unsteady two-dimensional flows in …...ﬂow include a hopf bifurcation from steady to oscillatory...

UNSTEADY TWO-DIMENSIONAL FLOWS IN COMPLEXGEOMETRIES: COMPARATIVE BIFURCATION STUDIES WITH

GLOBAL EIGENFUNCTION EXPANSIONS∗

ANIL K. BANGIA† , PAUL F. BATCHO‡ , IOANNIS G. KEVREKIDIS†‡ , AND

GEORGE EM. KARNIADAKIS§

SIAM J. SCI. COMPUT. c© 1997 Society for Industrial and Applied MathematicsVol. 18, No. 3, pp. 775–805, May 1997 007

Abstract. We present a bifurcation study of the incompressible Navier–Stokes equations in amodel complex geometry: a spatially periodic array of cylinders in a channel. The dynamics of theflow include a Hopf bifurcation from steady to oscillatory flow at an approximate Reynolds number Rof 350 and the appearance of a second frequency at approximately R ' 890. The multiple frequencydynamics include a substantial increase in spatial and temporal scales with Reynolds number ascompared with the simple limit cycle oscillation present close to R = 350. Numerical bifurcationstudies of the dynamics are performed using three forms of global eigenfunction expansions. The firstbasis set is derived through principal factor analysis (Karhunen–Loeve expansion) of snapshots fromaccurate direct spectral element numerical solutions of the Navier–Stokes equations. The second setis obtained from the eigenfunctions of the Stokes operator for this geometry. Finally eigenfunctionsare derived from a singular Stokes operator, i.e., the Stokes operator modified to include a variablecoefficient which vanishes at the domain boundaries. Truncated systems of (∼ 100) ODEs areobtained through projection of the Navier–Stokes equations onto the basis sets, and a comparativestudy of the resulting dynamical models is performed.

Key words. eigenfunction expansions, Galerkin method, bifurcation, continuation

AMS subject classifications. 58F39, 58F27, 58F21, 35B32, 35B60, 35B40

PII. S1064827595282246

1. Introduction. The analysis of hydrodynamic stability has classically beenbased on the linearization of the equations of motion about a given, usually stationary,solution [1, 2]. Methods for the study of secondary instabilities (e.g., [3]) have beenapplied to comparatively simple geometries and have often relied on the reduction ofthe problem to one-dimensional canonical or amplitude equations. Transitional flowsin complex geometries pose a much greater difficulty as there are neither analyticalstationary solutions nor simple analytical sets of spatial eigenfunctions available forthe geometry. Spatial discretizations result in large-scale dynamical models with toomany (∼ 105-107) ODEs or differential algebraic equations (DAEs) to make routinestability analysis possible, especially for time-dependent solutions. Traditional waysof studying such systems have been based on obtaining their low-dimensional repre-sentations near a particular marginally stable state (usually at steady states) usinga Lyapunov–Schmidt or center manifold reduction [4, 5, 6]. In a different approachto the same goal of low-dimensional reduction, rigorous proofs have been derived re-cently for several dissipative PDEs (e.g., [7]) regarding the existence of an inertialmanifold; inertial manifolds are finite-dimensional, smooth, invariant manifolds thatattract all trajectories exponentially in the (appropriate, infinite-dimensional) phasespace and contain the (global) attractor. The long-term dynamics arising from such

∗Received by the editors February 24, 1995; accepted for publication (in revised form) September29, 1995. This work was partially supported by the National Science Foundation and ARPA/ONR.

http://www.siam.org/journals/sisc/18-3/28224.html†Chemical Engineering, Princeton University, Princeton, NJ 08544 (akbangia@arnold.

princeton.edu).‡PACM, Princeton University, Princeton, NJ 08544 ([email protected]).§Applied Mathematics, Brown University, Providence, RI 02912. This author was supported in

part by AFOSR.

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776 BANGIA, BATCHO, KEVREKIDIS, AND KARNIADAKIS

a PDE should then, in principle, be represented by a small set of ODEs (called theinertial form) on this manifold. While the proof of the existence of such a manifoldfor the two-dimensional Navier–Stokes equations is not yet available, considerableprogress in its direction, including the proof of the existence of an inertial form, hasbeen made in the case of simple geometries [8]. No significant inroads have been madein the case of complex geometries, again because of the lack of analytical sets of globaleigenfunctions.

In this work, instabilities of flows in complex geometries are studied by low-dimensional projections of the Navier–Stokes equations onto suitable basis sets ofglobal, divergence-free functions. We expect such an approach to yield low-dimensionalrepresentations that capture the dynamical behavior of our model flows over a broadrange of Reynolds numbers (rather than just in the neighborhood of some particular“linearization” Reynolds number). In particular, we want to compare the perfor-mance, in this respect, of two different types of global eigenfunctions:

• sets derived a priori from the eigensystems of various differential operatorssuch as the Stokes and singular Stokes [9], [10] operators, and

• empirical eigenfunctions obtained from an appropriate principal componentanalysis of accurately computed (through direct simulation) solutions of theNavier–Stokes equations [11, 12, 13].

Each of these sets, through progressive truncation and Galerkin projection of theNavier–Stokes equations, provides a distinct hierarchy of systems of (an increasingnumber of) ODEs.

We analyze the solutions of these resulting systems of differential equations ob-tained from the various basis sets and study their ability to reproduce the dynamicsand transitions exhibited by the direct numerical simulation of the flow. As ourtestbed example we have selected a model complicated geometry: the “eddy pro-moter,” a two-dimensional, multiply-connected channel domain with a cylinder con-tained in it. With periodic boundary conditions in the streamwise direction, thiscorresponds to a spatially periodic array of cylinders in a channel. This flow is knownto exhibit, in two spatial dimensions, secondary transitions from periodic to quasi-periodic dynamics. The stability and bifurcations of the “base” limit cycle oscillationis studied via linearization around periodic solutions (Floquet analysis); such sta-bility studies for time-dependent solutions would be intractable without the use oflow-dimensional reduction methods.

The eddy-promoter domain is smooth, and thus highly accurate computations ofthe Navier–Stokes equations can be made with the spectral element method (SEM)[14, 15, 16]. Here simulations were performed with a ninth-order Gauss–Lobatto–Legendre polynomial in each spatial direction for the spectral element discretization.The spectral element skeleton mesh is given in Figure 1.1. The time integration usedfor the spectral element simulations was a semi-implicit second-order splitting scheme[17] composed of an Adams–Bashforth substep for the nonlinear term, a pressuresubstep, and an implicit Crank–Nicolson substep for the viscous terms.

