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Physica D 202 (2005) 218–237

Stability and accuracy of periodic ﬂow solutions obtained

by a POD-penalty method

S. Sirisup, G.E. Karniadakis∗

Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA

Received 14 May 2004; accepted 11 February 2005

Communicated by I. Mezic

Abstract

We develop a new penalty method to derive low-dimensional Galerkin models for ﬂuid ﬂows with time-dependent boundary

conditions. We then outline a procedure based on bifurcation analysis in selecting the proper values of the penalty parameter(s)

that yield asymptotically stable periodic solutions of the highest possible accuracy. We illustrate this new approach by studying

ﬂow past a circular cylinder using direct numerical simulation (DNS) data, and a wave-structure interaction problem using

particle image velocimetry (PIV) data.

© 2005 Elsevier B.V. All rights reserved.

MSC: 37E99; 65P99

Keywords: Penalty methods; Low-dimensional; Dynamical systems; Galerkin projections

1. Introduction

Low-dimensional systems for unsteady ﬂuid ﬂows, based on the proper orthogonal decomposition (POD), have

had mixed success in predicting the correct dynamics even at exactly the same set of parameters for which the POD

modes were obtained. Speciﬁcally, an erroneous state may be obtained after long-time integration even if the correct

periodic state is set to initialize the simulation—the numerical solution eventually drifts to a new erroneous state.

This has also been observed in other systems, for example in the Kuramoto–Sivashinsky equations [1]. Empirical

ﬁxes based on artiﬁcial dissipation, e.g. [2], can only correct the dynamics in short-term integration, and more

rigorous procedures need to be followed to guarantee asymptotic stability, e.g. see [3].

∗Corresponding author. Tel.: +1 401 863 1217; fax: +1 401 863 3369.

E-mail address: gk@dam.brown.edu (G.E. Karniadakis).

0167-2789/$ – see front matter © 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.physd.2005.02.006

S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 219

A distinction, however, should be made between autonomous low-dimensional dynamical ﬂow systems and

non-autonomous ones. We have observed, for example in many POD studies with diverse ﬂow systems [4,5], that

autonomous systems are more susceptible to this “drifting” while non-autonomous systems may reach asymptot-

ically stable and accurate states without incorporating any special treatment. An example of a non-autonomous

system is oscillatory ﬂow past a circular cylinder. We have observed through accurate numerical integration of the

corresponding POD system (for millions of time steps) that asymptotic stable and accurate states can be reached,

at least for an external frequency close to the natural frequency of the system, i.e. the vortex shedding frequency in

this case.

The implementation of complicated boundary conditions in Galerkin systems has historically been a matter

of some controversy [6]; see also [7–9]. An in-depth study of boundary conditions for Galerkin POD systems

was performed in [10]. Herein we introduce a penalty method, similar in spirit with the “tau” method in spec-

tral methods [11], but more ﬂexible in many aspects as we will see in this study. In particular, we will study

two different systems based on data obtained from direct numerical simulations (DNS) and experimental results

using particle image velocimetry (PIV) [12]. A new aspect of the current work is the use of the penalty param-

eter(s) as bifurcation parameter(s) in order to perform stability analysis using a standard package, e.g. AUTO

[13].

Penalty methods have been used in the past successfully in implementing boundary conditions for different

types of numerical discretizations. For example, ﬁnite difference schemes on complex geometries have been de-

veloped in [14] using a penalty method to impose Dirichlet boundary conditions. Also, a penalty method was

developed in [15] to enforce boundary conditions for shock-free compressible Navier–Stokes simulations. A

similar penalty method was used in imposing boundary vorticity constraints in [16]. In general, the penalty ap-

proach enforces the boundary conditions but also accounts for the governing equation at the boundary in a con-

tinuous manner, thus relaxing some of the numerical stiffness associated with very steep gradients at Dirichlet

boundaries.

From the fundamental point of view, we pose the following question:

•Is there a range in the penalty parameter τfor which the periodic solutions of the ﬂow system are asymptotically

stable, and are there any particular values of τfor which the solution is most accurate.

For full numerical discretizations of any type, the accuracy of the solution scales as the inverse of the penalty

parameter in well-resolved simulations. However, efﬁciency considerations require that a ﬁnite value for τmust

be used. As τ→∞we have a strong imposition of the boundary condition while for small values of τwe have a

weak imposition of the boundary conditions. The question then is how do low-dimensional discrete systems differ

from their “full-blown” counterparts in this respect? More importantly, should we impose the boundary conditions

for low-dimensional systems in a strong form or in a weak form. Intuitively, we expect to have stability above a

threshold value in τbut that does not imply good accuracy in predicting the ﬂow dynamics.

