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Sensor Placement for Closed-Loop Flow Control of a "D" Shaped Cylinder Wake

Kelly Cohen 1 Department of Fluid Mechanics and Heat Transfer, Faculty of Engineering, Tel-Aviv University

Tel Aviv 69978, Israel

Stefan Siegel2 Department of Aeronautics, U.S. Air Force Academy, Colorado Springs, CO 80840

Mark Luchtenburg 3

Department of Aeronautics, TU Delft, 2628 BL Delft, Netherlands

Thomas McLaughlin 4 Department of Aeronautics, U.S. Air Force Academy, Colorado Springs, CO 80840

and

Avi Seifert 5 Department of Fluid Mechanics and Heat Transfer, Faculty of Engineering, Tel-Aviv University

Tel Aviv 69978, Israel

The effectiveness of a sensor configuration, based solely on body surface pressure readings, for feedback flow control of the wake of a "D" shaped cylinder is investigated by DNS. The research is aimed at suppressing unsteady loads resulting from the von Kármán vortex shedding in the wake of bluff bodies at a Reynolds number range of 100-1000. The design of sensor number and placement was based on data from a laminar direct numerical simulation of the Navier Stokes equations for the baseline condition. A low-dimensional Proper Orthogonal Decomposition (POD) procedure was applied to the pressure and stream-wise velocity of the flow field. The sensor placement was based on the intensity of the spatial Eigen-functions obtained by applying POD to 286 surface pressures. The numerically generated data was comprised of 100 snapshots taken from the flow regime that corresponds to steady state vortex shedding. A Linear Stochastic Estimator (LSE) was employed to map the pressure signals from the body mounted sensors to the temporal coefficients of the reduced order model of the wake flow field in order to provide accurate yet compact estimates of the low-dimensional states. For a ten sensor configuration, results show that the root mean square estimation error of the estimates of the first two modes is within 1-3% of the desired values and for the third mode it is 12-20% accurate. This level of error is acceptable for a moderately robust controller required to close the loop, based on previous investigation.

I. Introduction NE 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. 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. The resulting wake behind the bluff body, known as the von-Karman vortex street2, exhibits alternating direction vortices leading to a sharp rise in drag, noise and fluid-induced vibration1. 1 Adjunct Researcher, Department of Fluid Mechanics and Heat Transfer, Faculty of Engineering, Tel-Aviv University, Senior Member AIAA. 2 Assistant Research Associate, Department of Aeronautics, USAF Academy, CO 80840, Senior Member AIAA. 3 Visiting Researcher, Department of Aeronautics / Graduate Student, TU Delft, Netherlands, Student Member AIAA. 4 Director of Research, Department of Aeronautics, USAF Academy, CO 80840, Assoc. Fellow AIAA. 5 Senior Lecturer, Department of Fluid Mechanics and Heat Transfer, Faculty of Engineering, Tel-Aviv University, Assoc. Fellow AIAA.

O

2nd AIAA Flow Control Conference28 June - 1 July 2004, Portland, Oregon

AIAA 2004-2523

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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The ability to control the wake of a bluff body could be used to reduce drag, increase mixing and heat transfer, and enhance combustion efficiency2-4. Shedding of counter-rotating vortices is observed in the wake of a two-dimensional cylinder above a critical Reynolds number (Re ~ 47, non-dimensionalized with respect to free stream speed and cylinder diameter). In Fig. 1, a picture taken at the center line of an unforced "D" shaped cylinder wake in a water tunnel, shows the von Kármán vortex street at Re = 150. The flow is from left to right. In this effort, the "D" shaped cylinder was selected so that the separation points will be fixed and to assure study of direct wake control and not separation control.

Drag, noise and vibration reduction are possible by controlling the wake of a bluff body. The flow may be influenced using several different forcing techniques and the wake response is similar for different types of forcing1. The following forcing methods have been employed: external acoustic excitation of the wake, longitudinal, lateral or rotational vibration of the cylinder, and alternate blowing and suction at separation points1.