The eddy-promoter domain was chosen due to the increased appearance of spatialand temporal scales in the long-term solution (the attractor) as a function of Reynoldsnumber R. Figure 1.2 illustrates the kind of solutions that this geometry exhibits forvarious Reynolds number ranges. The time history of the u velocity at the (arbitrary)point “A” marked in Figure 1.1 is also shown. The flow asymptotically approachesa stable stationary state below a Reynolds number of R ' 350. As the Reynoldsnumber is further increased beyond 350, the steady flow loses stability to a stable

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BIFURCATION STUDIES OF TWO-DIMENSIONAL FLOWS 777

0 1 2 3 4

-1

-0.5

0

0.5

1

FIG. 1.1. Spectral element mesh for the eddy-promoter geometry. The domain is composed of 88spectral elements, each with 9 × 9 collocation points. The flow is driven via a body force f = (2ν, 0)from left to right, and periodic boundary conditions are imposed in the streamwise direction. Thetop and bottom are walls with rigid boundary conditions.

time-periodic flow. At approximately R ' 890 the periodic solution loses stability,apparently to a stable quasi-periodic flow, which then persists well beyond R = 1000.The eddy-promoter domain has been previously studied both computationally [18]and, recently, experimentally [19]. These studies investigated a 50% longer domain,where the first bifurcation to unsteady flow occurs at a lower Reynolds number ofapproximately 150. In our simulations the shorter domain (smaller spacing betweencylinders in a periodic array) acts as a stabilizing influence. The experimental resultsindicate that the first bifurcation is two dimensional; a secondary three-dimensionalinstability is observed at a slightly larger Reynolds number with respect to the onsetof unsteady flow.

Here, we focus on the subsequent two-dimensional instability of the flow becauseof its comparative complexity and the associated modeling and computational issues.A numerical study of this instability requires the computation and computer-assistedlinearized stability analysis of limit cycle solutions of the Navier–Stokes equation. Thecomputational effort involved in such a task for sets of n coupled nonlinear ODEs in ashooting formulation involves (for each iteration) the time evolution of n2 + n ODEsaround the limit cycle. The efficient truncation of the Galerkin set of projected ODEsis therefore essential for making the computational analysis of secondary instabilityof periodic flows practical. Certain techniques like Krylov subspace methods [20, 21]have been suggested as potentially useful in tracking large systems and are currentlyunder investigation. This alternative method involves a technique for the constructionof a low-dimensional Krylov subspace in which short-term approximate solutions ofthe original large problem are found; it has been successfully applied to time evolutionand linear stability analysis of Couette–Taylor flow [22], computation of leading modesand stability of slide coating flow [23], and other applications involving solutions tosystems of differential equations [24, 25, 26].

This paper is organized as follows: in section 2 we discuss the procedures forthe construction of divergence-free orthogonal eigenfunctions as trial basis sets andD

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steady periodic quasi−periodic

0 450 900

. . . . . . o o o o o o o o o 0 0 0 0 0 0 0

l l1350

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l

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0 200 400t

0.0

0.2

0.4

u

0 200 400t

0.0

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0.4

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R=100 R=850 R=900

FIG. 1.2. Schematic bifurcation diagram of the eddy-promoter geometry as suggested by directnumerical simulation. The stationary solution becomes unstable at R ' 350 yielding a periodicsolution branch which then loses stability at R ' 890 to a quasi-periodic flow.

their properties. In section 3 we illustrate the procedure of Galerkin projection ofthe Navier–Stokes equations onto the basis sets to obtain nonlinear ODEs describingthe time evolution of the expansion coefficients. A brief discussion of computationalmethods for bifurcation and continuation analysis of stationary and periodic solutionsfor our nonlinear differential equations is included in the Appendix. We then presentresults for the eddy-promoter domain in section 4. Comparison, evaluation, anddiscussion of the results follows in section 5.

2. Basis sets.

2.1. Generalized Stokes eigensystem. The use of divergence-free expansionsin the numerical solution of the Navier–Stokes equations (in order to circumvent the

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incompressibility constraint) has been advocated in the past for simple geometries(e.g., [27, 28], where the basis was derived from combinations of Jacobi polynomials).Here one such type of basis is obtained from the solution of the following generalizedconstrained self-adjoint eigenproblem [9, 10]:

−∇Π + ∇ · [ρ(x)∇w] − q(x)w = λw in Ω,(2.1)∇ · w = 0 in Ω,(2.2)

Bw = 0 on ∂Ω,(2.3)

where the resulting set of w ∈ H 10 is a global spectral basis, H 1

0 is a Sobolev space ofdivergence-free vector functions, B is a linear boundary operator, and ρ(x) > 0 in theinterior of the domain Ω; ρ(x) ∈ C1 and may vanish on the domain boundary ∂Ω. Thescalar q(x) ∈ C0 can become infinite at the boundaries and is assumed to be positive.The boundary operator B can, in general, enforce any boundary condition suitablefor eigenproblems: homogeneous Robin, Neumann, Dirichlet, periodic or mixed. Heremixed Dirichlet and periodic cases are examined, since by performing divergence-freeprojections these boundary conditions lead to the elimination of the pressure gradientof the Navier–Stokes equations. The scalar Π(x) is a Lagrange multiplier used toimpose the solenoidal constraint, equation (2.2).

The operators of interest in this study are the Stokes (ρ(x) = 1, q(x) = 0) as wellas full singular Stokes operators [9]. The singular system is defined so that the scalarρ(x) vanishes at the domain boundaries and the scalar q(x) becomes infinite there.This will yield a complete set of orthogonal eigenfunctions that vanish at domainwalls. The singular system was motivated by its scalar counterpart which will exhibitexponential convergence rates in the expansion coefficients

∑∞n=0 anwn(x) for smooth

functions (u ∈ C∞). For nonsmooth functions, such as those resulting from thesolution of elliptic equations in domains with geometric singularities, the convergencerates are dictated by the degree of smoothness of the function, an ∼ O(λ−p

n ), wherep is the highest derivative for which the scalar version of equation(2.4) holds

0 = limε→0

[∮ρ(x) [∇pu · (n · ∇wi)] ds

].(2.4)

Here ε can be taken as a measure of distance from the wall, and the limit expressesthe value of the integral as the boundary ∂Ω is approached; the surface integral isgiven for the vector function case. The scalar ρ(x) is somewhat arbitrary as long asthe above properties are maintained. In this study ρ(x) was taken to be the solutionof a Poisson equation with a negative definite (constant) forcing

∇2ρ(x) = C < 0 in Ω,(2.5)Bρ(x) = 0 on ∂Ω.(2.6)

The positive definiteness of the scalar ρ(x) can be guaranteed by the minimum prin-ciple property of elliptic operators; in addition, the operators in complex geometriescan be easily inverted by standard techniques. The scalar q(x) is defined to be 4

ρ(x) ,and ρ(x) must vanish at a suitable rate to enforce the eigenfunctions to vanish at theDirichlet boundaries. The factor 4 was chosen because the first eigenfunction in thecase of a simple channel domain for this choice is exactly the stationary solution ofthe Navier–Stokes euqation. The computational methods for solving the eigensystemsemployed a spectral element method for spatial discretization and a Lanczos iterationprocedure for the discretized eigenproblem; for further details see [9] and [10].

