7 J our nal of H ydr odyna mics , Se r . B ,5 ( 2003 ) ,7 - 12 Chi na Ocea n Press , B ei ji n g - Pri nt e d i n Chi na

PROPULSIVE PERFORMANCE AND VORTEX S HEDD ING OF A FOIL IN Ξ FLAPPING MOTION Ya ng Ya n , L u Xi2yun , Yi n Xie2z he n D ep a rt me nt of Mec ha nics a n d Mec ha nical En gi nee ri ng , U nive rsit y of Scie nce a n d Tec h nol o gy of Chi na , Hef ei 230026 , Chi na , e 2mail : xlu @ ustc . e du. c n ( Receive d Sep t . 4 , 2002 ) ABSTRACT : 　The propulsive performance and vortex shedding of oscillating foil , which mimics biological locomotion , were nu2 merically investigated. The objectives of t his study were to deal wit h unsteady force , in particular t hrust force , exerted on t he foil in pitching and plunging motions , and to explore t he relation of t he propulsive performance wit h vortex structures near t he foil and vortex shedding in t he near wake. The two2dimensional in2 compressible Navier2Stokes equations in t he vorticity and stream2 function formulation were solved by fourt h2order essentially com2 pact finite difference schemes for t he space derivatives and a fourt h2order Runge2 Kutta scheme for t he time advancement . To reveal t he mechanism of t he propulsive performance , t he un2 steady force and t he shedding of t he trailing2 and leading2edge vortices of t he foil were analyzed. The effects of some typical fac2 tors , such as t he frequency and amplitude of t he oscillation , t he p hase difference between t he pitching and plunging motions , and t he t hickness ratio of t he foil , on t he vortex shedding and un2 steady force were discussed.

t o t he l oc o m oti o n p e rf or ma nce . I n p a rticula r , a n oscilla ti n g f oil has of t e n e m p l oye d as a t yp ical m odel t o deal wit h t he f la p p i n g m oti o n t o mi mic bi ol ogical l oc o m oti o n . The oretical a nal yses i ncludi n g t he vort e x p a nel met h od a n d t he unst ea d y lif ti n gli ne t he or y ha ve [1 ] bee n p e rf or me d f or bi r d f li ght a n d f is h swi m 2 [2 , 3 ] mi ng . A det aile d re view of va ri ous t he oretical met h ods has bee n give n by S mit h et al . [ 4 ] . Exp e ri 2 me nts o n oscilla ti n g f oils ha ve e xhi bit e d t he e xis 2 t e nce of op ti mal p a ra met e rs f or t he ge ne ra ti o n of [5 ] a n ef f ective t h rust . Tria nt af yll ou et al . st udie d e xp e ri me nt all y t he wa ke mec ha nis m f or t h rust ge n2 [6 ] e ra ti o n i n oscilla ti o n f oils . A n de rs o n et al . e x2 a mi ne d t he hi gh p r op ulsive ef f icie nc y of oscilla ti n g f oils a t a ce rt ai n f re que nc y. So me rece nt e xp e ri 2 me nt al w or k has bee n p e rf or me d t o i nvesti ga t e t he lea di ng2e dge vortices i n i nsect f li ght by Elli ngt o n et KEY WORDS : 　 vortex dynamics , vortex shedding , t hrust force , unsteady flow , flapping motion al . [ 7 ] , as well as t he wi n g r ot a ti o n a n d t he ae r od y2 na mic basis of i nsect f li ght by Dic ke ns o n et al . [ 8 ] . B y use of c o mp ut a ti o nal f lui d d yna mics met h 2 1 . 　INTROD UCTION ods , a t w o2di me nsi o nal h ove ri n g f light was nume r 2 I nsects a n d f is h ha ve e xp e rie nce d a billi o ns2 icall y c o mp ut e d by Gust afs o n et al . [ 9 ] , a n d t hei r yea r p r ocess of e voluti o n wit h na t ural selecti o n f or calcula t e d results we re qualit a tive a gree me nt wit h t hei r survival a n d ha ve de vel o p e d t hei r sup e ri or s o me a vaila ble e xp e ri me nt al da t a . L iu et al . [ 10 ] ap 2 a n d c o mp let e p e rf or ma nce of f li ght a n d swi m 2 p lie d a met h od of pse udo2c o mp ressi bilit y t o c o m 2 mi ng. Usuall y , a f lap p i ng m oti o n is a basic m ode p ut e visc ous f l ow a r oun d a t h ree 2di me nsi o nal ri gi d of l oc o m oti o n f or i nsects , bi r ds , a n d f is h . Th rust wi ng a n d e xa mi ne d t he a xial f l ows ass ocia t e d wit h a n d lif t a re ge ne ra t e d w he n t he f la p p i ng wi n gs or t he lea di n g2e dge vort e x as obse rve d i n t he e xp e ri 2 t ails i nt e ract wit h t he sur r oun di n g f lui ds . B eca use me nts by Elli ngt o n et al . [ 7 ] . Wa n g [ 11 ] p e rf or me d a of t he hi ghl y unst ea dy na t ure of visc ous f l ow t w o2di me nsi o nal c o mp ut a ti o n of f oil i n f la p p i ng a r oun d a f lap p i ng wi ng , it is f a r f r o m sa tisf act or y m oti o n t o re veal t he f re que nc y selecti o n i n f orwa r d t o un de rst a n d t he p hysical mec ha nis m a n d vort e x i nsect f lap p i ng f li ght . Rece ntl y , Sun et al . [ 12 , 13 ] s he ddi n g i n unst ea d y visc ous f l ow , w hic h is c rucial i nvesti ga t e d nume ricall y t he unst ea d y ae r odyna m 2 Ξ This work was supported by t he National Natural Science Foundation of China ( Grant Nos : 10125210 , 10072063) , t he Hun2 dred2 Talent Programme of t he Chinese Academy of Sciences , and t he Innovation Project of t he Chinese Academy of Sciences. ( Grant Nos : KJ CX2SW2L04 , KJ CX22SW2L2)