The Reynolds number regime studied in the current effort corresponds to the low Reynolds number, laminar, two-dimensional cylinder. Beyond this range the nature of the vortex shedding transitions to several three dimensional regimes as the Reynolds number increases5. When active open loop forcing of the wake is employed, the vortices in the wake can be "locked" to the forcing frequency. This also strengthens the magnitude of the vortices and consequently increases the drag. The cylinder wake may be controlled by forcing the flow. Open loop forcing has been successfully employed to delay boundary layer transition1. He at al6 applied this open-loop forcing approach to the boundary layer by rotation of the flow around a cylinder and show up to 60% drag reduction for Reynolds numbers of 200-1000.

As opposed to the open-loop approach, in this effort, the unsteady wake is controlled using a feedback controller. Active closed-loop flow control has been found to be an effective means of suppression of self-excited flow oscillations without geometry modification1.

Three of the popular approaches for feedback optimal control and control, based on low-dimensiintroduction of sensors in the wake and uses a simforces the flow. Experimental studies show that afeedback is able to delay the onset of the wake ins20% higher than the unforced case (Re = 47). Abomode but destabilizes other modes4. Li and Aubry7 using transverse displacements with the aim of maiof the flow show that this control algorithm is capviscous flow and, therefore, controlling vortex shed

The optimal control approach is more structustrategies, such as optimal control theory, to flowfunction that satisfies the Navier Stokes equatiodeveloped conditions for optimality which minimizdisadvantage of the optimal control approach is tcumbersome unsteady Navier Stokes-equations. Tproblem that has N time steps, the dimension ofdrawback adversely effects real time implementatseveral sub-optimal approaches surveyed by Li et a

Figure 1. Water Tunnel Flow Visualization Picture of the “D”-shaped cylinder wake, Re = 150

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control of a two-dimensional wake are: direct feedback control, onal modeling of the flow. Direct feedback control involves the ple control law, which produces a command to the actuator that linear proportional feedback control based on a single sensor tability, rendering the wake stable at a Reynolds number about ve Re = 60, a single-sensor feedback may suppress the original introduce a feedback controller capable of manipulating the flow

ntaining zero lift at all times. Direct numerical simulation studies able of keeping the lift close to zero in the impulsively started

ding7. red as it applies conventional and proven model-based control control problems. The central idea here is to minimize a cost ns that govern the flow. For example, Abergel and Temam8 es a cost function that represents the drag on a body. The main

he computational cost associated with the nonlinearities of the o illustrate this issue, Li et al9 point out that for a steady-state the unsteady problem is N times larger in comparison. This ion of optimal flow control and has led to the development of l9. Homescu et al.10 solve the open-loop optimal control problem

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that consists of finding the optimal angular velocity of the cylinder such that the Kármán vortex shedding into the wake of the cylinder is suppressed10.

Low-dimensional modeling is a vital building block when it comes to realizing a structured model-based closed-loop strategy for flow control. For control purposes, a practical procedure is needed to break down the velocity field, governed by the Navier Stokes partial differential equations, by separating space and time. A common method used to substantially reduce the order of the model is proper orthogonal decomposition (POD). This method is an optimal approach in that it will capture a larger 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 Gillies1.

The major building blocks of this structured approach are comprised of a reduced-order POD model, a state estimator and a controller. The desired POD model contains an adequate number of modes to enable reasonable modeling of the temporal and spatial characteristics of the large scale coherent structures inherent in the flow through which it may faithfully reproduce the flow. A POD procedure may be used to derive a set of reduced order ordinary differential equations by projecting the Navier-Stokes equations onto the modes11.Further details of the POD method may be found in the book by Holmes, Lumley, and Berkooz11. A common approach referred to as the method of “snapshots” introduced by Sirovich12 is employed to generate the basis functions of the POD spatial modes from flow-field information obtained using either experiments or numerical simulations. This approach to controlling the global wake behavior behind a circular cylinder was effectively employed by Gillies1 and is also the approach followed in the current research effort.

