Low-dimensional models for complex geometry flows ... · plex geometry flows studied here. Modeling such flows re- sults, after discretization, in large dynamical systems (typi- cally

Low-dimensional models for complex geometry flows: Application to groovedchannels and circular cylindersA. E. Deane, I. G. Kevrekidis, G. E. Karniadakis, and S. A. Orszag

Citation: Physics of Fluids A: Fluid Dynamics 3, 2337 (1991); doi: 10.1063/1.857881View online: https://doi.org/10.1063/1.857881View Table of Contents: http://aip.scitation.org/toc/pfa/3/10Published by the American Institute of Physics

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Low-dimensional models for complex geometry flows: Application to grooved channels and circular cylinders

A. E. Deane,@vb) I. G. Kevrekidis,a)*b) G. E. Karniadakis,“)sc) and S. A. Orszaga)sc) Princeton University, Princeton, New Jersey 08544

(Received 30 October 1990; accepted 18 June 199 1)

Two-dimensional unsteady flows in complex geometries that are characterized by simple (low- dimensional) dynamical behavior are considered. Detailed spectral element simulations are performed, and the proper orthogonal decomposition or POD (also called method of empirical eigenfunctions) is applied to the resulting data for two examples: the flow in a periodically grooved channel and the wake of an isolated circular cylinder. Low-dimensional dynamical models for these systems are obtained using the empirically derived global eigenfunctions in the spectrally discretized Navier-Stokes equations. The short- and long-term accuracy of the models is studied through simulation, continuation, and bifurcation analysis. Their ability to mimic the full simulations for Reynolds numbers (Re) beyond the values used for eigenfunction extraction is evaluated. In the case of the grooved channel, where the primary horizontal wave number of the flow is imposed from the channel periodicity and so remains unchanged with Re, the models extrapolate reasonably well over a range of Re values. In the case of the cylinder wake, however, due to the significant spatial wave number changes of the flow with the Re, the models are only valid in a small neighborhood of the decompositional Reynolds number.

I. INTRODUCTION

Close to the onset of a primary instability, systems pos- sessing a large (potentially infinite) number of degrees of freedom often exhibit simple dynamical behavior in the sense that the long-term motion is characterized by a small phase space dimension. This is the case for transitional flows exhibiting periodic or chaotic oscillations, such as the complex geometry flows studied here. Modeling such flows results, after discretization, in large dynamical systems (typically of the order of 10’ ODE’s for a two-dimensional problem). It is sometimes possible to obtain low-dimensional representations of such large dynamical systems locally in the neighborhood of a particular state, using a center manifold reduction.le3 It has been recently shown rigorously for several PDE’s, including models arising from flow instabilities (e.g., Foias et a1.,4 Constantin et al.,’ and Ternam ), that the dynamics is exponentially attracted onto a low-dimensional subset of phase space (an inertial manifold), which contains the global attractor. In principle, therefore, for systems known to possess low-dimensional dynamics, the behavior is obtainable by a small set of ordinary differen- tial equations (e.g., Ref. 7 and references therein). In prac- tice such a reduction is far from easy and often (particularly so for the Navier-Stokes equations) unavailable; such methods have sometimes been observed to yield accurate low- dimensional representations even when an inertial manifold is not formally known to exist (e.g., Ref. 8).

Here we follow an alternative model reduction approach, based on the Karhunen-Loke expansion. This method has been developed in the context of meteorology as the “method of empirical orthogonal eigenfunctions” (e.g.,

” Program in Applied and Computational Mathematics. ” Department of Chemical Engineering. ” Department of Mechanical and Aerospace Engineering.

Lorenz’ ) and in the context of fluid mechanics as the “proper orthogonal decomposition (POD)” by Lumley.i”*” The method has been found to be useful and efficient in approxi- mating the dynamics of flows characterized by large-scale spatially coherent structures. Consider the instantaneous spatial solution profiles of a PDE (“snapshots”). The basic premise of the POD (the orthogonal decomposition of the spatial covariance of these snapshots) is part of classical pattern recognition methodologies.‘2s’3 In the context of time- dependent flows and, in particular, turbulent hydrodyna- mics, Lumley’“‘6 proposed to use this method on spatial velocity correlations to identify coherent spatial structures. A combination of this hierarchy of structures with a Galer- kin weighted residual discretization of the fundamental model equations (the Navier-Stokes) then provides a spatially and temporally accurate model of the PDE dynamics, provided that a sufficient number of modes has been retained.

The straightforward application of this approach to highly resolved two- and three-dimensional flows can be a formidable computational task. Recently the “method of snapshots” introduced by Sirovich has made such calculations practical.‘7-20 During the last few years there has been renewed interest in and extensive application of the POD methodology. It has been applied by Sirovich and his research group at Brown University to a number of examples (thermal convection,21-24 the Ginzburg-Landau equa- tion2’ ); it has also been applied by the Cornell group (to the wall region of a turbulent boundary layer26 ), and by other researchers (turbulence in a channe1,27-29 Burgers’ equation,30 a mixing jet,3’-33 and a periodically forced mixing layer34 ) . These applications typically have simple geometries, while here we apply the technique to flows in complicated geometries that have only recently been simulated with a high order of accuracy.35 These highly resolved nu-

2337 Phys. Fluids A 3 (lo), October 1991 0899-8213191 II 02337-I 8$02.00 @ 1991 American Institute of Physics 2337

me&al simulations provide data for the application of the POD and have the advantage over experimental data that the entire spatial velocity and pressure fields are available. The global eigenfunctions constructed by the POD constitute an optimal set of orthogonal basis functions on which the full simulation data is projected. They are then used in a Galerkin projection to form a model set of dynamical equations. The resulting model system is kept low dimensional by retaining only a small number of POD modes (the most “energetic”) in the Galerkin projection. That the first few modes provided by the POD hierarchy are sufficient, is cor- roborated by the fact that the temporal signature of the flows in the regime studied is comparatively simple. Further, the flow fields themselves can be seen to possess large-scale organized spatial structures, identified by the POD eigenmodes. Thus the POD not only provides insight into the nature of the flow and its instabilities, but also permits reduction of the PDE’s. This reduction is crucial in allowing a detailed stability and bifurcation analysis to be performed; such analysis is virtually impossible for the size of the equivalent system of ODE’s resulting from an accurate spectral element or any other high resolution simulations of tIows in complex geometries.

Other alternative strategies for obtaining a hierarchy of global modes satisfying the incompressibility and the boundary conditions include the construction of Stokes eigenfunctions and the use of eigenfunctions of the linearization around a particular (nonlinear) flow. Stokes eigenfunctions have been extensively used in function-analytic studies of the Navier-Stokes equations (e.g., T6mam36 and Constan- tin3’ ) and recent studies have appeared on the relation between these and the POD modes.38 The relation between POD modes and the eigenfunctions of linear or linearized systems has also been recently put forward.39*40 The compu- tation of both the Stokes modes and the linearized eigenmodes close to a particular flow is, of course, feasible. Since the POD modes are by construction optimal in spanning the nonlinear attractor, they should account better for the dynamics close to it, especially in the actual parameter regime where they are obtained.

