Reduced-Order Modeling of Propulsive Vortex Shedding from a Free

Pitching Hydrofoil with an Internal Rotor

Phanindra Tallapragada Scott David Kelly

Abstract— The efficient development of control strategies forbiomimetic aquatic vehicles that exploit vortex shedding forpropulsion requires mathematical models that are sufficientlyrich to capture the necessary hydrodynamic phenomena whileremaining computationally tractable, and if possible while offer-ing some interface to analytical (in addition to computational)tools. In a sequence of recent papers, the authors have detailedan approach to modeling such systems that exploits certainidealizations to realize equations of motion that accommodatethe formalism of geometric mechanics on manifolds, andhave demonstrated the relevance of resulting models to actualrobotic systems. In the present paper, we describe an amendedapproach to modeling propulsive vortex shedding in the planethat retains critical features of previous models while providinga significant decrease in model dimension. We illustrate ourupdated approach by simulating the dynamics of a prototypicalfishlike swimmer comprising a free rigid hydrofoil that can beinduced to pivot about a body-fixed point through the oscillationof an internal rotor, shedding propulsive vorticity from itstrailing edge.

I. INTRODUCTION

The motion of a solid body through any fluid is influenced

by the fluid’s viscosity. The viscous boundary layer that

forms over the body’s surface reconciles the no-slip condition

at the surface with discrepancies between the velocity of the

body and that of the fluid nearby, and even a streamlined

body experiences drag as a result of shear in this boundary

layer. A body that isn’t entirely streamlined may experience

additional dynamic consequences from localized boundary

layer detachment. This phenomenon is essential to the de-

velopment of lift on a translating wing, for instance, which

requires boundary layer detachment to occur at the wing’s

trailing edge, imparting downward momentum to the air left

behind.

The forces associated with localized boundary layer de-

tachment from select contours on marine animals — for

instance, from the trailing edge of a tuna’s caudal fin or

from the lip of a medusa’s bell — are so significant that

qualitatively accurate models for marine animal locomotion

may be realized that prioritize these forces over all others

associated with fluid viscosity. Analogous models may be

used as the basis for the analysis and control of robotic

vehicles that rely on fin-like structures for propulsion and

steering.

This material is based upon work supported by the National ScienceFoundation under grant CMMI-1000652.

Phanindra Tallapragada and Scott David Kelly are with the Departmentof Mechanical Engineering and Engineering Science, University of NorthCarolina at Charlotte, Charlotte, NC 28223, USA.

Correspondence should be directed to scott@kellyfish.net.

At the trailing edge of a moving airfoil or hydrofoil, fluid

in the boundary layer is suddenly liberated both from the no-

slip condition and from the proscription on motion normal

to the foil. As a result, fluid rolls off the foil, potentially

(depending on the foil’s geometry) encountering fluid rolling

in the opposite sense off the foil’s opposite side. Fluid in the

wake of the foil thus acquires net vorticity, and the dynamic

influence of boundary layer detachment on the fluid as a

whole may be understood in terms of the dynamic creation

of this vorticity. The dynamic influence on the motion of

the body, meanwhile, may be understood in terms of the

forces required to reconcile the creation of vorticity with the

conservation of circulation around general material curves in

the fluid — according to Kelvin’s circulation theorem [1] —

and with the conservation of momentum overall.

The interpretation of fishlike swimming in terms of the

planar dynamics of a hydrofoil shedding vorticity from its

trailing point has a long history, dating at least to [2]. In

[3], [4], the second author and collaborators presented a

model for the fishlike swimming of a deformable foil realized

as the image of a circle under a time-varying Joukowski

transformation — a canonical conformal map generating a

foil with a cusp at its trailing point — and demonstrated

that a simple feedback law could be used to steer the foil

in the plane. The fluid surrounding the foil was assumed to

be inviscid and a Kutta condition was enforced periodically

in time to model boundary layer detachment at the trailing

point in a discretized way. This condition required that the

preimage of the trailing point correspond to a stagnation

point in the preimage of the flow under the Joukowski

map, and thus that the fluid separate smoothly from the

trailing point in the physical plane of the foil. With each

discrete application of the Kutta condition, a point vortex

was introduced to the fluid near the trailing point and an

impulsive change was made to the foil’s momentum so that

the desired fluid velocity was realized at the trailing point

without violating the conservation of momentum.

