[American Institute of Aeronautics and Astronautics 39th AIAA Fluid Dynamics Conference - San...

Modeling High Frequency Modes for Accurate

Low-Dimensional Galerkin Models

Imran Akhtar, Jeff Borggaard, Traian Iliescu

Interdisciplinary Center for Applied Mathematics

Virginia Tech, Blacksburg, VA 24061, USA

and

Calvin J. Ribbens

Department of Computer Science

Virginia Tech, Blacksburg, VA 24061, USA

Dynamical systems ideas have recently gained increased momentum in the study of

turbulent flows. The reason is that dynamical systems could be used to describe low

dimensional structures, such as the coherent structures in the turbulent boundary layer,

which play an important role in the dynamics of the flow. In this study, we perform three-

dimensional simulations of a turbulent flow past a circular cylinder at Re=1000. We record

1000 snapshots of the velocity field data for more than a dozen vortex shedding cycles. We

compute the POD modes and project the Navier-Stokes equations onto these modes. In

order to model higher frequency modes, we propose an LES-type approach and include an

additional term in the reduced-order model for closure.

I. Introduction

Most of the dynamical systems in fluid flows are described by partial-differential equations (PDEs). Atypical example is the control of fluid dynamical systems in which the Navier-Stokes equations are thestate equations. Due to the inherent nonlinearity in the Navier-Stokes equations and complexity of infinite-dimensional flow dynamics, one often reduces the PDEs to ordinary-differential equations (ODEs) to simplifythe dynamical system of the form

q = F(q). (1)

The real strength of reduced-order models lies in the predictive settings. The governing equations for mostof physical dynamical systems (e.g., fluid flows) comprise partial-differential equations (e.g., Navier-Stokesequations) which correspond to an infinite number of degrees of freedom. Such systems are solved numericallyusing various CFD methods, thereby reducing the system to a finite number of degrees of freedom. However,for time-varying, three-dimensional fluid flows, the number of degrees of freedom is of the order of millions.

One of the most successful dynamical systems ideas in the study of turbulent flows has been the ProperOrthogonal Decomposition (POD).1–4 POD starts with data from an accurate numerical simulation and thenextracts the most energetic modes in the system by using the singular value decomposition. One of the mainresearch areas in which POD has been used is to generate reduced-order models for the turbulent boundarylayer. The first such model was proposed by Aubry et al. in.5 This model has truncated the POD basis andhas used an eddy viscosity type approximation to model the effect of the discarded POD modes on the PODmodes kept in the model. The reduced-order model in5 has yielded good qualitative results, consideringthe coarseness of the approximation. The criterion used to assess the accuracy of the POD model was theintermittency of the bursting events in the turbulent boundary layer. In two subsequent papers,6,7 Podvinand Lumley have further investigated numerically the behavior of the POD model in.5 They found that themodel reproduced qualitatively well the physics of the turbulent boundary layer. Furthermore, by addingnew POD modes to the model, the accuracy of the model was increased.

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American Institute of Aeronautics and Astronautics

39th AIAA Fluid Dynamics Conference22 - 25 June 2009, San Antonio, Texas

AIAA 2009-4202

Copyright © 2009 by Authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Cazemier et al.8 developed a reduced-order model for the lid-driven cavity flow. They observed that theirmodel matched the CFD data only for short-time integration and diverged exponentially from their DNSresults later. They proposed a model with linear damping based on the average energy exchange betweenthe POD eigenfunctions. They argued that the energy content of a flow realization in the direction of PODeigenfunctions Φi is q2

i and on the average should be conserved. They added a linear damping term Diqi tothe dynamical system satisfying a constraint equation derived from energy conservation analysis.

