Hydrodynamic Stability Without Eigenvalues Author(S): Lloyd N

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Hydrodynamic Stability Without Eigenvalues Author(s): Lloyd N. Trefethen, Anne E. Trefethen, Satish C. Reddy, Tobin A. Driscoll Source: Science, New Series, Vol. 261, No. 5121 (Jul. 30, 1993), pp. 578-584 Published by: American Association for the Advancement of Science Stable URL: http://www.jstor.org/stable/2882016 . Accessed: 16/06/2011 09:32

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HydrodynamicStability Without Eigenvalues

LloydN. Trefethen, Anne E. Trefethen, Satish C. Reddy, Tobin A. Driscoll

Fluid flows that are smooth at low speeds become unstable and then turbulent at higher justlybe attributableto step (ii). It is a fact speeds. This phenomenon has traditionallybeen investigated by linearizing the equations of linear algebra that even if all of the of flow and testing for unstable eigenvalues of the linearized problem, but the results of such eigenvaluesof a linear system are distinct investigations agree poorly in many cases with experiments. Nevertheless, linear effects and lie well inside the lower half-plane, play a central role in hydrodynamic instability. A reconciliation of these findings with the inputs to that system may be amplifiedby traditional analysis is presented based on the "pseudospectra" of the linearized problem, arbitrarilylarge factors if the eigenfunctions which imply that small perturbations to the smooth flow may be amplified by factors on the are not orthogonalto one another. A ma- order of 105 by a linear mechanism even though all the eigenmodes decay monotonically. trix or operatorwhose eigenfunctionsare The methods suggested here apply also to other problems in the mathematical sciences orthogonalis said to be "normal"(10), and that involve nonorthogonal eigenfunctions. the linearoperators that arisein the Benard and Taylor-Couetteproblems are in this category. By contrast, Reddy et al. (11) discoveredin 1990 that the operatorsthat Hydrodynamicstability theory is the study two most studied examples of this kind: arisein Poiseuilleand Couette flow arein a of how laminarfluid flows become unstable, (plane)Couette flow, the flowwith a linear sense exponentially far from normal. At the precursorto turbulence.It is well known velocity profile between two infinite flat aboutthe sametime, the startlingdiscovery that turbulenceis an unsolvedproblem, but plates moving parallelto one another, and was madeby Gustavsson(12), Henningson not so well knownthat despitethe effortsof (plane) Poiseuille flow, the flow with a et al. (13), and Butlerand Farrell(14) that generationsof appliedmathematicians, be- parabolicvelocity profilebetween two sta- small perturbationsto these flows may be ginning with Kelvin, Rayleigh, and Rey- tionaryplates (Fig. 1). Other examplesfor amplified by factors of many thousands, nolds, manyof the presumablysimpler phe- which eigenvalueanalysis fails includepipe even when all the eigenvaluesare in the nomena of hydrodynamicstability also re- Poiseuilleflow (in a cylindricalpipe) and, lowerhalf-plane (Fig. 2). The elegantpaper main incompletelyunderstood (1, 2). to a lesser degree, Blasius boundarylayer by Butlerand Farrelldiscusses many details The traditional starting point of an flow (near a flat wall). omitted here and, together with a more investigation of hydrodynamicstability is For Poiseuille flow, eigenvalue analysis recent paper by Reddy and Henningson eigenvalue analysis, which proceeds in predicts a critical Reynolds number R = (15), forms the foundationof the present two stages: (i) linearizeabout the laminar 5772 at which instabilityshould first occur work (16). solution and then (ii) look for unstable (3), but in the laboratory,transition to The shadedregion in Fig. 2 has appeared eigenvaluesof the linearizedproblem. An turbulenceis observedat Reynoldsnumbers in many publications(17) and corresponds "unstableeigenvalue" is an eigenvalue in as low as R - 1000 (4). For Couette flow, to parametersfor which unstable eigen- the complex upperhalf-plane, correspond- eigenvalueanalysis predicts stability for all modesexist. The contoursoutside the shad- ing to an eigenmode of the linearized R (5), but transitionis observedfor Rey- problem that grows exponentially as a noldsnumbers as low as R 350 These (6). 3 function of time t. It is natural to expect anomaliesof "subcriticaltransition to tur- that a flowwill behave unstablyif and only bulence" have been recognizedfor many if there exists such a growingeigenmode, years,and the explanationhas traditionally and over the yearsmuch has been learned been attributedto step (i) above. If linear- about which flows possess such modes, a ization has failed, the reasoninggoes, one E ' distinction that dependson the geometry, must look more closely at the nonlinear the Reynoldsnumber, and sometimesoth- termsor perhapslinearize about a solution er parameters. other than the laminarone [the theory of 0 For some flows, notably those with in- "secondaryinstability" (7-9)]. stabilitiesdriven by thermalor centrifugal Recentlyit has emerged,however, that forces,the predictionsof eigenvalueanaly- the failureof eigenvalueanalysis may more 0 104 2x1 sis matchlaboratory experiments. Examples Reynolds number are Rayleigh-Benardconvection (a station- Fig. 2. Maximalresonant amplificationof 3D ary fluid heated from below) and Taylor- perturbationsin linearizedPoiseuille flow as a Couette flow (between a stationaryouter functionof Reynolds number R and xz wave- and a rotating inner cylinder). For other numbermagnitude k = N/'T2j2 (Eq.6). Inthe flows,notably those drivenby shearforces, shaded region, with leftmost point R = 5772, the predictionsof eigenvaluesanalysis fail to unstable eigenmodes exist and unbounded match most experiments.We considerthe amplificationis possible. The contoursoutside that region, from outer to inner, correspondto L. N. Trefethenis in the Departmentof Computer Fig. 1. Velocity profiles for two laminarflows finiteamplification factors of 103, 104 (dashed), Science, Cornell University, Ithaca, NY 14853 (independent of x and z). The geometry is an 105, 2 x 105, ..., 1.3 x 106. For example, ([email protected]).A. E. Trefethenis in the Cornell infinite3D slab of viscousincompressible fluid amplificationby a factorof 1000 is possible for TheoryCenter, CornellUniversity, Ithaca, NY 14853. bounded walls. The laminarsolu- all R - 549. In the transitionto S. C. Reddyis in the CourantInstitute of Mathematical by parallel laboratory, Sciences, New YorkUniversity, New York,NY 10012. tions satisfy the Navier-Stokesequations for all turbulenceis observed at R = 1000. The anal- T. A. Driscollis inthe Centerfor Applied Mathematics, Reynolds numbers, but for higher R, the flows ogous picturefor Couetteflow looks qualitative- CornellUniversity, Ithaca, NY 14853. are unstable and rapidlybecome turbulent. ly similarexcept thatthere is no shaded region.

