Freilich, Michael H.

Gollub, Jerry P.

English

Haverford College: Haverford Scholarship

Haverford College Haverford Scholarship Faculty Publications Physics 1974 Optical Heterodyne Study of the Taylor Instability in a Rotating Fluid Jerry P. Gollub Haverford College Michael H. Freilich Follow this and additional works at: https://scholarship.haverford.edu/physics_facpubs Repository Citation Gollub, Jerry P., and Michael H. Freilich. "Optical heterodyne study of the Taylor instability in a rotating fluid." Physical Review Letters 33.25 (1974): 1465. 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 33 16 DECEMBER 1974 NUMBER 25Optical Heterodyne Study of the Taylor Instability in a Rotating Fluid*J. P. Gollub and Michael H. FreilichPhysics DePa&ment, Have+os College, Have+os, Pennsylvania 19041(Received 16 September 1974; revised manuscript received 19 November 1974)The Taylor instability in a rotating Quid confined between two cylinders has been inves-tigated by light scattering. For rotation rates f asymptotically close to the critical rota-tion rate f, we find that the amplitude of the ordered flow varies as (f f ) 'I+ —3. Theaxial structure of the ordered flow is sinusoidal near f, but harmonics become substan-tial for f-f, large. The main features of the Landau approach to hydrodynamic instabil-ities are thus confirmed.A number of recent papers' ' have discussedthe behavior of fluids near hydrodynamic instabil-ities. One motivation for this work is the sugges-tive analogy between fluid instabilities and sec-ond-order phase transitions, both of which ex-hibit an order parameter that grows from zeroin the neighborhood of a critical point. Anotheris the hope of understanding the transition to tur-bulence, which occurs by means of a successionof progressively more complicated instabilities.Most of the previous work has been concernedwith the Rayleigh-Bd'nard or convective instabili-ty. %e have performed an experimental study ofthe Taylor instability~ in a rotating fluid, andfind excellent agreement with the Landau pic-ture" of hydrodynamic instabilities.The Taylor instability occurs when a fluid isconfined between an outer stationary cylinder andan inner rotating one. If the rotation rate f ex-ceeds a critical value f, , the radial pressuregradient and the viscous forces are not sufficientto provide the required centripetal accelerationof the fluid, and a new flow pattern perturbs thez-independent Couette flow. Superimposed on theoriginal azimuthal flow Ve(r), there is now (Fig.1) a toroidal roll pattern much like that of theRayleigh-Benard instability. Near f„ the veloc-ity components V„and V, are of course quiteSTREANILINESXg(INNER IROTATING) ~OUTER CYLINDERINCOMINGLIGHTSCATTEREOVLIGHTAPERTURECYLINDERPMT100-gm APERTURE1-mm APERTUREF.L. =4cm 8 EAMS P LITTERFLU IDATTENUATORIASER500-gm AP ERTUREF.L. = 10cm~gyyyyyyyyyyyyMIRRORFIG. 1. (a) Side view of the cylindrical apparatus andlight paths. The flow pattern shown schematically isactually superimposed on a much faster azimuthal Qowperpendicular to the page. The axis of the rotating cyl-inder is along the g direction. (b) Top view showingthe method used to mix the scattered and referencebeams. Each arm is 80 cm long.1465VOLUME 33, +UMBER 25 PHYSICAL REVIEW LETTERS 16 DECEMBER 1974small. If f is increased considerably beyond f, ,there are further instabilities in which the vorti-ces acquire circumferential waves, and eventual-ly a transition to a turbulent state (nonperiodic intime) occurs. The first instability is the subjectof this paper.According to the Landau theory' and subsequentwork' one expects that just above f„ the radialvelocity should be of the form V„=A,(r, s) cos(k,z),where e = (f—f,)/f, and the amplitude A, variesas E"'. Fluctuations may influence this depen-dence" for e &10 ', but this region is probablynot experimentally accessible. As c is increasedhigher harmonics should appear as a result of thethe nonlinear