 2휋 훼 − 훼퐿=0 퐶2 퐴푅 퐶 = 퐿 = 4휋 훼 − 훼 2 퐷,푖 휋퐴푅 2 + 퐴푅 2 퐿=0
퐶퐷,푖 2 4휋 훼 − 훼퐿=0 Prize exacted by the work needed to create the wake
퐴푅
Aerodynamics 120 Chapter 4: Thin airfoils and finite wings Elliptical wing planform
Untwisted wing:
no aero twist: 훼푙=0 constant no geom twist: 훼 constant
Tapez une훼푒푓푓équation= 훼ici.− 훼푖 constant
푐푙 = 푎0 훼푒푓푓 − 훼푙=0 : the section lift coefficient is constant along the span 푦
′ 1 2 Local lift:: 퐿 푦 = 휌 푉 푐 푦 푐 → the chord 풄 풚 is elliptical! 2 ∞ 푙
Aerodynamics 121 Chapter 4: Thin airfoils and finite wings Elliptical wing planform
2 2푦 Γ 푦 = Γ 1 − 0 푏
British Spitfire: the wing is formed from two ellipses of different minor axis. This shifts the major axis
푤 and the AC forward
Aerodynamics 122 Chapter 4: Thin airfoils and finite wings Other untwisted wings
Minimum induced drag but expensive to manufacture
Lift distribution far from optimum
Tapered planform (with 푐 taper ratio λ ≡ 푡 ) can be 푐푟 designed such that the lift distribution approximates the elliptic case
Aerodynamics 123 Chapter 4: Thin airfoils and finite wings Other untwisted wings
The rectangular wing 퐿′(푦) has the highest loading at the tip
The linearly tapered wing “unloads” the tip, reducing the flexural moment there, and looses only a fraction of 푦 the lift compared to the elliptic wing. Reduced construction costs
Aerodynamics 124 Chapter 4: Thin airfoils and finite wings Effect of geometrical wing twist
(wash-in)
(wash-out)
Spanwise loading of twisted elliptic wings
Aerodynamics 125 Chapter 4: Thin airfoils and finite wings General circulation distribution
5
Γ = 2 푏 푉∞ 퐴푛 sin 푛휃 푛=1 Symmetric loading
Γ
휃
푏 For an elliptic wing, using 푦 = cos 휃 , the circulation is 2 Γ = Γ0 sin 휃. It is thus natural to express a general circulation distribution as a sine series: ∞ Γ = 2 푏 푉∞ 퐴푛 sin 푛휃 푛=1
Aerodynamics 126 Chapter 4: Thin airfoils and finite wings General circulation distribution
The monoplane equation (slide 106) becomes: ∞
휇 훼 − 훼푙=0 sin 휃0 = 퐴푛 휇푛 + sin 휃0 sin 푛휃0 푛=1
푐 푎0 where 휇 = (do the steps for yourselves and verify the equation above, 4푏 u using Glauert’s first integral, slide 48) This equation must be evaluated at 푵 spanwise stations (i.e. at 푁 values of 휃0) and solved for the 푁 coefficients 퐴1, 퐴2, 퐴3, … 퐴푁. For symmetric loading only the odd coefficients are needed; typically, the first 4 or 5 coefficients are sufficient
Aerodynamics 127 Chapter 4: Thin airfoils and finite wings General circulation distribution
Once the coefficients 퐴푛 are available:
2 +푏/2 ∞ 휋 퐶퐿 = Γ 푦 푑푦 = 2 퐴푅 σ푛=1 퐴푛 푠푖푛 푛휃 sin 휃 푑휃 = 퐴1휋 퐴푅 푉∞푆 −푏/2 0 푪푳 depends only on the amplitude of the first harmonic!
푑Γൗ 1 +푏/2 푑푦 푁 sin(푛휃0) 훼푖(푦0) = 푑푦 = ⋯ = σ1 푛 퐴푛 4 휋 푉∞ −푏/2 푦0−푦 sin 휃0
2 2 +푏/2 2 푁 퐴푛 퐶퐷,푖= Γ 푦 훼푖 푦 푑푦 = ⋯ = 휋 퐴푅 퐴1 1 + σ2 푛 푉∞푆 −푏/2 퐴1 2 2 2 퐶퐿 푁 퐴푛 퐶퐿 = 1 + σ2 푛 = 1 + 훿 (Glauert’s integrals again!) 휋 퐴푅 퐴1 휋 퐴푅
Aerodynamics 128 Chapter 4: Thin airfoils and finite wings General circulation distribution
퐶2 퐶 = 퐿 1 + 훿 훿 is the induced drag factor 퐷,푖 휋 퐴푅 훿 ∈ 0, 0.2
푒 = 1 + 훿 −1 ≤ 1 span efficiency factor
퐶2 퐶 = 퐿 퐷,푖 휋 푒 퐴푅
For the elliptical lift distribution it is 훿 = 0 and 푒 = 1, which corresponds to minimum induced drag → the elliptical lift distribution characterizes the optimal planar wing planform
Aerodynamics 129 Chapter 4: Thin airfoils and finite wings General circulation distribution: moments
푧
푥 푀푦푎푤 푀푟표푙푙
푑푀푟표푙푙 = 푦 푑퐿 푦 푀푝푖푡푐ℎ 푑푀푦푎푤 = −푦 푑퐷
Aerodynamics 130 Chapter 4: Thin airfoils and finite wings General circulation distribution: moments
Pitching moment is available from 2D wing profile analysis (slide 65; careful, coefficients 퐴푖 are not the same!!)
