In the refining process of crystal growth, the fluid phys-ics of the melt is of great importance since the flow mayhave a strong impact on the quality of the final product. Thefluid motion is mainly caused by thermocapillary and buoy-ancy forces, and sometimes electromagnetic forces. Espe-cially, for the floating-zone technique, as it is suggested forspace processing under microgravity, the main interest hasbeen the thermocapillary convection.
One of the detrimental problems in the floating-zonetechnique are so called striations, inhomogeneous distribu-tion of chemical compounds, and dopants appearing in thefinal single crystal. Since this was found to be due to thetime-dependent oscillatory state of the thermocapillaryconvection,1 a number of extensive works have been re-ported.
A large part of the work reported in the literature hasfocused on simplified models where generic flows seen in thefloating-zone are realized. The most common model to dateis an axisymmetric geometry called the half zone, hereafterreferred to as HZ. In HZ, a liquid drop is held between twocoaxial rods maintained at different temperatures to imposean axial temperature gradient on the free surface. The firstexperimental detections of the oscillation were realized bySchwabe and Scharmann2 and Chun and Wuest1 in this typeof geometry which motivated works in linear stabilityanalyses,3,4 numerical simulations5,6 and experiments7 tostudy the onset instability characteristics and wave struc-tures. Later, supercritical behaviors of the oscillation wererevealed by both experiments and numerical simulations.8,9
One of the difficulties in experimental investigations ofHZ is the curvature of the free surface whose shape is knownto have a strong influence on the flow.10 For the sake ofbetter quantitative analyses and thus comparison with nu-
merical simulations, another axisymmetric geometry, annularconfiguration shereafter referred to as ACd was first sug-gested by Kamotaniet al.11 The system is an open cylindricalcontainer filled with a liquid to have a flat-free upper surface.A heated pipe with a prescribed temperature is located on theaxis of the container. Thermocapillary convection is thusdriven by imposing a radial temperature gradient on the flat-free surface. In this geometry, a flat-free surface can be easilyrealized and maintained. During the last decade, a series ofextensive studies was carried out by the group of Kamotaniand Ostrach including microgravity experiments inspace.12,13Lavalleyet al.14 reported detailed comparisons be-tween experiments and simulations, as well as velocity fieldsobtaind experimentally using particle image velocimetry.
Detailed experiments in microgravity and simulationsfor the AC has further elucidated the selection of the spatialstructure and the azimuthal wavenumber of the oscillatorymodes.15–17The Prandtl number was 6.8, but the aspect ratioof the annular domain was varied by one order of magnitude,to include very shallow layers. It is found that the azimuthalwavenumber and the spatial complexity of the modes in-creases as the depth of the layer is decreased. A somewhatdifferent geometry was studied by Sim and Zebib,18 whichstudied a dish filled with liquid, i.e., an AC without the cen-tral heated pipe. The thermal gradient is instead driven by anapplied heat flux to the free surface, and cooling at the rim.Results are given for a Prandtl number of 30, and free sur-faces with different curvatures. The results qualitatively re-semble those found for AC with the appearance of travelingor standing waves as the thermocapillary Reynolds numberexceeds a critical value.
Even if axisymmetric base flows in HZ and AC arequalitatively similiar, there are some differences. When thesurface flow is convective inertial, the vortex core is pulledtowards the hot corner in HZ whereas the core stays in thecenter or is pushed towards colder corner in AC. Kamotanietal.,13,19 from their observations, suggested that, in HZ andAC, the overall flow is mainly driven in the hot corner regionand the viscous bulk region, respectively. Based on this idea,
adPresent address: Department of Mechanical Engineering, School of Engi-neering, University of Tokyo, Japan.
bdAuthor to whom correspondence should be addressed. Electronic mail:[email protected]
they performed scaling analyses of characteristic variables inHZ and AC differently and saw good agreement with theresults from two-dimensionals2Dd numerical simulations. Itshould be noted that the analysis was carried out in the rangeof Marangoni numberssMad equivalent to the onset value foroscillatory flows.
The ultimate industrial interest here is to producestraition-free crystals, and one way would be to use feedbackcontrol to suppress the flow oscillations that are thought re-sponsible. With this industrial motivation there have been afew studies recently of control of the convective thermocap-illary instability.
