chaotic motion in an oscillatory boundary layermorse.uml.edu/articles.d/chaos.pdfchaotic motion in...

Chaotic motion in an oscillatory boundary layerV. Mehta, C. Thompson, A. Mulpur, and K. ChandraCenter for Advanced Computation and Telecommunications and Department of Electrical Engineering,University of Massachusetts, Lowell, Massachusetts 01854

~Received 25 July 1995; accepted for publication 30 July 1996!

The chaotic time oscillations in an incompressible fluid driven into motion by a harmonictime-varying pressure gradient is examined. Special attention is given to centrifugal destabilizationof the viscous boundary layer. The basic flow is shown to be linearly unstable. For increasingmodulation amplitude, the flow exhibits chaotic oscillations. The energy exchange betweensubharmonics and superharmonics of the least-stable spanwise wave number is considered. Thepresence of subharmonic Fourier modes are shown to accelerate the transition to temporally chaoticmotion. © 1996 American Institute of Physics.@S1054-1500~96!00204-2#

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Over the last 150 years, the problem of oscillatory motionof fluids has been examined by numerous investigatorsHowever, only recently have some of the causes for instability and chaotic motion in such flows been uncovered.Chaotic oscillations in wall-bounded oscillatory flowshave been linked to the nonlinear growth of vortical dis-turbances introduced into the viscous boundary layer.The temporal evolution of these disturbances gives rise tothe basic features evident in the resulting chaotic fluidmotion. Disturbances are generated by the environmentand consequently are comprised of numerous modes having a varied spatial wave number and amplitude. Oncethe fluid motion becomes unstable, according to lineartheory, the least linearly stable mode will prevail andchaotic oscillation will ensue at a threshold value of thecontrolling parameter. Nonlinear amplitude growth canyield alternative solutions, which display chaotic oscilla-tions at lower threshold values. To determine the leaststable disturbance, the sensitivity of the fluid motion tothe wave number composition and amplitude of an im-posed disturbance must be understood.

I. INTRODUCTION

Flows that are driven in total or in part by time periodexcitations find application in problems in heat and mtransfer. At a fluid–solid interface, oscillatory fluid motiohas been shown to give rise to an enhancement in the dission of passive contaminants1–3 and in heat transfer.4,5 Thisenhancement can be orders of magnitude greater thanafforded by molecular diffusion alone. The magnitude of tcited enhancement is a function of the dynamics and chater of the fluid motion in the viscous boundary layer. For threason effort has been directed toward understandingconditions under which oscillatory flows lose stability abecome turbulent.

Modifications of Stokes’ second problem have been uto model the viscous boundary layer. In the problem as ornally formulated by Stokes,6 the fluid adjacent to a rigidplanar surface is driven into motion by the in-plane timharmonic displacement of the surface. The magnitude ofresulting Stokes boundary layer decays exponentially w

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distance from the surface. The rate of decay is a functionthe viscosity of the fluid and frequency of oscillation. Modfied Stokes problems usually involve the considerationdifferent types of forcing and surface geometry. In wabounded flows, the oscillatory fluid motion results fromapplied pressure gradient. Deceleration of the fluid atwall results in the formation of an oscillatory boundary layadjacent to the surface.

Linear stability analyses of flow over planar boundarhave concluded that the Stokes boundary layer is linestable to two- and three-dimensional vortical distubances.8–10This body of work is based on the hypothesis ththe Stokes boundary layer becomes unstable at a crivalue of the streaming Reynolds numberRs . The streamingReynolds numberRs is defined asU0

2/~vn!, wheren is thekinematic viscosity of the fluid,v is the frequency of exci-tation, andU0 is the velocity amplitude. One may interprethe streaming Reynolds number asU0L/n, where the appro-priate length scaleL is taken to be the oscillatory particldisplacement~U0/v!.

The absence of a bifurcation point according to linetheory has motivated investigators to look to the effectsfinite-amplitude disturbances. Recently, Akahavanet al.10

concluded that instability of the secondary flow, generaby finite-amplitude linearly stable two-dimensional perturbtions of the flow, may be responsible for transition to turblence.

In the allied problem of an oscillatory boundary laygenerated by the transverse oscillation of a cylinder, Ha11

theoretically obtained a bifurcation point according to linetheory. This instability was earlier observed experimentaby Honji12 and corroborated by Sarpkaya.13 It was found thatconvex curvature of the boundary gives rise to centrifudestabilization of the boundary layer.

The influence of wall geometry on the stability of thoscillatory boundary layer has been investigated by Thomsonet al.14,15This work examined oscillatory flow in a twodimensional rigid walled channel with a width,H0(b2h),that varies slowly with changing axial position. The flumotion in the boundary layer was shown to be a functionRs and two additional nondimensional parameterse andR.The parametere is taken to reflect the degree of variation

6010 © 1996 American Institute of Physics

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602 Mehta et al.: Chaotic motion

the wall height with axial location. The parameterR is theoscillatory Reynolds number given byvH0

2/n and is based onthe channel heightH0.

At low amplitudes of excitation,e2Rs!1, a viscousstreaming flow occurs in the channel.14 This flow is secondorder in the oscillation amplitude and is the result of tgradient of the time-averaged Reynolds stress. Viscstreaming in this parameter range does not strongly intewith the basic oscillatory flow. However, as the oscillatiamplitude is increased, nonlinear inertial effects becomeportant. In such a case strong viscous streaming flow casuperceded by the occurrence of a boundary layer instabFor oscillatory flows where centrifugal instability occurs15

unstable modes are manifest by the appearance of courotating vortices and the generated vorticity is collinear wthe direction of the applied pressure gradient. The destabing centrifugal force is generated by the curvature ofboundary. The controlling stability parameter is the TaynumberT2, which is equal toe2Rs/R

1/2. The least stable vortical disturbances have a spanwise wave number equa0.35~v/n!1/2 and have been shown to bifurcate subcriticafrom stability at a Taylor numberT2 equal to 16.92. Scaledon the oscillatory boundary layer thickness, the critical spwise wave numberkc is equal to 0.35. Therefore, if boundary curvature is considered, the boundary layer loses stabat a finite value of the modulation amplitude, of the baflow. Moreover, for higher values ofT2 temporally chaoticoscillations ensue.16

The aforementioned analysis of the least linearly stavortex mode is expected to yield accurate results nearcritical Taylor number. However, experimental results12,17

from allied problems suggests that the amplitude of the sharmonics of the least stable wave number may be imporat higher values of the Taylor number. The extent to whthe presence of sub and superharmonics of the least swave number mediate the energy transfer from the bflow and the onset of chaotic oscillations remains an oquestion.

