Estimating the Oswald Factor from Basic Aircraft ......AERO –AIRCRAFT DESIGN AND SYSTEMS GROUP...

AERO – AIRCRAFT DESIGN AND SYSTEMS GROUP

Hochschule für Angewandte Wissenschaften Hamburg

Hamburg University of Applied Sciences

ESTIMATING THE OSWALD FACTOR

FROM BASIC AIRCRAFT GEOMETRICAL

PARAMETERS

Mihaela Niță Hamburg University of Applied Sciences

Dieter Scholz Hamburg University of Applied Sciences

Deutscher Luft- und Raumfahrtkongress 2012

German Aerospace Congress 2012

Berlin, Germany, 10.-12.09.2012 DLRK 2012

e

Scholz

Notiz

Download this file from http://OPerA.ProfScholz.de

German Aerospace Congress

Berlin, 10-12.09.2012

Dr.-Ing. Mihaela Nita

Estimating the Oswald Factor from Basic Geometrical Parameters

10.09.2012,

Aero - Aircraft Design and Systems Group

Hochschule für Angewandte

Wissenschaften HamburgHamburg University of Applied Sciences

Introduction

Airplane drag

zero-lift drag

drag due to lift

ESTIMATING THE OSWALD FACTOR FROM BASIC AIRCRAFT GEOMETRICAL PARAMETERS

)++=+=+= δππ

1(2

0,

2

0,,0, A

CC

Ae

CCCCC L

DL

DiDDD

Oswald factor e


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Method 1:

Oswald Factor e Calculated without Input of CDo

Application: Preliminary Sizing


MeDeFetheo kkkee ,,, 0⋅⋅⋅=


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Method 1: Calculate Oswald Factor e without Input of CDo



Me

De

Fe

theo

k

k

k

e

e

,

,

,

0

Oswald factor:

correction factor for the aspect ratio to calculate drag due to lift

theoretical Oswald factor, invisid drag due to lift only

correction factor: losses due to the fuselage

correction factor: viscous drag due to lift

correction factor: compressibility effects on induced drag


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Method 1: Oswald Factor e without Input of CDo



2

, 21

−=b

dk F

Fe

1;0

1,

=<

+

−=

ee

e

b

compeMe

ca

cM

Mak

e

3.0

1

82.10

00152.0

===

−=

comp

e

e

e

M

c

b

a

Aircraft category

All 0.114 0.974 -

Jet 0.116 0.973 0.873

Business Jet 0.120 0.971 0.864

Turboprop 0.102 0.979 0.804

General Aviation

0.119 0.971 0.804

bdF / Fek , 0,Dek

theoe see next pages


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Literature Study: HÖRNER

Corrections to induced drag for unswept wings as a function of taper ratio λ


Α/=) kf λ(

δ+=

1

1theoe

δ=k

)λ(f

λ Afetheo ⋅+

=)(1

1

λ


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Estimating a Theoretical Oswald Factor

� Approximation of Hörner‘s function:


Method 1: etheo for unswept Wings

0119.00706.01659.015.00524.0)( 234 +−+−= λλλλλf

From the derivative of thefunction f(λ), the optimumtaper ratio for unsweptwings is λopt = 0.357

λopt = 0.357


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Literature Study: NACA Report 921


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Method 1: etheo

Optimum Combination of Taper Ratio and Sweep Angle

� For each sweep, there is an optimal taper ratio that minimizes the induced drag.

� NASA Report 921 delivers a curve that relates the taper ratio to sweep for an approximate elliptical loading

� An equation that approximates this curve is

� The optimum taper ratio for an unswept wing is(this is not quite the value from Hörner)


250375.045.0 ϕλ ⋅−⋅= eopt

45.0=optλ


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Method 1: etheo for swept Wings

Shifting Hörners curve to a minimum as given by NASA

� The difference in the minimum between Hörner and NASA (sweep angle in degrees):

� Calculating the theoretical Oswald Factor from Hörner‘s now shifted curve:

Hörner‘s equation now applied as:


250375.045.0357.0 ϕλ ⋅−⋅+−=∆ e

Afetheo ⋅∆−+

=)(1

1

λλ

0119.0)(0706.0)(1659.0)(15.0)(0524.0)( 234 +∆−−∆−+∆−−∆−=∆− λλλλλλλλλλf


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Literature Study: KROO

Losses due to the Fuselage

� If the flow were axially symmetric and the fuselage were long, then mass conservation leads to

