Waves, vortices, and the gait of the water strider€¦ · problemathand. The key phenomenontobe understood is the global responseofa ßuid toa forceÞeld that is bothimpulsive and

Oliver BühlerCourant Institute of Mathematical SciencesCenter for Atmosphere Ocean Science

Waves, vortices, and the gait of the water strider

...and now for something completely different

Current paper submitted to JFM, 2005, 6

Under consideration for publication in J. Fluid Mech. 1

Impulsive fluid forcing and water striderlocomotion

By OLIVER BUHLERCenter for Atmosphere Ocean Science at theCourant Institute of Mathematical Sciences

New York University, New York, NY 10012, USA

(Submitted 19. September 2005)

This paper presents a study of the global response of a fluid to impulsive and localizedforcing; it has been motivated by the recent laboratory experiments on the locomotionof water-walking insects in Hu, Chan & Bush (2003). Theses insects create both wavesand vortices by their rapid leg strokes and it has been a matter of some debate whethereither form of motion predominates in the momentum budget. The main result of thispaper is to argue that generically both waves and vortices are significant, and that theytake up the momentum with share 1/3 and 2/3, respectively.

This generic result, which depends only on the impulsive and localized nature of theforcing, is established using the classical linear impulse theory, with adaptations to weaklycompressible flows and flows with a free surface. Additional general comments on exper-imental techniques for momentum measurement and the on the wave emission processare given and then the theory is applied in detail to the case of the water strider. Owingto its generality, the present theory should be useful for other fluid–structure interactionproblems in animal locomotion and elsewhere.

1. IntroductionThis paper has been motivated by recent laboratory experiments on the locomotion

of water-walking insects reported in Hu, Chan & Bush (2003). Specifically, the topic ofinterest is the exchange of momentum between the insect and the fluid underneath duringthe insect’s rapid leg stroke. During the stroke there is an equal-and-opposite exchange ofmomentum between the insect’s moving leg and the fluid underneath, and the insect usesthis recoil to propel itself forward. The opposite recoil felt by the fluid generates bothsurface waves and vortices in the fluid and it has been a matter of some debate whetherwaves or vortices predominate in their uptake of the recoil momentum (e.g. Bush & Hu2006, and references therein).

The present paper offers a synthesis of these competing hypotheses by arguing that thewaves and the vortices are both significant, and that they take up the recoil momentumwith share 1/3 and 2/3, respectively. This result is generic in the sense that it does notdepend on the details of the surface waves (such as whether surface tension is presentor not), rather it depends only on the spatially localized and impulsive nature of themomentum exchange. Therefore, this kind of result should be relevant to other fluid–structure interaction problems arising in animal locomotion or elsewhere.

The fluid-dynamical arguments leading to this generic result are adaptations of classicalarguments on fluid momentum and Kelvin’s hydrodynamic impulse, which can already be

26 John W.M. Bush and David L. Hu (LRH)

Figure 1: ARFL38. Bush JWM, Hu DL.

Figure 2: ARFL38. Bush JWM, Hu DL

Motivation by experiments:Water walking animals....

Surface statics is well understood (eg Keller 98). Dynamics much less so

Dynamics: waves and vortices(all images David Hu et al.)

Waves Vortices

Note importance of impulsive leg stroke

(20x slowed down) (real time)

Surface waves

0 1 2 3 4 5 6 70

20

40

60

80

100

120

140

160

k [cm-1

]

! [s

-1]

Dispersion relation

ω =√

gk + γk3

cp =ω

k

cg =dω

dk

Minimum phase velocity is 23 cm/s at wavelength 1.7 cm

Minimum group velocity is 18 cm/s at wavelength 4.4 cm

gravity surface tension

Drag computations for sources moving with fast but constant speed (ie not impulsive) (eg Raphael & de Gennes 96, Sun & Keller 01)

exp(i[kx− ωt])

Locomotion requires momentum exchange betweeninsect and water

Insect recoil Waves Vortices= +Momentum conservation

Hu, Chan, Bush, 2003

which was deduced independently by measuring the strider’s accel-eration and leaping height. The applied force per unit length alongits driving legs is thus approximately 50/0.6 < 80 dynes cm21. Anapplied force per unit length in excess of 2j < 140 dynes cm21 willresult in the strider penetrating the free surface. The water strider isthus ideally tuned to life at the water surface: it applies as great aforce as possible without jeopardizing its status as a water-walker.

The propulsion of a one-day-old first-instar is detailed in Fig. 3.Particle tracking reveals that the infant strider transfers momentumto the fluid through dipolar vortices shed by its rowing motion. Thewake of the adult water strider is similarly marked by distinct vortexdipole pairs that translate backwards at a characteristic speed Vv <4cms21 (Figs 3c and 4). Video images captured from a side viewindicate that the dipolar vortices are roughly hemispherical, with acharacteristic radius R < 0.4 cm. The vertical extent of the hemi-spherical vortices greatly exceeds the static meniscus depth24,120 mm, but is comparable to the maximum penetration depth ofthe meniscus adjoining the driving leg, 0.1 cm. A strider of massM < 0.01 g achieves a characteristic speed V < 100 cm s21 andso has a momentum P ¼ MV < 1 g cm s21. The total momentumin the pair of dipolar vortices of mass M v ¼ 2pR 3/3 isPv ¼ 2MvV v < 1gcms21, and so comparable to that of the strider.

The leg stroke may also produce a capillary wave packet, whosecontribution to the momentum transfer may be calculated. Weconsider linear monochromatic deep-water capillary waves withsurface deflection z(x,t) ¼ ae i(kx2qt) propagating in the x-directionwith a group speed cg ¼ dq/dk, phase speed c ¼ q/k, amplitude a,wavelength l ¼ 2p/k and lateral extent W. The time-averagedhorizontal momentum associated with a single wavelength,

Pw ¼ pjka2Wc21, may be computed from the velocity field andrelations between wave kinetic energy and momentum21,25. Ourmeasurements indicate that the leg stroke typically generates a wavetrain consisting of three waves with characteristic wavelengthl < 1 cm, phase speed c < 30 cm s21, amplitude a < 0.01–0.05 cm, and width L2 < 0.3 cm (see Fig. 3c). The net momentumcarried by the capillary wave packet thus has a maximum valuePw < 0.05 g cm s21, an order of magnitude less than the momen-tum of the strider.The momentum transported by vortices in the wake of the water

strider is comparable to that of the strider, and greatly in excess of thattransported in the capillary wave field; moreover, the striders arecapable of propelling themselves without generating discerniblecapillary waves. We thus conclude that capillary waves do not playan essential role in thepropulsionofGerridae, and therebycircumventDenny’s paradox. The strider generates its thrust by rowing, using itslegs as oars and its menisci as blades. As in the case of rowing boats,while waves are an inevitable consequence of the rowing action, theydo not play a significant role in themomentum transfer necessary forpropulsion. We note that their mode of propulsion relies on theReynolds number exceeding a critical value of approximately 100,suggesting a bound on the minimum size of water striders. Ourcontinuing studies ofwater striderdynamicswill follow thoseof birds,insects andfish11,15,16 in characterizing the hydrodynamic forces actingon the body through detailed examination of the flows generatedduring the propulsive stroke.We designed amechanical water strider, Robostrider, constructed

to mimic the motion of a water strider (Fig. 1c). Its proportions andMc value were consistent with those of its natural counterpart

Figure 3 The flow generated by the driving stroke of the water strider. a, b, The stroke of aone-day-old first-instar water strider. Sequential images were taken 0.016 s apart.

a, Side view. Note the weak capillary waves evident in its wake. b, Plan view. Theunderlying flow is rendered visible by suspended particles. For the lowermost image,

fifteen photographs taken 0.002 s apart were superimposed. Note the vortical motion in

the wake; the flow direction is indicated. The strider legs are cocked for the next stroke.

Scale bars, 1mm. c, A schematic illustration of the flow structures generated by the

driving stroke: capillary waves and subsurface hemispherical vortices.

letters to nature

NATURE |VOL 424 | 7 AUGUST 2003 | www.nature.com/nature 665© 2003 Nature Publishing Group

Ratio ?

Definition of wave and vortex momentum is non-trivialImpulsive leg stroke

1 0 ?0 1 ?

1/3 2/3 !

Impulsive forcing:localized in space and time

F (x, t) = F (x) δ(t)

∇× F "= 0

∇ · F != 0F

2 Buhler

found in the incompressible flow theory part of Lamb (1932). However, theses argumentsappear not to be very widely known, and thus it seems useful to draw the threadstogether in one place and present these arguments here together with their adaptationsto compressible and free-surface flows and with their application to the water striderproblem at hand.

The key phenomenon to be understood is the global response of a fluid to a force fieldthat is both impulsive and spatially localized (in a sense to be defined precisely later).Depending on the nature of the fluid and its boundary conditions (e.g. depending on itscompressibility and on the presence of free or rigid boundaries) the response of the fluidwill differ in detail, but it will always involve the propagation of pressure waves (possiblywith infinite speed) to large distances away from the forcing site and therefore it willalways involve the motion of fluid particles at large distances as well.

Moreover, the fluid motion typically involves both waves and vortices and it is a naturalquestion to ask how the structure of the force field affects the magnitude of the excitedwaves versus the excited vortices. As alluded to earlier, it will turn out that in many casesthe relative magnitudes of wave and vortex momentum do not depend on the structureof the force field at all, provided only that the force field is impulsive and localized.

The essence of this result can be understood based on a simpler statement, namelythat it turns out that the relevant momentum content in the waves and in the vortices isproportional to the first spatial moments of ∇ ·F and of ∇×F , respectively, where F (x)is the spatial structure of the impulsive force field. In general there is no link betweenthese first moments but for localized force fields (which vanish outside a finite region ofsupport) there is a trivial integration-by-parts identity∫

F dV = −∫

x ∇ · F dV =1

n− 1

∫x× (∇× F ) dV, (1.1)

where n is the number of spatial dimensions and the volume integral is extended over adomain that contains the support of F but is otherwise arbitrary.

The identity (1.1) illustrates clearly that if n > 1 then a localized vector field mustnecessarily have both a potential part as well as a non-divergent vortical part, and thatfor all values of n the first moments of these parts are matched in the ratio 1/(n − 1).By implication, the wave momentum and the vortex momentum are then matched in thesame ratio. Thus, for n = 1 all momentum is in the waves, for n = 2 there is equipartition,and for n = 3 wave momentum is 1/2 of the vortex momentum. The water strider casecan be shown to follow the three-dimensional ratio and together with global momentumconservation this gives the main result as stated above.

The theoretical steps leading to these results are straightforward provided attentionis paid to the usual difficulty of incompressible fluid momentum budgets in unboundeddomains, namely that the momentum integrals are not absolutely convergent and thusthat the momentum budget for infinite control region depends on the shape of the controlregion (e.g. Theodorsen 1941). As is well known, this ambiguity can be clarified byconsidering the limiting case of weak compressibility.

Furthermore, it should be pointed out that the momentum budget of an impulsivelyforced wave pulse is in many ways simpler than the momentum budget of a slowlyvarying wavetrain. The wave pulse contains a well defined momentum at first order inwave amplitude whereas the oscillatory wavetrain contains a well defined momentumflux at second order in wave amplitude (this plays a role in the water strider momentumbudget in §5.1 below). As is well known, to endow the wavetrain with a well definedsecond-order ‘wave momentum’ leads to the so-called pseudomomentum of a wavetrain,

Hints at the importance of dimensionality n=1,2,3

Compact force fields are very special

Impulsive forcing from rest: momentum budget for large domain D

Impulsive fluid forcing and water strider locomotion 3

and to the conceptual challenge of how to relate this pseudomomentum to the momentumand impulse budget of the entire flow (e.g. McIntyre 1981).