In this study, the temporal evolution of threedimensional vortical disturbances in an oscillatory boundlayer is examined in the limit as 1/R ande tend to zero. Thenew contribution of this study is the description of the roplayed by the wave number composition and amplitudedisturbances in the development of instability and chaooscillations. To this end, the spanwise variation in the disbance velocity comprised of Fourier modes having supermonics and subharmonics of the critical wave numberkc areexamined. The objective here is to determine the sensitiof the flow field to background disturbances. This will baccomplished by examining the temporal features of themerical solution of the local disturbance equations asT2 isvaried. The disturbance equations will be solved for two vues of the principal wave numberk, namely,kc and kc/4.These solutions will be used to determine how temporal ftures are modified, as the result of the energy balancetween the Fourier modes.

In Sec. II the problem of boundary layer stability inchannel with varying height is outlined. The basic flow a

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disturbance equations are presented. Section III is devotethe presentation of the method of solution of the disturbaequations. In Sec. IV the numerical results of the analysispresented. It is shown that the flow exhibits chaotic tempooscillations when sub- and superharmonics are present inFourier expansion. The onset of temporal chaos is eslished through calculation of the Lyapunov exponents fordisturbance kinetic energy. The growth in amplitude of thedisturbances gives rise to a secondary flow. In addition,presence of subharmonic modes is shown to yield chaoscillations at a lower Taylor number than for cases whonly superharmonics modes are present.

II. PROBLEM STATEMENT

The oscillatory fluid motion in a planar channel withslowly varying lower boundary and an upper boundaryuniform height will be considered~Fig. 1!. The height of thelower wall is defined byY5H0h(X/L0) and the upper wallis defined byY5H0b. The parametersL0 andH0 representthe typical wall wavelength and height of the channel lowwall. The slope of the lower wall isO~e!, wheree5H0/L0 .For the case of a slowly varying channel,e is much less thanunity. The fluid enclosed in the channel is considered tohave incompressibly. An imposed time harmonic pressgradient directed in theX direction will be used to generatetwo-dimensional basic flow in the channel. The deceleratof the flow at the boundaries gives rise to oscillatory bounary layers on the upper and lower wall of the channel. Ttwo-dimensional basic flow is subsequently perturbed bthree-dimensional vortical disturbance. The lower wall,virtue of its curvature, offers the least stable situation. Thefore, attention will be focused on the fluid motion adjacentthe lower wall. The temporal growth of the disturbance aplitude will be used to determine the stability of the basflow.

A. Basic flow

The asymptotic behavior for the basic flow follows frothe solution of momentum and continuity equations unthe limiting condition thate approaches zero. By invokingthis condition, the velocity can be determined in terms oregular expansion in gauge functions of the slope parame.14 As a result, the basic flow solution is globally valid i

FIG. 1. Oscillatory fluid motion in a planar channel with a slowly varyinlower boundary and an upper boundary of uniform height.

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603Mehta et al.: Chaotic motion

the spatial variables. Though a detailed derivation of goveing equations and their solution is given by Thompson,14,15

the basic assumptions, equations, and their solution wilgiven here for completeness.

The viscous boundary layer of the oscillatory flowtaken to have a small thickness ofO(dH0), whered is equalto 1/R1/2. The oscillatory particle displacement~U0/v! istaken to be small compared to the wall wavelengthL0. Thedimensional axial, vertical, and spanwise coordinatesgiven by the variablesX, Y, andZ, respectively. To determine the sensitivity of the basic flow to three-dimensiondisturbances, one must examine the fluid motion in thecous region, whereY2H0h is O(H0d) andZ is O(H0d).The axial coordinateX is scaled on the wall wavelengthL0.In the boundary layer region the nondimensional coordinaare given by

x5X/~Lo!,

y5~Y2H0h!/~H0d!, ~1!

z5Z/~H0d!.

Time is scaled on 1/v, wherev is the modulation frequencyThe velocities and pressure will be expressed as the sumdisturbance and a basic flow term. In the viscous regionbasic flow velocity component in the axial direction is takto beO(U0). Whereas by virtue of the no-penetration codition at the wall, the vertical component of the basic flowO(eU0). If the aforementioned scaling is applied to the dturbance quantities, the convective and centrifugal termthe momentum equations areO~e!. The aforementioned scaing results in a trivial solution for the disturbance velocityleading order ine. This result is consistent with observationof low-amplitude viscous streaming. Instability occurs whthe centrifugal force arising from the axial change in tmomentum flux~r0UV! is sufficiently large as to rende(e2Rs)5O(R1/2). Therefore the axial component of the diturbance velocity isO(U0), while the vertical and spanwisdisturbance velocities areO~@vn#1/2!. The scaling for the ba-sic and disturbance flow follows that of Thompson14–16 andis in line with that used by Hall11 for the case of centrifugainstability in the viscous region of a transversely oscillaticylinder. The quantity ~vn!1/2 can be rewritten as(eU0d

1/2)/T, whereU0 is the modulation amplitude. ThTaylor numberT25(e2Rs/R

1/2) is analogous to that givenby Hall. Applying the aforementioned scaling the veloccomponents and the pressure can be written as

U5U0@ u~x,y,t !1u~x,y,z,t !#,

V5~eU0!F @d v~x,y,t !1h8u~x,y,t !#

1S d1/2

Tv~x,y,z,t !1h8u~x,y,z,t ! D G ,

~2!W5~eU0!

d1/2

Tw~x,y,z,t !,

P5~r0vU0L0!S p~x,y,t !1e2d3/2

Tp~x,y,z,t ! D .

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The dimensional quantities (U,V,W,P) correspond to theaxial, vertical, and spanwise components of velocity, andpressure, respectively. The nondimensional basic flow teare denoted by~2!, whereas the nondimensional disturbanquantities are (u,v,w,p). The equations governing the basflow are obtained by substituting Eqs.~1!–~2! into theNavier–Stokes equations and setting the disturbance quties to zero. In the case of small fluid particle displacemethe basic flow at leading order ine is governed by the fol-lowing equations:

ut1 px5uyy1O~e2!, ~3!

py501O~e2!, ~4!

vy1ux50. ~5!

The basic flow velocity satisfies the no-slip and the zepenetration conditions aty50 andy5(b2h)/d. The solu-tion of the aforementioned equations can be obtained bregular perturbation expansion in the gauge functionen. Theaxial and vertical velocitiesu and v at the lowest order are

u5Re@2 iB~x!~12e2ay2e1aye2a~b2h!/d!e2 i t #1O~e2!,

~6a!

v5ReS 2 i

a

dB

dx~12e2ay1e1aye2a~b2h!/d!e2 i t1 iy

dB

dx

3e2 i t2 iB~x!h8

d~e1aye2a~b2h!/d!e2 i t D1O~e2!, ~6b!

wherea is equal to~~12i !/&!. The functionB(x) representsthe complex amplitude of the pressure gradient. Integrathe continuity equation across the channel cross-section,plying the condition thatv50 at y5(b2h)/d, and neglect-ing exponentially small terms yields the following equatiofor B(x):

d

dx@B~h2b!#1

2d

a

dB

dx50. ~7!

