Wingmills.doc

8/12/2019 Wingmills.doc

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Potential Flow2-Dimensional Vortex Panel Model:

Applications to Wingmills

by

John Moores

A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OF

BACHELLOR OF APPLIED SCIENCE

DIVISION OF ENGINEERING SCIENCE

FACULTY OF APPLIED SCIENCE AND ENGINEERINGUNIVERSITY OF TORONTO

S!er"#sor J$ D$ DeL%r#er

A!r#& '(()


2/54

Acknowledgements

The %*hor +o&, -e *o %.-no+&e,/e *he %#, o00ere, by *he s*%00 o0 *he Sbson#. F/h*12#n,*nne&3 L%b %* *he Un#"ers#*y o0 Toron*o Ins*#**e 0or Aeros!%.e S*,#es '(('4'(() %n, by Dr$ J%5es D$ DeL%r#er #n *he !ro,.*#on o0 *h#s *hes#s$ O0 !%r*#.&%r no*e

#s P%*r#.- 6,n#.h on +hose M%s*er7s *hes#s 5.h o0 *h#s +or- #s b%se,$ I re5%#n#n,eb*e, *o h#5 0or *he s//es*#ons he !ro"#,e, *o 5e %* *#5es o0 #5!%sse$ As +e&&8 *he%*hor +o&, -e *o %.-no+&e,/e *he 0#n%n.#%& s!!or* o0 *he N%*r%& S.#en.es %n,En/#neer#n/ Rese%r.h Con.#& #n *he !ro,.*#on o0 *h#s *hes#s$ B* 5os* o0 %&&8 I o+e %,eb* o0 /r%*#*,e M#.he&&e P%rsons 0or !roo0re%,#n/ *h#s 5%ns.r#!* %n, 0or *%-#n/ so5eo0 *he e9!er#5en*%& ,%*% ,r#n/ *#5es +hen I +%s #n.%!%.#*%*e, +#*h +or-8 %n, +#*ho*+hose he&!8 s!!or*8 %n, e9*r% .o5!*er *h#s *hes#s +o&, no* h%"e been !oss#b&e$

4 ii 4


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Abstract

USINGA'4DIMENSIONALTIME4MARCHINGVORTE:PANELMETHODDEVELOPEDBASEDONTHEAPPROACHBY6DUNICHANDDELAURIER8 APPLICATIONSTO2INGMILLS1OSCILLATINGAIRFOILPO2ER

GENERATORS3 AREE:AMINED$ INPARTICULAR8 THEMETHODOF6DUNICHISE:TENDEDTOAIRFOILSOFFINITETHIC;NESS$ FURTHERMORE8 ITISDETERMINEDTHATTHEATTACHEDFLO2PORTIONOFTHE

MODEL2HILENOTUSEFULFORPREDICTINGTHEAIR2INGMILLOFMC;INNEYANDDELAURIER8 MAYHAVEAPPLICATIONSINOTHERMEDIAORATSMALLERPITCHINGAMPLITUDESFOR2HICHFLO2SEPARATIONANDAIRFOILSTALLARENOTATISSUE$ ASSUCH8 ATHEORETICALBASISFORTHE

E:TENSIONOFTHEMODELTOACCOUNTFORVARIABLECAMBERINORDERTOMA:IMI6ETHEENERGYE:TRACTIONATLO2ERPITCHINGAMPLITUDESANDFREQUENCIESISDETERMINED$

4 iii 4


4/54

TAB! "F #"$T!$T%

No*%*#on


5/54

>$) D#s.ss#on


6/54

$otation

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t T#5es*e! 1se.on,s3

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Me,#5 Dens#*y 1-/5)3

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h T#5es*e! #n,e9

i Con*ro& Po#n* #n,e9

j Bon, Vor*e9 %n, "or*e9 !%ne& #n,e9

p,q 2%-e Vor*e9 #n,#.es

Ins*%n*%neos !#*.h 1r%,#%ns3

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H M%9#55 !&n/e %5!*,e 15e*ers30 M%9#55 !#*.h#n/ %5!*,e 1r%,#%ns3

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s*e! h 1D#5ens#on&ess3

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Bpqh In0&en.e .oe00#.#en* o0 +%-e "or*e9 on +%-e "or*e9 ! %* *#5e


Po*en*#%& %* % !o#n* #n *he 0, 0#e&,

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o0 *he %#r0o#&

u L#5#*#n/ "%&e o0 *he 0, !o*en*#%& %s +e %!!ro%.h *he !!er s#,e

o0 *he %#r0o#&

4 vi 4


7/54

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jh

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dh T#5es*e! s#e

4 vii 4


8/54

ist o. Fig)res and Tables

=$ M.;#nney %n, DeL%r#er7s 2#n/5#&&


9/54

#&apter '(ntrod)ction

'3' 4istor* and Moti5ation

To+%r,s *he en, o0 *he =(s *here +%s % s#/n#0#.%n* #n*eres* #n ,e"e&o!#n/ rene+%b&e

ener/y sor.es$ Th#s #n*eres* res&*e, #n rese%r.h #n*o se"er%& ne+ 0or5s o0 +#n,5#&&s

*o e9!er#5en* +#*h ,#00eren* 5e.h%n#s5s 0or re5o"#n/ ener/y 0ro5 0, 0&o+s$ For

#ns*%n.e8 *he !er#o, s%+ s#/n#0#.%n* %,"%n.es #n *he *heory o0 so5e non4.on"en*#on%&

0#9e, !#*.h +#n,5#&&s8 s.h %s *he "er*#.%& %9#s ,%rr#s ro*or *y!es =$ As +e&&8 *here +ere

se"er%& no"e& ,es#/ns ,e"e&o!e, 0ro5 ns*e%,y %ero,yn%5#. *heory$ For #ns*%n.e8

Je00ery',e"e&o!e, % 5o,e& 1&%*er re0#ne, by P%yne )%n, F%r*h#n/>3 0or % "er*#.%&&y4

5on*e, !#"o*#n/ +#n/ +h#.h 0&%!!e, #n *he +#n,$ A* %ron, *he s%5e *#5e 2#&son %n,

L#ss%5%n?%* *he N%*#on%& Rese%r.h Con.#&

,e"e&o!e, % .y.&o/ryo4*y!e ,e"#.e +h#.h8 *hro/h

%r*#.&%*#on o0 !#*.h +%s %b&e *o e9*r%.* ener/y 0ro5

*he 0&o+$

In .on*r%s*8 *he .on.e!* !ro,.e, by M.;#nney %n,

DeL%r#er@

4 b%se, !on *he +or- o0 A,%5-o

4 se,% r#/#, hor#on*%& %#r0o#& +#*h %r*#.&%*e, !#*.h#n/ %n,

!&n/#n/ *o e9*r%.* ener/y 0ro5 *he 0&o+$ S#n.e *he

,e"#.e .ons#s*s o0 % re.*%n/&%r !&%n40or5 +#n/

os.#&&%*#n/ h%r5on#.%&&y #n !#*.h %n, !&n/e 0#/re

= #* +%s ,bbe, % +#n/5#&&7$ Th#s .on.e!* !ro"#,e,

4 14

Figure 1: McKinney and

DeLauriers Wingmill

Reprinted From Ref 6


10/54

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+#n,5#&&s$ For #ns*%n.e8 +h#&e #* +%s obser"e, by M.;#nney *h%* *he +#n/5#&& %.h#e"es %

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4 24


11/54

'32 Approac&

The A!!ro%.h *%-en e5!&oye, % 5e*ho, b%se, !on *he +or- o0 6,n#.h$ S!e.#0#.%&&y8

by s#n/ *he e%*#ons ,er#"e, #n *h#s !re"#os +or- 0or *he 0&&y %**%.he, 0&o+ .%se8 %

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s#5#&%r *o *h%* e5!&oye, e9!er#5en*%&&y by M.;#nney %n, DeL%r#er$

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+#n/5#&&$

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4 34


12/54

#&apter 2

Panel Met&od T&eor*

Part ': %tead*-%tate Model

23' Vortex Panel Met&ods

2h#&e 5.h o0 *he +or- ,#s.sse, #n *h#s .h%!*er !%r%&&e&s *he +or- o0 6,n#.h =(8 #* #s

#5!or*%n* *o &%y *he /ron,+or- 0or *he !%ne& 5e*ho, +h#.h #s %* *he he%r* o0 *h#s

#n"es*#/%*#on$ As s.h8 #* #s ne.ess%ry *o !resen* *he *heory beh#n, !o*en*#%& 0&o+

"or*e9 !%ne& 5e*ho,s$ Bo*h s*e%,y %n, ns*e%,y .on,#*#ons %re e9%5#ne,

$

The s!e.#0#. %!!ro%.h se, %!!ro9#5%*es *he %#r0o#& by s#n/ % ser#es o0 #n0#n#*e8

,#s.re*e bon, "or*#.es 1%s

#n 0#/re '3 *o %!!ro9#5%*e %

.on*#nos ,#s*r#b*#on o0

"or*#.#*y$ In *he ns*e%,y

.%se8 *he %#r0o#& +%-e #s

s#5#&%r&y %!!ro9#5%*e,$ As

+e&&8 s#n.e %&& 5o*#ons

.ons#,ere, %re !&%n%r8 % '4

,#5ens#on%& %!!ro9#5%*#on

+%s e5!&oye,$

For a more complete discussion of discrete vortex panel methods, Zdunich 10andWinfield 9are excellent references.

4 44

Figure 2: Discrete Vortex fi laments (point Vortices in2D) Reprinted from Ref. 13


13/54

232 %tead* %tate Model T&eor*

2323' Vortex Flow

Vor*e9 F&o+8 %s ,e!#.*e, #n 0#/re )8 #s % !o*en*#%& 0&o+$ 2e5%y e9!ress *he !o*en*#%& ,e *o %ny s.h !o#n* "or*e9 1#n

% .yn,r#.%& re0eren.e 0r%5e .en*ere, %* *he "or*e9 sor.e3

%s

2

j

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j

j

jj r

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2

== '$'

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+h#.h %&&o+s s *o ,e.o5!ose *he !o*en*#%& #n*o *+o .o5!onen*s *he !o*en*#%& ,e *o

*he #n*er%.*#on o0 %&& *he bon, "or*#.es %n, *he !o*en*#%& %* #n0#n#*y

+=+= j

jVV '$)

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*h#n !&%*e or #n % +r%!!e, .on0#/r%*#on$ As +e&&8 *he %#r0o#& sr0%.e #s .ons#,ere, *o be#5!er5e%b&e *h%* #s %&& 0&o+s %re !re&y *%n/en*#%& *o *he sr0%.e$

A br#e0 no*e on .on"en*#ons #s re#re,$ 2h#&e 6,n#.h ses *he sbs.r#!* i*o ,eno*e

he solution to this pro!lem is provided in the discussion of "a#e convection in "hichthe free vortices are modeled usin$ cores such that the velocit% potential !ecomes &eroat the sin$ularit%

4 '4

Figure 3: Vortex Flow

Reprinted from Ref. 13


14/54

!%ne&s %n, .on*ro& !o#n*s8 #* +%s ,e.#,e, *o #ns*e%, re*%#n i0or se +#*h *he .on*ro&

!o#n*s +h#&e j#s se, 0or *he !%ne&s s#n.e *hey %re %sso.#%*e, +#*h *he bon, "or*#.es$

23232 Flow Tangenc* #ondition

S#n.e *he %#r0o#& #s so, #* #s re#re, *h%*8 %* *he sr0%.e8 *he 0&o+ be *%n/en*#%&$ Ths

#* #s .on"en#en* *o 0or5%&&y ,e0#ne *h#s

0)( =+= jstreamj nVn '$?