© 1995-2003 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

8 ics of a m odel f ruit f l y wi ng i n f lap p i ng m oti o n as well as t he lif t a n d p owe r re qui re me nts f or h ove r 2

w he re u a n d v rep rese nt t he vel ocit y c o mp o ne nts i n x a n d y di recti o ns , as s h ow n i n Fi g. 1 .

i ng f li ght . Alt h ough s o me w or k has bee n ca r rie d out base d o n nume rical met h ods , it is still nee de d t o use hi gh2or de r sc he mes i n nume rical c o m p ut a ti o ns , a n i mp r ove me nt ove r p re vi ous w or k , t o re veal un 2 st ea dy f l ow

mec ha nis m a n d vort e x

d yna mics ,

w hic h a re c rucial t o t he f orce ge ne ra ti o n o n t he m ovi ng f oil . I n t his st ud y , visc ous f l ow p ast a f oil i n p itc hi n g a n d p lungi ng m oti o ns , w hit h is ca p a ble of mi mici n g bi ol ogical l oc o m oti o n , is nume ricall y i nvesti ga t e d f or a wi de ra n ge of t he c o mp ut a ti o nal p a ra met e rs. The t w o2di me nsi o nal i nc o mp ressi ble N a vie r2St o kes e qua ti o ns i n t he vorticit y a n d st rea m 2f uncti o n f or mula ti o n a re s olve d b y a f ourt h2 or de r esse ntiall y c o mp act f i nit e dif f e re nce sc he me , [ 14 ] de vel op e d by E a n d L iu , f or t he sp ace de riva 2

tives a n d a f ourt h2or de r R un ge2 Kut t a sc he me f or t he ti me a dva nce me nt .