The possibility to stabilize the thermocapillary instabilityby locally altering the heat flux on the surface was first dem-onstrated experimentally by Petrovet al.20,21 A nonlinearcontrol was performed to stabilize the oscillation in a half-zone model by using local temperature measurements closeto the free surface and modifying the temperature at differentlocal locations. They have constructed a look-up table basedon the system’s response to a sequence of random perturba-tions. A linear control law using appropriate data sets fromthe look-up table was computed. The control law was up-dated at every time step to adapt the control law to the non-linear system. Using one sensor/actuator pair, a successfulcontrol was observed at the sensor location for Ma,17 750; however, infrared visualization revealed thepresence of standing waves with nodes at the feedback ele-ment and the sensor. This was resolved by adding a secondsensor/actuator pair which allows the control to damp outboth waves propagating clockwise and counterclockwise,thus standing waves. The performance of the control wasreported for a fixed Ma,15 000, where the critical valuewas Macr,14 000.
This was followed by Benzet al.22 who applied controlto a hydrothermal wave in a shallow fluid layer. The tem-perature signal and the phase information sensed by thermo-couples near the cold end of the layer was fed forward tocontrol a laser which heated the downstream fluid surfacealong a line.
For an annular configuration, Shiomiet al.23,24 appliedactive feedback control based on a simple cancellationscheme. Active control was realized by locally modifying thesurface temperature using the local temperature measured atdifferent locations fed back through a simple control law.Using two sensor/actuator pairs, a significant attenuation ofoscillation was observed in a range of Ma, with the bestperformance in the weakly nonlinear regime. Applying thecontrol to an oscillation with azimuthal wavenumber of 3smode 3d, in the regime with weak nonlinearity, the oscilla-tion was suppressed to the background noise level. The ex-periments also revealed the limit of the control. When Ma isabout 15% above the critical value, control fails to achievethe complete suppression of the oscillation, though a signifi-cant attenuation is still achieved. The loss of control is ac-companied with an increase in the amplitude of the first over-tones and a modulation in the controlled signal which maysuggest the appearance of another mode triggered by thecontrol.
Recently, with the similar idea of the cancellation
scheme, but in a half-zone model, linear and weakly nonlin-ear control of the oscillatory thermocapillary convection wasreported by Shiomiet al.25 The experiment utilizes an unitaspect ratio liquid bridge where the most dangerous modehas an azimuthal wavenumber of 2 with absence of control.The performance of control was quantified by analyzing lo-cal temperature signals and the flow structure was simulta-neously identified by flow visualization. With optimal place-ments of sensors and heaters, proportional control can raiseMacr by more than 40%. The amplitude of the oscillation canbe suppressed to less than 30% of the initial value up to 90%of Macr. The proportional control was tested for a perioddoubling state to stabilize the oscillation to a periodic state.Weakly nonlinear control was applied by adding a cubic termin the control law to improve the performance of the controland to alter the bifurcation characteristics of the system.
In the current study, we will study a few representativecases of control using numerical simulations. In addition towhat was found in the previous experiments, the presentsimulations also permit us to investigate the importance ofthe positioning of sensors and heaters, and the influence ofthe properties of the heaters. We will also study the spa-tiotemporal decomposition of the modes that appear, as Fou-rier decompositions of the modes are easily accessible in anumerical simulation, as opposed to the experiments. It willbe of particular interest to investigate how modes with dif-ferent spatial structures are suppressed or amplified, depend-ing on the heater positioning, heater size, etc.
II. GOVERNING EQUATIONS
As shown in Fig. 1, the geometry of the system is anopen cylindrical container with radiusR and depthH filledwith liquid, with a planar-free upper surface. A central co-axial cylinder of radiusRh provides an inner radial boundaryto the annular fluid volume. Thermocapillary convection isdriven by a radial temperature gradient on the horizontal freesurface, which is maintained by raising the temperature ofthe inner cylinder above that of the outer wall.
The aspect ratioAr =H /R, was kept at unity. The ratio ofthe inner to outer radius of the annulus,Hr =Rh/R, is Hr
=0.21.The bottom thermal boundary condition is adiabatic. The
FIG. 1. The geometry of the simulated annular configuration.
054107-2 J. Shiomi and G. Amberg Phys. Fluids 17, 054107 ~2005!
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surface tension is considered to be a linearly decreasingfunction of the temperature,
G = G0 − gsT − T0d, s1d
where the surface tension coefficientg has a positive con-stant value. This means that the flow will be driven outwardson the free surface.
