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 ). 2 cosδ - /
30! MAE 5420 - Compressible Fluid Flow! Pulling it All Together (9)! • Similarδ Reduction= wing half for angle Lift!!
' b ⋅ L * ' b ⋅ L * Lift = "(P2 + P4 ) − (P3 + P5 )$ ⋅ cos(α) − "(P2 + P3 ) − (P4 + P5 )$ tanδ ⋅sin(α) () # % 2 +, () # % 2 +, ⇓
"P2 ⋅[cosδ cos(α) − sinδ ⋅sin(α)] + $ "P2 ⋅ cos(δ + α ) + $ 0 1 0 1 b ⋅ L 0−P3 [cosδ cos(α) + sinδ ⋅sin(α)] +1 b ⋅ L −P3 cos(δ − α ) + = = 0 1 = 2 cosδ 0P cosδ cos(α) + sinδ ⋅sin(α) + 1 2 cosδ 0P cos δ − α + 1 0 4 [ ] 1 0 4 ( ) 1 0 1 #−P5 ⋅[cosδ cos(α) − sinδ ⋅sin(α)] % #0−P5 ⋅ cos(δ + α ) %1 b ⋅ L "(P2 − P) ⋅ cos(δ + α ) + (P4 − P3 )cos(δ − α )$ 2 cosδ # 5 %
[cosδ ⋅ cos(α) + sinδ sin(α)] = sin(δ − α ) → → Trigonometric Identity! [cosδ ⋅ cos(α) − sinδ sin(α)] = sin(δ + α )
31! MAE 5420 - Compressible Fluid Flow! Pulling it All Together (10)! • Lift/Dragδ = Coefficientswing half angle! !
b ⋅ L Drag = %(P2 − P5 )sin(δ + α ) + (P3 − P4 )sin(δ − α )' 2 cosδ & ( b ⋅ L Lift = %(P2 − P5 )⋅ cos(δ + α ) + (P4 − P3 )cos(δ − α )' 2 cosδ & (
!CD $ 1 !Drag$ b ⋅ L 1 ! (P2 − P5 )sin(δ + α ) + (P3 − P4 )sin(δ − α ) $ # & = # & = ⋅ # & = C γ 2 Lift 2 cosδ γ 2 P − P ⋅ cos δ + α + P − P cos δ − α " L % P M ⋅(b ⋅ L) " % P M ⋅(b ⋅ L) "( 2 5 ) ( ) ( 4 3 ) ( )% 2 ∞ ∞ 2 ∞ ∞
! - P2 P5 0 - P3 P4 0 $ - 0 # − sin(δ + α ) + − sin(δ − α ) & / P P 2 / P P 2 !CD $ / 1 1 2 # . ∞ ∞ 1 . ∞ ∞ 1 & # & = / ⋅ 2 C 2 cosδ γ 2 # & " L % - P2 P5 0 - P4 P3 0 / M ∞ 2 # & . 2 1 / − 2 ⋅ cos(δ + α ) + / − 2 cos(δ − α ) "#. P∞ P∞ 1 . P∞ P∞ 1 %&
32! MAE 5420 - Compressible Fluid Flow! Pulling it All Together (11)!
! + P2 P5 . + P3 P4 . $ δ+ = wing half. #angle−! sin(δ + α ) + − sin(δ − α ) & - P P 0 - P P 0 !CD $ - 1 1 0 # , ∞ ∞ / , ∞ ∞ / & # & = - ⋅ 0 C 2 cosδ γ 2 # & " L % + P2 P5 . + P4 P3 . - M ∞ 0 # & , 2 / - − 0 ⋅ cos(δ + α ) + - − 0 cos(δ − α ) "#, P∞ P∞ / , P∞ P∞ / %&
33! MAE 5420 - Compressible Fluid Flow! Pulling it All Together (12)!
! + P2 P5 . + P3 P4 . $ δ += wing half. angle# −! sin(δ + α ) + − sin(δ − α ) & - P P 0 - P P 0 !CD $ - 1 1 0 # , ∞ ∞ / , ∞ ∞ / & # & = - ⋅ 0 C 2 cosδ γ 2 # & " L % + P2 P5 . + P4 P3 . - M ∞ 0 # & , 2 / - − 0 ⋅ cos(δ + α ) + - − 0 cos(δ − α ) "#, P∞ P∞ / , P∞ P∞ / %& • Check against flat plate! Formulae … (δ=0)!
# - Pl Pu 0 - Pu Pl 0 & % − sin(α ) − − sin(α ) ( / P P 2 / P P 2 #P3 = P5 = Pu & #CD & 1 1 % . ∞ ∞ 1 . ∞ ∞ 1 ( → δ = 0 → % ( → % ( → ⋅ → P = P = P C 2 γ 2 % ( $ 2 4 l ' $ L ' - Pl Pu 0 - Pl Pu 0 M ∞ % ( 2 / − 2 ⋅ cos(α ) + / − 2 cos(α ) $%. P∞ P∞ 1 . P∞ P∞ 1 '(
# - Pl Pu 0 & % − sin(α ) ( / P P 2 #CD & 1 % . ∞ ∞ 1 ( → % ( = ...... Q.E.D C γ 2 % ( $ L 'δ =0 - Pl Pu 0 M ∞ % ( 2 / − 2 ⋅ cos(α ) $%. P∞ P∞ 1 '(
34! MAE 5420 - Compressible Fluid Flow! Example: Symmetric Double-wedge"
Airfoil … Drag (cont’d)!
Mach 2.0! Total drag!
Increasing t/c!
35! MAE 5420 - Compressible Fluid Flow! Example: Symmetric Double-wedge"
Airfoil … Drag (cont’d)!
Lift Coefficient Climbs ! • How About The! Almost Linearly with α! effect of angle of! attack on Lift !
+! =!
36! MAE 5420 - Compressible Fluid Flow! Example: Symmetric Double-wedge"
Airfoil … L/D! • For Inviscid flow! Supersonic! t/c = 0.0175! Lift to drag ratio! almost infinite! for very thin! M = 3.0! airfoil!
• But airfoils do not! +! fly in inviscid flows! =!
37! MAE 5420 - Compressible Fluid Flow! Example: Symmetric Double-wedge"
Airfoil … L/D (cont’d)! • Friction effects! M = 3.0! have small effect! on Nozzle flow! or flow in “large! + ! t/c = 0.0175! “ducts”! skin friction! • But contribute ! significantly! to reduce the ! performance of ! supersonic wings=!!
38! MAE 5420 - Compressible Fluid Flow! Next section: Overview of Turbulent Compressible Boundary Layers and Friction Induced Drag !
39! MAE 5420 - Compressible Fluid Flow! Section 7: Home Work
P"=26.436 kPa • Calculate Freestream Mach Number 3 1 5 10°
2 10° 7 4 6 • Assume γ=1.40
!=12° P04sensed=143.062kPa Shock wave P4=30.454 kPa
63 MAE 5420 - Compressible Fluid Flow