As *he /r%,#en* o0 *he !o*en*#%& #s re&%*e, *o *he "or*e9 s*ren/*hs *hro/h e%*#on '$'8

#* *rns o* *h%* *h#s .ons*r%#n* +#&& %&&o+ s *o so&"e 0or *he "or*e9 s*ren/*hs %..or,#n/

*o

=

+=+

j

j

jp

jp

j

j

jj nr

rknVnV 0

2

)(

2

'$@

Ths8 +e .%n 0#n, *he "e&o.#*y !o*en*#%& %* %ny !o#n* ! %&on/ *he %#r0o#&$

2323+ 6)tta #ondition

2h#&e 0&o+ *%n/en.y !ro"#,es s +#*h n e%*#ons #n n n-no+ns8 n%5e&y *he "or*e9s*ren/*hs8 +e h%"e no* ye* .ons*r%#ne, *he +%y #n +h#.h *he 0&o+ .o5es o00 *he %#r0o#&$

2e %..o5!sh *h#s by %!!&y#n/ % -**%4*y!e .ons*r%#n* 4 *h%* #s +e re#re *he 0&o+ *o

s5oo*h&y .o5e o00 *he *r%#n/ e,/e o0 *he %#r0o#&$ Ho+e"er8 *h#s .%ses *he sys*e5 *o

be.o5e o"er,e*er5#ne,8 s#n.e +e no+ h%"e n= e%*#ons #n n n-no+ns$ There0ore8

#* #s ne.ess%ry *o #/nore *he 0&o+ *%n/en.y .on,#*#on %* one o0 *he .on*ro& !o#n*s$

In *he .%se o0 *he 0&%* !%ne&8 *h#s #s en0or.e, by re#r#n/ *h%* *he 0&o+ *o be or*ho/on%&

*o *he n#* nor5%& o0 *he 0#n%& !%ne& %* *he *r%#n/ e,/e$ Ths8 *he .ons*r%#n* #s o0 *he

0o&&o+#n/ 0or5

(n the stead% state case Vstream= V his constraint "ill prove useful in )hapter ' as it allo"s us to simulate a cusped,cam!ered ed$e "ithout incurrin$ the computational overhead of varia!le $eometr%

4 *4


15/54

0

2

2 =

+

j

TE

jTE

jTE

jTE nr

rknV

'$

As 0or *he 0#n#*e *h#.-ness %#r0o#&8 An,erson=)s!e.#0#es *h%* n&ess +e h%"e % .s!e,

*r%#n/ e,/e 0ro5 +h#.h *he 0&o+ 5%y !ro.ee, s5oo*h&y8 *he "e&o.#*y !o*en*#%& 5s* /o

*o ero %* *he TE$ Th#s #s *he on&y +%y *o re.on.#&e *he o!!os#n/ !o*en*#%&s %* *h#s !o#n*$

As s.h8 *he -**% .on,#*#on 0or % 0#n#*e *h#.-ness %#r0o#& be.o5es

TElTEu = '$

2here TEu#s *he "or*e9 s*ren/*h on *he !!er !%ne& %* *he *r%#n/ e,/e %n, TEl#s *he

"or*e9 s*ren/*h on *he &o+er !%ne& %* *he *r%#n/ e,/e$ I* #s .r#*#.%& *h%* bo*h

s#n/&%r#*#es be e%& ,#s*%n.es 0ro5 *he *r%#n/ e,/e #n or,er *o .%n.e& e%.h o*her

!er0e.*&y$ As +e&& 4 %s #n *he .%se o0 *he 0&%* !&%*e %#r0o#& 4 +e 5s* #/nore one o0 *he

0&o+ *%n/en.y e%*#ons %* % !%r*#.&%r .on*ro& !o#n* #n or,er *o %!!&y *h#s .on,#*#on$

2323 Press)re and Aerod*namic Force Distrib)tion

In or,er *o .%&.&%*e *he 0or.es %n, 5o5en*s %.*#n/ on *he %#r0o#&8 #* #s ne.ess%ry *o

0#n, %n e9!ress#on 0or *he !ressre ,#s*r#b*#on o"er *he %#r0o#& by re&%*#n/ #* *o *he

"e&o.#*y 0#e&, ob*%#ne, 0ro5 *he #n*er%.*#on o0 *he "or*#.es$ I* .%n be sho+n 0ro5

Bernos e%*#on *h%* *he !ressre ,#00eren.e %.ross % !%ne& #s /#"en by

))((222

22

ulul

ul

luj VVVV

VVppp +=

==

'$

For % +r%!!e,4.on0#/r%*#on "or*e9 shee*8 *h#s e9!ress#on .%n be s#5!#e, so5e+h%*$

De *o *he #5!er5e%b#*y o0 *he %#r0o#&8 one o0 Vlor Vu+#&& %&+%ys be ero ,e!en,#n/

!on +he*her +e %re .ons#,er#n/ *he *o! or bo**o5 shee*$ Fr*her5ore8 s#n.e *he &o.%&

5! #n "e&o.#*y %.ross % !%ne& #s e%& *o *he %"er%/e "or*e9 s*ren/*h8 '$ s#5!#es

*o

2

2

=

j

j

jl

p

'$=(%

Ho+e"er8 0or % s#n/&e 0&%* shee*8 *he s5 o0 "e&o.#*y #s %&so re#re,$ Fro5 6,n#.h8 *h#s

4 +4


16/54

#s /#"en %s *+#.e *he "e&o.#*y 0on, %* *he .hor, ne$ Ths

jj

j

j

j Vtl

p )(

+

= '$=(b

Fro5 e#*her o0 '$=(% or '$=(b8 +e .%n ,er#"e *he 0or.e on e%.h #n,#"#,%& !%ne&

)( kplf jjj = '$==

Th#s .%n *hen be s55e, ! o"er *he !%ne&s *o /#"e *he *o*%&s 0or 0or.es %n, 5o5en*s

%.*#n/ on *he %#r0o#& %..or,#n/ *o

jfLjj

= '$='

if! jj= '$=)

jeajj frr" = )( '$=>

No*e *h%* *he 5o5en* #s .%&.&%*e, +#*h res!e.* *o *he e&%s*#. %9#s $

23+ %tead* %tate Model (mplementation

T+o ,#00eren* 0or5%*s +ere #n"es*#/%*e, 0or *he s*e%,y s*%*e .%se 0&%* !%ne&%n,

+r%!!e, !%ne&$ The 0&%* !%ne& .%se +%s s*,#e, #n /re%*er ,e*%#& s#n.e *he %#r0o#&s n,er.ons#,er%*#on +ere -no+n *o be *h#n 1=' *h#.-ness3 %n, sy55e*r#.$ As +e&&8 *he

&e%,#n/ e,/e s.*#on +%s -no+n8 #n *he .%se o0 M.;#nney7s +#n/5#&& *o be o0 **&e

#5!or*%n.e %s 0%r %s /ener%*e, !o+er +%s .on.erne,$

23+3' 7eometr* and (n.l)ence #oe..icients

his is the axis of pitchin$ "hich "e "ill use in the unstead% section and ma% !e setup as an% point on or off of the airfoil in the current code. flat panel "as the onl% investi$ated method due to the s%mmetr% of "in$millflappin$, ho"ever, the code could easil% !e modified and used for an% ar!itrar% shaped1 dimensional airfoil

4 -4


17/54

The /eo5e*ry o0 *he 0&%* !%ne& .%se #s !ro"#,e, #n F#/re >$ I* #s #5!or*%n* *o no*e *h%*

*he n5ber#n/ be/#ns %* *he &e%,#n/ e,/e$

Figure 4: Coordinate ystem used !or teady tate Flat "anel Code

#e$rinted !rom #e!% 1& wit' modi!ications to t'e model s'own in grey

(n!luence Coe!!icients

U!on #ns!e.*#on o0 e%*#on '$@8 #* #s %!!%ren* *h%* +e .%n re+r#*e *he *er5s #n"o&"#n/

*he "or*e9 s*ren/*h *o !ro,.e

=+j

jiji AnV 0

+here jji

ji

ij n

r

rkA

2

2

=

'$=?

Aij#s re0erre, *o %s %n #n0&en.e .oe00#.#en* %n, ,e!en,s n#e&y !on *he /eo5e*ry o0

*he %#r0o#&$ As s.h8 s#n.e +e h%"e n4= 0&o+ *%n/en.y .on,#*#ons %* n4= .on*ro& !o#n*s

%n, one -**% .on,#*#on %* *he TE8 +e !ro,.e *he 0o&&o+#n/ ne%r sys*e5

4 94


18/54

=

+

0

0

0

11

1

1

1111

TE

i

n

j

TEnTEjTE

iji

nj

nV

nV

nV

AAA

AA

AAA

'$=@

Th#s sys*e5 +%s so&"e, s#n/ *he M%*&%b 7 o!er%*or$ No*e *h%* *he 0#n%& ro+ o0 *he A7

5%*r#9 #s *he se* o0 e%*#ons .orres!on,#n/ *o *he -**% .on,#*#on 1#$e$ E%*#on '$3

%n, %s s.h %re ,eno*e, by TE$

One #n*eres*#n/ !o#n* #s *he .ons#s*en.y o0 *he obser"e, *o*%& .#r.&%*#on 1#$e$ *he s5

o0 %&& *he bon, "or*e9 s*ren/*hs3$ Th#s #s sr!r#s#n/ s#n.e *he *heore*#.%& ,#s*r#b*#on o0

"or*#.#*y #s 1*heore*#.%&&y3 #n0#n#*e %* *he &e%,#n/ e,/e$ As #* *rns o*8 *he "or*#.es

%!!e%r *o .o5!ens%*e 0or one %no*her8 %s s.h8 *he .#r.&%*#on es*#5%*e #s essen*#%&&y

!er0e.*+#*h on&y % 0e+ !%ne&s$ Ho+e"er8 *he /re%*er *he n5ber o0 !%ne&s *he be**er

*he es*#5%*e o0 *he .hor,4+#se .#r.&%*#on ,#s*r#b*#on %n, hen.e *he ,#s*r#b*#on o0 *

%n, 5o5en*$

A,,#*#on%&&y8 *he !re.#se 0or5 o0 *he -**% .on,#*#on re#res so5e e9!&%n%*#on$ In

se.*#on '$'$) #* +%s 5en*#one, *h%* on.e *he -**% .ons*r%#n* +%s %,,e,8 %n o"er4

,e*er5#ne, sys*e5 res&*e,$ The so&*#on ne.ess#*%*e, #/nor#n/ *he 0&o+ *%n/en.y

.on,#*#on %* one .on*ro& !o#n*$ B* +h#.h .on*ro& !o#n* *o #/nore

As #* *rns o* *h#s #s no* %n %&*o/e*her %rb#*r%ry ,e.#s#on$ Fro5 sy55e*ry8 #* #s

,es#r%b&e *o ne/&e.* e#*her *he LE or TE .on*ro& !o#n*8 ho+e"er8 on&y one o0 *hese !o#n*s

#s .orre.* *he LE !o#n*$ I0 #ns*e%, +e ne/&e.* *he Tr%#n/ E,/e !o#n*8 +e 0#n, *h%* *he

"or*#.#*y ,#s*r#b*#on #s *he re"erse %n, ne/%*#"e o0 *he e9!e.*e, *ren, +#*h *he

"or*#.#*y *en,#n/ *o+%r,s ne/%*#"e #n0#n#*y %* *he TE$ 2h#&e o0 5#nor #5!or*%n.e *o *he

s*e%,y s*%*e .%se8 re"ers#n/ *he .on*ro& !o#n* or,er#n/ 0or *he *#5e4s*e!!#n/ .%se +#&&

res&* #n erroneos res&*s *hro/h #n*er%.*#on +#*h *he n%s.en* +%-e "or*e9$

Fr*her5ore8 *h#s sy55e*ry %r/5en* h%s ser#os #5!.%*#ons 0or *he +r%!!e,

,#s*r#b*#on$

"ith onl% t"o panels "e achieve a circulation "ith / 10ms and an an$le of attac# of0.1 rads "e achieve the same circulation of 0.*2+2+19 "ith either 2 or '00 panels.