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Figure 2.1 displays the first four and a few higher Stokes eigenfunctions for theeddy-promoter geometry. The velocity vector plots of the first four singular Stokeseigenfunctions, for ‖ρ(x)‖∞ = 1.3, are given in Figure 2.2; here, the choice of 1.3gave the first eigenfunction closest in the L2 context to the Stokes flow solution forthe domain. The first four eigenfunctions of the two systems are considerably similarin spatial structure. The two eigensystems tend to depart from one another in thehigher modes where the zeros of the singular Stokes eigenfunctions begin to clusterat the Dirichlet boundaries. This clustering of zeros near Dirichlet boundaries maywell be responsible for the robustness of the singular Stokes eigensystem versus theStokes eigenfunctions in approximating complicated solutions, as discussed in [9]. Itis somewhat of an open question as to whether exponential convergence is found

(a) (e)

(b) (f)

(c) (g)

(d) (h)

FIG. 2.1. Velocity vector plots of Stokes eigenfunctions for the eddy-promoter domain. Athirteenth-order Gauss–Lobatto–Legendre polynomial was used in the spectral element discretizationin both spatial directions. The first four eigenfunctions are plotted in (a)–(d), respectively. Modes10, 20, 30, 40 are shown in (e)–(h), respectively.

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(a) (e)

(b) (f)

(c) (g)

(d) (h)

FIG. 2.2. Velocity vector plots of singular Stokes eigenfunctions for the eddy-promoter domain.A thirteenth-order Gauss–Lobatto–Legendre polynomial was used in the spectral element discretiza-tion in both spatial directions. The first four eigenfunctions are plotted in (a)–(d), respectively.Modes 10, 20, 30, 40 are shown in (e)–(h), respectively.

for the vector function analog to the scalar singular Sturm–Liouville problem. Thecomputational limitations are dictated by the need to resolve scales on the order ofthose in the eigenfunction associated with the largest eigenvalue. The singular Stokeseigensystem also exhibits large gradients near the wall region due to the clusteringof zeros there. The spectral element method can effectively deal with such cases byadding a thin layer of elements near the walls.

2.2. Empirical eigenfunctions. The above eigensystems have the advantageof providing an infinite number of eigenfunctions which can be solved a priori withsuitable numerical eigensolvers without any previous knowledge of the time-dependentsolutions of the Navier–Stokes equations. On the other hand, if the flow field isknown (or can be easily obtained) through either experiments or highly accurate nu-

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merical simulation of the Navier–Stokes at some particular Reynolds number valueor Reynolds number regime, then suitable global basis functions can be tailored fromthe flow field using an approach referred to as the “proper orthogonal decomposition”(POD) [29]. This is a standard statistical pattern recognition and image compressiontechnique (“principal component analysis”) used in the study of spatiotemporal pat-terns [30]. Lorenz [31] introduced this technique in the context of meteorology as the“method of empirical orthogonal eigenfunctions.” Lumley [29] proposed the use of thisprocedure to study “coherent structures” in the context of turbulent flow and coinedthe term POD. In other disciplines the same procedure is known as Karhunen–Loeve(KL) expansion and seems to have been independently rediscovered several times. Inrecent years this approach has been used in the modelling of a variety of systemsranging from turbulent wall layers [32, 33], turbulent Rayleigh–Benard convection[34], and time-dependent flows in complex geometries [35], to heterogeneous catalyticreaction-diffusion experiments [36, 37]. A detailed description of the mathematicalproperties of the method and its applications to turbulent flows can be found in arecent review by Berkooz, Holmes, and Lumley [11].

Application of this method first requires a priori storage of velocity fields (snap-shots) of the time-dependent flow; they constitute the data ensemble. Utilizing thespatial velocity correlations, the POD procedure identifies as dominant spatial struc-tures in the flow those with the most “energy,” i.e., mean square fluctuation. Thesestructures provide an orthogonal set of basis functions for a series representation ofthe ensemble data and are ordered in terms of their contribution to the total energyof the system. The basis is optimal in the sense that a truncated series representationof the data has a smaller mean square error than a representation in any other basisset of the same dimension for the same ensemble. The usual procedure can be sum-marized as follows [12]: given a spatiotemporal velocity field defined on a domain Ω,which in our case is the computational domain, we seek to maximize the mean squarefluctuation of the function ψi(x)

〈(u, ψi)(u, ψi)〉(ψi, ψi)

= λi.(2.7)

Here 〈〉 is the ensemble averaging operation which is typically taken to be the timeaverage for spatiotemporal data. The inner product is defined in the usual way,(f, g) =

∫Ω f(x)g∗(x)dx. This is a classical problem in the calculus of variations.

Maximization of the above quantity for a ψi(= ϕi) gives∫Ω〈u(x)u(x′)〉ϕi(x′)dx′ = λiϕi(x).(2.8)

The solutions to this equation, ϕi, are eigenfunctions of the autocorrelation oper-ator R(x, x′) = 〈u(x, t) u(x′, t)〉 with eigenvalues λi, i = 1, 2, . . .. These empiricaleigenfunctions are ordered according to their decreasing energy content (given by λi).The nonnegative definiteness of R(x, x′) assures that λi ≥ 0 and its eigenfunctionsform a complete orthogonal basis set. Any member of the ensemble may now bereconstructed from its modal decomposition on the eigenfunctions

u(x, t) =M∑i=1

ai(t) ϕi(x),(2.9)

where M is large enough such that modes (M + 1) and higher contain “negligible”energy. The modal amplitudes are orthogonal with respect to the chosen ensemble

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average

〈ai(t) aj(t)〉 = δij λi.(2.10)

Obtaining POD modes from the autocorrelation matrix can become computa-tionally very expensive for large spatially extended systems. The eigenmodes canalternatively be well approximated by using the “method of snapshots” described bySirovich [13, 12]. The ensemble averaged flow field is subtracted from the snapshots,so that the fluctuations about this mean field are decomposed into eigenmodes usingthe POD procedure. (We also performed a variant of this procedure by decomposingthe snapshots themselves into POD modes without subtracting the mean; the use ofthis second set of eigenfunctions gave very similar results to the first set—with meanfield subtracted—which we present.) Table 2.1 shows the first few eigenvalues of thePOD decomposition of the “reference” flow state at R = 1000. Direct numerical sim-ulation of the flow was performed using the spectral element method, and the timeevolution was continued until the startup transients decayed. The temporal dynamicsof the flow are apparently quasi periodic at this value of the Reynolds number. A totalof 120 snapshots were stored as the simulation ensemble over a temporal evolutionof 20 fundamental periods. The corresponding first four eigenfunctions are shown inFigure 2.3. A few higher modes and their eigenvalues are also included for comparisonof spatial scales.

Performing the POD procedure on fluctuations around the ensemble mean givesthe distribution of “energies” (eigenvalues) shown in the second column of Table 2.1.When the mean field is not subtracted from the data, the first POD mode, as onewould expect, appears very similar to the mean field and carries more than 93% ofthe total energy. Removing the energy of this first mean-like mode and renormalizingshows that the relative energies of the subsequent modes (fourth column in Table 2.1)compare closely with those in the second column (the fluctuation-POD modes). Thismore even distribution of energies in the second column modes leads to a slightly betterconditioning of numerical schemes for continuation and bifurcation analysis since nowno single mode is highly energetic compared with the rest. The eigenfunctions tend toappear in pairs, whose members are approximately phase shifted with respect to eachother in the streamwise direction. This result directly reflects the periodic boundaryconditions, and a linear combination of each pair can be approximately viewed as asingle “travelling wave” in the direction of the flow. The nature of the higher modesis not easy to describe; as we progress higher in the mode hierarchy, we find modeswith increasingly more complex spatial structure, as indicated by the number andcorresponding scale of “vortices” they contain.

