AIAA JOURNAL
Vol. 56, No. 10, October 2018
Feedback Flow Control: A Heuristic Approach
Jürgen Seidel,∗ Casey Fagley,† and Thomas McLaughlin‡
U.S. Air Force Academy, USAF Academy, Colorado 80840
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DOI: 10.2514/1.J056884
This paper describes a heuristic approach to feedback flow control and illustrates its use for two flowfields: the flow
around a tangent ogive at large angles of attack and the shear layer behind a backward-facing step. Both
investigations followed the feedback flow control approach laid out in this paper; however, because of the very
different flowfields and control goals, different techniques were selected for developing the most effective control
strategy. The flow instabilities on the tangent ogive made this flow a good candidate for effective flow control.
Unforced computational and experimental data showed the relevant flow features. With only four pressure sensors,
the relevant flow features were controlled, and a prescribed side force signal could be tracked. For the shear layer, the
control goal was to reduce the optical path difference, a control goal that cannot easily be observed in experiment or
simulation. Using a wavenet autoregressive exogenous model, a reduction in optical path difference of 40% was
achieved using a sensor array colocated with the aperture. Open-loop simulations showed the need to capture the
effect of forcing startup and shutdown. Adaptive feedback resulted in a new flow state that the open-loop data had not
captured.
Nomenclature
A
Aj
Aref
C
Cy
cy
C
Cμ
D
dp
d
Ff
Fn
Gs
hp
Kp , T z , T w , T d
kGD
L
n
n^
r
s
t
U∞
V^
X, Y, Z
xs
α
Δτ
δ99
θ
ρ
τ
=
=
=
=
=
amplitude of forcing input
jet exit area
model reference area
estimation coefficients
side-force coefficient
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
estimated side-force coefficient
momentum coefficient
model base diameter
disturbance pin diameter
disturbance input
frequency of forcing input
frequency (natural)
transfer function of the plant
disturbance pin height
reduced-order model parameters
Gladstone–Dale constant
model length
index of refraction
surface-normal vector
set-point reference
Laplace variable
time
freestream velocity
velocity vector
coordinate directions
sensor locations
angle of attack
nondimensional time step; Δt∕L∕U∞ boundary-layer thickness
momentum thickness
density
dimensionless time
I.
A
Overview
CTIVE closed-loop flow control (AFC) is a means to substantially
augment the aerodynamic performance of many systems
involving fluid flow. AFC is especially effective when exploiting
fluidic instabilities to manipulate large coherent laminar or turbulent
flow structures. Closed-loop flow control is a quickly growing,
multidisciplinary field combining control science, experimental and
computational fluid dynamics, and nonlinear dynamical systems.
Research efforts have the potential to improve performance of all
engineering systems in which a fluid flows (e.g., external flows around
air vehicles, ground-based systems such as bridges and buildings,
internal flows in pipes and propulsion systems, acoustical emissions and
mitigation, combustion and mixing problems, turbulence and transition
management, and alternative energy applications). Aerodynamic system
performance may be improved in the forms of drag reduction, increased
lift production, reduction of unsteady structural loading (vortex-induced
vibration), mixing optimization, etc. Both passive and active (which
incorporates both open- and closed-loop) flow control techniques have
been extensively studied (see e.g., [1,2]). Upon a survey of the literature,
implementation of closed-loop flow control to real-life applications is
mainly hindered by the design of control systems that are focused on a
point design (a single flow condition or a very limited scope of flow
conditions), usually based on a restricted parameter space. Extensions of
closed-loop flow control strategies to allow operation over a range
of parameters on three-dimensional (3-D) complex flowfields by
manipulation of (multiple) fluidic instabilities with distributed sensing
and actuation has proven to be exceedingly difficult to achieve.
Flow control hinges on the idea of injecting small local
disturbances into the flow, either passively or actively, resulting in a
direct and instantaneous modification of the flowfield. Ideally, the
perturbation of the flow becomes amplified through a natural, fluidic
instability, which influences the global flow state; thus, the control
power remains small by exploiting the fluidic instability. Although
this approach has long been recognized as the best way to achieve
flow control, practical implementations have been largely elusive.
The relationship of the introduced perturbation (e.g., body force,
pressure fluctuation, mass augmentation) and the mechanisms through
which its effect is transferred to the large coherent flow structures to be
controlled are not well understood; representing such interactions
using any dynamical model is therefore a research priority.
Flowfields that exhibit an exploitable instability typically have to be
described using the Navier–Stokes equations, which are characterized as
a coupled set of second-order, nonlinear partial differential equations.
