Physica D 239 (2010) 190–201

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Physica D

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

On continuation of inviscid vortex patches

Federico Gallizio a,b, Angelo Iollo b, Bartosz Protas c,⇤, Luca Zannetti aa Dipartimento di Ingegneria Aeronautica e Spaziale, Politecnico di Torino, 10129 Torino, Italyb Institut de Mathématiques de Bordeaux UMR 5251 CNRS, Université Bordeaux 1 and INRIA Futurs MC2, 33405 Talence cedex, Francec Department of Mathematics & Statistics, McMaster University, Hamilton, Ontario, Canada

a r t i c l e i n f o

Article history:Received 18 January 2009Accepted 19 October 2009Available online 28 October 2009Communicated by M. Vergassola

PACS:47.15.ki47.32.Ff

Keywords:Euler equationVortex patchesExistence of solutionsContinuationSteklov–Poincaré iterations

a b s t r a c t

This investigation concerns solutions of the steady-state Euler equations in two dimensions featuring fi-nite area regions with constant vorticity embedded in a potential flow. Using elementary methods of thefunctional analysis we derive precise conditions under which such solutions can be uniquely continuedwith respect to their parameters, valid also in the presence of the Kutta condition concerning a fixed sepa-ration point. Our approach is based on the Implicit Function Theorem and perturbation equations derivedusing shape-differentiationmethods. These theoretical results are illustratedwith careful numerical com-putations carried out using the Steklov–Poincaré method which show the existence of a global manifoldof solutions connecting the point vortex and the Prandtl–Batchelor solution, each of which satisfies theKutta condition.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

This work addresses certain fundamental properties of a flowmodel of interest in the study of massively separated flows pastbluff bodies. It is motivated by the field of flow control whereone of the recurring themes is the stabilization of unsteady vor-tex wakes with applications involving aeronautical or terrestrialvehicles [1]. Stabilization of such flows with a vanishingly smallcontrol effort is often based on the premise that there exist steadyunstable solutions characterized by recirculation regions withclosed streamlines for high enough Reynolds numbers. Therefore,in what follows the flow is considered effectively inviscid, so thatthe incompressible Euler equation is the relevant model. A furtherassumption is that the flow is two-dimensional (2D). Despite theirsimplicity, such models are capable of describing the large-scalevortex dynamics aswell as vortex equilibria, and these features areoften sufficient to allow for the use of suchmodels as a basis for de-velopment of effective control strategies [2]. With such problemsin mind, in the present investigation we use a combination of rig-orous mathematical analysis and careful numerical computationsto address an important theoretical question concerning the Euler

⇤ Corresponding author. Tel.: +1 905 525 9140x24116; fax: +1 905 522 0935.E-mail address: bprotas@mcmaster.ca (B. Protas).

equation, namely, the existence of a continuous family of solutionscharacterized by growing vortex patches.

To fix attention,we consider the flowdomain⌦ ⇢ R2 as shownin Fig. 1 which is motivated by the shape of high-lift devices usedin some experimental aeronautical applications (lift is increasedas a result of ‘‘trapping’’ a vortex in the cavity [3]; see Fig. 1).Points belonging to the domain ⌦ will be denoted by x = (x, y).While in such applications the domain⌦ is consideredunbounded,in order to avoid technical complications, in the mathematicalanalysis wewill assumewithout loss of generality that this domainis bounded. The numerical computations reported in this paperuse an unbounded domain. The domain boundary is assumedpiecewise smooth and not necessarily Lipschitz (i.e., cusps areallowed). Our domain ⌦ is constructed so that the upstream anddownstream boundaries coincide with the OX axis. As a result, thisdomain may also be used to model flows in the entire plane pastsymmetric obstacles where the OX axis is the axis of symmetry.Expressing the velocity u in terms of the streamfunction : ⌦ !R as u = [u, v] =

h@ @y , � @

@x

i, the 2D steady-state Euler equation

can be rewritten as the following boundary value problem [4]

r2 = F( ) in⌦, (1a) = b on @⌦, (1b)where F : R ! R is an a priori undefined function, and b :@⌦ ! R is a function corresponding to the boundary conditions

0167-2789/$ – see front matter© 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2009.10.015

F. Gallizio et al. / Physica D 239 (2010) 190–201 191

Fig. 1. Schematic of the flow domain: A represents the vortex patch with theboundary @A and with a constant vorticity ! embedded in a potential stream, Pis the separation point and @⌦ the boundary of the flow domain.

on the normal velocity component. On the parts of the boundary@⌦ where the normal velocity vanishes b is equal to a constantimplying that such parts of the boundary are streamlines. On theother hand, on the parts of the boundarywhere the normal velocitydoes not vanish (e.g., the inflow and outflow of the channel) bis obtained by integrating the relationship @ b

@s = �u · n, wheren represents the unit normal vector facing out of the domain⌦ , and s is the arc-length coordinate parameterizing the domainboundary @⌦ and having positive orientation. Eq. (1a) expressesthe fact that the vorticity ! = �F( ) has a constant valueon the streamlines in inviscid 2D time-independent flows. Thus,for regions with closed streamlines, this vorticity is not definedby the far-field boundary condition (1b) and, as a consequence,system (1)may admitmultiple solutions. In principle, each of thosevortex solutions could be adopted as a model for a separated flowin the infinite Reynolds number limit, and choosing the relevantone is a long-standing problem in theoretical hydrodynamics; werefer the reader to the monograph [5] for a survey of availableresults. This multiplicity of solutions is reflected in the differentdistributions of vorticity �F( ) which can be adopted for theregion with closed streamlines. Batchelor [6,7] argued that thelimiting solution for the viscous flowwith the Reynolds number Regoing to infinity is characterized by ! = const in the region withclosed streamlines, i.e., the finite areawake effectively reduces to avortex patch. Moreover, in general, solutions with vortex patchesmay also feature a jump 1h of the Bernoulli constant across thevortex sheet that separates the recirculating flow from the externalpotential flow. The present study addresses inviscid solutions thatin a broad sense belong to the family of such models, also referredto as the Prandtl–Batchelor flows [8]. Inviscid flows described by(1), but construed as limits of certain Navier–Stokes solutions asRe ! 1, can be made unique by taking into account ‘‘traces’’of viscous phenomena. Indeed, it was shown by Chernyshenko,see e.g. [9], that using the condition that the boundary layer becyclic removes one degree of freedom in the choice of the problemparameters. For the sake of simplicity, however, we will neglectthe vortex sheet at the patch boundary, which is equivalent tosetting1h = 0. A physical justification for this assumption is thata turbulent mixing layer at a high Reynolds number should cancelthe jump of the Bernoulli constant1h. Thus, the entire steady flowfield past a bluff body is then modelled as the coupling of twoinviscid flow regions (Fig. 1): an irrotational region exterior to thewake, and the wake with ! = const , which in terms of the right-hand side (RHS) in (1a) is expressed as

F( ) = �!H(↵ � ), (2)

whereH(·) denotes the Heaviside function,! is the (constant) vor-ticity, whereas ↵ 2 R is the value the streamfunction assumeson the boundary @A of the vortex patch A (cf. Fig. 1).

