Onset of wavy vortices in the finite-lengjth G~~~tt~~Tay~~r problem W. S. Edwards,” S. R. Beane,b) and S. Varmab) CenterforNonlinearDynamicsand Departmentof Physics,Universityof Texasat Austin.Aus& Texas73712 (Received22 October 1990;accepted19 February 199I ) The transition from Taylor vortex flow to wavy vortex flow in the Couette-Taylor problem, for finite annulus lengths, is studied. Using an accurate,fully resolvednumerical code, inflnitei cylinder nonlinear axisymmetric Taylor vortices are computed and their stability with respect to nonaxisymmetric disturbancescorrespondingto wavy vortices is determined.The present method computes Ieading.eigenmodesand eigenvaluesfor arbitrary axiaI Floquet exponentsk and arbitrary azimuthal wave numbers nz, so that the full complex dispersion relation w = w(k,m) is obtained. The linear coefficientsappearingin the Ginzburg-Landau equation are calculated and compared to previous experimental:‘results.It is found that for aspectratios (length to gap width) lbss than 30, the Ginzburg-La&au predictions are not reliable; for these aspectratios the transition is modeledas the superpositionof two Floquet modeswith k = -& rr/L, where L is the length of the annulus. This’model is verifled via a detailed ._ numerical and experimental study at aspectratios between8 and 34, for radius ratio 0.87, obtaining good agreementfor the critical Reynolds numbers?azimuthal wave numbers andwave speedsover the entire:aspect-ratiorange. I. INTRODlJCTlON modesand their critical Reynoldsnumbers and wave speeds The instabilities arising the Couette-Taylor problem takes us beyond the Ginzburg-Landau approximation of a have been well studied since the classic work of Taylor.’ ,quadratic dispersion relation. We obtain accurate predicWith the outer cylinder fixed and inner cylinder rotating, the tions for annuluslengthsas-shortasL = &i, =: 8, where& is primary transition from circular Couette flow to Taylor vor- the wavelengthof the critical Taylor vortex mode.tices is predictedwell by linear theory in the infinite-cyliiider The following sectionsstate the problem and set out our approximation. The observedsecondarytransition to wavy notation, describe our numerical methods, give numerical vortices is aIsoin rough agreementwith linear calculations,‘- results for linear Ginzburg-Landau coefficientscorrespond- ’ although it is well known that the finite length of the annu- ing to previously publishedexperimental work, describeour lus strongly influencesthe observedcritical Reynolds num- two-Floquet-mode model and our detailed numerical and ber and azimuthal wave number.“,” The problem has been experimental aspect-ratio study, and compare numerical analyzed with the help of a Ginzburg-Landau equation6,7 and experimenta results. A brief concluding section sug: with coefficientsobtained from experimental data, gestsan extensionof the model to the nonlinear case. Jones3F4 hasnumericalIy computedcritical linear modes for wavy vortices in the infinite-cylinder approximation. His calculations were limited to modes with the same wave- Il. PROBLEM STATEMENT length as the underlying Taylor vortex flow. Our numerical A. Geometry and dimensionless parameters study is in the samespirit; that is, we calculate fully nonlin- Figure 1 shows the geometry and coordinate system. ear axisymmetric Taylor vortices at a given Reynolds num: The flow of a Newtonian fluid of kinematic viscosity v takes ber R, and we linearize around this flow. However, we go pIacein the annular region betweencylindersoflengthL and beyond the assumption of axially periodic disturbancesby of radii a and b with b > a. The outer cylinder is at rest and consideringmodes with arbitrary Floquet exponentk iri the the inner cylinder rotates with angular speedil. We define axial direction. While it is true that the critical wavy vortex three dimensionlessparameters(v,r,R 1, which