Supersonic Airfoil Analysis - Virginia Techdevenpor/aoe3114/16 - Airfoil Analysis.pdf · Shapes...
Transcript of Supersonic Airfoil Analysis - Virginia Techdevenpor/aoe3114/16 - Airfoil Analysis.pdf · Shapes...
Supersonic Airfoil AnalysisSupersonic Airfoil Analysis
ShapesShapes
NACA‐0012Mach Number Contours
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DOUBLE WEDGE
BICONVEXBICONVEX
F‐35 Joint Strike FighterF 35 Joint Strike Fighter
TerminologyTerminology
pM∞, ρ∞, p∞, u∞
Angle of attack αo
p
g o
Span b
CoefficientsM∞,ρ∞,
p L
DCoefficients ρ∞,p∞,U∞
αoc
D
Pressure 221
∞∞
∞−≡
UppCp ρ
Lift 221
/
∞∞
≡UbcLCL ρ
Drag 221
/
∞∞
≡UbcDCD ρ
Shock‐Expansion TheoryShock Expansion TheoryM∞, p1
For geometric airfoil p2ρ∞,p∞,U∞
gsections in supersonic flow p3 p4
• Assume: (1) Each turn the supersonic flow makes can be computed separately from every other e g no wavecomputed separately from every other, e.g. no wave reflections back onto surface. (2) All waves are attached
• Method: (1) Break airfoil section into a sequence of implied l dd i bli h k t (2) Uor real sudden expansion or oblique shock turns. (2) Use
Prandtl Meyer or oblique shock theory to get pressures on flat airfoil sections. (3) Formulate results in terms of CL and CD
Example: M∞,ρ∞, p∞
Flat Plate Airfoilρ∞, p∞
αoo
/ bL2
21
/
/
∞∞
=
bcDMpbcLCL γ
221
/
∞∞
=MpbcDCD γ
Example M∞,ρ∞, p∞
Double Wedge Airfoilρ∞, p∞
α =0αo=0
/ bL2
21
/
/
∞∞
=
bcDMpbcLCL γ
221
/
∞∞
=MpbcDCD γ
Supersonic Thin Airfoil TheorySupersonic Thin Airfoil TheoryFor arbitrary airfoil sections in supersonic
M∞,ρ∞, p∞
flowρ∞ ∞
• Assumption: (1) Attached waves (2) Flow angles produced by airfoil are all smallangles produced by airfoil are all small.
• Allows all turns to be treated as isentropic. All i lifi d (li i d) i bAllows simplified (linearized) expression to be used to compute pressure changes.
Methods for computing a small angle turn at Mach 2Computing a Small Angle Turn at Mach 2
0.3
0.4Oblique-shock theoryPrandtl-Meyer theoryLinear theory
0.2
12
2
−
−=
MdM
pdp θγ
0
0.1
(p2-p
1)/(p 1)
-0.1
(
p1
0 3
-0.2θ
p2
-5 -4 -3 -2 -1 0 1 2 3 4 5-0.3
Turn angle degrees (positive convex)
Pressure CoefficientPressure CoefficientM∞,ρ∞, p∞
2− dMdp θγ ρ∞ ∞
12 −=
Mpp γ
Pressure CoefficientPressure CoefficientM∞,ρ∞, p∞
2− dMdp θγ ρ∞ ∞
12 −=
Mpp γ
Pressure CoefficientPressure Coefficient
M∞,2
=Cpα
ρ∞, p∞12 −MCp
α positive for netconcave turn from M∞
Decomposing y
dxyyd lu
c)(
21 +
=αdxyyd lu
t)(
21 −
=αp g
The Airfoilαo
x
dx
M∞
αo
Lift)(2)(2 00 +−
=++−
= CC tcpl
tcpu
ααααααLift 11 22 −− ∞∞ MM
plpu
Drag )(2)(2 00 +−=
++−= CC tc
pltc
puααααααDrag11 22 −− ∞∞ MM
plpu
Summaryy
dxyyd lu
c)(
21 +
=αdxyyd lu
t)(
21 −
=α
Summaryαo
x
dx
• Lift varies linearly with angle of 1
)(22
0 ++−=
MC tc
puααα
M∞
αo
y gattack
• The lift curve slope decreases 1
)(21
20
2
+−=
−∞
MC
M
tcpl
αααp
with Mach number• Camber and thickness have no
41
20
2
=
−∞
C
M
Lα
effect on lift, and only add drag• Drag goes as the square of )(4
1222
0
2
++=
−∞
C
M
tcD
ααα g g qangle of attack12 −∞M
D
Example y
⎟⎠⎞
⎜⎝⎛ −−=⎟
⎠⎞
⎜⎝⎛ −=
cx
cx
cy
cx
cx
cy lu 11.011.0p
Biconvex Airfoil
αox
⎠⎝⎠⎝ cccccc
Find CL and CD as f ti f
M∞
αofunctions of αoFind Cpu and Cpl as functions of x and αo
dxyyd lu
t)(
21 −
=α
dxyyd lu
c)(
21 +
=α
Example y
⎟⎠⎞
⎜⎝⎛ −−=⎟
⎠⎞
⎜⎝⎛ −=
cx
cx
cy
cx
cx
cy lu 11.011.0
Biconvex Airfoil
αox
⎠⎝⎠⎝ cccccc
Find CL and CD as f ti f
M∞
αo
0/2.01.0 =−= ct cx ααfunctions of αoFind Cpu and Cpl as functions of x and αo
1
420
−=
∞MCL
α
1
)(42
2220
−
++=
∞MC tc
Dααα
Variationsy
c/2 c/2Double Wedge Airfoil
αox
t
c/2 c/2
M∞
αo
Variations – Airfoil DesignVariations Airfoil Design