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 realistic to millustration of the various blocks within the Velocity, pressure or vorticity field data, providprocedure. The POD temporal coefficients determined by applying the least squares techniqthe estimation of the low-dimensional states ismeasurements may take the form of wake measensors. This process leads to the state and meaFor practical applications it is desirable to reduce

The requirement for the estimation schememodes. The main aim of this approach is to theredescribed by Balas13. Spillover has been the cafiltering was found to be an effective remedy14. by a certain configuration of sensors placed in tthe first two modes. The estimation scheme, bAdrian15, predicts the temporal amplitudes of thfrom either computational or experimental data. for control of the Ginzburg-Landau wake model.and determining locations that best enable the d

Figure 2. Low-Dimensional Modeling Strategy

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easure the modes directly and be able to close the loop. An low-dimensional modeling approach is presented in Fig. 2. ed from either simulation or experiment, is fed into the POD

time histories of the Pressure/Velocity/vorticity fields are ue to the spatial Eigen-functions and the unforced flow. Then, provided using a linear stochastic estimator (LSE). Sensor surements, or as in this effort, from body mounted pressure surement equations, required for design of the control system. the information required for estimation to the minimum. then is to behave as a modal filter that “combs out” the higher by circumvent the destabilizing effects of observation “spillover” as use for instability in the control of flexible structures and modal The intention of the proposed strategy is that the signals, provided he wake, are processed by the estimator to provide the estimates of ased on the linear stochastic estimation procedure introduced by e first two POD modes from a finite set of measurements obtained The LSE of POD modes was successfully applied by Cohen et al16 A major challenge lies in finding an appropriate number of sensors esired modal filtering. The need for modal filtering and the search

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for suitable sensor configuration is a problem common to the control of flexible structures, where a considerable research has been done17-18 and can be leveraged. Meirovitch14 provides a survey of some effective strategies.

The remainder of the paper is structured as follows: Section 2 describes the research objective and the uniqueness of the developed approach. Direct numerical simulation of the Navier Stokes equations, using a computational fluid dynamics (CFD) solver is described and presented in section 3. This is followed in section 4 by a description of the developed low-dimensional POD model based on the Pressure and Velocity (U component) in the flow-field. Then, in section 5, based on the POD model, a sensor configuration is determined to estimate the first four low-dimensional Pressure and Velocity modes. The sensor configuration design will be based on body mounted pressure signals obtained form the CFD model. Finally, section 6 provides conclusions of the current research and some recommendations for future work.

II. Motivation and Research Objective

Recent research on closed-loop control of the von Kármán wake instabilities19-20 have addressed the issue of sensor placement and number based on non-intrusive sensors in the wake. This approach may not always be implemented and it is important to develop an effective method for sensor placement and number based on body mounted sensors. These sensors may measure skin friction or surface pressures, as done in this effort. The advantages of surface mounted sensors are:

• Simple, relatively inexpensive and reliable • Essential for real-life, closed-loop flow control applications where the direct measurement of the wake

flow field is cumbersome (if not impossible) • Enable “nearly collocated” sensors and actuators, which eliminates substantial phase changes (affects

controller design) The main objective of this research effort is to demonstrate and validate a systematic approach, based on the

spatial Eigen-functions of the POD model, for determining sensor number and location for estimation of the truncated POD states of a cylinder wake.