In the POD model of the flow past a cylinder, Sirisup and Karniadakis9 showed that the inaccurate long-time integration depends on the number of modes in the Galerkin expansion, the Reynolds number, andthe flow geometry. They used a spectral vanishing viscosity (SVV) method,10 which adds a small amountof mode-dependent dissipation satisfying the entropy condition while retaining the spectral accuracy. Thus,Equation (1) is modified as

q = F(q) − H(q; ǫ,Qǫ), (2)

where ǫ → 0 is a viscosity amplitude and Qǫ is a viscosity convolution kernel. The SVV is typically applied tothe higher-order modes and the numerical value of ǫ depends on the number of modes for which the SVV isactivated. The parameters for the SVV model are found by an empirical method and a bifurcation analysis.However, their exact values are not known a priori and depend on the flow geometry and the number ofPOD modes.

The success of the reduced-order model in5 relies fundamentally on the energy cascade assumption,which states that energy flows from low index POD modes to higher index POD modes. The validity ofthe extension of the energy cascade concept to the POD setting is the main focus of the paper of Couplet,Sagaut, and Basdevant.11 The authors have investigated the energy transfer among POD modes in a non-homogeneous computational setting. By monitoring the triad interactions due to the nonlinear term in theNavier-Stokes equations, they have concluded that the transfer of energy among the POD modes is similarto the transfer of energy among Fourier modes. Specifically, they found that there is a net forward energytransfer from low index POD modes to higher index POD modes and that this transfer of energy is local innature (that is, energy is mainly transfered among POD modes whose indices are close to one another).

The studies in5–7,11 represent the motivation of our present study. These papers clearly suggest that theenergy cascade concept is also valid in a POD setting. Therefore, Large Eddy Simulation (LES) ideas basedon the energy cascade concept12 could also be used in devising reduced-order models in a POD context. Ourcurrent study proposes the use of such methodologies to improve the reduced-order POD model introducedin.5

We emphasize that our approach is complementary to that used in studies where the role of the modelingprocess is to provide increased numerical stabilization.13–19 Indeed, the modeling we are using in the presentstudy aims at providing the right physical behavior for the truncated POD model, i.e., to provide the energydissipation that the discarded POD modes would have yielded. The studies cited above, however, havea somehow different goal. They aim at providing the numerical stabilization necessary for a long termintegration of the truncated POD system while keeping the highest accuracy possible with the low numberof POD modes retained. The ultimate goal in POD model-reduction is, of course, to combine the numericalstability and accuracy of the truncated POD system with a faithful physical representation of the underlyingflow. The complex interplay between numerics and physics is, however, one of the main challenges that theCFD community currently faces.

It should also be noted that, in some cases of POD modes, while optimal in terms of energy representationof a data set, they may not be the best choice for reduced-order modeling,20 especially at higher Reynoldsnumbers. In POD based models, low-energy modes are generally truncated, therefore they do not accountfor observability and controllability of these modes.21 demonstrated such cases in which POD based modelsare unable to capture the dynamics of the system. The author proposed a balanced POD scheme in whichthe POD snapshot method was used to compute empirical Grammians.

In most of these applications, the POD-based approach is restricted to low Reynolds numbers. Thislimitation can be associated to the energy content present in the high frequency POD modes and can nolonger be neglected in the total energy of the dynamical system. In other words, the reduced-order model witha limited number of POD modes at high Reynolds number does not include the effect of higher POD modesanalogous to unresolved scales in large-eddy simulations (LES) framework. Thus, the dynamical system cannot dissipate enough energy through small scales. This leads to inaccurate long-time integration solution. Aquick solution to the problem could be to increase the number of snapshots and keep large number of POD

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modes in the Galerkin expansion. The drawback of this solution is the increase in dimensionality of the modeland complexity in the quadratic term which increases as cubic power of its dimension. Another approachis modeling the energy transport to the unresolved scales using the high frequency modes correspondingto higher POD modes neglected in the Galerkin expansion. In this paper, we will focus on modeling highfrequency modes in the reduced-order model and analyze the flow dynamics for a canonical problem, suchas flow past a circular cylinder. We intend to incorporate an LES-type term in the existing model, whichcan represent energy dissipation at smaller scales.

