Fraenkel, P. N.

Gollub, Jerry P.

Heutmaker, M. S.

English

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Haverford College Haverford Scholarship Faculty Publications Physics 1985 Convection Patterns: Time Evolution of the Wave-Vector Field M. S. Heutmaker P. N. Fraenkel Jerry P. Gollub Haverford College Follow this and additional works at: https://scholarship.haverford.edu/physics_facpubs Repository Citation Heutmaker, M. S., P. N. Fraenkel, and Jerry P. Gollub. "Convection patterns: time evolution of the wave-vector field." Physical review letters 54.13 (1985): 1369. This Journal Article is brought to you for free and open access by the Physics at Haverford Scholarship. It has been accepted for inclusion in Faculty Publications by an authorized administrator of Haverford Scholarship. For more information, please contact nmedeiro@haverford.edu. VOLUME 54, NUMBER 13 PHYSICAL REVIEW LETTERS 1 APRIL 1985Convection Patterns: Time Evolution of the Wave-Vector FieldM. S. Heutmaker, P. N. Fraenkel, and J. P. GollubDepartment ofPhysics, Haverford College, Haverford, Pennsylvania 19041, and Department ofPhysics,University ofPennsylvania, Philadelphia, Pennsylvania 19104(Received 6 February 1985)The evolution of Rayleigh-Benard convection patterns has been studied quantitatively with use ofdigital image processing methods to measure the time dependence of the wave-vector field. The re-lative importance of wave-number variations, roll curvature, defects, and sidewalls was determined,with the Swift-Hohenberg model as a framework for analysis. Deviations from the model becomepronounced for (R —R, )/R, & 2, where R, is the critical Rayleigh number.PACS numbers: 47.20.+m, 47.25.—cIn many nonlinear systems, a homogeneous statebecomes unstable to spatially periodic perturbationswhen the critical value of a parameter is exceeded.Hydrodynamic instabilities and morphological instabili-ties of growing crystals are examples. Stability theoryyields the onset and critical wave number of the insta-bility, but does not generally give unique predictionsfor the form or evolution of the resulting patterns at afinite distance above the onset. A particularly chal-lenging problem is that of the formation of Rayleigh-Benard convection patterns in a large layer. ' Theseare often textured structures containing defects; theycannot be described by a single wave number, but maybe specified by a two-dimensional wave-vector fieldq(r) that contains the full information about orienta-tions and spacings in the pattern.We show in this paper that q(r) can be measuredexperimentally using digital image processingmethods. We use this information, in conjunctionwith a two-dimensional models '0 for convective pat-tern evolution, to study various contributions to thedynamics, including defects, roll curvature, and the ef-fects of boundaries. Model equations are useful inpart because it is difficult to integrate the full hydro-dynamic equations over the long time scales character-izing pattern evolution. Various models have beenstudied numerically " and analytically, ' but noneof them have yet been tested experimentally. The par-ticular model we consider (due originally to Swift andHohenberg9) is a simplification of the hydrodynamicequations having the following interesting property: afunctional Fof the dynamical fields decreases monton-ically, in analogy to the minimization of the free ener-gy in equilibrium processes.We find that the validity of the model dependsstrongly on the distance e= (R —R, )/R, above theonset of convection. The experiments are consistentwith the model provided that e is neither too close toonset, nor larger than a cutoff e „=2.Using themodel as a framework for analysis, we find thatchanges in the magnitude of the wave-vector field dueto defect motion are particularly important, while rollcurvature and lateral boundaries contribute much lessE= 0.68(a) O.oop„ (b) O. l9gh(c) 0.6Zg„ (d) ig.