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Physica D 238 (2009) 1152–1160

Contents lists available at ScienceDirect

Physica D

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

A low-dimensional model of separation bubbles

R. Krechetnikova,∗, J.E. Marsdenb, H.M. Nagibc

aDepartment of Mechanical Engineering, University of California at Santa Barbara, Santa Barbara, United States

bControl & Dynamical Systems, California Institute of Technology, Pasadena, United States

cMechanical, Materials, and Aerospace Engineering Department, Illinois Institute of Technology, Chicago, United States

a r t i c l ei n f o

Article history:

Received 8 October 2007

Received in revised form

23 May 2008

Accepted 22 March 2009

Available online 8 April 2009

Communicated by M. Silber

Keywords:

Separation bubble

Flow separation control

Low-dimensional modeling

Singularity theory

Phenomenology

a b s t r a c t

In this work, motivated by the problem of model-based predictive control of separated flows, we identify

the key variables and the requirements on a model-based observer and construct a prototype low-

dimensional model to be embedded in control applications.

Namely, using a phenomenological physics-based approach and dynamical systems and singularity

theories, we uncover the low-dimensional nature of the complex dynamics of actuated separated flows

and capture the crucial bifurcation and hysteresis inherent in separation phenomena. This new look at

the problem naturally leads to several important implications, such as, firstly, uncovering the physical

mechanisms for hysteresis, secondly, predicting a finite amplitude instability of the bubble, and, thirdly,

to new issues to be studied theoretically and tested experimentally.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

1.1. Motivation and objective

In recent years there has been an increasing demand to extend

the range of aircrafts’ flight conditions to high angle-of-attack

regimes and high-amplitude maneuvers. The latter usually lead

to intense flow separation and dynamic vortex shedding which

in turn generate destructive pitching moments, sharp increases in

drag, and losses in lift. Therefore, the only way to enlarge the flight

envelope is to design efficient ways of controlling flow separation.

The classical approach is based on open-loop control, which is

achieved either by mechanical or fluidic actuation according to

operating schedules (lookup tables) constructed using extensive

and costly experimental studies. On the other hand, feedback

control schemes do not require operating schedules and, being

more efficient and reliable [1], also naturally allow one to address

the optimization issue.

However, feedback control is more demanding theoretically

since it requires an embedded model which predicts the behavior

of the physical system at hand. In the case of flow separation, the

∗Corresponding author.

E-mail address: rkrechet@cds.caltech.edu (R. Krechetnikov).

general equations of fluid motion – the Navier–Stokes equations

(NSEs) – are known and can, in principle, produce accurate

prediction of the flow structure, but because the real boundary

and initial conditions are noisy and cannot be precisely measured

and because the NSEs cannot be solved in real time in flight, this

approachisimpractical.However,anaccuratereal-timesolutionis

not actually necessary, since in reality one can use sensors on the

boundaryofliftingsurfaces,whichinturnreadoffacertainamount

ofextrainformationfromthephysicalsystemandthereforeshould

allow one to weaken the requirements on the model accuracy.

Thus, one is naturally led to look for coarse models, which should

also be low dimensional for computational real-time efficiency.

Construction of such models is the main objective of our work.

Historically, the importance of low-dimensional modeling of

unsteady aerodynamic characteristics – aerodynamic forces and

momentsactingonanaircraft–forcontrolpurposes,stabilityanal-

ysis,anddynamicsimulationshasbeenrealizedforalongtimeand

the appropriate models were developed; see, for example, [2–7].

However, the necessity to model the separation and vortex shed-

ding dynamics was realized just recently in view of the increas-

ing demand for high angle-of-attack regimes [2]. Early attempts to

develop dynamical models are based on the anzatz that the phe-

nomenon of flow separation behaves linearly for small variations

of the parameters involved [5–7], which has many limitations, as

willbeclearfromthesubsequentdiscussionwhenweestablishthe

requirements on a model aimed for robust control.

0167-2789/$ – see front matter © 2009 Elsevier B.V. All rights reserved.

doi:10.1016/j.physd.2009.03.017

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1.2. Previous works

In one of the early works, [2], the coordinate of a separation

point x ∈ [0,1] is taken as an internal state-space variable and

the ad hoc linear first-order equation is used to account for the

movement of the separation point for unsteady flow conditions:

dx

dt+ x = x0(α − τ2˙ α),

where we kept the original notations: τiare relaxation times, and

x0(α) is the stationary value of the separation point position for

a given angle of attack α. Then the value x = 1 corresponds

to attached flow, while x

=

separation. A similar approach was used in the construction of the

modelsforseparationphenomena,forexample,[5]andtheONERA

model [6]. In particular, the key feature of the state-of-the-art low-

dimensional model used in a closed-loop control of dynamic stall

with pulsed vortex generator jets due to Magill et al. [5] is a choice

of the governing physical parameters, such as lift Z and separation

state B with B = 0 corresponding to fully attached flow and B = 1

to fully separated flow. Steady states, Bs(α) and Zs(α), represent

the baseline case and the measured steady lift, respectively, as

functions of the angle of attack α. The experimentally measured

function Zs(α) may contain a hysteretic behavior and thus is an

empiricalwayofaccountingforahysteresis,assuggestedbyMagill

et al. [5]. Exploiting the physical arguments: (i) lift Z ∼ circulation

Γ(α); (ii) relaxation to a baseline state limt→+∞B(t) = Bs(α); (iii)

rise in lift when a dynamic vortex is shed Z ∼ Bt, one arrives at the

simplest low-order model with adjustable parameters,

τ1

0 corresponds to leading edge

Btt= −k1Bt+ k2[Bs(α) − B],

Zt= k3Btt+ k4[Zs(α) − Z] + Γααt.

As one can notice, all these models are linear.

In the case of aerodynamic models for forces and moments,

the addition of nonlinear terms is known to extend the range of

flight conditions to high angle-of-attack regimes and maneuvers

[4], which is usually done in an ad hoc manner, by simply adding

polynomial terms with unknown coefficients without any physical

insight or justification. Systematic application of local and global

bifurcationtheoriestotheaerodynamicmodelsisjustbeginningto

be appreciated [3]. To the authors’ knowledge there have been no

similar attempts in these directions for models of flow separation.

In fact, the systematic methods available to formulate such models

are very limited and the connection of known models to physics

is rather far from desired. A commonly used approach is to first

generate experimental data and then to extract the model by a

projection onto proper orthogonal decomposition (POD) modes

using, for example, balanced truncation or similar methods. This

techniqueisknowntobeincapableofcapturingthedynamicswith

a few modes for the wide range of governing parameters in view

of the open flow nature of the problem, in particular. In addition,

while POD models are based on the most energetic modes, they do

not provide deep insights into the physics of the flow.

(1a)

(1b)

1.3. Model requirements and methodology

In this work we develop a model which is not a mnemonic

device encoding the experimental observations, as ad hoc models

would be, but is physically motivated and thus is more robust in

reflecting the actual behavior for a wide range of flight and control

parameters.

In view of the necessary coarse nature of the model and the

application objectives, one has to decide which aspects of the

dynamics should be modeled reasonably accurately. With the

target of producing a model, upon which an observer in a closed-

loop control scheme can be based for a wide range of physical

parameters, in this work we identify the crucial elements of the

dynamics of separation bubble, namely bifurcation and hysteresis,

which need to be reflected in the model. These elements reflect

the fundamental nonlinear nature of the physical problem, which

apparently cannot be captured with linear models.

As follows from the above discussion, being nonlinear, low

dimensional, and physically motivated are key requirements for

a model. In this work, we shall make use of phenomenological

modeling,whichhasbeensuccessfulinmanyotherproblems,such

as Duffing’s equation for the buckling of elastic beams [8], simple

maps to describe a dripping faucet [9], which even capture the

observed chaotic behavior to a great extent, and bubble dynamics

in time periodic straining flows [10], to name a few. As will be

clear from the text later, besides appealing to a phenomenological

physically motivated analysis of empirical facts we also provide

a basis for it in dynamical systems and singularity theories. A

symbiosis of these two methodologies yields a coherent picture

of the phenomena. It should also be stressed that in view of the

coarse nature of the model that is sought, we effectively construct

an‘‘approximateglobalnormalform’’,whichreflectsthekeyglobal

features of the physical system, necessary for control purposes.

1.4. Paper outline

Toachievetheaboveobjectives,wefirstwillidentifythecentral

idea of our approach in Section 2, and then appeal to the tools of

the bifurcation and singularity theory [11], as will be made precise

in Section 3. The outline of the paper is as follows. In Section 3, we

discussthefirstnonlinearaspectofseparationbubbles,namelythe

bifurcation phenomenon and the way to model it. In Section 4, we

explore the basic physics of hysteresis phenomena, and suggest a

single model capable of capturing both bifurcation and hysteresis.

2. Central idea

A central notion and object, whose dynamics we study, is a

separation bubble, whose main features are as follows. First of

all, separation of the boundary layer develops due to an adverse

pressure gradient [12] which occurs when the angle of attack of an

airfoil is sufficiently large, see Fig. 1(a), and may be followed by re-

attachment as in Fig. 1(b), thus forming a typical flow around an

airfoil. The region encompassed by the boundary layer is termed

a separation bubble after the work of Jones [13] and, as shown in

Fig. 1, it can be closed or open. Classification of separation bubbles

concerns their laminar or turbulent nature, but at a coarse level

topologically laminar and turbulent bubbles do not differ, and thus

we will not be distinguishing between various cases, but rather

treat a generic case. It should be noted that, in certain physical

situations, a bubble needs to be understood in a time-averaged

sense [14]. Given the notion of a separation bubble, our dynamical

systems model will aim to capture its characteristics, which are

important for controlling separation phenomena.