The present authors expanded upon the control design

from [4] in [5] and demonstrated its applicability to the

fishlike swimming of a real aquatic robot in [6]. The authors

then presented adaptations of the modeling approach from

[4] to the context of a planar bluff body shedding point

vortices from an arbitrary surface point [7] and to that of

a spherical body shedding vortex rings from an oscillating

circular surface contour [8], [9], the latter system reminiscent

of a swimming jellyfish.

In each of the papers cited in the preceding paragraph,

the assumption of an ideal fluid subject to a purely localized

2013 American Control Conference (ACC)Washington, DC, USA, June 17-19, 2013

978-1-4799-0178-4/$31.00 ©2013 AACC 615

Fig. 1. Acceleration from rest of a rigid hydrofoil shedding point vorticesin accordance with the periodic application of a Kutta condition at the foil’strailing cusp, subject to the simultaneous conservation of momentum. Thefoil pitches because of sinusoidal oscillations in the relative orientation of aninternal rotor. Shed vortices may differ from one another in absolute strengthas well as sign, but the sign and strength of each remain constant overtime. Blue vortices correspond to counterclockwise vorticity, red vortices toclockwise vorticity.

mechanism for vortex shedding represented a substantial

simplification of the true viscous hydrodynamics of the

system being modeled. Between vortex shedding events, each

model benefitted further from the presence of a Hamiltonian

structure in the equations governing the interaction of the

body with the existing singular distribution of vorticity, as

detailed in [10] for the two-dimensional case and in [11],

[12] for the three-dimensional case. Nevertheless, each of the

models presented in these papers suffered from an analytical

intractability resulting from the rapid increase in dimension

associated with a large number of vortex shedding events.

Fig. 1 illustrates this issue. The vehicle depicted appears

in none of the preceding references, but comprises a rigid

Joukowski foil with zero camber coupled to an internal rotor

that’s driven to oscillate sinusoidally relative to the foil.

The figure represents a single frame from a simulation of

the foil accelerating from rest in a fluid initially devoid of

vortices. In accordance with the periodic application of a

Kutta condition at the foil’s trailing cusp, vortices are shed

at frequent regular intervals throughout the simulation, each

imparting an impulsive change in momentum to the foil at its

instant of creation and each interacting with the foil thereafter

according to the Hamiltonian equations from [10]. Hundreds

of vortices are present in Fig. 1, each requiring the addition

of two ordinary differential equations to the description of

the system’s dynamics.

The behavior of the discrete vortices in Fig. 1 reflects the

behavior observed of vorticity shed continuously from a real

oscillating foil in a viscous fluid in a critical way. Groups

of vortices of similar sign are shed over alternating intervals

of time; each group is observed to “roll up” thereafter into

a coherent structure approximating a single vortex. As more

vortices are shed with additional oscillations of the foil, the

wake of the foil will come to resemble the inverse Karman

vortex street highlighted in [2] as a hallmark of fishlike

swimming.

This observation suggests a strategy for decreasing the

dimension of the model represented by Fig. 1. If frequently

shed vortices with relatively low strength reliably roll up,

at lower frequency, into approximations of vortices with

relatively high strength, then vortices of the latter kind might

as well be shed directly at lower frequency. In particular,

instead of introducing a sequence of vortices with like

sign and individually fixed strength with each sweep of the

foil through the fluid, we might allow a single vortex of

time-varying strength to adjust the fluid velocity as needed

throughout each sweep at the foil’s trailing point. With each

reversal of the foil’s motion, a change in sign occurs in the

shed vorticity; this might provide a criterion for fixing the

strength of the most recently formed vortex, releasing this

vortex to the foil’s wake, and beginning the shedding process

anew.

