aero-B_ch4

Aerodynamics-B, AE2-115 I, Chapter 4Gerritsma & Van Oudheusden TUD Ad

1

Aerodynamics B - contents overview

Chap. 1 Introduction

Chap. 2 Fundamental Principles and Equations (basic concepts and definitions)

Chap.3+6 Inviscid, Incompressible Flow (Potential flows in 2D and 3D)

FUNDAMENTALS

Chap. 4 Incompressible Flow over Airfoils

Chap. 5 Incompressible Flow over Finite WingsAPPLICATIONS


2

Review of the results of Potential flow theory

Results for a closed body placed in a uniform flow:

• Drag = 0 (paradox of d’Alembert)

• Lift only when there is circulation: L = V (Kutta-Joukowski)

• Value of circulation is not unique (Kutta condition)

• solution for = 0 with source distribution on the contour

• solution for 0 with vortex distribution on the contour

Assumptions:• irotational• inviscid• incompressible• steady

Properties:• velocity field is governed by a

linear equation (Laplace)• superposition of solutions• pressure follows from Bernoulli


3

Chapter 4: Incompressible Flow over Airfoils

4.1-3 Introduction the Airfoil concept

4.4-6 Airfoil Theory principle: the vortex sheetthe Kutta conditionKelvin’s circulation theorem

4.7-8 Classical Thin Airfoil Theory for symmetrical and cambered airfoils

4.9 Lifting Flow over Arbitrary Bodies: the vortex panel method

4.11 Flow over an Airfoil - The Real Case: the effect of viscosity

ADDITIONAL MATERIAL: (see www.hsa.lr.tudelft.nl/~bvo/aerob)

4.A The Design Condition of an Airfoil

4.B Discrete Vortex Representation


4

The concept of the airfoil (wing section)Prandtl’s approach to the analysis of airplane wings:

(1) the study of the section of the wing (the airfoil)

(2) the modification of airfoil properties to account for the complete wing

What is an airfoil?

• an “infinite” wing in 2D flow

• the local section of a true wing

Motivation for looking at airfoils:

– the wing properties follow from the local airfoil properties

– a good model for slender wings (i.e. with large aspect ratio)

Airfoil section

x

y

z

V


5

Airfoil Nomenclature

• NACA method to generate standard “airfoil series”:

airfoil contour = mean camber line + thickness distribution

thickness

Leading edge

Trailing edgeMean camber line

Chord line

Chord c


6

Airfoil Characteristics

Attached flow:

cl ~

(inviscid) airfoil theory


7

Limitations of the (inviscid) airfoil theory

• Assumptions: - inviscid, irrotational flow

- incompressible

• What is correctly predicted: the pressure distribution over the airfoil

» lift and pitching moment

• What is absent: viscous effects: - boundary layer development- friction forces- flow separation

» no prediction of drag (D = 0!) or maximum lift

Conclusion: airfoil theory can reasonably predict lift and pitching moment as long as the flow does not separate


8

• Lift

• Pitching moment

Example: Results of the (thin) airfoil theory for the NACA 2412 airfoil

)(2 0 lc

1.20

23.0)( 0 lc

constant4/, cmc

053.04/, cmc


9

Theory: the vortex sheet

Basic idea: to reconstruct the lifting flow around a body (airfoil) by placing

many elementary vortices at convenient locations in the flow

(airfoil: on the contour, the camber line or the chord line)

point vortex

vortex sheet: distributed vorticity along a line

with variable strength (s)

A segment of length ds acts as a point vortex with strength (s).ds


10

Properties of the vortex sheet (1)

Induced Velocity: (vectorial addition)

Velocity Potential: (skalar addition)

A segment of length ds acts as a point vortex with strength: (s).ds

r

dsdVP

2

2

dsd P dsss

b

a

P )()(2

1


11


1. Total circulation around the vortex sheet

= total vortex strengthds

b

a

2. Across the vortex sheet there is a jump in the tangential

velocity that is equal to the local vortex strength

Proof:

circulation = total vortex strength dnvvdsuuds )()( 2121

21 uu let now: dn 0


12


3. There is a pressure difference across the vortex sheet proportional to the local vortex strength:

4. This pressure difference generates lift on the vortex sheet:

dsVdspdL '(=Kutta-Joukowski)