~~„FIG. 1. Digitally enhanced images from a pattern evolu-tion sequence at e= (R —R, )/R, =0.7; time is given inunits of the horizontal thermal diffusion time ~z.to J.The experiments were performed in a cylindricalconvection cell of diameter D = 86 mm and depthd = 3.0 mm (aspect ratio I = D/2d = 14.4). The work-ing fluid is water at 70'C where its Prandtl number is2.5, and the critical temperature difference AT, is0.720'C. The horizontal thermal diffusion time rhbased on the cell radius is 3.1 h. The structure of theconvective flow is determined by the refraction of anexpanded and collimated laser beam that enters thecell from above, traverses the fluid, reflects from thelower plate and impinges on a diffuse screen, where itis digitized (resolution of 320 pixelsx 240 pixelsx 8bits) by use of Newvicon camera as a transducer. Theapparatus will be described in detail elsewhere. '2 Digi-tized images are averaged to reduce noise, divided by areference image (with no convection), and then con-trast enhanced to reveal the weak convective struc-ture. 2Figure 1 shows four contrast-enhanced images froma sequence at e =0.7. Bright areas mark the cold (des-cending) fluid and dark areas mark the warm (ascend-ing) fluid. In order to obtain patterns that evolve sub-1985 The American Physical Society 1369VOLUME 54, NUMBER 13 PHYSICAL REVIEW LETTERS l APRIL 1985stantially, the system was prepared in a turbulent state,and then e was reduced to 0.7. As time passes the ini-tial cellular pattern quickly changes to bent convectionrolls with many point defects (similar to dislocationsand disclinations in solids) and line defects (similar tograin boundaries), as shown in Fig. 1(b). The defectsare slowly expelled from the pattern until a final time-independent state is reached in which the rolls alignnearly perpendicular to the sidewalls, and few defectsremain. The evolution process is slow; major changesoccur within the first horizontal thermal diffusiontime, but at least 10—20 thermal diffusion times are re-quired for the pattern to stabilize.In order to study the pattern evolution quantitative- FIG. 2. Wave-vector field q(r) on a grid of Points for theq(r) by dtgttal Pattern of Fig. I (c), as determined by digital image analysis.The background intensity is coded in proportion to the waveanalysis of the images. The role boundaries are firstnumber, while the line segments indicate the orientation offound by following contours of maximum or minimumintensity. A grid of evenly spaced points is then super-imposed over the pattern. The local roll orientationand spacing are interpolated at each grid point fromthe nearby roll boundaries in order to find the local orientation and magnitude of the wave vector. Small areasnear defects are excluded from this calculation since the wave vector is undefined there. Figure 2 shows the resultof this calculation for the pattern of Fig. 1(c). The background intensity is coded in proportion to the wavenumber q (r) = ~q(r) ~, while the line segments indicate the orientation of .q(r). The white excluded areas markthe locations of defects.Measurement of the wave-vector field allows us to use the Swift-Hohenberg model as a framework for analysis,and to estimate its range of validity. This model is an evaluation equation for a real order-parameter field p(x,y, t)that is proportional to the vertical velocity and temperature at a fixed height in the fluid layer. One of the most im-portant properties of this model is that it is variational; as P evolves a Lyapunov functional F decreases monotoni-cally:d rI 2t 2EQ + 2((p/4qp)[(7 +qp)Q] + 4gp ).In this equation gp is a coherence length scale, qp is areference wave number, and gis a coupling constant oforder unity. On the sidewalls of the cell, Q and its nor-mal derivative vanish. Far from the sidewalls, themodel favors a static pattern of straight rolls with wavenumber q = qp. Numerical simulationss and analysissuggest that the model is appropriate for small e.Near onset in a laterally large layer (e'~ I &) 1),Cross showed that one can separate the contribution toFdue to roll distortion in the bulk of the cell from thatdue to the suppression of convection near thesidewalls. 