The above are some of the issues we address in the current work. The outline of the paper is as follows: in the

next section, we describe the data sets based on which the POD-penalty system is derived. Then, the POD-penalty

formulation is given in some detail. Subsequently, the results of the bifurcation study using the penalty parameter are

presented for the two cases corresponding to DNS and experimental data. Finally, a summary and a brief discussion

is included in the last section.

2. Data gathering

We construct POD modes based on data obtained both from DNS as well as from experiments. We report here

on two prototype cases we have studied with the penalty method.

220 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237

Fig. 1. Computational domain for DNS.

2.1. Direct numerical simulation

We consider ﬂow past a circular cylinder for which both two- and three-dimensional POD models have been

constructedin[17]and[18],respectively.Inparticular,fortheconceptsdevelopedhere,weconsidertwo-dimensional

time-dependentinﬂowpastacircular cylinder at Reynolds number Re =100and500. The inﬂow velocity is uniform

but oscillatory in time, and is given by

U∞=(1.0+Asin(ωt),0).(1)

The amplitude of the forcing term A, is kept the same at A=0.1 for both Reynolds numbers. The forcing frequency

is chosen so that we have a lock-in (resonant) state; this is done by choosing the frequency to be close to the Strouhal

number which is 0.1667 for Re =100 and 0.22 for Re =500.

The computational domain is shown in Fig. 1. A time-dependent boundary condition is imposed at the inﬂow

boundary 1; periodicity is imposed on 3and 4while on 2the zero Neumann condition on velocity is im-

posed and the pressure is set to be zero. On the cylinder surface 5the no-slip boundary condition is prescribed.

Converged solutions were obtained using the spectral/hp element method [19]. The domain is discretized into

412 triangular elements while seventh-order Jacobi polynomial basis are used to obtain resolution independent

solutions.

2.2. Particle imaging velocimetry (PIV) experiment

Here we consider wave interaction with a vertical, surface-piercing cylinder, see Fig. 2. This ﬂow gives rise to

complex forms of wake structure due to the orbital particle trajectories of the incident wave, and the sweeping of

previously generated vortices past the cylinder due to the oscillatory nature of the wave. The cylinder is ﬁxed and the

ﬂow motion is sustained by the wave action. We will employ the experimental results obtained using particle image

velocimetry (PIV) by Yang and Rockwell [12]. For this ﬂow there are two important non-dimensional parameters

that need to be speciﬁed: First, the Keulegan–Carpenter number deﬁned by

KC =2πAo

D,

in which Aois either the displacement amplitude of the cylinder motion or the amplitude of the oscillatory ﬂow and

Dis the cylinder diameter. Secondly the Stokes number

β=fD2

ν,

in which fis the frequency of the motion and νis the kinematic viscosity.

S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 221

Fig. 2. Experimental set up [12].

Quantitative images were obtained using a technique of high-image-density particle velocimetry (PIV). There

are 13 phased-averaged snapshots available for a time period T=0.89. These images will be used to extract the

POD modes. Details of the experiments and the imaging approach are described in detail in [12]. A brief summary

is given next.

The vertical, rigidly suspended cylinder, which is shown in the schematic of Fig. 2, was maintained stationary

during all experiments. It had a diameter of D=12.7mm and a length of L=876mm. The submerged length

of the cylinder was 700mm. The value of the Keulegan–Carpenter number were KC =2πAo/D =13.9atthe

depth of the laser sheet, which is indicated in Fig. 2 as 51mm beneath the quiescent free-surface. The amplitude

Aocorresponds to the radius of the particle orbit of the wave, which was also determined at the depth of 51mm.

Furthermore, the value of the Stokes number was β=fD2/ν =164 for this experiment. The corresponding value

of Reynolds number is Re =KC ×β=2280.

3. POD-penalty systems

In order to employ time-dependent boundary conditions in low-dimensional models, we formulate a new method

to construct Galerkin systems. In particular, we incorporate the boundary conditions directly into the Navier–Stokes

equations as constraints, enforced via suitable penalty parameters. In the next section we will demonstrate how to

select the penalty parameters through bifurcation analysis in order to achieve asymptotically stable and accurate

periodic solutions.

222 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237

Herewe employ the hierarchical POD modes as a trial basis to represent the velocity ﬁeld. In addition, we employ

a Galerkin projection of the Navier–Stokes equations onto these modes to derive dynamical systems to simulate the

ﬂow. Let us decompose the total ﬂow ﬁeld Vas

V(x,t)=U0(x)+u(x,t),

where U0is the time-averaged ﬁeld. We express uas the linear combination of the POD modes as written in the

summation conventions:

u(x, y, t)=φu

j(x, y)aj(t),v(x, y, t)=φv

j(x, y)aj(t),

where aj(t) are the unknown coefﬁcients and φ=(φu,φ

v) deﬁnes the vector of the POD modal basis.