VdsVdsdLL ''

Vuuuu

uupp )(2

)( 21212

22

121

12

Vppp 12

(Bernoulli:)

Total lift:


13

Application of the vortex sheet to airfoil analysis

1. Arbitrary shape (thick airfoil):

vortex sheet on airfoil surface

2. Approximation for thin airfoil: vortex sheet on the camber line

- Task: determine vortex strength (s) such that airfoil surface becomes a streamline of the flow (numerical solution)

- The vorticity sheet can be seen to represent the (vorticity in the) thin boundary layer

- The lift follows from: dsVVL '


14

The Kutta condition

Potential flow with lift is not unique!

(Circulation may have any value)

The same happens for the flow around an airfoil

Which flow occurs in reality?

The flow that leaves smoothly at the trailing edge

The “Kutta condition”

Potential flow around a cylinder


15

The Kutta condition

• Be aware that the Kutta condition is an artificial, additional condition introduced to describe an effect that is the result of viscosity

• This does not mean that the entire effect of viscosity is included correctly, for example, there is still no drag!


16

Implementation of the Kutta condition

Consequences of the Kutta condition

21 VV Velocity at the trailing edge:

No pressure loading at the trailing edge:

021 VVTEStrength of the vortex sheet

at the trailing edge:

012 ppp


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The basic concept of the thin airfoil theory

• The airfoil is replaced by a vortex sheet along the camber line

• The (variable) strength of the vortex sheet is to be determined, such that the camber line is a streamline of the flow

(the flow tangency condition)

• The Kutta condition is imposed to fix the value of the circulation of the airfoil: TE = 0


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The flow-tangency condition (1)

The (variable) strength of the vortex sheet is to be determined, such that the camber line is a streamline of the flow

Simplification|:

For thin airfoil the effect of the vortex can be calculated as if the vortex sheet is along the chord:

For the total velocity component normal to the camber line:

0)(', swV n

normal component of the freestream

induced velocity of the vortex sheet

camber

)()(' xwsw


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The flow-tangency condition (2)

normal component of the freestream

velocity induced by the vortex sheet

)(, dx

dzVV n

)(2

)(

x

ddw

d

c

xw

0 )(2

)(

(x is fixed; is running variable)

slope of the camber line


20

Resume: the basic equations of the thin airfoil theory

)()(

)(

2

1

0 dx

dzVd

x

c

1. The fundamental equation of the thin airfoil theory:

the flow-tangency condition

(making the camber line z(x) a streamline)

0)( c

2. The relation that determines the circulation of the airfoil:

the Kutta condition

(making the flow smooth at the trailing edge)


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The symmetrical airfoil

Symmetrical airfoil:

Vd

0 0 )cos(cos

sin)(

2

1

)cos1(2

cCoordinate

transformation: d

cd sin

2)cos1(

2 0c

x

,0)( xz ,0dx

dz

Vd

c

x0 )(

)(

2

1

c0

0

Solution is given by:x

xcVV

2

sin

cos12)(


22

verification

is the solution?

Vd

0 0 )cos(cos

sin)(

2

1

sin

cos12)(

V

V

V

dV

dVd

)10(1

)cos(cos

cos11

)cos(cos

sin

sin

cos12

2

1

)cos(cos

sin)(

2

1

0 0

0 00 0

0

0

0 0 sin

sin

)cos(cos

cos

n

dn

Standard integrals:

(n=0,1,2…)


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The symmetrical airfoil (continued)

Vorticity distribution for the symmetrical airfoil x

xcVV

2

sin

cos12)(

Is the Kutta condition at the

trailing edge satisfied?

i.e.: = 0 for =

0cos

sin2

sin

cos1lim2)(

VV

L’Hopital’s rule

YES!

x

Note:the vorticity distribution is proportional to the lift distribution on the airfoil