Figure 2.4 is an attempt to compare the efficiency of the first few (here 20)eigenfunctions of each set in capturing a typical flow field. While the “error” (theunspanned component of the flow field) of the singular Stokes modes is concentratedin the viscous shear layer close to the cylinder, the corresponding error of the PODmodes has a much smaller norm and is spatially much more evenly distributed. Wewill see below that this efficiency ranking persists in the dynamic predictions of thecorresponding truncations.

3. Galerkin projections of the Navier–Stokes equations. The Navier–Stokes equations written in abstract form for a divergence-free function space aregiven as [38]

dvdt

+ νAv +B(v,v) = f ,(3.1)

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(a) (e)

(b) (f)

(c) (g)

(d) (h)

FIG. 2.3. Velocity vector plots of the empirical eigenfunctions for the eddy-promoter domain ata Reynolds number of 1000 are presented. A total of 120 snapshots obtained from direct simulationusing a ninth-order spectral element discretization with Gauss–Lobatto–Legendre polynomials weredecomposed. The first four eigenfunctions are plotted in (a)–(d), respectively. The mean field isshown in (e). Modes 10, 20, 40 are plotted in (f)–(h), respectively.

TABLE 2.1Eigenvalues of the POD decomposition of N = 120 snapshots at R = 1000.

Mode no. λAi , using mean λB

i , obtained using λBi after

i subtracted snapshots original snapshots renormalization0.93629

1 0.40605 2.59268 × 10−2 0.406952 0.37724 2.41673 × 10−2 0.379333 6.51934 × 10−2 4.17634 × 10−3 6.55523 × 10−2

4 6.02405 × 10−2 3.85905 × 10−3 6.05721 × 10−2

20 1.19736 × 10−3 7.28962 × 10−5 1.14419 × 10−3

40 5.89858 × 10−5 3.65147 × 10−6 5.73139 × 10−5

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(a)

(b)

(c)

(d)

(e)

FIG. 2.4. (a) A representative flow field at R = 1000 obtained from direct SEM simulation,(b) the same field reconstructed using the first 20 empirical modes, (c) the same field reconstructedusing the first 20 singular Stokes eigenfunctions, (d) the component of the flow field (a) not spannedby the first 20 empirical modes (difference of (a) and (b)), (e) the component of the flow field (a)not spanned by the first 20 singular Stokes modes (difference of (a) and (c)). This highlights theinability of a few (here 20) singular Stokes modes to capture the flow field structure, especially inthe viscous shear layer in the vicinity of the cylinder.

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where

A : D(A) ⊂ H → H , A = −P∇2,

B(v,v) = P (v · ∇v),

and P is the orthogonal projection P : L2(Ωd) → H . We have

H = v ∈ L2(Ω)| ∇ · v = 0,H 1

0 = ∇v ∈ L2(Ω)| ∇ · v = 0, Bv = 0 on ∂Ω.(3.2)

Periodic boundary conditions are also contained in H10, and solutions are obtained by

expanding the velocity field fluctuations about a constant “mean” U (0 in the caseof Stokes or singular Stokes) in terms of the trial basis

vN (x, t) = U(x) +N∑

i=1

ai(t)wi(x), i = 1, 2, 3, ..., N,(3.3)

where wi(x) is complete in H10. The basis and test functions are taken to be the

same for the Galerkin approximation. By taking the inner product of the governingequations with the test functions, we obtain a nonlinear system of ODEs for the time-dependent coefficients of the expansion. The use of solenoidal test functions removesthe pressure from the system through Green’s identity∫

Ωwi · ∇p =

∫Ω(∇ · wi)p−

∫∂Ωp(n · wi).(3.4)

The system of ODEs is truncated by retaining an N -term Galerkin projection and isgiven by

dai

dt= c1i + c2i,jaj + c3i,j,kajak + ν(c4i + c5i,jaj) + fi, i, j = 1, . . . , N,(3.5)

where

c1i = −(wi,∇ · (UU)),c2i,j = −(wi,∇ · (Uwj)) − (wi,∇ · (wjU)),

c3i,j,k = −(wi,∇ · (wkwj)),

c4i = (wi,∇2U),c5i,j = (wi,∇2wj),fi = (wi, f).

The existence of solutions has been proved for the case of d = 2, while for the d = 3case, existence is known only for finite time intervals [38, 39]. Here, only the d = 2case is considered for N on the order of 102.

The computation of the projection coefficients for the nonlinear term is responsiblefor the largest percentage of preprocessing time. There are N3 coefficients which mustbe computed and stored before time stepping begins. The computational complexityof evaluating the nonlinear projection coefficients can be significantly reduced byintegrating by parts and arriving at

c3i,j,k = −(wi,∇ · (wkwj)) = (∇wi, (wkwj)).(3.6)

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The evaluation of the nonlinear coefficients is therefore reduced to a single computa-tion of the gradient for each eigenfunction and a series of inner product operations.A further reduction in computational effort can be obtained by exploiting knownsymmetries of the coefficients

c3i,j,k = −c3j,i,k,(3.7)

c3i,i,k = 0.(3.8)

The total computational effort required is N vector gradient evaluations and 12 (N3 −

N2) inner product evaluations. The storage requirements are 12 (N3 − N2) as well.

All numerical procedures for the evaluations of the projection coefficients, such asspatial derivatives and inner products, were computed on the p-order Gauss–Lobatto–Legendre mesh for our model geometry, where p is the order of the discretization usedto compute the eigenfunctions.

The evaluation of the right-hand side of the set of ODEs (3.5) obtained throughGalerkin projection of the Navier–Stokes has a structure similar to a number ofmatrix–vector multiplications and vector additions. Consequently, the vector field andJacobian evaluation operations are completely vectorizable. The limit cycle shootingand continuation code, which involves time integration of the N2 +N ODEs has beentimed to run on a single Cray C90 processor at a peak speed of 535 MFLOPS. Thecode also has a natural potential for parallelization, as the evaluation of different linearand nonlinear terms in the right-hand side of equation (3.5) can be easily distributedover a number of processors.

4. Results. In this section we present a comparison of the solutions of truncatedmodels (based on different eigenfunction expansions) of the Navier–Stokes equationsin the eddy-promoter geometry. Recall that an accurate SEM simulation of the flowreveals that the stationary solution loses stability to a simply time-periodic flow (alimit cycle) at R ' 350 (refer to Figure 1.2). The second instability occurs approxi-mately at R ' 890 when the periodic solution apparently undergoes a Neimark–Hopfbifurcation to a torus. The quasi-periodic nature of the resulting dynamics manifestsitself as an invariant circle on the Poincare surface (see Figures 1.2 and 4.3). Figure4.1 shows the power spectrum of such a quasi-periodic attractor at R = 1000. Theprimary frequency (the one evolving from the original Hopf bifurcation at R = 350)is marked as f1, while the one associated with the secondary bifurcation f2 is approx-imately ten times slower. The calculations were repeated for several mesh sizes toverify that the second frequency is not an artifact of the numerical discretization. Asshown in Figure 4.1, the location of the principal peaks essentially converged (the vari-ation remained within 5%). The simplest conceivable set of bifurcations, which willexplain the transitions in the flow, is a series of two supercritical Hopf bifurcations,the first one of the steady solution at R ' 350 and the second one of the periodicsolution at R ' 890. Besides the critical parameter values at which the transitionsoccur, we are interested in how well the low-dimensional models capture the long-termattractor of the Navier–Stokes flow. There is no definite answer as to what is the bestway of doing such a comparison. We have chosen to compare our model differentialequations to the SEM solution in the following two ways.

