Dynamical control for capturing vortices near bluff bodies
Aron Pentek* and James B. KadtkeInstitute for Pure and Applied Physical Sciences, University of California, San Diego, 9500 Gilman Drive, La Jolla,
California 92093-0360
Gianni PedrizzettiDipartimento di Ingegneria Civile, Universita` di Firenze, via Santa Marta 3, 50139 Florence, Italy
~Received 20 January 1998!
We investigate the vortex dynamics near a translating and rotating circular cylinder in a two-dimensionaluniform viscous flow. In analogy with the point-vortex and Eulerian dynamics, there is an interesting scatteringeffect of vortices approaching the cylinder from far upstream. The vortex–boundary-layer interaction plays animportant role in the scattering processes. We implement a modified Ott, Grebogi, and Yorke chaos controlscheme, based on a low-dimensional Hamiltonian model of the flow, to capture and stabilize a concentratedvortex around the cylinder. This point-vortex-based control model can successfully be applied in a viscous flowwhen control is actuated by uniformly rotating the cylinder and actively changing the background flow velocityfar from the body. We demonstrate that such a control mechanism can simultaneously control the vortexdynamics, and also suppress the vortex shedding. An analysis of the vortex–boundary-layer interaction ispresented to explain the absence of vortex shedding during control simulations.@S1063-651X~98!10508-1#
The interaction of fluid flows and vortical structures wiembedded bodies is an important research area in fluidchanics, with widespread applications in hydrodynamicsaerodynamics, and structural engineering problems. In reyears, considerable effort has been made to control suchflows in order to improve the flow characteristics. The posible applications include wake stabilization, lift enhancment, drag and noise reduction, and mixing enhancemand are attracting increasing interest@1–5#.
The complexity of such control problems leads to tstudy of reduced low-dimensional flow models, which in ctain limits capture most of the qualitative features of vortebody interaction@6–11#. These models provide a framewowhere an active control algorithm can be easily develoand understood, before it is applied in a realistic fluid [email protected]., full Navier-Stokes~NS! equations#. More importantly,recent advances in control theory of dynamical systemsnaturally be applied in these reduced low-dimensional flsystems. In particular, the method developed by Ott, Gbogi, and Yorke~OGY! @12# has already proven to be sucessful in several applications, such as controlling a magtoelastic ribbon @13#, a thermal convection loop@14#,chemical reactions@15#, solid state devices@16#, and chaoticlasers@17#.
One of the simplest models for the interaction of a blbody with concentrated fluid vorticity is that of a singHamiltonian point vortex interacting with a two-dimension~2D! cylinder. This system has been extensively studied@18#,and it is known to exhibit several remarkable features incling a chaotic capturing phenomenon. It has been shown
*Also at Department of Physics, University of California, SDiego, La Jolla, CA 92093-0319.
PRE 581063-651X/98/58~2!/1883~16!/$15.00
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viously @8# that by a proper control algorithm such a vortecan be stably captured near the cylinder. A more interesresult is that the controlled capture can successfullyimplemented even for a distribution of vorticity, at corrsponding parameter regimes@8#. In this work a continuousvorticity distribution was evolved according to the NS equtions, coupled with inviscid ‘‘free-slip’’ boundary conditionon the cylinder surface. This approach, which essentisimulates an inviscid evolution and avoids some numerdifficulties, was aimed to study the qualitative correspodence of Hamiltonian dynamics for continuous fields.
Our aim in this paper is to similarly analyze the dynamof coherent vortical structures approaching a rotating cyder in a viscous fluid at Reynolds number around Re51000,and to develop a possible control mechanism to stably cture a vortex around the cylinder. First, we show that theran interesting vortex scattering effect, and even a vortex cturing phenomenon, in the case of a rotating cylinder. Thphenomena have been previously pointed out in the Hatonian vortex dynamics and in the inviscid flow. In the vicous case, however, they have a completely different phcal origin: the vortex–boundary-layer interaction, whicplays an important role in the vortex dynamics.
Secondly, we demonstrate that by proper perturbationthe flow it is possible to control a vortex passing by a cylder. The basic requirement we impose for such a conalgorithm is that it be implemented only through physicamotivated boundary conditions. This leads essentially to tmechanisms in the framework of our model system: oneeither rotate the cylinder and/or change the uniform baground flow velocity far from the body~i.e., the translationalvelocity of the cylinder!. The other parameters of the problem, such as the circulation around the body or the vorstrength, are not experimentally accessible parametersrealistic viscous flow, and thus they cannot be used for ctrol. Other forms of control, such as blowing and suctionthe cylinder surface, are not studied in this paper. They
however, likely candidates for alternative control actuatoas they have been used with success in other flow conproblems@7#. In this paper we use small changes in the uform background flow velocity far from the body as contractuator. We show that these perturbations combined wiuniform rotation of the cylinder can successfully control tvortex and simultaneously stabilize the boundary-layernamics.
In our analysis we first review the Hamiltonian vortedynamics, and implement the control algorithm for the poivortex system by using perturbations of the background flvelocity far from the body. In order to clearly understawhat the particular effects of the finite-size vortex patch,the diffusion of vorticity, and of the vortex–boundary-layinteraction on the vortex dynamics and control in a viscoflow are, we review in some detail the free-slip vortex dnamics. The free-slip numerical simulations are designeunderstand the vortex dynamics and control for a vorpatch in the presence of viscosity but without a boundlayer on the cylinder surface. Next, we analyze the vordynamics in a viscous flow around a rotating cylinder, ashow the influence of the boundary layer on the vortexnamics. Finally, these results are used to develop the coprocedure for the vortex dynamics.
This paper is organized as follows. In the next sectionvortex dynamics and control problem is formulated fHamiltonian point-vortex dynamics. Section III is devotedthe numerical procedure for solving the Navier-Stokes eqtions for free-slip and no-slip dynamics. In Sec. IV an anasis of the free-slip vortex scattering and control is presenSection V is devoted to the vortex scattering dynamics iviscous flow. The control algorithm for the viscous casedescribed in Sec. VI. A detailed analysis of the vortboundary-layer interaction and the stability of the boundlayer during control is presented in Sec. VII. Section Vcontains our concluding remarks.
II. THE HAMILTONIAN MODEL
The Hamiltonian model consists of a circular boundacentered at the origin of the coordinate system, embeddea uniform background flow of velocityu0 parallel to thexaxis and pointing in the negativex direction~Fig. 1!. We alsoassume a possible varying perturbationdu0 to the uniformbackground flow velocity. A point vortex of circulationk isadvected past the cylinder by the background flow, starupstream of the cylinder. In nondimensional polar coornates (r ,u), scaled by the cylinder radiusR0 and the back-ground flowu0, the Hamiltonian for the vortex dynamicsgiven by @18#
H52~11«!r S 121
r 2D sinu1s
2ln~r 221!, ~1!
where the first term is due to the background flow, andsecond term to the flow induced by an image vortex. Hs5k/2pR0u0 is the nondimensional vortex strength. Thparameter«5du0 /u0 represents the rescaled perturbationthe background flow. Note that the possible rotation ofcylinder is not included in Eq.~1!. This could be modeled byincluding a uniform circulation around the cylinder. Sinc
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however, in the viscous flow we are interested in simulatioon time scales much shorter than the typical spin-up timethe fluid around a rotating cylinder, as a first approximatithe effect of the cylinder rotation can be neglected if tvortex does not come very close to the cylinder surface.