The flows we study in this work are first, two-dimensional flow in a periodically grooved channel; and second, two-dimensional flow past a single circular cylinder. A detailed stability analysis of the first flow has been per- formed41*42 and companion experimental results have been reported. 43 Groove flows serve as a prototype in which the multipIe interactions of free shear layers, steady or unsteady vortices, and driven wall-bounded shear flows can be investi- gated in great detail. On the other hand, the fiow past a cylinder has served for nearly a century now as a model for fundamental studies of external flows and has indeed provided a “kaleidoscope of challenging fluid phenomena.“44 A quantification of all natural and forced states that the flow achieves in the Reynolds number regime (4O<Re$200) we examine here is given in Karniadakis and Triantafyllou.45 The application, therefore, of the POD technique to these flows will result first in a validation of the method in the low to modest Reynolds number range and complex geometries, and second in a better understanding of fundamental ques-

tions regarding these flows, i.e., frequency selection mecha- nism, transition, and bifurcation process.

The paper is organized as follows: In the next section we present the numerical simulations. Following this, in Sec. III we briefly describe the proper orthogonal decomposition procedure. In Sec. IV we present the resulting POD eigenfunctions for our prototype flows. In Sec. V the construction of a set of amplitude equations is described, and we conclude the paper with a discussion in Sec. VI. A comment on the notation: Superscripts refer to the Reynolds number while subscripts refer to the index of an eigenfunction. When there is no confusion about what Reynolds number is being referred to, the superscript is dropped.

II. DIRECT NUMERICAL SIMULATION OF FLOWS IN COMPLEX GEOMETRIES

In this section we briefly discuss the methodology employed to obtain the detailed databases for flow in a grooved channel and flow past a cylinder. We consider Newtonian fluids governed by the Navier-Stokes equations of motion and subject to the incompressibility condition

m x+Re-‘V’u+f in a, ot=-p (la)

Vu = 0 in 0, (lb) wherep is the static pressure, and the Reynolds number Re is defined separately for the two cases we consider; here D denotes total derivative, while f includes all forcing contributions. The solution domain 0 considered here is two dimensional.

To numerically solve ( la) we employ a high-order fractional spectral element scheme, according to which the solution at every time step is obtained after three substeps as follows: Considering first the nonlinear terms we obtain

e - I;J,&qy-q J--I ht

=-q~op,[ -N(u”-‘11, @a)

where N( u”) = 4 [ u”*Vu” + V*( u”u”) ] represents the nonlinear contributions written in skew-symmetric form at time level n At, and CI, and 6, are implicit/explicit weight coefficients for the stiffly stable scheme of order J (see Karniada- kis et aZ.46 ). The next substep incorporates the pressure equation and enforces the incompressibility constraint as follows:

(ii- ii)/At= -VP’+‘, Vl=O. (2b) Finally, the last substep includes the viscous corrections and imposes the boundary conditions, i.e.,

(you’+‘-d))/At=Re-‘V”u”+‘, (2c) where y0 is a weight coefficient of the backward differenti- ation scheme employed.“’

The above time treatment of the system of equations (2a)-( 2c) results in an efficient calculation procedure since it decouples the pressure and velocity equations as in (2b) and (2~). High-order time accuracy of this fractional scheme is achieved by solving the pressure equation (2b) in the form

V2pn+ ’ = V*(fi/At), (3)

2338 Phys. Fluids A, Vol. 3, No. 10, October 1991 Deane et al. 2338

k-lo--- 60 ,I

4 +---d=2

along with the consistent high-order pressure boundary con- dition46

J-I

-Reel C /?q[Vx(Vxii"-q)] >

, q=o

(4)

where II denotes the unit normal to the boundary I’. Equa- tion (3) is a Helmholtz equation with constant coefficients, which can be rewritten in the standard form

v*+ = g(x), (5) where we have defined + = p” + ‘, and g(x) = V*( B/At).

Spatial discretization of the Helmholtz equation in arbi- trarily complex domains is obtained using the spectral element methodology.47V48 For a two-dimensional geometry the computational domain R is broken up into several qua- drilaterals, which are mapped isoparametrically to canoni- cal squares. Field unknowns and data are then expressed as tensor products in terms of Legendre-Lagrangian interpo- lants. The final system of discrete equations is then obtained using a Galerkin variational statement, while efficient itera- tive algorithms (preconditioned conjugate gradients or spectral multigrid techniques) are used for the discrete system inversion. The advantage of the method and its success in simulating highly unsteady flows derives from its flexibil- ity to accurately represent nontrivial geometries while pre-

2339 Phys. Fluids A, Vol. 3, No. 10, October 1991

FIG. 1. (a) Flow in a grooved channel: Com- putational mesh. The domain is composed of 24 spectral elements each with 9 x 9 collocation points. The flow is from left to right, and periodic boundary conditions are imposed in the streamwise direction; bold lines represent walls with rigid boundary conditions. The geometry is two dimensional. (b) Flow past a circular cylinder: Computational mesh. The domain is composed of 58 spectral elements each with 7 X 7 collocation points. The flow is uniform at the inflow, periodic in the vertical direction, and outflow conditions described in the text have been applied.

(b)

serving the good resolution properties of spectral methods. Fast (exponential) convergence and parallel implementa- tions of the method make it also computationally competi- tive with low-order discretization techniques.49

The spectral element meshes for the grooved channel and the cylinder geometries considered here are shown in Figs. 1 (a) and 1 (b), respectively. Spectral quadrilateral elements of arbitrary size can be selected according to resolution needs. In the former case 24 elements of order 8 are used, while in the latter 58 elements of order 6 are used. Typical computations require 0.1 CPU seconds per time step on a single processor CRAY-Y/MP, while the time integration proceeds for approximately 5000 time steps (Ar~O.02). Extensive validation ofthese spectral element codes has been performed for similar flows as the ones considered here.4’p45

A. Grooved channel flow The first geometry considered is the periodically

grooved channel, a typical representative of a wall bounded flow with separation. The boundary conditions for the spectral element simulation therefore correspond to no slip at the rigid walls and periodicity in the streamwise direction. The flow is driven via a forcing term f = (2~,0). This is equivalent to the imposition of a constant mean pressure gradient, but leaves the pressure as written in (la) periodic in the horizontal direction. The equivalent pressure drop is scaled

Deane et a/. 2339

TABLE I. Mass flow rates and Reynolds number for flow in a grooved channel [Eqs. (6) and (7)).

r P Re

loo0 0.94 705 600 1.10 495 350 1.34 352

with the kinematic viscosity so that it maintains the flow rate (Q) approximately constant at different Reynolds numbers; variations in the Reynolds number are obtained through variations of the viscosity 1’. While such a scaling would elimi- nate any dependence of flow rate on the Reynolds number for a strictly parallel flow, the presence of the groove here results in a weak deviation (see Table I ) ; an empirical scaling of this flow rate variation as a function of the Reynolds number is given in Sec. V. The Reynolds number for this geometry is defined as

Re = 3&/4~. (6) Since the mass flow is not known precisely apriuri, we find it convenient to use the inverse viscosity as a bifurcation parameter and think of it as a reduced or modified Reynolds number:

r= l/v. (7) The stability properties of this flow have been studied in a series of papers. 4’s42,50 Briefly summarizing the results, it was found that for Reynolds number above a critical value, Reo, the flow reaches a steady periodic state, the frequency of which is determined by the ourer stable channel flow. The shear layer at the groove lip assists in destabilizing this outer flow component at a Reynolds number significantly lower than that corresponding to plane Poiseuille flow. Moreover, the nonlinear flow structure very closely resembles the linear channel eigenstructure (Tollmien-Schlichting waves) in spatial form as revealed by detailed linear stability analysis and comparisons with the corresponding plane Poiseuille flow.