578 SCIENCE * VOL. 261 * 30 JULY 1993 ed regionquantify the nonmodalamplifica- of a streamwisevortex may move fluidfrom ble amplificationover all real frequencies tion that may occur in these flows. The a region of higher to lower x velocity, or (Fig. 2) is possibilityof amplificationof perturbations vice versa, where it will appearas a large, sup T- ?)11 (6) of viscousflows by nonmodallinear mech- local perturbationin the x velocity (22). anisms has been recognizedfor a century Becausethese featuresconstitute 3D pertur- (18), but until the recent developments,it bations of the flow field, however, their An eigenvalueof ?Tis a numberX E C was not known that the magnitudesin- prevalencehas been difficultto reconcile such that _Tu= wu for some corresponding volved were huge. with the predictionsof eigenvalueanalysis. eigenfunctionu. Equivalently,it is a num- An essential feature of this nonmodal Nonmodal analysisoffers a linear explana- ber X with the propertythat perturbations amplificationis that it applies to three- tion of why these structuresare so common, with frequency w can be amplified un- dimensional(3D) perturbationsof the lam- for although streamwisestreaks are not boundedly:11I(xO - T) -'II = x' Generaliz- inar flow field (19). In much of the litera- eigenmodesof the linearizedflow problem, ing this definition, for any E > 0, an ture of hydrodynamicstability, attention they are pseudomodes. "E-pseudoeigenvalue"of ? is a number o has been restrictedto 2D (xy) perturba- To explainthis term, we mustdefine the such that 11(3 - >)-1jj 2 E-1, and a tions; in particular,the well-known Orr- evolutionoperator that is the mathematical corresponding"E-pseudomode" is any func- Sommerfeld equation is an eigenvalue basis of this article. Let uo = uO(x,y,z) tion u with IVCu- su11c EIIUII.If e is small, equationfor 2D perturbations.A principal denote the vectorvelocity fieldcorrespond- then an E-pseudomodeu may be excited to justificationfor this restriction has been ing to Poiseuilleor Couette laminarflow. a substantialamplitude by a small input Squire'stheorem, which assertsthat if a Let uo + U(t) = uO(x,y,z)+ U(x,y,z,t) be perturbation,possibly including noise in an flow has an unstable 3D eigenmode for the velocity field of a slightly perturbed experimentalapparatus. At R = 5000, for some R, then it has an unstable2D eigen- flow, that is, a nearby solution to the example, a streamwise streak is an mode for some lower value of R (20). The Navier-Stokesequations (uppercaseletters e-pseudomodeof the linearizedPoiseuille new resultsindicate that this emphasison distinguishquantities that are functionsof flow problemfor E 1.2 x 10-' and thus 2D perturbations has been misplaced. t). If we take U to be infinitesimal,then it can be excited by a streamwisevortex five When only 2D perturbationsare consid- satisfiesan equation ordersof magnitudeweaker in amplitude. ered, some amplificationcan still occur,but dU The set of E-pseudoeigenvaluesof an it is far weaker. (1) operatoris the "E-pseudospectrum"(23) The growing attention to 3D linear, dt t=-i9LU(t) nonmodalphenomena represents a signifi- where_T is a linearoperator that we call the A>(S) = {@E C: II(W4- ?)-1Il > E6-} cant change in the traditionalconception linearizedNavier-Stokes evolution opera- (7) of problemsof hydrodynamicstability. tor. As a measureof the size of solutionsto The pseudospectra{AE(?)} form a nested this equation, we define family of sets in the complex plane, with Streamwise Vortices and Streaks AO(Y)equal to the spectrumA(Y). If ? is IIU(t)II= (JIu(x,y,z,t)12dx dy dz) (2) normal, AE(2) is the set of all points at The flow featuresassociated with this am- distance?E fromA(2), but in the nonnor- plificationprocess have a distinctiveform: which we call the energynorm becauseits mal case, it may be much larger. A perturbationto the velocity field in the squareis sometimesinterpreted as an energy. formof a "streamwisevortex" evolves into a Suppose the linearizedfluid system is Spectra and Pseudospectra higher amplitude"streamwise streak" (Fig. drivenby a signalof the formV(t) = e-fttv 3). A streamwisevortex is an elongated for some frequencyX E C (C denotes the We numericallycalculated spectra and pseu- region of vorticity approximatelyaligned set of complexnumbers) and function v = dospectrafor Couette flow with R = 350 and with the x axis, and a streamwisestreak is v(x,y,z) 3,500 andfor Poiseuille flow with R = 1,000 an elongatedregion of high or low velocity dU and 10,000 (24). These Reynoldsnumbers mean roughlyspan the range in each case from (relative to the flow) approximately a + e for < t< x in the x direction.Streamwise vortices and = -iLcU(t) iwv -a occasionalturbulence (in someexperiments) streaksare persistentfeatures in laboratory (3) to unavoidableturbulence (even in experi- experimentsinvolving all kinds of internal It is easilyverified that the responsewill be ments under the most carefullycontrolled and shearflows conditions).To performour calculations, we boundarylayer (21). Phys- = = ically, they are not hard to explain: In a U(t) ie-u whereu ( --T)-Yv (4) firstFourier transform in x and z, reducing shearflow, a smallperturbation in the form (O denotes the identity operator). Thus, the calculationto one space dimension(y) the operator(ox9~ - - ) 1, known as the and two real parametersa and ti (wave resolventof X, transforms"inputs" v to the numbersin x and z). The determinationof linearizedfluid flow at frequencyX into the pseudospectrathen requiresa minimiza- corresponding"outputs" u. The degree of tion in the a(, plane (25). amplificationthat may occurin the process In the case of Couette flow, the spec- is equal to the operatornorm trumof ? is a continuousregion contained in the lower half-plane (Fig. 4) (for each a-,3 pair, the spectrum is discrete; the Fig. 3. Schematic illustrationof a structurethat 11( - ?)111 sup l-l (5) V?O- llvii union over a and 13is a continuum).This is appears in many shear flows. A low-amplitude true for all R, correspondingto the uncon- vortex or counterrotatingpair of vortices ap- (supdenotes supremumor maximum).Be- ditional stabilityof Couette flow according proximatelyaligned withthe flow entrainsfluid from regions of high to low x velocity and vice causean arbitrarytime-dependent perturba- to eigenvalue analysis (5). For Poiseuille versa. The entrainedfluid appears as a streak tion of the laminarflow can be reducedto flow, the spectrumlies in the lower half- of locally high or low streamwise velocity (not an integral over real frequencies by Fourier plane for small R, but as R increases, two shown). This process is linearbut unrelatedto analysis, values of X on the real axis llRare bumps appear that cross into the upper eigenmodes. of particular interest. The maximum possi- half-plane at R = 5772 (Fig. 5). These

