Simulations of three dimensional feedback control of a circular cylinder wake

American Institute of Aeronautics and Astronautics

1

Simulations of three dimensional feedback control of a circular cylinder wake

Jürgen Seidel*, Stefan Siegel†, Kelly Cohen‡, Volker Becker+,Thomas McLaughlin§ Department of Aeronautics, U.S. Air Force Academy, Colorado Springs, CO 80840, USA

Three-dimensional feedback controlled simulations of the wake behind a circular cylinder were performed. Feedback control was achieved by using a low dimensional model based on Proper Orthogonal Decomposition (POD) in conjunction with least square sensor based estimation to estimate the flow state. At a Reynolds numbers of 100, experimental and computational investigations have shown that although the vortex shedding is weakened by the introduction of closed-loop control, spanwise phase variations eventually lead to an instability which results in the loss of control. In the current simulations, a three-dimensional sensor array was placed in the wake to estimate the flow state based on two dimensional POD Modes, which were applied at multiple spanwise locations. A linear stochastic estimator was used to map sensor readings to the temporal coefficients of the POD modes. The simulations were aimed at investigating the efficacy of three-dimensional flow sensing to improve feedback control. Because the control input had only one degree of freedom (1 DOF), the mode amplitudes had to be combined into one actuator signal. The effect of selecting various spanwise sensing locations for feedback flow control was investigated.

Nomenclature ai = POD mode amplitude D = Cylinder diameter f = Frequency f0 = Natural vortex shedding frequency i = Index K = Feedback gain Kp = Proportional feedback gain Kd = Differential feedback gain Ka = Amplitude gain Kϕ = Phase gain M = Mach number Oy = Cylinder displacement Re = Reynolds number based on cylinder diameter t = Time U = Free stream velocity u = Mean velocity x,y,z = Streamwise, normal, and spanwise coordinate directions ∆t = Time step ϕ = Phase advance angle Φi = Spatial POD mode

* Visiting Researcher, Department of Aeronautics, Senior Member † Assistant Research Associate, Department of Aeronautics, Senior Member ‡ Contracted Research Engineer, Department of Aeronautics, Senior Member + Visiting Researcher, Department of Aeronautics § Director, Aeronautics Research Center, Department of Aeronautics, Associate Fellow

44th AIAA Aerospace Sciences Meeting and Exhibit9 - 12 January 2006, Reno, Nevada

AIAA 2006-1404

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

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 28

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

006-

1404


2

I. Introduction One of the main purposes of flow control is the improvement of aerodynamic characteristics of air

vehicles enabling augmented mission performance. An important area of flow control research involves the phenomenon of vortex shedding behind bluff bodies. These bodies often serve some vital operational function, however, their purpose is not to augment aerodynamic efficiency and often aerodynamic performance is sacrificed for functionality. Flow separates from large section of the bluff body’s surface and the resulting wake behind the bluff body, known as the von Karman vortex street, exhibits alternating direction vortices leading to a sharp rise in drag, noise and fluid-induced vibration.

Control strategies for two-dimensional bluff body wakes have been investigated for quite some time. In a two-dimensional cylinder wake, self-excited oscillations in the form of periodic shedding of vortices are observed above a critical Reynolds number of approximately 47. According to Ref. 1, the regime of laminar vortex shedding extends to a Reynolds number of approximately 180, above which three-dimensional instabilities lead to streamwise vortices in the wake. These streamwise vortices are distinctly different from the spanwise phase variations that occur in the vortex street at much lower Reynolds numbers.

Ongoing experimental and computational investigations have verified the spanwise phase variation of the natural Kármán vortex street at ReD=100 (Figure 1a).2,3 The non-linear oscillations of the vortex street lead to some undesirable effects associated with pressure fluctuations, which greatly increase the drag of the bluff body compared to the steady wake that can be observed at lower Reynolds numbers. Monkewitz (1996) showed that the von Kármán Vortex Street is the result of an absolute, global instability in the near wake of the cylinder.4 This absolute instability is causing the flow to behave as a self sustained oscillator, with internal positive feedback leading to temporal amplification of the oscillation by the recirculation region downstream of the cylinder. In the far wake the flow can be shown to be convectively unstable. Using open loop forcing, a distinct “lock-in” region of frequencies and amplitudes was demonstrated in Ref. 5. The lowest amplitudes, approximately Oy/D=0.05, are necessary when forcing is applied at the natural shedding frequency. This was verified recently in our experiments where “lock in” results in a uniform spanwise phase distribution (Figure 2).