terms in the Navier-Stokes equa-tions, so thatV„=Q,A, (r, ~) cos(Pk,z).The discussion above refers to the steady-statesituation. The response time of the system whenperturbed from equilibrium is expected to di-verge as ~ -0+. The major new result of thepresent work is the measurement of A~(e) for p= 1, 2.In order to study V„(r, z, e) we have utilized la-ser light (5 mW at 5145 A) scattered at an angleof 171' from the forward direction by a dilute sus-pension of 2-p, m polystyrene latex spheres in wa-ter, as shown in Fig. 1. The scattered light wasmixed with an unscattered but attenuated beam byuse of a Michelson-like interferometer, and thepower spectrum" of the photocurrent was ob-tained from a real-time spectrum analyzer. Thepower spectrum exhibits a peak approximately200 Hz wide at a frequency in the range 0 to10000 Hz. For our geometry the ratio of the lo-cal radial velocity to the mean frequency of thepeak is 1.94&&10 ' cm sec 'Hz '. The scatteringvolume was experimentally determined to beabout 1 mm long in the radial direction and 0.2mm in the orthogonal directions.The fIuid was contained between an inner blackaluminum cylinder of radius 1.555 cm, and a pre-cision-bore Pyrex tube of inner radius 2. 540 cm.The cylindrical region was 30 cm long, and wastemperature controlled to within 0.03'C. Temper-ature control was necessary because f, changesby 2% per degree because of the temperature de-pendence of the kinematic viscosity of the fluid.At 27.0'C, we found f, to be 0.0651+ 0.0002 Hz,which is consistent with the predictions of Chan-drasekhar. ' (A quantitative comparison is notpossible because the linear stability theory hasonly been evaluated for special geometries in+0.016 —e+0.012+0.008~ +0.004E 0-0.004-0.008-0,0120.5/~ ~ ~o0 0I I1.0 1.52 (cm)2.0FIG. 2. Axial variation of the radial velocity for(f-f, )/f, =0.014. The solid curve includes the funda-mental term and a small second-harmonic term.which the ratio R,/R, of radii either is —, or ap-proaches unity. However, previous work" hasestablished the correctness of theoretical predic-tions for f, and k, .) The rotation rate was mea-sured electronically to an accuracy of 0.02% andwas constant to within 0.o4%.The z dependence of V„was studied halfway be-tween the inner and outer cylinders by translat-ing the apparatus with the optics unchanged. InFig. 2 we present the results for a=0.014. Themaximum value of V„ is only 0.06VO, so that theperiodic structure is a fairly small perturbationon the azimuthal flow. As predicted for small e,the flow is nearly sinusoidal. However, there isa small second-harmonic term, which can be de-tected by the difference in magnitude of the posi-tive and negative peaks. The amplitudes of thefundamental and second-harmonic terms (ob-tained by computer Fourier analysis) are A,= 0.0155 cm/sec and A, = 0.0019 cm/sec, and thesolid curve shows that these two terms alone pro-duce an excellent fit, with a fundamental wavenumber k, =3.05 cm '. The product k]Rg is 7.75,which can be roughly compared to Chandrasek-har's prediction' k,R, =6.4 for the case R,/R, = &.As e becomes larger, the second and higherharmonies increase in importance relative to thefundamental term, because of their dependenceon higher powers of e. In Fig. 3, we present V„(z)for ~=0.465. Here the magnitude of V„has ahigher and sharper peak in the outward flowingregions than in the inward flowing ones. We findthat three terms are necessary to produce a fitto within the accuracy of the experiment, and theamplitudes are A, = 0. 1030 cm/sec, A, = 0.0356cm/sec, and As=0. 0072 cm/sec, The wave num-ber k, is 3.20 cm ', and is independent of e overVOLUME 33, NUMBER 25 PHYSICAL REVIEW LETTERS 16 DECEMBER 1974+0.12+0.08—I t I l I I I II I I~ I~ )0~A ~,m)CA~XI" +o.o4—E0-0.04-0.08—I0.5~O~-~—~-o-+1.02 (cm)I1.5E O.O1A.0,001 i'i Yi l0.01XA2 rX~ JX« t il0.10A, I1.00FIG. B. Axial variation of the radial velocity for(f—f~)/f~=0. 