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An empirical platform for optimal placement of open-loop microjet-in-crossflow actuators

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In this study, a novel technique to experimentally determine the most effective actuator locations in an active flow control (AFC) context is implemented and its efficacy demonstrated through detailed measurements of flow response to control. The platform developed provides an engineering solution to the problem of actuator placement with commonly available components, by using an optimizing algorithm to explore the parameter space with a wind tunnel in the loop. The optimizing algorithm, specifically a genetic algorithm, sequentially activates the microactuators in different patterns and evaluates a cost function related to each pattern through measurements in the flow field, which enables the exploration of thousands of actuator patterns, thereby rapidly converging to viable configurations. The platform is tested in a canonical bluff body flow, the cylinder with a slanted afterbody at a slant angle of 45º, which is known to produce a high drag wake. The optimization process found an actuator arrangement that is markedly different from those obtained through ``experience-informed'' actuator arrangements. This non-intuitive, but optimal solution reduced the vortex circulation - a surrogate for drag reduction - by 10.6%, which was 3x the maximum circulation reduction through traditional actuator arrangements. A posterior clean-up procedure revealed a pattern that can potentially be interpreted in terms of fundamental flow control physics. The substantial improvement in AFC-enabled performance makes this platform and approach worth exploring for other aerodynamic applications.
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An empirical platform for optimal placement of open-loop
microjet-in-crossflow actuators
Fernando Zigunov, Prabu Sellappan, and Farrukh S. Alvi
Florida Center for Advanced Aero-Propulsion (FCAAP), Department of Mechanical Engineering, FAMU-FSU College
of Engineering (Tallahassee, FL, 32310)
In this study, a novel technique to experimentally determine the most effective actuator
locations in an active flow control (AFC) context is implemented and its efficacy demonstrated
through detailed measurements of flow response to control. The platform developed provides
an engineering solution to the problem of actuator placement with commonly available com-
ponents, by using an optimizing algorithm to explore the parameter space with a wind tunnel
in the loop. The optimizing algorithm, specifically a genetic algorithm, sequentially activates
the microactuators in different patterns and evaluates a cost function related to each pattern
through measurements in the flow field, which enables the exploration of thousands of actuator
patterns, thereby rapidly converging to viable configurations. The platform is tested in a canon-
ical bluff body flow, the cylinder with a slanted afterbody at a slant angle of φ=45°, which
is known to produce a high drag wake. The optimization process found an actuator arrange-
ment that is markedly different from those obtained through “experience-informed” actuator
arrangements. This non-intuitive, but optimal solution reduced the vortex circulation - a surro-
gate for drag reduction - by 10.6%, which was 3×the maximum circulation reduction through
traditional actuator arrangements. A posterior clean-up procedure revealed a pattern that
can potentially be interpreted in terms of fundamental flow control physics. The substantial
improvement in AFC-enabled performance makes this platform and approach worth exploring
for other aerodynamic applications.
I. Nomenclature
A= Area of the vorticity integration [m2]
CD= Drag Coefficient [-]
Cµ= Microjet momentum coefficient [-]
Cπ= Microjet power coefficient [-]
®
ds = Differential vector along the path of integration [m]
D= Cylinder diameter [m]
f= Actuator frequency [Hz]
J= Cost function [-]
L= Slanted Section Length [m]
npt s = Number of points in the circulation contour [-]
P= Microjet mechanical power [W]
PB= Microjet back-pressure [psig]
r= Radius of the circulation contour [m]
ReD= Reynolds number based on cylinder diameter [-]
®
V= Velocity vector [m/s]
V= Free stream velocity [m/s]
x= Spanwise coordinate [m]
y= Vertical coordinate [m]
z= Streamwise coordinate [m]
Graduate Research Assistant, Mechanical Engineering, 2003 Levy Ave, AIAA Student Member
Research Faculty, Mechanical Engineering, 2003 Levy Ave, AIAA Member
Professor, Mechanical Engineering, 2003 Levy Ave, AIAA Associate Fellow
1
∆Γ = Change in vortex circulation [m2/s]
Γ= Vortex circulation [m2/s]
Γ1= Geometrical vortex core tracking quantity [-]
ωz= Dimensional vorticity [1/s]
φ= Slant Angle [deg]
II. Introduction
Active
flow control is a promising technology that displays a high potential to improve aircraft performance and
operational envelope, possibly playing a key role in the next generation of aircraft designs. It has been proven to
be highly effective in changing flow topology and integral quantities such as drag, for flow control scenarios ranging
from airfoils [1–3] to bluff body wakes such as the ones produced by simplified models of vehicles [4–6]. Active flow
control also presents desirable characteristics, such as the ability to be disabled during certain flight conditions, greatly
minimizing its negative effect on drag and lift, in contrast with passive flow control solutions such as vortex generators.
However, one of the many challenges that prevents wide usage of this technique in engineering applications is related
to the difficulty in identifying the most effective locations and patterns for actuator placement as well as their operating
conditions - amplitude and, for unsteady actuators, frequency and duty cycle - such that they have the most impact on
the flow. For the more complex flow topologies found in practical aeronautical applications, the problem is still open.
The challenge in predicting the outcomes produced by actuators is partly related to the complex interactions that occur
at higher Reynolds numbers, and the non-linearity of the Navier-Stokes equations. Theoretical approaches based on
stability analysis have been used until recently, in order to tackle this problem. Stability analysis of small perturbations
of flow fields [
7
], for example, has been shown to have some degree of success in explaining phenomenological
observations, especially at low Reynolds numbers. A computational approach that has gained traction in recent years is
the resolvent analysis [
8
,
9
] which, like other competing approaches, also relies on moving the non-linearity source away
from the formulation by considering it a “forcing term”, which limits the outcomes produced by the actuators to ones
that cause minimal changes to the mean flow. The recent effort by Liu et al.