The mathematical complexity of the governing equations provides
limited accessibility from a classical control design perspective, unless
substantial simplifying assumptions are invoked. The standard approach
to developing a dynamical model (and subsequently a control system)
Received 14 November 2017; revision received 12 April 2018; accepted for
publication 21 April 2018; published online 27 June 2018. This material is
declared a work of the U.S. Government and is not subject to copyright
protection in the United States. All requests for copying and permission to
reprint should be submitted to CCC at www.copyright.com; employ the ISSN
0001-1452 (print) or 1533-385X (online) to initiate your request. See also
AIAA Rights and Permissions www.aiaa.org/randp.
*Research Associate, Department of Aeronautics; jurgen.seidel@usafa.
edu. Associate Fellow AIAA.
†
Research Associate, Department of Aeronautics. Member AIAA.
‡
Director, Aeronautics Research Center, Department of Aeronautics.
Associate Fellow AIAA.
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SEIDEL, FAGLEY, AND MCLAUGHLIN
for a flowfield is to reduce the dimensionality of the discretized
system. This is typically accomplished through numerical decomposition techniques, system-identification techniques (linear, nonlinear,
exogenous, etc.), or a combination of both. To numerically reduce the
dimensionality of the data, the finite-dimensional system is projected
onto a set of spatial basis functions that satisfy certain energy [3],
controllability and observability [4], dynamical [5], or short-time
correlative [6] constraints. The temporal dynamics of these spatial basis
functions, or modes, are then obtained using system identification
techniques [7–10] or Galerkin projections [11] onto the Navier–Stokes
equations. Finally, the modeled dynamics, in conjunction with the
spatial modes, comprise the reduced-order model (ROM).
Another approach to arrive at a model is the application of a fully
“black-box” model, commonly implemented where the input–output
(a global, directly observable flow parameter) relationship is
identified [12]. Consequently, these black- or gray-box models are,
typically, developed at a single operating condition (Reynolds
number, angle of incidence, open-loop forcing condition, etc.) as well
as for a unique geometry. Research has shown that, once a
perturbation is introduced to any one of these parameters, the closedloop controller performance quickly declines. In other words, many
of the models in the literature suffer from the fact that the model basis
does not span a sufficiently large parameter space to allow model
validity for parameter variations (i.e., the model robustness is very
limited). Current research has shown that a family of linearized
operating points can be used for gain scheduling or linear matrix
inquality (LMI)–polytopic control schemes to improve the
robustness of feedback flow control [13,14].
Although these standard model-based approaches have struggled
with off-design conditions, they have provided tremendous insight
into the flow physics and have also enabled closed-loop flow control
at on-design conditions. For example, they have been employed on
the von Kármán vortex street (at low Reynolds numbers, Re ≈ 100)
behind a two-dimensional cylinder to reduce the resultant drag and
oscillatory lift force [6,15–20]. This was a tremendous research
accomplishment in the flow control community, which came in the
past 20 years. Successful control of this flow brought about a number
of findings that have shaped current research in flow control; for
instance, the efforts showed that modeling the transients from
unforced to successfully forced (actuated) states was critical for
successful control, leading to the realization that the analysis of
the flowfield using standard proper orthogonal decomposition
(projection onto an orthogonal basis to optimally represent the flow
from an energy standpoint) was insufficient. The introduction of a
shift mode [6,21], which captured this transient phenomenon from
the unactuated to the actuated state (lengthening of the recirculation
region), significantly improved the fidelity of the reduced-order
model. Indeed, Marxen et al. [22] recently observed best results of
flow control when the forcing frequency was based on a relevant
frequency of the controlled flowfield rather than an unstable
frequency of the uncontrolled flow.
The current research emphasis in flow control is still focused on
expanding ROM fidelity by spatial mode representation and selection
[23–27]. Although this increase in ROM fidelity provides more
physical insight into the fluid dynamics, the developed control
algorithms ultimately remain relatively elementary. For example,
proportional–integral–derivative [9,28–34] or adaptive/variable-gain
[35] control algorithms are typically adopted in the flow control
research community. Higher-order control schemes using model
predictive control, Smith predictors, or robust LMI–linear quadratic
regulator (LQR) approaches have also been successfully applied to
flow control problems, but a standardized methodology has yet to be
determined [14,36,37].
The method to develop a controller for a flowfield is described in
this paper. The approach has been used extensively in research at
the U.S. Air Force Academy Aeronautics Research Center. The
underlying philosophy is based on the notion that, only after
understanding the fundamental fluid dynamic processes in the flow
of interest, a successful flow control approach (with a predefined
goal) can be implemented. Such an implementation has to include
all aspects of feedback flow control: actuator location and
performance, sensor location and performance, and the design of an
appropriate controller.