The fluid is treated as a 3D incompressible Newtonianliquid. Therefore, the flow is governed by the incompressibleNavier–Stokes equations, energy equation, and continuityequation. In a zero-gravity environment, the governing equa-tions are
]u
]t+ su · = du = − = p +
Pr
Ma= · s=u + = uTd, s2d
]u
]t+ su · = du =
1
Ma= · s=ud, s3d
= ·u = 0. s4d
These equations have been nondimensionalized using thelengthR, temperature differenceDT, velocity scale from thethermocapillary stress balance,
U =gDT
ms5d
and time scale
t =R
U, s6d
wherem is the dynamic viscosity. The nondimensional pa-rameters appearing above are the Marangoni number Ma andthe Prandtl number Pr defined as
Ma =gDTR
ma, s7d
Pr =n
a, s8d
wherea andn are the thermal diffusivity and kinematic vis-cosity, respectively.
The boundary conditions for velocity are no slip on solidwalls and a thermocapillary stress on the horizontal free sur-face. The thermal boundary conditions are that the tempera-tures of the inner and outer cylinders are prescribed at dif-ferent values, thus enforcing a radial temperature graident.The bottom is assumed thermally insulated. At the free sur-face an insofar unspecified heat flux is present. This heat fluxq will below be set to account for the action of the heatingelements that provide the control:
u = 0, u = 1 at r = Hr , s9d
u = 0, u = 0 at r = 1, s10d
]v]z
=1
r
]u
]f, v = 0,
]u
]z=
]u
]r,
]u
]z= qsr,f,td at z= Ar ,
s11d
u = 0,]u
]y= 0 atz= 0. s12d
Here the equations have been formulated in terms of Ma.However, in the following we will frequently be referring tothe overcritical parameter«, defined as
« =Ma − Macr
Macr. s13d
Here Macr denotes the critical Marangoni number for onsetof oscillations. Once Macr is specified,« thus carries thesame information as Ma, but it is a more descriptive param-eter here where we are interested in properties of supercriti-cal oscillations.
III. NUMERICAL METHOD
A finite element method in thesr ,zd-plane combinedwith a pseudospectral method in the azimuthal direction wasused to solve the equations in cylindrical coordinates. AGalerkin approach is adopted to formulate the discrete equa-tions. The solutions are expanded in azimuthal Fouriermodes. For each mode, the equation system is solved in the2D sr ,zd plane using triangular elements with quadratic basefunctions for the velocity and temperature, and piecewiselinear functions for the pressuresP2P1d. On computing thenonlinear terms for theNf azimuthal planes, dealiasing wasdone by computing the nonlinear terms for 23Nf planes andfiltering the firstNf modes.
The time discretization is done using a semi-implicitscheme for the viscous terms and the nonlinear advectionterms, respectively. The pressure is decoupled from the ve-locity computations by using a projection method.26 Withthis implementation, the resulting linear equation system issolved by a conjugate gradient method. The explicit treat-ment of the convection of the nonlinear terms impose a re-striction on the timestep; a Courant–Friedricks–Lewey con-dition needs to be satisfied for numerical stability. The timenecessary to simulate 1 s inphysical time for Ma well overthe critical Marangoni number Macr is typically about 4 h ona PC with AMD Athlon MP2000.
The finite element method computation of thesr ,zdplane was coded using femLego,27 a symbolic coding tool.femLego is a toolbox for Maple which can generate com-plete finite element codes. Using the Maple work sheet as aninterface, the relevant system equations, Navier–Stokesequations in the current case, are simply typed in togetherwith the initial and boundary conditions, choice of exten-sional solvers, and optionally output formats for postprocess-ing. In a separate Maple worksheet, the type of finite ele-ments, triangular P2P1 in the present case, is specified.Executing the Maple worksheet, ready-for-compileFOR-
TRAN77 code is generated. Together with input parametersand the mesh information, the code can be immediately ex-ecuted.
As mentioned by Leypoldtet al.,9 this type of problemwith azimuthal periodicity is suited to adopt the pseudospec-tral method in the azimuthal direction. Furthermore, withmoderate strength of nonlinearity, the disturbance takes a
054107-3 Numerical investigation of feedback control Phys. Fluids 17, 054107 ~2005!
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form of periodic waves with a distinct fundamental modeand a few harmonic ones. In fact, some experiments showthat even for flow with a complex temporal oscillation, thespatial structure of the wave is still low dimensional.28 Thisallows the azimuthal structure to be resolved with sufficientaccuracy even with a limited number of Fourier modes.