The equations of motion for the vortex dynamics arethe Hamiltonian form:
r 51
r
]H
]u, and u52
1
r
]H
]r. ~2!
Equation~2! corresponds to a one degree of freedom automous Hamiltonian system that is always integrable. Wsmall, time-dependent perturbations introduced intobackground flow, however, Eq.~2! has the same structure athat of a driven one degree of freedom Hamiltonian systthat is known to generically exhibit chaos. Thus the vortapproaching the cylinder can be captured and exhibit coplicated, chaotic motion around the cylinder for afinite timebefore it is transported away downstream@18,19#. This be-havior is a hydrodynamic manifestation of chaotic scatter@20#, or more generally, of transient chaos@21#.
A detailed analysis of the phase space of the above sysreveals a simple flow topology@18#. To illustrate our controlstrategy, we choose a simple autonomous system with fis522.962 96, and«50 that produces a single saddle poiat (0,23) in Euclidean coordinates@Fig. 3~a!#. The controltechnique presented in this paper is independent of thecific choice of the parameters. The technique relies on thexistence of at least one saddle point which can be founany parameter regime. The solid lines in Fig. 3~a! are theconstant energy lines of the Hamiltonian~1!. Since the en-ergy is conserved during the motion, they also corresponthe vortex trajectories. We emphasize that these lines arethe stream lines of the flow. The stream-line pattern depeon the actual vortex position, and is changing in time asvortex is advected past the cylinder.
Our aim here is to capture and stabilize the passing voat this fixed point by utilizing the OGY method of chaocontrol. Briefly, the OGY method stabilizes one of the maunstable periodic orbits or fixed points which are inherenpresent in the dynamics of the system. For unstable fi
FIG. 1. Schematic diagram of the idealized flow.
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PRE 58 1885DYNAMICAL CONTROL FOR CAPTURING VORTICES . . .
points of hyperbolic character~also called saddle points!there exists a curve along which the fixed point can beactly reached, named the stable manifold@12#. The controlconsists of small perturbations applied in such a way adrive the actual trajectory close to the stable manifold ofdesired orbit~cf. Fig. 2!. In this way, we take advantage othe naturally attractive dynamics around the stable manito reach the unstable fixed point. Since the stable manifoltypically a set of measure zero, and it cannot be reacexactly, control is repeated at discrete time stepsDt to applycorrections. Here we will restrict ourselves to controllingsingle fixed point, and not the full transient chaotic dynaics. The method is nevertheless the same, and can be dirapplied for the chaotic case as well.
The control procedure described above is capable of ctrolling the vortex dynamics if the vortex initially starts ithe vicinity of the fixed point, where a linear approximatioof the dynamics is valid@12#. In open flows, however, theprobability that the vortex will pass by a close neighborhoof the fixed point is typically very small. Only a small fraction of initial conditions lying around the stable manifosatisfy this condition. To demonstrate this one could swith a large number of initial conditions and wait until onof them falls in the preselected neighborhood of the desorbit, as transient chaos has been first controlled@22#. Alter-natively one can implement a targeting algorithm@23# thatdrives the vortex to the fixed point, not necessarily wsmall perturbations.
Such a targeting algorithm has been proposed in Ref.@8#.It takes advantage of the robustness of the flow topologlarge perturbations that is observed in our model system,the saddle-point-like structure is preserved even for laperturbations. Ifr5A(r ,«) represents the dynamical systeof Eq. ~2!, the dynamics in the vicinity of the fixed point cabe approximated by the linearized equation
dr5Jdr1G«, ~3!
where the actual vortex position relative to the fixed poindr , and
FIG. 2. Schematic diagram of the saddle pointO and its stableand unstable manifolds~solid lines! in the phase space. Under smaperturbations the fixed point moves alongG to O8. The controlalgorithm requires a perturbation that brings the original pointr (t)close to the stable manifold. The perturbed system is showndashed line. Due to the applied control the trajectory approachestable manifold of the unperturbed system along the dotted line
-
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s
J5]A
]r Udr50,«50
~4!
is the Jacobian matrix evaluated at the saddle point.vector
G5]A
]« Udr50,«50
~5!
gives the effect of small perturbations on the dynamical stem. After a short evolution timeDt, the new vortex positionbecomes
dr ~ t1Dt !'@11JDt#dr ~ t !1GDt«~ t !. ~6!
To achieve control, the vortex dynamics is perturbedsuch a way that the vortex starting upstream of the cylinis driven along the stable eigendirection in the vicinity of tfixed point. Let es and eu denote the stable and unstabeigenvectors of the JacobianJ, respectively. We define thecovariant unstable eigenvectorfu by the relations
fu•eu51 and fu•es50.
The size of the perturbation is evaluated from the conditthat the projection ofdr on the covariant unstable eigenvetor fu of matrix J should decrease each time control is aplied @Fig. 1~b!#, i.e.,
fu•dr ~ t1Dt !5~12b!fu•dr ~ t !. ~7!
The parameterb is smaller than one, and is chosen to befunction of the distanceudr u in order to control the magni-tude of the perturbation during the targeting procedure. Trequired perturbations« can then be expressed as a functiof the positiondr relative to the fixed point in the followingway @8#:
«~ t !52S lu1b
Dt D fu•dr ~ t !
fu•G, ~8!
wherelu is the unstable eigenvalue ofJ. In the present papean exponential dependenceb5exp(2g udr u) has been usedalthough the qualitative results are not apparently depenon the particular functional form.
Figure 3~a! shows the dynamics of a point vortex undthe perturbations~8!. The parametersfu , G, and lu havebeen determined analytically from Eqs.~1! and~2!. The timeevolution of perturbations to the uniform background floare shown in Fig. 3~b!. One can clearly observe that thcontrol algorithm drives the vortex along the stable directto the saddle point. Except for the initial perturbations durithe targeting procedure, only extremely small perturbatioare needed to keep the vortex at the saddle point. Inexample shown in Fig. 3~b! the maximum value of the perturbation is quite large during the targeting procedure («max'1.6). This is due to the fact that if the dynamics is nchaotic, it is typically not possible to drive the vortex trajetory to the stable manifold by small perturbations only. Ocan, however, optimize this targeting procedure by propechoosing theb(udr u) function. Depending on the particulaobjective, one can attempt to either minimize the largest p
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1886 PRE 58PENTEK, KADTKE, AND PEDRIZZETTI
turbations or to achieve a smallest cost solution, e.g.,minimizing *«(t)dt. Alternatively, in a real fluid system oncan also impose restriction on the maximum allowed aceration of the cylinder, i.e., one can limitD«/Dt to reflectlimitations due to the inertia of the cylinder.