Nevertheless, a second motivation for the present paper is precisely this conceptualchallenge in the context of local wave–mean interaction theory, which is a theory thatseeks to explain how localized wavepackets interact nonlinearly with the mean flow inwhich they are embedded (Bretherton 1969; Buhler & McIntyre 2003, 2005). A resultthat has emerged from these studies is a certain conservation law for the sum of wavepseudomomentum and vortex impulse, which is another example in which seeminglyunrelated aspects of wave and vortex dynamics appear to be coupled.

The plan of the paper is as follows. The classical linear theory for the impulsive forcingof unbounded incompressible fluids with spatial dimension n = 1, 2, 3 is presented in §2(with specific connections to Kelvin’s impulse and experimental momentum measurementtechniques discussed in §2.5) and then adapted to weakly compressible flow in §3 and toincompressible flows with a free surface in §4. This involves a careful study of the waveemission process by which the flow adjusts to its steady vorticity-controlled end state.The water strider locomotion is then discussed in §5 and concluding remarks are givenin §6.

The general remarks given in this introduction are sufficient to allow readers who areprimarily interested in the water strider results to skip directly to §5.

2. Incompressible flow2.1. Incompressible momentum budget

We begin by noting some elementary facts about the momentum budget of an idealincompressible homogeneous fluid subject to an external force per unit mass F . For themost part we consider the fluid to be unbounded such that the position vector x ∈ Rn,where the number of spatial dimensions is n. The governing equations are

∇ · u = 0 andDu

Dt+

∇p

ρ= F , (2.1)

where u is the velocity, p is the pressure, and ρ is the homogeneous density, which is nowset to unity without loss of generality.

We are interested in the integral momentum budget for a Eulerian control regionD ∈ Rn with boundary ∂D, which is

ddt

∫D u dV +

∮∂D(uu · n + pn) dA =

∫D F dV, or

ddt M + Φ = R.

(2.2)

Here n is the outward unit normal vector on ∂D, M is the momentum contained in D, Φis the momentum flux across ∂D, and R is the momentum per unit time that the externalforce adds to D. This budget is unambiguous for finite-sized D, but typically the integralsare not absolutely convergent as D grows to infinity. Consequently, their limiting valuesas D →∞ typically depend on the shape of D (§119 in Lamb 1932; Theodorsen 1941).

It is clear from (2.2) that for given D and given external forcing F the fluid responsein D is partly momentum change as measured by dM/dt and partly momentum flux asmeasured by Φ. The sum of these two responses must equal R but it is not clear a priorihow R is partitioned between the two. This is the basic question to be discussed here inthe simple case of localized impulsive forcing.








∇ · u = 0 andDu

Dt+

∇p

ρ= F , (2.1)



ddt

∫D u dV +

∮∂D(uu · n + pn) dA =

∫D F dV, or

ddt M + Φ = R.

(2.2)



ρ = 1

4 Buhler

2.2. Localized impulsive forcingA well known classical problem (e.g. Lamb 1932) describes the response of a resting fluid(i.e. u = 0) to an impulsive force at t = 0. That is, we consider external forces of theform

F (x, t) = F (x)1

∆tg(t/∆t) (2.3)

in the limit of infinitesimal forcing duration ∆t → 0. Here the requirements on thefunction g(s) are that it is non-negative, zero for s < 0 and s > 1, and has unit integral.Therefore, in the limit ∆t → 0 the forcing becomes a delta function in time and thisis the definition of impulsive forcing. The classical argument then shows that duringt ∈ [0,∆t] the forcing magnitude and fluid acceleration are both large (i.e. |F | ∼ |ut| =O(∆t−1)), that the velocity is finite (i.e. |u| = O(1)), and therefore that the quadraticterm (u · ∇)u = O(1) in the material derivative is negligible compared to the forcingand pressure terms. By the same argument the impulsive momentum flux Φ is entirelydue to pressure. Overall, this scaling argument shows that the impulsive forcing problemreduces to a linear set of equations, which can be solved with elementary methods.

The classical impulse theory holds for arbitrary spatial structure F (x) of the impul-sive force. In the present context we restrict to localized forces, i.e. forces that havecompact support and hence are non-zero only within a finite-sized region around theorigin. Specifically, we require that there exists a length L <∞ such that

r = |x| > L ⇒ |F (x)| = 0. (2.4)

The restriction to localized forces is motivated physically and brings with it many math-ematical simplifications, e.g. it ensures that all spatial moments of F are finite provided|F | has at most integrable singularities, which we shall assume to be the case. It alsoensures that R converges to a well-defined limit as D →∞.

The impulsive momentum flux Φ requires the pressure distribution p, which as usual isdetermined by taking the divergence of the momentum equation and using (∇ ·u)t = 0.Because the nonlinear terms are negligible this yields the simple Poisson equation

∇2p = ∇ · F = ∇ · F 1∆t

g(t/∆t) (2.5)

together with the boundary condition ∇p → 0 as r → ∞. Therefore the impulsivepressure p has the same delta-sequence time dependence as F , i.e.

p = p(x)1

∆tg(t/∆t) and ∇2p = ∇ · F . (2.6)

By the assumptions on g(s), p equals the time integral of p over the forcing duration ∆t.Indeed, the linear momentum equation can be time-integrated from t = 0 to t = ∆t toyield the simple expression

t = ∆t : u(x,∆t) + ∇p(x) = F (x) (2.7)

for the velocity directly after the forcing. In the impulsive limit ∆t → 0 this expressionfurnishes the initial conditions at t = 0+ for the subsequent evolution of u withoutforcing. As is well known, (2.7) together with (2.6b) is equivalent to defining u(x,∆t) asthe least-square projection of F onto non-divergent vector fields.

Similarly, the time-integral of (2.2) over the forcing duration takes a simple form interms of the quantities

Φ =∫ ∆t

0Φdt and R =

∫ ∆t

0R dt, (2.8)

Advective flux negligible for impulsive forcing








∇ · u = 0 andDu

Dt+

∇p

ρ= F , (2.1)



ddt

∫D u dV +

∮∂D(uu · n + pn) dA =

∫D F dV, or

ddt M + Φ = R.

(2.2)



Waves Vortices

Ratio ?M(t = 0+) + Φ = R

One-dimensional case: n=1

ux = 0 ⇒ u(t) = 0 for any finite momentum input R

All response is in the wavelike pressure

Note absence of force curl if n=1


namelyt = ∆t : M(∆t) + Φ = R. (2.9)

Thus M(∆t) is the fluid momentum contained in D after the impulsive forcing andΦ is the time-integrated momentum flux across ∂D during the impulsive forcing. Forconvenience, we will often denote by M the value of M(∆t) as ∆t → 0. We shall beparticularly interested in the limit of (2.9) as D → ∞ whilst preserving its shape.† Inview of what has been said in the introduction, in the present case the wave field isassociated with the pressure response to the forcing. Therefore, its contribution to themomentum budget is Φ and the vortex contribution is M .

The basic question noted before can now be posed succinctly: for a given force structureF (x), how is the total impulsive momentum input R partitioned into fluid momentumM and time-integrated momentum flux Φ?

As demonstrated below, the answer does not depend on the structure of F (x), butit does depends both on the number of spatial dimensions n and on the shape of D(Theodorsen 1941). Specifically, for a spherical control volume D the answer is

M =n− 1

nR and Φ =

1n

R, (2.10)

i.e. in one dimension all momentum is carried away by the impulsive momentum flux,but only half of it is carried away in two dimensions, and only a third of it in threedimensions.‡ and We shall now demonstrate these results using elementary computationsand then move on to compressible flows.

2.3. One-dimensional incompressible flowIn the one-dimensional problem with position x the incompressibility condition is ux = 0and therefore u depends on t only. In an unbounded domain any nonzero u thereforeimplies infinite momentum. It follows that for finite momentum input R there can beno acceleration whatsoever and thus M = 0 and Φ = R for any interval D. Indeed, thepressure field with the correct boundary condition px → 0 as |x|→∞ is

p(x) = p(−∞) +∫ x

−∞F (x′) dx′ (2.11)

and if F is localized at the origin then the pressure is

p =R

2sgn(x) for |x| > L (2.12)

after adjusting the irrelevant global pressure constant. Clearly, the one-dimensional caseis very special because p does not decay with distance from the site of the forcing at theorigin. Therefore, if D is any interval between xL and xR such that xL < −L and xR > Lthen

Φ = p(xR)− p(xL) =R

2+

R

2= R (2.13)

† This limit can be formally defined in terms of a family of domains D(α) such that if theboundary ∂D(0) is the set of positions x satisfying f(x) = 0 for a suitable level-set function fthen the boundary ∂D(α) corresponds to f(x/α) = 0. The limit D →∞ then refers to D(α) asα→∞.

‡ The view that u is the projection of F onto non-divergent vector fields allows an interestingalternative derivation of (2.10a) based on the usual projection formula for Fourier transforms

u = (I − kk/|k|2) · F . Here the values of (u, F ) at k = 0 can be identified with (M , R) andfor n > 1 the projector is clearly indeterminate as k → 0, as it has to be. By averaging over allpossible orientations of k as k → 0 the expression (2.10a) is obtained.

For forcing at origin x=0:

M = 0 and Φ = R

px = F

1 0 n=1

Score so far.....

Pressure finder if n>16 Buhler

as anticipated. This shows that in the one-dimensional case all momentum input is un-ambiguously fluxed away by the pressure wave.

2.4. Two- and three-dimensional incompressible flowFor n > 1 the pressure is computed from the Poisson equation

∇2p = ∇ · F subject to ∇p→ 0 as r = |x|→∞ (2.14)

by using the Green’s function G(x,x′) that solves

∇2G = δ(x− x′) subject to ∇G→ 0 as r = |x|→∞. (2.15)

This Green’s function is

G(x,x′) =

{+ 1

2π ln |x− x′| n = 2

− 14π |x− x′|−1 n = 3

(2.16)

and the pressure p is then given by the convolution

p(x) =∫

G(x,x′)∇′ · F ′ dV ′ = −∫

F ′ · ∇′G(x,x′) dV ′. (2.17)

Here the integral extends formally over all x′ ∈ Rn and the primes indicate that thefunctions and differential operators relate to x′. The special second form follows fromthe divergence theorem together with the compactness of F ′(x′).

Actually, only the far-field asymptotics of p(x) is needed to compute the momentumflux

Φ =∮

∂Dpn dA (2.18)

in the targeted limit D → ∞. Specifically, only the dipolar part of p (which decays asr1−n) is relevant for Φ in the far field r & L. The dipolar part of p can be extractedfrom (2.17) by using the standard multipole expansion for the far-field region r & L. Inthis region the Green’s function is slowly varying in x′ and G can be Taylor-expanded as

G(x,x′) = G(x, 0) + x′ · ∇′G(x, 0) + . . . , (2.19)

which upon substitution in (2.17) yields the coefficients of the far-field multipole expan-sion. The first term does not depend on x′ and therefore makes no contribution in (2.17b)and the second term gives the dipolar term (cf. (1.1))

p ≈ −∫

F ′ dV ′ · ∇′G(x, 0) = −R · ∇′G(x, 0) as r →∞. (2.20)

Now, the symmetries of the Poisson equation in an unbounded domain imply thatG(x,x′) = G(|x− x′|) and therefore −∇′G(x,x′) = +∇G(x,x′). With the shorthand

G0(x) ≡ G(x, 0) =

{+ 1

2π ln r n = 2

− 14π r−1 n = 3.

(2.21)

the leading-order far-field pressure therefore takes the final form p ≈ R ·∇G0(x), whichis

p ≈ 12πr

R · xr

for n = 2 and p ≈ 14πr2

R · xr

for n = 3, (2.22)

both with error O(r−n) as r →∞. Higher-order terms in the multipole expansion affectthe momentum flux at finite r but become irrelevant as r →∞.