465. The solid line includes the fundamen-tal and the second- and third-harmonic terms.the range 0&& &0.5 to within the precision of themeasureinents (5'%%uo). The precision is limited bysmall thermally induced drifts in the flow patternnear ~=0.The dependence of the Fourier coefficients A~on & is shown in Fig. 4, Points marked by across were obtained by fixing the rotation rate,allowing the system to equilibrate (which re-quired at least 15 min when e &0.01), measuringV„(z), and then performing a complete Fourieranalysis. The first two amplitudes A, and A, canalso be obtained from a simpler procedure, inwhich only the magnitudes P, and P of the posi-tive and negative peaks in V„(z) at each value ofe need be measured. We utilize the combinations~(P, +P ) =A, +As+. . . and 2(P, -P ) =A2+A4+. . .to accomplish this. The amplitude A, is at most7% of A, in the range of interest, and can be re-moved as a small correction by using As(e) asobtained from Fig. 4 and other data. Removal ofA4 is unnecessary, as it was found to be negligi-ble over the range of ~ studied. Points for whichA, and A, were obtained in this manner are indi-cated by dots in Fig. 4. These points are consis-tent with those obtained by complete Fourier anal-ysis of V„(z).To test the prediction that A, varies as &"', aleast-squares analysis was performed, in whichthe weighted sum of the squares of the deviationsfrom D(f -f,) was minimized with D, P, and (ifdesired) f, as free parameters. The weightingwas determined by assuming the measurementerror hA, to be independent of e. We determinedexperimentally that 0.0649 &f, &0.0653 Hz by not-ing the rotation rate at which the photocurrentpower spectrum no longer has a peak at v &0.The uncertainty arises partly from the fact thatfrequency shifts of less than 200 Hz are not de-l: = (f — fg)/fgFIG. 4. Dependence of the Fourier coefficients A onthe rotation rate. The solid lines represent the resultsof least-squares analyses, with f,= 0.0651 Hz.tectable, and partly from what appear to be slightvariations in f, from run to run. We find that A,= (0.145 + 0.013)e'""'s cm/sec, where the uncer-tainties are due mainly to the uncertainty in f,.If we treat f, as a free parameter, the best fit oc-curs when f,=0.0649 Hz and P=0.52. We con-clude that the measurements are consistent withthe prediction A, —&'". The data of Berge andDubois, ' who obtained P & 2 for the convective in-stability, may have been influenced by the secondand higher harmonics, which were not measuredin their experiments.Applying the same approach to A, (e), we foundthat A, = (0.063+ 0.005)e'"""cm/sec representsthe data quite well. In order to make sure thatwe were actually observing asymptotic behavior,we repeated the analysis after eliminating pointsfor e&0.1, and found the exponent to be 0.76+0.06, unchanged except for a larger error. Wehave not considered As in detail because the dataare at present insufficient for an accurate deter-mination of the ~ dependence of A~.Our results confirm the Landau approach tofluid behavior asymptotically near an instabilityleading to a spatially periodic but time-indepen-dent flow. In particular, a single spatial Fouriercomponent dominates near the critical point, andits amplitude grows as e'". However, the expo-nent 0.77 + 0.03 of the second-harmonic term,which becomes important farther from the criti-cal point, is not understood at the present time.Hopefully this and other experiments will stimu-late theoretical work aimed at understanding thenonasymptotic region.We acknowledge with gratitude the help of Wil-liam Davidon with the statistical analysis and the1467VoLvME 3$, NvMBER 25 PHYSj. CAZ. REVIEW I.ETTZRS 16 DECEMBER 1974helpful experimental advice of A1bert Koenig andTom Davis.*Work supported in part by the National Oceanic andAtmospheric Administration under Grant No. 04-4-022-10 to Haverford College.