In the following we derive separately the low-dimensional system for the DNS data and the experimental

data.

3.1. DNS: POD-penalty system

The Galerkin projection of the Navier–Stokes equations with penalty terms included onto the jth POD mode is

φj·∂V

∂t +(V·∇)V+∇p−1

Re ∇2V+τ1ϒ(x)(V−U∞)dx=0,(2)

where τ1is the penalty parameter and U∞is the imposed velocity at the inﬂow boundary 1(see Fig. 1). The

function ϒ(x) is deﬁned as

ϒ(x)=1,ifxon 1

0,otherwise.(3)

We note here that on boundary 1we do not impose any boundary conditions as it is now treated as part of the

interior domain. The treatment of the pressure term is of particular importance, so we analyze the corresponding

Galerkin projection by using the Gauss’s theorem to obtain

φj·∇pdx=−

∇·φjpdx+∂

φj·npds. (4)

Obtaining the POD modes from DNS of an incompressible ﬂow ﬁeld leads to divergence-free eigenmodes, and

thus the pressure term inside the domain is eliminated (ﬁrst term in the above equation). On the side bound-

aries 3and 4we assume periodicity and hence the pressure boundary terms cancel each other. On the outﬂow

boundary 2the pressure is set to zero in the corresponding DNS. The inﬂow boundary 1should not be in-

cluded in the computation of the second term of Eq. (4) since we have already included it in the Navier–Stokes

equations. Therefore, there is no contribution from the pressure on this boundary in the integration by parts pro-

cedure. Finally, on the cylinder boundary the test function is zero and thus there is no pressure contribution there

either.

In summary, the Galerkin projection leads to the dynamical system:

daj

dt=fj(a)−Gj(a) (5)

S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 223

with a=[a1,a

2,...,a

M], where Mis the number of POD modes. The term fj(a) includes the convective and

viscous terms and has the form:

fj(a)=−

φj·((φi·∇)φk)dxaiak

−−1

Re

φj·∇

2φidx+

φj·((φi·∇)U0)dx+

φj·((U0·∇)φi)dxai

−

φj·((U0·∇)U0)dx−1

Re

φj·∇

2U0dx.

Also, Gj(a) is the boundary penalty term, which is written as follows (in summation convention):

Gj(a)=τ1ai1

φi(y|1)·φj(y|1)dy−1

(U∞−U0(y|1)) ·φj(y|1)dy,

where φm(y|n) means the function of yobtained by evaluating φmon n.

Since U∞(t) is time-dependent, we obtain a non-autonomous system. More details on the derivation of the above

formulation for the Navier–Stokes equation are presented in [20].

3.2. Experiment: POD-penalty system

The POD-penalty system for the experimental data is derived similarly. By referring to Fig. 3, we now employ

twopenalty terms τ1andτ2for the boundaries 1and 2, respectively.The Galerkin projection of the Navier–Stokes

equations with penalty terms onto the jth POD mode is now

φj·∂V

∂t +(V·∇)V+∇p−1

Re ∇2V+τ1ϒ1(x)(V−U1

∞)+τ2ϒ2(x)(V−U2

∞)dx=0,(6)

where the projection vector φjis deﬁned as previously, and U1

∞,U2

∞are the velocity vectors at the boundaries 1

and 2, respectively (see Fig. 3). The function ϒi(x) is deﬁned as

ϒi(x)=1,ifxon i

0,otherwise.(7)

Unlike the earlier DNS study where the ﬂow is two-dimensional, in this case the true ﬂow is three-dimensional

but only a two-dimensional slice is visualized via PIV. Correspondingly, imposing the divergence-free condition

on the two-dimensional POD modes is not appropriate. To this end, we will employ the divergent POD modes and

let the penalty terms “counteract” the divergent contributions (ﬁrst term of Eq. (4)); how accurate is this procedure

will be tested by the results presented in the next section. Intuitively, it can be justiﬁed as the penalty term controls

effectively the boundary mass ﬂuxes (on 1and 2), and thus by adjusting the value of τ1and τ2, respectively,

we can counteract any mass sources or sinks due to the pressure contributions in the domain interior. The pressure

contributions from the boundaries vanish due to periodicity and Dirichlet boundary conditions, similarly to DNS

case. Regarding the representation of the time-dependent velocity boundary condition at 1and 2, we have found

that it is accurate to use a Fourier series with 16 Fourier modes to represent the time-periodic forcing at those

boundaries. A systematic investigation of this has been presented in [5].

The Galerkin projection of the two-dimensional governing equations leads to the dynamical system:

daj

dt=fj(a)−G∗

j(a) (8)

224 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237

Fig. 3. Computational and PIV domains for KC =13.9.

with a=[a1,a

2,...,a

M], where Mis the number of POD modes. The term fj(a) has the same form as in Eq. (6).