IV. CODE VALIDATION
The code used here was developed during this work. Inorder to test the correctness and rule out possible codingerrors, various tests and detailed comparisons were madewith results from an older code available to ussLavalley etal.14d, with satisfactory results. An example with a compari-son for a subcritical case is shown in Fig. 2.
The more critical issue though is to show that the gridresolution used here is sufficient to accurately capture theflow in the particular case and parameter range we are inter-ested in. In Table I, results from three nonuniform grids withdifferent numbers of mesh points and resolution in theboundary layers were used. On constructing the grids, par-ticular care was given to resolving the thin boundary layersin the hot corner region, namely, the one developing alongthe vertical wall and the one on the free surface. They can becharacterized by the distance from the hot corner to the peaklocation of the surface radial velocityl and the minimumsurface boundary layer thicknessd. The region l 3d wasresolved by 536,637, and 7310 P1 grid points for the 2Dmeshes ofNr 3Nz=17315,21317, and 30330, for in-stance, in the axisymmetric computation for supercritical Mas=39 200d. As shown in Table I, the three grids result in very
small differences for the base-flow computation character-ized by the maximum radial velocities for both sub and su-percritical Ma. Carrying out 3D simulations, the change inMacr is still small especially betweenNr 3Nz=21317 and30330. Therefore we adopt the grid withNr 3Nz=21317for the present study.
For sNr ,Nzd=s21,17d, the 3D simulation was performedfor three cases with different numbers of Fourier modes. Thedependance of the results on azimuthal resolution is de-scribed in Table II with the amplitudes of the distinct Fouriermodes of the temperature at the midgap on the surface. Sincethe results ofNf=16 and 32 are almost identical, at least tocapture the behavior up to third harmonic mode of the fun-damental one,Nf=16 should be sufficient.
The mesh that is used in the simulations reported in thispaper is thus the 21316317 mesh on the second row ofTable I.
For the grids and the range of parameters adopted in thecurrent study, the critical Fourier modemc is 3. The resultingsaturated solution always exhibit a traveling wave, which isin agreement with the experiment of Shiomiet al.24 Thesefeatures of the oscillation can be clearly seen in Fig. 3, wherethe energy distribution in the spatiotemporal spectral decom-position atsr ,zd=s0.605,1d is shown using the presentationadopted by Leypoldtet al.9 For amth mode oscillation, thepeak frequencies can be written asfm= unufm,c, wheren is aninteger andfm,c is the fundamental frequency of themthmode oscillation. For an oscillation withsm, unud, clockwiseand counterclockwise propagating waves in the azimuthaldirection are denoted by positive and negative values ofn.
In the experiment in an annular configuration by Shiomiet al.,24 it was observed that the flow shows a supercriticalHopf bifurcation where the amplitude of the oscillation in-creases linearly with the square root ofe in the weakly non-linear regime. This can be confirmed in the result of thesimulation shown in Fig. 4, where the evolution of the am-plitudes A shows good agreement withA~em/2mc denotedwith the solid lines. The data points fall off from the lines as
FIG. 2. The radial temperature distribution for a steady subcritical case,with Pr=14 and Ma=17 290. Diamonds are results from the present code,stars are from Lavalleyet al. sRef. 14d.
TABLE I. Result on computation of two-dimensional base flow for different grids.u1 and u2 are radialvelocities for Ma=16 890 and 39 200.
GridsNr 3Nf3Nzd Drmin Dzmin u1,max u2,max Macr
17316315 0.0025 0.0042 0.0653 0.0592 30 548
21316317 0.0022 0.0042 0.0665 0.0593 28 840
30316330 0.0020 0.0033 0.0663 0.0594 29 078
TABLE II. Magnitudes of the outstanding Fourier modes of the tempera-
ture, ums3102d at the midgap on the surface,sr ,zd=s0.605,1d, for variousnumber of Fourier modes,Nf. Ma=39 200.sNr ,Nzd=s21,17d. m denotes theazimuthal wavenumber. The critical wavenumber ismc=3.
Nf u3 u6 u9 u12
8 0.4657 0.0115 ¯ ¯
16 1.0383 0.2142 0.0566 0.0131
32 1.0388 0.2147 0.0569 0.0131
054107-4 J. Shiomi and G. Amberg Phys. Fluids 17, 054107 ~2005!
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nonlinearity of the system becomes stronger, which was alsoobserved in the experiments form=mc.