An even simpler model of finite area wakes is provided by pointvortices. In this model, the region with closed streamlines is an

irrotational flow region with a vortex singularity. As observed byseveral authors [10–13], vortex patch solutions can be seen as thefinal result of an accretion process that starts from point vortices.Our point of departure for this work is the following

Conjecture 1 ([14]). If there is no point vortex in equilibrium thatsatisfies the Kutta condition, then the associated family of growingpatches, including the limiting Prandtl–Batchelor solution, also doesnot exist.

We note that this conjecture is in contradiction with some resultspresent in the literature. While it is well known that an invis-cid flow past a flat plate broadside to the oncoming stream doesnot admit any point vortex equilibria (we mention the review pa-per [15] for some interesting remarks concerning the history ofthis problem), Turfus [16] numerically detected finite area vortexpatches in such flow configurations. The existence of a closedwakein the inviscid flowpast normal plateswas also discussed by Turfusand Castro [17] who demonstrated that a cyclic boundary layer iscompatible with the finite area solution determined previously byTurfus. More recently, Castro [18] obtained computational resultssuggesting possible existence of a second branch in the graph rep-resenting the wake size versus the Reynolds number which wouldextrapolate to a finite area vortex in the limit of an inviscid flow.We also note that results indicating possible existence of solutionscontradicting Conjecture 1 were reported in [10]. In an attempt atresolving this conundrum, in the present work we make a step to-ward proving Conjecture 1 by demonstrating that vortex patch so-lutions of Euler equation (1)–(2) can in fact be locally continuedwith respect to parameters. We also remark that related questionsconcerning existence of continuous families of solutions of system(1)–(2) were addressed, albeit using rather different techniques,in [10]. The structure of this paper is as follows: first in Section 2wepresent a mathematically precise formulation of the problem, inSection 3we prove a theorem showing the conditions underwhichcontinuation of solutions with respect to parameters is possible, inSection 4 this result in generalized for the case of solutions sat-isfying the Kutta condition; since problem (1)–(2) is nontrivial tosolve numerically, a method based on the Steklov–Poincaré itera-tion is introduced in Section 5, whereas computational results arepresented in Section 6; summary and conclusions are deferred toSection 7.

2. Formulation of the problem

A fundamental property of system (1)–(2) is that the shape ofthe vortex patch A is not a priori determined and must be foundas a part of the solution of the problem. Thus, system (1)–(2)represents a free-boundary problem. Since we are going to needthis formulation in what follows, we now rewrite system (1)–(2)in a form that elucidates its free-boundary structure more clearly

r2 1 = �! in A(↵), (3a)

r2 2 = 0 in⌦ \ A(↵), (3b) 1 = 2 = ↵ on @A(↵), (3c)@ 1

@n= @ 2

@non @A(↵), (3d)

2 = b on @⌦, (3e)

where 1 = |A and 2 = |⌦\A are the restrictions of thestreamfunction to, respectively, the vortex patch A and its com-plement in⌦ .

Properties of solutions of (1)–(2), or (3), were studied com-putationally, and in some cases also analytically, by several re-searchers. It was observed that vortex patch solutions are in

192 F. Gallizio et al. / Physica D 239 (2010) 190–201

fact the result of an accretion process starting from a point vor-tex solution. The so-called ‘‘V-states’’ arising from desingulariza-tion of a single point vortex were studied by Wu, Overman andZabusky [19], and analytical results concerning the structure ofpossible singularities of the boundary of the resulting vortex patchwere derived in [20]. The case of desingularization of a pair of co-rotating vortices was first investigated by Saffman and Szeto [21].Two distinct continuous transformations of a co-rotating vortexpair into the Rankine vortex patch were discovered by Cerretelliand Williamson [22], and Crowdy and Marshall [23]. The case ofdesingularization of a pair of counter-rotating vorticeswas studiedby Pierrehumbert [24] with some details of his computations laterrectified by Saffman and Tanveer [25]. A limiting solution in thisfamily which also allows for a vortex sheet on the patch boundary(i.e., in which 1h 6= 0) is known as the ‘‘Sadovskii flow’’ and wasinvestigated by Sadovskii [26], and then by Moore, Saffman andTanveer [27]. In regard to desingularization of point vortices in theexterior of an obstacle (a circular cylinder), the paper by the Elcratet al. [13] is paradigmatic. In that paper the authors considered theFöppl curve pertinent to the flowpast a semicircular bumpwhich isthe locus of point vortices in a symmetric equilibriumwith the ob-stacle. They showed that each point vortex can be associated witha family of vortex patches with an increasing area |A| ,

R⌦H(↵ �

)d⌦ (the symbol , means ‘‘equal to by definition’’) and the samecirculation� as for the point vortex. Each element of the family canthus be regarded as an inviscid model of a finite area wake. In suchmodel, thewake is a regionwith closed streamlines boundedby thebody, the symmetry axis and the separatrix streamline separatingit from the exterior flow with open streamlines. The vorticity dis-tribution is given by ! = 0 in the exterior flow and ! = � /|A|inside the vortex patch. Assuming |A| to be the parameter definingan element of the family, the point vortex (Föppl) solution will bethe first element with |A| = 0, and the Prandtl–Batchelor solutionwill be the last element with ! = const in the entire region withclosed streamlines. In paper [14] the Föppl curve was generalizedfor the case of flow past a locally deformed wall yielding a locus ofpoint vortices in equilibriumwith an arbitrary obstacle. It was alsoargued in [14] that, as was the case for the semicircular bump, afamily of growing vortex patches can be associated with each suchpoint vortex configuration in equilibrium with the obstacle. It wasshown as well that when the obstacle has a sharp edge, then thenumber of such point vortex equilibria which additionally satisfythe Kutta condition is either null or finite. In regard to the first pos-sibility, Conjecture 1would imply that such obstacles do not admita finite area wake at high Reynolds numbers. Our present investi-gation seeks to shed some light on the problem of existence of suchfamilies of growing vortex patches from themathematical point ofview.Wewould like to understand the conditions under which so-lutions featuring vortex patches can be continuedwith respect to aparameter. Since parameterization of solutions of (1)–(2) in termsof |A| and/or � complicates somewhat themathematical structureof the problem, for the sake of transparency of our analysis, here-after we will assume that solutions depend on ! and ↵ which areexplicitly present as parameters in the governing system. The firstspecific question this work intends to answer is thus the following

Question 1. Given a solution = (↵0,!0) of problem (1)–(2)corresponding to the parameter values ↵ = ↵0 and ! = !0, underwhat conditions can this solution be continued with respect to one ofthe parameters ! and ↵ when the other one is held fixed?

In otherwords, wewant to characterizemathematically the condi-tions, sufficient or necessary, for the existence of unique neighborsolutions resulting from infinitesimal perturbations of the param-eters (Fig. 2).

ω0

α 0

ω0 α 0ψ( , )

Kutta line

α

ω

Fig. 2. Schematic of the dependence of the solution of system (1)–(2) on theparameters ↵ and !: (dashed line) when one of the parameters is held fixed, and(solid line) when Kutta condition (5) is imposed.

