Symmetry reduction in high dimensions, illustrated in a turbulent pipe
SSymmetry reduction in high dimensions, illustrated in a turbulent pipe
Ashley P. Willis, ∗ Kimberly Y. Short, † and Predrag Cvitanovi´c ‡ School of Mathematics and Statistics, University of Sheffield, S3 7RH, U.K. Center for Nonlinear Science, School of Physics,Georgia Institute of Technology, Atlanta, GA 30332-0430 (Dated: October 7, 2018)Equilibrium solutions are believed to structure the pathways for ergodic trajectories in a dynami-cal system. However, equilibria are atypical for systems with continuous symmetries, i.e. for systemswith homogeneous spatial dimensions, whereas relative equilibria (traveling waves) are generic. Inorder to visualize the unstable manifolds of such solutions, a practical symmetry reduction methodis required that converts relative equilibria into equilibria, and relative periodic orbits into peri-odic orbits. In this article we extend the fixed Fourier mode slice approach, previously applied1-dimensional PDEs, to a spatially 3-dimensional fluid flow, and show that is substantially moreeffective than our previous approach to slicing. Application of this method to a minimal flow unitpipe leads to the discovery of many relative periodic orbits that appear to fill out the turbulentregions of state space. We further demonstrate the value of this approach to symmetry reductionthrough projections (projections only possible in the symmetry-reduced space) that reveal the in-terrelations between these relative periodic orbits and the ways in which they shape the geometryof the turbulent attractor.
PACS numbers: 05.45.-a, 45.10.db, 45.50.pk, 47.11.4j
Chaotic dynamics can be interpreted as a trajectory instate space, where each coordinate corresponds to a de-gree of freedom. For higher-dimensional systems it canbe difficult to predict which coordinate choices will pro-vide the most instructive projections, given that plots ofthese trajectories are limited to displaying two or threedimensions at a time. To avoid clutter in the projectioncaused by families of orbits related by translations orreflections, symmetry-invariant measures such as spatialaverages are often favored. In practice, however, thereare only so many quantities that may be averaged and,in addition, information held in the spatial structure iswiped out in the averaging process. Often such averag-ing results in a largely uninformative projection of thedynamics.The study of turbulence is one example where sub-stantial progress has recently been made by viewing theflow as a dynamical system, but now a more informa-tive means of projection is required to comprehend theway in which the unstable manifolds of relative equi-libria and other invariant solutions shape the dynam-ics. These invariant solutions correspond to recurrentbut unstable motions [1] that share some characteris-tics with fully turbulent flows. Experiments [1, 2] andsimulations [3, 4] have identified transient visits to spa-tiotemporal patterns that mimic traveling wave solutions.Certain low-dissipation traveling waves of the Navier-Stokes equations have been shown to be important in thetransition to turbulence, where they lie in the laminar-turbulent boundary, separating initial conditions that ul- ∗ [email protected] † [email protected] ‡ [email protected] timately relaminarize from those that develop into tur-bulence [5]. Spatiotemporal flow patterns called ‘puffs’and ‘slugs’ are observed during the evolution to turbu-lence. Recently, spatially-localized solutions representa-tive of puffs have been discovered [6] and shown to belinked to spatially-periodic traveling waves in minimaldomains [7]. As traveling waves are steady in their re-spective co-moving frames, they are relative equilibria,solutions that do not exhibit temporal shape-changingdynamics. Their unstable manifolds, however, mold thesurrounding state space, carving pathways for relativeperiodic orbits, invariant orbits embedded in turbulencewhose temporal evolution captures dynamics of ergodictrajectories that shadow them. A detailed understandingof these recurrent motions is crucial if one is to system-atically describe the repertoire of all turbulent motions.With the removal of spatial translations, which obscurevisualizations of