TELE


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cVdcV

dc

VVdc

Vc

dLL

2

0

2

000

)cos1(

sin2sin

cos12)()('

The symmetrical airfoil: lift

sin

cos12)(

V

Calculation of the lift: d

cd sin

2

2)1.(

'2

21

cV

Lcl

Lift coefficient: Lift slope:

2d

dcl

dVdL


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The symmetrical airfoil: pitching moment

Calculation of the pitching moment about the leading edge:

22

1)cos1)(cos1(

2

sin2sin

cos12)cos1(

2

)()('

22

0

22

0

00

cVdcV

dc

Vc

V

dVdLMcc

LE

2)1.(

'22

21,

cV

Mc LE

LEmMoment coefficient

about leading edge:

dVdL


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The symmetrical airfoil: the center of pressure and the aerodynamic center

2)1.(

'22

21,

cV

Mc LE

LEm

Moment coefficient about leading edge:

LE

xCP

L

4

1, l

LEmCP

c

c

c

xCPLE xLM '' Center ofpressure:

2)1.(

'2

21

cV

Lcl

Lift coefficient:

04/, cmcMoment coefficient

about quarter-chord point:

quarter-chord point is also the aerodynamic center: is independent of !4/,cmc

xc

xcc

c

xxcc lLEm

CPlxm

,,


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The symmetrical airfoil: summary

2)1.(

'2

21

cV

LclLift coefficient:

Lift slope:

2d

dcl

Vorticity distribution (=lift distribution) x

xcVV

2

sin

cos12)(

04,4/, l

LEmcm

ccc

Moment coefficient

about quarter-chord point:

quarter-chord point is both the center of pressure:

and the aerodynamic center: is independent of 04/, cmc

4/,cmc


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4.8 The cambered airfoil

d

VV

w

dx

dz

0 0 )cos(cos

sin)(

2

1Condition to make the camber line z(x) a streamline of the flow

10 sin

sin

cos12)(

nn nAAV

The solution for this more general problem can be written as a

Fourier series:

“Basic solution” for the symmetrical airfoil:

A0 =

Additional terms

1

0 cos)(n

n nAAdx

dz

Substitution of the proposed solution in the upper equation

gives:

(use again standard integrals)

– the coefficients An (n=0,1,2,...) depend on the shape of the

camber line z(x)

– the coefficients A0 depends also on

Note: () = 0, so the Kutta condition

is satisfied


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The cambered airfoil: finding the coefficients An

1

0 cos)(n

n nAAdx

dz The solution can be interpreted as

a Fourier expansion of the function dz/dx

0

0

1d

dx

dzA

This Fourier series can be inverted to find the explicit relations for the individual coefficients An

0

cos2

dndx

dzAn

We can use these expressions in two ways:

1. Analysis: determine the coefficients An for a given camber line z(x)

2. Design: determine camber line z(x) for given coefficients An


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The cambered airfoil: the aerodynamic coefficients (1)

)2((...))(2

)1.(

'10

02

21

AAdc

cVcV

Lcl

The lift coefficient:

0

)1cos(1

2 ddx

dzcl

Note: for the lift coefficient only A0 and A1 required!

Independent of

Lift slope:

2d

dcl for every (thin) airfoil!

Zero-lift angle:

0

0 )1cos(1

ddx

dzL


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The cambered airfoil: the aerodynamic coefficients (2)

)2

(2

(...))(2

)1.(

' 210

0222

21,

AAAd

c

cVcV

Mc LE

LEm

Note: for the moment coefficient only A0, A1 and A2 required!