V. CONTROL METHOD
The control method is based on local modification of thesurface heat fluxq. The basic idea is to realize a local oppo-sition control using the knowledge of the structure of theoscillation. First, the dominant azimuthal wavenumber of theoscillation is checked. Knowing the wavenumber of the tar-get azimuthal mode, a sensor, and an actuator are placed indifferent positions. Then a simple linear cancellation schemecan be realized by feeding back the sensor signal to the ac-tuator.
The linear feedback control law can be expressed byrelating an imposed local heat flux to the surface temperatureat the point of the sensor as
qistd ; qsr0,fi + dfi,td = G1u8sr0,fi,1,td, s14d
u8sr,f,z,td = usr,f,z,td − usr,z,td. s15d
Herefi is the azimuthal position of theith sensor anddfi isthe angle between the sensor and the corresponding actuator,so that the azimuthal position of the actuator isfi +dfi. r0
=0.605 is the midgap location.The temperature fluctuationu8 is the difference between
the actual temperature andu, which is the time average ofthe spatial mean, i.e., the zeroth Fourier mode, over 5 s.G1 isthe linear control gain.
We consider two simple cancellation schemes usingpositive and negativeG1,
dfi =52jp
mc−
p
mc, G1 . 0,
2jp
mc, G1 , 0,6 s16d
where j is a positive integer. NegativeG1 is thus used whenwe expect the disturbances to be in phase at the sensor andactuator positions, and positiveG1 when they are at a 180°phase difference.
The finite size of the actuator is modeled by prescribingthe spatial distribution of the heat flux from one actuator asGaussian profiles in the radial and azimuthal direction,around the heater positionsr0,fi +dfid:
qsr,f,td = oi=1
p
qistdexpF− S r − r0
DrD2
− Sf − fi − dfi
DfD2G ,
s17d
wherep is the number of controllers.Dr and Df representthe thickness and length of the actuator.Dr =0.01, through-out the current study, while we have experimented with dif-ferent azimuthal lengths, to be specified below.
The choice of Gaussians as heat flux profiles is not in-tended as a precise model of the heater in the experiments byShiomi et al.,25 but rather as a means to explore the impor-tance of the overall heater size and aspect ratio. We do, how-ever, expect the control to be insensitive to the finer detailsof the heater geometry, since the amplified part of the wave-number spectrum is restricted to rather long waves, corre-sponding to the length of the heater or greater. Shorter wavesand higher wavenumber content in the heater model are ex-pected to be simply filtered out by the dynamical response ofthe flow. We thus claim that the important features of theheater are included in Eq.s17d, and that adding higher wave-number terms to Eq.s17d would not alter the results.
VI. RESULTS
Numerical simulations were carried out for the annulargeometry described above, with anAr =H /R=1 and Hr
=Rh/R=0.21. We assume microgravity conditions and disre-gard effects of gravity. The Prandtl number is chosen as Pr=14 to match that of the 1 cS silicone oil used in our previ-ous experiments. This choice of parameters makes it possiblefor us to compare results with those obtained experimentallyby Shiomi et al.25 In this paper we will focus on the influ-ence of the properties of the heaters and sensors, and we
FIG. 3. Full spectral decomposition of the surface temperature atsr ,zd=s0.605,1d for e=0.07. A denotes the amplitude of thesm,nd waves.
FIG. 4. Amplitudes of the critical azimuthal wavenumbermc=3 ssquaresdand the first and second harmonic modes,m=6 scirclesd and 9sstarsd over arange ofes,0.36d. The lines denoteA~em/2mc.
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have not attempted to cover other geometrical parameters orfluid properties. We have also neglected graivty, in order tofocus on the generic aspects of control of thermocapillaryconvection.
The critical value of the Marangoni number for onset ofoscillations Macr was determined by successive supercriticalsimulations. The oscillation amplitudes for these supercriti-cal Marangoni numbers were extrapolated to zero, to obtainMacr. The value obtained thus was Macr=28 840. Notice thatthis value of Macr differs from that found by Lavalleyetal.,14 since their simulations accounted for gravity, while thepresent ones assume zero gravity.