the dynamics, a far greater number ofprojections of chaotic trajectories is possible. In this arti-cle, we show that visualizations of the symmetry reduceddynamics can help us understand relationships betweendistinct families of periodic orbits and traveling wave so-lutions, which in turn lends support to the dynamicalsystems interpretation that relative periodic orbits formthe backbone of turbulence in pipe.Our approach is dynamical: writing the Navier–Stokesequations as ˙ u = v ( u ), the fluid state u at a particu-lar moment in time is represented by a single point instate space M [8]; turbulent flow is represented by an er-godic trajectory that wanders between accessible statesin M [9]. Essential to this analysis is that any two phys-ically equivalent states be identified as a single state:a symmetry-reduced state space ˆ M = M /G is formedby contracting the volume of state space representingstates that are identical except for a symmetry trans- a r X i v : . [ phy s i c s . f l u - dyn ] M a y FIG. 1. (Color online) Schematic of symmetry reductionby the method of slices. The blue point is the template ˆ x (cid:48) .Group orbits are marked by dotted curves, so that all pinkpoints are equivalent to ˆ x up to a shift. The relative periodicorbit (green) in the d -dimensional full state space M closesinto a periodic orbit (blue) in the slice ˆ M = M /G , a ( d − t (cid:48) .A typical group orbit crosses the slice hyperplane transver-sally, with a non-orthogonal group tangent t = t (ˆ x ). A slicehyperplane is almost never a global slice; it is valid up to theslice border, a ( d − x ∗ whose group orbits graze the slice, i.e. points whose tan-gents t ∗ = t (ˆ x ∗ ) lie in ˆ M . Beyond the slice border (dashed‘chunk’), group orbits do not cross the slice hyperplane locally. formation to a single point ˆ u . Only after a symmetry re-duction are the relationships between physically distinctstates revealed. In this article symmetry reduction is im-plemented with an extension of the ‘first Fourier modeslice’ method [10], a variant of the method of slices [11].The method of slices separates coordinates into phasesalong symmetry directions (‘fibers’, ‘group orbits’ thatparametrize families of physically-equivalent dynamicalstates) from the remaining coordinates of the symmetry-reduced state space ˆ M . The latter capture the dynami-cal degrees of freedom—those associated with structuralchanges of the flow.The Navier-Stokes equations are invariant under trans-lations, rotations, and inversions about the origin, andthe application of any of these symmetry operations toa state u ( x ) results in another dynamically equivalentstate. The boundary conditions for pipe flow restrictsymmetries to translations along the axial and azimuthaldirections, and reflections in the azimuthal direction. Inthe computations presented here,axial periodicity is as-sumed so that the symmetry group of the system is O (2) θ × SO (2) z . In order to illustrate the key ideas,we constrain azimuthal shifts, and focus on the familyof streamwise translational shifts { g } parametrized by asingle continuous phase parameter (cid:96) ,( g ( (cid:96) ) u )( z ) = u ( z − (cid:96) ) . If periodic axial symmetry is assumed, application of g gives a closed curve family of dynamically equivalentstates — topologically a circle, called a group orbit — in state space M . Were azimuthal (‘spanwise’) shifts in-cluded, equivalent states would lie on a 2-torus.Symmetry reduction simplifies the state space by re-ducing each set of dynamically equivalent states to aunique point ˆ u . The method of slices achieves thiswith the aid of a fixed template state u (cid:48) (see Fig. 1).A shift is applied so that the symmetry-reduced stateˆ u = g ( − (cid:96) ) u lies within the hyperplane orthogonal to t (cid:48) = lim (cid:96) → ( g ( (cid:96) ) u (cid:48) − u (cid:48) ) /(cid:96) , the tangent to the template u (cid:48) in the direction of the shift. For a time-dependentflow, one determines (cid:96) = (cid:96) ( t ) by chosing ˆ u to be thepoint on the group orbit of u closest to the template, (cid:104) ˆ u − u (cid:48) | t (cid:48) (cid:105) = 0 in a given norm. In this work we use theL2 or ‘energy’ norm E = (cid:104) u | u (cid:105) / (cid:82) u / dV .As traveling waves