II. Numerical Methodology

A parallel CFD solver is used to simulate the flow past a circular cylinder.22,23 The governing equationsare written as follows:

∂Um

∂ξm

= 0, (3)

∂(J−1ui)

∂t+

∂Fim

∂ξm

= 0, (4)

where the flux is defined as

Fim = Umui + J−1 ∂ξm

∂xi

p −1

ReD

Gmn ∂ui

∂ξn

. (5)

Here ReD = U∞Dν

; J−1 = det(

∂xi

∂ξj

)

is the inverse of the Jacobian or the volume of the cell; Um = J−1 ∂ξm

∂xjuj

is the volume flux (contravariant velocity multiplied by J−1) normal to the surface of constant ξm; andGmn = J−1 ∂ξm

∂xj

∂ξn

∂xjis the “mesh skewness tensor.”

An “O”-type grid is used to simulate the flow past a cylinder as shown in Fig. 1. Dirichlet and Neu-mann boundary conditions are used for the inflow and outflow boundary conditions, respectively. Details ofvalidation, verification, and parallel implementation are found in Refs. 22, 23. No-slip and no-penetrationboundary conditions are prescribed on the cylinder surface.

For three-dimensional flows, we performed a DNS on a 192 × 256 × 192 grid distributed over 64 (8 × 8)processors with a computational domain of 30 × 4π. The load per processor was 192 × 32 × 24 grid points.The spanwise length Lz of the cylinder was kept 4π to match the computational domain in the numericalstudy of Evangelinos and Karniadakis,24 as shown in Fig. 2 (left). A spanwise grid spacing of ∆z = 0.06545,inspired by the LES study of Kravchenko and Moin25 at ReD = 3, 900, is applied. This spanwise grid spacingis fine enough to capture the fine length scales generated along the cylinder span. At this Reynolds number,the wake is fully turbulent and is no longer ordered (see Fig. 2). Vortical structures such as ribs and hairpinvortices are observed in the wake.

We compute the mean drag coefficient and the Strouhal number for the two- and three-dimensional flowsat ReD = 1, 000 and tabulate these results in Table 1. These flow parameters compare well with the existingexperimental by Norberg26 and numerical results by Evangelinos and Karniadakis.24 Fig. 3 shows the 1-Denergy spectrum at (x/D, y/D) = (1.0, 0.5) in the flow. In addition to the peak at the shedding frequency0.20, the power spectrum exhibits a − 5

3 law in the inertial range, which extends about half a decade inwave number space. This proves the capability of the code in capturing turbulence. Grid and domainindependence studies have been performed for the current solver.22

Table 1. Flow parameters at ReD = 1, 000.

Data from CD St

Experiment26 1.0 0.21

3-D DNS24 1.02 0.202

3-D DNS (present) 1.11 0.205

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III. Reduced-Order Model

A. Galerkin Projection

POD has been widely used to identify the coherent structures27 in turbulent flows and examine their stabil-ity.4 The flow field data (u, v, w) is generated from an experiment or a numerical simulation and is assembledin a matrix W3N×S , as shown in Equation (6); each column represents one time instant or a snapshot andS is the total number of snapshots for N grid points in the domain. The vorticity field can also be used forPOD. In the case of the velocity field, however, the eigenvalues are a direct measure of the kinetic energy ineach mode.

W =

u(1)1 u

(2)1 . . . u

(S)1

......

...

u(1)N u

(2)N . . . u

(S)N

v(1)1 v

(2)1 . . . v

(S)1

......

...

v(1)N v

(2)N . . . v

(S)N

w(1)1 w

(2)1 . . . w

(S)1

......

