Section 7, Lecture 2: " Supersonic Flow on Finite Thickness Wings! • Concept of Wave Drag! • Anderson, Chapter 4, pp. 174-179 ! 1! MAE 5420 - Compressible Fluid Flow! Supersonic Airfoils! • Recall that for a leading edge in Supersonic flow there is! γ=1.1! a maximum wedge angle at which the oblique shock wave! remains attached ! γ=1.3! • Beyond that angle! γ=1.1! γ=1.05! γ=1.4! shock wave becomes! γ=1.2! detached from! =1.3! γ leading edge! γ=1.4! % 2 M 2 sin2 1 ' −1 { 1 (βmax ) − } θmax = tan ) * ) tan β %2 + M 2 %γ + cos 2β '' * & ( max )& 1 & ( max )(( ( % M 2 ( 1) 4 16( 1) 8M 2 ( 2 1) M 4 ( 1)2 ' 1 −1 ( 1 γ − + ) − γ + + 1 γ − + 1 γ + β = cos ) * max 2 2γ M 2 &) 1 (* 2! MAE 5420 - Compressible Fluid Flow! Section 6: Home Work 1(continued) g = 1.25 MAE 5420 - Compressible Fluid Flow Section 6: Home Work 1(Part 2 continued) g = 1.4 MAE 5420 - Compressible Fluid Flow Supersonic Airfoils (cont’d)! • Normal Shock wave formed off the front of a blunt leading! γ=1.1! causes significant drag! Detached shock waveγ!=1.3! Localized normal shock wave! Credit: Selkirk College Professional Aviation Program! 3! MAE 5420 - Compressible Fluid Flow! Supersonic Airfoils (cont’d)! • To eliminate this leading edge drag caused by detached bow wave ! Supersonic wings are typically quite sharp atγ=1.1 the! leading edge ! • Design feature allows oblique wave to attachγ=1.3 to !the leading edge ! eliminating the area of high pressure ahead of the wing. ! • Double wedge or “diamond”! Airfoil section! Credit: Selkirk College Professional Aviation Program! 4! MAE 5420 - Compressible Fluid Flow! Supersonic Airfoils (cont’d)! •When supersonic airfoil is at positive ! angleγ=1.1 of! attack shock wave ! at the top leading edge is weakened ! and the one at the bottom is intensified. ! γ=1.3! • Result is the airflow over the top ! of the wing is now faster.! • Airflow will also be accelerated ! through the expansion fans. ! • The fan on the top will be stronger ! than the one on the bottom. ! • Result is faster flow and therefore ! Credit: Selkirk College Professional Aviation Program! lower pressure on the top of the airfoil.! • We already have all of the tools we need to analyze the flow on this wing! 5! MAE 5420 - Compressible Fluid Flow! Supersonic Flow on" Finite Thickness Wings at zero α! Drag = b[ p2 − p3 ]t • Symmetrical Diamond-wedge airfoil, zero angle of attack! t / 2 D = 2b p l sin(ε) − p l sin(ε) ⇒ sin(ε) = rag [ 2 3 ] l Drag = b[ p2 − p3 ]t 6! MAE 5420 - Compressible Fluid Flow! Supersonic Flow on" Finite Thickness Wings at zero α! Drag = b[ p2 − p3 ]t • Look at pressure difference, p2-p3! • Oblique shock wave from region 1->2 .. p2 >p∞" ∞ Εξπανσιον φaν φροµ ρεγιον 2---> 3 ... π3 < π2! ... p >p > Δ > 0 ! 