At a Reynolds number of 120, an attempt to control the vortex shedding using closed loop feedback has been made experimentally.2 To estimate the instantaneous flow state, sensors were placed in one streamwise plane at the center of the model. Starting from the state after open loop forcing has been applied, i.e., with an essentially two dimensional flow, we were able to initially control the vortex shedding. However, the current closed loop control strategy, which is based on a model-independent two-dimensional high resolution truth simulation, is eventually unable to account for the three dimensional distortion of the flow (Figure 3).

POD analyses of the unforced three-dimensional CFD dataset show that, although a spanwise phase variation can be observed (see Figure 1b), the two-dimensional spatial modes look very similar at all spanwise locations (Figure 4). Experimental studies reported in Ref. 6 have shown that by using open loop forcing in the “lock-in” region, the phase variations can be minimized, rendering the wake nominally two dimensional. When this two dimensional wake is used as an initial condition for a feedback controlled flow simulation, it is conceivable that sensing the flow field using a three dimensional sensor array could result in an early detection of spanwise phase variations. The general hypothesis is that with this information and the development of an appropriate feedback control algorithm, a delay or suppression the onset of three dimensional wake distortions, which so far led to a loss of control, could be possible. In this paper, various combinations of the three-dimensional sensor information are analyzed for their ability to stabilize the wake behind a circular cylinder.

II. Computational approach

A. Flow Solver The computations are performed using Cobalt from Cobalt Solutions LLC. In the code, the full

compressible Navier-Stokes equations are solved based on the Finite Volume Formulation. The method is formally second order accurate in both space and time.

A structured, body fitted “O” grid with 326x198 points per spanwise plane is used for this investigation (Figure 5a). This grid is extruded to 31 planes along the cylinder axis for the three dimensional simulations

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 28

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

006-

1404


3

of a cylinder that is 96 diameters long (see Figure 5b). This resolution was shown to be sufficient to capture the large wave length of the spanwise phase variation.

Uniform flow boundary conditions at the far field are imposed using Riemann invariants. The cylinder is modeled using no-slip, no-penetration boundary conditions and the spanwise boundaries are modeled as periodic, which effectively simulates a cylinder of infinite aspect ratio. Grid point clustering is used in the cylinder wake region to accurately capture the Kármán vortex street. The wake is controlled by moving the cylinder normal to the free stream direction.

To ensure computational efficiency, the Mach number is set to M=0.1. The time step is ∆t=0.05D/U, where U is the free stream velocity and D the cylinder diameter. A previous study using this grid showed good comparison of the resulting Strouhal number with experimental results.7

B. Proper Orthogonal Decomposition Proper Orthogonal Decomposition is an optimal decomposition in the sense that it sorts the spatial

modes with respect to their energy content. The model is also referred to in the literature as the Karhunen-Loeve expansion. The method of snapshots introduced by Sirovich is implemented to reduce the order of the eigenvalue problem.8 For more detailed information about POD the reader is referred to the literature.9,10,11

In this study the u-velocity, which previously was found to be effective for feedback flow control of cylinder wakes, is analyzed by decomposition in its mean and fluctuating parts (Reynold's decomposition).6 The POD procedure is performed on the fluctuating part of the u-velocity:

∑=

Φ+=N

iii xtaxutxu

1)()()(),( , (1)

where u is the mean velocity, ai are the temporal coefficients, φi are the spatial Eigenmodes and N is the number of modes. To reduce the order of the control problem, the sum is typically truncated at a small number of modes which in turn relates to a specific threshold of the overall modal energy. E.g., N=4 contains more than 95% of the energy for the two-dimensional cylinder wake model.

C. Mode Estimation and Control Algorithm The basic control strategy, which is schematically illustrated in Figure 6, was applied in our water tunnel experiments and two-dimensional CFD simulations with reasonable success.6 For the closed-loop flow control simulations presented here, the approach was extended to allow for three dimensional sensor arrays. The basic elements of the feedback control system are the sensors for the streamwise velocity, the mode estimator based on the Linear Stochastic Estimation (LSE) technique, the closed-loop controller, and an optional low pass signal filter. The sensors are placed in an array of 7x5x31 downstream of the cylinder between 0.75<x<6.75, -1<y<1 at spanwise planes z/D=0,3.2,…,96 (see Figure 5b; a more detailed sensor placement investigation was reported in Ref. 12). An important aspect of the control strategy is the mode estimator that converts the u-velocity sensor signals at every spanwise plane, us(xs,ys,t;zs), to POD mode amplitudes ai(t) based on a set of spatial POD modes. This mode amplitude estimation is performed using a least-square algorithm that fits the sensor signals to the spatial POD modes obtained from a two-dimensional simulation (Figure 7). The main goal of the closed-loop flow control system is to suppress the vortex street. Based on previous studies reported in Ref. 6, the control law is based on the estimations of the temporal coefficients of POD modes 1 and 2. It was shown both computationally and experimentally for the two-dimensional wake that it is sufficient to feed back mode 1 to suppress the vortex street.6 To account for the spanwise phase variations, the instantaneous amplitude of POD mode 1 at every spanwise sensor plane is available, opening a large parameter space of possible data combinations and control strategies. The controller is a Proportional-Differential (PD) controller of the form

