American Institute of Aeronautics and Astronautics

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Simulation of a Feedback-Controlled Cylinder Wake Using Double Proper Orthogonal Decomposition

Stefan Siegel*, Kelly Cohen †, Jürgen Seidel‡, Selin Aradag§, and Thomas McLaughlin** Department of Aeronautics, United States Air Force Academy, CO, 80840

Feedback flow control on the wake of a circular cylinder at a Reynolds number of 100 is a challenging benchmark for controlling absolute instabilities associated with bluff body wakes. A two dimensional, high resolution simulation is used to develop low-dimensional models for estimator and controller design. Actuation is implemented as displacement of the cylinder normal to the flow. This control approach uses a low dimensional model based on a 15 mode Double Proper Orthogonal Decomposition (DPOD) applied to the velocity field. The control strategy involves the real-time estimation of the DPOD modes, obtained from a dynamic non-linear mapping algorithm using sensor readings, and a respective feedback command based on these estimates. Two control laws are developed. The first approach utilizes just a single mode feedback based on the fundamental von Kármán fluctuating DPOD mode. The second control law allows for dual inputs which include both the von Kármán fluctuating DPOD mode as well as its respective shift mode. The shift mode provides corrections for spatial variations in the POD Eigenfunctions which are caused by the feedback control-induced modification of the wake. The simulation results show that while both controller laws stabilize all 15 modes, the dual-input approach is more effective in reducing drag and fluctuating lift.

I. Nomenclature D Cylinder diameter Re=UD/ν Reynolds number St=fD/U Strouhal number f Frequency f0 Natural shedding frequency T Shedding Period = 1/f0 x Streamwise direction y Flow normal direction ν Kinematic viscosity ρ Density Δt Time step

U Mean flow velocity φ POD spatial mode a POD mode amplitude Kp Proportional controller gain Kd Differential controller gain ϕ Feedback phase ycyl Cylinder displacement normal to flow L0 Unforced lift force D0 Unforced drag force t Time

II. Introduction ne of the main purposes of flow control is the improvement of aerodynamic characteristics of air vehicles and munitions enabling augmented mission performance. An important area of flow control research involves the phenomenon of vortex shedding in the wake behind bluff bodies where the flow separates from the bluff

body’s surface. Shedding of counter-rotating vortices is observed in the wake of a two-dimensional cylinder above a

* Visiting Researcher, Department of Aeronautics, USAF Academy, and AIAA Senior Member. † Contracted Research Engineer, Department of Aeronautics, USAF Academy, and AIAA Associate Fellow. ‡ Visiting Researcher, Department of Aeronautics, USAF Academy, and AIAA Senior Member. § Visiting Researcher, Department of Aeronautics, USAF Academy, and AIAA Member. ** Director of Research, Department of Aeronautics, USAF Academy, and AIAA Associate Fellow.

O

37th AIAA Fluid Dynamics Conference and Exhibit25 - 28 June 2007, Miami, FL

AIAA 2007-4502

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

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critical Reynolds number of Rec~47, non-dimensionalized with respect to freestream velocity and cylinder diameter. This phenomenon is often referred to as the von Kármán vortex street, shown schematically in Figure 1. The vortex shedding leads to a sharp rise in drag, noise and fluid-induced vibration1-2. The ability to control the wake of a bluff body could be used to reduce drag, increase mixing and heat transfer, and enhance combustion3-4.

The Reynolds number regime studied in this effort, Re~100, corresponds to the range in which the wake is laminar and two-dimensional5. When active open-loop forcing of the wake is employed, the vortices in the wake can be "locked" to the forcing signal. This also strengthens the vortices and consequently increases the drag. As opposed to the open-loop approach, in this effort, the unsteady wake is controlled using a feedback controller. The feedback control law is designed using a reduced order model of the unsteady flow. A common method used to substantially reduce the order of the model is Proper Orthogonal Decomposition (POD). This method, as detailed in Holmes, Lumley, and Berkooz6, is an optimal approach in that it will capture the largest amount of the flow energy in the fewest modes of any decomposition of the flow. The two dimensional POD method was used to identify the characteristic features, or modes, of a cylinder wake as demonstrated by Noack, Tadmor, and Morzynski7. A common approach referred to as the method of “snapshots” introduced by Sirovich8 is employed to generate the basis functions of the POD modes from flow-field information obtained using either experiments or numerical simulations. Cohen, Siegel, McLaughlin and Gillies9 and Siegel, Cohen and McLaughlin10 have shown that using a low dimensional model, developed by Gillies1, the model of the cylinder wake flow can be successfully controlled using a relatively simple linear control approach based on the most dominant mode only. Recently, Siegel, Cohen, Seidel and McLaughlin11 developed an extension to the POD approach, referred to as ‘Double Proper Orthogonal Decomposition’ (DPOD), in which shift modes have been added to account for the changes in the flow due to transient forcing.