The critical Reynolds number for the appearance of the first (Hopf) bifurcation, Re,, depends on the dimensions of the groove, i.e., its depth and width as well as the streamwise periodicity length. For the dimensions considered here [Fig. 1 (a) ] we found that Re, z 300. At a higher Reynolds number, Re, _ -400, three-dimensional modes become excited for a selected spectrum of spanwise wave numbers, which eventually lead to further bifurcations. Our interest here is in the two-dimensional flow evolution and thus we restrict the di- sect simulations to two dimensions only. In this case, a steady periodic solution is obtained even at high Reynolds number Rez 1000 with increasing amplitude of oscillation, but without any further apparent bifurcations.

B. Cylinder flow

Here Xi refers to each collocation point and N is their total number. The times t, are the discrete times at which the snapshots are taken.

The flow past a circular cylinder is a classical example of The eigenvectors of this matrix #j have the property that a prototype external flow. Its most distinct feature is the they form a complete orthonormal set (when properly nor- vortex street that forms in the wake following the first super- malized) and have eigenvalues Aj that quantify the probabil- critical (Hopf) bifurcation and persisting to very high Reyn- ity of their occurrence in the flow so that

olds number in fully turbulent states. It has been determined both experimentally5’ and numerically45 that this vortex street is sustained by the near wake absolute instability that appears at Reynolds number Re, =: 37. Here the Reynolds number is based on the cylinder diameter d and the free- stream velocity U, ,

Re= U,d/v. (8) Above that Reynolds number a steady periodic state is

established similar to the one described in the grooved channel flow. This monochromatic oscillation persists up to Reynolds number Re z 200; according to recent finding$’ the flow then becomes three dimensional and reaches another stable state. Further increase of Reynolds number leads to a cascade of period doublings until the wake becomes chaotic at Rez 500.

This “fast” ordered transition suggests that this model flow can be represented faithfully using the POD methodology. As a first step, here we report on the two-dimensional states of the Row. The direct simulations are based again on spectral element discretizations [see Fig. 1 (b) 1. As regards boundary conditions, a uniform velocity profile is prescribed at the upstream boundary, while periodicity is imposed at the two side boundaries; finally, at the exit a fully developed profile is assumed corresponding to a modified Neumann/ viscous sponge boundary condition.46

Ill. PROPER ORTHC?GONAL DECOMPOSITION

A review of the method may be found in Preisen- dorfer.53 A more accessible discussion directed toward fluid mechanics is given by Sirovich.‘7-‘9 It is essentially a ration- al method for obtaining basis eigenfunctions for time-varying flows. We describe the procedure in the discrete time case, which is the relevant situation here. It is usual to invoke the ergodic hypothesis that time averages represent ensemble averages. In the parameter regimes and cases we study the flow is not deterministically chaotic. Therefore, to obtain a representative “coverage” of the low-dimensional manifold we wish to parametrize, at least in a neighborhood of the long-term flow (the attractor), we use several transients starting from several initial conditions to form an ensembie (this has been termed an “extended” POD, or EPOD34 ). Since the dara are obtained through direct numerical irite- gration we have available the velocity and pressure fields at several (M) instants in time; these constitute the ensemble of data. If we separate the data into mean U and time-varying parts tl, e.g.,

v(x,t) = U(x) -i- u(x,t), then the velocity covariance matrix is obtained as

(9)

R, = L * 2 “(Wm )u(x&? 1, ij= 1,2 ,...) N. (10) m I

2340 Phys. Fluids A, Vol. 3, No. 10, October 1991 Deane et a/. 2340

a, = tbb+,Yh (11) where the parentheses denotes the inner product and the angle brackets denote ensemble averages. The sum of these eigenvalues gives the total “energy” E of the system,

E= 2 a,. (12) m=l

The eigenvectors are by construction incompressible, v*l$J, = 0, (13)

and satisfy the boundary conditions. The velocity at any instant can be expanded in terms of these eigenvectors as

u(x,t) = 2 a, to+, (xl, (14) m=l

and forms the basis of a Galerkin expansion. The straightforward evaluation of the eigenstructure of

the matrix ( 10) is a very large computational task (in our examples the dimension of the matrix is 104-105). Using the method of snapshots” (expanding the solutions not on the original basis functions, but on the snapshots themselves) the calculation reduces to a much more tractable eigenprob- lem (in our examples, of dimension - 10’).

IV. EXTRACTION OF EMPIRICAL EIGENFUNCTIONS

w

0.6

1 0.55

0.5

100 120 140 160 180 200

A. Grooved channel flow t For the flow in the grooved channel we perform the

decomposition at r = 350, where the long-term attractor of the flow is a limit cycle. A representative snapshot on this limit cycle, u( x,t = to ) is shown in Fig. 2(a) and time traces of the horizontal (u ) and vertical (0) velocity as sampled at the point marked A in Fig. 1 (a) are shown in Fig. 2 (b) . As a consequence of the oscillatory instability, in the main body of the channel the flow develops a small oscillatory vertical component as compared to the unbifurcated steady flow for r < 300. We perform the decomposition using M = 52 snapshots from our simulation ensemble. Fewer ( -20) snapshots gave similar results for these temporally simple flows. As noted in the previous section we choose to decompose the time-varying part of the flow (9). In Table II we show the six largest eigenvalues corresponding to the most energetic modes. The modes are found to occur in pairs of comparable magnitude eigenvalues down the stack. Clearly, the first two modes dominate, comprising more than 99% of the “energy” of the motion. The first four modes collectively contain more than 99.96% of the energy. Thus within these modes we have almost completely captured the spatial structure of the flow field “responsible” for the limit cycle behavior.

t

FIG. 2. Flow in a grooved channel: (a) Instantaneous velocity field at r = 350 on the limit cycle. (b) Time traces of the horizontal and vertical velocity at point A marked in Fig. 1 (a). The dots refer to the instant of (a).

modes. The algebraic and geometric multiplicity 2 eigenvalue that would correspond to an “exact” traveling (Toll- mien-Schlichting) wave for a plane channel flow is broken by the inhomogeneity in the geometry. This yields two dis-

The spatial structure of the mean flow U350 and that of the first four modes is shown in Figs. 3 (a) and 3 (b) . We note that the mean flow appears similar to the steady flow for r c 300. This underscores the advantage of performing the decomposition on the time-varying part. The structure of the eigenmodes represents the perturbation field. The eigenmodes have been normalized to have unit L, norm. The first two modes, + , and I&, appear virtually as phase-shifted ver- sions of each other. The presence of the groove breaks the translational symmetry of a plane channel and hence the degeneracy of an infinite number of translationally shifted

TABLE II. Flow in a grooved channel. Normalized eigenvalues of the POD modes and their cumulative contribution to the total energy for r = 350.

Index R % energy

1 0.525 08 52.508 2 0.465 94 99.102 3 0.004 42 99.544 4 0.004 24 99.968 5 OX00 16 99.984 6 o.oal15 99.998

/ ; : $;i’i$Y?‘E;1 (a)

2341 Phys. Fluids A, Vol. 3, No. IO, October 1991 Deane et al. 2341

(4

t

FIG. 4. Flow in a grooved channel: Amplitudes of the projection of the velocity field obtained from the spectral element simulation onto the first four $p” (a) along the limit cycle at r = 350; (b) along the limit cycle at r= looo.