...

w(1)N w

(2)N . . . w

(S)N

(6)

Mathematically, we compute Φ for which the following quantity is maximum:

⟨

|u,Φ|2⟩

‖Φ‖2 , (7)

where 〈.〉 denotes the ensemble average. Applying variational calculus, one can show that Equation (7) isequivalent to a Fredholm integral eigenvalue problem represented as

∫

Ω

Rij(x,x′)Φj(x′) dx = λΦj(x), (8)

where i, j are the number of velocity components and R(x,x′) is the two-point space-time correlation tensor.4

B. POD Eigenfunctions

In the current study, we simulate the flow past a circular cylinder at different Reynolds numbers. Detailsof each simulation are given in Table 2. For ReD=100 and 200, we took 40 snapshots over a period ofone vortex-shedding cycle. For a relatively high Reynolds number of 525, 80 snapshots were recorded overtwo vortex-shedding cycles in order to capture more information of the flow field. Using the method ofsnapshots,28 we compute the POD eigenfunctions and their corresponding energy content in each mode.First four POD modes of the velocity fields for Cases I and IV are plotted in Figures 5-9.

Table 2. POD configurations.

Case ReD Domain Lz Grid size Snapshots (Period)

I 100 50D − 192 × 256 40(1)

II 200 50D − 192 × 256 40(1)

III 525 50D − 192 × 256 80(2)

IV 525 40D π 128 × 192 × 32 80(2)

In Figure 4, we plot the first 30 eigenvalues obtained from different cases. Each eigenvalue is normalizedas λi/

∑Si=1 λi. We observe that the two modes of each pair have values of the same order and they decrease

from one pair to the next approximately in a geometric progression. In Case I, the energy decays muchfaster as compared to other Reynolds numbers, indicating that fewer modes are required to represent the

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flow field. Case II shows a similar trend but with a less steeper decay in energy than Case I. In the othertwo cases III and IV, we simulate 2-D and 3-D flows at ReD=525 and compute the POD modes. The energydecay for the 2-D flow is higher than that for the corresponding 3-D flow.

In order to find the contribution of each mode to the total energy of the system, we define the cumulativeenergy as

EM =

∑Mi=1 λi

∑Si=1 λi

(9)

In Case I, the first ten modes contain more than 99.9% of the total flow energy. Case II shows a similartrend but requires more modes to capture the same amount of energy as in Case I. For the other two cases,we observe more than 30 modes are required to capture 99.9% of the energy. The energy content spanned bythe first 30 modes in a 2-D flow is higher than in a 3-D flow. In Figures 5-9, we plot the sreamwise, crossflow,and spanwise velocity modes for all of the cases (I-IV). Another observation is that as the Reynolds numberincreases the energy contained in high energy modes flattens. This behavior increases the number of PODmodes required to fully capture the flow physics in terms of the energy of the dynamical system.

C. Reduced-Order Model Validation

The POD eigenfunctions are used as a basis for a Galerkin projection of the incompressible Navier-Stokesequations. These POD eigenfunctions are orthogonal, divergence-free, and satisfy the boundary conditions.The velocity field is expanded as

u(x, t) ≈ u(x) +M∑

i=1

qi(t)Φi(x), (10)

where M is the number of POD modes used in the projection. We substitute Eqn. (10) into the Navier-Stokesequations, project these equations along the Φk, and obtain

qk(t) = Ak +M∑

m=1

Bkmqm(t) +M∑

m=1

M∑

n=1

Ckmnqn(t)qm(t), (11)

where

Ak =1

ReD

(Φk,∇2u) − (Φk, u · ∇u),

Bkm = −(Φk, u · ∇Φm) − (Φk,Φm · ∇u) +1

ReD

(Φk,∇2Φm),

Ckmn = −(Φk,Φm · ∇Φn).

where (a, b) =∫

Ωa · b dΩ represents the inner product between a and b. In Equation (11), A is an M × 1

vector resulting from the average flow field, B is the linear part of the dynamical system, and C is a tensorthat represents the quadratic nonlinearity,

We use the CFD data at ReD = 100 and choose first ten POD modes to develop a reduced-order modelfor the current flow configuration. We perform the Galerkin projection and compute Ak, Bkm, and Ckmn,where k,m, n = 1, 2, ..., 10 in Equation (11). Thus, the CFD problem with 192 × 256 degrees of freedom isreduced to a ten-dimensional dynamical system.