4 10 4


19/54

23+32 Finite T&ickness Air.oil

The #5!&e5en*%*#on 0or *he 0#n#*e *h#.-ness %#r0o#& !ro.ee,s #n 5.h *he s%5e +%y %s

0or *he 0&%* !&%*e$ Ho+e"er8 *here %re so5e s#/n#0#.%n* ,#00#.&*#es$ These ,#00#.&*#es

.en*er !r#5%r#&y on *he !rob&e5 o0 sy55e*ry$ I* #s e%sy *o re%son +#*h % 0&%* !%ne& %s

*o *he !re0erre, .on*ro& !o#n* *o e#n%*e #n or,er *o en0or.e *he -**% .on,#*#on8

ho+e"er8 +#*h % +r%!!e, sr0%.e8 *h#s .ho#.e #s &ess ob"#os$ Be0ore +e .%n %ns+er *h#s

es*#on #* #s ne.ess%ry *o ,es.r#be *he /eo5e*ry o0 *he +r%!!e, .on0#/r%*#on$

The 0#rs* *%s- #s *o .re%*e %n %#r0o#& sr0%.e$ De *o *he e9!e.*e, #n*eres* !r#5%r#&y #n

*he &e%,#n/ %n, *r%#n/ e,/e !or*#ons o0 *he %#r0o#&8 % s#nso#,%& ,#s*r#b*#on o0 !%ne&s #s

se, #ns*e%, o0 *he ne%r sys*e5 e5!&oye, on *he 0&%* !&%*e$ As +e&&8 5%ny

.o5b#n%*#ons o0 .on*ro& !o#n* %n, "or*e9 !%ne& !&%.e5en* +ere %**e5!*e, be0ore%rr#"#n/ on % /oo, .o5b#n%*#on o0 "or*#.es %* *he !o#n* o0 e%.h !%ne& %n, .on*ro&

!o#n*s %* *he !o#n*$ Th#s &e, *o % re&%*#"e&y e"en ,#s*r#b*#on o0 s#n/&%r#*#es %n,

.on*ro& !o#n*s o"er *he sr0%.e$ Th#s sr0%.e8 !ro"#,e, #n 0#/re ?8 sho+s *he "or*#.es %s

%s*er#9es %n, .on*ro& !o#n*s %s so, .#r.&es$

Figure ): Wra$$ed *ir!oil inusoidal +eometry , -*C* &&1. *ir!oil

/*sterixes are Vortices w'ile solid circles are control $oints0

I* #s .r#*#.%& 0#rs* *h%* *he *o! %n, bo**o5 !%ne&s %re 5#rror #5%/es o0 one %no*her %n,

%s s.h % .on*#nos +r%! #s no* !oss#b&e %* *he &e%,#n/ e,/e *he s#n/&%r#*#es %n,

.on*ro& !o#n*s s+%!!e, !&%.es on *he %#r0o#& *o *he %n, .hor,8 res!e.*#"e&y8 on *he

4 11 4


20/54

n,ers#,e$ Fr*her5ore #* #s #5!or*%n* *o %**e5!* *o 5%#n*%#n *he ,#s*%n.e be*+een

%,%.en* "or*#.es *o %"o#, e9.ess#"e s!#-#n/$

The sr!r#s#n/ res&* o0 %&& *h#s #s *h%* *he !o#n* +h#.h 5s* be o5#**e, #s *he = s*.on*ro&

!o#n* on *he !!er sr0%.e$ 2h#&e #* 5%y be %n*#.#!%*e, *h%* *h#s +o&, nb%&%n.e *he

%#r0o#&8 re.%&& *h%* *here #s no e"%&%*#on .on,.*e, %* *he *r%#n/ e,/e on *he &o+er

sr0%.e be.%se o0 *he +r%!!#n/ s+#*.h *h#s &e%"es bo*h *he 0#rs* %n, 0#n%& .on*ro&

!o#n*s o!!os#*e *o one %no*her %* *he s%5e 94.oor,#n%*e$

his distri!ution su$$estion comes from )ho" eference 1' !ut has also !eensuccessfull% used on sin$le panel airfoils such as in Win$field 9

4 12 4


21/54


22/54

-no+n %s % Von ;Kr5Kn S*ree* %s #n F#/re @$

As s.h8 #* #s ,es#re, *o *re%* *he +%-e s#5#&%r&y *o *he %#r0o#& 4 *h%* #s *o %!!ro9#5%*e

*he .on*#nos +%-e "or*e9 shee* +#*h % ser#es o0 ,#s.re*e "or*e9 0#&%5en*s$ The

0o&&o+#n/ se.*#ons ,es.r#be *h#s !ro.e,re$ L%s*&y8 +e +#&& +%n* *o .ons#,er ener/y$

De!en,#n/ !on +he*her !#*.h#n/ &%/s !&n/#n/ 0or %n os.#&&%*#n/ %#r0o#& or "#.e "ers%

+e .%n e#*her #5!%r* ener/y *o *he 0&o+8 *hs !ro,.#n/ *hrs* %n, *8 or .%n re5o"e

ener/y 0ro5 *he 0&o+8 %s #n *he .%se o0 % +#n/5#&&$

+323' Flow Tangenc* #ondition

As +#*h *he s*e%,y %#r0o#&8 #* #s ne.ess%ry *o h%"e 0&o+ *%n/en.y %* *he sr0%.e o0 *he

%#r0o#& *o en0or.e #*s #5!er5e%b#*y$ As be0ore +e h%"e

0)( =+= jstreamj nVn '$?

Ho+e"er8 *here %re *+o s#/n#0#.%n* .h%n/es$ F#rs*8 s#n.e *he %#r0o#& #s os.#&&%*#n/ #n !#*.h

%n, !&n/e %..or,#n/ *o

jft#h h)2sin( = )$=

kfth))2sin(( 0 += )$'

+e 5s* 5o,#0y *he "%&e o0 *he "e&o.#*y %* #n0#n#*y #n *he 5o"#n/ re0eren.e 0r%5e$

Ths

)(eaiairf$ilstream

rhVVVV +== )$)

+here jftf#h h)2cos(2 = )$>

%n, kftfh

)2cos(20

= )$?

As +e&&8 #* #s ne.ess%ry *o 5o,#0y *he ,#s*rb%n.e !o*en*#%& *o %..on* 0or *he 0ree

"or*#.es 0on, #n *he +%-e %n, *he she, "or*e9$ As s.h8 +e ,er#"e

No*e *h%* *he s#/n o0 #s ne/%*#"e be.%se +e %re %ss5#n/ % .%se +here ener/y #s

be#n/ e9*r%.*e, 0ro5 *he 0&o+ reai#s *he "e.*or 0ro5 *he e&%s*#. %9#s *o .on*ro& !o#n* i

4 14 4


23/54

%akeTE$und ++= )$@

P**#n/ *h#s %&& *o/e*her +e %rr#"e %* *he 0o&&o+#n/ 0or5&%*#on 0or *he 0&o+ *%n/en.y

.on,#*#on %* %ny .on*ro& !o#n* i%&on/ *he %#r0o#&

[ ] =

+

++

=+

j p

i

jp

pi

pi

jp

ji

jieai

istream

n

r

rkn

r

rknrhV

nV

0

2

2

)(

)(

22

)$

+3232 6)tta #ondition

I* #s or #n*en*#on #n *h#s 5o,e& *o 5%-e se o0 % ne%r -**% .on,#*#on o0 *he 0or5 se,

#n *he s*e%,y s*%*e .%se$ Th%* #s8 +e +#&& re#re *he 0&o+ *o be *%n/en*#%& %* *he TE$

2h#&e % -**% .on,#*#on b%se, on ero !ressre ,#00eren.e %* *he TE +%s *es*e, by

6,n#.h8 #* +%s no* 0on, *o be s#/n#0#.%n*&y 5ore e00e.*#"e *hen *h#s ne%r 5o,e&$ Ths

+e sh%&& 5o,e& *he ns*e%,y -**% .on,#*#on %s

[ ] =

+

++j p

TE

jTE

pTE

pTE

jTE

jTE

jTEeai n

r

rkn

r

rknrhV 0

2

2

)(

22

)$

As #n *he s*e%,y s*%*e .%se8 *h#s re#res s *o #/nore one .on*ro& !o#n* +hen

e"%&%*#n/ *he 0&o+ *%n/en.y .on,#*#on$

+323+ 6el5in8s #irc)lation T&eorem

In or,er *o n- *he %#r0o#& *o #*s +%-e %n, hen.e so&"e 0or *he n-no+n +%-e "or*e9

s*ren/*hs8 +e 5%-e se o0 ;e&"#n7s .#r.&%*#on *heore5$ S#5!&y !*8 *h#s re#res *h%*

*he s5 o0 *he bon, "or*#.#*y 1#$e$ *he .#r.&%*#on o0 *he %#r0o#&3 %n, *he "or*#.#*y o0 *he

+%-e be e%& %* e%.h *#5e s*e! s.h *h%* *he *o*%& "or*#.#*y #n *he 0&o+ #s .onser"e,$

S#n.e % "or*e98 on.e she,8 h%s % .ons*%n* 0#9e, s*ren/*h *he on&y n-no+n %n*#*#es

%re *he .#r.&%*#on o0 *he n%s.en* "or*e9 %* *he TE %n, *he .#r.&%*#on o0 *he bon,

"or*#.es$ As s.h8 *he "or*#.#*y &os* by *he %#r0o#& %* %ny *#5es*e! #s e#"%&en* *o *he

.#r.&%*#on o0 *h#s ne+ +%-e "or*e9$ Ths

4 1' 4


24/54

=+j

h

j

h

TEj

h

j )()( 1 )$

Th#s %,,s one 5ore n-no+n %n, one 5ore e%*#on *o or se*$

+323 ,nstead* Press)re Distrib)tion

I* .%n be sho+n *hro/h so&*#on o0 *he .o5!&e*e ns*e%,y !ressre e%*#on #n %

5o"#n/ re0eren.e 0r%5e *h%* *he !ressre ,#00eren.e be*+een *he !!er %n, &o+er

0%.es o0 *he "or*e9 shee* #s /#"en by

[ ] [ ] )(2

)( ululul

lu

tVVVVppp

++== )$=(

As s.h8 +e .%n sbs*#**e #n or e9!ress#ons 0or *he s5 %n, ,#00eren.e o0 *he "e&o.#*y

