A Galerkin Model of the Pressure Field in

Incompressible Flows

Imran Akhtar∗ and Ali H. Nayfeh†

Department of Engineering Science and Mechanics, MC0219

Virginia Tech, Blacksburg, VA 24061, USA

Calvin J. Ribbens‡

Department of Computer Science

Virginia Tech, Blacksburg, VA 24061, USA

Proper orthogonal decomposition (POD) has been used to develop a reduced-ordermodel of the hydrodynamic forces acting on a circular cylinder. Direct numerical simu-lations (DNS) of the incompressible Navier-Stokes equations have been performed usinga parallel computational fluid dynamics (CFD) code to simulate the flow past a circularcylinder. Snapshots of the velocity and pressure fields are used to calculate the divergence-free velocity and pressure modes, respectively. We use the dominant of these velocityPOD modes (a small number of eigenfunctions or modes) in a Galerkin procedure toproject the Navier-Stokes equations onto a low-dimensional space, thereby reducing thedistributed-parameter problem into a finite-dimensional nonlinear dynamical system intime. We investigate the stability of the reduced-order model by using long-time integra-tion and propose to use a shooting technique to home on the system limit cycle. We obtainthe pressure-Poisson equation by taking the divergence of the Navier-Stokes equation andthen projecting it onto the pressure POD modes. The pressure is then decomposed intolift and drag components and compared with the CFD results.

Nomenclature

D Diameter of the cylinderLz Spanwise length to diameter ratioU∞ Free-stream velocityν Kinematic viscosityRe Reynolds numberSt Strouhal numberCL Lift coefficientCD Drag coefficientCD Mean drag coefficientN Number of grid pointsM Number of POD modesS Number of snapshotsΦ(x) Velocity POD modeΨ(x) Pressure POD modeq(t) Velocity coefficient in Galerkin expansiona(t) Pressure coefficient in Galerkin expansion

∗Graduate Research Assistant and AIAA Student Member.†University Distinguished Professor and AIAA Fellow.‡Associate Professor.

1 of 27

American Institute of Aeronautics and Astronautics

46th AIAA Aerospace Sciences Meeting and Exhibit7 - 10 January 2008, Reno, Nevada

AIAA 2008-611

Copyright © 2008 by Authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

I. Introduction

Circular cylinders are extensively used in the study of bluff-body fluid dynamics due to their geometricsimplicity and common use in engineering applications.1–5 The flow past a circular cylinder undergoesvarious transition stages as the Reynolds number (Re) is increased, which makes the flow physics complexand provides rich phenomena for analysis. For example, the wake transitions from a steady to an unsteadystate around ReD ≈ 40 (based on the cylinder diameter D) and the von Karman vortex street is observed.As ReD exceeds 180, a second bifurcation occurs, and the wake transitions to a three-dimensional wake inthe form of vortex-loops and streamwise vortices. Around ReD = 105, the boundary layer becomes turbulentbefore separation.

Reduced-order modeling of incompressible flows, such as the flow past a cylinder, plays an importantrole in academic and industrial research. Several analytical approaches have been proposed to model vortexshedding over a circular cylinder. Bishop and Hassan6 were among the earliest to suggest using a self-excitedoscillator to represent the forces over a cylinder due to vortex shedding. Several other analytical modelswere extended to elastic and moving cylinders.7–11

Vortex shedding exerts oscillatory forces on the body, which are often decomposed into drag and liftcomponents along the freestream and crossflow directions, respectively. If the body is capable of flexingor moving rigidly, these forces can cause it to oscillate and the classical vortex-induced vibration (VIV)problem takes place. If the frequency of vortex shedding is close to a natural frequency of the body, theresulting resonance can generate large-amplitude oscillations, which may ultimately cause structural failure.Therefore, it is important to extend the reduced-order model of the velocity field to compute the pressureon the cylinder surface.

The proper orthogonal decomposition (POD) provides a tool to formulate an optimal basis or minimumdegrees of freedom (or modes) to represent a dynamical system. POD is also known as the Karhunen-Loeve expansion in statistics and principal component analysis or empirical orthogonal functions (EOF) inmeteorology. It has successfully been applied to many engineering and scientific systems, including low-dimensional dynamics modeling,12–16 image processing,14 and pattern recognition.17 These models allowan analytical insight into the physical phenomenon and enable application of dynamical system theory andcontrol methods.

POD was introduced in the field of turbulence by Lumley18 to identify the coherent structures in theflow and examine their stability.14 Later, Sirovich19 introduced the snapshot method to study the dynamicsof some turbulent flows. In this method, a set of instantaneous flow field solutions, or snapshots, is obtainedfrom either experimental data or a numerical simulation. This allows reduction of large data sets obtainedfrom computational fluid dynamics (CFD) or particle image velocimetry (PIV), while still preserving thedominant features of the flow represented by POD eigenfunctions or modes. The optimal basis functionscapture the dominant energy contents and makes the POD eigenfunctions a suitable candidate for thereduced-order model. Low-dimensional dynamical models consist of a set of nonlinear ordinary-differentialequations (ODE) of the form

q = F(q). (1)

This low-dimensional model can be used to effectively apply various strategies for flow control.The POD approach has been implemented successfully in the low-dimensional modeling of laminar cylin-

der wakes. Deane et al.20 modeled the dynamics of the flow past a cylinder with an eight-dimensionalGalerkin model. Ma and Karniadakis15 generalized the POD Galerkin models to predict the three-dimensionaltransition initiated by the Mode A instability5 at Re ≈ 180.

The results of the POD based reduced-order model, obtained in these studies, are in agreement with thenumerical simulations, at least for short-time integration. However, long-time integration of the reduced-order model might not produce the limit cycles obtained with the CFD code. The solution can drift to someerroneous state even if it is initialized with the correct periodic state. This instability is associated withthe presence of multiple spurious limit cycles. Foias et al.21 investigated the existence of multiple spurioussteady states in the Galerkin expansion of the Kuramoto-Sivashinsky equation. For a similar equation,Aubry et al.22 also found spurious states in their POD model that captured 99.99% of the system’s energy.They observed that the predicted solution is not the right limit cycle. In the POD model of the flow past acylinder, Sirisup and Karniadakis23 showed that the onset of divergence from the correct limit cycle dependson the number of modes in the Galerkin expansion, the Reynolds number, and the flow geometry. They used

2 of 27

American Institute of Aeronautics and Astronautics

a spectral vanishing viscosity (SVV) method,24 which adds a small amount of mode-dependent dissipationsatisfying 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 appliedto the higher 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. In the current study, we discuss a shooting method for homing on the correct limit cycle.