2 . 　GOVERNING EQUATIONS As s h ow n i n Fi g. 1 , a s ketc h of a f oil i n p itc h 2 i ng a n d p lungi ng m oti o ns t o mi mic bi ol o gical l oc o2 m oti o n is illust ra t e d . I n t his st ud y , t he f ra me is f ixe d wit h t he f oil m oti o ns , a n d t he t w o 2di me n 2 si o nal i nc o mp ressi ble N a vie r2St o kes e qua ti o ns i n t he vorticit y a n d st rea m2f uncti o n f or mula ti o n a re e mp l oye d . To no n2di me nsi o nalize t he gove r ni n g e2 qua ti o ns , t he c h or d le n gt h of t he f oil c is use d as t he le n gt h scale , t he f ree2st rea m vel ocit y U as t he vel ocit y scale . The no n2di me nsi o nal gove r ni n g e 2

Fig. 1 　Sketch of a foil in plunging and pitching motions

W he n we s olve t he N a vie r 2St o kes Eqs . ( 1 ) 2 ( 2 ) i n t he f ra me f ixe d wit h t he f oil m oti o ns , a n a ddi 2 ti o nal t e r m due t o t he Cori olis f orce occurs i n E q . ( 1 ) f or t he p itc hi n g m oti o n , a n d no f ictiti ous f orce ap p ea rs i n t he vorticit y Eq . ( 1 ) f or t he p lungi ng m oti o n c o nsi de re d he re . The no 2slip a n d no2p e ne 2 t ra ti o n boun da r y c o n diti o ns a t t he f oil a re e nf orce d e xp licitly t h r ough t he vorticit y boun da r y c o n diti o n a n d t he st rea m2f uncti o n boun da r y c o n diti o n , re 2 sp ectivel y . I n t he f a r f iel d , t he boun da r y c o n diti o n o n t he st rea m2f uncti o n is give n by t he p ot e ntial [ 11 ] f l ow , as use d b y Wa ng , t he det aile d f or mula 2 ti o n has bee n desc ri be d t he re . The m oti o n of t he f oil c o nsists of p itc hi n g a n d p lungi ng oscilla ti o ns . The p lungi ng oscilla ti o n is w rit t e n as h = A m sin ( 2πf t + θ)

( 4)

a n d t he p itc hi n g oscilla ti o n as

qua ti o ns a re t hus give n as ,

α = αm sin ( 2πf t + θ + <) 5ω 5 Ψ 5ω 5 Ψ 5ω 1 2 dΩ + = g ω- 2 5t 5y 5x 5x 5y Re dt

( 1)

g 2Ψ = - ω

( 2)

( 5)

w he re A m a n d αm a re t he a mp lit udes of t he p lungi ng a n d p itc hi n g oscilla ti o n , resp ectivel y , f rep rese nts t he f re que nc y of t he oscilla ti o n , a n d θa n d < de not e t he i nitial p hase a n d t he p hase dif f e re nce bet wee n t he p itc hi n g a n d p lungi ng oscilla ti o ns .

w he re ω a n d Ψ rep rese nt t he vorticit y a n d st rea m2 f uncti o n , resp ectivel y. Ω is t he r ot a ti o n sp ee d of 3 . 　 NUMERICAL METHODS t he p itc hi n g oscilla ti o n . Re is t he Re ynol ds num 2 To calcula t e visc ous f l ow a r oun d a m ovi n g be r , def i ne d as Re = Uc/ ν , a n d νis t he ki ne ma tic f oil , we e mp l oy a f ourt h2or de r esse ntiall y c o mp act visc osit y. B ase d o n t he st rea m 2f uncti o n , t he vel oc 2 f i nit e dif f e re nce sc he me , de vel o p e d by E a n d [ 14 ] it y is obt ai ne d b y L iu , t o s olve t he i nc o m p ressi ble N a vie r2St o kes e qua ti o ns. A n a dva nt a ge of t he sc he me is t ha t a t eac h ti me st ep , o nl y t w o Poiss o n s olve rs a re re 2 5Ψ 5Ψ ( 3) qui re d t o ac hie ve a f ourt h or de r s p a tial accurac y. u = , v =5y 5x B y usi ng t he N a vie r2St o kes e qua ti o ns i n t he vortici 2 © 1995-2003 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