III. Computational Model

A simple model of the flow-field was sought to design sensor configuration for feedback control algorithms. Numerical simulations were conducted on COBALT21 solver V.2.02. In this effort, the solver was used for direct numerical solution of the Navier Stokes equations with second order accuracy in time and space. An unstructured two-dimensional grid with 120,000 nodes and 115,000 elements was used. The cylinder geometry comprises of a semi ellipse with base height of 7mm and length of 61.25mm. The CFD grid developed for the "D" shaped cylinder is shown in Fig. 3. Additional simulation parameters are as follows: • Initial perturbation: AOA=1◦ (to kick off the vortex shedding) • Damping Coefficients:

o Advection = 0.01 o Diffusion= 0.00

• Reynolds Number: Re = 300 • Free stream velocity: Uinf = 34 [m/s] • Shedding Frequency 8.02 Hz => St = 0.165 • Time-step = 0.5 ms => 250 time steps per shedding cycle

r

Figure 3. CFD Grid for "D" shaped cylinde

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The simulation was triggered by skewing the incoming mean flow by α = 1 degrees to introduce an initial perturbation. The validity of the CFD model, used in this effort was established by comparing the Strouhal number from the simulation to a water tunnel experiment. The water tunnel experiment yielded St = 0.163, therefore there is good agreement since the difference between simulation and experiment is far less than the experimental uncertainty of 0.01.

IV. POD Modeling

Feasible real time estimation and control of the cylinder wake may be effectively realized by reducing the model complexity of the cylinder wake as described by the Navier-Stokes equations, using POD techniques. POD, a non-linear model reduction approach, is referred to in the literature as the Karhunen-Loeve expansion11. The ideal POD model will contain an adequate number of modes to enable modeling of the temporal and spatial characteristics of the large-scale coherent structures inherent in the flow, but no more modes than necessary.

In this effort, the flow field in the wake, represented by the Pressure and stream-wise Velocity variables are obtained from the numerical solution of the Navier-Stokes equations obtained using the CFD model. All the 100 snap-shots (~ 15 shedding cycles) were equally spaced in time. The snap-shots were taken after ensuring that the cylinder wake reached steady state. For control design purposes, the POD method enables the Navier-Stokes equations to be modeled as a set of ordinary differential equations (O.D.E.).11 At first, the flow field data is loaded and arranged from the CFD solver of the "D" shaped cylinder wake at Re = 300. The decomposition of this component of the velocity field (a similar representation may be made for the Pressure field) is as follows: ),,(),(),,(~ tyxuyxUtyxu += (1) where U denotes the mean flow and u is the fluctuating component that may be expanded as:

∑=

=n

k

kik yxtatyxu

1

)( ),()(),,( φ (2)

where ak(t) denotes the time-dependent coefficients and фi(x,y) represents the non-dimensional spatial Eigen-functions of the Velocity (see Fig. 4) and Pressure (see Fig. 5) determined from the POD procedure. From an ensemble of snapshots, the 'mean snapshot' is computed and then this mean is subtracted from each member of the ensemble. This is done primarily for reasons of scale; i.e. the deviations from the mean contain information of interest but may be small compared with the original signal.

Next, the empirical correlation matrix is computed using the inner product. Solving the Eigen-value problem, the Eigen-values and the orthogonal spatial Eigen-functions, фi(x,y), are obtained. Since the Eigen-values measure the relative energy of the system dynamics contained in that particular mode, they may be normalized to correspond to a percentage of the system energy. For the current working point (Re=300), the Eigen-values for the Pressure and U-Velocity are presented in Table 1. Note that the great majority of the energy, associated with the POD procedure, is located in the first two modes.

Finally, the time histories of the temporal coefficients of the POD model, ak(t), are determined using the extracted spatial modes and the snapshots of the unforced flow. However, it is not possible to obtain a direct measurement of ak, which is why it needs to be estimated from direct measurements such as body mounted sensors. An important aspect of reduced order modeling concerns truncation. How many modes are important and what are the criteria for effective truncation?

