Incompressible Flow over Airfoils - SNUaancl.snu.ac.kr/aancl/lecture/up_file/_1444613570_Chapter...
Transcript of Incompressible Flow over Airfoils - SNUaancl.snu.ac.kr/aancl/lecture/up_file/_1444613570_Chapter...
Aerodynamics 2015 fall - 1 -
Incompressible Flow over Airfoils
Road map for Chap. 4
Aerodynamics 2015 fall - 2 -
< 4.1 Introduction >
Incompressible flow over airfoils
Incompressible Flow over Airfoils
Prandtl (20C 초) Airfoil (2D)
Wind (3D)
Body
Airfoil : any section of the wing cut by a plane normal to y-axis
Aerodynamics 2015 fall - 3 -
< 4.2 Airfoil Nomenclature >
NACA (National Advisory Committee for Aeronautics) series
Incompressible Flow over Airfoils
Thickness
Camber
Leading edge
Mean camber line
Trailing edge
Chord lineUpper surface
Lower surface
Aerodynamics 2015 fall - 4 -
< 4.2 Airfoil Nomenclature >
NACA (National Advisory Committee for Aeronautics) series
Incompressible Flow over Airfoils
NACA 4-digit series
* NACA2412
2 : max. camber = 2% of the chord4 : the location of max. camber = 40% of the chord12 : max. thickness = 12% of the chordIf the airfoil is symmetric, it becomes NACA00XX
NACA 5-digit series
* NACA230122 : 2*0.3/2 = 0.3 design CL
30 : 30/2 % = the location of max. camber12 : max. thickness = 12% of the chord
Aerodynamics 2015 fall - 5 -
< 4.2 Airfoil Nomenclature >
NACA (National Advisory Committee for Aeronautics) series
Incompressible Flow over Airfoils
6-digit series laminar flow airfoil
* NACA65-218
6 : series designation5 : min. pressure location = 50% of the chord2 : design CL= 0.218 : max. thickness = 18% of the chord
Other notations
* SC0195
* VR12
Aerodynamics 2015 fall - 6 -
< 4.3 Airfoil Characteristics >
Incompressible Flow over Airfoils
* 1930~40 NASA carried numerous experiments on NACA airfoil characteristics(Measured Cl, Cd, Cm 2-D data)
* In the future, new airfoils should be designed and tested(consideration of aerodynamic, dynamic & acoustic limitation)
* Typical lift characteristics of an airfoilStall
Stall angle (12~18deg)
Zero lift angle
Maximum lift coefficient
: angle of attack
SepatationDynamic stall
How to measure Cl, Cd, Cm?
a0 =
Aerodynamics 2015 fall - 7 -
< 4.3 Airfoil Characteristics >
Incompressible Flow over Airfoils
[Def.] a, angle of attack : the angle between the freestream velocity and the chord
[Note] 1. a0 is not usually a function of Re.2. Cl,max is dependent on Re.
Aerodynamics 2015 fall - 8 -
< 4.3 Airfoil Characteristics >
Typical drag & pitching moment characteristics
Incompressible Flow over Airfoils
* Aerodynamic drag = Pressuredrag
Skin frictiondrag
(form drag)
Profile drag
* AC (Aerodynamic Center)[Def.] The point about which the moment is independent of AOA
Subsonic : AC=c/4Supersonic : AC=c/2
+
Sensitive to Re.
Aerodynamics 2015 fall - 9 -
< 4.4 Vortex Sheet >
Kutta-Joukowski Theorem
Incompressible Flow over Airfoils
* Kutta (German), Joukowski(Russia)
* Incompressible, inviscid flow
L = rvG
* G : positive clockwise
G
LiftG
Vortex filament of strength G
Aerodynamics 2015 fall - 10 -
< 4.4 Vortex Sheet >
Incompressible Flow over Airfoils
* g(s) = the strength of vortex sheet
per unit length along s
* From Biot-Savart Law
* Velocity potential for vortex flow
* Velocity potential at P
Aerodynamics 2015 fall - 11 -
< 4.4 Vortex Sheet >
Incompressible Flow over Airfoils
* Circulation around the dashed path
* If
(Note)
The local strength of the vortex sheet is equal to the difference (jump) in
tangential velocity across the vortex sheet
Aerodynamics 2015 fall - 12 -
< 4.4 Vortex Sheet >
(Note)
“Vortex sheet method” is more than just a mathematical device; it also has
a physical meaning
ex. : Replacing the boundary layer ( ) with a vortex sheet
Incompressible Flow over Airfoils
* “Vortex Sheet” - Application for inviscid, incompressible flow
* Calculate g(s) to form the streamlines with a give airfoil shape
Aerodynamics 2015 fall - 13 -
< 4.5 The Kutta Condition >
Incompressible Flow over Airfoils
* For a circular cylinder,
* For a given a, should have only one solution
?
