Isotropy subgroups: The 67-fold path to symmetric enlightenment

8/14/2019 Isotropy subgroups: The 67-fold path to symmetric enlightenment

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CHARTING THE STATE SPACE OF PLANE COUETTE FLOW:

EQUILIBRIA, RELATIVE EQUILIBRIA, AND HETEROCLINIC

CONNECTIONS

A ThesisPresented to

The Academic Faculty

by

Jonathan J Halcrow

In Partial Fulfillmentof the Requirements for the Degree

Doctor of Philosophy in theSchool of Physics

Georgia Institute of Technologyver. 1.4, October 24 2008

this documents contain 2 discussion of the symmetries of plane Couette flow:

1) a short one, from http://arxiv.org/abs/0808.3375 - first 9 pages2) a longer one, from the (updated) J Halcrow thesis - remaining pages

Predrag Cvitanovic, Feb 23 2009


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To be submitted to J. Fluid Mech. 1

Equilibrium and traveling-wave solutions ofplane Couette flow

By J. H A L C R O W, J. F. G I B S O N

AND P. C V I T A N O V I C

School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA

(Printed 23 February 2009)

We survey equilibria and traveling waves of plane Couette flow in small periodic cellsat moderate Reynolds number Re, adding in the process eight new equilibrium and twonew traveling-wave solutions to the four solutions previously known. Bifurcations un-

der changes of Re and spanwise period are examined. These non-trivial flow-invariantunstable solutions can only be computed numerically, but they are exact in the sensethat they converge to solutions of the Navier-Stokes equations as the numerical resolutionincreases. We find two complementary visualizations particularly insightful. Suitably cho-sen sections of their 3D-physical space velocity fields are helpful in developing physicalintuition about coherent structures observed in moderate Re turbulence. Projections ofthese solutions and their unstable manifolds from -dimensional state space onto suit-ably chosen 2- or 3-dimensional subspaces reveal their interrelations and the role theyplay in organizing turbulence in wall-bounded shear flows.

1. Introduction

In Gibson et al. (2008) (henceforth referred to as GHC) we have utilized exact equilib-rium solutions of the Navier-Stokes equations for plane Couette flow in order to illustratea visualization of moderate Re turbulent flows in an infinite-dimensional state space,in terms of dynamically invariant, intrinsic, and representation independent coordinateframes. The state space portraiture (figure 1) offers a visualization of numerical (or ex-perimental) data of transitional turbulence in boundary shear flows, complementary to3D visualizations of such flows (figure 2). Side-by-side animations of the two visualiza-tions illustrate their complementary strengths (see Gibson (2008b) online simulations).In these animations, 3D spatial visualization of instantaneous velocity fields helps elu-cidate the physical processes underlying the formation of unstable coherent structures,such as the Self-Sustained Process (SSP) theory of Waleffe (1990, 1995, 1997). Runningconcurrently, the -dimensional state space representation enables us to track unstablemanifolds of equilibria of the flow, the heteroclinic connections between them ( Halcrow

et al. 2009), and gain new insights into the nonlinear state space geometry and dynamicsof moderate Re wall-bounded shear flows such as plane Couette flow.Here we continue our investigation of exact equilibrium and traveling-wave solutions

of Navier-Stokes equations, this time tracking them adiabatically as functions of Re andperiodic cell size [Lx, 2, Lz] and, in the process, uncovering new invariant solutions, anddetermining new relationships between them.

The history of experimental and theoretical advances is reviewed in GHC, Sect. 2;here we cite only the work on equilibria and traveling waves directly related to thisinvestigation. Nagata (1990) found the first pair of nontrivial equilibria, as well as thefirst traveling wave in plane Couette flow (Nagata 1997). Waleffe (1998, 2003) computed

the short version, extracted from http://arxiv.org/abs/0808.3375


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4 J. Halcrow, J. F. Gibson, and P. Cvitanovic

approximately 100 wall units. These rolls, in turn, generate streamwise streaks of highand low speed fluid, by convecting fluid alternately away from and towards the walls.

The streaks have streamwise instabilities whose length scale is roughly twice the rollseparation. These coherent structures are prominent in numerical and experimentalobservations (see figure 2 and Gibson (2008b) animations), and they motivate our inves-tigation of how equilibrium and traveling-wave solutions of Navier-Stokes change withRe and cell size.

Fluid states are characterized by their energy E = 12

u2 and energy dissipation rateD = u2, defined in terms of the inner product and norm

(u, v) =1

V

dx u v , u2 = (u, u) . (2.3)

The rate of energy input is I = 1/(LxLz)

dxdz u/y, where the integral is taken overthe upper and lower walls at y = 1. Normalization of these quantities is set so thatI = D = 1 for laminar flow and E = ID. In some cases it is convenient to consider fieldsas differences from the laminar flow. We indicate such differences with hats: u = u yx.

3. Symmetries and isotropy subgroups

In an infinite domain and in the absence of boundary conditions, the Navier-Stokesequations are equivariant under any 3D translation, 3D rotation, and x x, u uinversion through the origin (Frisch 1996). In plane Couette flow, the counter-movingwalls restrict the rotation symmetry to rotation by about the z-axis. We denote thisrotation by x and inversion through the origin by xz. The xz and x symmetriesgenerate a discrete dihedral group D1 D1 = {e, x, z, xz} of order 4, where

x [u, v, w](x, y, z) = [u, v, w](x, y, z)

z [u, v, w](x, y, z) = [u,v, w](x,y, z) (3.1)

xz [u, v, w](x, y, z) = [u, v, w](x, y, z) .

The subscripts on the symmetries thus indicate which ofx, z change sign. The walls alsorestrict the translation symmetry to 2D in-plane translations. With periodic boundaryconditions, these translations become the SO(2)x SO(2)z continuous two-parametergroup of streamwise-spanwise translations

(x, z)[u, v, w](x, y, z) = [u, v, w](x + x, y , z + z) . (3.2)

The equations of plane Couette flow are thus equivariant under the group = O(2)x O(2)z = D1,x SO(2)x D1,z SO(2)z, where stands for a semi-direct product, xsubscripts indicate streamwise translations and sign changes in x, y, and z subscripts

indicate spanwise translations and sign changes in z.The solutions of an equivariant system can satisfy all of the systems symmetries, aproper subgroup of them, or have no symmetry at all. For a given solution u, the subgroupthat contains all symmetries that fix u (that satisfy su = u) is called the isotropy (orstabilizer) subgroup of u. (Hoyle 2006; Marsden & Ratiu 1999; Golubitsky & Stewart2002; Gilmore & Letellier 2007). For example, a typical turbulent trajectory u(x, t) hasno symmetry beyond the identity, so its isotropy group is {e}. At the other extreme isthe laminar equilibrium, whose isotropy group is the full plane Couette symmetry group.

