confomral mapping - wake

8/2/2019 Confomral Mapping - Wake

1/9

Physica D 240 (2011) 15741582

Contents lists available at SciVerse ScienceDirect

Physica D

journal homepage: www.elsevier.com/locate/physd

Wake structure of a deformable Joukowski airfoil

Adam Ysasi a, Eva Kanso a,, Paul K. Newton a,ba Department of Aerospace & Mechanical Engineering, University of Southern California, Los Angeles, CA 90089-1191, United Statesb Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1191, United States

a r t i c l e i n f o

Article history:

Available online 12 October 2011

Keywords:

Swimming

Vortex shedding

Unsteady point vortex

a b s t r a c t

We examine the vortical wake structure shed from a deformable Joukowski airfoil in an unbounded

volumeof inviscid and incompressible fluid. The deformable airfoil is considered to model a flapping fish.The vortex shedding is accounted for using an unsteady point vortex model commonly referred to as theBrownMichael model. The airfoilsdeformationsand rotationsare prescribed in termsof a Jacobi ellipticfunction which exhibits, depending on a dimensionless parameter m, a range of periodic behaviors fromsinusoidal to a more impulsive type flapping. Depending on the parameter m and the Strouhal number,one can identify five distinct wake structures, ranging from arrays of isolated point vortices to vortexdipoles and tripoles shed into the wake with every half-cycle of the airfoil flapping motion. We describethese regimes in the context of other published works which categorize wake topologies, and speculateon the importance of these wake structures in terms of periodic swimming and transient maneuvers offish.

2011 Elsevier B.V. All rights reserved.

1. Introduction

The vortical wake shed from a deformable Joukowski airfoil isconsidered as a simple model for the wake generated by a peri-odically swimming fish. Our main objective is to understand thedependence of the wake structure on the airfoil motion. A char-acterization of the wake structure as a function of shape defor-mations is relevant in many applications, such as understandingthe physics of fish swimming and bird flying and translating thisknowledge into engineering solutions in the form of biologically-inspired underwater and airborne vehicles. For experimental stud-ies of the wake structure see, for example, [13] for swimmingfish, [4] for flying birds, and [510] for oscillating rigid foils andcylinders.

In the mathematical modeling of swimming at large Reynoldsnumbers due to transverse flapping such as the swimming of

carangiform and anguilliform fish, different approaches have beenproposed to account for the wake dynamics. In the classical workof Wu on the swimming of a deformable plate [11], the authorused the assumption of small shape amplitudes which enablesone to solve analytically for the trailing vortex sheet. The vortexsheet is a surface across which the tangential component ofthe fluid velocity is discontinuous but the normal component iscontinuous. Lighthill, in his slender body theory, avoided solvingfor the complex wake dynamics altogether by considering the

Corresponding author. Tel.: +1 213 821 5788.E-mail address: [email protected] (E. Kanso).

momentum balance in a control volume containing the deformablebody and bounded by a plane attached at its trailing edge; seefor example [12]. Childress later improved on Lighthills theoryby providing an analytic approach to estimate the vorticity shedat the trailing edge [13]. Weihs modified Lighthills theory which focused on steady swimming for applications to transientmaneuvers and approximated the wake using classical methodsof unsteady aerodynamics in terms of pre-tabulated lift and dragcoefficients; see [14,15]. Lighthills reactive forcetheory is revisitedin [16] and the wake of a deformable body is modeled usingdiscrete point vortices introduced manually, so-to-speak, everyhalf cycle of the body flapping motion. In [17,18], the vortex sheetshed from the trailing edge of a deformable Joukowski airfoil ismodeled using discrete point vortices with a new point vortexintroduced at every time step to satisfy the KuttaJoukowskiregularity condition at the trailing edge. Resolving the dynamicsof a vortex sheet shed from a deformable body is quite challengingeven for prescribed or zero motion of the body; see [1921]. Directnumerical simulations of the NavierStokes equations have alsobeen used to resolve the wake dynamics; see for example [2224].

In this work, we consider a Joukowski airfoil with prescribedshape deformations and prescribed locomotion in the form of auniform translational velocity and periodically-varying orienta-tion. We follow the work of [25,26] in that we model the wakeusing an unsteady point vortex model, also referred to as theBrownMichael model. This model consists of introducing vortic-ityin the flow as point vortices of time-varying strength so that theKuttaJoukowski regularity condition is satisfied at the sharp edgeof thebody; formoredetails,see [25,26] and the references therein.