II.
Approach
As shown in Fig. 1, the first part of the flow control approach is the
interrogation of the uncontrolled flowfield to get a basic understanding
of the flow physics. The insights from this investigation provide valuable
information for many aspects of eventual feedback flow control
implementation such as initial actuator placement and dominant
frequencies, etc. Most importantly, the information collected in this step
allows for the definition of the basic flow state, which captures pertinent
characteristics of the flow. The data describing the flow state are then
used to determine observability, which drives the initial placement
of sensors, as well as initial actuator location and amplitude (and
frequency) of forcing, which is summarized as control authority in
Fig. 1. Although observability and controllability have a strict
mathematical definition in the context of control theory, for the purposes
of this paper, the terms are used in a qualitative manner. Sensor
observations (i.e., the ability to infer the behavior of the internal state by
external observations) are assessed by a correlation of sensor
measurements and the dynamic of the flow state. Also, control authority
(i.e., the ability to augment or move the internal state through an input) is
demonstrated by the open-loop response of the system to move the state
through the desired control trajectory. In addition, this initial step allows
for analysis of flow sensitivities to parameters such as freestream
velocity, angle of attack, etc. This information can enter the model and
control design to broaden the parameter space in which the resulting
controller can be employed.
With this information, the second step, open-loop (i.e.,
predetermined) forcing, can be executed. During this step, forcing is
introduced with varying input parameters, most importantly amplitude
and frequency. Results from this investigation, which can be
made experimentally, computationally, or as a combination, are then
scrutinized to understand the transient response of the flowfield to
control inputs and to update the flow state definition. In this regard,
experimental observations provide a straightforward way to
interrogate the flowfield for a wide variety of forcing parameters. On
the other hand, computational results can provide detailed insights into
the flow physics that are not readily available from experiments.
Therefore, the synergy between the experimental and computational
approaches arguably provides the breadth and depth of information
necessary to understand the flowfield and successfully implement
feedback flow control.
The data obtained from these open-loop forcing studies provide
important information to update flow sensor locations, which
Fig. 1 Road map for the development of closed-loop flow control
algorithms.
SEIDEL, FAGLEY, AND MCLAUGHLIN
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Fig. 2 Flow state on a tangent-ogive forebody as a function of angle of attack.
heuristically are located where the flowfield changes the most when
open-loop control is applied (although care has to be taken to avoid
“false positives”). In addition, it allows for the development of a
mapping between the sensor information, which in applications are
typically surface-mounted, and the flow states defined earlier, which
describe the whole flowfield. This mapping is captured in the flow state
estimator, which, in conjunction with the control algorithm, comprises
the flow control system.
The second use of the data sets is for the development of a reducedorder model. Typically, the data are analyzed using state-of-the-art
decomposition techniques (e.g., proper orthogonal decomposition
(POD), and its variants [3,6,25,28] or dynamic mode decomposition
(DMD) [38–40]) to gain further insight into the fluid-dynamic
interactions between the baseline flow and the control input. This
information describing the dominant changes in the flowfield can
influence sensor and actuator locations, but more importantly, after
the flow is decomposed and projected onto a set of suitable basis
functions (which are highly dependent upon the flow under
consideration), a reduced-order version of the flow can be developed.
The temporal behavior of the flow structures captured in the basis is
now given by a relatively small number of time-varying mode
amplitudes. Upon accurate identification of the physical or
dynamical modes of interest, relationships between actuation inputs
can be mathematically determined through system identification
techniques. Correctly identifying these relationships is crucially
important for follow-on controller development and design.
Finally, with the flowfield analysis complete and a model in place,
a controller can be developed using many theoretical descriptions.
However, it is important to not lose sight of the underlying flowfield
and return often to the flowfield of interest, either in experiment or
simulation, to determine the efficacy of the developed flow control
approach. Standard control algorithms (e.g., proportional–integral–
derivative, direct adaptive, linear genetic representation, model
reference, etc.) have shown great success in achieving the predefined
control goal for a variety of flows, but only after leveraging the
information contained in the reduced-order model of the flow.
This paper highlights the application of the heuristic approach
outlined previously on two flowfields: the flow over a tangent ogive at
high angles of attack and a free shear layer. The steps in the heuristic
approach illustrate that not a single decomposition method or control
design and implementation is suitable for a range of flowfields; on the
contrary, a solid understanding of the flowfield is necessary to apply the
appropriate tools for the flow control problem at hand. However, having
a unified strategy allows for a structured approach to robustly developing
flow control for a particular application. Continual improvements in
applied mathematics, signal processing, and computational power
provide innovative tools that can be inserted in the strategy, but it has to
be recognized that there likely will never be a single control approach for
all flow control applications. In both example problems described in this
paper, the particular application of the steps laid out previously will be
described along with the decisions that were made because of the results
obtained in each step to illustrate the need for an adaptable and hierarchal
strategy to implement flow control.