In between, the isotropy subgroup of the Nagata equilibria and most of the equilibria


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Equilibria and traveling waves of plane Couette flow 5

reported here is S = {e, s1, s2, s3}, where

s1

[u, v, w](x, y, z) = [u,v, w](x + Lx/2, y, z)s2 [u, v, w](x, y, z) = [u, v, w](x + Lx/2, y, z + Lz/2) (3.3)

s3 [u, v, w](x, y, z) = [u, v, w](x, y, z + Lz/2) .

These particular combinations of flips and shifts match the symmetries of instabilities ofstreamwise-constant streaky flow (Waleffe 1997, 2003) and are well suited to the wavystreamwise streaks observable in figure 2, with suitable choice of Lx and Lz. But S isone choice among a number of intermediate isotropy groups of , and other subgroupsmight also play an important role in the turbulent dynamics. In this section we provide apartial classification of the isotropy groups of , sufficient to classify all currently knowninvariant solutions and to guide the search for new solutions with other symmetries. Wefocus on isotropy groups involving at most half-cell shifts. The main result is that amongthese, up to conjugacy in spatial translation, there are only five isotropy groups in which

we should expect to find equilibria.

3.1. Flips and half-shifts

A few observations will be useful in what follows. First, we note the key role played bythe inversion and rotation symmetries x and z (3.1) in the classification of solutionsand their isotropy groups. The equivariance of plane Couette flow under continuoustranslations allows for traveling-wave solutions, i.e., solutions that are steady in a framemoving with a constant velocity in [x, z]. In state space, traveling waves either traceout circles or wind around tori, and these sets are both continuous-translation and timeinvariant. The sign changes under x, z, and xz, however, imply particular centers ofsymmetry in x, z, and both x and z, respectively, and thus fix the translational phasesof fields that are fixed by these symmetries. Thus the presence of x or z in an isotropygroup prohibits traveling waves in x or z, and the presence of xz prohibits any form

of traveling wave. Guided by this observation, we will seek equilibria only for isotropysubgroups that contain the xz inversion symmetry.

Second, the periodic boundary conditions impose discrete translation symmetries of(Lx, 0) and (0, Lz) on velocity fields. In addition to this full-period translation sym-metry, a solution can also be fixed under a rational translation, such as (mLx/n, 0) or acontinuous translation (x, 0). If a field is fixed under continuous translation, it is con-stant along the given spatial variable. If it is fixed under rational translation (mLx/n, 0),it is fixed under (mLx/n, 0) for m [1, n 1] as well, provided that m and n are rela-tively prime. For this reason the subgroups of the continuous translation SO(2)x consistof the discrete cyclic groups Cn,x for n = 2, 3, 4, . . . together with the trivial subgroup{e} and the full group SO(2)x itself, and similarly for z. For rational shifts x/Lx = m/nwe simplify the notation a bit by rewriting ( 3.2) as

m/nx = (mLx/n, 0) ,

m/nz = (0, mLz/n) . (3.4)

Since m/n = 1/2 will loom large in what follows, we omit exponents of 1 /2:

x = 1/2x , z =

1/2z , xz = xz . (3.5)

If a field u is fixed under a rational shift (Lx/n), it is periodic on the smaller spatialdomain x [0, Lx/n]. For this reason we can exclude from our searches all equilibriumwhose isotropy subgroups contain rational translations in favor of equilibria computed onsmaller domains. However, as we need to study bifurcations into states with wavelengthslonger than the initial state, the linear stability computations need to be carried out


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in the full [Lx, 2, Lz] cell. For example, if EQ is an equilibrium solution in the 1/3 =[Lx/3, 2, Lz] cell, we refer to the same solution repeated thrice in = [Lx, 2, Lz] as

spanwise-tripled or 3 EQ. Such solution is by construction fixed under the C3,x ={e, 1

/3x ,

2/3x } subgroup.

Third, some isotropy groups are conjugate to each other under symmetries of the fullgroup . Subgroup S is conjugate to S if there is an s for which S = s1Ss.In spatial terms, two conjugate isotropy groups are equivalent to each other under acoordinate transformation. A set of conjugate isotropy groups forms a conjugacy class.It is necessary to consider only a single representative of each conjugacy class; solutionsbelonging to conjugate isotropy groups can be generated by applying the symmetryoperation of the conjugacy.

In the present case conjugacies under spatial translation symmetries are particularlyimportant. Note that O(2) is not an abelian group, since reflections and translations along the same axis do not commute (Harter 1993). Instead we have = 1.Rewriting this relation as 2 = 1, we note that

xx(x, 0) = 1(x/2, 0) x (x/2, 0) . (3.6)

The right-hand side of (3.6) is a similarity transformation that translates the origin ofcoordinate system. For x = Lx/2 we have

1/4x x 1/4x = xx , (3.7)

and similarly for the spanwise shifts / reflections. Thus for each isotropy group containingthe shift-reflect xx symmetry, there is a simpler conjugate isotropy group in which xxis replaced by x (and similarly for zz and z). We choose as the representative of eachconjugacy class the simplest isotropy group, in which all such reductions have been made.However, if an isotropy group contains both x and xx, it cannot be simplified thisway, since the conjugacy simply interchanges the elements.

Fourth, for x = Lx, we have 1

x x x = x , so that, in the special case of half-cellshifts, x and x commute. For the same reason, z and z commute, so the order-16isotropy subgroup

G = D1,x C2,x D1,z C2,z (3.8)

is abelian.

3.2. The 67-fold path

We now undertake a partial classification of the lattice of isotropy subgroups of planeCouette flow. We focus on isotropy groups involving at most half-cell shifts, with SO(2)xSO(2)z translations restricted to order 4 subgroup of spanwise-streamwise translations(3.5) of half the cell length,

T = C2,x C2,z = {e, x, z, xz} . (3.9)

All such isotropy subgroups of are contained in the subgroup G (3.8). Within G, welook for the simplest representative of each conjugacy class, as described above.

Let us first enumerate all subgroups H G. The subgroups can be of order |H| ={1, 2, 4, 8, 16}. A subgroup is generated by multiplication of a set of generator elements,with the choice of generator elements unique up to a permutation of subgroup elements.A subgroup of order |H| = 2 has only one generator, since every group element is its owninverse. There are 15 non-identity elements in G to choose from, so there are 15 subgroupsof order 2. Subgroups of order 4 are generated by multiplication of two group elements.There are 15 choices for the first and 14 choices for the second. However, each order-4


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subgroup can be generated by 3 2 different choices of generators. For example, any twoofx, z, xz in any order generate the same group T. Thus there are (15 14)/(3 2) = 35

subgroups of order 4.Subgroups of order 8 have three generators. There are 15 choices for the first generator,

14 for the second, and 12 for the third. There are 12 choices for the third generator andnot 13, since if it were chosen to be the product of the first two generators, we wouldget a subgroup of order 4. Each order-8 subgroup can be generated by 7 6 4 differentchoices of three generators, so there are (15 14 12)/(7 6 4) = 15 subgroups of order8. In summary: there is the group G itself, of order 16, 15 subgroups of order 8, 35 oforder 4, 15 of order 2, and 1 (the identity) of order 1, or 67 subgroups in all (Halcrow2008). This is whole lot of isotropy subgroups to juggle; fortunately, the observations of 3.1 show that there are only 5 distinct conjugacy classes in which we can expect to findequilibria.