463

$4

FIG. 3. Flow in a grooved channel: (a) Mean velocity field at r = 350. (b) Spatial structure of the four most energetic eigenmodes at r = 350.

tinct “almost’‘-shifted eigenfunctions, corresponding to the “approximate” traveling nature of the solution in the body of the channel. The fact that the eigenvalues come in pairs of approximately equal magnitude (Table II) is consistent

with symmetry breaking of one double eigenvalue.54 The first pair of modes can be seen to closely resemble the Toll- mien-Schlichting waves plotted in Fig. 14 in Ref. 41 with corresponding wave numbera,,* = 1.257. The feature of the approximate phase shift character continues down the stack of eigenfunctions and is consistent with the eigenvalue pairing. Furthermore, the oscillation of the full flow can be un- derstood as the exchange of phase between the first and the second member of each pair. The next pair (& and & ) correspond to a half-wavelength mode [two counter-rotat- ing “eddies” constitute one wave, see Fig. 3 (b) 1; the corresponding wave number is a3,4 = 2.5 14. As regards temporal response of these modes, the second pair oscillates with twice the frequency of the first pair; the latter satisfies approximately the Orr-Sommerfeld dispersion relation for plane channel waves. The modes (b3 4, characterized by twice the wave number of $s,,*, oscillate’in time with twice the fundamental frequency so that modal phase velocity remains constant down the “mode stack.” The structure of the higher modes reveals that the “action” centers around the shear layer that develops between the channel and the groove. Near the upper wall the structure seen is identified as the so- called wall modes observed in the linear stability analysis of channel flow,55 and seems to be a universal feature of ail wall-bounded flows. 42 Having obtained the eigenfunctions, their individual contribution to simulation data on the limit cycle is obtained by the projection (u,+) and is shown in Fig, 4(a). This figure clearly shows how a particular realization (“snapshot”) is comprised of the instantaneous sum of the


TABLE III. Flow past a circular cylinder. Eigenvalues of the POD modes and their respectivecontributions to the total energy for the indicated Reyn- olds numbers.

Re= 100 Re = 150 Re=200 Index A % energy A % energy R % energy

1 0.499 10 49.910 0.499 50 49.950 0.505 17 50.517 2 0.473 79 97.289 0.474 86 97.436 0.467 72 97.289 3 0.00996 98.286 0.00798 98.234 0.008 15 98.104 4 0.009 62 99.247 0.007 58 98.992 0.007 66 98.870 5 0.00364 99.611 0.00480 99.472 0.00539 99.409 6 0.00348 99.959 0.004 58 99.931 0.00499 99.908 7 O.ooO 18 99.977 0.00029 99.960 0.00036 99.944 8 O.ooO 17 99.994 O.CCQ26 99.986 O.COO 31 99.975

POD modes and how the amplitude and phase variations develop.

What we have essentially accomplished up to this point is to represent the flow in terms of its most energetic components. The value of the procedure is based on the ability of the POD modes to serve with comparable accuracy in representing data obtained at Reynolds numbers different (either lower or higher) than the one used in obtaining them. To examine this, we show in Fig. 4(b) similar projections for parameters away from that of the decomposition (r = 1000.) These projections were performed by obtaining M = 20 realizations along a single cycle of the oscillation at the new Reynolds number. Note that away from the decomposition conditions, the modes are no longer eigenvectors of the correlation tensor formed at the new Reynolds number.lR

B. Cylinder flow

Turning to the flow past a cylinder we see similar features. The decomposition was performed using data from M = 20 realizations selected from an ensemble of transients approaching a stable lim it cycle. We performed the decomposition at three Reynolds numbers, Re = 100, 150, and 200. Table III shows the eigenvalues of the most energetic eight modes. Again the first two modes dominate the stack, and the modes appear in pairs. Concentrating on the decomposition at Re = 100, the spatial structure of the mean flow is shown in Fig. 5 (a), while that of the eigenmodes is shown in Fig. 5 (b). It has been shown both experimentally56 and numerically4’ that vortex streets are nondispersive traveling waves. However, due to dissipation their amplitude de- creases downstream and hence they are not translationally invariant. Thus, the observed proximity of the eigenvalue pairs (Table III) and the pairing of the eigenfunctions for the cylinder flow is analogous to that found in the grooved channel flow. As the index of the mode increases, the number of zero crossings in both the horizontal and vertical di- rections increases. As first pointed out by Sirovich,‘8*2’ it is useful in problems with geometric symmetries, like the symmetry around the centerline here, to augment the ensemble ofsnapshots by exploiting the symmetry; this would result in more accurate eigenfunction evaluation. The eigenfunctions we computed for the cylinder flow very closely respected


(4

(q l !!!@ m m @ @ @ @ (

FIG. 5. Flow past a circular cylinder: (a) Streamlines of the mean flow at Re = 100. (b) Spatial structure for the four most energetic eigenmodes at Re = 100. Shown are contours of dm, where (b = (4, ,q%, ) .

those symmetry properties without resorting to such a procedure. This is because the lim it cycle possesses the spatiotemporal symmetry that [u(x,y,t),v(x,y,t)] = [u(x, -y, c i- T/2), - U(X, - y,t + T/2)] where Tistheperiodofthe oscillation (Fig. 6), and our transient data were obtained very close to this cycle; therefore augmenting the data would essentially yield the same set.

As a consequence of the geometric symmetry, the individual eigenfunctions themselves are seen to possess a certain structure (their u and u components are either odd or even) about the m idplane (Sirovich’* ). This also implies that certain terms in the dynamical systems constructed using the POD modes will be identically zero. The role of symmetries in the POD methodology is currently being investi- gated by Aubry, Lian and Titi.57 If the u-component of a mode is odd in y (antisymmetric about the m idplane) then its u component is even by continuity. This is shown in Fig. 7, where “4, and ‘5~5~ are odd while “4, and “$2 are even. The components 5~5~ and “#4 are even while “& and “#4 are odd. It is, in general, not possible to a priori determine whether a particular mode will be even or odd. However, the reason that the primary two modes are odd in u and even in u is that individual snapshots appear approximately odd in u and even in v. The mean flow, on the other hand, is even in the x- velocity component and odd in the y-velocity component.

Figure 8 shows the variation of the first four mode amplitudes when the flow is projected onto these modes. The

Deaneet al. 2343

Q 2 4 d 8 10 0 0 2 4 6 83 10 2 4 6 83 10

0.6 , ( , , \ I _, , , , I P <

t t Q 2 4 6 a 10 to”“-t-L-L”’ 2 4 a a 10

t t

FIG. 6. Flow past a circular cylinder: Spatiotemporal symmetry of the limit cycle, (a) The flow at time s,. (b) The velocity fluctuation field at timer,,. (c) The flow at time f,, + T/2, where Tis the period. (d) The velocity fluctuation field at time I(, + T/2. (e)-(h) Time traces of the horizontal, u, and the vertical, U, velocities at the points A, A’, B, B’ shown in (a). The flow at time T/2 later [ (c),( d )] can be obtained by reflection in the midplane of the flow at time f [(a),(b)]. Consequently uA, is precisely uA translated in time by T/2, for instance, while u,, is precisely - oA translated in time by T/2.

0.05

0

-0.05

-“6 .-e____ vz

FIG. 7. Flow past a circular cylinder: Cross-section profiles of the first four eigenfunctions at x = 15 for Re = 200.