Thus only R, i.e. the zeroth spatial moment of F , affects the dipolar part of p in the far

6 Buhler







G(x,x′) =

{+ 1

2π ln |x− x′| n = 2

− 14π |x− x′|−1 n = 3

(2.16)


p(x) =∫

G(x,x′)∇′ · F ′ dV ′ = −∫

F ′ · ∇′G(x,x′) dV ′. (2.17)



Φ =∮

∂Dpn dA (2.18)


G(x,x′) = G(x, 0) + x′ · ∇′G(x, 0) + . . . , (2.19)


p ≈ −∫

F ′ dV ′ · ∇′G(x, 0) = −R · ∇′G(x, 0) as r →∞. (2.20)


G0(x) ≡ G(x, 0) =

{+ 1

2π ln r n = 2

− 14π r−1 n = 3.

(2.21)


p ≈ 12πr

R · xr


R · xr

for n = 3, (2.22)



6 Buhler







G(x,x′) =

{+ 1

2π ln |x− x′| n = 2

− 14π |x− x′|−1 n = 3

(2.16)


p(x) =∫

G(x,x′)∇′ · F ′ dV ′ = −∫

F ′ · ∇′G(x,x′) dV ′. (2.17)



Φ =∮

∂Dpn dA (2.18)


G(x,x′) = G(x, 0) + x′ · ∇′G(x, 0) + . . . , (2.19)


p ≈ −∫

F ′ dV ′ · ∇′G(x, 0) = −R · ∇′G(x, 0) as r →∞. (2.20)


G0(x) ≡ G(x, 0) =

{+ 1

2π ln r n = 2

− 14π r−1 n = 3.

(2.21)


p ≈ 12πr

R · xr


R · xr

for n = 3, (2.22)



6 Buhler







G(x,x′) =

{+ 1

2π ln |x− x′| n = 2

− 14π |x− x′|−1 n = 3

(2.16)


p(x) =∫

G(x,x′)∇′ · F ′ dV ′ = −∫

F ′ · ∇′G(x,x′) dV ′. (2.17)



Φ =∮

∂Dpn dA (2.18)


G(x,x′) = G(x, 0) + x′ · ∇′G(x, 0) + . . . , (2.19)


p ≈ −∫

F ′ dV ′ · ∇′G(x, 0) = −R · ∇′G(x, 0) as r →∞. (2.20)


G0(x) ≡ G(x, 0) =

{+ 1

2π ln r n = 2

− 14π r−1 n = 3.

(2.21)


p ≈ 12πr

R · xr


R · xr

for n = 3, (2.22)



First term in far-field multi-pole expansion (ie large r=|x-x’|):

Need to compute pressure to find momentum flux

∇× u = ∇× F ∇ · u = 0 u = F − ∇p

Far-field pressure determined by net momentum input R

Multi-dimensional case: n=2,3

Green’s function for pressure

1 0 n=11/2 1/2 n=21/3 2/3 n=3

6 Buhler







G(x,x′) =

{+ 1

2π ln |x− x′| n = 2

− 14π |x− x′|−1 n = 3

(2.16)


p(x) =∫

G(x,x′)∇′ · F ′ dV ′ = −∫

F ′ · ∇′G(x,x′) dV ′. (2.17)



Φ =∮

∂Dpn dA (2.18)


G(x,x′) = G(x, 0) + x′ · ∇′G(x, 0) + . . . , (2.19)


p ≈ −∫

F ′ dV ′ · ∇′G(x, 0) = −R · ∇′G(x, 0) as r →∞. (2.20)


G0(x) ≡ G(x, 0) =

{+ 1

2π ln r n = 2

− 14π r−1 n = 3.

(2.21)


p ≈ 12πr

R · xr


R · xr

for n = 3, (2.22)



6 Buhler







G(x,x′) =

{+ 1

2π ln |x− x′| n = 2

− 14π |x− x′|−1 n = 3

(2.16)


p(x) =∫

G(x,x′)∇′ · F ′ dV ′ = −∫

F ′ · ∇′G(x,x′) dV ′. (2.17)



Φ =∮

∂Dpn dA (2.18)


G(x,x′) = G(x, 0) + x′ · ∇′G(x, 0) + . . . , (2.19)


p ≈ −∫

F ′ dV ′ · ∇′G(x, 0) = −R · ∇′G(x, 0) as r →∞. (2.20)


G0(x) ≡ G(x, 0) =

{+ 1

2π ln r n = 2

− 14π r−1 n = 3.

(2.21)


p ≈ 12πr

R · xr


R · xr

for n = 3, (2.22)



6 Buhler







G(x,x′) =

{+ 1

2π ln |x− x′| n = 2

− 14π |x− x′|−1 n = 3

(2.16)


p(x) =∫

G(x,x′)∇′ · F ′ dV ′ = −∫

F ′ · ∇′G(x,x′) dV ′. (2.17)



Φ =∮

∂Dpn dA (2.18)


G(x,x′) = G(x, 0) + x′ · ∇′G(x, 0) + . . . , (2.19)


p ≈ −∫

F ′ dV ′ · ∇′G(x, 0) = −R · ∇′G(x, 0) as r →∞. (2.20)


G0(x) ≡ G(x, 0) =

{+ 1

2π ln r n = 2

− 14π r−1 n = 3.

(2.21)


p ≈ 12πr

R · xr


R · xr

for n = 3, (2.22)



Wavelike momentum flux

Far-field pressure dipole

Flux easy to compute for spherical control volume D

By “physical induction”:


namelyt = ∆t : M(∆t) + Φ = R. (2.9)

Thus M(∆t) is the fluid momentum contained in D after the impulsive forcing andΦ is the time-integrated momentum flux across ∂D during the impulsive forcing. Forconvenience, we will often denote by M the value of M(∆t) as ∆t → 0. We shall beparticularly interested in the limit of (2.9) as D → ∞ whilst preserving its shape.† Inview of what has been said in the introduction, in the present case the wave field isassociated with the pressure response to the forcing. Therefore, its contribution to themomentum budget is Φ and the vortex contribution is M .

The basic question noted before can now be posed succinctly: for a given force structureF (x), how is the total impulsive momentum input R partitioned into fluid momentumM and time-integrated momentum flux Φ?

As demonstrated below, the answer does not depend on the structure of F (x), butit does depends both on the number of spatial dimensions n and on the shape of D(Theodorsen 1941). Specifically, for a spherical control volume D the answer is

M =n− 1

nR and Φ =

1n

R, (2.10)

i.e. in one dimension all momentum is carried away by the impulsive momentum flux,but only half of it is carried away in two dimensions, and only a third of it in threedimensions.‡ and We shall now demonstrate these results using elementary computationsand then move on to compressible flows.

2.3. One-dimensional incompressible flowIn the one-dimensional problem with position x the incompressibility condition is ux = 0and therefore u depends on t only. In an unbounded domain any nonzero u thereforeimplies infinite momentum. It follows that for finite momentum input R there can beno acceleration whatsoever and thus M = 0 and Φ = R for any interval D. Indeed, thepressure field with the correct boundary condition px → 0 as |x|→∞ is

p(x) = p(−∞) +∫ x

−∞F (x′) dx′ (2.11)

and if F is localized at the origin then the pressure is

p =R

2sgn(x) for |x| > L (2.12)

after adjusting the irrelevant global pressure constant. Clearly, the one-dimensional caseis very special because p does not decay with distance from the site of the forcing at theorigin. Therefore, if D is any interval between xL and xR such that xL < −L and xR > Lthen

Φ = p(xR)− p(xL) =R

2+

R

2= R (2.13)

† This limit can be formally defined in terms of a family of domains D(α) such that if theboundary ∂D(0) is the set of positions x satisfying f(x) = 0 for a suitable level-set function fthen the boundary ∂D(α) corresponds to f(x/α) = 0. The limit D →∞ then refers to D(α) asα→∞.

‡ The view that u is the projection of F onto non-divergent vector fields allows an interestingalternative derivation of (2.10a) based on the usual projection formula for Fourier transforms

u = (I − kk/|k|2) · F . Here the values of (u, F ) at k = 0 can be identified with (M , R) andfor n > 1 the projector is clearly indeterminate as k → 0, as it has to be. By averaging over allpossible orientations of k as k → 0 the expression (2.10a) is obtained.

WaveVortex

Caveat: pressure integral is not absolutely convergent

6 Buhler







G(x,x′) =

{+ 1

2π ln |x− x′| n = 2

− 14π |x− x′|−1 n = 3

(2.16)


p(x) =∫

G(x,x′)∇′ · F ′ dV ′ = −∫

F ′ · ∇′G(x,x′) dV ′. (2.17)



Φ =∮

∂Dpn dA (2.18)


G(x,x′) = G(x, 0) + x′ · ∇′G(x, 0) + . . . , (2.19)


p ≈ −∫

F ′ dV ′ · ∇′G(x, 0) = −R · ∇′G(x, 0) as r →∞. (2.20)


G0(x) ≡ G(x, 0) =

{+ 1

2π ln r n = 2

− 14π r−1 n = 3.

(2.21)


p ≈ 12πr

R · xr


R · xr

for n = 3, (2.22)


Thus only R, i.e. the zeroth spatial moment of F , affects the dipolar part of p in the farp ∼ 1/rn−1 ∂D ∼ rn−1

Value of flux integral depends on shape of Deven if D goes to infinity

Corresponding statement true for vortex momentum M contained in D

Incompressible momentum budget is ambiguous!(Lamb, 1879)

Two-dimensional example(Theodorsen 1941)


F F F2b

2a

b! ab = ab" a

Φ→ 0 Φ→ R/2 Φ→ R

Figure 1. Two-dimensional schematic illustrating the dependence of Φ on the shape of D asD →∞. The force field F is indicated by the arrow and the circle has radius L and indicates thesupport of F . The dashed lines indicate different shapes of D. Left: a long and thin rectangleleads to Φ → 0. Middle: a square leads to Φ → R/2. Right: a short and fat rectangle leadsto Φ→ R. In all cases M = R −Φ.

field. In other words, the structure of F (x) is irrelevant, only its total integral matters.The limiting momentum flux Φ is easily computed for a spherically symmetric controlvolume D. For instance, in the case n = 2 we can align R with the x-axis and then usecylindrical coordinates such that (x, y) = (r cos θ, r sin θ) to evaluate (2.18) for a largecircle with radius r ! L as

Φ =∫ 2π

0

R cos θ

2πr(cos θ, sin θ) rdθ =

R

2(1, 0) =

R

2(2.23)

with error O(r−1). A precisely analogous computation in the case n = 3 using sphericalcoordinates yields Φ = R/3 in accordance with (2.10).

The previously mentioned dependence of Φ on the shape of D is easily demonstratedin the case n = 2 by using a rectangular control volume |x| ≤ a and |y| ≤ b (Theodorsen1941). The limit D → ∞ then corresponds to b → ∞ with a/b fixed. With R alignedwith the x-axis as before the limiting y-component of the momentum flux is zero whilstits x-component is

2∫ +b

−bp(a, y) dy =

2R

2π

∫ +b

−b

a

a2 + y2dy =

2R

πarctan(b/a). (2.24)

The previous result Φ = R/2 for a circular control volume agrees with this formula onlyif a = b, which corresponds to a square. If a %= b then the crucial 90o symmetry is lostand (2.24) differs from (2.23).

In particular, if b & a then Φ ≈ 0 whereas if b ! a then Φ ≈ R. In the light ofthe identity (2.9) this shows that for a long and thin rectangle D aligned with R all theexternal momentum input R appears as fluid momentum M and the pressure-inducedmomentum flux Φ to infinity is zero. On the other hand, if the same control rectangle isrotated by 90o then precisely the opposite partition would be observed. This is illustratedin figure 1.