For low-dimensional control schemes to be implemented, a real-time estimation of the modes present in the wake is necessary, since it is not possible to measure them directly, especially in real-time. Velocity field data, provided from either simulation or experiment, is fed into the DPOD procedure. The time histories of the temporal coefficients of the DPOD model are determined by mapping the unforced flow onto the spatial Eigenfunctions using a least squares technique. Sensor measurements may take the form of wake velocity measurements, as in this effort, or, for an application, can be based on surface-mounted pressure measurements or shear stress sensors. Then, the estimation of the low-dimensional states is provided using a nonlinear system identification approach12 with Artificial Neural Networks (ANN) and ARX models13-14. The controller acts on the flow state estimates in order to determine the actuator displacement (Figure 1 shows the overall setup of the controlled system). The goal of this effort is to develop closed-loop control laws using a 15 mode low dimensional DPOD model obtained from the full Navier-Stokes simulation of the flow field. High resolution simulations are then utilized to examine the effectiveness of the control laws from a global perspective.

Figure 1. Schematic of Feedback Control System Setup

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The remainder of the paper is structured as follows: Section II describes the numerical simulation of the Navier Stokes equations, using the Cobalt15 CFD solver. This is followed by a short description of the DPOD procedure in section III. Section IV provides a brief description of the sensor based estimation. Section V describes the control law. Section VI presents the closed-loop high resolution truth simulation results. Section VII provides conclusions of the current research and finally Section VIII provides the outlook for future work.

III. Computational Methodology Numerical simulations were conducted with Cobalt Solutions’ COBALT15 solver for direct numerical solution

of the Navier-Stokes equations with second order accuracy in time and space. A structured two-dimensional grid with 63,700 nodes and 31,752 elements was used (Figure 2). The grid extended from x/d=–16.9 to x/d=21.1 in the x (streamwise) direction, and y/d=±19.4 cylinder diameters in the y (flow normal) direction. Additional simulation parameters are as follows (see the Cobalt User’s Manual for more details on the numerical parameters15):

• Cylinder diameter d = 1m • Mean flow U = 34 m/s • Pressure P = 4.337 Pascal • Density ρ = 5.25·10-5 kg/m3 • Reynolds Number Re = 100 • Time step, ∆t = 0.00147s. • Non-dimensional time step, ∆t*=∆t·U/D= 0.05 • 3 Newton sub-iterations • Damping Coefficients: Advection = 0.01; Diffusion = 0.00, 32 Iterations for matrix solution scheme • Laminar Navier-Stokes equations, ideal gas • Vortex shedding frequency f = 5.55Hz.

For validation of the computations of the unforced cylinder wake at Re=100, the resulting value of the mean drag coefficient, Cd, was compared to experimental and computational investigations reported in the literature. Experimental data reported by Oertel16 and Panton17 point to Cd values between 1.26 and 1.4. Furthermore, Min and Choi18 report on several numerical studies that obtained drag coefficients between 1.34 and 1.35. The current simulations yield Cd =1.35, which compares well with the reported literature. Another important benchmark parameter is the non-dimensional shedding frequency (Strouhal number, St = f*D/U) for the unforced cylinder wake. Experimental results presented by Williamson5 point to values of St=0.167-0.168. The computations used in this effort result in St=0.163, which also compares well with the reported literature.

0 1 2 3 4 5

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/D

y/D

Figure 2 Computational Grid (.) and Sensor placement (o) for mode amplitude estimation.