We will in the remainder mainly study different attemptsto control oscillation in a weakly nonlinear case withe=0.07 si.e., Ma=30 859d. Without control, the oscillation inthis case shows an azimuthal wavenumber of 3. The resultingsaturated solution always exhibits a traveling wave, which isin agreement with the experiment of Shiomiet al.24 Thesefeatures of the oscillation can be clearly seen in Fig. 3, wherethe energy distribution in the spatiotemporal spectral decom-position atsr ,zd=s0.605,1d is shown using the presentationadopted by Leypoldtet al.:9 For amth mode oscillation, thepeak frequencies can be written asfm= unufm,c, wheren is aninteger andfm,c is the fundamental frequency of themthmode oscillation. For an oscillation withsm, unud, clockwiseand counterclockwise propagating waves in the azimuthaldirection are denoted by positive and negative values ofn.
A. G1<0
In this case, the controllers are configured so that thetemporal oscillations of the mode-3 waves are in phase at thesensor and the coupled actuator. This way, a proportionalfeedback control can be realized by feeding back the sensorsignal to the actuator with a negative control gain.
1. Single actuator control
Experimental observations have been reported that at-tempts to suppress the oscillation with a single controllermay result in a standing wave with nodes at the sensorlocation.21 This was investigated for the parametersse ,p,f1,df1,Df ,G1d=s0.07,1,0,2p /3 ,p /16,−500d. Afterthe initial transient state, the controlled system results in asaturated oscillatory state. In Fig. 5, a sequence of picturesvisualizing the isotherms on the free surface is plotted. Thepositioning of the sensor and heater is indicated in the smallinset to the left. It can be seen that the nodes and antinodesof the wave switch their positions through axisymmetricstates, which indicates that the oscillation is a mode-3 stand-ing wave as predicted by the experimental observations. Theoscillation escapes the control, since the sensor position is ata nodal line of the oscillatory mode, where the amplitude ofthe fluctuation is zero.
2. Multiple controller control
Now, following the idea of Petrovet al.,21 one morecontroller was added to cover two degrees of freedom in theazimuthal direction. The sensors and actuators were posi-tioned in pairs, with a 2p /3 angle between the corresponding
sensor and actuator, and ap /2 angle between the two sen-sors, is indicated in the inset in Fig. 6. In the notationadopted above the controller positions aresf1,f2d=s0,p /2d anddf1=df2=2p /3. This placement correspondsto that which gave the best results in the experiment of Shi-omi et al.24
When e is very small,,0.02, i.e., Ma=29 500, the os-cillation can be completely suppressed to a steady axisym-metric state. In Fig. 6, the time history of the controlledoscillation at sr ,f ,zd=s0.605,0,1d, is shown using a gainG1=−1000. The exponential decay of the oscillation showsthat, in this case, the control influences the linear property ofthe system without influencing the stability of other modes.
On increasinge slightly up to about 0.07si.e., Ma=30 859d, still in the weakly nonlinear regime discussed inShiomi et al.,24 the controlled oscillation begins to exhibitmore complexity depending on the value ofG1. WhenG1 issmall enough, on applying control, the oscillation still showsan exponential decay to an saturated oscillation with slightlysmaller amplitude. At this point, of course, the dominantmode is still 3. On increasingG1 for further suppression,although the overall magnitude of the oscillation is reduced,new modes are triggered by the control. The time history atthe sensor locationssf=f1,f2d are shown in Fig. 7 for avalue ofG1=−1500. The oscillations at both locations mono-tonically decay until they reach the minima beyond whichthe oscillation grows again and eventually saturates.
The active modes contributing to the controlled oscilla-tion can be identified by the full spectral decompositionshown in Fig. 8. The amplitude is normalized by the ampli-tude of the uncontrolled mode-3 traveling wavesFig. 3d anddenoted asg, the suppression ratio. A suppression of mode 3and amplification of other modes can be observed. The modewith the largest energy is mode 2 which is in agreement withthe implication of the experimental results. One interestingaspect of the controlled oscillation is that, even with manyactive modes, most of the waves are standing. The structureand the positioning of the oscillation seems to be selected foreach mode to have nodes close to the controllers, at leastwhene!1.
For the above case, a sequence of simulated isothermson the free surface is shown in Fig. 9. It can be observed thattrianglessa,fd and ellipsessdd appear in turns with distortedisothermal patterns in between. Maxima of the mode-3standing wave appear when the mode-2 standing wave is inaxisymmetric state and vice versa. From this, it can be un-derstood that mode-3 and mode-2 waves have similar fre-quency andp /4 phase difference.