A central idea of this work is to approach the modeling

of separation bubble phenomena by identifying the key crucial

elements of the bubble dynamics, namely bifurcations and

hysteresis, in the appropriate portion of configuration space, as

sketched in Fig. 2. In this figure we show the minimal dimension

of the configuration space, defined by the bubble size x, the angle

of attack α, and the actuation amplitude w; that is, we will be

looking for the minimal model determined by the dependence of

the bubble size on the angle of attack and actuation amplitude.

This minimal dimension is motivated by the fact that while

generally there are other parameters involved, such as the

Reynolds number Re, the critical angle of attack αc, and the airfoil

thickness h, the resulting model will still have wide applicability.

This can be understood based on the aerodynamic properties

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R. Krechetnikov et al. / Physica D 238 (2009) 1152–1160

(a) Separated flow: Open bubble.(b) Re-attached flow: Closed bubble.

Fig. 1. On the notion of separation bubble.

Fig. 2. A sketch of the key dynamic elements—bifurcation and hysteresis—to be

captured by the minimal number of parameters, namely the bubble size x, the angle

of attack α, and the actuation amplitude w.

of airfoils. To explain this, we draw critical curves, i.e., when

separation takes place depending upon Re, αc, and h in Fig. 3.

As illustrated by Fig. 3(a), in the case of real airfoils, separation

occurs at finite Reynolds numbers even at zero critical angle of

attack; the higher αcthe lower the critical Reynolds number Rec;

also, the thicker the airfoil, the lower Rec. Fig. 3(b) demonstrates

the fact that the thinner an airfoil, the larger the critical angle of

attack that is required to achieve separation at a given Reynolds

number Re∗. Finally, in theαc–Re plane in Fig. 3(c) one can observe

that, for fixed airfoil thickness h∗, separation can occur at zero αc,

whichrequireshighenoughReynoldsnumbers.Sinceinrealitythe

Reynolds numbers are huge (e.g. for real aircraft Re varies between

106and 1011), one concludes that limiting ourselves to ‘‘thick

airfoils’’, which can, in fact, be regarded as real airfoils since they

have to carry structural load and fuel, is not a serious restriction

in this first step towards low-dimensional modeling of separation

phenomena.

Having identified the key elements of the bubble dynamics

in the configuration space, we construct a model by successively

addressing bifurcation phenomena in Section 3 and hysteresis

phenomena in Section 4.

3. Bifurcation in the dynamics of separation bubble

3.1. On the notion of bifurcation

As was noted in Section 2, bubbles can be either in a closed

or open state. This allows us to introduce the first key element of

the low-dimensional modeling, namely it must capture this basic

bifurcation from an open to a closed state, as shown schematically

in Fig. 4, which is also known as bursting [15].

Notably, the fact that this is the primary bifurcation was

realized just recently [16]. While from the vast literature one can

get the impression that one separated flow is not like any other,

here we take a different point of view, i.e. we treat the coarse

behavior of separation bubbles as (generic) phenomena that can

be modeled by a single low-dimensional dynamical system.

3.2. Quantifying separation bubbles

To quantify the behavior of a separation bubble, consider

the coordinate x, measuring the distance along the airfoil from

the bubble onset to the bubble reattachment, as shown in

Fig. 4. The bubble dynamics in the first approximation can be

described by two parameters: the location of separation, xs, and of

reattachment, xr, which can move under the change of flight and

control parameters. In some cases, e.g. the Glauert Glas II airfoils,

the separation point xsremains fixed for all practical purposes.

Therefore, we will start by considering only the behavior of the

reattachment point, which experiences a primary bifurcation in

the above sense; extending the model to include variation of xs

will require the addition of a reliable separation criterion. As an

alternative to xr, one could also utilize the bubble area. From now

on we will use x as a variable representing the bubble state.

3.3. On the physical nature of bifurcation

The mechanism by which the excitation affects the flow lies

in the generation of instabilities, and thus of large coherent

structures [17] transferring high momentum fluid towards the

(a) Re vs. h for αc= 0 (solid

curve) and αc> 0 (dashed

curve).

(b) αcvs. h at fixed Re = Re∗. (c) Re vs. αcat fixed h = h∗.

Fig.3. TheplacementofrealairfoilsinthespacedefinedbytheReynoldsnumberRe,thecriticalangleofattackαc,andtheairfoilthicknessh:thecriticalcurvescorresponding

to the instant when separation occurs. Re∗and h∗are typical fixed values of these parameters.

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Fig. 4. Basic setup and primary bifurcation.