In general, the POD based approach models the velocity fields in the flow. However, in most of the en-gineering and industrial applications of fluid-structure interaction, the pressure distribution over the surfaceis essential for computing the hydrodynamics forces over the structure. These forces may lead the dynam-ical system to undesired vibrations, thus causing damage or even failure of the structure. It is thereforeimportant to model the pressure field in addition to the velocity field.

We present a pressure model for incompressible flows based on the Galerkin projection of the pressure-Poisson equation onto the POD modes. In incompressible flows, the pressure-Poisson equation is obtained bytaking the divergence of the Navier-Stokes equation in vector form and applying the continuity constraint.The model requires snapshots of the pressure field, in addition to snapshots of the velocity field, for computingthe pressure POD modes. We project the pressure-Poisson equation onto the pressure POD modes anddevelop a reduced-order model for the pressure field. The pressure is then integrated over the cylinder toobtain hydrodynamic forces acting on the cylinder.

The manuscript is organized as follows. In Section II, we discuss the governing equations and the method-ology for the numerical simulations. We validate the CFD solver and discuss its parallel implementation andperformance. We develop a reduced-order model for the velocity field of the flow past a circular cylinder inSection III. In Section IV, we discuss the stability of the reduced-order model and present a shooting methodto compute the physical limit cycle. Then we develop a reduced-order model for the pressure field in SectionV and compute the lift and drag forces on the cylinder.

II. Numerical Methodology

A. Governing Equations

The Navier-Stokes and continuity equations are the governing equations for the present problem. For in-compressible flow, they can be represented as follows:

∂uj

∂xj

= 0 (3)

∂ui

∂t+

∂

∂xj

(ujui) = −1

ρ

∂p

∂xi

+ µ∂2ui

∂xj∂xi

, (4)

where i,j=1,2,3; the ui represent the Cartesian velocity components (u,v,w); p is the pressure; ρ is the fluiddensity; and µ is the viscosity. Equations (3) and (4) are nondimensionalized using the diameter (D) of thecylinder as the length scale and the free stream velocity (U∞) as the velocity scale. Thus, the Reynoldsnumber is given by ReD = DU∞

ν, where ν = µ

ρ. The other important geometric quantity, spanwise length of

the body, is nondimensionalized using the cylinder diameter D and is denoted by Lz.In this study, a body conformal “O” type grid is employed to simulate the flow over a body, a planar

view is shown in Figure 1. We employ curvilinear coordinates (ξ, η, ζ) in an Eulerian reference frame. Forthe specific case of a circular cylinder, the generalized coordinates (ξ, η, ζ) can be represented by polarcoordinates (r, θ, z). However, to maintain curvilinear coordinates, we use the former notation.

Equations (3) and (4) are transformed into curvilinear coordinates in strong conservative form as follows:

∂Um

∂ξm

= 0 (5)

∂(J−1ui)

∂t+

∂Fim

∂ξm

= 0, (6)

3 of 27

American Institute of Aeronautics and Astronautics

where the flux is defined as

Fim = Umui + J−1 ∂ξm

∂xi

p −1

ReD

Gmn ∂ui

∂ξn

. (7)

Here 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; and Gmn = J−1 ∂ξm

∂xj

∂ξn

∂xj

is the “mesh skewness tensor”.

B. Discretization Scheme

A non-staggered-grid layout is employed to solve the transformed Navier-Stokes equations. The Cartesianvelocity components (u, v, w) and pressure (p) are defined at the center of the control volume in the com-putational space and the volume fluxes (U, V,W ) are defined at the mid points of its corresponding faces.All of the spatial derivatives are approximated with second-order accurate central differences except for theconvective terms. Using the same central differencing for the convection terms may lead to spurious oscil-lations in the coarser regions of the grid, thereby leading to erroneous results. In the present formulation,we discretize the convective terms using a variation of QUICK; that is, we calculate the face values of thevelocity variables (ui) from the nodal values using a quadratic upwinding interpolation. The upwinding

of QUICK is carried out by computing the positive and negative volume fluxes (Um+|Um|2 ) and (Um−|Um|

2 ),respectively, and using the generic stencil.

A semi-implicit scheme is employed to advance the solution in time. The diagonal viscous terms areadvanced implicitly using the second-order accurate Crank-Nicolson method, whereas all of the other termsare advanced using the second-order accurate Adams-Bashforth method. The Adams-Bashforth scheme waschosen because of its computational efficiency when coupled with the fractional step method. The discretizedequations are

∂Um

∂ξm

= 0 (8)

J−1 uk+1i − uk

i

∆t=

3

2(Ck

i + DE(uki )) −

1

2(Ck−1

i + DE(uk−1i ))

+ Ri(pk+1) +

1

2(DI(u

k+1i + uk

i )), (9)

where the ∂∂ξm

represent the discrete finite-difference operators in the computational space; the superscriptsrepresent the time step; the Ci represent the convective terms; the Ri are the discrete operators for thepressure gradient terms; DI is the discrete operator representing the implicitly treated diagonal viscousterms; and DE is the discrete operator for the explicitly treated off-diagonal viscous terms. Mathematically,these terms are defined as follows:

Ci = −∂

∂ξm

(Umui), (10)

Ri = −∂

∂ξm

(J−1 ∂ξm

∂xi

), (11)

DI =∂

∂ξm

(νGmn ∂

∂ξn

) for m = n, (12)

DE =∂

∂ξm

(νGmn ∂

∂ξn

) for m 6= n. (13)

It is important to note that, due to the orthogonality property for the specific case of a cylinder, the crossterms for the mesh skewness tensor Gmn are zero; that is, when m 6= n. Therefore, the terms in Equation(13) are identically zero and the stencil for the problem is a seven-point stencil for a 3-D problem and afive-point stencil for a 2-D problem.

C. Solution Algorithm

Commonly used numerical methods for solving the incompressible Navier-Stokes equations include the ar-tificial compressibility method25 and the fractional-step or projection method.26 The key feature of the

4 of 27

American Institute of Aeronautics and Astronautics

artificial compressibility method is that it allows the use of efficient compressible flow algorithms for com-puting incompressible flows. However, for a time-dependent problem, the system of equations may becomehighly stiff.27 On the other hand, a fractional-step method relies on the idea of operator splitting to un-couple the pressure computation from that of the velocity field, it is efficient for solving the incompressibleNavier-Stokes equations. The theory for this method is quite well developed28 and it has been successfullyused in a variety of flow configurations.29–33 In the present formulation, we apply a fractional-step methodto advance the solution in time. The fractional-step method splits the momentum equation into

(a) an advection-diffusion equation - momentum equation solved without the pressure term,(b) a pressure-Poisson equation - constructed by implicit coupling between the continuity equation and

the pressure in the momentum equation, thus satisfying the constraint of mass conservation.The governing equations are solved using a methodology similar to that employed by Zang et al.33

However, the algorithm is extended to parallel computing platforms and the 2-D domain decompositiontechnique is employed to distribute the problem among different processors. This time-splitting methodleads to the predictor-corrector solution. The predictor equation (advection-diffusion equation) is defined asfollows:

(

I −∆t

2J−1DI

)

(u∗i − uk

i ) =∆t

J−1

[

3

2(Ck

i + DE(uki ))

−1

2(Ck−1

i + DE(uk−1i )) + DI(u

ki )

]

in Ω , (14)

where Ω refers to the interior of the computational domain and I is the identity matrix. The variable u∗i is

called the “intermediate velocity”, which is not constrained by continuity. Unlike the solution procedure ofapproximate factorization adopted by Zang et al.33 to solve Equation (14), the line successive over relaxation(LSOR) scheme is employed to solve the advection-diffusion equation with a relaxation factor of 1.5.