9 t y2st rea m f uncti o n f or mula ti o n , t he vorticit y boun da r y c o n diti o n is e xp licitl y e nf orce d t o sa tisf y t he no2slip boun da r y c o n diti o n . To disc retize t he ti me de riva tive i n Eq . ( 1 ) , we use a f ourt h2or de r R unge2 Kut t a sc he me f or t he ti me a dva nce me nt . 4 . 　 RESUL TS AND D ISCUSSION I n t his st ud y , a n ellip tic f oil is use d , a n d a c o nf or mal ma p is e mp l oye d f or gri d ge ne ra ti o n wit h a gri d num be r 256 ×256 i n t he ra dial a n d ci r 2 cumf e re ntial di recti o ns , res p ectivel y. The c o mp u2 t a ti o nal do mai n is a bout 10 c i n t he ra dial di rec 2 ti o n , a n d ti me st ep is 0 . 0002 i n view of t he c o m p u2 t a ti o n st a bilit y c o n diti o n . The c o nve r ge nce c hec k wit h dif f e re nt gri d sizes a n d ti me st e ps has bee n e xt e nsivel y t a ke n . It has bee n det e r mi ne d t ha t t he c o mp ut e d results a re i n de p e n de nt of t he ti me st e p a n d t he gri d size use d i n t he p rese nt calcula ti o n . To deal wit h s yst e ma ticall y t he mec ha nis m of t he p r op ulsive p e rf or ma nce a n d vort e x s he ddi n g of t he f oil i n t he p itc hi n g a n d p lungi ng m oti o ns , t he c o mp ut a ti o n p a ra met e rs a re c h ose n as f oll ows . The 3 Re ynol ds num be r is Re = 10 , t he t hic k ness ra ti o of t he f oil λ = 0 . 0625 , 0 . 125 , 0 . 25 a n d 0 . 5 . I n Eqs . ( 4 ) a n d ( 5 ) , t he p lungi ng a mp lit ude A m = 0 . 08 , 0 . 16 , 0 . 24 a n d 0 . 32 , t he p itc hi n g a mp lit ude as αm = 5°, 10°, 15°a n d 20°, t he oscilla ti o n f re 2 que nc y f = 0 . 5 , 1 , 2 a n d 4 , t he p hase dif f e re nce < = 90°t o 270°, a n d t he i nitial p hase as θ = 180° . To i nvesti ga t e t he ef f ects of t h ose p a ra met e rs o n t he vort e x s he ddi n g a n d unst ea d y f orce , s o me t yp ical results a re mai nl y discusse d i n t he f oll owi n g. Fig. 2 s h ows t he ti me 2dep e n de nt dra g a n d lif t c oef f icie nts f or t he p lungi ng oscilla ti o n a t A m = 0 . 16 , f = 1 , λ = 0 . 125 . The lif t c oef f icie nt CL va ries s ym met ricall y a bout t he ze r o , t hus ti me 2a v2 e ra ge d lif t f orce is ze r o as e xp ect e d f r o m t he s ym 2 met ric f lap p i n g. The f re que nc y of CD is t wice t ha t of CL , beca use t he dra g is ge ne ra t e d i n bot h t he up a n d dow n oscilla ti o ns . The dra g c oef f icie nt CD va ries as ym met ricall y a bout t he ze r o value , be 2 ca use t he f ore2a n d2af t s ym met r y is br o ke n due t o t he f ree2st rea m vel ocit y. The ti me2a ve ra ge d dra g c oef f icie nt C gD is - 0 . 39 ap p r oxi ma t el y. He re , we def i ne a ne ga tive value of C gD as a t h rust i n t he mea n f orwa r d di recti o n . Thus t he mea n t h rust c o 2 ef f icie nt i n t his case is 0 . 39 . To a nal yse t he mec ha nis m of t he t h rust ge ne r 2 a ti o n , we e xa mi ne t he qualit a tive f ea t ures of vor 2 t e x s he ddi n g. Fr o m t he vorticit y st ruct ures , it is