SCIENCE * VOL. 261 * 30 JULY 1993 579 bumpsrepresent the modethat has received tures, but approximatelystreamwise fea- dU -t) = -iLTU(t), t 2 0, U(O) = v (8) most of the attention in the literature, tures with nonzero a and Rew are also dt known as a Tollmien-Schlichtingor TS stronglyamplified (28). wave. Judgingby the spectra alone, one The solution can be written U(t) = wouldconclude that Couette and Poiseuille Physical Implications exp(-it_T)v, where exp(-it-T) is the oper- flows are fundamentallydifferent because atorexponential. The factorby which such one has unstable eigenmodes for certain Pseudoresonrance.One interpretation of solutionscan growin time t is values of R and the other does not. nonnormalitywas describedabove in con- The pseudospectratell a differentstory. nection with Eq. 5. If a systemis govemed ||exp(-it.T)jI= sup (9) Forboth Couette and Poiseuilleflow, they by a linear operator?T that is normal or v*O llvll protrudesignificantly into the upperhalf- close to normal-familiarexamples include plane for all reasonablylarge values of R, musical instruments,vibrating structures, If ? were a normaloperator with spectrum implyingthat the correspondingevolution and molecules as described by quantum in the lower half-plane, we would have processesmust feature prominent nonmodal mechanics-then 11(o8l- ?T)- II is largeif I1exp(-it_T)jjc 1 for all t 2 0, but in effects (26). Qualitatively,the pseudospec- and only if w is close to an eigenvalue,and actuality,the growthis lIexp(-it-T)II= 0(R) tra in the Couette and Poiseuillecases look thus the frequenciesat which the system at timest = 0(R) (Fig.7). This dependence more alike than different,suggesting that resonatesare determinedby the spectrum. on R can be explained as follows: The these flows should behave similarly,as is Fora nonnormalsystem, however, 11( - streamwisevortex-streak interaction is invis- observedin experiments(27). _- `I may be large, and thus resonanceor cid and operateson a time scaleO(R) before One cannot see in Figs. 4 and 5 what "pseudoresonance"may occur even when w beingshut off by the effectsof viscosity(13). values of a and 3 are associated with is far from the spectrum.A plot of pseu- Such behavioris physicallystraightforward, variouspoints in the spectrumand pseu- dospectracan be interpretedas a plot of appearingcomplicated only when interpret- dospectra.In fact, the upperboundaries of contoursof equalresonance magnitude; the ed in the basisof eigenmodes. the spectracorrespond to modeswith ,B = real axis is of particularinterest because it The 0(R2) resonancesof Figs. 2 and 6 0, in keepingwith Squire'stheorem, where- correspondsto forcingat real frequencies. are approximatelyequal to the integrals as the upper parts of the pseudospectra Both Couetteand Poiseuilleflows exhib- underthe curvesin Fig. 7. Mathematically, correspondto pseudomodeswith I ? 0, it strong 3D pseudoresonancefor frequen- this can be seen fromthe formula indicatingthat the effectsof nonnormality cies w = 0 (Fig. 6). The magnitude is are predominantly3D. The highest points 0(R2), not 0(R) as one would have for a -1 = if'exp(-iLT)dt (10) of the pseudospectracorrespond to a = normal operatorwith spectraat the same Rew = 0, hence purelystreamwise struc- distanceO(R-1) below the real axis. This which implies and other asymptoticresults of our calculations are summarizedin Table 1. 11E-T-'l5 jjlexp(-it_T)jjdt (11) Transientgrowth. A second interpreta- 0. A Couette R= 350 tion of nonnormalityinvolves the transient In practice,this inequalityis typicallywith- growth of flow perturbationsthat may in a factorof 2 of equality.We can interpret evolve fromcertain initial conditions (11- the 0(R2) figurephysically by noting that 16). Considerthe initial-valueproblem the resonantamplification is a resultof the combinationof two effects: one (normal) factor 0(R) representingthe time scale 0.1 over which input energy can accumulate A Poimeufeb before it eventually decays, and another 0.1 R= 1000 )L CW (nonnormal)factor 0(R) representingtran- sient growth.