'p The bulk contribution can be written ap-proximately'3 in terms of the wave-vector field q(r) ofthe pattern:2" 2F~=—egp 'i d r[(q —qp) + (I/4qp )(V' q) ).This integral depends on two quantities: the square ofthe wave-number deviations (q —qp)2 and the squareof the wave-vector divergence (V q) /qp, which ispredominantly a measure of roll curvature. Thesidewall contribution is minimized when the rolls arealigned perpendicular to the sidewall:Fs = ,' J2~3t'&pI—Itl[(q s)/q]dl,!where s is a unit vector normal to the sidewall and theintegral is taken around the cell boundary.The amplitude of convection is suppressed near de-fects in the pattern, and this results in further contri-butions to F, which may be approximated as~D= 2 ~'&D~I;,2where XD is the number of point defects (disclina-tions) in the pattern and r, is the radius over which Qis suppressed. We assume that g & r, & 2g, where thecoherence length g is (pc t~2, and gp ——0.385d. Fur-ther theoretical work is required to specify it more pre-cisely. The sum of all these contributionsF= Fz+ F~+ I'D is predicted to decrease monotonical-ly during pattern evolution.For the reference wave number qo, we use the ex-perimentally measured wave number selected by a pat-tern of straight rolls. '4 This quantity decreases with in-creasing e, similar to the trend exhibited by the meanwave number in our textured stable patterns.Figure 3(a) shows the results of our experimentalmeasurement of the two parts of Fz for the evolutionsequence of Fig. 1, with a=0.7. The spatial averages1370VOLUME 54, NUMBER 13 PHYSICAL REVIEW LETTERS 1 APRIL 19850. 30E,= 0.68E =2.890. 200 10(a) 8.83' (b) g.si~„0. 00 '--1.5 -1.0 -0. 5 0. 0log(t/Th)(v.qPtq'1.0(c) 9.89 T'h (d) i l. I th30.(b) 6, = 0.68FIG. 4. Digitally enhanced images showing spontaneousdefect nucleation at e = 2.9.2010-1.5 -1.0I-0. 5 0. 0 0. 5 1.0 1.5to g (t/gh)FIG. 3. (a) Spatial averages of the wave-number devia-tions (q —qo)~ and wave-vector divergence (V' q)'/q$ vstime (logarithmically) at a=0.7. (b) Total Lyapunov func-tional F and the bulk, defect, and sidewall contributions vstime (see text).of (q —qo) and (V q)2/qo2 are shown as a functionof log(t/rh). (Lengths are measured in units of thecell depth. ) We find that the wave-vector divergencedecreases monotonically as the roll pattern becomesstraighter, but the wave-number deviation is muchlarger and fluctuates significantly as defects move.The total F and various contributions'3 to it areshown in Fig. 3(b). The sum is dominated by the bulkand defect terms Fz and FD, while Fs is much smaller,and nearly constant. To determine the possible rangeof FD, we assume that the effective core size is in therange g & r, & 2g. (It would be better to measure theamplitude suppression directly to determine FD, butthe shadowgraph images are not sensitive to thesuppression. ) The total F is consistent with a mono-tonic decline, but the uncertainty is large because ofthe relatively large range for Fz.We have performed this analysis on many other pat-terns both closer to and farther from onset. For smalle ( & 0.3), variations in Fz are small, and the declineof FD as defects are eliminated causes the only signifi-150.6= 2.89100.500.0. 5 1.0tog (&l'&h)1.5FIG. 5. Total Lyapunov functional F and bulk, defect,and sidewall contributions vs time at e = 2.9. The nucleationof defects leads to nonmonotonic variations in F.cant change in the total. At long times (up to about50vt, ) any variation in F is below our experimentalresolution (about 10'/o), but the patterns are noticeablytime dependent, ' substantially so for a=0.1. There-fore it is possible that the dynamics are not relaxation-al (monotonic) very close to onset. However, the evo-lution of F is qualitatively similar to that observed ate = 0.7 up to e = 2. Thus we conclude that the minim-ization of F is consistent with our data over a range;„&e & e,„,where e;„=0.3 and e,„=2.For e )e,„,on the other hand, defects nucleatespontaneously