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The aim is to develop an effective estimator of the low-dimensional states based upon flow-field velocity readings when subject to various forcing inputs within the lock-in region, where the frequency of the vortex shedding is identical to the forcing frequency. The emphasis is on the robustness of the estimation for off-design cases as depicted in Figure 2. Nine different data sets for the open loop forced cases, as marked in Figure 3, were obtained using forcing amplitudes of 10, 15, 20, 25 and 30 percent cylinder displacement. Some of the cases use a 5-10% lower or higher frequency at 30% displacement, which is still within the lock-in region. Lock-in is defined as the flow develping a fixed phase releationship between the actuation and the vortex shedding. The 25 percent cylinder displacement sinusoidal forcing serves as design point for model development (Design Case #1). On the other hand, the off-design cases, #2-#9, are utilized for model validation. In Figure 3, “Koopman” refers to experimental data described in his paper2, whereas the CFD data was generated using simulations described earlier in this section

Figure 3. “Lock-In” Envelope with Design and Off-Design Cases

IV. Double Proper Orthogonal Decomposition (DPOD) Modeling

POD is an efficient means to reduce spatially highly complex flow fields by representing them by a small number of spatial modes and their temporal coefficients7.

1

( , , ) ( ) ( , )K

k kk

u x y t a t x yφ=

= ∑ (1)

Equation 1 shows this decomposition, where a flow quantity u is represented by the spatial modes φk(x,y) and temporal coefficients ak(t). While this decomposition is well suited to time periodic flow fields, it faces problems for transient flows (see Siegel et al.19). Different additions to the basic POD procedure have been proposed, most notably the addition of a shift mode as introduced independently by Noack, Afanasiev, Morzynski and Thiele20 as well as Siegel, Cohen and McLaughlin.10 This shift mode originally only addressed changes to the mean flow, but this concept has been extended recently by Siegel, Cohen, Seidel and McLaughlin11 to adjust the fluctuating modes of transient flows as well. This modified POD procedure, referred to as Double POD (DPOD) procedure, provides shift modes for all physical modes of a transient flow field. A pictorial representation of the DPOD procedure is given in figure 4. Starting in the top left corner, the data is split into K bins and each bin is used as an input data set for its individual POD procedure.

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The resulting SPOD (Short Time POD) modes are then collected across the bins and POD is applied again to obtain the shift modes. The procedure is expressed mathematically in Equation (2), where the index i refers to the main (SPOD) modes of the first POD procedure, while the index j identifies the shift mode order,

, ,1 1

( , , ) ( ) ( , ).I J

i j i ji j

u x y t t x yα= =

= Φ∑∑ (2)

The mean flow mode M1,1 contains most of the energy, followed by the unsteady von Kármán modes, M2,1 and M3,1,. This DPOD formulation extends the original concept of the “shift mode”: we can now develop a “shift mode”, even a series of higher order shift modes, for all main modes i. The resulting mode set can be truncated in both i and j, leading to a mode ensemble that is IM × JM in size. After orthonormalization, the decomposition is again optimal in the sense of POD. In the limit of J=1, the original POD decomposition is recovered. While the different modes distinguished by the index i remain the main modes described above, the index j identifies the transient changes of these main modes: For J >1, the energy optimality of the POD decomposition in that direction leads to modes that are the optimum decomposition of a given main mode as it evolves throughout a transient data set. If J = 2, then modes Φ1,1 and Φ1,2 are the mean flow and its “shift mode” or “mean flow mode” as described by Noack et al.7 and Siegel et al.10, respectively. Thus the modes with indices j>1 can be referred to as first, second and higher order “shift” modes that allow the POD mode ensemble to adjust for changes in the spatial modes. We will refer to all of these additional modes obtained by the DPOD decomposition as shift modes, since they modify a given main mode to match a new flow state due to either a recirculation zone length or formation length change.

I x J DPOD Modes (x,y)

Truncationto I

Main Modes

�

Orthonormalization

POD(Sirovich)

Truncationto J

Shift Modes

n snapshots

Kbin

s

u(x,y,t)

POD(Sirovich)

POD(Sirovich)

Truncationto I

Main Modes

K bins

I SPOD modes

POD(Sirovich)

Truncationto J

Shift Modes

�

Time

J Shif

t Mod

es

Figure 4. Flow chart of DPOD decomposition process

These changes may be due to effects of forcing, a different Reynolds number, feedback or open loop control or similar events. Thus, in the truncated DPOD mode ensemble for each main mode, one or more shift modes may be retained based on inspection of energy content or spatial structure of the mode. Table 1 shows the normalized kinetic energy of the DPOD modes used in this investigation.