are the radisturbancein the infinite-cylinder approximation is always dins ratio 9 = a/b, the aspectratio l? 2::L /(b - a), and the found to be periodic, that is, to have k = 0, the ability to vary Reynolds number R = ,C&(b - al/v. Transition from k allows the computation of linear coefficientsof the Ginz- Zouette flow to Taylor vortices is brought about by increasburg-Landau equation. In addition, we have successfully ing R while keeping 77and r fixed; the critical Reynolds modeledthe transition in cyhnders of finite length L by as- number for~infinite-cylinder transition to Taylor vortices, suming that only two Floquet modes, with k = &r/L, are denoted R, is a strong function of ?j: sufficient to account for it. Direct computation of these B. Navier-Stokes equations, Couette flow, and Taylor “Present address: Laboratoire de Physique, E&e Normale Sup&ieure de vortices Lyon, 46 al& d’Italie, 69364 Lyon Cedex 07, France. b) Present address: Center for Particle Theory, CJniGksity of Texas, Austin, The Navier-Stokes equations in coordinate-freenotaTexas 787 12. tion are ?510 Phys. Fluids A 3 (6). June 1991 0899~8213191 /061510-a9$02.00 0 1991 American institute of Physics 1510 Downloaded 09 Dec 2003 to 128.83.156.151. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp is a neutrally stableeigenmodeassociatedwith the transition to Taylor vortices. This mode is axisymmetric and has one Fourier component in the axial direction with wavele,ngth a,. All other Fourier modes and nonaxisymmetric modes are damped.The transition to Taylor vortices is a supercritical bifurcation, and just beyondR, a pattern of toroidal vortices developswith approximately the wavelengthil, . In experiments the actual wavelength observed for nonlinear 1 H; Taylor vortices dependson the length L of the apparatusand also on the experimentalprotocol; we will return to this issue when wezdiscussour experimental study. The Taylor vortex velocity field UT, with pressureP, , I I breaksthe axial translational invarianceofcouette Bow. It is I I steady state and axisymmetric and reflection symmetric I I about vortex boundaries.There are two vortices per axial wavelength,and adjacentvortices are separatedby inflow or outflow boundariesalong which the tlow is radially inward or outward, respectively.Taylor vortices are periodic in thez direction, which we expressby the notation FIG. 1. Geometryand coordinatesystem. exp(/l,d,)U, = U,, (12) where the operator exp(&d, ) is an axial translation by ;1,. d,U = - (U*V)U + ImJ - VP, (1) C. Linearization v-u = 0, (2) The linear stability problem for the transition to wavy where we have chosenunits of masssuch that the fluid den- vortices is obtained by substituting U-+U, + u and sity is identically 1. We use cylindrical coordinates (r,B,z) P-+ P, + p into ( 1) and (2) and retaining only terms linear with velocity components ( U,., U, , U, ) . The no-slip condi- in the perturbation variables u and p. This yields tions at the inner and outer cylinder walls are U = ihe,, at r = a, (3) d,u = - (U,.V)u - (u*V)U, + vv*u - vp, (13) U = 0, at P = b, (4) VW = 0 (14) where e, is the azimuthal unit vector. The circular Couette No-slip conditions (3) and (4) for the total field imply, for flow solution to ( 1,2,3,4) for infinitely long cylinders is the perturbation, UC = (Ar + B/r)eg, (5) u = 0, at r = a,b. (15) where the constantsA and B are chosento satisfy the no-slip conditions (3) and (4) and have the values The symmetries of the Taylor vortex flow allow us to A= -fka*/(b*-a*), B==fla’b2/(b2-a2). The asso- seekeigenmodesof the form ciated pressurefield is u(r$,z,t) =~(~,zmodil,)e’““+‘~‘-““‘+c.c., (16) p,(r) =~&‘*E~+~B~~~ ;; I (6) Y where the azimuthal wave number m is restricted to be inThe field U, has all the symmetries of the equations and tegerby the requirement that the flow be periodic in 0, and w boundary conditions. Explicitly, these