The corrector equation is represented as follows:

uk+1i − u∗

i =∆t

J−1

[

Ri(φk+1)

]

in Ω (15)

∇ · uk+1i = 0 in Ω , (16)

where the function φ is related to the pressure p by

Ri(p) =

(

J−1 −∆t

2DI

)(

Ri(φ)

J−1

)

. (17)

The correction step in Equation (15) can be rewritten in terms of the volume flux as follows:

Uk+1m = U∗

m −∆t

J−1

(

Gmn ∂φk+1

∂ξn

)

, (18)

where U∗m = J−1 ∂ξm

∂xj(u∗

j )face is known as the intermediate volume flux. In the present work, QUICK is used

for interpolating the u∗j on the respective cell faces. Substituting Equation (18) into Equation (8) yields the

following pressure-Poisson equation for φn+1:

∂

∂m

(

Gmn ∂φk+1

∂ξn

)

=1

∆t

U∗m

∂ξm

. (19)

It can be noted that, in the pressure correction step, the coefficients consist only of the mesh skewness tensorGmn. This elliptic equation is solved using the bi-conjugate gradient stabilized method (BiCGStab).

Application of suitable and well-posed boundary conditions is crucial for any simulation. Appropriateboundary conditions are applied on different sections of the domain boundary. For the inflow boundarycondition, we use a Dirichlet boundary condition. This follows from the reasonable assumption that, farupstream of the body, the displacement effect of the body is negligible and thus the flow is sufficientlyclose to potential flow. This inflow boundary condition has been used in most of the simulations of flowsover bodies, varying in shape from streamlined to bluff, except for the cases when the effects of incomingturbulence are being analyzed. For the outflow boundary condition, we use a Neumann boundary condition.No-slip and no-penetration boundary conditions are applied on the cylinder surface. Moreover, periodicboundary conditions are applied in the η and ζ directions.

5 of 27

American Institute of Aeronautics and Astronautics

D. Code Validation

We performed a DNS of the flow over a circular cylinder at ReD=525 on a 128 × 192 × 32 grid. In thesimulation, the outer domain is 30D with Lz = π. A CFL (Courant-Friedrich-Levy) number based on theconvection term in curvilinear coordinates is used as a guideline in choosing the time step. The simulationsshow that a stable time stepping is achieved for a CFL ≈ 0.3, which for the grid used corresponds to anondimensional time step size ∆t = 2 × 10−3.

The fluid force on the cylinder is the manifestation of the pressure and shear stresses acting on the surfaceof the cylinder. The net force can be decomposed into two components, namely lift and drag forces. Theseforces are nondimensionalized with respect to the dynamic pressure. The coefficients of lift and drag canthus be written in terms of the pressure and shear stresses as follows:

CL = − 1Lz

Lz∫

0

2π∫

0

(

p sin θ − 1

ReD

ωz cos θ

)

dθdz (20)

CD = − 1Lz

Lz∫

0

2π∫

0

(

p cos θ + 1

ReD

ωz sin θ

)

dθdz, (21)

where p is the pressure and ωz is the spanwise vorticity component on the cylinder surface. The resultsobtained from this simulation are presented in Table 1, where CD represents the mean drag and St is theStrouhal number, the nondimensionalized frequency defined as St = fD/U∞.

Table 1. Computed flow parameters.

Data from CD St

Experiment34 1.15 − 1.2 −

Experiment1 − 0.21

3-D DNS35 1.24 0.220

3-D DNS (present) 1.25 0.1984

Grid and domain independence studies are critical in order to verify the accuracy of the computationalresults. Therefore, we perform a study to verify the grid/domain independence of our results at ReD=525.We compare some physical parameters, such as the mean drag coefficient and the Strouhal number, toestablish grid/domain independence.

E. Parallel Implementation and Performance

To implement the algorithm on a distributed-memory, message-passing parallel computer, we use a two-dimensional domain decomposition technique such that each processor gets a “slice” of the grid, as shown inFigure 2(a). In this figure, a two-dimensional view of the grid is shown and is divided among 8 processors.The spanwise dimension is also divided into domains such that each “slice” has some “depth”, as depictedin Figure 2(b). For example, if a 128 × 192 × 96 grid in (ξ, η, ζ) is decomposed into 8 × 3 processors in theη and ζ directions, then each processor will have a load of 128 × 24 × 32 grid points. This technique allowsfor a simple way to implement boundary conditions and keeps an equal load distribution for each processor.Moreover, the data points are exchanged only in two directions, the η and ζ directions, thereby reducing theinter-processor communication cost.

The speed-up and efficiency are typically defined as follows:

Speed-Up =Simulation time on 1 processor

Simulation time on Np processors(22)

(23)

Efficiency =Achieved speed-Up

Ideal speed-up. (24)

Several simulations were carried out to check the speed-up of the parallel code. For two-dimensional cases,the grid is decomposed only in the η direction. Figure 3(a) shows the speed-up curves for two different grid

6 of 27

American Institute of Aeronautics and Astronautics

sizes at ReD=200 along with the linear (ideal) speed-up curve. We note that the speed-up curves are withrespect to 2 processors. It is observed that the 256× 256 grid shows better performance than the 128× 128grid because the ratio of computation to communication (per processor) is greater for the large problem.The 256 × 256 case yields a parallel efficiency of 73% on 16 processors.

For the three-dimensional case, the speed-up is calculated with respect to 4 processors since the orga-nization of the code is inherently parallel and the boundary conditions are defined assuming at least fourprocessors, two in each of the η and ζ directions. The speed-up curve for the 64× 128× 48 grid at ReD=200is plotted in Figure 3(b). These results indicate significant savings in the computation time as the number ofprocessors increases from 4 to 64. We observe a parallel efficiency of 90% for this 3-D case on 64 processors.