7! MAE 5420 - Compressible Fluid Flow! 2 ∞ ραγ Supersonic Flow on" Finite Thickness Wings at zero α! Lift Fp α α Drag • Compare to Flat Plate! # p p & l − u % p p ( $ ∞ ∞ ' C = 0! CL = cosα D * γ 2 - M +, 2 ∞ ./ 8! MAE 5420 - Compressible Fluid Flow! Compare to wing in subsonic flow! 2! Mach 1! 3! A0 A1 A0 A0 A1 A0 • from Bernoulli! 2 2 _ dx " γ 2 % P p Cp q p 1 Cp M pdx = P0 0 = + max = $ + max ' ⇒ ∫ ∫ # 2 & 1 1 f [M ] P P 3 3 2 3 p = 0 = 0 dx dx pdx = P0 = −P0 ⇒ pdx =0 " γ 2 % f [M ] ∫ ∫ ∫ ∫ 1+ Cpmax M 2 2 f [M ] 1 f [M ] 1 #$ 2 &' 9! MAE 5420 - Compressible Fluid Flow! Compare to wing in subsonic flow (cont’d)! 2 2 dx • at zero lift .. Subsonic wings in! pdx = P ∫ 0 ∫ inviscid flow have No-drag!! 1 1 f [M ] 3 3 dx 2 dx 3 pdx = P = −P ⇒ pdx =0 ∫ 0 ∫ 0 ∫ ∫ 2 2 f [M ] 1 f [M ] 1 • Known as d'Alembert's Paradox … • Subsonic flow over wings … induced drag (drag due to lift) + viscous drag 10! MAE 5420 - Compressible Fluid Flow! Supersonic Wave Drag! • Finite Wings in Supersonic Flow have drag .. Even! at zero angle of attack and no lift and no viscosity…. “wave drag”! • Wave Drag coefficient is proportional to thickness ratio (t/c)! Drag p2 − p3 $ t ' C = = [ ] D wave _ & ) γ 2 % ( bcq p M c 2 ∞ • Supersonic flow over wings … induced drag (drag due to lift) + viscous drag + wave drag 11! MAE 5420 - Compressible Fluid Flow! Example: Symmetric Double-wedge" • 5° ramps, M∞ = 2.00! Airfoil! • Total chord: 2 meters! • Look at flow properties across ! each of the numbered regions! • Fineness ratio (t/c): 0.08749! 3! 5! 7! 1! 6! 2! 4! 12! MAE 5420 - Compressible Fluid Flow! Example: Symmetric Double-wedge" Airfoil (cont’d)! • Across region 1-2 … Oblique shock! δ2 = 5°>β2=34.302°, π2/π1=1.3154, M2=1.8213! M∞ = 2.00! 3! 5! 7! 1! 6! 2! 4! 13! MAE 5420 - Compressible Fluid Flow! Example: Symmetric Double-wedge" Airfoil (cont’d)! • Across region 2-4 … expansion fan! M2=1.8213! δ4 = -10°> π4/π2=0.5685, p4/π1=0.7478, M4=2.1848! M∞ = 2.00! 3! 5! 7! 1! 6! 2! 4! 14! MAE 5420 - Compressible Fluid Flow! Example: Symmetric Double-wedge" Airfoil (cont’d)! • Across region 1-3 … Oblique shock! δ3 = 5°>β3=34.302°, π3/π1=1.3154, M3=1.8213! M∞ = 2.00! 3! 5! 7! 1! 6! 2! 4! 15! MAE 5420 - Compressible Fluid Flow! Example: Symmetric Double-wedge" Airfoil (cont’d)! • Across region 3-5 … expansion fan! M3=1.8213! δ5 = -10°> π5/π3=0.5685, p5/π1=0.7478, M5=2.1848! M∞ = 2.00! 3! 5! 7! 1! 6! 2! 4! 