With the pressure now determined the velocity field u at t = 0+ can be computeddirectly from (2.7). Alternatively, u can be computed from ∇×u = ∇×F and ∇·u = 0.In two dimensions this is particularly simple, where it means finding the stream function

Φ =2R

πarctan(b/a)Momentum flux integral for rectangles (a,b):

How to resolve ambiguity?

All choices are consistent with

Two options:

(i) weak compressibility

(ii) Kelvin’s impulse

M + Φ = R

Weak compressibility (paper)

Introduce finite sound speed c

Solve a wave equation for pressure..

Fr = ct

disturbed

undisturbed

...expanding wave front

Selects spherical control volume by physical mechanismexpanding wave front

Kelvin’s impulse (paper)

8 Buhler

ψ such that

u = z ×∇ψ and ∇2ψ = z · ∇× F subject to ∇ψ → 0 as r →∞. (2.25)

This is identical to the pressure equation (2.14) after rotating the F in (2.14) clockwiseby 90o. Therefore the stream function ψ is equal to the pressure field p rotated clockwiseby 90o, i.e.

ψ(x, y) = p(−y, x) (2.26)with a corresponding dipolar far-field expansion for ψ. The momentum content M inD can be written as an integral of ψ over ∂D and the 90o symmetry between ψ and pthus makes clear why in the two-dimensional case M = Φ for all D that also have thissymmetry. An analogous statement holds in the three-dimensional case.

2.5. Kelvin’s impulse, trapped fluid momentum, and their use in experimentsThe ambiguity of the incompressible momentum budget makes it difficult to extract,say, an estimate for R from observations of impulsive fluid motion in an experiment.Fortunately, these difficulties can be side-stepped by using the theory of Kelvin’s hydro-dynamic impulse, which establishes an unambiguous link between R and the impulsivelyforced vorticity field. The vorticity is localized if F is, and it can be measured using parti-cle velocimetry techniques and is thus accessible to experiment. However, this a complexand expensive technique. A simpler and cheaper technique is flow visualization, whichallows direct observation not of the vorticity but of the movement of fluid trapped insidethe vortices. It turns out that the momentum of this trapped fluid can be linked to R aswell and this again allows estimation of R from flow data. Because of their importanceto experiments these concepts are briefly discussed in the present section.

By definition, for n > 1 Kelvin’s hydrodynamic impulse is the rotated first moment ofa localized vorticity field:

I(t) ≡ 1n− 1

∫D

x× (∇× u) dV. (2.27)

Here the requirement on D is that it be large enough to include the support of ∇ × u.Using integration by parts and decay conditions at infinity the rate of change of I subjectto a localized force F is computed as (e.g. Batchelor 1967)

dI

dt=

1n− 1

∫D

x× (∇× F ) dV =∫D

F dV = R. (2.28)

This result holds without ambiguity for n > 1 and for all shapes of D. It is also notrestricted to the linearized equations. In the special case of impulsive forcing from rest(2.28) implies that I(t) = R for all t > 0. From the earlier results it is clear that there isno unique relationship between I and M as D →∞, because M depends on the shapeof D whereas I does not. For spherical control volumes (n− 1)I = nM .

The Kelvin impulse theory shows that the momentum input by an external force canbe determined from the knowledge of the vorticity field alone, i.e. without having tocompute the pressure field at all. This is useful for experiments in which vortices mightbe easier to measure than waves. Using the vorticity distribution to deduce the forcingrecoil R is particularly easy if the vorticity takes a simple form such as a two-dimensionalpropagating dipole or a three-dimensional spherical vortex or vortex ring. For these formsanalytical expressions are known that relate I to a few parameters such as vortex sizeand circulation strength (e.g. Batchelor 1967, §7.2-7.3). For instance, this approach hasbeen used in Drucker & Lauder (1999) to estimate the force balance on a swimming fishby observing the vortex rings in its wake using digital particle image velocimetry.

8 Buhler

ψ such that

u = z ×∇ψ and ∇2ψ = z · ∇× F subject to ∇ψ → 0 as r →∞. (2.25)

This is identical to the pressure equation (2.14) after rotating the F in (2.14) clockwiseby 90o. Therefore the stream function ψ is equal to the pressure field p rotated clockwiseby 90o, i.e.

ψ(x, y) = p(−y, x) (2.26)with a corresponding dipolar far-field expansion for ψ. The momentum content M inD can be written as an integral of ψ over ∂D and the 90o symmetry between ψ and pthus makes clear why in the two-dimensional case M = Φ for all D that also have thissymmetry. An analogous statement holds in the three-dimensional case.

2.5. Kelvin’s impulse, trapped fluid momentum, and their use in experimentsThe ambiguity of the incompressible momentum budget makes it difficult to extract,say, an estimate for R from observations of impulsive fluid motion in an experiment.Fortunately, these difficulties can be side-stepped by using the theory of Kelvin’s hydro-dynamic impulse, which establishes an unambiguous link between R and the impulsivelyforced vorticity field. The vorticity is localized if F is, and it can be measured using parti-cle velocimetry techniques and is thus accessible to experiment. However, this a complexand expensive technique. A simpler and cheaper technique is flow visualization, whichallows direct observation not of the vorticity but of the movement of fluid trapped insidethe vortices. It turns out that the momentum of this trapped fluid can be linked to R aswell and this again allows estimation of R from flow data. Because of their importanceto experiments these concepts are briefly discussed in the present section.

By definition, for n > 1 Kelvin’s hydrodynamic impulse is the rotated first moment ofa localized vorticity field:

I(t) ≡ 1n− 1

∫D

x× (∇× u) dV. (2.27)

Here the requirement on D is that it be large enough to include the support of ∇ × u.Using integration by parts and decay conditions at infinity the rate of change of I subjectto a localized force F is computed as (e.g. Batchelor 1967)

dI

dt=

1n− 1

∫D

x× (∇× F ) dV =∫D

F dV = R. (2.28)

This result holds without ambiguity for n > 1 and for all shapes of D. It is also notrestricted to the linearized equations. In the special case of impulsive forcing from rest(2.28) implies that I(t) = R for all t > 0. From the earlier results it is clear that there isno unique relationship between I and M as D →∞, because M depends on the shapeof D whereas I does not. For spherical control volumes (n− 1)I = nM .

The Kelvin impulse theory shows that the momentum input by an external force canbe determined from the knowledge of the vorticity field alone, i.e. without having tocompute the pressure field at all. This is useful for experiments in which vortices mightbe easier to measure than waves. Using the vorticity distribution to deduce the forcingrecoil R is particularly easy if the vorticity takes a simple form such as a two-dimensionalpropagating dipole or a three-dimensional spherical vortex or vortex ring. For these formsanalytical expressions are known that relate I to a few parameters such as vortex sizeand circulation strength (e.g. Batchelor 1967, §7.2-7.3). For instance, this approach hasbeen used in Drucker & Lauder (1999) to estimate the force balance on a swimming fishby observing the vortex rings in its wake using digital particle image velocimetry.

∇× F "= 0

FExact law for unbounded flow

I(t = 0+) = R

Definition

∇× u = ∇× F

Impulse says nothing about wavesImpulse says everything about recoil

without ambiguity

Impulse can be used to compute recoil forces from vorticity measurements alone

2399Vortex wake and locomotor force in sunfish

plane orientation indicated that circulation magnitude does not

differ significantly between starting and stopping vortices

(d.f.=3 between vortices within planes; P=0.65). The average

of clockwise and counterclockwise vortex strength (Table 1,

!) was therefore used in each plane for calculations involving

total ring circulation (equations 3, 5). Vortex ring momentum

angle (Table 1, "), the angle of inclination of the line

connecting the two vortex centers, became acute within

200–300 ms of the start of the fin beat (Figs 3, 4). Mean

momentum angles measured at the end of the stroke in each

flow plane differed significantly from one another (Bonferroni

multiple-comparison test at #=0.0167), indicating an oblique

orientation of the vortex ring and its central water jet in space.

Velocity profiles and vortex dimensions

Both theory (Milne-Thomson, 1966) and experimental

studies of man-made vortex rings in fluid (Maxworthy, 1977;

Raffel et al., 1998) provide details of wake morphology with

Table 1. Vortex ring measurements from perpendicular flow-field sections of the wake during swimming at 0.5 L s$1

Flow-field plane

Measurement Frontal Parasagittal Transverse F

Ring momentum angle " (degrees) 23.04±3.24 39.70±5.06 81.02±3.47 64.5*** (16)

Jet angle % (degrees) $54.74±3.78 $49.49±6.38 $8.35±2.56 36.1*** (16)

Ring radius R&102 (m) 1.91±0.08 1.61±0.09 1.78±0.07 3.2 (18)

Ring area A&104 (m2) 11.57±0.96 8.29±0.89 10.22±0.89 2.9 (18)

Mean vortex core radius Ro&102 (m)‡ 1.06±0.11 1.13±0.09 1.16±0.13 0.21 (18)

Mean vortex circulation !&104 (m2 s$1)‡ 60.81±3.94 42.59±10.10 60.33±8.11 1.6 (7)

Ring momentum M&105 (kg m s$1) 711.04±140.98 352.29±61.19 616.43±63.22 3.8 (12)

All measurements made at the end of the stroke cycle and are reported as mean ± S.E.M.; N=3–8 per plane.

For " and %, positive and negative values indicate, respectively, angles above and below the horizontal.

‡Average magnitude of measurements for clockwise and counterclockwise vortices within each plane. Nested analyses of variance revealed

no significant differences between vortices within planes (see text).

F-statistics from one-way analyses of variance conducted on variables measured in each of three flow-field planes. d.f. = 2 among planes;

degrees of freedom within planes are shown in parentheses.

Significance was assessed at the Bonferroni-adjusted #=0.0071.

***P<0.001.

L, total body length.

Fig. 4. Fluid vorticity components in three

perpendicular wake planes (A–C) during

swimming at 0.5 L s$1. Vorticity posterior to

the left pectoral fin is shown at the time of

completion of the fin-beat cycle (orientation

and approximate position of flow planes with

respect to the fish are indicated by the axes

at the top; see Fig. 2). Red/orange

coloration represents positive vorticity or

counterclockwise fluid rotation, while

blue/purple colors indicate negative or

clockwise rotation. Green regions reflect a

lack of rotational motion. Arrows between the

pairs of counterrotating vortices show the

direction of the fluid jet present in all three

planes (see velocity vectors in Fig. 3B). This

pattern of vortex flow, generated during the

downstroke, indicates the production of a

single toroidal vortex loop in the wake. F1,

P1, T1, downstroke starting vortices; F2, P2,

T2, downstroke stopping vortices. The scale

bar applies to A–C.

2406 E. G. DRUCKER AND G. V. LAUDER

Fig. 8. Formation of the vortex ring wake over the course of the pectoral fin-beat period at 0.5L s!1. Left column: proposed mechanism for the developmentof wake circulation in the parasagittal plane. The pectoral fin is represented as a mid-chordwise section with estimated angles of attack. The triangle at theend of the section marks the dorsal surface of the leading edge. Red arrows indicate the direction of fin movement. Rotational flows visualized using DPIVare represented by black solid-line arrows. Dashed lines indicate predicted flows attached to the fin but not visualized in this study. The blue arrow signifiesthe orientation of central jet flow observed at the end of the upstroke. All flow structures are shown with the mean free-stream velocity subtracted. See textfor discussion of the dynamics of vortex development. Right column: hypothetical three-dimensional reconstructions of fluid flow based on flow patternsobserved in perpendicular planar sections of the wake (see Fig. 3). Labelled arrows indicate the direction of measured fluid flow. Each fin-beat cycle resultsin the introduction of a single, discrete vortex ring with a central fluid jet into the wake. Timings measured from the onset of fin downstroke (t=0). a,attached leading-edge vortex; b, clockwise flow around fin induced by the acceleration reaction (see text). Other abbreviations as in Figs 4, 6.