drift downstream without changingtheir spatial structure, the family of traveling wave states u ( t ) is dynamically equivalent (lies on the same grouporbit g ( (cid:96) ) u ) and may be represented by a single stateˆ u q . Thus all traveling waves are simultaneously reducedto equilibria in the slice, irrespective of their individualphase velocities, a powerful property of the method ofslices. Furthermore, all relative periodic orbits p , flowpatterns each of which recurs after a different time pe-riod T p , shifted downstream by a different (cid:96) p , close intoperiodic orbits in the slice hyperplane.Dynamics within the slice is given by˙ˆ u = v (ˆ u ) − ˙ (cid:96) (ˆ u ) t (ˆ u ) , (1)˙ (cid:96) (ˆ u ) = (cid:104) v (ˆ u ) | t (cid:48) (cid:105) / (cid:104) t (ˆ u ) | t (cid:48) (cid:105) , (2)where the expression for the phase velocity ˙ (cid:96) is knownas the reconstruction equation [12]. No dynamical infor-mation is lost and we may return to the full space byintegrating (2). In contrast to a Poincar´e section, wheretrajectories pierce the section hyperplane, time evolutiontraces out a continuous trajectory within the slice. Inprinciple, the choice of template is arbitrary; in prac-tice, some templates are preferable to others. While oneis concerned with the dynamics within the slice ˆ u ( t ), inpractice it may be simpler to record (cid:96) ( t ) and to post-process, or to process on the side, visualizations withinthe slice—slicing is much cheaper to perform than gath-ering u ( t ) from simulation or laboratory experiment.The enduring difficulty with symmetry reduction is indetermining a unique shift (cid:96) for a given state u , whileavoiding discontinuities in (cid:96) ( t ) that arise when multiple‘best fit’ candidates ˆ u = g ( − (cid:96) ) u to the template u (cid:48) oc-cur. A singularity arises if the group orbit g u grazesthe slice hyperplane (Fig. 1). At the instant this occurs,the tangents to the fluid state ˆ u and the template u (cid:48) are orthogonal, and there is a division by zero in the re-construction equation (2). In ref. [13] it was shown thatthe hyperplanes defined by multiple templates could beused to tile a slice, but while switching may permit thesymmetry reduction of longer trajectories, it is often notpossible to both switch templates before a slice border isreached and to simultaneously maintain continuity in (cid:96) .Furthermore, it is uncertain when to switch back to thefirst template, in order to produce a unique symmetry-reduced state. Our aim in this article is to avoid suchdifficulties through the use of a single template with dis-tant slice borders. The approach of Budanur et al. [10] forthe case of one translational spatial dimension fixes thephase of a single Fourier coefficient. This ‘Fourier’ sliceis a special case within the slicing framework, with theeffect of extreme smoothing of the group orbit. Here theapproach is extended to a spatially 3-dimensional case,that of turbulent pipe flow.For the case of a scalar field defined on one spatial di-mension [10] there is a unique Fourier coefficient appro-priate for determining the symmetry reduction. Here, forthe 3-dimensional turbulent flow, there are three compo-nents of velocity with a spatial discretization for each,and it is not obvious which coefficients to fix in order todefine an effective symmetry-reducing slice. In this pa-per we construct a template u (cid:48) ( r, θ, z ) = u c cos( αz ) + u s sin( αz ), where u c ( r, θ ) = (cid:82) L ˜ u cos( αz ) d z , u s ( r, θ ) = (cid:82) L ˜ u sin( αz ) d z , and L = 2 π/α , for some chosen state˜ u . This corresponds to (all of) the first coefficients inthe streamwise Fourier expansion for ˜ u . Arbitrary states u may then be projected onto a plane via a = (cid:104) u | u (cid:48) (cid:105) and a = (cid:104) u | g ( L/ u (cid:48) (cid:105) , respectively (see Fig. 2). Inthis projection, the group orbit g u of any state is a cir-cle centered on the origin, and the polar angle θ for thepoint ( a , a ) corresponds to a unique shift (cid:96) = θ ( L/ π ).The symmetry reduced state ˆ u = g ( − l ) u is the closestpoint on its group orbit to the template u (cid:48) . The slice isprojected onto the positive a -axis in this projection.Note that the approach is independent of discretiza-tion, and does not actually require a Fourier decomposi-tion. Note also that the inner-product gathers informa-tion from the full velocity field.As group orbits are circles crossing perpendicular tothe a -axis in this projection, (cid:104) t (ˆ u ) | t (cid:48) (cid:105) in (2) can only bezero if the circle shrinks to a point at the origin. This re-quires that both inner products (cid:104) u | u (cid:48) (cid:105) and (cid:104) u | g ( L/ u (cid:48) (cid:105) are zero at the same time, which has vanishing probabil-ity. While we thus avoid the slice border, there is a rapidchange in θ by ≈ π (in (cid:96) units by ≈ L/