For y beingO~1!, the velocities revert to the boundary laysolution of the planar oscillatory layer. This solution is otained from Eq.~6! in the limit asd tends to zero,

u5Re@2 iB~12e2ay!e2 i t #, ~8a!

v5ReF1 idB

dx S y21

a~12e2ay! De2 i t G . ~8b!

B. Disturbance equations

The equations governing the dynamic developmentthree-dimensional disturbances introduced into the baflow will be considered here. Of particular interest are thocases where the boundary slopee and boundary layer thick-nessd are of small magnitude. Instability in the viscous rgion is caused by the centrifugal acceleration of the fluwhich results from the curvature of the channel lower waThis occurs when the modulation amplitude rende2Rs5O(R1/2). In such a case the Taylor number isO~1!.This allows energy exchange between the basic flow and

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disturbance to take place. Applying the nondimensionalition and coordinate scaling given in Eqs.~1!–~2! to the equa-tions of motion, and subtracting the basic flow equatiofrom the result yield

vuy1Td1/2@~ uu!x1uux1 vuy#5L~u!2Qu1O~e!, ~9!

T2h9~2uu1u2!1Td1/2@ uvx1~ vv !y1uvx#

1T2 du vx1py5L~v !2Qv1O~e!, ~10!

Td1/2~ uwx1 vwy1uwx!1pz5L~w!2Qw1O~e!, ~11!

vy1wz1T d1/2ux50, ~12!

whereL(a) equalsayy1azz2at andQa equalsvay1waz .The temporal growth rate of the disturbance velocitiesgoverned by the magnitude of the Taylor numberT2.

In the aforementioned equations, the dependence ofdisturbance velocity amplitude on axial position is manifin O~d1/2! terms. Discarding these terms yields a solution tis locally valid in the axial position. One can consider tparameterT2h9u in Eq. ~10! as the local Taylor numbewhen u is replaced by the asymptotic axial velocity amptudeB(x) given in Eq.~7!. The magnitude of the local Taylor number changes with axial position and its value incates the degree of instability of the basic flow. Tboundary layer is least stable whenuh9 attains its minimumvalue. For an incompressible channel flow this situationcurs where the boundary attains its minimum convex curture and channel width. The axial position where this mimum occurs will be denoted by the positionx0. Energy isexchanged from the basic flow to the disturbance flow. Ttemporal growth of the amplitude of the disturbance isindication of instability of the basic flow. The axial component of the disturbance velocityu is responsible for this energy exchange, whereas the vertical velocity is driven byformer.

The disturbance velocities (u,v,w) and pressure are expanded in terms of a Fourier series inz, wherek representsthe principal spanwise wave number,

~u,v,w,p!5~u0,v0,w0,p0!1 (n51

N

~un,vn,wn,pn!einkz1c.c.

~13!

The expression given in Eq.~13! is substituted into Eqs~9!–~12!. By virtue of the Fourier expansion, and Eqs.~11!–~12! the dependence of theX andY momenta, Eqs.~9! and~10!, on p andw can be algebraically removed. Performinthe aforementioned operations results in the local disturbaequations for the coefficient of thenth Fourier harmonic inz,

2vmuyn2m2vnuy1

n2m

mvymun2mDm

1Dn~un!2utn50, ~14!

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2~nk!2T2h9~umun2m12uun!1~nk!2vmvyn2m

1~nk!2S n2m

m D vymvn2mDm1S n

n2mD ~vymvyy

n2m

1vmvyyyn2m!DnDn2m2

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m~vy

mvyyn2m1vyy

m vyn2m!

3DmDnDn2m2Dn~Dn~vn!2v tn!50, ~15!

where the functionDj represents the complement of the Krnecker delta and the operatorDn(a) is defined asayy2(nk)2

a. In the aforementioned equations, summation over thedexm is implied. The disturbance field is evaluated by soing for the Fourier coefficients ofv and u. The velocitycoefficients,un , vn , andvy

n satisfy homogeneous boundaconditions aty equal to zero and asy tends to`. The zcomponent of the velocity can be recovered using the conuity equation.

III. METHOD OF SOLUTION

The Galerkin method is employed to solve Eqs.~14! and~15!. Thenth Fourier coefficient of the velocity componenu andv are expanded in terms of the trial functionsCi(y),

un~y,t !5(i51

I

uin~ t !Ci~y!; vn~y,t !5(

i51

I

v in~ t !Ci~y!.

The trial functions in this series expansion are linear comnations of Chebyshev polynomialsTi ,

Ci~y!5 HTi~y!2T0~y!,Ti12~y!2T1~y!,

for i even,for i odd.

The computational domains:@21,1# was mapped to thephysical domain using the logarithmic coordinate transfmation y52y0 ln @~s11!/2#. The trial functionsCi(y) sat-isfy Dirichlet boundary conditions aty equal to 0 and . TheNeumann boundary condition forv at y equal to` is satis-fied by virtue of the mapping. The remaining Neumaboundary condition aty equal to zero is satisfied by imposing the constraint

~vyn!y5050,

on the trial series. The initial values for the coefficients ofvn

are chosen such that the Neumann boundary conditiosatisfied. The value of the scale factor of the transformaty0 was increased until the largest Floquet exponent ofsolution of the linearized equations reached a constant vaThis was found to occur aty0 equal to 3&. For this value ofy0, the mean square error in the approximation of the baflow is a minimum. The inner products are evaluated usthe Gauss-quadrature approximation. The resulting ordindifferential equations were integrated in time using a vaable time step backward difference formula scheme. Incalculations 30 trial functions iny were found to be suffi-cient. Numerical tests using up to 60 trial functions habeen performed to ensure invariance in the temporal beh

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ior of the solution. The second derivative of the boundashapeh9 is set equal to negative one in all calculationsported here.

IV. RESULTS AND DISCUSSION

The results reported in this study are based on obsetions of the temporal oscillations and the wave number sptrum of the kinetic energy of the disturbance velocity.describing the temporal behavior, the term cycle is usedrepresent the time interval 2p. This value corresponds to thperiod of the basic flow oscillation. The kinetic energy of tflow is the sum of the squares of each velocity componintegrated overy andz. The limits of integration are~0, ! iny and ~0,2pn/kc! in z. The value ofln52pn/kc corre-sponds to the wavelength of the fundamental spanwise wnumberkc/n . In the present studyn takes on the values of 1and 4. Integrating the disturbance kinetic energy with respto z yields the energy per wavelength in terms of the spwise Fourier amplitudes,

E~ t !5 (m50

N

Em/n~ t !5 (m50

N1

lnE0

`

dyuum~ t,y!u2

1uvm~ t,y!u21uwm~ t,y!u2; ~16!