0167-2789/$ see front matter 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2011.06.021
http://dx.doi.org/10.1016/j.physd.2011.06.021http://www.elsevier.com/locate/physdhttp://www.elsevier.com/locate/physdmailto:[email protected]://dx.doi.org/10.1016/j.physd.2011.06.021http://dx.doi.org/10.1016/j.physd.2011.06.021mailto:[email protected]://www.elsevier.com/locate/physdhttp://www.elsevier.com/locate/physdhttp://dx.doi.org/10.1016/j.physd.2011.06.021


2/9

A. Ysasi et al. / Physica D 240 (2011) 15741582 1575

Fig. 1. KuttaJoukowski transformation: (left) physical plane and (right) circle plane.

Roughly speaking, the unsteady point vortex model is a low-orderrepresentation of the roll-up of the vortical wake shed from thesharp edge due to boundary layer separation in a real fluid envi-ronment. Since it is physically impossible to unroll the shed wakeand therefore, in the unsteady vortex model, the magnitude of thestrength of the unsteady vortex is required to be monotonically in-creasing. That is to say, once the strength of the unsteady vortexreaches an extremum, it is kept at that value and a new unsteadyvortex is started from the sharp edge. A comparison of the vor-tex wakes behind oscillating plates based on the BrownMichael

model, a vortex sheet model and direct numerical simulations ofthe full NavierStokes equations is reported in [27]. Interestingly,the simpler BrownMichael model provides good approximationsof the wake structure and the strength of the shed vorticity.

A review of the dynamics ofN (finite and real) point vortices ofconstant strengths can be found in [28]. For the dynamic interac-tion of a rigid body with N point vortices of constant strengths, thereader is referred to [2932] and the references therein. The cou-pling of the latter body-vortex system to the unsteady point vortexwas first proposed in [26].

The outline of this paper is as follows. First, we prescribeperiodic shape deformations of a Joukowskiairfoil andassume thatthese prescribed deformations result in a uniform translationalvelocity and periodically-varying orientation of the airfoil. Wethen focus attention on the wake structure and its evolution intime. In particular, we examine the wake structure for a class ofprescribed motions in terms of a Jacobi elliptic function whichexhibits, depending on a dimensionless parameter m, a rangeof periodic behaviors from sinusoidal to impulsive flapping. Thisperiodic flapping motion can also be characterized by the Strouhalnumber St defined as the ratio of the distance between the tip-to-tip transverse motion of the trailing edge times the frequency offlapping to the translational velocity of the airfoil. Depending on mand St, one can identify five distinct wake structures, ranging fromarrays of isolated point vortices to vortex dipoles and tripoles shedinto the wake with every half-cycle of the airfoil flapping motion.This categorization is compared with known published work inother settings.

2. Problem setting

Consider a uniform and neutrally-buoyant planar body B, as-sumed to represent a cross-section of a fish body or fin, moving inan unbounded domain of incompressible, inviscid fluidFatrestatinfinity. For concreteness, letB take the form of a Joukowski airfoilwhose boundary can be conformally mapped to a circle of radiusrc . We introduce an orthonormal inertial frame {e1,2,3} where thedirections {e1, e2} span the physical plane of motion and e3 is theunit normal to this plane. The orientation ofB can be described bythe rotation angle about e3 and its location can be specified bythe coordinates of its centroid c (c1, c2) in the (e1, e2) plane;see Fig. 1. It is more convenient for what follows to introduce thecomplex variablez = x+iy (where i = 1) andits complex con-

jugate z = x iy. It is also convenient to parameterize the circleplane by the complex variable = + i, measured from a fixed

point O located at the center of the circle. The conformal transfor-mation from the physical plane to the circle plane can be writtenin the general form (see, e.g., [26])

z = c + g( )ei , (1)where the mapping g( ) is given by

g( ) = + c +a2

+ c. (2)

The parameters a and c=

c+

ic determine the shape of theairfoil (see, e.g., [18]), and the radius of the circle rc depends on aand c as follows

r2c = (a c )2 + 2c . (3)We prescribe shape deformations of the airfoil by varying theposition ofc (t) as a function of time twhilemaintaining the radiusof the circle rc constant which means that a also varies in order tosatisfy (3). In particular, we consider deformations of the form

c = o + ic (t), (4)where o is non-zero constant and c (t) varies periodically asdepicted in Fig. 1. These prescribed deformations are not areapreserving in general. That is to say, the area of the airfoil A =12i zdz variesin time. However, forall examplesconsidered in thispaper, the deviations from constant are quite small, A/Ao 0.01where Ao is the airfoil area at time t = 0.