Fig. 5. On the mechanism of actuation.

Fig. 6.

w and the frequency ω. The shaded region corresponds to a reattached flow

(closed bubble). Arrows indicate the change in location of the transition curve with

increasing Re and α.

Effect of time-varying actuation: criticality of the actuation amplitude

surface, see Fig. 5(b), and therefore leading to reattachment, as

indicated in Fig. 5(c). Since actuation exploits the instabilities of

the shear layer [18], the response to actuation depends on both

amplitudew andfrequencyω,andthereforeisnonlinear.Thelatter

again indicates, now from the point of view of actuation control

mechanisms, that the low-dimensional model must be nonlinear.

As follows from experiments with synthetic (zero mass flux) jet

actuation, the critical phenomena are as sketched in Fig. 6, where

the shaded region corresponds to a reattached flow (that is, a

closed bubble) and the arrows indicate a change in location of the

transition curve with an increase inα and Re, respectively. The size

of the bubble, x, has a specific dependence on the amplitude w

and frequency ω of actuation, i.e. ∂x/∂w < 0,∂x/∂ω < 0, when

moving away from the origin (w,ω) = 0 in Fig. 6. In this work we

focus on the case of time-invariant actuation, ω = 0, although the

time-varying case will be commented on later in this section.

Finally, it is notable that the criticality and hysteresis phenom-

ena depend on the connectedness of the flow domain: the bubble

experiences bifurcation only in the case of flow around an airfoil,

as in Fig. 7(b), while in the case of a hump model in Fig. 7(a), which

is frequently used in experiments, e.g. [19], there is no bifurcation.

Thus, there are two basic configurations in which the behavior of

theseparationbubblediffers:thehumpmodelandtheairfoilmodel.

Namely, in the hump case x(w) is smooth, while in the case of an

airfoilx(w)isdiscontinuous.Also,aswillbeimportantinSection3,

thehysteresisphenomenaarepresentonlyintheairfoilcase.Here,

in view of its practical importance, we naturally focus on the airfoil

case.

3.4. Modeling the bubble bifurcation

In developing a model, we are guided by the principle of

a minimal complexity together with the physical requirements

one has to meet. At the methodological level, there are two

basic ways to account for the form of x(w), which has both the

saturation and criticality shown in Fig. 7: (a) to design an algebraic

relation f(x,w) = 0, or (b) to introduce a dynamic description

f(x, ˙ x, ¨ x,...,w) = 0. The latter approach is better suited for

dynamics and control purposes, because in the case of active

feedback control one would need to deal with a few characteristic

times and transient effects, and thus the model should be

time dependent. The simplest possible way of introducing time-

dependent dynamics is a second-order oscillator model,

¨ x − µ˙ x = F(x,w),

where µ is a damping parameter. The justification for the latter

is the fact that both separation and reattachment points may

oscillate [14].

In what follows, we first formulate the mathematical require-

ments on a model in Section 3.4.1, then, by appealing to the ideas

of a potential function in Section 3.4.2 and a dynamic bifurca-

tion in Section 3.4.3, we identify a specific form of model (2) in

Section 3.4.4.

(2)

3.4.1. Mathematical requirements

Naturally, the bubble size x also depends on a flight parameter,

in our case the angle of attack α, which needs to be incorporated

in the model (2); thus, F in (2) in enhanced to F(x,w,α). Since

we want to minimize the functional complexity but retain the

true nonlinear features of the phenomena, the simplest form is a

quadratic nonlinearity,

F(x,w,α) = x2+ b(w,α)x + c(w,α),

(3)

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R. Krechetnikov et al. / Physica D 238 (2009) 1152–1160

Fig. 7. Two basic experimental configurations.

(a) Uncontrolled.(b) Controlled.

Fig. 8. Takens–Bogdanov bifurcation.

which will be justified by its physical implications and the poten-

tialfunctionapproachinSection3.4.2.Eq.(2)withthenonlinearity

givenby(3)possessesaTakens–Bogdanovbifurcation,asshownin

Fig. 8, when b2− 4c changes sign.

Indeed, the equilibrium points are given by x1,2 = −b

1

2

The eigenvalues of the linearization around x = x1are defined

by λ2

b2− 4c, while the eigenvalues of the

linearization around x = x2are λ2

when b2− 4c changes sign, one observes the transition from the

picture in Fig. 8(a) to the one in Fig. 8(b). The requirements on the

parameters in this model are dictated by the physics:

2∓

√

b2− 4c, so that F can be represented as (x − x1)(x − x2).

