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 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=aaas. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. American Association for the Advancement of Science is collaborating with JSTOR to digitize, preserve and extend access to Science. http://www.jstor.org ARTICLEr 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-