A low-dimensional model of separation bubbles
ABSTRACT 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.
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ABSTRACT: A low-order point vortex model for the two-dimensional unsteady aerodynamics of a flat plate wing section is developed. A vortex is released from both the trailing and leading edges of the flat plate, and the strength of each is determined by enforcing the Kutta condition at the edges. The strength of a vortex is frozen when it reaches an extremum, and a new vortex is released from the corresponding edge. The motion of variable-strength vortices is computed in one of two ways. In the first approach, the Brown–Michael equation is used in order to ensure that no spurious force is generated by the branch cut associated with each vortex. In the second approach, we propose a new evolution equation for a vortex by equating the rate of change of its impulse with that of an equivalent surrogate vortex with identical properties but constant strength. This impulse matching approach leads to a model that admits more general criteria for shedding, since the variable-strength vortex can be exchanged for its constant-strength surrogate at any instant. We show that the results of the new model, when applied to a pitching or perching plate, agree better with experiments and high-fidelity simulations than the Brown–Michael model, using fewer than ten degrees of freedom. We also assess the model performance on the impulsive start of a flat plate at various angles of attack. Current limitations of the model and extensions to more general unsteady aerodynamic problems are discussed.Theoretical and Computational Fluid Dynamics 09/2012; 27(5). · 1.75 Impact Factor
Physica D 238 (2009) 1152–1160
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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
Received 8 October 2007
Received in revised form
23 May 2008
Accepted 22 March 2009
Available online 8 April 2009
Communicated by M. Silber
Flow separation control
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.
1.1. 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 , 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
E-mail address: firstname.lastname@example.org (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
not actually necessary, since in reality one can use sensors on the
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
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 . 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
requirements on a model aimed for robust control.
0167-2789/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
R. Krechetnikov et al. / Physica D 238 (2009) 1152–1160
1.2. Previous works
In one of the early works, , 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:
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
model . 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.  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
et al. . 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,
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
, 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
be appreciated . 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
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.
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
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
as Duffing’s equation for the buckling of elastic beams , simple
maps to describe a dripping faucet , which even capture the
observed chaotic behavior to a great extent, and bubble dynamics
in time periodic straining flows , 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
features of the physical system, necessary for control purposes.
1.4. Paper outline
idea of our approach in Section 2, and then appeal to the tools of
the bifurcation and singularity theory , as will be made precise
in Section 3. The outline of the paper is as follows. In Section 3, we
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  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  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 . 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
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,
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
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 .
Notably, the fact that this is the primary bifurcation was
realized just recently . 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  transferring high momentum fluid towards the
(a) Re vs. h for αc= 0 (solid
curve) and αc> 0 (dashed
(b) αcvs. h at fixed Re = Re∗. (c) Re vs. αcat fixed h = h∗.
to the instant when separation occurs. Re∗and h∗are typical fixed values of these parameters.
R. Krechetnikov et al. / Physica D 238 (2009) 1152–1160
Fig. 4. Basic setup and primary bifurcation.
Fig. 5. On the mechanism of actuation.
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 , the response to actuation depends on both
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. , there is no bifurcation.
Thus, there are two basic configurations in which the behavior of
Namely, in the hump case x(w) is smooth, while in the case of an
in view of its practical importance, we naturally focus on the airfoil
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
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
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
F(x,w,α) = x2+ b(w,α)x + c(w,α),
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-
Fig. 8, when b2− 4c changes sign.
Indeed, the equilibrium points are given by x1,2 = −b
The eigenvalues of the linearization around x = x1are defined
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:
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
inequalities indicate that the physical behavior of the model is
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.
(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
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) =
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
2− c(w)x − d(w),
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.
R. Krechetnikov et al. / Physica D 238 (2009) 1152–1160
(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
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
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
to a certain extent as argued in , 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
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
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 .
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
implies that the effect of actuation depends upon the bubble size
x. As required, f(wc) = 2α1/2and the bubble shrinks as ∼ f−1
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
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
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 . Also, the fact that the boundary
layer is susceptible to finite-amplitude instabilities  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
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 . 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
< 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?
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
p0 → p0+ ∆p0, which is due to suction of a high pressure fluid
iand bubble length l correlate p?
2and l1< l2, respectively. For current purposes we neglect
cr, the pressure inside the bubble increases by a finite amount,
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 , and is reflected in the dependence wc(α).
R. Krechetnikov et al. / Physica D 238 (2009) 1152–1160
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
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
p∞− p0− ∆p0−ρ(u1
which produces a hysteretic behavior; see Fig. 13. Obviously, u1
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
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 .
cr, the pressure inside the bubble relaxes to its original value,
pressure dictated by Bernoulli’s equation, p = p∞− ρu2
cr(˙ umax> 0) : R0=
cr(˙ umax< 0) : R0=
cris consistent with physical observations.
cthen w → w + ∆w, since a too conservative
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
dashed line portion) in Fig. 11, a fact which has not been realized
Fig. 13. Schematics of model (6) for hysteresis.
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)
¨ 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
4However, this fact is known in other physical situations: e.g. a ferromagnetic
drop deforming in a magnetic field  and cavitating hydrofoils .
R. Krechetnikov et al. / Physica D 238 (2009) 1152–1160
other physical systems, e.g. a ferromagnetic drop deforming in a
magnetic field  and cavitating hydrofoils . 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  agree well with experiments .
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)
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.
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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