The moment coefficient

about the LE:

)(44 12,4/, AA

ccc l

LEmcm moment about the

quarter-chord point:

Independent of !For every (thin) airfoil the aerodynamic center is located at the quarter-chord point

The quarter-chord point is (in general) not the center of pressure:

l

cm

l

LEmCP

c

c

c

c

c

x 4/,,

4

10, CPmc


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The cambered airfoil: summary

)cos1(2

c

x

Vorticity distribution

(=lift distribution)

10 sin

sin

cos12)(

nn nAAV

Relation with the camber line shape z(x)

1

0 cos)(n

n nAAdx

dz

0

0

1d

dx

dzA

0

cos2

dndx

dzAn

Aerodynamic coefficients: )(2)2( 010 Ll AAc

)(4 124/, AAc cm

10

124/,

21

4

1

4

1

AA

AA

c

c

c

x

l

cmCP


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Additional Topics: A

4.A: The Design of an Airfoil4.A.1 The design of a camber line4.A.2 The design condition of an airfoil


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4.A.1: The Design of a Camber Line

Objective : to determine the camber line shape z(x) for given vorticity distribrution (=pressure or lift distribution)

1

0 cos)(n

n nAAdx

dz • For given choice of parameters A0, A1, … this equation is a 1st order dif.eq. for z(x)

•There are two boundary conditions: z(0)=z(c)=0 (both LE and TE on the x-axis)

Conclusions: 1. We cannot prescribe , but its value follows from the solution 2. For a different value of A0: the same z(x) and (- A0) 3. The value of A0 is not important for the shape of z(x)

The camber line is determined by the coefficients An, with n1


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4.A.2: The Design Condition of an AirfoilWhat is the reason for applying cambered airfoils?

cl

cd

airfoil with camberairfoil without camber

design condition

Application of a positive camber gives:• lower drag: the minimum drag occurs at positive lift• increase in the maximum lift• negative Cm,AC

= 0°

Cl = 0

Cm,AC = 0

= 0.5°

Cl = 0.51

Cm,AC = -0.106


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The “ideal angle of attack”

For the (theoretical inviscid) flow around the camber line :

• Trailing edge: smooth flow at every (Kutta condition)

• Leading edge: smooth flow only for a specific value of : opt

which is the ‘optimal’ or ‘ideal’ angle of attack

TE

LE

For any other angle of attack:

• the potential flow around the camber line gives infinitely large velocities

• the ‘real flow’ around the ‘true airfoil’ displays large velocity gradients near the nose


37

The design condition (1)

The design condition in terms of the thin-airfoil theory:compare:

– smooth flow at trailing edge (Kutta condition):

similarly:

– smooth flow at leading edge:

0TE

0LE

Implications for the vortex distribution of an arbitrary airfoil:

10 sin

sin

cos12)(

nn nAAV

Near the leading edge:

10

22)(

nn nAAV

)0( LE

00 A

Condition for: 0LE


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The design condition (2)Conclusion: The design condition (smooth flow at leading edge) occurs

for:

implying:0LE 00 A

0

0

1d

dx

dzA

0

cos2

dndx

dzAn

0

1d

dx

dzopt

)2( 10 AAcl

0

1 cos2)( ddx

dzAc optl

Note: for the symmetrical airfoil: 0opt 0)( optlc

With the earlier results:

10

12

21

4

1

AA

AA

c

xCP

1

2

42

1

A

A

c

x

opt

CP


39

The design condition of the parabolic camber line

0,...,4 3210 AAkAA • For the parabolic camber line we found (Problem 4.1a):

k=zmax /c is the maximum relative camberSo for the design condition we have :

0opt kAc optl 4)( 1 2

1

42

1

1

2

A

A

c

x

opt

CP

4

)(4/,

optlcm

cc

0...,, 6420 AAAA • For general camber lines that are

symmetrical w.r.t. the half-chord c/2:

Lift•

• the lift distribution at the design condition is symmetrical

• CP is then at half-chord

0opt

0)(when04/, optlcm cc

04/, cmcNote: longitudinal stability requires:


40

Design of a camber line with cl,opt>0 and cm,c/4=0

)0()( 1 KAc optl

0)(4 124/, AAc cm

Assume:

K

AA 21

2

00 )(81022coscosc

x

c

xKA

KA

dx

dz

Computation of the camber line z(x):

32 )(8)(157

3 c

x

c

x

c

xK

c

z

Integration with B.C.: z(0)=z(c)=0:

x

z

Note: negative camber at the tail of the airfoil


41


Numerical implementations of the vortex-sheet method

4.9: The Vortex Panel Method (4.10 in 3rd ed.)