In fact, the approach just described has a long history of

its own, traceable to [13], but has traditionally been invoked

to model vortex shedding from solid boundaries executing

prescribed motions, and not from bodies moving according

to dynamic interactions with their wakes. A recent exception

appears in [14], in which vortices with time-varying strength

— so-called Brown-Michael vortices, named for the authors

of [13] — are used to model the wake of the falling flat plate

famously considered by Maxwell [15]. The presentation in

[14] encompasses more than just Maxwell’s example, but

focuses on the explicit computation of forces and moments

in determining the influence of a vortical fluid on a moving

body.

In the remainder of the present paper, we develop a

model for the self-propulsion of the vehicle from Fig. 1

shedding Brown-Michael vortices rather than the more nu-

merous fixed-strength vortices of our prior work. Unlike the

falling body considered in [14], this vehicle is driven by the

actuation of an additional degree of freedom as a control

input, and we determine its motion not by computing forces

or moments explicitly but by imposing the condition that

momentum be conserved in the system overall throughout the

vortex shedding process. We expect it to be straightforward

for us to adapt the method we describe both to the fishlike

swimming of the flexing foil from [3], [4] and to the

jellyfishlike swimming of the three-dimensional body from

[8], [9].

II. MODELING

A. Physical Model

Consider a Joukowski foil surrounded by fluid of unit

density that extends to infinity in a two-dimensional plane,

represented by the complex domain C. The geometry and

motion of the foil are described by mapping its boundary ∂Bin the physical space from a circle C of radius rc centered

at the origin in the mapped plane through the Joukowski

transformation

z = F (ζ) = ζ + ζc +a2

ζ + ζc, (1)

where ζc ∈ C and a ∈ R. We refer to the plane of the foil’s

motion as the foil plane and the plane of the circle’s motion

as the circle plane. For the shape of the foil shown in Fig.

2, ζc ∈ R. The preimage of the sharp trailing edge of the

foil is given by ζt = a−ζc. The transformation is conformal

616

Fig. 2. The Joukowski transformation maps a circle of radius rc in thecomplex ζ plane to the foil in the z plane.

everywhere in the open set |ζc| > rc. The derivative of the

transformation

F ′(ζ) = 1−a2

(ζ + ζc)2

exists everywhere except at ζt.The foil has three degrees of freedom; it can translate

with velocity U = (U1, U2) and rotate about its center with

angular velocity Ω. We suppose the foil to be coupled to

a balanced rotor with no direct coupling to the fluid, as

depicted in Fig. 1. Let the combined mass of the foil and

the rotor be m and the moment of inertia of the rotor about

its center be Ic.

B. Motion of the Point Vortices

The fluid is assumed to be ideal almost everywhere, with a

singular distribution of vorticity modeled by N point vortices

with circulations Γn located at points ζn in the circle plane.

We note that each such vortex is mapped to a vortex with

circulation Γn at zn = F (ζn) in the foil plane. Following

[16], the complex potential describing the velocity of the

fluid in a frame of reference collocated with the foil may

be decomposed in terms of its dependence on the translation

of the foil, the rotation of the foil, and each of the N point

vortices in the form

w(ζ) = W (z) = U1w1(ζ)+U2w2(ζ)+Ωw3(ζ)+N∑

k=1

wkv (ζ).

On the boundary of the circle, the real components φkv(ζ),

φ1(ζ), φ2(ζ), φ3(ζ) of the complex potentials wkv (ζ), w1(ζ),

w2(ζ), w3(ζ) satisfy the boundary conditions

∇φkv ·n = 0, ∇φ1 ·n = 0, ∇φ2 ·n = 0, ∇φ3 ·n = 0, (2)

where n is a vector normal to the boundary. The rigid-body

potential functions are given by

w1(ζ) =−r2cζ

+a2

(ζ + ζc),

w2(ζ) = −ı

(

r2cζ

+a2

(ζ + ζc)

)

,

w3(ζ) = −ı

(

r2cζ

(

ζc +a2

ζc

)

−

(

kζc +a2

ζc

)

a2

ζ + ζc

)

.