2344 Phys. Fluids A, Vol. 3, No. 10, October 1991 Deane et al, 2344

FIG. 8. Flow past a circular cylinder: Amplitudes of the projection of the velocity field obtained from the spectral element simulation onto the first four 4:” along the limit cycle at Re = 100.

modes are such that a,,* have period Twhile a3,4 have period T/2. The spatiotemporal symmetry of the attractor, along with the symmetry (odd or even structure) of the modes, dictates the nature of the projections a, ( t). This symmetry is a particular case of what is sometimes referred to as “ponies on a merry go round” (POM) symmetry (see Aronson et al.‘* ) and is important because of the possible suppression of period doubling bifurcation from such states.59

V. CONSTRUCTION OF DYNAMICAL SYSTEMS

Once the spatial and temporal parts have been separated in the flow, a dynamical system is obtained by the projection of the Navier-Stokes equations onto the space of the eigenmodes. The Navier-Stokes equations can be written as

av -~“~o-v(p+~)+++f, (15) at where r is as defined in (7) and o = V Xv. For channel flow the Reynolds number is obtained through (6) while for the cylinder flow from (8). Expanding the velocity as in (9) we form evolution equations for the amplitudes in ( 14) through

au c&,---vxo+V(p+y)-$V’u-f), (16) at where the outer parentheses refer to the inner product. The term

c+i, VP), (17) by using the Green’s first identity, can be written

(18)

The first term on the right-hand side is zero because of incompressibility of the eigenmodes ( 13). The last term is zero along the rigid walls and vanishes under periodic boundary conditions. The channel flow is periodic in (v,p) and thus this term, and, in fact, the entire gradient term V(p + ]]~]]~/2), has no contribution.

The boundary conditions on the flow past the cylinder are periodic in the cross-flow direction and therefore there is

2345 Phys. Fluids A, Vol. 3, No. 10, October 1991 Deane et a/. 2345

no contribution at this boundary from the pressure in ( 18 ) . Furthermore, at the outflow boundary the pressure is zero,46 while at the inflow boundary the eigenvectors are zero by construction. Hence there is no contribution from the pressure in this geometry either. Note that there is a contribution from the gradient term V((]V](~/~), which we take into account. The reason that a model for pressure is needed in the work of Aubry et al.” and Zheng and Glauser’* is because there only part of the flow domain is analyzed, and addi- tional information is needed from the adjacent domain for the nonzero pressure contributions.

The above procedure leads to a system of Nequations by retaining N modes in the Galerkin expansion

da, - = cf + &a, + c:j,ka,a, dt

+ (l/r)(C4 + Cj,aj) + CT’f,, (19) where it has been assumed that a force, if present, acts only as a constant in the horizontal direction. The coefficients, c, in ( 19) are constants resulting from the various inner products. For the channel flow f, = 2/r while f, = 0 for the cylinder flow. Note that because the expansion treats the mean flow as a space-dependent constant, the equations have only quadratic nonlinearities. In (19) the constant terms arise from the contribution of the mean flow, the linear terms involve the time-varying part (including its interaction with the mean), and the quadratic terms are due to the interaction of the time-varying part with itself.

We are particularly interested in the ability of the model-reduction scheme to capture the long-term dynamics at Reynolds numbers different than those used for POD mode extraction (“model training”). Since the POD modes are used in a Galerkin projection of the Navier-Stokes equations, the resulting dynamical system dues contain a dependence on the Reynolds number and such predictions are, in principle, possible. We have not taken into account yet, however, the variation of the mean flow with the Reynolds number, as well as the possibility ofdestabilization of the flow in a phase-space direction not accurately spanned by the current set of empirical modes. We make a conscious effort to quantify these shortcomings below.

A. Grooved channel flow

We have studied the ODE’s using both phase space analysis softwarem and continuation and bifurcation algorithms. 6’ For the flow in the grooved channel, with training at r = 350 we find that four modes (N = 4) are the smallest set yielding stable oscillations at r = 350 [keeping more modes (N = 6, N = 8) gives very comparable results]. The result of an integration is shown in Fig. 9. Both the solutions of the model and that of the full simulation are shown in the figure. The curves for the simulation are obtained by the projection u350+350. The two sets are virtually identical, with relative errors increasing with the index of the mode.

In Fig. 10(a) we show the bifurcation diagram for the grooved channel flow four-equation model compared with full simulation results. The diagram depicts the norm of the oscillatory motion:

(4

z

0 5 10 15 20

(b) 0.02

2 0

-0.02

,’ -(I ,

>

: ‘. :

:

* : : #I

: : : ;

: ’ : ;

._.’ : ; *.*

0 5 10 15 20

t

FIG. 9. Flow in a grooved channel: Comparison of the amplitudes of the first and third modes from the four-equation model [ Rq. ( 19); dashed line] and those obtained by the projection II~%+I$‘~~ from the full simulation (solid line) for r = 350.

llall =+l,‘@, a3t))“2, (20)

where T is the period. In this four-equation system as r is increased from small values, a supercritical Hopf bifurcation occurs at Y = 320. Beyond this value, the system predicts a branch of stable limit cycles; this branch is accurate close to the decomposition Reynolds number of r, = 350; it can be

0.6 r,,,,,,,l,-l,l,,

(4 ii ii

a (31 -:

ii

” ’ 0 11 0 t-

200 400 600 800 1000 r

200 400 600 800 1000

r

FIG. 10. Flow in a grooved channel: Bifurcation diagrams of oscillatory solutions. In (a) amplitude [as calculated by Eq. (20) ] and in (b) period of the oscillation versus the reduced Reynolds number. The solid symbols correspond to the full simulation, the dashed line to ( 19), and the solid line to (22) and (Al).

seen to pass close to the “exact” limit cycle observed by full simulations at r,. For values away from r, the agreement between the model and the full system can be seen to deterio- rate. As Fig. 10(b) shows, the period of the model system remains virtually unchanged for r# 350. Thus this model is deficient in extrapolating the system behavior. We can identify a source of this deficiency by noting that the model described in ( 19) contains as a constant part the mean field evaluated at r = 350; this constant part does not vary with Reynolds number. At different values of r, the mean flow will be different from the mean flow at r, = 350. If this change can be accurately spanned by the c$~~‘, we may still expect the model to work without further modification; this is clearly not sufficient here. The issue of variation of the mean (constant part) of the flow as the parameters vary has been discussed by Sirovich. I9 Another solution is to perform system training using time series from several distinct parameter values, representing the entire regime of desired prediction validity. This would unfortunately mean that the derived eigenfunctions could not be identified with coherent structures at any particular parameter value.

For the grooved channel flow problem, we chose to account for the variation in the mean flow with the Reynolds number forming an ad hoc modification of the system, by writing

U(r) = yV50, where

@la)

y = a/r -I- 8. (216) Here a and /? are chosen such that the flow rates at r = 350 and r = 1000 match those from the simulation (Table I). The model (21b) reflects the fact that the flow rate drops slightly with the Reynolds number in the flow regime we examine. From (2 la) the flow rate at r = 600 is found to be Q = 1.08, while the actual value from the simulations (Table I) is Q = 1.10. This model strongly depends on the way equations ( la) are forced; according to the aforementioned scaling of the forcing the model in Eq. (2 1 b) is plausible. The model system can now be written symbolically as

2 = ?Cj + &,a, + c&a,a,

+ (l/r)(ycf+c$a, +Zcp). (22)

The system is explicitly written out in the Appendix (A 1). Figures IO(a) and 10(b) (solid line) show the corresponding ampfitude and period for this model system. The amplitude is predicted very well for Reynolds numbers away from the decomposition value. The period is predicted with a maximum error of 15% (at r = 1000) away from the decomposition value. The Hopf bifurcation location is captured fairly accurately, being r, = 292 in the model as compared to approximately r, = 300 obtained by the full simulation.

