# Cycle-to-cycle variations in cross-flow turbine performance and flow fields

Abigale Snortland*1, Isabel Scherl1, Brian Polagye1, Owen Williams2

###### Abstract

Cross-flow turbine performance and flow fields exhibit cycle-to-cycle
variations, though this is often implicitly neglected through time- and phase-
averaging. This variability could potentially arise from a variety of
mechanisms – inflow fluctuations, the stochastic nature of dynamic stall, and
cycle-to-cycle hysteresis – each of which have different implications for our
understanding of cross-flow turbine dynamics. In this work, the extent and
sources of cycle-to-cycle variability for both the flow fields and performance
are explored experimentally under two, contrasting operational conditions.
Flow fields, obtained through two-dimensional planar particle image
velocimetry inside the turbine swept area, are examined in concert with
simultaneously measured performance. Correlations between flow-field and
performance variability are established by an unsupervised hierarchical flow-
field clustering pipeline. This features a principal component analysis (PCA)
pre-processor that allows for clustering based on all the dynamics present in
the high-dimensional flow-field data in an interpretable, low-dimensional
subspace that is weighted by contribution to overall velocity variance. We
find that the flow-field clusters and their associated performance are
correlated primarily with inflow fluctuations, despite relatively low
turbulence intensity, that drive variations in the timing of the dynamic stall
process. Further, we find no evidence of substantial cycle-to-cycle
hysteresis. Cycle-to-cycle performance variability occurs earlier in the cycle
than flow-field variability, indicating the limits of co-temporal correlation
between performance and flow fields, but clustering reveals persistent
correlations between performance and flow-field variability during the
upstream portion of the turbine rotation. The approach employed here provides
a more comprehensive picture of cross-flow turbine flow fields and performance
than aggregate, statistical representations.

*Corresponding author email<EMAIL_ADDRESS>
Contributing authors email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
*1University of Washington, Mechanical Engineering, 3900 E Stevens Way NE, Seattle, WA 98195, USA

2University of Washington, Aeronautics and Astronautics, 3940 Benton Lane NE,
Seattle, WA 98195, USA

##  1 Introduction

Cross-flow turbines are able to harness the kinetic energy in wind, tidal
currents and rivers. Relative to axial-flow turbines, cross-flow turbines,
referred to a “vertical-axis” turbines in the wind sector, operate at lower
rotation rates, are insensitive to inflow direction, and may be able to
achieve higher power output per unit area within an array [1]. However,
because cross-flow turbines rotate perpendicular to the inflow, the blades
encounter a continually fluctuating angle of attack and relative inflow
velocity that leads to the unsteady, non-linear phenomenon of dynamic stall
[2, 3, 4, 5, 6, 7, 8, 9]. Similarly to stall on a static foil, flow separation
from the blade is eventually accompanied by a significant loss of lift and
increased drag. For cross-flow turbines, dynamic stall severity depends on the
ratio of the blade tangential velocity to the inflow velocity – the
dimensionless “tip-speed ratio”. While cross-flow turbine hydrodynamics are
functions of both the blade azimuthal position and the tip-speed ratio, cycle-
to-cycle variability in performance and near-rotor flow fields is also
observed. This is potentially caused by inflow fluctuations, hysteresis from
previous cycles, and the sensitive and stochastic nature of dynamic stall
itself. For example, any perturbations in the inflow velocity not only change
the kinetic energy available in the flow but also change the instantaneous
angle of attack and relative velocity encountered by the blade. This may, in
turn, appreciably affect the timing and severity of dynamic stall. Similarly,
Choudhry et al. [10] hypothesize that dynamic stall is influenced by the state
of the boundary layer, which suggests that hysteresis from previous cycles,
such as the extent of separated flow remaining on the blade at the beginning
of the next cycle, may affect future dynamic stall.

Cycle-to-cycle variability is commonly neglected in the cross-flow turbine
literature. RANS simulations, while often employed to study these flows, are
inherently unable to accurately model cycle-to-cycle variations, and the
computational expense of LES and DNS precludes their wider use to characterize
this variation over a large numbers of cycles [11]. For experiments, these
variations are implicitly neglected when data are time- or phase-averaged.
However, phase-averaging can remove information that would otherwise assist in
interpreting power production, vortex shedding, and stall, as well as
distorting the timing and character of the dynamic stall process by smearing
out non-linear phenomena and post-stall events [12, 13, 14, 15]. As such, more
sophisticated techniques that preserve cycle-to-cycle variability could
provide additional insight into the flow-field physics. Some cross-flow
turbine works do acknowledge the variability that is present in performance
and/or flow-field measurements, but simply treat it as experimental
uncertainty [16, 17, 18, 19, 20].

Our objective is to quantify the extent of cycle-to-cycle variation in cross-
flow turbine performance and flow fields, its sources, and the relationships
between performance and flow-field variability. To do so, we employ an
unsupervised clustering pipeline, comprised of well-developed, data-driven
methods, that identifies physically meaningful flow-field clusters with
differing dynamics relevant to the dynamic stall process. Clustering was
chosen since it can provide a basis for conditionally-averaging experimental
data using all the dynamics present, rather than resorting to hand-engineered
metrics (e.g., flow-field data based on phase-specific vortex separation from
the foil surface). We are also able to correlate these flow-field clusters
with performance and investigate the different sources of variability.

While cycle-to-cycle variability has not been explicitly studied for cross-
flow turbines, variability in dynamic stall is the subject of several recent
works [14, 21, 12, 15, 22, 11]. By using pressure tap measurements on a
flapping foil, Harms et al. [14] investigated variability in dynamic stall and
concluded that the phase-average and a measure of the spread (e.g., standard
deviation) were descriptive of the general dynamics for cases where bivariate
pressure distributions did not exhibit bimodal behavior. They hypothesized
that the cases exhibiting bimodal behavior were more sensitive to inflow
conditions and boundary layer unsteadiness. Tsang et al. [21] employed a
wavelet analysis on lift and drag data from a flapping foil, and showed that
non-stationarity between cycles increased with stall severity. They
hypothesized that this was the result of non-linear interactions between the
fluid force and the pitching motion. Lennie et al. [12] used a Covolutional
Neural Network and hierarchical clustering of pressure tap data to estimate
cycle-specific vortex convection speeds, and to understand variability in the
lift coefficient. They found that stall onset varied cycle-to-cycle, as did
vortex convection speed, most prominently post-stall. Because of this, phase-
averaging inadequately represented portions of the data set. Kuppers et al.
[11] showed that using hierarchical clustering on pressure tap data on a
flapping foil produced physically meaningful clusters (one exhibiting a higher
secondary lift peak and the other a lower one) even though no clear bimodal
behavior existed in the bivariate distributions. Ramasamy et al. [22] also
employed clustering to investigate cycle-to-cycle variation in pressure data
for a pitching foil. They focused on cases with bimodal behavior in bivariate
distributions and found that the clusters diverged in the post-stall region.
They also showed that cluster-conditional averages deviated substantially from
phase-averages, and that the clusters differed in shedding timing of the
dynamic stall vortex (inferred from the pressure data), lift production, and
flow recovery. They concluded that the clusters were associated with physical
processes. These studies highlight the presence and complexity of cycle-to-
cycle variability in systems that exhibit dynamic stall, and also demonstrate
that phase-averaging can generate misleading representations. However, because
none of these studies involve flow-field or inflow velocity measurements, they
could not quantify the sources of the observed variability, characterize the
flow-field variability beyond inference from force/pressure data, or directly
correlate force and flow-field variability.

An analysis that considers both forces and flow fields is necessary to
understand the extent and sources of the variability present and the
sensitivity of the dynamics. However, unlike forces and pressures, flow-field
data is high-dimensional in space and time, which poses apparent limitations
on the use of clustering. Specifically, a “curse of dimensionality” arises for
flow-field data because the algorithmic distances between all data points
become approximately equal as the number of coordinates grows. This makes
direct clusters difficult to define because all data points appear
increasingly similar to one another [23]. As such, clustering of high-
dimensional data requires adaptations for dimensionality reduction. Principle
component analysis, PCA, is a well-known technique useful for decoupling the
dynamics of complex, high-dimensional data sets, for feature selection, and
for dimensionality reduction [24, 25, 26, 27, 28, 29, 30]. Additionally,
several groups have demonstrated the use of PCA as a means of preserving
dynamics otherwise smeared out by phase-averaging of unsteady, vortex-
dominated flows [13, 31, 15]. Clustering in a PCA subspace is particularly
useful because the PCA basis optimally maximizes variance in the data [22].
Because of this, PCA and clustering are commonly used in conjunction [32, 33,
34, 35, 36]. Additionally, clusters in a PCA subspace are a useful basis for
producing probabilistic reduced-order models and for cluster-based feedback
control [37, 38].

We explore the topic of cycle-to-cycle variability in cross-flow turbines
using near-blade flow fields and performance (i.e., power output) for two
distinct operating conditions. Through this, we characterize the flow-field
variability and its correlation with turbine performance using unsupervised,
hierarchical clustering with a PCA pre-processor on the flow fields. The paper
is laid out as follows. Section 2 provides a theoretical background for flow-
field interactions with the moving rotor, then lays out the methodology for
turbine performance and flow-field measurement, flow-field clustering, and
correlations between flow-field clusters and performance. Section 3 quantifies
the extent of cycle-to-cycle performance and flow-field variability, then
explores how near-blade hydrodynamics, inflow velocity, and hysteresis from
previous cycles contribute to the observed variability.