The boundary penalty term G∗

j(a) is modiﬁed as follows:

G∗

j(a)=τ1ai1

φi(y|1)·φj(y|1)dy−1

(U1

∞−U0(y|1)) ·φj(y|1)dy

+τ2ai2

φi(y|2)·φj(y|2)dy−2

(U2

∞−U0(y|2)) ·φj(y|2)dy

3.3. Transformation to an autonomous system

In the next section we will show how to track periodic branches of the dynamical systems described by Eqs.

(5) and (8). However, in order to effectively use the AUTO dynamical system package [13] to track the periodic

branch, these non-autonomous systems need to be transformed to autonomous systems. To this end, we introduce

the nonlinear oscillators

dp

dt=p+ˆ

βq −p(p2+q2),dq

dt=q−ˆ

βp −q(p2+q2).

This particular system has an asymptotically stable solution given by

p(t)=sin(ˆ

βt) and q(t)=cos(ˆ

βt).

We then incorporate the nonlinear oscillator to the POD-penalty system in order to obtain an equivalent autonomous

system.

S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 225

3.3.1. DNS: POD-penalty autonomous system

WerecallthatinEq.(6)theonlytime-dependenttermisU∞,whichisgivenbyEq.(1).Theequivalentautonomous

system is

daj

dt=fj(a)−Gj(a,p),dp

dt=p+ωq −p(p2+q2),dq

dt=q−ωp −q(p2+q2),

where a=[a1,a

2,...,a

M], fj(a) is given by Eq. (6) and Gj(a,p) is now deﬁned as

Gj(a,p)=τ1ai1

φi(y|1)·φj(y|1)dy−1

((1.0+Ap, 0) −U0(y|1)) ·φj(y|1)dy.

Therefore, we have replaced the term sin(ωt)inEq.(1) with p(t).

3.3.2. Experiment: POD-penalty autonomous system

The transformation of the non-autonomous system to an equivalent autonomous one for the case of POD-penalty

system derived from experimental data is somewhat more complicated. For this POD-penalty system, we have the

representation of the velocity vectors at the boundaries 1and 2in the form of Fourier series as

Ui

∞(yj,t)=A(i, j)

0+

N

n=1

A(i,j )

ncosnπt

T+B(i,j )

nsinnπt

T,(9)

where Tis the period, Nthe number of Fourier modes (for this case N=16), yja grid boundary point, and i=1

or 2 for velocity vectors at the boundaries 1and 2, respectively.

We then can transform Eq. (8) into an equivalent autonomous system as follows:

daj

dt=fj(a)−G∗

j(a,p,q),dpn

dt=pn+nπ

Tqn−pn(p2

n+q2

n),dqn

dt=qn−nπ

Tpn−qn(p2

n+q2

n),

where with a=[a1,a

2,...,a

M], fj(a) is the same as Eq. (6),p=[p1,p

2,...,p

N], q=[q1,q

2,...,q

N] and

n=1...N. Note that here ˆ

β=nπ/ T .

Correspondingly, G∗

j(a,p,q) is deﬁned as

G∗

j(a,p,q)=τ1ai1

φi(y|1)·φj(y|1)dy

−τ1ny

k=1A(1,k)

0+

N

n=1

A(1,k)

nqn+B(1,k)

npn−U0(yk|1)·φj(yk|1)wk

+τ2ai2

φi(y|2)·φj(y|2)dy

−τ2ny

k=1A(2,k)

0+

N

n=1

A(2,k)

nqn+B(2,k)

npn−U0(yk|2)·φj(yk|2)wk.

Here nyis the number of grid points on 1and 2, and wkis the weight for the trapezoid integration.

226 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237

4. Results

4.1. DNS: Galerkin POD-penalty system

The Galerkin POD-penalty systems for Reynolds number Re =100 and 500 are derived by employing 100

snapshots per period for both cases. We ﬁrst present representative results of the stability of these solutions and

subsequently we investigate their accuracy.

4.1.1. Stability of periodic solutions

Here, we study stability of the solutions of the Galerkin POD-penalty model through bifurcation analysis. We

choose the bifurcation parameter to be the penalty constant τ1. In order to use the AUTO bifurcation tracking

package, for this case, the asymptotically stable periodic solution must be provided—this of course is not known a

priori. To this end, we assume a constant (typically large) value of τ1and obtain the corresponding solution of the

non-autonomous system. However, it is not certain that this solution will have the same period as the forcing period

orevenbeingperiodic [21–23]. Toovercomethis,we will studythe stability of theparticular solution for thatspeciﬁc

value of τ1using Poincar´

e maps, following the work of [24–26]. Speciﬁcally, we have obtained the stability of the

periodic solution using Poincar´

e maps, which we used to ﬁnd the return times of the periodic solution, following

procedures outlined in [24,25]. We then employed the algorithm in [26] in order to ﬁnd the Floquet multipliers of

the periodic solution.