11 ,y p d

aO K a Kt

∂= +

∂ (2)

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 28

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

006-

1404


4

where Oy is the cylinder displacement and a1 is the instantaneous amplitude of POD mode 1. For the current case, where a three-dimensional sensor array is used, a1 is determined as an average of the a1 determined at all participating spanwise planes. Kp and Kd are the proportional and differential gain, respectively. A more convenient way of expressing the PD controller as a function of a gain K and a phase advance ϕ is

0

sin( )cos( ), ,2p d

KK K Kfϕϕ

π= = (3)

where f0 is the natural vortex shedding frequency. Because of the observed large changes in the mean flow when control is applied, a variable-gain, variable-phase controller can be employed that incorporates these changes into the control law as follows:

( ) ( )5 5 5 50 0, ,A t t

K K a a K a aϕϕ= =

= − = − (4)

where KA is the amplitude gain, Kϕ is the phase gain, and a5 is the instantaneous amplitude of POD mode 5, the ‘shift’ mode. As for the amplitude a1 above, a5 is determined as an average the participating z planes. For the current simulations, the gains were set to KA=5·10-6 and Kϕ=0.75.2

III. Results In this section, a detailed comparison of representative closed loop simulations is shown. Starting from an unforced simulation in which the structures in the wake are highly three dimensional (Figure 8), open loop simulations were performed to establish a two-dimensional initial condition for the feedback controlled simulations. Compared to previously reported results, the spanwise coherence was drastically improved by using periodic boundary conditions in the spanwise direction. The parameters for the open loop forcing are f=f0=5.5Hz, A/D=0.15 where D is the cylinder diameter (Figure 5b). The comparatively large amplitude was chosen so that the transient time from the unforced to the open loop forced case is reduced and full spanwise coherence is achieved more quickly. Although a perfectly two-dimensional wake flow is not achievable in experiments, this flow field serves as an ideal starting point to explore the three dimensional development of the wake and the influence of feedback flow control.6 In our previous experimental and numerical investigations, it was shown that sensing the flow state in a single x,y-plane in the middle of the cylinder initially results in successful wake control in this sensor plane, but control is lost early at all other spanwise locations and eventually even in the sensor plane.3 A logical next step toward controlling a fully three dimensional cylinder wake is to place flow sensors not only in one spanwise plane, but distribute them in multiple parallel planes in the near wake, i.e. establish a three dimensional sensor array. This will allow for the observation of the development of the spanwise phase variation once the controller becomes effective in reducing the strength of the vortices and concomitantly their spanwise coherence. With our POD approach, the sensor signals of individual spanwise planes are used to estimate POD mode amplitudes that will be analyzed and combined to yield the input to the controller. For this paper, the investigation was limited to the effect of using averages of the POD time coefficients from various sensor planes. The sensor planes used for the computational cases reported are listed in Table 1.

Case # 1 2 3 Sensor locations z 48 All 0, 3.2, 48, 92.8, 96

Table 1: Summary of sensor planes for computational cases.

The first case duplicates the computations presented previously, but with the improved spanwise boundary condition.3 A comparison with the earlier data clearly shows the influence of this boundary condition on the spanwise coherence of the initial condition. Figure 10 shows the streamwise velocity component at (x,y)=(2.5,-0.5) as a function of time and spanwise location. In addition, the cylinder displacement Oy/D (the dashed lines mark ±5% forcing, the threshold for controller effectiveness found in two-dimensional simulations) and the resulting drag coefficient (the dashed line represents the value for the unforced wake13,14,15) are plotted. When started from the open loop controlled case, the controller quickly reached