Using this DPOD procedure, accurate transient spatial mode sets have been developed. The next step towards a dynamic model, which represents the temporal behaviour of the unsteady forced wake, is derived from the time coefficients obtained from the DPOD procedure. The low dimensional model that represents the time-dependent

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coefficients of the POD is then derived using a non-linear system identification approach based on artificial neural networks (ANN)25.

i

Energy(Mi,1)

[%]

Energy(Mi,2)

[%]

Energy(Mi,3)

[%]

1 94.89 0.83 0.20

2 1.22 1.06 0.28

3 0.33 0.27 0.08

4 0.09 0.04 0.02

5 0.04 0.02 0.01

Table 1. Normalized energy content of DPOD Modes

V. Sensor based estimation The amplitudes of the DPOD modes were used both for feedback, as described in the following section, as well

as for evaluation of the effectiveness of the controller. In order to be used for feedback control, the amplitudes of the modes that were used for feedback were estimated using a least square fit of the sensor data onto the spatial POD modes. The sensor locations, developed by Siegel et al10, were chosen to evenly cover the entire portion of the flow of interest for control, using 35 sensors as shown in Figure 2.

VI. Control Law Design The controller development is based on a Proportional and Differential (PD) controller. As mentioned earlier,

active forcing is introduced into the wake by displacement of the cylinder in the flow normal direction as shown schematically in Figure 1. The full state estimator provides estimates for all 15 DPOD amplitudes which may then be used as input to a full state controller. Since a neural network model of the flow has been developed, several indirect control designs can be employed13. The indirect design is very flexible and applicable in real-time for the problem at hand. Relevant concepts include approximate pole placement, minimum variance and predictive control. The approximation is based on instantaneous linearization, which is a popular method for control of ANN models13.

Following the effort by Siegel, Cohen and McLaughlin10, a similar PD feedback control strategy is employed for the single mode feedback control law:

2121 21 21cyl p d

dy K Kdtαα= ⋅ + ⋅ (3)

Instead of directly specifying the Kpand Kd gains, these can be expressed in terms of an overall gain K and a phase

advance ϕ for mode i:

cos( )

sin( )2

pi i i

i idi

K K

KKf

ϕ

ϕπ

= ⋅

⋅=

⋅

(4)

with f being the natural vortex shedding frequency. Equation (4) refers to the single-input closed-loop control based on feedback using an estimate of Mode (2,1). We will refer to this control law as a Single Input Single Output (SISO) controller in the following, since only one mode is used for determining the feedback response. The control law may be modified to enable dual-input from two modes, and a simple example is adding proportional control of the shift mode of the von Karman shedding mode, Mode (2,2). This control law which essentially constitutes a multi input single output (MISO) PD controller is:

21 2221 21 21 22 22 22cyl p d p d

d dy K K K Kdt dtα αα α= ⋅ + ⋅ + ⋅ + ⋅ (5)

where it is possible to substitute the proportional and differential gains in the same fashion as for the SISO controller, with an overall amplitude gain K and a phase shift ϕ for each of the two modes independently.

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VII. CFD feedback controlled simulation results This section presents the results both of single mode feedback, as well as multi mode feedback based on the

DPOD model. The effectiveness of the single mode feedback was evaluated by keeping the overall gain K constant at K=5e-4, while varying the phase ϕ of the feedback from 0 to 360 degrees. The controller amplitude was chosen such that the overall cylinder displacement would remain within the range of validity of model, i.e. limiting the maximum displacement y/D <0.3. The resulting closed loop lift and drag force are shown in Figure 5. It can be seen that for the range of feedback phases between about 30 and 250 degrees the drag is increased compared to the unforced flow field, while between 250 and 30 degrees a decrease in drag is observed. The normalized lift and drag forces for both ϕ=90º (drag increase) and ϕ=330º (drag decrease) is shown in Figure 6. In both cases the wake is stabilized at its new state, with about 10% drag decrease for the feedback phase of 330 degrees. The DPOD mode amplitudes for this case are shown in Figure 7. The amplitude of the mode used for feedback, M21, is greatly reduced by the effect of feedback control. However, the amplitude of its first shift mode, M22, is actually increased as a result of feedback. This observation led to the implementation of multi mode feedback, where both M21 and M22 are fed back with individual gains K and phases ϕ.