An important in practical applications class of problems (1)–(2)concerns the situation when the solution has a prescribed ‘‘sep-aration point’’, i.e., is subject to the so-called Kutta condition. Thiscondition, commonly used in aeronautical applications, followsfrom the observation that, for the Reynolds number above a certainvalue, the viscous flow separates on sharp edges. In the frameworkof the potential flow theory, the inviscid flow would be singular inthe absence of separation, with the flow velocities becoming un-bounded at sharp edges. By imposing a prescribed separationpoint,the Kutta condition simultaneously requires the flow to be regular.Mathematically, this condition is often expressed as follows

lims!s+0

@

@n(s) = � lim

s!s�0

@

@n(s) < 1, (4)

where s0 is the arc-length coordinate characterizing the location ofthe cusp. Relation (4) expresses the fact that, when the Kutta con-dition is satisfied, the tangential velocities on both sides of the cuspare in the limit the same and bounded. Since condition (4) is ratherawkward to handle in our analysis, we will use another, approxi-mate, formulation. Since the tangential velocity at the cusp is thuswell defined, we can extrapolate the value of the streamfunctionfrom the cusp P 2 @⌦ into the flow domain, i.e.,

|P+✏t = b, (5)

where ✏ > 0 is a small number, whereas t is the unit tangent vec-tor at P , so that (P + ✏t) 2 ⌦ . We add that in the actual numer-ical computations, original form (4) of the Kutta condition will beused (see Section 5 for details). Evidently, imposing the Kutta con-dition constrains the two-parameter family of solutions, so that aone-parameter family could be expected, although the existenceof such families in certain important cases is still an open prob-lem [14]. Thus, the second question we would like to answer inthis work can be framed as follows:

Question 2. Given a solution of problem (1)–(2)which in additionsatisfies also the Kutta condition (5), under what conditions can thissolution be continued with respect to ↵ or !?

Locus of solutions constrained by (5) is indicated as the ‘‘Kutta line’’in Fig. 2.

We will address these two questions using a combination ofsome elementary methods of functional analysis and the theory ofelliptic partial differential equations (PDEs). More specifically, wewill do this in Section 3 in the following steps:

F. Gallizio et al. / Physica D 239 (2010) 190–201 193

(1) use a suitable weak formulation of system (1)–(2) to constructan implicit function of the parameters ↵ and !,

(2) employ the Implicit Function Theorem in the Banach space todetermine conditions under which continuation is possible;use the ‘‘shape-differential’’ calculus to determine the Jacobian(perturbation equation) required in the statement of this the-orem,

(3) use the Lax–Milgram Theorem together with some standardestimates to determine sufficient conditions under which theJacobian of the implicit function is invertible.

In addition, one more step will be required in order to answerQuestion 2 in Section 4, namely:(4) linearize Kutta condition (5) and use the maximum principle

to show that this condition can be always satisfied.

3. Continuation with respect to parameters

We begin by stating a weak formulation of Euler equation(1)–(2). For problems with inhomogeneous boundary conditionsfirst we need to perform ‘‘lifting’’ to transform the problem to aformwith the homogeneous boundary conditions. We do this hereby introducing an auxiliary function ⇥ : ⌦ ! R defined as asolution of the following problem

r2⇥ = 0 in⌦, (6a)⇥ = b on @⌦. (6b)Expressing the streamfunction as = � + ⇥ in (1)–(2) yields anequivalent boundary value problem for the function � : ⌦ ! Rwith homogeneous Dirichlet boundary conditions, namely

r2� = �!H(↵ � � �⇥) in⌦, (7a)� = 0 on @⌦. (7b)Assuming now that the function � and the test function ' belongto the Sobolev space H1

0 (⌦) of functions with square-integrablegradients and bounded support in⌦ [28], the corresponding weakformulation of (7) becomes

� 2 H10 (⌦),

Z

⌦

r� · r'd⌦ � !

Z

A'd⌦ = 0,

8' 2 H10 (⌦). (8)

The existence of solutions of problems of this type was considered,for example, in [29]; it is also supported by ample computationalevidence which was reviewed in Introduction. We will thusassume that for some parameter values ↵ = ↵0 and ! = !0 thereexists a solution �0 = �(↵0,!0) of problem (8), and we are nowinterested in the conditions under which this solution �0 can beuniquely continued with respect to one of the parameters (Fig. 2).We also emphasize that the auxiliary function⇥ does not dependon the parameters ↵ and !. To focus attention, we will thereforefix the value of vorticity !0, and will consider the solution to bea function ↵ only, i.e., � = �(↵). Weak formulation (8) can thenbe represented using an implicit function G : H1

0 ⇥ R ! R asG(�,↵) = 0. Local existence of such one-parameter family ofsolutions �(↵) is addressed by the Banach space version of theImplicit Function Theorem [30].

Theorem 1 (Implicit Function Theorem). If X , Y , Z are Banach spaces,U ✓ X ⇥ Y is an open set, (x0, y0) 2 U, f : U ! Z is a continuousdifferentiable function, f (x0, y0) = 0 and Dyf (x0, y0) 2 L(Y ; X) isinvertible with a continuous inverse, then there exist neighborhoodsU1 of x0 and U2 of y0, such that U1 ⇥ U2 ✓ U and a uniquecontinuously differentiable function g : U1 ! U2, such that

f (x, g(x)) = 0, 8x 2 U1 (9)

and

Dg(x) = �[Dyf (x, g(x))]�1Dxf (x, g(x)), 8x 2 U1, (10)

where L(Y , X) is the vector space of all bounded linear operatorsfrom Y into X, whereas Dxf and Dyf denote the partial Fréchetderivatives with respect to the first and second variable.

Theorem1 can be applied to our problemby identifying the Banachspace X with our solution space H1

0 (⌦), and the Banach spacesY , Z with R. The implicit function f will then represent the weakformulation (8), i.e.,G(�,↵) = 0. Clearly, for this theorem to apply,we have to ensure that the Jacobian D�G(�,↵) of the implicitfunction is an invertible operator with a continuous inverse. Thefirst step towards this end is to identify the form of the JacobianD�G(�,↵). As is evident from formulation (3), our system is of thefree-boundary type, and therefore its differentiation with respectto a parameter must be carried out with care. One reason is thatwhen the vortex patch boundary @A is perturbed, this also affectsthe location of where the boundary (interface) conditions (3b)and (3c) are imposed. These issues are addressed by the shape-differential calculus which is a suite of mathematical techniquesallowing one to differentiate PDEs defined in variable domains.Below we summarize the main facts only which are relevantto our problem, and refer the reader to the monograph [31]for further details. Let �0