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 28

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

006-

1404


5

large amplitudes and achieved control for all spanwise locations. Around t=1.8s after controller activation (approximately ten shedding cycles), a drag minimum is reached when spanwise coherence is still fairly well maintained. This minimum coincides with a phase change in the Oy/D signal. At later times, control is compromised and the wake returns to a state of low spanwise coherence. This loss of control could in part be due to the fact that the forcing amplitude drops below the threshold found by Blevins, Oy/D<5÷10%.5 However, it should be noted that as in previous computations, control is maintained in the sensor plane z=48 until t=3.8s, when the amplitude of the fluctuations starts to increase again and control is ultimately lost due to three-dimensional disturbances. Scrutinizing the phase information of the sensor signal relative to the forcing (Figure 11), it becomes obvious that the actuation phase changes rapidly at t=1.8s from -90o to +90o in response to the changes in the flow field. At t=4.2s, the inverse phase change takes place. In the second case investigated in this study, the forcing input is computed as the average of the POD mode amplitudes from all sensor planes. In this case (Figure 12), the forcing amplitude decreases more slowly than in Case 2. This results in a longer time interval in which the drag is lower than the unforced baseline value, but eventually the control breaks down because the forcing amplitude is reduced below the threshold value. Figure 13 shows a zoom-in of the POD modes at all sensor planes and the resulting average signal that was fed to the controller. The figure illustrates how the average amplitude drops mainly due to phase shifts in the mode amplitudes resulting in partial signal cancellation. Note also that control is lost within three shedding cycles. Even though the forcing amplitude increases again at later times, control was not regained because of the low forcing amplitudes. During the successful control, a slow phase change that starts at t=2s and spans -360o can be seen for almost all spanwise locations (Figure 14). This is indicative of a delay of the vortex formation. The final case presented in this investigation uses sensor information from the spanwise edges of the domain in addition to the sensors located in the center plane. This configuration was studied because it was observed that often the three dimensional disturbances originate at the spanwise edge of the integration domain. Figure 15 shows u(t,z), Oy/D(t) and cd(t) (note the shorter time span of the simulation). Initially, the controller increases the forcing amplitude and the velocity fluctuations are successfully reduced. In contrast to the cases presented above, however, this does not lead to a significant decrease in amplitude of mode 1 because the flow at the sensor planes stays in phase (Figure 16). Consequently, the forcing amplitude stays above 5% and the controller is effective for a longer period of time. Note also that the loss of control occurs more slowly than in the other cases. Interestingly, there is no phase change in the forcing signal throughout the duration of the simulation, possibly also due to the larger amplitude levels.

IV. Conclusion The effect of averaging various combinations of two-dimensional spanwise mode amplitudes for feedback control of a cylinder wake has been investigated. Other strategies, e.g. control based on the signal furthest from the instantaneous mean, are possible put outside the scope of the current investigation. Starting from an idealized, highly two-dimensional open loop case, the three-dimensional feedback controlled simulations show that, independent of the number and location of the sensor planes, control is initially successful for the whole spanwise extent. For approximately two seconds or ten vortex shedding cycles, the controller is able to significantly reduce the vortex shedding, resulting in a reduction of the drag coefficient of more than ten percent. For later times, once the three-dimensional phase variations develop in the wake, controller effectiveness is reduced and the drag coefficient returns to a value close to the one for the unforced wake. Part of the explanation for this loss of control lies in the fact that once the controller becomes effective, the feedback amplitude drops below a certain threshold level necessary to maintain controllability. This threshold was inferred from two-dimensional data to be approximately 5-10 percent, which seems to also hold in three-dimensional wakes based on the observations from the simulations presented here. A controller that utilizes data from the spanwise edges of the computational domain as well as the center plane was shown to be most effective, in a large part due to the fact that POD mode amplitudes stay in phase and therefore the amplitude supplied to the controller is not reduced below the threshold value given above. However, even for this controller control is eventually compromised. Within the framework of feedback flow control, this study highlights the necessity for a better approach to controlling three-dimensional flow fields. First, it seems that the three-dimensional dynamics of the cylinder wake are not captured by the two-dimensional POD model used for feedback flow control in this effort. Toward this end, it will be necessary to develop a truly three-dimensional POD model of the

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 28

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

006-

1404


6

cylinder wake so that the development of the three-dimensional flow structures can be accurately captured. Based on this model and the ability to place a three-dimensional sensor array in the wake, a new controller will have to be developed. If it is possible to establish observability with this model/sensor approach, further investigations are necessary to determine if a single actuator, as used in this investigation, is sufficient to achieve controllability.

Acknowledgments The authors would like to acknowledge funding by the Air Force Office of Scientific Research, Dr. Sharon Heise. We would also like to thank Dr. Jim Forsythe of Cobalt Solutions, LLC for their support.