We integrate Equation (11) simultaneously using the ode45 function in Matlab and compute the qk(t).The temporal evolutions of the first four velocity coefficients represent an oscillatory solution, as shownin Figure 10. The pairs (q1, q2) and (q3, q4) have a frequency of ω and 2ω respectively, with a 90 phasedifference within each pair. We also project each snapshot onto the POD modes to compute the velocitycoefficient qi(t) as follows:

qi(t) = (u′(x, ti),Φ(x))Ω. (12)

For the ten-dimensional Galerkin model, we perform a two-dimensional projection of the phase portraiton the plane (q1, q2) and compare it with the projection obtained on the (q1, q2)-plane. We observe thatthe two projections compare well as shown in Figure 11. The Galerkin approximation of the system canalso be validated by reconstructing the velocity field and comparing it to the numerical simulations. Theapproximated flow field is in good agreement with the CFD results. Details of validation can be found inRef. 22.

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IV. Closure Model Approach

It is evident from Figure 4 that as the Reynolds number increase, the number of POD modes requiredto capture the energy of the dynamical system also increases. For Reynolds number flows where three-dimensional effects are significant, more POD modes are required than those required for the two-dimensionalflow at the same Reynolds number. Thus, truncation in the number of POD modes in the reduced-ordermodel may lead to inaccurate prediction of the flow physics. It is important to note that the modelingprocedure presented here is not meant to overcome the stability issues of the reduced-order models. It israther an attempt to model the effect of neglected energy dissipation effects.

The success of the reduced-order model in5 relies fundamentally on the energy cascade assumption,which states that energy flows from low index POD modes to higher index POD modes. The validity of theextension of the energy cascade concept to the POD setting is the main focus of the paper of Couplet et al.11

The authors have investigated the energy transfer among POD modes in a non-homogeneous computationalsetting. By monitoring the triad interactions due to the nonlinear term in the Navier-Stokes equations, theyhave concluded that the transfer of energy among the POD modes is similar to the transfer of energy amongFourier modes. Specifically, they found that there is a net forward energy transfer from low index PODmodes to higher index POD modes and that this transfer of energy is local in nature (that is, energy ismainly transfered among POD modes whose indices are close to one another).

In this study, we perform DNS of the flow past a circular cylinder with Lz

D= 2 over 144×192×16 grid at

ReD = 1000. At this Reynolds number, the wake is fully turbulent and the flow is not periodic anymore, asobserved at lower Reynolds number range. Unlike low Reynolds number flows, the lift and drag coefficientsare not periodic anymore as shown in Fig. 12 and we observe frequency and amplitude modulation. It is alsoevident from the power spectra of the lift and drag forces (see Fig 13) in which we observe noise representingturbulence in the wake.

We record 1000 snapshots over 14 shedding cycles with over 60 snapshots per shedding cycle. We place aprobe at (1.0,0.5,1.0) and record the velocity field. We plot streamwise velocity in Fig. 18 where the verticallines indicate the time interval ver which snapshots have been taken. Using the method of snapshots, wecompute the POD modes and plot the streamwise, normal, and spanwise velocity modes in Figures 15, 16,and 17, respectively. We observe that the 3-D modes are not symmetric anymore, as observed in the 3-Dmodes for ReD = 525. At present we use only the first 150 snapshots due to limitation in computing thePOD modes in which we capture few non-periodic shedding cycles.

The first 10 POD modes capture about 92% of the system’s energy while 20 modes capture about 98% ofthe energy. We project these snapshots onto the POD-modes and obtain q for first 10 and 20 POD modes.We use these projected q and reconstruct the velocity field at the probe location as shown in Fig. 19. Asexpected the velocity field constructed from 20 modes is in good agreement with the DNS data as comparedto velocity data constructed from the first 10 POD modes. Although the agreement is close, there is someenergy which is deficient and requires closure for energy dissipation.