Em/n is the kinetic energy per wavelengthln of the Fouriercomponent having the spanwise wave numbermkc/n.

The influence of sub and/or superharmonic Fourier coponents of the spanwise wave numberkc on the disturbancekinetic energy are considered here. A disturbance is denas being of type PS when only the linearly least stable mand its integral superharmonic Fourier componentspresent in the disturbance. In such a casen is equal to 1.When sub and superharmonics ofkc are present, the disturbance will be termed type SPS. In this casen is greater thanunity. The square root of the Taylor numberT2, which isproportional to the amplitude of the basic flow, will be usto parameterize our observations. As the value ofT is in-creased, both PS- and SPS-type disturbances exhibit perand chaotic oscillations in time. The post-transient solutiocomputed for the initial disturbance fields of type PS aSPS are markedly different. The notion that variations ininitial conditions yield multiple solutions is supported by eperimental observations in Taylor–Couette flows.18 InTaylor–Couette flows the wavelength of the vortices fofixed supercritical Taylor number is established by the iniconditions. Initial conditions may yield a vortex wavelengthat is smaller than, equal to, or larger than the critical walength.

Considering the decay rate of the disturbance enewith increasing wave number, the number of Fourier moin Eq. ~13! is chosen. The disturbance energy wave numspectrum denotes the amplitude distribution of the spanwFourier components, comprising the disturbance kineticergy, as a function of the spanwise wave number. The ccontribution to the disturbance is made by a limited numof Fourier modes. The corresponding range in wave numis denoted as the passband of the energy spectrum. The

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trum decays sharply for wave numbers to the left and righthis range. The passband of the energy spectrum is controby the number of active Fourier modes in the unstable regof the T22k plane. At a fixed value of the Taylor numbethe range of wave number values is delimited by the neustability curve given in Fig. 2. This curve segments theT22kplane into stable and unstable regions. Disturbanceshave wave numbers lying in the unstable region of theT22kplane generate the active modes.

The neutral stability curve is determined by applyinFloquet theory to the linearized disturbance equations.Floquet exponents are computed by solving the linearidisturbance equations for values ofT2 andk. The locus ofpoints for which the magnitude of the Floquet exponentequal to unity yields the neutral stability curve.

Numerical experiments for Taylor numbers near criticconfirm that linear theory successfully predicts the cutwave numbers for the active region or passband. Howefor larger values ofT2 as unstable modes grow in amplitudenergy is exchanged between the active and stable Fomodes. In order to appropriately capture the dynamics offlow, a sufficient number of stable Fourier modes mustretained. In such a case the number of modes is increauntil the solution is invariant to a further increase in tnumber of Fourier terms. Numerical experiments have bperformed using this criterion. For the PS case, ten termthe Fourier series provide ample bandwidth for modelingenergy exchange between unstable and dissipative moFor SPS-type disturbances, 30 terms are required. For vaof T greater than~52!1/2, SPS disturbances require additionsubharmonic modes belowkc/4. It appears that this numbeincreases with the Taylor number. This state of affairs mbe the result of the onset of spatial bifurcations in the floIn Taylor–Couette flows a bifurcation to a doubly period

FIG. 2. Neutral stability curve obtained from Floquet analysis of linestability equations.

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FIG. 3. ~a! Bifurcation diagram for the evolution of an initial disturbancefield composed of the linearly least stable wave number and its superhmonics.~b! Bifurcation diagram for the evolution of an initial disturbancefield composed of the linearly least stable wave number and its sub- asuperharmonics.~c! Convergence of the Lyapunov exponents.

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vortex flow typically follows singly periodic flow as the Taylor number is increased.19 In the present problem spatial bfurcations of this type would yield waves in the axial coodinate. This would require the axial variation of thdisturbance to be considered in the governing equations

A. Classification of the temporal behavior usingbifurcation diagrams and Lyapunov exponents

An overview of the dynamic regimes for varying Taylonumber are presented succinctly by the bifurcation diagragiven in Figs. 3~a! and 3~b!. The results for the PS-type floware summarized in Fig. 3~a!. In this case the interaction between the basic flow and Fourier components havingfundamental spanwise wave numberkc and ten of its integralmultiples are considered. The bifurcation diagram is cstructed using 15 time cycles of the total kinetic energy

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fixed values of the Taylor number. Each point on the dgram represents a sample of the total kinetic energy sigtaken at intervals of one half-cycle after transients insolution have dissipated. The results given in Fig. 3~a! indi-cate that the disturbance kinetic energy is time periodic foTbelow ~50!1/2. This is determined by the return to the samamplitude at the end of every half-cycle. For values ofTbetween~50!1/2 and ~60!1/2 the disturbance kinetic energundergoes a single period doubling bifurcation. This is edent from the appearance of two amplitude points that reevery other half-cycle. AtT equal to~70!1/2 the bifurcationdiagram shows a clustering of points about four energy aplitudes. These semiperiodic oscillations mark the onsechaos. The wide range of energy amplitudes present aTequal to~80!1/2 indicates strongly chaotic oscillations.

No. 4, 1996

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Current results differ from those previously reportedThompsonet al.16 in which the first two Fourier modes arused to model chaotic PS-type disturbances. Thoughmodel was successful in gleaning the basic features intime evolution of the disturbance velocity, the interactibetween higher modes was not considered. In comparwith present results one finds that the first period doublbifurcation occurs at a value slightly higher than theT equalto ~50!1/2 reported, whereas the onset of chaotic behaviofound to occur at a lower value ofT. It appears that thepresence of additional superharmonic Fourier modes indisturbance velocity lowers the amplitude threshold at whtemporally chaotic oscillations occur.

The summary of results or SPS-type disturbancesgiven in Fig. 3~b!. In this case the interaction between Forier modes havingkc/4 and 30 of its integral multiples isconsidered. For SPS-types disturbances five Fourier harmics are required to generate temporally chaotic behavHere the energy transfer to the subharmonic modes frombasic flow is mediated by the least stable wave numbHowever, five Fourier modes are not sufficient for describthe energy exchange between the unstable modes andpative modes for high Taylor numbers.

For SPS-type disturbances, the disturbance kineticergy oscillates periodically for values ofT between~30!1/2

and~45!1/2. Period doubling oscillations appear atT equal to~47!1/2 and are manifest by the presence of two recurrvalues for the disturbance kinetic energy. ForT greater than~50!1/2 the disturbance kinetic energy exhibits temporachaotic oscillations. The energy levels for the PS and Scases are comparable forT below ~50!1/2.