푏/2 Roll moment: 푀푟표푙푙= 휌 푉∞ න 푦 Γ(푦) 푑푦 = … −푏/2 ∞ 휌 푉2 푏3 휋 휋 퐴 휌 푉2 푏3 = ∞ 퐴 න sin 푛휃 sin(2휃) 푑휃 = 2 ∞ 4 푛 8 푛=1 0
휋 퐶 = 퐴 퐴푅 푀 푟표푙푙 4 2
푪푴 풓풐풍풍 depends only on the amplitude of the second harmonic!
Aerodynamics 131 Chapter 4: Thin airfoils and finite wings General circulation distribution: moments
Yaw moment (accounting only for induced drag): 푏/2 푏/2 푀푦푎푤= − න 푦 푑퐷푖 = − න 푦 훼푖 휌 푉∞ Γ 푦 푑푦 = ⋯ −푏/2 −푏/2 ∞ ∞ 휌 푉2 푏3 휋 = − ∞ 푛 퐴 퐴 න sin 푛휃 sin 푚휃 cos(휃) 푑휃 = 2 푛 푚 푛=1 푚=1 0 ∞ 휋 휌 푉2 푏3 = − ∞ (2푛 + 1) 퐴 퐴 8 푛 푛+1 푛=1 ∞ 휋 푪 depends on the 퐶푀 푦푎푤 = − 퐴푅 (2푛 + 1) 퐴푛 퐴푛+1 푴 풚풂풘 4 amplitude of all harmonics! 푛=1
Aerodynamics 132 Chapter 4: Thin airfoils and finite wings Generic wing
The span efficiency factor 푒 for a 퐶2 퐶 = 퐿 nonplanar wing can be larger than 퐷,푖 휋 푒 퐴푅 one, i.e. the induced drag can be less than the ideal value
Another “efficiency factor”, called the Oswald efficiency factor, 푒0, takes into account the variation with 퐶퐿 of the total drag, including the viscous profile drag. It is defined in practice by curve fitting a known total drag polar
2 퐶퐿 퐶퐷 = 퐶퐷푚푖푛 + 휋 푒0 퐴푅
Aerodynamics 133 Chapter 4: Thin airfoils and finite wings Generic wing
퐶퐷푚푖푛 ≈ 퐶퐷0
퐶퐷0: zero-lift drag coefficient
퐶퐿 C
퐶퐷
Aerodynamics 134 Chapter 4: Thin airfoils and finite wings Rectangular wing: effect of 퐴푅
퐶퐷,푖 푝:
푝 = 5
2 푝 = 퐴푅 휋
Measured drag polar for a rectangular wing with 푝 = 5 (left) and for rectangular wings of varying aspect ratio (right)
Aerodynamics 135 Chapter 4: Thin airfoils and finite wings Rectangular wing: effect of 퐴푅
퐴푅 varies typically from 6 to 22 for standard subsonic airplanes and gliders; 푝: 퐶 it has a strong influence on 퐶퐷,푖 퐷,푖
Let us consider two finite wings with same profile but different aspect ratio: 2 퐶퐿 퐶퐷 1 = 푐푑 + 휋 푒 퐴푅1 2 퐶퐿 퐶퐷 2 = 푐푑 + 휋 푒 퐴푅2 휋 Drag polar converted to 퐴푅 = 5 2
Aerodynamics 136 Chapter 4: Thin airfoils and finite wings Rectangular wing: effect of 퐴푅
Assume that the two wings operate at the same 퐶퐿; also, since the airfoil 푝: section is the same for both wings, 푐 푑 퐶 is the same. The variation of 훿 (and 푒) 퐷,푖 between the two wings is small P: Hence 2 퐶퐿 1 1 퐶퐷 1 = 퐶퐷 2 + − 휋 푒 퐴푅1 퐴푅2 i.e. the data of a wing with 퐴푅2 can be scaled to the case of wings of 휋 Drag polar converted to 퐴푅 = 5 any other aspect ratio 2
Aerodynamics 137 Chapter 4: Thin airfoils and finite wings Lift slope for a wing of general planform
푎0 푎 = 푎 휏 is the induced lift factor 1 + 0 (1 + 휏) 휋 퐴푅 휏 ∈ 0, 0.25
푝:
Prandtl’s (1921) rectangular wing data
Aerodynamics 138 Chapter 4: Thin airfoils and finite wings Lift slope for a wing of general planform
Downwash → effective angle of attack of finite wing and lift
If a section at 푦 = 푦0 of a finite wing were to have the same lift as the same profile of an infinite wing at angle of attack 훼2퐷, its angle of attack would have to be increased by 퐶퐿 푁 퐴푛 sin(푛휃0) 퐶퐿 훼푖(푦0) = 1 + σ2 푛 = 1 + 푔(푦0) . 휋퐴푅 퐴1 sin 휃0 휋퐴푅
Neglecting the correction factor 푔(푦0), two wings (with different 퐴푅’s) have the same lift 퐶퐿 of an infinite wing if their angles of attack are 퐶퐿 퐶퐿 훼1 ≈ 훼2퐷 + and 훼2 ≈ 훼2퐷 + 휋 퐴푅1 휋 퐴푅2
Aerodynamics 139 Chapter 4: Thin airfoils and finite wings Lift slope for a wing of general planform
푝: 푝:
퐶퐿 1 1 Lift data converted to 푝 = 5 using 훼1 = 훼2 + − 휋 퐴푅1 퐴푅2
Aerodynamics 140 Chapter 4: Thin airfoils and finite wings When can Prandt’s LLT be applied?