1. We compare how well the models capture the limit cycle attractor at a referencevalue of R = 750, and we do this in the phase space defined by the coefficients ofPOD (empirical) eigenfunctions. The solutions of both the SEM and the models areprojected onto the first three POD modes, and the phase projections are compared.

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0.00 0.02 0.04 0.06 0.08 0.10frequency

10-5

10-4

10-3

10-2

10-1

100

pow

er

7x7 mesh11x11 meshf1

f2

f1+f2

f1-f2

FIG. 4.1. Power spectrum of the quasi-periodic velocity field at Re = 1000 for two differentmesh sizes. The solid line corresponds to a mesh with 7×7 collocation points per element, while thedotted line, to 11 × 11 collocation points per element. The primary and the secondary frequenciesare marked as f1 and f2, and they match to within 0.5% and 5%, respectively.

2. At R = 1000, where the quasi-periodic long-term SEM solution lies on a torus,we make a similar phase space representation in POD coefficients and also look at itsPoincare map section for comparison.

Figure 4.2 illustrates the dynamics of the flow on a limit cycle at R = 750 forthe SEM solution. Here a1, a2, a3 are the coefficients of the projection of the SEMsolution on the first three POD modes, respectively. The temporal variation of the a1and a2 is shown in Figure 4.2(a). Figure 4.2(b) presents two phase space projectionsof the limit cycle. The Poincare section is defined by the instant when the projectiononto the second mode (i.e., a2) passes through zero in the increasing direction. ThePoincare sections are marked in Figures 4.2(a) and 4.2(b). The instantaneous flow fieldon the Poincare surface is shown (with the POD-ensemble mean flow subtracted) inFigure 4.2(c). The SEM data at R = 1000 are similarly presented in Figure 4.3.Figure 4.3(a) shows the temporal dynamics of projections of the solution onto thefirst two POD modes. The phase space projection of the torus is presented in Figure4.3(b). The Poincare cut of the torus appears as an invariant circle in the phase spaceas illustrated by Figure 4.3(c). Henceforth, the long-term solution of our models con-structed from Stokes, singular Stokes, and empirical eigenfunctions will be compared

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0 50 100 150t

-0.2

-0.1

0.0

0.1

0.2

a2

-0.2

-0.1

0.0

0.1

0.2

a1

(a) (b)

(c)

-0.2 -0.1 0.0 0.1 0.2a1

-0.04

-0.02

0.00

0.02

0.04

a3

-0.2 -0.1 0.0 0.1 0.2a1

-0.2

-0.1

0.0

0.1

0.2

a2

FIG. 4.2. Limit cycle solution at R = 750 obtained from direct simulation. (a) Time history ofprojections of the solution on the first two POD modes. (b) Two different phase space projectionsof the limit cycle attractor. (c) Instantaneous flow field (with ensemble mean subtracted) on thePoincare surface. The Poincare intersections are marked in (a) and (b).

with the SEM solution at R = 750 and R = 1000 by presenting them in a similarformat as we did for the direct simulation results.

4.1. Stokes eigenfunctions. Table 4.1 shows the critical values of the Reynoldsnumber at the primary bifurcation for Stokes eigenfunction models at various trun-cation levels. The primary instability is accurately captured by the first 100 eigen-functions. The instability is a supercritical Hopf bifurcation, and a pair of complexeigenvalues of the linearized flow cross into the right half plane at the critical Reynoldsnumber. Figure 4.4 shows the long-term dynamics of the 150 Stokes modes modelat R = 750. Reconstructions of the limit cycle projections are compared with thedirect simulation results (in dotted line) in Figure 4.4(b). The agreement between the

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−0.1−0.05

00.05

0.1

−0.2

−0.1

0

0.1

0.2−0.1

−0.05

0

0.05

0.1

a4a2

a3

-0.06 0.00 0.06a3

-0.06

0.00

0.06

a4

0 200 400 600 800 1000t

-0.2

-0.1

0.0

0.1

0.2

a2

-0.2

-0.1

0.0

0.1

0.2

a1

(a)

(b)

(c)

FIG. 4.3. Flow on a torus at R = 1000 obtained from direct simulation. (a) Time history ofprojections of the solution on the first two POD modes. (b) A phase space projection of the torusattractor. (c) The Poincare section of the torus at a2 = 0. The Poincare intersections are markedin (a) and (b).

TABLE 4.1Stokes eigenfunctions: critical Reynolds number values at the primary Hopf bifurcation.

N (Stokes) Rcr ± 5SEM 35020 26050 370100 325150 305

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0 50 100 150t

-0.22

-0.11

0.00

0.11

0.22

a2

-0.22

-0.11

0.00

0.11

0.22

a1

(a) (b)

(c)

-0.2 -0.1 0.0 0.1 0.2a1

-0.04

-0.02

0.00

0.02

0.04

a3

-0.2 -0.1 0.0 0.1 0.2a1

-0.2

-0.1

0.0

0.1

0.2

a2

FIG. 4.4. Limit cycle solution at R = 750 obtained using 150 Stokes eigenfunctions. (a) Timehistory of projections of the solution on the first two POD modes. (b) Two different phase spaceprojections of the limit cycle attractor. The phase space projection of the SEM solution is shownin dotted line for comparison. (c) Instantaneous flow field (with ensemble mean subtracted) on thePoincare surface (compare with Figure 4.2(c)). The Poincare intersections are marked in (a) and(b).

model and SEM is relatively poor, and the error in the Strouhal period is approxi-mately 12%. The flow field at the Poincare intersection is displayed in Figure 4.4(c),and there is significant deviation from SEM results in the vicinity of the cylinder.When the Reynolds number was increased to 1000, models using up to 150 Stokesmodes failed to reproduce the quasi-periodic dynamics of the flow. Therefore, the first150 Stokes eigenfunctions are able to accurately capture just the primary instabilityin the model eddy-promoter geometry.

4.2. Singular Stokes eigenfunctions. The critical Reynolds number valuesfor the primary Hopf instability is shown in Table 4.2 for different order truncation

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TABLE 4.2Singular Stokes eigenfunctions: critical Reynolds number values at the primary Hopf bifurcation.