During the control part, the size and frequency of tmaximum perturbations@the three spikes in Fig. 3~b!# de-pend on the precision of measurement ofdr . Higher preci-sion in dr results in more frequent and smaller magnituperturbations. If the absolute precision isz, the maximumperturbation scales as«max;z(lu11/Dt), and the time in-terval between spikes asDtspikes;(ln z)/lu .
Our aim in this paper is to analyze the correspondvortex dynamics and control as above in a viscous flow. Dto the complexity of the problem, we first analyze the vortdynamics with inviscid free-slip boundary conditions, anda second step we concentrate on the viscous case. Thesection details the numerical procedure for both the free-and no-slip simulations.
III. NUMERICAL SOLUTION OFNAVIER-STOKES EQUATIONS
In our numerical analysis we use a pseudospectral evtion scheme discussed previously in Ref.@24#. The NS equa-tion is solved using a vorticity-stream function representat(v,c) of the flow implemented on a polar grid centeredthe body. The scheme uses a finite-difference approximain the radial direction and spectral decomposition in thegular direction. The velocity field in polar coordinates (r,f)is given by
FIG. 3. ~a! Typical flow topology: vortex flow lines with arrowsindicate direction of the vortex motion. Small circles indicate timevolution of a point vortex under the dynamics of the control atargeting scheme. Initial vortex position is~3.0, 24.0), ands522.962 96. The control time interval isDt50.1 and the targetingparameterg51. ~b! The perturbation to the background flow vlocity during control.
y
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x
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u-
n
n-
vr51
r
]c
]fand vf52
]c
]r~9!
for the radial and tangential velocities, respectively. Theequations rendered in dimensionless form are
]v
]t5
1
rS ]c
]r
]v
]f2
]c
]f
]v
]r D11
Re¹2v, ~10!
v52¹2c, ~11!
where the Reynolds number Re5u0R0 /n andn is the kine-matic viscosity.
To ensure high grid density in the physically interestiregion ~close to the body! a radially stretched grid is introduced. Such a grid maintains high resolution close tocylinder yet can still extend to large distances, to simulatreal open flow. In particular, we useh5 ln@(r211a)/a# asthe new radial coordinate witha being a stretching parameter.
The main component of the flow field is due to the bacground flow, and can be derived from the inviscid streafunction:
c0~r,f!52~11«!r S 121
r2D sinf. ~12!
To simplify numerical calculations we rewrite the NS equtions to evolve only the corrections to this flow. The resuing equations are somewhat lengthy and are omitted herebrevity. They are, however, identical to those in Ref.@24#and the reader is referred there. The time evolution is coputed using a third-order Runge-Kutta scheme with fixstep size@24#.
On the outer boundary of the computational domainboundary conditions are set to match the inviscid solutionfrom the body, i.e.,
c~rmax,f!5c0~rmax,f!. ~13!
Details of how these boundary conditions are numericaimplemented can also be found in Ref.@24#.
A. Details of the free-slip simulation
The physical boundary conditions on the cylinder surfaare, in the free-slip case,
]c
]f Ur51
50. ~14!
Numerically this is implemented asc(r51,f)50. Addi-tionally, as suggested in Ref.@25#, the numerical boundarycondition ]v/]r50 is imposed on the cylinder boundarwhich correctly reproduces an inviscid dynamics.
As the initial vorticity field for the control simulations, weused a localized Gaussian distribution of vortici;exp@(r2r0)2/2d2#centered on a pointr0 far from the cyl-inder, of typical transversal sized. The initial position of theGaussian vortex was in the vicinity of the stable manifoldthe fixed point of the background point-vortex model,avoid artificially large initial perturbation during the targe
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PRE 58 1887DYNAMICAL CONTROL FOR CAPTURING VORTICES . . .
ing procedure. The extensiond of the distribution rangesfrom 0.2 to 0.6, so the size of the vortex is comparable wthat of the cylinder.
The positionr0(t) of this extended vortex, required fothe input to the control algorithm, is computed as the cenof vorticity in a domainD(t) around the vortex:
r0~ t1DT!5
*D~ t !r 8v~r 8!u„2v~r 8!…d2r 8
*D~ t !v~r 8!u„2v~r 8!…d2r 8
, ~15!
whereDT is the time step for numerical integration andD(t)is a disk of radius 2d centered on the vortex position at timt, i.e., r0(t). The functionu(v)51 for v.0 andu(v)50for v,0, is used to weight the positive vorticity only, avoiding confusion with the opposite-signed shed vorticity~whichappears only in the no-slip simulations!. In a realistic con-troller such an input should come from instantaneous phcal measurements, such as multiple pressure observaand/or velocity monitoring. It has been shown previouthat the vortex dynamics can be entirely reformulated inpressure-measurement space, by recording two or threepressures on the cylinder surface@26#. Then a control algo-rithm analogous to the one described here can be appliethe pressure space without relying on actual vortex posi@27#.
B. Details of the no-slip simulation
The physical boundary conditions on the cylinder surfaare typical no-slip conditions, i.e.,
]c
]fUr51
50 and]c
]rUr51
5V, ~16!
whereV is the angular velocity of the cylinder, measuredthe counterclockwise direction. These boundary conditican be used to specify the stream function and the vorticitthe boundary. As numerical boundary condition for tstream function,c50 is used. The value of the wall vorticitis obtained by inserting conditions~16! in Eq. ~11!, specifiedat the wall and using a standard second-order estimate@25#.
To ensure physically self-consistent initial conditions fthe vortex dynamics, and to avoid the transient effectswake generation during the control process, we first compthe solution for a fully developed wake with no controllevortex present. To break the initial symmetry of the wa~which is unavoidable for symmetric initial conditions! thecylinder was impulsively rotated back and forth during tfirst few time units and stopped afterwards. The relaxattoward a periodic solution is checked by monitoring the elution of forces on the cylinder@cf. Fig. 15~a!#.
After this, a Gaussian distribution of vorticity is superimposed on the vortex street solution centered at a pointr0 farupstream of the cylinder. In the numerical scheme this cresponds to a vortex that suddenly enters the finite comptional domain. In response to this event the total circulatcontained in the domain grows by a quantity equal tovortex circulation.