The potential function wkv (ζ) due to a point vortex located

at ζk outside a circular cylinder can be constructed according

to the Milne-Thomson circle theorem [17] in terms of an

image vortex of circulation −Γk located inside the cylinder

at r2c/ζk. Thus

wkv (ζ) =

Γ

2πı

(

log (ζ − ζk)− log

(

ζ −r2cζk

))

.

The image vortex inside the cylinder introduces a net circula-

tion around the cylinder, consistent with Kelvin’s circulation

theorem. This development of net circulation is essential to

the propulsion of the foil.

The kth vortex in the foil plane is simply advected under

the influence of the remaining N−1 vortices, of the N image

vortices, and of the motion of the foil itself, governed by the

potential function

Wk(z) = W (z)−Γ

2πılog (z − zk).

The corresponding motion of the vortex in the circle plane

is given by

ζk =

(

dWk

dz− (U1 + ıU2 + ıΩzk)

)

1

F ′(ζ). (3)

C. Motion of the Foil

If the fluid is initially at rest, the linear and angular

impulse in the system, which can be interpreted loosely as

the combined linear and angular momenta of the foil and the

fluid, are conserved [16]. The equations of motion for the

foil and vortices together assume the form of Lie-Poisson

equations, a class of noncanonical Hamiltonian equations. It

was shown in [10] that these take the form(

d

dt+Ω×

)

L = 0,

dA

dt+V × L = 0,

where L and A are the linear and angular impulse in foil-

fixed coordinates. The linear impulse is the sum

L = MV + Lv

of the impulse due to rigid-body motion of the foil and the

impulse due to the vortices. Here M is the total effective

mass of the foil, including added mass due to the fluid, and

Lv =

∮

∂B

r× (nb ×∇φv)ds+N∑

n=1

Γnrn × e3.

The angular impulse in the foil-fixed frame is given by

A = −1

2

N∑

n=1

Γn||rn||2

−1

2

∮

∂B

||r||2(nb ×∇φb + nb ×∇φv)ds. (4)

In the preceding equations, r = (Re(z), Im(z)) and nb is

the normal vector on ∂B pointing into the fluid. Equation (4)

can be simplified considerably — for details see [10], [18]

— to obtain a conservation law of the form

I

Vx

Vy

Ω

+

(

Lv

Av

)

=

(

RTL0

A− Icφ

)

, (5)

617

where L0 is the linear impulse of the system in spatially

fixed coordinates, R is the rotation matrix that transforms

body-fixed coordinates to spatially fixed coordinates, U =RV, and I is the total effective inertia tensor for the body,

including the added inertia due to the fluid’s motion. The

linear impulse Lv and angular impulse Av are functions of

the positions of the vortices alone and do not depend on their

velocity. They are given by

Lv = −

N∑

n=1

Γn

2πı

(

ζn −r2cζn

)

(6)

and

Av =1

2π

(

2r2c (ζc +a2

ζc)

ζn−

2a2(r2c − ζcζc)

ζc(ζn + ζc)−

2a4ζc(r2c − ζcζc)

(r2c − ζcζc)(ζn + ζc)

)

−1

2

N∑

n=1

ΓnZZn . (7)

The term Icφ has been added to right-hand side of (5)

to account for the spinning of the rotor. The exchange of

momentum between the foil and the vortices governs the

evolution of the system overall.

The velocity at any point in the fluid away from a vortex

in the circle plane is given by

ζ =

(

dW

dz− (U1 + ıU2 + ıΩz)

)

1

F ′(ζ).

The boundary conditions (2) allow fluid to slip along the

surface of the foil, but at the trailing edge of the foil, which is

a singularity of the Joukowski transformation, the derivative

F ′(ζ) vanishes, causing the velocity of the fluid in the circle

plane to become undefined. The only physically allowable

value for the fluid velocity at the trailing edge in the foil

plane is zero; this is the Kutta condition. The Kutta condition

requires that at the pre-image ζt of the trailing edge,

dw(ζ)

dζ

∣

∣

∣

∣

ζ=ζt

= 0.