Figure 5 Non-dimensional RMS amplitudes of the stabilized lift and drag forces using single mode feedback (L0 and D0 are the RMS amplitudes of the unforced flow’s lift and drag force, respectively)

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Figure 6 Lift and Drag for feedback using Phase angles 90 degrees (left) and 330 degrees (right), SISO feedback

Figure 7 DPOD Mode Amplitudes for SISO feedback with phase 330 degrees (The controller is activated at t/T=0. The mode amplitudes are labeled according to Mij)

Figure 8 shows a parameter scan varying the feedback phase of M22, while keeping all other gains as well as the

phase of M21 constant. Again a range of detrimental phases exists, where the drag is compared to the new baseline of SISO feedback with 330 degrees phase. However, for a small range of feedback phases between ϕ=310º and

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ϕ=50 º, a further reduction in drag beyond SISO control can be observed. This demonstrates the benefit of using MISO control over SISO control. However, the detailed analysis of the MISO control run the led to the largest reduction in drag (Figure 9) shows that there is a lack of stabilization in this type of feedback control. While the drag initially is decreased by more than 6% compared to the SISO level, it is followed by an increase in drag to a stable level about 3% below the baseline. Analyzing the DPOD mode amplitudes for this case (Figure 10), one can see that control of M22 actually destabilizes M21 later in the simulation.

Figure 8 MISO feedback of Modes 21 and 22, with K=0.5e-3 for both modes and a fixed phase of 330 deg for M21. (The feedback phase of M22 is varied, and the resulting lift and drag forces are normalized by those of the SISO feedback of M21 with K=5e-4 and phase = 330.)

Figure 9 Lift and Drag for MISO feedback with Modes M21 and M22, K21= K22=5e-4, ϕ21=330º, ϕ22=20 º

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Figure 10 DPOD Mode Amplitudes for MISO feedback of modes M21 and M22 with gains K21= K22=5e-4 and phases ϕ21=330º, ϕ22=20 º. The modes are labeled as Mij

Figure 11 Lift and Drag for MISO feedback with Modes M21 and M22, K21= 0.5e-3, K22=0.375e-3, ϕ21=330º, ϕ22=20 º

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Figure 12 DPOD Mode Amplitudes for MISO feedback of modes M21 and M22 with gains K21= 0.5e-3, K22=0.375e-3, and phases ϕ21=330º, ϕ22=20 º. The modes are labeled as Mij

With some fine tuning of the gains applied to M21 and M22, the flow field can be stabilized using multi mode feedback. Figures 11 and 12 demonstrate this, where the flow is stabilized at a lift fluctuation level more than one order of magnitude smaller than the unforced flow, and at a reduction of drag of more than 10%. However, in this situation the fluctuating amplitude of the shift modes is increased as well – not just for the von Kármán modes, but also for the higher harmonic modes. From a flow physics perspective, this indicates a shift of the vortex formation further downstream as the controller becomes effective in stabilizing the near wake. At the same time, the actuation which remains at the cylinder location becomes less and less efficient in controlling this remaining vortex shedding far away from the cylinder.

VIII. Conclusion We demonstrate CFD results of feedback control of the two-dimensional circular cylinder at Re=100 based on one or two modes of a DPOD model developed using transient data truncated to 15 Modes. The objective of the control is stabilization of the von Kármán vortex street. Two different controllers are investigated, a Single Input Single Output (SISO) feeding back the von Karman mode amplitude, and a Multi Input Single Output controller (MISO), feeding back the von Kármán mode and its shift mode. The SISO control leads to a reduction of the overall drag force of about 10 % for the best combination of proportional and differential gains. It is worth mentioning that all

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investigated SISO feedback gains lead to a stabilized wake, which is not the case for simpler POD models that do not include transient data. However, while the amplitude of the von Kármán mode used for feedback is reduced, as a result of feedback, the amplitude of its shift mode is increased. This observation leads to the use of the MISO controller, where in addition to PD feedback with the best combination of gains found in SISO control the shift mode, M21, is used for feedback as well. The MIMO control is able to further reduce the drag compared to the best SISO case temporarily by more than 6 percent, in its stabilized state by more than 3 percent. The investigated MISO gains so far did lead to some partial destabilization of the flow. The overall conclusion of this investigation when compared to the results achieved by Siegel et al.10 is that a DPOD model is able to greatly increase the stability and performance of the feedback controller. This demonstrates the importance of obtaining a model that covers the entire parameter range that the feedback controller covers, rather than a point design type of a model that is only valid for the time periodic uncontrolled flow. While demonstrated for the benchmark case of a laminar circular cylinder wake, we expect similar trend for more complex flows which we will investigate in the future.