Mode

Pressure POD

Eigen-Values [%]

Velocity (U) POD

Eigen-Values [%]1 2 3 4

48.21 41.30 4.73 4.68

49.31 43.14 2.92 2.85

Total 98.92 98.22

Table 1: Eigen-Values of Wake Flow-Field

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0 5 10

−2

0

2

Mode1

0 5 10

−2

0

2

Mode2

0 5 10

−2

0

2

Mode3

0 5 10

−2

0

2

Mode4

0 5 10

−2

0

2

Mode1

0 5 10

−2

0

2

Mode2

0 5 10

−2

0

2

Mode3

0 5 10

−2

0

2

Mode4

Figure 4. Eigen-Functions of stream-wise Velocity of the “D”-shaped cylinder Wake Flow Field Solid lines are positive, dashed lines are negative isocontours

Figure 5. Eigen-Functions of Pressure of wake Flow Field Solid lines are positive, dashed lines are negative isocontours

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The answers to the above questions have been addressed by Cohen et al.22 That effort demonstrated that control of the POD model of the von Kármán vortex street in the wake of a circular cylinder at low Reynolds numbers (Re~100) is enabled using just the first mode. Furthermore, feedback based on the first mode alone significantly attenuated all the other modes in the four-mode POD model.

In view of the above result, in this effort, truncation of the POD model takes place after the first three modes, which contain 95-97% of the energy, as seen in Table 1. At this point, it is imperative to note the difference between the number of modes required to reconstruct the flow and the number of modes required for effective low-dimensional modeling for control design purpose. In this effort, we are interested in estimating only those modes required for closed-loop flow control. On the other hand, a more accurate reconstruction of the velocity and pressure field, based on a low-dimensional model, may be obtained using between 4-8 modes.

The quintessential question is whether an effective estimate of the states, of the 3 mode low-dimensional model coefficients, ak, can be estimated based on body mounted pressure sensors. The answer is positive and the details that provides the estimate of the first three modes, a1-a3, are presented in the next section.

V. Estimation, Sensor Configuration and Results

The time histories of the temporal coefficients of the flow field Velocity and Pressure variables are determined by introducing the POD spatial Eigen-functions into the flow field data using the least squares technique. The intent of the proposed strategy is that the pressure measurements provided by the body mounted pressure sensors are processed by the estimator to provide the estimates of the first three temporal modes of the flow field variables stream-wise Velocity and Pressure. The estimation scheme, based on the linear stochastic estimation (LSE) procedure introduced by Adrian15, predicts the temporal amplitudes of the first three POD modes from a finite set of pressure measurements obtained from the CFD solution of the uncontrolled cylinder wake. All the measurements were taken after ensuring that the cylinder wake flow regime converges to steady state vortex shedding. The mode amplitudes, a1-a3, will be mapped onto the extracted sensor signals from the pressure sensors, us, as follows:

∑=

=m

ss

nsn tuCta

1)()( (3)

where m is the number of sensors and Cns represents the coefficients of the linear mapping. The effectiveness of a

linear mapping between for velocity measurements and POD states has been experimentally validated by Cameron et al24. The coefficients Cn

s (n =1-3; s = 1, m) in Equation (3) are obtained via the linear stochastic estimation method from the set of discrete sensor signals and temporal mode coefficients, a1 – a3.

The issue of sensor placement and number has been dealt with in an ad-hoc manner in published studies concerning closed-loop flow control. For effective closed-loop control system, the following questions need to be answered:

• How many sensors are required? • Where should the sensors be placed? • What are the criteria for judging an effective sensor configuration? • What are the robustness characteristics of a given sensor configuration?

In this effort, an attempt will be made to emulate some of the proven successes from the field of structural control. Heuristically speaking, when some very fine dust particles are placed on a flexible plate, excited at one of its natural frequencies, after a short while the particles arrange themselves in a certain pattern typical of those frequencies. The particles will be concentrated in the areas that do not experience any motion (the nodes). On the other hand, at the areas where the motion is large (the internodes) will be clean of particles. It is at the internodes that the vibrational energy of that particular mode is at a maximum and sensors placed at these locations are extremely effective in estimating that particular mode24.