Let us examine a speciﬁc case to illustrate this approach. We consider the low Reynolds number Re =100 case

with the number of POD modes M=6 and integrate Eq. (5) for a few values of the penalty parameter, say in

the range of τ1∈[2000,3000]. We found, through the Poincar´

e map, that in this range an asymptotically stable

periodic solution does indeed exist. Hence, we can choose any value of τ1∈[2000,3000] to apply AUTO in order

to study stability of periodic solutions more systematically. We also found that the period of the asymptotic state

is T=5.9988 while the corresponding period from the full DNS is identical to this value. As we will show in the

next section, agreement in the period does not imply agreement in the ﬂow ﬁeld dynamics between the DNS and

the low-dimensional system.

Repeating this procedure with higher truncations at M=10 and 20 at the same Reynolds number, we found

similarly asymptotically stable periodic states, which can be used as starting points for the AUTO bifurcation

analysis. A similar study was performed for the POD-penalty system at Reynolds number Re =500 for M=6,

10 and 20 POD modes. With the penalty parameter τ1=3000, the system posses an asymptotically stable solution

with period T=4.5454, which is in agreement with the results from the full DNS. Other large values of the penalty

parameter yield similar results. After obtaining the asymptotically stable periodic solution, we use it as a starting

point for AUTO, and track the stability of the periodic solution by decreasing τ1until loss of stability is detected.

This produces the lowest value of the penalty parameter that guarantees stability. Speciﬁcally, for all the systems

examined here loss of stability shows bifurcation into a torus. The results of this analysis for both Reynolds numbers

are listed in Table 1, where in the third column the minimum values of τ1for stability are presented.

Table 1

Values of penalty parameter for stability and accuracy requirements

Re Modes Lowest penalty parameter for stability Best penalty parameter for accuracy

Re =100 6 5.24127 ≈5200

10 5.23467 ≈6000

20 3.20842 ≈6500

Re =500 6 1.39258 ≈9000

10 1.26819 ≥2.5×105

20 1.27168 ≥3.0×106

S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 227

Fig. 4. Simulation using M=10 POD modes for Re =100, τ1=5×105. DNS: ; POD-penalty simulation: solid line.

4.1.2. Accuracy of periodic solutions

We now turn our attention to the accuracy of the ﬂow dynamics predicted by the low-dimensional system at

different values of the τ1parameter. In Fig. 4 we plot the phase portraits predicted by the POD-penalty system for

τ1=500,000 against the DNS corresponding results for the system with truncation at M=10. At this value of

τ1an asymptotically stable state is obtained with the correct time period but as can be observed in this plot the

accuracy in the ﬂow dynamics is poor. In order to improve this accuracy we ﬁrst deﬁne a relative error for each

penalty parameter by

Eτ1=M

i=1(a∗

i−Q∗

i)2

M

i=1(Q∗

i)2,(10)

where a∗

iand Q∗

iare the maximum of the predicted POD modal coefﬁcients corresponding to the low-dimensional

system and DNS, respectively.

In Fig. 5 we plot the results of the bifurcation analysis for Re =100 and M=10 for the ﬁrst four POD modes.

We see that for the higher values of the penalty parameter τ1the accuracy of a∗

icompared to Q∗

iis worse than for

228 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237

Fig. 5. Bifurcation diagram using M=10 POD modes for Re =100. a∗

icorresponds to solid line and denotes the maximum of the POD-penalty

coefﬁcients. Q∗

i, denoted by dash line, is the corresponding maximum coefﬁcient of the DNS predictions.

the lower values of τ1. Similar results are also observed for the case of Re =100 with M=6 and 20, and also at

Re =500 with M=6. However, this trend is not universal. For example, for the truncation M=20 at Re =500

(see results in Fig. 6), the agreement in the ﬂow dynamics is good for large values of τ1but there is a lower bound

of the penalty parameter τ1below which this agreement is lost. Similar results were obtained at Re =500 with

M=10. Therefore, it is the combination of the penalty parameter and truncation parameter for certain complexity

in the ﬂow dynamics (here governed by the Reynolds number) that determines the quality of the prediction in the

POD-penalty system.

InFig.7,therelativeerror, deﬁned in Eq. (10), is plottedagainstthepenaltyparameterinordertodeterminethebest

penalty parameter for the case Re =100 with M=10. The best penalty parameter is found to be approximately

6000. A qualitatively different result is provided in Fig. 8, where the relative error for the case Re =500 with

Fig. 6. Bifurcation diagram using M=20 POD modes for Re =500. The legend is as in Fig. 5.