%s ,#s.sse, #n *he !re"#os .h%!*er 5o,#0#e, by or !,%*e, ,e0#n#*#ons 0or *hese

%n*#*#es8 #n *he %!!ro!r#%*e re0eren.e 0r%5e$ As 0or *he *#5e4,er#"%*#"e o0 *he

!o*en*#%& #* #s *e5!*#n/ *o /o s*r%#/h* *o % 0#n#*e4,#00eren.e %!!ro%.h8 ho+e"er8 *he

!roo0 *h%* *h#s #s #n,ee, "%, #s so5e+h%* #n"o&"e,%n, *here0ore +e !resen* *he

en*#re ns*e%,y !ressre e%*#on +#*ho* !roo0

[ ]

1

1

22

2

2

)(

+

+

+

+

=

hhh

j

h

j

j p

i

jp

pi

pi

jp

ji

j

ieai

h

j

j

ttn

r

rkn

r

rk

nrhV

tl

p

)$==

On.e *h#s !ressre ,#s*r#b*#on h%s been ,e*er5#ne, *he 0or.es %n, 5o5en*s %re

.%&.&%*e, %s #n .h%!*er ' %..or,#n/ *o e%*#ons '$== *hro/h '$=>$

ri$orous proof of this can !e found in eference 14, adapted for this $eometricalsituation !% Zdunich. he proof of this can !e found in Zdunich, section 3.3.4

4 1* 4


25/54


26/54

+3+ Time %tepping model (mplementation

The o"er%&& #5!&e5en*e, *#5e s*e!!#n/ .o,e h%, *he 0o&&o+#n/ or,er o0 o!er%*#on /#"en

*he ser4,e0#ne, #n!* !%r%5e*ers o0 s*e! s#e8 0reen.y8 "e&o.#*y %* #n0#n#*y8 !#*.h &%/8

!&n/e %n, !#*.h %5!*,e8 n5ber o0 !%ne&s %n, &o.%*#on o0 *he e&%s*#. %9#s

=$ C%&.&%*e *#5e4"%ry#n/ /eo5e*ry 1se.*#on )$)$=3

'$ C%&.&%*e s*%*#. #n0&en.e .oe00#.#en*s be*+een *he bon, "or*#.es on *he

%#r0o#& %n, *he n%s.en* +%-e "or*e9 1se.*#on )$)$'3

)$ De*er5#ne *he #n#*#%& ,#s*r#b*#on o0 "or*#.#*y on *he %#r0o#& s#n/ *he s*e%,y

s*%*e .%se +#*h no +%-e "or*#.es !resen* 1se.*#on '$)3

>$ De*er5#ne *he B %n, U #n0&en.e .oe00#.#en*s 0or *he !%r*#.&%r *#5e s*e! #n

es*#on 1se.*#on )$)$'3?$ So&"e *he sys*e5 o0 e%*#ons /ener%*e, *o y#e&, *he ne+ bon, "or*e9 %n,

n%s.en* +%-e "or*e9 s*ren/*hs

@$ C%&.&%*e *he Pressre %n, For.e D#s*r#b*#ons 0ro5 *he ne+ /eo5e*ry

1E9%.*&y %s #n )$'$>3

$ Con"e.* *he +%-e 0or+%r,s *o *he ne9* *#5es*e! 1se.*#on )$)$)3 %n, re!e%*

0ro5 s*e! > n*#& *he !re4,e0#ne, n5ber o0 *#5e !o#n*s h%"e been

e"%&%*e,$

+3+3' 7eneration o. Time-Var*ing 7eometr*

As *he 5o*#on o0 *he %#r0o#& #s #5!ose,8 #* +%s re&%*#"e&y e%sy *o /ener%*e *he /eo5e*ry$

S!e.#0#.%&&y8 /#"en *he 0reen.y %n, s#e o0 *he *#5e s*e!8 #* +%s !oss#b&e *o !re4

/ener%*e *he *#5e "%ry#n/ /eo5e*ry s#n/ % 5o,&e !ro/r%5$ S#5!&y !*8 *h#s .o,e

#5!&e5en*e, e%*#ons )$= %n, )$' on *he !%ne&$ As s.h8 +h#&e #* +%s on&y e"er

%!!e, *o *he 0&%* !%ne& .%se8 #* +o&, s#5!&y be % 5%**er o0 0ee,#n/ *h#s !ro/r%5 %

,#00eren* s*%r*#n/ /eo5e*ry #n or,er 0or #* *o !ro!%/%*e *hro/h *#5e %ny /eo5e*ry o0

"or*e9 shee* ,es#re,$

4 1- 4


27/54

Fig)re 9:Coordinate ystem used !or ime2Marc'ing Flat "anel Code

#e$rinted !rom #e!% 1& wit' modi!ications to t'e model s'own in grey

As .%n be seen 0ro5 0#/re 8 *he /eo5e*r#. .o,e #s s/h*&y ,#00eren* 0ro5 *h%* o0

6,n#.h s#n.e %n e&%s*#. %9#s h%s been #n.or!or%*e, +h#.h .%n be e%s#&y 5o"e, %bo* *o

s#5&%*e ,#00eren* .on,#*#ons$ The &o+er4.%se .oor,#n%*e %9es .orres!on, *o *he

/eo5e*r#. %9es +h#.h %re h%n,e, *o *he /r#, /ener%*or +h#.h #n *rn !roe.*s *he5 %s

% 0n.*#on o0 *#5e on*o *he !!er .%se #ner*#%& %9es$

As +e&&8 *he !os#*#on o0 *he "or*e9 she,,#n/ !o#n* +%s se&e.*e, *o be h%&0 % !%ne& &en/*h

1#$e$ l' 3 beh#n, *he *r%#n/ e,/e$

+3+32 #alc)lation o. (n.l)ence #oe..icients and!n.orcement o. 6)tta and 6el5in #onditions

As #n,#.%*e, %* *he be/#nn#n/ o0 se.*#on )$)8 *h#s *%s- +%s %..o5!she, #n se"er%&

s*e!s$ A&& *o&, 0#"e ,#00eren* "%r#e*#es o0 #n0&en.e .oe00#.#en*s +ere /ener%*e, o0 +h#.h

4 19 4


28/54

*hree %re #5!or*%n* 0or *h#s se.*#on %n, *he o*her *+o 0or +%-e .on"e.*#on$ These *hree

%re

=$ *he #n0&en.e o0 bon, "or*#.es on %#r0o#& .on*ro& !o#n*s

j

ji

ji

ij nr

rkA

2

2

=

'$=?

'$ *he #n0&en.e o0 nbon, +%-e "or*#.es on %#r0o#& .on*ro& !o#n*s

j

jp

pih

ip nr

rkB

2

2

=

)$=

)$ *he s*re%5 "e&o.#*y 10or en0or.#n/ *he F&o+ T%n/en.y Con,#*#on3

[ ]

[ ] i

eaihh

ieaii

nrkftfjftf#V

nrhVU

)2cos(2)2cos(2

)(

0 =

+=

)$=

2e no+ se *hese *o ,e*er5#ne *he sys*e5s o0 e%*#ons re#re, *o !ro.ess *he *#5e4

5%r.h#n/ s.he5e$ F#rs*&y8 *he -**% %n, ;e&"#n .on,#*#ons %re .o5b#ne, +#*h *he 0&o+

*%n/en.y .on,#*#on #n % 0or5 s#5#&%r *o *h%* o0 *he s*e%,y .%se #n or,er *o .%&.&%*e *he

s*ren/*hs o0 *he bon, "or*#.es %n, o0 *he s#n/&e n%s.en* "or*e9

0

000

1111

1

1

11

11

11111

1

1

1

1

1

11111

=

+

+

h

p

TEhTEpTE

ihipi

hp

j

h

j

n

j

h

TE

h

n

h

i

h

TEV()TTEnTEjTE

iV()Tiniji

V()Tnj

BBB

BBB

BBB

U

U

U

AAAA

AAAA

AAAA

)$=

As 5en*#one, #n *he be/#nn#n/ o0 se.*#on )$)8 *he ne9* s*e! #s *o .o5!*e *he !ressre

,#s*r#b*#on 0ro5 e%*#on )$==$ Fo&&o+#n/ *h#s #* #s ne.ess%ry *o .on"e.* *he +%-e

4 20 4


29/54


30/54

*he s#n/&%r#*y s.he5%*#.%&&y #n*o *+o re/#ons ,ene%*e, %* *he r%,#s $-no+n %s *he

"or*e9 .ore r%,#s$ The 0or5&%*#on /#"es ,#00eren* e9!ress#ons 0or *he "e&o.#*y !o*en*#%&

,e!en,#n/ !on +he*her *he !o#n* o0 #n*eres* es +#*h#n *he .ore r%,#s$

For r%,## /re%*er *h%n $+e h%"e *he 0%5#%r res&*

r

rkV jj 2

== '$'

+h#&e 0or r%,## +#*h#n *he .ore r%,#s +e se *he e9!ress#on

2

2

rkV

jj

== )$')

Figure : #anine Core Velocity "otential c'ematic#e$rinted !rom #e!erence 1&

4 22 4


31/54

#&apter

Panel Met&od Model Veri.ication/0es)lts and Disc)ssion

3' Veri.ication o. %tead* %tate Model

3'3' i.t #)r5e

The *%s- o0 "er#0y#n/ *he s*e%,y s*%*e 5o,e&s 4 bo*h 0&%* !&%*e %n, +r%!!e,

.on0#/r%*#ons 4 #s s*r%#/h*0or+%r,$ In bo*h .%ses #* #s ne.ess%ry *o .o5!%re *he

/ener%*e, *4.r"e *o *h%* e9!e.*e, 0ro5 *he *heory o0 *h#n %#r0o#&s

2=l+ >$=

+here+q

L+l

= %

>$'

%n, =j

jjsVL % >$)

No*e *h%* +h#&e *he * !er n#* s!%n .%n be ob*%#ne, #n e#*her *he s*e%,y or ns*e%,y

.%se by #n*e/r%*#n/ *he 0or.e ,#s*r#b*#on8 e%*#on >$) #s "%, on&y 0or *he s*e%,y s*%*e

.%se$

The res&*s ob*%#ne, #n *he 0&%* %#r0o#& .%se %re !resen*e, #n 0#/re +h#&e *hose 0or *he

+r%!!e, .on0#/r%*#on %re !ro"#,e, #n 0#/re =($

4 23 4


32/54

Figure 5: Flat "late *$$roximation to t'e Li!t Cur6e !rom t'in air!oil t'eory

/num7er o! $anels 8 )&0

Th#s !%r*#.&%r /r%!h +%s *r#e, %* se"er%& ,#00eren* !%ne& s#es %n, sho+e, no .h%n/e

o+#n/ *o *he "or*#.#*y !ro!er*#es ,#s.sse, #n se.*#on '$)$=$

Figure 1&: Full 'icness *ir!oil *$$roximation

to t'e Li!t cur6e !rom t'in *ir!oil 'eory /num7er o! $anels $er sur!ace 8 )&0

4 24 4


33/54

In .on*r%s*8 *he "%&es 0or *he .oe00#.#en* o0 * ob*%#ne, 0or *he 0&& *h#.-ness %#r0o#&