III. Reduced-Order Model of the Velocity Fields

A. Numerical Simulation

We simulate a two-dimensional flow past a cylinder at ReD = 100. In the simulation, we employ a 192× 256grid with an outer diameter of 50D. The flow data or snapshots of the steady-state velocity and pressurefields are sampled with a constant time interval (∆Ts). The velocity field data (u, v) are assembled in amatrix W2N×S , as follows:

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

(25)

Each column represents one time instant or a snapshot and S is the total number of snapshots for N gridpoints in the domain. The vorticity field can also be used for POD, however, in the case of the velocityfield, the eigenvalues of W are a direct measure of the kinetic energy in each mode. Deane et al.20 observedthat 20 snapshots are sufficient for the construction of the first eight eigenfunctions at ReD = 100− 200. Ingeneral, numerical studies36 suggest that the first M POD modes, where M is even, resolve the first M/2temporal harmonics and require 2M number of snapshots for convergence. In the current study, we take 40snapshots of the flow field over one vortex shedding cycle and assemble the data in W. We write the velocityfield as the sum of the average component (u) and the fluctuation component (u′). Here, u = 〈u〉, where〈〉 is the time average of the assembled data, is subtracted from W. Then, the fluctuations are expanded interm of the Φi as follows:

u(x, t) ≈ u(x) +

M∑

i=1

qi(t)Φi(x), (26)

where M is the number of POD modes used in the projection. We perform the singular value decomposition(SVD) of the data ensemble; that is, we represent W as W = UΣVT , where U represents the POD basis, Σcontains the singular values σi, and V is the right singular matrix. We compute the first S POD modes usingstandard LAPACK routines. The POD modes are orthogonal, divergence-free and satisfy the boundaryconditions in the domain. The eigenvalue λi of WTW is a measure of the energy contained in the ith modeand is equal to the square of the corresponding singular value σi. In Figure 5, we plot the first 20 eigenvaluesnormalized as λi = λi/

∑Sj=1 λj . We observe that eigenvalues occur in pairs and decrease from one pair to

the next approximately in a geometric progression. The eigenvalue λi is proportional to the energy containedin the ith mode of the expansion. The first ten modes contain more than 99.9% of the total flow energy. Thisproperty of the POD modes makes the Galerkin approximation a suitable candidate for the reduced-ordermodeling.

We plot the first eight POD modes of the streamwise and crossflow velocity components in Figures 6 and7, respectively. We observe that the streamwise component φu

i (i = 1, 2, 5, 6) are antisymmetric with respectto the x -axis; that is,

φui (x,−y) = −φu

i (x, y), (27a)

7 of 27

American Institute of Aeronautics and Astronautics

while φui (i = 3, 4, 7, 8) are symmetric; that is,

φui (x,−y) = φu

i (x, y). (27b)

On the other hand, the crossflow component φvi is symmetric for i = 1, 2, 5, 6 with respect to the x -axis; that

is,φv

i (x,−y) = φvi (x, y), (28a)

and antisymmetric for i = 3, 4, 7, 8; that is,

φvi (x,−y) = −φv

i (x, y). (28b)

B. Galerkin Projection

We substitute Equation (26) into Equation (6), project this Equation onto Φk, and obtain

qk(t) = Ak +

M∑

m=1

Bkmqm(t) +

M∑

m=1

M∑

n=1

Ckmnqn(t)qm(t), (29)

where

Ak =1

ReD

(Φk,∇2u) − (Φk, u.∇u),

Bkm = −(Φk, u.∇Φm) − (Φk, φm.∇u) +1

ReD

(Φk,∇2φm),

Ckmn = −(Φk, φm.∇Φn),

and (a, b) =∫

Ωa · b dΩ represents the inner product between a and b.

We note that using Green’s theorem and the divergence-free property, the pressure term drops out fromEquation (29)15 for the case of p=0 on the outerflow boundary (Ωso). The POD eigenfunctions are identicallyzero on the inflow boundary because the average flow is subtracted from the total flow. However, in caseof Neumann boundary conditions on Ωso, the contribution of the pressure term is not exactly zero for thecylinder wake. The outer domain is intentionally kept at 25D from the cylinder to minimize the pressureeffects. Hence, the pressure is negligible on the outflow boundary so the pressure term vanishes in thereduced-order model.36

In Equation (29), A is an M × 1 vector, resulting from the average flow field and B is the linear part ofthe dynamical system. It is an M ×M matrix comprising three terms: the first two terms are the byproductof the nonlinearity in the Navier-Stokes equation, (convection term) and originate from the interaction of theaverage field with the eigenfunctions, whereas the third term results from the linear dissipative operator andis a function of ReD. Thus, we can write B as the sum of a linear component and a component originatingfrom the nonlinearity as shown:

B = BNL +1

ReD

BL. (30)

From linear stability analysis, we can compute the eigenvalues of B and hence ascertain the stability of thetrivial solution. Moreover, C is a tensor that represents the quadratic nonlinearity in the dynamical system.

C. POD Simulation

We choose the first ten POD modes to develop a reduced-order model for the current flow configuration. Weperform the Galerkin projection and compute Ak, Bkm, and Ckmn, where k,m, n = 1, 2, ..., 10 in Equation(29). Thus, the CFD problem with 192× 256 degrees of freedom is reduced to a ten-dimensional dynamicalsystem.

The eigenvalues of B consist of five complex conjugate pairs. For small Re, all eigenvalues are in the leftplane. As Re is increased past the critical value Rec ≈ 40, a pair of these eigenvalues cross transversely theimaginary axis; that is the system undergoes a Hopf bifurcation, consequently, vortex shedding is initiatedleading to a periodic solution. It would be interesting to determine the individual contribution of the linearand nonlinear components on the eigenvalues of B. The eigenvalues of BL/ReD are real and in the left-half

8 of 27

American Institute of Aeronautics and Astronautics

plane. Thus, the instability arises from the terms generated due to the nonlinear interaction of the averageflow and the POD eigenfunctions from the Navier-Stokes projection. The instability due to the linear part ofthe dynamical system would lead to the growth of the velocity with time. However, the nonlinear componentCijk in the reduced-order model plays a stabilizing role and limits the growth, giving rise to a limit cycle.

We integrate Equation (29) using a Runge-Kutta scheme to compute the qi. The temporal evolution ofthe first eight velocity coefficients represent an oscillatory solution as shown in Figure 8. The pairs (q1, q2),(q3, q4), (q5, q6), and (q7, q8) have a frequency of ω, 2ω, 3ω, and 4ω, respectively, with a 90 phase differencewithin each pair. We also project each snapshot onto the POD modes to compute the velocity coefficientqi(t) as follows:

qi(t) = (u′(x, ti),Φ(x))Ω. (31)

For the ten-dimensional Galerkin model, we perform a two-dimensional projection of the phase portrait onthe plane (q1, q2) and compare it with the projection obtained on the (q1, q2)-plane. We observe that thetwo projections compare well as shown in Figure 9. The Galerkin approximation of the system can alsobe validated by reconstructing the velocity field and comparing it to the numerical simulations. In Figure10, we compare the instantaneous velocity field from the CFD results with the velocity field obtained fromEquation (26). The approximated flow field is in good agreement with the CFD results.