##  2 Methods

We begin with a discussion of the theoretical blade-level hydrodynamics in
Section 2.1, then discuss the experimental acquisition of the simultaneous
performance and PIV measurements in Sections 2.2-2.4, and conclude with a
detailed description of the PCA analysis and flow-field clustering pipeline in
Section 2.5.

###  2.1 Theoretical Blade-level Hydrodynamics and Performance

To contextualize experimental flow fields and performance, it is instructive
to consider the kinematics and dynamic stall theory relevant to the
hydrodynamics of cross-flow turbines. A schematic of the blade geometric
definitions pertinent to the kinematics is given in 1a,b. Two key factors
which govern the near-blade hydrodynamics are the nominal angle of attack,
$\alpha^{*}$ (affecting lift and drag coefficients), and the nominal incident
velocity, $U^{*}_{rel}$, (affecting the magnitude of the lift and drag
forces). In the absence of any induced flow (i.e., proximate changes in
streamwise and cross-stream velocities as a consequence of interaction with
the turbine), the nominal angle of attack, defined as the angle between the
chord line and $U^{*}_{rel}$ at the quarter chord, $C/4$, is

$\alpha^{*}(\lambda,\theta)=tan^{-1}\bigg{[}\frac{sin(\theta)}{\lambda+cos(\theta)}\bigg{]}+\alpha_{p}$
(1)

. Here $\alpha_{p}$ is the blade preset pitch angle, $\theta$ is the blade
azimuthal position ($\theta\>=\>0^{\circ}$ defined as a turbine blade pointing
directly upstream), and $\lambda$ is the tip-speed ratio. The tip-speed ratio
is a non-dimensional representation of the turbine rotation rate, defined as

$\lambda\>=\>\frac{r\omega}{U_{\infty}}$ (2)

where $r$ is the turbine radius and $\omega$ is the rotation rate. The nominal
incident velocity (relative velocity to $C/4$) is the vector sum of the
tangential velocity, $r\omega$, and the freestream velocity, $U_{\infty}$,
such that its non-dimensional magnitude is

$\frac{||U^{*}_{rel}(\lambda,\theta)||}{U_{\infty}}=\sqrt{\lambda^{2}+2\lambda
cos(\theta)+1)}.$ (3)

Azimuthal variations in $||U^{*}_{rel}||$ and $\alpha^{*}$ over one turbine
rotation are shown in Figure 1c,d. For negative $\alpha^{*}$, on the upstream
sweep, the lift vector points inward to the center of rotation and, therefore,
the suction side of the blade is the inner surface. Conversely, for positive
$\alpha^{*}$ on the downstream sweep, the suction side of the blade is the
outer surface. We refer to these as _nominal_ quantities because induction,
manifesting as deceleration of the flow through the turbine rotor, is
appreciable but unknown.

Because $\alpha^{*}$ and $U^{*}_{rel}$ depend on $\lambda$, the phase,
duration, and severity of dynamic stall are influenced by this parameter. A
decrease in $\lambda$ reduces $||U^{*}_{rel}||$ and increases the range of
$\alpha^{*}$ encountered during a cycle, which corresponds to earlier vortex
shedding, increased stall severity, and delayed flow recovery. In severe, or
“deep” dynamic stall cases (lower $\lambda$, larger $\alpha$ ranges), the
near-blade flow field is characterized by the formation and roll-up of an
energetic dynamic stall vortex that is on the order of the blade chord. This
vortex grows to maturity and sheds before the maximum angle of attack is
reached. After shedding, the blade experiences a sharp drop-off in lift and an
increase in drag. In contrast, any vortex growth in “light” dynamic stall
cases (higher $\lambda$, smaller $\alpha$ ranges) is prematurely terminated
and shedding is induced by downward flow entrainment as $\alpha$ begins to
decrease [39, 40].

Figure 1: (a,b) Blade geometric definitions, (c) normalized nominal relative
velocity trajectories, and (d) nominal angle of attack for the two tip-speed
ratios. The tangential velocity is defined tangent to the chord line (dashed
line through blade). A positive pre-set pitch angle is depicted in (a), and
(b) shows the angle of attack definition. The static stall angle in (d) is for
a foil in rectilinear flow at a similar Reynolds number
($Re_{c}\>=\>1.5x10^{5}$, [41]). We note that because of the rapidly varying
angle of attack and appreciable induction, the comparison between $\alpha^{*}$
and the steady-state stall angle is qualitative.

Provided turbine geometry and all other relevant non-dimensional parameters
(e.g., Reynolds number, Froude number, blockage) are held constant, phase-
averaged hydrodynamic power, $P$, and the global velocity fields,
$\vec{\boldsymbol{V}}$, are functions of $U_{\infty}$, $\lambda$, and $\theta$
[16, 18]. Within a single turbine cycle, $n$, hydrodynamic power is non-
dimensionalized as the coefficient of performance
$\eta(\lambda,\theta,n)\>=\>\frac{P}{\rho U^{3}_{\infty}Lr}$ where $\rho$ is
fluid density and $L$ is the blade span. The coefficient of performance is
often presented as $C_{P}$, but $\eta$ is used here for notational simplicity
and does not imply a “water-to-wire” efficiency. The upstream sweep
($\theta\>=\>0^{\circ}\>-\>180^{\circ}$) is commonly referred to as the “power
stroke” of the turbine as it produces most of the hydrodynamic power, while
the downstream sweep ($\theta\>=\>180^{\circ}\>-\>360^{\circ}$) is
characterized by parasitic drag, post- and secondary-stall events, and
boundary layer reattachment. The deviation between nominal and true values for
$\alpha$ and $U_{rel}$ (Figure 1) is most pronounced on the downstream sweep
because of momentum extraction by the upstream sweep.

###  2.2 Experimental Facility

Experiments were performed in the Alice C. Tyler flume at the University of
Washington, a rendering of which is shown in Figure 2a. The data presented in
this paper utilized a mean dynamic water depth, $h$, of 0.5 m, resulting in a
channel cross-sectional area $A_{C}$ of 0.375 m2 (0.75 m width). The water
temperature was maintained at $36.3\pm 0.2$ $\circ$C, giving a $\rho$ of 993.5
kg/m3, and a kinematic viscosity, $\nu$, of $7.1x10^{7}$ m2/s. An acoustic
Doppler velocimeter (Nortek Vectrino) operating at a 100 Hz sampling rate and
positioned approximately 5 diameters upstream of the turbine rotor, measured
an average $U_{\infty}$ of 0.69 m/s with a turbulence intensity, $TI$, of
1.8-2.1% . These conditions corresponded to a Froude number,
$Fr\>=\>\frac{U_{\infty}}{\sqrt{gh}}$, of 0.31 where the gravitational
constant $g$ is 9.81 m/s2.

Figure 2: Annotated PIV experimental setup in the flume (a,b) and turbine free
body diagram (c). The turbine is one-bladed but utilizes a top strut designed
to support two blades.

###  2.3 Cross-flow Turbine

####  2.3.1 Experimental Setup and Measurements

These experiments utilized a one-bladed (NACA 0018 foil) turbine. The turbine
has a radius of 8.6 cm, blade span of 23.4 cm, a blade chord length, $C$, of
4.06 cm and a $6^{\circ}$ preset pitch. The support structure comprised of a
NACA 0008 foil strut at the top of the blade span and a large acrylic plate
(40 cm diameter) at the bottom. The plate facilitates PIV imaging of the mid-
span from the cameras positioned below the flume (Figure 2a,c). Because
turbine torque is measured in line with the drive train, a one-bladed turbine
is necessary to directly tie performance variations to the near-blade flow
fields (i.e., with a two-bladed turbine, the torque contribution from each
blade is ambiguous). The blockage ratio, $\beta\>=\>\frac{2Lr}{A_{C}}$ is 8.9%
and the Reynolds number, $Re_{C}\>=\>\frac{U_{\infty}C}{\nu}$ is
$3.95x10^{4}$.

As shown in Figure 2c, the turbine rotation rate, $\omega$, is regulated by a
servomotor (Yaskawa SGMCS-02B servomotor, Yaskawa SGDV-2R1F01A002 drive). In
these experiments, we are operating under constant speed control holding
$\omega$ constant, which requires the servomotor to apply a variable control
torque, $\tau_{c}$, which is measured by a hollow reaction torque cell (Futek
FSH02595) rigidly coupled to the motor and flume cross beam. An air bearing
(Professional Instruments Company Block-Head 4R low-inertia) absorbs the
thrust moment while imparting a minimal bearing torque.

MATLAB Simulink Desktop Real-Time was used for data collection and turbine
control. For each control set point (i.e., constant $\omega$), torque cell
data were acquired for $60+$ seconds at 1 kHz. Blade position and $\omega$
were measured by the servomotor encoder with a resolution of $2^{18}$
counts/rotation, also sampled at 1 kHz.