III.
Applications
A. Forebody
The flowfield around a slender axisymmetric forebody varies
dramatically with its angle of attack. Typically, as the angle of attack
is increased, four flow regimes are observed: attached flow
(α 0 → 15 deg), symmetric vortex flow (α 15 → 40 deg),
asymmetric vortex flow (α 40 → 60 deg), and unsteady wakelike
flow (α 60 → 90 deg), each of which is shown in Fig. 2 [41–43].
Each flow regime has distinct fundamental fluid dynamic
characteristics, which are briefly discussed in the following sections.
It should be noted that the exact angle of attack at which these diverse
flow characteristics are observed depends on many factors, including
the detailed geometry, surface quality, etc., and Fig. 2 is only intended
as an illustration of the types of flow states on a tangent-ogive
forebody.
As shown in Fig. 2, when the angle of attack is between 15 and
60 deg, the flow on the leeward side of the forebody develops into two
primary vortices, one on the port and starboard side, respectively. As
long as these vortices remain symmetric with respect to the centerline
of the forebody, the pressure distribution around the forebody will
also remain symmetric. However, once these vortices become
asymmetrically distributed around the forebody, a large side force
and yawing moment result from the asymmetric pressure distribution
around the model. When this occurs, the magnitude of the side force
can equal the magnitude of the normal force on the body, resulting in
wing rock, a coning motion, or complete loss of control.
Once the angle of attack increases beyond approximately 40 deg,
a convective instability (along with the previous global vortex
instability) exists. This instability amplifies any geometric perturbation
of fluidic disturbance to naturally provoke an asymmetric vortex
configuration (i.e., side force, yaw moment, increase in drag).
Although passive and open-loop active flow control efforts [44–56]
have focused on delaying the onset of any asymmetry in the primary
vortices, Bernhardt and Williams [57] hypothesized that, through
closed-loop flow control, not only could the side force be regulated
(i.e., a symmetric vortex configuration could be achieved), but the
convective instability could also be used to improve the agility and
maneuverability of the forebody in flight regimes where standard
control surfaces are useless. Patel et al. [58] also closed the loop on an
axisymmetric vortex phenomenon in this angle-of-attack range using
a frequency-based proportional–integral–derivative control law with
the employment of deployable flow effectors (vortex generators) to
manipulate the vortex state.
The geometry used in this investigation is shown in Fig. 3. The
flow conditions were chosen such that the Reynolds number based on
base diameter D was ReD 156;000 and the angle of attack was set
at α 50 deg. A small, pin-shaped disturbance was added at the
starboard side of the model. The pin had a diameter of dp ∕D 0.001
and was hp ∕D 0.00045 tall. The center of the pin was placed at
X∕D 0.04. This geometric disturbance was sufficient to initiate a
natural, deterministic asymmetric vortex state. The model shown in
Fig. 3 was mounted in the wind tunnel on a force balance with a
rear sting.
Unsteady, laminar DDES simulations of the flow were performed
using COBALT V5.2 from Cobalt Solutions, LLC. Cobalt is an
unstructured, cell-centered finite volume code that solves the
compressible Navier–Stokes equations [59]. The grid spacing was
designed such that the spacing near the tip of the model is set at 0.01%
of the base diameter and transitions to 1.0% of the base diameter at the
base of the model. The initial point spacing normal to the surface in
the boundary layer resulted in an average y 4 × 10−2 . The Mach
number was set to M 0.1, and the time step was set to
Δτ 4.34 × 10−4 , which gave a Courant–Friedrich–Levy (CFL)
number of approximately 4.
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Fig. 3 Model geometry of a von Kármán ogive with fineness ratio of 3.5: a) schematic of the model [22], and b) definition of mass blowing angle
(δ 30 deg) [60].
Fig. 4 Open-loop forcing inputs (left), resulting side-force response with model prediction (right). Solid line: −Cy , dotted line: Cy , dashed line: predicted
response from the LTI model [60].