This fact cannot be accounted for by the boundary contion ~13! which imposes the irrotational field, with zero ci
h
r
i-ns
aint
inn
e
sat
fte
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culation, on the outer edge of the domain. The artificial costraint of condition~13! would require creation of additionacirculation. The only mechanism by which additional circlation can be created is by vortex shedding from the cylinsurface. Thus the placement of the vortex in the computional domain would result in an impulsive vorticity genertion and consequent impulsive shedding even if the vorteplaced far from the cylinder. To maintain the circulation baance, condition~13! should be replaced by a condition whethe actual value of circulation around the domain is imposexplicitly. Instead, for numerical convenience, this is doby introducing a second vortex of equal strength and opsite sign far downstream which maintains the vorticity bance in the computational domain. This additional vortdoes not influence directly the vortex dynamics~it is far fromthe upstream vortex!, but successfully balances the circultion in the computational domain. The position of the vortplaced upstream is monitored in analogy with the free-scase using Eq.~15!.
The drag and lift coefficientsCD andCL are computed as@28#
CD5E0
2p
df~2 p cosf2srfsinf!,
~17!
CL5E0
2p
df~2 p sinf1srfcosf!,
where the first and second terms describe the contributionthe pressure and shear stress forces, respectively. The disionless quantitiesp and srf are defined as
p5p
ru02R0
51
ReE0
f
dw]v
]r~r51,w! ~18!
and
srf5s rf
ru02R0
5R0
Re@v~r51,f!1V#. ~19!
Here r is the density of the fluid.To test the numerical procedure, several runs were car
out for different values ofV at Re5500 with no controlledvortex present. The time evolution of the drag and lift coficients was monitored. We have found good agreement@27#with the coefficients measured by Chewet al. using a hybridvortex scheme@28#.
IV. FREE-SLIP DYNAMICS AND CONTROL
Before applying the control scheme in a NS flow, we hato determine whether the basic features of the point-vorflow topology are still preserved in the NS flow. Suchanalysis has been presented previously for the free-slip@19#. Here, we briefly reexamine some of the importantsults with a control implemented by changing the unifobackground flow velocity at infinity.
Previous results on vortex scattering around a cylinusing free-slip dynamics have shown a strong correspdence to the Hamiltonian vortex dynamics@19# if the vortex
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1888 PRE 58PENTEK, KADTKE, AND PEDRIZZETTI
patch size is small (d,0.5). It has been shown analyticalin Ref. @19# that the correction to the Hamiltonian dynamicequations is of orderh0
2, whereh0 is a small parameter characterizing possible internal degrees of freedom of the vo~e.g., ellipticity!. This can also be observed in Fig. 4 whethe time evolution of the vorticity is shown for an initiapatch size ofd50.25. The Reynolds number is Re51000.The vortex maintains coherence as it is advected pastcylinder and the center of vorticity closely follows thHamiltonian trajectory. If the vortex is extended (d.0.5) theinteraction is significantly more complex. This can be oserved in Fig. 5, where the vortex dynamics in the case offree-slip boundary conditions is plotted for a typical initicondition for a large vortex patch of sized50.6. During thevortex-cylinder interaction process the initially strongly cherent vortex is stretched, and some low-vorticity filamedetach from the vortex and remain around the cylinder eafter the core of the vortex leaves the domain shown in F5. In this case only the core of the vortex patch of sized50.6 maintains coherence and follows roughly the Hamtonian trajectory@cf. Fig. 6~a!#.
Thus we expect that for spatially coherent vortices mof the features of the point-vortex control can be readimplemented for NS free-slip dynamics. The only importainviscid process that cannot be modeled with a simple povortex description is the vortex breakdown, when stroshearing fluid motion breaks away a large extended voror some strongly stretched vortex filaments. As we will sbelow @cf. Fig. 6~d!#, however, there are no regions of stroshear in the vicinity of the target saddle point, and theeffects do not play an important role during control on timscales shorter than the viscous time.
To achieve control, external perturbations are introdu
FIG. 4. Time evolution of a small vortex of sized50.25, withfree-slip dynamics. The vorticity field is shown at time intervals2. Constant vorticity lines are shown in increments ofvmax/10. Theinitial vortex position is~3.0,24.0),s522.962 96, and Re51000.One grid cell corresponds to a unit square.
x
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-
t
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e
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to the background flow velocity at infinity in the same spiras for the point-vortex case. The position of the vortex rquired to evaluate the perturbation is obtained as the ceof vorticity of the controlled vortex~15!. Thus the action ofthe controller is determined solely as though a point vortexpresent at the center of vorticity of the extended vortex.
To test the validity of the control scheme in the free-slframework, we place a blob of vorticity with initial condition(x0 ,y0)5(3,24) andd50.6 for Re51000. A grid of 1283128 is used with stretching parametera50.2, domain sizermax5100, and the time step for numerical integration0.01. We first evolve the vorticity profile without the controscheme, as shown in Fig. 6~a!. One can observe that the corof the blob advects past the cylinder essentially following tHamiltonian flow lines in spite of the significant shape ditortion and detachment of the low-level vorticity. Then wrepeat the same numerical experiment, now with the contler turned on@Fig. 6~b!#. The vortex slowly approaches thHamiltonian fixed point and remains there for at leastcharacteristic flow times. In spite of the fact that the cotrolled vortex has a slightly distorted shape, the algorithresults in the stabilization of the vortex very close to thfixed point. Figure 6~c! displays the applied perturbation tothe uniform background flow as a function of time. After thshort targeting period, the required perturbation is smaabout 5% of the background flow velocity. With a perfemodel of the flow the perturbations would go to zero in thlong time limit. The discrepancy is due to the fact that thshape distortions are not included in the point-vortex modon which the control algorithm is based. Note that whileelliptic vortex like the one in the final stages of controlexpected to rotate, this one does not have a rotational mtion. The absence of rotation can be explained from the
FIG. 5. Time evolution of a large, extended vortex of sized50.6, with free-slip dynamics. The vorticity field is shown at timintervals of 2. Constant vorticity lines are shown in incrementsvmax/10. The initial vortex position is~3.0, 24.0), s522.962 96,and Re51000.
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PRE 58 1889DYNAMICAL CONTROL FOR CAPTURING VORTICES . . .
pology of the stream lines@Fig. 6~d!#, which is essentiallysimilar to the stream-line pattern for a vortex pair@29#. Thevortex takes on the shape of the stable stream-line patteFigure 6~d! also shows that there is no shearing motioaround the controlled vortex and thus the only mechanismwhich vorticity can be lost is by diffusion beyond the circular stream-line pattern. On the time scales of our simulatiothis is minimal and thus there is no significant loss of voticity during control. This numerical experiment demonstrates that the topology of the underlying Hamiltonian dynamics is still exhibited in the free-slip case. The vorteitself remains stable under the applied perturbations in spof the shape distortion and diffusion of the original Gaussiavorticity profile.