As the airfoil deforms, the location of its centroid in the physicalplane changes. The centroid position c is dictated by both theprescribed deformations and the orientation of the airfoil suchthat

c = zc ei , (5)where zc is defined by Azc = 14i

z2dz. When expressed as

function of the prescribed deformations c (t) (and in turn a(t)) inthe circle plane, one has

zc =a6c + c (r2c c c )3

(r2c

c c )

[(r2c

c c )2

a4

]

. (6)

We assume that the prescribed cyclic deformations induce auniform translational velocity (Vx, Vy) and a periodic rotational

velocity (t) = (t) of the airfoil in the physical plane. In otherwords, we do not solve the coupled body-fluid system to obtainthe locomotion of the airfoil due to the prescribed deformations asdone in [33,16]. Rather, we impose on the airfoil a net locomotionconsisting of a steady translation and periodic rotations. To thisend, the velocity of the centroid is given by

vc = (Vx + iVy) (zc izc ) ei . (7)The resulting free stream velocity in the circle plane is

U = vc ei = zc izc Vx iVy

ei . (8)

The Joukowski airfoil sheds a vortical wake from its sharpedge as it deforms in the fluid domain. We account for the shed
http://dx.doi.org/10.1016/j.physd.2011.06.021http://dx.doi.org/10.1016/j.physd.2011.06.021http://dx.doi.org/10.1016/j.physd.2011.06.021http://dx.doi.org/10.1016/j.physd.2011.06.021http://dx.doi.org/10.1016/j.physd.2011.06.021http://dx.doi.org/10.1016/j.physd.2011.06.021http://dx.doi.org/10.1016/j.physd.2011.06.021http://dx.doi.org/10.1016/j.physd.2011.06.021http://dx.doi.org/10.1016/j.physd.2011.06.021http://dx.doi.org/10.1016/j.physd.2011.06.021http://dx.doi.org/10.1016/j.physd.2011.06.021http://dx.doi.org/10.1016/j.physd.2011.06.021http://dx.doi.org/10.1016/j.physd.2011.06.021http://dx.doi.org/10.1016/j.physd.2011.06.021http://dx.doi.org/10.1016/j.physd.2011.06.021http://dx.doi.org/10.1016/j.physd.2011.06.021


3/9

1576 A. Ysasi et al. / Physica D 240 (2011) 15741582

vorticity using the unsteady point vortex model which consists,as explained in Section 1, of introducing vorticity in the flowas point vortices of monotonically-varying strength so that theKutta condition is satisfied at the sharp edge; see [25,26]. Forconcreteness, the unsteady vortex is identified by its variablestrength 1(t) and location z1(t) while the previously shed pointvortices have frozen strengths 2, . . . , N and time-dependentpositions z

2(t

) , . . . ,z

N(t

).

3. Fluid velocity and complex potential

The complex potential function is defined as F(z) = (x,y) +i(x,y), with being the real potential function and thestream function. The existence of and is guaranteed byirrotationality (except at the location of the point vortices) andby the continuity equation div(u) = 0, respectively; see forexample [34]. By linearity of the problem, the complex potentialF(z) can be written as a superposition of the contributions Fb(z)due to translational and rotational velocities of the airfoil, Fs(z)due to shape deformations and Fn(z) due to the presence of pointvortices in the wake,

F(z) = Fb(z) + Fs(z) +N

n=1nFn(z). (9)

The potentialFb dueto therigidmotion of theairfoil is harmonicin the fluid domain with proper decay at infinity while satisfyingthe boundary conditions on the airfoil surface C

Re [iFb]C = Re[ivc (z c)

1

2|z c|2

]. (10)

That is, Fb(z) has the form of a RiemannHilbert problem (see,e.g., [35,36]) and it can be transformed, using the conformalmapping (1) and its inverse, into a Dirichlet problem in the circleplane whose solution is given in terms of the Schwarz formula. To

this end, one gets

Fb( ) = U( g()) +r2c

U ir, (11)

where

r( ) = 12

r2c + 2c

r2c

+ c c

+ 2a2

r2c + c

( + c )+ a

4( c )( + c )(r2c c c )

. (12)

The complex potential due to body deformations is constructed

as a linear superposition of the two components,Fs( ) = F ( )c + Fa( )a. (13)F and Fa are also harmonic in the fluid domain with proper decayat infinity while satisfying the following boundary conditions atthe airfoil surface C and the circle surface