We then integrate the dynamical system for long time to study the stability of the limit cycle. We observethat the states qi diverges from the physical limit cycle to a spurious limit cycle. In Figure 11, we plot thefirst four states and observe that the divergence is gradual for some of the states, whereas it is abrupt anddevelops an offset in the other states. This long-time integration divergence occurs after approximately 400shedding cycles, however, it is found to depend on the number of modes being used in the reduced-ordermodel. Increasing the number of modes, which correspond to approximately 100% of the energy, we foundthat the solution of the reduced-order model still drifts to a non-physical limit cycle. Because experimentsand CFD simulations yield only one limit cycle corresponding to the correct solution of the physical problem,the appearence of spurious limit cycles is a maifestation of the Galerkin projection.

To investigate convergence of the solution of the reduced-order system, we use a direct approach in thetime domain. We apply a shooting37 method to adjust the initial conditions so that the system does notdrift to a spurious limit cycle. We discuss this method in detail in the next section.

IV. Shooting Method

In this method, the initial-value problem (1) is converted into a two-point boundary-value problem asfollows:

q = F(q) (32a)

q(0) = α (32b)

q(T ) = α, (32c)

where α is the initial guess and T is the period of the limit cycle. In other words, we seek an initial conditionq(0) = α and a solution q(t;α) with a minimal period T such that

q(T,α) = α. (33)

In other words, starting from a point on the limit cycle (i.e., q(0) = α) and integrating Equation (32a) overthe period T of the limit cycle should bring back the trajectory to its initial condition. Because we do notknow α and T exactly, we start with a guess of α0 and T0 and use a Newton-Raphson scheme to correctthis guess; that is, we calculate

δα = α − α0 (34a)

δT = T − T0, (34b)

such that Equation (33) is satisfied within a tolerance ε; that is,

q(T0 + δT,α0 + δα) − (α0 + δα) ≤ ε. (35)

9 of 27

American Institute of Aeronautics and Astronautics

We expand Equation (35) in a Taylor series expansion about (T0,α0) and obtain

[

∂q

∂α

(T0,α0) − I

]

δα −∂q

∂T(T0,α0)δT = α0 − (T0,α0), (36)

where ∂q

∂αis an M × M matrix, I is an M × M identity matrix, and ∂q

∂Tis an M × 1 vector. Moreover it

follows from Equation (1) that

∂q

∂t(T0,α0) = F(α0). (37)

In order to compute ∂q

∂α, we differentiate Equation (1) and the initial conditions with respect to α, and

obtaind

dt(∂q

∂α

) = DqF(q)∂q

∂α

(38a)

∂q

∂α

(0) = I, (38b)

where DqF(q) is the Jacobian of F(q).

We solve Equations (32a), (32b), (38a), and (38b) to obtain q(t,α0) and ∂q

∂αat (T0,α0) simultaneously.

However, Equation (36) constitutes a system of M with M + 1 unknowns. Since the phase is arbitrary in aperiodic solution of an autonomous system, we impose a phase condition to obtain an extra equation. Thereare several ways to specify the phase condition. In the present case, we require the corrections δα to benormal to the vector field F; that is, FT δα = 0. Thus, we obtain the following system:

[

∂q∂α

(T0,α0) − I F(α0)

FT (α0) 0

][

δα

δT

]

=

[

α0 − q(T0,α0)

0

]

. (39)

For the reduced-order model of the flow past a cylinder, developed in Equation (29), the Jacobian is computedas

∂qk

∂ql

= Bkl +

M∑

m=1

Cklmqm +

M∑

n=1

Cklnqn. (40)

We then differentiate Equation (29) with respect to the αi(t) to obtain the matrix ∂q

∂αas follows:

∂qk

∂αi

=

M∑

m=1

Bkm

∂qk

∂αi

+

M∑

m=1

M∑

n=1

Ckmn

∂qn

∂αi

qm +

M∑

m=1

M∑

n=1

Ckmnqn

∂qm

∂αi

. (41)

Thus, we solve Equation (41), which is a set of M2 equations and the M equations in the original dynamicalsystem. Therefore, in terms of computational cost, we solve M2 + M equations in the shooting method tocompute the new initial conditions and time period T of the limit cycle. We note that the shooting methodis sensitive to the initial time period T0 and requires a “good” guess for convergence.

In the present study, the first snapshot data provides a reasonable initial guess α0 for α. We solve110 equations to compute the new initial conditions. We then integrate Equation (29) with the modifiedinitial conditions. In Figure 12, we compute the time response of the first four states for approximately4000 vortex-shedding cycles and observe a stable solution. The figure shows the envelope of the oscillatingresponse of the states for 20000 time units. The reduced-order model thus obtained is stable and does notrequire any additional term to stabilize the dynamical system. It is important to note that this iterativeprocedure of correcting the initial conditions works well when almost 100% of the energy is contained in themodes of the Galerkin expansion. However, it is in no way a substitute of modeling higher frequency scalesdue to the truncated modes and should not be confused with viscosity models employed at higher Reynoldsnumbers.

10 of 27

American Institute of Aeronautics and Astronautics

V. Reduced-Order Model of the Pressure Field

In the Galerkin projection of Navier-Stokes equations onto the POD modes and application of the Green’stheorem, the pressure term drops out from the model. The divergence-free property of the POD modes isessential for this step. However, this is true only in case of the Dirichlet pressure boundary condition (p=0)on the outflow boundary. Infact, there is some contribution of the pressure, though negligible, in caseof the Neumann boundary condition ( ∂p

∂n= 0) on the outerflow boundary. In incompressible flows, the

pressure-Poisson equation is the governing equation for the pressure; that is,

∂2p

∂x2i

= −∂ui

∂xj

∂uj

∂xi

, (42)

where i and j refer to the Cartesian components of the vector.Noack et al.36 discussed the contribution of the pressure term in the Galerkin models. They developed

a low-dimensional Galerkin model for spatially evolving laminar and transitional shear layers, based on 2Dand 3D Navier-Stokes simulations. They observed that the effect of the pressure term is significant fora two-dimensional mixing-layer, is less pronounced for the three-dimensional analogue, and is small in anabsolutely unstable wake flow. Subsituting the Galerkin approximation (26) into Equation (42) yields