This condition can be enforced at (frequent) regular intervals

through the shedding of new vortices, fixing the strengths of

these vortices in such a way that the total linear and angular

impulse are simultaneously conserved, as

I

∆Vx

∆Vy

∆Ω

+ ΓN+1

(

∆Lv

∆Av

)

= 0,

where (∆Vx,∆Vy,∆Ω) are impulsive changes in the veloc-

ity of the foil and (∆Lv,∆Av) are changes in the fluid

impulse corresponding to the addition of a vortex of unit

circulation. It follows that whenever a new vortex is added

to the fluid, the foil receives an impulsive kick. The addition

of each new vortex increases the dimension of the system

by two.

D. Reduced-Order Model

In simulations like the one that produced Fig. 1, the

dimension of the system increases into the thousands, pre-

senting computational challenges and obscuring qualitative

aspects of the system’s dynamics that could be exploited

for controlled propulsion. It’s possible, however, to model

the shedding of a point vortex with unsteady circulation

that adjusts continuously to enforce the Kutta condition and

the conservation of impulse. It’s intuitively clear that ΓN ,

the strength of the last shed vortex, might be adjusted in

a continuous fashion to satisfy the Kutta condition in a

persistent way, but it can be shown that merely changing ΓN

violates the conservation of momentum for the fluid enclosed

in a control volume around the points zt and zN [14]. To

compensate for the change in the momentum of the fluid

within this control volume, the position of the vortex must

evolve as well.

It was shown in [14] that the dynamics of an unsteady

vortex that satisfies the conservation of momentum of the

fluid can be described by the equation

zN +ΓN

ΓN

(zN − zt) =dWN (z)

dz, (8)

where zN is the location of the unsteady vortex. The addition

of the term Γ

Γ(zN −zt) conserves the momentum of the fluid

enclosed by any branch cut around zt and zN .

Because the rollup of a vortex sheet is permanent and

cannot be undone, the circulation of the unsteady vortex

described above should not decrease in magnitude. This

places an additional constraint on the evolution of ΓN .

When the absolute value of ΓN reaches a maximum, the

unsteady vortex is released and its circulation is held constant

thereafter. Vortex shedding occurs in the instant at which

ΓN ΓN < 0,

after which the shed vortex is treated as a steady vortex with

dynamics governed by (3).

As noted, the evolution of the circulation of the unsteady

vortex has to continuously satisfy the conservation of impulse

and the Kutta condition simultaneously, i.e.,

d

dt

dw(ζ)

dζ

∣

∣

∣

∣

ζ=ζt

= 0,

Id

dt

Vx

Vy

Ω

+d

dt

(

Lv

Av

)

=

(

0

−Icφ

)

.

(9)

Thus the evolution of the unsteady vortex and the motion

of the foil are governed by (1), (8), and (9) simultaneously

while the motion of the other (N − 1) steady vortices is

governed by (3).

III. SIMULATION

We consider two simulations of a rigid Joukowski foil

propelling itself through vortex shedding. In the first case

the foil sheds steady point vortices from its trailing edge at

regularly spaced instants in time, in the second the foil sheds

618

0 0.002 0.004 0.006 0.008 0.01 0.012−5

0

5

10

15x 10

−4

y

x

(a)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

θ(t

)

t

(b)

0 5 10 15 20 25 30 35 400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Γ(N

)

N

(c)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

1

2

3

4

5

6

7

8

9

10

ΣΓ

(t)

t

(d)

Fig. 3. (a) Trajectory of the foil in the (x, y) plane, (b) heading angle ofthe foil versus time, (c) circulation of the most recently shed vortex versustime, and (d) total shed vorticity versus time for the model in which vorticityis shed discretely.

one Brown-Michael vortex. In each case the foil is driven by

a rotor that rotates such that

φ =π

6(1− cos (ωt)),

where ω ∈ 10, 20, 30, 40, 50.