*en,e, *o .on"er/e *o % "%&e .&ose *o *he *h#n %#r0o#& %!!ro9#5%*#on %s *he n5ber o0

!%ne&s #n.re%se,$ Th#s #s !erh%!s ,e *o *he #5!er0e.* ,#s*r#b*#on o0 "or*e9 !%ne&s

o"er *he sr0%.e % *o!#. +h#.h +#&& be e9!&ore, #n *he ne9* se.*#on$

O"er%&&8 *he %/ree5en* o0 bo*h 5o,e&s +#*h *he *h#n4%#r0o#& *heory +h#.h *hey %re

e9!e.*e, *o e5&%*e #s "ery /oo,$

3'32 Distrib)tion o. Vorticit*

As %n %,,#*#on%& *es* o0 *he "%,#*y o0 *he 0&%*4!&%*e %!!ro9#5%*#on8 #* #s ne.ess%ry *o

!&o* *he obser"e, ,#s*r#b*#on o0 "or*#.#*y o"er *he !%ne&$ The *heory 0or *h#s .o5es0ro5 Cho+ +ho s*%*es *h%* on % 0&%*4!%ne& %#r0o#&

,

,+V,

= )sin(2)( >$>

As s.h8 *he ,#s*r#b*#on o0 *he .#r.&%*#on +%s /r%!he, %/%#ns* *h#s 0n.*#on #n F#/re

== %* %n %n/&e o0 %**%.- o0 ($= r%,#%ns %n, *hen ,#"#,e, by %n %!!ro!r#%*e nor5%#n/

0%.*or$ As #n 6,n#.h8 /oo, %/ree5en* +%s 0on, 0or % ,#s*r#b*#on o0 ?( !%ne&s8 %

s#*%*#on +h#.h +%s no* #5!ro"e, s#/n#0#.%n*&y %* h#/her !%ne& n5bers$

As +#*h *he * .r"e s&o!e8 *he ,%*% %re #n /oo, %/ree5en* +#*h *he *heore*#.%&

.#r.&%*#on$

4 2' 4


34/54

Figure 11: Distri7ution o! 7ound circulation on a !lat $late air!oil%

/num7er o! $anels 8 )&0

3'3+ #oe..icient o. Press)re

A be**er 5e%sre o0 *he !er0or5%n.e o0 *he +r%!!e, 5o,e& +#*h 0#n#*e *h#.-ness #s *he

.%&.&%*#on o0 *he !ressre .oe00#.#en* ,#s*r#b*#on o"er *he %#r0o#& on *he !!er %n,

&o+er sr0%.es$ As s.h8 0#/re =' sho+s *he ,#s*r#b*#on ob*%#ne, .o5!%re, %/%#ns* %

.on*#nos ,#s*r#b*#on !%ne& .o,e !ro"#,e, by Jones$

As .%n be seen8 *he &o+er sr0%.e %s +e&& %s 5.h o0 *he !!er sr0%.e #s #n e9.e&&en*

%/ree5en*$ A ,#s.re!%n.y #s obser"e, %* *he *r%#n/ e,/e -e&y ,e *o *he 0%.* *h%* *he

!%ne& .o,e h%s no .on*ro& !o#n*s here *hey h%"e been se, *o 0or5&%*e *he -**%

.on,#*#on$ O0 /re%*er #n*eres* #s *he s#*%*#on *o+%r,s *he LE$ Here8 sho+n #n *he b&o+4

! o0 0#/re =)8 *he ,%*% be.o5e s!r#os$ I* #s n-no+n +hy *h#s o..rs8 ho+e"er8

s#n.e *he /eo5e*ry h%, !ro"e, "ery 0#n#.-y #n *he !%s* *h#s %s!e.* o0 *he 5o,e& #s *he

-e&y .&!r#* on.e %/%#n$

he author learned Fortran so that this code could !e translated into matla! forcomparaison

4 2* 4


35/54

Th#s bee0 #s re#n0or.e, by 0#/re =>$ Th#s #s % /r%!h o0 *he !ressre ,#s*r#b*#on n,er

% ne%r ,#s*r#b*#on o0 !o#n*s %n, sho+s no s.h %r*#0%.* %* *he &e%,#n/ e,/e$ Des!#*e

*h#s8 ,/#n/ 0ro5 *he * .r"e /r%!h ob*%#ne,8 *here #s so5e .o5!ens%*#on *%-#n/

!&%.e s#n.e *h#s s!r#os ,%*% ,oes no* h%"e % s#/n#0#.%n* e00e.* on *he *o*%& .#r.&%*#on

o0 *he %#r0o#&$

As s.h8 +h%* #s nee,e, #s % ro*#ne *o /ener%*e %n e%& /r#, #n *+o ,#5ens#ons *o

%**e5!* *o re/&%r#e *he s#e o0 *he !%ne&s$ For bo*h *he s#nso#,%& %n, ne%r

,#s*r#b*#ons *he +ors* !er0or5%n.e #s #n &o.%*#ons +here *he /eo5e*ry %!!ro%.hes *he

"er*#.%& %n, *hs +e %re br,ene, e#*her +#*h !%ne&s +h#.h %re *oo s5%&& or +h#.h %re

*oo b#/$

Figure 12 : omparison of wrappe! airfoi l "s# a mo!ern ontinuous Vorticit$Distri%ution &anel o!e (num%er of panels ' per surface* gri! spacing:

sinusoi!al )

4 2+ 4


36/54

Figure 13: 9low2$ o! L; section o! Figure 1.

Figure 14: omparison of wrappe! airfoi l "s# a mo!ern ontinuous Vorticit$Distri%ution &anel o!e (num%er of panels ' per surface* gri! spacing:

linear)

4 2- 4


37/54

32 Veri.ication o. Time-Marc&ing Model

323' Flow Vis)aliation

The 5os* ob"#os 5e*ho, *o /e* so5e #,e% o0 ho+ +e&& *he .o,e #s 0n.*#on#n/ #s *o se0&o+ "#s%%*#on *e.hn#es$ As s.h8 .o,e +%s +r#**en *o %n%&ye *he +%-e %n, *o

/r%!h#.%&&y !resen* *he *r%#n/ "or*#.es +h#.h res&* 0ro5 *he %#r0o#& 5o*#on$ As seen #n

0#/re @8 +%-e ro&&4! #s %sso.#%*e, +#*h *hrs*#n/8 +h#&e % s5oo*h +%-e #s #n,#.%*#"e o0

ener/y be#n/ e9*r%.*e, 0ro5 *he 0&o+$ I* #s !oss#b&e *o "#e+ *hese s#*%*#ons #n 0#/res

=? %n, =@8 ho+e"er #* #s 0#rs* ne.ess%ry *o #n*ro,.e so5e 0&o+ !%r%5e*ers

Re,.e, Freen.y

=V

f+k

>$?

A,"%n.e R%*#o+

tV= >$@

The re,.e, 0reen.y ,es.r#bes *he 0&o+

re/#5eW 0or - ($= *he 0&o+ #s ns*e%,y

%n, .%nno* be %!!ro9#5%*e, by s*e%,y4

s*%*e or %s#4s*e%,y s*%*e 5e*ho,s$ The

"%s* 5%or#*y o0 *he 5o*#on #n +h#.h +e

%re #n*eres*e, e9#s*s +#*h#n *he r%n/es

($= *o =8 %s s.h8 % *es* .%se +#*h - X($?

+%s se&e.*e, +#*h ($= r%,#%n 5%9#55

!#*.h %n, ($?. !&n/e$

The %,"%n.e r%*#o #s % 5e%sre o0 ho+

%..r%*e *he 0&o+ #s be#n/ re!resen*e, by

*he *#5e4s*e!!#n/ 5o,e&$ 2h#&e #* +%s

0on, ,r#n/ *he .ons#,er%*#on o0 In,#.#%&

5o*#on *h%* %n %,"%n.e r%*#o o0 ($(? +%s

se0&8 *h#s !ro"e, *#5e .ons5#n/ *o

!ro.ess #n 5%*&%b 1on *he or,er o0 se"er%&

hors !er 0&%!!#n/ .y.&e3$ As s.h8 ,e *o

*#5e .ons*r%#n*s8 *he 5%or#*y o0 *he

5e%sre5en*s *%-en s#n/ *he .o,e +ere

4 29 4

Figure 1 : +a,e Visualization oft-rust

Figure 16 : +eig-ting of Vortices (.e!an! /lue "ortices are of opposite sign)


38/54

.o5!&e*e, 0%r 0ro5 *he #,e%& %,"%n.e r%*#o%s#,e 0ro5 *he #n,#.#%& .%ses 1#n 0%.*8 *he

!o+er .r"es +ere .o5!&e*e, +#*h %n %,"%n.e r%*#o o0 on&y ($> no* so 5.h 0or

%..r%.y %s *o ,e*er5#ne *he n,er&y#n/ *ren,s3$ A /r%!h#.%& re!resen*%*#on o0 *h#s

*r%,eo00 #s !ro"#,e, #n 0#/re =$

Figure 10 : ra!e offs associate! wit- increase! panel an! t ime steppingaccurac$

I* #s #n*eres*#n/ *o no*e *he r%!#, 0%&&o00 o0 *he !o+er +#*h *he #n.re%se #n !%ne& n5ber$

As %n %s#,e *h#s +%s #n"es*#/%*e, %n, %s #* *rns o* *he ,e.re%se #n !o+er %s 5ore

%n, 5ore !%ne&s %re %,,e, #s ,e 5%#n&y *o *he 0%.* *h%* *he n%s.en* "or*e9 /e*s .&oser

%n, .&oser *o *he TE %s *he s#e o0 e%.h !%ne& ,e.re%ses$ 2hen se"er%& *r#%&s +ere

.on,.*e, +#*h 0#9e, "or*e9 she, !o#n*s #* +%s ,#s.o"ere, *h%* *he !o+er %.*%&&y

#n.re%se, s/h*&y %s *he n5ber o0 !%ne&s #n.re%se,$

3232 (ndicial Motion

A .&%ss#. 5e%ns o0 .%br%*#n/ ns*e%,y "or*e9 !%ne& 5e*ho,s #s *o se % +%/ner7 .%se

n%5e, %0*er *he %ero,yn%5#.#s* +ho 0#rs* !ro!ose, % *heore*#.%& so&*#on 0or *he

For the computation time reuired for various advance ratios, please see ppendix

4 30 4


39/54

#n0&en.e o0 +%-e "or*#.es %s *hey %re .on"e.*e, ,o+ns*re%5$ The *heory #s s#5!&e %*

*#5e X(8 *he %#r0o#& 5%-es % s,,en 1#n,#.#%&3 5o"e5en*W *h#s #n,.es % .h%n/e #n *he

bon, "or*#.#*y +h#.h re#res % +%-e "or*e9 *o be re&e%se, %s !er *he ;e&"#n Theore5$

Figure 13 : +agner ase

The res&*s o0 *h#s *es* %re

!resen*e, #n 0#/re = %n, *he

res&*#n/ +%-e #s "#s%e, #n

0#/re =$ I* +%s 0on, *h%* %

.o5b#n%*#on o0 ?( !%ne&s %n,

%n %,"%n.e r%*#o o0 ($(?