2408

high speeds may be necessary to counter the growing negative

lift associated with the fish’s anterior body profile or positive

lift generated posteriorly by oscillation of the tail (Lauder et

al., 1996).

A consistent balance of forces computed from wake

measurements has been elusive for animals locomoting in

fluid. Calculations of lift based on the estimated wake

geometry of birds, for example, have often fallen short of

values necessary to maintain steady flight against the force of

gravity (but see Spedding, 1987). In reconstructing three-

dimensional vortex lines in the wake of flying pigeons

(Columba livia) and jackdaws (Corvus monedula), the elegant

studies of Spedding et al. (1984) and Spedding (1986) showed

that shed vortex rings were paradoxically too small and too

weak to provide weight support. Since ring radii measured in

orthogonal planar sections of the wake were comparable, the

assumption of a more-or-less symmetrical ring is justified.

Non-midline and oblique transections of the vortex loop,

however, are possible errors that could result in deviations in

R and ! to which calculations of momentum are sensitive. Such

types of error are the unavoidable consequence of studying

unsteady fluid flows produced by freely moving animals in

which the shape, orientation and path of travel of the wake are

out of the experimenter’s control. For a fish swimming by axial

undulation, Müller et al. (1997) recently found that thrust

power calculated from flow velocities in frontal-plane

transections of the tail’s wake exceeded that expected from

theory by a factor of three. In this case, circular ring geometry

was selected in the absence of complete three-dimensional

flow-field information. Since many variations of the vortex

ring morphology are theoretically possible from oscillating

biofoils (Brodsky, 1994; Rayner, 1995; Dickinson, 1996), the

assumption of a particular wake structure based on single-plane

views of downstream flow may lead to inaccurate estimations

of propulsive force.

For some swimming animals, the locomotor force balance

has been proposed to involve an unsteady ‘squeeze’

mechanism for the supplementation of thrust. Acceleration of

the animal’s propulsor against the body at the end of the

propulsive stroke is thought to eject a rearward jet of water

whose reaction provides a forward force (Daniel and

Meyhöfer, 1989). This mechanism has been cited for the

pectoral fins of fish as a possible means of increasing the total

thrust generated per fin beat (Geerlink, 1983). For the bluegill

sunfish, however, the pectoral fins are adducted against the

body at the end of the upstroke at comparatively low speed. In

addition, direct observations of particle motion in the wake of

the pectoral fins at the end of the upstroke do not reveal the

presence of a strictly downstream-oriented jet (Fig. 3B). For

these reasons, we conclude that, for pectoral fin propulsion in

sunfish, the squeeze force does not substantially augment thrust

produced by vortex ring shedding.

In the present study, locomotor forces were calculated as

averages over the course of the fin-stroke duration. Upon

completion of the upstroke, remaining bound vorticity is shed

from the fin in the form of stopping vortices that migrate away

from their starting vortex counterparts (e.g. Fig. 9B,C, P3 and

P4). By the end of the kinematic pause period, free

counterrotating vortices are no longer in close proximity.

Accordingly, the mutual antagonism of regions of opposite-

sign vorticity in the wake, as described by the Wagner effect,

is much reduced, and circulation attains its full steady-state

magnitude (Dickinson and Götz, 1996, p. 2103). Forces

estimated over T are therefore the most informative for

resolution of the overall force balance on the animal. However,

forces acting on a swimming fish are undoubtedly out of

equilibrium at any particular moment during the stride. At high

swimming speeds, for example, sunfish generate positive lift

on the pectoral fin downstroke and negative lift on the

upstroke, as indicated by the tilted orientation of vortex rings

in the wake (Fig. 6D). DPIV analysis of flow-field patterns

over increasingly small time intervals throughout the fin-beat

period would allow estimation of instantaneous locomotor

forces (equation 4), a necessary step in determining the time

course of force development for labriform swimming.

Comparisons with other animals moving through fluids

Animals locomoting in air must continuously generate lift,

a constraint apparent in the form of their wake. For many

vertebrates (Kokshaysky, 1979; Spedding et al., 1984;

Spedding, 1986) and insects (Grodnitsky and Morozov, 1992,

1993; Dickinson and Götz, 1996), vorticity is shed

predominantly on the wing downstroke. Vortex ring

momentum angles are quite shallow (e.g. Spedding et al.,

1984), reflecting an earthward fluid jet. The upstroke of these

animals is thought to perform little or no useful aerodynamic

function (Rayner, 1979a; but see Brodsky, 1994; Ellington,

1995). In contrast, buoyant animals swimming in water are

E. G. DRUCKER AND G. V. LAUDER

Fig. 10. Summary of empirically determined hydrodynamic force

balance on bluegill sunfish swimming at 0.5 L s"1. Reported thrust

and lift values are the mean reaction forces experienced by both left

and right pectoral fins together averaged over the fin-stroke period.

These propulsive forces calculated from wake velocity data are not

significantly different from the measured resistive forces of total

body drag and weight (Table 2). A large medially directed reaction

force (reported per fin) may be involved in maintaining body

stability during locomotion.

Weight

3.37 mN

Lift3.24 mN

Thrust

11.12 mN

Drag

10.58 mN

Medial6.96 mN

Measuring the forces exerted by organisms on the fluids

through which they move is a difficult proposition. On land,

devices such as force plates allow direct measurement of the

force applied by locomoting animals to the ground, but in the

aquatic and aerial media, such devices are not applicable. As

a result, a number of alternative approaches have been used in

an effort to understand the mechanisms by which organisms

propel themselves in fluids. For example, considerable effort

has been expended in developing models of the process by

which swimming and flying organisms generate locomotor

force. Early theoretical models involved steady- or quasi-

steady-state analyses that employed time-invariant lift and

drag coefficients (e.g. Weis-Fogh, 1973; Blake, 1983). For

swimming animals, such locomotor mechanisms have been

termed ‘lift-based’ or ‘drag-based’ (for reviews, see Webb

and Blake, 1985; Vogel, 1994). More recent models have

elaborated a vortex theory of propulsion based on time-

dependent circulatory forces rather than estimated force

coefficients (Rayner, 1979a,b; Ellington, 1984b,c). Unsteady

mechanisms, such as the ‘clap and fling’ (Lighthill, 1973;

Weis-Fogh, 1973), take into account rotation as well as

translation of the propulsor and acknowledge the complex time

history of force development (see also Dickinson, 1994;

Dickinson and Götz, 1996). Recently, the use of

supercomputers to solve the unsteady Navier–Stokes equations

for numerous grid points around a model animal has provided

insights into the dynamics of locomotor force production

(Carling et al., 1994, 1998; Liu et al., 1997, 1998). By

deforming the model animal, a time-dependent estimate of the

forces exerted on the fluid can be calculated. Despite these

advances in computational approaches, however, the difficulty

in empirically measuring forces exerted by freely moving

2393The Journal of Experimental Biology 202, 2393–2412 (1999)

Printed in Great Britain © The Company of Biologists Limited 1999

JEB2130

Quantifying the locomotor forces experienced by

swimming fishes represents a significant challenge because

direct measurements of force applied to the aquatic

medium are not feasible. However, using the technique of

digital particle image velocimetry (DPIV), it is possible to

quantify the effect of fish fins on water movement and

hence to estimate momentum transfer from the animal to

the fluid. We used DPIV to visualize water flow in the

wake of the pectoral fins of bluegill sunfish (Lepomis

macrochirus) swimming at speeds of 0.5–1.5L s!!1, where L

is total body length. Velocity fields quantified in three

perpendicular planes in the wake of the fins allowed three-

dimensional reconstruction of downstream vortex

structures. At low swimming speed (0.5L s!!1), vorticity is

shed by each fin during the downstroke and stroke reversal

to generate discrete, roughly symmetrical, vortex rings of

near-uniform circulation with a central jet of high-velocity

flow. At and above the maximum sustainable labriform

swimming speed of 1.0L s!!1, additional vorticity appears on

the upstroke, indicating the production of linked pairs of

rings by each fin. Fluid velocity measured in the vicinity of

the fin indicates that substantial spanwise flow during the

downstroke may occur as vortex rings are formed. The

forces exerted by the fins on the water in three dimensions

were calculated from vortex ring orientation and

momentum. Mean wake-derived thrust (11.1 mN) and lift

(3.2 mN) forces produced by both fins per stride at 0.5L s!!1

were found to match closely empirically determined

counter-forces of body drag and weight. Medially directed

reaction forces were unexpectedly large, averaging 125 %

of the thrust force for each fin. Such large inward forces

and a deep body that isolates left- and right-side vortex

rings are predicted to aid maneuverability. The observed

force balance indicates that DPIV can be used to measure

accurately large-scale vorticity in the wake of swimming

fishes and is therefore a valuable means of studying

unsteady flows produced by animals moving through

fluids.

Key words: swimming, pectoral fin locomotion, flow visualization,

hydrodynamics, digital particle image velocimetry, bluegill sunfish,

Lepomis macrochirus.

Summary

Introduction

LOCOMOTOR FORCES ON A SWIMMING FISH: THREE-DIMENSIONAL VORTEX

WAKE DYNAMICS QUANTIFIED USING DIGITAL PARTICLE IMAGE

VELOCIMETRY

ELIOT G. DRUCKER* AND GEORGE V. LAUDER

Department of Ecology and Evolutionary Biology, University of California, Irvine, CA 92697, USA

*e-mail: [email protected]

Accepted 14 June; published on WWW 25 August 1999

PIV measurements and vortex ring approximation

Summary for unbounded flows with spherical control volumes

1 0 n=11/2 1/2 n=21/3 2/3 n=3

Wave Vortex

Φ M

Now: adapt classical impulse theory to free-surface flow and apply to water

strider

Free-surface flow

hF

z = 0p = 0 at z = 0

Flat surface is at constant air pressure during impulsive forcing

Consider only horizontal force at depth h

Vertical force is a little different (see paper)

∇× u = ∇× F as before, but pressure finder must be modified for new boundary condition

Simplest case: perfectly localized force field

14 Buhler

front arrives, and then immediately jumps down to negative infinity. Thereafter it grad-ually asymptotes towards its final positive value (cf. figure 2). Of course, these localizedinfinities are caused by the delta function form of the initial conditions and they dis-appear after convolution with smooth initial data, as shown in the figure. Nevertheless,they are indicative of the non-uniform and oscillatory manner in which the momentumis distributed in the two-dimensional case.

4. Free surface flowWe return to three-dimensional incompressible flow but restrict its extent to be semi-

infinite. Specifically, we add a free surface whose undisturbed rest location in Cartesiancoordinates (x, y, z) is at z = 0, say, were z is increasing upwards and gravity pointsdownwards. There may also be surface tension but neither surface tension nor gravitycan affect the impulsive response of the fluid to localized forcing because both are relatedto fluid particle displacements, which are negligible during the impulsive forcing.

On the other hand, the impulsive fluid response is profoundly affected by the dynamicalboundary condition at the free surface, which for a flat surface is p = 0. In the case ofhorizontal fluid forcing (i.e. forcing parallel to the free surface) it will be seen that thepressure field now decays sufficiently fast with distance from the forcing site to make therelevant momentum flux integral absolutely convergent and approaching zero as D →∞.This implies that M(0+) = R for horizontal forcing without ambiguity. The case ofvertical forcing is different and more similar to the forcing of an unbounded fluid in thatthe momentum budget again depends on the shape of D as D →∞.