A simple procedure outlined by Sirovich” allows us to quantify the relation of the eigenfunctions of the flow at one Reynolds number to those at another. Briefly stated, the eigenfunctions obtained at a Reynolds number r, form a complete set of orthogonal functions. At a different value r, it is

2346 Phys. Fluids A, Vol. 3, No. IO, October 1991 Deane et al. 2346

FIG. 11. Flow in a grooved channel: The most energetic eigenfunction ( I$‘“) at r = loo0 obtained by “rotation” (see the discussion in the text).

therefore possible to express these new 4 as linear combina- tions of the old +. We perform such an assessment for the grooved channel flow. Projecting the flow snapshots obtained at r = 1000 onto the first four +:‘” and then extracting the eigenfunctions at r = 1000, $f” we find

‘q), 1ca

$2 43

\ 4 4 i

=

[ - - 0.0411 0.3304 0.9454 0.0434 - - - - 0.0288 0.0326 0.3207 0.9462 - 0.0166 0.2222 0.9738 0.0460 - 0.0032 0.2214 0.0555 0.9736 1 $2

x+3 *

0 +I 350

$4

An inspection of the matrix elements shows that there has been some energy exchange between the first two modes (of the order of 10%) and between the third and fourth modes, the change in the latter being smaller (around 5% ). This exchange of energy, which can be thought of as a “rotation” of the coordinates of the +‘s, is discernible in Fig. 11, which shows 6, . Visual inspection of the primary eigenfunction shown in Fig. 3(b) shows the “phase shift” character this coordinate rotation has caused: The pattern has been approximately translated longitudinally along the body of the channel.

If the “old” set of eigenfunctions spans the data obtained at a new Reynolds number, this method of obtaining new eigenfunctions as rotations of the old ones should be equivalent to obtaining them directly (through a POD performed at the new parameter value). The new data, in prac- tice, will not beperfectly spanned by the old eigenfunctions. Consequently the eigenfunctions 4:” are not completely aligned with #“. The inner products are


0.9801 0.1321 0.0146

0.1507 0.9819 0.0154 0.0157 ’ 0.1121 0.1201 0.8744

0.1220 0.1019 0.3710 0.0150 1 0.3661 0.8713

The component of ulOoo that is not spanned by the $350 is represented by the quantity

( JIuLMx) - zq= * (u’“~9i”“)46’“11> = o*0787.

(11u’“11) (23)

We see that on the average there is about 7.9% not captured. With the inclusion of more modes, a greater fraction of energy is captured. For instance retaining six modes does not account for 7.5%. Even though the data at r = 1000 are not perfectly captured by the modes obtained at Y = 350, the - 8% difference still yields a dynamical system with reason- able long-term properties. This is in marked contrast with the cylinder results discussed below, where extrapolation de- teriorates very rapidly with the Reynolds number.

B. Short-term dynamics

The ultimate test of the low-dimensional models obtained through the POD procedure is their ability to accurately capture the stability and bifurcation behavior of the underlying detailed simulations. Nevertheless, even though the long-term dynamic predictions of such models may be sometimes inaccurate-i.e., they do not predict the correct attractors-their short-term prediction capabilities may still be useful, for example, in real-time flow control applications. For this purpose, in addition to the bifurcation diagrams presented above (quantifying the long-term predictive accuracy of the models), we also studied their short-term ability to represent the full simulations.

Figures 12(a) and 12(b) show the results for the grooved channel at r = 350. Two distinct phase-space projections of the four-dimensional reduced model space are shown; the solid line shows a transient away from the lim it cycle, while the solid triangles denote the projections (in this space) of the full spectral element transient simulation. These points are recorded at constant time intervals Ar ( Ar is of the order of 30 times the simulation time step). It is obvious that, even for this transient away from (but approaching) the final lim it cycle, the short-term prediction capabilities of the model are very satisfactory. The long-term attractor of the model is also shown in the figure and is practically indistinguishable from the projection of the full simulation attractor on the four-POD mode subspace. Figure 12(c) illustrates the (almost linear) growth of the prediction error in time between the model integration and the full spectral element simulation along the same transient. The symbols have the same meaning as in the phase portrait pic- tures: squares indicate prediction error at Ar, and stars prediction error at.5 AT. Figure 13 shows the same situation at r = 1000. Here (because of the model shortcomings we discussed previously), the long-term attractor is well captured for the first two POD modes, but is not so accurate in the third and fourth modes [Fig. 13(b) 1. Nevertheless, the short-term prediction (squares) ofthe model close to the full simulation attractor is obviously good, even though it dete-

Deane et a/. 2347

(4

‘iY

P-9

d

0 C

8

ii

8

P al

s 2 Ia

-0.2 0 0.2

FIG. 12. Flow in a grooved channel; short- and long-term prediction accuracy: (a) and (b) Phase space projections of full simulations and model integrationsat r = 350. Thesolid linesshow a transient, away from the limit cycle, obtained through full simulation. Broken lines indicate long-term prediction: the model attractor is the short-dash limit cycle, while the full simulation attractor is shown with the long-dash curve. Triangles are projections of points obtained at intervals Ar = &, of the oscillation period. The open squares show the result of model integration for Ar, while the stars show the result of model integration for 5 Ar. (c) shows the rms relative phase-averaged (over one period) model prediction error.

FIG. 13. Flow in a grooved channel; short- and long-term prediction accuracy: (a) and (b) Phase space projections of full simulations and model integrations at r = 1000. The solid lines show the attracting limit cycle obtained through full simulation. The model attractor is the short-dash limit cycle. Triangles are projections of points Ar apart. The open squares show the result of model integration for AT, while the stars show the result of mode/integration for 5 AT. (c) shows therms relative phase-averaged prediction error.

2348 Phys. Fluids A, Vol. 3. No. IO, October 1991 Deane et a/. 2348

0 a

a?

(b)

CT

0 c

ii fs

I! F Q1 L-4 43 PC

0.4

0.2

0

-0.2

-0.4 # -0.4 -0.2 0 0.2 0.4

ai

0.1, I1 1, I1 I, ,,,I, I I \ I,, I I, 1 I ‘_I .-.

1

: : ,’ i : : / :

: : ,’ :

I n, I : ,*’ 0 0 :

6 k, 1

1 I

: / : ,* ‘, - ; : ,* : : : ; ‘. : t.* I ‘.m/’ 1 I...‘...

5 -e -0.l'I""' r"""" m"'n"n'

-0.6 -0.4 -0.2 0 0.2 0.4 0.t

al

10 20 t

riorates with time (stars). Figure 13(c) quantifies prediction error growth in time.

C. Cylinder flow For the flow past a cylinder with training at Re = 100

we find that at least six modes need to be retained in order to obtain stable oscillations in ( 19). The system with four modes displays an oscillation, which, however, slowly grows without bound but with the addition of modes & and &, the oscillations stabilize. A comparison of the model system and the full system shows that the amplitude and the period are obtained quite well at the decomposition Reynolds number Re = 100 (Fig. 14).

Figure 15 compares model bifurcation calculations with the full simulation. As discussed, when the decomposition is performed at Re, = 100, the model correctly predicts a stable limit cycle at Re,. This limit cycle branch is predicted by the model to start at a (subcritical) Hopf bifurcation at Rez80, and persists until Rez 120, where it turns backward at a limit point (one Floquet multiplier equal to unity). Thus the accuracy of the model predictions rapidly deterio- rates as we move away from the decomposition value. Figure 15 also shows the results of performing the decomposition at Re = 150 and 200: The predictions are excellent when the the model is integrated at the decompositional value Re, = 100, 150, and 200.