� Aspect ratio A = b²/S is kept constant by convention, so the losses are included into the span efficiency e. The reduction in e is expressed by the factor ke,F . So the factor on the span efficiency ke,F is

� In practice losses are bigger and experience has shown that induced drag increment is about twice the simple theoretical value, so


222Fdbb −=′

2

2

22

, 1

−=−=b

d

b

dbk FF

Fe

2

, 21

−=b

dk F

Fe


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� Fuselage correction factor

for the Oswald factor e

to correct missing or reduced

lift due to presents of fuselage


2

, 21

⋅−=b

dk F

Fe

ke,F

ke,F

2

, 1

−=b

dk F

Fe

Method 1: ke,F Fuselage Correction


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Method 1: Influence of Mach Number: B737 and MPC75


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k_e,M_PG

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0,40 0,50 0,60 0,70 0,80 0,90

k_e,M

M

ke_M

� Mach number correction for Oswald factor - from Prandtl-Glauert (ke,M,PG) comparedwith own factor ke,M (using data for the A320)

� ke,M fits the data better than Prandtl-Glauert (ke,M,PG) (see next page)

2,, 1 Mk PGMe −=

1;0

1,

=<

+

−=

ee

e

b

compeMe

ca

cM

Mak

e

Method 1: Influence of Mach Number - Comparison with Prandtl-Glauert-Correction


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Oswald factor – A320 data and own method. Own Mach number correction ke,M

Method 1: Influence of Mach Number and Results for A320

0,5

0,6

0,7

0,8

0,9

1,0

0,40 0,50 0,60 0,70 0,80 0,90

e, k

_e

,M

M

e_A320,DBD

e_calc_A320

ke_M


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Method 1: Summary: Influence of Parameters

> 0,9 . Driven by wing parameters: taper, sweep, aspect ratio. Small influence!

> 0,9 . Driven by fuselage diameter to span

around 0,85 . Influenced by zero lift drag of the aircraft, but CD0 not required

0,5 … 1,0. Strongest influence on final result of e for cruise of jet aircraft !

theoe

Fek ,

0,Dek

Mek ,


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Method 1: Results: The Polar of the Boeing 737-800


Aircraft Flight Manual Calculated


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Method 2:

Oswald Factor e Calculated with Input of CDo and Twist

Application: Conceptual Design


APQ

ke Me

π+= ,

Fetheo keQ

,

1

⋅= K = 0,380,DKCP =


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Method 2: Oswald Factor e Calculated with Input of CDo and Twist

APQ

ke Me

π+= ,

0,2

2)(D

L

L

L

L KCC

wC

C

vCP +

⋅+

⋅=

θθαα

Fetheo keQ

,

1

⋅=

Twist θ = iW,o – iW,i (is generally negative)v and w from DUBSK = 0,38

For A > 4:

( ) 4tan12

22

5022 +−+⋅+

=MA

ACL

ϕπ

α

K = 0,380,DKCP =

Including the Effect of Twist:

( )( ) ( )22

2

0006.010051.00088.0

0037.03.00134.0

Aw

v

−⋅−=

−−=

λλλλ

v gets negative when λ< 0.33

from Method 1


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The lift-dependent drag term has two components:

� an inviscid part, also called vortex drag

� a viscous part

The term Q covers the inviscidpart of the induced coefficient,

The term P is used to express the viscous part of induced drag coefficient.

Theoretical Background


2, LiD CP

A

QC ⋅

+=π

APQe

π+= 1

Qeinviscid /1=

+= PA

Q

Ae ππ1

PA

QAe

+=

π

π 1

wCCvCCKCCA

QC LLLLDLiD

220,

2, )( θθ

π αα+++=


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( ) 20037.03.00134.0 λλ −−=v 22 0051.00088.0)0006.01/( λλ −=− Aw


Literature Study: Including the Effects of Twist: v and w from DUBS


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Extension of Method 1 and 2:

Oswald Factor e for Nonplanar Configurations


2

,

,,

,,,,

,

21

0

+=

⋅+

=

⋅⋅⋅⋅=⋅=

b

h

kk

kAPQ

ke

kkkkee

kee

NPNPe

NPeMe

NP

NPeMeDeFetheoNP

NPeNP

π


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Estimating the Oswald Factor for Non-Planar Configuratio ns