Further increase ofG1 results in more excitation ofbroad spectral components which leads to an increase of theoverall magnitude of the oscillation as shown in Fig. 10. Inthe experiment of Bárcena,29 the broadening of the spectracould be prevented by increasing the azimuthal length of theactuator. The idea is that, the longer the actuator is, the lessenergy is distributed on higher wavenumber components,therefore the excitation of the higher modes should be re-duced. This aspect is explored by settingDf to 3p /16, threetimes the original length. Correspondingly,G1 is set to 2000,one third of the previous case, in order to roughly maintain
054107-6 J. Shiomi and G. Amberg Phys. Fluids 17, 054107 ~2005!
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the total output from the actuators. The spectrum shown inFig. 11 shows a clear reduction of the broadening of bothtemporal and spatial spectra. Note that, in this case, the am-plification of the distinct destabilized modess1 and 2d areeven larger than the case shown in Fig. 10.
B. G1>0
In order to reduce the new wavenumber 2 mode weadopt a different configuration of the controllers,sf1,f2d=s0,p /6d, andsdf1,df2d=s5p /3 ,p /3d for the first and sec-
ond actuator locations, as sketched in the inset in Fig. 12.This allows the sensors to be placed only an anglep /6 apart,too small to allow a wavenumber 2 mode to go undetected.This choice was also guided by the experiments in the half-zone and annular geometries by Shiomiet al.25 and Bárcenaet al.,29 which show that the amplification of the new appear-ing mode, mode 2 for this case, is weaker for the cancellationscheme withG1.0 than withG1,0.
A simulation for e=0.07 shows that the change in thecontrol method results in reducing the amplification of mode2. This leads to an almost complete suppression of the oscil-
FIG. 5. A time sequence of surface isotherms from 0 to 1 of the oscillation with single actuator control. The inset on the upper left shows the positions of thesensorS and the heaterA. Parameter values arese ,p,f1,df1,Df ,G1d=s0.07,1,0,2p /3 ,p /16,−500d.
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lation at the sensor locations as show in Fig. 12. Yet, modu-lated signals with finite amplitudes can be detected at thesensors. The modulation has a period of about 30 s, and theyare out of phase atf1 andf2. The global suppression of theoscillatory flow can be observed by the spectral decomposi-tion shown in Figs. 13sad and 13sbd which are time-averageddata for time of 30–40 s and 50–60 s, respectively. The twotime windows correspond to whenurms8 sf1d@urms8 sf2d andurms8 sf1d!urms8 sf2d, whereurms8 denotes the root mean squareof u8. The switch in the direction of wave propagation inboth of the dominant modess2 and 3d can be observed. Inboth pictures, mode-2 and mode-3 waves have equivalentenergy. The avarage suppression ratio of the dominant modeis about 15% which is in agreement with previous experi-ments.
The gainG1 in the control law, Eq.s14d, has so far beendiscussed only in nondimensional terms. In order to compare
with the gains found efficient in the experiments of Bárcenaet al.,29 we write down the corresponding dimensional ex-pression for the gain:
G1,exp= kDTRG1E0
2p EHr
1
expF− S r − r0
DrD2
− Sf − fi − df
DfD2Grdr df, s18d
whereG1,exp is the amplification, so thatG1,expu8 is the netpower output at the heater.G1,exp is thus measured in units ofwatts.k is the thermal conductivity of the fluidsW/mKd, R isthe radius of the container.Dr and Df are the nondimen-sional heater widths, as used in Eq.s17d. The experiments ofBárcenaet al.29 show that, for the current range ofe, thenecessary magnitude of the linear control gainG1,exp was inthe order of 0.1 W. With the valuesk=0.1 W/m K for the 1cS silicone oil used in the experiment, and the temperaturedifferenceDT=20 K from the experiment,G1=3000 from
FIG. 6. Temporal signals at the sensorpositionsf=f1,f2. sr ,zd=s0.605,1d.The inset to the left shows the posi-tions of the sensorssS1,S2d and theheaterssA1,A2d. Parameter values arese ,p,Df ,G1d=s0.02,2,p /16,−1000d.
FIG. 7. Temporal signals atf=f1,f2. sr ,zd=s0.605,1d. Sensor and heaterpositions are the same as indicated in Fig. 6se ,p,Df ,G1d=s0.07,2,p /16,−1500d.
FIG. 8. Spectral decomposition atsr ,zd=s0.605,1d for the saturated oscil-lation in Fig. 7.
054107-8 J. Shiomi and G. Amberg Phys. Fluids 17, 054107 ~2005!