For the free-slip control the capture time is determined bthe viscosity. On viscous time scales the concentrated vorpatch diffuses away and eventually will be broken away bshearing fluid motion. Since this time scale is on the order1000 time units we have never been able to reach this stain our numerical simulations.
V. VORTEX SCATTERING IN A VISCOUS FLOW
We now examine the vortex scattering in the case of nslip boundary conditions on the cylinder surface, when thcylinder is rotating counterclockwise with uniform angulavelocity V. This rotation is characterized by the dimension
FIG. 6. ~a! Evolution of vortex core for the dynamics of anuncontrolled vorticity distribution with free-slip dynamics shown indetail in Fig. 5. The corresponding point-vortex trajectories ashown by thin dashed line.~b! shows the controlled dynamics. Thevortex is stably captured at the unstable fixed point of the Hamtonian flow situated at (0,23.0) in Euclidean coordinates. As initialconditions,~3.0, 24.0), s522.962 96, and sized50.6 have beenused. One contour level corresponding tovmax/2 is plotted everyunit of time. Thick dashed lines correspond to additional vorticitcontours of 0.3 and 0.1 ofvmax at t530. The figure clearly indi-cates that the control scheme is stable even for a large vorticdistribution, and a close correspondence with the underlying Hamtonian system.~c! Time evolution of the control parameter: theperturbation to the background flow velocity.~d! Stream lines of theflow in the controlled case att530.
rn.
y
n-
-
ten
yexyfge
-e
-
less parametera5VR0 /u0, i.e., the ratio of the velocity ofthe cylinder surface to that of the background flow at infiniThe vortex shedding, drag, and lift coefficients have beextensively analyzed fora values ranging from 0 to 6@28#.In this section we focus our attention on the dynamics ofexternal vortex that approaches the cylinder from upstreinteracts with it, and then is advected downstream. Sucvortex could originate from vortices shed by other bodplaced far upstream. This vortex dynamics can be regaras a kind of scattering process, with a very simple dynam~uniform advection! far from the cylinder, and a highly nontrivial interaction close to the cylinder. With the initial vortex coordinatex0 fixed, y0 can be regarded as an impaparameter that characterizes the scattering process.
There is an additional parameter in the problem, the phof the beginning of the simulation relative to the periodvortex shedding. Since the vortex approaching the cylinstrongly disturbs the vortex shedding, we have found thatphase is not an important parameter in the simulation.
Figure 7 displays vortex trajectories with different impaparameters ata50 ~no rotation!. For comparison Fig. 3~a!displays the vortex dynamics for the Hamiltonian systeThe basic difference comes from the vortex–boundary-lainteraction. Figure 8 shows the time evolution of the vortity field for the trajectory marked in Fig. 7. As the vorteapproaches the cylinder on a trajectory similar to the Hamtonian dynamics, at a critical distance from the cylinder sface a secondary vortex of opposite sign is induced inboundary layer, which pairs with it and is advected awSince the strength of the two vortices is not the same, ttravel on curved trajectories. The vortex trajectories can tycally intersect themselves or other trajectories, since thelocity field depends not only on the actual vortex position brather on the entire trajectory, i.e., the whole history of t
e
l-
tyil-
FIG. 7. Vortex scattering with no-slip boundary conditionsRe51000. The trajectory of the center of vorticity for vortex scatering with different impact parameters is shown fora50. As ini-tial vortex coordinatex055, 26<y0<0 has been used. For comparison solid lines in Fig. 3~a! display the vortex trajectories foHamiltonian dynamics.
and.5 andis 2. A
arameter
1890 PRE 58PENTEK, KADTKE, AND PEDRIZZETTI
FIG. 8. ~Color! Time evolution of the vorticity field for the trajectory marked in Fig. 7. The vorticity contours are shown in greenblue for positive and negative vorticity, respectively. The vorticity increment between consecutive level lines is 5, starting with 222.5 for green and blue lines, respectively. The cylinder is shown in red. The time between consecutive instants of vorticity field2563200 grid has been used in angular and radial direction, respectively, with a step size for integration of 0.001. The stretching pwas a50.2 and the radius of the computational domainrmax5100. The size of the initial Gaussian profile wasd50.25 and the vortexstrengths522.962 96. One grid cell corresponds to a unit square.
a
ut
Toongofe
teor
derthethee
tatem
thee-
onong
time evolution. Interestingly, Fig. 7 suggests that there isenvelope of these trajectories that defines a region aroundcylinder that is not accessible for vortices coming from oside.
Next, we focus on the case of a rotating cylinder.understand the scattering process in the presence of a ring cylinder, we recall that the spin up of the surroundifluid at radiusr from the origin takes place on a time scaleorder Re(r21)2. Thus the typical time scale can be assumto be of order of Re. This is much larger than the timevortex spends around the cylinder before it is transporaway by the background flow. Therefore, on relatively sh
nthe-
tat-
dadt
time scales, and for distances not too close to the cylinsurface, the vortex dynamics is not directly affected bycylinder rotation. The rotation does, however, influencevortex dynamics through its effect on the stability of thboundary layer. We note here that the cylinder starts to roimpulsively at timet50, when the vortex is placed upstreaof the flow at coordinates (x0 ,y0). Apart from numericalconvenience, this seemingly arbitrary relation betweenstart of the spin and the position of the vortex will be rgarded as part of the control action~see next section!. Thedetails of the actual vortex dynamics may subtly dependthis choice, but we observe no qualitative differences as l
c
yltey
org.eornd
ilthgil-ore
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t.ionn-ntSi-nd-ac-
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PRE 58 1891DYNAMICAL CONTROL FOR CAPTURING VORTICES . . .
asur0u!Re ~herer0 denotes the initial vortex position! @27#.Figure 9 shows the vortex scattering for different impa
parameters for a rotating cylinder ata54. One can observethat the vortex can now come significantly closer to the cinder surface. The trajectory marked with an arrow indicathe existence of a saddle-point-like structure below the cinder at some instant of time. The time evolution of the vticity field corresponding to this trajectory is shown in Fi10. The boundary layer is stable when the vortex is relativfar from the cylinder surface, but a large positive-signed vtex is shed as the scattered vortex comes close to the cylisurface.
Figure 11 displays the vortex scattering ata510. Herethe overall flow structure has some similarities to the Hamtonian case, as vortex trajectories typically penetratewake of the cylinder. There is a limiting curve for incominvortex trajectories similar to the separatrix in the Hamtonian case, that divides trajectories passing below and abthe cylinder. One of the significant differences is that seveinitial conditions lead to a finite-time vortex capture. Thtime evolution of the vorticity field for one of these capturtrajectories is shown in Fig. 12. In the point-vortex dynaics, such finite-time vortex capture has been observed fooscillating cylinder only. In the Hamiltonian dynamics thorigin of such capture is the explicit time dependence ofequations of motion~2! that leads to the formation of a chaotic saddle in the vicinity of the cylinder@20,30,31#. In thecase of a viscous flow there is an implicit time dependencthe dynamics due to the vortex–boundary-layer interactMoreover, since the captured vortex comes close to theinder surface, it is affected by the fluid that is spinning upthe vicinity of the rotating cylinder surface. This also cotributes to the fast rotation of the vortex around the boNote that there is no significant vortex shedding during sa capture process.