4.B: Discrete Vortex Representation of the camber line (Additional Topic)


42

Arbitrary shape (thick airfoil):

vortex sheet on airfoil surface

Numerical implementation• Approximate the true contour by n straight panels:

i=1,2,…n

• Describe the vortex strength on each panel, e.g., by a constant value of i.

• Take on each panel a control point where the flow-tangency condition is to be satisfied, e.g., the center of each panel;

• Evaluate this condition, for each control point:

here, Ai,j is the contribution of panel # j on the velocity in control point # i

• This system of n equations for n unknowns (i) is singular (the circulation is undetermined), and one of the equations is to be replaced by (a form of) the Kutta condition, (TE) = 0.

0)()( , inin VV 0cos1

,

j

n

jjii AV

The Vortex Panel Method (principle)


434.B: Discrete Vortex Representation of the camber line (Additional Topic)

12 3

Continuous vortex representation of the camber line

Simplification:

Discrete vortex representation of the camber line:

• each panel has one vortex 1

and one control-point

• How must the locations of the vortex and the control point be chosen?


44

Discrete vortex representation of the symmetrical airfoil

• Approximation with a single vortex

Exact thin-airfoil theory: 04/, cmc

2)1.(

'2

21

cV

Lcl

This result can be

reproduced by

a single vortex of strength: placed at x = c/4cVV

L

'

Considerations:

• Because of the simplification with one vortex, the flow-tangency is no longer satisfied at every point: the camber line is no longer a streamline!

• At what point is the flow-tangency condition satisfied, i.e., what point would have served as control point?


45

One-vortex representation of the symmetrical airfoil

cV

x = c/4V

w

x

The velocity induced by this vortex at point x on the camber line is:

)4/(2)4/(2 cx

cV

cxwi

The total velocity normal to the camber line is:

cx

cxVwVw i

41

43

Conclusion:

• the flow-tangency condition is satisfied only at the 3/4-chord point;

• we obtain the correct lift and pitching moment point by placing the vortex at the 1/4-chord and taking the control point at 3/4-chord.


46

• Flow-tangency at: the 3/4-chord points of each panel!

Two-vortex representation of the symmetrical airfoil

Approximation with two vortices:• Take two vortices, 1 and 2 ,

placed in x1 and x2 , respectively.

• Divide the camber line in panels and place a vortex in the 1/4-chord point of each panel:

1

x1V

w

x

2

x2

cxcx 85

281

1 and

• Require lift and moment to be in agreement with the thin-airfoil theory:

cV 21

0)4

()4

( 2211 c

xc

x 4

124

31 and

))((

))((

85

81

87

83

cxcx

cxcxVwVw i • Determine normal velocity component:

cxcx 87

83 and


47

Discrete vortex representation: Recipe for the General Case

A generalization of the previous results for the symmetrical airfoil leads to

the following ‘recipe’ to treat an arbitrary camber line:• Divide the chord line in n panels: j=1,2,…n

• Place a vortex j on the 1/4-chord point of each panel j;

• Choose the 3/4-chord point of each panel as a control point, x j

• Evaluate the condition that in each control point the flow must be tangent to the camber line:

• This gives n equations for the n unknowns, j

• From the values of j the lift and pitching moment can be calculated

(Kutta-Joukowski)

V

w

dx

dz i

i

)(

slope of the camber line at control point xi

Velocity induced in control point xi , by all the vortices


48

Chapter 4: Final remarks

BASIC MATERIAL (2nd ed.):• Study thoroughly:

Sections 1 to 5 and 7 to 8 + 14 (summary)

• Read very carefully (be familiar with the contents):Section 6 Kelvin’s circulation theoremSection 9 The Vortex Panel Method Section 11 The Flow over an Airfoil - The Real Case

ADDITIONAL MATERIAL: (see www.hsa.lr.tudelft.nl/~bvo/aerob)

4.A The Design Condition of an Airfoil

4.B Discrete Vortex Representation

Make the Related Problems from the set of Exercises!

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