Lifting Line

8/13/2019 Lifting Line

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y = = 4s V n=1

Ansin n .

*he re+uired strenth of the distri!ution coefficients (An) for a i%en eometry and set of free-streamconditions can !e calculated !y applyin a surface flow !oundary condition. *he e+uation used is !ased onthe usual condition of ero flow normal to the surface.

Vn = 0

$or - wins this condition is applied at se%eral span-wise locations. /y matchin flow and surface anles,the normal %elocity !oundary condition can !e restated as a re+uirement for the flow anles at the section to!e in !alance. nlie 2- section flow where flow anles are set !y freestream direction and surface anlesonly, in - win flow an e&tra flow anle component is introduced !y the shed %ortices that are producedand trail !ehind the win. *his trailin %orte& sheet produces a downwash.

$or a - win, the local section flow anle must !e e+ual to the sum of the win#s anle of attac, thesection twist and the downwash induced flow anle.

2D = i twhere

i =

wi

Vwhere is the - win anle of attac,

t is the win twist anle and wi is the %elocity induced !y trailin %orte& sheet.

*he downwash %elocity is caused !y the shed %orticies trailin !ehind the win. *he %orte& strenth in thetrailin sheet will !e a function of the chanes in %orte& strenth alon the win span. *he mathematicalfunction descri!in the %orte& sheet strenth is thus o!tained !y differentiatin the !ound %orte& distri!ution.

d = 4s V n=1

n An cosn d

*hus the downwash at any span position on the win can !e found !y interatin the influence of indi%idualelements of the trailin sheet. ach sheet element !eha%es lie 1'2 of an infinite %orte& line so

d w i = d

4 r

where r is the distance across the span !etween the %orte& element and the point at which downwash is!ein calculated. r = ( y yi). *he full downwash will !e,

w i = 1

4s

s1

yy id .

3n su!stitution of the mappin %aria!les and interation, the result is,

w i = V n=1

n Ansin n

sin .


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Also, the 2- section lift coefficient is a function of the local flow incidence and the !ound %orte& strenth atthis span location. o that,

Cl = a02D0 =

V c

where a0 is the section lift slope (dCl

d2D), 0 is the section ero lift anle and cis the chord lenth.

*his can !e rearraned in terms of %orte& strenth,

= 1

2a 0Vc i t0

and su!stitutin for %orte& strenth and induced anle produces the followin !oundary condition e+uation,

n=1

An sin n sin n a0c8s = a

0c

8s sin t0

*his final !oundary e+uation contains all the unnown coefficients of the win model#s %orte& distri!utionalon with the win#s eometry and the stream conditions. 5t can !e used to find coefficientsA1, A2, A3, ...Assumin that hiher order coefficients !ecome increasinly small and mae nelii!le contri!ution to theresult, one method of solution is to truncate the series at term A6. /y applyin the !oundary condition at Nspan locations a set of simultaneous linear e+uations can !e constructed. *his set can !e sol%ed forcoefficientsA1toAN.

A cosine distri!ution of span-wise locations should !e used for the !oundary conditions to match theassumed win loadin distri!ution. 7learly the num!er of coefficients used will determine the accuracy of thesolution. 5f the win loadin is hihly non-elliptical then a larer num!er of coefficients should !e included.*his occurs when analysin wins with part span flaps. *hese cause a discontinuity in the spawise loadinand hence re+uire a much larer num!er of coefficients to accurately descri!e the distri!ution. here thethe win loadin is symmetric a!out the win root, the contri!ution of e%en functions will !ecome ero.7oefficientsA2,A4,A6,... are all ero and can !e dropped from the analysis.

3nce the coefficients of the load distri!ution are nown the total lift of the win can !e found !y interation.

Lift = Vs

s

. dy

which reduces to a lift coefficient of

CL = . A R . A1

whereARis the aspect ratio of the win

AR = b

2

S =

4s2

S.

A conse+uence of the downwash flow is that the direction of action of each section#s lift %ector is rotatedrelati%e to the free-stream direction. *he local lift %ectors are rotated !acward and hence i%e rise to a liftinduced dra. hile the o%erall o%ernin e+uations are potential flow and hence do not i%e rise to frictionor pressure dra, this lift induced dra will !e a sinificant component of the o%erall dra of the win.

*he downwash %elocity induced at any span location can !e calculated once the strenth of the win loadinis nown. *he %ariation in local flow anles can then !e found. /y interatin the component of section liftcoefficient that acts parallel to the free-stream across the span, the induced dra coefficient can !e found.


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Di = Vs

s

sin i . dy

which produces the followin induced dra coefficient ,

CDi = . AR. n=1

n . An2

6o information a!out pitchin moment coefficient can !e deduced from liftin line theory since the liftdistri!ution is collapsed to a sinle line alon the 1'4 chord. All forces will !e centered at the 1'4 chord in thismodel irrespecti%e of aerofoil section eometry.

pecial !ase of Purely "lliptical #ing Loading

5f the win planform is elliptical, c=c0cos then it can !e shown that the win load distri!ution is also apurely elliptical function and all coefficients e&cept A1will %anish.

y = = 0sin = 4sVA1sin

Also for simple straiht, untwisted wins, this solution may !e used as a first appro&imation as for theseeometries all other coefficients will !e much less than A1 and the loadin will !e appro&imately elliptical.

5n this case, a sinle eneral !oundary condition e+uation results containin only one unnown, the %orte&line strenth at the win root.

0 = 4s VA1

*he solution of the !oundary condition in this case leads to a constant downwash across the span and thefollowin simple answers for lift coefficient and induced dra coefficient.

d CLd

= a0

1 a0

. AR

,

0 (2-D section) = 0 (3-D wing),

CDi = CL

2

. AR.

$eference

Houghton & Carruthers "Aerodynamics for Engineering Students", Arnold Ed 3 19!.

oftware % Prandtl Lifting Line Progra&

*he followin computer proram allows the user to define win plan-forms (without sweep) and to definewin root and win tip section properties.


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*he proram assumes a linear %ariation of section properties !etween win root and tip and that the winloadin will !e symmetric a!out the win root.

ection lift data in terms of a0, 0 , ie. CL 2D=a02D0 are re+uired for the win root and thewin tip sections. *his information can !e o!tained from pu!lished 2- e&perimental data or theoreticaltechni+ues such as thin aerofoil theory.

*he proram uses the a!o%e liftin line e+uations to et solutions for lift coefficient %ersus anle of attacand induced dra coefficient %ersus lift coefficient2. $or a i%en anle of attac the proram will display theresultin distri!ution of section lift coefficient across the span, the distri!ution of downwash at the win and alistin of the solution $ourier coefficients fo this anle.

A flapped section can also !e input. *he percentae of win span with flap must !e input to create a flappedwin section. *he flap section properties are assumed to !e those entered for the win root section. *hesesection properties will !e ept constant across the flapped portion of the span. *he section properties usedout!oard of the flap will also !e constant and assumed to !e e+ual to those of the win tip.

oftware :"randtl iftin ine ol%er for unswept wins

eturn to *a!le of 7ontents
http://web.aeromech.usyd.edu.au/aero/liftline/liftline.phphttp://web.aeromech.usyd.edu.au/aero/liftline/liftline.phphttp://sites.google.com/site/aerodynamics4students/table-of-contentshttp://sites.google.com/site/aerodynamics4students/table-of-contentshttp://web.aeromech.usyd.edu.au/aero/liftline/liftline.php

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Lifting Line