R =3500

0 - U 110 a B ~~~~~~~~~Poiseduie E 1041 R=10 ~1o2 R=350 -1 0[1 Realpart of frequency R=00 10 0 4. and in the Fi. 5. Sam as Fi.4btfrPie _l lw o 0.2 0.4 Fig. Spectrum E-pseudospectra F? 5772, tw bup in th spctu exen Frequency complex w-plane of the linearized Navier- Stokes evolutionoperator for Couetteflow. The Fig. 6. Pseudoresonance curves for Couette int th upe half-plane. spectrum is the shaded region, and the solid flow at various Reynolds numbers. Each curve curves,reaad frommainRyaesouter topare inner, are scle the boundariesdiferetly shows 11(uO- ?)-111, the maximum possible of the E-pseudospectrafor E = 10-2, 10-2.5, amplification of perturbations to the laminar 10-3 , and1_ The liesin the lower flow field as a function of o. An spectrum Real partof frequency frequency half-plne,nibu t then cpseudninospetraextendr szig- analogous plot for Poiseuilleflow looks qualitatively the same except that for F?> 5772, an additionalresonance of magnitude X appears for X 0.2.

580 SCIENCE * VOL. 261 * 30 JULY 1993 ARTICLE

A rigorous connection can be made where the asteriskdenotes conjugatetrans- spectrumof ? to perturbations,however, between transientgrowth factors and the pose and v and u are the functions that raises the question of whether this effect geometry of the pseudospectra.One can achieve the supremumin Eq. 5 for X = 0 mightbe better understoodas a symptomof prove [in matrix terminology,u and v are the the nonnormalityof the unperturbedprob- principal singularvectors in the singular lem than as a descriptionof how transition sup ||exp(-it.)lj >? sup e-1Cr(.T) (12) value decomposition(SVD) of ?T(29); the to turbulence actually takes place under t20 E>0 phrase "conjugatetranspose" is also from naturalconditions (31). = where cr,(-) SupP,,E(A) Imo (15). In matrix algebral. This operatortransforms words, if the pseudospectraprotrude far into streamwisestreaks into streamwisevortices, Favored Structures the upper half-plane, then substantial tran- closingthe loop so that transientgrowth for sient growth must be possible. For the finite t can feed back to become modal Unlike eigenmodes,the pseudomodesof a Poiseuille and Couette flows considered growth for all t. Of course, there is no linear operator are not uniquely deter- here, this estimate falls short of the actual reasonto expect such a perfectlycontrived mined; the precise structureexcited in a growth by a factor 1.4. perturbationto arise under naturalcondi- highly nonnormallinear system will depend Destabilizingperturbations. A third inter- tions; yet our calculationsshow that even on the details of the excitation. But it is pretation of nonnormality is based on an random perturbationsof ? often have noteworthythat the three physicalmecha- altemative, equivalent definition of the much the same effect (30). Such perturba- nismsconsidered above lead to similarpre- pseudospectra of an operator ? tions mightbe introducedin the laboratory, dictions of what flow structuresshould be forexample, by imperfectionsin the bound- prominentin shear flows at high Reynolds A (-) = closure{w E C: X E A(_T + t) arywalls. numbers.Consider the initial-value prob- for some Z with 11Zf1' E} (13) A different prospective application of lem of Eq. 8 with initial flow field U(O) = In words, the E-pseudospectrumof ?Tis the the idea of destabilizingoperator perturba- v and solutionexp(-it_S)v. The amplitude union of the spectra of all perturbed opera- tions may be to the theory of "secondary history of this solution is given by c torsX + with 11j11< E, togetherwith any instability"as an explanationof subcritical exp(-it.T)vjj, and the operator norm limit points of this set (a technicality of transitionto turbulence(7-9). This theory Iexp(-it.T)IIis the upper envelope of all little importance). It is interesting to recon- is founded on the observationthat when such curves correspondingto all initial sider Figs. 4 and 5 in the light of this certainlaminar shear flows are perturbedby functionsv. Figure9 indicateshow closely alternative characterization. The pseu- certainphysically motivated waves of large this envelope is approachedby three phys- dospectral contours in these figures imply amplitude,the resultingproblem is eigen- icallyinteresting choices of v. Eachof these that although the flows in question are value unstable.The greatsensitivity of the functions can be characterizedmathe- eigenvalue stable (for R < 5772 in the maticallyas the function on which a cer- Poiseuille case), exceedingly small pertur- tain linear operatorattains its norm; the bations to the evolution operator, of order 10-2~~~~~~ operators are ?- 1, exp(-itop0T), and O(R-2), suffice to make them eigenvalue exp(-iO+LT),respectively, with the O+no- unstable (Fig. 8). The norm of the minimal tation indicatinga limit as t -> 0. The first destabilizing perturbation is Emin = function, v1, is the one that excites a '10-4 \ (sup.E,-Rj( - _ ))1II) ', the inverse of maximalresonant response, or equivalent- the maximal resonance of Eq. 6. For exam- ly, induces a minimal destabilizingpertur- ple, although Poiseuille flow with R = 5000 bation. The second,v2, is the Butler-Farrell 10-6. is eigenvalue stable, there exists a perturba- "optimal" that achieves maximal total tion W of norm 1.2 x 10' that renders it growthat some time t = topt.The third,V3, unstable. 102 103 104 is the perturbationwith maximal Reynolds number growth This raises the question of the physical rate at t = 0, which has been studiedby Fig. 8. Minimalnorm Emin = O(R-2) of an meaning of operator perturbations and their Lumleyand others (14, 32). We computev, relevance to instabilities observed in the operatorperturbation Z that can destabilize an numericallywith the aid of the SVD ap- shear flow laboratory. The minimal destabilizing per- eigenvalue-stable at Reynoldsnum- plied to discreteapproximations of the as- ber R. The dependence is so nearly quadratic turbation Z with = Emin is 11%11 easily char- that the curves appear straightto plottingac- sociatedoperators (33). acterized: It is the rank-1 operator EminVU* curacy. The dots mark the approximateRey- Although the physicalideas behind vp, nolds numbersat which Couette and Poiseuille v2, and v3 are different, v1 and v2 achieve flows are typicallyobserved to undergo transi- comparable and near-maximal transient 15CU tionto turbulence(350 and 1000, respectively). growth (and also comparableresonance, R 4000 R=Jo