We attempt to model a closure term based on an LES-type approach. We introduce an additional termin the reduced-order model (11)

qk(t) = Ak +M∑

m=1

Bkmqm(t) +M∑

m=1

M∑

n=1

Ckmnqn(t)qm(t) +M∑

m=1

Dkmqm(t), (13)

whereDk = (Φk, Cs |S|∇

2Φ).

Here |S| is the Frobenius norm of symmetric strain rate tensor. Integrating equation (13) is not straight-forward because the norm in the equation makes the integration of the above system complex. However, inorder to compute the norm, we substitute the q with q obtained from the projections for each snapshot. Ofcourse, this will limit integration time to the time for which snapshots are obtained. Nevertheless, it providesa suitable starting point in our attempt at addressing closure models in the POD-based reduced-order modelframework. The term D is stabilizing, since its eigenvalues are negative and Cs acts as an additional weightto this term. As of now we keep Cs = 2 to control the effect of modeled term. We expect that the closureterm will be effective to model the effect of missing modes.

In this study, we keep M=10 and integrate the reduced-order models obtained in Eqns. (11) and (13).

Since we are using (q) to compute D term, we integrate the model only upto the time interval over which

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snapshots are available. We compute the time evolution of the qi obtained with and without the closuremodel. First six temporal coefficients are plotted in Fig. 20. In these figures, we observe that the standardreduced-order model tends to blow-up due to inappropriate treatment of energy transfer in the system. Thisbehavior is especially clear for q3 and q4. Inclusion of closure model term tends to contain the blow-upby adding diffusion to the system. We expect that the the model will be able to predict the flow field instatistical sense.

V. Conclusion and Future Work

We proposed an LES-type closure model for POD-based reduced-order models. We investigated the flowpast a circular cylinder at Re=1000 where the wake is turbulent and recorded the snapshots of the velocityfield. We computed the POD modes for all the velocity components and observed spatially unsymmetricbasis functions. We projected the snapshot data onto few of these POD modes and compared the velocitiesat a probe location in the wake of the cylinder. We then developed a closure term for the reduced-ordermodel similar to the one used in the subgrid scale modeling. We presented our initial results and investigatedthe effect of the modeled term. In future studies, we will further investigate the effect of the parametersin the model from a more physical point of view. We will also perform long-time integration to computestatistical quantities associated with the flow.

VI. Acknowledgments

This research was partially supported by the Air Force Office of Scientific Research under contractsFA9550-07-1-0273 and FA9550-08-1-0136 and the National Science Foundation under contract DMS-0513542.Numerical simulations were performed on Virginia Tech Advanced Research Computing - System X. Theallocation grant and support provided by the staff is also gratefully acknowledged.

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pp. 561–590.

(a)

X

Y

Z

z-partition =1

z-partition =2

z-partition =3

(b)

Figure 1. (a) A 2-D layout of “O” grid in the (r, θ)-plane, showing the inflow and outflow directions and (b)a 3-D layout of the complete domain and the grid is plotted for only one processor, indicating the load perprocessor in a 24 (8×3) processor platform. The grid is plotted only for the region of processor “1”.

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Figure 2. Geometry of the cylinder with arrows showing the flow direction and checkerboard pattern indicating8 processors in the spanwise direction (left), isosurfaces of ωz at t = 0 (center) and t ≈ Ts/2 (right).

10−2

10−1

100

101

10−10

10−8

10−6

10−4

10−2

100

Frequency

Ene

rgy

spec

trum

slope = − 5/3

Figure 3. 1-D energy spectrum at ReD=1,000.

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Mode

λ

5 10 15 20 25 3010-6

10-5

10-4

10-3

10-2

10-1

100

Figure 4. Normalized eigenvalues. Square: Case I; triangle: Case II; circle; Case III, and diamond; Case IV.