As noted in the previous paragraph the model obtained using decomposition data at Re, = 100 does not yield an (even inaccurate) attractor when integrated at Re = 150 (Fig. 16). However, when the correct mean flow from the working Reynolds number is used along with the eigenfunctions at the decomposition value (Re, = 100) the range of predicted stable (not necessarily accurate) oscillations ex- tends beyond the turning point at Re = 120. For instance,

0 5 10 15 20

(b) t (4

d

E 2 d

85 P f3> B

A 41 23 o-“““““““““‘- 50 100 150 200 250

Re

FIG. 15. Flow past a circular cylinder: Bifurcation diagram of oscillatory solutions shown as the amplitude [as calculated by Eq. (ZO)]. The solid squares correspond to the full simulation. Open squares are the predictions of models using both mean and basis functions at the decompositional values. The triangles refer to predictions of models with the “correct” mean flow and the POD modes obtained at the decompositional value of Re = 100. The solid line is the periodic branch obtained from the model using mean and basis functions at Re, = 100. This last model does not predict stable oscillations for Re > 120.

the mean velocity U”’ from Re = 150 used with the eigenfunctions +‘@ from Re, = 100 yields stable oscillations (Fig. 16). Even though the predicted limit cycle is reasonably accurately captured for the first and second modes [Figs. 16(a) and 16(c) 1, the prediction rapidly fails in the higher modes both in the short-term [Fig. 16(b) ] and in the shape of the attractor [Fig. 16(d)]. Since the first two modes dominate in amplitude, the norm shown in Fig. 15 also appears well approximated. However, the detailed features of the predicted flow are significantly different from the full simulation.

The component of u 15’ that is not spanned by the +‘O” is represented by the quantity

FIG. 14. Flow past a circular cylinder; short- and long-term predictions. Com- parison between the amplitudes of (a) the first and (b) third modes from a short integration of the six-equation model (dashed line) and those obtained by the projection u”‘L+‘@’ from the full simulation (solid line) on the limit cycle for Re = 100. Projections of the long- term attractor from the model (solid line) and the full simulation (triangles) areshowninthe(c)a,-a,and(d)a,-a, planes.


(4

(d”

(b>

d

0 5 10 15 20 -6 - -6 -4 -2 0 2 4 6

4~~ 0 5 10 15 20

t

(4

47

a1

1

c . .

0 .A

-1

B

A. . A .

rlr

FIG. 16. Flow past a circular cylinder; extrapolation to a nondecompositional Reyn- olds number. Comparison between the amplitudes of (a) the first and (b) the third modes from a short integration of the six- equation model (dashed line) and those obtained by the projection u’*“-~‘c” from the full simulation (solid line) on the limit cycle for Re = 150. The model used the “correct” mean, W? Projections of the long-term attractor from the model (solid line) and the full simulation (triangles) are shown in the (cl a, -uz and (d) a, -(I) planes.

-6

(((IP - zp= * (U’“08tP9Qtl’“ll) = 0 6468

(ll~15011) . . (24)

We see that on the average there is some 65% not captured. With the inclusion of 8, 14, and 20 modes from Re = 100 the corresponding fractions in the above are 0.6465, 0.6458, and 0.6403. Thus the flow changes with Reynolds number cannot be simply thought of as an “excita- tion ofthe next higher mode(s)” in the Re = 100 hierarchy.

In order to explore this rapid deterioration in the ability of the model to accurately extrapolate to nondecompositional Reynolds numbers we show in Fig. 17 eigenfunctions obtained by three separate procedures. Figure 17(a) shows 4, i”” obtained by performing the decomposition at Re, = 150. Figure 17(b) shows the corresponding vertical velocity profile along the centerline. Figure 17 (c) shows the first “rotated” eigenfunction ~$1”” obtained by projecting simulation data u’“~I--+~‘” (as discussed in Sec. V A for the grooved channel flow). The inability of the 6”’ [and of the original t+‘O”, e.g., Fig. 17 (e), of which these are linear com- binations] to capture the flow at Re = 150 is apparent upon comparison of the spatial structure of Figs. 17 ( c ) and 17 (a). The change in the streamwise wave number [approximately 8%, as can be seen in the corresponding profiles, Figs. 17(b) and 17 (d) ] cannot be accommodated by linear combina- tions of the +‘O”. Keeping 8, 14, and 20 modes from the Re = 100 hierarchy is not capable of spanning even the first mode of the Re = 150 data well. The dimension of the attractor has not changed between the two cases; rather, the subspace where the attractor lives has “rotated” in the full space and cannot therefore be well spanned by the old modes. If, on the other hand, POD modes at Re = 150 are obtained directly, the first six perform very well. The two basis sets span appreciably different subspaces. At Re = 200


-4 -2 0 2 4 6

ai

the changes are so pronounced (approximately 12% in the streamwise wave number, with some 9 1% of the energy lying out of the subspace spanned by the 4’“) that even gross features of the predicted attractor (like the norm in Fig. 15) are inaccurate.

The ability of the POD modes to effectively span the ensemble data does not necessarily imply that the POD- based dynamical system will accurately reproduce the attractor (long-term as opposed to just short-term dynamics) of the original simulation. Transients of this dynamical system starting on the projection of the attractor of the full simulation, could wander close to this projection for several periods, but eventually drift away. If transients in the process of approaching the full-simulation attractor are included in the original ensemble, this should allow the POD modes to span not only the attractor itself, but also the dynamics in its neighborhood in the full space. The stability and accuracy of the long-term attractor predicted by the POD-bused model is then also enhanced.

For most of the cases we studied (all grooved channel results, as well as the cylinder for Re = 100 and 150) the POD/Galerkin model attractors did not critically depend on our choice of transient simulation ensemble. Even when the simulation data lay on a single transient practically on the full attractor, the decomposition gave excellent short- term predictions and satisfactory attractors in the resulting models. For the cylinder Row case of Re = 200, however, data on a single transient, while still yielding excellent short- term predictions in the model, failed to predict a stable attractor close to the full-simulation one. An ensemble consist- ing of snapshots along several transients (five, shown in Fig. 18) in the neighborhood of the full attractor did yield the correct long-term dynamics (Fig. 19, where 12 POD modes were retained in the Galerkin projection). This data at

Deane et al. 2350

&Yx, Y) (4

4t”(x, Y) (4

0.06

= Ii-

r 2:. -0.

-0.07

0.12

G . .5 f, -e-

-0.11 2

Re = 200 was also augmented by including data obtained tive bifurcation analysis of these flows; such analysis (for through use of the geometric symmetries of the flow. In most example, stability calculations for limit cycles) are not prac- POD applications in the literature it is assumed that the tical in the fully discretized systems. We also note that it was chaotic or noisy nature of the data helps sample enough of not necessary to resort to models of energy dissipation to the neighborhood of the full simulation attractor. In tempor- higher modes (eddy viscosity24 or Heisenberg closure26 ). ally simple (laminar) flows, such as the ones considered This is due, perhaps, to the nonturbulent nature of the flows here, some care must be taken in constructing an ensemble we are considering, where the neglected energy is only a few that can capture the dynamics close to (and not only on) the tenths of one percent of the total at the decomposition val- full attractor. ues.