~R. Graham, Phys. Rev. Lett. Bl, 1479 (1978).W. A. Smith, Phys. Rev. Lett. 82, 1164 (1974).H. Haken, Phys. Lett. 46A, 199 (1979).P. Berge and M. Dubois, Phys. Rev. Lett. 32, 1041(1974).'G. Ahlers, Phys. Rev. Lett. 88, 1185 (1974).J.B. McLaughlin and P. C. Martin, Phys. Rev. Lett.SS, 1189 (1974).See, for example, S. Chandrasekhar, Hydrodynamicand IIyChomagnetic Stability (Clarendon Press, Oxford,England, 1961).L. D. Landau and E. M. Lifshitz, Izuid Mechanics(Pergamon, London, 1959).For a review, see R. C. DiPrima and E. H, Rogers,Phys. Fluids, Suppl. II, 155 (1969).~OFor a review of light-beating spectroscopy, seeH. Z. Cummins and H. L. Swinney, in Progress in Op-tics, edited by E. Wolf (American Elsevier, New York,1970), Vol. 8.R. J. Donnelly and D. Fultz, Proc. Roy. Soc., Ser. A258, 101 (1960).Breaking the Roton Barrier: An Experimental Study of Motion Faster than the LandauCritical Velocity for Roton Creation in He IlfA. Phillips and P. V. E. McclintockDepartment of Physics, University of Lancaster, Lancaster, England(Received 9 September 1974)We report the first observation of objects moving through He II with equilibrium driftvelocities p beyond the Landau critical velocity o for roton creation. With T 0.4 K, P=25 bar, Il=2 kV cm, v-p~ for negative ions is larger than a theoretical prediction bya factor of 10 . The vortex-ring nucleation rate is found to decrease with Eabove 300V cm, thus resolving apparent inconsistencies between earlier experiments.In his celebrated explanation of superfluidity,Landau' showed that the kinetic energy of a liq-uid flowing at velocity v through a tube (or thatof a. heavy object moving through the liquid) can-not be dissipated through the creation of an exci-tation of energy e and momentum p in the liquidunless v ~ «/p. For He 11 the minimum value ofe/p is nonzero, occurring close to the roton min-imum in the elementary excitation spectrum, sothat a, critical velocity v, =(e/P);„=50 m sec 'exists, below which dissipation ought not to oc-cur in the superfluid. Measured critical veloci-ties are usually orders of magnitude smaller,because of the onset of vortex formation at lowervelocities, but Hayfield' reported that the driftvelocity v of negative ions in He II under pres-sure P&12 bar below 0.6 K appeared to reachand to be limited by v, when the applied field wasraised to about 70 V cm '. This has remainedthe only known situation to which Landau's origi-nal criterion for the breakdown of superfluidityappears to be relevant.Takken' has considered roton creation by nega-tive ions moving at velocities slightly greaterthan v, on the basis of a wave radiation model inwhich each ion is assumed to generate a conicalwave of coherent roton radiation, much like thedisturbance produced by an airplane breaking thesound barrier. By analogy with the aerodynamiccase, a rapid increase in drag is expected as thevelocity increases past v, : Takken concludedthat an upper bound on v is given by v„b=v, (1+10 "F'), where the electric field F is in Vcm '. A 1% increase of v above v, would therefore require I ) 10' V cm ', implying that anyincrease of v beyond v, ought to be almost impos-sible to observe experimentally.An attempt by Neeper and Meyer4 to test thisremarkable assertion was thwarted by an unex-pected increase in the vortex nucleation rate vwith falling temperature, such that at 0.3 K onlyvortex rings, and no bare ions, arrived at theircollector. Our recent observation' that the fieldemission current at 0.3 K in He II increasesdramatically with P above 12 bar, and is temper-ature independent below 0.4 K, seemed to be in-consistent with Weeper and Meyer's result. Ap-proximate values of v deduced' from the field-emission measurements apparently indicated thefeasibility of our present experiment to test Tak-1468

Optical Heterodyne Study of the Taylor Instability in a Rotating Fluid

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Optical Heterodyne Study of the Taylor Instability in a Rotating Fluid

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