In this section we analyzed the vortex scattering onrotating and translating cylinder in a viscous flow. While tdynamics far from the body is described qualitatively by t
FIG. 9. Vortex scattering with no-slip boundary conditions. Ttrajectory of vortices with different impact parameters is showna54.
t
-sl--
ly-er
-e
veal
-an
e
ofn.l-
.h
a
Eulerian dynamics, the interaction with the boundary layerthe vortex comes close to the cylinder induces some ninteresting effects, i.e., the presence of a region not acsible for vortices approaching from upstream, and also a vtex capturing effect@27#. We plan to present a detailed analsis of this scattering process in future.
VI. VORTEX CONTROL IN THE VISCOUS CASE
The numerical experiments in the preceding section sgest that the boundary-layer dynamics depends stronglythe rotation parameter. The larger thea, the longer is thetypical time scale on which a large secondary vortex formAt large a values the boundary-layer dynamics does ncouple with the vortex dynamics. Even ata54, there is nosecondary vortex shed as long as the vortex approachingcylinder is about one cylinder radius away from the cylindsurface. On the other hand, on time scales much smallerRe, the velocity field at the target fixed point is not affectby the cylinder rotation, and thus our Hamiltonian model~1!is expected to be valid in the vicinity of the fixed poinThese arguments have resulted in the following formulatof the control algorithm: let the cylinder be rotating at costanta throughout the control procedure in order to prevesecondary shedding of strong opposite-signed vorticity.multaneously, the perturbation to the uniform backgrouflow velocity at infinity du0 is changed as a control parameter. The required perturbation is evaluated based on thetual vortex position, assuming the ideal Hamiltonian floapproximation, Eqs.~1!, ~2!, and ~8!. Such a numerical ex-periment is presented in Fig. 13. The corresponding conperturbation in the background flow velocity field is showin Fig. 14.
In Fig. 13, as control is applied the vortex approachestarget saddle point~marked by a cross! along the stableeigendirection. Due to the constant rotation of the cylind~a54!, the vortex shedding gradually diminishes and finadisappears aroundt512 @plate~g!#. The vortex settles downon the fixed point and remains stable throughout the simtion. Correspondingly, the magnitude of the perturbatslowly decreases, but asymptotically does not reach thevalue. This effect is due to the simple form of the contrmodel of the flow, which does not take into account tadditional vorticity in the boundary layer, and to the smbut non-negligible effect of the cylinder rotation on the vlocity field at the target point. These additional effects leada renormalized unperturbed background flow velocity thaslightly smaller than 1, as also suggested by the small netive asymptotic value of the perturbation of20.3. In fact byestimating the position of the fixed point and the eigenvaland eigenvectors from the Navier-Stokes scattering simtions, rather than from the Hamiltonian model, one can fther reduce the magnitude of the perturbations. Such a silation is presented in Ref.@27#, where the control parameterare extracted from the marked trajectory of Fig. 9. Thequired perturbations for control are decreased by an ational 50%@27#.
During the control procedure the overall controlled voticity is conserved. The vorticity in a large domain arouthe controlled vortex changes less than 1% during the con
r
the
1892 PRE 58PENTEK, KADTKE, AND PEDRIZZETTI
FIG. 10. ~Color! Time evolution of the vorticity field witha54 for the trajectory marked by arrow in Fig. 9. The parameters ofnumerical solution are the same as those used in Fig. 8.
leol
so1
tisnd-dickift
b
e
s
gin-of
redheee
f atlyive
simulation. The original compact Gaussian vorticity profihowever, diffuses away considerably as shown by the evtion of the vorticity contour lines in Fig. 13.
The time evolution of the drag and lift coefficients is almonitored throughout the numerical experiment. Figuredisplays these coefficients for various values ofa for boththe controlled and the uncontrolled state. As a reference,time evolution witha50 and no controlled vortex presentalso shown@Fig. 15~a!#. These are essentially the drag alift coefficients for a cylinder uniformly translating in a viscous flow. The periodic oscillations are due to the periovortex shedding. When the cylinder is rotating counterclowise with a54 a Magnus effect is observed and the lcoefficient is significantly increased@cf. Fig. 15~b!#. Note,however, that the lift cannot be increased without bound
,u-
5
he
c-
y
increasing a. Figure 15~c! shows practically the samasymptotic value for the lift coefficient ata510 as for theone ata54. Also, the ratio of the lift to drag coefficient iactually decreasing froma54 to a510 in accordance withprevious observations by Chewet al. @28#.
Figure 15~d! shows the time evolution of the lift and dracoefficients during the control procedure. There are twoteresting observations. First, the mean lift coefficient isnegative sign, e.g., the net lift force is toward the captuvortex. If the captured vortex would be of positive sign, ttarget fixed point would lie symmetrically just above thcylinder and correspondingly, the lift would be toward thvortex and of positive sign. This is similar to the case ocaptured vortex over an airfoil, that is known to significanincrease the lift. Secondly, the drag coefficient is posit
rathn-a
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ithyhi
anaeae. 1d
obeorr
le
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e-
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PRE 58 1893DYNAMICAL CONTROL FOR CAPTURING VORTICES . . .
and small, which means the body translates practically dfree and it is even subject to a small thrust. We note thatactual asymptotic value of the lift coefficient in the cotrolled case is smaller than the one created by a simple Mnus effect at corresponding angular velocities@cf. Figs. 15~b!and 15~d!#. The ratio of the lift to drag coefficient is, however, significantly higher. We note that the controlled vortecylinder system shown in Figs. 13~k!–13~p! has a very simi-lar stream function pattern to that of a translating dipole wthe controlled vortex being one of the vortices and the cinder playing the role of the opposite-signed vortex. Tforces we obtained in the NS simulations are consistent wsuch a picture.
Figure 15~e! displays the drag and lift coefficient forcontrolled vortex when the cylinder is rotating at constaa510. This shows that the forces do not reach a stationvalue, but are oscillating periodically around the value msured in Fig. 15~c! for a54. This is due to the fact that thvortex shedding does not disappear completely, as in Figand there is a small shedding of low-level vorticity that leato the oscillations in the lift and drag coefficients.
The rotation of the cylinder is essential for the successthe control. If the cylinder is not rotating, the vortex canstabilized for extremely short times only. Soon after the vtex reaches the target saddle point, an opposite-signed vois shed from the boundary layer, pairs with the controlvortex, and subsequently the control fails.