From unforced simulations, it was observed that the flow separates
approximately at the widest point of the body (90 deg in the
azimuthal direction) for the symmetric vortex state; the asymmetric
vortex state shifted this line somewhat. All this information entered
the state definition for this flowfield (see Fig. 1). In addition, the
location of the separation line allowed for a heuristic sensor
placement (observability in Fig. 1): four pressure taps at 80 deg
from the windward meridian (slightly upstream of separation) at
X∕D 2 and X∕D 3 to estimate the side force on the body. For
this particular geometry, a large body of research was available to
supplement the flow state information to satisfy the observability
constraint. In the next step, open-loop forcing was applied in the
simulations to gather information for the flow state mapping as well
as the reduced-order model construction identified in Fig. 1.
To force the flow, two mass blowing patches were placed at
90 deg from the windward meridian; each patch began at
X∕D 0.1 and ended at X∕D 0.3 with a width of Z∕D 0.0015
(see Fig. 3). Forcing was applied in two forms, as a step input and as
impulse forcing. The step input is shown in Fig. 4, in which inputs
were applied to both port and starboard actuators. Note that, for
visualization purposes, portside actuation is shown as negative values.
This open-loop forcing investigation allowed for the construction of
the map of achieved side force Cy as a function of forcing amplitude
shown in Fig. 5. The momentum coefficient Cμ is defined as
Z
^ V^ ⋅ n
^ dAj ∕ρ∞ U2∞ Aref
Cμ ρV
A
where V^ is the mean jet exit velocity, Aj is the jet exit area, U∞ is the
freestream velocity, and Aref is the model reference area (here, the
ogive base area).
As identified in Fig. 5, there are different regimes of the response
to forcing in this flowfield. For very small forcing amplitudes,
a small deadband was observed, followed for intermediate
amplitudes by a linear relationship between forcing amplitude and
side-force production. For large amplitudes, the effect of forcing
saturated.
The remaining steps in the development are flow state estimation
and reduced-order model development. For flow state estimation,
Fig. 5 Comparison of the steady-state response from the open-loop CFD
data and the second-order LTI model [60].
the side-force coefficient was estimated from the four pressure
sensors on the ogive body in the form
cy t C1 P1 t C2 P2 t C3 P3 t C4 P4 t C ⋅ Pxs ; t
C
(1)
where the coefficients Ci were determined in a least-squares sense
from the open-loop flow state database.
Analyzing the response to the forcing inputs shown in Fig. 4
showed that a linear time-invariant (LTI) parametrization was well
suited to model the system response. Therefore, the input–output
relationship is
Cy s Gs sCμ s
(2)
The structure of the model in continuous time has the form
Gs s Ns
sm am−1 sm−1 am−2 sm−2 : : : a1 s a0
Keθs n
Ds
s bn−1 sn−1 bn−2 sn−2 : : : b1 s b0
(3)
for a linear system with m zeros, n poles, and a pure time delay eθs.
For more details of the model development, see [60]. Many different
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SEIDEL, FAGLEY, AND MCLAUGHLIN
system identification techniques were now available to determine the
coefficients of the polynomials in the numerator and denominator.
The technique for time-domain identification used in this effort was
the prediction error method (PEM) [61,62]. One of the advantages
was that the PEM directly identifies a model structure of the form of
Eq. (3). Training the model using the flow state database resulted in
the following LTI model:
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Gs s K p
1 Tzs
e−T d s
1 2ζT w s T w s2 (4)
which is represented by the solid line in Fig. 5 in comparison with
the data.
Finally, this LTI model can be used evaluate the dynamics of a
feedback control system. The system is described using the transfer
functions for the plant model Gs s, the control system Gc s, and the
transfer function Gd s of the input disturbance, which can be written as
cy C
Gs Gc
1Gs Gc
Gd
1Gs Gc
r
d
(5)
where r and d are the set-point reference and disturbance inputs,
respectively. The controller is specifically designed to track a set-point
side force while rejecting input disturbances.
Although a range of control algorithms were tested on the LTI
model, a predictive proportional–integral (PI) control law was derived
in the frequency domain. The predictive aspect of the PI control was
implemented by a Smith prediction in hopes to reduce the convective
time delay of the actuator response. The controller performance
illustrated in Fig. 6 for both experiments and computations showed
a fast control response of the asymmetric configuration [60].
In particular, the presence of multiple instabilities (i.e., separated shear
structures along the axis of the body) were unable to be attenuated due
to the non-co-located actuator and sensor as well as a fluidic instability
that resulted in unsteady fluctuations about the set-point.