####  2.3.2 Blade-level Performance Calculation

Hydrodynamic power produced by the turbine is the product of the hydrodynamic
torque, $\tau_{h}$, and $\omega$. At the turbine level, $\tau_{h}$ is the net
hydrodynamic torque produced by the blades, less the parasitic torque from
support structure drag. For constant $\omega$ and negligible bearing torque,
the torque balance reduces to $\tau_{h}=\tau_{c}$.

Because the measured torque is dominated by the parasitic torque incurred by
large bottom plate, “full-turbine” performance is not a meaningful metric. To
this end, “blade-level” $\eta$ is calculated by subtracting phase-averaged
performance for the turbine support structure (no blade), $<\eta_{S}>$, at the
same inflow conditions from the full-turbine performance measurements,
$\eta_{T}$ (Figure 3). Here the $<>$ brackets denote the phase-average which
is conditional on $\lambda$ and $\theta$, or, in other words, an average for a
single operating condition and azimuthal position across multiple cycles.
Blade-level $\eta$ is used throughout to describe cycle-to-cycle performance
variability. This approach requires that secondary interactions between the
blades and support structures are minimal (demonstrated in [42, 43]) and that
cycle-to-cycle fluctuations are dominated by variation in blade performance
and not variation from the support structure.

All performance measurements were filtered with a low-pass, zero-phase,
Butterworth filter to remove high-frequency electromagnetic interference from
the servomotor. Both the 75 and 30 Hz cutoff frequencies used for the turbine
and support structure performance data, respectively, are $10+$ harmonics
faster than the blade passage frequency. Additionally, the Butterworth filter
has no effect on the time-averaged performance. Therefore, it unlikely the
filter is removing any hydrodynamic power.

In calculating $\eta$ we note there is some ambiguity in the choice of
$U^{3}_{\infty}$. For example, one could attempt to use instantaneous velocity
measurements, but there is a temporal mismatch between the time the freestream
velocity is measured and when it interacts with the rotor. Additionally, we
cannot use PIV to estimate inflow since our field of view does not extend far
enough upstream to measure the undisturbed inflow. Our notional choices are
(1) to use a time-average of all of the cubed freestream velocity measurements
acquired at a tip-speed ratio set point, or (2) apply an advection correction,
as in Polagye et al. [44], to compute the instantaneous freestream velocities
and calculate cycle-specific kinetic power. The advection correction is based
on a cross-correlation of $U_{\infty}$ with measured power and the application
of Taylor’s frozen flow hypothesis. Option (2) performs poorly in these
experiments, in that it generates greater relative variability in $\eta$ than
measured power, possibly because of the single-bladed configuration.
Consequently, we utilize option (1), but in Section 3.3, we employ a hybrid
approach to estimate “cluster-specific” kinetic power as an explanatory factor
for cycle-to-cycle performance variation.

Figure 3: Schematic representations of the calculation of blade level
performance, $\eta$, by subtracting support-only “performance”, $\eta_{S}$,
from the full-turbine performance, $\eta_{T}$, at the same conditions.
$\eta=\eta_{T}\>-<\eta_{S}>$. Figure 4: (a) A simple timing diagram for the
PIV acquisition. Field of view locations captured for (b) CCW rotation and (c)
CW rotation. Example blade and support structure shadows are shown. (d,e) PIV
segments captured for the different fields of view and visual representation
of corresponding $A$ matrix assignments for input into the PCA pre-processor.

###  2.4 Flow Fields

####  2.4.1 PIV Data Acquisition

Two-dimensional, two-component, phase-locked, flow-field measurements were
obtained in a streamwise plane at the mid-span of the turbine, simultaneously
with turbine performance measurements. While the performance measurements were
continuously recorded over the entire turbine rotation, each PIV acquisition
only captured flow-field data during each cycle at discrete, prescribed
$\theta$ positions. PIV acquisition was controlled by TSI Insight and
acquisition for each cycle commenced upon receipt of trigger pulses sent at a
specified $\theta$ from the Simulink model controlling the turbine. The PIV
system returned pulses at each image pair capture instance (Figure 4a), the
timing of which were logged by the Simulink model. Using these timing signals,
the PIV snapshots were aligned with performance in post-processing. We note
that this does not produce perfect phase-locking, but “phase jitter” between
image pairs at the same $\theta$ was on the order of $0.009^{\circ}$, which we
deem insignificant for the current analysis.

The general arrangement of the PIV laser and cameras is shown in Figure 2a. A
dual cavity, Nd:YLF laser (Continuum Terra PIV) capable of a repetition rate
of 10 kHz, illuminated the flow with an approximately 2 mm thick light sheet
in the field of view, FoV. A high-speed camera (Vision Research Phantom v641)
with 2560 x 1600 resolution and a 105 mm lens at f# 16, and a calibration of
11.67 pixels/mm resulted in a FoV of 21.9 x 13.7 cm [$5.4C$ ($1.3D$) x $3.4C$
($0.8D$)]. The flow seeding (10 $\mu$m silver-coated hollow-glass beads)
produced particle images of approximately 3 pixels in diameter. To minimize
laser reflections at the blade surface, matte black paint was applied.

As shown in Figure 4b,c, the combination of the camera lens focal length and
streamwise extent of the laser sheet necessitated multiple FoVs to capture the
important stages of dynamic stall and flow recovery. Each FoV outline is
color-coded to denote the flow segment, $S_{k}$ (where $k$ denotes the segment
letter), used in the clustering analysis detailed in Section 2.5. Sequences of
9 image pairs for $S_{a}$, $S_{b}$ and $S_{e}$, and 6 image pairs for $S_{c}$
and $S_{d}$ were acquired per rotational cycle with prescribed angular
displacements between frames ranging from approximately $5-6.5^{\circ}$,
depending on desired phase resolution. A total of 139 image pairs ($N\>=\>139$
turbine cycles) were acquired at each rotational phase. FoV positioning relied
on a combination of camera and turbine movement. A motorized, three-axis
gantry positioned the camera relative to the laser sheet and provided the
dominant adjustment for cross-stream FoV positioning, as well as fine
adjustments in the streamwise direction. The limited streamwise extent of the
laser sheet (Figure 4b,c) necessitated shifting the turbine by
$\approx\frac{1}{2}D$ upstream to illuminate and capture the downstream blade
sweep. Both the turbine shaft and blade cast shadows in the laser sheet.
Therefore, to obtain data adjacent to the suction and/or pressure sides of the
foil at all phases of interest, PIV measurements were repeated with the
turbine spinning in both clockwise and counter-clockwise directions as
depicted in Figure 4b,c. By changing the direction of rotation, we exploit the
symmetry of the system to minimize the impact of blade shadows on the
following analysis.

PIV processing was performed in LaVision DaVis (version 10.1.1). Background
subtraction using a Butterworth filter on phase-matched images mitigated
residual reflections and background illumination variation. The shadowed
regions, visible turbine structures, and remaining reflections were manually
masked prior to PIV processing. The phase-specific masks applied are functions
of $S_{k}$, $\lambda$, and $\theta$, but constant for all $n$. Processing
utilized a multi-grid, multi-pass cross-correlation algorithm with adaptive
image deformation at a 75% overlap and 32 x 32 pixel final interrogation
window size resulting in a 0.69 mm vector spacing (approximately 60 vectors
per blade chord). Spurious vectors (less than 2%) were removed with a
universal outlier median filter utilizing a 9x9 filter region.

All vector field post-processing was performed in MATLAB (R2020b). To align
the different flow segments and limit the contribution of blade rotation in
the PCA pre-processing, each FoV was translated from the flume reference frame
to a blade reference frame. Figure 5 provides an overview of this process.
First, we located and aligned the center of rotation between the different
FoVs. Tracking a small dot on the end of the shaft and registering all
resultant PIV vector fields to a common shaft location corrected phase-
dependent shaft procession (slight run-out on the cantilevered turbine shaft).
Because of parallax and differences in the index of refraction between air and
water, the imaged center of the turbine shaft does not correspond to the
center of rotation of the turbine at the imaging plane. Therefore, we
determined a best-fit location of the center of rotation in each FoV by
manually fitting blades to the masked regions. Second, with the center of
rotation located, the flow fields were rotated to the blade-centric coordinate
system by locating the turbine blade in each $\vec{\boldsymbol{V}}$ field and
rotating the entire field to a common blade position. Third, after rotation, a
constant crop boundary and common, segment-specific, mask were applied to each
frame and the relative velocity fields with respect to the blade were computed
as the vector sum of $\vec{\boldsymbol{V}}$ with the blade tangential velocity
component, $r\omega$. The segment-specific mask was a function of $\lambda$
only and was formed by combining all the phase-specific masks in a specific
segment. Finally, the cropped fields were interpolated to a common grid
relative to $C/4$. The cropped relative velocity fields,
$\vec{\boldsymbol{\Phi}}(\lambda,\theta,n,x,y)\>=\>[u_{rel}(\lambda,\theta,n,x,y)\>,\>v_{rel}(\lambda,\theta,n,x,y)]$
within each $S_{k}$ are functions of, $\lambda$, $\theta$, and $n$.

Figure 5: Overview schematic of the flow-field rotation pipeline. The center
of rotation between the different fields of view is aligned, then the fields
are rotated and cropped into a common blade-centric reference frame. A common
mask between all the fields of view in a flow segment is applied and the
velocity fields relative to the blade are calculated.