Aerodynamics 2015 fall - 14 -
< 4.5 The Kutta Condition >
Incompressible Flow over Airfoils
* From the experiments, we know that the velocity at the trailing-edge in
finite. Kutta Condition
* The circulation around the airfoil is the value to ensure that the flow
smoothly leaves the trailing edge.
g(TE)=V1-V2=0
V(TE)=finite
Aerodynamics 2015 fall - 15 -
< 4.6 Kelvin’s Circulation Theorem >
Incompressible Flow over Airfoils
* Assume)
The time rate of change of circulation around a closed curve
consisting of the same fluid elements is zero
1. Inviscid
2. Incompressible
3. No body forces
Ex) Starting vortex
[ at rest ] [ after the start ]
Aerodynamics 2015 fall - 1 -
< 4.7 Classical Thin Airfoil Theory >
The Symmetric Airfoil
Incompressible Flow over Airfoils
* Assumptions
i) The camber line is one of the streamlinesii) Small maximum camber and thickness relative to the chordiii) Small angle of attack
i) Find g(s)ii) Use Kutta-Joukowski theorem, L’=rVG
* Purposes
Aerodynamics 2015 fall - 2 -
< 4.7 Classical Thin Airfoil Theory >
The Symmetric Airfoil
Incompressible Flow over Airfoils
* The component of free-stream velocity normal to the mean camber line at P
From small angle assumption
Aerodynamics 2015 fall - 3 -
< 4.7 Classical Thin Airfoil Theory >
The Symmetric Airfoil
Incompressible Flow over Airfoils
* If the airfoil is thin,
: velocity normal to the camber lineinduced by the vortex sheet
: velocity normal to the chord lineinduced by the vortex sheet
* The velocity at point x by the elemental vortex at point x
* The velocity at point x by all the elemental vortices along the chord line
Aerodynamics 2015 fall - 4 -
< 4.7 Classical Thin Airfoil Theory >
The Symmetric Airfoil
Incompressible Flow over Airfoils
* The sum of the velocity components normal to the surface at all point along the vortex sheet is zero
The fundamental equation of thin airfoil theory
Aerodynamics 2015 fall - 5 -
< 4.7 Classical Thin Airfoil Theory >
The Symmetric Airfoil
Incompressible Flow over Airfoils
* Sysmmetric airfoil no camber,
* Transform variable x into q
, ,
Aerodynamics 2015 fall - 6 -
< 4.7 Classical Thin Airfoil Theory >
The Symmetric Airfoil
Incompressible Flow over Airfoils
* Check Kutta condition
By L’Hospital’s rule
Indeterminant form
Aerodynamics 2015 fall - 7 -
< 4.7 Classical Thin Airfoil Theory >
The Symmetric Airfoil
Incompressible Flow over Airfoils
* Since we get g(q), now calculate G, L
* Lift :
* Lift coefficient :
* Lift slope :
Lift coefficient is linearly proportional to angle of attack.
Aerodynamics 2015 fall - 8 -
< 4.7 Classical Thin Airfoil Theory >
The Symmetric Airfoil
Incompressible Flow over Airfoils
* The moment about the leading edge
Aerodynamics 2015 fall - 9 -
< 4.7 Classical Thin Airfoil Theory >
The Symmetric Airfoil
Incompressible Flow over Airfoils
* The moment coefficient
* Aerodynamic center is located at c/4 for incompressible, inviscid and symmetric airfoil (true in real world)
* Center of pressure : the point at which the moment is zeroAerodynamic center : the point at which the moment is independent of aoa
Aerodynamics 2015 fall - 10 -
< 4.8 The Cambered Airfoil >
Incompressible Flow over Airfoils
* From thin airfoil theory,
* For cambered airfoil,
* The solution becomes
Fourier series term due to camber
Leading term for symmetric airfoil
Transformx into q
…… (a)
…… (b)
…… (c)
Aerodynamics 2015 fall - 11 -
< 4.8 The Cambered Airfoil >
Incompressible Flow over Airfoils
* Substitute (c) into (b)
By using the integral standard form
Aerodynamics 2015 fall - 12 -
< 4.8 The Cambered Airfoil >
Incompressible Flow over Airfoils
For Fourier cosine series,
[Note] given Determine g(q) to make the camber line a streamline with A0, An
+ Kutta condition, g(p)=0
Aerodynamics 2015 fall - 13 -