B. Shear Layer
The flowfield under consideration in this section is the shear layer
forming behind a backward-facing step. The flow acts as a prototype of
the shear layer separating from turrets mounted on the fuselage of
airborne systems. When light travels through these shear layers, the
large-density variations, especially when looking back through the
separated shear layer (e.g., [63,64]), produce significant wave-front
distortions that are detrimental for system performance. Although the
dynamics of coherent structures in shear layers are complex (see for
example [65]), from an optical point of view, the large coherent
structures (Kelvin–Helmholtz vortices) are of most interest because of
their reduced core pressure [66]. This pressure well, and its concomitant
density well, are responsible for the optical aberrations because, for air,
the index of refraction n is a linear function of the density ρ:
nx; y; z; t 1 kGD ρx; y; z; t
(6)
where kGD is the Gladstone–Dale constant. From the index of
refraction, the optical path length (OPL) is calculated as its path integral
through the medium (see Fig. 7):
Z
OPL ns ds
(7)
c
To measure the wave-front distortion, the optical path difference
(OPD), which is the difference between the OPL at one time instant
Fig. 6 Representations of a) closed-loop experimental results showing set-point tracking of the measured side-force, and b) closed-loop Navier–Stokes
simulations showing side-force set-point tracking capability for varying references, Cry .
Fig. 7 Example of scattering of light through density fluctuations (or pressure wells) due to flowfield variations.
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SEIDEL, FAGLEY, AND MCLAUGHLIN
and the mean OPL over the complete aperture [67], is typically
computed and experimentally measured using wave-front sensors.
For this problem, the origin of the coordinate system is at the step with
the x axis pointing downstream, the y axis normal to the flow, and the
z axis in the spanwise direction (see Fig. 7). Assuming that the light
beam propagates along the y direction, the OPD is then
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OPDx; z; t; y OPLx; z; t; y − OPLx; z; t; yx;z
(8)
The OPD quantifies the phase change of the wave front
propagating through a medium with variable index of refraction.
Usually, the metric used to assess the severity of the aberrations is the
root-mean-square value, OPDrms . The goal of feedback flow control
for this problem is to minimize the density fluctuations and therefore
to minimizee the optical aberrations.
Many experimental studies of aerooptics have been conducted
with geometries ranging from backward-facing ramps to round
turrets (see [64,66]). The findings of these studies suggest that openloop flow control has only a marginal effect on the optical distortions.
Thus, feedback flow control must be used to improve the mitigation
of optical abberations due to complex flow structures over the
previous attempts. The approach laid out in Sec. II was directly
followed and is described next.
Initially, the unforced flow dynamics were explored. A typical
instantaneous result of the unforced simulations is shown in Fig. 8a,
where the flow structures are visualized using the Q vortex
identification criterion [68,69]. At this time instant, the shear-layer
(Kelvin–Helmholtz) vortices are visible starting approximately one
step height downstream of the separation point. In addition, the OPD
is plotted in color above the flow structures, showing the strong
correlation of the location of the flow structures with the largest
optical aberrations. The separating boundary layer has a thickness of
δ99 ∕H 0.047, and for a turbulent boundary layer, the corresponding
Reynolds number based on momentum thickness θ is Reθ ≈ 4500.
Analysis of the three-dimensional density data showed that
small-scale, three-dimensional flow features do not have a
significant effect on the OPD (Figs. 8a and 8c). The large, spanwise
coherent structures have the greatest influence on the optical quality
of the flow, as one would expect. When performing POD on the
density field, the fluctuating modes separate the spanwise coherent
fluctuations in one mode pair (modes 1 and 2), and higher mode
pairs capture the spanwise distortions (Figs. 8b and 8d). Because the
most energetic modes, which correspond to the large, spanwise
coherent structures, correlate very nicely with the OPD shapes, it
can be posited that the higher-order spanwise distortions do little to
influence the optical properties of the flow. Thus, using this analysis
of the unforced flow dynamics, it was deemed that the large-scale,
coherent structures had the most dominant effect on the OPD, and
the flow state was defined by the temporal behavior of the spanwisecoherent POD modes.
To influence the behavior of the free, unstable shear layer, mass
injection was used at the corner of the step. The actuator location was
colocated with the origin of the fluid dynamic instability to maximize
the control effectiveness. An open-loop parameter study was
performed varying jet velocity and frequency. The resulting flow is
shown in Fig. 8c for a single frequency and amplitude. When
periodic, spanwise uniform forcing is applied, the spanwise
coherence is increased, and the first spatial mode pair is even more
dominant, as shown in Fig. 8d. The shear layer is highly susceptible to
disturbances and locks in for a range of frequencies and amplitudes
[70]. Although the response due to actuation does exhibit lock-in, the
interaction between forcing and state behavior is highly nonlinear.
An initial transient is observed, in which vortex merging occurred
before lock-in. Also, when forcing was turned off, a similar ending
transient phenomenon was observed. It was seen from work on the
cylinder wake [6] that these transient dynamics are highly important
for successful employment of feedback flow control.
