Fluid Mechanics of Fish Swimming 1

GENERAL  ARTICLE Fluid Mechanics of Fish Swimming 1. Lift-based Propulsion

Jaywant H Arakeri

F ish sw im b y c o o rd in a te d m o tio n o f th e ir b o d y a n d ¯ n s. T h is a rtic le d iscu sse s th e lift-b a se d p ro p u lsio n a d o p te d b y fa st sw im m e rs lik e d o l- p h in s. In tro d u c tio n Jaywant H Arakeri is a M an y of u s w ou ld h ave m arveled at th e seem in gly e® ort- professor at the Depart- ment of Mechanical less an d gracefu l w ay in w h ich ¯ sh sw im , tu rn , acceler- Engineering and the ate, an d b rake. F rom a m ech an ics p ersp ective, q u estion s Center for Product are often asked h ow go o d th ey are as sw im m ers.1 A re Development and th ey m ore e± cien t th an a p rop eller d riven u n d er-w ater Manufacture, Indian veh icle? W h ere d o es th e en ergy ex p en d ed b y th e m u s- Institute of Science, Bangalore. His research cles go? W h at is th e d rag force d u e to th e h igh ly u n - is mainly on instability stead y ° ow on th e u n d u latin g ¯ sh b od y ? Is th e w ake of and turbulence in fluid a w h ale qu ieter th an th at of a su b m arin e? A n sw ers to flows. m an y of th ese q u estion s are n ot kn ow n an d are top ics of cu rren t research . In th is article som e b asic issu es related to self p rop ellin g sy stem s are d iscu ssed , an d w e w ill lo ok at in som e d etail a ty p e of p rop u lsion ad op ted by m any fast sw im m in g ¯ sh ,w h ich is essen tially b ased on th e `lift' 1 Lighthill has made major con- forces gen erated by th eir ° ap p in g tails. A n gu illiform tributions to the understanding sw im m in g, seen in eels, w h ere th e w h ole b o d y seem s to of many aspects of fish locomotion using his unique ability of take p art in p rop u lsion w ill b e d iscu ssed in a later arti- combining physical insight into cle. F ish sw im m in g is gen erally at h igh R ey n old s n u m - mathematical models. b ers (> 1000), im p ly in g th at p ressu re an d in ertia forces d om in ate (see B ox 1). F igure 1 sh ow s th e m ain p arts of ¯ sh th at are relevant to sw im m in g. A ll of th ese featu res w ill n ot exist in all th e sp ecies. M otion is ach ieved b y coord in ated m ovem en t Keywords of th e b o d y an d som e or all of th e ¯ n s. Propulsion, lift, drag, self- propelling bodies. F ish th at gen erate th ru st p rin cip ally v ia b o d y an d /or

32 RESONANCE  January 2009 GENERAL  ARTICLE

Figure 1. Morphological features of a fish. Pectoral and pelvic fins are paired. Anal and dorsal fins are median fins.

cau d al tail ¯ n (B C F ) m otion s are k n ow n collectively as B C F sw im m ers. T h e sw im m in g m otion of B C F sw im - m ers m ay b e fu rth er classi¯ ed b ased on th e ty p e of m ovem en t of th e b o d y an d th e tail (F igu re 2). A n guilliform lo com otion in volves u n d u lation of th e en tire b o d y, w ith am p litu d e grow in g tow ard th e tail. A s on e go es from an gu illiform to thu n n iform m o d es th e b o d y u n d u - lation red u ces. T yp ical caran giform sw im m ers (jack s, m ackerel, an d sn ap p er) h ave a n arrow p ed u n cle an d a tall forked cau d al ¯ n . T h ese are am on g th e sw iftest of sw im m ers. T h e fastest are thun n iform sw im m ers. T h ese ¯ sh , in clu d in g tu n a an d som e sh ark s, h ave very low -d rag b o d y sh ap es, n arrow p ed u n cles, an d tall lu n ate (crescen t-sh ap ed ) cau d al ¯ n s. In an guilliform lo com o- tion , ad op ted b y eels for ex am p le, th e en tire b od y is th e p rop u lsor; th ru st is p ro d u ced b y u n d u lation , i.e., p ass- Figure 2. Different modes in g a tran sverse w ave from h ead to tail. In thu n n iform of BCF swimming. lo com otion , alm ost all of th e th ru st is b y th e cau d al ¯ n . A – Anguilliform. (e.g., eel) B – Subcarangiform. (e.g., trout) undulatory  oscillatory C – Carangiform. (e.g., Euthynnus affinis) D – Thunniform. (e.g., shark) Redrawn from Review of Fish Swimming Modes for Aquatic Locomotion, IEEE Journal of Oceanic Engineering, Vol.24, No.2, pp. 237–252, 1999, D M Lane, M Sfakiotakis and J B C Davies, Heriot-Watt University.