∆p = −

3∑

i=1

3∑

j=1

∂Φmi

∂xj

∂Φnj

∂xi

qmqn, (43)

where m and n denote the expansion modes. They expanded the solution of Equation (43) as a function ofthe velocity field in the form

p =

N∑

m=0

N∑

n=0

pmnqm(t)qn(t), (44)

where N is the number of POD modes and the pmn represent the partial pressures and satisfy the Neumanncondition (i.e., n · ∇p = 0). However, the objective of their work was to model the pressure gradient termas a function of the qm(t). Therefore, they let

(Φi,−∇p)Ω = −

(

Φi

N∑

j=0

N∑

k=0

∇pjkqj(t)qk(t)

)

Ω

=

N∑

j=0

N∑

k=0

C′ijkqj(t)qk(t), (45)

where C′ijk = (Φi,−∇p)Ω for i = 1, 2, ..., N . Thus, this pressure model leads to an additional quadratic term

in the Galerkin system (29). In another empirical pressure model, they neglected the quadratic terms qjqk

and kept only the linear terms. We note that q0q0 = 1 and q0qi = qi constitute the linear terms in theGalerkin expansion. They record pressure snapshot data pm and compute the corresponding qm(t) from thesnapshot data. The pressure gradient term thus obtained is given by

(Φi,−∇p)Ω =

N∑

j=0

B′ijqj(t), (46)

where the coefficients of B′ij are determined from linear regression of the pm and qm(t).

In another numerical study, Cohen et al.38 investigated feedback flow control of the wake of a “D” shapedcylinder using the POD approach, also known as Karhunen-Loeve expansion.14 They obtained the steady-state streamwise velocity u and pressure p data from 100 equally-spaced snapshots over approximately 15vortex-shedding cycles at Re=300. The flow field (u, p) is decomposed as the sum of an average componentand a fluctuation component using the temporal coefficients, given in Equation (26). Using the inner product,they computed the empirical correlation matrix and solved the eigenvalue problem to obtain the Φ(x). Theq(t) are obtained by projecting the snapshots onto the POD modes. For their closed-loop control design,the qi(t) could not be measured directly and therefore they designed an estimator to compute the states ofthe system using the Linear Stochastic Estimator (LSE) approach. The q1−3 are mapped onto the extractedsensor signals from the pressure signals ps as qm(t) =

∑ns=1 Cm

s ps(t), where n is the number of sensorsand Cm

s represents the coefficients of the linear mapping. This approach of measuring the pressure field issensitive to the number of sensors and their configurations and depends on the accuracy of the estimator.

11 of 27

American Institute of Aeronautics and Astronautics

We use the reduced-order model of the velocity field to develop a model for the pressure field. Knowledgeof the velocity field enables us to construct a pressure model from the pressure-Poisson equation. A naiveapproach would be to reconstruct the velocity field from the reduced-order model at every time step and solvethe pressure-Poisson equation over the entire domain to compute the pressure at every instant. However,the computational cost of this approach would be of the same order of magnitude as solving the completeNavier-Stokes equations in CFD.

An alternate solution is to compute the POD eigenfunctions of the pressure field and use a similarGalerkin procedure for the pressure-Poisson equation. The pressure field is written in a matrix PN×S , eachcolumn representing one time instant as follows:

P =

p(1)1 p

(2)1 . . . p

(S)1

......

...

p(1)N p

(2)N . . . p

(S)N

(47)

Similar to the velocity field, we write the pressure field as a sum of a mean component (P ) and a fluctuationscomponent (p′). The average pressure P = 〈p〉 is subtracted from P. We perform the SVD of the resultingP and compute the pressure POD modes (Ψi). The first eight pressure POD modes are plotted in Figure14. Next, the pressure is expanded in terms of the Ψi(x) as follows:

p(x, t) ≈ P (x) +M∑

i=1

ai(t)Ψi(x). (48)

We note that in the Galerkin expansion (48), the temporal coefficients ai(t) are different from the qi(t). Wealso note that integrating the average pressure over the cylinder surface yields zero lift and the mean dragcorresponding to its Reynolds number. We substitute Equations (26) and (48) into Equation (42) and obtain

∂2P

∂xi2

+

N∑

k=1

ak(t)∂2Ψk

∂xi2

= −

[

∂ui

∂xj

+

N∑

m=1

qm(t)∂Φ(m)i

∂xj

]

.

[

∂uj

∂xi

+

N∑

n=1

qn(t)∂Φ(n)j

∂xi

]

. (49)

Equation (49) is then transformed into curvilinear coordinates using a transformation similar to that usedin the CFD code where the Φi(x) and the qi(t) are known a priori. We project the transformed equationonto Ψk(x) and obtain the following governing equations for the ak(t):

Dkmak(t) = Ek +

M∑

m=1

Fkmqm(t) +

M∑

m=1

M∑

n=1

Gkmnqn(t)qm(t), (50)

where

Dkm = (Ψk,∇2Ψi)

Ek = −(Ψk,∇2P ) − (Ψk,∇u : ∇u)

Fkm = −(Φk,∇u : ∇Ψm) − (φk,∇Ψm : ∇u)

Gkmn = −(Φk,∇Ψm : ∇Ψn).

Equations (50) constitute a set of algebraic equations quadratic in terms of the qi. We compute the qi

by integrating the dynamical system in Equation (29). Using these coefficients, we compute the ai fromEquation (50). Similar to the qi, we observe that the pairs (a1, a2), (a3, a4), (a5, a6), and (a7, a8) have thefrequencies ω′, 2ω′, 3ω′, and 4ω′, respectively, with a 90 phase difference within each pair, as shown inFigure 15. Due to the nonlinearity in the system, ω′ is slightly different from ω in the velocity reduced-ordermodel. We note that ω′ corresponds to the Strouhal number in the CFD simulation. Thus, we recreate thepressure field using Equation (48). We compute the mean-pressure distribution over the cylinder surfacefrom the POD simulation and compare it with the CFD simulation in Figure 16(a). We observe a goodagreement between the two results.

12 of 27

American Institute of Aeronautics and Astronautics

The pressure is then integrated over the surface to compute the lift and drag forces on the cylinder usingEquation (21). It should be noted that Equation (21) contains both of the pressure and shear components.The shear component is computed from the modeled velocity field. In Figure 16(b), we plot the lift coefficientcomputed from the POD approximation along with lift calculated from the CFD solver. We also observegood agreement in the mean and fluctuating drag components.

VI. Conclusion and Recommendations

A low-dimensional model of the lift and drag forces on a circular cylinder has been developed. Numericalsimulation of the flow past the cylinder at ReD = 100 is performed to obtain snapshots of the flow field(velocity and pressure). We computed the eigenfunctions for the velocity and pressure by using the PODapproach. The model is obtained from a Galerkin projection of the Navier-Stokes equations onto the velocitymodes. Although the POD based models lack robustness away from the reference simulation, they providean analytical insight into the physical phenomenon. Moreover, these models, being a set of ODE, enableapplication of dynamical systems theory and control methods. We also investigated the stability of themodel and presented a shooting method to compute initial conditions on the limit cycle and its period.The pressure model is then developed from a Galerkin projection of the pressure-Poisson equation onto thepressure modes. The lift and drag coefficients are then computed by integrating the pressure and shear forceson the cylinder surface.