The bifurcation diagram provides qualitative evidenof chaotic oscillations. Exponential sensitivity to initiaconditions must be verified if one is to show that timwaveforms are indeed chaotic. To this end the largLyapunov exponent is used. For the PS cases the larLyapunov exponent was determined for Taylor numbe~18,20,30,40,50,70,72,80! and for SPS cases Taylor numbers:~30,40,45,47,50,52,55,60!.

The largest Lyapunov exponent was calculated fotwo-dimensional phase space comprised of the kineticergy of the axial and spanwise disturbance velocity comnentsEu andEw . Given an initial range of energies in thEu2Ew plane, the largest Lyapunov exponent is evaluaby monitoring the expansion of the state-space trajectowith increasing time.20 When the magnitude of this largeLyapunov exponent is positive the signal is termed chao

The trajectory in the phase planeEu2Ew is described bythe vector positionz(t). Two points at the time instancest0andt1 are chosen such that their distanced05uz(t0)2z(t1)uis less than a prescribed toleranceb. After a time tm haselapsed the distance between the trajectod15uz(t01tm)2z(t11tm)u is determined. The estimate othe Lyapunov exponent for this first iteration is givenl15ln(d1/d0)/tm .

On the next iteration, a replacement vector is foundz(t11tm) and t0 is updated ast05t01tm . The new vectorsatisfies the condition thatd05uz(t0)2z(t2)u is less thanb.

CHAOS, Vol. 6,

ise

ong

is

eh

is-

n-r.her.gssi-

n-

g

S

stest:

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ds

.

s

r

In addition, z~t2! is chosen such that the angle betwez(t11tm) and the replacement vectorz~t2! is less than 0.3radians. This ensures that the ensemble of locations uderive from a similar set of initial conditions. The distanafter tm has elapsed isd25uz(t01tm)2z(t21tm)u. Hencethe estimate of the Lyapunov exponent at the second ittions is given by l25ln(d2/d0)/tm The procedure is re-peated, yielding a series of estimates$l j , j51,2,...% for theLyapunov exponent. These estimates are then averageevaluate the largest Lyapunov exponent. For the cases exined, tm equal to 3.33p was used.

The case of PS-type disturbances withT5~80!1/2 is con-sidered first. From the analysis of 160 cycles of potransient data and with the distance thresholdb50.01, theestimate of the Lyapunov exponent was determined usin

maximum of 95 l j estimates. The mean value of thLyapunov exponent for increasing number of estimatesshown in Fig. 3~c!. The mean value of the Lyapunov exponent converges to the value 0.63. The variance in this emate was 2.931023. The positive value of the Lyapunoexponent supports the claim that chaotic oscillations occuthis value of the Taylor number. For the Taylor numbeconsidered the Lyapunov exponent was positive onlyTaylor numbers greater than 70.

For SPS-type disturbances a typical Lyapunov exponresult is shown in Fig. 3~c! for the valueT equal to~52!1/2. Inthis case 125 cycles of post-transient data and a valuebequal to 0.05 were used. The mean value of the Lyapuexponent for this case was found to be 0.22. The variancthis estimate was 9.931023. For the cases considered thLyapunov exponent remained positive for Taylor numbgreater than 50.

It may be concluded that both PS- and SPS-type disbances become chaotic. However, SPS-type disturbancecome chaotic at a lower value of the Taylor number. Thefore the presence of subharmonic spanwise wave numbethe velocity disturbance accelerates the onset of chaoticcillations. The spanwise wave number composition offlow field for given value ofT will be considered next. Be-cause of the lower threshold for chaotic oscillations estlished for the subharmonic case, assessing the degree ofpling between the spanwise harmonic components for bthe PS and SPS cases is in order. Such interactions wilconsidered in the forthcoming sections. In addition to ththe structure of the velocity field will also be presented.

B. Disturbances with primary and superharmonicmodes

1. Temporal and spatial features of the disturbancekinetic energy

In this section the time behavior of the total kinetic eergy is presented along with the distribution of the eneamong its Fourier wave number components. The reshighlight the relationship between the temporal and spacharacter of the disturbance velocity field. Modes havingtegral multiples of the primary wave numberkc are exam-ined first.

No. 4, 1996

ldnics.e


FIG. 4. ~a! Energy wave number spectrum for an initial disturbance fiecomposed of the linealy least stable wave number and its superharmo~b! Time variation of the energy for an initial disturbance field with thlinearly least stable wave number and its superharmonics.

ivc

intru

-

o

erases

pidthelorfer.oral.n-

The disturbance energy wave number spectra are gin Fig. 4~a!. These spectra were constructed from a 15 cytime average of each of the modal energiesEm/1(t). Theamplitude of each mode is plotted versus their correspondwave number. The three separate wave number speccurves corresponding to the cases whereT is equal to~40!1/2,~60!1/2, and~80!1/2 are given. The maximal energy contribution corresponds to the wave numberkc for each case. Thisfeature is reflected in the velocity field by the prominencevortex cells with wavelength equal to 2p/kc . In all cases theenergy wave number spectra decay as 1/ka for k greater than

CHAOS, Vol. 6,

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gm

f

5kc . The tails of the spectra are found to decay at a slowrate as the Taylor number is increased. For the three cshown,T equal to~40!1/2, ~60!1/2, and~80!1/2, the exponentais approximately 14, 9.8, and 9.44, respectively. This radecay in the amplitude denotes low spatial complexity inflow. Although the wave number spectra for these Taynumbers are similar, their temporal character is quite difThis reflects a weak coupling between spatial and temptransitions over an intermediate range of Taylor numbers

The temporal oscillations of the disturbance kinetic eergy over five cycles for each of the aforementionedT values

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swerhalf-t oior

e

d ofen-

ur.te-

en


is given in Fig. 4~b!. It should be noted that these waveformare obtained after transient oscillations have decayed. Ascan see in the figure, the temporal behavior varies widwith changing Taylor number. The disturbance energy foTequal to~40!1/2 reaches a quasisteady equilibrium state texhibits time periodic oscillations with a period of one hacycle. Since the disturbance kinetic energy is the resulsquaring disturbance velocity components, the oscillatfrequency of the kinetic energy is twice that of the distubance velocity. AsT is increased to~60!1/2, the disturbancepasses through a single temporal bifurcation. The kinetic

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ergy sustains a higher amplitude and repeats with a perioone cycle. Chaotic oscillations in the disturbance kineticergy are evident in the case whereT is equal to~80!1/2. Forthis case, no long time periodic behavior was found to occThis fact was verified by observing the solution after ingrating the disturbance equations for 400 cycles.

2. The disturbance velocity field

The time evolution of flow observed in a plane taktransverse to the axial direction~i.e., y2z plane! is depicted

FIG. 5. ~a! Velocity field for an initial disturbance containing the linearly least stable wave number mode and its superharmonics forT2560. ~b! Velocity fieldfor an initial disturbance containing the linearly least stable wave number mode and its superharmonics forT2580.