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B o x 1 . R e y n o ld s N u m b e r a n d S tro u h a l N u m b e r

Reynolds number and Strouhal number are two important non-dimensional numbers in relation to swimming of ¯sh. Reynolds number is de¯ned as U l U l Re = ½ = : ¹ º

½ is °uid density, ¹ is °uid dynamic viscosity, and º = ¹ =½ is kinematic viscosity. For 3 3 2 6 2 water, ½ = 1000 kg/m , ¹ = 1 10¡ N.s/m , and º = 1 10¡ m /s; for air, under £ 3 £ 5 2 standard atmospheric conditions, ½ = 1:2 kg/m , ¹ = 1:8 10¡ N.s/m and º = 1:5 5 2 £ £ 10¡ m /s. U is a reference velocity, e.g., the forward velocity of a ¯sh, and l is reference length, e.g., the length of the ¯sh body. Reynolds number is a relative measure of the importance of inertial forces to viscous forces. Thus in a high Reynolds number °ow, the inertial and pressure forces dominate and viscous forces may be expected to be negligible. However, as Prandtl showed, the e®ect of viscosity can be large even when its absolute value is very small. For low Re (< 1), viscous forces dominate. In swimming creatures, low Reynolds numbers are obtained for extremely small sizes.

For most °ows in water and air, the Reynolds numbers tend to be high. For a blue whale with a length of 30 m and swimming at 10 m/s the Reynolds number is 300 million; for a 30 cm long mackerel swimming at 3 m/s, the Re is about 1 million. At such high Reynolds numbers, at least part of the boundary layer on the body will be turbulent, and the wake will de¯nitely be turbulent. In contrast, for a sea urchin sperm with a length of 0.15 mm and speed = 0.2 mm/s, the Re 0.03. ' A commonly used non-dimensional measure of frequency for periodic °ows is Strouhal number, de¯ned as f H St = : U For a ¯sh, f is the frequency of °apping, of for example, the tail; H is a length scale, usually the amplitude of the tail motion; U is the forward velocity. Another commonly used non-dimensional frequency is the reduced frequency, 2¼ f l ¾ = : U

A lth ou gh th e cau d al ¯ n s are u sed in m ost of ¯ sh sp ecies, th ere are m an y ty p es of ¯ sh , k n ow n as M ed ian an d /or P aired F in (M P F ) sw im m ers, th at gen erate th ru st u sin g p rin cip ally m ed ian (e.g., d orsal an d an al) an d p aired (e.g., p ectoral) ¯ n s. B C F sw im m ers also m ay u se M P F m od es for m an o evrin g an d stab ilization .