N (S.Stokes) Rcr ± 5SEM 35020 29550 270100 330150 395

models using singular Stokes eigenfunctions. Again the first 100 eigenfunctions givean accurate representation of the primary instability. The dynamics of the flow atR = 750 obtained using the first 100 modes are plotted in Figure 4.5 and using the first150 modes in Figure 4.6. The phase space projections of the limit cycle presented inFigures 4.5(b) and 4.6(b) very well approximate the SEM data shown as a dotted line.The error in the Strouhal period atR = 750 is within 5%. A close inspection of Figures4.5 and 4.6 and Table 4.2 seems to indicate that after some “optimal” truncation size(here ∼ 100 modes), including more modes does not ameliorate the approximationcapabilities of the model. We believe that the (very slight) deterioration of the 150mode model compared with the 100 model is due to the progressive loss of accuracyin the computation of the higher eigenmodes themselves, and their inner productsneeded to evaluate the right-hand side of the ODEs. A similar phenomenon will beseen (and discussed) in the case of the empirical eigenfunction models.

The flow field at the Poincare map section is shown in Figures 4.5(c) and 4.6(c).The small structures in the shear layer region near the cylinder are very well pickedup by the singular Stokes modes, while the Stokes modes failed to do so. This is amanifestation of the “robustness” of singular Stokes eigensystems due to the clusteringof zeros in the higher modes near Dirichlet boundaries as discussed before. At R =1000, however, models with up to 150 singular Stokes modes fail to reproduce thelong-term temporally quasi-periodic nature of the flow, even though their short-termprediction capabilities are excellent. The periodic limit cycle ultimately exhibited atthis Reynolds number, however, is seen to live in the same “ballpark” in phase spaceas the torus obtained through SEM (see Figure 4.7(a)). On further continuation of theperiodic solution branch, the model eventually exhibited a secondary instability viaa supercritical Hopf bifurcation at R ≈ 1400. The bifurcation scenario is illustratedin Figure 4.7(b). The L2 norm of the coefficients of the modes at the Poincareintersection is plotted against the Reynolds number. The stable periodic solutionbranch (solid line) loses stability to a quasi-periodic solution (solid circles) via a Hopfbifurcation when a pair of complex conjugate Floquet multipliers cross out of the unitcircle in the complex plane.

4.3. Empirical eigenfunctions. The empirical eigenfunctions were obtainedfrom a POD of solution snapshots obtained from direct numerical simulation of theflow at R = 1000. Low-dimensional models obtained by Galerkin projection of theNavier–Stokes equations on the first 20, 40, 80, and 120 POD modes were studied(the dimension of the discretized SEM system is ∼ 2 × 104). Table 4.3 shows thecritical Reynolds number values at which the models exhibit the first Hopf bifurcationfrom the stationary solution. The first instability seems to be well captured by thefirst 40 empirical modes, even though these modes are obtained at the much higherdecompositional Reynolds number of 1000. Time integrations in the critical regionconfirm that the Hopf bifurcation is supercritical.

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0 50 100 150t

-0.2

-0.1

0.0

0.1

0.2

a2

-0.2

-0.1

0.0

0.1

0.2

a1

(a) (b)

(c)

-0.2 -0.1 0.0 0.1 0.2a1

-0.04

-0.02

0.00

0.02

0.04

a3

-0.2 -0.1 0.0 0.1 0.2a1

-0.2

-0.1

0.0

0.1

0.2

a2

FIG. 4.5. Limit cycle solution at R = 750 obtained using 100 singular Stokes modes. (a) Timehistory of projections of the solution on the first two POD modes. (b) Two different phase spaceprojections of the limit cycle attractor. The phase space projection of the SEM solution is shownin dotted line for comparison. (c) Instantaneous flow field (with ensemble mean subtracted) on thePoincare surface (compare with Figure 4.2(c)). The Poincare intersections are marked in (a) and(b).

Figure 4.8 shows the long-term dynamics on the limit cycle for the 120 modesmodel at R = 750. Figure 4.8(a) shows the temporal behavior of the coefficients ofthe first two eigenmodes. Two phase space projections of the limit cycle are plottedin Figure 4.8(b). The actual limit cycle from SEM solution is plotted as a dottedline. It is clearly seen that the actual limit cycle is very well approximated by thereconstructed limit cycle from the 120 POD modes. The error in Strouhal periodis approximately 5%. The flow field at the Poincare map intersection is displayedin Figure 4.8(c) and compares very well to the actual field in Figure 4.2(c). Thisdemonstrates the robustness of the POD eigenmodes in accurately capturing solu-

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0 50 100 150t

-0.2

-0.1

0.0

0.1

0.2

a2

-0.2

-0.1

0.0

0.1

0.2

a1

(a) (b)

(c)

-0.2 -0.1 0.0 0.1 0.2a1

-0.04

-0.02

0.00

0.02

0.04

a3

-0.2 -0.1 0.0 0.1 0.2a1

-0.2

-0.1

0.0

0.1

0.2

a2

FIG. 4.6. Limit cycle solution at R = 750 obtained using 150 singular Stokes modes. (a) Timehistory of projections of the solution on the first two POD modes. (b) Two different phase spaceprojections of the limit cycle attractor. The phase space projection of the SEM solution is shown ina dotted line for comparison. (c) Instantaneous flow field (with ensemble mean subtracted) on thePoincare surface (compare with Figure 4.2(c)). The Poincare intersections are marked in (a) and(b).

tions of the Navier–Stokes away (here R = 750) from the decompositional Reynoldsnumber (here R = 1000). Figure 4.9 illustrates the long-term quasi-periodic dynamicsof the same 120 modes model at R = 1000. Figure 4.9(a) shows the time evolutionof the coefficients of the first two modes, Figure 4.9(b) shows the phase space repre-sentation of the invariant torus, while the Poincare cut of this torus at (u, φ2) = 0is shown in Figure 4.9(c). The Poincare map invariant circle obtained from directsimulation is also shown in unfilled squares for comparison, and the agreement is verygood. Decreasing the number of modes in the model resulted in declining accuracyin predicting the long-term behavior of the flow, even though retaining its qualitativequasi-periodic nature. The short-term dynamics, however, are accurately reproduced

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800 1000 1200 1400 1600Reynolds number, R

0.55

0.60

0.65

0.70

||a||

-0.1-0.05

00.05

0.1

-0.2

-0.1

0

0.1

0.2-0.1

-0.05

0

0.05

0.1

a4a2

a3

(a)

(b)

FIG. 4.7. (a) Limit cycle solution at R = 1000 obtained using 150 singular Stokes modes. It livesin the neighborhood of the torus (dotted lines) obtained from direct simulation. (b) The bifurcationdiagram for the 150 singular Stokes modes exhibiting a supercritical Hopf bifurcation at R ' 1400.

by as low as 20 POD modes; these give an error of ∼ 7% after time evolution of anentire fundamental period.

In order to locate and characterize the secondary instability, numerical continu-ation of the periodic solution branch was performed in the Reynolds number range750–1000. Figure 4.10 illustrates the complete bifurcation scenario for the 80 POD

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TABLE 4.3Empirical eigenfunctions: critical Reynolds number values at the primary Hopf bifurcation.