No. 4, 1996


FIG. 5. ~Continued.!

denns

rytruio

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wthonrcl-

otiosesrlb’srg

loryeraran-be-thetur-ion

.

fifymethe

merti-ondon

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by velocity vectors in Fig. 5. The orientation and magnituof the velocity vectors are denoted by the direction alength of the arrows. The velocity field is given in a rectagular cross section extending one-half a wavelength, baon the principal spanwise wave numberk equal tokc . In thespanwise directionz extends@0, p/kc#, and extends to aheight of 30 boundary layer units above the rigid boundaThe height of the cross section is chosen such that the sture of the flow can be clearly shown. The spanwise variatof the velocity field is periodic in space with period 2p/kc .To the left of each frame, the velocity profile of the axicomponent of the basic flow is shown. Maximum responof the vertical component of the disturbance velocity tycally occurs within 4.25 to 20 boundary layer units~n/v!1/2.The thickness of the disturbance layer as predicted by lintheory is approximately 12 boundary layer units.15 In thepresent study the thickness of the disturbance layerfound to increase with increasing Taylor number. This isresult of a centrifugal force arising from the acceleratiover the curved boundary. The momentum generated fofluid away from the boundary. The flow is brought into baance by the pressure, which is manifest in the viscous termvertical component of the disturbance equation. Therefthe boundary layer thickness of the disturbance is a reflecof the location where this balance in achieved. For valueTaylor number nearTc the disturbance layer thickness rissharply from the value predicted by linear theory to neatwice that value. Subsequent increases in the Taylor numyield little variation in the bounding value of disturbancevertical span. However, the disturbance experiences la

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d-ed

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e-

ar

ase

es

ofrenof

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er

fluctuations in its vertical extent over time, as the Taynumber is increased. The growth in the disturbance lathickness to nearly twice the value predicted by linetheory, is owed to the nonlinear interaction between spwise modes. Truncating the number of spanwise modeslow the current number results in an underprediction ofthickness of the disturbance layer. The extent of the disbance is small at high frequencies. For a 10 Hz oscillatthe boundary layer is 0.013 cm. Therefore, 30d for this caseis only 4 mm, and scales as 1/v1/2 with increasing frequency

The velocity vectors and profiles forT equal to~60!1/2

are given in Fig. 5~a!. Each frame is drawn at intervals oone-fourth the excitation period. The flow vectors identthe evolution of two vortices. The legends above each fraindicate the observation time scaled on the time period ofexcitation.

As the basic flow begins to decelerate, the first fralabeled time equal to zero shows the presence of two voces. The first vortex spans the entire frame while the secvortex having a smaller magnitude appears superposedthe first vortex. The effect of the second vortex can be sby the weaker flow that appears slightly below and to theof the frame’s center. The basic flow achieves its minimuacceleration magnitude at 0.25 of a cycle. As the acceleraof the basic flow increases, the vortices collapse into a sinvortex, which intensifies near the boundary during the tiinterval between 0.25 and 0.5 of a cycle. A lateral displament of a single vortex from the center of the frame to tlower left corner of the frame can be seen during the fi

No. 4, 1996

e

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inity

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half-cycle of excitation. The vortex is also skewed with rspect to a vertical line atz equal top/2kc . This spanwisedisplacement and the skewed orientation of the vortex resfrom the contribution of the superharmonic Fourier modFrames corresponding to time 0.5 and 0.75 cycles showthe vortex strength continues to intensify, even as the bflow passes through the maximum acceleration magnitudthe basic flow. Time equal to one marks the end of the ficycle, where the flow returns to the same state as seen atequal to zero. These features repeat periodically in time.trajectory of the vortices in they2z plane is also found torepeat over successive cycles. An examination of the acomponent of vorticity has found a vorticity concentrationthe Stokes layer region for times that coincide with mamum acceleration in the axial basic flow. Deceleration ofaxial basic flow is accompanied by advection of vorticaway from the boundary. A suppression of vorticity advetion away from the boundary is expected for times whereaxial basic flow experiences maximum shear.

For T less than~50!1/2 the vertical component of thedisturbance velocity oscillates at twice the external forcfrequency. The axial component of the disturbance velocon the other hand, oscillates at a period of 2p. Hence, thevertical component of the disturbance velocity is foundoscillate at twice the frequency of the axial component. Tcan be explained by considering the disturbance equat~9! and ~10!. The termsvuy and 2T2h9uu on the left-handside of these equations constitute the linear forcing termschannel energy from the basic state having the particlelocity u to the disturbance velocity components (u,v). Con-sider a temporal variation of the basic flow and the axcomponent of the disturbance velocity with time-harmofrequenciesv andvu , respectively. Under this assumptiothe linear forcing term in Eq.~10! gives rise to the frequenciesuv2vuu anduv1vuu. For values ofT less than~50!1/2 theaxial component of the disturbance velocityu oscillates atthe same frequency asu. Hencevu is equal tov. A harmonicbalance can be established in Eq.~10! between the linearforcing term, 2T2h9uu and linear terms inv, L(v), if v isrequired to oscillate at 2v. This time dependence forv alsosatisfies the harmonic balance between the linear forcterm,vuy and the linear terms inu, L(u), in Eq. ~9!.

The period doubling behavior atT equal to~60!1/2 is theresult of the nonlinear contribution of the Reynolds streterms. These terms result in a temporal bifurcation inamplitude of the vertical component of the disturbancelocity. As a result, the fluid motion in the transverse plandue to the velocity componentsv andw, oscillates with atime period of 2p.

The velocity field for the chaotic case whenT is equal to~80!1/2 is presented in Fig. 5~b!. At time equal to zero theflow field consists of a single vortex. This vortex is concetrated near the lower right corner of the frame. The baflow decelerates from time zero to time 0.25. The vortexadvected vertically, away from the boundary during this timinterval. As the vortex moves away from the rigid boundaa weak recirculation zone forms adjacent to the boundDuring the time interval between time 0.25 to time 0.50, t

CHAOS, Vol. 6,

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basic flow accelerates. The vortex diminishes in strenover this interval. The vertical position of the vortex appeaunchanged. The flow vectors at time equal to 0.50 reveala second vortex has begun to form near the boundary. Ftime equal to 0.50 to time 1.0 the first vortex continuesdiminish in strength, while the second vortex increasesstrength. As the basic flow decelerates from time 1.0 to ti1.25, the second vortex is advected in the vertical directiThe extent of the vertical displacement is less than theplacement seen between time zero and time 0.25. The diences in the velocity field for the time interval zero to 0.and the time interval 1.0 to 1.25 reflect the chaotic naturethe disturbance field atT equal to~80!1/2. The spatial trajec-tories of the vortices do not repeat over successive cycowing to the temporally chaotic nature of the disturbanceaddition, a concentration of vorticity in the oscillatorboundary layer may not coincide with maximum shear inaxial basic flow.