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F o rc e s o n B o d ie s M o v in g w ith C o n sta n t V e lo c ity It is u sefu l to lo ok at som e gen eral p rin cip les w ith regard to `self-p rop ellin g' b o d ies (see B ox 2). A p erson w alkin g, rid in g a b icy cle or m on o cycle (F igu re 3a), a m otor ve- h icle, an airp lan e, a ¯ sh are ex am p les of self-p rop ellin g b o d ies. In th is article w e w ill con ¯ n e ou rselves to ¯ sh m ov in g w ith con stan t sp eed in a straigh t lin e. It m u st b e m en - tion ed th at even in th is case th e oscillatory m otion s of th e b o d y an d ¯ n s resu lt in sm all p erio d ic variation s of sp eed an d lateralp osition , an d th e m otion is stead y on ly in an average sen se. A p p lication of N ew ton 's 2n d law in th e vertical an d h or- izontal d irection s to a b od y m ovin g w ith con stan t sp eed in a straigh t lin e, i.e., a n on -acceleratin g b o d y gives

L W = 0; ¡ T D = 0: ¡ T h e w eigh t (W ) h as to b e su p p orted b y an u p w ard force (L ), an d an y friction al force (D ) h as to b e overcom e by a forw ard force (T ). In ad d ition , th e n et m om en t in each d irection h as to b e zero. F or an aircraft ° y in g level w ith con stan t sp eed , L is th e aero d y n am ic lift force w h ich su p p orts th e w eigh t, T is th e en gin e th ru st w h ich overcom es th e d rag or friction al

w (a) w (b) U Figure 3. Examples of bodies moving with constant U speed on level ground. a) Monocycle; b) cart being pulled. For the T L monocycle (a), there is no D force in the horizontal di- L rection.

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B o x 2 . S e lf-P ro p e llin g B o d ie s

For a self-propelling body moving with constant velocity, the net force on the body in the vertical and horizontal directions is zero.

In natural systems, like a human walking or a ¯sh swimming, there is always unsteadiness. Even during steady walking, the body is not really moving with a constant velocity; there is acceleration and deceleration during each cycle. When we put a foot down on the ground, friction acts in a direction opposite to the direction of motion; when we raise a foot, friction on the rear foot is in the direction of motion. These two opposing frictional forces cancel over one cycle.

Analysis of walking, though extremely interesting, is complicated. A cart being pulled with constant velocity (F igu re 3b) is relatively easier to understand. A horizontal force (T ) has to be applied to overcome the wheel (rolling) frictional force (D ). Similiarly in a bicycle, the ground at the rear wheel applies a forward force to overcome the rearward frictional force being applied by the ground on the front wheel. But a monocycle (F igu re 3a), which would seem to be conceptually simple, is more complicated. The total weight is supported by a normal component of the reaction force from the ground. Is there a frictional force applied by the ground in the horizontal direction? What is its direction? The person is certainly doing work; where is the energy going? The reader can think about answers to these questions.

Aircraft and ¯sh propelled primarily by their caudal ¯ns are somewhat like the bicycle. The thrust, which is from the engine or the ¯sh tail, and the frictional force (the drag), which is on the main body, are acting at distinct points. However, in the case of certain other ¯sh, like eels, and birds the situation is more like the monocycle: it is di±cult or impossible to distinguish between forward and rearward forces.

In self-propelling bodies moving with constant velocity, there is no gain in mechanical energy, potential or kinetic, as would happen when climbing a mountain or when accel- erating. All the work goes into heat energy through non-elastic deformation of surfaces and viscous dissipation in the surrounding °uid.

force (D ) of th e air (F igure 4a). O n a sw im m in g ¯ sh , b u oyan cy force (F B ) an d an y lift force (L ) gen erated by th e ¯ n s cou nteract th e w eigh t (F igure 4b ). A s m en tion ed ab ove, in m ost of th e sy stem s fou n d in N atu re, th e velo city is n ot really con stan t an d T = D on ly w h en averaged over on e cy cle of th e gen erally p eri- o d ic m otion . A lso, th ere is a su b tle d i® eren ce b etw een n atu ral sw im m ers an d ° yers an d m an -m ad e on es like

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Figure 4. a) An aircraft fly- (a) U ing with constant velocity. L Thrust (T) from the engine D balances the drag (D); lift (L) mainly from the wings balances the weight (W). T b) For the fish shown here W the thrust is mainly from the flapping tail. Lift from (b) fins and buoyancy (F ) bal- U + L B ance the weight. T D

aircraft. In th e case of an aircraft th e th ru st p ro d u cin g p ortion (p rop eller or jet en gin e) an d th e d rag p ro d u cin g p ortion (p rim arily th e airp lan e b o d y an d th e w in gs) are sep arate en tities. In th e case of ¯ sh (esp ecially an gu illiform sw im m ers w h ere th e w h ole b od y u n d u lates an d p ro d u ces `th ru st') b oth th ru st an d d rag are p rod u ced by th e sam e m ov in g su rface. B ird an d in sect ° igh t is even m ore com p licated ; lift, th ru st an d som e of th e `d rag' are gen erated from th e ° ap p in g w in gs. P ro p e lle r-ty p e o r L ift-b a se d P ro p u lsio n T h e airfoil sh ap e (F igure 5a) { lon g an d th in w ith a rou n d ed n ose an d sh arp trailin g { is a rem arkab le geom - etry th at p ro d u ces a large lift force (L ) an d ex p erien ces very low d rag force (D ). (T h is sh ap e w ork s for su b - son ic, h igh R ey n old s n u m b er ° ow s. S u p erson ic airfoils h ave p oin ted lead in g ed ges to m in im ize sh o ck p rod u ced In the case of fish d rag. A t low R ey n old s n u m b ers th e force gen eration both thrust and m ech an ism is v iscou s an d sh ap e is n ot so im p ortan t.) drag are produced L ift is perpen dicu lar to th e relative velocity b etw een th e by the same ° u id an d th e airfoil. D rag, like friction , is op p osite in moving surface. d irection to th e relative m otion .

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(a) (b)

Figure 5. a) The airfoil T h e airfoil form s th e b asis for a large nu m b er of su r- shape with rounded lead- faces w h ere lift or th ru st n eed s to b e p rod u ced e± ciently. ingedge and sharp trailing W in g of an aircraft is th e m ost v isib le ap p lication of air- edge gives a large lift per- pendicular to the direction foil. B u t w in gs of b ird s, m arin e p rop eller, steam tu rb in e ofmotion,andexperiences an d h elicop ter rotor b lad es, tails of ¯ sh (F igu re 5b ) all a relatively small drag op- h ave cross section s w ith an airfoil sh ap e. posingthemotion. Uisthe relative velocity between T h e lift force on an airfoil of p lan form area A m ov in g the fluid and the airfoil. w ith velocity U in station ary ° u id is u su ally w ritten as b)Fishtailshavesymmetri- 1 2 cal airfoil cross-sections. L = C µ ¶½ U A : (1) L 2 F or sy m m etrical airfoils of th e typ e fou n d in ¯ sh tails, th e lift co e± cien t C 2¼ ® : (2) L ¼ T h e an gle of attack, ® , b etw een th e airfoil an d th e rela- 2 For generation of fluid me- tive velo city 2 d irection u su ally is less th an ab ou t 15 o to chanical forces what matters is the relative velocity between the p revent stall (B ox 3). T h e m ax im u m lift co e± cien t for fluid and the body. Velocity of stead y ° ow is ty p ically arou n d 1.5. N ote th at lift varies the fluid U past a stationary air- lin early w ith d en sity an d area, an d as th e sq u are of ve- foil or theairfoil withvelocity U in lo city. T h e ab ove relation s are u sefu l to estim ate lift a stationary fluid will produce forces. A 1 m 2 airfoil w ith a lift co e± cien t of 1, m ov in g identical lift forces. at 1 m /s in w ater gen erates a lift of 500 N or ab ou t 50 k g. In air w ith 1000 tim es low er d en sity, to gen erate th e sam e force w ou ld req u ire a velo city of ab ou t 30 m /s.

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(a) (b) Figure 6. a) Propeller blades have curved airfoil cross-sections. b) The relative motion of any section of the blade Ωr consists of forward velocity U and velocity ΩΩΩr due to the blade rotation.