(1)= x1− x2 = −√

(2)= x2− x1=√

b2− 4c. Thus

(a) The stability of equilibria points should obey

α < αc: b2− 4c > 0

α > αc: w > wc(α),b2− 4c > 0

(stability: no separation),

w < wc(α), b2− 4c < 0

where stability implies that one equilibrium is stable (λ is

imaginary),andthesecondoneisunstable(λisreal).Theabove

inequalities indicate that the physical behavior of the model is

alsogovernedbythecriticalangleofattackαc(whichisdefined

when the flow separates at w = 0), and the critical control

amplitude wc(α), when flow reattaches at α fixed.

(b) The critical actuation amplitudewcshould grow with(α−αc),

since the higher the angle of attack, the more control input is

required to make the flow reattach.

(stability),

(instability: separation),

(c) The bubble size x, which is a stable equilibrium, should shrink,

x → 0, as w increases. At the same time, the domain of

attraction of this equilibrium should shrink too, so that the

bubblebecomessusceptibletoafinite-amplitudeinstability,as

is known to be the case from experiments; see the upper part

of Fig. 6.

3.4.2. Potential function approach

In order to get better insight in the model construction, let us

assign a potential function V(x), where dependence on w and α

is suppressed for the sake of this discussion, such that V?(x) =

−F(x):

V(x) = −x3

which is physically determined by the effective elastic properties

of a bubble coming from its interaction with the outer flow. The

elasticity of the separation bubble is evidenced by introducing

disturbances outside the bubble and observing the changes in

the bubble characteristics, i.e. shape and pressure inside.1Since

the effective tension of the shear layer tends to maximize the

bubble boundary by making it open and only the external energy

input (excitation) counterparts this effect and makes the bubble

closed, the separation bubble boundary elastic properties should

be modeled with negative interfacial tension −? σ.2The latter is

Given this justification for the origin of the potential energy,

we can next observe that a finite bubble corresponds to V(x) as in

Fig. 9(a), and an infinite bubble corresponds to Fig. 9(b). Therefore,

now from an energetic point of view, we can conclude that a

3− b(w)x2

2− c(w)x − d(w),

(4)

opposite to the behavior of real bubbles with positive interfacial

tension, which usually tends to minimize the interfacial area.

1Both elasticity and a non-trivial state equation of the separation bubble have

been confirmed experimentally (personal communication: John Kiedaisch, IIT).

2Alternatively, one can use a non-trivial state equation for the pressure inside

the bubble, which can in principle be measured experimentally.

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(a) Potential function for a finite bubble.(b) Potential function for an infinite bubble.

Fig. 9. Potential function V(x); d = 0, c = 0 in (4).

Fig. 10. Critical curve in the (x,w)-plane: on the dynamic bifurcation; the solid

black line represents stable equilibria, the dot–dash line is a dynamic bifurcation

when the bubble grows indefinitely with time. Phase portraits in rectangles

correspond to the ones in Fig. 8.

quadratic nonlinearity (or cubic potential V) is adequate for the

descriptionofthebifurcationphenomenon(seealsothediscussion

in Section 3.4.3).

Without loss of generality, we can assume that d

Considering w as a control parameter, the requirements on the

coefficients in V(x) are such that the equilibria, V?(x) = −x2−

bx − c = 0, obey

w > wc : two equilibria (stable and unstable),

V??(x1) > 0,

w ≤ wc: only one equilibrium point, which is unstable

(marginally stable).

=

0.

V??(x2) < 0;

The stability conditions can also be reformulated in terms of

eigenvalues, as indicated in Fig. 10. For illustrative purposes, in the

particular case when c = 0 the equilibria points xi(w,wc) are

easily computable: x1 = −b and x2 = 0. The stability criterion

(second variation) for these equilibria is given by the sign of the

second derivative, V??(x) = −2x − b, which at the equilibria

points assumes the values b and −b, respectively. Besides the

stability conditions, one needs to impose dx1/dw < 0, since the

bubble shrinks when the control amplitude increases. Thus, the

bifurcation from the state in Fig. 9(a) to the one in Fig. 9(b) is

obviously associated with the condition when b(wc) = 0.

In general, as one can further infer, in the space of curves in

(w,wc) there is an infinite number of solutions b = b(w,wc),

c = c(w,wc). In practice, a systematic procedure would be as

follows: depending on the particularities of the experimental data,

one expands b and c in terms of some basis functions of w, wc,

etc., and then determines coefficients in that expansion through

a calibration procedure based on experimental data. The latter

procedure is, in fact, used in aerodynamic models (i.e. connecting

forces and moments) [4,20].