+or-e, +e&& +#*h *he

#n.re5en*%& .os* o0 %,,#*#on%&

%..r%.y beyon, *h%* !o#n*

be#n/ n0%"or%b&e$

he data for this "a$ner test "ere retrieved from a ta!le provided !% 5arric# 1'

4 31 4

Figure 14 : +a,e Visualization for +agner ase


40/54

323+ Wingmill Power and !..icienc*

The 0#rs* s*e! #n *he "er#0#.%*#on #s *o

!&o* *he .oe00#.#en*s o0 5o5en* *

1F#/re '(3 %n, !o+er 1F#/re '=3 *o

5%-e sre *h%* *hey h%"e % 0or5

s#5#&%r *o *h%* e9!e.*e, 0ro5

M.;#nney 0or *he 2#n/5#&&1F#/re

''3$

These %/ree so5e+h%* +e&& #n *h%*

+e see % s#5#&%r *y!e o0 s#*%*#on$The %"er%/e o0 *he !o+er /r%!h #s

/re%*er *h%n ero8 *hs *he %#r0o#& #s

,e0#n#*e&y re5o"#n/ ener/y 0ro5 *he

0&o+$ As +e&&8 *he !o+er os.#&&%*#on

o..rs %* *+#.e *he 0reen.y o0 *he

os.#&&%*#on #n *he *$

Un0or*n%*e&y *here +%s #ns00#.#en*

*#5e *o .on,.* % r#/oros ser#es o0

*es*s on *he !o+er /ener%*e, by *he

s#5&%*e, +#n/5#&&$ Th#s +%s no*

%#,e, by *he 0%.* *h%* *he /re%*er

!%r* o0 *he e9!er#5en*%& ,%*% 0ro5

+h#.h so5e o0 *he /r%!hs +ere

,er#"e, %!!e%rs *o be 5#ss#n/ 0ro5 *he %!!en,#9 o0 M.;#nney7s == *hes#s$ Ho+e"er8

e"en #0 #* +ere %"%#&%b&e8 *he *#*y o0 *h#s ,%*% #s so5e+h%* es*#on%b&e ,e *o *he

0&o+ .on,#*#ons$

As s.h8 #* +%s ,e.#,e, *o s#5!&y !er0or5 %n e9!er#5en*%& #n"es*#/%*#on %* h#/h

%,"%n.e r%*#o 1X($>8 +h#.h %&&o+e, re%son%b&e .o5!*%*#on *#5es3 *o ,e*er5#ne *he

n,er&y#n/ *ren,s$ As s.h8 !o+er /r%!hs %re !ro"#,e, 0or *he M.;#nney 2#n/5#&&

10#/re ')3 #n % 0&o+ %* 5s os.#&&%*#n/ %* ? H 1- X ($))3 s#n.e *h#s +%s *he 5os*

4 32 4

Figure 2 : oefficients of 5ift an! omenta%out t-e -alf c-or! for 7irfoil

Figure 21 &ower 8enerate!


41/54

e00#.#en* .%se *es*e, #n ==$ To "er#0y *h#s8 %n e00#.#en.y !o&%r #s %&so !ro"#,e, 1F#/re

'>3$ 2h#&e +e h%"e b%, %/ree5en*8 *he n5bers %.h#e"e, %re o0 *he s%5e or,er o0

5%/n#*,e

Figure 22 : 9xpecte! &ower* ormal force an! moment oefficients.eprinte! from .eference 6

4 33 4

Figure 2; :9fficienc$

Variat ion wit-pitc- lag angle

Figure 2< :&ower 8enerat ion

Variat ion wit- pitc-lag angle

(="e is a flut tercase* "e is a t-rust

case)


42/54

3+ Disc)ssion

2h#&e *he %**%.he, 0&o+ "or*e9 !%ne& 5o,e& ,es.r#be, #n *h#s !%!er ,oes h%"e so5e

#&&s*r%*#"e "%&e8 *he o"er%&& *#*y 0or !re,#.*#n/ *he !er0or5%n.e o0 +#n/5#&&s #s

es*#on%b&e$ Ho+e"er8 *h#s #s no* *o s%y *h%* % 5o,#0#e, "ers#on +o&, no* be %b&e *o

o"er.o5e s.h ,#00#.&*#es8 b* s#5!&y *h%* #* #s #5!or*%n* *o re.o/n#e *he #*%*#ons o0

*he .rren* 5o,e&$

F#rs* o0 %&&8 +e .ons#,er *he /r%!hs ob*%#ne,$ The %*y o0 *he +%/ner .%se ob*%#ne, #s

/oo, %n, .orres!on,s 0%"or%b&y +#*h *he res&*s ob*%#ne, by 6,n#.h8 ho+e"er8 +e rn

#n*o ,#00#.&*#es +hen .ons#,er#n/ !o+er$ F#/re = ,e5ons*r%*es *h%* *here #s %

.o5!&e9 re&%*#onsh#! #n *he 5o,e& be*+een s*e! s#e8 !%ne& n5ber %n, e9*r%.*e,

!o+er +h#.h s*#&& h%s no* 5%n%/e, *o .on"er/e e"en %* reso&*#ons h#/her *hen *he

s..ess0& #n,#.#%& .%se$

2hen .ons#,er#n/ *he /r%!hs ob*%#ne, %s !%r* o0 *he +#n/5#&& .%se8 +h#&e *he /ener%&

0or5 #s /oo,8 *he *ren,s obser"e, #n 0#/res %&so see5 *o be 0&%+e,$ For #ns*%n.e #n *he

.%se o0 VX5s8 0re X ?8 5%9 !#*.h#n/ %n/&e X )( 4 % .%se o0 !%r*#.&%r #n*eres* *o

M.;#nney +e obser"e *h%* *he 5%9#55 e00#.#en.y #s ob*%#ne, #n *he *hrs*#n/ r%n/e

1ener/y #n!* *o 0&o+3 %n, no* #n *he 0&**er#n/ r%n/e 1ener/y o*!* 0ro5 0&o+3$

Fr*her5ore8 !on e9%5#n#n/ on&y *he &%**er r%n/e8 +e ob*%#n % !e%- #n bo*h e00#.#en.y%n, !o+er %* ='(4=)( r%*her *hen *he e9!er#5en*%& ( 0or e00#.#en.y %n, ==( 0or

!o+er$ I0 +e #ns*e%, .ons#,er *he *hrs*#n/ .%se8 *h#s &%/ ,oes %!!e%r *he !o+er !e%-s

%* 4>(8 +h#&e *he e00#.#en.y ,oes so %* .&ose *o 4($

Se.on,&y8 *here #s *he #sse o0 %**%.he, 0&o+ .on,#*#ons$ These %re %ss5e, *o be

!resen* #n *h#s "or*e9 !%ne& 5o,e& e"en *ho/h *he 5%#n *es* .%ses %&& h%"e % !#*.h#n/

%5!*,e o0 '> or /re%*er$ In 0%.* 0ro5 *he !o#n* o0 "#e+ o0 *he +#n/5#&& %s % !o+er

/ener%*#on *oo&8 *he .%ses #n +h#.h *he 0&o+ #s no* %**%.he, %re o0 /re%*es* #n*eres*$ In

0%.*8 M.;#nney %n, DeL%r#er +ere %b&e *o sho+ *h%* /re%*er !#*.h %n/&es on *he or,er

o0 )( &e, *o s#/n#0#.%n* #5!ro"e5en*s bo*h #n *er5s o0 !o+er /ener%*e, %n,

e00#.#en.#es$ I* #s #n *h#s %re% *h%* o!er%*#ons %re ,es#r%b&e$

As s.h8 #* #s no* sr!r#s#n/ *h%* *he ,%*% ob*%#ne, 0ro5 *he 5o,e& ,#s%/ree +#*h *he

4 34 4


43/54

,%*% o0 M.;#nney$ O0 /re%* #5!or*%n.e #s *he obser"e, hys*eres#s &oo! #n *he M.;#nney

,%*% +h#.h8 +h#&e 5#&, 0or *he &o+er s!ee, *r#%&s8 #s %n #5!or*%n* 0%.*or #n *he h#/her

s!ee, .%ses$ Th#s s*ron/&y s//es*s *h%* e"en *ho/h #* .%n be sho+n 0ro5 /eo5e*ry

*h%* *he re&%*#"e +#n, "e.*or ne"er e9.ee,s *he s*%*#. s*%&& %n/&e o0 *he %#r0o#&8 s*%&n/

#s #n 0%.* o..rr#n/$ As s.h8 #* #s so5e+h%* 5e%n#n/&ess *o .o5!%re *he 5o,e& *o *h#s

e9!er#5en*%& ,%*%$

Ho+e"er8 *here 5%y be s#*%*#ons #n +h#.h *h#s 5o,e& .%n be se0&$ The 5os* ob"#os

.%se #s one +#*h s5%&&er !#*.h#n/W ho+e"er8 %s be0ore *h#s #s -e&y o0 **&e #n*eres*$

Un0or*n%*e&y8 *h#s s#*%*#on ,oes no* #5!ro"e 0or o*her .o55on 5e,#5s$ For

#ns*%n.e8 +h#&e *he Reyno&,s n5ber 0or *he s#*%*#on ,es.r#be, #n %#r #s Z =9=(?+e

0#n, *h%* +#*h +%*er %s % 5e,#5 +e /e* % Reyno&,s n5ber o0 Z =$@9=(@,e *o *he

#n.re%se #n ,ens#*y by % 0%.*or o0 (( +#*h on&y % ?@40o&, #n.re%se #n 0, "#s.os#*y$ Th#s

s//es*s *h%* no* on&y 5#/h* % +%*er +#n/5#&& be !&%/e, by 0&o+ se!%r%*#on b* 5#/h*

%&so h%"e .%"#*%*#on #sses #0 *he 0reen.y o0 os.#&&%*#on +%s .o5!%r%b&e *o %#r$

Th#s o*nes % se.on, +e%-ness #n *he 5o,e& n%5e&y *h%* s#n.e *he 5e*ho, +%s

#n#*#%&&y ,e"e&o!e, *o ,e%& +#*h %.*#"e8 #5!ose, 0&%!!#n/ #n or,er *o /ener%*e *hrs*

*here #s no !ro"#s#on *o ,e*er5#ne *he e#br#5 0reen.y 0or % +#n/5#&&$ In or,er *o

%..o5!sh *h#s8 #* #s ne.ess%ry *o #n*ro,.e %n %eroe&%s*#. .o5!onen* *o *he 5o,e&

+h#.h #s beyon, *he s.o!e o0 *h#s !%!er$ Ho+e"er8 *here #s re%son *o bee"e *h%* *he

s#*%*#on 5%y no* be %s ,#re %s &%s* !%r%/r%!h s//es*e,$

For #ns*%n.e8 *here #s no re%son *o bee"e *h%* *he .on,#*#ons #n +%*er +o&, be

.o5!%r%b&e *o *hose #n %#r$ For #ns*%n.e8 *he #n.re%se #n ,ens#*y no* on&y #n.re%ses *he