T h e lift can also b e w ritten in term s of th e circu lation a ro u n d th e a irfo il, L = ½U ¡. Circulation (¡)isde- ¯ned as a lineintegralof°uid velocity around a closed lo o p , ¡ = ¡!U:d ¡!S ,where S isdistance m easured along H th e loop . ¡ is on e m easu re of th e stren gth of a vortex. M an y of th e vortical stru ctu res (trailing vortices an d starting vortices) h ave rou gh ly sam e values of circulation as around the w ing. T he thrust from a propeller (Figure 6 a ) is e sse n tia lly from the liftgenerated by the rotating blades. C onsider the case of a propeller m oving w ith constant velocity U and producing thrust,T . A t any rad ial cross-section th e airfoil w ill see a relative velocity d u e to th e forw ard velocity U an d th e rotation al velocity − r (Figure 6b). − is th e ro ta tio n a l v e lo c ity in ra d ia n s/ s a n d r is th e rad ial d istan ce of th e airfoil cross-section from th e ax is ofrotation. T he liftand drag forces can be decom posed into a com p on en t in th e forw ard d irection contribu ting to th e th ru st an d tan gential com p on en t con tribu ting to thetorque required to turn thepropeller(Figure 7). T hethrustfrom the°appingtail(thecaudal¯n)ofa ¯sh The thrust from is very sim ilar to th at from a rotating p rop eller b lad e. the flapping tail B oth are based on liftforce generated from airfoiltype (the caudal fin) of cross-section s. T h e cau d al ¯ n oscillates laterally as th e a fish is very ¯sh m oves forward. It has two oscillatory m otions { heave or lateralm otion w ith am plitude H ,and a rotary similar to that from or pitching m otion around an axisw ithin the airfoil. a rotating propeller T he heavingm otion and the constant forward velocity blade.

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Figure 7. For a propeller blade or a fish tail, the forward velocity U and the airfoil velocity Uf due to either blade rotation or tail flapping gives a resultant velocity UR. Lift La is per- pendicular to UR and drag

Da is in the direction of UR.

The resolution of La and Da in the forward direction contributes to thrust. resu lts in w ave-like p ath of th e ¯ n (F igure 8a); th e p itch - Figure 8. a) Motion of a tail in g m otion en su res a p rop er an gle of attack of th e airfoil airfoil section consists of at each location . F igu re 8b sh ow s th e h eave an d p itch forward motion and a up m otion s of th e tail if on e rid es w ith th e ¯ sh . A gain and down heaving motion F igure 7 sh ow s h ow th ru st is gen erated . U n like in th e with amplitude H and fre- p rop eller case, U f varies p erio d ically w ith tim e. quency f. The tail continu- ously re-orients or pitches A n im p ortan t p aram eter is th e freq u en cy of oscillation to maintain a proper angle of th e cau d al ¯ n , f . O p tim al th ru st gen eration seem s of attack. The wavelength to o ccu r w h en th e n on -d im en sion al freq u en cy, S trou h al of motion in the direction n u m b er, S t = f H = U is in th e ran ge 0.25{0.4 (see also of motion is U/f. b) The heaving and pitching mo- B ox 1). H igh er forw ard sp eed is ach ieved b y a h igh er tions ofthe airfoilduring½ freq u en cy of ° ap p in g, q u icker b eatin g of th e tail. a cycle. The airfoil quickly W a k e S tru ctu re re-orients at the two ex- treme points.(Adapted T h e d istu rb an ce created b eh in d a m ov in g b o d y in a sta- from Lighthill [1].) tion ary ° u id is k n ow n as th e w ake. F igure 9 sh ow s th e

(a) (b) U

Uf Da La H

U/f

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Figure 9. a)Adragproduc- (a) ing body drags the fluid in the forward directionin the wake. b)A thrust producing body pushes the fluid behind. (b) In both cases, the bodies areshownmovinginasta- tionary fluid. tion ary ° u id is k n ow n as th e w ake. F igure 9 sh ow s th e ° u id b ein g d ragged alon g b eh in d a (d rag p ro d u cin g) b o d y ; th e ° u id in fron t an d sid es is p u sh ed aw ay by essentially a n on -v iscou s m ech an ism . T h is p h en om en on is read ily exp erien ced as a gu st of air on th e sid e of a road as a fast m ov in g b u s p asses u s. A th ru st p ro d u cin g b o d y (F igure 9b ) p u sh es th e ° u id in a d irection op p osite to d irection of m otion .