Figs. 3 and 4 compare the two cases when ω = 50. The

first observation we make is that the total shed vorticity —

given byN∑

n=1

Γn in the first case and by the strength of the

first Brown-Michael vortex in the second — evolves similarly

in both cases. The time at which the first Brown-Michael

vortex reaches maximum circulation and is frozen is t1 =0.034. For the foil shedding steady vortices the time at which

the foil stops shedding positive vortices and begins shedding

negative vortices is t2 = 0.037.

The foil’s heading evolves almost identically in both cases,

but differences are apparent in the trajectories of the foil in

the (x, y) plane. The net displacement and the translational

velocity of the foil have a direct proportional dependence on

the positions of the shed vortices. Since the time intervals

we consider are small, the net displacement is small and

small variations in the positions of the vortices relative to

the foil can lead to large relative errors in the displacement

of the foil. Here we remark that the unsteady Brown-Michael

vortex preserves the constraint of the Kutta condition con-

tinuously in time with a small error. The discrete steady

vortex shedding mechanism preserves the constraint of the

Kutta condition only on a measure zero set of specific (but

frequent) time instants.

The angular motion of the rotor acts as a control input to

generate vorticity. The total circulation of the vortices and

their positions in turn completely determine the position,

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

0.2

0.4

0.6

0.8

1

1.2x 10

−3

y

x

(a)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

θ(t

)

t

(b)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.0350

1

2

3

4

5

6

7

8

ΣΓ

(t)

t

(c)

Fig. 4. (a) Trajectory of the foil in the (x, y) plane, (b) heading angle ofthe foil versus time, and (c) total shed vorticity versus time for the modelin which vorticity is shed continuously.

orientation, and velocity of the foil. The position of the

Brown-Michael vortex in the second case and the center of

the corresponding cluster of steady vortices in the first are

almost the same; discrepancies in the total vorticity shed are

the key to discrepancies in the other simulated quantities.

This is clear from (6) and (7), which show that the total

vorticity has a significant influence on the values of Lv

and Av. We therefore proceed to compare the total vorticity

shed in the two cases when the rotor oscillates at different

frequencies ω.

This comparison is shown in Fig. 5. We observe that

in every case, the total circulation of the steady vortices

eventually exceeds the circulation of the unsteady vortex.

This discrepancy results in part from the fact that the Kutta

condition is satisfied continuously in one case but only

discretely in the other, and in part from a difference in the

way the location at which vorticity is added to the fluid

is determined in each case. For the steady-vortex model,

each new vortex is introduced midway along a circular arc

between the last shed vortex and the foil’s trailing edge, while

in the Brown-Michael model, new vorticity is added to the

fluid collocated with previously shed vorticity.

The computational advantage of the Brown-Michael model

is apparent in a comparison of the total number of vortices

shed in each case. When ω = 10, for instance, 185 vortices

are shed in the steady-vortex model, increasing the dimension

of the system’s state space over time to 373, while only the

first unsteady vortex develops in the Brown-Michael model,

for which the dimension of the state space remains 5.

ACKNOWLEDGMENT

The authors thank Jeff Eldredge for fruitful suggestions.

619

0 0.01 0.02 0.03 0.04 0.050

1

2

3

4

5

6

7

8Σ

Γ(t

)

t

(a) ω = 40

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

1

2

3

4

5

6

ΣΓ

(t)

t

(b) ω = 30

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

3

3.5

4

ΣΓ

(t)

t

(c) ω = 20

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ΣΓ

(t)

t

(d) ω = 10

Fig. 5. Total shed vorticity versus time for different rotor frequencies ω.The red curve in each case is the circulation of the unsteady Brown-Michaelvortex and the blue curve is the running sum of the circulations of the shedsteady vortices. The running sum is shown only until the instant at whichvortices with negative circulation begin to be shed.

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