###  2.5 Correlating Turbine Performance and Flow-field Variability with
Hierarchical Clustering

Cycle-to-cycle variations are present in performance and flow fields, but the
correlation between the two is not obvious a priori. For example, if flow
fields are observed in isolation, the significance of the observed flow
structures on turbine performance is unknown. The flow fields are high-
dimensional and a reduced order representation of $\vec{\boldsymbol{\Phi}}$ is
needed to compactly describe cycle-to-cycle variation. Clustering with a PCA
pre-proccesor allows us to do so while considering all of the flow-field
dynamics (and the interplay between them) in an unsupervised manner. Here we
describe the flow-field clustering pipeline used for each flow segment. Using
this pipeline, we correlate the variability between the simultaneously
captured performance and flow fields.

####  2.5.1 Flow-field PCA Pre-processor

Figure 6: Overview of the flow-field clustering pipeline. Everything upstream
of the hierarchical clustering is considered an element of the PCA pre-
processor. This involves performing a singular value decomposition on the
relative velocity fields in the blade-centric reference frame. The weighted
activations (multiplication of the singular values and the phase-varying
weights) are then separated by cycle. For each flow segment, the hiearchical
clustering algorithm identifies two clusters from this population of cycles.

Figure 6 describes the flow-field clustering pipeline. Here the PCA pre-
processor enables clustering of all the dynamics using a low-dimensional
subspace that is interpretable and weighted by contribution to overall
velocity variance.

PCA Setup: PCA represents the dynamics of complex data sets as the linear
summation of orthogonal modes ranked by the amount variance in the data they
describe [24]. The singular value decomposition (SVD) is used to compute the
PCA modes for a data matrix $\boldsymbol{A}_{\\#}$ as

$\boldsymbol{A}_{\\#}=\boldsymbol{\phi}\boldsymbol{\sigma}\boldsymbol{a}^{*}.$
(4)

where the PCA modes are columns of $\boldsymbol{\phi}$, the singular values,
which quantify the contribution of each mode, are diagonals of
$\boldsymbol{\sigma}$, and the phase evolution of the modes make up the rows
of $\boldsymbol{a}^{*}$. Here $*$ denotes the matrix transpose and $\\#$
specifies a specific data matrix. The percentage of the flow-field variance
described by a specific mode, $\phi_{j}$, is quantified using the
corresponding singular value $\sigma_{j}$, as
$\Psi_{j}\>=\>\frac{\sigma_{i}}{\sum_{j=1}^{J}{\sigma_{j}}}$ where $J$ is the
total number of modes (smallest dimension of $\boldsymbol{A}_{\\#}$).

Here, each $\boldsymbol{A}_{\\#}$ is made up of reshaped
$\vec{\boldsymbol{\Phi}}$ column vectors, $u_{rel}$ stacked on $v_{rel}$. Any
missing values (from the removal of spurious PIV vectors during processing)
are linearly interpolated. The columns of each $\boldsymbol{A_{\\#}}$ were
sorted chronologically
[$(n=1,\theta_{1}),(n=1,\theta_{2})...(n=N,\theta_{end})$] with the row-wise
mean subtracted prior to SVD decomposition. The presence of the shadowed
regions meant a common mask across all $\theta$ would occlude the majority of
the FoV from analysis, so different segments, $S_{k}$, at each $\lambda$ were
strategically grouped to maximize data yield. This results in five distinct
PCA computations of $\boldsymbol{A_{\\#}}$ (Figure 4). While a few of the
$\boldsymbol{A_{\\#}}$ contained multiple $S_{k}$, each $S_{k}$ was clustered
separately after PCA pre-processing. This is because each $S_{k}$ is an
independent data set (e.g., $n=1$ in $S_{a}$ is unrelated to $n=1$ in
$S_{b}$).

Computing Weighted Activation Pictures: To compactly describe how the dynamics
evolve within a rotation, “weighted activation pictures” in the PCA subspace
are constructed as $\boldsymbol{\sigma}$$\boldsymbol{a}^{*}$. This
multiplication may also be interpreted as the projection of each
$\vec{\boldsymbol{\Phi}}$ onto the PCA basis, and encodes both the dominance
and phase evolution of all the dynamics (PCA modes) of the system into lower-
dimension representations for the hierarchical clustering algorithm. The
result of this multiplication is 139 weighted activation pictures (one for
each $n$) in each $S_{k}$. Each “picture” is composed of column vectors of the
individual weighted activation profiles across all of the computed PCA modes.
The individual weighted activation profiles are functions of $\theta$, and
describe the dominance of each mode across the range of $\theta$ included in
each $S_{k}$. The weighted activation profiles associated with a mode,
$\phi_{j}$, are denoted as $a_{j}$.

####  2.5.2 Flow-field Clustering

After the PCA pre-processor, each $\vec{\boldsymbol{\Phi}}$ field is
represented in a low-dimensional form suitable for clustering. The weighted
activation pictures in each $S_{k}$, were reshaped into column vectors and
sorted into two clusters (the highest number of clusters that produces unique
phase-averaged weighted activation profiles) via a hierarchical clustering
algorithm with a Ward linkage method [45] to minimize variance and a Euclidian
distance metric. We utilized the implementation within the MATLAB
_clusterdata_ function.

####  2.5.3 Cluster Assignments

After flow-field clustering, we have a set of cluster assignments in each
$S_{k}$ (Figure 6). We can use these cluster assignments to evaluate the
correlation between the flow field, $\vec{\boldsymbol{\Phi}}$, and
performance, $\eta$, associated with the $\vec{\boldsymbol{\Phi}}$ clusters.
This basis also allows us to investigate ties between the performance and
flow-field variability and the potential sources of this variability by
considering both conditional-averages and individual trajectories of different
parameters based on flow-field cluster assignment. As shown in Figure 6,
cluster assignments are denoted as $\Phi^{k,c}_{\lambda=l}$, where $k$ is the
segment designation (a-e), $c$ is the cluster number (1 or 2), and $l$ is the
tip-speed ratio set point. Conditional-averages across $n$, based on cluster
assignment, for any variable $X$ (e.g., $\eta$, $\vec{\boldsymbol{\Phi}}$)
within one $S_{k}$ are expressed as $<X|\Phi^{k,c}_{\lambda=l}>$. In cases
where examining individual trajectories of $X$ within a cluster is preferred,
an equivalent set notation, $\\{\\}$, denotes the subset of cycles of $X$
assigned to the specified cluster. For example, a set of individual $\eta$
trajectories based on $\vec{\boldsymbol{\Phi}}$ clusters is written as

$\eta_{\Phi}(\lambda,\theta,n,c)=\\{\eta(\lambda,\theta,n)|\Phi^{k,c}_{\lambda=l}\\}.$
(5)

For all segments, $c=1$ always corresponds to the cluster with a higher
associated time-averaged performance, $\bar{\eta}$, for full revolutions. To
quantify the statistical significance of the identified clusters, we utilized
a two-sided Wilcoxon Rank Sum test [46] (MATLAB _ranksum_ function) with a 5%
significance level. The null hypothesis is that the populations of $X$
contained in each cluster are drawn from the same continuous distribution and
have equal medians. Rejection of the null hypothesis means the populations of
$X$ contained in each cluster are statistically significant.

Considering how individual trajectories or cluster conditional-averages differ
from the phase-averages over all $n$ can illuminate cluster-specific
characteristics. This is done for both performance and flow fields.
Conditional performance perturbations, $\eta^{\prime}_{\Phi}$, are defined as
the differences between the individual performance trajectories in each
cluster and the phase-averaged performance, $<\eta(\lambda,\theta,n)>$, within
the same $S_{k}$. They are computed as

$\eta_{\Phi}^{\prime}(\lambda,\theta,n,c)=\eta_{\Phi}(\lambda,\theta,n,c)-<\eta(\lambda,\theta,n)>$
(6)

These perturbation quantities help highlight where the clusters are performing
better or worse than the phase-average.

Conditional difference fields, $\Phi^{\prime}$, highlight how the conditional-
averaged velocity magnitude fields,
$||<\vec{\boldsymbol{\Phi}}|\Phi^{k,c}_{\lambda=l}>||$, differ from the phase-
averaged fields, $||<\vec{\boldsymbol{\Phi}}>||$. As with the phase-averaged
fields, the conditional-averaged velocity magnitudes are computed from the
component averages (i.e., $||<u_{rel}>,<v_{rel}>||$). The conditional
difference fields, computed at each point in space ($x$,$y$), are defined as

$\Phi^{\prime}_{\Phi}(\lambda,\theta,c,x,y)=||<\vec{\boldsymbol{\Phi}}|\Phi^{k,c}_{\lambda=l}>||-||<\vec{\boldsymbol{\Phi}}>||.$
(7)

##  3 Results

We explore the three potential sources of cycle-to-cycle performance
variability: variations in near-blade flow fields, variations in inflow
velocity, and hysteresis from previous cycles. Section 3.1 describes the
statistical variability in performance and flow fields, highlighting the
influence of tip-speed ratio and the blade’s phase in the rotation. In Section
3.2, we utilize flow-field clustering to correlate cycle-to-cycle performance
and flow-field variability. We conclude with a discussion of the impact of the
freestream velocity perturbations in Section 3.3 and hysteresis in Section
3.4.

