Thin Airfoil Theory, Lý thuyết cánh, tài liệu dành cho sinh viên Kỹ thuật Hàng không Vũ trụ.Tính toán, thiết kế và vẽ biên dạng cánh. Giúp tạo cơ sở kiến thức về cánh để phục vụ cho việc thiết kế cánh trong máy bay, máy thủy khí,...
1 • Lift coeffcient: • Moment coeficient: • Center of pressure: • Aerodynamic Center: x y V l m cVlC l 2 2 1 / 22 2 1 / cVmC m Point about which moment is zero Point about which moment does not change with angle of attack Thin Airfoil Theory 2 Thin Airfoil Theory – Setup Non-penetration condition? Kutta condition? Bernoulli? Assumptions: 1. Airfoil is thin << c 2. Angles/slopes are small e.g. sin , cos 1, slope angle 3. Airfoil only slightly disturbs free stream u', v' << V V u t c l (<0) u=V cos +u' v=V sin +v' x Chord c Camber tcl tcu lut luc or )( )( 2 1 2 1 ½ Thickness y Symbols: (cusped TE) 3 Thin Airfoil Theory - Simplifications Bernoulli: Kutta Condition: u=V cos +u' v=V sin +v' x c V(c,0 + ) V(c,0 - ) y Assumptions: 1. Airfoil is thin << c 2. Angles/slopes are small e.g. sin , cos 1, slope angle 3. Airfoil only slightly disturbs free stream u', v' << V 4 Thin Airfoil Theory - Simplifications V u t c l (<0) u=V cos +u' v=V sin +v' x tcl tcu y V x y Exact: Linearized: y=0 + y=0 - u(c,0 + )=u(c,0 - ) VxuxC p /)0,(2)0,( Non-Penetration Condition: 5 A Source Sheet Jump in normal velocity component (numerically equal to half the sheet strength) A Vortex Sheet Jump in tangential velocity component (numerically equal to half the sheet strength) Solving for the Flow V x y Linearized problem: y=0 + y=0 - u(c,0 + )=u(c,0 - ) VxuxC p /)0,(2)0,( dx d dx d V xv tc )0,(' V x iy Proposed Ideal Flow Solution: y=0 + y=0 - x 1 dx 1 z Source sheet + vortex sheet 7 Solving for the Flow V x y Linearized problem: y=0 + y=0 - u(c,0 + )=u(c,0 - ) VxuxC p /)0,(2)0,( dx d dx d V xv tc )0,(' V x iy Proposed Ideal Flow Solution: y=0 + y=0 - x 1 dx 1 Source sheet + vortex sheet V xq dx xx x VV xv c 2 )( )( )( 2 1)0,(' 1 0 1 1 V x dx xx xq VV xu c 2 )( )( )( 2 1)0,(' 1 0 1 1 Kutta condition: 0)(so 2 )( )( )( 2 1 2 )( )( )( 2 1 so )0,(')0,(' 1 0 1 1 1 0 1 1 c V c dx xc xq VV c dx xc xq VV cu V cu cc Non- penetration: Pressure: Pressure Difference: 8 General Algebraic Solution V xq dx xx xi Vdx d dx d c tc 2 )( )( )( 2 1 1 0 1 1 V xq dx xx xi Vdx d dx d c tc 2 )( )( )( 2 1 1 0 1 1 Non-penetration V x 0 c 0 )cos1(/ )cos1(/ 1 2 1 1 2 1 cx cx 9 General Algebraic Solution Fourier Series Solution gives: 1 )sin(2 sin cos1 )2(/)( n no nBBV where: 0 )cos()( 2 dn dx d B c n V x 0 c 0 )cos1(/ 2 1 cx p C m O l 10 Transferring the moment - Conclusions 0 )cos()( 2 dn dx d B c n V x 0 c 0 )cos1(/ 2 1 cx m O l m x )(2 10 BBC l )( 21 4 1 4 1 BBCC lmO Now: 1 0 )sin(4 sin cos122 )( n np nB B C [...]... cos C p ( ) 4 Bn sin( n ) sin n 1 16 Comparison with Exact NACA 0012 NACA 0012 1.5 2 2o angle of attack o Cp at 2 4o angle of attack 1 o Cp at 4 1.5 Change in Cp Cp Cp 0.5 Thin airfoil theory 1 0 0.5 -0.5 -1 0 0.2 0.4 0.6 0.8 0 0 1 0.2 0.4 0.8 x/c x/c 0.05 0.05 y/c y/c 0.6 0 -0.05 0 -0.05 0.1 0.2 0.3 0.4 0.5 x/c 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 x/c 0.6 0.7 0.8 0.9 17 ... B1 ) x C mx ( c 1 )Cl 1 ( B1 B2 ) 4 4 16 Comparison with data Summary of airfoil data Abbott, Ira H Von Doenhoff, Albert E Stivers, Louis, Jr Cl Cmc/4 http://naca.larc.nasa.gov/re ports/1945/naca-report-824/ 15 Example 4 – Helicopter Rotor The loading distribution Cp is measured on a helicopter rotor airfoil section as a function of angle of attack Estimate the change in loading produced... Foil An airfoil has a straight camber line defined as: c / c 0 Determine the aerodynamic characteristics and vortex sheet strength t c u l 0 V x / c (1 cos ) 1 2 c 0 x 2 d Bn c ( ) cos(n )d 0 dx Cl 2 ( B0 B1 ) x C mx ( c 1 )Cl 1 ( B1 B2 ) 4 4 1 cos ( ) / V (2 Bo ) 2 Bn sin(n ) sin n 1 11 Example 2 – Parabolic Foil An airfoil. .. aerodynamic characteristics x / c (1 cos ) 1 2 c 0 x 2 d Bn c ( ) cos(n )d 0 dx Cl 2 ( B0 B1 ) x C mx ( c 1 )Cl 1 ( B1 B2 ) 4 4 12 Example 3 – NACA 2412 A NACA 2412 airfoil has a camber line given by the equations: c c 1 10 1 90 x / c ( x / c) 2 2 45 x / c 36 ( x / c ) 2 2 16 2 4 10 4 10 4 x c 1 10 0 x / c (1 cos ) 1 2 c 0 x V 0 x/c 4 . change with angle of attack Thin Airfoil Theory 2 Thin Airfoil Theory – Setup Non-penetration condition? Kutta condition? Bernoulli? Assumptions: 1. Airfoil is thin << c 2. Angles/slopes. Thickness y Symbols: (cusped TE) 3 Thin Airfoil Theory - Simplifications Bernoulli: Kutta Condition: u=V cos +u' v=V sin +v' x c V(c,0 + ) V(c,0 - ) y Assumptions: 1. Airfoil is thin <<. small e.g. sin , cos 1, slope angle 3. Airfoil only slightly disturbs free stream u', v' << V 4 Thin Airfoil Theory - Simplifications V u t c l (<0) u=V cos +u' v=V sin +v' x tcl tcu y V x y Exact: Linearized: y=0 + y=0 - u(c,0 + )=u(c,0 - ) VxuxC p /)0,(2)0,( Non-Penetration