2) whenever thetrajectory ( a , a )( t ) sweeps past the origin, see the insetto Fig. 2. Rapid phase shifts notwithstanding, this choiceof template has made possible the discovery and analysisof the many relative periodic orbits discussed below.‘Minimal flow units’ [15], which capture much of thestatistical properties of turbulence, have been invaluablein analyzing fundamental self-sustaining processes [16].Here, the fixed-flux Reynolds number for all calcula-tions is Re = DU/ν = 2500, where lengths are non-dimensionalized by diameter D and velocities are nor-malized by the mean axial speed U . The minimal flowunit is in the m = 4 rotational subspace, such that( r, θ, z ) ∈ [0 , ] × [0 , π ] × [0 , π . ]. The size of the domainis more usefully measured in terms of wall units, ν/u τ ,where u τ = − ν ( ∂ r u z ) | wall , which allows comparison withflow units used in other geometries. In these units, thedomain is of size Ω + ≈ [100 , , tL /2 c t FIG. 2. ( top left ) Schematic of the first Fourier mode slice,with a , a defined in the text. In this projection the sliceborder is a zero-measure ‘point’ at the origin. ( bottom ) For ageneric ergodic trajectory the phase velocity c = ˙ (cid:96) ( t ) appearsto encounter singularities whenever it approaches the sliceborder, which, however, is never reached [10]. Closer inspec-tion reveals a rapid but continuous change in the shift ( topright ) by ≈ L/ (cid:96) ( t ) − ct Galilean frame, moving (forour parameter values) at c = 1 . D c D KY µ ( max ) ω or θ TW N L/ . . . . N U/ . . . . . . . . . . . . . T . Listedare mean dissipation D , mean down-stream phase velocity c , the number of unstable eigen-directions (two per each com-plex pair), Kaplan-Yorke dimension D KY , the real part ofthe largest stability eigenvalue/Floquet exponent µ ( max ) , andeither the corresponding imaginary part ω ( max ) for travelingwaves, or the phase θ of the complex Floquet multiplier forrelative periodic orbits, or its sign, if real: -1 indicates inversehyperbolic. FIG. 3. (Color online) Projection of 32 relative periodicorbits and traveling waves using symmetry-invariant coor-dinates,
I/I lam , D/D lam where I lam = D lam are energyrates for the laminar flow. All discovered traveling wavesare included: (B) TW . , (C) TW . , (D) TW . , (G)TW . , (H) TW . , (F) TW . and (J) TW . , except forTW N L/ . , TW N U/ . and TW . , which lie far outsidethe ergodic cloud (grey dots). spanwise and streamwise dimensions, respectively. Ourflow unit compares favorably with the minimal flow unitsfor channel flow [15] Ω + ≈ [ > , , − + ≈ [68 , , + ≈ [68 , , Re τ = ( D/ u τ /ν = 100 ± Openpipeflow.org , along withthe open source code used to calculate these orbits.Visualizations of high-dimensional state space trajec-tories are necessarily projections onto two or three dimen-sions. A common choice is to monitor the flow in termsof the rate of energy dissipation D = ρν (cid:82) u ·∇ u dV and FIG. 4. (Color online) Projection of the symmetry-reducedinfinite-dimensional state space onto the first 3 PCA princi-pal axes, computed from the L2-norm average over the naturalmeasure (the gray ‘cloud’) in the slice. 