###  3.1 Performance and Flow-field Variability

The tip-speed ratio affects both time-average performance and the properties
of cycle-to-cycle variation around that average, as visualized in Figure 7.
Mean performance and its variability both increase with tip-speed ratio but
performance reaches a maximum near $\lambda\>=\>2.9$, while variability
continues to increase at higher tip-speed ratios. For the remainder of this
work, we focus on two cases: $\lambda\>=1.5$ (sub-optimal performance) and
$\lambda\>=2.5$ (near-optimal performance) which exhibit differences in both
time-averaged performance and cycle-to-cycle variability. Both cases are
relevant to turbine operation because maximizing efficiency (optimal tip-speed
ratio) is a control objective when the freestream velocity is below the
turbine’s rated speed, while the sub-optimal case corresponds to a strategy
for shedding power above the rated speed. We note that the larger variation
associated with $\lambda\>=\>2.5$ is due in part to the larger performance and
address this later by normalizing by the time-averaged performance as the
coefficient of variation.

Figure 7: Time-averaged coefficient of performance for the one-bladed turbine
investigated. The dashed line represents the time-average at each $\lambda$.
Near-optimal (green dot) and sub-optimal (purple dot) cases are highlighted.
The bivariate distribution (yellow to blue shading) describes the range of
time-average performance over individual cycles at each $\lambda$.

A defining feature of cross-flow turbine performance is periodic variation
within a cycle (i.e., performance variation with $\theta$). The polar
representations of performance and flow fields for both tip-speed ratios in
Figure 8 show that the largest cycle-to-cycle variation in phase-specific
performance occurs in the upstream sweep around the performance peak
($\theta\>=\>30^{\circ}\>-\>135^{\circ}$ for $\lambda\>=\>1.5$ and
$\theta\>=\>60^{\circ}\>-\>165^{\circ}$ for $\lambda\>=\>2.5$). In agreement
with time-averaged performance (Figure 7), the $\lambda\>=\>1.5$ case exhibits
less cycle-to-cycle variability during the smaller, narrower performance peak
(lower time-averaged performance) than the $\lambda\>=\>2.5$ case. The phase-
specific variation in performance is appreciable, but secondary in comparison
to the range in performance over a cycle. For both tip-speed ratios, the
bivariate distributions of performance are unimodal at each $\theta$,
suggesting that this local variation is adequately described by the mean and a
descriptor of the spread [14].

The phase-averaged flow fields of global velocity magnitude around a cross-
flow turbine (Figure 8) depend strongly on blade position and tip-speed ratio.
Overall, dynamic stall severity and turbine performance are inversely
proportional for the two cases (i.e., the higher tip-speed ratio decreases
stall severity and increases performance). Specifically, coherent structures
from dynamic stall are more apparent for the sub-optimal case,
$\lambda\>=\>1.5$, than for the near-optimal case, $\lambda\>=\>2.5$. In
comparing the flow fields with turbine performance, we see that for
$\lambda\>=\>1.5$, the performance peak coincides with formation, growth, and
shedding of the dynamic stall vortex. On the other hand, for
$\lambda\>=\>2.5$, the flow remains primarily attached until maximum
performance, beyond which we observe increasing separation at the trailing
edge. For both tip-speed ratios, performance is at a minimum during the
downstream sweep, consistent with lower incident velocities due to extraction
of momentum from the flow during the “power stroke”. The flow fields are in
qualitative agreement with dynamic stall theory (Section 2.1), as well as
prior experiments [5, 7, 2] and simulation [47, 3] for cross-flow turbines.
Because the lower tip-speed ratio leads to a wide oscillation in angle of
attack, the $\lambda\>=\>1.5$ case experiences “deep” dynamic stall. This is
evidenced by a strong dynamic stall vortex that is shed before the maximum
nominal angle of attack (Figure 8a), as well as prolonged post-stall flow
separation. In contrast, for $\lambda\>=\>2.5$, the foil experiences only
“light” dynamic stall with smaller vortex structures, limited flow separation
until near the maximum nominal angle of attack, and faster post-stall flow
recovery.

Figure 8: Bivariate distribution of phase-specific coefficient of performance
(blue-to-purple shading with 0.05 $\Delta\eta$ in the radial direction and
$2^{\circ}$ $\Delta\theta$ in the $\theta$ direction) for a deep dynamic stall
case at $\lambda\>=\>1.5$ and a light dynamic stall case at $\lambda=2.5$. The
orange line corresponds to $\eta\>=\>0$, demarcating the power producing from
power consumptive phases. Inset are phase-averaged global velocity magnitude
fields (normalized by the freestream velocity) [NOTE: the blade path (blue
dashed line) is not to scale]. The velocity fields are presented in the fixed
global reference frame. Shaded regions at the periphery denote the radial
extent of flow segments for PIV, $S_{k}$. Figure 9: Coefficient of variation
fields (colorbar has been truncated at 50% for visualization) at select
$\theta$ for (a) $\lambda\>=\>1.5$ (b) and $\lambda\>=\>2.5$ and their
accompanying relative velocity fields (normalized by the freestream velocity).
The location of the $\theta$ positions in (a-b) are labeled in (c-d) which
show the (c) sum of the coefficient of variation fields (normalized by the
number of data points) (d) and the coefficient of variation for performance
for both $\lambda$.

To characterize the cycle-to-cycle variation in the flow fields, we compute
the standard deviation fields, $s_{\Phi}$. These are presented normalized by
the mean, as the coefficient of variation, to facilitate comparisons with
performance variability. Since the flow-field data is sparse in $\theta$, we
cannot calculate an accurate time-average, so we define the flow-field
coefficient of variation as
$CV_{\Phi}(\lambda,\theta,x,y)=\frac{s_{\Phi}}{\tilde{\phi}}$, where
$\tilde{\phi}$ is the mean of all the PIV vectors collected at a given phase.
Figure 9 demonstrates the spatial extent of flow-field variability as a
function of $\theta$ and $\lambda$. In this figure, $CV_{\Phi}$ ranges from
$1\%$ to $150\%$. During the early portion of the cycle (locations i and ii,
$S_{a}$), variability is confined to a region on the order of the foil
thickness close to the blade and in the wake (Figure 9a,b) for both tip-speed
ratios. The two cases diverge around $\theta$ = 90∘ (beginning of $S_{b}$).
There, for $\lambda\>=\>1.5$, flow-field variability expands in the vicinity
of the dynamic stall vortex. In contrast, at the same $\theta$, variability
for $\lambda\>=\>2.5$ remains confined to the blade wake and the region of
separated flow at the trailing edge. By the end of the cycle (location v,
$S_{e}$), for $\lambda\>=\>2.5$, variation is once again primarily limited to
the blade wake while for $\lambda\>=\>1.5$, we observe high variability in a
large region of separated flow (locations iv.-v., $S_{c}-S_{e}$). We can
compactly visualize the trends across phase by summation of ${CV_{\Phi}}$ in
space (i.e., $\sum{CV_{\Phi}}$). As shown in Figure 9c, $\sum{CV_{\Phi}}$ is
higher at all resolved phases for $\lambda\>=\>1.5$ than $\lambda\>=\>2.5$.
For $\lambda\>=\>1.5$, $\sum{CV_{\Phi}}$ increases dramatically during the
growth of the dynamic stall vortex ($S_{b}$) and, for the resolved phases of
the power stroke, reaches a maximum at location iii. This is in agreement with
prior work that has consistently demonstrated large increases in cycle-to-
cycle variation post-stall [12, 11, 22]. In contrast, for $\lambda\>=\>2.5$,
$\sum{CV_{\Phi}}$ remains relatively low during the power stroke, and the
maximum variability likely occurs in a portion of the flow field not included
in these experiments. For both cases, the highest observed flow-field
variability occurs in the downstream sweep. In summary, cycle-to-cycle flow-
field variability increases with stall severity and is concentrated around
coherent structures originating from the blade or in the blade wake.

The increase in flow-field variability for the lower tip-speed ratio is
expected given the higher stall severity, but is inverted from the trends in
time-average and phase-specific performance variability, which are higher for
$\lambda\>=\>2.5$ (Figure 7 and 8). However, the coefficient of variation for
performance, $CV_{\eta}=\frac{s_{\eta}}{\bar{\eta}}$ (Figure 9d), is higher
for $\lambda\>=\>1.5$ across most of the cycle, consistent with flow-field
variability. In other words, while $\lambda\>=\>2.5$ exhibits higher absolute
performance variability, the $\lambda\>=\>1.5$ case has higher relative
variability.

Finally, it is instructive to consider the phase correlation between flow-
field and performance variability. For $\lambda\>=\>1.5$, $\sum{CV_{\Phi}}$
lags $CV_{\eta}$, particularly in $S_{a}$ (Figure 9c,d), which suggests that
corresponding flow-field variability present during maximum performance
variability likely occurs too close to the blade to be resolved by these PIV
measurements. Similarly, for $\lambda\>=\>2.5$, the $\sum{CV_{\Phi}}$ is
minimal during the phases of maximum performance variability. For both tip-
speed ratios, the observed high flow-field variability in the downstream sweep
is associated with lower relative performance variability (we address this
contradiction further in Section 3.2).

Figure 10: Clustering analysis overview for for (a) $\lambda\>=\>1.5$ (b) and
$\lambda\>=\>2.5$. (i,ii) Cluster conditional difference fields and (iii)
phase averages for selected $\theta$, with (iv) corresponding performance
trajectories and (v) performance perturbations. The grid spacing in (i), (ii)
and (iii) is $C/4$. In (iv), each line is colored by cluster assignment and
the black lines represent the cluster conditional-averages for performance or
performance perturbations. The dashed rectangles denote the $\theta$ range
($S_{b}$) where the flow fields in (i),(ii) and (iii) were captured.

###  3.2 Flow-field Clusters and Their Associated Performance

Due to the non-coincidence between the performance and flow-field variability,
we must employ a different approach to gain insight into how the observed
flow-field variability is correlated with performance variability in other
portions of the cycle. By utilizing the flow-field clusters and their
associated, cycle-specific performance trajectories, $\eta_{\Phi}$, we are
able to better characterize the flow-field and performance variability, as
well as identify correlations between them. This is done in an unsupervised
manner, without relying on hand-engineered metrics. For this analysis, we
focus on segment $S_{b}$, which encompasses the end of the power stroke for
both tip-speed ratios and the shedding of a dynamic stall vortex for
$\lambda\>=\>1.5$.

Figure 10 summarizes the comparison of the flow-field clusters and their
corresponding performance trajectories for the two tip-speed ratios within
segment $S_{b}$. The conditionally-averaged difference fields (i and ii)
highlight the deviation between the cluster conditionally-averaged fields and
the phase-averaged fields (iii). The performance trajectories (iv) and their
perturbations from the phase-average (v) for each cycle reveal correlations
between performance and the flow-field clusters. For both tip-speed ratios,
the conditionally-averaged difference fields reveal opposing behaviors between
the clusters (e.g., regions of lower-than-average relative velocities in one
cluster are coincident with regions of higher-than-average relative velocities
in the other). In line with the coefficient of variation fields (Figure 9),
the differences are more pronounced for $\lambda\>=\>1.5$ than for
$\lambda\>=\>2.5$. Yet, for both tip-speed ratios, performance trajectories
are separated by their associated flow-field clusters (iv and v). More
significantly, the flow-field clusters for both tip-speed ratios are
correlated with differences in maximum performance (Figure 10iv), even though
$S_{b}$ does not fully span the performance peak. This suggests that the flow-
field variability in $S_{b}$ stems from the growth and shedding of
hydrodynamic structures which originate from but are unresolved in segment
$S_{a}$ due to their proximity to the blade. Finally, we note that, for
$\lambda\>=1.5$, the cluster-averaged performance perturbations reveal that
cluster 1 and cluster 2 alternate phases of superior performance over the
cycle, even though cluster 1 out-performs cluster 2 on a time-averaged basis.

Considering all segments, the difference between time-averaged performance
associated with the two clusters is highest in $S_{a}$ (suggesting the
greatest correlation between flow fields and performance is found here) and
declines in segments later in the cycle for both tip-speed ratios The higher
correlation between time-averaged performance and segment-specific flow fields
around the performance peak is expected, as the majority of the power is
produced there. Similarly, given the low performance throughout the downstream
region, resolvable flow-field variability in this region would have a limited
impact on time-averaged performance. Additionally, beyond the performance
peak, the hydrodynamic features associated with power production advect,
diffuse, and are diluted in the post-stall region, thereby obscuring their
connection to performance earlier in the cycle. The obscuring of structures
originating from the power stroke is further exacerbated in the downstream
sweep by the switching of the pressure and the suction sides of the blade.
Thus it is unlikely that the hydrodynamics in the downstream sweep retain any
clear correspondence to those in the upstream sweep.

We now dive deeper into the two tip-speed ratio cases for segment $S_{b}$ to
describe how the flow fields differ between clusters and their relationship
with performance. We primarily focus on the $\lambda\>=\>1.5$ case since it
exhibits strong flow-field variability. We then utilize the $\lambda\>=\>2.5$
case to highlight how, despite the lower flow-field variability and limited
near-blade resolution, we are still able to extract useful insight from the
flow-field clusters.

####  3.2.1 $\lambda\>=\>1.5$

For this tip-speed ratio, the deep dynamic stall results in growth and
shedding of a dynamic stall vortex in segment $S_{b}$. Difference fields for
the cluster with higher time-averaged performance, cluster 1, highlight
primarily lower-than-average velocities, with only small regions of higher-
than-average velocity. The opposite behavior is seen for the poorer performing
cluster, cluster 2. These velocity differences suggest several possible
mechanisms: the dynamic stall vorticies for the two clusters differ in
strength, in their location with respect to the blade (a consequence of
different shedding timing), or both.

The difference in time-averaged performance between the two clusters is
comparatively small (1.5% with respect to the time-average), but, as apparent
in Figure 10a-iv, the performance peak for cluster 1 is slightly higher in
amplitude and occurs slightly earlier in the cycle. The shift in phase of
maximum performance is highlighted in Figure 11a. On average, maximum
performance occurs at $\theta\>\approx\>83^{\circ}$ for cycles in cluster 1
and at $\theta\>\approx\>83.9^{\circ}$ for cycles in cluster 2. The
distributions of the phase of maximum performance are statistically
significant, as per the rejection of the null hypothesis for the Wilcoxon rank
sum test (Section 2.5.3). These differences in the timing and amplitude of the
performance peak between the clusters counter the phase-average trends over
tip-speed ratio (Figure 11b) where a performance peak occurring earlier in the
cycle has lower maximum and time-averaged performance, and is associated with
operation at a reduced tip-speed ratio.

Figure 11: Histograms of cycle-specific phase associated with maximum
performance for (a) $\lambda\>=\>1.5$ (c) and $\lambda\>=\>2.5$. (b) Phase-
averaged coefficient of performance for select tip-speed ratios.

To gain further insight into how vortex dynamics differ between the two
clusters, we first consider modal analysis. A key benefit of the PCA pre-
processor is the ability to interpret the resulting clusters through their
modes and modal coefficients. The directions of the largest variance in the
data are described by the modes and sorted by the singular values. Thus the
first mode describes the hydrodynamics with the highest variability and the
variation described decays as mode number increases. By considering the modal
flow fields and their accompanying weighted activation profiles, we can
identify the dynamics most associated with the variation and how their
contribution changes with blade position and/or between clusters.

Figure 12: (left) Weighted activation profiles across each cycle, colored by
the flow-field cluster assignments for $\lambda\>=\>1.5$ are presented for
$S_{a}$ and $S_{b}$. The opaque thick lines are the conditional-averages for
each cluster and the black dashed line is the phase-average over all cycles.
(right) Magnitude fields for the first three modes with variance explained for
each mode noted.

Figure 12 shows the first three modes and the variability in their weighted
activation profiles within segments $S_{a}$ and $S_{b}$. These three modes
describe nearly $14\%$ of the flow-field variance within these segments. We
interpret modes 1 and 2 as related to dynamic stall vortex shedding while mode
3 is attributed to near-blade vortex dynamics occurring earlier in the cycle.
In this interpretation, mode 1 represents shedding dynamics that occur once
the vortex core is at least a foil thickness away from the blade while mode 2
occurs earlier in the shedding process. The high variance explained by mode 1
is consistent with the velocity magnitude (Figure 9a and 10a-iii) and
coefficient of variation fields (Figure 9a), which are dominated by the
shedding of the dynamic stall vortex.

We observe that the weighted activation profiles for these three modes are
well-converged in $S_{a}$ but diverge between the flow-field clusters in
$S_{b}$. In $S_{b}$, the better-performing cluster, cluster 1, has larger
weighted activations in mode 1 and the weighted activations are shifted
earlier in phase for mode 2. For mode 3, both clusters have similar mean-
weighted activations everywhere except for
$\theta\>=\>90^{\circ}\>-\>120^{\circ}$. This is consistent with low flow-
field variability in segment $S_{a}$ and increased variability in $S_{b}$
(Figure 9c and Figure 10). At $\theta\>=\>119^{\circ}$, where the difference
fields are most distinct, we see that the higher-than-average velocities
closer to the blade for cluster 1 are captured by the higher mode 1 weighted
activation. On the other hand, the lower mode 2 weighted activation for
cluster 1 at this position captures the lower-than-average velocities farther
from the blade.

To establish some physical intuition for the differences between the two
clusters in segment $S_{b}$, we explore the chord-wise and chord-normal
position of the dynamic stall vortex, ($DSV_{C}$ and $DSV_{\perp}$
respectively), and the reversed flow fraction, $U^{rev}_{\Phi}$ (Figure
13a-c). The position of the vortex core is defined as the location of the
maximum value of the swirling strength in individual flow fields reconstructed
with 30 PCA modes. The truncation resulted in cleaner swirling strength fields
which significantly improved tracking performance. There is no statistical
correlation between the clusters and chord-wise DSV position. For
$\theta\>=\>107^{\circ}-137^{\circ}$ however, the chord-normal separation
distance is statistically higher for cluster 1. The larger chord-normal
separation distance combined with the earlier change in the $DSV_{\perp}$
slope for cluster 1 is indicative of earlier vortex shedding. These results
are consistent with [22] who showed for a pitching foil that clusters based on
pressure data were able to reveal earlier shedding of a dynamic stall vortex.