The above heuristic approach has been used by Bayon de Noyer24 in locating effective sensor placement for acceleration feedback control to alleviate tail buffeting of a high performance twin tail aircraft. Note the usage of the term “effective sensor configuration” as it is based on validated heuristics as opposed to “optimal sensor configuration” that results from a mathematically optimal pattern search for a sensor configuration. So, what needs to be done to determine an effective sensor configuration is to find the areas of energetic modal activity.

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CFD data provided simulated pressure signals at 286 locations distributed at equidistances all along the surface of the cylinder (see the cylinder surface grid in Fig. 6). A three- step procedure is proposed for determining sensor placement and number as follows:

• Run the POD procedure on the 286 pressure signals. Examine the frequencies of the energetic modes. In this effort, three energetic modes are found and their frequencies correspond to the fundamental von Kármán shedding frequency (first two modes) and the next higher frequency (third mode).

• The spatial Eigen-functions obtained from the POD procedure provides information concerning the locations where the modal activity is at its highest (see Fig. 8). Examine the maxima/minima of the spatial Eigen-functions of the 286 Pressure signals.

• Place sensors at the energetic maxima and minima of each mode as shown in Fig. 6. The locations of the sensors in Table 2 are referenced in terms of the coordinates, non-dimensionalized with the model base height H, namely, X/H and Y/H. Three sensors are placed at the upper surface of the cylinder near the trailing edge and three more sensors placed symmetrically at the lower surface. These six sensors target modes 1 and 2. Finally, four sensors are located at the base of the cylinder, targeting mode 3. The time histories of the 10 sensors are presented in Fig. 7. Note the two distinct frequencies picked up by the sensors.

• Run the LSE procedure to obtain the transformation matrix Cns and to obtain the estimates of the U-

Velocity/Pressure POD mode coefficients. The estimated versus desired mode amplitude plot, for the above sensor configuration is presented in Fig. 9. The estimates resulting from the 10 sensor configuration are very accurate as seen in Table 2.

For convenience, this RMS error (defined as the difference between the RMS of the desired extracted mode amplitudes and the estimates obtained form the LSE procedure) is normalized with the RMS of the desired extracted mode amplitudes, presented as a percentage. The resulting error and the number of sensors may be integrated together into a cost function. The purpose of the design process would then be to select the configuration that minimizes this cost function. For this configuration, the RMS estimation errors are provided in Table 2. Considering the fact that the LSE is providing U-Velocity/Pressure estimates of the wake flow field, the RMS values in Table 2 are low and can be used for closed-loop flow control using a moderately robust controller. Also, provided in Table 2 are the RMS errors for sensor configurations consisting of 2, 4 and 6 sensors. Note that the three latter configurations enable the estimation of modes 1 and 2 quite well but not of the 3rd since they do not include a sensor that targets the higher frequency.

Number of

Sensors

Sensor Locations (x/H, y/H)

Mode 1 RMS Error

[%]

Mode 2 RMS Error

[%]

Mode 3 RMS Error [%]

POD Coeff. Estimated (in Wake)

2 (-0.07, 0.50), (-0.10,-0.50) 22.74 60.27 Only Modes 1 & 2 Targeted

Pressure

4 (-0.07, 0.50), (-0.10,-0.50) (-0.10, 0.50), (-0.07,-0.50)

2.99 8.07 Only Modes 1 & 2 Targeted

Pressure

6 (-0.07, 0.50), (-0.10,-0.50) (-0.10, 0.50), (-0.07,-0.50) (-0.13, 0.50), (-0.13,-0.50)

0.78 2.17 Only Modes 1 & 2 Targeted

Pressure

10 (-0.07, 0.50), (-0.10,-0.50) (-0.10, 0.50), (-0.07,-0.50) (-0.13, 0.50), (-0.13,-0.50)

(0, 0.42), (0,-0.01) (0, 0.24), (0,0.08)

0.77 2.10 20.90 Pressure

10 Same 10 sensor configuration as above

2.30 2.30 12.5 U_Velocity

Table 2: Sensor Coordinates and RMS Estimation

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−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0−5