C. Disturbances with primary, sub-, andsuperharmonic modes

1. Temporal and spatial features of the kinetic energy

In this section the energetics of the velocity field fmodes that are integral multiples of the principal wave nuber k equal tokc/4 will be examined. The initial conditionsfor the disturbance are chosen such that all of the Foumodes are excited. In doing so, we ensure that the domispatial features become evident. It is shown that chaoticcillations occur at lower Taylor numbers when subharmoFourier modes are part of the disturbance.

The energy wave number spectra for the casesT equal to~30!1/2, ~47!1/2, and~52!1/2 are given in Fig. 6~a!. For Taylornumbers less then 52, modes with wave numbers thateven integral multiples ofkc/4 dominate the spectrum. Thcutoff wave number for the superharmonics occurs near 2kc .Beyond this value of the wave number, the energy wanumber spectra decay as 1/ka, where the exponenta is ap-proximately equal to 10.9, 8.4, and 8.3 forT equal to~30!1/2,~47!1/2, and~52!1/2, respectively. This decay in the amplituddenotes higher spatial complexity than that of the PS-tdisturbance. The reduction in slope over the PS case givereason to believe that with the additional subharmonic Frier modes, small-scale structures maybe generated at hiTaylor numbers. Maximum amplitude of the disturbance eergy spectra occurs atkc/2 rather than atkc , as in the PScase. At corresponding Taylor numbers, the total kineticergy in the presence of SPS sustain lower amplitudes tthe PS type disturbances.

Time variations in the total disturbance kinetic enerfor each of the aforementioned Taylor numbers are givenFig. 6~b!. For increasing Taylor number, the disturbancenetic energy undergoes periodic, period doubling, and cotic oscillations. Linear analysis predicts that a disturbanwith a spanwise wave number ofkc is the least stable and ia dominant contributor to the flow. This is indeed the cawhen the Taylor number is nearTc

2. For higher Taylor num-bers nonlinear features in the dynamics must be conside

No. 4, 1996

lduper-ldnics.


FIG. 6. ~a! Energy wave number spectrum for an initial disturbance fiecomposed of the linearly least stable wave number and its sub- and sharmonics.~b! Time variation of the energy for an initial disturbance fiewith the linearly least stable wave number and its sub- and superharmo

etv

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The energy wave number spectrum forT equal to~30!1/2

will be considered first. The spectrum consists of discrspanwise components that achieve high amplitudes at einteger multiples ofkc/4. The Fourier mode atkc/4 has smallmagnitude. This mode falls outside the unstable rangescribed by the linear stability curve. At this value of thTaylor number the energy oscillates periodically.

At T equal to~47!1/2 the disturbance has undergonefirst period doubling bifurcation. This is evident from thtime signature given in Fig. 6~b!. The energy spectra of th~30!1/2 and ~47!1/2 cases are similar. Results at intermedia

CHAOS, Vol. 6,

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e-

e

Taylor numbers suggest that the transitions in the tempbehavior of the energy are not strongly connected towave number content of the flow field.

The energy exhibits temporally chaotic oscillations atTequal to~52!1/2. For values ofT greater than~52!1/2 the en-ergy wave number spectrum was found to have a broadistribution in its Fourier amplitudes. This broadening indcates that subharmonic Fourier modes belowkc/4 should beconsidered if one is to resolve the energy exchange at higvalues of T. Nonlinear interaction between the Fourimodes produces a state wherekc/2 is the wave number of the

No. 4, 1996

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dominant Fourier mode. This state of affairs is in agreemwith experimental observations12 at values of the Taylornumber near transition to turbulent fluid motion.

The dynamic behavior of the disturbances for PS aSPS cases for a fixed value ofT differ in temporal and spatial structure. Numerical calculations for the SPS moaboveT equal to~30!1/2, where the initial wave number spectrum contained all components demonstrate that thekc/2Fourier mode ultimately dominates the flow. Thereforeflow shows sensitivity to subharmonic wave number comnents in the wave number spectrum. As a result, SPS disbances are likely to be the preferred route to chaos.chaotic behavior for PS and SPS disturbances is charaized by a low number of modes. Temporally chaotic osciltions have been found with disturbance equations using~PS! or five ~SPS! Fourier modes, while neglecting axiavariations in the disturbance quantities. Low-mode modare adequate for describing basic temporal features. Hever, accurate parametrization of temporal and spatialtures requires the consideration of higher Fourier modes

2. Relationship of subharmonic response to alliedproblems

The shift in the maximal energy response from Fourmodes having primary to those having subharmonic wnumbers has been reported for other problems involvingcillating flows over curved boundaries. These include ctrifugal destabilization of the boundary generated by cylders undergoing transverse and torsional oscillation. Hon12

in experiments using a transversely oscillating cylindfound that the spanwise wavelength increases with incring modulation amplitude. The geometry in Honji’s expement differs from that being considered in the current invtigation. Under the assumptions of large values ofoscillatory Reynolds number and small displacements,case can be brought into agreement with the current probThis can be done by performing a local analysis ofcylinder flow about the point of maximum tangentivelocity. Honji reports measured wave numbers rangbetween ~0.27–0.19! for values of T ranging between~~24!1/2–~44!1/2!. From his description it is not clear whethethe wave number of the disturbance varies abruptly or graally with changes in the Taylor number. A theoretical acount of the linear stability of oscillating cylinder flowsgiven by Hall.11 The critical oscillation amplitudes for unstable motion given by Hall are in agreement with the eperimental observation of Honji. A comparison of the criticwave number is not given since Honji’s paper does notport wave number data near the critical Taylor number. Uing the nondimensional scheme employed here, the critwave number of Hall takes the value of 0.35. Therefore, aTequal to~44!1/2 the measured wavelength is nearly one-hthat given by the linear stability analysis. Similar experimetal observations regarding this behavior has been reporteTatsuno and Bearman.17

Experimental results for the case of torsionally oscilling cylinders have been reported by Seminara and Ha21

CHAOS, Vol. 6,

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and Park, Barenghi, and Donnelly.22 Both studies find thatmodes having subharmonic spanwise wave number incrin amplitude as the Taylor number is increased. Seminet al. find the mode to be 0.2 times the critical wave numbobtained from the linear analysis. Parket al.do not provide awave number measurement. However, they indicatetransitions from Taylor vortex flow to chaotic flow occuabruptly. A theoretical study of subharmonic destabilizatiin the aforementioned case for small departure from the ccal Taylor number is given by Hall.23 The analysis considerthe secondary stability of the Taylor vortex flow to a distubance wave number with a spanwise 0.5kc . This subhar-monic mode was found to be unstable.