D ep en d in g on th e ty p e of b o d y, a w ake can h ave in ter- Figure 10. The two oppo- estin g stru ctu res. T w o cases are relevan t to ¯ sh p rop u l- site signed trailing vortices sion . A liftin g b od y, as a w in g, m ov in g w ith con stan t from the tips of a lift gener- velocity p ro d u ces a p air of trailin g vortices (F igure 10). ating wing. The circulation T h e in d u ced d rag (see B ox 3) is related to th e en ergy around the vortices is pro- carried b y th ese vortices an d is a p en alty for gen erat- portional to that around the in g lift. T h e p rop eller b lad es (sin ce th ey in d iv id u ally wing. The flow behind the gen erate `lift') also leave b eh in d trailin g vortices. H ow - wing is downward. The kinetic energy in the vortices ever, sin ce th e tip is rotatin g, th e trailin g vortex from is related to the induced each rotatin g b lad e of th e p rop eller traces a h elical p ath drag, which goes like the (F igure 11). square of the lift.

Figure 11. Trailing vortices of the type seen in Figure 10 trace helical paths from each rotating propeller blade.

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B o x 3 . L ift a n d D ra g 

Any body moving steadily forward with speed U experiences a force (which in general can be time varying) that is usually decomposed into drag force (D ) opposite to the direction of motion, and lift force (L ) normal to the direction of motion. (We are not concerned with the lateral component of the force.) These forces are due to viscous stresses, which are generally tangential to the body surface, and °uid pressure, which is normal to the surface. Integral of the stresses and pressure over the body surface area gives the force acting on the body. At high Reynolds numbers, lift is mostly due to pressure and drag is due to both viscous stresses and pressure.

The lift force is usually written in terms of the lift coe±cient, C L ,

1 2 L = C L µ ½ U ¶A : 2

For symmetrical airfoils of the type found in ¯sh tails, the lift coe±cient C 2¼ ® . L ¼ The angle of attack, ® , between the airfoil and the relative velocity direction usually is less than the stall angle, which typically lies between 10 and 15 degrees. Stall happens due to °ow separation and results in sudden loss of lift and increase in drag; stall is to be generally avoided. The lift coe±cient is generally less than about 1.5 under steady conditions, though much higher lift coe±cients (and higher stall angles) are obtained under unsteady conditions, as on a °apping ¯sh tail.

Lift generation can be explained by the Bernoulli principle: relatively higher °uid speeds and lower pressure on the upper surface of an airfoil compared to those on the lower surface. In terms of circulation around the airfoil, lift can be written as L = ½ U ¡.

An important feature of a wing that generates lift are two trailing vortices (F igu re 10) roughly in line with the tips of the wing; these are sometimes visible as white trails (condensed water vapour in the low temperature cores of the vortices) in the sky behind an aircraft. The strength of these vortices is proportional to the lift. These vortices represent lost energy left behind and cause an additional drag called induced drag.

Drag force is generally due to contributions from °uid pressure and tangential (viscous) stress acting on the surface of the body. For streamlined bodies, pressure drag is negligible compared to the skin friction drag; for blu® bodies skin friction drag is negligible compared to the pressure drag.

For lifting bodies, like a wing, it is common to write another component of drag called the induced drag, D i, which may be considered to be a penalty for producing lift.

 See article by S P Govinda Raju, Resonance, Vol.14, No.1, pp.19–31, 2009. continued...

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Box 3. continued...