32 relative periodicorbits, and a subset of the 7 shortest relative periodic orbits,together with traveling waves (A) TW N U/ . , (B) TW . ,(C) TW . , (D) TW . , (E) TW . , (F) TW . . WhileTW . appears to lie in the very center of the ( I, D ) pro-jection Fig. 3, it is revealed in this state space projection tolie far from the ergodic cloud, outside the box plotted, asare (E) TW N L/ . and (F) TW . . Due to a ‘rotate-and-reflect’ symmetry, each solution appears twice, with the ex-ception of (A) TW N U/ . (and the far-away TW N L/ . ),which belong to the ‘rotate-and-reflect’ invariant subspace.Our relative periodic orbits capture the regions of high natu-ral measure very well. The symmetry-invariant subspace hasa strong repulsive influence, separating the natural measureinto two weakly communicating regions. The inset shows theergodic cloud from another perspective. the external input power required to maintain constantflux I = Q ∆ p , where Q = (cid:82) u · dS is the flux at anycross-section and ∆ p and is the pressure drop over thelength of the pipe. As the time-averages of I and D arenecessarily equal, traveling waves and orbits, which maybe well-separated in state space, are contracted onto ornear the I = D line, a drawback of the 2-dimensional( I, D ) projection. Fig. 3 shows that the orbits appear tooverlap with the ergodic region, but reveals little of therelationships between solutions; we use D values only todistinguish traveling waves solutions listed in table I.In the symmetry-reduced state space it is possible toconstruct coordinates that are intrinsic to the flow it-self, using spatial information that would otherwise besmeared out by translational shifts. To obtain a globalportrait of the turbulent set, Fig. 4, we project solu-tions onto the three largest principal components ˆ e i ob-tained from a PCA of N =2000 independent ˆ u (cid:48) i = ˆ u i − ¯ˆ u ,where ¯ˆ u is the mean of the data, using the SVD method(on average the square of the projection p i = (cid:104) ˆ u (cid:48) ( t ) | ˆ e i (cid:105) equals the i th singular value of the correlation matrix R ij = N − (cid:104) ˆ u (cid:48) i | ˆ u (cid:48) j (cid:105) ).The lower / upper branch pair TW N L/ . /TW N U/ . were obtained by continuation from asmaller ‘minimal flow unit’ [13]. In table I and inthe ( I, D )-projection Fig. 3 the upper branch travelingwave TW N U/ . appears to be far removed from tur-bulence, unlikely to exert influence. The PCA projec-tion of the symmetry-reduced state space, however, re-veals the strong repelling influence of TW N U/ . whose30-dimensional unstable manifold acts as a barrier tothe dynamics, cleaving the natural measure into two‘clouds’, forcing a trajectory to hover around one neigh-borhood until it finds a path to the other, bypassingTW N U/ . . The two ergodic ‘clouds’ are related bythe ‘rotate-and-reflect’ symmetry ( π/ N U/ . is invariant (for symmetries of pipeflow see ref. [13]).The symmetry-reduced state space projections revealsets of relative periodic orbits with qualitatively simi-lar dynamics. The short-period orbits are well spreadover the dense regions of natural measure, and the longrelative periodic orbits in (a) appear to ‘shadow’ shortorbits in (b), but also exhibit extended excursions thatfill out state space. While sets of relative periodic orbitsoften share comparable dissipation rates and Floquet ex-ponents (table I and Openpipeflow.org data sets), it isthe state space projections that are essential to establish-ing genuine relationships.In summary, we have shown that symmetry reductioncan be applied to a dynamical system of very high dimen-sions, here turbulent pipe flow. An appropriately con-structed template renders the method of slices substan-tially more effective for projecting the dynamics and for Newton searches for invariant solutions. The method isgeneral and can be applied to any dynamical system withcontinuous translational or rotational symmetry. Projec-tions of the symmetry-reduced space reveal fundamentalproperties of the dynamics not evident prior to symmetryreduction. In the application at hand, to a turbulent pipeflow, the method has enabled us to identify for the firsttime a large set of relative periodic orbits embedded inturbulence, and to demonstrate that the key invariant so-lutions strongly influence turbulent dynamics. To followthis demonstration of the power of symmetry reduction,work is now underway to determine the relationship be-tween relative periodic orbits [17]. Analysis of their un-stable manifolds are expected to reveal the intimate linksbetween traveling waves and relative periodic orbits, al-lowing for explicit construction of the invariant skeletonthat gives shape to the strange attractor explored by tur-bulence.