Reyno&,7s n5ber8 b* %&so #n.re%ses *he 5%/n#*,e o0 *he 0or.es %.*#n/ on %

sb5er/e, %#r0o#& %&on/ +#*h *he %!!%ren* 5%ss$ As s.h8 #* #s re%son%b&e *o e9!e.*

*h%* *he 0reen.#es en.on*ere, +o&, be s#/n#0#.%n*&y &o+ere, %n, *he %bo"e .r#s#s

5#/h* be %*o5%*#.%&&y %"er*e,$

Irre/%r,&ess8 #* 5%y be ,es#re, *o +or- %* &o+er !#*.h#n/ %5!*,e8 %n, %s s.h #*

+o&, be bene0#.#%& *o 5%9#5#e *he ener/y e9*r%.*#on$ One +%y o0 %..o5!sh#n/ *h#s

#s *o se % "%r#%b&e4/eo5e*ry %#r0o#& *o o!*#5#e *he %ero,yn%5#. 0or.#n/ on *he %#r0o#&$

To %..o5!sh *h#s8 "%r#%b&e .%5ber +o&, be se,$ A,,#*#on%&&y8 %s #n *he .%se o0 %

h#/h4* se.*#on8 *h#s %,,#*#on o0 .%5ber 4 !ro!er&y !h%se, 4 5#/h* be %b&e *o %&&o+

4 3' 4


44/54

%**%.he, 0&o+ %* h#/her !#*.h %n/&es$

Th#s .%5ber .o&, %r#se #n one o0 *+o +%ys #* .o&,8 o0 .orse8 be %r*#.&%*e,8 *hs

%,,#n/ % *h#r, ,e/ree o0 0ree,o5 *o *he %#r0o#&8 ho+e"er8 #* .o&, %&so be %..o5!she,

*hro/h %eroe&%s*#. *%#&or#n/ o0 *he %#r0o#& s*r.*re 5.h #n *he +%y *h%* *he

orn#*ho!*er +#n/ *+#s* %/5en*s *he /ener%*e, *hrs*$ In e#*her .%se8 *he .on.e!*%&

0or5&%*#on #s s#5#&%r$

As s.h8 #* #s % +or*h+h#&e *ho/h* e9er.#se *o .ons#,er *he ne.ess%ry 5o,#0#.%*#ons

+h#.h +o&, nee, *o be 5%,e *o *he .rren* "or*e9 !%ne& 5e*ho, #n or,er *o %..on*

0or *h#s "%r#%b&e .%5ber$ Ths8 *he %*hor h%s en,e%"ore, *o !ro,.e %n %,,#*#on%&

se.*#on #n +h#.h *h#s *heore*#.%& b%s#s #s !ro"#,e, %n, #*s 5er#*s ,#s.sse,$ The

#5!&e5en*%*#on %n, %,,#*#on o0 *h#s .o5!onen* *o *he .o,e ,es.r#be, #n *h#s !%!er

+o&, be %n #n*eres*#n/ 0*re e9er.#se es!e.#%&&y #0 e9!er#5en*%& ,%*% 0or %r*#.&%*e,

+#n/5#&&s #n +%*er .h%nne&s be.o5es %"%#&%b&e$

(n this case "e are perhaps more in the spirit of the !endin$ of a fish6s tail, ho"everthe analo$% holds.

4 3* 4


45/54

#&apter 1

!xtension o. Panel Met&od to +-Degree o.Freedom Wingmills

13' !xtension o. t&e T&eor*

In *h#s .h%!*er8 +e e9*en, *he +#n/5#&& "or*e9 !%ne& 5o,e& *o %..on* 0or % )4,e/ree o00ree,o5 %r*#.&%*e, +#n/5#&&$ Th#s 5o,#0#.%*#on 5%y be #n.&,e, #n % 0*re "ers#on o0

*he !%ne& .o,e$ There %re *hree 5e%ns o0 %..o5!sh#n/ *h#s e%.h 5ore

.o5!*%*#on%&&y e9h%s*#"e *he &%s*$

The s#5!&es* 5e*ho, o0 s#5&%*#n/ "%r#%b&e .%5ber #s by %!!&y#n/ % 5o,#0#e, ;**%

Con,#*#on %* *he *r%#n/ e,/e$

Perh%!s % 5ore re%s*#. +%y o0 #5!&e5en*#n/ *h#s "%r#%b&e /eo5e*ry #s +#*h % .on*ro&

sr0%.e s#n.e *he 5e.h%n#.s o0 *h#s *y!e o0 %r*#.&%*#on %re +e&& n,ers*oo, %n, .o&,

be 5e.h%n#.%&&y %..o5!she, 0%#r&y e%s#&y

L%s*&y8 +e sh%&& .ons#,er % 0&&y %r*#.&%*e, .%5bere, sr0%.e +hose /eo5e*ry #s 0&&y

his su$$estion came out of a conversation "ith 7r. 7e8aurier on the matter.

4 3+ 4


46/54

.%!%b&e o0 ,e0or5#n/$

132 Modi.ied 6)tta #ondition

1323' T&eor*

In *he !%ne& .o,e8 *he -**% .on,#*#on #s se, *o ,e*er5#ne *he +%y #n +h#.h *he 0&o+

&e%"es *he %#r0o#& sr0%.e8 *hs #* .%n be se, *o s#5&%*e % .s!e, *r%#n/ e,/e *o /#"e

*he #&&s#on o0 .%5ber %* *he TE$ As s.h8 #n *he /ener%e, ns*e%,y .%se8 +e 5%y

re!resen* *he -**% .on,#*#on %s

[ ] =

+

++j p

TE

jTE

pTE

pTE

jTE

jTE

jTEeai n

r

rkn

r

rknrhV 0

2

2

)(

22

)$

s#n.e *he !os#*#on o0 *he %#r0o#& #n *he #ner*#%& re0eren.e 0r%5e #s .ons*%n*&y .h%n/#n/8

*he n#* nor5%& %* *he TE #s e9!resse, %s % 0n.*#on o0 *#5e$ As s.h8 *o s#5&%*e

.%5ber8 +e 5s* ro*%*e *h#s "e.*or %..or,#n/ *o

h

TE

h

TE

h

TE nn-n

100

0)cos()sin(

0)sin()cos(

)(

==

?$=

+here )2sin(0 ++= hft ?$'

No*e *h%* #0 X( 1#$e$ *he &%/ be*+een !#*.h#n/ %n, .%5ber ro*%*#on3 *hen *he !h%se o0

.%5ber 5%*.hes *he !h%se o0 *he !#*.h#n/ #$e$ +hen *he %#r0o#& #s %* #*s h#/hes* %n/&e

o0 %**%.- #* #s %* #*s 5os* .%5bere,$

13232 (mplementation

In or,er *o #5!&e5en* *he 5o,#0#e, ;**% .on,#*#on #* #s ne.ess%ry *o %&*er *he

.%&.&%*#on o0 = e&e5en*s o0 *he RHS o0 *he sys*e5 o0 e%*#ons$ As +e&&8 one ro+

1i=TE3 o0 *he 0or5er&y .ons*%n* A45%*r#9 no+ be.o5es *#5e4s*e! ,e!en,%n*$ Ho+e"er8

#0 +e re*%#n % 5o,e& #n +h#.h *he /eo5e*ry #s #5!ose, %n, !re.%&.&%*e, 1#$e$ *he

.rren* 5o,e&3 #* #s re%son%b&e *o h%"e %"%#&%b&e % 5%*r#9 o0 *hese *#5e ,e!en,%n*

4 3- 4


47/54

ro+s$ S.he5%*#.%&&y8 on.e *hese .h%n/es h%"e been 5%,e +e %.h#e"e *he 0o&&o+#n/

0or5 0or e%*#on )$=

?$=)

The %&*ere, ro+s %re h#/h/h*e, #n /rey$

13+ #ontrol %)r.ace

13+3' T&eor*

Con.e!*%&&y *h#s "ery s#5#&%r *o .h%n/#n/ *he *r%#n/ e,/e -**% .on,#*#on e9.e!* *h%*

se"er%& !%ne&s %re ro*%*e,$ I* #s bes* 0or *h#s ro*%*#on o..r %* % .on*ro& !o#n* r%*her

*hen #n *he 5#,,&e o0 % !%ne& so %s no* *o .o5!.%*e *he .%&.&%*#on o0 0or.es %n,

5o5en*s &%*er on$ 2e ,e0#ne % h#n/e !o#n* #n 5.h *he s%5e +%y %s +e ,e0#ne, *he

e&%s*#. %9#s8 *he *o*%& ,#s!&%.e5en* %* % !o#n* #n 0ron* o0 *he h#n/e 1or %* *he h#n/e3 #s

n.h%n/e, +h#&e %&& !o#n*s beyon, h%"e %n %,,#*#on%& ,#s!&%.e5en* %s /#"en by

#.&/Eir ?$?

+here kfth))2sin(( 0 ++= ?$>

%n, kftf h))2cos(2 += ?$?

hese are currentl% calculated usin$ vector method and as such a nonflat $eometr%"ill not confuse their calculation

4 39 4

0

000

1111

1

1

11

11

11111

1

1

1

1

1

11111

=

+

+

h

p

TEhTEpTE

ihipi

hp

j

h

j

n

i

h

TE

h

n

h

j

h

h

TEV()T

h

TEn

h

TEj

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iV()Tiniji

V()Tnj

BBB

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U

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48/54

%s s.h8 *he 0&o+ *%n/en.y %n, !ressre 5! *er5s 5s* #n.&,e *h#s *he*% *er5 0or

%&& # *he H#n/e !o#n*$ Ths

0or i HINGE

j#.&/Eieaij nrrhVU )( ++=

?$@

No*e *h%* *he -**% .on,#*#on #s n.h%n/e, s#n.e *he 0&o+ #s s*#&& *%n/en*#%& *o *he 0#n%&

%#r0o#& se/5en* 1e"en *ho/h *h%* se/5en* #s no+ &o.%*e, on % 5o"#n/ .on*ro&

sr0%.e3$ Ho+e"er8 *he A4*y!e -**% #n0&en.e .oe00#.#en*s %re %&so *#5e ,e!en,%n*

s#n.e *he ,#s*%n.e be*+een *he5 %n, *he res* o0 *he %#r0o#& #s .h%n/#n/$

13+32 (mplementation

A&on/ +#*h *he .h%n/es ,es.r#be, %bo"e8 *he ro+s %n, .o&5ns o0 *he A 5%*r#9 beyon,

*he H#n/e !o#n* be.o5e *#5e4,e!en,%n* s#n.e *he .on*ro& !o#n*s %n, bon, "or*#.es on

*he .on*ro& sr0%.e .h%n/e !os#*#on +#*h res!e.* *o *he re5%#n,er o0 *he %#r0o#&$

S.he5%*#.%&&y

?$

4 40 4

0

000

11111

1

1

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11

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1

1

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1

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111111

=

+

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p

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ihipi

hp

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h

j

n

#

j

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h

n

h

#

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TEVTEnTE#TEjTE

#V#n###j#

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Vn#j

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AAAAA

AAAAA

AAAAA

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49/54

13 F)ll* Artic)lated #ambered %)r.ace

De *o *he /ener%& n%*re o0 *h#s "%r#e*y o0 .ons*r%#n* #* #s #5!oss#b&e *o /#"e *hes!e.#0#.s +h#.h .o&, be se* o* #n *he o*her *+o .%ses$ Ins*e%,8 #n %,,#*#on *o

s!e.#0y#n/ *he *#5e4"%ry#n/ /eo5e*ry #* +#&& be ne.ess%ry *o s*%*e *he ,er#"%*#"es o0