VI. SUMMARY AND DISCUSSION

We have used the POD procedure to obtain low-order models describing two distinct two-dimensional flows in complicated geometries. We find that the resulting model dynamical systems are capable of capturing both the short- and long-term dynamics of the full simulations close to, and often away, from the parameter values at which they were obtained (and where, hence, they are strictly valid). This feature-the robustness of the POD modes-has not been previously explored in detail for the type of flows we have considered here. The reduction in dynamical system size resulting from the POD has allowed us to perform a quantita-

Our results for the grooved channel flow show that eigenfunctions obtained at some reference parameter value can be fairly accurate in describing the system behavior at new parameter values, provided that some care is taken to account for the mean flow at the new conditions. Apparently this can sometimes be accomplished in an approximate man- ner; in some situations (e.g., wall-bounded flows) the mean flow does not significantly change its shape (spatial dependence) over the range of Reynolds numbers considered. In the grooved channel flow it remains fairly parabolic in the dy- namically significant channel portion of the geometry. For this example, matching the flow rate by a simple Reynolds number model based on simulations at a few distinct condi-


FIG. 17. Flow past a circular cylinder; change of the POD eigenspaces with Reynolds number. Contours ofthe vertical velocity 4, are shown in (a), (c), and (e) for the primary eigenfunction while (b), (d), and (f) show the corresponding profiles along the centerline. In (a) and (b) the mode +I”” is shown, obtained by decomposition at Re = 150. In (c) and (d) the “rotated” mode @ ’ is shown, obtained by projecting data at Re = 150 onto the modes +‘O”, the first of which is included as (e) and (f) for comparison.

0 a 10 11111111111~~~1~~~1

5-

d O-

1 -5 -

““‘c’lt”‘ll’l”’ -10 -5 0 5 10

1

(6” 0

-1

-10 -5 0 5 10

al FIG. 18. Flow past a circular cylinder; sampling the neighborhood of the attractor at Re = 200. Projectionsin (a) thea, -a, and (b) thea, -a, planes of five distinct transients forming the ensemble. The solid circles are the projections of the transient lying closest to the attractor.

~~~~~

0 5

04

(6" ~~~

0 5 10 15 20

t

(4

tions yields a more accurate dynamical system. Thus, it appears possible that methods to compute the mean flow at different Reynolds numbers (e.g., transport models for turbulent flows6’), coupled with the eigenmodes obtained at one representative base state, could significantly enhance the predictive capabilities of POD models.

In other cases (e.g., external Bows), however, the flow may undergo significant change in shape away from the decomposition value. For the wake of the circular cylinder, for example, the length of the recirculation region as well as the streamwise wave number in the low Reynolds number regime depend strongly on the Reynolds number.45 This renders an ad hoc parametrization of the mean flow (based on simple amplitude scaling) useless. Furthermore, as discussed, even though the dynamics remain low dimensional, the POD eigenspaces change significantly with Reynolds number. It is not surprising therefore that the models we derived are so sensitive to the decomposition Reynolds number. We are investigating a systematic way of parametrizing the POD modes with the Reynolds number (smooth inter- polation between sets of basis vectors).

The Galerkin procedure employed here is “flat,” in the sense that a number of POD modes were retained and the remaining ones discarded. A particularly interesting direction is to combine the approximate inertial manifold methodology with the POD eigenmode hierarchy, and approximate the solution component on the higher POD modes as a function of its components on the lower, more energetic ones. It might, for example, be possible to use only four inde- pendent ODE’s in the cylinder case with the same accuracy, if the (necessary) fifth and sixth modes were approximated as functions of the first four in a “nonlinear Galerkin-POD” model. This is currently being pursued.

In this paper we have restricted our attention to two- dimensional flows that have simple temporal signature, ex- ploring the POD model-reduction capability in complex

(4

b?

FIG. 19. Flow past a circular cylinder; short- and long-term predictions at Re = 200. Comparison between the amplitudes of (a) the first and (b) the third modes from a short integration of the 12- equation model (dashed line) and those obtained by the projection u*%-M$*~ from the full simulation (solid line) on the limit cycle. The model used the “correct” mean, UzM. Projections of the long-term attractor from the mode1 (solid line) and the full simulation (triangles) areshown in the (c) a, -u2 and (d) Q, -a3 planes.

2352 Phys. Fluids A, Vol. 3, No. 10, October 1991 Deane et at. 2352

geometries. The results presented here indicate that, with a well-approximated mean field, orthogonal eigenfunctions that are optimal at a particular parameter value (but less SO at other parameter values) are very useful in obtaining accurate reduced model systems. These systems can be used for short- and long-term prediction, as well as stability and bifurcation calculations, infeasible for traditional discretizations in such geometries. These properties, particularly the short-term prediction accuracy obtained at low computational cost make this approach a powerful tool for real-time flow control applications. As we mentioned above, the sec- ondary instabilities in both of the flows considered are three dimensional; we are currently working toward application of the POD to these fully three-dimensional states.

Note added in proo$ M. Kwak has very recently proved

J

that the two-dimensional Navier-Stokes equations with periodic boundary conditions do possess an inertial manifold.

ACKNOWLEDGMENTS

This work is partially supported by the National Science Foundation (Grants No. EC%717787 and No. CTS8957213), the David and Lucile Packard Foundation, and the Office of Naval Research (Contracts No. NOOO14- 90-J-1312 and No. N00014-86-K-0759).

APPENDIX: FOUR-EQUATION MODEL FOR FLOW IN A GROOVED CHANNEL

For the flow in a grooved channel accounting for the variation of the mean flow as in (2 la), the four mode equations written explicitly are

ir, = 0.006 47y + O.O484ya, + 0.000 09~: - 0.625 14ya, + 0.049 37a, a2

- 0.002 05a: + 0.118 85ya, 0.144 1 la, a3 0.070 19a,a, + 0.032 98a: - - + r- ‘( - 0.646 66 - 1.177 86~ - 13.3801a, - 2.082 33a, + 3.043 49a, - 2.755 98~~)

- 0.096 52ya, + 0.026 65a, a4 0.104 16a, a4 + 0.036 05a, u4 0.032 72u:, - - ci, = 0.000 02f + 0.567 65ya, - 0.047 77~: + 0.037 75yu, + O.O05a, a2

- 0.000 02a: + 0.119 96yu, + 0.064 63a, u3 +‘O. 146 8u,u, + 0.007 58~:

+ 0.050 66ya, - 0.167 83u, u4 + 0.007 53u,a, + 0.058 78u,u, + 0.029 79~:

+ t- ‘(0.349 14 - 0.5957~ - 1.996 17a, - 14.3301a, + 4.655 84a, + 1.714 59a,),

ii, = 0.005 24y + 0.140 81~: - 0.004 02u, a, - 0.146 23~: - 0.020 66u, a3

- 0.003 12a,u, + 0.0026a: + y( 0.011 41u, 0.011 81u, 0.141 21u, 1.219 Olu,) - - - - + r- ‘( - 0.242 44 - 0.835 95~ + 3.011 38a, + 4.674 34a, - 53.369a, - 0.819 16~ )

- 0.182 37u, a4 + 0.147 42u, u4 0.117 49a, a4 0.008 74u:, - - ti, = 0.002 32f + 0.007 22yu, - 0.026 71~: - 0.008 56yu, + 0.280 84a, a,

- 0.004 1 la: + 1.229 64ya, + 0.142 29a,u, 0.221 28u,u, + 0.121 52~: -

+ r- ‘(0.003 92 - 0.616 91~ - 2.6049u, + 2.009 66u, - 1.038 29u, - 49.4938a, 1

- 0.132 52ya, + 0.027 92u, u4 0.026 59u, a4 + 0.009 13a, u4 0.000 79a:. - -

(AlI

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Low-dimensional models for complex geometry flows ... · plex geometry flows studied here. Modeling such flows re- sults, after discretization, in large dynamical systems (typi- cally