S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 229

Fig. 7. Left: relative error for the POD-penalty model with respect to DNS with M=10 POD modes for Re =100, and a close-up on the right.

M=20 is plotted against the penalty parameter. Here we ﬁnd that the error decreases monotonically with the

penalty parameter but above some value, much greater than the stability bound, the accuracy saturates.

A summary of our studies to determine the best values of the penalty parameter τ1for the best possible accuracy

is presented in the last column of Table 1. Following the results of the bifurcation analysis, we then performed

integration of the POD-penalty system using the parameters in Table 1. The corresponding results for two typical

cases are shown in Fig. 9 for Re =100 and Fig. 10 for Re =500. For the latter case we observed the following: for

the simulation of the POD-penalty system with M=10, the system can predict correctly up to the ﬁfth mode while

the prediction of higher modes is erroneous. When we increase the number of modes in the system to M=20, the

accuracy of prediction of the dynamics is much better, i.e. good accuracy is now obtained up to 15th mode, see

Fig. 11. We recall that these two cases correspond to a lower bound in the penalty parameter for best accuracy as

determined by the bifurcation analysis. This ﬁnding could possibly suggest that for cases with a lower bound for

the most effective penalty parameter a higher truncation is required to achieve better accuracy; it is not clear if this

result will be true in other ﬂow problems, however. We also note that integrating the POD-penalty Galerkin system

Fig. 8. Relative error for the POD-penalty model with respect to DNS with M=20 POD modes for Re =500, and a close up on the right.

230 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237

Fig. 9. Simulation using M=10 POD modes for Re =100, τ1=6000. DNS: ; POD-penalty simulation: solid line.

requires approximately 10 periods to reach an asymptotically stable state even if the “exact” DNS conditions are

used in the initialization process.

In summary, the lesson learned from the DNS study is that above a certain threshold in the value of penalty

parameter, stability of the periodic solution is obtained. However, the best accuracy may be obtained for speciﬁc

values of the penalty parameter that seem to depend strongly on the ﬂow dynamics and the truncation in the

POD-penalty low-dimensional system.

4.2. Experiment: Galerkin POD-penalty system

In this section, we will present the results for the POD-penalty for the experimental data. We have used two

truncations in the number of modes here, M=6 and 12. The number of Fourier modes that represents the periodic

time-dependent boundary condition is set to N=16. For this POD-penalty model, there are two penalty parameters

τ1andτ2that need to be speciﬁed. However,wewill adopt here a procedure where we trackthese penalty parameters

by ﬁxing one of them at a point that the asymptotically stable periodic solution for the POD-penalty system is

obtained. As in the previous study with DNS data, this asymptotically stable periodic solution might not be accurate

S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 231

Fig. 10. Simulation using M=20 POD modes for Re =500, τ1=4×106. DNS: ; POD-penalty simulation: solid line.

but it will be used as a starting point for AUTO to track the minimum value of either τ1and τ2for asymptotic

stability of the periodic solutions.

4.2.1. Stability of periodic solutions

From preliminary numerical experiments for both M=6 and 12 we have determined that at τ1=1000 and

τ2=1000 the corresponding POD-penalty systems possess an asymptotically stable solution; see Fig. 12 for these

speciﬁc parameters. As in the previous case with the DNS data, the periodic solution is then studied through the

Poincar´

e map to examine its stability. We have found that the periodic solution for these speciﬁc penalty parameters

is indeed asymptotically stable with period of T=0.89, which is in agreement with the data from PIV.

Havingdeterminedthe starting pointfor AUTO,thetracking of stabilityof the periodicsolution is then performed

by ﬁxing τ1=1000 and decreasing τ2until loss of stability is detected. We then switch the role of τ1and τ2and

perform an analogous analysis. This study produces the lowest values of the penalty parameter for stability, and the

corresponding results are presented in Table 2. We have found that for both values of M=6 and 12 the periodic

solution loses its stability when one of the Floquet multipliers crosses the unit circle at −1. This was also observed

232 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237

Fig. 11. Higher modes from the simulation using M=20 POD modes for Re =500, τ1=4×106. DNS: ; POD-penalty simulation: solid

line.

for M=6 with τ2=1000 ﬁxed while varying τ1. However, in the case of the truncation with M=12 the periodic

solution loses its stability when one of the Floquet multipliers crosses the unit circle at 1.