The spectrum of application of the reduced-order models can be broadened by combining the snapshotdata in different flow regimes and computing the POD eigenfunctions. The modes thus obtained may beused to model the flow characteristics over a wider range of Reynolds number. Moreover, control techniquescan effectively be applied to the reduced-order model to suppress vortex shedding.

VII. Acknowledgments

This work was supported by Vetco Gray and Starmark Offshore Inc. Numerical simulations were per-formed on Virginia Tech Advanced Research Computing - System X. The allocation grant and supportprovided by the staff is also gratefully acknowledged.

References

1Roshko, A., “On the development of turbulent wakes from vortex streets,” NACA Rep. 1191 , Vol. (unpublished), 1954.2Tomboulides, A. G., Israeli, M., and Karniadakis, G. E., “Efficient removal of boundary divergence errors in time-splitting

methods,” Journal of Scientific Computing, Vol. 4, No. 3, 1989, pp. 291–308.3Karniadakis, G. E. and Triantafyllou, G. S., “Three-dimensional dynamics and transition to turbulence in the wake of

bluff objects,” Journal of Fluid Mechanics, Vol. 238, 1992, pp. 1.4Wu, J., Sheridan, J., and Welsh, M. C., “An experimental investigation of the streamwise vortices in the wake of a bluff

body,” Journal of Fluids and Structures, Vol. 8, 1994, pp. 621–635.5Williamson, C. H. K., “Vortex dynamics in the cylinder wake,” Annual Review of Fluid Mechanics, Vol. 28, 1996,

pp. 477–539.6Bishop, R. and Hassan, A., “The Lift and Drag Forces on a Circular Cylinder in Flowing Fluid,” Proceedings of the Royal

Society Series A, Vol. 277, 1963, pp. 32–50.7Hartlen, R. and Currie, I., “Lift-Oscillator Model of Vortex-Induced Vibration,” ASCE Journal of Engineering Mechanics,

Vol. 96, 1970, pp. 577–591.8Iwan, W. and Blevins, R., “A Model for Vortex-Induced Oscillation of Structures,” Journal of Applied Mechanics, Vol. 41,

No. 3, 1974, pp. 581–586.9Landl, R., “A Mathematical Model for Vortex-Excited Vibration of Cable Suspensions,” Journal of Sound and Vibration,

Vol. 42, No. 2, 1975, pp. 219–234.10Skop, R. and Griffin, O., “On a theory for the vortex-excited oscillations of flexible cylindrical structures,” Journal of

Sound and Vibration, Vol. 41, No. 3, 1975, pp. 263–274.11Krenk, S. and Nielsen, S., “Energy Balanced Double Oscillator Model for Vortex-Induced Vibrations,” ASCE Journal of

Engineering Mechanics, Vol. 125, No. 3, 1999, pp. 263–271.12Berkooz, G., Holmes, P., and Lumley, J. L., “The proper orthogonal decomposition in the analysis of turbulent flows,”

Annual Review of Fluid Mechanics, Vol. 53, 1993, pp. 321–575.13Deane, A. E. and Mavriplis, C., “Low-dimensional description of the dynamics in separated flow past thick airfoils,”

AIAA Journal , Vol. 6, 1994, pp. 1222–1234.14Holmes, P., Lumley, J. L., and Berkooz, G., Turbulence, coherent structures, dynamical systems and symmetry, Cam-

bridge University Press, Cambridge, UK, 1996.

13 of 27

American Institute of Aeronautics and Astronautics

15Ma, X. and Karniadakis, G., “A low-dimensional model for simulating three-dimensional cylinder flow,” Journal of Fluid

Mechanics, Vol. 458, 2002, pp. 181–190.16Noack, B. R., Afanasiev, K., Morzynski, M., and Thiele, F., “A hierarchy of low-dimensional models for the transient

and post-transient cylinder wake,” Journal of Fluid Mechanics, Vol. 497, 2003, pp. 335–363.17Sirovich, L. and Kirby, M., “Low-dimensional procedure for the characterization of human faces,” J. Opt. Soc. Am. A.,

Vol. 4, No. 3, 1987, pp. 529–524.18Bakewell, H. P. and Lumley, J. L., “Viscous sublayer and adjacent wall region in turbulent pipe flow,” The Physics of

Fluids, Vol. 10, No. 9, 1967, pp. 1880–1889.19Sirovich, L., “Turbulence and the dynamics of coherent structures,” Quart. Appl. Math., Vol. 45, 1987, pp. 561–590.20Deane, A. E., Kevrekidis, I. G., Karniadakis, G. E., and Orsag, S. A., “Low-dimensional models for complex geometry

flows: Application to grooved channels and circular cylinder,” Phys. Fluids A, Vol. 3, No. 10, 1991, pp. 2337–2354.21Foias, C., Jolly, M. S., Kevrekidis, I. G., and Titi, E. S., “Dissipativity of the numerical schemes,” Nonlinearity, Vol. 4,

1991, pp. 591–613.22Aubry, N., Lian, W. Y., and Titi, E. S., “Preserving symmetries in the proper orthogonal decompostion,” SIA J. Sci.

Comput., Vol. 14, No. 2, 1993, pp. 483–505.23Sirisup, S. and Karniadakis, G. E., “A spectral viscosity method for correcting the long-term behavior of POD models,”

Journal of Computational Physics, Vol. 194, 2004, pp. 92–116.24Tadmor, E., “Convergence of spectral methods for nonlinear conservation laws,” SIAM Journal of Numerical Analysis,

Vol. 26, No. 1, 1989, pp. 30.25Chorin, A. J., “Numerical solution of incompressible flow problems,” Mathematics of Compututation, Vol. 22, 1968,

pp. 745–762.26Chorin, A. J., “A numerical method for solving incompressible viscous flow problems,” Journal of Computational Physics,

Vol. 2, 1967, pp. 12–26.27Steger, J. L. and Kutler, P., “Implicit Finite-Difference Procedures for the Computation of Vortex Wakes,” AIAA Journal ,

Vol. 15, No. 4, 1977, pp. 581–590.28Chorin, A. J., “A Numerical Method for Solving Incompressible Viscous Flow Problems,” Journal of Computational

Physics, Vol. 135, No. 2, 1997, pp. 118–125.29Orszag, S. A. and Kells, L. C., “Transition to Turbulence in Plane Poiseuille and Plane Couette Flow,” Journal of Fluid