3.4.3. Dynamic bifurcation

The transition from one potential to another is controlled by

a bifurcation parameter, such as angle of attack α or actuation

amplitudew.Infact,thelattertwoparametersareinterchangeable

to a certain extent as argued in [21], since a change in α or in w

leads to a change in circulation around an airfoil, and thus to a

change in the flow structure. Apparently, this transition of x from

finite to infinite is dynamic in the sense that the bubble becomes

infiniteinFig.9(b)astimet → ∞.Thisdynamicbifurcationcanbe

clarified using phase portraits in Fig. 10, and should be compared

to the standard static bifurcation, which is of algebraic nature,

resulting from the condition of vanishing vector fields. As one can

learn from Fig. 10, at the critical value of the actuation amplitude

wcboth equilibria coincide and are unstable with respect to any

infinitesimal disturbances (marginally stable), so that the bubble

grows with time and becomes unbounded for t → ∞. Forw > wc

there are two equilibria points: one is unstable and the other is

stable.Thelattercorrespondstothesituationwhenthebubbleisof

finite size, and this state has a finite domain of attraction. Note that

the potential energy shape, as in Fig. 9, is crucial in allowing the

‘‘dynamic’’ bifurcation: a V-shaped potential function apparently

would not allow this type of bifurcation; in addition the domain of

attractionwouldbemodeledinconsistentlywithphysics.Asimilar

type of argument will be applied to the hysteresis phenomena

in Section 4. In conclusion, having identified, based on physical

arguments, that the potential should be of the shape as in Fig. 9

in order to allow a dynamic bifurcation, the problem reduces to

determination of the coefficients in (4). This general procedure is

the subject of the singularity theory [11].

3.4.4. The model and its interpretation

For simplicity, restricting ourselves to the case of thick airfoils

when separation occurs at αc= 0 without actuation, with one of

infinitely many admissible choices of b and c we get

¨ x + µ˙ x = −Vx(x;α,w) = (x − α)2+ f(w)x.

Here f(w) = a1w + a2w2+ ··· represents a nonlinear response of

theseparationregiontoactuatorexcitationsw.Theproductf(w)x

implies that the effect of actuation depends upon the bubble size

x. As required, f(wc) = 2α1/2and the bubble shrinks as ∼ f−1

(5)

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R. Krechetnikov et al. / Physica D 238 (2009) 1152–1160

Fig. 11. Experimental effect of time-varying actuation: Hysteresis phenomenon as

it depends on the amplitude w.

for w → ∞. While this is the simplest possibility, from the above

descriptionitisclearthatthereisenoughflexibilitytocalibratethe

model through the fitting functions, F(x,w,α), and parameters,

(b,c,...) within the bounds given above.

By construction, the model (5) reflects the basic generic

dynamic behavior of separation bubbles. In the conservative time-

invariant case the parametric space is just(α,f(w)). When control

is absent, f(w) = 0, the bubble is open, which corresponds to an

unstable phase portrait in Fig. 8(a); that is, any initial conditions

lead to an unbounded bubble size x. When sufficient control is

applied(considerfirstα fixed),thebubblecloses,whichisreflected

in the change of the phase portrait as shown in Fig. 8(b). In this

case there are two equilibrium points: one is a saddle, which is

unstable and thus not physically observable, and the other is a

stable center. Therefore, there exists a non-zero basin of attraction

which leads to a finite bubble size, x < ∞. Fig. 8(b) also suggests

that the system is susceptible to a finite-amplitude instability for

w > wc, a fact which is conceivable but has never been studied in

experiments systematically. Nevertheless, it is known empirically

that the bubble opens if the actuation amplitude becomes large

enough, as in Fig. 6; see also [22]. Also, the fact that the boundary

layer is susceptible to finite-amplitude instabilities [23] suggests

that the separation bubble formed out of it may also be finite-

amplitude unstable. The inclusion of dissipation in the model (5)

does not change the nature of the phase portrait; however, it does

change the basin of attraction.

Finally, the inclusion of time-varying effects in the control,

w = w0cosωt with ω ?= 0, also demonstrates that the bubble

transforms from an open to a closed state. Thus, as required, the

model (5) captures the primary bifurcation and dynamic behavior

of the separation bubble, except for the phenomenon of hysteresis.

In the rest of this paper we will explain how the model (5) can be

enhanced to account for the hysteresis shown in Fig. 11. While the

model (5) is illustrated for a particular choice of coefficients in (3),

there is obviously enough freedom to choose parameters, which is

important in order to fit the model to a particular application via

calibration.

It is notable that a model of a similar form was deduced in

an ad hoc way for a real bubble deforming in a straining flow of

Taylor’s four-roll mill studied by Kang and Leal [10]. This system

experiences a bifurcation from a deformed but closed state to an

open tip-streaming state, when the original bubble forms pointed

open ends that emit tiny bubbles.

4. Hysteresis in the dynamics of separation bubble

4.1. Experimental observations: The model objectives

The basic effects of time-varying control were discussed in

Section 3.3 and reflected in Fig. 6. However, the effect of changing

amplitude and frequency is not trivial in view of the presence of a

hysteresis [24,25] in all variables(α,w,ω), as illustrated in Fig. 11

for the dependence of the bubble size on the actuation amplitude,

x(w). Experiments demonstrate that hysteresis is present no

matter how slowly the actuation amplitude w is changed, which

suggests that the model should depend only on the sign of the

rate ˙ w.