100 Table 1. Leading-order behavior of various quantities as R -X 0 (numbers accurate to 1% for R > 100) and the associated wave numbers a and P. The results for Poiseuille flow pertain to the highly R 2000 nonnormal part of the problem, ignoring the R > 5772 mode (TS wave).

=50 t Couette Poiseuille

Value a ( Value a

0 250 500 Distance of spectrum from real axis (R/2.47)-1 0 0 (R12.47)-1 0 0 nlme Maximum resonance sup R,Il(wJ - Y)-1II (R/8.12)2 0 1.18 (RI/17.4)2 0 1.62 Transient growth > fR129.1 35.7/R 1.60 R/71.5 0 2.04 Fig. 7. Nonmodal transient growth of flow per- supt 0 Ijexp(-itT)fI Optimal time for above turbationsfor Couette flow. For any finite F?, top R/8.52 35.7/R 1.60 R/13.2 0 2.04 Lower bound based on pseudospectra R/42.6 0 1.62 RI/103. 0 2.04 viscous effects shut off the growth on a time Transient growth (a = 0) R/29.3 0 1.66 R/71.5 0 2.04 scale 0(R).

SCIENCE * VOL. 261 * 30 JULY 1993 581 not shown). These functionsare also simi- Table 2. "Transitionto turbulence"of a 2 x 2 lar in structure,both having approximately 12. nonlinearmatrix model problem (Eq. 14). Al- the formof streamwisevortices that evolve though the growth factor M of the linearized into streamwisestreaks. problem is O(R), nonlinearmixing effectively ~8 : cubes this figure so that the thresholdampli- tude E is O(R?-3). Nonlinear Bootstrapping and Transitionto Turbulence 4V3 R M E Ratio Kelvinwrote in 1887 (34): 12.5 3.18 3.41 x 10-3 0 40 80 120 8.1 It seems probable, almost certain indeed, that nlme 25 6.28 4.22 x 10-4 ... the steady motion is stable for any viscosity, 8.0 however small; and that the practical unsteadiness Fig. 9. Initialgrowth and eventual decay of 50 12.5 5.27 x 10-5 pointed out by Stokes forty-fouryears ago, and so Ijexp(-itY)vIlresulting from three initialpertur- 8.0 admirablyinvestigated experimentally five or six bations v for Couette flow, R = 350. The 100 25.0 6.58 x 10-6 years ago by Osborne Reynolds, is to be explained dashed curve is the operatornorm as in Fig. 7. by limits of stability becoming narrowerand nar- rowerthe smaller is the viscosity. growing solution components has the po- This view of instabilityis still standard,but tential for inducing a threshold amplitude E to this day, it has never been confirmedin = O(R-3) or O(R-4) with respect to initial detail. In this final section, we speculate =102 or forcing data, respectively; a random per- aboutwhat the eventual confirmationmay turbation, for example, will often suffice. look like-about how nonlinearand linear =10-4 Higher order nonlinearities lead to similar mechanismsinteract to bring about transi- effects, though the exponents may be low- tion to turbulence. ered, for example, to E = O(R-2) and Consider the 2 x 2 nonlinear model O(R-3) for a cubic nonlinearity. problem The Navier-Stokes equations are more complicated than our 2 x 2 model. One du (-R' 1\ 0 100 200 difference is that instead of 2-vectors, they d = Au + |IuIIBu,A = ( - I lime Fig. 10. IIu(t)IIfor solutionsto the nonlinear2 x act on functions with infinitely many de- 2 model problem of Eq. 14 with initialampli- grees of freedom, most of which do not B=(1 ? ) (14) tudes IIu(0)1I= 10-7, 10-6, 10-5, 10-4, 4 x experience nonnormal linear growth. 10-4, 5 x 10-4, 10-3, and 10-2. The threshold There will always be some energy in the where R is a large parameter. The linear amplitudeis Ilu(0)II= 4.22 x 10-4. growing pseudomodes, however, and in a term, involving the nonnormal matrix A, pipe or channel of substantial extent, ran- amplifies energy transiently. The nonlinear dom fluctuations can be expected to raise term, involving the skew-symmetric matrix grown to order RE by the linear growth the energy levels in such components local- B, rotates energy in the ulu2 plane but does mechanism but moved into a direction that ly well above the statistical average (37, not create or destroy energy, -for it acts no longer excites growth (the correspond- 38). Another difference is that the nonlin- orthogonally to the motion (35). Thus, we ing left singular vector). Meanwhile, how- ear interactions in the Navier-Stokes equa- have linear amplification coupled with en- ever, the nonlinear term has had the effect tions act across different wave numbers ox ergy-neutral nonlinear mixing, a situation of transferring some of this energy back to and 13(12, 13), making the quadratic non- that holds also for the equations of fluid the original direction, with amplitude linearity of Eq. 14 perhaps unrealistically mechanics (36). R(Re)2 = R3E2 because the nonlinearity is strong. Notwithstanding these qualifica- We considerthe norms IIu(0IIfor solu- quadratic and the time scale is 0(R). If R3F2 tions, we conjecture that transition to tur- tions to Eq. 14 with R = 25 starting from is of order less than e, the process is not bulence of eigenvalue-stable shear flows eight different initial vectors u(O) = (0, self-sustaining and the energy decays. On proceeds analogously to our model in that const)T (Fig. 10), where the T denotes the other hand, if R3E2 is of order greater the destabilizing mechanism is essentially transpose.For IIu(0)1I< 10-4, the curvesare than E, there is more energy than at the linear in the senses described above and the approximately translates of one another on start and sustained growth may occur. amplitude threshold for transition is O(R-) this log scale, indicating that the evolution Thus, the threshold amplitude is E = for some -y < -1 (39). is effectivelylinear. At IIu(O)II= 4 x 10-4, O(R-3). A similar experiment shows that if the nonlinearity has a pronounced effect. the same nonlinear equation is driven by a Conclusion For iIu(O)II= 5 x 10-4 and higher ampli- forcing oscillation e"Zv instead of an initial tudes, the curves do not decay but blow up vector u(O), the threshold amplitude be- We have discussed three linear approaches to a critical point of amplitude = 1. comes e = O(R-4). to the phenomenon of instability of shear These calculations reveal a remarkable It may appear that these results indicate flows: (i) pseudoresonance, (ii) transient phenomenon: The amplitude growth is far the great power and importance of nonlin- growth of flow perturbations, and (iii) de- greater than that of the linearized problem ear effects. Yet in two senses, these energy stabilizing operator perturbations. These du/dt = Au. We find that e is of order R-3, growth scenarios are essentially linear. ideas are by no means independent. Math- not R` (Table 2). This "bootstrapping" First, as mentioned above, the nonlinear ematically, all are related to the pseu- effect can be explained as follows. Suppose term in Eq. 14 does not add energy but dospectra of the operator ?, and physically, the solution at t = 0 consists of a vector of merely redistributes it. Second, the appear- all depend on the same mechanisms of amplitude e in a direction that excites ance of the bootstrapping phenomenon extraction of energy from the mean flow by growth of the linear problem du/dt = Au does not depend on the precise nature of structures such as streamwise vortices. One (the principal right singular vector of A).- the nonlinearity. Any quadratic nonlinear should not expect that one of these ideas At a later time, of order R, the solution has term that transfers energy from decaying to will prove to be right and the others wrong.