Fig. 2 　 Time2dependent lift and drag coefficients for t he plunging oscillation at A m = 0 . 16 , f = 1 , λ = 0 . 125

f oun d t ha t a re ve rse Ka r ma n vort e x 2st reet is f or me d i n t he wa ke of t he f oil , w he re t he vortices i n t he wa ke r ot a t e i n t he op p osit e di recti o n c o m 2 p a re d t o a classic Ka r ma n vort e x wa ke . He nce t he i n duce d f l ow has a c o m p o ne nt m ovi n g bac kwa r d wit h resp ect t o t he f oil t o ge ne ra t e t he t h rust . To un de rst a n d vort e x e voluti o n m ore clea rl y nea r t he f oil , Fi g. 3 s h ows t he vorticit y c o nt ours a t se ve ral p hases duri n g o ne p e ri od . W he n t he f oil is m ovi n g dow nwa r d , a ne ga tive lea di n g2vort e x L V N 1 is f or me d a n d p ai re d wit h a not he r p ositive lea di n g2 vort e x L V P0 t o f or m a vort e x p ai r ( i . e . , L V P0 + L V N 1 ) as s h ow n i n Fi g. 3 ( a ) , w he re t he p ositive vort e x rep rese nts a c ount e r2cl oc kwise vort e x a n d t he ne ga tive o ne is a cl oc kwise vort e x. Mea nw hile , a ne ga tive vort e x ( L V N 0 ) f or me d i n t he p re vi ous p e ri od m oves dow nst rea m al o n g t he l owe r2si de of t he f oil a n d a p ositive t raili n g2vort e x TV P1 is ge ne r 2 a t e d. N ot e t ha t t he vort e x p ai r ( L V P0 + L V N 1 ) r ot a t es a n d sep a ra t es a gai n i n Fi g. 3 ( b ) . The n , it is i nt e resti n g t o f i n d t ha t t he vort e x L V N 1 is p assi ng ove r t he lea di n g2e dge of t he f oil f r o m t he up p e r2 si de t o t he l owe r2si de of t he f oil , a n d its e voluti o n is helpf ul t o ge ne ra t e a t h rust f orce due t o t he f or 2 ma ti o n of t he l owe r2p ressure dist ri buti o n i n duce d by t he vort e x. W he n t he f oil is oscilla ti n g up wa r d , as e xp ect e d , t he vort e x e voluti o n is re p ea t e d i n t he

© 1995-2003 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

10 op p osit e di recti o n . I n Fi g. 3 ( c ) , t he lea di n g2vor 2 t e x L V N 0 c oalesces wit h t he t raili n g2vort e x TV N 1 t o s he d dow nst rea m . Mea nw hile , t he ne ga tive vort e x L V N 1 f or me d i n t he up p e r2si de of t he f oil is p ai re d wit h t he vort e x L V P1 t o f or m a vort e x p ai r ( i . e . , L V P1 + L V N 1 ) i n Fi g. 3 ( c ) . The vort e x p ai r ( L V P1 + L V N 1 ) is sep a ra t e d i n Fi g. 3 ( d ) . The vort e x L V N 1 m oves dow nst rea m al o n g t he l owe r2 si de of t he f oil a n d c or res p o n ds t o t he vort e x L V N 0 i n Fi g. 3 ( a ) . The vort e x L V P1 m oves ove r t he lea d 2 i ng2e dge f r o m t he l owe r2si de t o t he up p e r2si de of t he f oil a n d c or resp o n ds t o t he vort e x L V P0 i n Fi g. 3 ( a ) . I n t he f oll owi n g oscilla ti o n p e ri od , t he vor 2 t e x e voluti o n is rep ea t e d as desc rip ti o n a bove .