Re

[i dF

dd

]

= Rei g( )

cdz

C

,

Re

[i dFa

dd

]

= Rei g( )

adz

C

. (14)

Upon integrating both sides of(14) for F and Fa and ensuring that

Kelvins conservation of circulation around the airfoil is satisfied(see [18] for more details), one gets that

F( ) = i[ r

2c

+ a

2r2c

2c+ a

2

a

4 2c

4

2a4r2c c

6log

rc

a2r2c

3c a

4r2c

c 4

+ 2a2r2c 1

3c+ c a

2

6 log

+ 2a

4r2c x

6log

rc

],

Fa( ) = 2a

r2c

c a

2c

2 a

2r2c

4log

rc

+ r2c (

4 a2 2c ) 2c

4log

,

(15)

where = + c and 2 = r2c c c .The complex potential due to the presence of N point vortices

is given by the well-known MilneThomson theorem (also knownas the circle theorem; see [37])

Fn( ) =1

2 i[

log ( n) log r2c

n]

, (16)

where r2c /n is the position of the image vorticity.The complex fluid velocity u = ux iuy at a point z = x + iy

in the physical plane that does not coincide with a point vortex isobtained from the relation

u = dF(z)dz

= dF(z())d

d

dz, (17)

whereas, due to a theorem by Lin [38,39], the regularized velocityinduced at a point vortex zn is given by

un =d

dz [F(z) n

2 ilog (z zn)]zn

. (18)

In particular, Eq. (17) takes the form

u = vc +ei

g( )

U r

2c

2U i r

+ F

c +

Fa

a

+N

n=1

n

2 i

1

n n

n r2c

. (19)

Eq. (19) is used to compute the velocity at the trailing edge ofthe airfoil denoted by 1,0 = a c , as shown in Fig. 1. It isimportant to observe that the velocity in (19) admits a singularityat the trailing edge 1,0. This is apparent by noting that g

( ) =g/ = 1 a2/( + c )2 which at 1,0 = a c is equal tozero. To eliminate this singularity, we impose the KuttaJoukowskiregularity condition by introducing an unsteady point vortex inthe flow as mentioned in Section 2 that makes the quantity in thebracket of Eq. (19) vanish as discussed below.

4. Equations of motion

The equations of motion for the system of vortices is expressedmore compactly in the physical plane and consists of the followingset of 2N ordinary differential equations

zn + (zn zn,0)n

n= un, (20)

where the complex conjugate of un is given by (18) and the term

containing n is known as the BrownMichael correction term.Here, it is non-zero only for the unsteady vortex 1 and zero for


4/9


t

m=0.995

m=0

cn(4Kt/T) 0.2

1

0

1

0.2

0.4

0.6

0.8

0.8

0.6

0.4

0 10.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

t

m=0

m=0.995

d[cn(4Kt/T)]/dt

10

8

6

4

2

0

2

4

6

8

10

0 10.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(a) cn(4Kt/T). (b) d (cn(4Kt/T)) /dt.

Fig. 2. Plots of (a)the cnoidal function cn(4Kt/T) and(b) itstime derivativeversus time tfor m = 0, 0.8, 0.95, 0.995. Note that for m = 0, onerecovers thecosine function.The period is set to T = 1.

Fig. 3. Geometry of Joukowski airfoil over a half cycle 2 K/T for prescribed flapping c = o + io cn(4Kt/T) and orientation (t) = o cn(4Kt/T). The case ofm = 0 isshown in dashed line and m = 0.995 in black line. The parameter values are set to o = 1/8 o = 1/2, o = 1/2, rc = 1 and T = 1.

(a) streamlines. (b) n versus t.

(c)

n versus t.

Fig. 4. (a) Snapshot of the streamlines at t

=5.002, (b) strength of n versus time and (c) n versus time for prescribed motions c = 0.125 i0.2 cn(4Kt/T), = 0.2 cn(4Kt/T) and parameters m = 0.8 and St = 0.1 which correspond to Region I in Fig. 9.


5/9


(a) streamlines. (b) n versus t.

(c)

n versus t.