are the three contin- is complex with Im(w) being the growth rate. The aziuous symmetries: muthal wave speedof the mode is given by Re(w)/m. The Floquet parameter k is taken to be real and without loss of translational invariance: J,Uc = 0, (7) generality we may consider only k ‘s within the first Brillouin axisymmetry: J,Uc = 0, (8) zone defined by - ?r/il, = [ U,(r,O, -z,t),U,(r,O, -G), (r,z) domain, periodic in z with period il,; and that this ‘is true of both the Taylor vortex nonlinear computations and - Uz(r,B, -z,t,]. (11) the solution of the eigenproblem.Thesecomputations allow There exists a critical Reynolds number R, = R, (7) such full investigation of the complex dispersion relation that Couette flow is stable only for R CR,. At R = R, there w = w(m,k;R,v,A, ). 1511 Phys. Fluids A, Vol. 3, No. 6, June 1991 Edwards, Beane, and Varma 1511 Downloaded 09 Dec 2003 to 128.83.156.151. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp 111.FINITE ASPECT RATIO: GINZBURG-LANDAU The 1LK’correction (241~t.o the critica Reynoldsnumber is EQUATIONS in good agreementwith experiment? -in a long-aspect-ratio A. Linear coefficients and the dispersion relation system (iT;;;JO?of radius ratio 9 = 0.89, when the values {;- 2.6 and Rw = 1.09OR, are used to fit the data. We Experimental evidence (Ref. 7 and referencescited note, however,that each azimuthal wave number m has its therein) suggeststh-at the dynamics of wavy vortices near own infinite-cylinder critical-R Iy, its own valuesof the linear transition may be modeledby the one-dimensionalcomplex coefficients,and thus its own thresholdand frequencyshifts. Ginzburg-Landau equation The critical azimuthal wave number for annulus length L is r,d,A=E(l+ic,)ii~~~(l+ic,)dfd--gfAI?~~, the one whoseshifted R w is leastat that L. The experimental data cited included observationsof m w - 2, 3, 4 and thus (17) provide a go& test of these threshold-shift predictions. where E= (R - RIY)/RW, with R wmbeing the critical ‘However, the analysisof Ahlers7 assumedthat the various Reynoldsnumber for onset of wavesin the infinite-cylihder azimuthal wave numbershad similar infinite-cylinder R w’s case.The complex amplitude A = 4 (z,t) is assumedto be and /&‘s, and thus only one valuefor K w and one valuefor &, slowly varying in both z and t, such that the disturbance were determinedfrom the data. Our numerical calculations velocity field is approximatedby for this radius ratio are shown in Table-1and Fig. 2. We do not iind that m,. = 4 is predicted by Ginzburg-Landau nd(z,t)B(r,z mod A,)exp(im,B - iw,t)-+ C.C. thresholdshifts in the rangeof aspectratios reported, so its (1%) appearancein the experimentalresults is anomalousfrom Here ii(l;zmodR, )exp(im,B - iwwt) is the critical infi- this point of view. In Sec.1V we will proposea model which nite-cylinder wavy-vortex eigenmodewith kw = 0. We as- appearsto accuratelypredict R w, m w, and o w over a wide sumethat in ( 17) the nonlinearcoefficientg haspositive real range of aspectratios. part so that the bifurcation is supercritical.The linear coeffi- The fact that azimuthal wave numbersother than 1 are cientsare obtainedfrom the dispersionrelation via the Tay- observedat Dnsetof wavy vortices is due to the finite length lor seriesapproximation of the annulus. ?Venote that the analysisof Walgraef et aL9 ul(m,k;t-) -aI wzddptw) +$W%&J. (19) predicts that higher azimuthal wave numbers will be observeda%lower aspectratios. SubsUtutingA = exp[ikz -- i(w - ow) t] into ( 171, ignor- To our knowledgeexperimentalresults for the frequening the nonlinear term, evaluating’d, and a z and equating cy correction (27) have not beenpreviously reported. coefficientsof E and k ’ betweenthe resulting equationand (:19) leadsto the following explicit expressionsfor comput- C. Extrapolation to infinite aspect ratio ing 7-*,c,, c(: and c1 from w(m,k;~): Pfister and Rehbergqnoted