↵ , @ @↵

���↵=↵0

denote the perturbation

variable obtained by varying the parameter ↵ while keeping theother parameter ! constant (it corresponds to the quantity Dg(x)appearing in the statement of the Implicit Function Theorem).We assume there exists a vector field Z : ⌦ ! R2 such thatZ ·n|@⌦ ⌘ 0. When the parameter ↵ is perturbed, i.e., ↵ = ↵0 +↵0,the resulting perturbed vortex patch boundary @A(t, Z) can berepresented as follows

x(t, Z) = x + tZ for x 2 @A(0), t 2 R, (11)

where @A(0) is the boundary of the unperturbed patch. Integralsdefined on such variable domains, e.g., the second term in weakformulation (8), are shape-differentiated as follows [31]✓Z

A(t,Z)�d⌦

◆0=

Z

A(0)� 0d⌦ +

I

@A(0)� (Z · n)ds, (12)

where � : A(t, Z) ! R denotes the integrand expression and � 0

is its shape derivative. Applying this result to differentiate (8) weobtaindd↵

G(�(↵),↵) = [D�G(�,↵)]�0↵ + D↵G(�,↵)

=Z

⌦

r�0↵ · r'd⌦

�!0

I

@A'(Z · n)ds, 8' 2 H1

0 (⌦). (13)

We nowproceed to relate the perturbation field (Z·n) appearing inthe second integral in (13) with the perturbations of the parameter↵. We do this by shape-differentiating boundary condition (3c)which defines the position of the vortex patch boundary

d d↵

����@A(t,Z)

= 0↵|@A(0) + @

@n

����@A(0)

(Z · n)

= �0↵

��@A(0) + @

@n

����@A(0)

(Z · n) = d↵d↵

= 1, (14)

where we used the fact that⇥ 0↵ ⌘ 0, so that

Z · n = 1 � �0↵|@A(0)

@ @n

���@A(0)

, (15)

where we also make the assumption, to be formalized later, that

194 F. Gallizio et al. / Physica D 239 (2010) 190–201

@ @n

���@A(0)

6= 0. Using (15) in (13) we can transform the latter

expression to

�0↵ 2 H1

0 (⌦),Z

⌦

r�0↵ · r'd⌦ + !0

I

@A

✓@

@n

◆�1

�0↵'ds

| {z }[D�G(�,↵)]�0

↵

= �!0

I

@A

✓@

@n

◆�1

'ds| {z }

D↵G(�,↵)

, 8' 2 H10 (⌦), (16)

which we symbolically express in the ‘‘strong’’ for as"

�r2 + !0

✓@

@n

◆�1�����@A(0)

�(x|@A(0) � x)

#

�0↵

=✓@

@n

◆�1�����@A(0)

�(x|@A(0) � x) in⌦, (17a)

�0↵ = 0, on @⌦. (17b)

Eqs. (16) and (17) represent, respectively, the weak and strongform of the perturbation of Euler equation (7) resulting from vari-ations of the parameter ↵. Using the same techniques it can beshown that in the case when we vary the vorticity !, and the pa-rameter ↵ = ↵0 is held fixed, the perturbation equation involvesthe same operator [D�G], and takes the form

�0! 2 H1

0 (⌦),

Z

⌦

r�0! · r'd⌦ + !0

I

@A

✓@

@n

◆�1

�0!'ds

=Z

A'ds 8' 2 H1

0 (⌦), (18)

which we symbolically express in the ‘‘strong’’ for as"

�r2 + !0

✓@

@n

◆�1�����@A(0)

�(x|@A(0) � x)

#

�0!

= H(↵0 � ) in⌦, (19a)

�0! = 0, on @⌦ (19b)

where �0! , @�

@!

���↵=↵0

.

Given perturbation equations (16) and (18), the next step isto determine the conditions under which the associated Jacobianoperator [D�G(�,↵)] is invertible. To this end we will employ theLax–MilgramTheoremwhich provides the sufficient conditions forexistence of solutions of elliptic boundary value problems of thetype (16) and (18). Assuming now that H is a real Hilbert spacewith k · k denoting its norm, (·, ·) its inner product, and h·, ·i thepairing with its dual space, we have the following

Theorem 2 (Lax–Milgram, [32]). Assume that

B : H ⇥ H ! R

is a bilinear mapping for which there exist constants ⇠ , ⌘ > 0 suchthat

|B[w1, w2]| ⇠kw1k kw2k, 8w1, w2 2 H, (20a)

⌘kwk2 B[w, w], 8w 2 H. (20b)

Finally, let T : H ! R be a bounded linear functional on H. Then,there exists a unique element w0 2 H such that

B[w0, w] = hT , wi, 8w 2 H.

In our problem the Hilbert space H can be identified with thesolution space H1

0 (⌦), whereas the bilinear form B with the weakform of the Jacobian (16) [resp. (18)] regarded as a function of boththe perturbation variable �0

↵ (resp. �0!) and the test function ',

i.e., B[�0↵,'] = [D�G](�0

↵,') [resp. B[�0!,'] = [D�G](�0

!,')].Thus, in order to establish invertibility of the Jacobian, we needto demonstrate boundedness (20a) and coercivity (20b) of thisbilinear form.

To fix attention we will focus on problem (16). As regards bou-ndedness,we first apply the Cauchy–Schwarz inequality to the firstterm on the left-hand side (LHS) in (16)����

Z

⌦

r�0↵ · r'd⌦

���� Z

⌦

(r�0↵)

2d⌦�1/2 Z

⌦

(r')2d⌦�1/2

k�0↵kH1

0 (⌦)k'kH10 (⌦). (21)

Next we introduce the following

Assumption 1. There exist constants Umax � Umin > 0 such that

sign(!0)@

@n Umax a.e. on @A, (22a)

sign(!0)@

@n� Umin a.e. on @A. (22b)

The constants Umin and Umax thus represent an upper bound and alower bound on the tangential velocity component on the vortexpatch boundary @A. By examining a simple vortex system, e.g., asingle point vortex, it is evident that vortices with positive circula-tion induce positive azimuthal (tangential) velocity, and vice versa,hence Assumption 1 is justified. Using assumption (22a), the sec-ond term on the LHS in (16) can be bounded from above as follows

!0

I

@A

✓@

@n

◆�1

�0↵'ds !0

Umin

I

@A�0↵'ds. (23)

The term on RHS in (23) can be further bounded by applying theCauchy–Schwarz inequality combined with the obvious estimatekf kL2(⌦) Ckf kH1

0 (⌦) with some C > 0I

@A�0↵'ds k�0

↵kL2(@A)k'kL2(@A) C2k�0↵kH1

0 (⌦)k'kH10 (⌦). (24)

Combining inequalities (21), (23) and (24) we obtain for the bilin-ear form��[D�G](�0

↵,')��

✓1 + !0

UminC2

◆k�0

↵kH10 (⌦)k'kH1

0 (⌦), (25)

which shows that boundedness condition (20a) is satisfied.As regards coercivity (V-ellipticity) condition (20b), we proceed

as follows

8� 2 H10 (⌦),

[D�G](�,�) =Z

⌦

(r�)2d⌦ + !0

I

@A

✓@

@n

◆�1

�2ds

�Z

⌦

(r�)2d⌦ + |!0|Umax

I

@A�2ds

�Z

⌦

(r�)2d⌦ � ⇠k�kH10 (⌦), (26)

where ⇠ > 0 is a constant, and we employed assumption (22b)together with the Poincaré inequality. Estimate (26) shows that,subject to assumption (22b), the coercivity condition (20b) issatisfied, and therefore the Jacobian [D�G] of the implicit functionis invertible. Finally, we need to show that the inverse Jacobian[D�G]�1 is a continuous operator. We do this by combining

F. Gallizio et al. / Physica D 239 (2010) 190–201 195

inequality (20b) with the estimate B[w0, w0] = hf , w0i kf kH�1(⌦)kw0kH1

0 (⌦), w0 2 H10 (⌦), which yields

kw0kH10 (⌦) = k[D�G]�1f kH1

0 (⌦) 1⌘kf kH�1(⌦). (27)

Estimate (27) shows that the inverse Jacobian [D�G]�1 is a contin-uous, and therefore also bounded [30], operator. Combining Theo-rems 1 and 2 with inequalities (25) and (26) we thus have a proofof the following

Theorem 3. Assume conditions (22) hold. Then, there is a neighbor-hood of the point (↵0,!0) in which there exist smooth families = (↵,!0) and = (↵0,!) of solutions to problem (1)–(2) depend-ing, respectively, on the parameters ↵ and !.

4. Continuation in the presence of Kutta condition

Our next goal is to consider conditions under which continua-tion of solutions is possible subject to Kutta condition (5) which re-stricts the set of solutions to a one-parameter family. Beforewe cando this,weneed to address one technical difficulty, namely, the factthat Kutta condition (5) requires the solution of problem (1)–(2)to be defined at any, in principle arbitrary, point (P + ✏t) 2 ⌦ ,whereas so far we considered weak solutions only which do notnecessarily possess this property. Furthermore, in our subsequentdevelopment we will need to employ the maximum principle forthe Laplace equation which requires the solutions to be at leastC2. With this in mind, we need to establish that weak solutions ofthe perturbation equations, whose existence and uniqueness wasproved in Theorem 3, are in fact smooth in ⌦ \ A, i.e., in the partof the flow domain where Kutta condition (5) may be imposed. Asregards the original Euler equation, we note that the solution 2 isdefined by (3b), (3c) and (3e), i.e., it satisfies the Laplace equationin⌦ \Awith the Dirichlet boundary conditions. Regularity of suchproblems was considered for instance in [33], where it was proved(in Section 4.5) that weak solutions of the Laplace equation in gen-eral domains possess in fact the required C2 regularity in the inte-rior of the domain. This result allows us to justify complementingsystem (1)–(2)with Kutta condition (5). Since solutions of this aug-mented system represent a one-parameter family, perturbation ofthe Kutta condition yields to the first order |P+✏t + 0

!|P+✏t�! + 0↵|P+✏t�↵

+ O�(�↵)2 + (�!)2

�= b, (28)

where! = !0+�! and↵ = ↵0+�↵. Assuming to fix attention that�! = �!(�↵), using (5) and neglecting quadratic terms, relation(28) is simplified to the form

�! = 0↵

0!

����P+✏t

�↵, (29)

which implies that, to the leading order, the vorticity perturbations�! can be adjusted to the perturbations of the independentparameter ↵, so that the Kutta condition is satisfied, provided that

0!|P+✏t 6= 0. (30)

In the same spirit, assuming that the vorticity ! serves as theindependent parameter, i.e., �↵ = �↵(�!), we obtain the condition

0↵|P+✏t 6= 0. (31)

Since in the domain ⌦ \ A 3 (P + ✏t) the perturbation variables 0↵ ⌘ �0

↵ and 0! ⌘ �0

! satisfy the Laplace equations with thehomogeneous Dirichlet boundary conditions imposed on @⌦ [cf.(17b) and (19b)], it follows from themaximumprinciple for ellipticPDEs [32] that conditions (30) and (31) are satisfied. With this wehave thus proved the following theorem

Theorem 4. Assume the conditions of Theorem 3 are satisfied. Then,there are neighborhoods of the points ↵0 and !0 in which thereexist one-parameter families = (↵) = (↵,!(↵)) and = (!) = (↵(!),!) of solutions to problem (1)–(2)subject to additional condition (5) which depend, respectively, on theparameters ↵ and !.

5. Numerical method

Our objective in this section is to introduce a method for thenumerical solution of problem (1)–(2) in a semi-infinite domainbounded by a wall with a protruding obstacle (Fig. 1). The idea ofthe proposed method is to approximate system (1)–(2) on a suit-able grid. The streamline = ↵, which separates the rotationalregion A(↵) from the surrounding irrotational domain⌦ \ A(↵), isnot explicitly tracked, but is detected as a jump of the vorticity !represented on the grid. In general, resolving accurately this sepa-ratrix would require a refined grid. Furthermore, since the physicaldomain extends to infinity, the truncated computational domainshould be large in comparison to the size of the vortex patch A(↵),and a straightforward application of any standard solutionmethodto the problem in the entire domain would require a very largenumber of grid points. In addition, some artificial far-field bound-ary conditions would have to be adopted at the outer boundary tomodel the inflow/outflow from the truncated domain. On the otherhand,when > ↵, Eqs. (1a) and (2) reduce to the Laplace equationwhich, in general, should not require such a significant computa-tional effort.

The method proposed here overcomes these difficulties bycombining a conformalmapping of the physical domain into a suit-able transformed plane and the decomposition of the transformeddomain into two subdomains, namely:

• a small interior subdomain⌦i which includes the image of therecirculating flow, and

• an exterior subdomain⌦e comprising the remainder of the flowfield which extends to infinity.

Briefly, system (1)–(2) is solved numerically by combining afinite-difference approach in the interior subdomain ⌦i with ananalytical expression for the solution of the Laplace equation inthe exterior subdomain⌦e. The two solutions are coupled throughthe boundary conditions on the interface �AB separating the twosubdomains: for the interior subproblem we use the Dirichletboundary condition i = e, whereas for the exterior subproblemthe Neumann boundary condition @ e/@n = @ i/@n, where thesubscripts i and e refer to the solutions defined on the interiorand the exterior subdomains, i.e., i , |⌦i and e , |⌦e .Repeated solution of such two coupled problems in known as theSteklov–Poincaré iteration which is a well-known approach in thedomain decomposition literature [34]. As regards computationalefficiency, the fact that one has to perform iterations is offset bya modest number of grid points required to solve the interiorsubproblem. In the following subsections we describe the two keyenablers of the proposed method, namely, the conformal mappingand the solution technique in the transformed domain.

5.1. Conformal mapping

To fix attention, we consider flows past a wall extending to in-finity in the upstream and downstream directions, and featuringa cusped obstacle. An example of such a flow domain is shown inFig. 3(a) with the interior subdomain⌦i bounded by a segment ofthe solid wall and an interface �AB connecting points A and B. Wewill show here how, by combining two conformal mappings, sucha domain can be transformed to a domain with a simple geome-try in which our problem can be solved using standard techniques.