The 15 order-2 groups fall into 8 distinct conjugacy classes, under conjugacies betweenxx and x and zz and z. These conjugacy classes are represented by the 8 isotropy

groups generated individually by the 8 generators x, z, xz, xz, zx, x, z, andxz. Of these, the latter three imply periodicity on smaller domains. Of the remainingfive, x and xz allow traveling waves in z, z and zx allow traveling waves in x. Onlya single conjugacy class, represented by the isotropy group

{e, xz} , (3.10)

breaks both continuous translation symmetries. Thus, of all order-2 isotropy groups, weexpect only this group to have equilibria. EQ9, EQ10, and EQ11 described below areexamples of equilibria with isotropy group {e, xz}.

Of the 35 subgroups of order 4, we need to identify those that contain xz and thussupport equilibria. We choose as the simplest representative of each conjugacy class theisotropy group in which xz appears in isolation. Four isotropy subgroups of order 4

are generated by picking xz as the first generator, and z, zx, zz, or zxz as thesecond generator (R for reflect-rotate):

R = {e, x, z, xz} = {e, xz} {e, z}

Rx = {e, xx, zx, xz} = {e, xz} {e, xx} (3.11)

Rz = {e, xz, zz, xz} = {e, xz} {e, zz}

Rxz = {e, xxz, zxz, xz} = {e, xz} {e, zxz} S .

These are the only isotropy groups of order 4 containing xz and no isolated translationelements. Together with {e, xz}, these 5 isotropy subgroups represent the 5 conjugacyclasses in which expect to find equilibria.

The Rxz isotropy subgroup is particularly important, as the Nagata (1990) equilibriabelong to this conjugacy class (Waleffe 1997; Clever & Busse 1997; Waleffe 2003), as do

most of the solutions reported here. The NBC isotropy subgroup ofSchmiegel (1999) andS ofGibson et al. (2008) are conjugate to Rxz under quarter-cell coordinate transforma-tions. In keeping with previous literature, we often represent this conjugacy class withS = {e, s1, s2, s3} = {e, zx, xxz, xzz} rather than the simpler conjugate group Rxz.Schmiegels I isotropy group is conjugate to our Rz; Schmiegel (1999) contains manyexamples of Rz-isotropic equilibria. R-isotropic equilibria were found by Tuckerman &Barkley (2002) for plane Couette flow in which the translation symmetries were brokenby a streamwise ribbon. We have not searched for Rx-isotropic solutions, and are notaware of any published in the literature.

The remaining subgroups of orders 4 and 8 all involve {e, i} factors and thus involve


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states that are periodic on half-domains. For example, the isotropy subgroup of EQ7 andEQ8 studied below is S {e, xz} R {e, xz}, and thus these are doubled states of

solutions on half-domains. For the detailed count of all 67 subgroups, see Halcrow (2008).

3.3. State-space visualization

GHC presents a method for visualizing low-dimensional projections of trajectories in theinfinite-dimensional state space of the Navier-Stokes equations. Briefly, we construct anorthonormal basis {e1, e2, , en} that spans a set of physically important fluid statesuA, uB, . . . , such as equilibrium states and their eigenvectors, and we project the evolv-ing fluid state u(t) = u yx onto this basis using the L2 inner product (2.3). It isconvenient to use differences from laminar flow, since u forms a vector space with thelaminar equilibrium at the origin, closed under addition. This produces a low-dimensionalprojection

a(t) = (a1, a2, , an, )(t) , an(t) = (u(t), en) , (3.12)

which can be viewed in 2d planes {em, en} or in 3d perspective views {e, em, en}. Thestate-space portraits are dynamically intrinsic, since the projections are defined in termsof intrinsic solutions of the equations of motion, and representation independent, sincethe inner product (2.3) projection is independent of the numerical or experimental repre-sentation of the fluid state data. Such bases are effective because moderate-Re turbulenceexplores a small repertoire of unstable coherent structures (rolls, streaks, their mergers),so that the trajectory a(t) does not stray far from the subspace spanned by the keystructures.

There is no a priori prescription for picking a good set of basis fluid states, andconstruction of{en} set requires some experimentation. Let the S-invariant subspace bethe flow-invariant subspace of states u that are fixed under S; this consists of all stateswhose isotropy group is S or contains S as a subgroup. The plane Couette system athand has a total of 29 known equilibria within the S-invariant subspace four translated

copies each of EQ1 - EQ6, two translated copies of EQ7 (which have an additional xzsymmetry), plus the laminar equilibrium EQ0 at the origin. As shown in GHC, thedynamics of different regions of state space can be elucidated by projections onto basissets constructed from combinations of equilibria and their eigenvectors.

In this paper we present global views of all invariant solutions in terms of the or-thonormal translational basis constructed in GHC from the four translated copies ofEQ2:

x z xz

e1 = c1(1 + x + z + xz) uEQ2 + + +

e2 = c2(1 + x z xz) uEQ2 + (3.13)

e3 = c3(1 x + z xz) uEQ2 +

e4 = c4(1 x z + xz) uEQ2 + ,

where cn is a normalization constant determined by en = 1. The last 3 columns indicatethe symmetry of the basis vector under half-cell translations; e.g. 1 in the x columnimplies xej = ej .

4. Equilibria and traveling waves of plane Couette flow

We seek equilibrium solutions to (2.1) of the form u(x, t) = uEQ(x) and traveling-waveor relative equilibrium solutions of the form u(x, t) = uTW(x ct) with c = (cx, 0, cz).


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Gibson, J. F. & Cvitanovic, P. 2009 Periodic orbits of plane Couette flow. In preparation.

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APPENDIX C

SYMMETRIES AND ISOTROPIES OF SOLUTIONS

C.1 Isotropy subgroups: The 67-fold path to symmetric enlightenment

[A shorter (but more carefully proofread) version of the material covered in this appendixcan be found in ref. [30].]