ti me2a ve ra ge d dra g c oef f icie nt values a t A m = 0 . 08 a n d 0 . 24 a re calcula t e d a n d a re a p p r oxi ma t el y 0 . 12 a n d - 0 . 3 , resp ectivel y , f or t he p lungi ng oscilla 2 ti o n a t f = 1 , λ = 0 . 125 . B y c o mp a ri ng C gD value i n Fi g. 2 , a n ef f ective t h rust f orce is ge ne ra t e d a t t he a mp lit ude A m = 0 . 16 ap p r oxi ma t el y. This be ha vi or is c o nsist e nt wit h t he calcula t e d results b y [ 11 ] Wa n g . He re , s o me a nal ysis is brief l y give n t o illus 2 t ra t e t he ef f ect of t he f re que nc y of t he oscilla ti o n o n t he f orce . Fi g. 4 s h ows t he ti me2dep e n de nt dra g c oef f icie nt f or t he p lungi ng oscilla ti o n a t f = 0 . 5 a n d 2 , resp ectivel y , f or A m = 0 . 16 a n d λ = 0 . 125 . The t h rust f orce is ge ne ra t e d beca use t he ti me 2a v2 e ra ge d dra g c oef f icie nt is ne ga tive . It is als o f oun d t ha t a re ve rse Ka r ma n vort e x2st reet is f or me d i n t he wa ke of t he f oil f or t he c or res p o n di n g c o n di 2 ti o ns i n Fi g. 4 .

Fig. 4 　Time2dependent drag coefficient for t he plunging os2 cillation at A m = 0 . 16 , λ = 0 . 125

Fig. 3 　Instantaneous vorticity contours at A m = 0 . 16 , f = 1 , λ = 0 . 125 . Solid lines represent positive values and dashed lines negative values. Increment of t he contours is 5

To illust ra t e t he ef f ect of t he a m p lit ude of t he oscilla ti o n o n t he f orce a n d vort e x s he ddi n g , t he

To illust ra t e t he ef f ect of t he t hic k ness ra ti o of t he f oil o n t he f orce a n d vort e x s he ddi n g , se ve r 2 al dif f e re nt t hic k ness ra ti os a re calcula t e d . As s h ow n i n Fi g. 2 , t he ti me2a ve ra ge d dra g c oef f icie nt C gD is - 0 . 39 ap p r oxi ma t el y a t λ = 0 . 125 . W he n t he t hic k ness ra ti o i nc reases , base d o n our calcula 2 ti o n f or λ = 0 . 25 , A m = 0 . 16 a n d f = 1 , C gD is 0 . 065 ap p r oxi ma t el y w hic h is not a t h rust f orce . As a t yp ical case , it is well k now n t ha t a dra g f orce is alwa ys f or me d f or a t ra nsve rsel y oscilla ti n g ci rcu2

© 1995-2003 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

11 [ 15 ] la r c yli n de r ( i . e . , λ = 1 ) i n a unif or m f l ow . Thus , t he t hic k ness ra ti o must be li mit e d a reas o n 2 a ble ra n ge t o ge ne ra t e a n ef f ective t h rust f orce . Visc ous f l ow a r oun d a f oil wit h p itc hi n g oscil 2 la ti o n a r oun d its mi d2c h or d is calcula t e d . The ti me 2 a ve ra ge d dra g c oef f icie nt C gD is alwa ys p ositive f or dif f e re nt a mp lit udes of t he oscilla ti o n . Cor re 2 sp o n di n gly , a classic Ka r ma n vort e x2st reet is f or me d i n t he wa ke of t he p itc hi n g f oil . Furt he r m ore , a n oscilla ti n g f oil wit h c oup le d t he p itc hi n g a n d p lungi ng m oti o ns is i nvesti ga t e d . As a t yp ical case , we f i rst a nal ysis t he f orce be ha v 2 i or f or A m = 0 . 16 , αm = 5°, < = 180°, f = 1 , λ = 0 . 125 . Fi g. 5 s h ows t he ti me2dep e n de nt dra g c oef 2 f icie nt. The ti me2a ve ra ge d dra g c oef f icie nt C gD is ne ga tive t o f or m a t h rust f orce . To e xa mi ne t he f ea t ure of vort e x s he ddi n g , a re ve rse Ka r ma n vor 2 t e x2st reet is f or me d i n t he wa ke .