The typical capture time for the no-slip simulationsabout 50 time units. On this time scale the boundary lathickens gradually and eventually a vortex is shed. At tpoint the control is lost since the shed vortex usually pawith the controlled vortex and the vortex pair formed cannbe stabilized any longer.
VII. THE VISCOUS RESPONSE OFTHE BOUNDARY LAYER
Our goal in this section is to show that there is a steasolution for the boundary-layer equations in the presenc
FIG. 11. Vortex scattering with no-slip boundary conditionThe trajectory of vortices with different impact parameters is shofor a510.
g-e
g-
-
l-eth
try-
3,s
f
-texd
rsst
yof
an external vortex, as suggested by the viscous simulatin the preceding section. Previous results on the vorteboundary-layer interaction show that there are two batypes of response of the boundary layer when a vortex paclose to a wall@32#. In the first case, when the vortex speis relatively low, there is no steady viscous solution. Secoary vortical structures develop on the wall, leading toeruption of the boundary layer@33,34#. In the second casewhen the vortex moves fast relative to the wall, there isstable solution, although the boundary layer ‘‘thickengradually@35,36#. Here we show that the long-term stabilitof the boundary layer observed in the experiments in Secis due to an effect more subtle than the one leading totypical stable behavior mentioned above.
When analyzing the viscous response of the boundlayer we assume that the control is active, as shown in F13~k!–13~p!. Therefore the vortex remains stationary, athe cylinder is rotating with a constant angular velocity. Fthe boundary-layer dynamics, this means that the boundconditions can be considered as time independent for splicity.
Let us first introduce the following notations for the vlocity components:
w5v r , u5vf ~20!
for the radial and angular components, respectively, andfine the new boundary-layer scaled radial variables
y5~r21!Re1/2, w5wRe1/2. ~21!
The Prandtl equations, governing the evolution of the useparated boundary layer, result as
]u
]t1w
]u
] y1u
]u
]f52
]p`
]f1
]2u
] y2, ~22!
]w
] y1
]u
]f50. ~23!
Herep` denotes the pressure outside of the boundary ladue to the inviscid solution at the cylinder boundary in tpresence of the background flow and the vortex. The Bnoulli equation implies, for a pressurep` at the outer edge othe boundary layer,
dp`52u`du` . ~24!
Therefore Eq.~22! expressed in terms of the inviscid veloity u` on the cylinder surface can be written
]u
]t1w
]u
] y1u
]u
]f5u`
]u`
]f1
]2u
] y2. ~25!
The boundary conditions are
u5a, w50 at y50 ~26!
and
u~ y,f,t !→u`~f!, as y→`. ~27!
n
ters
1894 PRE 58PENTEK, KADTKE, AND PEDRIZZETTI
FIG. 12. ~Color! Time evolution of the vorticity field fora510 in the case of the trajectory marked by arrow in Fig. 11. The parameof the numerical solution are the same as those used in Fig. 8.
ioe-
ardhe
e
cidedwn-re.ts.
erse-
To increase precision in the physically interesting regnear the cylinder wall, we introduce the radially stretchcoordinateh5 ln@(y1b)/b#, with b being a stretching parameter. Then the governing equations~25! and~23! in the newcoordinates become
]u
]t1
w
beh
]u
]h1u
]u
]f5u`
]u`
]f1
1
~beh!2F ]2u
]h22
]u
]hG ,
~28!
1
beh
]w
]h1
]u
]f50. ~29!
nd
We solve these equations numerically with a standfinite-difference method. The only input necessary for tproblem isa andu`(f). The angular velocity of the cylin-der a is constant, whileu`(f) depends implicitly on thevortex position and background flow. Figure 16~a! shows thetypical inviscid velocity profile when the vortex is on thtarget fixed point and the cylinder is not rotating, i.e.,a50.This profile has been obtained as a solution for the invisproblem with the controlled vortex placed on the target fixpoint, and a second opposite-signed vortex placed far dostream, to maintain vorticity balance, as explained befoThis profile shows the existence of four stagnation poinOne can observe that there is one region with strong advpressure gradient~around 1.6&f/p&1.7) where the secondary vortex is expected to develop@37#.
the initial
PRE 58 1895DYNAMICAL CONTROL FOR CAPTURING VORTICES . . .
FIG. 13. ~Color! Time evolution of the vorticity field in the case of the controlled vortex dynamics witha54. The overall controlledvorticity does not change significantly during the control procedure. The decrease of the number of level lines is due to the fact thatcompact Gaussian profile diffuses away, i.e.,d increases gradually.
3te
by
ethrcnd
one
ese
heb-the
this
gen
hense
The numerical calculations are performed on a grid364 in radial and angular direction, respectively. The ouboundary is set at a large but finite valuehmax54.0 with thestretching parameterb50.5. The convergence is checkedrepeating the simulation on a larger grid 723128. No sig-nificant differences have been observed.
Figure 16~b! shows the solution to Eqs.~23! and ~24! attime t50.48. As an initial condition for the velocity field, wassume that the inviscid solution is valid throughoutboundary. One can observe the newly developed backcilating structure which later leads to an eruption of the bouary layer as seen in Fig. 8~c!. Note that the ‘‘spiky’’ appear-ance of Fig. 16~b! is due to the presence of the stagnatipoints on the cylinder surface, and the position of flows p
6r
eu--
r-
pendicular to the surface correspond approximately to thstagnation points.
Figure 16~c! shows the result of the simulation att52.0for a rotating cylinder, at the typical value used during tcontrol processa54.0. One can observe the complete asence of the secondary vortical structures, in contrast tocase when the cylinder is not rotating. To understandremarkable stability of the boundary layer@38#, let us trans-form our problem to a frame comoving with the rotatinboundary. The inviscid velocity and pressure profile sefrom this frame will be similar to the one in Fig. 16~a!, withone difference: in this comoving frame the vortex and tbackground flow is rotating around the cylinder. This meathat the velocity and pressure profile is changing in tim
uri-
a
radi-inhe
onnstnt
entedn-
s.esionsd aonbi-us
hedy-S
d.yl-re-
k-
trd
re
s the
yer
1896 PRE 58PENTEK, KADTKE, AND PEDRIZZETTI
periodically. The region with positive streamwise pressgradient from Fig. 16~a! will have a negative pressure gradent just after a period approximatelyt0;p/2a. If the cylin-der is rotating fast enough, the outer edge of the bound
FIG. 14. Time evolution of the control parameter« during thecontrol procedure shown in Fig. 13.