Mechanics, Vol. 96, No. 1, 1985, pp. 159–205.30Kim, J. and Moin, P., “Application of a Fractional-Step Method to Incompressible Navier-Stokes,” Journal of Computa-

tional Physics, Vol. 59, 1985, pp. 308–323.31Mittal, R. and Balachandar, S., “Effect of three-dimensionality on the lift and drag of nominally two-dimensional cylin-

ders,” Physics of Fluids, Vol. 7, No. 8, 1995, pp. 1841–1865.32Street, C. L. and Hussaini, M. Y., “A Numerical Solution of the Appearance of Chaos in Finite Length Taylor-Couette

Flow,” Applied Numerical Mathematics, Vol. 6, 1991, pp. 123–139.33Zang, Y., Street, R., and Koseff, J., “A Non-staggered Grid, Fractional Step Method for Time-Dependent Incompressible

Navier-Stokes Equations in Curvilinear Coordinates,” Journal of Computational Physics, Vol. 114, 1994, pp. 18–33.34Weiselsberger, C., NACA TN 84 , Vol. (unpublished), 1922.35Mittal, R. and Balachandar, S., “Direct numerical simulation of flow past elliptic cylinders,” Journal of Computational

Physics, Vol. 124, 1996, pp. 351–367.36Noack, B. R., Papas, P., and Monkewitz, P. A., “The need for a pressure-term representation in emperical Galerkin

models of incompressible shear flows,” Journal of Fluid Mechanics, Vol. 523, 2005, pp. 339–365.37Nayfeh, A. H. and Balachandran, B., Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Meth-

ods, Wiley, New York, 1995, pp. 449–454.38Cohen, K., Seigal, S., McLaughlin, M. L. T., and Seifert, A., “Sensor placement for closed-loop flow control of a “D”

shaped cylinder wake,” 2nd AIAA Flow Control Conference, AIAA Paper No. 2004–2523, 2004.

14 of 27

American Institute of Aeronautics and Astronautics

Figure 1. A 2-D layout of an “O” grid over a circular cylinder, showing the inflow and outflow directions.

(a)

X

Y

Z

z-partition =1

z-partition =2

z-partition =3

(b)

Figure 2. (a) A 2-D layout of the 128 × 192 × 96 “O” grid distributed among 8 processors in the η direction.The grid is plotted only for the region of processor “1”. (b) A 3-D layout of the complete domain and the gridis plotted for only one processor, indicating the load per processor in a 24 (8×3) processor platform. The gridis plotted only for the region of processor “1”. (Note: grid is drawn course clarity.)

15 of 27

American Institute of Aeronautics and Astronautics

(a)

Number of Processors

Spe

ed-U

p(W

RT

to4

Pro

cess

ors)

0 8 16 24 32 40 48 56 640

2

4

6

8

10

12

14

16

(2 x 2)

(4 x 2)

(8 x 2)

(16 x 2)

(4 x 3)

(16 x 3)

(16 x 4)

(b)

Figure 3. Speed-up trends of the (a) 2-D and (b) 3-D CFD code relative to two and four processors fordifferent grid sizes. The dashed line represents the ideal speed-up.

(a) (b)

Figure 4. Average flow fields: (a) streamwise and (b) crossflow velocity components at ReD=100.

16 of 27

American Institute of Aeronautics and Astronautics

Mode

λ

5 10 15 20

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Figure 5. Normalized eigenvalues at ReD = 100.

17 of 27

American Institute of Aeronautics and Astronautics

(a) Mode 1 (b) Mode 2

(c) Mode 3 (d) Mode 4

(e) Mode 5 (f) Mode 6

(g) Mode 7 (h) Mode 8

Figure 6. The streamwise velocity modes (φu

i, i = 1, 2, ..., 8) at ReD=100.

18 of 27

American Institute of Aeronautics and Astronautics

(a) Mode 1 (b) Mode 2

(c) Mode 3 (d) Mode 4

(e) Mode 5 (f) Mode 6

(g) Mode 7 (h) Mode 8

Figure 7. The crossflow velocity modes (φv

i, i = 1, 2, ..., 8) at ReD=100.

19 of 27

American Institute of Aeronautics and Astronautics

Time

q 1,2

360 370 380

-2

0

2

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

Timeq 3,

4360 370 380

-0.1

0

0.1

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

Time

q 5,6

360 370 380

-0.1

0

0.1

(c) q5 (solid) and q6 (dashed)

Time

q 7,8

360 370 380

-0.05

0

0.05

(d) q7 (solid) and q8 (dashed)

Figure 8. The velocity coefficients qi = 1, 2, ..., 8 at ReD=100.

20 of 27

American Institute of Aeronautics and Astronautics

q1

q 2

-2 0 2

-2

0

2

Figure 9. Projection of the POD phase portrait on (q1, q2)-plane (solid) and snapshot portrait (triangle).

21 of 27

American Institute of Aeronautics and Astronautics

(a) u velocity (CFD) (b) u velocity (POD)

(c) v velocity (CFD) (d) v velocity (POD)

Figure 10. Instantaneous velocity fields from CFD and POD simulations.

22 of 27

American Institute of Aeronautics and Astronautics

(a) q1 (b) q2

(c) q3 (d) q4

Figure 11. Divergence of the qi, i = 1, 2, ..., 4 to spurious limit cycles.

23 of 27

American Institute of Aeronautics and Astronautics

(a) q1 (b) q2

(c) q3 (d) q4

Figure 12. Long-time integration of the qi, i = 1, 2, ..., 4 showing the envelope of the oscillating response.

(a)

Figure 13. Average pressure field.

24 of 27

American Institute of Aeronautics and Astronautics

(a) Mode 1 (b) Mode 2

(c) Mode 3 (d) Mode 4

(e) Mode 5 (f) Mode 6

(g) Mode 7 (h) Mode 8

Figure 14. The pressure modes (Ψi, i = 1, 2, ..., 8) at ReD=100.

25 of 27

American Institute of Aeronautics and Astronautics

Time

a 1,2

360 370 380

-0.5

0

0.5

(a) a1 (solid) and a2 (dashed)

Timea 3,

4360 370 380

-0.1

0

0.1

(b) a3 (solid) and a4 (dashed)

Time

a 5,6

360 370 380

-0.02

0

0.02

(c) a5 (solid) and a6 (dashed)

Time

a 7,8

360 370 380

-0.01

0

0.01

(d) a7 (solid) and a8 (dashed)

Figure 15. The pressure coefficients ai = 1, 2, ..., 8 at ReD=100.

26 of 27

American Institute of Aeronautics and Astronautics

(a) (b)

Figure 16. Comparison of (a) the mean pressure and (b) the lift coefficients; POD (solid) and CFD (triangle).

27 of 27

American Institute of Aeronautics and Astronautics