Fig. 5. (a) Snapshot of the streamlines at t = 5.002, (b) strength of n versus time and (c)

n versus time for prescribed motions c = 0.125 i0.2 cn(4Kt/T), = 0.2 cn(4Kt/T) and parameters m = 0.8 and St = 0.3 which correspond to Region II in Fig. 9.

all n 2. These equations can be written in the circle plane bysubstituting (1) and (18) into (20). The resulting expression takesthe form

g(n)n + (g(n) g(n,0))n

n

= ig(n) g(n)c g(n)

aa + 1

g(n)

U r2c

2nU + i r(n)

n+ F(n)

nc +

Fa(n)

na

j =n

j

2 i

1

n j j

jn r2c

+n

2 i n

nn r2c +g

(n)

2g(n)

. (21)

The Kutta condition is used to ensure that the complex velocity(19) remains finite at the trailing edge of the airfoil 1,0 = a c .This amounts to evaluating the quantity in the bracket at 1,0 andsetting it to zero, which upon some simplifications gives

2Im

U1,0 1,0r1 i1,0 u,0c + ua,0a

+N

n=1

n

2

1 + 2Re

1,0

n 1,0

= 0, (22)

where

r1 =r

=1,0, u,0 =

F

=1,0,

ua,0

=

Fa

=1,0. (23)

Eqs. (21) and (22) form an algebraic-differential set of 2N + 1equations which we solve for 1(t) and zn(t), n = 1, . . . , N.

It is important to note here that this system of equations is ill-posed when a new unsteady vortex is started from the trailingedge because the right-hand side of(21) is singular. Indeed, at theonset of shedding, the vortex and its image conjugate are at thesame point. To overcome this difficulty, we follow the approachin [25,40,26] and propose an analytic solution valid only for smalltime which we use to obtain the initial behavior of the system; seeAppendix for details.

5. Wake structure

In this section, we study the behavior of the wake as a func-tion of the prescribed deformations and prescribed translationaland rotational motions of the airfoil. We particularly examine pre-scribed deformations, orientation and translational velocities ofthe Joukowski airfoil of the form

c = o + io cn

4K

Tt

, = o cn

4K

Tt

,

Vx = Vo, Vy = 0. (24)Here, o, o, o and Vo are non-zero constants and cn(4Kt/T) is the

cnoidal elliptic function with K(m) given by K(m) = /20

d/(1m sin2 )1/2 and T the period of flapping or shape cycle. It is worth

noting that m is a dimensionless parameter that determines theprofile of the cnoidal function as shown in Fig. 2. If m = 0, one


6/9



n versus time for prescribed motions c = 0.125 i0.2 cn(4Kt/T), = 0.2 cn(4Kt/T) and parameters m = 0.995 and St = 0.3 which correspond to Region IV in Fig. 9.

recovers the sinusoidal cosine function cos(2 t/T) and ifm = 1,one gets the hyperbolic secant. Note that as m increases, the time

derivative of the cnoidal function (shown in Fig. 2(b)) changesrapidly when the airfoil reaches its maximum, zero and minimumflapping angles. It is this rapid change in the shape deformationsthat we refer to as impulsive flapping. The airfoil motion in thephysical plane is illustrated in Fig. 3 for two parameter valuesm = 0 and m = 0.995. This motion can be characterized by twoparameters: m which dictates the shape of the cnoidal function asdepicted in Fig. 2 and the Strouhal number St defined as

St = LVoT

. (25)

In (25), the length scale L is defined by the lateral motion of thetrailing edge. Our objective is to map out the dependence of thewake structure on the Strouhal number St and the dimensionless

parameter m which dictates the profile of the periodic deforma-tions, from sinusoidal to impulsive.

The conformal mapping approach presented in this papertransforms the problem of wake generation behind the deformingairfoil into that of vortex formation behind a cylinder oscillatingin a free stream. Vortex formation behind a cylinder oscillating ina free stream is a classical example that is known to afford a greatvariety of wake structures depending on the cylinders oscillations;see the seminal work of Williamson & Roshko in [10]. WilliamsonandRoshkomapped out the wake structures behind the oscillatingcylinder as function of the oscillation parameters, namely, theamplitude and frequency (or Strouhal number) of the harmonicoscillations. A simple mathematical model was later developedin [41] to explain the important wake transitions. Examples of the

wakes reported in [10] include von Krmn-type wakes, referredto as 2S wakes, in which two vortices of opposite sign are shed

per oscillation period and 2P wakes in which two vortex pairs areshed per oscillation period. We use this terminology to categorize

the wake structures obtained here in the context of the simplifiedBrownMichael vortex model.