Fig. 5 　Time2dependent drag coefficient for t he pitching oscil2 lation at A m = 0 . 16 , αm = 5°, < = 180° f = 1,λ= 0 . 125

To deal wit h t he ef f ects of t he p itc hi n g a n d p lungi ng a mp lit udes ( i . e . , αm , A m ) a n d t he p hase dif f e re nce < o n t he dra g f orce ( or t h rust f orce ) , Fig. 6 s h ows t he dra g c oef f icie nt ve rsus ti me f or αm = 5°a n d 10°, A m = 0 . 08 a n d 0 . 16 , a n d < = 90°, 135°a n d 180° . A t < = 135°, as s h ow n i n Fi gs . 6a a n d 6c , a n ef f ective t h rust f orce is ge ne ra t e d , w hic h is c o nsist e nt wit h t he p re vi ous e xp e ri me nt al w or k [ 6 ] . Howe ve r , a t t he p lungi ng oscilla ti o n a m 2 p lit ude A m = 0 . 08 ( Fi g. 6 d ) , t he ti me2a ve ra ge d dra g c oef f icie nt is p ositive . I n a li mit e d case , w he n A m →0 , t he f oil m oti o n is o nl y t he p itc hi n g oscilla ti o n , a n d a dra g f orce is alwa ys ge ne ra t e d as discusse d i n t he a bove . Thus , t o ge ne ra t e hi gh p r op ulsive ef f icie nc y , a n op ti mal p a ra met e r2c o m2 bi na ti o n is nee de d . 5 . 　CONCL UD ING REMARKS The p r op ulsive p e rf or ma nce a n d vort e x s he d 2 di ng of a f oil wit h t he m oti o ns of p itc hi n g a n d

Fig. 6 　Time2dependent drag coefficient at f = 1 , λ = 0 . 125

p lungi ng oscilla ti o n a re i nvesti ga t e d by s olvi n g t he t w o2di me nsi o nal i nc o mp ressi ble N a vie r2St o kes e 2 qua ti o ns i n t he vorticit y a n d st rea m2f uncti o n f or 2 mula ti o n . To accura t el y p re dict t he f orce a n d vor 2 t e x e voluti o n , t he f ourt h 2or de r esse ntiall y c o mp act f i nit e dif f e re nce sc he mes a re e m p l oye d t o dis 2 c retize t he sp ace de riva tives a n d t he f ourt h 2or de r

© 1995-2003 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

12 R unge2 Kut t a sc he me t o a p p r oxi ma t e t he ti me a d 2 va nce me nt . I n t his st ud y , a wi de ra n ge of p a ra me 2 t e r is c h ose n t o re veal t he f orce c ha ract e ristics a n d vort e x e voluti o n . The ti me 2dep e n de nt dra g a n d lif t f orces , i n p a rticula r t h rust f orce , a n d t he rela ti o ns of t he f orce be ha vi or wit h vort e x st ruct ure nea r t he f oil a n d vort e x s he ddi n g i n t he nea r wa ke a re dis 2 cusse d . B ase d o n our calcula t e d results , a n o p ti mal p a ra met e r2c o m bi na ti o n t o ge ne ra t e a hi gh p r op ul 2 sive ef f icie nc y ma y be i n t he f oll owi n g ra n ge . The t hic k ness ra ti o of t he f oil is λ = 0 . 062520 . 15 , t he p lungi ng a mp lit ude A m = 0 . 1520 . 3 , t he p itc hi n g a mp lit ude αm = 5° 215°, t he oscilla ti o n f re que nc y f = 122 , t he p hase dif f e re nce < = 90° 2180° . O n t he ot he r ha n d , a ni mal l oc o m oti o n is ce rt ai nl y f a r m ore c o mp le x a n d dive rse t ha n t he si m p le m odel c o nsi de re d he re . It is obvi ousl y t ha t our w or k is still helpf ul t o un de rst a n d t he unst ea d y ef f ect a n d f un da me nt al l oc o m oti o n mec ha nis m i n bi ol o gical f li ght a n d swi m mi n g. I deall y , t h ree2di me nsi o nal c o mp ut a ti o n a r oun d a n elasticall y f le xi ble wi n g is desi ra ble a n d is a t a r get i n our f urt he r w or k .

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