In long-time simulations of turbulence, symmetry considerations can be disposed off ina few lines: In absence of boundary conditions, the Navier-Stokes equations are equivariantunder any 3D translation, 3D rotation, and x x, u u inversion through theorigin that we shall refer to as xz. The pair of counter-moving plane Couette flow wallsrestricts the translation symmetry to 2D in-plane translations. The streamwise shearingmotion restricts the 3D rotation symmetry to a single discrete symmetry operation, thewall interchange wall-normal-streamwise rotation by about z-axis that we shall referto as x (symmetry referred to as centrosymmetry by Tuckerman and Barkley [84]).Together with inversion xz, this symmetry generates a discrete dihedral group D1,xD1,z ={e, x, z, xz} of order 4,

z [u,v,w](x,y,z) = [u,v,w](x,y,z)

x [u,v,w](x,y,z) = [u,v, w](x,y, z) (107)

xz [u,v,w](x,y,z) = [u,v,w](x,y, z) ,

where z is the spanwise reflection in z, and and xz = xz. It would be more correct torefer to the subgroup of rotations generated by x as C2,x (or D1,x, the integers modulo

2 notation very frequently used in applied mathematics literature [36]) but we use theisomorphic group D1,x here to distinguish this group from the Cn,x subgroups ofSO(2) inwhat follows. The subscripts on x, z, and xz are chosen to indicate the sign changes inx, y and z under the action of each symmetry.

With periodic boundary conditions, the two translations are restricted to the compactSO(2)x SO(2)z continuous two-parameter group of streamwise-spanwise translations,

(x, z)[u,v,w](x,y,z) = [u,v,w](x + x, y ,z + z) . (108)

SO(2) is isomorphic to the circle group S1, the group of rotations in plane. The equationsof plane Couette flow with boundary conditions (13) are thus equivariant under the groupGPCF = O(2)xO(2)z = D1,xSO(2)xD1,zSO(2)z, where subscript x indicates stream-

wise translations and wall-normal-streamwise inversion, z indicates spanwise translationsand inversion, and stands for a semi-direct product. As far as a generic turbulent stateis concerned, the symmetry discussion can stop here.

However, for compact time-invariant solutions (equilibria, relative equilibria, periodicsolutions) and their instabilities the symmetries play a paramount role [34, 28], both fortheir classification and for understanding the dynamics connecting them, so here we shallneed a much finer grained description of symmetries.

The key observation is that the G-equivariance of the equations does not imply G-invariance of their solutions. A solution of an G-equivariant equation is in general mapped

64

The long version
http://en.wikipedia.org/wiki/Centrosymmetrichttp://www.warwick.ac.uk/~masax/Research/papers.htmlhttp://www.warwick.ac.uk/~masax/Research/papers.htmlhttp://en.wikipedia.org/wiki/Centrosymmetric


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into a group orbit of solutions under the action ofG, that is, a set of solutions that map intoeach other under the action of G. A solution may be invariant under an isotropy subgroup(or stabilizer subgroup) H G; that is, gu = u for g H. At one extreme, a genericturbulent solution has the trivial isotropy subgroup Hturb = {e}. The other extreme is thelaminar solution whose isotropy group is the full GPCF. The full classification of the lattice

of intermediate isotropy subgroups is a laborious undertaking, even for a set of symmetriesas plain as the ones considered here; fortunately, the set of dynamically important isotropysubgroups is relatively small. We refer the reader to the immense literature on the subject(we enjoyed reading [?] and [?] most), and give here a briefest of summaries, sufficient forthe classification of the time-invariant solutions encountered so far.

Why are the equilibria for moderate Re born equipped with non-trivial isotropy groups?The physical intuition is that while the nonlinear convective terms amplify unstable fluidmotions, the dissipation damps out fluid states with rapidly varying velocity fields. Formoderate Re the balance of the two smooths out the long-time solutions, and equilibriawith most symmetry tend to be the smoothest time-invariant solutions. A bit of historyillustrates this. The laminar solution is the smoothest, least dissipative solution. Nagatadiscovered the EQ1 and EQ2 equilibria in 1990 by continuing a known solution from Taylor-

Couette flow to plane Couette. Waleffe calculated the same solutions in a different way,noted that they are invariant under shift-rotate and shift-reflect isotropy subgroup thatwe refer to below as the subgroup S, and utilized this symmetry to reduce the dimensional-ity of his PDE discretizations. Busse and Clever [?] also noted this invariance. Schmiegel [?]undertook an investigation of several isotropy subgroups, and in what follows we shall notethe subgroups that he identified. We started our exploration of plane Couette dynamicsaround Nagata/Waleffe equilibria, and focused our searches for new equilibria within theS-invariant subspace, since this symmetry restriction fixes the x, z phase of solutions andreduces the dimensionality of their unstable manifolds. In what follows we find primar-ily S-invariant equilibria because we initiated our guesses within that invariant subspace.However, as the subspace is unstable, errors accrued by long numerical simulations had

sufficient deviation from the S-invariant subspace that the Newton search detected close-byequilibria whose isotropy subgroup is S3 (discussed below). One would do well to look forsolutions with other symmetries by initiating searches within other invariant subspaces.

A state with no symmetry: A turbulent state space trajectory isotropy group is Hturb ={e}. Such trajectory maps into a different trajectory under the action of the group,with translations generating a torus, and the inversions generating four tori of symmetry-equivalent solutions.

Relative equilibria: Let us first dispose of continuous symmetries. Equivariance of planeCouette flow equations under continuous translations allows for relative equilibria solutions,i.e., solutions that are steady in a frame moving with a constant velocity in the [x, z] plane.A generic relative equilibrium solution traces out a circle or torus in state space; this setis both continuous translation and time invariant. The sign changes under reflections /rotations x, z imply particular centers of symmetry and thus fix the phase of a field withrespect to x and z, respectively. Hence the presence of x in an isotropy group rules outtraveling waves in x, and likewise for z. The isotropy subgroups with relative equilibriasolutions are the streamwise-spanwise traveling SO(2)x SO(2)z , the streamwise travelingSO(2)x D1,z Cm,z , and the spanwise traveling D1,x Cm,x SO(2)z (we explain theemergence of cyclic subgroups Cm next).

Equilibria: Velocity fields invariant under the inversion xz (i.e., under both x and z

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flips) cannot b e traveling waves, so we search for equilibria in subspaces whose isotropygroup includes the inversion among its group elements.

O(2) is not an abelian group, since reflections and translations along the same axisdo not commute [?]: = 1. Rewriting this relation as 2 = 1, we note that

xx(x, 0) = 1(x/2, 0) x (x/2, 0)

zz(0, z) = 1(0, z/2) z (0, z/2) . (109)

This conjugation relation implies, in terms of coordinate transformations, that a shift ofx in x followed by a flip across x = 0 is conjugate to a flip across x = 0. For x = Lx/2

we have xx = 1/4x x

1/4x . Hence in classifying isotropy groups we can trade half-shift

and flip xx in favor of a flip x in the shifted coordinate system. However, if both xand xx are elements of a subgroup, neither can be eliminated, as the conjugation simplyinterchanges them.