Therefore, the model should reflect the fact that there are two

stable steady state solutions for the range of the control parameter

wc1

< w< wc2, as in Fig. 11, which is an experimental

fact. Mathematically, this means that the selection between these

two solutions is due to the placement of initial conditions in the

corresponding domain of attraction. Also, for w > wc2, there

should be only one stable solution, while for w < wc1the bubble

should ‘‘bifurcate’’ to infinity in a dynamical manner, as described

in Section 3.4. The challenge of modeling the hysteresis, similar to

that of bifurcation, comes from the fact that there are no analytical

results on the behavior of the bubble. In particular, the domain

of attraction of stable solutions is not well characterized from the

existing empirical data.

Therefore, in order to model hysteresis, one first needs

to understand its physical origin, which is addressed below,

in Section 4.2, where we suggest the physical mechanisms

of hysteresis. This together with the dynamical systems and

singularity theory allow us to modify the model (5) to account for

hysteresis, which is the subject of Section 4.3.

4.2. On the physics of hysteresis

Physically, separation bubbles are caused by a strong adverse

pressure gradient, which makes the boundary layer separate from

the curved airfoil surface. Actuation with w > wc effectively

reduces the adverse pressure gradient3and makes the bubble

closed, as in Fig. 4(b). This can be seen from Bernoulli’s equation,

since the velocity drop is related to the pressure rise, p+ρu2/2 =

p0, where p is a dynamic pressure, and p0is the fluid pressure at

rest. From Bernoulli’s equation and Fig. 4 we can conclude that

the pressure rise p?

p?

the second-order effects of vorticity and thus assume constant

pressure inside the bubble, p1= p2= const, which is the standard

assumption when modeling the separated flow about an airfoil by

Kirchhoff’s zone of constant pressure [26,2].

Given the fact that the separation bubble boundary possesses

elastic properties, see Section 3.4.2, we can provide a simple

mechanistic explanation of the origin of the hysteresis. For

simplicity, consider the two-dimensional situation depicted in

Fig. 12: a hemispherical bubble having variable size with the left

end fixed and with its right end free to move, thus modeling a

separation bubble with moving reattachment point. The bubble

size changes depending upon the free-stream velocity umax, which

is chosen to be the control parameter. When umaxincreases and

the right end of the bubble reaches the trailing edge at R0at critical

u2

p0 → p0+ ∆p0, which is due to suction of a high pressure fluid

i− p??

iand bubble length l correlate p?

1− p??

1<

2− p??

2and l1< l2, respectively. For current purposes we neglect

cr, the pressure inside the bubble increases by a finite amount,

3Notethatforsomeairfoilsthesameeffectcanbeachievedbychangingtheangle

of attack α, i.e., the larger the angle of attack α, the stronger the adverse pressure

gradient: this ‘‘interchangeability’’ of the effects of the actuation amplitude and the

angle of attack is well known [21], and is reflected in the dependence wc(α).

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R. Krechetnikov et al. / Physica D 238 (2009) 1152–1160

1159

Fig. 12. Mechanical model of separation bubble hysteresis.

from the lower side of the airfoil. Hence the bubble size increases

abruptly by some amount. Conversely, when umaxdecreases and

the bubble reaches the trailing edge at R0at a different critical

u1

p0→ p0− ∆p0. The jump in pressure at the critical point – when

reattachment is at the trailing edge – has the following physical

explanation. It is known that the lift drops when the bubble opens,

which effectively means that the pressure balance between the

lower and upper surfaces of an airfoil has changed: some amount

of pressure at the lower surface has leaked into the upper surface,

namely into the bubble. The latter is allowed by unsteadiness of

the process, i.e., the unsteady Kutta–Joukowsky condition.

Therefore, the mechanical analog of a separation bubble is p =

p0+ ? σ/R,p > p0, so that the bubble grows when the ambient

? σ

u1

p∞− p0− ∆p0−ρ(u1

which produces a hysteretic behavior; see Fig. 13. Obviously, u1

u2

In the light of the above, one can account for the hysteresis in

Fig. 11 in model (5) as follows. When ˙ w < 0 andw passes through

wcthe transformation w → w − ∆w with ∆w > 0 is applied,

since physically the effectiveness of control drops by ∆w. When

˙ w > 0 and w = w?

amountofcontrolhasbeenappliedbeforereachingw = w?

altogether lead to the desired hysteretic behavior. The same can

be done to account for the hysteretic dependence on the angle of

attack α.