4.2.2. Accuracy of periodic solutions

In order to ﬁnd the most effective values of the penalty parameters that produces the most accurate asymptotically

stable periodic solution compared to the data from PIV, we also track the periodic branch by increasing τ1or

Table 2

Values of penalty parameters for stability limit and accuracy requirement for the POD-penalty system corresponding to experiment with

Re =2280

Modes Lowest penalty parameter for stability Best penalty parameters for accuracy

τ1=1000 τ2=1000 τ1=1000 τ2=1000

M=6 201.237 27.935 ≈2×105,∗≈7.5×104

M=12 142.652 601.109 ≥2×107≥7.5×107

∗Corresponding asymptotically stable periodic solution has a relative error greater than 5%.

S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 233

Fig. 12. Simulation using M=12 POD modes for PIV data, τ1=1000, τ2=1000. PIV: ; POD-penalty simulation: solid line.

τ2while ﬁxing the other one at 1000. Then, we compute the relative error in order to ﬁnd the best value of

the penalty parameter for accuracy. In Fig. 13, the relative error is presented for the case of M=12 with ﬁxed

τ2=1000. A summary of best values of the penalty parameter for all cases is presented in the last column of Table

2. Using the values from the table we then integrate in time the POD-penalty system to reach the asymptotic stable

periodic states. The results of such simulations are presented in Figs. 14 and 15 as phase portraits. There is very

good agreement with the corresponding experimental data with the higher truncation giving higher accuracy, as

expected.

5. Summary

We have developed a Galerkin POD-penalty method to construct low-dimensional dynamical systems for un-

steady ﬂuid ﬂows with time-dependent boundary conditions. Penalized boundaries are incorporated directly in the

Galerkin statement of the Navier–Stokes equations, and thus information about the pressure ﬁeld on such bound-

aries is not required. The resulting dynamical system is non-autonomous, so we couple it to an equivalent nonlinear

oscillator in order to study its stability using standard bifurcation analysis.

234 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237

Fig. 13. Relative error for the POD-penalty model from PIV data. Here, M=12 with ﬁxed τ2, and a close up on the right.

Fig. 14. Simulation using M=6 POD modes for PIV data, τ1=7.5×104,τ2=1000. PIV: ; POD-penalty simulation: solid line.

S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 235

Fig. 15. Simulation using M=12 POD modes for PIV data, τ1=8×107,τ2=1000. PIV: ; POD-penalty simulation: solid line.

We study the stability and accuracy of periodic solutions using the penalty parameter(s) as bifurcation param-

eter(s). We consider two prototype ﬂows based on results from direct numerical simulations (ﬂow past a circular

cylinder), and from experiments (wave–structure interaction). The results from both studies are qualitatively similar.

We ﬁnd that there is a threshold value of the penalty parameter above which asymptotic stability of the periodic

solution is guaranteed. This is an expected result, similar to what is known for numerical discretizations of Navier–

Stokes equations. The surprising, however, ﬁnding is that the accuracy of the solution predicted by the Galerkin

POD-penalty system does not improve as the penalty parameter increases, as it is the case for full numerical dis-

cretizations. Instead, there is a speciﬁc range within which the solution is accurate. In particular, depending on the

number of modes (i.e. truncation) and the ﬂow complexity (i.e. Reynolds number) the best solution may correspond

to a speciﬁc value of the penalty parameter or a range well above the threshold value for stability.

In numerical discretizations that employ the penalty approach to impose Dirichlet or other type of boundary

conditions,asthe penalty parameter approachesa very large number (e.g. inverseofmachineprecision)theboundary

conditions are imposed exactly, i.e. in a strong form. Correspondingly, the error in the solution scales inversely

proportional to the penalty parameter. Our ﬁndings here suggest that for low-dimensional systems, imposing the

boundary conditions in a strong form may lead to an erroneous solution. Similar trends have been observed in

spectral penalty methods for simulations of high Reynolds number turbulence at relatively low resolution [27]. This

236 S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237

can have great consequences in constructing effective low-dimensional dynamical systems as well as in formulating

proper boundary conditions in large-eddy simulations. However, generalization of this conclusion to other ﬂow

problems has to be tested very carefully.

Acknowledgments

The ﬁrst author gratefully acknowledges the Development and Promotion of Science and Technology Talents

(DPST) project from Thailand for providing his scholarship during his graduate studies at Brown University. The

authors would like to thank Dr. Y. Yang and Prof. D. Rockwell for providing PIV data. This work was supported

by ONR and NSF, and computations were performed at the facilities of NCSA (University of Illinois at Urbana-

Champaign) and at DoDs NAVO MSRC.

References

[1] N. Aubry, W.Y. Lian, E.S. Titi, Preserving symmetries in the proper orthogonal decomposition, SIAM J. Sci. Comput. 14 (2) (1993)

483–505.

[2] W. Cazemier, R.W. Verstappen, A.E. Veldman, Proper orthogonal decomposition and low-dimensional models for driven cavity ﬂows,

Phys. Fluids 10 (7) (1998) 1685–1699.