< 4.8 The Cambered Airfoil >
Incompressible Flow over Airfoils
* The total circulation due to the entire vortex sheet
,By using
Aerodynamics 2015 fall - 14 -
< 4.8 The Cambered Airfoil >
Incompressible Flow over Airfoils
Lift slope,
* Lift coefficient for a cambered thin airfoil
Aerodynamics 2015 fall - 15 -
< 4.8 The Cambered Airfoil >
Incompressible Flow over Airfoils
[Note]
* Lift slope is 2p for any shape airfoil
* Zero lift angle :
* Lift coefficient for a cambered thin airfoil
Aerodynamics 2015 fall - 16 -
< 4.8 The Cambered Airfoil >
Incompressible Flow over Airfoils
* The total moment about the leading edge
* Moment coefficient
A1 & A2 both are independent of aoa The quarter-chord is the aerodynamic center for a cambered airfoil
Aerodynamics 2015 fall - 17 -
< 4.8 The Cambered Airfoil >
Incompressible Flow over Airfoils
* The center of pressure
Not a convenient point
Aerodynamics 2015 fall - 18 -
< 4.8 The Cambered Airfoil >
The influence of camber on the thin airfoil
Incompressible Flow over Airfoils
* The cambered airfoil * The symmetric airfoil
Aerodynamics 2015 fall - 1 -
< 4.10 The Vortex Panel Method >
Incompressible Flow over Airfoils
* Thin airfoil theory
- Closed form- Limited to thin airfoil,
* Exactly same idea of thin airfoil theory, but no closed form g(s) solve numerically
* Panel method
- Vortex panel- Source panel non-lifting cases
Aerodynamics 2015 fall - 2 -
< 4.10 The Vortex Panel Method >
Incompressible Flow over Airfoils
* The velocity potential at P due to j-th panel
Controlpoint
Boundarypoint P(x,y)
(xj,yj)
J-1 j
J+1
x
y
qj
* Let’s put point P at the control point of i-th panel
Aerodynamics 2015 fall - 3 -
< 4.10 The Vortex Panel Method >
Incompressible Flow over Airfoils
* At the control points, the normal component of velocity is zero.
- The component of V normal to i-th panel
- The normal component of induced velocity at (xi, yi)
= : f (panel geometry)
Aerodynamics 2015 fall - 4 -
< 4.10 The Vortex Panel Method >
Incompressible Flow over Airfoils
* Boundary condition :
+ Kutta condition :
i
i+1
* Now, we have (n+1) eq. with n unknowns ignore one of control points
* Total circulation :
* Lift :
Inside the solid surface
* The flow velocity tangent to the surface = g
ui,1
ui,2
Aerodynamics 2015 fall - 5 -
< 4.12 The Flow over an Airfoil – the Real Case >
Stall
Leading-edge stall
Incompressible Flow over Airfoils
Flow separation takes place
over the entire top surface
of the airfoil after occurring
at the leading edge
Aerodynamics 2015 fall - 6 -
< 4.12 The Flow over an Airfoil – the Real Case >
Stall
Trailing-edge stall
Incompressible Flow over Airfoils
α = 5° α = 10° α = 15° α =22.5°
Flow separation takes place from the trailing edge at
thicker airfoils than leading-edge stall
Aerodynamics 2015 fall - 7 -
< 4.12 The Flow over an Airfoil – the Real Case >
Stall
Thin airfoil stall
Incompressible Flow over Airfoils
Leading-edge stall
Flow separation takes place
over the entire surface of
the airfoil after occurring at
the leading edge
Aerodynamics 2015 fall - 8 -
< 4.12 The Flow over an Airfoil – the Real Case >
Stall
Lift-coefficient curves
Incompressible Flow over Airfoils
10 20
1.0
0.5
1.5
Lif
t co
effi
cien
t
α, degrees
Leading-edge stall
Trailing-edge stall
Thin airfoil stall
Aerodynamics 2015 fall - 9 -
Incompressible Flow over Airfoils
< 4.12 The Flow over an Airfoil – the Real Case >
High-lift devices
Leading edge slat
Trailing edge flap
Aerodynamics 2015 fall - 10 -
Incompressible Flow over Airfoils
< 4.12 The Flow over an Airfoil – the Real Case >
High-lift devices
Trailing-edge flap (plain type)
More camber → Higher lift
Aerodynamics 2015 fall - 11 -
Incompressible Flow over Airfoils
< 4.12 The Flow over an Airfoil – the Real Case >
High-lift devices
Effect of slats and flaps