that the stationary nonlinTV= [Im(a,wlw)] -I, (20) ear equation ( 17) with c, = C, I’ 0 may be integrated using c,, = - ToRe(c?,wlw), (21) Jacobi elliptic functions; this result was fitted to experimen5: = - i~r,iZ)Im(d~&~,), tal measurementsof IA(z) 1 for various values of R above (22) R w. The procedureallowede to be determinedfrom R, from c, = (7-,-#2{i>Re(dEwlw). (231 which extrapolation to e ==0 yielded Rw for the infIniteWe evaluatethesederivativesat criticality (E = 0, k - Of by cylinder problem.This providesus with a good experimental finite differencingin E (for ~~and c(j) and k (for 5 i and c , ) . caseagainst which to test our numerical code with k = 0. Their experimentalradius ratio was T,?- 0.5066, for which B. Threshold and frequency shifts they found R, -I 455 for m - 1. The annulus length was A simple consequenceof (17) is a shift in the critical ,Reynoldsnumber for onset of wavy vortices in a system of finite length L relative to that for infinitely long cylinders, Assuming boundary conditions A ( &~L /2,t) := 0, and sub- TABLE I. Ntimerical results for radius ratio q = OS9 corresponding toexstituting (jlC-t - ioL with wL real, theJivstei@fimode-ofthe periments of Dominguez-Lerma et ai.’ Otiset of Taylor vortices occurs at linearization of ( 171, that is, the eigenmodewith-smallest RR;= 125.67-w&h 2, = 2.0080fb - a). Here I?, refers to the critical fz=!zj-,is Reynolds number for the onset of waves for infinite cylinders, r,, and &, are .~~ as in (20) and (22 ) ~The column &,/FWx is shown for comparison with A,(z,t) = cos(m/L)exp( - iw,t). (34) Fig. 22 of Ahlers,’ which defines $ differtintly from our convention. The values reported in that paper were R&I?, --- 1.09 and &,$~W/tr, = 2.7, Substituting this in ( 17), evaluatingthe derivativesand di- which wereassumed tobeapproximatelyindependent ofm. ‘I‘hevalue 26: viding through by A (z,tf producesthe relation is the coefficient of the f/L ’ threshold shift (26). See also Fig. 2. --h-&Q, =+(I +ic,) --&(l +ic,)(lr’/L”). (25) m R&i, TIJ 6, &pGz 4s The real part of this equation yields the threshold shift 1 1.080 0.553 4.9 4.7 237 EL-’ ?fr$i/L * (261 2~ I.089 0.159 2.60 2.72 66.7 and the imaginary part, with (26) givesthe frequencyshift 3 1.104 0.085 1.88 1.98 34.9 4 1.127 0.058 1.55 1.64 23.7 WL = 7-f) -‘[n-yj(c, -c,)/L”]. (27) 1612 Phys. Fluids A, Vol. 3, No. 6, June 1891 Edwards, Bean%, and Varma 1512 Downloaded 09 Dec 2003 to 128.83.156.151. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp ( 19) to first order in c and secondorder in k, valid only in the neighborhoodof criticality, that is, near E = 0, k.- 0. Numerically, we find that the quadratic approximation in k is valid only for very small k. Figure 3 showsfor 7 = 0.8703 and m = 3 the numerically determined growth rate as a function of k within the first Brillouin zone - T/I, < k < + n-/,X,. The axial periodicity and reflection symmetry of the Taylor vortex baseflow.imply that CL) is a periodic function of k with period 21r/il,) and that d, w must be zero at the origin and at the zone boundaries.Thus the quadratic approximation must fail well before the zone boundary is reached.Also shown in the figure is a parabola fit to the numerical data at k = 0, which correspondsto the quadratic G inzburg-Landau approximation. To investigatethe relevanceof the two Floquet modes 1.09 i --I-- I.---.-.