The presence of the free surface admits surface waves, which are excited because theimpulsively forced fluid has a nonzero vertical velocity at z = 0 and therefore disturbsthe flat surface. Just as in the compressible case the impulsively generated initial stateadjusts to a final steady state via the emission and outward propagation of surface waves,which carry momentum away from the site of the initial forcing. The only differencecompared to the compressible case is the dispersive nature of these waves. However, thefinal adjusted state does not depend on the detailed nature of the waves (which implies,for instance, that surface tension modifies the wave pattern but not the final adjustedstate).

The general solution to impulsive forcing with a free surface will be described in thefollowing sections, which sets the scene for the discussion of water strider locomotion in§5.

4.1. Impulsive pressure response with free surfaceWe consider an impulsive force field localized at x = y = 0 and depth z = −h whereh > 0. Specifically,

F (x, t) = F (x)δ(t) = Rδ(x)δ(y)δ(z + h)δ(t). (4.1)

The impulsive pressure field p = p(x)δ(t) satisfies

∇2p = ∇ · F = R · ∇ δ(x)δ(y)δ(z + h) subject to

{p = 0 at z = 0

∇p→ 0 as r →∞ . (4.2)

The corresponding free-surface Green’s function G(x,x′) is the solution to (4.2) withright-hand side equal to δ(x−x′). It is equal to the unbounded Green’s function G(x,x′)for n = 3 in (2.16) provided an equal-and-opposite image source term is added at themirrored location (x′, y′,−z′) to ensure that the boundary condition G(x, y, 0,x′) = 0 is

14 Buhler

front arrives, and then immediately jumps down to negative infinity. Thereafter it grad-ually asymptotes towards its final positive value (cf. figure 2). Of course, these localizedinfinities are caused by the delta function form of the initial conditions and they dis-appear after convolution with smooth initial data, as shown in the figure. Nevertheless,they are indicative of the non-uniform and oscillatory manner in which the momentumis distributed in the two-dimensional case.

4. Free surface flowWe return to three-dimensional incompressible flow but restrict its extent to be semi-

infinite. Specifically, we add a free surface whose undisturbed rest location in Cartesiancoordinates (x, y, z) is at z = 0, say, were z is increasing upwards and gravity pointsdownwards. There may also be surface tension but neither surface tension nor gravitycan affect the impulsive response of the fluid to localized forcing because both are relatedto fluid particle displacements, which are negligible during the impulsive forcing.

On the other hand, the impulsive fluid response is profoundly affected by the dynamicalboundary condition at the free surface, which for a flat surface is p = 0. In the case ofhorizontal fluid forcing (i.e. forcing parallel to the free surface) it will be seen that thepressure field now decays sufficiently fast with distance from the forcing site to make therelevant momentum flux integral absolutely convergent and approaching zero as D →∞.This implies that M(0+) = R for horizontal forcing without ambiguity. The case ofvertical forcing is different and more similar to the forcing of an unbounded fluid in thatthe momentum budget again depends on the shape of D as D →∞.

The presence of the free surface admits surface waves, which are excited because theimpulsively forced fluid has a nonzero vertical velocity at z = 0 and therefore disturbsthe flat surface. Just as in the compressible case the impulsively generated initial stateadjusts to a final steady state via the emission and outward propagation of surface waves,which carry momentum away from the site of the initial forcing. The only differencecompared to the compressible case is the dispersive nature of these waves. However, thefinal adjusted state does not depend on the detailed nature of the waves (which implies,for instance, that surface tension modifies the wave pattern but not the final adjustedstate).

The general solution to impulsive forcing with a free surface will be described in thefollowing sections, which sets the scene for the discussion of water strider locomotion in§5.

4.1. Impulsive pressure response with free surfaceWe consider an impulsive force field localized at x = y = 0 and depth z = −h whereh > 0. Specifically,

F (x, t) = F (x)δ(t) = Rδ(x)δ(y)δ(z + h)δ(t). (4.1)

The impulsive pressure field p = p(x)δ(t) satisfies

∇2p = ∇ · F = R · ∇ δ(x)δ(y)δ(z + h) subject to

{p = 0 at z = 0

∇p→ 0 as r →∞ . (4.2)

The corresponding free-surface Green’s function G(x,x′) is the solution to (4.2) withright-hand side equal to δ(x−x′). It is equal to the unbounded Green’s function G(x,x′)for n = 3 in (2.16) provided an equal-and-opposite image source term is added at themirrored location (x′, y′,−z′) to ensure that the boundary condition G(x, y, 0,x′) = 0 is


satisfied. The result is

G(x,x′) = G(x, x′, y′, z′)−G(x, x′, y′,−z′), (4.3)

which depends on the horizontal components of x−x′ and on both z and z′. Convolutionwith the specific source term in (4.2) yields

p =∫

G(x,x′)R · ∇′ δ(x′)δ(y′)δ(z′ + h) dV ′ = −R · ∇′ G(x, 0, 0,−h), (4.4)

where the last term denotes the gradient of G with respect to x′ at the source location.We consider separately the pressure field due to horizontal forces and that due to

vertical forces. Let Rh = Rhx and Rv = Rvz denote the horizontal and vertical partsof R = Rh + Rv and similarly let the pressure be p = ph + pv. For horizontal forcing(4.4) can be simplified by using the symmetry −∇′G = +∇G and the short-hand (2.21),which leads to

ph = +Rh∂

∂xG(x, 0, 0,−h) = Rh

∂

∂x(G0(x, y, z + h)−G0(x, y, z − h)). (4.5)

For vertical forcing the z′-derivative of the Green’s function G leads to

pv = Rv∂

∂z(G0(x, y, z + h) + G0(x, y, z − h)). (4.6)

Specifically, the three-dimensional pressure ph is

ph = −Rh

4π

∂

∂x

{1√

x2 + y2 + (z + h)2− 1√

x2 + y2 + (z − h)2

}(4.7)

and pv is

pv = −Rv

4π

∂

∂z

{1√

x2 + y2 + (z + h)2+

1√x2 + y2 + (z − h)2

}. (4.8)

It is important that ph and pv differ markedly in the far field r " h. The far-field formof ph is computed from (4.7) by Taylor-expanding in h, which yields

ph ≈ −h3Rh

2π

xz

r5. (4.9)

This decays rapidly as O(r−3) and therefore the net momentum flux Φ = 0 as D →∞.In other words, the impulsive fluid response to horizontal forcing in the presence of afree surface is entirely confined to fluid acceleration, i.e. M(0) = Rh for sufficiently largecontrol volumes D. This hold without ambiguity for all shapes of D.

On the other hand, pv has a dipolar far-field form

pv ≈ 2Rv

4π

z

r3, (4.10)

which does not depend on the forcing depth h. This pressure field is exactly twice thethree-dimensional pressure due to impulsive forcing in an unbounded domain as com-puted in (2.22). Integrating pv over a hemispheric control volume D in the lower halfplane thus yields the same result as before, namely Φ = Rv/3. This result also dependson the shape of D. In summary, we have obtained that for hemispherical control volumes

M(0+) = Rh +23Rv and Φ = 0Rh +

13Rv (4.11)

as D →∞.



G(x,x′) = G(x, x′, y′, z′)−G(x, x′, y′,−z′), (4.3)


p =∫




ph = +Rh∂

∂xG(x, 0, 0,−h) = Rh

∂

∂x(G0(x, y, z + h)−G0(x, y, z − h)). (4.5)


pv = Rv∂

∂z(G0(x, y, z + h) + G0(x, y, z − h)). (4.6)


ph = −Rh

4π

∂

∂x

{1√

x2 + y2 + (z + h)2− 1√

x2 + y2 + (z − h)2

}(4.7)

and pv is

pv = −Rv

4π

∂

∂z

{1√

x2 + y2 + (z + h)2+

1√x2 + y2 + (z − h)2

}. (4.8)


ph ≈ −h3Rh

2π

xz

r5. (4.9)



pv ≈ 2Rv

4π

z

r3, (4.10)


M(0+) = Rh +23Rv and Φ = 0Rh +

13Rv (4.11)

as D →∞.



G(x,x′) = G(x, x′, y′, z′)−G(x, x′, y′,−z′), (4.3)


p =∫




ph = +Rh∂

∂xG(x, 0, 0,−h) = Rh

∂

∂x(G0(x, y, z + h)−G0(x, y, z − h)). (4.5)


pv = Rv∂

∂z(G0(x, y, z + h) + G0(x, y, z − h)). (4.6)


ph = −Rh

4π

∂

∂x

{1√

x2 + y2 + (z + h)2− 1√

x2 + y2 + (z − h)2

}(4.7)

and pv is

pv = −Rv

4π

∂

∂z

{1√

x2 + y2 + (z + h)2+

1√x2 + y2 + (z − h)2

}. (4.8)


ph ≈ −h3Rh

2π

xz

r5. (4.9)



pv ≈ 2Rv

4π

z

r3, (4.10)


M(0+) = Rh +23Rv and Φ = 0Rh +

13Rv (4.11)

as D →∞.

Method of images for free-surface Green’s function:

Result Far-field expansion

...decays very fast at infinity!

Φ = 0 All momentum is in the vortex at t=0+

Heuristic explanation: opposing image force system

16 Buhler

hF

F ′

z = 0

F

F ′

Figure 3. Image force system for computing the impulsive pressure field due to horizontal andvertical forcing. The physical force F centred at z = −h is paired with an image force F ′ centredat z = h such that ∇ · F is equal and opposite to ∇ · F ′. Left: the image of a horizontal forcepoints in the opposite direction and the sum is zero. Right: the image of a vertical force pointsin the same direction and the sum is twice the physical force.

The differences in the fluid response to horizontal and vertical forcing can also beunderstood qualitatively by inspection of the image sources that were used in the con-struction of G. More precisely, the image sources can be thought of as arising from animage force field in the upper half plane z > 0 that has the required equal-and-oppositedivergence field when compared to the physical force field in z < 0; see figure 3. Theimpulsive pressure in z < 0 can then be thought of as arising from the combined imageand physical force system in a an unbounded fluid.

By inspection, the correct image force field has horizontal components that are sign-reversed compared to the horizontal components of the physical force field and verticalcomponents that are identical. Therefore the integral over both the physical and theimage force field has zero horizontal component and doubled vertical component. It is thisintegral which controls the far-field pressure field. This makes obvious why ph = O(r−3)whilst pv has the dipolar, O(r−2) form that is twice the expression for unbounded flow.These considerations also explain why the far-field of ph is proportional to h and vanishesas h→ 0. This is because h→ 0 corresponds to a coalescence of the opposing horizontalcomponents of the image and physical force fields.

4.2. Impulsive velocity field and flow adjustmentThe impulsive velocity field at t = 0+ is again given by (2.7), i.e. u(x, 0+) = F −∇p.Of particular interest is the velocity at the free surface z = 0 because it determines thegeneration of surface waves. The horizontal components u and v at z = 0 are in fact zerobecause the pressure is horizontally homogeneous by the free-surface boundary condition.However, the vertical velocity w is not zero at the free surface. Specifically,

w = −∂p

∂z= −∂ph

∂z− ∂pv

∂zat z = 0, (4.12)

and substitution of (4.7) and (4.8) yields the contributions

− ∂ph

∂z

∣∣∣∣z=0

=3Rh

2π

hx

(x2 + y2 + h2)5/2(4.13)

This image force ensures p=0 at z=0

Net physical + image momentum equals zero,and this determinesfar-field pressure

0 1 ??

Wave adjustment to balanced state

But: vertical velocity at surface is not zero at t=0+

16 Buhler

hF

F ′

z = 0

F

F ′

Figure 3. Image force system for computing the impulsive pressure field due to horizontal andvertical forcing. The physical force F centred at z = −h is paired with an image force F ′ centredat z = h such that ∇ · F is equal and opposite to ∇ · F ′. Left: the image of a horizontal forcepoints in the opposite direction and the sum is zero. Right: the image of a vertical force pointsin the same direction and the sum is twice the physical force.