References 1. Williamson, C.H.K., “Vortex Dynamics in the Cylinder Wake”, Ann. Rev. Fluid Mech., 1996, 28:477-539 2. Siegel, S., Cohen, K. & McLaughlin, T., 2004, Experimental Variable Gain Feedback Control of a Circular

Cylinder Wake, AIAA 2004-2611. 3. Seidel, J., Siegel, S., Cohen, K. & McLaughlin T., 2005, Three dimensional simulations of a feedback controlled

circular cylinder wake, AIAA 2005-0295. 4. Monkewitz, P. A., “Modeling of self-excited wake oscillations by amplitude equations”, Experimental Thermal

and Fluid Science, Vol. 12, 1996, pp. 175-183. 5. Koopmann, G., “The Vortex Wakes of Vibrating Cylinders at Low Reynolds Numbers”, Journal of Fluid

Mechanics, Vol. 28 Part 3, 1967, pp. 501-512. 6. Luchtenburg, M., Siegel, S., Cohen, K. & McLaughlin, T., 2004, Comparison of 2D POD modes extracted from

two- and three-dimensional simulations of a cylinder wake, AIAA 2004-1269. 7. Siegel, S., Cohen, K. & McLaughlin T., 2003, Feed-back control of a circular cylinder wake in experiment and

simulation (invited), AIAA 2003-3569. 8. Sirovich, L., 1987, Turbulence and the Dynamics of Coherent Structures Part I: Coherent Structures,Quarterly

of Applied Mathematics 45, No. 3, 561-571. 9. Holmes P., Lumley, J. L. & Berkooz, G., 1996, Turbulence, Coherent Structures, Dynamical Systems and

Symmetry, Cambridge University Press. J. Fluid Mech. 322, 215-241. 10. Newman, A. J., 1996, Model Reduction via the Karhunen-Loeve Expansion Part I: An Exposition, Technical

Research Report 96-32 ISR. 11. Newman, A. J., 1996, Model Reduction via the Karhunen-Loeve Expansion Part II: Some elementary examples,

Technical Research Report 96-32 ISR. 12. Cohen, K., Siegel,S., McLaughlin, T., 'Sensor Placement Based on Proper Orthogonal Decomposition

Modeling of a Cylinder Wake', 33rd AIAA Fluid Dynamics Conference, Orlando, AIAA 2003-4259, 2003 13. Oertel, H. Jr., “Wakes Behind Blunt Bodies”, Annual Review of Fluid Mechanics, Vol. 22, 1990, pp. 539-564. 14. Panton, R.L., “Incompressible Flow”, 2nd Edition, John Wiley&Sons, New York, 1996, pp. 384-400. 15. Min, C., Choi, H., “Suboptimal Feedback Control of Vortex Shedding at low Reynolds Numbers”, Journal of

Fluid Mechanics, Vol. 401, 1999, pp. 123-156.

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 28

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

006-

1404


7

Figure 1: Re=100, unforced. a) Water tunnel flow visualization, b) CFD simulation

Figure 2: Successful lock-in of Kármán vortex street, Re=100, phase lag 0o, t=47s. Note the flow distortion near the end of the cylinder.

a) b)

c) d)

Figure 3 Flow visualization for feedback with 115o phase lag. a) t=10s, b) t=20s, c) t=30s, d) t=33.3s after controller activation. Re=100.8

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 28

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

006-

1404


8

a) b)

c) d)

Figure 4 POD modes at spanwise location a) z=16d, b) z=32d, c) z=48d, d) z=64d.

a)

b)

Figure 5: a) 2D slice through computational domain, b) Spanwise grid planes (u-velocity component shown in color)

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 28

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

006-

1404


9

Figure 6: Control strategy and controller components.

Figure 7: Spatial POD modes obtained from 2D simulation. Mode 5 is the “shift mode”.

Figure 8: Unforced simulation. ωz=5 isosurface and pressure at the center and the spanwise boundary (x,y) planes.

Figure 9 Open loop simulation. ωz=5 isosurface and pressure at the center and the spanwise boundary (x,y) planes.

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 28

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

006-

1404


10

Figure 10: Case 2 u(t,z), forcing Oy/D(t) and Cd(t).

Figure 11: Case 2 Phase of sensor signal relative to forcing.

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 28

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

006-

1404


11


Figure 13: Zoom-in of POD mode amplitudes for Mode 1 at participating sensor planes and their mean (red).

Figure 14: Case 3 Phase of sensor signal relative to forcing.

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 28

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

006-

1404


12


Figure 16: POD mode amplitudes for Mode 1 at participating sensor planes and their mean (red).

Figure 17: Phase of sensor signal relative to forcing.

Dow

nloa

ded

by U

NIV

ER

SIT

Y O

F C

INC

INN

AT

I on

Nov

embe

r 28

, 201

4 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

006-

1404

Simulations of three dimensional feedback control of a circular cylinder wake

Documents

Transcript of Simulations of three dimensional feedback control of a circular cylinder wake