It is worth noting that the idea of mapping out the dependenceof the wake structure on the oscillation parameters was recentlyreconsidered in [8] for a symmetric rigid airfoil oscillating in atwo-dimensional free stream created by a vertically flowing soapfilm. These soap-film experiments focused on sinusoidal pitchingoscillations andreporteda wide variety of simpleand exoticwakesfor various amplitude and frequency of oscillations. The focus ofour study, as mentioned earlier, is to understand the dependenceof wake structure on the swimming velocity (as reflected byStrouhal number St) and the profile of the periodic deformations,from sinusoidal to impulsive (as dictated by the dimensionlessparameter m).

In Figs. 48, we show the wake structure obtained by varyingthe parameters m and St while the remaining parameters areheld at o = 1/5, o = 1/8, o = 1/5, rc = 1, andT = 1. In other words, the Strouhal number St is varied byvarying the translational velocity Vo. Panels (a) in Figs. 48 showsnapshots of the streamlines of the velocity field at t = 5.002while panels (b) and (c) show the strength of the shed vorticesand the total circulation in the wake (sum of strength) versus time,respectively. Each of these figures represent one of five distinctstructures of the wake that we identified depending on the twoparameters m and St; see Fig. 9. The results in Fig. 9 are obtainedby discretizing the parameter space (St, m) and computing thewake of the deformable airfoil at each of these discrete points.In region I where the Strouhal number is low St < 0.2, that is

to say, the translational velocity Vo is large, the wake, after sometransientbehavior, is an aligned 2S wake that resemblesan array of


7/9



n versus time for prescribed motions c = 0.125 i0.2 cn(4Kt/T), = 0.2 cn(4Kt/T) and parameters m = 0.8 and St = 0.5 which correspond to Region III in Fig. 9.

isolated point vortices of equal and opposite circulations, with twovortices of equal and opposite strength shed per oscillation period

as evidenced in Fig. 4.As the Strouhal number increases 0.2 < St < 0.4, the

structure of the wake bifurcates depending on the value m: inregion II for m < 0.95 the wake structure is similar to a reversevon Krmm wake or 2S wake (Fig. 5) whereas in region IV form > 0.95, that is to say, for impulsive motion, one gets a 2Pwake with predominantly two point vortices shed in every halfcycle of the airfoil oscillations and pair-up to form vortex dipolesthat propagate obliquely in the flow (Fig. 6), with their directionbeing determined by initial conditions. The increased sheddingperiodicity is evident in Fig. 6(b). Numerically, three vortices areshed each half-cycle where one vortex has negligible effect giventhat its intensity is much smaller than the other two (see Fig. 6(b)and (c)).

As the Strouhal number increases further 0.4 < St < 0.6,the von Krmm structure observed in region II disappears as thevortices shed in two consecutive cycles of oscillations begin topair-up and propagate obliquely as dipoles resulting in a P wake(Fig. 7). Finally, the dipoles shed every half cycle of oscillations inregion IV are now replaced by tripoles also shed every half-cycleof oscillations as evidenced in Fig. 8(b) and (c). More specifically,the previously negligible vortex in region IV now has significantlyhigher strength, resulting in what initially sheds as a 2P+2S wake.Due to downstream vortex interactions, the wake structure breaksup into tripoles and more complex configurations.

6. Conclusions

A deformable Joukowski airfoil is proposed as a model for theflapping of a fish body or tail in an otherwise quiescent fluid.

The wake structure is investigated as a function of m and Stwhere m indicates the profile of the flapping motion, from smooth

sinusoidal motion to more sharply peaked, but still periodicmotion. The Strouhal number St indicates the ratio of the flappingspeed to the translational motion of the fish. For almost sinusoidalflapping m < 0.95 in the range 0.2 < St < 0.4, which is therange of Strouhal numbers identified in the experimental work [3]as the optimal range in terms of swimming efficiency, we find thatthe wake structure corresponds to a reverse von Krmn street.For lower St, the wake structure consists of an array of isolatedpoint vortices with no fluid jetpropagatingdownstream.This wakestructure presents little advantage to the flapping airfoil in termsof momentum thrust induced by the wake. For higher St, the pointvortices shed in twoconsecutive cycles of flapping begin to pair-upto form vortex dipoles that propagate obliquely to the direction ofmotion, also reducing the locomotory advantage of a thrust wake.

For impulsive flapping m > 0.95, we find that as the Strouhalnumber increases, the airfoil begins to shed three point vorticeswith every half-cycle of its flapping motion. For 0.2 < St < 0.4,the third vortex shed when the airfoil approaches its straight-outconfiguration has negligible magnitude and the wake structureconsists of dipoles shed every half-cycle of flapping.For 0.4 < St

confomral mapping - wake

Documents