Periodic boundary conditions on equilibrium solutions identify the fields on opposingboundaries and restrict their isotropy subgroups to discrete cyclic subgroups Cm,x Cn,x SO(2)x

SO(2)z along the span-, stream-wise directions [?]. For rational shifts x/Lx =

m/n we simplify the notation a bit by rewriting (108) as

m/nx = (mLx/n, 0) , m/nz = (0, mLz/n) , 0 < m < n , (110)

and, as m/n = 1/2 will loom large in what follows, omit 1/2 exponents:

x = 1/2x , z =

1/2z , xz = xz . (111)

If a solution is invariant under a cyclic shift

u(x,y ,z) = 1/nx u(x,y,z) = u(x + Lx/n,y,z) , (112)

it is the [Lx/n, 2, Lz ]-cell solution repeated n times downstream the cell. Invariance under1/nx indicates that a [Lx, 2, Lz ] is redundant, in the sense that a cell of length Lx/n captures

the same state, so it quicker and easier to compute it in the small cell. However, as weneed to study bifurcations into states with wavelengths longer than the initial state, thelinear stability computations need to be carried out in the full [Lx, 2, Lz ] cell. SupposeEQ is an equilibrium solution in the 1/3 = [Lx/3, 2, Lz ] cell. Then we shall refer to thesame solution repeated thrice in = [Lx, 2, Lz ] as 3 EQ. This solution is invariant underC3,x = {e, 1/3x , 2/3x } subgroup.

Which subgroup is realized as an equilibrium isotropy subgroup is a physical question.As the shearing is streamwise, the earliest bifurcations (as function of increasing Re) havesomething to do with streamwise instabilities. As the structure of Navier-Stokes solutions is

a sensitive function of spanwise / streamwise / wall-normal nonlinear interactions and ratiosof cell and wall-unit scales, which solutions arise first is a highly nontrivial issue [ ?, ?]. Inaddition, for small cells the existence of solutions is a sensitive function of (and an artifactof) the periodicities imposed by the cell aspect ratios [Lx, 2, Lz ].

A velocity field constant along the streamwise or spanwise direction is always dampedby the flow to the laminar solution [?, ?, ?], so equilibrium solutions have to wiggle inboth directions [?]. The more wiggles a solution has, the higher its dissipation rate.Solutions invariant under the combinations of reflections (107) and half-cell shifts (111)allow for the gentlest variation in velocity fields across the cell, so these generally have the

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lowest dissipation rates, are the least unstable, and consequently are the ones to emergefirst as equilibrium solutions as Re is increased. So we commence our discussion of equilib-rium isotropy subgroups by SO(2)x SO(2)z translations restricted to order 4 subgroup ofspanwise-streamwise translations of half the cell length,

T = C2,x

C2,z ={

e, x, z, xz}

. (113)

The order-16 isotropy subgroup

G = D1,x C2,x D1,z C2,z GPCF (114)

of the full symmetry group contains all half-shift isotropy subgroups within it.For spanwise full-cell z = Lz and half-cell z = Lz/2 shifts (109) implies

1z z z = z (115)

1/4z z 1/4z = zz , (116)

and similarly for the streamwise shifts / reflections.

Let us first enumerate the subgroups H G. From (115) we see that the half-shiftsand reflections commute, so G is an abelian subgroup of O(2)x O(2)z . Subgroupscan be of order |H| = {1, 2, 4, 8, 16}. A subgroup is generated by multiplication of a set ofgenerator group elements, with the choice of generator elements unique up to a permutationof subgroup elements. A subgroup of order |H| = 2 has only one generator, since every groupelement is its own inverse. There are 15 non-identity elements to choose from, so there are15 subgroups of order 2. Subgroups of order 4 are generated by multiplication of two groupelements. There are 15 choices for the first and 14 choices for the second. Equivalentsubgroup generators can be picked in 3 2 different ways, so there are (15 14)/(3 2) = 35subgroups of order 4. Subgroups of order 8 have three generators. There are 15 choices forthe first, 14 for the second, and 12 for the third (If the element were the product of the

first two, we would get a subgroup of order 4). There are 7 6 4 ways of picking equivalentgenerators, i.e., there are (15 14 12)/(7 6 4) = 15 subgroups of order 8. In summary:there is the group itself, of order 16, 15 subgroups of order 8, 35 of order 4, 15 of order2, and 1 (the identity) of order 1, or 67 subgroups in all.This is whole lot of subgroups tojuggle; fortunately, the following observations reduce this number to 5 isotropy subgroupsthat suffice up to conjugacies to describe all equilibria studied here.

(a) The isotropy subgroups for which we seek equilibrium solutions include xz within itselements.

(b) The right hand side of (116) is a product of two group elements, but the left hand sideis a similarity transformation that moves the coordinate frame origin by a quarter-

shift. Hence we can always trade in a single spanwise (or streamwise) shift-reflectcombination in favor of a reflect operation . (If both j and jj are elementsof a subgroup, neither can be eliminated, as the similarity transformation simplyinterchanges them.)

(c) By (112) any state whose isotropy subgroup includes a factor {e, i} is a doubled state.

In what follows we label subgroups H, S, T, S3, . . . , in a somewhat arbitrary fashion, forlack of a systematic labeling scheme more compact than listing the group elements.

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(a) (b)

Figure 26: (a) The isotropy group of equilibrium EQ8 is Axz given in (121). According to(112), factor {e, xz} means that the state tiles the [Lx, Lz ] cell twice. How so? Transportthe spanwise / wall-normal section distance Lx/2 along the x direction, and then shift itspanwise by Lz/2 along the z direction. The equilibrium solution in this half-cell, invariant

under the R isotropy subgroup (124) tiles the cell twice. (b) x-average of EQ8 velocityfield u(y, z) is C2,z = {e, z} (spanwise half-cell shift) invariant. The R invariance (isotropysubgroup (124)) under discrete reflections is immediate. W03 cell, Re = 270.

The conjugation of xx to x (and similarly for z) allows the 15 order-2 groups to bereduced to the 8 order-2 groups {e, g} where g is one ofx, z , xz, xz, zx, x, z, andxz. Of these, the latter three imply periodicity on smaller domains. Of the remaining five,x and xz support traveling waves in z, z and zx support traveling waves in x. Onlya single subgroup of order |H| = 2, generated by xz, supports equilibria:

S3 = {e, xz} . (117)

EQ11 is an example of an equilibrium with S3 isotropy subgroup.Four distinct isotropy subgroups of order 4 are generated by picking xz as the first

generator, and z, zx, zz or zxz as the second generator (R for reflect-rotate):

R = {e, x, z, xz} = S3 {e, z}Rx = {e, xx, zx, xz} = S3 {e, zx} (118)Rz = {e, xz, zz, xz} = S3 {e, zz} = IRxz = {e, xxz, zxz, xz} = S3 {e, zxz} = S .