The differences in the fluid response to horizontal and vertical forcing can also beunderstood qualitatively by inspection of the image sources that were used in the con-struction of G. More precisely, the image sources can be thought of as arising from animage force field in the upper half plane z > 0 that has the required equal-and-oppositedivergence field when compared to the physical force field in z < 0; see figure 3. Theimpulsive pressure in z < 0 can then be thought of as arising from the combined imageand physical force system in a an unbounded fluid.

By inspection, the correct image force field has horizontal components that are sign-reversed compared to the horizontal components of the physical force field and verticalcomponents that are identical. Therefore the integral over both the physical and theimage force field has zero horizontal component and doubled vertical component. It is thisintegral which controls the far-field pressure field. This makes obvious why ph = O(r−3)whilst pv has the dipolar, O(r−2) form that is twice the expression for unbounded flow.These considerations also explain why the far-field of ph is proportional to h and vanishesas h→ 0. This is because h→ 0 corresponds to a coalescence of the opposing horizontalcomponents of the image and physical force fields.

4.2. Impulsive velocity field and flow adjustmentThe impulsive velocity field at t = 0+ is again given by (2.7), i.e. u(x, 0+) = F −∇p.Of particular interest is the velocity at the free surface z = 0 because it determines thegeneration of surface waves. The horizontal components u and v at z = 0 are in fact zerobecause the pressure is horizontally homogeneous by the free-surface boundary condition.However, the vertical velocity w is not zero at the free surface. Specifically,

w = −∂p

∂z= −∂ph

∂z− ∂pv

∂zat z = 0, (4.12)

and substitution of (4.7) and (4.8) yields the contributions

− ∂ph

∂z

∣∣∣∣z=0

=3Rh

2π

hx

(x2 + y2 + h2)5/2(4.13)

w(x, y, z = 0, t = 0+) =3Rh

2π

hx

(x2 + y2 + h2)5/2

This leads to surface waves, which propagate away from the forcing site.They leave behind a balanced vortex flow


− ∂pv

∂z

∣∣∣∣z=0

= −Rv

2π

1(x2 + y2 + h2)3/2

+3Rv

2π

h2

(x2 + y2 + h2)5/2, (4.14)

where Rh is aligned with the x-axis as before. Thus, horizontal forcing leads a dipolarstructure with rising motion ahead of the forcing region and sinking motion behindit. Vertical forcing leads to a monopolar, axisymmetric structure that changes sign athorizontal distance

√x2 + y2 =

√2h such that for negative Rv (i.e. downward forcing)

there is sinking motion in a small inner region and rising motion elsewhere. In the presenceof both horizontal and vertical forcing the far-field w is dominated by the term withslowest decay, which is the monopolar term proportional to Rv.

The presence of nonzero w at the surface implies the impulsive generation of surfacewaves, which subsequently propagate away from the site of the forcing. Just as in thecompressible case these waves can carry momentum out of a control volume D centredat the forcing site. The only difference is that surface waves are dispersive whereas soundwaves are not. This raises the question whether all waves will propagate away from theforcing site as t → ∞. This will be the case if the group velocity for all wavenumbersis nonzero and this is indeed satisfied for surface waves under the influence of gravityand surface tension (see (4.20)). Thus, the flow can be expected to adjust to a steadybalanced flow by the emission and propagation of unsteady surface waves. This steadyflow is entirely controlled by the vorticity distribution and it is therefore independent ofthe details of the waves such as their dispersion relation (apart from the nonzero groupvelocity requirement). For instance, the presence of surface tension modifies the unsteadywave field but not the steady balanced flow.

The balanced flow can be computed directly by writing the velocity field explicitly asthe sum

u = ub + uw (4.15)of a balanced and a wave part such that ∇×ub = ∇×F and ∇×uw = 0. In addition, theboundary condition for the balanced flow is wb = 0 at z = 0. With this definition, plussuitable decay conditions at infinity, the balanced flow is a well-defined steady solutionof the linear equations. As before, ub can be viewed as the least-square projection of Fonto non-divergent vector fields subject to the additional constraint wb = 0 at z = 0. Fora localized force field the balanced flow is essentially that of a vortex ring centred at theforcing position and aligned with R.

To compute directly the net momentum transported away by the surface waves ismore complicated than in the compressible case because of wave dispersion. However,this complication can be side-stepped by considering a thought experiment in which thebalanced flow itself is set up impulsively and therefore its momentum can be computeddirectly by the methods of §2.4. Since the momentum of a given flow does not dependon how the flow has been set up this establishes the momentum of the balanced flow.

To this end the boundary condition at z = 0 is changed from free-surface type (i.e.p = 0) to rigid-lid type (i.e. w = 0). For horizontal forcing this is equivalent to an imageforce in the upper half plane that points in the same direction as F . This is as in the leftpanel of figure 3, but with both arrows now pointing in the same direction. This makesclear that the impulsive pressure in the far field is twice that of a horizontal F alonein an unbounded domain and therefore the net flux of horizontal momentum across alarge hemispheric control volume is equal to the usual Rh/3 in three dimensions. Thisconstruction shows that the momentum of the balanced flow due to horizontal forcinghas the value 2Rh/3 relative to a large hemispherical control volume.

For vertical forcing it is easiest to imagine a solid wall at z = 0 during the impulsiveforcing. By continuity it is then kinematically impossible to have any vertical movement


− ∂pv

∂z

∣∣∣∣z=0

= −Rv

2π

1(x2 + y2 + h2)3/2

+3Rv

2π

h2

(x2 + y2 + h2)5/2, (4.14)


√x2 + y2 =










− ∂pv

∂z

∣∣∣∣z=0

= −Rv

2π

1(x2 + y2 + h2)3/2

+3Rv

2π

h2

(x2 + y2 + h2)5/2, (4.14)


√x2 + y2 =









Balanced boundary condition:

F = Rhxδ(x)δ(y)δ(z + h)

From pressure solution:

Irrotational surface wave dynamics

Surface elevation:

18 Buhler

of the centre of mass of the incompressible fluid in z < 0, and this implies that the mo-mentum of the balanced flow due to vertical forcing must be precisely zero. In summary,we have obtained that

M(0+) = Rh +23Rv and M(+∞) =

23Rh + 0Rv. (4.16)

In other words, during the adjustment process the surface waves carry away 1/3 of thehorizontal momentum Rh and 2/3 of the vertical momentum Rv.†

4.3. Dynamics of surface wavesFor completeness it is briefly described how the detailed surface-wave dynamics duringthe adjustment process can be computed. This is useful for flow observations and for theexperimental water strider considerations discussed in §5.3.

The irrotational velocity component related to the waves is governed by the verticalsurface undulation ζ(x, y, t) defined such that ζ = 0 corresponds to a flat surface. Theinitial-value problem for the surface waves after impulsive forcing can be written in termsof the already computed vertical velocity at the surface as

ζ(x, y, 0) = 0 and ζt(x, y, 0) = (4.13) + (4.14). (4.17)

The wave solution in time is computed using a two-dimensional Fourier transform for ζsuch that

ζ(x, y, t) =1

(2π)2

∫ ∫ζ(k, l, t) exp(i[kx + ly]) dkdl. (4.18)

For a plane wave solution ζ ∝ exp(i[kx + ly − ωt]) with given wavenumber vector k =(k, l) the frequency ω(k) is determined by the dispersion relation. For surface wavesthere are two branches ±ω(k), where ω(k) = ω(−k) ≥ 0 is the positive branch of thedispersion relation. This choice is convenient because for surface waves the dispersionrelation depends only on κ = |k| and thus using the positive dispersion relation ω(κ) ≥ 0is natural. Using the reality condition the general solution for ζ is

ζ(k, t) = ζ(k, 0) cos ωt + ζt(k, 0)1ω

sinωt (4.19)

Specifically, for deep-water waves with surface tension the positive dispersion relation is

ω(κ) = +√

gκ + γκ3 and thereforedω

dκ=

12

g + 3γκ2√gκ + γκ3

(4.20)

where g is gravity and γ is the surface tension divided by the water density. As notedbefore, this dispersion relation has the required nonzero group-velocity property for allwavenumbers.

5. Water strider locomotion5.1. Observations and theory

The water strider is a small insect that can stand and move on water by exploiting thesurface tension of the air–water interface. The typical insect has a core body size of about1 cm and supports its weight on thin long hydrophobic legs that bend the water surface.It propels itself forward by a rapid sculling motion of its central pair of legs. The staticforce balance of a water strider standing on water is very well understood (e.g. Keller

† In the two-dimensional version of (4.16) the 2/3 is replaced by 1/2.

18 Buhler


M(0+) = Rh +23Rv and M(+∞) =

23Rh + 0Rv. (4.16)




ζ(x, y, 0) = 0 and ζt(x, y, 0) = (4.13) + (4.14). (4.17)


ζ(x, y, t) =1

(2π)2



ζ(k, t) = ζ(k, 0) cos ωt + ζt(k, 0)1ω

sinωt (4.19)


ω(κ) = +√


dκ=

12


(4.20)





w(x, y, z = 0, t = 0+)Initial conditions

18 Buhler


M(0+) = Rh +23Rv and M(+∞) =

23Rh + 0Rv. (4.16)




ζ(x, y, 0) = 0 and ζt(x, y, 0) = (4.13) + (4.14). (4.17)


ζ(x, y, t) =1

(2π)2



ζ(k, t) = ζ(k, 0) cos ωt + ζt(k, 0)1ω

sinωt (4.19)


ω(κ) = +√


dκ=

12


(4.20)





18 Buhler


M(0+) = Rh +23Rv and M(+∞) =

23Rh + 0Rv. (4.16)




ζ(x, y, 0) = 0 and ζt(x, y, 0) = (4.13) + (4.14). (4.17)


ζ(x, y, t) =1

(2π)2



ζ(k, t) = ζ(k, 0) cos ωt + ζt(k, 0)1ω

sinωt (4.19)


ω(κ) = +√


dκ=

12


(4.20)





18 Buhler


M(0+) = Rh +23Rv and M(+∞) =

23Rh + 0Rv. (4.16)




ζ(x, y, 0) = 0 and ζt(x, y, 0) = (4.13) + (4.14). (4.17)


ζ(x, y, t) =1

(2π)2



ζ(k, t) = ζ(k, 0) cos ωt + ζt(k, 0)1ω

sinωt (4.19)


ω(κ) = +√


dκ=

12


(4.20)





Surface gravity waves with surface tension

Nonzero group velocity: all waves get away & balanced vortex flow is left behind

Other wave details are irrelevant! (eg surface tension,...)

Flow adjustment problem

Impulsive forcing t=0+

M = R_h

Wave emission and

propagation

M changes

Adjusted balanced state

M = ?

very fastt=0.01 sec

fastt=0.1 sec

slowt=1 sec

Wave picture using typical water strider parameters

forcing depth h=0.1 cm(meniscus depth)

standard gravity

standard water density, surface tension, and viscosity

colormap not constant

time sequence up to t=0.04 sec

horizontal domain 6x6 cm

Detailed snapshotImpulsive fluid forcing and water strider locomotion 21

x

y

Surface elevation at t = 0.04

-2 -1 0 1 2 3

-2

-1

0

1

2

3

-0.01

-0.005

0

0.005

0.01

-3 -2 -1 0 1 2 3-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

x

!