16! MAE 5420 - Compressible Fluid Flow! Example: Symmetric Double-wedge" Airfoil … Drag! • Calculate Drag Coefficient! π2/π1=1.3154! M∞ = 2.00! π3/π1=1.3154! p /π =0.7478! 3! 5! 7! 4 1 1! p5/π1=0.7478! 2! 4! 6! 1.3154 + 1.3154 0.7478 + 0.7478 $ % 0.08749 )1 " p2 p3 % 1 " p4 p5 % , " − # + $ + ' − $ + ' . 2 2 *2 # p∞ p∞ & 2 # p∞ p∞ & - " t % C = = 1.4 2 D wave $ ' γ 2 # c& 2 M 2 =0.01774! 2 17! MAE 5420 - Compressible Fluid Flow! Example: Symmetric Double-wedge" Airfoil … Drag (cont’d)! • Plot wave drag! Mach 2.0! versus thickness ratio! 0° α! Drag p2 − p3 $ t ' C = = [ ] D wave _ & ) γ 2 % ( bcq p M c 2 ∞ Thickness ratio! 18! MAE 5420 - Compressible Fluid Flow! Example: Symmetric Double-wedge" Airfoil … Drag (cont’d)! • Look at mach number ! Effect on wave drag! Increasing mach! • Mach Number tends! to suppress wave drag! Thickness ratio! 19! MAE 5420 - Compressible Fluid Flow! Example: Symmetric Double-wedge" Airfoil … Drag (cont’d)! • How About The! Compression! effect of angle of! “Induced” drag! attack on drag ! Wave drag! +! α=0°' =! 20! MAE 5420 - Compressible Fluid Flow! Example: Symmetric Double-wedge" Airfoil … Drag (cont’d)! Mach 2.0! Total drag! Increasing t/c! 21! MAE 5420 - Compressible Fluid Flow! Example: Symmetric Double-wedge" Airfoil … Lift! • Calculate Zero α Lift Coefficient! π2/π1=1.3154! M∞ = 2.00! π3/π1=1.3154! p4/π1=0.7478! p /π =0.7478! 3! 5 1 5! 7! 1! 6! 2! 4! )1 " p2 p4 % 1 " p3 p5 % , + + − + . L 2 #$ p p &' 2 #$ p p &' C = ift! = !* ∞ !!!!!∞ ∞ ∞ - cos[5o ] = 0 •Makes sense! L wave _ γ ( ) bcq M 2 2 22! MAE 5420 - Compressible Fluid Flow! Pulling it All Together (1)! Consider airfoil! = wing half angle! @ positive! δ angle of attack, ! … length … L! • Lower forward! Ramp Angle ! … θ=δ + α" • Upper Forward Ramp Angle … θ=δ α" L! θ > 0 “oblique shock”! θ < 0“expansion fan”! 23! MAE 5420 - Compressible Fluid Flow! Pulling it All Together (2)! • Angle from! δ = wing half angle! 3-5 and 2-4! Is always convex! (expansion), 2δ" • Use Shock-Expansion Theory to calculate Ramp pressures Upper: { P3, P5} Lower: {P2, P4} 24! MAE 5420 - Compressible Fluid Flow! Pulling it All Together (3)! δ = wing half angle! • Upper: { P3, P5} Lower: {P2, P4} • Calculate Normal And Axial Pressure Forces on Wing … Normal Force! *$ L / 2 L / 2 ' $ L / 2 L / 2 ' - Fnormal = b ⋅ ,& P2 ⋅ − P3 ⋅ ) cosδ + & P4 ⋅ − P5 ⋅ ) cosδ / +% cosδ cosδ ( % cosδ cosδ ( . * L L - b ⋅ L = b ⋅ (P2 − P3 ) + (P4 − P5 ) = +*(P2 + P4 ) − (P3 + P5 ).