Consequently, a low-dimensional dynamic model was needed to
accurately represent the evolution of the mode amplitudes aj t of the
Fig. 8 3-D simulation data: a–b) unforced, c–d) Ff ∕Fn 1.33, A∕U∞ 0.10. Instantaneous flow structures (Q-isosurface, gray) and OPD (color) at
y∕H 0.5 (Figs. 8a, 8c). Isosurface of 3-D POD mode pairs of density (Figs. 8b, 8d).
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SEIDEL, FAGLEY, AND MCLAUGHLIN
truncated POD model for a given forcing signal. Previous work used
artificial neural network autoregressive exogenous (ANN-ARX)
systems to identify the dynamical behavior of the time coefficients in
the forcing parameter space [6]. Neural networks are widely used in
the scientific community for process modeling, artificial intelligence,
pattern recognition, machine learning, etc. This nonlinear system
identification technique has been argued to be a universal approximator,
capable of representing any type of data trend [71]. However,
some inherent problems of ANN models exist. First, there is no
straightforward method for determining the number of hidden neurons,
number of layers, or parameters of the regression vector. Training relies
heavily on trial and error to find a combination of parameters that yields
acceptable results. Second, the convergence of these networks depends
heavily upon the initialization of the weighting matrices. This can lead
to drastically different results when training a single network with a
given set of parameters twice because of the initial random generation
of weights. Third, a properly trained network will behave as a black box
in which little mathematical/physical insight can be gained. And fourth,
training times are extremely long due to multimodal error surfaces that
tend to trap the solution in local minima.
For this flowfield, wavelets were combined with an artificial neural
network (ANN) architecture (i.e., a wavelet basis function was used
as the transfer function of a neuron to create a wavelet neural network
or wavenet, WN). These wavenets were first introduced by Zhang
et al. [72,73], Chen and Bruns [74], and Polycarpou and Weaver [75]
and have been applied in many areas such as functional
approximation, system identification, adaptive control, and nonlinear
modeling and optimization. Multiple techniques exist to design the
architecture of such wavenets. One technique is to replace the
existing transfer function of a neural network (usually sigmoid or
signum functions) with a wavelet basis function. Another approach
for integrating these two ideas is to use the wavenet as a preprocessing
filter for the nonlinear artificial neural net identifier. In this project, a
combination of these two methods was applied. The model structure
was decomposed into three blocks: a linear, a preprocessing scaling
function, and a wavelet function block. The model was then trained to
3831
accurately estimate the frequency-rich, highly nonlinear POD modal
amplitudes.
The WNARX system presents a mathematical model that relates
the evolution of time coefficients of the numeric decomposition
over the open-loop forcing parameter space. WNARX system
identification techniques allow for prediction and simulation of the
POD mode amplitudes and are not limited to single-input/singleoutput systems. WNARX are strictly causal systems, depending on
current and past time histories of chosen inputs. For the development
of the WNARX model, a regression vector is formed such that the
previously estimated mode amplitudes and current and past actuation
inputs are compiled in a vector. This regression vector serves as the
input to the low-dimensional WNARX model. A nonlinear function
relates the regression vector to the time coefficient at the future time.
Once a model structure is chosen, the simulation or prediction error is
minimized over a training data set in a least-squares sense. The
number of input/output neurons, time history sequences of input/
outputs, and training data sets are carefully selected for adequate
model performance. For more information on the training algorithm
and ANN architecture, the reader is referred to [76,77].
The WNARX model was validated for an off-design flow case for
which the forcing signal was turned on at t 0 s, at which point the
flow went through a transient period before locking into the forcing
frequency. The forcing was then turned off at t 0.025 s to
reestablish the unforced flow state. As shown in Fig. 9, the model
captures the lock-in region of the periodic forcing very well. Once the
forcing was turned off at t 0.025 s, the model accurately predicts
the type of nonlinear signal in the unforced flow. Expecting an exact
replication of the unforced time coefficients is unrealistic because
the original flow dynamic is extremely nonlinear and only quasiperiodic. However, the important point is that the model of the
unforced flow did not decay to zero over time. This indicates
that there is a periodic attractor to the nonlinear function for the
WNARX system. Thus, the attractor was assumed to be near the
solution of the unforced state, which is shown by the similarities in
periodic trends.
Fig. 9 Mode amplitudes for off-design validation of four-mode WNARX model for flow case of F 600 Hz and A∕U∞ 0.1. WNARX output (solid),
POD model (dashed).