Wing with winglets

The following relations can be derived from geometry:


eb

be

A

Ae

Ae

C

eA

CC

Ae

CC

b

h

b

b

effeffWL

WL

L

eff

LWLiD

LiD

eff

⋅

=⋅=

==

=

+=

2

22

,,

2

,

21

ππ

π

eb

heWL ⋅

+=2

21


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Estimating the Oswald Factor for Non-Planar Configurations

Wing with wingletsCorrection of the simple geometrical consideration via the factor


WLk

WLeMeDeFetheoWLeWL

WL kkkkeekeb

h

ke ,,,,,

2

0

21 ⋅⋅⋅⋅=⋅=⋅

+=

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0,0 0,1 0,2 0,3 0,4 0,5

h/b

1/k_

e,W

L

22

,2

1

==

+=

b

b

A

A

b

h

kk effeff

WLWLe


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Real wings with winglets: Boeing and Airbus data


ekeb

h

ke WLe

WLWL ⋅=⋅

+= ,

22

1


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Wing with dihedral

The following relations can be written:


( )

−Γ

=

+=⇒−=

Γ=⇒Γ⋅=

1cos

1

2

1

212

1

cos

1cos

22

b

hb

h

b

bbbh

b

bbb

effeff

effeffe

A

Ae eff ⋅=Γ

Γ⋅⋅⋅⋅= ,,,, 0 eMeDeFetheo kkkkee

22

, 1cos

111

21

−Γ

⋅+=

⋅+=Γ

WLWLe kb

h

kk


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Wing with dihedral

From the simple geometrical consideration:

A more accurate evaluation is achieved by penalizing the relation with the factor


22

, cos

121

Γ=

⋅+=Γ b

hke

WLk

22

, 1cos

111

21

−Γ

⋅+=

⋅+=Γ

WLWLe kb

h

kk


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Various Non-Planar Configurations

KROO


Span efficiency for various optimally loaded non-planar systems (h/b = 0.2)

Literature Study: KROO


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Non-Planar Configurations in GeneralThe following relations can be written: (this time via a penalty factor called

The factor for wings with winglets and dihedral, investigated above, becomes now a

particular case of the factor kNP.

Having the from KROO, can be calculated for each configuration


NPk

ekeeb

h

ke NPeNP

NPNP ⋅=⇔⋅

+= ,

22

1

NPkNPek ,

1

12

, −⋅=

NPeNP

kb

hk


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Non-Planar Configurations in General


NPeMeDeFetheo

NPeNP

NP

kkkke

ekeb

h

ke

,,,,

,

2

0

21

⋅⋅⋅⋅=

⋅=⋅

+=

h/b = 0.2 general


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The Box-Wing Aircraft

; ;


ke

e

box

refD

D

refi

boxi ==,

,

bhkk

bhkk

e

e

e

e NP

ref

box

/

/

21

43

⋅+⋅+==

bhkk

bhkkk

D

D

refi

boxi

/

/

43

21

,

,

⋅+⋅+==


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Estimating the Oswald Factor for Non-Planar Configurations



0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

h/b

k

(c) k Prandtl

(d) k Rizzo

(a) k Prandtl (Biplane)

(b) k Prandtl (Biplane 2)

k from iDrag results fit to

"box wing equation" with

parameters from (f)k from iDrag results fit to

"winglet equation"

k DeYoung

k iDrag results

k from wind tunnel data


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0,0

0,5

1,0

1,5

2,0

2,5

0,0 0,2 0,4 0,6 0,8 1,0

h/b

e_bo

x / e

_ref

e_box / e_ref from iDrag results fit to'box wing equation' parameters from (f)

e_box / e_ref from iDrag results fit to 'winglet equation'


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Summary and Conclusions

� Simple, physics based method for both conventional and unconventionalconfigurations given

� Treats more in depth the special cases of wings with winglets, wings with dihedraland box wings

� Information provided to assist aircraft designers in making a sufficient accurate estimation of the span efficiency factor e during preliminary aircraft design and aircraft design optimization.



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Thank you!

http://AERO.ProfScholz.de

ESTIMATING THE OSWALD FACTOR

FROM BASIC AIRCRAFT GEOMETRICAL PARAMETERS

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Wissenschaften Hamburg

Hamburg University of Applied Sciences

Estimating the Oswald Factor from Basic Aircraft ......AERO –AIRCRAFT DESIGN AND SYSTEMS GROUP...

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