Incompressible Flow over Airfoils - SNUaancl.snu.ac.kr/aancl/lecture/up_file/_1444613570_Chapter 4.pdf · Aerodynamics 2015 fall - 3 - < 4.2 Airfoil Nomenclature > NACA (National

Aerodynamics 2015 fall - 1 -

Incompressible Flow over Airfoils

Road map for Chap. 4


< 4.1 Introduction >

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


< 4.2 Airfoil Nomenclature >

NACA (National Advisory Committee for Aeronautics) series


Thickness

Camber

Leading edge

Mean camber line

Trailing edge

Chord lineUpper surface

Lower surface





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





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


< 4.3 Airfoil Characteristics >


* 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 =




[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.



Typical drag & pitching moment characteristics


* 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.


< 4.4 Vortex Sheet >

Kutta-Joukowski Theorem


* Kutta (German), Joukowski(Russia)

* Incompressible, inviscid flow

L = rvG

* G : positive clockwise

G

LiftG

Vortex filament of strength G




* 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




* 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



(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


* “Vortex Sheet” - Application for inviscid, incompressible flow

* Calculate g(s) to form the streamlines with a give airfoil shape


< 4.5 The Kutta Condition >


* For a circular cylinder,

* For a given a, should have only one solution

?


< 4.5 The Kutta Condition >


* 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


< 4.6 Kelvin’s Circulation Theorem >


* 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 ]


< 4.7 Classical Thin Airfoil Theory >

The Symmetric Airfoil


* 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





* The component of free-stream velocity normal to the mean camber line at P

From small angle assumption





* 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





* 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





* Sysmmetric airfoil no camber,

* Transform variable x into q

, ,





* Check Kutta condition

By L’Hospital’s rule

Indeterminant form





* Since we get g(q), now calculate G, L

* Lift :

* Lift coefficient :

* Lift slope :

Lift coefficient is linearly proportional to angle of attack.





* The moment about the leading edge





* 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


< 4.8 The Cambered Airfoil >


* 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)




* Substitute (c) into (b)

By using the integral standard form




For Fourier cosine series,

[Note] given Determine g(q) to make the camber line a streamline with A0, An

+ Kutta condition, g(p)=0




* The total circulation due to the entire vortex sheet

,By using




Lift slope,

* Lift coefficient for a cambered thin airfoil




[Note]

* Lift slope is 2p for any shape airfoil

* Zero lift angle :

* Lift coefficient for a cambered thin airfoil




* 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




* The center of pressure

Not a convenient point



The influence of camber on the thin airfoil


* The cambered airfoil * The symmetric airfoil


< 4.10 The Vortex Panel Method >


* 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




* 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




* 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)




* 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


< 4.12 The Flow over an Airfoil – the Real Case >

Stall

Leading-edge stall


Flow separation takes place

over the entire top surface

of the airfoil after occurring

at the leading edge



Stall

Trailing-edge stall


α = 5° α = 10° α = 15° α =22.5°

Flow separation takes place from the trailing edge at

thicker airfoils than leading-edge stall



Stall

Thin airfoil stall


Leading-edge stall

Flow separation takes place

over the entire surface of

the airfoil after occurring at

the leading edge



Stall

Lift-coefficient curves


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




High-lift devices

Leading edge slat

Trailing edge flap




High-lift devices

Trailing-edge flap (plain type)

More camber → Higher lift




High-lift devices

Effect of slats and flaps

Documents

Incompressible Flow over Airfoils - SNUaancl.snu.ac.kr/aancl/lecture/up_file/_1444613570_Chapter 4.pdf · Aerodynamics 2015 fall - 3 - < 4.2 Airfoil Nomenclature > NACA (National