N (PODs) Rcr ± 5SEM 35020 71040 38080 360120 405

0 50 100 150t

-0.2

-0.1

0.0

0.1

0.2

a2

-0.2

-0.1

0.0

0.1

0.2

a1

(a) (b)

(c)

-0.2 -0.1 0.0 0.1 0.2a1

-0.04

-0.02

0.00

0.02

0.04

a3

-0.2 -0.1 0.0 0.1 0.2a1

-0.2

-0.1

0.0

0.1

0.2

a2

FIG. 4.8. Limit cycle solution at R = 750 obtained using 120 POD modes. (a) Time historyof projection coefficients of the first two modes. (b) Two different phase space projections of thelimit cycle attractor. The phase space projection of the SEM solution is shown in dotted line forcomparison. (c) Instantaneous flow field (with ensemble mean subtracted) on the Poincare surface(compare to Figure 4.2(c)). The Poincare intersections are marked in (a) and (b).

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(a)

(b)

(c)

0 200 400 600 800 1000t

-0.2

-0.1

0.0

0.1

0.2

a2

-0.2

-0.1

0.0

0.1

0.2

a1

-0.06 0.00 0.06a3

-0.06

0.00

0.06

a4

−0.1−0.05

00.05

0.1

−0.2

−0.1

0

0.1

0.2−0.1

−0.05

0

0.05

0.1

a4a2

a3

FIG. 4.9. Flow on a torus at R = 1000 obtained using 120 POD modes. (a) Time historyof projection coefficients of the first two POD modes. (b) A phase space projection of the torusattractor. (c) The Poincare section of the torus at a2 = 0. The invariant circle computed usingSEM is shown in unfilled squares for comparison. The Poincare intersections are marked in (a)and (b).

modes model. The L2 norm in the POD space is plotted against the Reynolds num-ber. The bifurcation scenario for 20 and 40 mode system is qualitatively similar tothis. At a critical value of the Reynolds number, a complex conjugate pair of Floquetmultipliers crosses out of the unit circle (see Figure 4.10(b)). The secondary instabil-ity is a Hopf bifurcation giving birth to a stable quasi-periodic branch of solutions.Table 4.4 summarizes the critical values of the Reynolds number for modal expansionsof various orders.

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Im(W) Im(W) Im(W)

Re(W) Re(W) Re(W)

R=760 R=940 R=1000

750 800 850 900 950 1000Reynolds number, R

0.150

0.154

0.158

0.162

0.166

||a||

(a)

(b)

FIG. 4.10. (a) Bifurcation diagram for the 80 POD modes model exhibiting a supercritical Hopfbifurcation at R = 940. (b) At the critical value of Reynolds number, a pair of complex Floquetmultipliers cross out of the unit circle.

When we increase the size of the model further, however, we find a slight deteri-oration in the predicted dynamics, comparable with that seen before for the singularStokes system. Incorporating all 120 empirical eigenmodes in the model changes thebifurcation scenario as shown in Figure 4.11. A unique stable limit cycle exists belowR ' 870 and a stable torus above R ' 915. However, the stable limit cycle branch

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TABLE 4.4Empirical eigenfunctions: critical Reynolds number values at the secondary Hopf bifurcation.

N (PODs) Rcr ± 5SEM 89020 96540 91580 940

750 800 850 900 950 1000Reynolds number, R

0.150

0.155

0.160

0.165

0.170

0.175

||a||

a

b

FIG. 4.11. Bifurcation diagrams for models based on 120 POD modes. (a) Straight (linear)Galerkin model: here the stable torus branch is born through a global bifurcation involving a saddle-type unstable limit cycle branch of solutions at R ' 910. (b) 40 master–80 slave (nonlinear Galerkin)model: here the stable torus branch appears when the limit cycle branch of solutions becomes unstablevia a supercritical Hopf bifurcation at R ' 890.

(solid line) now first undergoes a saddle-node bifurcation at R = 916, and a brief re-gion of bistability (coexistence of stable periodic and stable quasi-periodic solutions)is observed. A careful investigation shows that the torus now arises from a global bi-furcation, involving a homoclinic crossing of the invariant manifold of a saddle type,unstable limit cycle.

The behavior of this model is qualitatively different from its lower-order trunca-tions in two ways. First, this model displays hysteresis and multiple stable solutionsin a certain narrow region of Reynolds number. Second, this model does not ex-hibit a direct secondary Hopf bifurcation from the periodic solution branch. It isprecisely the low dimensionality of our truncated model that allows us to continuearound folds in the limit cycle branches, compute unstable (saddle type) limit cycles,and approximate and follow their unstable manifolds, thus explaining this bifurcationscenario.

Even though no evidence of such a global bifurcation was found in direct spec-tral element simulation of flow through the eddy-promoter domain, such a bifurcationscenario is a viable alternative by which a periodic flow can lose stability to a stable

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temporally quasi-periodic flow. In this case the energy content of the high modes is solow that one might consider them as “noise” and filter them out. Furthermore, theirenergies are practically indistinguishable, and so the sense of hierarchy so obviousat the top of the eigenvalue list is lost. Changing the data ensemble and includingin it transient time series approaching the attractor will change the lower part (hereroughly the last 20) of the current hierarchy. We believe that in this case the slightdeterioration for the largest set is, in effect, the result of eventually including mean-ingless higher modes. A measure of the noise level and error in the data set as well asthe sensitivity of the basis functions to reasonable variations in the ensemble (differentsampling on the attractor, inclusion of transient data) is desirable.

4.4. Nonlinear Galerkin methods. While we observe that ∼ 100 modes arenecessary to span the solution and reproduce the dynamics, the temporal behaviorof the flow is relatively simple and could in principle be found using a much smallernumber of equations. Continuing our search for further reduction of the numberof independent degrees of freedom capable of representing dynamics of the flow, wenow briefly turn toward techniques motivated by the theory of approximate inertialmanifolds (AIM). All the approaches we described above yield a hierarchy of globalmodes. The idea here is to consider the first few (lower) modes of the hierarchy tobe the master modes which govern the dynamics of the flow. We think of the highermodes as slave modes in the sense that they “follow” the slow oscillations of thelower (master) modes. These faster moving slave modes quickly relax onto a manifoldparameterized by the master modes. An approximation to this manifold (the so-called steady manifold) is obtained by setting the time dependence of the slave modesto zero, a procedure known to be a valid approximation for certain evolutionary PDEs[40, 41]. This formal procedure gives a set of algebraic equations expressing the slavemodes as a function of the master modes:

dum

dt= f(um(t) + zm(t)) master,

dzm

dt≡ 0 = g(um(t) + zm(t)) slave.

The second equation above can be recast as zm = φ(um), which is the definition ofthe steady manifold.

We applied this master–slave reduction method built on the empirical eigenfunc-tion hierarchy to investigate the bifurcation sequences of the resulting differential–algebraic systems. Computing limit cycle solutions for the above system can be donethrough the application of a shooting technique to just the master modes (see theAppendix). Savings in the computational effort can be substantial compared withthe full Galerkin method, as the number of variational equations are reduced to m2

(where m is the number of master modes). The bifurcation sequence of the 40 master–80 slave model is compared with that of a straight (linear) Galerkin method in Figure4.11. The stable quasi-periodic branch of solutions arises from a supercritical Hopfbifurcation of the limit cycle branch at R ' 890. Table 4.5 displays the criticalReynolds number values at which various master–slave ratio models exhibit a sim-ilar Hopf bifurcation. The number of independent degrees of freedom necessary toobtain a converged transitional Reynolds number is therefore significantly reducedcompared with “linear” traditional Galerkin truncations. This might be expected,given the low-dimensional nature of the flow in the Reynolds number regime studied;the higher modes cannot be ignored without loss of accuracy, but it seems appropriateto treat them passively as a function of the “active” lower modes.