- +, 2 2 ./ 2 25! MAE 5420 - Compressible Fluid Flow! Pulling it All Together (4)! δ = wing half angle! • Upper: { P3, P5} Lower: {P2, P4} • Calculate Normal And Axial Pressure Forces on Wing … Axial Force! *$ L / 2 L / 2 ' $ L / 2 L / 2 ' - Faxial = b ⋅ ,& P2 ⋅ − P4 ⋅ ) sinδ + & P3 ⋅ − P5 ⋅ ) sinδ / +% cosδ cosδ ( % cosδ cosδ ( . * L L - b ⋅ L = b ⋅ (P2 − P4 ) + (P3 − P5 ) tanδ = +*(P2 + P3 ) − (P4 + P5 ).- tanδ +, 2 2 ./ 2 26! MAE 5420 - Compressible Fluid Flow! Pulling it All Together (5)! δ = wing half angle! b ⋅ L Fnormal = "(P2 + P4 ) − (P3 + P5 )$ # % 2 b ⋅ L Faxial = "(P2 + P3 ) − (P4 + P5 )$ tanδ # % 2 • Transform to wind axis! To calculate Lift, Drag! Drag = Faxial cos(α) + Fnormal sin(α) Lift = Fnormal cos(α) − Faxial sin(α) 27! MAE 5420 - Compressible Fluid Flow! Pulling it All Together (6)! δ = wing half angle! b ⋅ L Fnormal = "(P2 + P4 ) − (P3 + P5 )$ # % 2 b ⋅ L Faxial = "(P2 + P3 ) − (P4 + P5 )$ tanδ # % 2 • Transform to wind axis! To calculate Lift, Drag! Drag = Faxial cos(α) + Fnormal sin(α) Lift = Fnormal cos(α) − Faxial sin(α) ( b ⋅ L + ( b ⋅ L + Drag = "(P2 + P3 ) − (P4 + P5 )$ tanδ ⋅ cos(α) + "(P2 + P4 ) − (P3 + P5 )$ ⋅sin(α) )* # % 2 ,- )* # % 2 ,- ( b ⋅ L + ( b ⋅ L + Lift = "(P2 + P4 ) − (P3 + P5 )$ ⋅ cos(α) − "(P2 + P3 ) − (P4 + P5 )$ tanδ ⋅sin(α) )* # % 2 ,- )* # % 2 ,- 28! MAE 5420 - Compressible Fluid Flow! Pulling it All Together (7)! • Collectδ = Termswing half on Drag angle Equation! ! ( b ⋅ L + ( b ⋅ L + Drag = "(P2 + P3 ) − (P4 + P5 )$ tanδ ⋅ cos(α) + "(P2 + P4 ) − (P3 + P5 )$ ⋅sin(α) )* # % 2 ,- )* # % 2 ,- ⇓ # b ⋅ L & # b ⋅ L & # b ⋅ L & # b ⋅ L & Drag = P tanδ ⋅ cos(α) + P ⋅sin(α) + P tanδ ⋅ cos(α) − P ⋅sin(α) $% 2 2 '( $% 2 2 '( $% 3 2 '( $% 3 2 '( # b ⋅ L & # b ⋅ L & # b ⋅ L & # b ⋅ L & + P ⋅sin(α) − P tanδ ⋅ cos(α) − P tanδ ⋅ cos(α) − P ⋅sin(α) = $% 4 2 '( $% 4 2 '( $% 5 2 '( $% 5 2 '( # b ⋅ L & # b ⋅ L & P P $% 2 2 '( $% 3 2 '( [sinδ ⋅ cos(α) + cosδ sin(α)] + [sinδ ⋅ cos(α) − cosδ sin(α)] + cosδ cosδ # b ⋅ L & # b ⋅ L & P P $% 4 2 '( $% 5 2 '( [cosδ ⋅sin(α) − sinδ cos(α)] − [sin(δ)cos(α) + cos(δ)⋅sin(α)] cosδ cosδ 29! MAE 5420 - Compressible Fluid Flow! Pulling it All Together (8)! • Collectδ Terms = wing! half angle! " b ⋅ L % " b ⋅ L % P P #$ 2 2 &' #$ 3 2 &' Drag = [sinδ ⋅ cos(α) + cosδ sin(α)] + [sinδ ⋅ cos(α) − cosδ sin(α)] + cosδ cosδ " b ⋅ L % " b ⋅ L % P P #$ 4 2 &' #$ 5 2 &' [cosδ ⋅sin(α) − sinδ cos(α)] − [sin(δ)cos(α) + cos(δ)⋅sin(α)] cosδ cosδ [sinδ ⋅ cos(α) + cosδ sin(α)] = sin(δ + α ) → → [cosδ ⋅sin(α) − sinδ cos(α)] = sin(δ − α ) Trigonometric Identity! b ⋅ L 2 Drag = P2 sin(δ + α ) + P3 sin(δ − α ) − P4 sin(δ − α) − P5 sin(δ + α ) = cosδ b ⋅ L ,(P2 − P5 )sin(δ + α ) + (P3 − P4 )sin(δ − α ).