5o*#on o0 %&& *he .on*ro& !o#n*s 0or *he !r!oses o0 .%&.&%*#n/ *he !ressre ,#s*r#b*#on

%n, 0&o+ *%n/en.y .on,#*#on$ As s.h8 e"ery e&e5en* o0 *he A45%*r#9 +#&& be *#5e

,e!en,%n* +#*h *he e9.e!*#on o0 *he 0#n%& ro+$

4 41 4


50/54

#&apter

%)mmar*

3' #oncl)sions

In s55%ry8 *he obe.*#"es o0 *h#s !%!er +ere 5e* +#*h 5#9e, s..ess$ I* +%s !oss#b&e

*o e9*en, *he s*e%,y s*%*e 5o,e& 1%n, hen.e *he *#5e4"%ry#n/ 5o,e&3 *o %#r0o#&s o00#n#*e *h#.-ness by +r%!!#n/ % shee* o0 ,#s.re*e "or*e9 0#&%5en*s$ In *h#s .%se8 % ,e.en*

res&* +%s ob*%#ne, +#*h bo*h s#nso#,%& %n, ne%r ,es.r#!*#ons o0 *he sr0%.e

/eo5e*ry$ Ho+e"er8 +e %re 0or.e, *o .hoose +he*her +e !re0er or %no5%&y %* *he

&e%,#n/ or *r%#n/ e,/e %s s.h8 #* 5%y be +or*h+h#&e *o #n"es*#/%*e 0r*her

,#s*r#b*#ons #n or,er *o #5!ro"e *h#s res&*$

In *er5s o0 % *#5e4s*e!!#n/ 5%r.h#n/ "or*e9 5o,e& /oo, res&*s 0or *he 0or.es %n,

5o5en*s on *he %#r0o#& +ere ob*%#ne, %s sho+n by *he +%/ner .%se o0 #n,#.#%& 5o*#on$

As +e&& 0&o+ "#s%%*#ons sho+ *h%* *he +%-e %n, seen.e o0 "or*e9 she,,#n/ #s %s

+e e9!e.* %n, %/rees +e&& +#*h 0&o+ "#s%%*#on o0 % *hrs*#n/ bo,y$ Des!#*e *h#s8 #n

*he .ons#,er%*#on o0 *he !o+er o0 % +#n/5#&&8 +h#&e *he /r%!hs o0 *8 5o5en* %n,

!o+er h%"e *he .orre.* 0or58 *hey ,o no* y#e&, *he .orre.* res&*$ Th#s #s -e&y ,e *o

se!%r%*e, 0&o+ e00e.*s %n, *hs #s % 0%#&re o0 *he *es* .%se se&e.*#on %n, *he /ener%&

4 42 4


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&%.- o0 +#n/5#&& e9!er#5en*%& ,%*% r%*her *hen *he 5o,e& !er se$

In or,er *o %,, 0r*her ,e!*h *o *he 5o,e& by #5!ro"#n/ *he ener/y e9*r%.*#on %* &o+

!#*.h#n/ %n/&es %n, *o #n.re%se *he 5%9#55 !#*.h %n/&e 0or +h#.h *he 0&o+ re5%#ns

%**%.he,8 % *heore*#.%& ,#s.ss#on o0 *he 5o,en/ .ons#,er%*#ons o0 "%r#%b&e .%5ber

+%s n,er*%-en$ Us#n/ *he *heory o0 "or*e9 !%ne& 5e*ho,s8 e%*#ons +ere ob*%#ne,

+h#.h .o&, be se, *o /ener%*e % 0*re "ers#on o0 *he !%ne& .o,e !resen*e, here

#n.or!or%*#n/ *h#s %s!e.*$

32 F)t)re #onsiderations

There %re 0or re.o55en,%*#ons 0or 0*re +or- on *h#s sbe.* +h#.h 0%&& o* o0 *h#s

*hes#s$ F#rs*&y e#*her 5ore !o+er0& .o5!*#n/ 0%.#*#es sho&, be se, or *he !%ne&

.o,e sh&, be #5!&e5en*e, #n &o+er&4&e"e& &%n/%/e s.h %s C or FORTRAN$ Th#s

+o&, %&&o+ s5%&&er *#5e s*e!s *o be se,8 *hs #5!ro"#n/ or es*#5%*es o0 !o+er$

J,/#n/ by 0#/re =8 #* %!!e%rs *h%* *he !&o* .on"er/es +e&& %* s5%&&er *#5e s*e!s e"en

*ho/h *he ,e!en,en.e on *he n5ber o0 !%ne&s #s so5e+h%* !n/$ Ths8 5ore

,%*% #s re#re, #n *h#s 5ore .o5!*%*#on%&&y e9!ens#"e r%n/e$

Se.on,&y8 *h#s 5o,e& +#&& re#re %n %eroe&%s*#. .o5!onen* #n or,er *o ,e*er5#ne *he

.orre.* 0reen.y o0 0&%!!#n/ 0or se #n *he *#5e4"%ry#n/ !%ne& .o,e$ As +e&&8 #0

%n%&yses %re ,es#re, 0or *he M.;#nney +#n/5#&&8 #* +#&& be ne.ess%ry *o se % 5o,e&

+h#.h *%-es #n*o %..on* se!%r%*e, 0&o+s %n, ,yn%5#. s*%&n/ e00e.*s$

F#n%&&y8 *he 5o,e& +o&, bene0#* 0ro5 *he .o&&e.*#on o0 0r*her e9!er#5en*%& ,%*%$ S#n.e

*here #s .rren*&y "ery **&e +#n/5#&& ,%*%8 %n, no ,%*% %* %&& 0or %**%.he, 0&o+ .%ses

1s#n.e8 %,5#**e,&y8 *hese *en, *o be %* &o+ e00#.#en.#es 0or %#r3 #* #s ,#00#.&* *o "er#0y *he

5o,e& #n *he .on*e9* o0 +#n/5#&&s$ Des!#*e *h#s8 +h#&e *he 5o,e& 5%y h%"e /oo,!re,#.*#"e "%&e on.e *he .%5bere, .%ses h%"e been #n.or!or%*e,8 n*#& e9!er#5en*%&

e"#,en.e #s ob*%#ne, 0or *h#s .%se8 *he 5o,e& .%n no* be "er#0#e, #n *h#s .%se$

4 43 4


52/54

0e.erences

= S*r#.-&%n,8 J$H$ 1=?3 The Darrius Turbine: A Performance Prediction Model usin

Multiple !tream Tubes$ Un#*e, S*%*es De!%r*5en* o0 Ener/y8 S%n,r#%

L%bor%*or#es$ %"%#&%b&e onne h**!+++$!ro,$s%n,#%$/o"./#4

b#n*e.hb%..ess4.on*ro&$!&=??(>)=$!,0

' Je00ery8 J$R$ 1=3 "scillatin Aerofoil Pro#ect$ Re!or* 0ro5 *he Po.-n/*on S.hoo&

Des#/n Cen*re8 2es* Green8 Po.-n/*on8 Yor-8 En/&%n,$

) P%yne8 P$R$ 1=3 The Aeolian $indmill and other oscillatin ener% e&tractors $

Ann%& Mee*#n/ o0 *he A5er#.%n Che5#.%& So.#e*y$

> F%r*h#n/ S$ 1=3 !r#"%*e .o55n#.%*#on *o 2#&%5 M.;#nney$ As re!or*e, #n

Re0eren.e @$

? 2#&son8 R$E$ %n, L#ss%5%n8 P$B$S$ 1=>3 Applied Aerod%namics of $indmills$

NTIS PB4')8 ??$

@ M.;#nney8 2 %n, DeL%r#er8 JD 1==3 [The 2#n/5#&& An Os.#&&%*#n/42#n/

2#n,5#&&$\'ournal of Ener%"o& ?8 n'8 !!=(4==?$

4 44 4
http://www.prod.sandia.gov/cgi-bin/techlib/access-control.pl/1975/750431.pdfhttp://www.prod.sandia.gov/cgi-bin/techlib/access-control.pl/1975/750431.pdfhttp://www.prod.sandia.gov/cgi-bin/techlib/access-control.pl/1975/750431.pdfhttp://www.prod.sandia.gov/cgi-bin/techlib/access-control.pl/1975/750431.pdf


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A,%5-o8 D$A$ %n, DeL%r#er8 J$D$ 1=3 [An e9!er#5en*%& s*,y o0 %n Os.#&&%*#n/

2#n/ 2#n,5#&&$\ Proceedins of the !econd (anadian $or)shop on $ind

Enineerin!! @>4@@$

Ten/8 N$H$ 1=3 The de*elopment of a computer code for the numerical solution

of unstead%+ in*iscid and incompressible flo, o*er and airfoil$ M%s*er7s Thes#s8

N%"%& Pos*/r%,%*e S.hoo&$

2#n0#e&,8 J$F$ 1=(3A Three-Dimensional nstead% Aerod%namic Model $ith

Applications to Flappin-$in Propulsion.M%s*er7s Thes#s$ Un#"ers#*y o0

Toron*o Ins*#**e 0or Aeros!%.e S*,#es$ Toron*o8 On*%r#o8 C%n%,%$

=( 6,n#.h8 P$ 1'(('3A Discrete /orte& Model of nstead% !eparated Flo, About a

Thin Airfoil for Application to 0o*erin Flappin $in Fliht. M%s*er7s Thes#s$

Un#"ers#*y o0 Toron*o Ins*#**e 0or Aeros!%.e S*,#es$ Toron*o8 On*%r#o8

C%n%,%$

== Jones ;$D$ 8 D%"#,s8 S$ %n, P&%*er8 M$F$ 1=3 [Os.#&&%*#n/ 2#n/ Po+er

Gener%*#on\ ASME P%!er 4(?( #n Proceedins of the third A!ME'!ME 'oint

Fluids Enineerin (onference.

=' Br%-e8 A8 6r#-e58 68 %n, M#r8 A 1'(('3 [Mo,]s%*#on ,e &7E9*r%.*#on ,e &7Ener/#e

Eoenne !%r ne A#&e Os.#&&%n*e$\ In Proc2d2 de la Forum nternationale sur

les Eneries Renou*elables 4FER5 77.

=) An,erson8 J$D$ 1'((=3 Fundamentals of Aerod%namics$ )r,E,#*#on$ Pbshe, by

M.Gr%+ H#&& Pbsh#n/$

=> Jones8 ;$D$ 1'(()3 Onne Con*#nos Vor*e9 P%ne& So&"er onne

h**!+++$%%$n!s$n%"y$5#&Zones

=? G%rr#.-8 I$ E$ 1=)3 "n some reciprocal relations in the theor% of nonstationar%

flo,s.NACA Te.hn#.%& Re!or* n@'$

4 4' 4
http://www.aa.nps.navy.mil/~jones/http://www.aa.nps.navy.mil/~jones/


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=@ Cho+8 C$Y$ 1=3 ntroduction to (omputational Fluid Mechanics$ Pbshe, by

John 2#&ey %n, Sons Pbsh#n/$

Wingmills.doc

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