[10]
on solving a full LES cavity model
with synthetic jet-like forcing based on the resolvent modes does suggest, however, that the technique has potential to be
useful in tackling the spatial sensitivity problem.
Despite the obvious advantages of computational approaches to solving the problem of effective flow control
actuator placement, such as high scalability and reduced number of iterations in experimental testing, the aerodynamics
community also needs an increased variety of practical solutions that together can move the active flow control technology
forward to the engineering context. To satisfy these needs, many experimental researchers employed approaches that
are loosely inspired by theory, which in this study will be denoted as “informed ad-hoc”. These techniques attempt to
exploit well-known simple instabilities a flow might possess and are based on intuition and experience of the researchers
involved. They have proven to be very useful in advancing the field of active flow control and demonstrated many
times that it is possible to achieve net drag/power reduction [
11
,
12
] and improved performance of other aerodynamic
quantities such as jet noise [
13
,
14
] or operational envelope [
3
,
15
]. However, there is still a high degree of trial and error
involved, since only a handful of actuator patterns can be deployed in any single study. Therefore, this approach could be
greatly enhanced with emerging technologies like machine learning and model-free optimization, as long as a computer
has access to changing the actuator pattern and its parameters with the wind tunnel in the loop. Very recent experimental
studies [
16
,
17
] showed that machine learning and optimization techniques can be applied directly in the experimental
context, enabling the exploration of a large parameter space with thousands of data points in a short directed study. This
type of experimental approach is currently more viable, in the economic sense, than its computational counterpart (i.e.,
simulating thousands of actuator combinations). One of the reasons for the cost imbalance is related to the enhanced
multi-scale nature of the microactuator placement problem; in order to accurately simulate the effect of the microactuator
array in a given high Reynolds number flow field, it is critical to spatially resolve both the large flow scales related to the
model scale as well as the microactuator scales, which currently is prohibitively expensive and a research topic by itself.
In this study, the authors propose a novel approach, expanding on the technique used by Duriez et al.
[17]
, that is
capable of experimentally exploring thousands of actuator array patterns experimentally in a short period of time. This
enables the timely attainment of an engineering solution to the flow control problem that can be used in many complex
flow fields where the most effective active actuator locations are not intuitive. In contrast to past studies using machine
learning approaches [
16
,
17
], which do not tackle the actuator placement problem, this study will focus on the problem
of experimentally selecting the most effective spatial pattern of microactuators by using optimization. The cylinder with
a slanted afterbody, an aerodynamic model with a known, actively studied bluff body wake will be employed as the test
2
subject to demonstrate this novel technique.
III. Experimental Details
A. Facility and Model Details
The experiments described in this study were performed in the Low Speed Wind Tunnel facility (LSWT) at the
Florida Center for Advanced Aeropropulsion (FCAAP) of FAMU-FSU College of Engineering. The facility is an
open-circuit wind tunnel with a square test section of 0
.
762 m side and is capable of free stream velocities of
V=
2
.
5
m/s to
V=
70 m/s. The model employed in this study is a cylinder with a slanted afterbody, displayed in Figure 1 (a).
This model was chosen because it presents a complex bluff body wake that contains a strong pair of counter-rotating
vortices.
The model diameter was
D=
0
.
146 m, with a straight section of length 2
D
and a blunt nose with an ellipsoid shape
of 2:1 ratio. The model rear slant angle was
φ=45°
, which is known to produce a very strong counter-rotating vortex
pair and a drag coefficient of
CD
0
.
6[
18
]. The free stream velocity was chosen to be
V=
10
.
3m/s, which results in
a Reynolds number based on the cylinder diameter of
ReD=
1
×
10
5
. This Reynolds number is very close to the one
used for the drag measurements reported by Morel [18].
The slanted cylinder was supported by a steel tube, necessary to pass the cables and pneumatic tubing to control and
feed the microjets. The tube was shrouded with a NACA0020 airfoil of thickness 22 mm, in order to minimize its impact
in the wake of the model. The wake of this model is described in detail with three-dimensional PIV measurements in
Zigunov et al. [19].
B. Actuator Array Construction and Optimizer Setup
The platform developed in this study was designed to enable the control computer used for optimization to have
access to changing the actuator configuration with as much freedom as possible. In order to do so, the system displayed
schematically in Fig. 1(b) was built. The optimization computer, running a Matlab
®
genetic algorithm script, sends
serial commands to a 2-D CNC traversing mechanism that holds a Turbulent Flow Instrumentation four-hole (“Cobra”)
probe. The four-hole probe is, for every case, traversed around the right (as seen from upstream of the model) vortex
core centerline in a circular path on a spanwise plane with 40 points, as displayed in Fig. 1 (d). The choice of number of
points will be detailed in Section IV.A. The probe was positioned at 1
D
downstream of the trailing edge. Only one
vortex was traversed, to reduce the experimental time necessary to compute the cost function. Each data point comprises
of 1.6 seconds of data, and the four-hole probe pressure measurements allow for the computation of the time-averaged
three-dimensional velocity vector at each traversed point. This enables for the measurement of the vortex circulation, by
integrating:
Γ=C
®
V·®
ds (1)
Circulation was shown by Bulathsinghala et al.
[20]
to be directly proportional to the drag coefficient for the slanted
cylinder model for various slant angles with a negligible zero-crossing of
CD0=
0
.
01, meaning changes in circulation
should translate directly to changes in drag. The circulation was elected as the cost function variable, instead of more
direct choices (such as force balance measurements), due to the higher accuracy of the measurements offered by the
four-hole probe. The measurement uncertainty in circulation is estimated to be
±
0
.
5%. The cost function then was
defined as:
J=
Γje xel Γb asel ine
Γbas eli ne
=
∆Γ
ΓCD
CD
(2)
Where
Γbas eli ne
is the vortex circulation with all “jexels” off and
Γje xel
is the circulation for the particular
“jexel” pattern being analyzed. The optimization computer had access to the air supply pressure through a digitally
controlled pressure regulator (Bellofram 961-085-000). It also had access to 100 solenoid valves (Matrix Pneumatix
DCX.321.1E3C2.24) connected to the pressurized air manifold. The valves are capable of toggling the air flow at
frequencies up to 200 Hz, which was set as the upper limit of the frequency range accessible to the optimizer. Each
solenoid was controlled by an individually generated, parameterized square waveform produced by the custom electronics
board shown in Fig. 1 (a1). The board, driven by a PIC32MZ2048EFH144 microcontroller, was custom designed to
3
Probe circles
around vortex core
z
y
t
(a)
(a1) Signal generator (a3) Solenoid manifold
(a6) Jexel array:
59 individually
addressable
actuators
(a5) Tubing
(a2) Cabling to solenoids
(a4) Air Inlet
D
D 2.4D
Vortex core
centerline
DAQ
Solenoid Manifold
(100 valves)
SigGen/Driver
USB/Serial
Air
Supply
(b) D
(c)
Jexel (4x 0.4mm jet)
6 mm
SPIV plane
r4-hole probe
path
(d)
Fig. 1 (a) Model drawing and coordinate system. (b) Schematic of the arrangement of components used in this
experiment. (c) Detail of the 3D printed “jexel”. (d) Position of the PIV plane and detail of the 4-hole probe
path displayed over a sketch of the known baseline flow field
fit inside the aerodynamic model, given the necessity of an individual power cable for every solenoid that would be
impossible to route through the supporting strut tube. The microcontroller is capable of simultaneously generating 108
independent digital waveforms sampled at a rate of 50 kS/s that can be parameterized through a USB-Serial connection
to the optimization computer.
A back-plate containing the micro-actuators was manufactured through the SLA 3D printing process and is shown
in Fig. 1 (a6). It contained 59 pneumatic channels, each of which contains four 0.4 mm diameter microjets arranged in a
square pattern of 6 mm side. The plate was manufactured on a Formlabs Form 2 3D printer, and extreme care was taken
to ensure every single jet was completely clear of resin prior to post-cure. Each pneumatic channel will be referred to in
this manuscript as a “jexel”, short for “jet pixel”, in a reference to how computers build image patterns in a similar way
we seek to define the most effective pattern of jets in the model surface. A detail of the “jexel” is given in Fig. 1 (c).
The “jexel” is therefore defined as a group of four microjets, which was a trade-off design decision taken to cover a
larger surface of the aerodynamic model with the same number of pneumatic channels.
C. Genetic Algorithm Overview
An optimizer based on the Genetic Algorithm (GA) was implemented in this study. The usage of a GA instead
of other optimization techniques was motivated by the very large parameter space to be covered and the presence of
significant uncertainty in the measurements. The GA is, in general, an algorithm that is robust to these aspects of the
parameter space. A population size of 30 individuals was chosen due to the long time necessary to evaluate the cost
function. The standard operations of the GA, i.e., Elitism, Mutation and Crossover were applied every generation. An
elite fraction of 40% was chosen and the mutations randomly added/removed/shuffled “jexels” and their properties. The
details of the implementation are given in Appendix A.
4
20 min. warm-up
0.1 0.2 0.3 0.4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
/ (V D)
0 50 100 150 200
Npts
0.76
0.765
0.77
0.775
0
-0.77
-0.768
-0.766
Experiment Time [hours]
63
(a) (b) (c)
Fig. 2 Convergence of circulation as a function of (a) core radius r/Dfor npt s =
60
and (b) number of
measurement points npt s for r/D=
0
.
274
(vertical scale is magnified in (b) and (c)). Blue points indicate
parameters chosen for optimization experiments. (c) Long-term stability of the circulation of the model. Dashed
line in (a) is the half-distance between vortex cores
The problem explored in this study is of a combinatorial nature: choosing any number of “jexels” between 5 and
25, out of the available 59 positions, yields
86
×
10
16
possible patterns. By including the possible values for the
remaining variables, a parameter space of the size of
10
21
was expected. For the experimental time available, it was
expected that about
3000 patterns would be viable to explore - a tiny fraction of the parameter space. To deal with
this issue, the parameters on the GA (described in Appendix A), were on the aggressive side to foster breadth of the
search. The results obtained, however, suggest that the parameter space is somewhat smooth and that some degree of
superposition might be applicable - i.e., certain features of the patterns found seem to have separable functionality.
The cost function was also kept simple - greedily maximizing the changes in circulation. A posterior clean-up
process was applied to improve the interpretability of the results obtained. The clean-up procedure is described in
Appendix B.
D. Stereoscopic Particle Image Velocimetry (SPIV)
For the best configuration obtained through the GA and a few of its variants, SPIV was performed at a spanwise
plane at the same downstream location as the four-hole probe (1
D
from the trailing edge, as shown in Fig. 1 (d)).
SPIV served the purpose of confirming that changes observed with the four-hole probe were accurate depictions of the
changes in the vortex characteristics and provide a supporting measurement for the results obtained. It was performed
with a LaVision PIV system, with two sCMOS cameras looking at the plane of interest from upstream at an angle of
44°
. The cameras were fitted with a Nikkor 50 mm lens and a Scheimpflug adaptor. The laser sheet was produced by
a Quantel Evergreen Nd:YAG double-pulse laser with a pulse energy of 200mJ, and the PIV images were acquired at 15
Hz. Acquisition, storage, de-warping, stereo self-calibration and post-processing of the images into vector fields was
performed in the LaVision 8.4 software. Sets of 500 vector fields were obtained for each case examined, achieving a
vector spacing of 0.62 mm/vector with a 32
×
32 px correlation window size and 75% overlap. The uncertainty of the
mean vectors obtained was <2%.
IV. Baseline Measurements and Characterization of Low Frequency Oscillations
A. Circulation Convergence
To determine the number of points and the radius of the circle where the circulation is being evaluated in the physical
domain, a set of experiments was performed to assess the convergence of this metric as a function of these two variables
(
npt s
and
r
). The measurements of baseline circulation are shown as a function of these parameters in Fig. 2. As can be
observed in Fig. 2 (a), convergence of the circulation as a function of radius occurs at about
r/D
0
.
25. As a function
5
of number of points, the circulation shown in Fig. 2 (b) is already close to its converged value even for
npt s =
20, as can
be noted by the magnified circulation scale in (b) and the 0.5% error bar shown in blue. The number of points
npt s
in
which circulation is computed should be minimized to increase the number of experiments that can be accomplished
during a given amount of time, and therefore the parameters chosen for the experiments presented in this study are
r/D=0.274 and npt s =40.
B. Low Frequency Oscillations
When performing optimization in real systems where long term experiments will be conducted, extra care must be
taken to ensure the conditions under which the cost function is being computed are not changing too much over large
time periods. If this is the case, the optimization algorithm cannot differentiate between changes in the cost function
achieved by real improvements from variations in the environment that also affect the cost function. Obviously, this can
negatively affect the optimization process and lead to null results. Therefore, it is imperative to assess this effect and the
viability of implementing the four-hole pitot probe setup to measure circulation. An experiment to quantify the drift in
the facility over time was therefore performed. The wind tunnel was started at the fan frequency that produced the
desired Reynolds number (
ReD=
1
×
10
5
) and the traversing mechanism was set up to acquire 40 samples of velocity
evenly distributed along a circumference of radius
r/D=
0
.
274 for the computation of circulation, since it was observed
(Section IV.A) that these parameters are sufficient for the convergence of the quantity of interest.
A complete traversal of the 40 points took about 126 seconds per case, where 1.6 seconds per point corresponded
to data acquisition and the remaining delay (
1.5 s) was introduced to let the mechanism reach the target position,
preventing acquisition while in motion. This interval affords 685 data points per day of experimentation, which was
deemed sufficient for reaching an engineering solution to this problem. In this experiment, no “jexels” were activated,
instead simply observing the variations in the baseline over the course of 7 hours. The results obtained are displayed in
Fig. 2(c). A warm-up period of about 20 minutes is observed, where the data is believed to have increased variability
while the mechanical and electrical components of the fan and its driver electronics reach thermal equilibrium, slightly
changing the free stream velocity. After the warm-up period, the measured circulation has a standard deviation of 0.15%
of the mean value (excluding the shaded zone), meaning an uncertainty bar of 0.5% covers three standard devpiations
associated with the low frequency variations in the facility. The circulation mean value of the baseline case obtained by
the four-hole probe measurement,
Γ4H/(VD)=
0
.
767, was very close to the value later obtained in PIV measurements
ΓPI V /(VD)=0.79, where ΓP IV =AωzdA was obtained by vorticity integration.
V. Manually Selected “jexel” patterns
In order to initialize the GA with a meaningful starting point and ensure the patterns selected were free of hindsight
bias, the first experiment performed was a parametric scan with manually selected “jexel” patterns. The parameter scan
was coarse, including five frequencies (
f={
5
,
50
,
100
,
150
,
200
}
Hz) and two back-pressures (
PB={
5
,
10
}
psi) for
each pattern. The duty cycle was kept fixed at 50% and the phase angle was fixed at
0°
. A total of 17 patterns were
examined, and the 10 best patterns are displayed in Fig. 3 organized by their relative circulation reduction with respect
to the baseline case. In the plot area, each experiment is represented by a black dot, where the uncertainty is shown as a
blue errorbar for reference. The gray box highlights the extents of the changes observed by varying the parameters
previously described. The best two patterns potentially excited the shear layer prior to separation at the sharp edge of the
model. The configuration M1 had the best performance for the actuation frequency of
f=
200 Hz and back-pressure of
PB=
10 psi, however the performances of M1 and M2 are indistinguishable given the blue uncertainty bar of
±
0.5%.
The configurations M3 and M4, however, only accomplish half as much as M2, indicating it is their combined effect that
leads to M2’s performance. The configurations M8 and M10, where the blowing occurs at the boundary layer in a line
of “jexels”, had a less significant effect. M5 and M7, on the other hand, seemed to be slightly more effective than M8
and M10 with their staggered arrangement. It is possible that their effect on the boundary layer can be compounded to
improve M2.
In general, the higher frequency cases scored larger circulation reductions, however it was later found (as described
in Section VII) that the most likely reason for the improved reduction in circulation is the increased momentum flux
of each “jexel”, as the solenoid reaction time became an increasingly larger fraction of the cycle period. At this
stage, however, a steady blowing case was not examined. It is worthwhile mentioning, however, that even this manual
optimization phase can be extremely useful in the wind tunnel testing context, since hundreds of configurations can be
planned and rapidly deployed in a short experimental campaign. From the 170 cases explored, the best 30 configurations
(including their frequencies and back-pressures) were fed as the initial generation of the GA.
6
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
Flow
[%]
-1
0
1
2
3
4
- /
Fig. 3 Relative reduction in circulation measured in a parametric scan (frequency, back-pressure) for manually
selected “jexel” patterns. Gray bars represent the range of changes obtained for all parameters, each dot being
one experiment. Below the x-axis, the corresponding 10 best “jexel” patterns are shown. Grayed dots in the
inset figures represent “jexelsthat were not enabled, and blue dots represent enabled “jexels”
VI. Results from the Genetic Algorithm
The GA described in Section III.C was deployed in this phase of the experiments. The evolution of the results for
51 generations, corresponding to a total of 1530 “jexel” patterns, is summarized in Fig. 4. The manually optimized
seed population of Generation 1 (G1) was re-evaluated, this time observing a circulation change of 3.5% for the seed
configuration M2. As discussed, this is the same change observed in the manual optimization step - given the uncertainty
in the measurements. This configuration persisted as the best configuration from G1 to G5. In G6 the first “breakthrough
was observed, and subsequent “breakthroughs” were observed in G15, G19, G20, G27 and G30. Due to the random
nature of the GA, the patterns produced look very noisy and indistinguishable from each other at a first glance. A
few observations can be made, however: First, the GA seemed to favor patterns that involve activating some of the
“jexels” just upstream of the slant edge, along the cylindrical surface, mostly at the center plane. Secondly, “jexels” just
after the slant edge, farther from the center plane, also seemed to be deactivated more often. Finally, the GA preferred
actuation frequencies close to the upper limit (200 Hz). Even though the patterns look noisy, there is a clear motivation
to understand them further, since their performance was three times better than the best manually selected configuration,
with a similar number of active “jexels”.
VII. Clean-up Process and Interpretability of the Results
A clean-up step is as important as the GA optimization step when the cost function does not incentivize the algorithm
to seek solutions with a lower actuator count. Here, deactivation of single “jexels” is iteratively performed, starting with
the best pattern found at G51 (which is the same pattern of G30). Since the initial configuration had 21 “jexels” active,
a total of
21
n=1n=
231 evaluations were necessary to complete the clean-up procedure. Further details on this process
are given in Appendix B. The iterative deactivation allows for definition of which “jexels” are effectively contributing
to the results obtained, versus “jexels” that were left active because their contribution to the overall performance is
neither positive nor negative - in which case they are only consuming actuation power.
Prior to the clean-up, it is important to address the fact that the GA preferred higher frequencies and higher duty
cycles in the solution. This observation led the authors to believe the algorithm exploited the bandwidth limits of the
solenoid valve setup. If the momentum flux was the underlying variable that increased performance, it is far more likely
- given the randomness of the GA - that the GA will accomplish increases in average momentum flux by increasing
both the frequency and duty cycle than by associating the blowing frequency to exactly 0 Hz. Since the frequency was
defined as an integer-valued variable, there was a 1/200 chance of using a steady blowing frequency, which is much
7
0 10 20 30 40 50
-2
0
2
4
6
8
10
12
- / [%]
Best Indiv.
All Individuals
G1-G5
-3.5%
Flow
G6
-5.0%
G7-G15
-5.7%
G16-G18
-5.8%
G19
-7.2%
G20-G26
-8.8%
G27-G29
-10.2%
G30-G50
-11.2%
Flow
Fig. 4 Evolution of reduction in circulation as a function of GA generation. Inset figures are top views of the
model, showing the breakthroughs in the jexel patterns as generations advanced. Breakthroughs also indicated
in the chart as red arrows. Grayed dots in the inset figures represent jexels that were not enabled, and blue dots
represent enabled jexels.
Flow
16
1
2
3
5
6
4
7
8
9
10
11
13
14
15
17
18
19
20
21 12
1 5 10 15 20
-1
0
1
2
3
Effectiveness [%]
Flow Flow
Most effective 8 jx
ΔΓRR=-10.4%
(c)(b)(a)
ΔΓLL=-6.8%
L R
L R
Symmetric config. (12jx)
ΔΓRR=-10.6%
ΔΓLL=-9.5%
Fig. 5 (a) Clean-up process of the “jexel” patterns obtained in Generation 51. Each “jexel” was associated
with the measured change (b) in circulation ∆Γ incurred by its deactivation. In (c), elected configuration for
SPIV and measurements obtained for left (L) and right (R) vortex circulation
8
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-1.1
-1
-0.9
-0.8
-0.7
(a) Baseline
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-1.1
-1
-0.9
-0.8
-0.7
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-1.1
-1
-0.9
-0.8
-0.7
(b) Asymmetric 8jx
(c) Symmetric 12jx
Flow
D/V
(d) Right Core Vorticity Distribution
-30 -20 -10 0 10 20 30
|
D/V
|
Fig. 6 (a-c) Vorticity fields observed through SPIV for the three configurations examined in more detail. Black
stars indicate vortex core tracked by peak Γ1[21]. Circled yellow dots show baseline case vortex core location
superposed on (b) and (c) for reference. Red dashed line indicates position of the cylindrical body. In (d), a
comparison of the vorticity distribution as a function of radius for the right core between baseline (a) and the
symmetric “jexel” pattern (c) is shown
lower than the
1/4 chance of being in the upper end (150 to 200 Hz) where this phenomenon has a significant impact.
A posterior study with a mass flow meter confirmed that the the mass flow rate
Û
m200 Hz =
0
.
85
Û
mDC
at 50% duty cycle,
where Û
mDC =2.26 SLPM/jexel at PB=7.5psi.
Due to this observed behavior, the blowing frequency was reset to 0 Hz for the remaining part of this study. The
circulation change measured with the 4-hole probe with DC blowing was
∆Γ/Γ=
11
.
5%, which is effectively no
change compared to the original GA solution, given the measurement uncertainty (See Fig. 4). The results obtained by
the clean-up procedure are summarized in Fig. 5. The “jexel” pattern obtained in Generation 51 is shown in more detail
in Fig. 5(a), where the “jexels” are numbered according to their effectiveness (lowest number = highest effectiveness).
A high effectiveness “jexel” is defined as one that causes a large loss in overall performance when deactivated. For
example, the “jexel” ranked #1 (ID = 1) caused an increase in circulation of 3.8% when turned off, meaning it is the
most important “jexel” in the configuration. The “jexels” were then organized by this metric, as shown in Fig. 5(b).
The lowest ranked “jexels” were found to have a slightly negative contribution to the performance beyond measurement
uncertainty, i.e., their effect is of increasing circulation. Curiously, their location is very close to the most effective
“jexels”, which is a clear indicator of how the microjet-in-crossflow actuator placement problem is not straightforward.
The 8 most effective “jexels” were then picked for further study with Particle Image Velocimetry (PIV). Their
pattern is highlighted in both Fig. 5(a) as a thin outline and plotted explicitly in Fig. 5(c). An alternative symmetric
configuration was also evaluated with PIV, whose pattern is also shown in Fig. 5(c). The circulation values in Fig.
5(c) summarize the PIV observations, where the asymmetric pattern did affect the right vortex to a similar extent as
9
measured by the four-hole probe - to a 1% difference. The left vortex in the asymmetric configuration, however, was not
affected nearly as much. This evidences how the GA biased its solution to maximize the effect where the measurement
was performed - a cautionary of performing optimization with partial measurements that do not capture the complete
and relevant flow response. The symmetric pattern, however, corrects for that effect and the measured changes in
circulation are effectively the same for both vortices. Symmetry could have been enforced in the hardware design step;
however, the authors also were interested in confirming that asymmetric configurations are indeed less attractive.
The mean PIV vector field of the baseline case is presented in Fig. 6 (a) as streamlines overlaid on vorticity contours.
It is presented against the two actuated cases previously discussed in Fig. 6 (b) and (c). The PIV fields show a significant
displacement of the vortex core, as can be inferred from comparing the black stars against the circled yellow dots
(baseline core locations). For the asymmetric actuator case (b), the right vortex core (on the actuator side) was displaced
significantly more than the left core. As expected, the symmetric case presented a similar displacement of the core for
both sides. The displacement of the core was directed upwards by about 0
.
05
D
, which corresponds to
7
.
5mm in
physical space. It is consistent with the observed weakening of the vortices, since the induced velocity of one vortex
on the other is reduced, causing less vertical displacement of the cores as they move downstream. Interestingly, the
reduction in vortex strength does not seem to be related to increased diffusion of the core, since in the comparison of
vorticity distribution shown in Fig. 6 (d) an overall reduction in vorticity as a function of radius is observed. In other
words, the vortex is not being weakened by increasing the core size through diffusion, but by prevention of vorticity
production at the base of the body.
Up to the current stage of this study, however, it is not clear what exactly was the effect of the microjets in the flow
field that prevented vorticity from being produced. It is fair to conjecture that the jets in the cylindrical portion of the
model are manipulating the boundary layer. The jets downstream of the edge, at the flat section of the slanted surface,
might be substantially interfering with the separation bubble dynamics, perhaps shrinking it. Since the separation
bubble has been shown to be connected to the pair of vortices in this flow field [
19
], it is likely that changes in its
topology might affect the dynamics of the vortex pair, possibly even its formation process. The actuation strategy found
by the GA, however, is completely novel in the context of bluff body wake control and is not straightforward to define a
priori, especially when one considers that “jexel” IDs #18 and #20 in Fig. 5(a), which are also inside the separation
bubble, had measurably the opposite effect as “jexels” #1 and #3. More importantly, the solution obtained has proven to
be much more effective than the paradigm of line-shaped arrays of jets, which has become a de-facto standard when it
comes to “informed ad-hoc” studies [4, 6, 22–28].
It is important to acknowledge, however, that the solution obtained might still not be ideal from an engineering
standpoint. Due to the way the cost function was defined, the algorithm greedily reduced circulation, with complete
disregard to the amount of power necessary to operate the jets. The momentum and power coefficients (
Cµ
and
Cπ
) of
each “jexel” are given below:
Cµ=Jexel Momentum
Model Drag =
ρjet u2
jet Aj et
0.5CDρV2
Amode l
=5.3×103/jexel (3)
Cπ=Jexel Power
Model Drag Power =
0.5ρjet u3
jet Aj et
0.5CDρV3
Amode l
=1.93 ×102/jexel (4)
Where the values provided were obtained through a mass flow meter measurement to determine the “jexel” mass
flow rate, and correspond to a back-pressure of
PB=
7
.
54 psi, which was the back-pressure found by the GA. The
normalization by the model drag and drag power, respectively, is used to improve the physical interpretability. For
Cπ
, for example, the value obtained means every single “jexel” (i.e., group of 4 microjets) would have to reduce the
drag power by 1.93% to pay for its energy cost. Unfortunately, the solution found in this study did not attain net
positive power savings, since its cleaned-up, symmetric configuration employs 12 “jexels”, meaning it requires 23.2%
of the drag power to be operated, while only reducing drag by 10.6% (inferred from SPIV circulation measurements).
Therefore, the “jexel” configuration obtained in this study has an expected power
increase
of
P/P= +
12
.
6%. It
might be possible to further reduce the power requirement by having finer control over each of the four microjets within
each “jexel”, which would be feasible with increased complexity in the initial study or with a follow-up, targeted study.
Since the cost function defined at the start of this study did not include power savings as an objective, the fact that
the optimal configuration obtained did not achieve net power savings is not surprising. The optimization technique
demonstrated in this study, however, is likely capable of finding a net positive power savings solution provided that was
the initial objective of the algorithm.
10
VIII. Conclusions
In this study, a new type of test bed was developed and deployed to find a solution to the microjet-in-crossflow actuator
placement problem. A genetic algorithm was employed to explore the high dimensional, combinatorial parameter space.
About
2000 different “jexel” configurations were explored, along with an additional
600 configurations on auxiliary
studies to confirm the variables that were observed. The number of spatial “jexel” patterns explored in this study is by
far the largest ever experimentally accomplished in a single study, to the knowledge of the authors, which evidences that
this technique can add a new perspective to practical studies in actuator placement.
The initial focus of this study was an implementation in the slanted cylinder wake problem at an angle of
45°
, and
the results are very encouraging. The reduction in circulation measured, which is a proxy for drag in this model, was
of 10
.
6% as measured by PIV. The solution found by the Genetic Algorithm was about three times more effective
than the most effective solution (3.5%) defined manually with a similar number of active microactuators. This result
demonstrates not only the strength of the optimization approach, but also how important it is to have flexibility in the
location of the microactuators in this type of study. The physical mechanisms that the GA exploited to produce the result
observed are yet to be fully understood. It is possible that the main mechanisms are boundary layer energization and
modification of the separation bubble which, due to its strong coupling to the vortex pair could have prevented vorticity
from being generated at the base of the model. The solution found in this study did not generate net power savings,
as that was not the objective defined in this study. However, an appropriately defined cost function or objective will
likely yield such an outcome. More generally, the technique presented herein can be very useful for finding practical
engineering answers to the actuator placement problem, as well as improving the general understanding of the flow
control problem when complex base geometries are involved.
Appendix
A. Details on the Genetic Algorithm Implementation
The particular implementation of the GA used in this study defined a case genome with one continuously varying
parameter for the “jexel” back-pressure, 0
<PB<
10 psi, and 4 arrays of parameters of variable length:
JxAddresses
stores the addresses (1-59) of the “jexels” activated for that case;
JxPhases
stores the individual phase delays with
respect to a reference time,
JxDuties
stores the duty cycle used for each “jexel”. A single frequency,
JxFrequency
,
was used for all “jexels. All arrays had the same length, which corresponded to the number of “jexels” activated for
the corresponding case. A minimum length of 5 and a maximum length of 25 was defined, limited by the maximum
current capacity of the electronic board designed.
The genetic algorithm was then fed an initial population of 30 “jexel” configurations, which were the best cases
obtained after a parametric scan of manually selected patterns that followed the classical “informed ad-hoc” approach.
These patterns are further detailed in Section V. The population was kept constant at 30 individuals per generation. For
each individual, the cost function described in Equation (2) was measured and evaluated.
The baseline circulation,
Γbas eli ne
, was re-evaluated with all “jexels” deactivated after every 10 cases to make sure
any low-frequency trends in the baseline case were mitigated. A moving average of 10 baselines was used to reduce
sudden changes between the individuals’ cost functions.
Every generation, the genetic algorithm performed the operations described below. The specific values of the
probabilities and probability distributions were arbitrary and were kept unchanged during the experiment.
Elitism:
40% (12) of the best individuals (least
J
) were selected as “elite” individuals and carried over to the next
generation.
Mutation: Each elite individual was then mutated. Mutation would perform the following operations:
Change jexel count:
With a probability of 30%, the number of “jexels” would be changed. If change
occurred, one “jexel” would be added to or removed from all arrays with 50% chance.
Change jexel addresses:
With a probability of 30%, the addresses of
N
“jexels” would be changed. The
random number
N
, between 1 and 6, would be picked from a uniform distribution and random entries of the
JxAddresses array would be swapped to unassigned addresses.
Change jexels frequency:
With a probability of 50%, the frequency
f
of all “jexels” would be added to a
Gaussian random number of zero mean and standard deviation of 30 Hz. To ensure 0
f
200 Hz, another
random number would be picked if the frequency was out of bounds until 0f200 Hz.
Change jexel phases:
With a probability of 50%, the phases of
N
“jexels” would be changed. The random
number
N
, between 1 and 6, would be picked from a uniform distribution and random entries of the
JxPhases
11
array would be added to a Gaussian random number of zero mean and standard deviation of 60 deg. Proper phase
wrapping was performed.
Change jexel duty cycles:
With a probability of 50%, the duty cycles of
N
“jexels” would be changed. The
random number
N
, between 1 and 6, would be picked from a uniform distribution and random entries of the
JxDuties
array would be added to a Gaussian random number of zero mean and standard deviation of 30%. The
duty cycle was enforced to be between 0 and 100%.
Change back-pressure:
With a probability of 30%, the backpressure
PB
of all “jexels” would be added to a
Gaussian random number of zero mean and standard deviation of 3 psi. To ensure 0
PB
10 psi, another
random number would be picked if the back-pressure was out of bounds.
Crossover:
The remainder 60% (18) individuals were “children” of the mutated elite individuals generated with
the two previous steps. Crossover would be performed by following the steps below:
Choose parents:
Two random individuals from the 12 mutated elite individuals would be selected for the
crossover operation.
Jexel count definition:
Since each parent is expected to have a different number of “jexels”, one of the
parents was picked randomly to define the length of the child’s genome (i.e., the length of the JxXXX arrays).
Cross-over jexels:
The genome of the selected parent was copied to the child’s genome. Then, half of the
“jexels” would be randomly selected and swapped by the other parent’s jexels. Non-repeatability of the “jexel”
addresses was enforced.
Back-pressure interpolation:
The back-pressure was interpolated between the two parents with a randomly
chosen weight 0<w<1, such that PB,C=wPB,P1+(1w)PB,P2.
Evaluation:
All the new individuals in the current generation would be evaluated by measuring
Γ
and computing
J.
This algorithm was performed every generation. From Generation 30 onwards, however, in order to prevent the
aggressively random parameters from changing the population too much, the best 3 elite individuals stopped being
mutated by the algorithm.
B. Details on the “jexel” Clean-up Procedure
The inherent randomness of the GA is known to produce results that are difficult to interpret. To improve
interpretability and draw better conclusions about what the solution found by the GA is doing, a clean-up procedure was
performed after convergence of the GA. This clean-up procedure is necessary because there could be many “jexelsin
the best configurations that do not have a significant effect on the cost function, but did not get removed by the GA due
to random chance or the limitation in the number of iterations performed. This step, therefore, is crucial to draw a
deeper understanding from the results.
Starting with the “BestIndividual” configuration, the following pseudocode was used:
CurrentIndividual BestIndividual;
for j1to BestIndividual.nJexels do
for i1to CurrentIndividual.nJexels do
TestIndividual CurrentIndividual;
TestIndividual.RemoveJexel(i);
Cost(i)=TestIndividual.EvaluateCostFunction();
end
Find Kthat minimizes Cost(K);
CurrentIndividual.RemoveJexel(K);
end
Following this procedure enabled assessment of the contribution of each individual “jexel” to the obtained solution,
slowly removing one “jexel” at a time and observing how relevant its contribution was. As discussed in Section VII,
many of the “jexels” produced by the GA could be removed without reducing the solution fitness, unveiling a reasonably
interpretable pattern.
12
Funding Sources
This research is supported by the Air Force Office of Scientific Research (AFOSR), grant no. FA9550-17-1-0228,
"Dynamics of Unsteady Flow past Bluff Bodies with Lofted Bases”, under program manager Dr. Gregg Abate.
Acknowledgments
The authors would like to thank the master machinist Jeremy Phillips, for his expertise and craftsmanship in preparing
the actuator parts required for this study.
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14
... Experimental studies on curved bluff body surfaces at the current stage of development must be parametric in spatial dimensions, which is deemed to be expensive due to the necessity of changing the model, restarting the wind tunnel, among other practical issues. A recent study by the authors [21], however, demonstrates that enabling a computer to access different actuator locations via multiple channels of pneumatic solenoids is feasible and affordable with state-of-the-art technology. In this context, further advancements in measurement, processing and channel count increase (for example, via MEMS actuators [22]) can empower experimental studies to bring active flow control technology to engineering applications by better tackling the spatial sensitivity problem. ...
... Motivated by this unresolved issue, a recent study by the authors [21] demonstrated the development of an experimental system that is capable of optimizing the actuator spatial pattern utilizing a genetic algorithm (GA) approach, inspired by the recent experimental applications of GAs in flow control [36][37][38]. The device developed in [21] was able to achieve a drag reduction of 10.5% in the wake of a cylinder aligned with the flow with a slanted afterbody. ...
... Motivated by this unresolved issue, a recent study by the authors [21] demonstrated the development of an experimental system that is capable of optimizing the actuator spatial pattern utilizing a genetic algorithm (GA) approach, inspired by the recent experimental applications of GAs in flow control [36][37][38]. The device developed in [21] was able to achieve a drag reduction of 10.5% in the wake of a cylinder aligned with the flow with a slanted afterbody. The wake of the cylinder with a slanted afterbody displays two possible flow regimes with different topologies, conjectured by Morel [39] based on drag measurements and observed experimentally with flow visualization techniques by Zigunov [40], with the presence of hysteresis at low Reynolds numbers (Re D < 60 000). ...
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Experiments have been undertaken to study the formation of afterbody vortex flows from cylindrical bodies with a slanted base, whose upsweep angle was varied between 24° and 32°. Vortex roll-up is mostly completed in the first half of the upswept section, where the vortex causes largest suction on the surface. Towards the trailing-edge the vortices become more axisymmetric and stronger with increasing upsweep angle. Although there is some delay in vortex roll-up at lower Reynolds number, the main features of the vortex flow are similar to those at higher Reynolds number. The strength of the vortices at the trailing-edge was proportional to the time-averaged drag coefficient, which increased by nearly 50% in the range of upsweep angles tested. The vortex was more coherent with reduced meandering and a smaller core radius towards the trailing-edge. This reduction in meandering along the streamwise direction had not been observed previously with other external vortex flows in aerodynamics. Proper Orthogonal Decomposition revealed that the helical displacement mode with azimuthal wavenumber m = 1 was the dominant mode towards the trailing-edge, suggesting that the afterbody vortices bear much similarity with the more widely studied wing tip vortices and delta wing vortices. The instantaneous vortex pair exhibits time-dependent asymmetry; however, there is virtually no correlation between the displacements of the vortex centers.
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This is the first textbook on a generally applicable control strategy for turbulence and other complex nonlinear systems. The approach of the book employs powerful methods of machine learning for optimal nonlinear control laws. This machine learning control (MLC) is motivated and detailed in Chapters 1 and 2. In Chapter 3, methods of linear control theory are reviewed. In Chapter 4, MLC is shown to reproduce known optimal control laws for linear dynamics (LQR, LQG). In Chapter 5, MLC detects and exploits a strongly nonlinear actuation mechanism of a low-dimensional dynamical system when linear control methods are shown to fail. Experimental control demonstrations from a laminar shear-layer to turbulent boundary-layers are reviewed in Chapter 6, followed by general good practices for experiments in Chapter 7. The book concludes with an outlook on the vast future applications of MLC in Chapter 8. Matlab codes are provided for easy reproducibility of the presented results. The book includes interviews with leading researchers in turbulence control (S. Bagheri, B. Batten, M. Glauser, D. Williams) and machine learning (M. Schoenauer) for a broader perspective. All chapters have exercises and supplemental videos will be available through YouTube.
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