582 SCIENCE * VOL. 261 * 30 JULY 1993 ARTICLE

More likely, each may prove relevant to a 10. T. Kato, Perturbation Theory for Linear Operators most of our figures requires an optimization with particularclass of (Springer-Verlag, New York, 1976); A. Pazy, Semi- respect to a or ,Bor both. The largest calculations experiments, for it is groups of Linear Operators naturalto and Applications to are required for the pseudospectra of Figs. 4 and speculatethat transitionto tur- Partial Differential Equations (Springer-Verlag, 5, which are determined by evaluating 11(wO- bulencemay be triggeredin distinctcircum- New York, 1983). .T)-11jon a grid in the w-plane of typical size 64 by stancesby distinct causes, such as, respec- 11. S. C. Reddy, P. J. Schmid, D. S. Henningson, 64 and plotting contours of the resulting data SIAM J. Appl. Math. tively, (i) laboratoryvibrations, (ii) initial 53, 15 (1993). array. For each value of w on the grid, 11(wl - 12. L. H. Gustavsson, J. Fluid Mech. 224, 241 (1991). ?)-111 is computed by numerical minimization of 13. - or inlet disturbances,or (iii) deviationsof D. S. Henningson, in Advances in Turbulence, A. 11( ?.,)0-11jj with respect to a and 3, and the pipe or channel geometry from the V. Johansson and P. H. Alfredsson, Eds. (Spring- each evaluation of this function entails a matrix Poiseuilleor Couette ideal. In er-Verlag, New York, 1991), pp. 279-284; D. S. singular value decomposition (SVD). the coming Henningson, A. Lundbladh, A. V. Johansson, J. 26. The pseudospectral contours in Fig. 5 bend decade, as numerical simulations of the Fluid Mech. 250, 169 (1993). around the TS bumps but come very close to nonlinearNavier-Stokes equations become 14. K. M. Butler and B. F. Farrell, Phys. Fluids A 4, them, revealing that the TS wave is oriented at a routine, much progresswill be madein the 1637 (1992). sizable angle to the remainder of the system. The 15. S. C. Reddy and D. S. Henningson, J. Fluid Mech. angle is e- 16.40 as measured in the energy elucidation of these details and a fuller 252, 209 (1993). inner product, corresponding to an eigenvalue picture will emerge of the interaction of 16. Additional related developments, both theoretical condition number K = 1/sin - 3.52. Thus the linear and nonlinear effects in fluid me- and experimental, are described in K. S. Breuer small bump in the e = 10-35 pseudospectral and J. H. Haritonidis, J. Fluid Mech. 220, 569 contour of Fig. 5B lies at a distance 3.52 x 10-3O5 chanics. (1990); Y. Bun and W. 0. Criminale, ibid., in press; from the spectrum. Besides hydrodynamicstability, there B. G. B. Klingmann, ibid. 240,167 (1992) 27. One difference between the Couette and Poi- are other fields in which nonorthogonal 17. This region has appeared in many places, includ- seuille problems is that the spectra and pseu- eigenfunctionsarise and ing figures 4.11, 4.15, and 5.5 of (1). These and dospectra are broader in the latter case. This is a eigenvaluesmay be other figures in the literature have the x wave consequence of the fact that the mean flow ve- misleading. Examplesin fluid mechanics number a as the ordinate, not the xz wave- locity is zero for Couette flow but nonzero for include the instabilityof magneticplasmas number magnitude k = (a2 + p2)1/2, and thus Poiseuille flow, so that a plane wave of wavenum- (40) and the formationof cyclones (41). correspond to 2D perturbations only. This chang- ber a traveling with the mean flow will have ce 0 es the upper boundary in such a way that the for Couette flow but X = O(a) for Poiseuille flow. Examplesin numericalanalysis include the width of the shaded region shrinks to zero as R Figures 4 and 5 would look different if the Navier- stiffnessand numerical instability of discret- Stokes problem were formulated in a moving izationsof differentialequations (42) and 18. W. M'F. Orr, Proc. R. Ir. Acad. Sect. A 27, 9 reference frame, though the behavior- along the (1907); ibid., p. 69; K. M. Case, J. Fluid Mech. 10, imaginary axis would be essentially unchanged. the convergenceof iterativealgorithms for 420 (1960) and Phys. Fluids 3, 143 (1960); G. 28. For details on structures with a ? 0 see (14) and nonsymmetricmatrix problems(43). The Rosen, ibid. 14, 2767 (1971); T. Ellingsen and E. B. F. Farrell and P. J. loannou, Phys. Fluids A 5, recurringtheme in these and other applica- Palm, ibid. 18, 487 (1975); M. T. Landahl, J. Fluid 1390 (1993). Mech. 98, 243 (1980); D. J. Benney and L. H. 29. G. H. Golub and C. F. Van Loan, Matrix Compu- tions is that althoughthe long-timebehav- Gustavsson, Stud. Appl. Math. 64,185 (1981); L. tations (Johns Hopkins Univ. Press, Baltimore, ior of an evolvingsystem may be controlled S. Hultgren and L. H. Gustavsson, Phys. Fluids MD, 1989); R. A. Horn and C. R. Johnson, Topics by nonlinearities,some importantphenom- 24,1000 (1981); A. D. D. Craik and W. 0. Crimi- in MatrixAnalysis (Cambridge Univ. Press, Cam- ena are of a short-time nale, Proc. R. Soc. London Ser. A 406, 13 (1986); bridge, 1991). nature and are B. F. Farrell, Phys. Fluids 31, 2093 (1988). 