FIG. 15. Time evolution of drag~bold line! and lift ~thin line!coefficients for~a! a50, ~b! a54, and~c! a510 when no con-trolled vortex is present.~d! and ~e! show the drag and lift coeffi-cients during control fora54 anda510, respectively. Note, thathe sign of the coefficients corresponds to the choice of the coonate system in Fig. 1, i.e., positive drag or lift coefficient corsponds to a force pointing in the positivex or y direction, respec-tively.
e
ry
layer experiences an average pressure, without strong gents. The actual criteria for stability could be formulatedthe following way: the typical time scale associated with tgeneration of the vortical structures in the casea50, that ison the order oftv;0.5, should be larger thant0. In terms ofthe cylinder rotation, this means that the angular velocityashould be larger than 1.6. In the limit of largea, the inviscidvelocity at the wall seen in the comoving frameu8 (f,t) canbe replaced by its time averaged value^u8 (f,t)& t , that inthe first approximation is constant and no longer dependsf. In this way, the flow is similar to one occurring betweetwo concentric cylinders, when the inner cylinder is at rewhile the outer one is rotating with a relatively low, constaangular velocity.
VIII. CONCLUSIONS
In this paper we studied the interaction of a large cohervortex with a translating and rotating cylinder. We showthat in a viscous flow, there is an interesting and highly notrivial scattering effect of advecting vortical structureMoreover, in the case of a rotating cylinder, these vorticcan be captured for long periods. Based on the observatof vortex scattering in viscous flows on the one hand, ancontrol scheme previously implemented in inviscid flowsthe other, we developed a simple control algorithm to stalize an external vortex near a moving cylinder in a viscoflow.
As a main result, we demonstrated that control of tvortex dynamics based on a low-dimensional reducednamical model, previously reported in Hamiltonian and N‘‘free-slip’’ simulations, can be achieved in a viscous fluiWhile previous studies used the circulation around the cinder as a control parameter, here we introduced a morealistic perturbation: small changes in the uniform bacground flow velocity at infinity, combined with a uniform
i--
FIG. 16. ~a! Inviscid velocity~solid! and pressure~dashed! pro-file on the cylinder surface for«50, with vortex centered on thetarget fixed point. The pressure is plotted in units ofru0
2/2, andp0
is the pressure at infinity. These profiles have been obtained asolution to the ‘‘free-slip’’ dynamics with a vortex of sized50.25 placed on the fixed point. Stream lines of the boundary-lasolution in the laboratory frame:~b! for a50 at t50.48; and~c! fora54.0 att52.0.
nthbetema-n
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PRE 58 1897DYNAMICAL CONTROL FOR CAPTURING VORTICES . . .
rotation of the cylinder. The perturbation in the backgrouflow velocity corresponds to the change of the velocity ofcylinder translating in a uniform fluid at rest, and canimplemented even in an experiment. The model sysshown in this paper is a remarkable example of how socomplex fluid flows described by partial differential equtions, which are inherently infinite dimensional, can be cotrolled using a simple low-dimensional model of the fluflow.
The success of the control is due to the disappearancthe vortex shedding which results from the stability of tboundary layer. We studied this boundary-layer dynamicdetail, and showed numerically that under the applied perbations the boundary layer is stable, and thus no significvortex shedding is expected during control. We note hthat the same reasoning which leads to stability of the bouary layer in the presence of the controlled vortex cancarried out without the presence of an external vortex, incase of a rotating and translating cylinder. This suggeststhe vortex shedding should disappear ata above some criti-cal valueac . In fact such disappearance of the vortex shding has been pointed out in experiments by JaminetVan Atta @39# at low Reynolds number flows around a rotaing cylinder. Their observation shows that the critical valof the rotation parameter increases with the Reynolds nber but has a plateau ofac52 starting at Re'80, abovewhich the critical rotation velocity no longer depends on tReynolds number. This result is in qualitative agreemwith our explanation, since the boundary-layer argument psented in this section is essentially Reynolds number inpendent.
In all the successful chaos control experiments presethus far @13–17#, control has been performed without thexplicit knowledge of the dynamical equations. The unstaperiodic orbits, and their eigenvalues and eigenvectors hbeen obtained by reconstructing the dynamics directly frtime series. In simple open systems like the one studiethis paper, a few scattering trajectories can furnish enoinformation to reconstruct the dynamics around the sadpoint. For example, the marked trajectory in Fig. 9 incorprates all this information with sufficient precision to be usin the control dynamics.
In the present study we restricted our investigation to twdimensional flows. An interesting question is whetherdynamics of a vortex filament in a three-dimensional flowstable or not under the control perturbations we apply,whether instabilities will bend and fold the vortex intocomplicated structure. Clearly, more study is needed toswer this question.
Another interesting problem is whether the flow field c
,
de
me
-
of
inr-nted-eeat
-d
-
te-e-
ed
leve
inh
le-
-e
r
n-
be controlled for more generally shaped bodies. In particuthe flow over low-speed airfoils with an attached free vorthas attracted recent interest@40–42#. These works havemainly focused on the stability analysis of the vortex dynaics. Others have studied this problem using a leading-eflap @43–45# or blowing and suction on the airfoil surfac@46# for flow modifications. These attempts, however, areactive control schemes. It is known that unstable fixed poof the vortex dynamics can be found in the Hamiltonimodel of a vortex over a Joukowski airfoil@47#. While con-trol of vortex dynamics in such states is possible in tHamiltonian model system, direct numerical simulationneeded to study the corresponding control problem in vcous flows. In some applications, a vortex stably attacover an airfoil is of primary interest since relatively higlevels of lift can be achieved@48#. In other applications,however, the formation and trapping of vortices over airfsurfaces is not desired, since the detachment of the vofrom the airfoil leads to a sudden decrease of lift, sometimknown as the ‘‘dynamic stall effect’’@32#. In these casesactive control methods could play another role, namely,prevent metastable capture by driving the vortex towardunstable direction instead of the stable one.
We must emphasize that, while most point-vortex andviscid results can be readily extended to more complex bies using conformal mappings, the viscous results cannoimmediately generalized. The stability of the boundary lay~which is the key to our successful control method! dependsstrongly on the particular geometry considered. The esseobservation is that if a proper mechanism is found that sbilizes the boundary layer~suppresses the vortex shedding!,the control of the vortex dynamics in a viscous flow cansuccessful. At present, there is no reliable way to extrinformation from the boundary layer which could be useddesign a simple controller that stabilizes the boundary layThis problem remains a continuous challenge for the fldynamics and aerodynamics community.
ACKNOWLEDGMENTS
The authors wish to thank L. Cortelezzi, E. Novikov,Tel, C. W. Van Atta, and A. Wiegand for very useful discusions. A.P. and J.K. wish to acknowledge partial supportthis project by a grant from the Office of Naval Researunder Grant No. N00014-96-1-0056, the U.S.–HungarScience and Technology Joint Fund under Project JF N286 and 501, and by the Hungarian Science Foundationder Grant Nos. OTKA T17493 and F17166. G.P. wishesacknowledge partial support by a travel grant from tNATO Office of Scientific Affairs, under Grant No. SA.52-05 ~CRG.931160!, and the Italian MURST.
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