-L I _-__LI__L__ with k = + n-/L, we conducteda numericaland experimen2n 30 20 50 60 70 80 90 tal study of the onsetof wavy vorticesas a function of aspect LIP - ~1 ratio. The radius ratio 0.8703 was available in an existing experimentalapparatusdesignedfor anotherpurpose,” and FIG. 2. Comparison of calculated Ginzburg-Landau threshold shifts with previous experimental results. Solid curves represent thresholds predicted thus we chose to examine this case.We determined from from the first eigenmode of the G-L equation using the coefficients shown linear calculationsfor the stability of Couette flow that the in Table I. The dotted line is the fit to the experimental data of Dominguez- infinite-cylinder Taylor transition occurredat R, = 116.096 Lerma et al.,5*7 which included m = 2,3,4. It agrees well with our m = 2 with a critical wavelengthR, = 2.0076(b - a). Since it is neutral curve, but there is no obvious reason for the agreement. However, the experimentally observed m = 4 results do not agree with the G-L well known that stablenonlinearTaylor vorticesexist over a threshold shifts in this range of aspect ratios. finite range of wavelengths,and since we did not wish to study the effect of Taylor vortex wavelengthon the wavy 18.5(b - rt) and their baseTaylor vortex flow had nine pairs of vortices,for an averagewavelengthof 2.056(b - a). Using this wavelengthand radius ratio we find R w = 445. The _--2.2% discrepancybetween the critical Reynolds numbers may be due to end effectsin the experiments,the approxima0 l---i tions inherent in the G inzburg-Landau analysisincluding -. the assumptionc0 = c, = 0, or errors introduced in fitting --‘2-'-. Im(w) I//(&- ay - 1'1 the data to the Jacobielliptic functions and extrapolating;no . error analysiswaspresentedin their paper.Another possible -.. . explanationis that at theseaspectratios there exist distinct -10 ----- \::__;,=I---'- . ---modeswith similar critical Reynoldsnumbers;our computed critical eigenmode’slargestamplitude was on the outflow boundary, which agreeswith descriptionsof the so-called ‘rjet-flow” mode. - -20 ,J. IV. FLOQUET THEORY FOR THE ASPECT-RATIO DEPENDENCE OF THE ONSET OF WAVY VORTICES A. Floquet modes and the axisymmetric end -30 -____ I -boundaries 0 0.5 I V(*lU The simple cosineeigenmode(24) of the linear G inzburg-Landau equationmay be rewritten trivially as FIG. 3. Dispersion relation within the first Brillouin zone for parameters cos(m/L) = i(eikz + e - “‘), 7 = 0.8703, /1, = 2.0076, m = 3, R = 13 1.025. Thisolid curves are growth (28) rates of the four least stable modes. There is a countable infinity of more where k = IT/L. This suggeststhat the linear eigenmodeof stable modes which are not shown. Only the leading mode, the one with the finite-length problem may be viewedasthe superposition maximum growth rate, is important for determining stability. The Reynolds number for this figure is critical for the onset of m = 3 waves, so that of two eigenmodesof the form (16) with k = f n-/L. The the leading eigenmode has Im(w) = 0 for kw = 0; this mode is associated advantageof this viewpoint is that it avoids the approxima- with the uppermost solid curve. The dotted curve is a parabola fitted to the tion to the dispersionrelation that the G inzburg-Landau top curve at k = 0. The horizontal bar indicates the range of Floquet expomodel makes [ Eq. ( 19) 1. The linear coefficientsr0 and 5 g nents k = r/L expIored in our numerical and experimenta study for L = 41, to 17/1,. For this restricted range of Floquet exponents, the top occurring in ( 17) and used to predict threshold shifts in curve does not intersect any other curve, so we may assume that only this (26) are calculated on the basis of the Taylor expansion mode is important. 1513 Phys. Fluids A, Vol. 3, No. 6, June 1991 Edwards, Beane, and Varma 1513 Downloaded 09 Dec 2003 to 128.83.156.151. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp transition, we chose to consider in both the numerics and V. ~X~E~l~~~TAL VERIFICATION experiment only those annulus Iengths that are integer mul- A. Description of the apparatus tiples of A,, that is, L = L, --7n/2,, and we further insisted that in the experimental Taylor vortex flow for length L, We had available in our laboratory an apparatus designed and built by flirst to study the aspect-ratio depenthere were exactly n pairs of Taylor cells between the end dence of the attractor ~dimensionin weak Couette-Taylor rings. The end rings in the experiment were fixed, giving rise turbulence.” The systemconsistedof two independently roto Ekman pumping that results in end cells of slightly longer tating&inders. The inner cylinder ofbrass was machined to axial extent that those in the bulk of the commn, which were then, of course,slight1yshorter than A,; we will discussthis a radius of a = 6.635 -& 0.005 cm and the outer cylinder of effect in a subsequentsection when we analyze our experi- precision bore glass tubing had an inner radius of b = 7.620 f 0.005 cm. For our study the outer cylinder was mental results. kept fixed. The cylinders were housedin a water bath which The experimental apparatus had a maximum aspectramaintained the temperature at 24.5 “C to within 0.1 “C. The tio sufficient to accommodate 17 pairs of Taylor vortices. inner cylinder was driven by a computer-controlled microWe studied the set of aspectratios L,, n = 4,.,.,17. The horistepper motor through a system of timing be&s and pulleys. zontal bar in Fig. 3 covers the corresponding range of FloEnd boundary conditions were establishedby two nanquet exponentsff/( DA, ) for k- very careful design and construction of Don Hirst’s apparawithin the first Rrillouin zone defined by the spatially peri- tus which expedited the experimental work. We also thank odic Taylor vortex flow. We find that the quadratic approxi- the refereesfor several veryvaluable suggestionsthat have mation to this dispersion relation near k = 0 deviatessignifi- improved both the content andorganization of the paper. cantly from the actual dispersion relation when k is of the This researchis supported in part by the Oflice of Naval order of s-/15/2,. This suggeststhat the first eigenmode-of ResearchNonlinear Dynamics Program, and by NSF Grant the Ginzburg-Landau equation, used to predict threshold No. DMS-8901767. Computational resourcesfor this work shifts, will not be reliable for aspectratios of approximately were provided by the University of Texas SystemCenter for 30 and below. IIigh Performance Computing. For aspect ratios between 42, --,8 and 17& ~34 wehaveconducteda detailed numerical and experimentalstudy which indicates that the onset of wavy vortices may be accurately predicted by assuming that the finite-cylinder eigen- ‘G. I. Taylor, Philos. Trans. R. So?. London Ser. 223,289 ( 1923). mode is the superposition of two infinite-cylinder Floquet ‘J. A. Cole, J, Fluid Mech. 75, 1 ( 1976). “C. A. Jones, J. Fluid Mech. 102,249 II981 1. modes with k = f r/L. We know of no other theoretical ‘?. A. Jones, J. (Zemput. Phys. 61,321 (1985). model in which critical Reynolds numbers, azimuthal-wave- 5M. A. Dominguez-Lerma, G. AhIers, and D. S. Cannel1 (unpublished). numbers, and wave speedsof wavy vortices agree in detail “G. Pfister and I. Rehberg, Pbys. Lett. A 83,19 ( 198 1) . with experiment over this range of aspectratios. ‘G. Ahlers, in lectures in rhe Sciencesof Complexity, SF1 Studies in the Sciencesof Complexity, edited by D. Stein (Addison-Wesley, Reading, Although our work is-computationally more intensive MA, 1,989), pp. 175-224. than perturbation analysessuch asthat of Walgraef et al.” or sW. S. Edwards, L. S. Tuckerman, R. A. Friesner, and D. (7. Sorensen (in infinite-cylinder stability calculations such as those of preparation). “D. Walgraef, P. BB&mans, and G. Dewel, Phys. Rev. A 291514 (19841. Jones,3%4 it is important to note that our calculations were all ‘“D. A I-fist, Ph.D. thesis, Yniversity of Texas, 1987. performed on a small two-dimensional (r,z) domain, period- “CT I). Andereck, S. S. Liu, and H. L. Swinney, J. Fluid Mech. 164, 155 ic in z with period A,. No attempt was made to accurately (i986). 1518 Phys. Fluids A, Vol. 3, No. 6, June 19$1 Edwards, Beane, and Varma 1518 Downloaded 09 Dec 2003 to 128.83.156.151. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp