LINEAR AND NONLINEAR BEHAVIOR OF

LINEAR AND NONLINEAR BEHAVIOR OF TRANSIENT LIFT RESPONSE TO PULSE-ACTUATION

BY XUANHONG AN

Submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical and Aerospace Engineering in the Graduate College of the Illinois Institute of Technology

Approved Advisor

Chicago, Illinois July 2014

ACKNOWLEDGMENT I would like to thank my advisor, Professor David R. Williams for his guidance and help as I was working for him and writing my thesis. I would like thank the thesis committee members, Professor Candace E. Wark and Ankit Srivastava for their commence and feedback for improvement. Jeremy Michael Weirich, David Stuart, Lou Grimaud, Simeon Iliev, Ghazaleh Kheradmand, thanks for your help improving my experience in the laboratory. Finally, I would like to thank the Support from Air Force Office of Scientific Research Grant FA9550-12-1-0075.

iii

TABLE OF CONTENTS Page ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . . . . . . . . .

iii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . .

1

1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Literature Review . . . . . . . . . . . . . . . . . . . . 1.3. Objective . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3

2. EXPERIMENTAL SETUP

. . . . . . . . . . . . . . . . . .

4

3. RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

3.1. Lift coefficient response dependence on single-pulse control signal input . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Multi-pulse lift response dependence on pulse interval . . .

8 18

4. LOW-DIMENSIONAL MODELS OF THE SYSTEM . . . . . .

25

4.1. Black box model . . . . . . . . . . . . . . . . . . . . . 4.2. Convolution method with single-pulse kernel . . . . . . . . 4.3. G-K type model . . . . . . . . . . . . . . . . . . . . .

25 30 41

5. DISCUSSION OF RESULT . . . . . . . . . . . . . . . . . .

47

5.1. Comparison of different low-dimensional models . . . . . . 5.2. G-K type model prediction for AFC-pitching combinationopen loop controlled . . . . . . . . . . . . . . . . . . .

47 50

6. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . .

60

6.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Future work . . . . . . . . . . . . . . . . . . . . . . .

60 60

APPENDIX

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

A. DATA PROCESSING . . . . . . . . . . . . . . . . . . . . .

62

iv

B. CODE OF G-K TYPE MODEL FOR AFC . . . . . . . . . . .

69

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

v

LIST OF FIGURES Figure

Page

2.1

Wing and actuators location.

. . . . . . . . . . . . . . . . . .

5

2.2

Piezoelectric actuators. . . . . . . . . . . . . . . . . . . . . .

5

2.3

Experimental setup for the jet speed measurement. . . . . . . . .

6

2.4

Input signal definition.

7

3.1

∆CL for single-pulse, ∆tp = 0.12t+ , pulse amplitude is 60V, α = 20o .

8

3.2

Peak value of ∆ CL dependence on single-pulse pulse width, α = 20o , input signal amplitude=60V. . . . . . . . . . . . . . . . . . .

9

Indication of the alignment of ∆ CL curve-aligned at the beginning of the pulse signal. . . . . . . . . . . . . . . . . . . . . . . .

10

Indication of the alignment of ∆ CL curve, aligned at the end of the pulse signal. . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

Single-pulse lift response for different pulse width aligned at the beginning of the pulse signal, α = 20o , input signal amplitude=60V.

11

Single-pulse lift response for different pulse width aligned at the end of the pulse signal, α = 20o , input signal amplitude=60V. . . . . .

12

Peak value of ∆ CL dependence on single-pulse amplitude, α = 20o , ∆ CL = 0.12t+ . . . . . . . . . . . . . . . . . . . . . . . . . .

13

3.8

∆ CL dependence on pulse amplitude α = 20o ∆tp = 0.12t+ . . . .

13

3.9

∆CL dependence on α, ∆tp = 0.12t+, pulse amplitude is 60V.

. .

14

3.10 Peak value of ∆CL dependence on α, ∆tp = 0.12t+, pulse amplitude is 60V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3.11 Uj vs pulse width for single-pulse input signal amplitude=60V, ∆tp = 0.01t+ to 0.31t+ . . . . . . . . . . . . . . . . . . . . . . . . .

16

3.12 comparison of Uj peak value and ∆ CL peak value vs pulse width for single-pulse, α = 20o , input signal amplitude=60V. . . . . . .

17

3.13 comparison of Uj peak value and ∆ CL peak value vs amplitude for single-pulse, α = 20o , ∆tp = 0.12t+ . . . . . . . . . . . . . . . .

17

3.14 Peak value of ∆ CL vs T, pulse amplitude is 60V, α = 20o . . . . .

18

3.3 3.4 3.5 3.6 3.7

. . . . . . . . . . . . . . . . . . . . .

vi

3.15 ∆ CL for 5-pulse case, pulse amplitude is 60 V, α = 20o . . . . . .

19

3.16 ∆CL dependence on pulse interval, α = 20o , T = 1.25t+ . . . . . .

20


21

3.18 ∆CL dependence on pulse interval, α = 20o , T = 3.5t+ . . . . . . .

22


23

3.20 ∆CL dependence on pulse interval, α = 20o , T = 7t+ . . . . . . . .

24

4.1

PRBS input and output data for system identification, α = 12o .

.

26

4.2

bode plot for the linear system, the dash black line is the nominal model defined by the mean of family model parameters, the blue and red lines are the bode plot for all the cases with different input amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

∆CL prediction for open loop control k=0.064, triangle wave, α = 12o , the loop is going counterclockwise. . . . . . . . . . . . . .

28


29


30

4.6

integral kernel. . . . . . . . . . . . . . . . . . . . . . . . . .

31

4.7

Comparison of convolution model prediction and measured data for single-pulse, ∆tp = 0.12t+ , amplitude=60V, α = 20o . . . . . . . .

32

Comparison of convolution model prediction and measured data for multi-pulse T = 1.25t+ , α = 20o . . . . . . . . . . . . . . . . . .

33

Comparison of convolution model prediction and measured data for multi-pulse T = 1.75t+ , α = 20o . . . . . . . . . . . . . . . . .

34

4.10 Comparison of convolution model prediction and measured data for multi-pulse T = 3.5t+ , α = 20o . . . . . . . . . . . . . . . . . .

35


36


37

4.13 Comparison of convolution model prediction and measured data for a random input signal T = 3.5t+ , α = 20o . . . . . . . . . . . . .

38

4.3 4.4 4.5

4.8 4.9

vii

4.14 convolution model prediction of ∆CL for k=0.064 triangle wave, α = 12o , the loop is going counterclockwise. . . . . . . . . . . .

39


39


40

4.17 x0 with 3m/s free stream speed and 12 degree angle of attack.

. .

42

4.18 ∆CL k=0.064 triangle wave G-K type model prediction, amplitude is 60V, α = 12o , the loop is going counterclockwise wise. . . . . .

43

4.19 ∆CL k=0.128 triangle wave G-K type model prediction, amplitude is 60V, α = 12o , the loop is going counterclockwise. . . . . . . . .

44

4.20 ∆CL k=0.257 triangle wave G-K type model prediction, amplitude is 60V, α = 12o , the loop is going counterclockwise. . . . . . . . .

45

4.21 random signal, pulse amplitude=60V, α = 12o . . . . . . . . . . .

46

5.1

∆CL k=0.064 triangle wave, amplitude is 60V, α = 12o .

. . . . .

48

5.2


. . . . .

49

5.3


. . . . .

50

5.4

Diagram of combined G-K type model. . . . . . . . . . . . . . .

51

5.5

An example of open loop control input signal(1.6s delay case). . .

52

5.6

∆CL prediction for 0s delay between pitching and excitation. . . .

53

5.7

∆CL prediction for 0.4s delay between pitching and excitation. . .

54

5.8


54

5.9


55

5.10 ∆CL prediction for 1.6s delay between pitching and excitation. . .

56

5.11 periodic pitching input signal with feed-forward control . . . . . .

57

5.12 periodic pitching input with feed-forward control . . . . . . . . .

57

5.13 random pitching input signal with feed-forward control . . . . . .

58

5.14 random pitching input with feed-forward control . . . . . . . . .

59

viii

LIST OF SYMBOLS Symbol

Definition

f

Frequency

c

The chord length of the wing

x

Distance from the location of the actuator to the trailing-edge of the wing

U

Free stream speed

F+

Reduced frequency F + =

Aactuator

The area of actuator slot

Awing

x U

The area of the wing

ρ

Air density

Uj

Actuator jet speed

Cµ

Dimensionless ρUj2 Aactuator 0.5ρU 2 Awing

excitation

moment

t

Time

t+

Dimensionless time which is

∆tp

Pulse interval

CL

Lift coefficient

α AF C

Cµ

=

t c/U

Pulse width

T

∆CL

coefficient,

Lift coefficient difference from the baseline(steady lift coefficient without actuation) Angle of attack Active flow control

ix

πf c U

k

Reduced frequency K =

v

Input voltage to the actuators

x

The proportion of flow attached over the wing

x0

The stationary value of x

τ1

The relaxation time constant

τ2

Total time delay

CLtotal

Total lift coefficient of AFC-pitching combined motion

CLpitching

Lift coefficient for pithing motion

∆CLAF C

Lift coefficient difference from the baseline(steady lift coefficient without actuation for AFC)

x

ABSTRACT The transient lift coefficient for a NACA0009 airfoil with pulsed jet actuation type of active flow control at stall conditions is investigated. The experimental measurements show the lift coefficient dependence to a single-pulse with different pulse widths and amplitudes. Based on the single-pulse with optimal parameters, multipulse and continuous pulse actuation cases are studied, which indicate that the pulse interval is the major factor of the active flow control system capability. Linear and nonlinear model are used to predict the lift coefficient variation for different input signals to the actuators. A combined nonlinear model is introduced to predict lift coefficient change due to arbitrary unsteady pitching motion with active flow control.

xi

1 CHAPTER 1 INTRODUCTION 1.1 Motivation High angle of attack flight condition could cause a stall which will significantly decrease the lift on the wing suddenly and increase the drag at the same time. On the other hand, rapidly maneuvering a high performance flight and rotating blades of helicopters can lead to a so called dynamic stall. Many investigations have been done with active flow control applied to the quasi-steady flight condition, but transient active flow control for unsteady flow condition is still under investigation. 1.2 Literature Review Previous work has been done with several types of actuator, different test models and objectives. Margalit, et al.[1] showed that burst mode with short duty cycle excitation is more effective than sinusoidal. Glezer and Amitay [2] gave a comprehensive review for the synthetic (zero-net mass flux) jet for the burst mode excitation. The effect of different parameters for a single-pulse actuation were analyzed by Albrecht, et al [3]. Greenblatt, et al. [4] showed that a reduced frequency of 0.3 ≤ F + ≤ 4 is effective for a two-dimensional airfoil, where F + =

fc U∞

is the re-

duced frequency, f is the frequency in Hz, c is the distance from the actuator to the trailing-edge of the wing, U∞ is the free stream velocity. While the periodic excitation effective momentum addition was in the range 0.01% ≤ Cµ ≤ 3% and the experimental data indicated that excitation at the shoulder of a deflected flap was particularly effective even up to large deflection angles of δf = 60o and was effectively independent of Reynolds number. Another investigation done by A. Seifert, et al. [5] showed that unsteady flow control using periodic excitation with a reduced frequency is slightly higher than the natural vortex shedding frequency is a more efficient way. Woo, et

2 al. [6] also investigated the optimal excitation frequency for a 2-D stalled wing, and demonstrated that smaller pulse separation will lead to a larger lift increment when the pulse separation is no smaller than 0.625t+ . Seifert, et al. [7] also showed that increasing Reynolds number did not have an adverse effect on the modulated blowing. The optimal excitation frequency as a function of the incline angle was studied by Cierpka, et al. [8] using POD analysis and a vortex detection algorithm. They concluded that optimal excitation frequency must be lower for larger angle of attack. Darabi and Wygnanski [9] investigated the forced reattachment process of active flow control for naturally separated flow over a solid surface which and concluded that the Minimum reattachment time occurred at an optimal excitation frequency + + of Fopt , which was independent of amplitude and flap inclination, where, Fopt is the

reduced frequency at the maximum lift increment. Woo, et al. [10] [11] studied the flow mechanisms associated with momentary attachment and relaxation processes of pulsed actuation over a stalled airfoil to mimic control of transitory stall on helicopter blades with a combustion actuator. They concluded that the dynamic response of the stalled flow to a single-pulse actuation is following the actuation, the separated vorticity layer on the suction surface of the airfoil is severed, becomes detached, and then rolls into a large-scale. They also determined the delay and relaxation time of the flow response for single-pulse actuation. This is investigated with more detail in Section 3.1 of this thesis. Instead of using a jet excitation application, LePape, et al. [12] used retractable vortex generators located at the leading edge of an airfoil to mitigate the adverse effects of dynamic stall. Williams, et al. [13] created a linear model to predict the lift variation output for the input voltage signal using a convolution approach. Kerstens, et al. [14] introduced a black-box model identified from measurement of the lift coefficient for a random binary input signal. Then, based on the linear approach of the plant model, Kerstens, et al. [15] introduced a control system to suppress the lift coefficient variation caused by the surging flow. Karim, et

3 al. [16]and Alrefai, et al. [17] devolved a technique to suppress the dynamic stall due to pitching motion. They used the steady suction to prevent the reverse-flow fluid accumulating at the leading-edge region which can cause a dynamic stall. 1.3 Objective The first objective is to determine optimal pulse input to actuators for single and multi-pulse to achieve the maximum lift increment. The second objective of the current investigation is to compare low-dimensional models that predict the time varying transient lift coefficient response of separated flow over a NACA0009 wing, that would occur for an arbitrary input signal to the actuators of an active flow control system. A black-box model identified with the Prediction Error Method(PEM), a convolution integral model and a nonlinear model were built and analyzed. Further, a combined lift coefficient prediction model for a pitching airfoil with active flow control excitation was combined with another nonlinear lift coefficient prediction model developed by Grimaud [18] for a pitching wing. The combined model would be able to provide a feed-back signal for a future closed loop control system. As a preliminary step, it is necessary to characterize the lift response to the piezo-actuator input signal. Single pulse cases for different pulse width, pulse amplitude, angle of attack were investigated. And multi-pulse cases with different pulse interval were studied.

4 CHAPTER 2 EXPERIMENTAL SETUP The Andrew Fejer Unsteady Wind Tunnel which is located at the Fluid Dynamic Research Center of IIT was used for the experimental data acquisition. It is a closed-circuit, low-speed facility driven by an axial-vane fan powered by a 40 hp synchronous motor with 30m/s maximum flow speed. The wind tunnel test section is 0.61m × 0.61m in cross section and 3.1m in length. The free stream speed is maintained at a constant 3m/s as measured by a pitot tube for all measurements. A two dimensional NACA0009 wing with a 0.245m chord and 0.6m span was used as the test article. Force and moments are directly measured by an ATI Inc., Nano-17 transducer attached to the center span of the wing located at 1/3 of the chord. A plate holding the force transducer is supported by two servos, which can plunge or pitch the airfoil by moving up and down. The command signal to the two servos was sent from a dSPACE 1104 system running at 10,000 cycles/sec. Eight piezo actuators are located at x/c=0.077 from the leading edge on the upper surface of the wing separated by 0.073m as shown in Figure 2.1. The piezoelectric actuators driven by a PiezoDrive PDL200 amplifier are shown in Figure 2.2. The mechanism of the actuator jet production was well described by Cattafesta [19] and Glezer, et al. [2]. The excitation jet points to the trailing edge with 30o angle relative to the local surface plane of the wing. The maximum Cµ =

ρu2 Aact 0.5ρU 2 Awing

= 0.021 is

obtained at 80V input. The input signal to the amplifier is sent from a dSPACE 1104 system running at 10,000 cycles/sec.

5

Figure 2.1. Wing and actuators location.

Figure 2.2. Piezoelectric actuators.

The actuator jet speed measurement is done by a hotwire anemometer located at the center of the slot, and at 0.3mm from the plane of the opening. The experimental setup of the jet speed measurement is shown in Figure 2.3. Based on Margalit [1] and Glezer’s work [2], a burst mode actuation signal with variable pulse width, pulse amplitude and pulse interval was built. The modulated input signal sent to the amplifier includes two parts, the base signal is a pulse wave

6

Figure 2.3. Experimental setup for the jet speed measurement. with 50% duty cycle at 360 Hz, which is the same as the resonant frequency of the piezo actuator used to create the maximum possible excitation velocity [19]. A lower frequency square wave, treated as the control signal with variable pulse width, pulse interval and amplitude, is modulated on top of the resonant frequency wave. The definition of the pulse input signal is shown in Figure 2.4, where ∆tp is the pulse width, T is the pulse interval and A is the amplitude of the pulse signal. Here, all the time variables are in t+ unit, nominalized by

c . U∞

7

Figure 2.4. Input signal definition.

8 CHAPTER 3 RESULTS 3.1

Lift coefficient response dependence on single-pulse control signal input In this section, the lift coefficient variation dependence on pulse width and

amplitude of the single-pulse control signal is studied. As shown in Figure 3.1 the lift coefficient initially decreases right after the pulse signal has been triggered, and it reaches the minimum value at 1.1t+ . The lift coefficient then goes up to the maximum at 2.5t+ , and goes back to the initial condition after 17t+ . ∆CL is the difference between the forcing lift coefficient and the base line. From the fluid mechanical point of view, the lift variation will reach its maximum value when the down stream edge of the large scale forcing vortex approaches the trailing edge [3]. 0.35 ∆ CL 0.3

input signal

0.25

∆ CL

0.2 0.15 0.1 0.05 0 −0.05 −0.1

0

10

20

30

40

50

60

70

t+

Figure 3.1. ∆CL for single-pulse, ∆tp = 0.12t+ , pulse amplitude is 60V, α = 20o . 3.1.1 Lift coefficient response dependence on the pulse width.

The lift

response curve, including the peak value of the lift and the delay of the lift peak

9 appearance from the end of the pulse signal, is affected by the pulse width ∆tp . Figure 3.2 shows that the peak value of ∆CL increased with larger pulse width from 0.01t+ to 0.12t+ pulse width before the saturation occurs at 0.12t+ . Related to the structure of the flow field, a longer excitation time produces a stronger vortex, which will lead to a higher lift amplitude [3].

0.345 0.34

Peak value of ∆ CL

0.335 0.33 0.325 0.32 0.315 0.31 0.305 0.3

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

∆ tp (t+)

Figure 3.2. Peak value of ∆ CL dependence on single-pulse pulse width, α = 20o , input signal amplitude=60V.

To analyze the time delay of the lift coefficient dependence on pulse width, all ∆CL curves aligned with the beginning and the end of the pulse control signal are investigated. Since the input pulse signal and ∆CL are synchronized automatically, alignment of the ∆CL curves with a specific point can be achieved by aligning the input signal at that point. The alignment with the leading edge of the pulse and the trailing edge of the pulse is shown in Figure 3.3 and Figure 3.4, respectively.

10

0.4 input signal ∆ tp=0.01t+ 0.35

input signal ∆ tp=0.06t+ input signal ∆ tp=0.12t+

0.3

input signal ∆ tp=0.18t+

Voltage/240

0.25


0.2 0.15 0.1 0.05 0 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

t+

Figure 3.3. Indication of the alignment of ∆ CL curve-aligned at the beginning of the pulse signal.


Voltage/240

0.3


0.25


0.2



0.15

0.1

0.05

0 0

0.2

0.4

0.6

0.8

1

t+

Figure 3.4. Indication of the alignment of ∆ CL curve, aligned at the end of the pulse signal.

11

0.35

∆ CL 0.01t+ ∆ CL 0.06t+

0.3

∆ CL 0.12t+ ∆ C 0.18t+

0.25

L

∆ CL 0.24t+

∆ CL

0.2

∆ CL 0.31t+

0.15 0.1 0.05 0 −0.05 −0.1 1

2

3

4

5

6

7

8

t+

Figure 3.5. Single-pulse lift response for different pulse width aligned at the beginning of the pulse signal, α = 20o , input signal amplitude=60V.

If the ∆CL curves are aligned with the beginning of the pulse input signal, shown in Figure 3.5, the delay is not strongly dependent on the pulse width. In Figure 3.6, all the ∆CL curves are aligned with the end of the pulse signal. It shows the delay time from the end of the pulse input signal to the start point of the lift increasing becomes smaller when the pulse width is larger, the time differences between the delays are the same as the time differences between the pulse width. This is because all the plots are aligned at the end of the pulse signal. The longer duty cycle of actuation starts to create the vorticity earlier, as a consequence, the downstream edge of the vortices will reach the trailing edge earlier.

12

0.35 ∆ CL 0.01t+ ∆ CL 0.06t+

0.3

∆ CL 0.12t+

0.25

∆ CL 0.18t+ ∆ CL 0.24t+

0.2

∆ CL 0.31t+

∆ CL

0.15 0.1 0.05 0

−0.05

0

1

2

3

4

5

6

7

t+

Figure 3.6. Single-pulse lift response for different pulse width aligned at the end of the pulse signal, α = 20o , input signal amplitude=60V. 3.1.2 Lift response dependence on the pulse amplitude.

Several cases for

different pulse amplitudes from 10V to 80V were investigated with the data taken in the wind tunnel experiment. The lift response is also strongly dependent on the input pulse amplitude. Higher input voltage leads to a larger peak value of the lift coefficient, as shown in Figure 3.7. A larger time delay from the end of the input pulse to the appearance of the peak lift is shown in Figure 3.8 with increasing pulse amplitude. The explanation for this behavior is higher input voltage can generate a larger excitation flow velocity which creates a stronger vortex [3]. More details about this mechanism will be discussed in section 3.1.4.

13

0.45 0.4


0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 10

20

30

40 50 Voltage(V)

60

70

80

Figure 3.7. Peak value of ∆ CL dependence on single-pulse amplitude, α = 20o , ∆ CL = 0.12t+ .

DCL 10V DCL 20V DCL 30V DCL 40V DCL 45V DCL 50V DCL 55V DCL 60V DCL 70V DCL 80V

0.4

0.3

∆ CL

0.2

0.1

0

−0.1 0

2

4

6

8

10

12

14

t+

Figure 3.8. ∆ CL dependence on pulse amplitude α = 20o ∆tp = 0.12t+ .

14 3.1.3 attack.

Single-pulse lift coefficient response dependence on the angle of The difference of the lift coefficient from the base line is affected by the

angle of attack, as shown in Figure 3.9. To make the relationship between the ∆CL peak value and the incline angle more clear, Figure 3.10 is also shown here. The ∆CL increment for 4o and 6o incline angle is almost zero, because the flow remains attached in these two cases. In 10o and 12o incline angle cases, the maximum value of the negative part of ∆CL is the largest compared with others, but the positive increment for 10o is much smaller than 12o . The largest lift increment appears at about 15o case. A conclusion can be made that the lift increment produced by AFC is zero for attached flow, and then keeps rising with incline angle, until it reaches 15o , and then, decreases from 15o all the way to 20o because the actuation is no longer able to fully reattach the flow.

0.5

α=4o α=6.1o

0.4

α=8o α=10o

0.3

α=12o α=13o

∆ CL

0.2

α=13.9o α=14.9o

0.1

α=16.1o α=17.1o

0

α=18.1o α=19.1o

−0.1

α=20o

−0.2 5

10

15

20

t+

Figure 3.9. ∆CL dependence on α, ∆tp = 0.12t+, pulse amplitude is 60V.

15

0.5 0.45 0.4

peak value of ∆ CL

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

4

6

8

10

12 α

14

16

18

20

Figure 3.10. Peak value of ∆CL dependence on α, ∆tp = 0.12t+, pulse amplitude is 60V. 3.1.4 Lift coeffiecient dependence on actuator jet speed. The bridge between input control signal and the output lift coefficient variation is the jet exciting from the slot located at the leading edge. Velocity measurements for the jet created by the actuation are necessary for establishing a more detailed relationship between input signal and the output lift change. The velocity, generated by a single-pulse input with variable pulse width, and fixed 60V amplitude, as measured by a hot-wire, is shown in Figure 3.11.

16

6

∆ tp=0.01t+ ∆ tp=0.06t+ ∆ tp=0.12t+

5

∆ tp=0.18t+ ∆ tp=0.24t+

jet speed(m/s)

4

∆ tp=0.31t+

3

2

1

0

−1

0.4

0.5

0.6

0.7

0.8

0.9

t+

Figure 3.11. Uj vs pulse width for single-pulse input signal amplitude=60V, ∆tp = 0.01t+ to 0.31t+ . Figure 3.11 shows the total response time of the jet speed variation is about 1t+ , which is much shorter than that for the lift response comparing with Figure 3.6. The trends of the jet speed amplitude and the ∆CL peak value are shown in Figure 3.12 and Figure 3.13, which indicate that the excitation jet speed and ∆CL are almost linear correlated.

6

0.34

5

0.33

peak value of ∆ CL 4 peak value of Uj

0.32

3

0.31

2

0.3

0

0.05

0.1

0.15

0.2

0.25

0.3

j

0.35

peak value of U (m/s)


17

1 0.35

+

pulse width(t )

Figure 3.12. comparison of Uj peak value and ∆ CL peak value vs pulse width for single-pulse, α = 20o , input signal amplitude=60V.

0.5

10 peak value of ∆ CL

0.4

8

0.3

6

0.2

4

0.1

2

0 10

20

30

40 50 pulse amplitude(V)

60

70

peak value of Uj(m/s)


peak value of Uj

0 80

Figure 3.13. comparison of Uj peak value and ∆ CL peak value vs amplitude for single-pulse, α = 20o , ∆tp = 0.12t+ .

18 3.2 Multi-pulse lift response dependence on pulse interval The results presented in section 3.1.1 show that 0.12t+ is the saturation point for ∆CL curve, so we keep ∆tp = 0.12t+ for single-pulse input signal, and vary the pulse separation T to determine the lift response dependence on the pulse interval for multi-pulse case. The result is shown in Figure 3.14. The result of multi-pulse case shows that the peak value of ∆CL increases with the pulse interval increasing at first, and then, decreases. The maximum lift gain occurs at T = 3.5t+ pulse interval or excitation reduced frequency at F + = 0.28. This result agrees with Greenblatt’s [4] conclusion that most efficient excitation frequency is 0.3 ≤ F + ≤ 4 [4].

0.55

peak value of ∆ CL

0.5

0.45

0.4

0.35

1

2

3

4

5

6

7

T(t+)

Figure 3.14. Peak value of ∆ CL vs T, pulse amplitude is 60V, α = 20o .

Figure 3.15 shows that the ∆CL initial decrease and increasing lift is independent of the pulse interval at the early stage(from 0t+ to 3t+ ), even the T = 1.25t+ case and T = 7t+ case are identical in this time region.

19

0.6

T=1.25t+ T=1.75t+ T=3.5t+

0.5

T=5.25t+ T=7t+

∆ CL

0.4

0.3

0.2

0.1

0 0

5

10

15

20

25

30

35

t+

Figure 3.15. ∆ CL for 5-pulse case, pulse amplitude is 60 V, α = 20o . To make the relationship between the input control signal and ∆CL more clear, each case with different pulse interval is also shown from Figure 3.16 to Figure 3.20. Figure 3.16 shows the T = 1.25t+ pulse signal is triggered before the ∆CL peak produced by the previous pulse, which means it will interrupt the single-pulse mechanism, this leads to the smaller peak value than the single-pulse case.

20

∆ CL Voltage

0.3

60 40

0.1

voltage(V)

∆ CL

0.2

20 0

0

0

2

4

6

8

10

12

14

16

t+

Figure 3.16. ∆CL dependence on pulse interval, α = 20o , T = 1.25t+ . In T = 1.75t+ case shown in 3.17, the ∆CL curve decreases substantially after the pulse signal. The peak value of ∆CL is also smaller than the single-pulse case.

21

∆ CL 0.3

Voltage

∆ CL

60 40

0.1

voltage(V)

0.2

20 0

0

0

2

4

6

8

10

12

14

16

18

t+

Figure 3.17. ∆CL dependence on pulse interval, α = 20o , T = 1.75t+ . The T = 3.5t+ case has the most interesting phenomenon, it shows that a larger ∆CL peak value than the single-pulse case is obtained. Comparing Figure 3.18 with Figure 3.1, we can tell that the pulse signal goes in to the actuator 3t+ before the second peak created by the single-pulse mechanism. Instead of preventing the large scale vortex formation as the shorter pulse interval cases shown above, it enhances this process.

22

0.4

0.2 60 0.1

voltage(V)

∆ CL

0.3

40 20

0

∆ CL

0

Voltage 0

5

10

15

20

25

30

35

t+

Figure 3.18. ∆CL dependence on pulse interval, α = 20o , T = 3.5t+ . The pulses are excited 0.2t+ before the peak of the first secondary bump in T = 5.25t+ case, shown in Figure 3.19, and the amplitude of the inverse part is two times larger than that caused by single-pulse. The maximum ∆CL increment is smaller than the single-pulse case.

23

∆ CL Voltage

0.3

60 40

0.1

voltage(V)

∆ CL

0.2

20 0

0

0

5

10

15

20

25

30

35

40

t+

Figure 3.19. ∆CL dependence on pulse interval, α = 20o , T = 5.25t+ . For the T = 7t+ pulse interval case, shown in Figure 3.20, the pulses are triggered when the flow field almost recovers to its initial state, but ∆CL is still 0.1 larger than the base line, so the peak value of ∆CL is slightly larger than the single-pulse case.

24

0.5 ∆ CL Voltage 0.4

0.2 60 0.1

voltage(V)

∆ CL

0.3

40 20

0

0

0

10

20

30

40

50

t+

Figure 3.20. ∆CL dependence on pulse interval, α = 20o , T = 7t+ . Comparing all the cases with different pulse interval, T = 3.5t+ or F + = 0.28 has the largest lift increment, but a higher excitation frequency is desired without sacrificing to much lift increment for control purpose. So the actuation should be focused on is chosen to be T = 1.56t+ or F + = 0.63 pulse interval, with 0.12t+ pulse width.

25 CHAPTER 4 LOW-DIMENSIONAL MODELS OF THE SYSTEM In this chapter, three low-dimensional models are introduced to predict the ∆CL dependence on the input control signal to the actuators. The main purpose here is to predict the ∆CL response to an input control signal modulated on a F + = 0.63 excitation frequency (T = 1.56t+ ) with 0.12t+ pulse width actuation signal at 12o angle of attack. The first model is a classical first order transfer function with a time delay, that was identified using system identification techniques. The second model is based on the convolution integral, the advantage of this model is it can predict more detailed ∆CL variation using the information for each input pulse signal. The third is a nonlinear model that attempts to obtain a more precise model for the low frequency control signal variation. 4.1 Black box model 4.1.1 System identification.

The ∆CL response to a family of pseudo random

binary signal(PRBS) inputs with different pulse amplitudes from 10V to 80V is used for the system identification. For example, the 15V amplitude is shown in Figure 4.1. The prediction-error method [14] is used on the input and output signal to get a first order linear model with a time delay as shown in equation 4.1.

Gp (s) =

Ktf −θs e Ttf + 1

(4.1)

Where Ktf , Ttf and θ are the parameters need to be determined by the system identification process. By repeating the same procedure for all the family of PRBS,a first order system was identified for each amplitude of the whole family of input signals. Taking the mean of the transfer function family a mean model is obtained. Here Ktf = 1.2407, Ttf = 0.2857 and θ = 0.0699. The bode plot is shown in Figure

26 4.2. experimental data model prediction Voltage/20

0.3 0.25 0.2

∆ CL

0.15 0.1 0.05 0 −0.05 −0.1

0

100

200

300

400

500

600

700

800

t+

Figure 4.1. PRBS input and output data for system identification, α = 12o .

27

Magnitude [dB]

20 0 −20 −40 −60 −1 10

0

1

10

2

10

10

f [Hz]

Phase [deg]

0 −200 −400 −600 −800 −1

0

10

1

10

10

2

10

f [Hz]

Figure 4.2. bode plot for the linear system, the dash black line is the nominal model defined by the mean of family model parameters, the blue and red lines are the bode plot for all the cases with different input amplitude. 4.1.2 Model validation. The model is tested with three triangle wave control signals, and α = 12o . The amplitude of all the three signals is 60V, but the frequency for each of them is k=0.064, 0.128 and 0.257 or F + =0.02, 0.039 and 0.078. where k is the reduced frequency defined by k =

πf c U∞

The argument here is that if the black box model

is capable to track the low frequency component of the ∆CL curve. It should be pointed out that the bumps in the measured ∆CL curve indicate the short duration lift response for each pulse signal to the actuators. Figure 4.3 shows the result of the black-box model ∆CL prediction for k=0.064 case. Since this linear model does not account for the nonlinear relationship between ∆CL and the voltage, the black-box model is not able to predict the experimental data.

28

0.25

0.2

0.15

∆ CL

0.1

0.05

0 experimental data 1st order tf with delay −0.05

−0.1

0

10

20

30

40

50

60

70

Voltage

Figure 4.3. ∆CL prediction for open loop control k=0.064, triangle wave, α = 12o , the loop is going counterclockwise. In k=0.128 case, shown in Figure 4.4 the experimental data for ∆CL is actually more linear than k=0.064 case. This leads to a better ∆CL prediction for the blackbox model.

29

0.25

0.2

0.15

∆ CL

0.1

0.05

0

experimental data 1st order tf with delay

−0.05

−0.1

0

10

20

30

40

50

60

70

Voltage

Figure 4.4. ∆CL prediction for open loop control k=0.128, triangle wave, α = 12o , the loop is going counterclockwise. For a even higher frequency case, k=0.257, the black-box model is not capable of predicting the ∆CL variation. Because the nonlinear effect is significant in this high frequency case.

30

0.3 experimental data 1st order tf with delay 0.25

0.2

∆ CL

0.15

0.1

0.05

0

−0.05

0

10

20

30

40

50

60

70

Voltage

Figure 4.5. ∆CL prediction for open loop control k=0.257, triangle wave, α = 12o , the loop is going counterclockwise. 4.2 Convolution method with single-pulse kernel One of the advantages of the convolution method is that it can predict the lift response for each single-pulse. So, the convolution integral method is investigated [13].

Z

+∞

(f ∗ g)(t) =

f (τ )g(t − τ )dτ

(4.2)

−∞

In this case, the kernel f (τ ) is the single-pulse lift response with 0.12t+ pulse width which is shown in Figure 4.6, and g(τ ) is the input voltage to the actuator. Then, formula 4.2 becomes: Z

+∞

CLsingle (τ )v(t − τ )dτ

CL (t) =

(4.3)

−∞

While, in discrete case, CL (k) =

X

CLsingle (j)v(k − j)

(4.4)

31 In Section 4.2, the angle of attack is 20o for single and multi-pulse cases, and 12o for the high frequency excitation cases, which means the two kernels are used in this section, one for 20o case and one for 12o case. 0.35 ∆ CL 0.3


0.25

∆ CL

0.2 0.15 0.1 0.05 0 −0.05 −0.1

0

10

20

30

40

50

60

70

+

t

Figure 4.6. integral kernel.

4.2.1 Convolution approach for single-pulse. First, the model was tested with a single-pulse signal, which is the same as the one that produces the kernel of the convolution. Here, ∆tp = 0.12t+ and the pulse amplitude is 60V.

32

∆ CL experimental data ∆ CL convolution model prediction

0.3

Voltage/200

0.25

∆ CL

0.2 0.15 0.1 0.05 0 −0.05

0

5

10

15

20

25

t+

Figure 4.7. Comparison of convolution model prediction and measured data for singlepulse, ∆tp = 0.12t+ , amplitude=60V, α = 20o . Figure 4.7 shows the convolution model prediction is accurate, and even the harmonics are predicted so well. The single-pulse prediction is not enough to demonstrate the capability of the convolution method for arbitrary signal input, since the interaction between pulses is not accounted for in the case. 4.2.2

Convolution approach for multi-pulse and continuous pulse.

In

order to validate the linear model and determine the limitation of this method, the convolution method’s lift variation prediction for 10 multi-pulses with different pulse intervals was plotted on top of the wind tunnel measured data. The results are shown in Figure 4.8 to Figure 4.12. Here ∆tp = 0.12t+ .

33

0.9

∆ CL experimental data

0.8

∆ CL convolution model prediction Voltage

0.7

∆ CL

0.5 0.4 0.3

60

0.2

40

0.1

20

0

voltage(V)

0.6

0

0

10

20

30

40

50

60

70

80

t+

Figure 4.8. Comparison of convolution model prediction and measured data for multipulse T = 1.25t+ , α = 20o . Figure 4.8 shows that the convolution model ∆CL prediction is good before 3t+ , but the prediction keeps rising to a maximum 0.77, which is about 2.6 times larger than the measured ∆CL , at 9t+ . The experimental data reaches the maximum value of ∆CL , 0.33, at 3t+ and then, fluctuating and recovering. Therefore, the convolution approach over predicts the measurement by a large amount.

34


0.9

∆ CL concolution model prediction

0.8

Voltage

0.7

0.5

60

0.4

40

0.3

20

voltage(V)

∆ CL

0.6

0.2 0 0.1 0 0

2

4

6

8

10

12

14

16

18

t+

Figure 4.9. Comparison of convolution model prediction and measured data for multipulse T = 1.75t+ , α = 20o In T = 1.75t+ case, the ∆CL prediction of the convolution model, shown in Figure 4.9 with light blue line, deviates from the experimental data after 4t+ , however, this is better than T = 1.25t+ case, and the linear approach is able to simulate each peak of ∆CL in the measured data, although the absolute value is still not correct.

35 ∆ CL experimental data ∆ CL convolution model prediction Voltage 0.5

voltage(V)

0.4

∆ CL

0.3

0.2 60 0.1

40 20

0

0

0

5

10

15

20

25

30

35

t+

Figure 4.10. Comparison of convolution model prediction and measured data for multi-pulse T = 3.5t+ , α = 20o . The ∆CL convolution model prediction and experimental data for T = 3.5t+ are shown in Figure 4.10. One can tell that the model prediction and the measured data are almost identical, which means linear model works in this case. The assumption can be made that a good ∆CL prediction can be given by the convolution model for all the cases if the pulse interval is larger than 3.5t+ , but an exception exists. Figure 4.11 shows that the convolution model failed to predict the experimental ∆CL curve in T = 5.25t+ case. Since the pulse signal is triggered exactly at the peak of the second bump created by the previous pulse excitation, this behavior could related to some vortex interaction, and more investigations such as PIV with POD, DMD are needed in the future.

36 ∆ CL experimental data ∆ C convolution model prediction L

Voltage 0.4

0.2 60 0.1

voltage(V)

∆ CL

0.3

40 20

0

0

0

10

20

30

40

50

60

t+

Figure 4.11. Comparison of convolution model prediction and measured data for multi-pulse T = 5.25t+ , α = 20o . In the T = 7t+ the pulse signal is triggered when ∆CL almost comes to the initial condition. The convolution model prediction does a good job.

37


0.5

∆ CL convolution model prediction Voltage

0.4

0.2 60 0.1

voltage(V)

∆ CL

0.3

40 20

0

0

0

10

20

30

40

50

t+

Figure 4.12. Comparison of convolution model prediction and measured data for multi-pulse T = 7.00t+ , α = 20o . 4.2.3

Convolutionl approach for random input control signal with low

excitation frequency.

Since the T = 3.5t+ case has both an accurate linear

prediction and the largest lift increment, a continuous random pulse input signal with 3.5t+ pulse interval and 0.12t+ pulse width was used to test the linear model. Figure 4.13 shows the ∆CL prediction in this case is reasonably good, but some errors exist, because some nonlinear effects(the interaction between pulses) occur when the modulated low frequency control signal is changing.

38

0.6

∆ CL experimental data ∆ CL convolution model prediction

0.5

Voltage/600 0.4

∆ CL

0.3

0.2

0.1

0

−0.1

0

200

400

600

800

1000

1200

t+

Figure 4.13. Comparison of convolution model prediction and measured data for a random input signal T = 3.5t+ , α = 20o . 4.2.4 Convolutionl approach for triangle input control signal modulated on a high frequency excitation signal. The validation of the convolution method is also tested with the triangle waves input with k=0.064, 0.128, 0.257 in frequency and 60V in amplitude, the actuators is operated at F + = 0.63(T = 1.56t+ ) excitation. The angle of attack is 12o in this case. A compensator, which has a scaling factor equal to 1/2.8 is applied to the convolution method, since the original one will not work with F + = 0.63(T = 1.56t+ ) excitation. Also, the kernel used in this case is the single-pulse lift response at 12o angle of attack. Figure 4.14 to Figure 4.16 show that the convolution method does predict the lift variation for every single pulse, but it fails to track the nonlinear response of the low frequency ∆CL curve.

39

0.25

0.2

0.15

∆ CL

0.1

0.05

0 measurement convolution with single pulse kernel

−0.05

−0.1

0

10

20

30

40

50

60

70

Voltage

Figure 4.14. convolution model prediction of ∆CL for k=0.064 triangle wave, α = 12o , the loop is going counterclockwise.

0.25

0.2

0.15

∆ CL

0.1

0.05

0

−0.05 experimental data convolution with single pulse kernel −0.1

0

10

20

30

40

50

60

70

Voltage


40

0.3 experimental data convolution with single pulse kernel 0.25

0.2

∆ CL

0.15

0.1

0.05

0

−0.05

0

10

20

30

40

50

60

70

Voltage


4.2.5

Bandwidth for convolution method.

The convolution integral will

fail to predict lift variation when the pulse interval approaches 1.75t+ from 3.5t+ . This means the bandwidth is somewhere between 1.75t+ and 3.5t+ pulse interval representing 7.1Hz to 3.6Hz for frequency or 1.8 to 0.9 for reduced frequency k(F + = 0.56 to F + = 0.28). However that means the lift response is not linear any more when the pulses are too close. From the fluid mechanics point of view, the second pulse is triggered before the lift increment produced by the previous pulse reaches its maximum, the formation of the vortex for single-pulse mechanism is interrupted, so that the convolution method based on single-pulse does not work for high frequency pulse excitation. In other words, the forcing is too fast to allow for the formation of more than one vortex per period [20]. The mechanism of the leading edge bubble built up and broken down is the major factor to limit the band width.

41 4.3 G-K type model Since the separation point is pushed towards the leading edge by increasing the angle of attack slowly(quasi steady state)[21], one can consider the excitation as a method to pull the separation point back towards the trailing edge, if increasing the mean value of ∆CL is the goal and the lift response for each single-pulse is not a major concern for control. Based on Goman and Khrabrov’s work [21], one can describe the movement of the separation point for unsteady flow conditions with the following equation,

τ1

dx + x = x0 (v(t − τ2 )) dt

(4.5)

Where τ1 = 0.2s(2.5t+ ) is the relaxation time constant, τ2 = 0.16s(2t+ ) defines the total time delay of the lift variation, τ1 and τ2 are defined from the single-pulse lift response curve without considering about the harmonics. x is the proportion of flow attached over the wing, x = 0 means the flow is fully separated, x = 1 represents fully attached flow. x0 (v) is the stationary value of x for different input voltage which is obtained from a low frequency(0.017Hz) voltage variation with an range from 0V to 80V .

42

1 0.9 0.8 0.7

X

0.6 0.5 0.4 0.3 0.2 0.1 0

0

10

20

30

40 Voltage(V)

50

60

70

80

Figure 4.17. x0 with 3m/s free stream speed and 12 degree angle of attack. Since ∆CL is just a function of x, when the angle of attack is fixed,

∆CL = x∆CLmax

(4.6)

∆CLmax is the maximum stationary ∆CL (V, α). Three triangle waves with different frequency and a random binary signal input modulated at 0.7F + (k=2.14) pulse frequency and 0.12t+ pulse width excitation were used to test the nonlinear model for ∆CL prediction, the angle of attack was set to be 12o . The result of the model prediction is shown in Figure 4.18 to Figure 4.21. It is clear that the G-K type model ∆CL prediction fits the low frequency component of the experimental data well, which is shown in Figure 5.1. The low frequency value of ∆CL in experimental data is tracked by the model. At the beginning, the model prediction is not accurate because of the initial condition transient.

43

0.25

0.2

0.15

∆ CL

0.1

0.05 experimental data G−K model prediction

0

−0.05

−0.1

0

10

20

30

40

50

60

70

Voltage

Figure 4.18. ∆CL k=0.064 triangle wave G-K type model prediction, amplitude is 60V, α = 12o , the loop is going counterclockwise wise. The G-K type model also gives a good ∆CL prediction in 0.5Hz(k=0.13) triangle wave case, shown in Figure 4.19, which indicates accurate lift variation prediction can be obtained by the G-K type model for unsteady condition.

44

0.25 experimental data G−K model prediction 0.2

0.15

∆ CL

0.1

0.05

0

−0.05

−0.1

0

10

20

30

40

50

60

70

Voltage

Figure 4.19. ∆CL k=0.128 triangle wave G-K type model prediction, amplitude is 60V, α = 12o , the loop is going counterclockwise. Figure 4.20 shows that 1 Hz or k=0.26 is not the limitation of the G-K type model. The predicted ∆CL curve fits the mean value of the experimental data well.

45


0.2

∆ CL

0.15

0.1

0.05

0

−0.05

0

10

20

30

40

50

60

70

Voltage

Figure 4.20. ∆CL k=0.257 triangle wave G-K type model prediction, amplitude is 60V, α = 12o , the loop is going counterclockwise. As shown in Figure 4.21, the input voltage is also plotted on top of the ∆CL curves to demonstrate the delay and relaxation. Comparing the model prediction and the experimental data, we can tell that the delay and relaxation stage of the ∆CL variation is predicted so well, on the other hand, the value of τ1 and τ2 is chosen properly.

46 random binary input 0.5 experimental data voltage/200 model prediction

0.4

0.3

∆ CL

0.2

0.1

0

−0.1

−0.2

0

100

200

300

400

500

600

700

800

time(t+)

Figure 4.21. random signal, pulse amplitude=60V, α = 12o . The bandwidth for G-K type model is at least higher than k=0.257 since the model gives a good prediction in this case.

47 CHAPTER 5 DISCUSSION OF RESULT 5.1 Comparison of different low-dimensional models

The lift coefficient

predictions for the triangle wave signals with k=0.064, 0.128 and 0.257 frequency are used to compare the three models. The actuators are operated at F + = 0.63 excitation frequency with ∆tp = 0.12t+, which is the same as the triangle wave case mentioned at the beginning of Chapter 4. It should be pointed out that the judgment of the model validation is different for these three approaches, because the goal for black-box model and G-K type is to predict the low frequency lift variation, but not for the high frequency bumps produced by each single-pulse input signal. The convolution method is capable of predicting the ∆CL curve details. In k=0.064 case, shown in Figure 5.1, the G-K type model prediction gives the best fit for the experimental data curve. While, the black-box model and the convolution method do not. The black-box model over predicts the ∆CL curve at the maximum value of voltage and under predicts the ∆CL when the voltage is going up or down. While the prediction accuracy of convolution method is in between.

48

0.25

0.2

0.15

∆ CL

0.1

0.05

0 experimental data G−K model prediction 1st order tf with delay convolution with single pulse kernel

−0.05

−0.1

0

10

20

30

40

50

60

70

Voltage

Figure 5.1. ∆CL k=0.064 triangle wave, amplitude is 60V, α = 12o . The G-K type model still gives a good prediction with no surprise in k=0.125 case, which is shown in 5.2. But the black-box model gives a better prediction than the k=0.064 case. The reason is that ∆CL in k=0.125 case is more linear with voltage than that in k=0.064 case. It is hard to tell if the prediction of convolution is better for k=0.125 than k=0.064 case.

49

0.25

0.2

0.15

∆ CL

0.1

0.05

0 experimental data G−K model prediction 1st order tf with delay convolution with single pulse kernel

−0.05

−0.1

0

10

20

30

40

50

60

70

Voltage

Figure 5.2. ∆CL k=0.128 triangle wave, amplitude is 60V, α = 12o . In Figure 5.3, the result indicates that the G-K type model still has an advantage compared with the other two approaches, although the convolution model can not track the lift curve precisely, the trend predicted by it is better than the black-box model.

50

0.3

experimental data G−K model prediction 1st order tf with delay convolution with single pulse kernel

0.25

0.2

∆ CL

0.15

0.1

0.05

0

−0.05

0

10

20

30

40

50

60

70

Voltage

Figure 5.3. ∆CL k=0.257 triangle wave, amplitude is 60V, α = 12o . The black-box model does not work well for the triangle wave, because ∆CL does not increase linearly with the voltage, as was shown in section 4.3. Consequently, the over prediction and under prediction happens. The convolution does a better job than the black-box model, because the kernel contains the single-pulse lift information which makes it a linear but higher order system, or in other words, some nonlinear behavior information of the lift response for single-pulse is included in the kernel. The G-K type model fits the low frequency part of measured data better than any of the linear models. Because the nonlinear steady case behavior of ∆CL is taken care of by the function x0 (v), the delay and transient nonlinear behavior is handled by the state space function. 5.2 G-K type model prediction for AFC-pitching combination-open loop controlled Another nonlinear G-K type model to predict CL variation during pitching motion of the wing was introduced by Lou Grimaud [18]. The CL variation due

51 to the pitching motion can be calculated using angle of attack data for the wing. However, this thesis does not focus on the details of this model here, instead we will just use the result of this pitching G-K type model to investigate the relationship between pitching motion and AFC. Assuming the x0 curve is the same as the 12o angle of attack for different angle larger than the minimum stall angle, and the transient lift response for the actuation is independent of the pitching motion, then the total lift response could be the sum of the two components,

CLtotal = CLpitching + ∆CLAF C

(5.1)

Figure 5.4. Diagram of combined G-K type model.

Here, α(t) is the angle disturbance, and Gd is the angle disturbance model. The comparison of the model prediction and the experimental measurement is shown in Figure 5.6 to Figure 5.10. Here, the excitation input signal is a triangle wave while the pitching input is a sine wave. The frequency of both is the same, but we can vary the time delay between them manually. The angle and voltage input are plotted in Figure 5.5,

14

80

13

60

12

40

11

20

10

0

1

2

3 time(s)

4

5

6

voltage(V)

angle of attack(degree)

52

0

Figure 5.5. An example of open loop control input signal(1.6s delay case). With 0s delay between actuation input signal and pitching maneuver, shown in Figure 5.6, CL predicted by the combined G-K type model fits the experimental data reasonably well. Also, comparing to the other cases with different delay time, the actuation in the 0s delay case shrinks the hysteresis loop.

53

1.5 experimental data model prediction 1.4

1.3

CL

1.2

1.1

1

0.9

0.8 9.5

10

10.5

11

11.5

12 α

12.5

13

13.5

14

14.5

Figure 5.6. ∆CL prediction for 0s delay between pitching and excitation. Figure 5.7 shows the result of 0.4s delay time case. The combined G-K type model also gives a good CL prediction in this case, but the experimental data does not match the model prediction around 11o to 12.5o when the wing is pitching up. This also happened in 0.8s delay case, shown in Figure 5.8. Figure 5.9 and Figure 5.10 show that the model prediction fits the experimental data well. Since the only difference between these cases is a phase shift between the two signals, and correlated is with Figure 3.10.

54

1.25 experimental data model prediction

1.2 1.15 1.1

CL

1.05 1 0.95 0.9 0.85 0.8 9.5

10

10.5

11

11.5

12 α

12.5

13

13.5

14

14.5

Figure 5.7. ∆CL prediction for 0.4s delay between pitching and excitation.

1.5 experimental data model prediction 1.4

1.3

CL

1.2

1.1

1

0.9

0.8 10

10.5

11

11.5

12

α

12.5

13

13.5

14

14.5


55

1.4

1.3

1.2

CL

1.1

1


0.8

0.7 10

10.5

11

11.5

12

α

12.5

13

13.5

14

14.5


56


1.3

1.2

CL

1.1

1

0.9

0.8

0.7 10

10.5

11

11.5

12

α

12.5

13

13.5

14

14.5

Figure 5.10. ∆CL prediction for 1.6s delay between pitching and excitation. Starting with the G-K type model for pitching motion, a feed-forward controller was built to suppress the lift variation due to a disturbance of angle of attack using AFC. The objective of the controller is to maintain a desired CL = 1.1. Using the acquired incline angle data and input voltage to the actuators, shown in Figure 5.11 and Figure 5.13, CL predicted by combined G-K type model is plotted in Figure 5.12 and Figure 5.14. The result shows that the model is good even for random pitching with feed-forward control case.

20

100

10

50

0

0

5

10

15

voltage(V)


57

0 25

20

time(t+)

Figure 5.11. periodic pitching input signal with feed-forward control

1.25

model prediction experimental data

1.2

1.15

CL

1.1

1.05

1

0.95

0.9 10

10.5

11

11.5

12

α

12.5

13

13.5

14

14.5

Figure 5.12. periodic pitching input with feed-forward control

14

80

13

60

12

40

11

20

10

0

20

40

60

80

100

120

voltage(V)


58

0 140

+

time(t )

Figure 5.13. random pitching input signal with feed-forward control

59


CL

1.15

1.1

1.05

1 120

140

160

180

200

220

240

260

time(t+)

Figure 5.14. random pitching input with feed-forward control In this section, the G-K type model used for calculating ∆CLAF C is the model defined at fixed 12o angle of attack. But based on the results in section 3.1.3, function x0 (v), the steady-map of separation proportion versus input voltage, is not the same for different angle fo attack, so the G-K type model for the actuation is also different for different angle of attack. So the assumption made at the beginning of this section, x0 is independent of angle of attack can cause some prediction error for a pitching wing, although the error is small in these pitching-AFC combined cases.

60 CHAPTER 6 CONCLUSIONS 6.1 Summary Single-pulse actuation signal experiment shows ∆CL is strongly dependent on pulse amplitude and pulse width, but the pulse amplitude has more influence than pulse width on CL variation. The measured data shows that the peak value of ∆CL saturated at 0.12t+ pulse width for a fixed voltage, and at 80V for the fixed pulse amplitude case. The maximum ∆CL that can be obtained by single-pulse actuation is 0.342 when the angle of attack is 20o . The pulse interval or equivalently excitation frequency is a major factor in determining the lift increase, the maximum lift increase occurs between F + = 0.19 to F + = 0.56, the maximum ∆CL should be equal or larger than 0.5. The convolution integral model is not able to predict high frequency excitation and some relatively low frequency cases, depending on when the pulses are triggered. The G-K type model for active flow control can give a good prediction of the low frequency part ∆CL , even for k=0.26 triangle wave and the random binary case. Since x0 is obtained from experimental data, a specific x0 should be determined for each different active flow control system. The combined G-K type model prediction for CL is useful for the unsteady aerodynamic case, implying that the independent relation ship between pitching and active flow control is reasonable. 6.2 Future work In section 3.1.4, the jet speed is measured by a hot-wire, but pressure measurement is needed in future for the jet speed, because the hot-wire can only measure

61 the absolute value of the velocity but not the direction. Overall, the combined G-K type model makes a good CL prediction for most actuation-pitching combined cases. The assumption made in section 5.2, that the CL variation produced by active flow control is independent of the angle of attack, is reasonable for relatively small amplitude pitching motion. However, a more general G-K type model based on a function of both α and voltage(x = x(α, v)) need to be built for the arbitrary pitching-AFC combined motion.

62

APPENDIX A DATA PROCESSING

63 Servos position and the input control signal to the actuator of the AFC system were acquired with Matlab. The angle can be calculated by using the servo position data. Phase averaging method using a synchronized signal is applied to remove the random motion from the data and all the force data is filtered at 300Hz to remove the mechanical vibration produced by the AFC actuation. The processing code for single-pulse case with fixed α is listed below,

close all; clear all; clc; % define the parameters aoa=22.1; filter cutoff=300;

% angle of attack % multiplied by sampling frequency in filtering

filter cutoff2=300; f=2000;

% sample rate

% load data FB steady=daqread('FB u=3 aoa=20p0 pulse=0p31(steady) volt=60 afc'); FB off a=daqread('FB u=3 aoa=20p0 pulse=0p12(single)(3) volt=60 a'); FB off b=daqread('FB u=3 aoa=20p0 pulse=0p31(single) volt=60 b'); BNB off a=daqread('BNB u=3 aoa=20p0 pulse=0p12(single)(3) volt=60 a'); BNB off b=daqread('BNB u=3 aoa=20p0 pulse=0p31(single) volt=60 b'); raw FB=daqread('FB u=3 aoa=20p0 pulse=0p12(single) volt=60 afc'); BNB=daqread('BNB u=3 aoa=20p0 pulse=0p12(single) volt=60 afc');

% export pitot raw data and input voltage to the actuator

pitot=BNB(:,2); V=BNB(:,3);

64

% filter data [b,a] = butter(3,filter cutoff2/f,'low'); FB=filtfilt(b,a,raw FB);

% processing pitot off=mean((BNB off a(:,2)+BNB off b(:,2))/2); pit volt=(mean(pitot)); U=sqrt(abs((pit volt-pitot off)*15.275)*2/1.225); coef=0.1495*0.5*1.2041*(mean(U))ˆ2;

[L1,D1,M1]=transform17new(FB,aoa); [L off,D off,M off]=transform17new(FB off a,aoa); [L steady,D steady,M steady]=transform17new(FB steady,aoa);

DCL raw=(L1-mean(L steady))/coef; CL raw=(L1-mean(L off))/coef;

% phase average and alignment [phase1 pulse,L phased,X,Y,pot wing Array,FB Array]= SIPA(BNB(:,4),L1,0,10);

%%BNB(:,3) for 0.01t+ else BNB(:,4)

[phase1 pulse,V phased,X,Y,pot1 wing Array,FB6 Array]= SIPA(BNB(:,4),BNB(:,4),0,10);


[phase1 pulse,CL p phased,X,Y,pot1 wing Array,FB6 Array]= SIPA(BNB(:,4),CL raw,0,10);


for i=1:length(V phased) %

if V phased(i)>0.01*(max(V phased))%%%%%%%%%%for e if V phased(i)>0.01*(max(V phased))%%%%%%%%%%for b m=i;

%

break end

%%%%%%%%%%%%%%%%%%%%%%no break for e ,

break for b

65

end %% Lc phased=circshift(L phased,(length(V phased)-m)+40); Vc phased=circshift(V phased,(length(V phased)-m)+40); CLc p phased=circshift(CL p phased,(length(V phased)-m)+40); Lc phased=Lc phased-mean(L steady);

% plot data figure(1); plot((1:length(CL m))/(2000*0.08),CL m); hold on; plot((1:length(CL m))/(2000*0.08),Vc phased/10/4,'r'); xlabel('t+'); ylabel('\Delta C L'); title('\Delta C L 20deg singal pulse'); legend('\Delta C L 0.31t+');

The single-pulse code can be used For the multi-pulse and triangle wave case just changing the loaded files. For the pitching combined with AFC cases,

clear all;close all;clc;

%define the parameters f pitch=0.5; f=2000; filter cutoff=10; filter cutoff FB=100; alpha mean=12.0; U=3;

66

coef=0.1495*0.5*1.2041*(mean(U))ˆ2; offset angle=0.4; %0.45 for 1hz

%data loading u='3';

%mean tunnel velocity [STRING]

aoa='12(1st order)'; amp='2';

%mean wing angle of attack [STRING, NO PERIOD]

%actuation pulse width in

non-dimensional t+ (t+=t/(chord/velocity)) freq='0p5'; dCl='0p17';

%pulse amplitude in volts %maximum voltage

AFC='on';

name=strcat('u=',u,' meanaoa=',aoa,' amp=',amp,' freq=',freq, ' dCl=',dCl,' AFC=',AFC);

BNB=daqread(strcat('BNB ',name,' pitch')); BNB off a=daqread(strcat('BNB ',name,' a')); BNB off b=daqread(strcat('BNB ',name,' b'));

FB=daqread(strcat('FB ',name,' pitch')); FB off a=daqread(strcat('FB ',name,' a')); FB off b=daqread(strcat('FB ',name,' b'));

% calculate angle of attack %% y1=BNB(:,6); y2=BNB(:,7); Dy v=y1-y2; Dy=0.0292*(Dy v+offset angle);%0.465 Dy=Dy-0.01088;

alpha=interp1(delta y,alpha ref,Dy);

67

[b,a] = butter(3,filter cutoff/f,'low'); alpha=filtfilt(b,a,alpha);%%%%%%%%%%%%%%%%%%%% %% y1 off a=BNB off a(:,6); y2 off a=BNB off a(:,7); Dy v off a=y1 off a-y2 off a; Dy off a=0.0292*(Dy v off a+offset angle); Dy off a=Dy off a-0.01088;

alpha off a=interp1(delta y,alpha ref,Dy off a);

[b,a] = butter(3,filter cutoff/f,'low'); alpha off a=filtfilt(b,a,alpha off a); %% y1 off b=BNB off b(:,6); y2 off b=BNB off b(:,7); Dy v off b=y1 off b-y2 off b; Dy off b=0.0292*(Dy v off b+offset angle); Dy off b=Dy off b-0.01088;

alpha off b=interp1(delta y,alpha ref,Dy off b);

[b,a] = butter(3,filter cutoff/f,'low'); alpha off b=filtfilt(b,a,alpha off b);

% lift calculation [L1,D1,M1]=transform17new(FB,alpha+2); [L off a ,D off a ,M off a]= transform17new(FB off a(1:length(alpha off a),:),alpha off a+2); % [L off a0,D off a0,M off a0]= transform17new(FB off a0,alpha mean);

68

[L off b ,D off b ,M off b]= transform17new(FB off b(1:length(alpha off b),:),alpha off b+2);

%phase averaging ref=filtfilt(b,a,BNB(:,1)); ref a=filtfilt(b,a,BNB off a(:,1)); ref b=filtfilt(b,a,BNB off b(:,1));

[phase1 pulse,L1 phased,X,Y,pot1 wing Array,L Array]=SIPA(ref,L1,0,2); [phase1 offa pulse,L off a phased,X,Y,pota wing Array,L offa Array]= SIPA(ref a,L off a ,0,10); [phase1 ofba pulse,L off b phased,X,Y,potb wing Array,L offb Array]= SIPA(ref b,L off b ,0,10); [phase1 pulse,V phased,X,Y,pot1 wing Array,V Array] =SIPA(ref,BNB(:,4),0,10);

[phase1 pulse,alpha phased,X,Y,apot1 wing Array,a Array]= SIPA(ref,alpha,0,5); [phase1 offa pulse,alpha off a phased,X,Y,apota wing Array,a offa Array] =SIPA(ref a,alpha off a,0,10); [phase1 ofba pulse,alpha off b phased,X,Y,apotb wing Array,a offb Array] =SIPA(ref b,alpha off b,0,10);

L phased=L1 phased(1:m)-L off phased(1:m);

Lc phased=circshift(L phased,151); alphac phased=circshift(alpha phased,151); Vc phased=circshift(V phased,151);

69

APPENDIX B CODE OF G-K TYPE MODEL FOR AFC

70

close all; clear all; clc; U=3;

%air spped

c=0.245;%chord fs=2000; %sample rate aoa=12; % mean angle of attack p i=0.14;%pulse interval (s) period=2;%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% period in seconds change with different period nc=8;

%number of cycle

load('60s base triangle 80V'); %load steady data for AFC

%linespace npoints=nc*period*fs;

f=1/period;

% frequency in Hz

t=linspace(0,nc/f,npoints);

dt=mean(diff(t));

alpha=12-A*cos(2*pi*f*(t+0.8))'; % creating angle of attack data

% creating AFC voltage input data for j=0:period*fs:600000-period*fs for i=(1+j):(period*fs+j) if i