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TABLE 4.5Critical Reynolds number values at the secondary Hopf bifurcation for various size master–slavemodels based on the POD hierarchy.

Master/slave Rcr ± 5SEM 89020/60 93040/80 89080/40 890

5. Discussion. The use of suitable global divergence-free eigenfunction expan-sions appears to be a powerful tool for modeling Navier–Stokes flow through com-paratively complex geometries. In this work, we demonstrated the use of Stokesand singular Stokes eigenfunctions as well as POD-based modes as “geometrically fit-ted” basis sets for approximating Navier–Stokes solutions. We compared the abilityof these eigenfunctions to approximate the long-term attractors of the periodic andquasi-periodic flows and their accuracy in reproducing the transitions between thesestates in an objective manner. Note that the secondary transition to a quasi-periodicstate is not related to the onset of vortex shedding behind the cylinder.

Obtaining accurate models of spatially extended systems that have a minimalnumber of degrees of freedom is essential for a detailed computer-assisted study of theirdynamics; this is because dynamical systems techniques for characterizing the stabilityand bifurcations of time-dependent solutions can be practically applied to systemswith up to ∼ 100 degrees of freedom. Galerkin projections of the Navier–Stokesusing global basis sets onto a system of ODEs and reduction in dynamical systemsize through truncation of the projections allows us to perform stability calculationsfor limit cycle solutions in our model flow that would not be practical with the fullydiscretized system.

Singular Stokes modes perform markedly better than Stokes modes in terms ofcapturing both the short- and the long-term dynamics of the flow in the periodicregime (for a comparable truncation size). For wall-bounded flows in smooth domains(such as our example) the empirical eigenfunctions prove the “best” performers (inthe sense of accuracy and low dimensionality); they also seem to be quite robust in ap-proximating solutions to the Navier–Stokes away from the decompositional Reynoldsnumber. Obtaining empirical eigenfunctions, however, requires the knowledge of theentire flow field as a statistical input to the proper orthogonal decomposition pro-cedure. Another shortcoming of the POD procedure is the sensitivity of the “high”modes to the particular data ensemble as well as the gradual loss of the meaning ofmode hierarchy as the eigenvalues recede to very low levels, close to numerical noise.At the high end of the spectrum, it becomes difficult to identify meaningful low-energymodes (that may become more important in nearby values of the Reynolds number).

On the other hand, the arbitrariness in the scalars ρ(x) and q(x) makes thedefinition of an “optimal” singular Stokes operator (for our purposes) somewhat ofan open question. The idea of combining both empirical and theoretical approachesto modeling the flow in an effective way seems then also worth investigating. Someapproaches have been put forward by other researchers; Batcho [48] proposed theuse of the eigenfunctions of a Reynolds–Orr operator, while Poje and Lumley [47]proposed a procedure which requires the mean velocity field as statistical input, toextract spatial modes for turbulent flows that maximize the growth rate of volume-averaged coherent kinetic energy.

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The motivation to modeling flows of the type we investigate here is preciselytheir inherent low dimensionality (along with nontrivial dynamic behavior). Theasymptotic dynamics of such flows should, in principle, be completely characterizedthrough the knowledge of a small set of modes. The higher modes in our hierarchiesare often too important to be truncated away, but after the death of initial transientsthey should, at least in principle, be uniquely determined as functions of a few lowerones, through some sort of a slaving function (or AIM). The methods we discuss in thispaper, by providing us with a meaningful mode hierarchy, make the application of suchmaster–slave approaches possible. As a first computational attempt in this directionof further reduction of the model of independent degrees of freedom, we found that a“steady manifold” built on the POD hierarchy was distinctly successful in capturingboth the periodic and the quasi-periodic flow dynamics and the transition betweenthe two. Our calculations also suggest the existence of an “optimal cutoff” in themodal hierarchy, above which passive treatment of the “slaved” high modes becomesappropriate. In this work we focused on the successful reduction of Navier–Stokesflow models with emphasis on the minimal system size that would retain accuracy.The computational efficiency of these schemes is a subject of current research.

Appendix. Stability and continuation. The Galerkin projection of theNavier–Stokes equations on our basis sets results in systems of nonlinear coupledODEs of the form

dx(t)dt

= f(x(t), R), x(·), f(·, ·) ∈ Rn,(A.1)

where R is the Reynolds number. The steady states are computed using Newton’smethod, and the curve of a steady state as a function of the Reynolds number istraced using standard pseudoarclength continuation techniques [42, 43]. The stabilityof the steady states is governed by the eigenvalues of the Jacobian matrix Dxf ; theyare stable if all the eigenvalues lie in the left half plane.

Limit cycles (T-periodic solutions) of (A.1) are computed as fixed points of thePoincare return map using a shooting formulation. Details of the Floquet theory andthe general relation between Poincare maps and linearized flows can be found in manytextbooks, for example, in Hartman [44]. For a given state x0 on the limit cycle, theNewton iteration is performed to solve

R = x0 − x(T ) = 0,(A.2)

where x(T ) is the state of the system after some appropriate time interval T (a guessfor the period of the limit cycle). An additional constraint to single out a point on alimit cycle is provided by the Poincare surface “S,” in our case defined by recording thestate of the system every time it passes through a select value of one of the dependentvariables xi

S = xi(t = T ) − C = 0,(A.3)

where C is a constant. Equations (A.2) and (A.3) constitute a system of n equationswith n unknowns, namely, the dependent variables x1, x2, . . . , xi−1, xi+1, . . . , xn andthe period of the limit cycle T . The Jacobian matrix

∂R∂x0 = I − ∂x(T )

∂x0(A.4)

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of the system involves the n × n state transition matrix describing the sensitivity ofthe final state x(T ) to the initial condition x0 obtained by time integration of the n2

variational equations

V =∂f(x)∂x

V, V(t = 0) = I(A.5)

to give

V(t = T ) ≡ ∂x(T )∂x0 .(A.6)

For differential-algebraic systems, as in the case of master–slave models, equation(A.2) needs to be solved only for the ODEs (the master modes). Time integration ofthe sensitivity/variational equations were performed using algorithms like ODESSA[45] for ODEs and DDASAC [46] for DAEs. A pseudoarclength continuation schemeis employed to trace the periodic solution branch as a function of Reynolds number.The inhomogeneous sensitivity equations with respect to the Reynolds number R andthe period T are also used to complete the Jacobian calculation for the continuationscheme. The eigenvalues of the monodromy matrix V(t = T ) (Floquet multipliers)dictate the stability of the limit cycle; a change in stability occurs when one or a pairof complex conjugate multipliers cross out of the unit circle in the complex plane.

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unsteady two-dimensional flows in …...ﬂow include a hopf bifurcation from steady to oscillatory...

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