[3] S. Sirisup, G.E. Karniadakis, A spectral viscosity method for correcting the long-term behavior of POD models, J. Comp. Phys. 194 (1)

(2004) 92–116.

[4] X. Ma, G.E. Karniadakis, H. Park, M. Gharib, DPIV-driven ﬂow simulation: a new computational paradigm, Proc. R. Soc. London A 459

(2031) (2003) 547–565.

[5] S. Sirisup, G.E. Karniadakis, Y. Yang, D. Rockwell, Wave-structure interaction: simulation driven by quantitative imaging, Proc. R. Soc.

London A 460 (2043) (2004) 729–755.

[6] X. Zhou, L. Sirovich, Coherence and chaos in a model of turbulent boundary layer, Phys. Fluid A 4 (12) (1992) 2855–2874.

[7] G. Berkooz, P. Holmes, J.L. Lumley, N. Aubry, E. Stone, Observations regarding ‘Coherence and chaos in a model of turbulent boundary

layer’ by X. Zhou and L. Sirovich [Phys. Fluids A 4 (1992) 2855], Phys. Fluid 6 (4) (1994) 1574–1578.

[8] L. Sirovich, X. Zhou, Reply to ‘Observations regarding ‘Coherence and chaos in a model of turbulent boundary layer’ by X. Zhou and L.

Sirovich, [Phys. Fluids A 4 (1992) 2855], Phys. Fluid 6 (4) (1994) 1579–1582.

[9] P. Holmes, J.L. Lumley, G. Berkooz, Turbulence, Coherence Structures, Dynamical Systems and Symmetry, Cambridge University Press,

1996.

[10] J.F. Gibson, Dynamical Systems Models of Wall-Bounded, Shear-Flow Turbulence, PhD thesis, Cornell University, 2002.

[11] D. Gottlieb, S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM-CMBS, Philadelphia,

1977.

[12] Y. Yang, D. Rockwell, Wave interaction with a vertical cylinder: spanwise ﬂow patterns and loading, J. Fluid Mech. 460 (2002) 93–

129.

[13] E.J. Doedel, R.C. Paffenroth, A.R. Champneys, T.F. Fairgrieve, Yu.A. Kuznetsov, B. Sandstede, X. Wang, Auto 2000. Continuation and

bifurcation software for ordinary differential equations (with homcont). Technical Report, Caltech, 2001.

[14] S. Abarbanel, A. Ditkowski, Asymptotically stable fourth-order accurate schemes for the diffusion equation on complex shapes, J. Comp.

Phys. 133 (2) (1996) 279–288.

[15] J.S. Hesthaven, D. Gottlieb, A stable penalty method for the compressible Navier–Stokes equations. I. Open boundary conditions, SIAM

J. Sci. Comput. 17 (3) (1996) 579–612.

[16] J. Trujillo, G.E. Karniadakis, A penalty method for the vorticity–velocity formulation, J. Comp. Phys. 149 (1999) 32–58.

[17] A.E. Deane, I.G. Kevrekidis, G.E. Karniadakis, S.A. Orszag, Low-dimensional models for complex geometry ﬂows: application to grooved

channels and circular cylinders, Phys. Fluids A 3 (10) (1991) 2337–2354.

[18] X. Ma, G.E. Karniadakis, A low-dimensional model for simulating 3D cylinder ﬂow, J. Fluid Mech. 458 (2002) 181–190.

[19] G.E. Karniadakis, S.J. Sherwin, Spectral/hp Element Methods for CFD, Oxford University Press, 1999.

[20] S. Sirisup, Issues in low-dimensional modeling of unsteady ﬂows: convergence, asymptotic stability and reconstruction procedures, PhD

thesis, Division of Applied Mathematics, Brown University, 2005.

[21] T.A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, 1985.

[22] T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Springer-Verlag, 1975.

S. Sirisup, G.E. Karniadakis / Physica D 202 (2005) 218–237 237

[23] G.Makay,On some possibleextensionsofMassera’s theorem, ProceedingsoftheSixthColloquium on the QualitativeTheoryofDifferential

Equations (In EJ Qualitative Theory of Differential Equations), vol. 16, 2000, pp. 1–8.

[24] M. Hannon, On the numerical computation of Poincar´

e maps, Physica D 5 (1982) 412–414.

[25] W. Tucker, Computing accurate Poincar´

e maps, Physica D 171 (3) (2002) 127–137.

[26] R. Seydel, Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, Springer-Verlag, 1994.

[27] P.J. Diamessis, J.A. Domaradzki, J.S. Hesthaven, A spectral multidomain penalty method model for the simulation of high Reynolds

number localized incompressible stratiﬁed turbulence, J. Comput. Phys. 202 (1) (2005) 298–322.