The presence of the trailing vortices can be explained in two di®erent ways. It is due to a relatively higher pressure existing on the lower surface of the wing compared to that on the upper surface. The °uid from the lower surface thus tries to move towards the upper surface around the two corners of the wing creating the wing tip vortices. The strength of these vortices is directly related to the pressure di®erence and thus to the lift force,

L 2 D = ; i ½ U 2 b2 where b is the span of the wing. Thus wide wings, as found in gliders, have low induced drag.

Like lift, drag is usually written in terms of a drag coe±cient C D , 1 D = C ½ U 2 A : D 2

The reference area is proportional to wetted area for streamlined bodies, and the frontal area for blu® bodies. C D is typically 0.01 for streamlined bodies and around 1 for blu® bodies.

T h e secon d case is related to th e fact th at a vortex is also p ro d u ced w h en ever th e lift arou n d a b o d y is ch an ged . L ift arou n d an airfoil can ch an ge d u e to a ch an ge in th e relative velo city or a ch an ge in an gle of attack (see equ ation s (1) an d (2)). A n airfoil su d d en ly started from rest p ro d u ces a d istin ct vortex term ed as th e startin g vortex (F igu re 12). T h e circu lation arou n d th e startin g vortex is th e sam e as th at arou n d th e airfoil.

Figure 12. An impulsively started airfoil creates a starting vortex behind it. Any change in lift either due to change in angle of attack or change in speed creates such vortices. The flow visualization picture isfrom Prandtl and Tietjens [2].

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Figure 13. The wake of a flapping fish tail consists ofinterconnected, alternat- ing signed vortex rings. The flow created by these ringsis anundulating back- ward jet.

B oth th e an gle of attack an d th e m agn itu d e of th e relative velo city b etw een th e tail an d ° u id ch an ge cyclically in tim e. T h u s th e lift on th e tail (w h ich contrib u tes to th e th ru st) ch an ges cy clically ; in fact, th e d irection of lift w ith resp ect to th e airfoil ch an ges sign d u rin g on e cy cle. T h e cyclical ch an ge in lift u su ally resu lts in tw o op p osite sign ed vortices b ein g sh ed from th e trailin g ed ge of th e tail ¯ n in each cycle. T h ese togeth er w ith th e tip vortices (th e eq u ivalen t of th e h elical vortices from a p rop eller) form con tin u ou s vortex lo op s (F igu re 13). T h e vortex lo op s are essen tially a series of in tercon - n ected in clin ed vortex rin gs. (V ortex rin gs in com m on p arlan ce are k n ow n as `sm oke' rin gs.) T h e axial com p o- n ent of th e im p u lse in each rin g is con n ected w ith th e th ru st gen erated by th e tail. M o m e n tu m a n d E n e rg y in th e W a k e E arlier, w e h ad com m en ted th at a d rag p rod u cin g b o d y p u lls th e ° u id alon g w ith it in th e w ake an d a th ru st p ro d u cin g on e p u sh es th e ° u id in th e b ack w ard d irection (F igures 9, 11). F rom m om entu m p rin cip le th e velocity The starting in th e w ake can b e related to th e force on th e b o d y, vortices and the tip F N ½ U Z w n d A : (3) vortices together ¼ form a series of interconnected w n is th e velocity com p on en t in th e forw ard d irection in vortex rings. th e w ake an d th e in tegral is over th e cross-section al area of th e w ake. F or d rag p rod u cin g b o d y F N = D ;w n > 0;

44 RESONANCE  January 2009 GENERAL  ARTICLE

Figure 14. The three types Wake structures behind moving bodies of wakes behind an airfoil: a) drag producing with a (a) Karman vortex street; b) thrust producing obtained from appropriate oscillatory motion of the airfoil with a reverse Karman vortex street; (b) c) oscillation with a Strouhal number ~ 0.3 pro- duces zero force on the airfoil,a momentum less wake andvorticesalignedonthe line of motion. (c) Figure 3