The survey of experimental and theoretical work sugests that subharmonic destabilization is a common feain problems involving centrifugal instability and oscillatorflow. In particular, these studies suggest that the submonic mode with wave numberkc/2 plays a key role intransition to chaotic motion.

3. The disturbance velocity field

The velocity in a transverse section with a spanwwidth of 4p/kc and a height of 30 boundary layer units abothe rigid boundary will be considered. It should be noted tthe spanwise frame width used here is four times that ufor the PS cases. The velocity field is periodic in the spwise direction with period 8p/kc .

The velocity field forT equal to~47!1/2 is given in Fig.7~a!. For this value of the Taylor number the flow has udergone a period doubling bifurcation. The basic flow pfiles are presented to the immediate left of each frametime period of 0.75 cycle is highlighted by four time frameThe relative size of the velocity vectors delineate the velocamplitude. The first frame at time equal to zero begins wthe basic flow entering the deceleration phase of thehalf-cycle. The transverse disturbance flow field is compoof two vortices having maximum vertical velocity along thvertical linesz equal to zero and 4p/kc . This structure of theflow field is consistent with the result that thekc/2 is thedominant mode. The vortex gains energy as time increafrom zero to 0.25 of a cycle. As the vortex intensifies,center is displaced in the positive vertical direction. The vtices diminish in strength as the basic flow accelerates frtime 0.25–0.5 cycles. Experimental observations have afound the vortex to remain coherent as it is advected awfrom the rigid boundary for Taylor numbers in the temprally periodic regime.12 For time 0.5–0.75 of a cycle thevortex intensity is increased, though not to the level seentime equal to 0.25. From the time evolution of the velocfield we observe that the gains in the disturbance enecoincide with the deceleration phase of the basic flow cyand the losses coincide with the acceleration phase. Thisservation has also been verified by examining the tempvariation of disturbance energies against the time variationthe basic flow. Similar observations have also been madePS-type interactions by comparing the energies with thesic flow variation for values ofT less than~70!1/2. We also

No. 4, 1996

yeicontioineled

e-

vervemetionimedi-s.on-ewser


note that the vortical flows extend several boundary launits above the boundary. The flow field is time periodwith a period of one cycle. This is reflected in the repetitiof spatial trajectories of vortices over successive excitaperiods. The flow also exhibits a concentration of vorticitythe oscillatory boundary layer at times of maximum acceration in the axial basic flow. A similar feature was notfor T equal to~60!1/2 for PS-type disturbances.

The structure of the velocity field once the flow becomchaotic is given in Fig. 7~b!. The velocity vectors and profiles are presented forT equal to~52!1/2. The frame at time

CHAOS, Vol. 6,

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n

-

s

zero reveals two vortices along the edges of the frame. Othe first quarter-cycle the two vortices intensify as they moupward and the vortex center shifts away from the fraedge. This time interval corresponds to the deceleraphase of the basic flow. As the basic flow accelerates to t0.50, the vortices diminish in strength. Meanwhile two adtional vortices have formed below the original two vorticeAs the original vortices dissipate, their vertical drift is alsdiminished. Over the next half-cycle the new vortices intesify while the older ones dissipate. The center of the nvortices experiences a downward motion as it moves clo

FIG. 7. ~a! Velocity field for an initial disturbance containing the linearly least stable wave number mode and its sub- and superharmonics forT2547. ~b!Velocity field for an initial disturbance containing the linearly least stable wave number and its sub- and superharmonics forT2552.

No. 4, 1996


FIG. 7. ~Continued.!

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to the edge of the frame, between time 0.50 and time 0Over the next half-cycle, time 1.0 to time 1.5, the new vtices move upward. The vertical drift of the vortices is fasbetween time 1.0 and 1.25. The vortices are found to remcoherent as they are advected upward. This feature is cmon among periodic and chaotic disturbances of type PSSPS. The structure of the velocity field as well as the potion of the vortex centers does not repeat after an oscillaperiod, forT equal to~52!1/2. The nonperiodic behavior othe vorticity distribution and vortex spatial trajectory reflethe chaotic nature of the flow.

V. CONCLUDING REMARKS

The transition to chaotic motion in an incompressibfluid, driven by a harmonic time-varying pressure gradiehas been presented. It was shown that both PS and SPSturbances become temporally chaotic, though at differmodulation amplitudes. The presence of subharmonic wnumber components in the disturbance accelerate the trtion to chaos. The dynamic behavior of the disturbancesPS and SPS cases at like values of the Taylor number diThe disturbance field was found be sensitive to the wnumber content of the initial disturbance field. PS-tymodes are the least stable nearTc while the SPS mode become chaotic more rapidly for an increasing Taylor numb

ACKNOWLEDGMENTS

This work was partially supported by U.S. Departmeof Energy Grant No. DE-FG02-91ER14179. The auth

CHAOS, Vol. 6,

2.-rinm-ori-n

tdis-ntvesi-rr.e

r.

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also wish to thank members of the Center for AdvancComputation and Telecommunications at the UniversityMassachusetts—Lowell for their valuable comments.

1C. A. Dragon and J. B. Grotberg, ‘‘Oscillatory flow and mass transpora flexible tube,’’ J. Fluid Mech.231, 135 ~1991!.

2M. K. Sharp, R. D. Kamm, A. H. Shapiro, E. Kimmel, and G. E. Kaniadakis, ‘‘Dispersion in a curved tube during oscillatory flow,’’ J. FluMech.223, 537 ~1991!.

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5B. G. Woods, ‘‘Sonically enhanced heat transfer from a cylinder in crflow and its impact on process power consumption,’’ Int. J. Heat MTransfer35, 2367~1992!.

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8C. Von Kerczek and S. H. Davis, ‘‘Linear stability theory of oscillatorStokes layers,’’ J. Fluid Mech.62, 753 ~1974!.

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10R. Akahavan, R. D. Kamm, and A. H. Shapiro, ‘‘An investigationtransition to turbulence in bounded oscillatory Stokes flows,’’ J. FluMech.225, 423 ~1991!.

11P. Hall, ‘‘On the stability of unsteady boundary layer on a cylinder osclating transversely in a viscous fluid,’’ J. Fluid Mech.146, 347 ~1984!.

12H. Honji, ‘‘Streaked flow around oscillating a circular cylinder,’’ J. FluiMech.107, 509 ~1981!.

13T. Sarpkaya, ‘‘Force on a circular cylinder in viscous oscillatory flowlow Keulegan–Carpenter numbers,’’ J. Fluid Mech.165, 61 ~1986!.

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ng

to

anlow

lo

ir-

id

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ar

ion

bi-


14C. Thompson, ‘‘Acoustic streaming in a waveguide with a slowly varyiheight,’’ J. Acoust. Soc. Am.75, 97 ~1984!.

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No. 4, 1996

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