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the current simulation is equivalent toG1,exp=0.111 W, ingood agreement with theG1,exp=0.1 W found in the experi-ment.
Actuation with heaters
In reality, it is not easy to construct an actuator whichallows both heating and cooling. Petrovet al.21 demonstratedthe possibility to use a Peltier device which heats or coolsdepending on the direction of the applied current. However,the magnitude of the maximum power output is limited if thetime response of the actuation is to be kept within an allow-
able range. Therefore, in the previous experiments,29 weadopted heaters as actuators, which results in a restrictedlinear control law,
qsfi + dfd = HG1u8sfid, u8sfid ù 0,
0, u8sfid , 0,J s19d
whenG1.0. The influence of the restriction on the actuationwas examined by simulating the same case as in Fig. 12, butlimiting the actuation to heating only. Here,G1=6000, twicethe value in the previous case, in order to roughly maintainthe correspondence in terms of the total power output of theactuators. The solutions of these two cases are compared in
FIG. 9. A time sequence of surface isotherms from 0 to 1 of the saturated oscillation in Fig. 7. The inset to the left shows the positions of the sensorssS1,S2dand the heaterssA1,A2d.
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Fig. 14, where the temporal signals atf=f1,f2 for G1
=3000 with alternate heating and coolings—d and G1
=6000 with only heatings——d are shown. The results showgood agreement between the two cases, which encouragesthe use of the heaters in the experiments.
VII. CONCLUSIONS
We have studied a few representative cases of control ofthermocapillary convection using numerical simulations. Inaddition to what was found in the previous experiments, thepresent simulations gives us access to the complete thermaland velocity fields and their spectra. It permits us to investi-gate the importance of the positioning of sensors and heaters,and the influence of the properties of the heaters, in a con-trolled manner that complements the previous experimentalfindings.
On applying the cancellation scheme with negative valueof G1 for a very small«s,0.02d, the control influences thelinear properties of the target mode in the system without
destabilizing other modes. With a slight increase in«s=0.07d, the performance of the control is limited due to theappearance of new modes, mainly mode 2. In order to studyhow the spatiotemporal features of the oscillation is affectedby the control we have studied azimuthal and temporal Fou-rier decompositions of the oscillations, easily accessible inthese numerical simulations. An excess gainG1 results in abroadening of the spectra. One of the facts revealed by thenumerical investigation is that the control certainly inducesspatial complexity to the oscillation which increases the di-mension of the problem. On the other hand, it was shownthat the complexity, i.e., the width of the broadened spectradepends on the azimuthal length of the actuator. This impliesthat adjustment of this parameter can improve the controlperformance.
Between the two configurations with opposite sign ofG1, the control works better for the one with positiveG1,where the two sensors can be placed only an anglep /6apart. With this configuration, for«=0.07, the destabilization
FIG. 11. Spectral decomposition atsr ,zd=s0.605,1d for the saturated oscil-lation with control, using a heater three times as long as in Fig. 10.se ,p,Df ,G1d=s0.07,2,3p /16,−2000d.
FIG. 10. Spectral decomposition atsr ,zd=s0.605,1d for the saturated oscil-lation with control with high gain and short heater.se ,p,Df ,G1d=s0.07,2,p /16,−6000d
FIG. 12. Temperature signals at thetwo sensor positions for the sensor po-sitioning with positive gain. The twosensorssS1,S2d are placed at an angleof p /6, as indicated in the inset.sr ,zd=s0.605,1d. se ,p,Df ,G1d=s0.07,2,p /16,3000d.
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of new modes can be delayed and the oscillation can besuppressed significantly. A global suppression was confirmedin the entire temperature field. The obtained suppression ra-tio, g,0.15, matches with that obtained in the experiment.
In the previous experiments, we have blamed the low signalto noise ratio for the inferior control performance in annularconfiguration to that in a half zone. However, the currentresult suggests that the reason may lie in the difference in thephenomena themselves instead of the sensitivity of the ex-periment. Since the two geometries show certain differencesin the flow characteristics, it is not surprising if they reactdifferently being subjected to the same control methods.
In our previous experiments, the actuation was limited toheating only. The influence of this limitation was examinedby comparing the control performance with alternate heatingand cooling to that with only heating, The good agreementbetween the results from the two cases confirms that, evenwith this limitation on the actuation, the control performs ina similar manner and achieves the same extent of suppres-sion.
ACKNOWLEDGMENT
Financial support from The Swedish Research Council isgratefully acknowledged.
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