IX. Outlook In this publication, we developed simple PD controllers that perform either single or multi mode feedback. The

use of an accurate nonlinear dynamic model for flow state estimation enables mode based feedback. While the performance of these controllers is greatly improved over those using a less accurate POD or other models, the overall controller development method employed is empirical in nature. The process of tweaking controller gains is certainly less efficient than what a state based feedback controller can achieve. Thus, our future research direction is towards nonlinear multi input single / and or multi output controllers. Further work will also investigate the use of dynamic non-linear estimators using smaller number of sensors in closed-loop truth simulations.

X. Acknowledgments The authors thank Lt. Col. Scott Wells (AFOSR), and Dr. James Myatt (AFRL) for their support and assistance.

The authors acknowledge the assistance of Dr. Jim Forsythe of Cobalt Solutions, LLC.

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11S. Siegel., K. Cohen., J. Seidel and T. McLaughlin, “State estimation of transient flow fields using double proper orthogonal decomposition (DPOD)”, presented at the International Conference on Active Flow Control, Berlin, Germany, September 27-29, 2006.

12Nelles, O., Nonlinear System Identification, Springer-Verlag, Berlin, Germany, 2001, Chap. 11. 13Nørgaard, M., Ravn., O., Poulsen, N.K., and Hansen, L.K., Neural Networks for Modeling and Control of Dynamic

Systems, 3rd printing, Springer-Verlag, London, U.K., 2003, Chap. 2. 14Ljung, L., System Identification: Theory for the User, 2nd Edition, Prentice Hall, Upper Saddle, NJ, USA, 1999, Chap.4. 15Cobalt CFD, Cobalt Solutions, LLC, Available: http://www.cobaltcfd.com. 16H. Oertel, Jr.., “Wakes behind blunt bodies”, Annual Review of Fluid Mechanics, vol. 22, pp. 539-564, 1990. 17R. L. Panton, Incompressible Flow. 2nd Edition, New York: John Wiley & Sons, 1996.

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18C. Min and H. Choi, “Suboptimal feedback control of vortex shedding at low Reynolds numbers”, J Fluid Mechanics, vol. 401, pp.123-156, 1999.

19Siegel, S., Cohen, K., Seidel, J., McLaughlin, T., 'Short Time Proper Orthogonal Decomposition for State Estimation of Transient Flow Fields', 43rd AIAA Aerospace Sciences Meeting, Reno, AIAA2005-0296, 2005

20B. R. Noack, K. Afanasiev, M. Morzynski and F. Thiele “A hierarchy of low dimensional models for the transient and post-transient cylinder wake”, J. Fluid Mechanics, vol. 497, pp. 335–363, 2003.

21O. Stalnov, V. Palei, I. Fono, K. Cohen, and A. Seifert, "Experimental validation of sensor placement for control of a "D" shaped cylinder wake", presented at the 35th AIAA Fluid Dynamics Conference and Exhibit, Toronto, Canada, June 6-9, 2005, AIAA Paper 2005-5260.

22Cohen, K., Siegel, S., Seidel, J., and McLaughlin, T., “Low dimensional modeling, estimation and control of a cylinder wake”, Invited lecture at Invited Session organized by Dr. James Myatt of AFRL/VA called "Order Reduction and Control for Aerodynamics", 45th IEEE Conference on Decision and Control, San Diego, 13-15 December 2006.

23Y. Shin, K. Cohen, S. Siegel, J. Seidel, and T. McLaughlin, “Neural network estimator for closed-loop control of a cylinder wake”, Presented at the AIAA Guidance, Navigation, and Control Conference and Exhibit, Keystone, Colorado, 21 - 24 Aug 2006, AIAA Paper 2006-6428.

24K. Cohen, S. Siegel, J. Seidel, and T. McLaughlin, “ "Reduced order modeling for closed-loop control of three dimensional wakes", Presented as an Invited lecture at the 3rd AIAA Flow Control Conference, San Francisco, California, 5 - 8 June 2006, AIAA Paper 2006-3356.

25 Siegel, Stefan; Cohen, Kelly; Seidel, Jürgen; Luchtenburg, Mark; McLaughlin, Thomas, 'Low Dimensional Modeling of a Transient Cylinder Wake Using Double Proper Orthogonal Decomposition', under review for publication in J. Fluid Mechanics, June 2006

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