30. At R = 5000, for example, approximately 90% of essentially linear (44). If the linearized 19. By 3D, we mean z-dependent, as is conventional all random perturbations ' to _Twith jl'gll= 10-3 problem is far from normal, eigenvalues in this field. Thus, a perturbation that depends on render a Poiseuille flow eigenvalue unstable, may be preciselythe wrongtool for analyz- y and z but not x is said to be 3D, not 2D. where by a random perturbation we mean an 20. H. B. Squire, Proc. R. Soc. London Ser. A 142, 621 elementwise random perturbation of the 20 x 20 ing it, for eigenvaluesdetermine the long- (1933). matrix that represents the projection of _Tonto the time behaviorof a nonnormallinear pro- 21. R. Breidenthal, J. Fluid Mech. 109, 1 (1981); L. P. space spanned by its 20 dominant eigenmodes cess, not the transient. Bernal and A. Roshko, ibid. 170, 499 (1986); J. C. (with fixed ct = 0 and , = 1.62). Lasheras and H. Choi, ibid. 189, 53 (1988); M. 31. The validity of the secondary instability model of Asai and M. Nishioka, ibid. 208, 1 (1989); S. K. natural transition to turbulence has been ques- REFERENCESAND NOTES Robinson, Tech. Memo. NASA TM-103859 (Na- tioned for other reasons; alternatives are some- tional Aeronautics and Space Administration, times known as models of "bypass transition" [M. 1. P. G. Drazin and W. H. Reid, Hydrodynamic Washington, DC, 1991). V. Morkovin, in Viscous Drag Reduction, C. S. Stability (Cambridge Univ. Press, Cambridge, 22. This process is sometimes called "lift-up";see M. Wells, Ed. (Plenum, New York, 1969)], pp. 1-31. 1981). T. Landahl, SIAM J. Appl. Math. 28, 735 (1975). 32. J. Serrin, Arch. Ration. Mech. Anal. 3,1 (1959); F. 2. This article is adapted from L. N. Trefethen, A. E. 23. The term pseudospectrum was introduced by L. H. Busse, Z Angew. Math. Phys. 20, 1 (1969); J. Trefethen, S. C. Reddy, Tech. Rep. TR 92-1291 N. Trefethen [in Numerical Analysis 7991, D. F. L. Lumley, in Developments in Mechanics, L. H. (Department of Computer Science, Cornell Uni- Griffiths and G. A. Watson, Eds. (Longman, Lon- N. Lee and A. H. Szewczyk, Eds. (Notre Dame versity, Ithaca, NY, 1992) and , T. A. don, 1992), pp. 234-266]. Essentially the same Press, South Bend, IN, 1971), vol. 6, pp. 63-88; D. Driscoll, Tech. Rep. 92TR1 15 (Cornell Theory idea has been applied over the years by various D. Joseph, Stability of Fluid Motions I (Springer- Center, Cornell University, Ithaca, NY, 1992). authors, including J. M. Varah, J. W. Demmel, and Verlag, New York, 1975). Readers interested in additional details are urged especially S. K. Godunov and his colleagues. 33. To compute v3, we make use of its characteriza- to contact the first author for copies of these Possibly its earliest application is by H. J. Landau tion as the principal eigenfunction of (.T - 2*)12i, reports. [J. Opt. Soc. Am. 66, 525 (1976)]. with growth rate at t = 0 equal to the correspond- 3. S. A. Orszag, J. Fluid Mech. 50, 689 (1971). The 24. Spectra of for particular values of a and , ing eigenvalue, or equivalently, to the distance Reynolds number is defined by R = LVp4l, where have been published by various authors (3), and that the numerical range of T protrudes into the L is the half-width of the channel, V is the maxi- pseudospectra of T appear in (1 1, 15). Neither upper half-plane. This is essentially the Hille- mum velocity, p is the density, and , is the spectra nor pseudospectra of - itself have been Yosida theorem of functional analysis (10). viscosity. The distance, velocity, and time vari- plotted previously. 34. Lord Kelvin (W. Thomson), Philos. Mag. 24, 188 ables are nondimensionalized by L, V, and LIV, 25. Our computations have been carried out on a (1887). respectively. Connection Machine System CM-5 in Fortran and 35. This can be seen algebraically from the absence 4. S. J. Davies and C. M. White, Proc. R. Soc. London on Sun workstations in Matlab, with numerical of B in the easily verified formula diJujI2Idt = uT(A Ser. A 119, 92 (1928); V. Patel and M. R. Head, J. methods adapted from those of Reddy and Hen- + Ar)u. Fluid Mech. 38, 181 (1969); D. R. Carlson, S. E. ningson (11-13, 15). First, the problem is Fourier 36. The equation for dllull2Idt in fluid mechanics is Widnall, M. F. Peeters, ibid. 121, 487 (1982). transformed in x and z, and the working variables called the Reynolds-Orr equation (1). Our 2 x 2 5. V. A. Romanov, Funct. Anal. Appl. 7,137 (1973). are taken as the y components of velocity and model is not so far from the interaction of vortex 6. A. Lundbladh and A. V. Johansson, J. Fluid Mech. vorticity; a weighted norm is introduced for these 'tilting' and "stretching" described in Orzsag and 229, 499 (1991); N. Tillmarkand P. H. Alfredsson, variables that is equivalent to the energy norm 11J11. Patera (8); the first and second components of u ibid. 235, 89 (1992). The 1 D problem is discretized by a Chebyshev are analogous to the normal components of ve- 7. P. S. Klebanoff, K. D. Tidstrom, L. M. Sargent, ibid. hybrid spectral method and then projected onto a locity and vorticity, respectively. 12, 1 (1962). space spanned by a set of dominant eigen- 37. We have not discussed the relation between 8. S. A. Orszag and A. T. Patera, ibid. 128, 347 modes, leading to matrices typically of size about spatially global and local results, which is some- (1983). 80 x 80. Inthe Poiseuiilecase, the odd and even what obscured by modal or pseudomodalanaly- 9. B. J. Bayly, S. A. Orszag, T. Herbert, Annu. Rev. solutionswith respect to ydecouple, and we take sis, because differentwave numbersa,,B decou- Fluid Mech. 20, 359 (1988). advantage of this separation.The generationof ple orthogonally.A relatedmatter is the investiga-