An increase in reversed flow fraction can indicate more flow separation on the
blade and therefore more severe stall. We define this quantity as

$U^{rev}_{\Phi}(\lambda,\theta,n,c)=\frac{\sum{}{}[\\{u_{rel}|\Phi^{s,c}_{\lambda=l}\\}<0]}{NxNy},$
(8)

where the $[$ $]$ brackets represent Iverson Bracket Notation which, similar
to the Kronecker Delta, returns a value of 1 when the statement in the
brackets is true and a value of 0 when false. Cluster 1 exhibits more reversed
flow everywhere in $S_{b}$ (statistically significant). This is consistent
with earlier separation and the possibility that cluster 1 cases have stronger
dynamic stall vorticies, as evidenced by the higher-than-average velocities
near the blade.

In summary, we see clear differences between the clusters for modes that
represent various periods of the dynamic stall process, and have shown that
the dynamic stall vorticies have different velocity distributions and are shed
at different times. Specifically, cycles in cluster 1, have better
performance, more reversed flow, higher near-blade velocities in the dynamic
stall vortex, and dynamic stall vortex cores that are further from the blade.
These hydrodynamics are indicative of an earlier and potentially stronger
stall which is consistent with the maximum performance occurring earlier in
the cycle (Figure 11).

Figure 13: Location of the dynamic stall vortex core in the chord-wise
direction, $DSV_{C}$, (a) and in the chord-normal direction, $DSV_{\perp}$ (b)
for both clusters. Vortex positions are normalized by the chord length, $C$.
(c) Reversed flow fraction. The solid lines are the conditional-averages
associated with each cluster and the violin plots (_violinplot_ MATLAB
function from [48]) at each $\theta$ combine box plots and smoothed histograms
to highlight the underlying distribution of the populations. The $\mathbf{*}$
denote phases where the result of the Wilcoxon rank sum test show we cannot
reject the null hypothesis that the clusters are samples from continuous
distributions with equal medians at the 5% significance level (i.e., the
difference between the two distributions may not be statistically
significant).

####  3.2.2 $\lambda\>=\>2.5$

The higher tip-speed ratio case presents an opportunity to test the flow-field
clustering method when there is less apparent flow-field variability due to
the lighter dynamic stall and weaker vortex shedding that remains closer to
the blade. Despite this, the difference in time-averaged performance between
the two flow-field clusters is 2% with respect to the total time-average. This
is actually slightly higher than for $\lambda\>=\>1.5$ and consistent with the
higher absolute performance variability for $\lambda\>=\>2.5$. Referring once
again to Figure 10, the amplitude of the performance peak for the better-
performing cluster, cluster 1, is slightly higher, but, unlike for
$\lambda\>=\>1.5$, there is no apparent phase shift (Figure 11c). This is
consistent with the convergence in the phase of maximum performance at higher
tip-speed ratios in Figure 11b. Despite the relatively low variability, the
flow-field clustering results in lower-than-average velocities at the trailing
edge of the blade and slightly higher-than-average velocities near the leading
edge for cluster 1, while the opposite is true for cluster 2.

Figure 14: (left) Weighted activation profiles across each cycle, colored by
the flow-field cluster assignments for $\lambda\>=\>2.5$ are presented for
$S_{a}$ and $S_{b}$. The opaque thick lines are the conditional-averages for
each cluster and the black dashed line is the phase-average over all cycles.
(right) Magnitude fields for the first three modes with variance explained for
each mode noted.

The first three modes capture nearly 14% of the total variance and depict
spatial structures on the order of a foil thickness that remain close to the
blade and in the wake (Figure 14). We interpret mode 1 as pertaining to
attached flow on the blade throughout segment $S_{a}$, followed by an increase
in reversed flow at the trailing edge as $\sigma_{1}a_{1}$ crosses zero
(representing a change in sign of the mode), and the change in direction and
magnitude of the relative velocity (Figure 10b-iii). Mode 2 represents
increased flow along the entire blade when the weighted activations are
positive, but enhanced separated flow at the trailing edge when negative.
Similarly, mode 3 describes subtle changes in velocities along the blade. The
discontinuity in mode 3 between segments is not likely the result of a rapid
change in the dynamics. It is instead most likely a consequence of uncertainty
in establishing the blade position between segments, which manifests as a
slight misalignment between flow fields of different segments during the
translation to the blade-centric reference frame. The $\lambda\>=\>2.5$ case
is more sensitive to this error because the energetic dynamics are located
adjacent to the blade, and therefore more susceptible to occlusion by the
common mask, much more so than for $\lambda\>=\>1.5$. The weighted activation
profiles reveal minimal deviations between the clusters, with the exception of
mode 2 in $S_{a}$ and mode 3 in $S_{b}$. For mode 3, the relatively lower
weighted activation for cluster 1 captures the lower-than-average trailing
edge velocities. These lower velocities are indicative of more flow separation
(theoretically limiting lift production) which contradicts the higher time-
averaged performance for the cluster.

In summary, the $\lambda\>=\>2.5$ case exhibits less flow-field variability in
comparison to $\lambda\>=\>1.5$, as well as more muted conditional-average
difference fields and modes, and less distinct cluster specific weighted
activation profiles. Despite this, the flow-field clusters and their
associated performance are distinct and meaningful.

###  3.3 Impact of Freestream Velocity Perturbations

For both tip-speed ratios, we observe a dichotomy between the flow-field and
performance trends. The flow fields for cluster 1 show evidence of earlier and
potentially stronger stall, but cluster 1 has higher time-averaged
performance. While the flow-field and performance differences between the
clusters could be the result of the stochastic dynamic stall process, an
alternative hypothesis is that these variations are the result of freestream
velocity perturbations. Any changes in the freestream velocity impact both the
kinetic energy available in the flow and the instantaneous tip-speed ratio.
Cycle-to-cycle variation in the rotation rate is negligible (standard
deviations in $\omega$ are 0.02-0.03% of the time-average), so any
perturbations in the freestream will result in cycle-specific tip-speed ratios
that differ from the average. Figure 15 demonstrates that for both tip-speed
ratios, the instantaneous freestream velocities (advection-corrected as in
Section 2.3.2) are correlated with the flow-field clusters. The difference
between the mean of each cluster, is 1.45% and 1.2% of the time-averaged
velocity for $\lambda\>=\>1.5$ and $\lambda\>=\>2.5$, respectively. These
differences were found to be statistically significant. For both tip-speed
ratio cases, cluster 1 has a higher conditionally-averaged inflow velocity. As
a result, the blade is effectively operating at a lower tip-speed ratio for
cluster 1 cycles while also encountering more kinetic energy in the flow.

Figure 15: Histogram of freestream velocities for clusters derived from flow
fields in $S_{b}$ at (a) $\lambda\>=\>1.5$ and (b) $\lambda\>=\>2.5$. Figure
16: (a) $\lambda\>=\>1.5$ and (b) $\lambda\>=\>2.5$. Individual (i)
coefficient of performance and (ii) cluster-specific coefficient of
performance trajectories around the peak for the two clusters. The solid lines
are the phase-average for cluster 1 and the dashed lines are the phase-average
for cluster 2. Cluster-specific histograms of (iii) cycle-specific, time-
averaged coefficient of performance and (iv) time-averaged, cluster-specific
coefficient of performance. The $\mathbf{*}$ denotes that results of the
Wilcoxon rank sum test show we cannot reject the null hypothesis that the
clusters are samples from continuous distributions with equal medians at the
5% significance level.

Since the coefficient of performance, $\eta_{\Phi}$, is defined using the
time-average of the cubed freestream velocity measurements acquired at the
tip-speed ratio set point, it does not account for cycle-to-cycle inflow
variability. Consequently, we consider a cluster-specific kinetic power,
proportional to the conditional-average of the cubed freestream velocities
associated with each cluster. Utilizing this, a cluster-specific coefficient
of performance, $\eta^{*}_{\Phi}$, is defined as

$\eta^{*}(\lambda,\theta,n,c)=\frac{P}{\frac{1}{2}\rho\overline{\\{U_{\infty}^{3}|\Phi^{s,c}_{\lambda=l}\\}}LD}.$
(9)

A comparison between the performance trajectories around the peak and
histograms of cycle-specific, time-averaged performance for $\eta_{\Phi}$ and
$\eta^{*}_{\Phi}$ are presented in Figure 16. For $\lambda\>=\>1.5$, both the
$\eta_{\Phi}$ and $\eta^{*}_{\Phi}$ trajectories around the peak are
statistically significant between the clusters (Figure 16a-i,ii). However, the
locally higher $\lambda$ cluster, cluster 2, subtly outperforms cluster 1 at
maximum $\eta^{*}_{\Phi}$. Similarly, the time-averaged distributions for both
$\eta_{\Phi}$ and $\eta^{*}_{\Phi}$ are statistically significant between the
clusters, but cluster 2 now outperforms cluster 1. These observations are now
consistent with phase-averaged (Figure 11a) and time-averaged (Figure 7)
performance trends across tip-speed ratios.