−4

−3

−2

−1

0

1

2

3

4

5

x/H [−]

y/H

[−]

286 possible pressure taps

Sensors targeting modes 1 & 2On trailing edge (Sensors # 1− # 6)

Sensors targeting mode 3On Base (Sensors # 7− # 10)

Figure 6. 10 sensor configuration out of 286 possible ports

2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.518.52

18.53

18.54

18.55

18.56

18.57Time Signals of Pressure Taps

Time [s]

Pres

sure

Sensor 1Sensor 2Sensor 3Sensor 4Sensor 5Sensor 6Sensor 7Sensor 8Sensor 9Sensor 10

Figure 7. Time Histories of Pressure taps for the 10 Sensors

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−9 −8 −7 −6 −5 −4 −3 −2 −1 0−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2Eigen−Functions along X axis

x/H [−]

Mod

al E

igen

−Fun

ctio

ns

Mode 1Mode 2Mode 3

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

y/H [−]

Mod

al E

igen

−Fun

ctio

ns

Eigen−Functions along Y axis

Mode 1Mode 2Mode 3

Figure 8. Spatial Surface Eigen-functions for 286 Pressure sensors (note that the mode amplitudes are plotted against x/H – in the top Figure; and against Y/H in the bottom Figure)

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2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5−2000

−1500

−1000

−500

0

500

1000

1500

2000Comparison Mode Amplitudes based on Sensors (*) and Full Flow Field (−)

Time [s]

Mod

e Am

plitu

de

Mode 1Mode 2Mode 3Mode 1−EstMode 2−EstMode 3−Est

2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5

−3

−2

−1

0

1

2

3Comparison Mode Amplitudes based on Sensors (*) and Full Flow Field (−)

Time [s]

Mod

e Am

plitu

de

Mode 1Mode 2Mode 3Mode 1−EstMode 2−EstMode 3−Est

Figure 9. Estimation of POD Time Coefficients for 10 Sensor configuration (U_Velocity – Top Figure; Pressure – Bottom Figure)

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VI. Conclusions and Recommendations A heuristic procedure, based on proper orthogonal decomposition modeling, was developed to determine the

placement and number of body mounted pressure sensors for feedback flow control of a “D” shaped cylinder wake. The design used surface pressure readings obtained from CFD simulations to estimate POD mode coefficients of the wake flow field (both U-Velocity and Pressure) at a Reynolds number of 300. Results show that the estimates of the two first modes of the wake flow field are very accurate (1-3% RMS Error). This level of error is acceptable for a moderately robust closed-loop flow controller. To the best of our knowledge, this represents the first time a systematic procedure has been offered and numerically validated to effectively address the issue of sensor placement and number towards closed-loop flow control based on body mounted sensors.

Further research will aim at examining sensitivity of the developed heuristic procedure for sensor number and placement to noise and variations in Reynolds number. In addition, the design for sensor placement and number will be verified using experimental data from actual body mounted sensors for real-time estimation of the POD mode coefficients. Furthermore, the effectiveness of the sensor configuration will be examined for variable open-loop steady-state and transient forcing conditions. Finally, it required to examine the sensitivity of the developed strategy for different Reynolds numbers.

Acknowledgments The authors would like to acknowledge the support and assistance provided by Lt. Col. Sharon Heise (AFOSR).

The authors would like to thank Dr. Jim Forsythe of COBALT Solutions, LLC, for assistance with the CFD grid, as well as the Modeling and Simulation Research Center at USAFA for providing the high performance computational facilities used for this investigation.

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41, No. 8, 2003, pp. 2163-2176. 8Abergel, F., and Temam, R., "On some Control Problems in Fluid Mechanics", Theor. Comput. Fluid

Dynamics, Vol. 1, 1990, pp.303-325. 9Li, Z., Navon, I.M., Hussaini, M.Y., Le Dimet, F.X., "Optimal control of cylinder wakes via suction and

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