SCIENCE * VOL. 261 * 30 JULY1993 583 tion of spatial rather than temporal evolution of flow Technology, Stockholm, Sweden, 1993)]. L. Reichel, L. N. Trefethen, ibid., p. 796. disturbances, which according to results of P. J. 40. W. Kerner, J. Comp. Phys. 85,1 (1989). 44. There is evidence that this statement applies even to Schmid (personal communication), leads to results 41. B. F. Farrell, J. Atmos. Sci. 46, 1193 (1989). fullydeveloped turbulence; see M. J. Lee, J. Kim, P. qualitatively analogous to those reported here. 42. S. C. Reddy and L. N. Trefethen, Numer. Math. 62, Moin, J. Fluid Mech. 216, 561 (1990) and K. M. 38. Numerical experiments indicated that even ran- 235 (1992); D. J. Higham and L. N. Trefethen, BIT, Butlerand B. F. Farrell,Phys. FluidsA 5, 774 (1993). dom fluctuations tend to excite growing pseudo- 33, 285 (1993). The importance of a proper treat- 45. Supported by an NSF grant [DMS-9116110 modes. A theory of the nonnormal response of ment of nonnormality in numerical analysis, and a (L.N.T.)] and a graduate fellowship (T.A.D.). We linearized flows to stochastic forcing data has corresponding proper definition of stability, was acknowledge the assistance over several years of been developed by B. F. Farrelland P. J. loannou detailed by H.-O. Kreiss in the 1960s [ibid. 2, 153 D. Henningson and P. Schmid and, more recently, (in preparation). (1962); R. D. Richtmyer and K. W. Morton, of K. Butler, B. Farrell,and P. loannou. In addition, 39. The bounds -21/4 c ey -1 are derived by G. Difference Methods for Initial-ValueProblems (Wi- we acknowledge the helpful comments of B. Kreiss, A. Lundbladh, and D. S. Henningson [Tech. ley-lnterscience, New York, 1967)]. Fornberg, P. Holmes, R. Scott, and E. Siggia, and Rep. TRITA-NA-9307(Department of Numerical 43. N. M. Nachtigal, S. C. Reddy, L. N. Trefethen, SlAM also Thinking Machines Corporation for the dona- Analysis and Computing Science, Royal Instituteof J. MatrixAnal.Appl. 13, 778 (1992); N. M. Nachtigal, tion of extensive CM-5 computer time.

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