Surface elevation on centre line y=0 at time t = 0.04

Figure 4. Numerical solution of deep-water system with surface tension subject to impulsiveforcing using the initial conditions (5.1). The parameters are Rh = 0.5 cm4 s−1 and h = 0.1 cm.Left: surface elevation ζ at time t = 0.04 in cgs units, showing large-scale dipolar structure andsmall-scale ripples that act as forerunners. Right: elevation on y = 0 at the same time (cf.figure 2).

surface tension divided by water density γ = 74 cm3 s−2, and the kinematic viscosity ofwater ν = 0.01 cm2 s−1. The viscosity leads to dissipation of kinetic energy and hence todamping of plane surface waves with exponential rate 2νκ2, where κ =

√k2 + l2 is the

magnitude of the horizontal wavenumber vector k = (k, l) (e.g Lamb 1932, page 625).For weakly dissipating plane waves (i.e. waves whose damping rate is small compared totheir frequency) the only modification in the solution to the non-dissipative initial-valueproblem described in §4.3 is to multiply the undamped spectral solution ζ(k, t) by thedecay factor exp(−2νκ2t). This is sufficient for the problem at hand. Roughly speaking,the surface wave evolution occurs over the time scale of a few hundredth of second andover this short time scale viscous dissipation acts significantly only on waves with scalesof a millimetre or less.

The initial condition for the surface waves is an undisturbed surface and a verticalsurface velocity given by (4.13), which yields

ζ(x, y, 0) = 0 and ζt(x, y, 0) =3Rh

2π

hx

(x2 + y2 + h2)5/2(5.1)

if the leg stroke direction is aligned with the x-axis. Here Rh is the net momentumtransfer per unit mass into the fluid. The insect recoil is therefore −ρRh, where ρ is thedensity of water, which is unity in cgs units.

The initial-value problem is solved numerically using a doubly periodic square domainwith side length 6 cm and up to 512 spectral modes, which is sufficient to computequantities such as the maximum surface elevation with one per cent error. A typicalimage of the surface elevation ζ(x, t) is illustrated in figure 4. The dominant structureis an expanding dipolar surface wave with raised surface in the direction of forcing. Thehorizontal scale of the dipole is much bigger than h and the dipole is preceded by small-scale ripples, which have appreciable magnitude on the scale of h. For the time shown,these small-scale ripples have been significantly affected by the viscous dissipation butthe dipole has not. The overall dynamics of the wave pattern is dominated by surfacetension and gravity plays a lesser role. This is important for the maximum wave heightdiscussed next.

Structure of balanced flow


x

y

Balanced surface stream function at time t = 0.5s

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

-1.5

-1

-0.5

0

0.5

1

1.5

x

y

Balanced surface velocity potential at time t = 0.5s

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Figure 5. A numerical solution to (5.9) is evaluated for the horizontal balanced flow in thefree surface z = 0 at time t = 0.5 s. The physical parameters are as before. Displayed are thestream function ψ and velocity potential φ such that (ub, vb) = (φx − ψy, φy + ψx) at z = 0.Left: the stream function ψ exhibits a typical vortex-pair structure that corresponds to vortexpropagation to the right. Right: the velocity potential φ indicating convergence and hencesinking motion just below the surface in x > 0 and a compensating divergence region and risingmotion in x < 0.

the three-dimensional rolling motion induced by an infinitesimal vortex ring submergedat depth h. Despite this consistency, however, such a vortex flow structure is not observedin experiments.

This is because the vortex flow is observable for a fraction of a second and this is a longenough time interval such that nonlinear and viscous effects must be taken into account.Indeed, in nonlinear viscous theory the vortex ring self-advects itself nonlinearly in the x-direction whilst its vorticity and velocity structure rapidly diffuses outward. These effectsare easily observable in the water strider experiments in Hu et al. (2003). Nonlinearitycan not be treated here, but viscous diffusion is easily incorporated into the balancedflow evolution by letting ub diffuse according to

∂ub

∂t= ν∇2ub and

(∂ub

∂z,∂vb

∂z, wb

)= 0 at z = 0. (5.9)

These boundary conditions correspond to a “slippery” rigid lid that does not supporttangential stresses. Thus, during a fraction of a second ub rapidly diffuses outwards fromz = −h and this quickly changes the horizontal flow in the free surface. The evolving ub

has a three-dimensional structure but in the surface z = 0 it takes the simple form of avortex pair together with a divergent source–sink flow.

This is illustrated in figure 5, where the balanced flow in the surface z = 0 at timet = 0.5 s is visualized using a Helmholtz decomposition for the horizontal velocities suchthat (ub, vb) = (φx − ψy,φy + ψx). This decaying surface flow resembles a cross-sectionof Hill’s hemispherical vortex and the trapped fluid momentum can be estimated on thisbasis as in Hu et al. (2003).

6. Concluding remarksThe logic behind the basic results presented in this paper can be summarized as fol-

lows. Localized fluid forcing creates a global fluid response with corresponding globalmomentum fluxes, and these momentum fluxes decay with distance r from the forcingsite such that the only relevant component of the momentum flux as r →∞ is its dipolarpart. In turn, the magnitude of this dipolar part depends only on the integrated mo-


x

y

Balanced surface stream function at time t = 0.5s

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

-1.5

-1

-0.5

0

0.5

1

1.5

x

y

Balanced surface velocity potential at time t = 0.5s

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Figure 5. A numerical solution to (5.9) is evaluated for the horizontal balanced flow in thefree surface z = 0 at time t = 0.5 s. The physical parameters are as before. Displayed are thestream function ψ and velocity potential φ such that (ub, vb) = (φx − ψy, φy + ψx) at z = 0.Left: the stream function ψ exhibits a typical vortex-pair structure that corresponds to vortexpropagation to the right. Right: the velocity potential φ indicating convergence and hencesinking motion just below the surface in x > 0 and a compensating divergence region and risingmotion in x < 0.

the three-dimensional rolling motion induced by an infinitesimal vortex ring submergedat depth h. Despite this consistency, however, such a vortex flow structure is not observedin experiments.

This is because the vortex flow is observable for a fraction of a second and this is a longenough time interval such that nonlinear and viscous effects must be taken into account.Indeed, in nonlinear viscous theory the vortex ring self-advects itself nonlinearly in the x-direction whilst its vorticity and velocity structure rapidly diffuses outward. These effectsare easily observable in the water strider experiments in Hu et al. (2003). Nonlinearitycan not be treated here, but viscous diffusion is easily incorporated into the balancedflow evolution by letting ub diffuse according to

∂ub

∂t= ν∇2ub and

(∂ub

∂z,∂vb

∂z, wb

)= 0 at z = 0. (5.9)

These boundary conditions correspond to a “slippery” rigid lid that does not supporttangential stresses. Thus, during a fraction of a second ub rapidly diffuses outwards fromz = −h and this quickly changes the horizontal flow in the free surface. The evolving ub

has a three-dimensional structure but in the surface z = 0 it takes the simple form of avortex pair together with a divergent source–sink flow.

This is illustrated in figure 5, where the balanced flow in the surface z = 0 at timet = 0.5 s is visualized using a Helmholtz decomposition for the horizontal velocities suchthat (ub, vb) = (φx − ψy,φy + ψx). This decaying surface flow resembles a cross-sectionof Hill’s hemispherical vortex and the trapped fluid momentum can be estimated on thisbasis as in Hu et al. (2003).

6. Concluding remarksThe logic behind the basic results presented in this paper can be summarized as fol-

lows. Localized fluid forcing creates a global fluid response with corresponding globalmomentum fluxes, and these momentum fluxes decay with distance r from the forcingsite such that the only relevant component of the momentum flux as r →∞ is its dipolarpart. In turn, the magnitude of this dipolar part depends only on the integrated mo-

Includes diffusionon slower time scale

Vortex dipolein surface

Hemispherical vortex in 3d

Momentum ?

No tangential stress at free surface

Momentum of adjusted balanced state

Can show that adjusted balanced state has momentum:

18 Buhler


M(0+) = Rh +23Rv and M(+∞) =

23Rh + 0Rv. (4.16)




ζ(x, y, 0) = 0 and ζt(x, y, 0) = (4.13) + (4.14). (4.17)


ζ(x, y, t) =1

(2π)2



ζ(k, t) = ζ(k, 0) cos ωt + ζt(k, 0)1ω

sinωt (4.19)


ω(κ) = +√


dκ=

12


(4.20)





Precisely the same as in unbounded 3d case!

Waves have propagated the remaining 1/3 of the momentum away to infinity!Wave details are irrelevant (e.g. surface tension)

Generic momentum result in 3d

Waves 1/3 Vortices 2/3

Last slide

24 Buhler

mentum input R of the localized force F . Therefore, the far-field momentum flux canbe computed as if the impulsive force were perfectly localized, i.e. as if F = Rδ(t)δ(x),and it is this simple generic form that makes possible the generality of the momentumdecomposition results summarized in (2.10).

The generality of these results should make them useful in a broader range of fluid–structure interaction problems, and also in the wave–mean interaction theory contextalluded to in the beginning. Indeed, the effect of small-scale waves on large-scale meanflows is often described using the concept of localized effective wave-induced forces (e.g.Reynolds-stress convergences). The present work then makes clear that the mean-flowmomentum budget will necessarily contain wave components as well as vortex compo-nents; an example of such a mean-flow wave component is the far-field remote recoildescribed in Buhler & McIntyre (2003).

The specific application of these results to water strider locomotion shows that recoilforces can be computed either by measuring the waves or by measuring the vortices: eachkind of fluid motion carries an imprint of the recoil force in it. Thus, in this example atleast, waves and vortices do not appear as unrelated and nearly independent parts of thefluid motion, but rather waves and vortices appear as flip sides of the same coin.

I would like to thank J. W. M. Bush and D. L. Hu for the very interesting discussionsof their experimental work on water walkers that motivated this theoretical work, as itshould be. The non-uniqueness of the global momentum budget for incompressible flowswas first pointed out to me a long time ago by M. E. McIntyre, and it is a pleasure toacknowledge many stimulating discussions on this and related topics ever since. D. G.Andrews kindly provided me with a reference to (and a copy of) the important workof Theodorsen (1941). Financial support for the wave–vortex theoretical aspects of thiswork under NSF grant OCE-0324934 is gratefully acknowledged. This paper was writtenup largely at the NSF-sponsored 2005 summer program in Geophysical Fluid Dynamicshosted by the Woods Hole Oceanographic Institution (USA), and I would like to thankthe 2005 GFD directors for their support.

REFERENCES

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge: University Press.Bretherton, F. P. 1969 On the mean motion induced by internal gravity waves. J. Fluid

Mech. 36, 785–803 (referred to herein as B69).Buhler, O. & McIntyre, M. E. 2003 Remote recoil: a new wave–mean interaction effect. J.

Fluid Mech. 492, 207–230.Buhler, O. & McIntyre, M. E. 2005 Wave capture and wave–vortex duality. J. Fluid Mech.

534, 67–95.Bush, J. W. M. & Hu, D. L. 2006 Propulsion at the interface. Ann. Rev. Fluid Mech. to

appear.Denny, M. W. 1993 Air and water: the biology and physics of life’s media. Princeton Univ.

Press.Drucker, E. G. & Lauder, G. V. 1999 Locomotor forces on a swimming fish: three-

dimensional vortex wake dynamics quantified using digital particle image velocimetry. J.Experimental Biology 202, 2393–2412.

Hu, D. L., Chan, B. & Bush, J. W. M. 2003 The hydrodynamics of water strider locomotion.Nature 424, 663–666.

Keller, J. B. 1998 Surface tension force on a partly submerged body. Phys. Fluids 10, 3009–3010.

Lamb, H. 1932 Hydrodynamics, 6th ed . Cambridge: University Press.Longuet-Higgins, M. S. 1977 The mean forces exerted by waves on floating or submerged

Documents

Waves, vortices, and the gait of the water strider€¦ · problemathand. The key phenomenontobe understood is the global responseofa ßuid toa forceÞeld that is bothimpulsive and