196 F. Gallizio et al. / Physica D 239 (2010) 190–201

a

b

c

Fig. 3. Domains used in the numerical solution of system (1)–(2): (a) z-plane (thephysical domain), (b) �-plane (the computational domain for the exterior problem),and (c) ⇣ -plane (the computational domain for the interior problem); in the figuresthe solid lines represent (the images of) the domain boundaries, whereas the dottedlines represent the interface �AB separating⌦i and⌦e in the three planes.

According to the Riemann mapping theorem [35], any arbitrarysimply-connected region, such as the one shown in Fig. 3(a), canbe conformallymapped onto the upper half plane of a transformeddomain. Let us denote z-plane the complex physical plane, wherez = x+ iy and i = p�1, and �-plane the transformed plane. Thereexists, thus, a mapping function z = z(�)whichmaps the real axisof the�-plane onto an arbitrarily shaped line in the z-plane extend-ing in both directions to infinity, such that the upper half plane inthe �-plane (Fig. 3(b)) is mapped onto the region above the wall inthe z-plane.

The flow domain shown in Fig. 3(a) is obtained from the halfplane shown in Fig. 3(b) using a simple variation of Ringleb’s snow-cornice mapping [36], namely

z = �

a� b + �21

�a � b � �1

, (32)

where the complex parameter �1 defines the shape of the corniceand the two real parameters a and b are such that zA = z(�1),zB = z(1) (zA and zB are the positions of the points A and B in the z-plane, see Fig. 3(a)). As shown in Fig. 3(b), in the�-plane the interiorsubdomain is mapped into the region inside the unit semi-circle,while the exterior subdomain is its complement in the upper halfplane.

The second conformal mapping transforms the interior of therectangle in the ⇣ -plane (Fig. 3(c)) into the upper half of the �-plane (Fig. 3(b)) and is based on the Jacobi elliptic sine-amplitudefunction Sn. In the ⇣ -plane the interior subdomain correspondsto the lower half of the rectangle and the exterior subdomain

corresponds to the upper half of the rectangle (Fig. 3(c)). Themapping function is

� = Sn(⇣ ,m)

d, (33)

where the real parameters m and d are the elliptic modulus andthe scaling factor, respectively. The elliptic modulus m defines theaspect ratio of the rectangle in the ⇣ -plane through the equation|⇣D � ⇣C |2|⇣A � ⇣C |

= K(m)

K 0(m),

where K(m) is the complete elliptic integral of the first kind andK 0(m) = K(1 � m), whereas ⇣A, ⇣C and ⇣D are the coordinates ofthe points A, C and D in the ⇣ -plane (Fig. 3(c)). The scaling factord is a free parameter and its choice determines the coordinates zCand zD of the points C and D in the physical plane and the locationin the ⇣ -plane of the image ⇣P of the cusp P .

As described below, system (1)–(2) is solved numerically on anequispaced Cartesian grid defined inside the lower rectangle in the⇣ -plane. For an accurate enforcement of the Kutta condition (4), itis convenient to select a value of the parameter m such that thepoint ⇣P coincides with a grid node. Discretizing the side CA of therectangle with p grid points and collocating the image of the pointP with the nth node gives⇣P = d[�K(m) + i(n/p)K 0(m) + i(n/p)K 0(m)/2],so that (33) becomes

�P = Sn[�K(m) + i(n/p)K 0(m)/2,m]Sn[�K(m) + iK 0(m)/2,m]

which makes it possible to obtain a suitable value of the ellip-tic modulus m via a trial-and-error approach. An example of thecomputational grid generatedwith the procedure described above,albeit coarser than the ones used in the actual computations, isshown in Fig. 4. The figure also features a magnification of the re-gion showing the point P coinciding with a grid point.

5.2. Solution in the transformed domain

System (1)–(2) is transformed to the ⇣ -plane with ⇣ = ⇠ + i⌘acting as the independent variable. Defining r2

⇣ , @2⇠ + @2⌘ , Eqs.(1a) and (2) become

r2⇣ = 1

J[�!H(↵ � )], (34)

where J , |dz/d⇣ |2. The interior subdomain⌦i coincides with therectangle ABCD whose upper, lower, left and right boundaries aredenoted, respectively, �AB, �CD, �AC , �BD. We will denote⌦v the ro-tational portion of the interior subdomain (⌦v ⇢ ⌦i). In the inte-rior subdomain the problem is defined by the following boundaryconditions i = 0 on �AC [ �CD [ �DB, (35a) i = e on �AB, (35b)where e is the Dirichlet boundary condition expressed in terms ofthe solution in the exterior subdomain ⌦e. The proposed methodis based on the Schauder fixed point theorem [30]. Let the stream-function be defined by

i = 0 + ! 1, in⌦i, (36)where 0 satisfies the systemwith a homogeneous RHS and inho-mogeneous boundary conditions, i.e.,

r2⇣

0 = 0 in⌦i, (37a)

0 = 0 on �AC [ �CD [ �DB, (37b)

0 = e on �AB, (37c)

F. Gallizio et al. / Physica D 239 (2010) 190–201 197

a b

Fig. 4. Examples of a coarse grid generated by the conformal mapping on the z-plane: (a) in the interior subdomain ⌦i , and (b) magnification of the neighborhood of thecusp P .

whereas 1 satisfies the system with an inhomogeneous RHS andhomogeneous boundary conditions, i.e.,

r2⇣

1 = 0 in⌦i \⌦v, (38a)

r2⇣

1 = 1J

in⌦v, (38b)

1 = 0 on �AC [ �CD [ �DB [ �BA. (38c)

The vorticity! is computed by imposing the Kutta condition at thecusp ⇣P . We note that condition (4) implies the following relationin the transformed ⇣ -plane

@

@⇠

����⇣P

= 0. (39)

We also observe that discretizing the derivative in (39) with afinite-difference formula yields approximate relation (5). In the in-terior subdomain, the algorithm consists in iterating the equation

r2⇣ n+1 = 1

J[�!nH(↵ � n)] , n = 1, . . . , (40)

where the subscripts represent the iteration numbers. System (37)is not affected by this iterative procedure, thus 0 is computedonly once. With 0 fixed and ⌦v given, 1 and ! can be deter-mined at any given nth iteration. Then, at the (n + 1)th iteration,a new shape of the vortex region ⌦v is determined by the levelset n+1 = ↵ and the process is repeated until convergence is at-tained.

On the other hand, the potential flow in the exterior subdomain⌦e is computed in the �-plane. Let the complex potential we bedefined as

we(�) = Q1�+1X

j=1

aj��j (41)

with

Q1 = lim�!1

✓dwe

dz

◆ ✓dzd�

◆= q1

a,

where q1 is the asymptotic velocity in the physical plane. In the �-plane, the exterior subdomain⌦e corresponds to the region exte-rior to the unit circle in the upper half plane. The portion of the realaxiswith 1 |�| < 1 is the image of the solidwalls upstreamanddownstream of the cavity. We see that in order to ensure the im-permeability of the solid wall, exterior potential (41) must be suchthat = const on the real axis, and as a result the coefficients aj,j = 1, . . . , must be real. Noting that � = ⇢ exp(i'), the problem isclosed by enforcing theNeumann boundary condition (@ e/@⇢) =(@ i/@⇢) on the common boundary �AB, with (@ i/@⇢) expressedin terms of the interior solution i. We observe that the derivatives

in the directions normal to the interface �AB in the �-plane and ⇣ -plane are related through the following identity

@ e

@⇢= @ i

@⌘

����d�d⇣

�����1

.