Each of these isotropy subgroups of GPCF has three other isotropy subgroups conjugate

to it under quarter-cell shifts (116). For example, the three groups {e, xx, z , xzx},{e, x, zz, xzx}, and {e, xx, zz, xzxz}, are conjugate to R under quarter-cell coor-dinate shifts in x, z, and x, z respectively. Thus the isotropy subgroups given by (118)account for 12 of the 35 order-4 subgroups of G. Note that R, Rx, Rz and Rxz are theonly inequivalent isotropy subgroups of order 4 containing xz and no isolated translationelements. Together with S3 these yield the total of 5 non-conjugate isotropy subgroups inwhich we can expect to find equilibria.

The order-8 subgroups of G are generated as in our combinatorial count above, bymultiplying out all distinct products of three Z2 subgroups, {e, ij}{e, k}{e, mn},

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i,j,k,,m,n {0, x , z}, and 0 = 0 = e;{e, x, z , xz , xz, xxz , zxz, xzxz}{e, x, z , x, xz, xx, zx, xzx}{e, x, z , z , xz , xz, zz, xzz}{e, x, x, xx, zz, zxz, xzz, xzxz}{e, x, x, z , xz, xx, xz , xxz}{e, x, xz, zx, zz, xxz, xzx, xzz}{e, x, z, xz, zx, zxz, xzx, xzxz}{e, xz , xz, xx, xz, zx, zz, xzxz}{e, z, x, xz, zx, xxz, xzz, xzxz}{e, z, x, z, xz , zx, zz, zxz}{e, z, xz , xx, xz, zxz, xzx, xzz}{e, z, z , xx, zz , xxz, xzx, xzxz}

{e, x, xz, xz, zz, xxz, zxz, xzx}{e, x, z, xz , xz , xzx, xzz, xzxz}{e, z , xz , xx, zx, xxz, zxz , xzz}

(119)

They all involve factors of {e, j} and are thus repeats of a smaller fundamental domainhalf-cells. The 15 order-8 isotropy subgroups can be reduced by coordinate transformationto the following 9 subgroups: R {e, x}, R {e, z}, R {e, xz}, Rx {e, z}, Rz {e, x}, Rx {e, xz} = Rz {e, xz}, T {e, x}, T {e, z}, and T {e, xz}. As eachof these subgroups contains pure translation elements, each can be reduced to a simplersymmetry subgroup on a smaller domain. R {e, xz} is the isotropy subgroup of EQ7 andEQ8.There are 4 subgroups of order 8 which by (112) tile the cell twice: A? is generated byelements

A? = xxz, zxz, z . (120)

Ax = {e, x, z, xz , x, xx, zx, xzx} = R {e, x}Az = {e, x, z, xz , z, xz, zz , xzz} = R {e, z}Axz = {e, x, z, xz , xz, xxz , zxz, xzxz} (121)

= R {e, xz}A? =

{e, x, z, xz , ?, x?, z?, xz?

}.

As illustrated by figure 26, Axz is the isotropy subgroup of equilibria EQ7 and EQ8 discussedbelow (after a spanwise quarter-shift (116) is applied to eliminate some half-shifts). Thereis one subgroup which appears with 4 translational copies, generated by x, z and xz.It contains copies of all R subgroups (118). There are 3 subgroups of order 8 which areinvariant under the translational similarity transform. These all contain T and its productswith either x, z, or xz.

Finally, the laminar solution EQ0 isotropy group |H| = 16 is the full subgroup |G| = 16.Its multiplicity is m = 1, i.e., it appears only once.

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What about G = D1,x Cm,x D1,z Cn,z subgroups of O(2)x O(2)z for m,n > 2?By (112), a solution in [Lx, 2, Lz ] repeated n times across the cell of size [Lx, 2, nLz ] is

also a solution, but with a larger isotropy group, invariant under Cn,z . Already for smallaspect cells studied here we encounter solutions (such as 2 EQ1) with C4 symmetry,invariant under quarter shifts 1/4. They are important, as they can bifurcate into states

with wavelengths longer than the initial state.Now that we have enumerated the five distinct isotropy subgroups, it is helpful tointroduce notation for the corresponding invariant subspaces of solutions of Navier-Stokesequations (by an invariant subspace we mean that a state of the fluid initiated withinsuch a subspace remains in it for all times). We denote invariant subspaces of solutions of(13) as follows. Let S denote the order-4 isotropy subgroup Rxz (118), and S3 = {1, xz}the order-2 isotropy subgroup (117) inversion. Let U be the space of velocity fields thatsatisfy the boundary and zero-divergence conditions of plane Couette flow, and let US andUS3 be the S and S3-invariant subspaces ofU respectively: US = {u U | su = u, s S},and US3 = {u U | xzu = u}. As xz changes the sign of both x and z, US and US3 donot admit relative equilibrium solutions.

Remark: Nomenclature. It is unfortunate that the group theory nomenclature oftencollides with the nonlinear dynamics and fluid dynamics terminology. Here we refer to apoint in the dynamical systems dd state space where a = v(a) = 0 as an equilibriumpoint, and the point in a 3D fluid flow where the Eulerian velocity vanishes as a stagnationpoint point. The corresponding notions for symmetries of flows under continuous Lie grouptransformations we usually preface by a group-theoretic qualifier, such as G. We referto a group orbit in contradistinction to orbit, the set of state space points related bytime evolution, and to Gfixed point space in contradistinction to dynamically stationaryequilibrium point.

Example: Order 8 invariant subgroups describe doubled states. The isotropy subgroup

of EQ7 and EQ8 is

{e, s1, , s7} = {e, zx, xxz, xzz , zz, xzx, xz, x}.

Using spanwise quarter-shift along z we get rid of some half-shifts

s1 = zx zxz, s3 = xzz xz , s4 = zz z , s6 = xzx xzxz,

This results in the canonical form of this isotropy subgroup

R {e, xz} = {e, x, z, xz , xz , xxz, zxz, xzxz}.

Example: Isotropy subgroups generated multiplicatively. The abelian structure ofG lendsitself to a binary labeling of its |G| = 16 elements:

g1234 = 1x

2z

3x

4z . (122)

In this form the group multiplication table is simply mod 2 addition of j; for example,g1011g1100 = g0111.

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In the Cn-invariance case the translational label 4 in (122) is n-ary, i.e., in groupmultiplication it is composed by addition mod n.

Associated with action of G are projection on 16 irreducible representations, symmet-ric/antisymmetric under action of g1234 :

P =

1

24 (1 x)(1 z)(1 x)(1 z) . (123)We shall find these projection operators handy for visualization of state space flows.