The above mechanistic model of hysteresis captures the

physics and proves that the separation bubble has a non-trivial

potential function associated with it. It should be noted that such

discontinuous modeling of hysteresis based on the rate sign ˙ w

is still widely used in applications and known as play and stop

(classical Prandtl model) models; see Visintin [27].

cr, the pressure inside the bubble relaxes to its original value,

pressure dictated by Bernoulli’s equation, p = p∞− ρu2

decreases:

max/2,

u2

cr(˙ umax> 0) : R0=

p∞− p0−ρ(u2

cr)2

2

,

(6a)

cr(˙ umax< 0) : R0=

? σ

cr)2

2

,

(6b)

cr<

cris consistent with physical observations.

cthen w → w + ∆w, since a too conservative

c.These

4.3. Accounting for hysteresis in model (5)

However, for the purpose of deriving a universal model which

combines both the bifurcation and the hysteresis in a dynamic

manner, i.e. suitable for control purposes, it makes sense to follow

another way of modeling hysteresis phenomena, based on the

construction of an appropriate potential function V(x), similar to

what was done in Section 3.4.

The grounding thesis is that the true curve of states in Fig. 11

is not the solid discontinuous one, but rather the ‘‘true’’ picture for

separationbubblescorrespondstothesmoothcurve(includingthe

dashed line portion) in Fig. 11, a fact which has not been realized

Fig. 13. Schematics of model (6) for hysteresis.

Fig. 14.

functions; solid black lines represent stable equilibria, while dashed lines are

unstable equilibria; the dot–dash line represents a dynamic bifurcation (the bubble

size grows with time unboundedly). The V(x) plots in rectangles show the shape of

the potential for w < wc1, w ∈ [wc1,wc2], and w > wc2, respectively.

Hysteresis curve in the (x,w)-plane and corresponding potential

in the literature on separation bubbles before.4This smooth curve

corresponds to the equilibria states of an appropriate potential

function V(x); the dashed curve is not physically observable in

view of the instability of the corresponding equilibrium states.

From Section 3.4 we know that the potential function should be

of special shape, i.e. when x → ±∞, then V(x) → ∓∞, that

is the highest-order terms in α should be odd. Then, as a natural

generalization of the picture in Fig. 10, we arrive at Fig. 14.

At the technical level, the lowest-order potential suitable for

achieving the picture in Fig. 14 is of the fifth order, so model (5)

becomes

¨ x + µ˙ x = −Vx(x;α,w),

with V being the fifth-order polynomial of x. The existence of

such a potential is apparent, and its coefficients in the polynomial

representation can be found with the help of linear programming

given a set of inequalities and equalities based on the calibration

requirements. By design, model (7) accounts for the observed

hysteretic behavior; see Section 4.2.

In conclusion, the fact that the hysteresis originates from

the particularity of the potential function is well known from

(7)

4However, this fact is known in other physical situations: e.g. a ferromagnetic

drop deforming in a magnetic field [28] and cavitating hydrofoils [29].

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R. Krechetnikov et al. / Physica D 238 (2009) 1152–1160

other physical systems, e.g. a ferromagnetic drop deforming in a

magnetic field [28] and cavitating hydrofoils [29]. For example, the

ferromagnetic drop shape is a simple balancing between magnetic

and interfacial energy of the drop: the former tends to elongate

the drop, while the latter tends to make the drop spherical, which

leads to hysteretic behavior. In the case of a cavitation bubble on a

hydrofoil, the hysteresis can be explained with the help of inviscid

free-streamline theory [30,31] since the boundary of the bubble

is well defined physically: the predictions of the free-streamline

theory [32] agree well with experiments [33].

5. Conclusions

This work has focused on the fundamental aspects—the most

important physical and dynamic behavior—of a generic separation

bubble using thick airfoils as a paradigm. Given an incomplete

experimental knowledge of the complex phenomena of separation

bubble, we applied the deduction based on bifurcation and

singularity theories and thus (i) filled in incomplete pieces in

the dynamical picture of the phenomena, (ii) advocated that this

dynamical picture is finite dimensional at the coarse level, (iii)

developedaconstructivewayofbuildingamodel,and(iv)deduced

a model.

The model can be further enhanced in particular by (a)

incorporating a non-trivial state equation of a bubble, (b)

accounting for separation at non-zero angle of attack ac ?= 0,

and (c) calibrating the model for a given airfoil according to the

procedure outlined in Sections 3.4.2 and 4.3. The approach taken

here allows one to generalize models (5) and (7) hierarchically:

e.g. to account for the three-dimensionality of a separation bubble

one would need to introduce a second bubble size variable and

to come up with physical reasonings on its dynamics. These

are the potential future directions of this study and will require

considerable theoretical and experimental efforts. We also expect

that this approach to low-dimensional modeling will be helpful in

real-time flow control.

Acknowledgements

R.K. would like to thank Prof. Anatol Roshko for helpful and

encouraging discussions. This work was supported in part by NSF-

ITR Grant ACI-0204932.

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