(a) Mode 1 (b) Mode 2

(c) Mode 3 (d) Mode 4

Figure 5. The streamwise velocity modes (φu

i, i = 1, 2, 3, 4) at ReD=100.

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Figure 6. The crossflow velocity modes (φv

i, i = 1, 2, 3, 4) at ReD=100.

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Figure 7. Isosurfaces of the streamwise velocity modes (φu

i, i = 1, 2, 3, 4) at ReD=525 with 25% transparency.

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Figure 8. Isosurfaces of the crossflow velocity modes (φv


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Figure 9. Isosurfaces of the spanwise velocity modes (φw


Time

q 1,2

360 370 380

-2

0

2

(a) q1 (solid) and q2 (dashed)

Time

q 3,4

360 370 380

-0.1

0

0.1

(b) q3 (solid) and q4 (dashed)

Figure 10. The velocity coefficients qi = 1, 2, 3, 4 at ReD=100.

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q1

q 2

-2 0 2

-2

0

2

Figure 11. Projection of the POD phase portrait on (q1, q2)-plane (solid) and snapshot portrait (triangle).

150 200 250 300 350 400 450 500

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Time

CL

150 200 250 300 350 400 450 500

1

1.05

1.1

1.15

1.2

1.25

Time

CD

Figure 12. Time histories of the lift and drag coefficients at Re=1000.

0 0.5 1 1.5 210

−8

10−6

10−4

10−2

100

f

Pow

er

fs

3fs

0 0.5 1 1.5 210

−10

10−8

10−6

10−4

10−2

100

f

Pow

er

fs

4fs

2fs

Figure 13. Power spectra of the lift and drag coefficients at Re=1000.

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(a) Streamwise Mean Mode (b) Normal Mean Mode

(c) Spanwise Mean Mode

Figure 14. Isosurfaces of the mean velocity modes at ReD = 1000.

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(e) Mode 5 (f) Mode 6

Figure 15. Isosurfaces of the streamwise velocity modes (φu

i, i = 1, 2, ..., 6) at ReD = 1000.

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Figure 16. Isosurfaces of the normal velocity modes (φv

i, i = 1, 2, ..., 6) at ReD = 1000.

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Figure 17. Isosurfaces of the spanwise velocity modes (φw

i, i = 1, 2, ..., 6) at ReD = 1000.

60 80 100 120 140 160−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

T ime

up

Figure 18. Time history of streamwise velocity at probe location (1.0,0.5,1.0). Vertical dashed lines indicatethe time of snapshots.

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64 66 68 70 72−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

T ime

up

(a) u − velocity

64 66 68 70 72−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

T ime

vp

(b) v − velocity

64 66 68 70 72−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

T ime

wp

(c) w − velocity

Figure 19. Comparison of projected velocity components with DNS data at probe location (1.0,0.5,1.0); DNS(solid,red), M=10 (dash,black), and M=20 (dashdot,blue)

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0 2 4 6 8 10 12 14 16 18 20 22−3

−2

−1

0

1

2

3

T ime

q 1

(a) q1

0 2 4 6 8 10 12 14 16 18 20 22−3

−2

−1

0

1

2

3

T ime

q 2

(b) q2

0 2 4 6 8 10 12 14 16 18 20 22−3

−2

−1

0

1

2

3

T ime

q 3

(c) q3

0 2 4 6 8 10 12 14 16 18 20 22−3

−2

−1

0

1

2

3

T ime

q 4

(d) q4

0 2 4 6 8 10 12 14 16 18 20 22−3

−2

−1

0

1

2

3

T ime

q 5

(e) q5

0 2 4 6 8 10 12 14 16 18 20 22−3

−2

−1

0

1

2

3

T ime

q 6

(f) q6

Figure 20. Time histories of the qi (i=1,2,...,6) with (dashed) and without (solid) model.

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