There is only one isotropy group of order 2: S3 = {g0000, g1010}. Multiplication with 7group elements of form g0234 yields 7 isotropy subgroups of order 4

S3 g0010 = {g0000, g0010, g1000, g1010} = RS3 g0110 = {g0000, g0110, g1010, g1100} = RxS3 g0011 = {g0000, g0011, g1001, g1010} = RzS3 g0111 = {g0000, g0111, g1010, g1101} = Rxz , (124)

and half-cell doubled states

S3 g0100 = {g0000, g0100, g1010, g1110} = DxS3 g0001 = {g0000, g0001, g1010, g1011} = DzS3 g0101 = {g0000, g0101, g1010, g1111} = Dxz . (125)

Group elements of form g1234 all appear in the above subgroups and contribute no furtherorder 4 subgroups.

Multiplication ofRs with group elements not already within them generates 6 isotropysubgroups of order 8 (all names temporary, for now)

A =

{g0000, g0110, g1010, g1100, g0001, g0111, g1011, g1101

}B = {g0000, g0110, g1010, g1100, g0011, g0101, g1001, g1111}C = {g0000, g0011, g1001, g1010, g0100, g0111, g1101, g1110} (126)

Ux = {g0000, g0010, g0100, g0110, g1000, g1010, g1100, g1110}Uz = {g0000, g0010, g1000, g1010, g0001, g0011, g1001, g1011}

Uxz = {g0000, g0010, g1000, g1010, g0101, g0111, g1101, g1111} (127)

and a multiplication table

R

g0100 = Ux , R

g0001 = Uz , R

g0101 = Uxz

Rx g0001 = A , Rx g0010 = Ux , Rx g0011 = BRz g0001 = Uz , Rz g0100 = C , Rz g0101 = B

Rxz g0001 = A , Rxz g0010 = Uxz , Rxz g0011 = C (128)

Multiplication of Ds with group elements not already within them generates further sub-groups.

Example: {e, xz} invariant equilibrium: EQ11

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S3 isotropy subgroup {1, xz} is order 2. The smaller group means a bigger quotientedstate space. There are eight copies of any S3-invariant field, generated by the four half-celltranslations and by reflection about the z = 0 plane.

Let u be s3-invariant: u = s3u. s3 commutes with all fourelements ofT = {e, x, z, xz}.Thus s3( u) = s3u = u, and so u is s3 symmetric for all four choices of .

Moreover, s1u is S3-invariant for S3-invariant u (and generally s1u = u for {x, z , xz}. We have u = s3u, so s1u = s1s3u = s3(s1u). That then gives us eightcopies: s u for s {e, s1} and T. Trying to expand by s2 multiplication doesnt pro-duce any new fields, since s2u = s1s3u = s1u. So in general, each S3-invariant field has eightcopies in the s3 invariant subspace, unless these copies coincide due to further symmetries(e.g. fields with additional s1 symmetry total only four copies). Figure 27 shows the eightS3-invariant copies of EQ11.

Translations of half the cell length in the spanwise and/or streamwise directions com-mute with S. These operators generate T, a discrete subgroup (113) of the continuoustranslational symmetry group SO(2) SO(2) : Take EQ2 as an example of an S3-invariantfield. It has four distinct copies within US, and USis a subset of the S3-invariant subspace.So we have an example of an S3-invariant field with at least four copies in the S3-invariant

subspace.EQ7 and EQ8 have multiplicity 2, as their isotropy subgroup is (121).

C.2 Action of the s1, s2 elements on the Gibson basis

The basis elements of Gibson basis which are in the s1 and s2 subspaces are given by:

jujkl(y) + v

jkl(y) + kwjkl(y) = 0, (129)

where

gj(x) = sin jx j < 0

1 j = 0

cosjx j > 0

(130)

and where u,v,wjkl (y) is given by Rl(y) or a multiple of Ql(y) defined as

Rl(y) =

1+y2 l = 0

1y2 l = 1

12(2l+1)

(Pl2(y) Pl(y)) l 2, (131)

Ql(y) =

y1

Rl(y)dy .

To start consider the effect of s1 and s2 on gj(x) and gk(z). Substituting z =

z or

x = x gives:

gj(x) =

gj(x) j < 0gj(x) j 0

(132)

Translation by a half period is given by

gj(x + ) =

gj(x) j = 0

gj(x) otherwise(133)

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Then for the y-dependent parts of the basis:

Rl(y) =

R1(y) l = 0

R0(y) l = 1

Rl(y) l > 1 and l is even

Rl(y) l > 1 and l is odd

(134)

Ql(y) =

Q0(1) Q1(y) l = 0Q1(1) Q0(y) l = 1Q2(1) Q2(y) l = 2Ql(y) l 3 and l is oddQl(y) l 3 and l is even

, (135)

With the action of the symmetries given as

s1u

vw

(x,y ,z) = u

vw

(x + Lx/2, y, z) (136)

s2

uv

w

(x,y ,z) =

uv

w

(x + Lx/2, y, z + Lz/2) (137)

s3

uv

w

(x,y ,z) =

uv

w

(x, y, z + Lz/2) , (138)

each element of the basis set is either symmetric or antisymmetric with respect to eachsymmetry, see table 15.

C.2.1 Structure of the ODE representation

Note the following identity for Legendre polynomials:Pl(y) =

Pl+1(y) Pl1(y)2l + 1

(139)

This means we can alternatively define Ql(y) as:

Ql(y) =

y1

Rl(y) =

y1

Pl2(y) Pl(y) (140)

= Pl1(y) Pl3(y)2l 3 P

l+1(y) Pl1(y)2l + 1

(141)

=Rl+1(y)

2l + 1 Rl1(y)

2l 3 (142)

Since (Pm, Pn) = mn/(2n +1), (Rm, Rn) = mn(1/(2m + 1 ) + 1/(2m 3))+ m,n+2/(2m +1) + m,n2/(2m + 1).