ACKNOWLEDGMENTS
We are indebted to M. Farazmand, N. B. Budanur,J.F. Gibson, X. Ding, F. Fedele, E. Siminos, M. Avila,B. Hof, and R. R. Kerswell for many stimulating discus-sions. A. P. W. is supported by the EPSRC under grantEP/K03636X/1. K. Y. S. was supported by the NationalScience Foundation Graduate Research Fellowship underGrant NSF DGE-0707424. P. C. thanks the family of lateG. Robinson, Jr. and NSF DMS-1211827 for support. [1] B. Hof, C. W. H. van Doorne, J. Westerweel, F. T. M.Nieuwstadt, H. Faisst, B. Eckhardt, H. Wedin, R. R.Kerswell, and F. Waleffe, Science , 1594 (2004).[2] D. J. C. Dennis and F. M. Sogaro, Phys. Rev. Lett. ,234501 (2014).[3] R. R. Kerswell and O. Tutty, J. Fluid Mech. , 69(2007), arXiv:physics/0611009 .[4] A. de Lozar, F. Mellibovsky, M. Avila, and B. Hof, Phys.Rev. Lett. , 214502 (2012).[5] Y. Duguet, A. P. Willis, and R. R. Kerswell, J. FluidMech. , 255 (2008), arXiv:0711.2175 .[6] M. Avila, F. Mellibovsky, N. Roland, and B. Hof, Phys.Rev. Lett. , 224502 (2013).[7] M. Chantry, A. P. Willis, and R. R. Kerswell, Phys. Rev.Lett. , 164501 (2014).[8] J. F. Gibson, J. Halcrow, and P. Cvitanovi´c, J. FluidMech. , 107 (2008), arXiv:0705.3957 .[9] E. Hopf, Commun. Pure Appl. Math. , 303 (1948).[10] N. B. Budanur, P. Cvitanovi´c, R. L. Davidchack,and E. Siminos, Phys. Rev. Lett. , 084102 (2015), arXiv:1405.1096 .[11] E. Cartan, La m´ethode du rep`ere mobile, la th´eorie desgroupes continus, et les espaces g´en´eralis´es , Expos´es deG´eom´etrie, Vol. 5 (Hermann, Paris, 1935). [12] C. W. Rowley and J. E. Marsden, Physica D , 1(2000).[13] A. P. Willis, P. Cvitanovi´c, and M. Avila, J. Fluid Mech. , 514 (2013), arXiv:1203.3701 .[14] P. Frederickson, J. L. Kaplan, E. D. Yorke, and J. A.Yorke, J. Diff. Eqn. , 185 (1983).[15] J. Jim´enez and P. Moin, J. Fluid Mech. , 213 (1991).[16] J. M. Hamilton, J. Kim, and F. Waleffe, J. Fluid Mech.287