Let us set g(') = (@ i/@⇢)⇢=1. We remark that the interiorsolution along the commonboundary �AB defines the function g(')in the interval 0 ' ⇡ . To ensure impermeability of the solidwalls we therefore need g(0) = g(⇡) = 0. Thus, g(') can becontinued on the interval⇡ ' 2⇡ , i.e., on the entire unit circleof the �-plane, by assuming that g(⇡ + �) = �g(⇡ � �), with 0 � ⇡ . Since Im[(dw/d�)�]⇢=1 = (@ e/@⇢)⇢=1, Eq. (41) yields

g(') = Q1 sin' �1X

j=1

jaj sin(j') =✓@ i

@⇢

◆

⇢=1. (42)

Truncating the series in (42) at some number N , we can determineall the unknown coefficients aj, j = 1, . . . ,N by using a suitablenumber of collocation points in [0,⇡].

The interior and exterior flow computations are iterated until i and e converge to the same values on the interface �AB.A linear relaxation approach n+1

e = (1 � fr) ̃n+1e + fr n

ehas was adopted for the solution values on the interface �AB[cf. (37c)] with the under-relaxation factor fr 2 [0, 1] chosenheuristically. Continuous families of solutions of (1)–(2) aretracked by modifying the value of the parameter, i.e., ↵, or !0, andthen solving the problem again using the solution obtained for theprevious parameter value of the initial guess.

5.3. Benchmark tests

The accuracy of the method was analyzed by comparing theresults of the numerical computations to the analytical solutionavailable when ↵ ! �1, that is when the vortex patch A(↵)shrinks to a point vortex (see, for instance, [14]). In this case, theinterior solution i is obtained by modifying Eqs. (36), (37b) and(37c) as follows

i = 0 + � 1 in⌦i,

0 = � v on �AC [ �CD [ �DB, 0 = e � v on �AB,

where � is the vortex circulation and v = � �2⇡ log |z � zv| is

the streamfunction induced by a point vortex located at zv in anunbounded domain.

The streamline pattern obtained in such a point vortex solutionis shown in Fig. 5(a). The L2 errors of the numerical solution withrespect to the analytical solution computed based on the flow ve-locity along the separatrix streamline�AB and the vortex circulationare in both cases O(10�6). Fig. 5(b) represents the corresponding

198 F. Gallizio et al. / Physica D 239 (2010) 190–201

a b

Fig. 5. Streamline patterns for (a) the point vortex equilibrium solution and (b) the corresponding Prandtl–Batchelor solution.

Fig. 6. The interior domain⌦i in the Batchelor flow.

Table 1

CPU times required to obtain the solution of system (1)–(2) with ↵ = �0.05 on a1500 ⇥ 1500 grid with different values of the under-relaxation factor fr .

fr Iterations CPU time (s)

0.25 37 6300.40 10 3000.50 7 2500.60 10 5700.75 19 1270

Prandtl–Batchelor flowwith a constant nonzero vorticity in the en-tire recirculation zone computedwith the presentmethod. A close-up of the interior region in the Prandtl–Batchelor flow is shown inFig. 6.

We close this subsection by commenting on the computationalefficiency of the proposedmethod. Themain computational cost isdue to solution of the interior Poisson problem (38) at each iter-ation. A fast Poisson solver was adopted from the Fishpack90 li-brary [37]. The computations were done on a PC with the AMDAthlon 64 3000+ 1.81 GHz CPU and with 1 Gb RAM. Examplesof the CPU times required to solve the full problem are shown inTable 1 for a 1500 ⇥ 1500 grid and for different values of the re-laxation factor fr . Fig. 7 shows the rates of convergence obtained inthe solution of this problem with different values of fr .

6. Numerical computations of continuous families of solutions

In this section we present results of numerical computationsto illustrate Theorems 3 and 4. We will first analyze the casewhere the Kutta condition is not imposed and solutions of (1)–(2)can be continued simultaneously with respect to ↵ and !. Thenwe will consider the case with Kutta condition (5) imposed, sothat solutions of (1)–(2) can be continued with respect to oneparameter only. As was argued in Introduction, existence of pointvortex equilibria and of the associated families of steady vortexpatches satisfying also the Kutta condition is a nontrivial problem.It becomes particularly involved when the domain boundarypossesses a fore-and-aft symmetry. In this regard, it was shownin [14] that in such cases the existence of equilibrium point vortexconfigurations depends on the specific form of the conformal mapthat transforms the domain boundary into the real axis in the

Fig. 7. Rates of convergence of the Steklov–Poincaré algorithm described inSections 5.1 and 5.2 and obtained in the solution of system (1)–(2) with ↵ =�0.05 on a 1500 ⇥ 1500 grid with different values of the under-relaxation factorfr indicated in the legend. The horizontal line without symbols represents thetolerance used in the termination condition.

transformed domain. Thus, we choose this conformal map in theform

z = �

a+ b � 1

5��a + b + i

�5 , (43)

where, as explained in Section 5.1, the parameters a and b aredetermined by requiring that zA = z(�1), zB = z(1). In accordancewith the criteria derived in [14], formula (43) ensures the existenceof equilibrium vortex configurations. In view of the controversysurrounding the question of existence of vortex equilibria indomains with symmetries (cf. Section 1), we find this particularconfiguration an interesting one. We also add that conformal map(43) is closely related to Ringleb’s snow cornice mapping (32), sothe Steklov–Poincaré method developed in Section 5 can be usedhere.

6.1. Continuation in the absence of the Kutta condition

As a reference, we consider the solution of problem (1)–(2)corresponding to ↵0 = �0.5 which, without loss of generality,also satisfies Kutta condition (5). The streamline pattern of thissolution is illustrated in Fig. 8(a). Since the vortex patch boundaryis regular, Assumptions (22) are clearly satisfied and, by Theorem3,this solution can be continued with respect to the parameters. Wethus consider two different continuations of the reference solution,namely:• holding the vorticity ! fixed, and perturbing the boundary

value of the streamfunction as ↵ = ↵0 ± �↵, where �↵ >0 is a perturbation (different in each of the two cases); the

F. Gallizio et al. / Physica D 239 (2010) 190–201 199

Kutta line

a b

c d

e f

Fig. 8. Streamline patterns in the solutions of system (1)–(2): (a) The referencesolution obtained with ↵ = �0.5, (c), (d) solutions with unchanged ! andperturbed ↵, (e), (f) solutions with unchanged ↵ and perturbed !; thicker linesmark the streamlines bounding the recirculation regions; figure (b) representsschematically the reference solution and the loci of the perturbed solutions in the↵–! plane.