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j k l ujkl(y) vjkl(y)

wjkl(y) s1 s2 s3

0 0 0 {2, . . . , L}e Rl(y) 0 0 S A A0 0 0 {2, . . . , L}o Rl(y) 0 0 S S S1 0 0

{2, . . . , L

}e 0 0 Rl(y) A S A

1 0 0 {2, . . . , L}o 0 0 Rl(y) A A S0 j 0 {3, . . . , L1}e Rl(y) jQl(y) 0 A S A0 j 0 {3, . . . , L1}o Rl(y) jQl(y) 0 A A S0 j 0 {3, . . . , L1}e Rl(y) jQl(y) 0 A A S0 j 0 {3, . . . , L1}o Rl(y) jQl(y) 0 A S A1 j 0 {2, . . . , L}e 0 0 Rl(y) S S S1 j 0 {2, . . . , L}o 0 0 Rl(y) S A A1 j 0 {2, . . . , L}e 0 0 Rl(y) S A A1 j 0 {2, . . . , L}o 0 0 Rl(y) S S S0 0 k {2, . . . , L}e Rl(y) 0 0 A A S0 0 k {2, . . . , L}o Rl(y) 0 0 A S A0 0 k {2, . . . , L}e Rl(y) 0 0 S A A0 0 k {2, . . . , L}o Rl(y) 0 0 S S S1 0 k {3, . . . , L1}e 0 kQl(y) Rl(y) A A S1 0 k {3, . . . , L1}o 0 kQl(y) Rl(y) A S A1 0 k {3, . . . , L1}e 0 kQl(y) Rl(y) S A A1 0 k {3, . . . , L1}o 0 kQl(y) Rl(y) S S S0 j k {2, . . . , L}e kRl(y) 0 -jRl(y) S A A0 j k {2, . . . , L}o kRl(y) 0 -jRl(y) S S S0 j k {2, . . . , L}e kRl(y) 0 -jRl(y) A A S0 j k {2, . . . , L}o kRl(y) 0 -jRl(y) A S A0 j k {2, . . . , L}e kRl(y) 0 -jRl(y) S S S0 j k {2, . . . , L}o kRl(y) 0 -jRl(y) S A A0 j k {2, . . . , L}e kRl(y) 0 -jRl(y) A S A0 j k {2, . . . , L}o kRl(y) 0 -jRl(y) A A S1 j k {3, . . . , L1}e kRl(y) jkQl(y) jRl(y) S A A1 j k {3, . . . , L1}o kRl(y) jkQl(y) jRl(y) S S S1 j k {3, . . . , L1}e kRl(y) jkQl(y) jRl(y) A A S1 j k {3, . . . , L1}o kRl(y) jkQl(y) jRl(y) A S A1 j k {3, . . . , L1}e kRl(y) jkQl(y) jRl(y) S S S1 j k {3, . . . , L1}o kRl(y) jkQl(y) jRl(y) S A A1 j k {3, . . . , L1}e kRl(y) jkQl(y) jRl(y) A S A1 j k {3, . . . , L1}o kRl(y) jkQl(y) jRl(y) A A S

Table 15: The polynomial triplets for a complete set of linearly independent basis functionsjkl up to a given Fourier discretization |j| J, |k| K and polynomial degree l L.A indicates that a member of the basis is anti-symmetric with respect to the symmetryoperator si and S indicates symmetric. Indices are j and k are strictly positive unlessexplicitly stated otherwise. e and o indicate the even and odd members of a set, respectively.Note we could drop one of the columns since s1s2 = s3, so if one is S the other two areeither S,S or A,A and if one is A then the others are S, A or A,S.

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TABLE OF CONTENTS

SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

I INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Poincare, Turbulence, and Hopfs Last Hope . . . . . . . . . . . . . . . . 1

1.2 The Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 What is New in This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 3

II NAVIER-STOKES EQUATIONS AND PLANE COUETTE FLOW . . . . . . 4

2.1 In the Beginning... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Scale Invariance and the Reynolds number . . . . . . . . . . . . . . . . . 5

2.3 Plane Couette Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3.1 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.2 On 2D Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Turbulent plane Couette flow . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4.1 Reynolds Decomposition . . . . . . . . . . . . . . . . . . . . . . . 9

2.4.2 Wall Units and The Law of the Wall . . . . . . . . . . . . . . . 10

2.5 A Brief History of Turbulent Shear Flows . . . . . . . . . . . . . . . . . . 11

III SYMMETRY AND PLANE COUETTE FLOW . . . . . . . . . . . . . . . . . 15

3.1 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.2 Representation Theory . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.3 Equivariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.4 Irreducible Representations and Equivariance . . . . . . . . . . . . 19

3.2 Discrete Symmetry: The Lorenz Attractor . . . . . . . . . . . . . . . . . 20

3.3 Symmetries of Plane Couette Flow . . . . . . . . . . . . . . . . . . . . . . 21

IV METHODOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1 Discretization and the Method of Lines . . . . . . . . . . . . . . . . . . . 24

4.1.1 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1.2 Temporal Discretization . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 A Survey of Methods for Integrating the Navier-Stokes Equations . . . . 26

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4.2.1 Primitive Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 Krylov Subspace Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3.1 Arnoldi Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3.2 GMRES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4 Newton Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4.1 Termination of Newton Algorithms . . . . . . . . . . . . . . . . . 30

4.4.2 Line Search Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 30

4.4.3 Model Trust Region Algorithms . . . . . . . . . . . . . . . . . . . 31

4.5 Finding Invariant Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.6 State Space Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

V EQUILIBRIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.1 Equilibria and Relative Equilibria . . . . . . . . . . . . . . . . . . . . . . 37

5.2 Bifurcations under Variation ofRe . . . . . . . . . . . . . . . . . . . . . . 42

5.3 Bifurcations under Variation of Spanwise Width . . . . . . . . . . . . . . 43

VI HETEROCLINIC CONNECTIONS . . . . . . . . . . . . . . . . . . . . . . . . 46

6.1 Three Heteroclinic Connections for Re=400 . . . . . . . . . . . . . . . . . 46

6.1.1 A Heteroclinic Connection from EQ4 to EQ1 . . . . . . . . . . . . 46

6.1.2 Heteroclinic Connections from EQ3 and EQ5 to EQ1 . . . . . . . . 47

6.2 Heteroclinic Connection from EQ1 to EQ2 at Re = 225 . . . . . . . . . . 48

VII SPECULATION AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . 49

7.1 Turbulence as a walk about exact coherent states . . . . . . . . . . . . . . 49

7.2 Refinement of Heteroclinic Connections . . . . . . . . . . . . . . . . . . . 51

7.3 For the Experimentalist . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

VIII CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

APPENDIX A EQUILIBRIA DATA . . . . . . . . . . . . . . . . . . . . . . . . . 56

APPENDIX B HETEROCLINIC CONNECTION DATA . . . . . . . . . . . . . 62

APPENDIX C SYMMETRIES AND ISOTROPIES OF SOLUTIONS . . . . . . 64

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

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VITA

Jonathan Halcrow was born in 1981 in Port Jefferson, NY, and started roaming the coun-tryside between New York and Virginia for 14 years. Eventually he settled in Georgia andattended North Gwinnett High School. After graduation in 1999, it was off to Georgia Techto get a B.S. in Math and another in Physics in 2003. Realizing that theres no place likehome, he stayed there for graduate school.

In 2008 he got a haircut, cleaned his desk and the access corridor to it, started eat-ing fruits and vegetables, started bicycling to work, stopped smoking, lost weight, all inpreparation to accession to yuppiedom and long AARP years that are to follow. At somepoint in all this he managed to find the time to write a thesis.

Isotropy subgroups: The 67-fold path to symmetric enlightenment

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