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3832
SEIDEL, FAGLEY, AND MCLAUGHLIN
Fig. 10 OPD calculation of closed-loop CFD simulation with adjusted controller. Periodic forcing for 0 < t < 0.025 s, closed-loop simulation for
0.025 < t < 0.06 s, and unforced for t > 0.06 s.
Although the previous flow control example had directly measurable
quantities (i.e., the force or the surface pressure measurements), the
application of the shear layer used an unobservable quantity to drive the
cost function on the controls approach [76]. Moreover, the OPD, which
is the line integral of the index of refraction through the density field, is
a direct function of the state definition of the flow. A state estimator was
used to predict the temporal coefficients, and the OPD was computed
and used as the cost function of the adaptive control algorithm. The
sensor array was chosen as a surface pressure array at the base of the
backward-facing step near the notional aperture. The time series of the
surface pressure measurements were correlated to the flowfield state (i.
e., time coefficients), and the placement was selected based upon
maximum correlation [78]. A dynamic mapping function was used to
predict the state of the flow at a given instant. Very good results were
observed, which indicates that the flow state is highly observable using
surface pressure measurements.
Once the estimation algorithm and control algorithm were
identified, the full closed-loop system was tested. A range of control
techniques were applied to the ROM, including a linear quadratic
regulator, H ∞ control, and direct adaptive control. The direct
adaptive control showed the best performance (i.e., the results
showed that the optical abberations of the shear layer were mitigated
most effectively). The results are briefly shown in Fig. 10, where
open-loop forcing was prescribed during 0 < t < 0.025 s, at which
point closed-loop forcing was turned on, followed by the unforced
state for t > 0.06 s. As Fig. 10 indicates, a 40% reduction of the OPD
was achieved using this feedback control approach. The interesting
finding was that the adaptive control provided a forcing input that
minimized the optical distortions but was not represented by openloop transients. In other words, the adaptive feedback took the flow
to a new state that the traditional open-loop trajectories had not
captured.
IV.
Conclusions
This paper laid out an approach to applying feedback flow control
and illustrates its use for two example flowfields: the flow around an
tangent ogive forebody at a large angle of attack and the shear layer
behind a backward-facing step. In both investigations, the steps of the
approach to feedback flow control outlined in Fig. 1 were followed;
however, because of the very different flowfields and control goals,
different techniques were selected for determining the most effective
control strategy.
For the tangent ogive, the goal was to control the side force.
Unforced simulation and experimental data showed the basic flow
features and allowed for determining the sensor placement, as well as
control effectiveness. With only four pressure sensors, the flowfield
was adequately described and ultimately controlled (i.e., it was
possible to track a prescribed side-force signal). This flowfield
provided a strong instability and was therefore a good candidate for
effective flow control.
For the shear layer, the control goal was to reduce the optical path
difference (OPD), which presented a much more “indirect” goal (i.e.,
a quantity that is not easily observed in a fluid dynamic experiment or
simulation). Using a wavenet–ARX model of the highly nonlinear
dynamics of the flowfield, a reduction in OPD of 40% was achieved
using a sensor array colocated with the notional aperture. Open-loop
simulations showed that capturing the nonlinearity of the flow
response to the forcing input was crucially important because the
salient flow physics (i.e., vortex pairing) was significantly influenced
by turning the forcing on and off. In the end, the adaptive feedback
took the flow to a new state that the traditional open-loop trajectories
had not captured, which showed the robustness of the approach and
highlights the performance improvements that are achievable when
using feedback flow control.
The heuristic approach described in this paper relies on a thorough
understanding of the dynamics in the flowfield of interest. With this
understanding, the flow states can be defined accurately, and a model
that covers the relevant dynamics while reducing the system
complexity can be defined. Arguably, the model development is the
crucial step in the process of developing a feedback flow control
method because a broad range of tools is available for controller
development. In the end, the controller effectiveness hinges on the
fidelity of the underlying model.
Acknowledgments
This work was supported in part by the U.S. Air Force Office of
Scientific Research with program manager Douglas Smith. This work
was supported in part by a grant of computer time from the U.S.
Department of Defense High Performance Computing Modernization
Program at the U.S. Air Force Research Laboratory. This material is
based on research sponsored by the U.S. Air Force Academy
under agreements FA7000-13-2-0002 and FA7000-13-2-0009. The
U.S. Government is authorized to reproduce and distribute reprints
for Governmental purposes notwithstanding any copyright notation
thereon. The views and conclusions contained herein are those of the
authors and should not be interpreted as necessarily representing
the official policies or endorsements, either expressed or implied, of the
U.S. Air Force Academy or the U.S. Government.
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D. Greenblatt
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