Table 2

Parameters of the reference solution (R) and the perturbed solutions (A1, A2, B1, B2)for the case when the Kutta condition is not enforced.Solution ↵ ! |A| �

R �0.5 �25.481 0.189 �4.811A1 �0.646 �25.482 0.221 �5.628A2 �0.392 �25.481 0.163 �4.156B1 �0.5 �22.456 0.227 �5.103B2 �0.5 �28.593 0.156 �4.563

corresponding solutions A1 and A2 are shown in Fig. 8(c), (d),and their locus is represented by the horizontal line in Fig. 8(b),

• holding the boundary value ↵ of the streamfunction fixed, andperturbing the vorticity as ! = !0 ± �!, where �! >0 is a perturbation (different in each of the two cases); thecorresponding solutions B1 and B2 are shown in Fig. 8(e), (f), andtheir locus is represented by the vertical line in Fig. 8(b).

All relevant parameters, i.e., |A| and � in addition to ↵ and ! ofthe reference and perturbed solutions are collected in Table 2. Wenote that while the reference solution has the separation point atthe cusp of the obstacle, this is no longer the case for the perturbedsolutions.

Fig. 9. Family of vortex patch solutions of system (1)–(2) subject to Kuttacondition (5) and connecting the point vortex solution (↵ ! �1) with thePrandtl–Batchelor solution (↵ = b = 0).

Fig. 10. Streamline pattern of the Prandtl–Batchelor solution (↵ = b) in whichthe vortex patch coincides with the recirculation region.

6.2. Continuation in the presence of the Kutta condition

Subject to Kutta condition (5), solutions of system (1)–(2)represent a one-parameter family. In the limit ↵ ! �1 theyapproach a point vortex solution, whereas for ↵ = b = 0 therecirculation region and the vortex region coincide resulting inthe Prandtl–Batchelor solution. Boundaries of the vortex regionfor solutions belonging to this family and corresponding to theintermediate values of ↵ are shown in Fig. 9. Since the vortex patchboundaries in each of those solutions are regular, Assumptions (22)are clearly satisfied and, by Theorem 4, each of these solutions canbe continued. In Fig. 10 we also show the entire streamline patternfor the terminal Prandtl–Batchelor solution obtained for ↵ = b.The locus representing this solution family in the ↵–! plane, i.e.,the Kutta line, is shown in Fig. 11(a), whereas the correspondinglocus in the |A|–� plane is shown in Fig. 11(b). In both figuresone can clearly see the transition from the point vortex to thePrandtl–Batchelor solution.

7. Conclusions

In this paper we identified the conditions under which vortexpatch solutions of Euler equations (1)–(2) can be continued withrespect to both the vorticity ! and value ↵ of the streamfunctiondefining the vortex boundary. These conditions, given by (22), aresatisfied as long as the vortex patch boundary is smooth. In otherwords, if there exists a smooth vortex patch in equilibrium withthe obstacle and characterized by given values of the vorticity !and the streamfunction ↵ (or, equivalently, the circulation � andthe patch area |A|), then there also exist nearby perturbed solutionsin the↵–!, or |A|–� , plane. A situation inwhich conditions (22) arenot satisfied may arise when the vortex patch boundary has a sin-gularity in the formof a corner, or a cusp (it is known from [20] thatthese are the only two singularity types possible). Given the struc-ture of the proofs of Theorems3 and4, conditions (22) are sufficient,but not necessary. Therefore, it might in principle happen that vor-tex patcheswith singularities of their boundaries could still be con-tinued. On the other hand, however, all computational results weare aware of and which we reviewed in Section 2 indicate that sin-gular patches represent in fact terminalmembers of solutions fam-ilies: in addition to our results presented in Section 6.2 (e.g., Fig. 9),this was also observed in [27] for the Sadovskii flow and in [23] fora system of two vortex patches touching at three cusp points. Wealso add that our analysis did not include the case of a patch witha vortex sheet on the boundary which would require a slight gen-eralization of our approach. While computational evidence for the

200 F. Gallizio et al. / Physica D 239 (2010) 190–201

a b

Fig. 11. Loci of solutions of system (1)–(2) satisfying also Kutta condition (5) in (a) !–↵ parameter space, and (b) � –|A| parameter space.

existence of perturbed vortex patch solutions has been known for along time, in this investigation we derived, for the first time to thebest of our knowledge,mathematically precise conditions allowingone to predict when such a continuation is possible. Furthermore,we also show that if the original solution satisfies the Kutta condi-tion, then there exists a nearby vortex patch having, for example,a different area, but still satisfying the Kutta condition.

Themethodologywedeveloped in this paper does not allowoneto construct neighboring solutions, but only establish their exis-tence. Our results are based on linearizations of governing system(1)–(2), hence they are valid only locally in the neighborhood ofthe reference solution. In principle, one could attempt to constructneighboring solutions by representing them as a Taylor series withrespect to a parameter, and then summing up perturbation vari-ables of increasing orders. However, in addition to the need to es-tablish analyticity of solutions of (1)–(2)with respect to the param-eter, this would require the summation of an infinite series whichis impractical. Therefore, in actual computations it is much morepractical to use methods of numerical continuation, and indeedwith such techniques we were able to determine a whole fam-ily of vortex patch solutions connecting the point vortex and thePrandtl–Batchelor solution.

We emphasize that Eqs. (17) and (19) satisfied by the pertur-bations variables �0

↵ and �0! are the main objects of our analysis.

Given the free-boundary nature of governing equation (1)–(2), sys-tematic derivation of these perturbation equations required theuse of a special technique, namely, the shape-differential calculus.We remark that these perturbation equations are also of indepen-dent interest and may be used to study, for instance, the stabilityof solutions of (1)–(2).

One of the motivations for this work was to understand the ex-tent to which the existence of a vortex patch solution satisfyingthe Kutta condition implies the existence of the limiting point vor-tex solution also satisfying the Kutta condition, or, equivalently,the extent to which the non-existence of such a point vortex so-lution implies that no steady vortex patch can be found to satisfythe Kutta condition for a specific boundary configuration (cf. Con-jecture 1). We made a step towards solving this problem by iden-tifying sufficient conditions for the existence of continuous fami-lies of solutions characterized by vortex patches. These conditions,given by Eqs. (22), are satisfied as long as the vortex patch bound-ary remains regular, i.e., free from geometric singularities such ascusps or corners. While extending our findings to include in a rig-orous manner the limiting case of a point vortex (correspondingto ↵ ! �1 and ! ! �1) remains an outstanding challenge,the results of computations seem to support the ‘‘accretion’’ sce-nario. Finally, we note that there is also a range of interesting ques-tions concerning the global structure of the solution manifold. Ourhope is thatmethods of nonlinear functional analysis, such as Fred-holm’s degree theory, might shed some light on these problems.

Acknowledgements

The authors benefited from fruitful discussions with WalterCraig, David Lannes, Haysam Telib and Oleg Volkov. The third au-thor acknowledges the support through an NSERC Discovery Grant(Canada) and the hospitality of Institut de Mathématiques de Bor-deaux where a part of this work was accomplished. The first, sec-ond and fourth authors acknowledge the support through the EUSixth Framework—STREP Project VortexCell2050.

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