as shown in Fig. 4, causing large non-monotonic variations in F (Fig. 5). A large increase inF~, due primarily to substantial wave-number varia-tions, occurs when new defects enter the pattern.These major pattern changes seem to be precipitatedby the climbing of newly created dislocations into re-gions of large roll spacing near defects in the bulk ofthe cell. We have observed the nucleation of these1371VOLUME 54, NUMBER 13 PHYSICAL REVIEW LETTERS 1 APRIL 1985dislocations in each of three runs at e = 2.9.The final-state patterns are clearly not unique (atany fixed e), a result that was previously noted for rec-tangular3 and smaller circular2 cells. The various tex-tured final patterns obtained from different initial con-ditions can have values of F differing by about 25%.We conclude that this functional must have many localminima.Furthermore, we note significant qualitative differ-ences in the way that the patterns adapt to thegeometry at small and large ~. For e ( 1, grain boun-daries are the dominant type of defect, while at higher~ disclinations are present. This is consistent with thedifferent e dependence of the various contributions toF, as well as with the increase with e of the size of theband of stable wave numbers.In summary, measurement of the time-dependentwave-vector field provides a useful way to characterizepattern evolution quantitatively. This can be accom-plished by digital image analysis for evolving convec-tion patterns. The Swift-Hohenberg model provides afratnework for estimating the relative importance ofdefects, roll curvature, and wave-number variations.The largest changes in the Lyapunov functional Farisefrom wave-number variations associated with defectmotion, while roll curvature and boundary effects aremuch smaller. Deviations from the model becomepronounced above e = 2, though it is useful for a sig-nificant range of e. A better understanding of pointdefects would allow a more precise comparisonbetween theory and experiment. It would be desirableto test other models, some of which include nonrelax-ational effects, and to investigate the effect of chang-ing the Prandtl number.This work was supported by the National ScienceFoundation through Grant No. MEA-8310933. Weappreciate helpful discussions with G. Ahlers,M. Cross, and P. Hohenberg. One of us (J.P.G.) ack-nowledges the support of a Guggenheim fellowship.&V. Croquette, M. Mory, and F. Schosseler, J. Phys.(Paris) 44, 293 (1983).V. Steinberg, G. Ahlers, and D. S. Cannell, to be pub-lished. These authors also used digital image processingmethods.3J. P. Gollub, J. F. Steinman, and A, R. McCarriar, J.Fluid Mech. 125, 259 (1982); J. P. Gollub and A. R. McCar-riar, Phys. Rev. A 26, 3470 (1982).4M. C. Cross and A. C. Newell, Physica (Utrecht) 10D,299 (1984).5M. C. Cross, P. G. Daniels, P. C. Hohenberg, and E. D.Siggia, J. Fluid Mech. 127, 155 (1983).6Y. Pomeau and P. Manneville, J. Phys. (Paris) 42, 1067(1981).7P. Manneville, J. Phys. (Paris), Lett. 44, L903 (1983).8H. S. Greenside and W. M. Coughran, Jr. , Phys. Rev. A30, 398 (1984).9J. Swift and P. C. Hohenberg, Phys. Rev. A 15, 319(1977).tOM. C. Cross, Phys. Rev. A 25, 1065 (1982).~A. Zippelius and E. D. Siggia, Phys. Fluids 26, 2905(1983).&2M. S. Heutmaker and J, P. Gollub, to be published.3The expression for F& has been simplified for purposesof presentation. The actual equation we use to determine F~has a small additional contribution. We also include a smallterm FGB due to grain boundaries in the experimental deter-mination of F [Figs. 3(b) and 5].t4A. Pocheau and V. Croquette, J. Phys. (Paris) 45, 35(1984) .~5Persistent time dependence over even longer times hasbeen convincingly demonstrated by G. Ahlers, D. S. Can-nell, and V. Steinberg, following Letter [Phys. Rev. Lett. 54,1373 (1985)].1372

Convection Patterns: Time Evolution of the Wave-Vector Field

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Convection Patterns: Time Evolution of the Wave-Vector Field

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