The results are much the same for the $\lambda\>=\>2.5$ case with the
exception that the $\eta^{*}_{\Phi}$ trajectories at the performance peak are
no longer statistically significant between the clusters (Figure 16b-ii),
though the time-averaged $\eta^{*}_{\Phi}$ distributions remain distinct. In
comparison to $\lambda\>=\>1.5$, we observe a better collapse in
$\eta^{*}_{\Phi}$ for the $\lambda\>=\>2.5$ case despite its higher
performance variability. This is likely because the angle of attack profiles
become less dependent on the tip-speed ratio as the tip-speed ratio increases
(Figure 1). This means that, nominally, the sensitivity of the blade
kinematics to inflow perturbations is inversely proportional to the tip-speed
ratio. As a consequence, the hydrodynamics for $\lambda\>=\>2.5$ are
potentially less sensitive to inflow perturbations than for $\lambda\>=\>1.5$.

In summary, for both tip-speed ratios, performance based on a cluster-specific
freestream kinetic power, $\eta^{*}_{\Phi}$, confirms the hypothesis that the
observed flow-field and performance differences between clusters are primarily
caused by inflow velocity variations. This performance dependency on the
inflow velocity is in agreement with the axial-flow literature which has
consistently shown that turbine power output is correlated with inflow
conditions [49]. Here, the increase in $\eta_{\Phi}$ observed for cluster 1 is
a consequence of not accounting for the increased inflow kinetic energy, which
increases turbine power output even as the actual efficiency is degraded by
the lower cycle-specific tip-speed ratio. While these results suggest that it
might be preferable to calculate efficiency on a cycle-specific basis, we
found that this approach increases performance variability in this data set.
Consequently, we find that cluster-specific performance is more robust to
uncertainty in the advection correction (Section 2.3.2).

###  3.4 Cycle-to-Cycle Hysteresis

Figure 17: Cluster conditionally-averaged flow fields in $S_{e}$ for
$\lambda\>=\>1.5$ as the blade transitions from the downstream to upstream
sweep.

As previously noted (Figure 9c-d, Section 3.2), the cycle-to-cycle flow-field
variation during the downstream sweep does not coincide or correlate with
performance variability. However, it is conceivable that flow-field variation
at the end of one cycle could affect future cycles. To explore this
possibility, we consider cluster conditionally-averaged flow fields in Figure
17, focusing on $S_{e}$ for $\lambda\>=\>1.5$, a case with distinct
differences between the two clusters. Flow recovery in cluster 2 appears
slower, with more separated flow and lower velocities near the trailing edge
at $\theta\>=\>356^{\circ}$. Despite this, we observe three factors that
support a hypothesis of limited hysteresis between cycles. First, as the blade
enters the next cycle ($\theta\>=\>368^{\circ}$), differences between the
cluster 1 and cluster 2 flow fields has considerably diminished. Second, there
is limited cycle-to-cycle performance variability at the beginning of the
cycle ($\theta\><\>40^{\circ}$,(Figure 9). Third, there is limited variation
in the weighted activation profiles for the flow fields at the beginning of
the next cycle (Figure 12, $S_{a}$). If hysteresis from the flow-field
variability in the prior cycle was present, we would expect flow-field
variability to persist through $\theta\>=0^{\circ}$ and to see attendant
performance variability at the start of the cycle. Thus, it is unlikely that
hydrodynamics in the previous cycle contribute to the observed variability in
future cycles, especially in comparison to variation associated with the in
freestream velocity.

We note that this result is likely influenced by our choice of control scheme.
Specifically, if turbine torque, rather than rotation rate, was the regulated
quantity, the instantaneous rotation rate would vary within and between
cycles, such that the blade kinematics would no longer be deterministic [44].
Under such a control scheme, we might see greater variation in the flow fields
between cycles and stronger impacts on future cycles.

##  4 Discussion and Conclusion

Cycle-to-cycle performance and flow-field variability for cross-flow turbines
are often implicitly neglected through time- and phase-averaging, but, as
demonstrated here, are statistically significant. This variability is
potentially caused by dynamic stall’s stochastic nature, freestream velocity
perturbations, and hysteresis from previous cycles. In this work, we explored
the extent and sources of cycle-to-cycle variability using near-blade flow
fields and performance metrics for sub-optimal, $\lambda\>=\>1.5$, and near-
optimal, $\lambda\>=\>2.5$, tip-speed ratios. The flow-field clustering
technique developed for this purpose proved sensitive to cycle-to-cycle
variations and highlighted correlations between performance and flow-field
variability. This technique was effective for both tip-speed ratios despite
the lower flow-field variability for $\lambda\>=\>2.5$, where dynamic stall is
weaker.

Overall, performance and hydrodynamic variability were found to be non-
coincident in phase. Across all phases, the coefficient of variation in
performance ranges from $4\>-\>20\%$. While an imperfect comparison, the
phase-specific flow-field coefficients of variation are as high as $150\%$, an
order of magnitude greater. Performance variability is highest around the
maximum performance within a cycle, but, because of limits to near-blade flow-
field resolution, observable flow-field variability is minimal until beyond
maximum performance. For $\lambda\>=\>1.5$, flow-field variability increases
during the growth and shedding of the large dynamic stall vortex, and, for
$\lambda\>=\>2.5$, during the growth of the separated flow region at the
trailing edge. Despite this lag in the flow-field variability, clusters based
on observed flow fields throughout the upstream sweep are correlated with
time- and phase-averaged performance for both tip-speed ratios. This is not
evident when considering only aggregate, statistical measures of variability.
In contrast, the flow-field variability during the downstream sweep has little
correlation with performance.

Hysteresis and dynamic stall stochasticity may contribute to variability, but
freestream velocity perturbations dominate the observed cycle-to-cycle
variation in both the flow fields and performance. Given that the high flow-
field variability at the end of the turbine rotation is not accompanied by
high variability at the beginning of the rotation, it is unlikely that
hysteresis impacts are substantial. While we cannot completely disentangle the
impacts of freestream velocity perturbations and dynamic stall stochasticity,
the cycle-specific freestream velocities are correlated with the flow-field
clusters. These velocity perturbations are shown to impact the kinetic energy
available in the flow and to perturb $\lambda$ enough to influence the dynamic
stall process. For $\lambda\>=\>1.5$, the better-performing cluster for
$\theta=95\>-\>143^{\circ}$ exhibits higher inflow velocities, a higher
reversed flow fraction, earlier dynamic stall vortex shedding, and a
performance peak occurring slightly earlier in the cycle. All of these
behaviors are consistent with the locally lower $\lambda$, however, the higher
performance is counter to the trend where a lower $\lambda$ is associated with
lower maximum and time-averaged performance. This contradiction is also
present for the $\lambda\>=\>2.5$ case. When performance is calculated with a
cluster-specific kinetic power, the apparent contradiction is resolved and the
locally higher tip-speed ratio cluster outperforms the other. This suggests,
for these conditions, that the change in the available kinetic power in the
inflow has a greater influence than the perturbation to the local TSR.

A clear performance and hydrodynamic dependence on assigned cluster is
observed, despite the relatively low freestream turbulence intensity of
1.8-2.1%. That said, the differences in time-averaged performance between
clusters are small (1-3% relative to the time-average of all data). Therefore,
for these conditions, phase-averaging flow fields and performance is an
effective way to investigate general trends. This work demonstrates that
clustering is useful for more nuanced analyses that seek to understand the
connections between observed flow fields and turbine performance. In field
settings or laboratory conditions with higher turbulence, or the use of torque
control, cycle-specific tip-speed ratios could be perturbed further from the
phase-average. For such conditions, performance and hydrodynamic variability
may increase, making a cluster-specific analysis a more necessary alternative
to phase-averaging. Additionally, the performance dependency on cluster points
to the potential extension of this work to developing probabilistic reduced-
order models for prediction and control.

We must note that, in future studies, it would be important to consider the
number of clusters as a free parameter. Here, two clusters proved appropriate,
but this may not be optimal for all cases. For the current data set, two
clusters produce unique phase-averaged weighted activation profiles in the
principle component analysis. While we utilize clusters based on the flow
fields, PIV data collection is time intensive and generally confined to
laboratory settings. Investigating the benefits of clustering on the basis of
performance could be a direction for future research.

The flow-field clustering approach contributes to our understanding of the
mechanisms responsible for the performance and hydrodynamic variability of
cross-flow turbines. It provides a more comprehensive picture of the phase-
varying flow fields than aggregate, statistical representations, and provides
conditionally averaged groups that are not based on subjective, hand-
engineered metrics. We observe physically meaningful clusters representing a
series of distinct flow-field evolutions. These clusters are correlated with
performance when based on flow fields captured during the turbine power
stroke, and reveal variations in timing of the dynamic stall process.

##  5 Acknowledgments

The authors thank the Alice C. Tyler Charitable Trust for supporting the
research facility and acknowledge the substantial contributions by Benjamin
Strom, Hannah Ross, Aidan Hunt, Carl Stringer, Erik Skeel, and Craig Hill for
their contributions to the development and upgrades of the experimental setup
and code base. We also thank our colleagues in the Pacific Marine Energy
Center for their continued support. Financial support was received from the
United States Department of Defense Naval Facilities Engineering Systems
Command and through the National Science Foundation Graduate Research
Fellowship Program.

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