Prandtl’s classical lifting line theory yields reasonable results for straight wings at moderate to high aspect ratio. However, it is inappropriate for
- low-aspect-ratio straight wings (퐴푅 < 4, as a rule of thumb) - swept wings - delta wings
Aerodynamics 141 Chapter 4: Thin airfoils and finite wings Empirical corrections
Low-aspect-ratio straight wings
푎0 푎 = 푎 푎 2 푎 1 + 0 + 0 휋 퐴푅 휋 퐴푅
H.B. Helmbold (1942)
퐴푅
Aerodynamics 142 Chapter 4: Thin airfoils and finite wings Empirical corrections
Swept wings
푎 cos Λ 푎 = 0 푎 cos Λ 2 푎 cos Λ 1 + 0 + 0 휋 퐴푅 휋 퐴푅
D. Kuchemann (1978)
Aerodynamics 143 Chapter 4: Thin airfoils and finite wings Numerical methods
For low-aspect-ratio wings, swept wings, and delta wings, lifting-surface theory must be used. In modern aerodynamics, such lifting-surface theory is implemented by the vortex panel or the vortex lattice techniques.
Another approach is aerodynamic strip theory, which treats each section of the wing as 2D. It takes information from a 3D panel code or from LLT, and uses the effective angle of attack; thus, in some way, it includes the effect of the three- dimensionality of the wing.
Aerodynamics 144 Chapter 4: Thin airfoils and finite wings Numerical methods: vortex panel
Aerodynamics 145 Chapter 4: Thin airfoils and finite wings Numerical methods: vortex lattice
Aerodynamics 146 Chapter 4: Thin airfoils and finite wings Vortex lattice method (Thomas, 1976)
푥/푐
푥/푐
Aerodynamics 147 Chapter 4: Thin airfoils and finite wings Far field calculations
The same analytical results already obtained for lift and induced drag can be recovered by applying the integral form of the momentum equation over a large 푉∞ control volume (assuming inviscid, steady flow, with no body forces)
Tapez une équation ici. න 휌 푽 (푽 ∙ 풏) 푑푆 + න 푝 풏 푑푆 = 푭 푆 푆
Aerodynamics 148 Chapter 4: Thin airfoils and finite wings Far field calculations
Aerodynamics 149 Chapter 4: Thin airfoils and finite wings Far field calculations
For the simplest case (elliptic lift distribution), the Trefftz’s plane approach yields the same lift and drag already found, i.e. 휋 푏 휋 2 퐿 = 휌 푉∞ Γ0 퐷 = 휌 Γ 4 푖 8 0
However, the approach allows consideration of more complex configurations
Aerodynamics 150 Chapter 4: Thin airfoils and finite wings Vortex filaments and lifting line theory
Exercises
1. The rectangular vortex filament in the figure has strength G = 200 m2/s. The rectangle is in the plane A-B-C-D. Find the magnitude of the velocity in P.
2. Revise all the worked-out examples in the book by Anderson at the end of section 5.3.
3. Revise the completely solved problem proposed in the slides which follow. Then repeat the exercise using the excel worksheet provided by Dr. Joel Guerrero.
Aerodynamics 151 Chapter 4: Thin airfoils and finite wings Solved exercise
(slide 127; 휃0 = 휙)
Aerodynamics 152 Chapter 4: Thin airfoils and finite wings Solved exercise
slide 127
푠 = 푏/2
Aerodynamics 153 Chapter 4: Thin airfoils and finite wings Solved exercise
Aerodynamics 154 Chapter 4: Thin airfoils and finite wings Solved exercise
Aerodynamics 155 Chapter 4: Thin airfoils and finite wings Solved exercise
Aerodynamics 156 Chapter 4: Thin airfoils and finite wings Solved exercise
Aerodynamics 157 Chapter 4: Thin airfoils and finite wings Solved exercise
퐶퐷,푖
Aerodynamics 158 Chapter 4: Thin airfoils and finite wings Solved exercise
Aerodynamics 159 Chapter 4: Thin airfoils and finite wings