Airfoil Characteristics
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- Ὀρφεύς Γερμανός
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1 Thin Airfoi Theory
2 Airfoi harateristis y m Lift oeffient: Moment oefiient: enter of ressure: Aerodynami enter: ρ /
3 Thin Airfoi Theory Setu Symbos: y u (< v sinv' t hord u osu' Assumtions:. Airfoi is thin <<. Anges/soes are sma e.g. sin, os, soe ange 3. Airfoi ony sighty disturbs free stream u', v' << Kutta ondition? (used TE amber ½ Thikness t ( u ( u or u t t Non-enetration ondition? ernoui? 3
4 Thin Airfoi Theory - Simifiations ernoui: ( u / [( os u' ( sin v' u' v' u' v' os sin v ]/ Linearized Pressure oeffiient Assumtions:. Airfoi is thin <<. Anges/soes are sma e.g. sin, os, soe ange 3. Airfoi ony sighty disturbs free stream u', v' << v sinv' Kutta ondition: (, (, y u osu' (, (, -
5 Thin Airfoi Theory - Simifiations Eat: Eat: y v sinv' u osu' Sma disturbanes and anges: d dt v'(, u d d v'(, u d dt d d u (< t u t t Airfoi thin: v'(, d d dt d Linearized: y (, Likewise for ower surfae: v '(, d dt d d y Or: v '(, d dt d d y - u(, u(, - 5
6 A Soure Sheet Jum in norma veoity omonent A orte Sheet Jum in tangentia veoity omonent 6
7 Soving for the Fow Linearized robem: y y (, u(, / v '(, d dt d d Proosed Idea Fow Soution: iy y z y - u(, u(, - y - d Soure sheet vorte sheet ome veoity at z q( iγ ( d due to eement d : ( z ome veoity at z due to whoe sheet: W '( z q( iγ ( d ( iy ome veoity at y due to sheet: W' (, q( iγ ( d ( K So what? N.. Karamheti defines the vorte sheet strength with the oosite sign v'( q(, W '(, γ ( Im d ( u'(, W Re, '( q( γ d m ( ( 7
8 Soving for the Fow Linearized robem: y y (, u(, / v '(, y - u(, u(, - d dt d d Proosed Idea Fow Soution: y iy v' (, u' (, γ ( d ( q( q( γ d m ( y - d Soure sheet vorte sheet ( Kutta ondition: u' (, u'(, so q( γ ( q( γ ( d so ( ( d γ ( Nonenetration: d dt iγ ( q( d d d ( Pressure: Pressure Differene: q( γ ( (, d ( Soution order? Need q(? 8
9 9 Genera Agebrai Soution ( γ q d i d d d d t ( ( ( γ How to sove for γ( w/o seifying? q d i d d d d t ( ( ( γ add ( ( ( d i d d γ Kutta ( d d Non-enetration os ( / os ( / Write γ as Sove integra for eah term in Reate to for d d
10 Genera Agebrai Soution Fourier Series Soution gives: γ ( / os sin ( o n n sin( n m O / ( os Δ where: and we have that n d ( os( n d d Δ ( γ ( / ρ Substitute for γ( and evauate: ( mo m ρ O ρ Δd Δ ( d Δ ( ( os sind Substitute for γ( and evauate: mo ( (
11 Transferring the moment - onusions Δ Now: ( ( ( mo ( ( os n sin( n n sin So, Lift varies ineary with Lift urve soe is amber ony ats to infuene the zero ift ange of attak Thikness has no effet on ift and moment m O n / ( os m d ( os( n d d Lift ats Aerodynami enter is m enter of ressure is
12 Eame Symmetri Foi An airfoi has a straight amber ine defined as: / Determine the aerodynami harateristis and vorte sheet strength. / γ ( / u ( os n t ( d ( os( n d d ( ( m os ( o n sin( n sin n os ( / sin( os / ( /
13 Eame Paraboi Foi An airfoi has uer and ower surfaes defined as: /.5( / ( / u /.3( / ( / Determine the aerodynami harateristis / u ( os n t ( d ( os( n d d ( ( m 3
14 Eame 3 NAA A NAA airfoi has a amber ine given by the equations: 9 / 5 6 / ( / 36 ( / Determine the aerodynami harateristis / / / u ( os n t ( d ( os( n d d ( ( m d d 5 9 d d ( os ( os. ( os.77rad
15 ( os d ( os ( os os d ( os d.9 os d.85 ( os os d ( os os d. 38 Use Matab! (.79 m / o (.53.8 / u ( os n t ( d ( os( n d d ( ( m 5
16 omarison with data Summary of airfoi data Abbott, Ira H on Doenhoff, Abert E Stivers, Louis, Jr m/ htt://naa.ar.nasa.gov/re orts/95/naa-reort-8/ 6
17 Eame Heioter Rotor The oading distribution Δ is measured on a heioter rotor airfoi setion as a funtion of ange of attak. Estimate the hange in oading rodued by a degree hange in ange of attak. / u ( os n t ( d ( os( n d d ( ( m 7
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